Theme

Small perturbations, large consequences

The corrections that were supposed to be negligible: Neptune found in the residuals, Mercury's extra arcseconds, and orbits that are stable only for a while.

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

Orbits

The orbit is an ellipse, and the Sun is not in the middle of it

Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

$What an eccentricity does to the shape, and what it does to the offset. The fractional flattening 1 − b/a and the fractional focal offset c/a, against eccentricity. The offset is first order in e and the flattening is second, so at Earth's e = 0.0167 the outline is 0.014% from a circle while the Sun sits 1.67% of the semi-major axis off the centre — a factor of 120.$ Orbits

An orbit can look exactly like a circle and still not be one

Earth's orbit departs from a circle by fourteen parts in a hundred thousand. The Sun's offset from its centre is a hundred and twenty times larger, and everything interesting is in the offset.

The osculating semi-major axis of a perturbed orbit. The semi-major axis a test particle would have if the perturber vanished, computed from its position and velocity at every step of an integration over 26 orbits of the perturber. It is not constant: a short-period ripple rides on a slow trend, and only the trend accumulates.

Orbits

Elements that do not stay constant

Six numbers fix an orbit for all time, and the phrase is only true in a universe containing two bodies. Add a third and the six start moving — some of them wandering and returning, one or two of them drifting in one direction forever, and the difference between those two behaviours is the whole of celestial mechanics after Newton.

Orbits

Aiming at a plane instead of at a planet

A spacecraft arriving at a planet is not aimed at a periapsis distance. It is aimed at a point in a plane perpendicular to its own incoming asymptote, because that is the one coordinate in which the miss distance responds linearly to a correction — and every navigation product ever published for a flyby is written in it.

Four averages of one distance, and the two of them that are the semi-major axis. The average distance of a body from its primary, against eccentricity and in units of the semi-major axis, computed four ways: averaged over time, over true anomaly, over eccentric anomaly, and as the harmonic mean in time. Every curve is a quadrature over the orbit — 2,048 panels uniform in eccentric anomaly, with Kepler's equation supplying the time weight — and not a closed form. Two of the four are exactly a at every eccentricity, which is why they are drawn as one line: the eccentric-anomaly average, because the mean of cos E over a turn is zero, and the harmonic mean in time, because the time weight cancels 1/r at every node before the sum begins. The other two are not: the time average is a(1 + e²/2), which rises to 1.4050 a at e = 0.9, and the true-anomaly average is a√(1−e²) — the semi-minor axis — which falls to 0.4359 a there. So a is the average distance in two senses out of four, and the ordering b ≤ a ≤ ⟨r⟩ₜ holds at every eccentricity with equality only on the circle. At Earth's e = 0.0167 the four agree to 0.014%, and at Mercury's e = 0.2056 the spread is 2.14%. The distinction is invisible for the planets and unavoidable for a comet, and it is the reason a quoted "mean distance" has to say which mean.

Orbits

The average depends on what is being averaged

Four ways of averaging one orbit's distance from its primary give four different numbers, and only two of them are the semi-major axis. Which two is not a matter of convention, and the same arithmetic decides how much sunlight a planet receives in a year.

The angle that has nowhere to be measured from. An eccentricity vector carried round a circle of radius 0.034 centred at 0.031 — which is what a secular perturbation does to one, a forced eccentricity with a free one turning about it. Below, the two components e cos ϖ and e sin ϖ, which are smooth, bounded and perfectly ordinary throughout. Above, the longitude of pericentre read off them, which is not: as the eccentricity passes its minimum of 0.0030 the pericentre sweeps through most of a circle, at up to 4080° per unit time against the 12° the free vector itself turns in the same interval. Nothing has happened to the orbit. The pericentre is a place on the orbit, and a nearly circular orbit does not have one — so ω, and Ω with it at zero inclination, are angles measured from a feature that is not there. The equinoctial elements are the pair drawn below, and a propagator written in them steps through this instant without noticing it.

Orbits

The elements that stop existing

An orbit needs six numbers, and three of the usual six are angles measured from features a perfectly ordinary orbit may not have. At zero eccentricity there is no pericentre to measure from, and the arithmetic knows it.

Orbits

An inclination that turns into an eccentricity

A distant companion cannot change an orbit's size or its energy. It can take a circular orbit tilted past 39.23 degrees and drive it to an eccentricity near one, and back, over and over — and the companion's mass and distance set only the clock.

The same thrust is worth 2.08 times more at perigee, and out of plane it is worth nothing at all. The rate of change of the semi-major axis under a unit acceleration in each of the three directions, against position around an orbit of eccentricity 0.35, from Gauss's variational equations. The along-track curve carries the factor p/r = 1 + e cos f and therefore peaks at perigee, where the same impulse is worth 2.08 times what it is worth at apogee — the whole of the Oberth effect, arriving as a term in a differential equation rather than as an argument about kinetic energy. The radial curve is antisymmetric about apoapsis and integrates to exactly zero over a revolution: pushing outwards for half an orbit and being pushed back for the other half changes the energy by nothing, which is checked here by quadrature and comes out at -9.0e-18. And the out-of-plane response is identically zero at every point of the orbit, because W is perpendicular to the velocity and does no work. An orbital plane can be rotated without touching the energy, and that is why a plane change is so expensive: none of what is spent goes anywhere useful.

Orbits

Which direction moves which element

Resolve a small force into three components and Gauss's equations say exactly what each one does. The out-of-plane component can rotate an orbit and can never change its energy; the along-track component owns the semi-major axis outright and is worth more at perigee than at apogee by a factor that is pure geometry.

Earth's eccentricity is a sum of 8 sinusoids. The eccentricity of Earth over 800 thousand years, from the Laplace–Lagrange solution for all eight planets — the secular matrix built from the JPL masses and semi-major axes, symmetrised, and diagonalised by Jacobi rotations. It runs between 0.0035 and 0.0436, and it has no period, because it is a sum of 8 incommensurable frequencies. The two largest contributions to this planet are the modes at 3.73 and 7.33 arcseconds per year, drawn as the flat lines: those are constants, and everything moving in the figure is their beat. The check is the quadratic form ½ΣΛe², which a symmetric secular matrix conserves exactly and which drifts by 6.7e-16 across the whole interval — computed from the same curves the figure draws and from nothing the eigenvalues were fitted to. The true angular momentum deficit, Σ Λ(1 − √(1−e²)), drifts by 7.1e-4, and that difference is not an error either: the two agree only to fourth order in e, and Mercury at 0.206 supplies almost all of the gap.

Orbits

No planet has an eccentricity of its own

Strip the short-period terms out of the planetary equations and what is left is a linear system. Its eigenvectors are modes of the whole solar system, and the number a catalogue quotes for a planet's eccentricity turns out to be a reading of a clock rather than a property of the planet.

One admissible root, 0.01% from the truth. Gauss's reduction of three directions to a distance, drawn as the two relations whose intersection it is. Three observations of Ceres on days 0, 20, 40 of an arc, generated from its elements and used only as sight directions — no range, no radial velocity. The rising curve is geometry: the heliocentric distance a candidate at geocentric distance ρ₂ would have, r₂² = ρ₂² + 2ρ₂(R₂·L̂₂) + R₂², which contains no dynamics at all. The falling curve is dynamics: ρ₂ = A + µB/r₂³, with A and B built from the three sight vectors, the three observer positions and the three times, and containing no orbit. Eliminating ρ₂ between them gives r₂⁸ + a r₂⁶ + b r₂³ + c = 0 — an eighth-degree equation, from a problem with exactly as many equations as unknowns. Here they cross once at a positive ρ₂, at r₂ = 2.5893 AU against the true 2.5890. The other 2 real roots are rejected not by fitting but by sign: the ρ₂ each implies is negative, and an object behind the observer was not the thing observed.

Orbits

Three observations and no orbit at all

Three directions in space give six numbers for the six elements of an orbit, which sounds like a solved problem. The algebra that solves it is of the eighth degree, and for a near-Earth asteroid three perfect observations can be consistent with three different orbits.

Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft.

Orbits

The table that is a fit

A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics.

$Every model curve has slope −1, and four measurements agree on κ to 1.5×. Semi-major-axis drift against body diameter, for a thermal recoil in which a fraction κ = 0.085 of the absorbed sunlight comes back out along-track. The three curves are the same expression at 1, 1.6, 2.5 astronomical units, and each has a slope of exactly −1: the acceleration is the absorbed power divided by the mass, which is a cross-section over a volume, so it falls as one over the size and nothing else on this axis changes it. A kilometre-wide body drifts a few metres a year; a ten-metre one drifts hundreds. The four filled marks are the bodies whose drift has actually been measured as a fitted parameter in an orbit solution, and they do not lie on any single curve because each carries its own density, distance and obliquity. What they agree about is the number beside each: solve every measured drift for the efficiency that would produce it and the four answers are 0.084, 0.085, 0.089, 0.129 — a factor of 1.5 apart, for a quantity that could in principle have been anything from zero to a fifth. That agreement is the evidence that the mechanism is understood, and it is the only evidence there is, because the thermal conductivity that sets κ has never been measured for any of them.$ Orbits

An orbit moved by heat

A rotating body re-radiates absorbed sunlight from the hemisphere that has had time to warm, so the recoil is not aimed at the Sun. The resulting force is a few parts in ten billion of gravity, it is the only orbital force whose sign depends on which way the body spins, and it has been measured to four figures.

The divisor is 0.129″/yr and the theory's own error is 0.24. Six frequencies of the secular solar system on one logarithmic axis, in arcseconds per year. The top two are the pair whose near-equality is the whole story: the perihelia of Mercury and Jupiter separate at 1.333″ a year, the nodes of Mercury and Venus at 1.462, and the difference of those two differences is 0.129 — a resonant argument that turns once every 10.0 million years. A term with that argument in the disturbing function acts in one direction for five million years at a stretch, which is what pumps Mercury's eccentricity, and it is the reason the inner solar system's Lyapunov time is what it is. The bottom three bars are why this figure exists. The divisor is smaller than the corrections the theory that computes it leaves out. Relativity contributes 0.4298″ a year to g₁ alone — the same 43 arcseconds a century that broke Newtonian gravity — which is 3.3 times the divisor; the fourth-order terms in the eccentricity that Laplace–Lagrange truncates come to about 0.24″, which is 1.8 times it; and the second-order solution computed on this page gets 0.35″, missing the published value by more than the value itself. A theory cannot bound what it cannot resolve. Laplace's proof that the eccentricities stay bounded is a proof about a system whose frequencies are constants, and the frequency that decides the question is not one.

Orbits

The bound that holds only in the linear theory

Laplace proved the planetary eccentricities bounded, and the proof is a proof about a linearised system with constant frequencies. One combination of those frequencies is nearly zero — and it is smaller than the terms the linearisation threw away, which is why the stability of the solar system is a probability rather than a theorem.

Orbits

A collision dated by a scatter plot

Nothing in the solar system carries a date. A collisional family does — because a force that depends on a body's size has been pushing its fragments apart ever since, so the cloud is a V whose slope is an elapsed time, and one of those dates is confirmed by fossil meteorites in Swedish limestone.

Orbits

A comet that arrives a day early

Encke's comet returned two and a half hours ahead of prediction, every revolution, for decades before anybody could say what was pushing it. The force is a rocket — a few tonnes a second of vapour leaving the sunward side of a rotating nucleus, delivered almost entirely in the few weeks around perihelion, and it accumulates in the arrival time as the square of the number of returns.

The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.

Gravitation

The tide is a difference, which is why there are two of them

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

$The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio.$ Gravitation

Five places that keep station, in a problem with no solution

Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.

The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart.

Gravitation

The distance at which a moon stops holding together

The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it.

Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run.

Gravitation

Three bodies, and what "no solution" actually means

The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks.

$The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye.$ Gravitation

Resonance clears a gap in one place and locks a moon in another

When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

$The surface a star stops at. The equipotential through L₁ — the Roche lobe — at mass fractions 0.50, 0.20, 0.05, in the frame that rotates with the pair. Each is a level set of exactly the same function the zero-velocity curves are level sets of, at exactly the critical value, so nothing here is a new construction: the Roche lobe is the last closed equipotential, and it is closed only because the two lobes touch at a single point. Material that reaches that point is no longer bound to the star it came from, and it leaves through an opening of zero area. The lobes are drawn in the orbital plane; in three dimensions each is a teardrop, and its volume-equivalent radius is what "the size of a Roche lobe" means. As the mass ratio becomes extreme the smaller star's lobe shrinks towards it, which is why a white dwarf accreting from a companion has a lobe smaller than the Sun.$ Gravitation

The surface a star stops at

Around each star of a close pair there is a last closed equipotential, and the two touch at a single point. A star that swells to reach it hands its outer layers to its companion through an opening of zero area — and the transfer, once started, makes itself worse.

Mercury's perihelion, term by term. The observed advance of Mercury's perihelion is 5600 arcseconds per century against the equinox. Almost all of it is the equinox: the coordinate frame itself turns, and removing it leaves 574. Subtracting the perturbations of the other planets — computed by Le Verrier in 1859 and refined many times since — leaves 42.98 arcseconds a century that nothing in Newtonian gravitation accounts for. The bars are logarithmic in nothing; they are the real proportions, which is why the residual is barely visible beside the frame term.

Gravitation

Forty-three arcseconds, after everything else

Mercury's perihelion moves through 5,600 arcseconds a century. Nearly all of that is the coordinate system turning, and almost all of the rest is the other planets. What was left over was 43 — under one per cent of the raw number, and the most consequential residual in the history of the subject.

Gravitation

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

PSR B1913+16: 45 years of periastron arriving 88 seconds early. The cumulative shift of periastron passage for PSR B1913+16, Δt = ½(Ṗ_b/P_b)T², over 45 years from its discovery in 1975. The orbital period is 0.322997449 days and is measured to be shortening at 2.423e-12 seconds per second — a change in the twelfth decimal place, which over a career accumulates to 87.5 seconds. That is the whole reason the effect is measurable. Ṗ_b is taken here as a measured quantity and no radiated power is derived from it; the parabola is the general-relativistic prediction as published, and the points are that prediction scaled by the published ratio of observed to predicted decay, 0.997 ± 0.002 (Hulse & Taylor 1975; Weisberg & Huckins 2016), which is how the agreement is quoted. The error bars are drawn: at the last point the bar is 1.25 px tall against a dot 9 pixels across, so they are invisible, and their invisibility is the result. The campaign has run 45 years.

Gravitation

An orbit measured to be shrinking

A 7.75-hour orbital period that shortens by 68 nanoseconds each turn is beyond any single measurement, and unmissable after fifty thousand of them, because the shift accumulates as the square of the elapsed time. Forty-five years of pulse arrival times have made it 87.5 seconds, which is why the rate is a measured quantity rather than an inferred one.

A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get.

Gravitation

A prediction with an expiry date

The inner solar system's Lyapunov time is about five million years, so a centimetre of error becomes an orbit in a hundred million. The ephemeris dies while the system survives — because the elements stay bounded when the phase does not, and only one of those is what stability means.

The contours a planetary encounter cannot cross. Contours of the Tisserand parameter with respect to Jupiter in the plane of semi-major axis and eccentricity, drawn for a coplanar orbit. An encounter with Jupiter moves a comet along one of these curves and never across one, because T is what the encounter conserves. The heavy contours are at T = 3 and T = 2, and they are the boundaries the comet families are defined by: T > 3 means no encounter is possible at all, since v∞²/v_J² = 3 − T and a negative squared speed is not a trajectory; 2 < T < 3 is the Jupiter family; below 2 the approach speed exceeds Jupiter's own orbital speed and the orbits are the nearly isotropic ones. The shaded boundary is the crossing condition — an orbit whose pericentre is outside Jupiter's, or whose apocentre is inside it, never meets the planet whatever its T. The five comets are placed at their JPL elements and labelled with the T the literature quotes; all five with inclination sit off the coplanar contours by exactly the cos i in the definition, which is why 1P/Halley's is negative — a retrograde orbit meets Jupiter at nearly twice Jupiter's speed.

Gravitation

The number that survives the encounter

A comet that passes Jupiter comes away with every orbital element changed. One combination of them is not changed, and it is enough to recognise the comet afterwards, to sort the comet families, and to say where a spacecraft can and cannot go.

$A thousandth of the field, and all of the precession. Left: the Earth's figure against a sphere of the same equatorial radius, with the flattening drawn 28× its true value. The real difference between the equatorial and polar radii is 21.4 km on 6378 — 1 part in 298 — which at this size would be 0.5 pixels and invisible, so the drawing is a schematic and the number is here instead. Right: the two components of the J₂ perturbation at the surface, each as a fraction of the monopole μ/r², both differentiated from the potential rather than quoted. The radial one strengthens the inward pull by 1.62×10⁻³ over the equator, where the extra mass is, and weakens it by 3.25×10⁻³ over the poles, vanishing at ±35.26° where P₂ does. The transverse one is zero at the equator and at both poles and peaks at ±45°, at 1.62×10⁻³ — and that is the component that does the work. It pulls an inclined orbit back towards the equatorial plane, which is a torque about the line of nodes, and a torque applied to something already turning moves it sideways rather than back. Averaged over an orbit the pair leave a, e and i untouched and turn the whole plane instead, which is why a term a thousandth of the field is the largest single perturbation on almost every satellite ever flown.$ Gravitation

The Earth's shape, read off a satellite's node

The Earth is a thousandth of a part from being a sphere, and that thousandth turns every satellite's orbital plane. Vanguard 1 measured it in 1959 — and one retrograde inclination turns the plane at exactly the rate the Sun moves, which is a perturbation used as a design constraint rather than corrected for.

The wake, computed from the streamlines that make it. Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. The overdensity is behind the mass, and that is the entire mechanism: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from.

Gravitation

A drag with nothing to drag against

A massive body moving through a sea of light ones raises an overdensity behind itself and is pulled back by it. The force does not depend on the masses of the background bodies at all — and it is strongest at one particular speed.

The last curve across the cylinder, before and after it breaks. Surfaces of section for the standard map at K = 0.5, 0.9716, 1.4, each 22 trajectories iterated 41 times from a column of starting values. Nothing here is placed: every dot is an iterate. At K = 0.5 the islands are separate and the space between them is filled with curves that run all the way round in θ — a trajectory cannot get from one island to the next, and one started beside the hyperbolic fixed point wanders 2.91 in p — 0.46 of a cylinder — and stops. At K = 0.9716, which is Greene's threshold to four figures, the last of those curves is on the point of going and the same trajectory still only reaches 4.84, which is 0.77 of a cylinder. At K = 1.4 it covers 6.1 cylinders: the barrier is gone and there is nothing left to stop it. That transition is what a chaotic zone in the asteroid belt is, drawn without any asteroids.

Gravitation

Where the chaos comes from

A Lyapunov time says how long a prediction lasts. It does not say what destroyed it. The mechanism is two resonances whose libration widths overlap, and the transition can be watched happening on a surface of section as one number is turned up.

255 km sees degree 154; 35786 km sees degree 6. The same field spectrum, multiplied by the upward continuation factor (R/r)ˡ⁺¹ for orbits at 255 km, 450 km, 800 km, 35786 km. A harmonic of degree ℓ has ℓ bumps around the planet, so it falls off with height like a wave of that wavelength — fast, and faster the finer it is. The horizontal rule is a measurement floor; where each curve crosses it is the highest degree that orbit can feel at all, and the answer is 154 at 255 km, 101 at 450 km, 65 at 800 km, 6 at 35786 km, or 130 km, 198 km, 308 km, 3340 km of horizontal resolution on the ground. Nothing in that arithmetic is an instrument. It is why GOCE flew at 255 kilometres with drag compensation rather than at a comfortable altitude with a better gradiometer, why the Moon's mascons were not seen until something orbited low over them, and why a geostationary satellite's ephemeris needs a field with four terms in it.

Gravitation

The field a satellite is allowed to feel

Past the flattening, a planet's gravity is a sum of harmonics whose sizes follow a rule with no physics in it. How much of that sum a spacecraft can measure is decided by its altitude and by nothing else — which is why one mission flew at 255 kilometres and had to push itself along.

Gravitation

A feeding zone, and the spacing it forces

The radius that decides what a planet may keep also decides what it could reach while it was growing — and measuring the gaps between planets in that unit turns a distribution spanning three orders of magnitude in astronomical units into a band a factor of ten wide, with a floor that is partly a theorem and partly a fit.

Io's measured heat needs k₂/Q = 0.016, and Enceladus's needs 0.011. Tidal surface heat flux against orbital eccentricity, from Ė = (21/2)(k₂/Q)GM_p²R⁵ne²/a⁶ evaluated at each satellite's own orbit, drawn at a common k₂/Q of 0.015. Every line has slope 2 because the dissipation is quadratic in e and nothing else on this axis varies. The filled marks are each body at its actual eccentricity; the two ringed ones are the bodies with a measured surface heat flux, and they are the only points here that are observations. Solving each of those for k₂/Q gives 0.016 for Io and 0.011 for Enceladus — within a factor of 1.5 of one another, for a warm silicate body and a 500-kilometre ball of ice, which ought to be a coincidence and is instead the sharpest problem in the subject: nothing about Enceladus's ice can plausibly dissipate at 0.011, and the number is what the heat requires all the same. The dashed line is the Earth's measured surface heat flux, 0.087 W/m², which Io exceeds by a factor of 28 — the most volcanically active body in the solar system is the fourth largest moon of the fifth planet, and the reason is entirely in the orbit.

Gravitation

A moon heated by not being allowed to relax

Tidal dissipation goes as the square of an eccentricity that tides themselves destroy, so a moon radiating tidal heat is spending something it cannot have saved. Io's would be gone in a hundred and forty thousand years, and the resonance that keeps putting it back is the reason there are volcanoes.

Gravitation

A step that must not be adapted

A symplectic integrator's bounded energy error is a property of a fixed step. It is conserving a Hamiltonian a step-size away from the intended one, and changing the step changes which Hamiltonian — so refining the step at a close encounter, which is the first thing anybody does, destroys the only property the method was chosen for.

Gravitation

A ring weighed by the wave crossing it

Saturn's rings are a few tens of metres thick and spread over an area larger than the Earth, made of pieces nobody can resolve, and nothing has ever landed on them. Their mass per unit area is nevertheless known to a few per cent — from the rate at which the crests of a wave crowd together as it travels outwards.

One measured number, and every pair of masses that produces it. The plane of the two component masses of an inspiralling binary, with three curves of constant chirp mass across it. The middle one is GW150914's value of 28.7 solar masses, and the chirp mass recovered from the coordinates of the drawn curve varies along its whole length by 7.4e-14 per cent — which is the point: every binary on that line radiates the same frequency sweep at leading order, so the early inspiral cannot tell them apart. Two of them are marked. An equal pair of 33.0 and 33.0 solar masses and a lopsided pair of 63.6 and 18.4 sit on the same contour, and their total masses differ by a factor of 1.24. What separates them is the mass ratio, which enters the phasing only at the first post-Newtonian order, suppressed by the square of the orbital speed in units of the speed of light — small through the hundreds of cycles that carry most of the signal, and appreciable only in the last few, where that speed approaches a third of c. So the chirp mass is a measurement and the individual masses are an inference from the end of the signal, which is exactly the part a detector's high-frequency noise eats first.

Gravitation

One number where two masses were

The hundreds of orbits an inspiralling binary completes inside a detector's band depend on its two masses only through one combination of them. Every pair on that contour radiates an identical sweep, so the early signal — which carries nearly all the signal-to-noise — cannot say which pair it was.

7 decades of relaxation time, and a Hubble time between them. Crossing time and two-body relaxation time for 5 self-gravitating systems, against the number of bodies in each. The lower marks are the crossing time R/σ, which spans 3 decades; the upper ones are t_relax ≈ 0.1N/ln N times it, which spans 8. The horizontal rule is a Hubble time, and the same expression puts these systems on opposite sides of it. Two of them relax — an open cluster and a globular cluster — so their stars have exchanged energy, their heavy stars have sunk, and their present structure is not the one they were born with. The rest do not: a dwarf spheroidal, an elliptical galaxy, a cluster of galaxies, the elliptical missing it by a factor of 10⁶, which is why a galaxy is modelled as a smooth potential with no stars in it at all. The dependence is on N and hardly at all on anything else, because the Coulomb logarithm makes every decade of impact parameter contribute equally. A cluster that relaxes also evaporates: about a stellar mass in a hundred and forty leaves per relaxation time, so the globular cluster empties in 6.1·10¹¹ years.

Gravitation

A cluster that boils itself away

A star cluster has no thermostat. Encounters between its members push a few of them above escape speed, the cluster loses them, and losing them makes it contract — which makes it hotter, which makes more of them escape. A self-gravitating system heats up as it loses energy, and the process ends by destroying the system.

The Sun's altitude through the day at latitude 52°. Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.

The observed sky

The Sun's path, and the tilt that makes the seasons

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.

The observed sky

The pole star has a shelf life, and the sky has a slow hand

The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more.

The observed sky

The eclipse that repeats a third of a world away

Three lunar periods nearly share a multiple after 6,585 days. The word "nearly" is what makes eclipses predictable, and the leftover third of a day is what moves each repeat a third of the way round the Earth.

The observed sky

The wobble inside the wobble

Precession and nutation are the same torque. The difference is that the Moon's orbital plane turns once in 18.6 years, so part of the pull oscillates instead of accumulating — and a catalogue position is not a direction until it says which pole it is measured from.

The observed sky

Six kinds of second

The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

$The Metonic cycle slips a day in 219 years, and the next rule needs 334. Accumulated disagreement between a lunisolar rule and the sky, against elapsed time, for the continued-fraction convergents of the 12.368266 synodic months in a tropical year. Each line is one historical cycle: 2 years to 25 months, 3 years to 37 months, 8 years to 99 months, 11 years to 136 months, 19 years to 235 months, 334 years to 4131 months. Every line has slope exactly 1, because an error made once per cycle accumulates linearly, so the only thing that distinguishes the rules is where they start. The two that were actually used are the octaeteris — eight years, 1.59 days out per cycle, useless within a generation — and Meton's nineteen, which is out by 2.1 hours per cycle and therefore takes 219 years to slip a single day. The reason nineteen is so much better than eleven is a number in the continued fraction rather than anything about the Moon: the partial quotient that follows 235/19 is 17, and a convergent's error is bounded by one over the next quotient times the square of its denominator — so a large quotient there is exactly a good approximation here, and the next improvement costs 334 years of cycle for a rule nobody could keep. What follows from the 219 years is the whole character of a lunisolar calendar: it is a table rather than an observation. The ecclesiastical moon that fixes Easter is computed from a Metonic cycle, not looked at, and the Julian version of that computus — still used to date Easter in the Eastern churches — has slipped four to five days from the sky since it was fixed in the fourth century, exactly as this plot says it must. A calendar's job is agreement, and agreement and accuracy are different requirements that diverge at a rate the arithmetic predicts.$ The observed sky

A month that has to be tabulated

A lunisolar calendar reconciles two periods that share no common multiple, so every historical cycle is a convergent of one continued fraction. Meton's nineteen years is wrong by two hours, which is a day in two hundred and nineteen years — and that is why the moon that fixes Easter is a table rather than the sky.

The observed sky

A star that blinked before it should have

In March 1977 three teams watching a star pass behind Uranus recorded it dimming five times before the planet arrived — and then, symmetrically, five times again on the way out. The symmetry is the whole argument — nothing but a set of rings concentric with the planet produces a mirror image about closest approach.

Starlight

The same width for three different reasons

Thermal motion, rotation and collisions each widen an absorption line, and they can be tuned to areas that agree to a part in a thousand. What separates them is the shape, and the shape carries a rotation speed from one profile and a surface gravity from another.

Starlight

The direction a photon count throws away

A photometer records how many photons arrived. It discards a two-component quantity that survives every attenuation on the way, and that quantity carries a magnetic field direction, a grain size, and the shape of an exploding star nobody can resolve.

$A colour term of -0.059 magnitudes per magnitude, and one star that will not obey it. Synthetic photometry of blackbodies from 3,000 to 42,000 K through two V bands: the standard one, and a natural system whose effective wavelength is 9 nm longer and whose width is 1.12 times as large. The vertical axis is the difference between the two magnitudes for the same star — not a constant, because a wider redder band collects a different fraction of a hot spectrum than of a cool one. Fitting a straight line against B − V gives a colour term of -0.0585 magnitudes per magnitude and leaves a residual of 3.1 millimagnitudes, which is why a linear transformation is the standard reduction and why it works. The mark off the line is a cool star with molecular absorption bands in the red, drawn from the same blackbody with three synthetic bites taken out of it: it sits 27 millimagnitudes from the fit, 9 times the blackbody scatter. A colour term knows one number about a star and a spectrum has a shape, and that gap is the reason all-sky photometry stops at a per cent while differential photometry on one field reaches a millimagnitude.$ Starlight

The same star through two telescopes

A magnitude is defined by a response curve, and no two telescopes have the same one. The difference between two observatories' measurements of one star is not a constant to be subtracted but a function of the star's colour — and for a star whose spectrum has structure, not even that.

Starlight

A field strength read off a line that will not split

A few hundred gauss splits a spectral line by a ten-thousandth of its own Doppler width, which no spectrograph will ever resolve. The two components are circularly polarised with opposite handedness, so the split survives in the difference of two polarisations — where it is linear in the field rather than quadratic.

Starlight

A mean dominated by the gaps

The opacity in the equation of radiative transport is not an average of the opacity. It is a harmonic average weighted by the temperature derivative of the Planck function, which makes it a measurement of the transparent windows between the lines rather than of the lines — and that single fact decides what adding a million spectral lines to a table does.

$Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find.$ Stars

A clock read against a model of everything in between

A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image.

4 cycles of wings, and the polarity reverses at every boundary. Sunspot latitude against date, one mark per spot group, over 4 cycles of 11 years. The pattern is the reason the plot is called a butterfly diagram, and both of its features are laws with names. Spörer's law is the downward slope: spots emerge near ±28° at the start of a cycle and near ±7° at the end, so each wing narrows towards the equator and never crosses it. Hale's law is what the two mark shapes say: the leading spot of a pair has one magnetic polarity in the north and the other in the south, and both reverse when a new cycle starts — so a diagram that repeats every 11 years in appearance repeats only every 22 in magnetism. The wings overlap: the first high-latitude spots of a cycle appear about 1.6 years before the last low-latitude spots of the one before, which is why counting spots gives a cycle length slightly different from measuring one between polarity reversals. The dot density in time is the sunspot number itself, drawn from the standard skewed fitting function — the rise to maximum takes about four years and the decline about seven, in every cycle ever recorded.

Stars

A magnetic clock read off a butterfly

Plot sunspot latitude against date and the marks form wings that open at thirty degrees and march to the equator. The polarities reverse between wings, so the magnetic period is twenty-two years and the famous eleven is an artefact of counting spots rather than fields.

A gravity assist with a 70° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

Spaceflight

Stealing speed from a planet, which does not notice

A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.

$How small a sphere of influence is. Each planet's sphere of influence as a fraction of its own orbital radius, against that radius, both logarithmic. The largest belongs to Jupiter at 6.19% and the smallest to Mercury at 0.194%. The patched-conic method treats a trajectory as heliocentric everywhere outside these, and the figure is the argument for why that costs so little: they are thousandths of the journey.$ Spaceflight

One trajectory, stitched from three two-body problems

An interplanetary flight is a problem with no closed solution. It is flown by cutting it into pieces that each have one, and the seams are places where the model is knowingly false.

Down by 280 km, and faster by 164 m/s. A circular orbit at 400 km with a ballistic coefficient of 100 kg/m², integrated down to 120 km through a tabulated atmosphere at solar minimum and solar maximum. At solar min it takes 1.2 years; at solar max it takes 147 days — a factor of 2.9 for the same satellite in the same orbit, decided by an eleven-year cycle nobody controls. The rising curves are the orbital speed on the right-hand scale, and they are the point: the drag force is opposite the motion and takes energy out, and the body goes faster, from 7673 to 7836 m/s. There is no contradiction in it. The specific energy is −μ/2a, so removing energy shrinks a, and the circular speed √(μ/a) rises when a falls; the kinetic energy gained is exactly half the potential energy lost, and the other half is what the air took. Every point on every curve was integrated from da/dt = −(ρ/β)√(μa), and the speed at each point is √(μ/a) at that point rather than a separate model.

Spaceflight

An orbit that speeds up as it is slowed down

Drag takes energy out of a satellite and the satellite goes faster. There is no paradox in it, only a sign — and the same sign makes a re-entry date a space-weather forecast rather than an orbital computation, which is why Skylab was predicted for 1983 and came down in 1979.

three revolutions, on a turning Earth. The ground track of a circular orbit at 420 km and 51.64° inclination, over 3 revolutions, on an equirectangular graticule. The latitude is a sine wave bounded by ±51.64° exactly — sin φ = sin i sin u, so the inclination is the highest latitude the orbit ever passes over, and it is reached twice per revolution. Each successive pass is displaced west by (ω⊕ − Ω̇) × 92.90 min = 23.61°, of which 0.32° is the orbital plane's own regression and the rest is the planet turning underneath: the vehicle comes back to nearly the same place in inertial space and the place has moved. The period used is the nodal one, 92.899 min against the Keplerian 92.970: J₂ makes the two differ by 4.31 s, which is 0.018° of walk per revolution and 102° in a year — the difference between a repeat track and a track that used to repeat. The map is equirectangular and therefore wrong about area everywhere; what it is right about is longitude difference, which is the whole of what this figure measures.

Spaceflight

The line under a satellite

A ground track is an orbit seen from a frame that is turning, so every pass lands west of the last one. The track closes only when two periods are commensurable — which turns "look at the same place every day" into a condition on the altitude.

Two outcomes, and a boundary with no width. 26 trajectories launched from one point beyond L₂, all at the one speed the Jacobi constant C = 3.5124 permits there, differing only in the direction they set off in. The heavy curve is the zero-velocity boundary at that constant — the region no trajectory of this energy may enter — and it is open at L₂ by the neck the trajectories are aimed at. 11 of the 26 pass through into the secondary's realm and 15 turn back, and they are not interleaved — sweeping the launch direction through 65° finds one changeover and nothing in between. Bisecting the first of them pins it to 2.6e-12 radians, and the integrator runs out of digits before the boundary runs out of sharpness. That surface is the tube. It is the stable manifold of the periodic orbit about L₂, it separates transit from non-transit everywhere and not only in this fan, and a mission that wants to arrive for nothing has to be put inside it.

Spaceflight

The tube that leads out of a neck

Below a certain energy the forbidden region opens at a Lagrange point, and a trajectory may pass. Which ones do is decided by a surface with no width at all — and two tubes that meet give a transfer that costs nothing at the join.

Where the fuel goes, and it is not where a satellite points. Left, the orbit pole of a geostationary satellite, in degrees from the Earth's. The Sun and the Moon between them carry it round a circle of radius 7.4° in 53 years, and a satellite launched into the equatorial plane starts on the rim of that circle rather than at its centre — so its inclination climbs from zero at 0.88° a year, reaches 14.8° after 27 years, and comes back. Right, what holding it costs. A plane change of 0.88° at 3.07 km/s is 47.1 m/s a year; holding the longitude against the equatorial bulge, computed from the same resonant term that makes the longitude a pendulum, is 1.8 m/s a year. North–south is 96% of the budget, and a satellite that gives up on it does not fail — it starts tracing a figure of eight on the sky 1.8° tall in the first year, which a fixed dish cannot follow and a steerable one can. Retiring at the end of the propellant is therefore a choice about which service ends first.

Spaceflight

The orbit that has to be paid for every year

A geostationary satellite is not in equilibrium in any direction. The Sun and Moon tilt its plane by 0.85 degrees a year, the Earth's equatorial ellipticity makes two longitudes stable and two unstable, and the end of a satellite's life is the end of its propellant.

A burn along the track moves the chaser 8330 m backwards. Three 0.5 m/s impulses from rest alongside a target in a 400 km circular orbit, followed for 2 revolutions in the frame riding on the target. Along-track distance runs across the page with the direction of travel to the left, and radial distance up. The prograde burn ends 8330 metres behind after one revolution — exactly 6πΔv/n, and it is behind rather than ahead because the burn raised the orbit and a higher orbit takes longer. The retrograde burn ends 8330 metres ahead by the same arithmetic with the sign reversed. The radial burn is the third case and the strange one: it opens a closed loop and returns exactly to where it started after a revolution, having gone nowhere at a cost of 0.5 m/s. That is not a curiosity but the basis of the R-bar approach, in which a vehicle closes on a station from below along a path that costs nothing to abandon.

Spaceflight

A burn that moves the wrong way

In the frame riding on an orbiting target, a thrust along the direction of travel leaves a chaser eight kilometres behind after one lap, a radial thrust returns it exactly to where it started, and every free relative orbit is the same ellipse — twice as long along the track as it is across.

A quadratic and a linear, crossing at 149 objects. The two rates that decide whether a shell at 900 km is stable, against how many objects are in it. Production goes as N² — every collision needs two objects, so the number of collisions is proportional to the square of the population, and each one is taken here to make 1600 trackable fragments. Removal goes as N, because drag acts on each object independently and takes 1,195 years to do it at this altitude. A quadratic and a linear cross exactly once, at 149 objects in this shell, and above that crossing the population grows with nothing launched. The shell presently holds about 2,280, which is 15 times the crossing. Every number on the production side is uncertain by a factor of a few — the fragment yield most of all, and the cross-section is calibrated against an observed collision rate rather than measured — so the position of the crossing carries that uncertainty with it. The shape does not, and the shape is the argument: a quadratic overtakes a linear once and never comes back, the crossing falls as the altitude rises because the lifetime is in the denominator, and what results is a threshold rather than a trend.

Spaceflight

A collision rate that needs no collision

The flux through an orbital shell is a gas-kinetic calculation with no orbits in it. Production goes as the square of the population and removal goes as the first power, so a quadratic overtakes a linear once and never comes back — and which side of that a shell is on is decided by its altitude.

Exoplanets

A planet that was the star's own rotation

A dark spot rotating across a star removes light from the approaching limb and then the receding one, and the line centroid moves. That is several metres a second at the rotation period, from no planet at all — and two of the most celebrated nearby planets were withdrawn on exactly this evidence.

Exoplanets

A gap in a histogram that says how planets are built

Between the super-Earths and the sub-Neptunes there is a radius at which planets are markedly rarer. The gap is not a gap in what can be detected, and its slope with orbital period names the process that made it.

A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.

Exoplanets

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

Exoplanets

A planet where one cannot form

A Jupiter at four days orbits inside a region that was too hot to hold ice and too small to hold the material. It did not form there — and the distribution of eccentricities says which of two journeys brought it in.

Libration and circulation of a resonant angle. The critical angle of a resonance over time, integrated from the pendulum equation it obeys. Below the separatrix the angle oscillates about a fixed value — the body is locked; above it, the angle runs without bound and the body is not.

Exoplanets

A chain that could not have been assembled in place

Seven planets whose successive period ratios are all close to small whole numbers. Capture into a resonance requires the orbits to converge slowly, and converging slowly is something planets can only do in a disc.

$The same wind strips a mini-Neptune inside 0.06 au and leaves a hot Jupiter intact. The fraction of a planet's hydrogen envelope removed in 5 billion years by an energy-limited wind, against orbital distance, at a heating efficiency of 0.15 and an extreme-ultraviolet fluence integrated over the star's own history — saturated at L_XUV/L_bol = 3.2·10⁻⁴ for the first 100 million years and declining as t to the power −1.23 after, which comes to 4.9·10¹⁵ J m⁻² at one astronomical unit and is some 7 times what today's flux would give over the same span. This is a different mechanism from the tail of a Maxwellian, not a correction to it. Close to a star the upper atmosphere absorbs more extreme ultraviolet than it can radiate away, expands, and flows off as a wind whose rate is set by the energy arriving — Ṁ = ηπR³F/GM — so the exponential in the Jeans parameter vanishes entirely and what remains is a ratio of radius cubed to mass, which is one over the density. That is why the three curves are ordered as they are. The hot Jupiter is dense enough to lose 0.0087 of itself even at 0.047 au, where HD 209458 b sits and where its escaping hydrogen makes a transit fifteen per cent deep in Lyman α against one and a half per cent in the optical — an exosphere filling and overflowing the Roche lobe, and still costing the planet almost nothing. The mini-Neptune loses its whole envelope anywhere inside 0.06 au, and what is left when it does is a bare core about 1.5 Earth radii across. That is one of the two standard accounts of the gap in the radius histogram, and this figure is what it looks like before any of the observations are brought in.$ Exoplanets

A planet ten times larger in one colour

A hot Neptune that blocks one and a half per cent of its star's light in the optical blocks fifteen per cent of it in the ultraviolet line of hydrogen. No bound atmosphere can be that large — the material is well outside the planet's Roche lobe — so the observation is not a measurement of an atmosphere but of one leaving.

Galaxies

An arm that cannot be made of stars

A galaxy's disc turns differentially, so any feature made of a fixed set of stars is wound to invisibility within a couple of rotations. Spiral arms are ten billion years old and still open, which means an arm is a place rather than a thing.

Where a 2-armed pattern is allowed to exist in the Milky Way. Angular speed against radius, with the two combinations that matter for a pattern of 2 arms. A wave turning at 25 km/s per kpc corotates with the stars at 8.7 kpc, has an inner Lindblad resonance at 1.9 kpc and an outer one at 13.8 kpc. Between the Lindblad radii the wave propagates; at them it is absorbed. Every one of these radii is found by bisection on the curves as drawn, not quoted — and none of them is a property of the pattern alone, because κ comes from the rotation curve, which comes from the mass.

Galaxies

Where a pattern is allowed to turn

A spiral wave has one pattern speed, and that single number decides where in the disc it can exist at all. The boundaries are set by the frequency at which a star rocks radially about its circular orbit — a frequency the rotation curve already contains.

Three mass models, 2.3× apart in the disc, agreeing to 3.1 km/s everywhere. NGC 3198's rotation curve, decomposed three ways. The stellar disc has been scaled by 0.2, 0.65, 1.1 times its photometric mass, and for each scaling the halo's asymptotic speed and core radius have been fitted — not chosen — to reproduce the same total. The heavy curve and the marked points are that total: the three models agree with it to 3.08 km/s at every radius, well inside a measurement error of 4.5. The light curves below are the disc's own contribution, and at 17 kpc they differ by a factor of 2.3 — from 36 to 85 km/s — with the halo taking up exactly the slack, 126 down to 97. The curve is one function and the decomposition asks for two. The free parameter is the mass-to-light ratio of the stars, which the kinematics never measures, and it is why a "maximum disc" fit and a halo-dominated fit are both published for the same galaxy. The degeneracy does have one hard edge: scaling the disc to 1.35 times its photometric mass cannot be fitted by any halo in the family — the best leaves 5.6 km/s rms — because past the maximum-disc solution the stars alone already overshoot the curve and a halo cannot have negative mass. That is the one thing a rotation curve says about M/L on its own, and it is an upper limit. What separates the rest has to come from somewhere else: the vertical velocity dispersion of the disc, which weighs the stars alone; gas-rich dwarfs where there is scarcely a disc to argue about; or the baryonic Tully–Fisher relation, which ties the halo's speed to the baryons and would be a coincidence if the two were independent.

Galaxies

The same curve, two galaxies

A rotation curve is one function of radius. Decomposing it asks for two — how much of the speed is the stars and how much is the halo — and the mass-to-light ratio that trades one against the other is not measured by anything in the kinematics. A maximum disc and a halo-dominated fit run through identical points.

A bridge and a tail, integrated. A disc of 180 massless particles on circular orbits, and a companion of 1 times the primary's mass on a parabolic orbit with a pericentre of 1.6 disc radii, integrated from before the encounter to well after it. Times are in units of the disc's own outer orbital period, measured from pericentre. The outermost ring, drawn separately, is the one that produces both the bridge and the tail: at 3 its furthest particle is 31.4 disc radii from the centre, having started at 0.9; the panels are scaled to hold ninety per cent of the particles, so the very end of the tail is outside them. Nothing has been ejected and no material is new — every particle is on the orbit its own initial conditions and the two masses give it.

Galaxies

A bridge and a tail drawn by one force

The long thin streamers coming out of interacting galaxies look like debris thrown off by a collision. They are nothing of the kind — a simulation with no collisions in it, no gas and no self-gravity produces them from the tidal field alone, and only when the encounter runs the same way the disc turns.

Galaxies

The cloud that cannot hold itself up

A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star.

Cosmology

The same constant, measured twice, five sigma apart

Cepheids and supernovae give an expansion rate of about 73 kilometres per second per megaparsec. The microwave background gives 67.4. Both quote errors near one per cent, both have been rebuilt from scratch by rival teams, and the gap between them has grown as the measurements have improved.

Distance modulus against redshift, for three universes. The distance modulus μ = 5 log₁₀(D_L/10 pc) against redshift for three universes, all with H₀ = 67.36 km/s/Mpc, with 60 model supernovae drawn from the ΛCDM curve with 0.15 magnitudes of scatter. The point of the figure is how little difference there is: across two decades of redshift the three curves stay within a few tenths of a magnitude, and at z = 0.5 the accelerating and decelerating cases differ by 0.387 mag. A cosmology is not read off this plot. It is read off the residual, which is the next figure.

Cosmology

An expansion that was supposed to be slowing

Gravity is attractive, so an expanding universe full of matter must be decelerating, and the only question was by how much. Two teams set out to measure the deceleration and both found a quarter of a magnitude of extra faintness at redshift half — which is the wrong sign.

The acoustic peaks, and where the geometry says they should be. The temperature angular power spectrum of the microwave background. The drawn curve is a monotone interpolation through the published positions and heights of the six peaks and five troughs of the Planck 2018 TT measurement — it is a representation of data, and nothing between two extrema is claimed. The marks along the top are not: they are computed from this collection's own cosmology as ℓₐ(m − 0.267), where ℓₐ = π × 13866 / 144.43 = 301.6 — π times the comoving distance to last scattering divided by the sound horizon there — is the angle the sound horizon subtends at last scattering turned into a multipole. The two agree to 2.8 per cent at worst across six peaks, which is the whole of what makes this a measurement of geometry: a wave of known physical wavelength, seen at a known distance, is a protractor. The first peak at ℓ = 220 corresponds to about 0.82 degrees on the sky — roughly twice the width of the full Moon, which is the largest hot and cold patch the sky has.

Cosmology

A standing wave frozen at one instant

The temperature of the microwave background varies across the sky by one part in a hundred thousand, and the sizes of the patches are not random. There is a preferred angular scale near one degree, and it is a sound wave that stopped ringing four hundred thousand years after the beginning.

One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out.

Cosmology

Two coincidences with one mechanism

Opposite sides of the microwave sky were never in causal contact and have the same temperature to one part in a hundred thousand. The total density sits on the knife edge of flatness, which is an unstable equilibrium. Two fine-tunings of quite different kinds, and one epoch of accelerated expansion removes both.

A slice 1,000 megaparsecs across. 872 galaxies in a wedge 1000 comoving megaparsecs deep, generated as a Thomas process — Poisson centres at 0.00012 per square megaparsec, 13 galaxies each on average, scattered about their centre with a Gaussian of 22 Mpc. That is not a simulation of how structure formed and does not pretend to be; it is the simplest clustered process whose correlation function is known exactly, which is what lets the next figure check a measurement against something rather than against itself. What it does reproduce is the impression: at this scale the distribution is obviously not uniform, there are groups and there are gaps, and the eye finds patterns in it readily — including several that are not there, because the eye finds patterns in a Poisson field too. The question the next two figures ask is the only one that settles it: at what separation does the clustering stop, and is there a scale above which a box of this universe looks like any other box?

Cosmology

Homogeneous above a hundred megaparsecs

Every cosmological calculation in this field assumes the universe is the same everywhere. Looked at on any scale a person can picture, it plainly is not — it is stars in galaxies in groups in clusters in filaments around voids. The assumption is not a claim about appearance; it is a claim about a statistic, and the statistic has a scale attached.

Fingers 3.1 times long, a large scale squashed to 0.92, and a test of gravity. Left: a 360-megaparsec slice of a clustered universe, as the galaxies actually sit. Right: the same galaxies as a redshift survey records them, with the line of sight up the page. Nothing has moved sideways, because an angle is an angle; every displacement is along the line of sight, because that coordinate came from a redshift and a redshift is the expansion plus whatever the galaxy is doing on its own. Two effects, opposite in sign and separated by scale. Inside a cluster the motions are virial and random, 720 kilometres a second of them, which at H₀ = 67.36 is 11 megaparsecs of smearing on an object a few across: the clusters become fingers 3.1 times longer than they are wide, all pointing at the observer, which is the one structure in cosmology that is definitely not real. On the scale of a supercluster the motions are coherent — everything is falling in, so the far side is approaching and the near side receding — and the structure is compressed rather than stretched, to 0.92 of its true extent here, measured on cluster centroids so the fingers have already averaged away. Below: why that compression is worth having. Its amplitude is the rate at which structure is currently growing, and the growth rate is Ωₘ(z) raised to a power that general relativity fixes at about 0.55. A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes that exponent and nothing else, and the two curves drawn — γ = 0.55 and γ = 0.68 — differ by only 4 per cent at redshift a half, against error bars of 12 per cent on the points beside them. The worst systematic in a redshift survey is the measurement — and it is a hard one, because a quarter of a change in the exponent that governs how gravity assembles structure moves the observable by less than the width of the curve it is drawn on. What the picture cannot show is the degeneracy that limits it: what is measured is fσ₈, a product, and separating the growth rate from the amplitude of clustering needs something else entirely.

Cosmology

A map that is not of positions

One axis of every redshift survey is not a distance but a velocity, and the difference is not noise. Inside a cluster it smears the galaxies into a finger pointing at the observer; on the scale of a supercluster it compresses the structure — and the amount of that compression is a test of gravity.

Cosmology

A map stretched by the thing it measures

A redshift survey plots galaxies at distances derived from their redshifts, and a galaxy's redshift contains its own motion as well as the expansion. So the map is systematically distorted — squashed on large scales, drawn out into radial spikes on small ones — and both distortions are caused by the gravity the survey exists to measure.

Stars

The flow that narrows its own channel

Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand.

A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.

Galaxies

A stream is not the orbit it came from

A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one.

The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

Gravitation

An answer obtained along a path that was not taken

Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all.

$Dynamical friction is strongest at 0.97 times the dispersion, and weaker either side. The dynamical friction on a body moving through a Maxwellian sea of stars, against its own speed in units of the square root of two times the velocity dispersion. The dashed curve is the fraction of the field moving slower than the body, which is the only part of the field that contributes: a star overtaking from behind pulls the body forward exactly as often as one being overtaken pulls it back, so the fast tail cancels out entirely. That fraction rises from nothing to one. The drag is that fraction divided by the square of the speed, and the quotient of a saturating numerator and a growing denominator has a single maximum, here at 0.97. The consequences run in both directions. A body moving much faster than the stars around it is barely slowed at all, which is why a galaxy passing through a cluster at a thousand kilometres a second does not sink; and a body already at rest with respect to the field feels nothing either, because there is no wake to be behind it. Sinking is therefore fastest in the middle of the process rather than at the start of it.$ Gravitation

A drag that is strongest in the middle

Dynamical friction is not like air resistance. Only the stars moving slower than the body contribute at all, so the drag rises from nothing, peaks when the body has slowed to about the speed of everything around it, and falls away again — which is why a fast satellite is never captured and a slow one sinks in a hurry.

An edge where two torques balance, 21 kilometres from the shepherd. Two torques on the edge of a ring, against distance from a shepherding moon, both axes logarithmic and both scaled by the same combination of surface density, radius and orbital rate so that only their shapes are being compared. The flat line is the viscous torque, which comes from collisions between ring particles and does not care how far away anything is; it is drawn for a kinematic viscosity of 12 square centimetres a second, within the range ring seismology gives. The falling line is the moon's, summed over the first-order resonances that crowd together as the gap narrows, which makes it an inverse cube. A flat curve and an inverse cube cross once, and the crossing is where an edge can sit: closer in the moon wins and pushes the material back, further out viscosity wins and the ring spreads. For Daphnis and the Keeler gap the balance lands 20.8 kilometres out against a measured half-width of 21, which is agreement to well inside the uncertainty on the viscosity — and it is the only handle anybody has on that viscosity, since the quantity being inferred is the collision rate among particles a metre across, a billion kilometres away.

Gravitation

An edge is a balance, not a boundary

A ring of particles spreads, because collisions move angular momentum outward. Something has to push back, and the something is a small moon whose torque falls as the inverse cube of the gap. A flat curve and an inverse cube cross once, and the crossing is the sharp edge — which is why an edge exists at all rather than a gradient.

Exoplanets

Capture is a direction, not a strength

A resonance holds a body that drifts into it from one side and lets go of one that drifts out the other way, and the asymmetry is not about how strong the resonance is. It is the sign of a derivative — whether the trapped region is growing or shrinking — which is why a chain of planets in resonance is direct evidence that they migrated toward each other.

Relativity switches the cycle off, halving its reach at a ratio of 0.80. The greatest eccentricity a Kozai–Lidov cycle reaches, against the strength of the orbit's own relativistic pericentre precession, measured in units of the cycle's own precession rate at zero eccentricity. The horizontal axis is logarithmic and spans three decades. At the left the relativistic term is negligible and the cycle reaches 0.838, which is the closed-form value for a start at 65 degrees and is what the integration is checked against. At the right it is gone. The mechanism depends on the pericentre staying put while the outer body pulls on the same side of the orbit for a whole cycle, and the relativistic precession is a competing rotation of that same pericentre; when it is faster, the pull averages away. The threshold sits near one by construction and the transition is sharp rather than gradual, with the reach halved at 0.80. What makes it matter is where the relativistic term is largest: it grows as the pericentre falls, so it strengthens exactly as the cycle drives the orbit inward, and it therefore sets a floor on the pericentre distance that this mechanism can deliver a body to.

Orbits

The precession that switches the cycle off

A distant companion can trade an orbit's inclination for its eccentricity, over and over, and drive a pericentre almost onto the central body. General relativity's own precession competes with the mechanism for the same pericentre, and when it wins the cycle stops — sharply, at a ratio of one, which puts a floor on how close anything can be delivered.

Exoplanets

A torque that nearly cancels

A planet embedded in a gas disc pulls on the material inside its orbit and outside it, and the two torques are almost equal and opposite. What survives the subtraction is a per cent of either, and it is still enough to carry a planet from where it formed to its star in less time than the disc lasts.

$The Sun's internal rotation: differential above 0.693 R, rigid below, and the 15 nHz between two modes. Angular velocity against fractional radius, in nanohertz, from the inversion of hundreds of thousands of measured rotational splittings. Two features had to be discovered rather than deduced. The convection zone rotates differentially in latitude — 473 nanohertz at the equator, 330 at sixty degrees — and that latitude dependence persists all the way down through it, in surfaces that are very nearly radial rather than the cylinders a rotating convective fluid was expected to produce. And below about 0.693 of the radius the latitude dependence stops: the radiative interior turns as a single rigid body at 430 nanohertz, which is a remarkable thing for a fluid with no strength to do, and requires something — most likely a weak internal magnetic field — to be enforcing it. The two regimes are joined by a shear layer a few per cent of the radius thick, the tachocline, and it is where the solar magnetic cycle is generally thought to be generated, because it is the only place in the Sun with the shear a dynamo of that strength needs. The horizontal marks are what individual modes would report: the rotation averaged over the cavity each one occupies, weighted by the time the wave spends at each radius. A mode of degree 100 is trapped near the surface and reports 473 nanohertz; one of degree 1 passes through the deep interior and reports 458. The 15-nanohertz difference between them is the measurement, and splittings are measured to about a nanohertz — which is why the profile can be resolved at all. The kernels used here have the right support and the right sense; a real inversion uses computed eigenfunctions, and its resolution below 0.2 R is poor for the reason the previous figure gives.$ Stars

A fluid that turns as one piece

The Sun's surface turns faster at its equator than at its poles, and everyone expected the inside to do something similar. Below seven-tenths of the way down it does not — the radiative interior rotates as rigidly as a bell, and nothing in hydrodynamics makes a fluid do that.

Gravitation

How much a world gives

A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside.

Drift against thermal inertia: a peak at Γ = 93, in the same place for every size. How fast an asteroid's orbit drifts under its own re-radiated heat, against the thermal inertia of its surface, at a rotation period of 4.3 hours and 1.13 astronomical units. Both axes are logarithmic, and the curves are four diameters. The non-monotonic shape is the content. A surface that conducts nothing re-radiates its heat the instant it receives it: the emission is then symmetric about the sub-solar point and the transverse push cancels exactly. A surface that conducts perfectly is isothermal, has no temperature contrast at all, and again pushes nowhere. The force lives between those two nothings, and peaks where the surface's thermal time constant is comparable to the rotation period — here at Γ = 93 in SI units, and at the same place on every curve, because the size scales the drift without moving the optimum. That separation is what makes the effect a measurement. A drift rate on its own is a single number with several unknowns in it; a drift rate together with a size from radar, a spin from a light curve and a density from a flyby leaves the thermal inertia as the only thing not measured, and solving for it says what the surface is made of. Fine dust sits near 50, bare rock in the thousands, and the values measured for the bodies spacecraft have visited — Bennu at 310, Ryugu at 225, Itokawa at 700 — straddle the peak, with the two rubble piles a factor of two or three above it and the Moon's dust well below. Being past the optimum is not a small effect but it is a gentle one: the curve falls as one over the thermal inertia on that side, so a surface three times more conductive than optimal still drifts at a third of the best rate, while one three times more insulating drifts at a third as well. The shape is symmetric in the logarithm, which is why the measurement is a good one for telling dust from pebbles and a poor one for telling pebbles from boulders. The curve is one-dimensional linear theory for a rotating half-space: it has the right limits and the right peak, and it omits the body's shape, which for an irregular asteroid changes the answer by tens of per cent.

Orbits

A drift rate that says what the surface is made of

The thermal recoil that moves an asteroid's orbit depends on how long its surface holds heat, and the dependence is not monotonic — a perfect insulator and a perfect conductor both push nothing. The peak in between means a measured drift is a measurement of thermal inertia, which is a measurement of grain size.

$Where each gravity harmonic gets its signal: J₂ from the bulk, J₁₀ from the outer 12 per cent. Why a spacecraft that never enters a planet can say something about its depth. Each zonal harmonic of the external field is an integral over the interior density weighted by r to the power of the degree plus two, and the curves here are those integrals accumulated outward: the fraction of each coefficient that has been contributed by the time the integration reaches a given fractional radius, for an interior of polytropic index 1. The weighting climbs steeply with degree, so the curves separate. Half of J₂ comes from inside 73 per cent of the radius, and half of J₁₀ from inside 88 per cent — the higher coefficients barely know the deep interior exists. That ordering is the whole basis of gravity science as a probe. A single coefficient is one number and constrains almost nothing; a series of them, each weighted differently, is a coarse depth profile, and it is how Jupiter's core turned out to be smeared over half the planet rather than sitting as a distinct sphere at the middle. Two limits are worth stating with it. The information falls off fast: by degree ten the kernel is concentrated in a shell so thin that measuring the coefficient says little about anything below it. And every curve here assumes north–south symmetry, under which the odd harmonics vanish identically — so a measured J₃ or J₅ is not a deeper probe of the same thing but a measurement of something else entirely, which at a giant planet is how fast and how deep the winds run.$ Gravitation

A core weighed by something that never went in

The external gravity field of a planet is a series, and each term of it is an integral over the interior density weighted by a different power of radius. Measure enough terms and the series becomes a coarse depth profile — which is how Jupiter's core turned out to be smeared over half the planet rather than sitting at the middle.

Dissipation against viscosity for Io: a peak at 10^13.5 Pa s, and two solutions at the observed rate. The imaginary part of the Love number — the part that turns tidal work into heat — against the viscosity of the body's interior, for a Maxwell rheology at Io's size, density and forcing period. Both axes are logarithmic and the curve is not monotonic, which is the whole content of the figure. At high viscosity the body is elastic: it stores the energy the tide puts in and gives it back, and dissipates nothing. At low viscosity it is fluid: it deforms all the way and does so in phase with the forcing, and dissipates nothing again. Everything happens in between, at viscosities for which the Maxwell time — viscosity divided by rigidity — is comparable to the orbital period, and the peak here is 0.735 at 10^13.5 pascal seconds. Two consequences follow, and they pull in opposite directions. The peak is an upper limit: a homogeneous body of this size cannot dissipate more than that however its viscosity is chosen, so a measured heat flow above it would refute the model rather than constrain it. And below the peak the observed value is met twice — at 10^11.5 and at 10^15.5 pascal seconds — so a heat flow alone does not say which side of the peak the interior is on. Breaking that degeneracy needs a second observable, and the usual one is the phase of the response rather than its size. The picture treats the body as one homogeneous Maxwell solid, which is certainly wrong for a moon with a molten layer; a partial melt concentrates the dissipation and shifts the peak.

Gravitation

One heat flow, and two viscosities

Io radiates a hundred thousand gigawatts of tidal heat, and that number is supposed to say something about the rock inside it. It does, and not what one would expect — because dissipation vanishes at both extremes of viscosity, the measured heat is produced by two different interiors and cannot choose between them.

An oscillation that does not matter, on a ramp that does. Stored angular momentum in a reaction wheel over 160 days at 550 kilometres, with a capacity of 25 newton metre seconds. The total environmental torque is 1.75e-4 newton metres, of which 35 per cent is taken to survive averaging over an orbit. The fast oscillation is the part that does not survive: it has the 95.6-minute orbital period, reaches 0.10 newton metre seconds, and returns to where it started every revolution, so it consumes capacity and nothing else. The ramp under it is the secular part, and its slope measured between two instants a whole number of orbits apart is 6.117e-5 newton metres, which is the secular torque and is how the figure checks itself. The wheel fills in 4.7 days and has to be emptied 33 times in the span drawn. Every attitude-controlled spacecraft in the collection lives on this sawtooth, and the vertical drops are the only part of it that costs anything: the store can be moved between wheels for nothing, and taken out of the vehicle only by pushing against something outside it.

Spaceflight

The spin that has to be put somewhere

A spacecraft holding an attitude is not resisting a force. It is absorbing a slow, one-directional trickle of angular momentum from the gradient of gravity across its own body, from sunlight, from the last of the atmosphere — and every store it has for that trickle fills up.

The observed sky

A tilt that is not a constant

The Earth's axis leans by 23.4 degrees, and that lean is what makes the seasons. It is also a dynamical variable with its own equilibria, its own resonances and its own chaos — and on Mars the same variable has swung between nearly zero and sixty degrees without anything having to happen.

A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.72:0.5, with the principal axes of the surrounding tidal field drawn across it at 30 degrees to its own. The torque is proportional to the difference of the patch's principal moments times the sine of twice that angle, so it is exactly zero when the two sets of axes agree — checked at both alignments — and largest at forty-five degrees. A spherical patch takes no torque whatever the field around it does, which is why the spin of every galaxy begins as a statement about its shape. Right, the angular momentum against time in units of the turnaround time: in linear theory the torque acts on a patch still expanding with the universe and the angular momentum grows as the first power of time, measured off the drawn curve as t^1.000. At turnaround the patch detaches from the expansion, its quadrupole shrinks, and the torque switches off — so a galaxy's spin is fixed before it has collapsed at all, by neighbours it will never interact with again. What the figure cannot show is the sign: the same mechanism gives no preferred direction, and the observed near-absence of alignment between neighbouring galaxies' spins is the check on that.

Cosmology

Spin acquired before there was anything to spin

Every galaxy turns, and nothing in a smooth expanding universe turns. The rotation was applied while the material was still a mildly overdense patch spread across megaparsecs — torqued by the tidal field of its neighbours, growing steadily with time, and switching off the moment the patch stopped expanding.

Orbits

A cut-off period that is an age

Plot eccentricity against orbital period for the binary stars of one cluster and the picture has a wall in it. Below a certain period every orbit is circular; above it, the original spread survives untouched. The wall moves outward as the cluster ages, and where it stands is a measurement of how stars dissipate a tide.

A shear layer that should have spread

The Sun's convection zone turns differentially — its equator laps its poles about once every three months — and the radiative interior below turns as one rigid piece. Between them is a transition four per cent of the radius thick. Nothing in hydrodynamics keeps a velocity discontinuity that thin for four and a half billion years.

Galaxies

An arm that is undone by the work it does

A spiral pattern takes angular momentum from the inner disc and delivers it to the outer, which lets mass move inwards without violating anything. It is paid for in random motion, and a disc with too much random motion cannot carry a wave — so the pattern destroys the conditions it needs to exist.

Exoplanets

A misalignment only cool stars forget

A third of hot Jupiters orbit at a large angle to their star's equator, and some go round backwards. Sort the same planets by the temperature of their host and the picture changes — below about 6,250 kelvin almost all are aligned, and above it almost none are. The boundary is not about the planets.

The observed sky

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

A vanishing field that changes the answer completely. The growth rate of the magnetorotational instability against wavenumber, both in units the orbital frequency sets: the vertical axis is the growth rate divided by Ω, the horizontal is the wavenumber multiplied by the Alfvén speed and divided by Ω. The maximum is exactly three quarters of the orbital frequency and it does not depend on the field strength at all — it occurs at kv_A = √15Ω/4, so a weaker field simply moves the fastest-growing wavelength to a longer one. The curve vanishes above kv_A = √3Ω, which is the only thing the field decides: modes shorter than that are stabilised by magnetic tension. An unmagnetised Keplerian disc is Rayleigh-stable and would never accrete; a disc with a field a millionth of the strength needed to matter dynamically grows a mode that doubles in about a fifth of an orbit. The limit is discontinuous, which is the reason this was found late and by algebra rather than early and by observation.

Gravitation

The weakest field changes the answer

A Keplerian disc is stable by every hydrodynamic test there is. Add a magnetic field of any strength whatever — a millionth of what would matter dynamically — and it becomes violently unstable, at a growth rate of three quarters of an orbit per radian that does not depend on the field at all.

Three motions, three decades apart, and that is the point. The three periods of a trapped 1-MeV electron's motion against the shell it is trapped on, on a logarithmic time axis. It gyrates about a field line in 0.22 milliseconds, bounces between its mirror points in 0.33 seconds, and drifts right round the planet in 16 minutes — ratios of 1498 and 3015 at L = 4. Each motion carries a conserved quantity: the magnetic moment, the longitudinal invariant, and the magnetic flux the drift shell encloses. The separation is what makes them conserved. An invariant survives anything that changes slowly compared with its own period, so a disturbance lasting minutes destroys the third and leaves the first two untouched — and a particle that keeps its magnetic moment while being moved inward to a stronger field must gain energy. That is not a loophole; it is how the belts are filled.

Spaceflight

Three clocks and nothing to fall onto

A charged particle in a dipole field gyrates, bounces and drifts, on timescales a millisecond, a second and a quarter of an hour. The three periods are three decades apart, and that separation is not a curiosity — it is the reason each motion has a conserved quantity, and the reason a magnetic storm can accelerate particles rather than merely stir them.

One step of memory, and it is kept at the poles. Two predictors of a solar cycle's amplitude, each tested against the seven cycles for which both quantities exist, with amplitudes scaled so that cycle 21 is one. Left: the polar field measured at the minimum before the cycle begins, which correlates with what follows at r = 1.00. Right: the amplitude of the previous maximum, which correlates at r = 0.34 — that is, not at all. The cycle is therefore not a pendulum with momentum; it is a process with exactly one state variable, and the variable is the poloidal field that the decay of the previous cycle's spots leaves behind at high latitude. That is the observational core of the flux-transport picture: spots emerge tilted, their trailing polarity drifts poleward, and what accumulates there is the seed the next cycle's shear will wind up. It is also the only prediction in solar physics with a lead time of years, and it was what said, correctly and unpopularly, that cycle 24 would be the weakest in a century.

Stars

One step of memory kept at the poles

The amplitude of a solar maximum tells almost nothing about the next one. The polar field measured at the minimum in between tells a great deal — so the cycle is not a pendulum with momentum but a process with exactly one state variable, and the variable is a field of a few gauss at latitudes nobody can see well.

The observed sky

The flare that arrives from somewhere else

The solar wind blows radially outward and the Sun rotates, so the field frozen into the wind is wound into a spiral making forty-five degrees to the radius at the Earth. Energetic particles follow the field rather than the line of sight, which is why the flares that deliver particle storms are the ones on the western limb rather than the ones facing the planet.

A resonance with the planet's rotation, and it sorts by size. Above: the radii at which a charged grain's orbital frequency is commensurate with the planet's spin, in planetary radii, with the measured edges of Jupiter's halo ring drawn over them. The synchronous radius is 2.24; the 3:2 resonance falls at 1.71 and the 2:1 at 1.41. The halo's outer boundary is at 1.71 and its inner extent near 1.4 — the ring ends where the resonances are, and both numbers were measured by a spacecraft camera with no reference to this arithmetic. Below: which grains care. The charge on a grain is proportional to its radius and the mass to the cube, so the force per unit mass goes exactly as the inverse square of the size, and the resonance grips sub-micron dust while leaving anything larger on a Keplerian orbit. That is why the halo is a cloud of fine dust puffed a thousand kilometres out of the ring plane while the coarse material stays flat: the same field acting on the same orbit sorts the material by size, which no gravitational resonance can do.

Orbits

A resonance with the planet itself

Every other resonance in celestial mechanics is a commensurability between two orbits. A charged dust grain has a third clock available — the planet's rotation, which sweeps its magnetic field past the grain — and the commensurability with that selects by charge-to-mass ratio, which means by grain size.

The two halves of what is left over. The disturbing potential of a perturber on a test particle, against the difference in longitude between them, at a semi-major axis ratio of 0.62. The upper curve is the direct term — the perturber's own attraction, which peaks at conjunction where the separation is smallest and falls to 1/(1+α) half a turn later. The lower one is the indirect term, which exists only because the coordinates are centred on a primary that is itself being accelerated, and which is a pure cosine of the longitude difference. The indirect term is the larger of the two over 3 per cent of the circle, and it averages to exactly zero while the direct term averages to something positive. Everything that happens slowly in a planetary system comes from that asymmetry: the part that survives averaging is not the part that dominates the instantaneous force.

Orbits

The series that is subtracted

The two-body problem is solved, so nobody solves it twice. Every planetary theory since Newton begins by taking that solution away and asking what is left — and what is left is an infinite series whose terms are stacked in a hierarchy that makes the first half-dozen of them enough.

A branch that adds nothing above 1 km and everything below it. Cumulative crater counts on a 3.5-billion-year-old surface, with the population split into the craters made by objects arriving from outside and the craters made by blocks thrown out of larger ones on the same surface. The two are indistinguishable in a photograph and completely different as a statistic. Secondaries stop at about 1 kilometre, because that is the largest crater a block leaving at a few hundred metres a second can excavate, so the upper half of the plot is unaffected. Below it they are steeper — slope -3.2 against the primaries' -2 — and by the smallest diameter drawn they outnumber the primaries 292 to one. A count taken at 100 metres and read through the primary production curve returns an age of 4.34 billion years for ground that is 3.5, and it returns it with a small formal error, because the counting statistics are excellent. The error is not in the counting.

Orbits

The craters that were not primary

Counting craters dates a surface, and the method works because impacts from space arrive at a known rate. Some of the holes were not made from space. They were made by rock thrown out of the larger holes on the same surface, and they are far more numerous than anything that arrived.

Orbits

An acceleration that was the spacecraft's own heat

Two probes leaving the solar system were tracked for thirty years and both drifted from their predicted paths by a tenth of a nanometre per second squared. The residual was real, it was constant, and it was the same on both. It was also the waste heat of the reactors that powered them, radiating slightly more one way than the other.

Orbits

The drag that sorts a disc by size

Starlight does three different things to a dust grain depending on how big it is — blows it out of the system, drags it inward over millennia, or ignores it entirely. Which one happens is decided by a single length, and whether it happens at all is decided by how crowded the disc is.

A moment of inertia that is a shape plus an assumption. The Darwin–Radau relation: the polar moment of inertia a body would have, given the ratio of its rotation parameter to its flattening, if it were a hydrostatic fluid. The curve passes through exactly 0.4 at q/f = 0.8, which is the uniform sphere and the one point that needs no interior model, and falls as the body becomes more centrally condensed. Four of the five worlds drawn sit on it within a few per cent of their independently measured moments, which is what makes the relation usable at all. The world that does not appear on the plot is the Moon, whose q/f is 0.025 — far outside the window in which the relation has a real solution, because its shape is a fossil frozen in when it was much closer to its primary and has nothing to do with its present rotation. That failure is the useful one. A relation that returns a wrong answer quietly is dangerous; this one returns no answer at all.

Gravitation

The part of a shape the spin cannot explain

A rotating planet bulges by a predictable amount. Subtract that amount from the shape actually observed and something is usually left over — a few parts in a hundred thousand for the Earth, and ninety-seven per cent of the whole bulge for the Moon. The residue is not an error. It is the only remote measurement of what a planet's interior is doing that does not average over the whole body.

A step size below which a smaller step is worse. The error left in a long integration against the step size, for methods of three different orders, with both contributions drawn. The falling lines are truncation error, whose slope on these axes is exactly the order of the method. The rising line is round-off, identical for all three because it is a property of the arithmetic and not of the algorithm: every operation loses a few bits, the losses are independent, and they accumulate as the square root of the number of steps — which is why its slope is −1/2 and why it rises as the step shrinks. Each method's total has a minimum, at a step of 1.0e-6, 5.0e-6, 9.7e-4 for orders 1, 2, 4. Below that minimum every halving of the step costs time and makes the answer worse. That is the practical reason a solar-system integration is not run at an arbitrarily fine step, and it is a reason with nothing to do with computer time.

Gravitation

An error that grows like a random walk

A long integration accumulates two errors with opposite habits. One falls when the step is made smaller and grows in proportion to the time; the other grows when the step is made smaller and accumulates as a square root. Which of the two dominates decides whether a billion-year integration means anything.

$One measurement, one line, and every point on it is an interior. The Love number against the quality factor, both logarithmic. An orbital measurement — a moon observed to be receding, a spin observed to be slowing — determines only the ratio of the two, so it picks out a diagonal band rather than a point, and every interior along that band reproduces the observation exactly. A body that deforms twice as easily and dissipates half as efficiently is indistinguishable from one that does the opposite. What breaks it is a measurement of the deformation on its own: a spacecraft tracking the body's gravity field through a tidal cycle measures k₂ directly, which is a vertical line here, and the intersection gives Q = 5.36·10⁴. The uncertainty on that answer is the two fractional errors added in quadrature, 20 per cent, and it is dominated by whichever of the two was worse — which for every body in the solar system is the orbital rate rather than the Love number.$ Gravitation

A heat flow that depends on a number nobody can compute

Every tidal rate in astronomy — a moon receding, a spin slowing, an orbit circularising, a satellite melting — is proportional to one combination of two quantities that no orbital measurement can separate. One of them describes how much a body deforms and the other how badly it leaks, and only a spacecraft can tell them apart.

$A star drawn out into 3.0 arcseconds of spectrum. Atmospheric refraction relative to its value at 550 nanometres, against wavelength, at four zenith angles. The air's refractive index rises towards the blue, so the blue image of a star sits above the red one and the object is smeared into a short vertical spectrum. At 60 degrees from the zenith the separation across an optical band is 2.96 arcseconds — several times the size of the image at a good site, and comparable to the width of a spectrograph slit. Every curve here is the same curve multiplied by the tangent of the zenith angle, which is why one corrector with an adjustable strength works at every airmass. The practical consequences are three: a slit aligned other than vertically loses blue light or red light depending on where it was centred, a photometric aperture contains a different fraction of the light in each band, and an astrometric position depends on the colour of the star it is measured from.$ The observed sky

The atmosphere is a prism as well as a lens

Refraction lifts a star towards the zenith, and everybody corrects for that. It lifts blue light further than red, and the difference is a short vertical spectrum a few arcseconds long — larger than the image, larger than a spectrograph's slit, and quietly present in every ground-based measurement not taken straight overhead.

The observed sky

A solar radius measured past a mountain range

The most accurate way to measure the Sun's diameter is to time an eclipse. What is timed is the moment sunlight vanishes behind the Moon's edge — and the Moon's edge is a horizon with mountains on it, so the measurement is a difference between the Sun's limb and a lunar landscape that has to be supplied from somewhere else.

Two longitudes a satellite falls towards, and 0.5 m/s a year to stay elsewhere. The along-track potential a geostationary satellite feels, against longitude. The Earth's equator is slightly elliptical — about seventy metres between its long and short axes — and that one harmonic of the gravity field gives the geostationary ring two minima and two maxima. A satellite left unattended drifts towards the nearer minimum at 75° or 255° east, overshoots, and librates about it with a period of a couple of years. The peak acceleration accumulates 0.5 metres a second of velocity change a year, and an operational east–west budget is a small multiple of that once the correction cycle is accounted for — a fixed cost of every commercial slot in the ring, paid forever. The two minima are the graveyard of the geostationary population: uncontrolled satellites accumulate there, which is why they are the most crowded longitudes in the belt and why an uncontrolled object is most likely to be found near one.

Spaceflight

A satellite that drifts to one of two longitudes

The Earth's equator is elliptical by about seventy metres. That one harmonic of the gravity field gives geostationary orbit a potential with two minima, so an unattended satellite slides towards the nearer one and stays — and every operational slot in the ring is paid for with fuel, every year, forever.

A density that spans a factor of 25 at 400 kilometres. Thermospheric density against altitude, for three levels of solar activity, with the model's own uncertainty band drawn around the middle curve. The extreme ultraviolet output of the Sun heats the upper atmosphere, so the scale height rises with activity and the density at a fixed altitude rises with it — by a factor of 25 at 400 kilometres between solar minimum and maximum. Superposed on that are a diurnal bulge of about a factor of two, semiannual variations, and geomagnetic storms that raise the density by tens of per cent within hours. The best empirical models reproduce past conditions to about 15 per cent, and orbital lifetime is inversely proportional to density, so a re-entry predicted a year ahead carries that error and the far larger one of not knowing what the Sun will do.

Spaceflight

A density model wrong by a factor of two

Everything about a low orbit's future depends on the density of the air at four hundred kilometres, and that density varies by a factor of twenty-five over the solar cycle, by two within a day, and by tens of per cent during a storm nobody predicted. Every model of it is an empirical fit, and re-entry dates are quoted with the honesty that implies.

A patch that throws away 2.5 to 436 metres a second. The velocity error a patched-conic approximation makes at each planet's sphere of influence, on a logarithmic scale, for an approach at 3 kilometres a second. The sphere of influence is where the two ways of writing the problem — planet-centred with the Sun as a perturber, or Sun-centred with the planet as one — become equally bad, and at that radius the neglected solar tide is exactly two times the fifth root of the planet's mass ratio times the planet's own pull. That is drawn beside each planet, it runs from 0.09 to 0.50, and it is not the same for all of them — a factor of 5.6 that the definition does not remove. Because both neglected terms are largest exactly at the surface where the switch is made, the trajectory has a discontinuity in its acceleration and an accumulated velocity error of metres a second. That is negligible for a mission design and enormous for a navigation solution, which is why patched conics are used to find a trajectory and never to fly one: the real trajectory is obtained by numerically integrating the full n-body problem, differentially corrected onto the patched-conic solution as a starting guess.

Spaceflight

The discontinuity a patched conic hides

An interplanetary trajectory is designed as two exact solutions glued along a surface where neither is valid. At that surface both neglected forces are at their largest, so the stitched path has a kink no real trajectory has — and the size of the kink is metres a second, which is a rounding error for a mission design and a catastrophe for a navigation solution.

$Three perturbing masses drawing one curve. Transit-timing residuals for three systems whose perturbing planets are 12, 8, 5 Earth masses — a factor of 2.4 apart — each given the inner-planet eccentricity that the near-resonant theory says will compensate: 0.0000, 0.0166, 0.0363. The three curves have amplitudes of 1.7, 1.7, 1.7 minutes, within 0 per cent of each other, and they are drawn by three separate integrations that were told nothing about the theory used to pick the eccentricities. The eccentricity enters the near-resonant term divided by Δ, the fractional distance from exact resonance — here 0.0267 — so a hundredth of an eccentricity does the work of a factor of two in mass. This is why a transit-timing mass is not a mass until something else fixes the eccentricity, and why the masses that came out of the first years of such fits were systematically lower than the radial-velocity masses of the same planets.$ Exoplanets

A mass that is only a mass once the eccentricity is known

The near-resonant part of a transit-timing signal carries the perturber's mass and the pair's free eccentricity in the same bracket, divided by the distance from resonance. A hundredth of an eccentricity therefore does the work of a factor of two in mass, and three quite different systems draw one curve.

Galaxies

A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

Gravitation

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

June sunlight at 65°N, against where perihelion sits. Daily-mean insolation at latitude 65 degrees north on the June solstice, at an obliquity of 23.44 degrees, against the longitude of perihelion measured from the March equinox — the angle that precesses right round in about twenty-one thousand years. Three eccentricities are drawn. The swing is ±10.0 per cent at e = 0.05 and ±1.0 per cent at e = 0.005, in proportion to the eccentricity, because the Sun–Earth distance on a fixed date carries e cos of the precession angle and that is first order. Over the same range of eccentricity the annual mean at this latitude moves by 0.12 per cent, because the annual mean carries 1/√(1−e²) and that is second order. The two together are the whole of the precession term in Milankovitch's theory: the eccentricity does almost nothing to how much sunlight the Earth receives and a great deal to when it arrives, and the ice sheets of the northern hemisphere respond to the summer they might melt in rather than to the year's total. It also explains why the precession signal disappears when the orbit is nearly circular: multiply a large angular swing by a vanishing eccentricity and there is nothing left, which is what the innermost curve here is.

Orbits

An average that precession cannot move

Sunlight arrives as the inverse square of the distance and time passes as its square, so the two cancel exactly in a year's integral. The longitude of perihelion therefore changes the annual mean insolation at every latitude by precisely nothing — and changes June at 65°N by ten per cent.

$A family of 276, or of 582, depending on one number. Membership of a synthetic family against the cutoff velocity used to define it, for a family of 260 genuine fragments ejected at 15 metres a second sitting in a background of 417 unrelated bodies. The clustering is single linkage in the standard proper-element metric, started from one object and grown until nothing more is within the cutoff of anything already absorbed. Three curves: the number of objects claimed, scaled to its largest value; the fraction of the real family recovered; and the fraction of the claim that is background. The shape is the whole difficulty. At low cutoff the family is fragmented and only its core is found. There is then a plateau — near 58 metres a second here, giving 276 members of which 6 per cent are background — and that plateau is what every published family list is chosen at. Past it the count runs away, because single linkage absorbs an object and then searches from the object it has just absorbed: one chance interloper bridges the family to the belt and the algorithm returns half the main belt. Completeness and contamination rise together, so there is no cutoff at which both are good and the plateau is a compromise rather than a discovery.$ Orbits

A family whose size is a choice

Nothing observable distinguishes a collisional fragment from an asteroid that happens to be nearby. Membership is assigned by clustering at a cutoff velocity, and the same family has 276 members or 582 according to which cutoff is used — with completeness and contamination rising together, so no cutoff is good.

Why the Moon holds the tilt still. Spin-axis precession rate against obliquity, with the solar system's own secular frequencies drawn across it. The rising-and-falling curves are α cos ε for three precession constants: the Earth as it is, with the Moon supplying about two thirds of the torque; and a Moonless Earth at two plausible rotation rates. The horizontal lines are the nodal eigenfrequencies of the planetary secular system — s₆, s₃, s₄, s₂, s₁, s₇, s₈ — which are the rates at which the Earth's own orbit plane wobbles. A crossing is a resonance: the axis is precessing about a plane that is itself turning at the same rate, the resonant angle stops circulating, and the obliquity librates instead of holding still. The present Earth precesses at 50.3 arcseconds a year, faster than every secular frequency in the list, and its nearest crossing is at 61.3°, which is 37.8 degrees from the obliquity drawn. Take the Moon away and α falls by a factor of three; the crossings come down with it, 3 of them land below 85°, and where several overlap the obliquity has no stable value at all. The Moon does not hold the axis by pulling on it. It holds it by making the precession too fast to resonate with anything.

The observed sky

A tilt held still by being too fast to resonate

The Moon does not hold the Earth's axis by pulling on it. It triples the rate at which the axis precesses, which lifts that rate clear of every frequency at which the Earth's own orbit plane wobbles — and a precession with nothing to resonate with cannot wander.

A factor of 100 in the wind, 8 degrees in the aurora. The latitude of the last closed field line against the solar wind's dynamic pressure, for a dipole of 0.31 gauss at the equator. The chain is short and every link is exact. Pressure balance puts the magnetopause at a distance going as the sixth root of the field pressure over the ram pressure; a dipole line reaching equatorial distance L returns to the surface at colatitude arcsin(1/√L); so the polar cap's edge is that, and the aurora sits just equatorward of it on the outermost closed lines. Across the 100-fold range of pressure drawn — which covers everything from a quiet wind to a severe storm — the magnetopause moves from 13.1 to 6.1 planetary radii and the oval from 74.0 to 66.1 degrees. The shaded band is where the oval is actually observed. The open magnetic flux is exactly the reciprocal of the standoff distance, so it rises by a factor of 2.15 over the same range — which is the quantity that actually matters for a storm, and it is the only one of the three that changes by much.

The observed sky

An aurora that is a sixth root inside an arcsine

The magnetopause distance fixes which field lines are open, the open ones map to a cap around each pole, and the cap's edge is where the aurora is. A hundredfold change in the solar wind moves that edge by eight degrees — and across planets whose fields differ by four orders of magnitude it moves by ten.

Spaceflight

The fragments nobody can see and cannot shield against

A catastrophic collision in low orbit makes about a thousand pieces big enough to track and sixty thousand big enough to destroy a spacecraft. The catalogue is under two per cent of the hazard, the gap is not one better radars will close, and the fragments' lifetimes span a factor of twenty-five within a single event.

Stars

Three clocks and a runaway

Whether mass transfer between two stars is stable is a comparison of two logarithmic derivatives. How fast it runs is a separate question with three possible answers fourteen orders of magnitude apart — and the answer decides whether the companion accretes, is buried, or is swallowed.

$Where the 9.4 km/s should be divided between two unequal stages. Overall payload fraction against the share of the 9.4 km/s given to the first stage, for a vehicle whose two stages are not alike: kerosene and oxygen, first stage at ε = 0.06 and I_sp 300 s, so vₑ = 2.942 km/s; hydrogen and oxygen, second stage at ε = 0.09 and I_sp 450 s, so vₑ = 4.413 km/s. The curve falls to zero at both ends, because a stage asked for too much Δv has a negative payload fraction and the vehicle does not close. The maximum is at 26.8 per cent — 2.52 km/s in the first stage and 6.88 in the second — and it delivers 5.130 per cent of the lift-off mass as payload against 4.240 per cent for an equal division. The gain from optimising is 21.0 per cent of the payload — worth having on a vehicle whose payload is five per cent of its mass, and small next to the difference the second stage's exhaust speed makes. What the shape says is more useful than where the peak is: moving ten points either side of the optimum still delivers 4.905 per cent, so the penalty for getting the split wrong is 4.4 per cent of the payload. A penalty that small is why launch vehicles are staged on structural and operational grounds — where the tank domes go, which engines exist, what can be transported by road — and the calculus is run afterwards to check that nothing has been left on the table. Note also which way the optimum leans: the better stage is the second, and it is given more of the work, because a share of Δv bought at a higher exhaust speed costs less mass.$ Spaceflight

The split that is not an equal split

Two stages sharing a velocity budget do not share it evenly, and the calculus that divides it hands more of the work to whichever stage has the better exhaust speed. The optimum is interior, it is worth about a fifth of the payload, and at this mission it is flat enough that nobody designs to it.

Exhaust speed against the molecular weight of the exhaust. The ideal exhaust speed of a converging–diverging nozzle, √(2γ/(γ−1) · R_uT_c/M · [1 − (p_e/p_c)^((γ−1)/γ)]), against the mean molar mass of the exhaust, at three chamber temperatures — 2200, 3000, 3600 K — with γ = 1.2 and an expansion to 1 per cent of chamber pressure. Every choice a propellant makes enters through two symbols, and the speed goes as the square root of their ratio: four times the chamber temperature doubles it, and a quarter of the molar mass doubles it too. That symmetry is the point. A cooler flame with a lighter exhaust beats a hotter one with a heavier, and the marked combinations show it — hydrogen + oxygen at M = 10, T_c = 3500 K, 441 s; methane + oxygen at M = 20.5, T_c = 3550 K, 310 s; kerosene + oxygen at M = 23, T_c = 3670 K, 298 s. The hottest flame drawn belongs to kerosene + oxygen and the fastest exhaust to hydrogen + oxygen, which are not the same entry. Hydrogen's advantage is not that it burns hot; it burns slightly cooler than kerosene. Its advantage is that the mixture is run fuel-rich on purpose, so the exhaust carries unburnt hydrogen and its mean molar mass falls to about 10 rather than water's 18 — buying more in the denominator than it loses in the numerator.

Spaceflight

Choosing a propellant is choosing a molecular weight

An exhaust speed is the square root of a chamber temperature divided by a molecular weight, so a cooler flame with a lighter exhaust beats a hotter one with a heavier. Hydrogen wins the rocket equation and loses the tank, and the two cannot be optimised separately.

Where the missing kilometres a second go. The two losses along a gravity-turn ascent, against the vehicle's thrust-to-weight ratio at lift-off, from an integration of the trajectory rather than from a table. Every run spends the same 9400 m/s of ideal Δv and every one is flown as the same manoeuvre: one pitch kick, solved by bisection so that the vehicle is horizontal at burnout, and thereafter zero angle of attack so that gravity alone turns it — which makes the steering loss identically zero and the comparison a fair one. What differs is how much of the Δv survives as speed. The gravity loss is ∫g sin γ dt, the part of the thrust spent holding the vehicle up rather than accelerating it, and it falls as the thrust rises because a vehicle that leaves quickly spends less time doing it: 1263 m/s at T/W = 1.15 against 381 at 2.2. The drag loss is ∫(D/m) dt and it rises, because the same haste means reaching high speed lower down where the air is: 119 m/s against 1874. The sum is least at T/W ≈ 1.5, at 1198 m/s, and the minimum is shallow — which is why real vehicles cluster between 1.2 and 1.5 and none of them is there because of this curve. At the marked 1.5 the 9400 m/s of ideal Δv leaves the vehicle at 8202 m/s and 42 km, against a circular speed of 7884 m/s there, so the 1198 m/s of loss is the whole of the answer to why orbit costs about 9.4 km/s when orbital speed is under 7.9.

Spaceflight

Orbit costs 7.8 and a launch buys 9.4

The gap between orbital speed and the velocity change a launcher spends is not overhead. It is three integrals along the ascent, only one of which can be reduced by flying better, and the two that can be traded move in opposite directions.

$The best split of a plane change between perigee and apogee, for three turns. How much is saved by moving part of the plane change into the perigee burn, against how much is moved, for turns of 15°, 28.5°, 51.6°. Doing the whole rotation at apogee is the standard answer and it is not the cheapest one: at perigee the turn is bought as a small correction to a burn that is happening anyway, so the first fraction of a degree is nearly free while the apogee saving is linear. Each curve therefore rises to an interior maximum — 10 m/s at 1.35° for a 15° turn, 25 m/s at 2.23° for a 28.5° turn, 40 m/s at 2.88° for a 51.6° turn — and falls back through zero at about twice that split. The saving is small against a 4.78 km/s budget, and it is free.$ Spaceflight

A rotation split between two burns

The standard answer is to do the whole plane change at apogee, where the vehicle is slowest. It is not the cheapest answer, and the reason is that a small part of the turn bought at perigee is a second-order correction to a burn that is happening anyway.

Rotating an orbit by flying away from it first. Total Δv against the angle turned, in units of the circular speed of the starting orbit, for two ways of rotating an orbital plane at a fixed radius. The single combined burn does everything at once and costs √(v₁² + v₂² − 2v₁v₂cos Δi); the three-burn route raises the apoapsis to 200 starting radii, turns there where the speed is only 0.7 per cent of what it was, and comes back down. Below 48.9° the single burn is cheaper and the two extra burns are not worth paying for. Above it the three-burn route wins, and it wins by more the larger the angle: at 90° it costs 0.8314 against 1.4142, a saving of 41 per cent. The mechanism is the one thing worth carrying away. A plane change costs 2v sin(Δi/2) and is therefore proportional to the speed at which it is done, so the cheapest place to turn is the slowest place available — and a vehicle can make a slow place by climbing, at a cost that does not grow with the angle while the rotation's cost does. That is why the crossover is an angle rather than a distance, and why it exists at all. What the figure does not price is time: the round trip to 200 starting radii takes 2015 times the period of the starting orbit, which for a low Earth orbit is months.

Spaceflight

Flying further away in order to turn

A plane change costs the speed at which it is done, so the cheapest place to turn is the slowest place available — and a vehicle can make a slow place by climbing. Above about thirty-nine degrees the round trip pays for itself, and the crossover is an angle rather than a distance.

A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 90-day event at impact parameter 0.3, computed with the observer's orbital motion included at three values of the microlens parallax. The dashed curve is π_E = 0 and is exactly symmetric in time, because a straight-line track past a point lens has to be. The others are not. The Earth's displacement over the months the event lasts adds a term π_E times its projected orbital motion to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away, and the curve acquires an asymmetry of 16 per cent at π_E = 0.15 and 34 per cent at π_E = 0.35. That asymmetry is the whole measurement. An ordinary event gives one dimensioned number, t_E, which mixes the lens mass with two distances and a proper motion and therefore weighs nothing; the parallax gives a second, and two constraints on the same lens are what a mass requires. It is only available on long events — the Earth has to move appreciably while the magnification is changing — which is why parallaxes are measured for the timescales above about fifty days and not for the short ones that a low-mass lens produces.

Exoplanets

An asymmetry that the Earth's own orbit puts in

A microlensing event delivers one number with dimensions, and one is not enough to weigh anything. The Earth's motion over a long event distorts the light curve, and the distortion is the second constraint — after which the mass follows with no distance in it at all.

The extreme sunset and sunrise, neither of them on the December solstice. Sunset and sunrise in local mean solar time through the ninety days around the December solstice, at latitude 52°, computed from Kepler's equation and the tilt. Each is the solar noon — which is 12:00 minus the equation of time — plus or minus half the day arc, and the two ingredients have different stationary points. The day length is stationary at the solstice, exactly, because the declination is. The equation of time is not: it is changing at about 0.50 minutes a day there, moving solar noon steadily. So the sum is still changing after the day length has stopped, and the earliest sunset comes on 14 Dec — 9 days before the solstice on 23 Dec — while the latest sunrise comes on 31 Dec, 9 days after it. The shortest day is the solstice, and neither of its ends is extreme there. Nothing in this is an approximation or a correction: it is a sum of two functions with different stationary points, and the stationary point of a sum is not the stationary point of either.

The observed sky

The earliest sunset is not the shortest day

A sunset time is solar noon plus half the day length, and the two have different stationary points. The day length stops shortening at the solstice; the equation of time does not stop moving, so the sum keeps falling — and the earliest sunset comes days before the shortest day.

The figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.

Spaceflight

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

The observed sky

A clock whose zero is moving

Sidereal time is counted from the equinox, and the equinox does not stay put. Its steady drift makes the sidereal day eight milliseconds short, its acceleration puts a quadratic term into the formula for sidereal time, and the Moon makes it nod by a second every nineteen years. The Earth rotation angle removes all three by counting from a point defined not to move along the equator.

Exoplanets

The envelope that doubles a planet lasts longest

The gap in the radii of small planets is not merely a place where planets are rare — it is nearly empty, and the reason is a peak. The time a young star needs to strip a planet's hydrogen is longest for the envelope that swells the planet to a little over twice its core's size. Anything thinner runs away to nothing, and anything thicker settles back towards the peak.

The expansion rate and the acceleration, and which of them reaches inside an orbit. Two quantities per unit distance through cosmic time, both in units of today's H₀². The square of the expansion rate, H², falls steeply from the big bang and is 1.00 today by definition. The acceleration of the expansion, ä/a, is the sum of a matter-and-radiation part, −Ωₘ/(2a³) − Ωᵣ/a⁴, drawn dashed, and the cosmological constant's part, ΩΛ = 0.685, which is the same at every time. The sum was negative — the expansion decelerating — until the universe was 7.7 Gyr old, at a = 0.614 or redshift 0.63, and is 0.527 today. The equation of motion of anything orbiting inside a bound system carries ä/a and never H: the rate at which distant galaxies recede does not appear in it at all. Of ä/a, the matter part is the mean density of the universe, which inside a galaxy or a planetary system is already counted in the mass that is doing the holding, many million times over. What is left is the constant: a fixed outward acceleration per unit distance, ΩΛ H₀², that does not grow with time and does not care how fast the universe is expanding.

Cosmology

An orbit feels the acceleration and never the rate

If space expands, it is natural to ask why the Earth's orbit does not. The answer is not that gravity resists the stretching. It is that the expansion rate never appears in the equation of motion of a bound orbit at all — only the acceleration does, and of that only the cosmological constant's part survives, as a fixed outward push that moves the Earth's orbit once, by twelve picometres, and never again.

Galaxies

The debris that returns fastest lights up last

The debris of a star torn apart by a light black hole comes back within two weeks, in greater excess of what the hole can swallow than for any heavier hole. It should make the promptest flare, and it may make the slowest — because to shine, the returning stream has to crash into itself, and where it does so is decided by how far relativity swings its orbit round. Round a light hole the swing is a few degrees, and the streams meet only near the far end of their orbit, moving slowly.

Galaxies

The stars a black hole eats come from a narrow band

A star is torn apart only if its orbit happens to point almost exactly at the hole — within a millionth of the possible directions of its angular momentum. Stars on those orbits are gone within one orbit, and the only way new ones arrive is by the slow random walk of encounters with other stars. Close to the hole that walk is too slow; far from it the orbits take too long; and most of the stars a hole destroys come from a narrow band between.

The colour of the zenith at twilight, with and without ozone. The colour of the zenith sky relative to sunlight — the 450 nm brightness against the 650 nm brightness, in magnitudes, bluer upward — against the Sun's depression, computed by single scattering with a 300 Dobson-unit ozone layer and again with none. Scattering alone favours blue by λ⁻⁴, which would make the sky 1.60 magnitudes bluer than sunlight if nothing were removed on the way; that is the dotted line. But after sunset every ray has travelled a long grazing path, and Rayleigh scattering removes blue from that path faster than red, so the two effects fight. Without ozone they very nearly cancel: the zenith is 0.01 magnitudes from sunlight's own colour at sunset and 0.06 at 6° — a pale, colourless sky — and only turns bluer, −0.13 at 10°, once the lit layer has climbed above most of the air the grazing ray used to cross. With ozone the zenith is −0.34 at sunset, −0.67 at 6° and −0.76 at 10°: bluer than without by 0.72 magnitudes at 6°, because the grazing ray also crosses the ozone layer near its tangent point, and ozone's Chappuis band absorbs orange and red rather than blue. The ozone cross-sections are approximate, and the conclusion does not depend on them to better than a factor of two.

The observed sky

Ozone keeps the twilight zenith blue

After sunset the sky overhead turns a deep blue, and scattering alone cannot explain it. The light that reaches the zenith has first grazed hundreds of kilometres of air, which strips blue out of the sunlight as fast as scattering puts it back, and the two very nearly cancel. What tips the balance is a gas that makes up a few parts in ten million of the atmosphere and absorbs the orange and red end of the spectrum.

Exoplanets

The stripping runs ahead of the starlight that drives it

If young stars carve the radius valley with their X-ray light, the valley should be finished early — half of it before the star has delivered a third of that light, and nine tenths of it within a few hundred million years. If the planets' own cooling cores carve it instead, planets should still be crossing it billions of years later. The two accounts put the valley in the same place, and they are separated by the one thing a histogram cannot show — when.

Exoplanets

A smaller star puts the valley lower

Round stars of half the Sun's mass, the gap between bare rocky cores and sub-Neptunes should sit at a smaller radius than round the Sun, because at the same orbital period their planets receive a tenth of the light. Their stars also stay young and active far longer, which pushes the other way. Photoevaporation weighs those two against each other in a definite proportion, and the answer is a valley that scales as the star's mass to about the power three tenths.

The XUV energy that reached one astronomical unit, for three young Suns. The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth's distance from the Sun, against age, for three histories that differ only in how long the young Sun stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator, 300 Myr for a fast rotator. After saturation each declines to a common track by 1 Gyr, as rotation histories are observed to converge. By 4.5 Gyr the totals are 2.06·10¹⁵ J/m² for the slow rotator, 3.66·10¹⁵ J/m² for the medium rotator, 6.47·10¹⁵ J/m² for the fast rotator: the fast rotator delivered 3.1 times as much as the slow rotator, nearly all of it in the first few hundred million years. The Sun's own rotation at that age is not measured; it is inferred from the spread of rotation periods in young clusters, and all three histories are consistent with a Sun that ends up rotating as it does now.

Exoplanets

The young Sun's spin decides what the Earth kept

The calculation that strips sub-Neptunes applies just as well to a planet with a trace of hydrogen. An Earth that captured a few hundredths of a per cent of its mass from the gas it formed in — several times the hydrogen now in its oceans — would have lost all of it, or kept most of it, depending on something nobody has measured — how fast the Sun was spinning in its first few hundred million years.

m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end.

Cosmology

The tilt knows the slope and not the height

The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it.

Starobinsky: e-folds before the end, against how reheating went. N, the number of e-folds between the pivot scale leaving the Hubble radius and the end of inflation, for the Starobinsky potential, against the temperature at which reheating finished, for 3 equations of state during it. All the lines meet on the right at instant reheating, 2.6 × 10¹⁵ GeV, where N = 55.6. The left edge is 5 MeV, below which nucleosynthesis would not have happened. w = 0, oscillating field: 42.0 at 5 MeV; w = ⅓, like radiation: 55.6 at 5 MeV; w = 1, kination: 69.0 at 5 MeV. The reason is how far the universe stretches while the energy density falls: an oscillating field dilutes like matter, as a⁻³, so for the same fall in density it expands further than radiation would, more of the growth of today's scales happens after inflation, and fewer e-folds of inflation are needed to put them where they are. A stiff epoch with w = 1 dilutes as a⁻⁶, stretches less, and needs more. A radiation-like epoch changes nothing. None of this epoch has been observed; the lines are the arithmetic of energy and entropy, and the spread between them is how much an unobserved history moves a quantity the spectral index depends on.

Cosmology

The epoch nobody saw moves the tilt

Between the end of inflation and the hot universe that made the light elements lies an interval nothing has observed, in which the energy of the inflaton became radiation. How long that took changes how many e-folds before the end the observed scales left — by as many as fourteen — and that moves every model's predicted spectral index by more than the measurement's uncertainty. A potential is never tested by the tilt alone; a potential and a reheating history are tested together.

One sky with and without a local non-Gaussianity of fNL·σ = 0.3. The same scale-invariant random field drawn twice, from one seed: on the left as it is, Gaussian, and on the right after the local transformation Φ → Φ + fNL(Φ² − ⟨Φ²⟩) with fNL·σ = 0.3. Solid contours are one and two standard deviations above the mean, dashed ones below, and each map is measured against its own mean and spread. The transformation adds to every value in proportion to its square, so peaks are pushed up and troughs are pulled back towards the mean: the area above +2σ goes from 1.8 to 4.7 per cent of the map and the area below −2σ from 2.4 to 0.0, and the skewness of the values rises from −0.073 to 1.484. This is exaggerated by a factor of about 2200. The primordial potential varies by about 3 × 10⁻⁵, so even fNL = 5 — the size of the current uncertainty — makes fNL·σ about 10⁻⁴.

Cosmology

A test that can only fail one way

Single-field inflation predicts a local non-Gaussianity of 0.015, a skewness in the primordial potential of a few parts in a million. The measurement is −0.9 ± 5.1. A detection at the level of one would eliminate every model with a single clock at once; a null result at any reachable precision confirms nothing, because the prediction lies below anything the sky has enough independent modes to measure.

An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist.

Exoplanets

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

Exoplanets

One timing curve and five planets that could draw it

A transiting planet whose times wander at a 58-day period, by just over a minute, is being pulled by something — but the period says only how far from some resonance the pull comes, not from which. Perturbers inside and outside the orbit, near four different commensurabilities, each with its own mass, reproduce the same curve to a fraction of a per cent. Timing alone returns a list, and even the detail that shortens it hides a coincidence of its own.

The path of a spin across the sphere of fixed momentum. A body with principal moments 1, 8, 8.6, started about its axis of least inertia with a 3° wobble and an internal energy sink strong enough that the whole motion fits in 80 turns and each circuit of the path can be seen. The disc is the near hemisphere of the sphere of fixed angular momentum, seen from a direction between all three body axes, with the near end of each axis marked. Each thin curve is a contour of kinetic energy on that sphere — a polhode, one of the paths the angular momentum can follow in the body with no dissipation — and the thick curve is the separatrix through the intermediate axis, which divides motions that circle the axis of least inertia from motions that circle the axis of greatest. The coloured path is what the integration did: solid on the near hemisphere, dashed where it passes behind. It starts at the open dot and ends at the filled one.

Spaceflight

A spin that left the axis it was given

The first American satellite was spun about its long axis, like a rifle bullet, and soon after launch it was tumbling end over end. Nothing outside it had pushed. A body that cannot change its angular momentum but can lose energy has exactly one place to end up, and a long body spun about its length is as far from that place as a spin can be.

$A spin about the middle axis of a 1:2:3 body, turning over every 2.9 turns. A body with principal moments of inertia 1, 2, 3, spun about its intermediate axis with a hundredth of its angular momentum knocked onto the axis of least inertia, integrated with no dissipation and no external torque. The curves are the components of the angular momentum along the three body axes, as fractions of its fixed size. The intermediate component stays near one for 1.5 spin periods, then swings through zero to minus one — the body turns over, end for end — and keeps doing so every 2.9 periods, 14 times in the span drawn. Energy and angular momentum are both conserved throughout, the energy to better than one part in a billion; nothing is being lost and nothing drives the flips. A spin about the intermediate axis is an equilibrium like a pencil balanced on its point, and the smallest disturbance grows exponentially, here by a factor of e every 0.28 spin periods, until it carries the body to the opposite equilibrium and back.$ Spaceflight

A wingnut that turns over on its own

Spin a rigid body about the axis whose moment of inertia is neither the largest nor the smallest and it turns end over end, again and again, with nothing pushing it and nothing lost. The flip was noticed aboard a space station in 1985 and was already implicit in equations written in 1765. How long it waits is a logarithm, and no care in setting up the spin can make the logarithm infinite.

Where the gradient of gravity holds a spacecraft still. The plane of the two inertia ratios that decide whether a spacecraft pointing at the Earth is held there by the gravity gradient: k₁ — the pitch moment of inertia less the yaw moment, divided by the roll moment — across, and k₃ — the pitch moment less the roll moment, divided by the yaw moment — up, with roll along the velocity, pitch normal to the orbit and yaw towards the Earth. Shaded points satisfy all three conditions of the linear theory — pitch is stable when k₁ > k₃, and roll and yaw together when k₁k₃ > 0 and 1 + 3k₁ + k₁k₃ > 4√(k₁k₃). The large region at upper right, 12.4 per cent of the square, is the one in which the pitch moment is the largest and the yaw moment the smallest, the arrangement of a long boom hanging towards the Earth. The small region just left of the vertical axis and below the horizontal one, 2.0 per cent, is a second, narrow island of stability with the moments in a different order, found by DeBra and Delp in 1961. There the orientation is a maximum of the potential in roll and yaw rather than a minimum, held only by the gyroscopic coupling of the two, and a damper — the very thing the long-boom region needs — destroys it: with damping of 0.05 of the orbital rate, a swing of a hundredth of a radian at (−0.10, −0.21) grows to a full radian within 11 orbits, while the same swing on the long boom shrinks 435-fold in 40. A boom along the vertical, pitch moment largest sits at (0.97, 0.40) and is stable; the same boom with roll and pitch moments swapped sits at (0.93, −0.40) and is unstable: the same boom, with two nearly equal moments exchanged, crosses from one side of an axis to the other.

Spaceflight

A boom held upright by a difference in gravity

A long spacecraft in orbit is pulled into line with the vertical for nothing — its near end feels slightly more gravity than its far end, and the difference is a torque. The torque restores and never dissipates, so the vehicle swings like a pendulum whose clock is the orbit. Whether it is held at all comes down to three inequalities between its moments of inertia, and one region that satisfies all three is destroyed by the damper every such spacecraft needs.

A tumble removed with a coil and a compass, at 500 km. The rotation rate of a small spacecraft — principal moments 0.0067, 0.041, 0.043 kg m², the proportions of a three-unit cubesat — tumbling at 8.8° a second after release, against orbits at 500 km, with nothing to control it but magnetic coils driven by the B-dot law: a dipole opposite to the rate of change of the field measured aboard, capped at 0.2 A m². The field is a dipole tilted 9.2° from the Earth's axis and turning with the Earth. In a polar, 97.4°, orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.58 orbits; in a 51.6° orbit the rate settles at 0.12° a second over the last orbit drawn, passing 1° a second after 0.32 orbits; in an equatorial orbit the rate settles at 2.44° a second over the last orbit drawn. The law needs no knowledge of the spacecraft's attitude: a tumbling body sees the Earth's field swing round in its own frame, and a dipole opposing that swing produces a torque that removes the part of the spin perpendicular to the field. The polar orbit does not reach zero. It settles at 0.94 of twice the orbital rate, 0.13° a second, and twice the orbital rate is how fast the field direction itself turns round a polar orbit: a body turning with the field sees little change to oppose. An equatorial orbit keeps the field pointing nearly the same way all the way round, so the spin about it is reached only through the dipole's tilt and the Earth's turning, and 28 per cent of the starting rate is still there at the end.

Spaceflight

A tumble stopped by the field it tumbles through

A small satellite leaves its deployer tumbling, and the first thing most of them do is stop, using nothing but a magnetometer and three coils. The law they run needs no idea where the satellite is pointing. What it cannot do, at any instant, is touch the spin about the local field line — so how much tumble survives is decided by how much the field's direction changes along the orbit, and the stillness it reaches is defined by the field rather than by the stars.

Orbits

Only two force laws let an orbit come back

That a planet returns to the same point of its own path after one lap is not a fact about orbits. It is a fact about the exponent in the force, and out of the whole continuum of attractions only two — the inverse square, and a spring — bring every bound orbit back to where it started.

The same well after the primary has lost 45 per cent of its mass. Two effective potentials for one body: the solid curve before the primary loses mass and the faint one after, both at the same angular momentum, because a central force of any strength exerts no torque. The well shallows and its floor moves out from r = 1.00 to 1.82. The body's own level moves with it — from E = -0.420 to -0.127 — and the two horizontal lines are drawn where the radial action is conserved, which puts the turning points at 0.71–1.67 before and 1.30–3.03 after. The ratio between them is 2.3333 in both, so the orbit is the same shape at a larger size: everything about the body's path has scaled and its eccentricity of 0.4 has not moved. That is what a slow change leaves behind, and it is not what a sudden one leaves.

Orbits

The well moves, and the body keeps its share of it

When the Sun becomes a white dwarf it will throw away half its mass, and every planet's orbit will swell by the same factor. Their eccentricities will not change at all — provided the loss is slow, and the only meaning "slow" has here is slow compared with one orbital period.

Starlight

The continuum is made by one atom in ten thousand

The Sun's light leaves through an ion that exists only because a hydrogen atom will hold a second electron by three-quarters of an electronvolt. The electrons it holds come almost entirely from magnesium, silicon and iron — so the level against which every solar line depth is measured is rationed by elements present at one part in ten thousand.

The one number a cloud cannot change by squeezing. The mass-to-flux ratio in units of its critical value, against column density, for clouds threaded by fields of 3, 10, 30, 100 microgauss. λ below one is subcritical, and the field alone holds the cloud up, and no amount of compression changes that, because squeezing raises the magnetic and the gravitational energy at the same rate and leaves their ratio exactly where it was. λ above one is supercritical and the field is irrelevant to whether the cloud collapses. Each line has slope exactly one because λ is proportional to the column density at fixed field, and each crosses the boundary at 3.9·10²⁰ cm⁻² for 3 µG, 1.3·10²¹ cm⁻² for 10 µG, 3.9·10²¹ cm⁻² for 30 µG, 1.3·10²² cm⁻² for 100 µG. The Jeans mass is a threshold a cloud can cross by contracting; this one is a label it is born with, and the only way past it is to let the field leak out.

Galaxies

Support that cannot be squeezed away

A cloud held up by pressure can always be defeated by compressing it, because gravity gains faster than heat does. A cloud held up by a magnetic field cannot — squeezing raises both energies at exactly the same rate, so the ratio a cloud is born with is the one it keeps, and the only way out is to let the field leak.

Turbulence stops being supersonic at 0.04 parsecs. The velocity dispersion of molecular gas against the size of the region it is measured over, from σ = 1 (R/pc)^0.5 km/s, with the isothermal sound speed of 10 K gas drawn flat beneath it. The two cross at 0.035 parsecs, which is solved for here rather than quoted: below that scale the motions are subsonic and the gas is supported by its own pressure, above it they are supersonic and nothing thermal is relevant. At ten parsecs the Mach number is 17 and at a hundredth of a parsec it is 0.53. The crossing is the scale at which turbulent support runs out, and it is within a factor of two of the size of the dense cores that actually form stars — which is either the most important coincidence in the subject or the reason cores are the size they are.

Galaxies

The support and the seed are the same motions

A molecular cloud's lines are ten times wider than its temperature allows, and the width grows with the size of the region measured. The motions that widen them hold the cloud up as a whole and make the dense lumps inside it — so the same turbulence that delays star formation is what decides where it happens.

What Io dissipates depends on how fast the tide is applied. The dissipative part of the Love number, −Im k₂, against the period of the forcing, for Io at an interior viscosity of 1e+16 Pa s. Two rheologies are drawn: a Maxwell solid, a spring and a dashpot in series, and an Andrade solid, which adds the anelastic creep every real material shows in the laboratory and a Maxwell body does not, with an exponent of 0.3. They agree for tides slower than the Maxwell time of 1.9 days, where the body has time to flow and the transient is irrelevant. They part completely for faster ones: at the shortest period drawn the Andrade body dissipates 20 times what the Maxwell body does. A quality factor quoted without a period is half a number, and which half is missing depends on a rheology measured in a laboratory rather than derived.

Gravitation

A quality factor quoted without a period is half a number

The tidal response of a solid body is not a constant. It is a function of how fast the tide is applied, and two rheologies that agree perfectly about a slow tide disagree by orders of magnitude about a quick one — so the same moon has one Love number at its orbital period and a different one at the period of its own libration.

$A mass measured by the structure it prevents. The fractional suppression of the small-scale matter power spectrum against the sum of the neutrino masses. The relic density follows from the thermal history with no astrophysics in it — the neutrinos’ share of the critical density, times h², is the mass sum divided by 93.14 eV exactly — and the suppression is about eight times the neutrinos' share of the matter, because a particle moving at a large fraction of the speed of light streams out of an overdensity while it is still growing and takes its own gravity with it. The two mass orderings put a floor under the sum at 58 meV and 100 meV; the cosmological upper bound is near 0.12 eV, which is 7.2 per cent of suppression. The floor and the ceiling are within a factor of two of each other, so this is a measurement about to happen or a model about to break, and the quantity it returns is a sum rather than any individual mass.$ Cosmology

A mass measured by what it stopped from forming

Neutrinos were relativistic in the early universe and are not now, so they are the one entry in the cosmic budget that changes category. What cosmology measures is not their density but the hole they leave — they stream out of a growing clump and take their gravity with them, and the missing structure bounds a particle mass more tightly than any laboratory has.

Exoplanets

A composition that dates a formation rather than placing it

The ice line in a young disc starts six astronomical units out and sweeps inward to under three as the disc drains. A body at four astronomical units therefore formed dry or wet depending only on when — so what it is made of is a clock, and reading it as a map is the mistake the moving line makes easy.

The astrometry an occultation campaign has to have. How far the shadow lands from where it was predicted, against the angular error in the positions it was predicted from, for a Centaur at 15 AU, a Kuiper belt object at 40 AU, Uranus at 19 AU. The conversion is one line — an angle times a distance — and one milliarcsecond at one astronomical unit is 0.7255 kilometres. The horizontal bands are each body's own shadow width, which is its diameter, and the crossing is the accuracy at which a campaign stops being a lottery: a Centaur needs 23.0 mas, a Kuiper belt object needs 4.1 mas, Uranus needs 3701.0 mas. Before the all-sky astrometric surveys the typical error was tens of milliarcseconds, which is thousands of kilometres at these distances, so events by small bodies were found by accident and not by appointment. The same event then measures the body's position to a few milliarcseconds or better, which improves the ephemeris that predicts the next one.

The observed sky

Each event pays for the prediction of the next

An occultation is predicted from two positions and lands where the arithmetic says. Recording it then measures the occulting body's position to a few milliarcseconds — better than a year of imaging — so the observation that the prediction made possible improves the ephemeris the next prediction comes from.

$The drag coefficient of a sphere runs from 2.03 to 2.79, and 2.2 is a convention. The free-molecular drag coefficient of a sphere against the accommodation coefficient — the fraction of striking molecules that thermalise with the surface and leave in a cosine distribution rather than bouncing — at speed ratios 2, 4, 8, with the surface at 0.3 times the flow's temperature. Specular reflection gives 2.469 at the lowest speed ratio drawn and 2.001 in the hypersonic limit, where every molecule delivers exactly twice its own momentum. Accommodation adds the re-emitted flux, which leaves at the wall temperature in a direction the flow did not choose, and it adds most where the speed ratio is smallest — which is high up, where the light species dominate. The conventional 2.2 lies outside this family at both ends: at s = 8 a sphere reaches only 2.112 even at full accommodation, and at s = 2 it is already 2.469 with none. That is not a defect of the arithmetic — 2.2 is a fitted average for satellite shapes, whose flat panels have a higher coefficient than a sphere of the same projected area, and the sphere is drawn because it is the one geometry with a closed form. What survives the shape is the dependence: a satellite's drag coefficient is an assumption about its surface chemistry and its attitude, and every density inferred from drag carries it in inverse proportion.$ Spaceflight

A coefficient that belongs to the surface, not the satellite

Every density ever inferred from satellite drag was divided by a drag coefficient, and that coefficient is not a property of the spacecraft. It is a property of what happens when an oxygen atom at eight kilometres a second strikes a surface it has already coated — and the conventional 2.2 is a convention.

Spaceflight

A weather forecast made out of orbits

A density model fitted to fifty years of satellite drag is a climatology, and its error does not shrink with more data. Updating it from the drag observed on objects in orbit right now is the manoeuvre a weather forecast makes — and it buys a factor of several for a day and nothing at all for a fortnight.

A corridor 0.2 km wide, and a density known to a factor of 2. The apoapsis a vehicle is left on after a single atmospheric pass, against the periapsis altitude it aimed at, for ballistic coefficients of 60, 130, 300 kg/m² arriving at 3 km/s. The energy removed is the density at periapsis times an effective path length of √(2πrₚH), divided by the ballistic coefficient — so it falls exponentially with altitude and the curve is steep. Hitting a 1000 km apoapsis to ±10 per cent requires a periapsis inside 0.2 kilometres. Getting the atmosphere wrong by a factor of 2 moves the aim point by 7.6 kilometres, which is 30.7 times the corridor's own width — so a ballistic vehicle aiming at a planet whose density is known to a factor of two misses by more than the tolerance allows, and the manoeuvre has to be flown rather than aimed.

Spaceflight

A manoeuvre that has never been flown once

Arrive on a hyperbola, dip once through the atmosphere, leave on a bound orbit having spent no propellant. The saving is a kilometre a second or more, the physics is the same as an entry corridor, and nobody has done it — because the corridor is a tenth of a kilometre wide and the density is known to a factor of two.

Galaxies

The same census, taken in two places

Fit a Schechter function to a rich cluster and to the field around it and the two come back with different slopes and different knees. Both differences are real, and neither can be read as a cause — a cluster's galaxies are also redder, and the same photometry measures both.

Starlight

Three shifts larger than the error bar, and two that cancel

An unresolved companion, a reddening and a metallicity each move a colour index by more than any modern photometer's precision. Two of them move it in opposite directions, so the three together can return the right temperature by cancellation rather than by accuracy.

Orbits

Two damping times, one crossing, and the slope that separates them

The equilibrium tide gives a damping time going as the sixteen-thirds power of the period and the dynamical tide as the seventh. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose.

At 15 keV the decrement is 9 per cent shallower and the null has moved to 224.4 GHz. The thermal Sunyaev–Zel'dovich distortion at one fixed Compton parameter, 10⁻⁴, computed with the relativistic kinetic equation expanded to second order in kTₑ/mₑc² for gas at 5, 10, 15 keV, against the non-relativistic shape that is the same for every temperature. All are scaled to the non-relativistic curve's largest excursion. Heating the gas at fixed y does two things to the spectrum. The decrement becomes shallower — by 9.3 per cent at its deepest point for 15 keV — and the increment becomes lower and broader, by 17.3 per cent at its peak, because fast electrons scatter photons over a wider spread of frequencies than slow ones and some of the boost is carried to frequencies above the drawn range. The crossing moves up, from 217.5 GHz to 224.4 GHz. A cluster's temperature is therefore written into the shape of its distortion and not only its amplitude, which is what a thermometer needs; and a Compton parameter read off one frequency with the non-relativistic shape is biased low by the drawn amount, which is what a mass estimate does not need. The expansion is good to well under a per cent below 15 keV; above 20 it has to be replaced by the exact integral.

Cosmology

A null that moves with the temperature

The frequency at which a cluster's hot gas vanishes from the microwave sky was derived for slow electrons. The electrons in a massive cluster move at a quarter of the speed of light, the null drifts half a gigahertz per keV, and what is left at the old frequency reads as a velocity as large as the ones being sought.

Exoplanets

A thermostat that only halves the error

The carbonate–silicate cycle is credited with keeping a planet's water liquid across the whole width of its habitable zone. Written down with its own measured exponents, it is a proportional controller that cancels about three-fifths of a change in sunlight, takes half a million years to do it — and reaches the published outer edge only on a planet with an order of magnitude more volcanism than the Earth.

Exoplanets

A year too short to feel its own eccentricity

A planet on an eccentric orbit can have a comfortable average and murderous extremes, and the habitable zone is drawn from the average. Whether the surface lives on the average or on the extremes is not decided by the flux at all — it is the ratio of how long the surface takes to change temperature to how long the year lasts, and the star sets the year.

Galaxies

A gradient the old stars have walked away from

The gas in a disc galaxy is richer in metals near the centre than at the edge, by about six-hundredths of a dex per kiloparsec in the Milky Way. Two ingredients of disc growth make that slope, a disc that grows from the inside out makes it flatten with time — and the old stars that should carry the steeper history have moved several kiloparsecs from where they were born.

$A power law has no timescale: half the Type Ia supernovae by 740 Myr and a tail to 13.7 Gyr. The fraction of all the Type Ia supernovae a single burst of star formation will ever produce that have exploded by a given delay, on a logarithmic time axis, for t⁻¹ from 40 Myr; t⁻¹·⁴ from 40 Myr; single delay of 1 Gyr; Gaussian, 3 ± 1 Gyr. The power law is what rates measured against host-galaxy ages and against the cosmic star-formation history both favour, and its cumulative fraction rises as the logarithm of the delay — equal numbers per decade of time. Half have exploded by the geometric mean of its limits, 740 Myr, 55 per cent by 1 Gyr, and the last are still exploding after a Hubble time. A single delay turns the whole population on at once; a Gaussian concentrates it at a characteristic age. The power law's shape has a physical reading: if white dwarfs explode when a pair of them merges by emitting gravitational waves, the merger time goes as the fourth power of their separation, and a broad distribution of separations becomes a distribution of delays with no preferred scale. What the drawing cannot say is which progenitors are involved — the measured rates constrain the shape and the normalisation, about one Ia per thousand solar masses of stars formed, and not the mechanism.$ Galaxies

The iron clock has no single delay

The α-element knee is drawn as though Type Ia supernovae switched on a billion years after the stars that made them. Measured rates say otherwise — the delays are spread evenly over every decade from forty million years to a Hubble time, as a power law with no timescale in it — and a clock with no timescale bends where a clock with one would break.

The observed sky

A triangle of meetings that turns in eight centuries

Jupiter and Saturn meet every twenty years, and each meeting falls about two-thirds of the way round the sky from the last, so the meetings trace a triangle. The triangle turns a third of a turn in 838 years because five of Jupiter's years almost equal two of Saturn's — and that same near-fit is the largest perturbation in the solar system, the one that made Saturn appear to be slowing down.

At the Sun–Earth L₂ the cheapest correction is every 23 days, and it costs e σ per e-folding. The annual station-keeping cost at the Sun–Earth L₂ point, against the interval between corrections, on logarithmic axes, for velocity errors of 0.5 cm/s, 2.0 cm/s, 5.0 cm/s along the unstable direction at each correction. Correcting often costs a lot because every correction carries its own error σ; correcting rarely costs a lot because the error has grown by e^(T/τ) in between, with an e-folding time τ = 23.4 days set by the point's growth rate of 2.484 times the orbital mean motion. The product (365.25/T) σ e^(T/τ) has its minimum at exactly T = τ, where the annual cost is 365.25 e σ/τ: 0.21 m/s a year for σ = 0.5 cm/s, 0.85 m/s a year for σ = 2.0 cm/s, 2.12 m/s a year for σ = 5.0 cm/s. The minimum is broad, so an operator can correct at a convenient interval near the e-folding time for little penalty, and the cost scales linearly with how well the spacecraft's velocity is known and executed. This is a one-dimensional caricature: a real halo orbit's correction also removes a stable component it need not, and solar radiation pressure on a large sunshield is a steady error source of its own. The figure's claim is the structure — an unstable equilibrium is cheap to hold if the instability is caught while it is still small, and its cost is a navigation budget rather than a force budget.

Spaceflight

An unstable point that costs less to hold than a stable orbit

A spacecraft at the Sun–Earth L₂ point sits on an equilibrium that throws it away, doubling any error every sixteen days. It holds station for a few metres per second a year — a twentieth of what a geostationary satellite pays to stay on an orbit that is stable. The difference is what is being paid for — an instability caught small costs a navigation budget, and a steady torque costs a force budget.

Starlight

The comparison stars are part of the measurement

Measuring a star against others on the same frame cancels everything the atmosphere and the instrument do to all of them at once — a two per cent change in transparency vanishes completely. What does not vanish is the comparison stars' own noise, which the target inherits, and the variability of any comparison that is not constant, which the target reports as its own.

A wall at 2.33 hours that bends into a slope below 598 metres. The fastest rotation period a body of bulk density 2 g/cm³ can hold against its own spin, against its diameter, on logarithmic axes, for cohesive strengths of 0, 10, 100, 1000 pascals. With no cohesion the limit is the density-only barrier √(3π/Gρ) = 2.33 hours at every size. A cohesion C adds a stress that does not depend on size to a gravitational stress that goes as the square of the radius, so small bodies are held mostly by cohesion and can spin faster in proportion to their smallness: below the corner the limiting period goes as the diameter. The corner is where the two stresses are equal, at a diameter of 189 m for 10 Pa, 598 m for 100 Pa, 1.9 km for 1000 Pa. The scaling is the strength-regime form with a single coefficient of order one; detailed limits depend on the angle of friction and the shape. What the figure makes plain is why the observed spin barrier is sharp for kilometre-sized asteroids and fades below a few hundred metres, and why a handful of fast rotators a few hundred metres across can be rubble piles with a few tens of pascals of cohesion — a strength far below that of any rock — rather than monoliths.

Orbits

A spin barrier with a corner in it

A rubble pile cannot spin faster than a period set by its density alone — 2.3 hours for most asteroids — at any size. Give it a cohesion of a few tens of pascals, weaker than any rock, and the barrier bends into a slope below a corner at a few hundred metres. The thermal torque that drives bodies to the barrier doubles their spin in a time that grows as the square of their size, so the bodies it pushes hardest are the ones that can go past.

$A rubble pile that splits below a mass ratio of 0.204 can lose its piece; above it, the piece stays. The total energy of two spherical components of equal density in contact, spinning together at the rate at which their mutual gravity just holds them against the spin, against the mass ratio of the smaller to the larger, in units of G m₁²/R₁. The energy is the kinetic energy of the rotating pair minus their mutual gravitational binding. When the smaller piece is a small fraction of the whole, the spin carries more energy than the binding and the total is positive: a body spun to breakup that sheds a fragment of that size has enough energy for the fragment to escape entirely, becoming a separate asteroid on a nearly identical orbit. The total changes sign at q = 0.204. Above that ratio the pair cannot separate without an energy source; it stays as a binary, orbiting and eventually synchronising, or re-accretes. As q goes to zero the energy tends to 0.2 G m₁²/R₁, the rotational energy of the primary alone at its breakup rate. Nothing in the threshold depends on the size or the density of the body — it is a pure number from the geometry of two touching spheres — and asteroid pairs sharing an orbit have been found overwhelmingly with estimated mass ratios below it.$ Orbits

A split that decides whether the piece can leave

A rubble pile spun past its limit splits in two, and whether the smaller piece escapes or stays in orbit is not a matter of luck. Two touching spheres spinning at their shared limit have positive total energy only when the smaller is less than 0.204 of the larger's mass — a number with no size and no density in it. Below it the pieces can become a pair of asteroids on nearly identical orbits; above it, a binary. And the larger the piece that leaves, the slower the body left behind.

$A gauge good to 7 per cent with a tenth of the load left, and to 23 per cent with three hundredths. The uncertainty in the propellant remaining in a spacecraft tank, as a percentage of what remains, against the fraction of the 450-kilogram load still in the tank, on a logarithmic uncertainty axis with the tank emptying to the right. Bookkeeping — summing every thruster firing through a flow-rate model — carries an error common to all burns of 2 per cent of the mass used, plus an independent 5 per cent per burn that averages down over 2000 firings; its absolute error grows with the mass used. Gauging by pressure and temperature infers the empty volume of the tank from the gas law applied to a known mass of pressurant, with a combined 0.66 per cent uncertainty in n R T / P and a 0.2 per cent uncertainty in the tank's volume; its absolute error grows as the gas fills the tank. The two methods are independent and are combined by inverse variance. With a tenth of the load left the combined estimate is uncertain by 2.9 kg, 7 per cent of what remains; with three per cent left, by 3.1 kg, 23 per cent. Near empty the absolute error barely changes, so halving what is left doubles the relative error — the gauge is at its worst exactly when the last manoeuvre has to be planned from it.$ Spaceflight

A fuel gauge that is worst when it is needed

A spacecraft's tank has no float and no dial. The propellant left is estimated by adding up every burn or by reading the pressure and temperature of the gas above the liquid, and both methods' errors grow with the propellant used. Relative to what remains, the error doubles every time what remains halves — so a geostationary satellite has to hold back months of station-keeping as a margin against a gauge that cannot see the last few kilograms.

$The quantity a capture has to change. The Jacobi constant along three trajectories through the same encounter with the secondary, integrated in the rotating frame at mass fraction 0.0009543. The undissipated one holds its value to 3e-11 over the whole passage — that is not an approximation but an identity of the equations of motion, and the residual is the integrator's. A body that enters through the neck at L₁ with the neck open leaves through it, because C has not changed and the neck is still open; a purely gravitational two-body encounter is time-reversible and cannot end in a bound orbit. The other two runs remove energy in the rotating frame. A drag proportional to velocity raises C continuously at a rate 2k v², and a single impulse against the motion raises it in a step. Both finish above C(L₁) = 3.039, which is the line at which the neck shuts and the body is trapped in the secondary's lobe. Which mechanism operated for a given satellite is not recoverable from its orbit today — but it is partly recoverable from the population, because the three leave different distributions behind.$ Gravitation

A moon that two bodies could not have caught

A body that falls into a planet's Hill sphere under gravity alone leaves it again, because the quantity the rotating frame conserves is the same on the way out as on the way in. Every captured moon is therefore a record of something that removed energy — and which of the three candidates did it is written in the inclinations rather than in any one orbit.

A limit with a notch cut in the middle of it. The stability limit for a satellite, in units of its planet's Hill radius, against the inclination of its orbit to the planet's. The two flat stretches are the coplanar answers — 0.5 prograde and 0.7 retrograde, the difference between them being the Coriolis asymmetry that a rotating frame imposes and no potential can hold. Between them the limit is not an interpolation. Inside the shaded band, from 39.23° to 140.77°, the Sun's averaged perturbation exchanges the orbit's inclination for its eccentricity, and a satellite whose semi-major axis is comfortably inside the lobe has an apocentre that is not — so the boundary is set by a(1 + eₘₐₓ) rather than by a, and it falls to exactly half the coplanar value at ninety degrees. The band edges are cos²i = 3/5 exactly, with no fitted quantity anywhere in them, and the notch is the reason a quoted stability limit in Hill radii is a statement about a coplanar orbit and not about a satellite.

Gravitation

A stability limit with a notch cut in it

The two numbers usually quoted for how far out a moon may orbit are coplanar numbers. A tilted orbit is not somewhere between them: inside a band of inclination fifty-six degrees wide the Sun converts the tilt into an eccentricity, the apocentre rather than the semi-major axis has to stay inside the lobe, and the limit falls to half. The band is empty in the sky.

The relative orbit when the target's own is not circular. Free relative motion of a chaser about a target, over one revolution, for four eccentricities of the target's orbit — drawn by propagating two exact Keplerian orbits and differencing them in the target's own rotating frame rather than by any linearised solution. The chaser differs from the target by 0.0035 in eccentricity, so the two semi-major axes are identical. Along-track distance runs across the page. At e = 0 the path is the familiar closed ellipse, 2.00 times as long along the track as across it — recovered here from an exact propagation, which is the check that the calculation is right. Raise the target's eccentricity and the path stops being an ellipse and starts breathing: the separation varies by a factor of 2.00 over one revolution at e = 0 and 2.26 at e = 0.7, because the frame itself is now turning at a rate that changes through the orbit and the chaser is swept around it fastest at periapsis. Every one of them still closes exactly, and that is the one statement of the circular theory that survives — two orbits of equal semi-major axis have equal periods and cannot drift apart, whatever else differs between them.

Spaceflight

The relative orbit that breathes

Every result in the close-approach frame — the two-to-one ellipse, the burn that moves the wrong way, the eight kilometres behind — assumes the target's own orbit is circular. Around an eccentric target none of the geometry survives. One statement does, and it is about energy rather than shape.

A cycle and a radiation, racing for the same pericentre. Two integrations of the same compact triple, in units of the cycle's own timescale at the starting separation. The dashed curves are the semi-major axis and the solid ones the eccentricity. Radiation alone shrinks the orbit slowly and smoothly, reaching a tenth of its starting size after 1136 cycle times — which the closed form for a circular inspiral predicts to within a few per cent, and which is the check that the integrator is right. With the cycle running, the distant companion drives the eccentricity to 0.9104, the pericentre falls by two orders of magnitude, and the radiation — whose rate goes as the inverse seventh power of one minus the square of the eccentricity — empties the orbit in 18. That is a factor of 62, obtained without changing a single parameter of either mechanism. The two do not merely add: the cycle manufactures, at its own peak, exactly the configuration in which the radiation is most efficient, and the radiation then shrinks the orbit until the relativistic precession switches the cycle off.

Orbits

A cycle that builds what the wave needs

Gravitational radiation empties an orbit at a rate that goes as the inverse seventh power of one minus the square of the eccentricity. A distant companion drives exactly that eccentricity, over and over. The two are not independent mechanisms that happen to act on the same binary — the first manufactures, at its own peak, the configuration the second is most efficient in.

Orbits

A ceiling the outer orbit imposes

The closed form for how far a cycle can drive an eccentricity assumes the inner orbit is a test particle, so the angular momentum it gives up goes nowhere. A real inner binary of comparable mass has to be paid for: the outer orbit absorbs what the inner one loses, tilts, and changes the very inclination the cycle depends on.

Stretched towards the mass, sheared across it. Two rings of galaxies around the same concentration of mass. Left, galaxies physically beside it: its tidal field stretches each one along the line to the centre, so their long axes point radially and their mean tangential ellipticity is −0.30. Right, galaxies far behind it: their light is deflected past the mass and each image is sheared tangentially, with mean tangential ellipticity +0.30. The two patterns are perpendicular. A weak-lensing survey correlates the shapes of foreground and background galaxies to measure the shear, and when a foreground galaxy was aligned by the very structure that lenses the background one, their product is negative — −0.090 here — and subtracts from the signal rather than adding noise to it. What the picture exaggerates is the size: real intrinsic alignments are a per cent or two in the mean, buried under a random shape scatter of about 0.3, and are found only statistically.

Cosmology

The shape a galaxy was given before it was lensed

Weak lensing assumes that galaxies point in random directions, so that a coherent stretch can only have been put there by the light's journey. The tidal field that set every galaxy's spin also stretched its shape — towards the very mass that lenses the galaxies behind it — and the resulting error is not noise but a signal of the opposite sign.

A spin that prefers the axis in the middle. Where the spin of a tidally torqued patch points, measured against the three principal axes of the tidal field acting on it: the distribution of the absolute cosine between the spin and each axis, from 20,000 patches with random shapes in one field of principal values −0.75, 0.09, 0.66. Principal values are those of the second derivative of the potential, so a positive one compresses. An isotropic spin would give three flat lines at one. The spin avoids the most compressive and the most stretching axes and prefers the intermediate one, because the torque about each axis is proportional to the difference of the field's principal values about the other two, and the intermediate axis is the one whose two neighbours differ most: squared, those weights are 0.11, 0.66, 0.24 of the total. Fitted as Lee and Pen's alignment parameter, the directions give a = 0.617. The preference is real and it is weak — the densest bin is 3.37 times the isotropic value — which is the whole difficulty of using spins to map a field.

Cosmology

A spin that remembers the field and not its sign

Tidal torque theory says where a galaxy's spin should point, not only how large it should be. It points along the middle axis of the tidal field, it is exactly perpendicular to an ideal filament, and it is an even function of the field — so a map of spins can find the directions of the cosmic web and can never say which of them is collapsing.

Exoplanets

The gas that turns round beside the planet

The two spiral wakes a planet raises nearly cancel, and what decides the direction it drifts is a third, smaller torque from the gas sharing its orbit. That gas makes a U-turn in front of the planet and behind it, and whether the turn pushes the planet outward depends on how the disc was heated and on a viscosity nobody has measured.

Exoplanets

A planet circularised at twice the distance that would destroy it

A giant planet thrown onto an orbit of eccentricity 0.97 by a distant companion loses its orbital energy to tides at every close passage and keeps almost all of its angular momentum. The orbit therefore shrinks to a circle at exactly twice its closest approach — and since a closest approach inside the Roche distance destroys the planet, the hot Jupiters delivered this way should begin at twice the Roche distance, measured in units of each planet's own size.

Galaxies

The halo a galaxy drags inward

The cusp that dark-matter simulations predict is a halo with no galaxy in it. When gas cools and settles into a disc at the centre, the dark matter is pulled in after it, and a quantity conserved on slowly changing orbits says by how much. In a large disc galaxy the pull triples the dark matter at the centre and multiplies tenfold the energy needed to dig a core; in a dwarf it barely registers.

$A band and a strip, and the constant where they cross. The plane of the two quantities a planetary ephemeris cannot separate: the fractional rate of change of the gravitational constant, vertically, and of the Sun's mass, horizontally, both in units of 10⁻¹⁴ per year. The orbits respond only to their sum, the rate of change of GM☉, so a measurement of that sum — drawn here as −9.0 ± 3.0 × 10⁻¹⁴ per year, the precision recent ephemerides reach, not any one published value — is a diagonal band. On its own it allows any Ġ/G at all, provided the Sun's mass is changing to compensate. The vertical strip is the Sun's mass loss computed from its luminosity and its wind, −9.9 to −8.8 × 10⁻¹⁴ per year. Where the two overlap, Ġ/G lies between −3.2 and 3.9 × 10⁻¹⁴ per year. The bound on a change in a fundamental constant is therefore set as much by how well the solar wind is measured as by how well the planets are.$ Gravitation

A constant measured through a Sun that is getting lighter

Whether the gravitational constant changes with time is tested by watching the planets, and the planets report only the product of G with the Sun's mass. The Sun is losing mass — four million tonnes a second as light, a million and a half more as wind — at nearly a ten-trillionth of itself a year, and every bound on a changing G is a statement about the solar wind as much as about gravity.

A doubling rule, and the planet that broke it. Each planet's real semi-major axis against the value the Titius–Bode rule 0.4 + 0.3 × 2ⁿ gives it, both in astronomical units on logarithmic axes; points on the diagonal are fitted exactly. The rule gives the Earth 1.0 by construction and fits Venus, Mars, Jupiter and Saturn within a few per cent. It needed the value n = 3 left empty until Ceres was found in its slot in 1801, and it predicted Uranus, discovered in 1781, within 2%. Neptune, discovered in 1846 by a calculation that assumed the rule, sits 22% inside its predicted place, and Pluto occupies the place Neptune should have had. Mercury is given n = −∞, which is a free choice made to fit it. The labels give each body's departure from the rule.

Orbits

A spacing law that random orbits obey

The planets' distances roughly double from one to the next, and a rule written down in 1766 predicted the asteroid belt and Uranus before either was found. It failed on Neptune, and the reason it worked at all turns out to be that almost any set of orbits forbidden to crowd each other looks geometric on a logarithmic scale — as a test against thousands of random systems shows.

Orbits

A hotter summer that is also a shorter one

When the Earth passes closest to the Sun in northern summer, the northern summer is brighter — and, by Kepler's second law, shorter by exactly the right amount to deliver almost the same energy. Measured by the energy it delivers above a melting threshold rather than by its brightest day, a summer barely feels the precession cycle at all and feels the tilt of the axis several times more strongly.

The weakest bodies are a few hundred metres across. The energy per unit mass needed to shatter a body and disperse half of it, against the body's diameter, for basalt and water ice, from fits to impact simulations calibrated on laboratory fragmentation. Small bodies are held together by material strength, which falls with size because a larger body contains larger flaws; large bodies are held together by their own gravity, which rises with size because the fragments have to be thrown apart as well as broken. The two regimes meet in a minimum at 235 m for basalt and 204 m for water ice, where a body needs the least energy per kilogram to destroy of any size — 127 J/kg and 57 J/kg, the kinetic energy per kilogram of motion at only 16 and 11 metres a second. A collisional population is therefore not self-similar: it has a preferred size, and its size distribution records it.

Orbits

The size at which a body is easiest to break

A pebble is held together by the strength of its rock and a moon by its own gravity, and somewhere between them is a size that has too little of either. Bodies a few hundred metres across are the weakest in the solar system — sixteen metres a second of impact energy per kilogram is enough to scatter one — and a population grinding itself down records that weakness as a knee in its size distribution.

$A square field imaged as a squashed one. A square grid of stars 1° across, centred 70° from the zenith, and how refraction moves each one relative to the field's centre after the displacement common to the whole field has been removed. The arrows are magnified 200 times. Refraction lifts every star towards the zenith, but it lifts the lower ones more because refraction grows towards the horizon, so the field is compressed vertically by 2377 parts per million; and every star moves along its own vertical circle, which converge at the zenith, so the field is compressed horizontally by 286 parts per million. The corners move by up to 4.4 arcseconds against the centre — a hundred times the precision of a modern relative astrometric measurement, and a distortion that changes as the field crosses the sky.$ The observed sky

A square field imaged as a squashed one

Relative astrometry measures positions against neighbouring stars so that the atmosphere's refraction, common to them all, cancels. It does not cancel. Refraction changes across the field, so a square patch of sky seventy degrees from the zenith is imaged squashed by a few parts in a thousand — and stars of different colour are moved by different amounts, by more than the parallaxes being measured.

The weak spot in the field. Contours of the strength of the Earth's magnetic field at the surface, in microtesla, over longitude and latitude, from an eccentric dipole: a dipole tilted to pass through the geomagnetic poles and displaced about 570 km from the Earth's centre towards the western Pacific. The field is strongest near the poles and weakest near the equator, as any dipole's is, but the displacement makes the side of the Earth furthest from the dipole's centre weaker still: the minimum, 22.9 µT, lies at 8° S, 40° W, over the South Atlantic. The measured anomaly sits further south and west, near 26° S and 55° W, and is slightly deeper, because the real field has higher harmonics that no dipole carries; the eccentric dipole gets the side of the planet and the size right and the exact place approximately. Trapped particles mirror where the field reaches a fixed strength, so wherever the field is weak they mirror closer to the ground — and over this region the inner belt reaches down into the upper atmosphere.

Spaceflight

A belt emptied over one ocean

The Earth's magnetic field is not centred on the Earth. Its dipole sits five or six hundred kilometres towards the western Pacific, so the opposite side of the planet — the South Atlantic — has the weakest field at any altitude, and a trapped particle that mirrors safely above the atmosphere everywhere else dips into it there. The inner edge of the inner radiation belt is drawn by that one region, and it is moving.

Exoplanets

An escape that stops following the starlight

The standard estimate of how fast a planet loses its atmosphere assumes the loss grows in proportion to the ultraviolet light it absorbs. It does, up to a point. Beyond about ten thousand erg per square centimetre per second the escaping gas is fully ionised, its heat leaves as Lyman-α light instead of lifting anything, and the loss grows only as the square root of the flux — so the one efficiency every calculation needs is really a function of where the planet sits.

A disc galaxy walked down the plane by three dry mergers. Specific angular momentum against stellar mass, both logarithmic. The solid line is the relation disc galaxies follow, j proportional to M to the two-thirds, normalised to 1000 kpc km/s at 10¹¹ solar masses; the dashed line is the spheroids', a fifth as high. The points follow the median galaxy built from discs that start on the line at 2.51·10¹⁰ solar masses by successive equal-mass mergers, each between two galaxies with the same history as each other, adding spin vectors in random directions and keeping none of the orbital angular momentum. Each merger doubles the mass and moves the point right by 0.3 of a decade; the vector sum keeps on average only two thirds of the specific angular momentum while the line rises by the two-thirds power of two, so the point falls below it: to 0.45 of the line after one merger, 0.18 after two and 0.08 after three. Two dry major mergers carry a disc to where spheroids are measured.

Galaxies

Two spins that cancel, and an orbit that leaves

Disc galaxies and ellipticals of the same stellar mass differ by a factor of five in how much angular momentum each gram of their stars carries. Adding spin vectors in random directions explains most of it — two mergers of equals, with the orbit's angular momentum handed to the dark halo, carry a disc down to where ellipticals sit — and whether a merger makes a spheroid or rebuilds a disc turns on one number the vectors do not supply.

The observed sky

A substorm timed by webers and volts

The solar wind opens the Earth's magnetic field at a rate that can be quoted as a voltage — tens of kilovolts, which is webers of flux per second — and the tail closes it again. Half a gigaweber loaded at fifty kilovolts takes three hours, and three hours is how often substorms recur. The aurora's position is the open flux, its surges are the unloading, and above a certain driving the ionosphere starts to limit the circuit that drives it.

When a target can be observed from a telescope that cannot look near the Sun. The days of the year (across) on which a target at ecliptic longitude 0° and a given ecliptic latitude (up) can be observed by a telescope allowed to point only between 85° and 135° from the Sun, the constraint of a large infrared telescope at the Sun–Earth L2 point. The Sun moves along the ecliptic once a year, and the allowed band sweeps round with it. A target on the ecliptic is inside the band twice a year, for about 51 days each time, a little before and a little after its opposition — never at it, since opposition is 180° from the Sun. The two windows widen with latitude and merge into one above 45°, where the elongation at opposition no longer exceeds 135°; within 5° of either ecliptic pole, where the elongation at conjunction no longer falls below 85°, the target is never outside the band at all, which is the continuous viewing zone. The pattern depends on ecliptic, not equatorial, coordinates: the season of a space telescope is set by the Sun's path, and the Earth's axis has no part in it.

The observed sky

A season without a night

A telescope in space has no horizon and no night, and it can still only look at most of the sky for part of the year. Its season is a band of solar elongation swept round the ecliptic by the Sun, so an object on the ecliptic can be seen for two stretches of about seven weeks and never at opposition, while the few degrees round the ecliptic poles can be watched all year. From low orbit the Earth takes the night's place, and the zones it never hides are carried round the sky every eight weeks by the planet's equatorial bulge.

One surface, two ages, and a kink between them. The cumulative number of craters per square kilometre larger than a given diameter, both on logarithmic axes, for a surface that formed 3.9 billion years ago and was flooded 3.3 billion years ago by a layer 200 m thick. The dashed lines are the isochrons of the two ages under the lunar chronology. Craters with rims taller than the flood — larger than 5.6 km, since a simple crater's rim stands about 0.036 of its diameter high — survived it, so above the kink the count follows the 3.9-billion-year isochron. Everything smaller was buried, and the craters counted there formed after the flood, so below the kink the count follows the 3.3-billion-year isochron. Fitting the branches separately returns 3.90 and 3.30 billion years. The kink is not noise and not a change in the impactors: its diameter is a measurement of how thick the flood was.

Orbits

A kink that measures how deep the lava was

A surface that was flooded partway through its life carries two crater populations — the old craters too big to bury, and the young ones made since — and its crater count follows one age above a certain diameter and another below it. The diameter of the kink is the thickness of the flood, read through the height of a crater's rim, and a single age fitted across it describes nothing that ever happened.

The spike, and where one passage through the planets sends it. The distribution of the reciprocal semi-major axis 1/a of long-period comets, in units of 10⁻⁴ AU⁻¹ and each curve scaled to its own peak. The narrow shaded peak is the Oort spike: comets arriving from the Oort cloud for the first time, with 1/a of about 4·10⁻⁵ AU⁻¹ and a spread of a few times 10⁻⁵. The broad curve is where those same comets go after one passage through the planetary region, whose perturbations change 1/a by a typical 5·10⁻⁴ AU⁻¹ — ten times the spike's width. 47 per cent end with negative 1/a, on hyperbolic orbits that leave the Solar System; only 4 per cent remain inside the spike's width. A comet in the spike is therefore on its first passage, almost by definition: the planets cannot put a comet back into the spike once it has been through them.

Orbits

A spike only first-time visitors can be in

Long-period comets arriving from the Oort cloud have orbital energies packed into a spike a few hundred-thousandths of an inverse astronomical unit wide. One passage through the planets changes that energy by ten times as much, so no comet can be in the spike twice — and the spike's position, near thirty thousand astronomical units, is where the Galaxy's own tide is just strong enough to lift a comet's perihelion over Jupiter and Saturn in a single orbit.

$The same porkchop with Mars's orbit tilted, and the ridge that appears. Contours of departure energy C₃ for transfers from the Earth to Mars, over departure date and flight time, with Mars's orbit inclined by its real 1.85° to the ecliptic and every point a solved three-dimensional Lambert problem. The dashed line is where the transfer angle is exactly 180°. Along it the contours pile up into a ridge reaching C₃ of 346 km²/s² and more, against 7.9 at the cheapest point: a transfer half-way round the Sun has to lie in a plane through the Sun and two nearly opposite points, and reaching a target a fraction of a degree out of the ecliptic tilts that plane towards the pole. The ridge splits the plot into two basins — Type I transfers, less than half-way round, below and to the left, and Type II, more than half-way, above — and the cheapest transfers sit in one basin or the other, never on the line between them.$ Spaceflight

A wall across the porkchop

Draw the cost of going to Mars with Mars's orbit in the Earth's plane and the porkchop plot is a smooth pair of basins. Tilt Mars's orbit by its real 1.85 degrees and a wall appears between them, hundreds of times higher than the valley floor, along the line where the transfer goes exactly half-way round the Sun. The wall is a plane change forced by geometry, its height changes from one launch window to the next, and a single burn half-way to Mars removes it.

Cosmology

An emptiness that expands faster

A cosmic void is not a hole in the universe but a region with a fifth of the average density, and gravity makes it behave like a small open universe: it expands faster than its surroundings, every galaxy in it streams outward, and in a redshift survey it is stretched along the line of sight — the opposite of what the same gravity does around a cluster. The stretch measures how fast structure grows, and the average void's roundness measures the geometry of the universe.

$A clumped plasma recombines earlier. The fraction of hydrogen still ionised against redshift, decreasing to the right, on a logarithmic axis, from Peebles's three-level atom: for a smooth plasma (solid), and for one whose baryons are clumped into two equal volumes at 1.71 and 0.29 times the mean density, a clumping factor of 0.5 (thick dashed, averaged by mass; thin dashed, each zone alone). Recombination goes as the density squared, so the dense zone recombines first and carries most of the mass; the average ionisation falls through one half at redshift 1292 instead of 1273, and at redshift 1100 it is 0.123 against 0.145. The photons cannot see the individual patches, which are smaller than their mean free path; they see only the average, and the average recombines earlier.$ Cosmology

A field that would have moved the ruler

A magnetic field left over from the early universe, too weak to matter to any galaxy, could still have pushed the baryons into small clumps before the universe became transparent. Clumped hydrogen recombines earlier, the sound waves in the plasma stop sooner, and the ruler that the microwave background is read with comes out shorter — which raises the Hubble constant the background implies, in the direction of the disagreement that has resisted every other explanation.

Gravitation

A disc that throws its spin upward

A disc does not need turbulence to accrete. A magnetic field threading it can fling gas off its surface along field lines tilted more than thirty degrees from the vertical, and each gram flung out leaves with several times the angular momentum it had — so a wind carrying a few per cent of the inflow can drive the rest onto the star. Because it pulls on the disc's faces rather than across its edge, a wind outpulls the turbulence as soon as the field's pressure is a few ten-thousandths of the gas's.

$Supply and demand in a cluster's core. The rate at which hard binaries heat a star cluster's core, against the core's central density, for binary fractions of 1%, 5%, 20%, both on logarithmic axes and in units that put the balance for five per cent at one. The heating rate per binary does not depend on how hard the binary is — the energy released per encounter is proportional to its binding, and the chance of an encounter to its size, so the two cancel — and summed over a core whose radius shrinks as its density rises, the total grows as the square root of the density. The horizontal line is the demand: the energy the rest of the cluster carries away by relaxation through its half-mass radius, which the core has no say in. Each supply curve meets it at one density: 25 for 1%, 1.0 for 5%, 0.063 for 20%. The core contracts or expands until it produces exactly what is demanded, and a core rich in binaries does so at a lower density.$ Gravitation

A core that makes what the cluster asks for

After core collapse, a star cluster's binaries heat it — but how much they heat it is not theirs to decide. The rest of the cluster carries energy outward at a rate fixed by its own size and mass, and the core contracts or swells until its binaries supply exactly that. So the core's structure follows from the halo's demand, and the cluster expands at a rate that forgets how it was born.

A chaotic orbit that waits, and then moves. The momentum of one trajectory of the standard map at K = 1.2, unwrapped and counted in whole turns of 2π, against the iteration number, over 39,892 iterations. The motion inside each band is chaotic from the first step — the trajectory's Lyapunov time is 3.7 iterations — but it lingers near one integer resonance for thousands of iterations before reaching the next: the waits were 3,098, 1,464, 9,461, 15,957, 6,058, 3,854 iterations. Each arrival at the next integer resonance, drawn dashed, is marked. Between the integer resonances lie the remnants of the broken invariant curves, which no longer block the trajectory but still hold it, and each passage is quick compared with the wait before it.

Gravitation

A barrier that leaks at a rate that can be computed

When the last invariant curve between two resonances breaks, it does not vanish. It becomes a cantorus — a curve full of gaps — and a chaotic orbit has to find the gaps to get through. The time that takes grows as the cube of how close the system is to the threshold, and it has nothing to do with the Lyapunov time, which is why an orbit can be hopelessly unpredictable and still stay put for the age of the solar system.

One stream after three different populations of dark subhaloes. The density of stars along the middle sixteen kiloparsecs of a thin stream, relative to its mean, after four billion years of encounters with dark subhaloes drawn from a mass function falling as the −1.9 power of the mass between 10⁵ and 10⁸ solar masses. Top: the abundance cold dark matter predicts — in this realisation 26 encounters, 4 of them above a million solar masses — which leave 32 per cent of the stream's length more than 30 per cent below the mean density. Middle: 0.3 of that abundance, 8 encounters and 7 per cent of the length in gaps. Bottom: none at all, where the only variation is the counting noise of the 60,000 model stars. No single dip in the top trace is a clean gap: they overlap, and a large one and several small ones are superposed at every point. What differs between the three is the statistics of the whole trace, not any feature in it.

Galaxies

A spectrum that counts what no single gap can

One gap in a stellar stream measures a subhalo's mass multiplied by an unknown time. A stream crossed by dozens of them measures something better — how many subhaloes of each size there are — but only as a statistic of the whole stream's density, where no feature can be pointed to and one stream is one throw of the dice.

$How often Easter falls on each date, over the Gregorian computus's whole cycle. The fraction of years in which Gregorian Easter falls on each of its 35 possible dates, 22 March to 25 April, computed year by year over 5.7 million years — the length after which the computus repeats exactly, because the golden number, the weekday and the two century corrections only then return to the same combination. The distribution rises from 0.48 per cent on 22 March through a plateau near 3.3 per cent a date to a spike of 3.87 per cent on 19 April, then falls to 0.74 per cent on 25 April. The ramps at each end come from adding a Sunday to a full moon: the earliest dates need both the earliest full moon and the right weekday. The spike on 19 April has no astronomical meaning at all; it is made by a rule in the table that forbids the paschal full moon to fall on 19 April and moves it to the 18th, so the 18th is a full-moon date half as often again as its neighbours.$ The observed sky

A full moon computed from the year number

The date of Easter is defined by the Moon and computed without looking at it — ten lines of integer arithmetic on the year number, a finite machine that repeats itself exactly every 5.7 million years. Its output has a spike on 19 April that no sky produces, a table that has kept pace with the real Moon for four centuries, and a Julian twin that is now four days behind it.

How far a small error at departure moves the arrival at Mars. The distance by which a craft on the Hohmann transfer from the Earth's orbit to Mars's misses its aim point at the nominal arrival time, against an error in its departure speed, on logarithmic axes, integrated numerically for the 259-day flight. An error along the direction of motion (solid) moves the arrival by 76 km for every millimetre a second; one across it (dashed) by 23 km. The relation is linear over the whole range drawn: a metre a second at departure — the precision of a good launch — is 76 thousand kilometres at Mars, several times the planet's own radius and a thousand times what a landing needs. The ratio is the corresponding element of the state transition matrix, and it is the number every navigation plan is built around.

Spaceflight

A millimetre a second at launch, seventy-six kilometres at Mars

A craft on its way to Mars arrives seventy-six kilometres from where it was aimed for every millimetre a second its departure speed was wrong. No launch is that good, so no trajectory is flown as designed; it is flown as corrected, by Newton's method run on the matrix that says how the arrival depends on the departure — which doubles its correct digits at every step and makes a two-body sketch into a real trajectory in two iterations.

Two clocks for a radiation-belt electron, at Kp 2. Two times against drift shell L, on a logarithmic scale, for electrons of about 1 MeV. The first is how long radial diffusion takes to move an electron by one Earth radius, the inverse of an empirical diffusion coefficient that grows as the tenth power of L — at Kp 2, 550 years at L = 2, 10 years at L = 3, 21 days at L = 5. The second is how long the electron survives before it is scattered into the loss cone. Inside the plasmapause, here at L = 4.7, that is set by plasmaspheric hiss: years at L = 1.3, 14 days at L = 2.5, 5.2 days at L = 3; outside it a single lifetime of 30 days stands for every other loss. Inside L = 4.8 the electron is lost before diffusion can carry it further in, which is the shaded region; outside, diffusion wins and the belt is fed. The two curves cross at a single place, and that crossing is the inner edge of the outer belt.

Spaceflight

The gap between the belts is a race between two clocks

Electrons reach the inner magnetosphere by diffusing inward from the outer edge, and they are lost there to waves that scatter them into the atmosphere. The diffusion slows as the tenth power of distance and the loss does not, so the two clocks cross at one sharp place. Inside it lies the slot, and at megaelectronvolt energies the inner electron belt is not a steady population at all but what the last great storm left behind.

Gravitation

Stars lost to a kick that adds no energy

A tidal shock changes each star's energy by two terms. One is the heating every textbook integral computes, always positive and second order in the shock. The other is the product of the star's own velocity and its kick, first order in the shock and exactly zero on average — and for the weak shocks a cluster meets most often it decides who leaves, doubling the losses without adding any energy at all.

$Debris discs of very different masses, fading to one line. The fractional luminosity of a debris belt at 50 AU — the share of its star's light the dust absorbs and re-radiates — against age, from a million to ten billion years, for belts that start with 1, 10, 100, 1000 Earth masses of planetesimals up to 60 km across. Each stays near its starting brightness until its largest bodies have had time to collide, 16.5 billion years, 1.7 billion years, 165 million years, 17 million years respectively, and then fades as one over the age. A heavier belt starts brighter and grinds itself down faster, and the two effects cancel: after their collisional times all belts lie on the same line, the dashed maximum, whatever their mass. At ten billion years the two heaviest differ by 1 per cent.$ Orbits

A ceiling that forgets how much there was

A belt of planetesimals around a star shows itself only through its dust, and the dust is ground from the belt's largest bodies by collisions. Give the belt more mass and it makes more dust — but it also collides more often, and grinds itself away faster. At late times the two effects cancel exactly, and every belt at a given distance and age lies on the same line of maximum brightness whatever it started with. A disc above that line cannot be a steady cascade, and it is telling of a recent catastrophe.

Orbits

An orbit that turns over

A distant third body makes an inner orbit trade inclination for eccentricity in slow, regular cycles, and one quantity — the inner orbit's angular momentum along the outer orbit's axis — keeps the trade bounded and forbids the orbit from ever tipping past ninety degrees. That conservation holds only if the outer orbit is a circle. Make it eccentric and the quantity drifts, the cycles stop repeating, and at a moment of extreme eccentricity the inner orbit turns over, from prograde to retrograde, with its pericentre pressed a thousand times closer than the regular cycle allowed.

Mars's tilt over fifty million years, and a steadier twin. The obliquity of a Mars-like planet against time over 50 million years, integrated from 25 degrees with its orbital plane wobbling in five secular modes. With Mars's precession constant, 8.26 arcseconds a year, the axis wanders between 1 and 70 degrees, lingering near some values and lurching between them. The same planet precessing 6 times faster — about the Earth's rate, which the Moon's pull makes possible — stays between 23.2 and 30.8 degrees, because its precession is too fast to resonate with any of the orbital-plane frequencies. The orbital perturbations are illustrative, of the size Mars's own are; the chaos is computed from them.

The observed sky

A past that can only be given as odds

Mars's axis is tilted by twenty-five degrees today, and nothing holds it there. Its slow precession falls among the frequencies at which its orbital plane wobbles, the resonances overlap, and the tilt wanders chaotically between nearly zero and sixty degrees. Two calculations that start a hair apart part company within a few tens of millions of years — so Mars's obliquity a billion years ago is not a number anybody will compute. What can be computed is how much of its history the planet spent at each tilt, and that distribution is a prediction the ice at its poles can test.

Spaceflight

Fifty objects that hold most of the future

Low Earth orbit holds thousands of dead satellites and spent rocket stages, and the debris of the future will come mostly from their collisions. But not in equal shares. A derelict's contribution is the chance it is hit, times the fragments it would make, times how long they would stay up — and all three favour the heaviest objects in the most crowded, highest orbits. Ranked that way, a few tens of rocket bodies near 850 kilometres carry nearly half the risk, and the arithmetic of cleaning space becomes the arithmetic of choosing them.

Spaceflight

Braking on a moon before braking with the engine

A gravity assist is usually told as a way to gain speed, but the same exchange runs the other way. A spacecraft falling toward Jupiter can pass behind one of its moons on the way in, lose a few hundred metres per second of its speed to the moon's orbit, and arrive at its capture burn needing a fifth less propellant. Which moon brakes best depends on three things at once — its mass, how fast it is met and how fast it moves — and the best one sits in radiation no spacecraft wants to fly through.

$The chance that a candidate is a planet, against how strong its signal is. The reliability of a transit candidate — the fraction of candidates at a given detection statistic that are planets rather than false alarms — against the multiple-event statistic, the combined signal-to-noise of all its transits, for orbital periods of 10, 100, 350 days. Below the 7.1 threshold nothing is a candidate. At a statistic of 9 the reliability is 100 per cent, 80 per cent, 23 per cent; at 12 it is 100 per cent, 95 per cent, 44 per cent. The true planets and the false alarms are modelled, not fitted: the planets as a falling power law above the survey's completeness curve, the false alarms as an exponential tail above the threshold that grows longer and taller with period, because a long-period signal is two or three transits and two or three transits average away little of the non-Gaussian noise that makes false alarms. At short periods a candidate near threshold is nearly always real; at a year, a candidate near threshold is usually not.$ Exoplanets

At a year, most candidates are noise

A transit survey's threshold is set so that Gaussian noise yields less than one false planet in the whole catalogue. The noise is not Gaussian, and at year-long periods the candidates just above the threshold are more often artefacts than planets — so the rate of Earth analogues is partly a measurement of the noise.

Which hierarchical triples survive, against the outer orbit's pericentre and eccentricity. Each cell is one integration of a triple: an equal-mass inner binary on a circular orbit of unit separation, and a third body of 1 times the binary's mass on an outer orbit of the given eccentricity with its pericentre at the given distance from the binary's centre of mass, all in one plane and going the same way round. Cells coloured as survivors kept the binary intact and ejected nothing for 30 outer orbits — 37 of 50; the others broke up, shaded darker the longer they lasted. The curve is the Mardling–Aarseth stability criterion, a pericentre of 2.8 inner separations times ((1 + q)(1 + e)/√(1 − e))^0.4, derived from where the resonances between the outer orbit and the inner one begin to overlap, and meant for survival over very many orbits. Every triple outside it survived. Inside it, 17 of 30 also survived this run; the outermost break-up is at a pericentre of 3.0 for a circular outer orbit and 2.6 at e = 0.6. The boundary drawn from a finite integration is therefore a lifetime rather than a line — triples between the break-ups and the curve survived a run of this length and are not thereby shown to be stable; the criterion is a statement about how the resonances are arranged, and a finite integration can only fail to refute it.

Gravitation

A third body kept at a safe distance

Most triple stars last for billions of years because they are hierarchical — a close pair and a distant third. How distant is distant enough has a formula, the formula comes from overlapping resonances, and an integration of any finite length can only fail to refute it.

The figure-eight with equal masses, and with one mass changed. The paths of three bodies in the figure-eight choreography over 100 periods, with equal masses (left) and with the third body's mass changed by 0.0001 (right). With equal masses the three paths lie on one another: each body runs round the same figure-of-eight, a third of a period behind the next. With the mass changed the three bodies stay bound but leave the single curve, filling a broad band round where it was. The choreography is an exact solution only for exactly equal masses, and the island of stable orbits near it is narrow.

Gravitation

A periodic orbit that the sky cannot keep

Three equal masses can chase one another for ever round a single figure of eight. The orbit exists, it is stable, and it will almost certainly never be seen — because its island of stability is a hundredth of a per cent wide in the masses, and nothing in nature makes three stars equal to that precision.

A cloud of fragments that pinches where it was born. 240 fragments thrown from a satellite on a circular orbit 800 kilometres up, with kicks of about 100 metres a second in random directions in the orbital plane, each then following its own Kepler ellipse, drawn at 0.5, 1, 4.5, 20 orbital periods of the parent, with the Earth's centre marked and the break-up point at the top. Radial offsets from the parent's circle are exaggerated 2.5 times. Half a period after the break-up the fragments are spread 28 degrees along the orbit and 1374 kilometres in altitude, the kicks having carried some to higher apogees and others to lower perigees. After one period the along-track spread has grown to 53 degrees but the whole cloud is only 112 kilometres thick, and the 83 fragments within eight degrees of the break-up point span 35: every fragment's orbit passes through the break-up point, and the cloud pinches there once a period. By 4.5 periods it is 231 degrees long and 863 kilometres thick overall; by 20 it is 352 degrees long, still pinched to 11 kilometres where it began.

Orbits

Every fragment comes back through the same point

When a satellite breaks up, its pieces fly off in every direction and every one of them returns, an orbit later, to the place where it broke. The cloud pinches there once a period, stretches along the orbit at a rate set by one component of each kick, and becomes a ring in hours.

The observed sky

A clock allowed to leave the Sun

Twenty-seven leap seconds have kept civil time within a second of the Earth's rotation since 1972. By 2035 they will stop, and the clock will be allowed to drift from the Sun. How far it drifts, and how soon a minute accumulates, depends on a length of day that no one can forecast.

The three terms of the minimum disposal height. The minimum height above the geostationary ring to which a satellite should be raised at the end of its life, by the international guideline ΔH = 235 km + 1000 · Cᵣ · A/m, broken into its three terms, for reflectivity times area-to-mass ratios of 0.005, 0.01, 0.02, 0.03, 0.05 square metres per kilogram. The first 200 km is the protected zone itself; the next 35 km is the most that the Moon, the Sun and the Earth's uneven gravity lower an orbit's perigee over time; the rest is the eccentricity that sunlight raises every year, which a derivation from first principles puts at 942 km per unit of Cᵣ·A/m for an orbit started circular — the guideline's round thousand. Totals: 240 km, 245 km, 255 km, 265 km, 285 km. The term that depends on the satellite is the one sunlight sets, and for a light satellite with large solar arrays it is the largest.

Spaceflight

A graveyard height with a term for sunlight

A geostationary satellite at the end of its life is raised about three hundred kilometres and switched off. The height is written as a formula — 235 kilometres plus a thousand times its reflectivity and area-to-mass ratio — and each term is a separate piece of orbital mechanics, the last one sunlight bending the orbit a little every year.

Phase-space density during a storm fed from inside. Electron phase-space density at fixed first and second adiabatic invariants, against the drift-shell parameter L, before a two-day storm (grey) and 0.5, 1, 2 days after it begins, from a model of radial diffusion with a diffusion rate rising as L¹⁰ and a loss lifetime of 4 days. The boundary is held fixed and a source centred at L = 4.5, standing for acceleration by whistler-mode chorus waves, acts for the storm's two days; the density grows a peak at L = 4.5, well inside the boundary, and diffusion then spreads it both inward and outward. A peak in phase-space density inside the outer boundary cannot be produced by diffusion from outside while the boundary is rising — diffusion only smooths gradients — so its presence then is the signature of local acceleration, and its absence is consistent with transport.

Spaceflight

A peak that says where the electrons were made

The outer radiation belt refills with electrons of millions of electronvolts within hours of a storm. Whether they were carried inward from the edge of the magnetosphere or accelerated where they are found shows in one place only — the density of electrons in the coordinates their motion conserves, which a detector does not measure directly.

$A velocity error that grows with distance. The uncertainty in one galaxy's peculiar velocity, v = cz − H₀d, against its distance in Mpc/h, when the distance comes from an indicator with a fractional error of 5 per cent, 20 per cent, 25 per cent (type Ia supernovae, 5%; Tully–Fisher, 20%; fundamental plane, 25%). The redshift is exact to a few kilometres a second, so the whole error is the distance error times H₀, and it grows in proportion to the distance. The horizontal line is the rms velocity of a galaxy along one direction in linear theory for the model drawn, 302 km/s. Each galaxy's own velocity is measured to better than its own size only inside 60 Mpc/h for 5 per cent, 15 Mpc/h for 20 per cent, 12 Mpc/h for 25 per cent; beyond that a single velocity is mostly noise, and a velocity survey is useful only as an average over many galaxies.$ Cosmology

Velocities that see past the edge of the map

A redshift survey infers the motions of galaxies from the distortions of its own map. A peculiar-velocity survey measures them directly, one galaxy at a time, by subtracting a distance from a redshift — and every one of those measurements is dominated by its own error beyond a few tens of megaparsecs. What makes the survey worth doing is that a velocity answers for structure far larger than the volume it is measured in.

Cosmology

A void is not short of dwarf haloes

The centre of a large void holds a fifth of the average density of matter and almost no galaxies, and it is natural to read the two facts as one. They are not. The theory of how haloes form says a void turns its matter into dwarf-sized haloes at least as readily as anywhere else — it is short of big haloes, not small ones — so a void empty of dwarf galaxies is a statement about which haloes were allowed to light up.

$One history of mass loss, three orbits, three outcomes. The eccentricity of three planets' orbits against the fraction of the star's mass lost so far, as the star sheds 45 per cent of its mass in 6 equal pulses at irregular intervals. Each planet starts with e = 0.05. The pulses are the same events in the star's own time, and the planets differ only in their periods, so each pulse lasts 3, 0.3, 0.03 orbital periods respectively. For the innermost planet, to which every pulse is slow, the eccentricity ends at 0.050: the orbit swells and keeps its shape. For the outermost, to which every pulse is sudden, each pulse kicks the eccentricity by an amount set by where the planet happened to be, and it ends at 0.298. The middle planet ends at 0.090. The same total loss leaves three different shapes, and only one of them is predicted by either limit alone.$ Orbits

The size forgets how the mass was lost, and the shape does not

A dying star loses half its mass, and a planet round it keeps its orbit's shape if the loss is slow and has it scrambled if the loss is sudden. Real stars lose mass in pulses — each one sudden or slow depending on the planet — and what survives a train of pulses is neither answer: the orbit's size settles to the slow value quickly, while its eccentricity random-walks and keeps a record of the history.

$How much of a population is driven how close. The fraction of an isotropic population of inner orbits whose Kozai–Lidov cycling brings 1 − e below a given value within 150 secular timescales, against that value, both logarithmic, for octupole strengths ε = 0, 0.01, 0.03; each curve is 128 integrated orbits with random pericentres and nodes. The thin line is the quadrupole closed form, √((3/5)(1 − (1 − q)²)), which the ε = 0 population follows. At 1 − e = 0.01 the fractions are 9.4, 20.3, 40.6 per cent; at 0.001, 3.1, 18.0, 38.3. The octupole matters most where the quadrupole is weakest — at the extreme closeness a planet needs to be captured by tides, or a pair of compact objects needs to merge — because it adds orbits whose quadrupole cycle stopped far short of it.$ Orbits

Eleven in a hundred by geometry, and the rest by waiting

A distant companion drives an inner orbit through cycles of eccentricity, and whether a given system reaches a dangerously close approach depends on its inclination. Counted over a population with random orientations, the answer at quadrupole order is pure geometry — about eleven per cent reach one per cent of their semi-major axis, all at the same inclination. An eccentric outer orbit multiplies that share, spreads the inclinations, and makes the count depend on how long the systems have been waiting.

The same tilt, and ice ages of two different lengths. Top: the obliquity anomaly in degrees from the five largest terms of the standard series, over 2 million years of model time; it is dominated by the 41,000-year cycle and has no power at all near 100,000 years. Bottom: ice volume in a threshold model driven by it — the ice grows steadily, and it collapses whenever volume plus a multiple of the obliquity anomaly crosses a threshold, marked by dots. For the first 1 million years the threshold is low and the ice collapses at nearly every obliquity maximum, on average every 43 thousand years. At the vertical line the threshold is raised, and nothing else changes: the ice now survives one or two obliquity maxima and collapses at the second or third, every 108 thousand years on average. The forcing is identical on both sides; the ice sheet counts it differently.

Orbits

An ice sheet that skips a beat of the tilt

The last eight hundred thousand years of ice ages come roughly every hundred thousand years, and the Earth's orbit has no forcing of any strength at that period. A model in which an ice sheet collapses only when it is both large and warmed by a high-obliquity summer reproduces the rhythm with nothing but the forty-one-thousand-year tilt — the ice sheet grows through one or two maxima and falls on the next — and a slow rise in how large the ice must grow turns the older forty-one-thousand-year world into the newer one without any change in the orbit.

Each kilometre of aim error, turned into a velocity error. The error in the outgoing velocity per kilometre of error in where a spacecraft passes a moon, in metres per second per kilometre, against the flyby altitude on a logarithmic scale, for the four large moons of Jupiter at an encounter speed of 6 km/s. The flyby turns the velocity through an angle δ with tan(δ/2) = aₕ/b, where b is the aim point's distance from the moon's centre and aₕ = μ/v∞²; missing the aim by a kilometre along b changes δ, and missing it by a kilometre across b rotates the plane of the turn, and both move the outgoing velocity by about 2v∞aₕ/b² per kilometre. At 300 km above Ganymede that is 0.32 m/s per km along and 0.32 across. The closer and more massive the flyby, the more it turns — and the more it amplifies whatever error the approach carried.

Spaceflight

A flyby that multiplies its own aiming error

A close flyby of a moon turns a spacecraft's path by ten degrees for nothing, and it turns every error in the aim with it. A kilometre of miss at three hundred kilometres above Ganymede becomes a third of a metre per second of velocity error, and a week later two hundred kilometres of miss at the next moon. A tour of flybys is an amplifier that must be re-zeroed after every pass, and whether a small spacecraft can use one in place of an engine burn depends on how many kilometres it can aim to.

What it costs to follow a turning grapple point. The velocity a chaser must spend each minute to hold station on a grapple point carried round by a flat spin, ω²r × 60 s, against the spin rate in degrees per second, for points 1, 3, 10 m from the spin axis, both axes logarithmic. The dashed line is what holding station 10 m from a target in a 400 km orbit costs in the same units, from the orbital dynamics alone: 2.3 mm/s per minute. At 1° per second, a point 3 m out costs 5 cm/s per minute — 24 times the orbital term; at 5° per second, 1.4 m/s per minute, 595 times. The cost rises as the square of the rate, so a target spinning twice as fast is four times as expensive to follow, and above a few tenths of a degree per second the orbit — the whole subject of the approach up to this point — has become the small correction.

Spaceflight

A grapple point that moves faster than the orbit

Every rendezvous problem up to the last few metres is orbital mechanics: two bodies in nearly the same orbit, drifting apart and together under gravity. A derelict satellite tumbling at a few degrees a second changes the problem entirely. Holding station on a point of its surface costs more each minute than a week of orbital station-keeping, the moment to grab it comes and goes in seconds, and a free tumble never repeats — so the orbit becomes a small correction to a problem in rotation.

The orbit that turns across the bar. The radial displacement with which a nearly circular orbit answers a weak rotating bar, against radius in kiloparsecs, in linear theory, scaled to its largest value: positive means the orbit's long axis lies along the bar, negative across it. The disc's rotation curve rises within 0.35 kpc to 220 km/s, the bar ends at 3 kpc and turns with corotation at 3.6 kpc, and the disc then has two inner Lindblad resonances, at 0.37 and 0.78 kpc (vertical lines). Without dissipation (dashed) the response diverges at each resonance and changes sign across it: orbits between corotation and the outer resonance are elongated along the bar — the x1 family that builds the bar itself — and orbits between the two resonances are elongated across it — the x2 family. Gas, which dissipates (solid, damping 8 km/s/kpc), cannot follow the flip abruptly; its orbits turn through ninety degrees over a band round the resonance, where streams of gas on neighbouring orbits cross, collide and shock. That band is where a nuclear ring forms. Near corotation the linear theory itself fails, and the curves stop short of it.

Galaxies

The bar leaves its gas where the orbits turn

A rotating bar in a disc galaxy drives gas inwards from kiloparsecs out, and the gas does not fall to the centre. It stops in a ring a few hundred parsecs across and forms stars there at rates the rest of the galaxy cannot match. The ring sits at the bar's inner Lindblad resonance, where the orbits that support the bar give way to orbits turned across it — so its existence says there is mass concentrated at the centre, and its size says how fast the bar turns.

The polarisation peaks sit in the temperature troughs. A tight-coupling sketch of the two acoustic sources against multipole, each scaled to its own maximum: the temperature, from the compression of the photon–baryon fluid at last scattering plus its Doppler term, and the E-mode polarisation, which is driven by the fluid's velocity gradient. The sound horizon and the distance to last scattering put the acoustic scale at ℓ = 302, the measured phase of the peaks is 0.267 and the damping scale is ℓ = 1400; the baryons' weight, which changes the heights of the temperature peaks and not their places, is left out. The temperature peaks fall at ℓ ≈ 218, 515, 813, 1110, 1408; the polarisation peaks at ℓ ≈ 130, 391, 681, 975, 1272, between them, because a compression is at its extreme when the fluid is momentarily still, and the velocity is largest a quarter of a cycle later. The model has no projection, no decaying potentials and no thickness, so it places the peaks and says nothing about their shapes.

Cosmology

Polarised only where the glow was lopsided

An electron bathed in light from every side scatters it unpolarised, and so does one bathed in light brighter on one side. Only a glow brighter along one axis than across it leaves a polarisation behind, and in the microwave background that glow existed for a short time, in a thin part of a thin shell, because the plasma was moving.

The spectrum ahead is the spectrum behind, shifted. The difference between the local power spectrum of the microwave background measured in the direction of the observer's motion and in the opposite direction, as a percentage of the average, against multipole, at 369.8 km/s. Ahead, the pattern is compressed and every feature moves to a higher multipole by a factor 1 + β; behind, to a lower one by 1 − β, with β = 1.234e-3. The difference is largest where the spectrum is steepest — 2.30 per cent at most in this range — and passes through zero at every peak and trough, where a small shift changes nothing. The spectrum is an interpolation through the measured peaks and troughs, so its steepness between them is known only to that resolution. A per cent of a quantity estimated from millions of independent modes is a measurable number.

Cosmology

A speed read from the finest spots in the map

The dipole in the microwave sky is taken to be the observer's motion and subtracted. The same motion also squeezes the small-scale pattern towards the direction of travel by a part in a thousand, and that squeeze measures the speed a second time without using the dipole at all — which is the only way to ask how much of the dipole is the universe's own.

Exoplanets

A model of the star that can carry the planet

Starspots are modelled as correlated noise with the star's rotation period built in, and the model is good enough to remove signals that would otherwise be announced as planets. It is also good enough to remove planets. Near the rotation period no amount of data in one season separates the two, and away from it a flexible noise model quietly takes most of the evidence a planet would have had.

Exoplanets

A planet has one velocity in every colour

A Doppler shift moves every line in a spectrum by the same velocity, because it is the whole star that moves. A starspot does not move anything; it removes light, and how much light it removes depends on the colour it is seen in. Measuring the velocity in the blue and the red at once separates the two in a single night — until the star's magnetic field, which grows more visible towards the red, takes part of the separation back.

Exoplanets

A slope that is a whole curve of companions

A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside.

Galaxies

The one radius where the orbits stop mattering

A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.

The tide's spectrum, with every frequency set by the sky. The equilibrium tide-raising potential broken into its largest constituents, each a sinusoid whose frequency is a sum of whole-number multiples of six astronomical rates — the Earth's turning relative to the Moon, the Moon's and the Sun's motions round the sky, and the slow turning of the lunar perigee and node — and whose height is its equilibrium amplitude relative to M₂. They fall in three bands: near two cycles a day (M₂ at 1.9323, S₂ at exactly 2, N₂, K₂), near one (K₁, O₁, P₁, Q₁), and near zero (fortnightly, monthly, semi-annual). Every frequency is known to many decimal places from astronomy; what a harbour does to each, its own amplitude and phase, is not, and is the only thing a tide table has to measure.

Gravitation

A tide table is astronomy with the harbour fitted

Nobody can compute the tide at a harbour from the Moon and the Sun, because the ocean's response depends on coastlines and depths nobody models well enough. Tides are predicted to centimetres years ahead anyway, because the frequencies are pure astronomy and only the response to each one has to be measured. The method needs a record long enough to tell neighbouring frequencies apart, and a correction for a cycle of 18.6 years.

Gravity per kilometre of topography, by wavelength. The free-air gravity anomaly per kilometre of topography — the admittance — on Earth, against the wavelength of the topography, for elastic plates 5, 25, 100 km thick, with crust of density 2800 kg/m³ over mantle of 3300 and compensation at 35 km. Short wavelengths are uncompensated and give the full attraction of a slab of crust, 117 mGal per km. Long wavelengths are compensated and give nearly nothing, because the mass of the mountain is cancelled by the root beneath it. Where the curve falls between the two is set by the plate's thickness, so a measured admittance spectrum is a measurement of how strong the lithosphere was when the load was applied.

Gravitation

A mountain's gravity says what holds it up

A mountain can stand because the rock beneath it is strong, or because it floats on a deep root of light crust like an iceberg. The two leave different gravity behind them, and the difference depends on the mountain's size — so the ratio of gravity to topography, measured wavelength by wavelength, is a measurement of how stiff a planet's outer shell was when the mountain was built.

$A slow shock still heats the cluster. The energy a Plummer star cluster gains from a compressive tidal pulse, as a fraction of what the same impulse delivered instantaneously would give, against the pulse's duration times the orbital frequency at the half-mass radius. The points come from integrating 1200 stars through the pulse one by one. The lines are prescriptions applied to the same stars: an oscillator's exponential suppression and a power law, each applied star by star at the star's own orbital frequency, and the shortcut of one power-law factor for the whole cluster, evaluated at the half-mass radius. At the longest pulse drawn the cluster keeps 0.28 of the impulsive energy; the star-by-star rules give 0.24 and 0.36, bracketing it, and the half-mass shortcut gives 0.017, 17 times too little. The total is set by the outermost stars, which carry most of the impulsive energy and are always slower than the shock.$ Gravitation

A slow shock heats the halo and robs the core

The textbook treatment of a slow tidal shock multiplies the instantaneous answer by a suppression factor that falls steeply as the shock lengthens. Integrated star by star through shocks of intermediate length, a cluster behaves differently. Its total heating falls slowly, because the outer stars are always slower than any shock and carry most of the energy, and its core does not merely escape the heating: on balance its stars lose energy, which no factor that multiplies a positive number can describe.

The observed sky

Every star in the catalogue is bent by the Sun

The bending of starlight by the Sun is remembered as an eclipse measurement of 1.75 arcseconds at the solar limb. The same bending reaches every direction in the sky, falling only slowly away from the Sun, and at right angles to it is still four milliarcseconds — two hundred times the precision of a modern star position. Every catalogue corrects every star for it, and the correction has become the most precise test of how much space a mass curves.

Two ways out of a transfer orbit. The eccentricity against the semi-major axis of a spacecraft spiralling out from a geostationary transfer orbit — perigee 300 km up, apogee at geostationary height — under a continuous thrust of 0.2 millimetres a second squared, integrated through hundreds of revolutions, for two steering laws. Thrust always along the velocity (red) raises the orbit efficiently but leaves its shape almost unchanged, reaching the geostationary semi-major axis after 64 days still with an eccentricity of 0.59 — an orbit swinging between 17,000 and 67,000 km. A law that points the thrust to reduce a weighted distance from the target in both elements (blue) circularises as it raises, and reaches a circular geostationary orbit in 129 days.

Spaceflight

The thrust that rounds an orbit turns once a revolution

An electric engine pushes gently for months, and the obvious way to point it — along the velocity — raises an orbit as fast as possible and leaves its shape almost untouched. Starting from an elongated transfer orbit, that is the wrong job. Rounding the orbit off needs the thrust forward at apogee, backward at perigee and sideways in between, a full turn of the thrust vector every revolution, and the law that does it can be written down from the equations for how each element changes.

Two critical mass ratios that move with where the mass goes. The logarithmic response of the donor's Roche lobe to its own loss of mass, ζ, against the mass ratio donor over accretor, for conservative transfer (dashed) and for all of the transferred mass lost from the accretor, from the donor and to a ring. The horizontal lines are two donors' own responses: −1/3 for a deep convective envelope, which swells as it is stripped, and +2 for a radiative one, which shrinks. Transfer is stable where the donor's line lies above the lobe's curve, so each crossing is a critical mass ratio — conservatively 0.63 and 1.72. Re-emission from the accretor moves them to 0.96 and 2.25, a wind from the donor to 1.24 and beyond the range drawn; and with everything fed to a ring the convective donor is unstable at every mass ratio drawn and the radiative one only below 0.19. Nothing about the donor has changed between the curves; only the question of where a gram that is not accreted leaves from.

Stars

Mass that leaves takes the orbit's spin with it

Most of the gas a star pours onto its companion never stays there. Where it leaves from decides how much angular momentum it carries, and that one bookkeeping entry moves the largest donor that can transfer stably by a factor of two — without anything about either star having changed.

How far a turbulent disc bends the field it drags. The tangent of the angle by which an accretion flow bends a vertical magnetic field at the disc's surface, against the magnetic Prandtl number Pm — the ratio of the turbulent viscosity that drives the inflow to the turbulent resistivity that lets the field slip — for discs of aspect ratio 0.03, 0.05, 0.1. In a steady state the inflow drags the field in at exactly the rate it diffuses back, and that balance sets tan i = (3/2) Pm H/R. The dashed line is 30° from the vertical, the inclination a field needs before gas is flung off along it. Turbulence whose own Prandtl number is of order one bends the field by about the disc's aspect ratio, a few degrees; reaching 30° needs Pm of 13, 8, 4. The same turbulence that carries the gas in lets the field out almost as fast, because both are the same eddies.

Gravitation

A field the turbulence will not let in

A disc that accretes through its own turbulence drags its magnetic field inward, and the same eddies let the field slip back out almost as fast. A disc of ordinary thickness bends its field by a few degrees instead of the thirty a wind needs, so a turbulent disc cannot gather the flux that would let it stop being turbulent.

Gravitation

A disc that stores its inflow and spends it in a century

Where only a thin ionised skin of a young disc can transport angular momentum, the gas arriving from the cloud cannot be passed on, and it piles up in the dead midplane until gravity stirs it and the whole column ignites. The star then gains most of its mass in bursts of a century every ten thousand years — which is why almost every protostar ever observed is fainter than its growth requires.

Orbits

A tilt the Moon should not have

A moon's orbit precesses about a plane that sits on its planet's equator close in and on the planet's own orbit far out. A Moon born in the Earth's equator and carried outward by the tides should have followed that plane all the way to the ecliptic, because the angle it keeps to it is an adiabatic invariant. It is inclined by five degrees instead, and slow recession cannot have put them there.

$Four numbers that carry a position and a velocity round an orbit. The Lagrange coefficients f, g, ḟ and ġ — g and ḟ scaled by the mean motion so that all four are pure numbers — against time as a fraction of the period, for an ellipse of eccentricity 0.6 and periapsis 0.5 AU, each computed from one universal-variable solve starting at periapsis. The new position is f times the old position plus g times the old velocity, and the new velocity ḟ times the old position plus ġ times the old velocity: the state at any time is a linear combination of the state at the start, with no semi-major axis, eccentricity or anomaly formed. f and ġ return to one and g and ḟ to zero after a full period, which is the orbit closing. At every one of the 401 instants the combination f ġ − ḟ g equals one to 6·10⁻¹⁵, which is the conservation of angular momentum written as the area of a parallelogram.$ Orbits

Four numbers that carry a state without an orbit

Once the universal variable is solved, a body's position and velocity at any later time are a linear combination of its position and velocity now, with four coefficients and no orbital elements formed at all. The four obey one identity exactly — the conservation of angular momentum, written as an area — and its failure measures the error a calculation has made without knowing the answer.

How fast an error in the departure grows, near a neck and on a conic. How much a small error in a spacecraft's velocity is magnified in its later position — the largest singular value of the block of the state-transition matrix that maps velocity errors to position errors, in the Earth–Moon system's units — against time in days, integrated from the variational equations of the restricted three-body problem. A trajectory held at the L1 point between the two bodies (blue) amplifies errors exponentially, by a factor of e every 1.48 days — the rate the linearised dynamics there predict, 2.932 in the system's units, which the integration reproduces to within two per cent. A two-body circular orbit at half the separation (green), integrated the same way with the second body's mass switched off, amplifies them only secularly — an error in speed is an error in period, and that grows along the track in proportion to time. Over 14 days the neck has multiplied an error by 2,359; the conic by 9.

Spaceflight

A trajectory with no good first guess

A trajectory designed from conics is corrected onto the real one by Newton's method, and the method works because the arrival depends on the departure only polynomially. Near a Lagrange point the dependence is exponential: every day and a half near the Earth–Moon neck multiplies an error by e. The first guess has to be exponentially good, the digits of the departure run out in weeks, and the only cure is to stop treating the trajectory as one piece.

$Removals a year, set almost entirely by one percentage. The number of derelicts that must be removed each year to hold a shell at 50 per cent of its critical count, against the fraction of disposals that fail, for a 650-satellite constellation at 800, 1000, 1200 km. Because a dead satellite at these heights stays up for centuries, almost every failure has to be removed by hand, and the removal rate is very nearly the failure rate: at 1200 km, 13 a year if ninety per cent of disposals succeed and 1.3 if ninety-nine do. The removal budget is therefore set by an engineering reliability rather than by the orbital environment, and a percentage point of reliability is worth a fixed number of removal missions a year for as long as the constellation flies. Five removals a year is enough for this shell only above 96.2 per cent reliability.$ Spaceflight

A removal rate set by one percentage

How many dead satellites have to be taken out of a high orbit each year is decided less by the orbit than by how often a satellite fails to dispose of itself. Above eight hundred kilometres a derelict stays for centuries, so the removal rate needed to keep a shell stable is very nearly the failure rate, and moving disposal reliability from ninety per cent to ninety-nine cuts the work tenfold.

Fitted to one curve, three halo histories agree at the Sun and not at the centre. The dark-matter density from 1 to 15 kpc from the Galactic centre, on logarithmic axes in GeV per cubic centimetre, for three halos fitted to the same flat rotation curve with the same disc of 5·10¹⁰ solar masses: one as dark-matter-only simulations produce it, and two contracted by the disc under different assumptions about the dark matter's orbits. At the Sun's 8.2 kpc they give 0.45, 0.41, 0.40 GeV/cm³, a spread of 12 per cent. At 2 kpc they give 3.2, 4.6, 4.6, a spread of 44 per cent. Inside the region the curve constrains, a fit has no freedom to move dark matter about, and the history drops out; inward of it, where the disc and bulge dominate the curve and the dark matter is a small part of the total, the history decides the answer.

Galaxies

A history the rotation curve absorbs

Whether the Milky Way's dark halo was pulled inward by its disc changes the dark matter at the Sun by half again — if the halo is taken from a simulation. Fit it to the Galaxy's own rotation instead, and the history drops out at the Sun's radius and survives only near the centre. The local density that detection experiments depend on is set by the disc's mass, not by the halo's past.

$The mass at which a growing core cuts off the pebbles drifting past it. The pebble-isolation mass — the mass at which a core perturbs the gas around it enough to raise a pressure maximum outside its orbit, where drifting solids stop — against distance from the Sun, for a passively heated disc whose thickness grows as the distance to the 2/7, with α = 10⁻³. The mass goes as the cube of the disc's aspect ratio, because a planet has to push the gas by a fraction of its own pressure, and so it grows outward: 7 Earth masses at 1 AU, 30 at 5 AU where Jupiter's core formed, 138 at 30. A core reaching this mass stops growing from pebbles, starts accreting gas, and becomes a dam: everything drifting inward from outside its orbit is held there.$ Exoplanets

A core that shut the door on the pebbles

Solids in a gas disc drift inward, and a growing planet eats them as they pass until it is heavy enough to bend the gas around it. Then a pressure maximum forms outside its orbit, the drift stops there, and everything beyond is cut off from everything within. Jupiter's core reached that mass early, and the solar system's meteorites still fall into two families that never mixed.

$The fraction of a spin alignment a disc can show. The ratio of a disc population's shape alignment — its mean ellipticity along the projected tidal template — to the Lee–Pen alignment of its spins, against the discs' thickness as a fraction of their diameter, each point from 30,000 discs in a field of shape 0.2. The thinnest disc drawn, 0.05 of its diameter, passes 0.204 per unit of spin alignment; a disc a fifth as thick, typical of a spiral's stellar disc, 0.171; a body as thick as it is wide passes nothing, because a sphere has no shape to align. The ratio is the conversion between what a spin survey measures and what a lensing survey fits, and it is small for two reasons that multiply: a spin alignment is a weak preference among directions, and a disc shows its direction only in proportion to how far it is tipped towards edge-on.$ Cosmology

An alignment a disc shows only when it is tipped

A disc galaxy's outline on the sky is its spin seen in projection, so the torque alignment of spins has to appear in shapes too. It does, multiplied by a factor near a sixth and carried almost entirely by the discs seen close to edge-on — which is how a lensing survey can fit the term and find nothing while a survey of spins finds it.

Exoplanets

A spread the projection does not hide

A single projected spin–orbit angle says little about the true one, because the star's tilt is unknown. A population of nearly aligned systems is different: for small tilts the projected angle is one component of a two-component vector, so the rms true obliquity is exactly √2 times the rms projected angle — no star's inclination needed.

Gravitation

A few thousand clouds outrun a hundred billion stars

Encounters between stars relax the Galactic disc in a time hundreds of thousands of times the age of the universe, so in practice they do nothing. Put a fraction of the same mass into a few thousand molecular clouds and the time falls to the age of the disc — because relaxation counts the mass of each perturber, not their number. The disc's present thickness is the record of that, and of a bound on anything heavier hiding in the halo.

Gravitation

Torques that add because the orbits hold still

Near a massive black hole every star's orbit is almost a fixed ellipse, repeated thousands of times before it precesses. The torques the other stars exert on it therefore point the same way orbit after orbit, and they add in proportion to time instead of cancelling in a random walk — so angular momentum near the hole relaxes hundreds of times faster than ordinary encounters allow, and orbital planes faster still.

Orbits

A distant planet felt only as a stretch

An undiscovered planet far beyond Neptune pulls the Sun and Saturn almost equally, and an ephemeris, which measures only where Saturn is relative to the Sun, cannot see a pull that moves both together. What it can see is the difference — a tidal stretch falling as the mass over the cube of the distance — and most even of that is absorbed by refitting Saturn's orbit. What is left decides which planets the solar system can be hiding.

$The Moon pushed towards the Sun, once a month. The change in the Earth–Moon distance, in millimetres, if the Earth's gravitational self-energy fell towards the Sun differently from the rest of its mass by a fraction η = 0.001 — the term that would be present at the synodic period, the cycle of lunar phases, in an integration of the Moon's orbit in the Sun's full field. The amplitude is 11.4 mm, which is 11.4 metres times η: the Moon's orbit would be stretched towards the Sun at new moon and squeezed at full, or the reverse, depending on the sign. The shaded band is ±2 mm, roughly the precision of a modern lunar laser ranging normal point. A violation a thousandth of the size of the effect it tests would be twice that precision.$ Orbits

A Moon that would lean towards the Sun

Part of the Earth's mass is its own gravitational binding energy — a few parts in ten billion. If that part fell towards the Sun even slightly differently from the rest, the Earth and the Moon would fall at different rates and the Moon's orbit would be pushed sunward once a month by about eleven metres for every unit of the difference. Fifty years of laser ranging to the Moon have looked for that push at the millimetre level, against a term the Sun's ordinary gravity puts at the same frequency a hundred kilometres high.

A storm's depression at the equator, as an energy. The depression of the horizontal magnetic field at the Earth's surface against the total kinetic energy of the particles trapped in the ring current, on logarithmic axes. The Dessler–Parker–Sckopke theorem makes the two proportional — ΔB/B₀ = −⅔ E/Eₘ, with Eₘ = 8.4·10¹⁷ joules the energy of the Earth's dipole field above the surface — whatever the particles' energies, pitch angles or distribution in radius. The marked storms are read off the line: a moderate storm of 50 nT is 2·10¹⁵ J of trapped particles, an intense one of 100 nT 4·10¹⁵ J, the storm of March 1989 at 589 nT 2.4·10¹⁶ J, and the lowest estimate for the Carrington storm of 1859 3.4·10¹⁶ J — a few hours of the solar wind's power flowing past the magnetosphere.

The observed sky

A storm weighed by a compass needle

During a geomagnetic storm the horizontal field at the equator drops by tens or hundreds of nanotesla, and a theorem says the drop is proportional to the total energy of the particles trapped around the Earth — whatever their energies, their pitch angles or their distances. A ground magnetometer therefore weighs a population tens of thousands of kilometres up, in joules, and the shape of the record says what that population is made of.

Ions picked up from Mars and swept off on loops. The paths of a hydrogen and an oxygen ion created at rest just above the dayside of Mars and picked up by the solar wind, flowing left to right at 420 km/s through an interplanetary field of 3 nT perpendicular to the page. In the planet's frame each moves on a cycloid: it is accelerated by the wind's electric field, turns in the magnetic field, comes momentarily to rest and repeats, drifting downstream at the wind's speed and reaching twice it at the top of each loop — four times the wind's energy per unit mass. The loop height is twice the gyroradius: 2923 km for hydrogen, 46774 km for oxygen, against a planetary radius of 3390 km. The oxygen is thrown clear of the planet in a single plume to one side, which is how a planet without a magnetic field loses its heavy atmosphere to the wind.

The observed sky

A planet that borrows the wind's field

Venus and Mars have no magnetic field of their own, yet the solar wind does not reach their atmospheres. It is stopped by their ionospheres, whose thermal pressure balances the wind's at an altitude that moves logarithmically with every gust, and the interplanetary field piles up and drapes around the obstacle to make a magnetosphere on loan. The arrangement holds only while the ionosphere out-pushes the wind, and it leaks heavy ions on loops wider than the planet.

What a departure loses to taking time. The gravity loss of a departure burn — the velocity a rocket spends beyond what an instantaneous impulse would need — against its initial thrust-to-weight ratio, on logarithmic axes, for departures from a 300-km orbit to hyperbolic excess speeds of 0.5, 3, 6 km/s, an exhaust speed of 4.41 km/s and thrust held along the velocity. The impulsive costs are 3.21, 3.60, 4.74 km/s. At a thrust-to-weight of 1.2 the losses are 4, 5, 11 m/s; at 0.1, 298, 396, 708; at 0.03, 1485, 1785, 2546. For a stage of high thrust the loss falls as about the inverse square of the thrust (slope -1.98 at the high end), because the burn arc shrinks in proportion and the loss grows as the square of the arc.

Spaceflight

A burn that cannot happen at one point

Every transfer cost in orbital mechanics is computed as though the engine fired for an instant. A real departure burn runs for minutes or hours, and the vehicle travels a long arc while it does — climbing, slowing, and delivering thrust ever further from the point where it would have done the most good. The extra velocity that costs is the gravity loss, and it grows as the square of the arc: negligible for a strong stage, ruinous for a weak one, and recoverable by splitting the burn at the price of time.

$The atmosphere's delay, by how high the spacecraft stands. The one-way delay of a radio signal through the troposphere, in metres of range, against the elevation of the line of sight, for an atmosphere whose refractivity falls exponentially with a hydrostatic scale height of 8 km and a wet one of 2 km. At the zenith it is 2.42 m. Integrated along a straight ray through a curved atmosphere (solid), it is 5.0 m at 30°, 13.4 m at 10° and 24.6 m at 5°. The flat-atmosphere rule, zenith delay over the sine of the elevation (dashed), agrees above about twenty degrees and then runs away, giving 46.2 m at 3° where the curved atmosphere gives 35.9: a ray at a low angle leaves the atmosphere through the top of a shell, not through the side of a slab. Ray bending, ignored here, adds centimetres below about five degrees.$ Spaceflight

Two metres overhead and thirty at the horizon

A radio signal crossing the lower atmosphere is delayed by about two and a half metres of range straight up and by more than ten times that near the horizon. Tracking a spacecraft means converting the first number into the second, and the conversion — the mapping function — is where almost all of the troposphere's error lives: under a millimetre above thirty degrees, centimetres to decimetres in the last ten. And the last ten degrees are exactly the observations the geometry cannot do without.

$Where relativity overtakes the margin: 306 Schwarzschild radii for 10⁵ solar masses. Two things against the size of a contracting star, measured in its own Schwarzschild radius, on logarithmic axes. The flat lines are the margin by which the star's pressure-weighted adiabatic exponent exceeds 4/3 — the Newtonian condition for stability — for stars of 1000, 10⁴, 10⁵, 10⁶, 10⁷, 10⁸ solar masses with mean molecular weight 0.61; it does not change as the star shrinks, because it is fixed by the gas fraction of the pressure, and it falls with mass from 0.036 to 1.2·10⁻⁴. The diagonal is what general relativity adds to the requirement, 1.1245 × 2GM/(Rc²), which in these units is the same line for every mass. Contraction moves each star leftward along its flat line, and where the line meets the diagonal the star becomes dynamically unstable: at 31, 97, 306, 966, 3,052, 9,649 Schwarzschild radii respectively. Every one of those crossings lies between 31 and 9,649 Schwarzschild radii out, where the star is not remotely compact — the relativistic correction is tiny, and it decides the matter only because the margin it is compared with is tinier.$ Stars

A star tipped over at a thousand times its horizon

A star of a million suns is held up almost entirely by its own light, which leaves it stable by about a thousandth. General relativity raises the bar by an amount that is tiny until the star is compact, and a thousandth is tiny too — so the star collapses at nearly a thousand times the radius of the black hole it becomes, at a mean density below that of air, and with no fuel yet burning.

The centre goes soft at 19 solar masses; the whole core only at 27. Two criteria for the pair instability, against the mass of the oxygen core, each the deepest value reached as the core contracts through the pair-producing range. The orange curve is local: the lowest Γ₁ − 4/3 at the core's centre. It goes negative at 19.1 solar masses — the centre of every such core passes through gas that, taken by itself, cannot hold itself up. The blue curve is the criterion that actually decides the matter, the pressure-weighted mean over the whole core, and it stays positive until 27.5 solar masses of oxygen. In the shaded band between the two the centre goes soft and the core does not go unstable, because the stiffer gas around the centre outvotes it. The threshold is computed for a core built as an n = 3 polytrope with the pairs ignored in its structure; detailed evolutionary models find the first pair-instability pulses in stars whose helium cores exceed roughly thirty solar masses, so the construction lands in the right neighbourhood rather than on a precise number.

Stars

A soft centre the rest of the core outvotes

Above a billion kelvin, light in a star's core starts turning into electron–positron pairs, and gas that spends its heat on making mass cannot push back hard enough to hold itself up. But stability belongs to the whole core, not to its centre. An oxygen core goes soft in the middle from about nineteen solar masses and holds; it gives way only near twenty-eight, when the soft gas carries two-thirds of the pressure-weighted volume.

The kick speeds read back out of the periods: median 100 m/s against the 100 m/s put in. The distribution of kick speeds recovered from the fragments' periods alone, by p(v) = −2v g′(v) applied to the along-track kicks, for 300 and 3000 fragments (points), against the log-normal distribution actually used, median 100 m/s and width 0.3 dex (line), on a logarithmic speed axis with the density per logarithmic interval. With 3000 fragments the recovery follows the shape of the truth and puts the median at 100 m/s, though differentiating a measured distribution twice smooths its peak and roughens its tails; with 300 it is noisy but already in the right place. The inversion uses one assumption, that the kicks were isotropic, and one measurement per fragment, its period — the quantity tracking networks determine first and best.

Orbits

A kick speed read off the slope of a histogram

Throw fragments at one speed in random directions and the component of each kick along the orbit is spread evenly between plus and minus that speed — Archimedes' theorem about the area of a sphere, applied to debris. Each fragment's period records that component, so the histogram of periods is a stack of flat bands, one per speed, and its slope gives the speeds back: a break-up weighed by its own violence from a catalogue of orbits, if the event was isotropic, and fragment by fragment if it was not.

Two clouds with the same kicks: the explosion's centre at −2.3 m/s, the collision's at 10.9. The along-track kicks read from the periods of 1500 fragments, for an explosion (blue) and for a collision (orange) whose fragments received the same random kicks plus a shared velocity change of 12 m/s along the track — the centre-of-mass motion a projectile's momentum leaves behind. The two histograms have the same width and shape; the collision's is displaced. The number-weighted mean of the explosion's kicks is −2.3 ± 2.6 m/s, consistent with zero, and the collision's 10.9 ± 2.5 m/s. The width of a cloud says how violent the event was; its centre says whether anything arrived from outside.

Orbits

A collision moves the cloud's centre, and an explosion cannot

However violently a satellite blows itself apart, its fragments' momenta add up to the parent's: the centre of the cloud keeps moving exactly as the satellite did. A collision is different, because the projectile brings momentum in, and a collision energetic enough to shatter its target needs a projectile heavy enough to move the centre by at least 80,000 joules per kilogram divided by the impact speed — eight metres a second in low orbit. The width of a debris cloud measures how violent the event was. Its centre measures whether anything arrived from outside.

Spaceflight

The one axis the field will not take

A satellite that keeps one face towards the Earth can empty its reaction wheels into the Earth's magnetic field with coils, using no propellant. Its own rotation, once an orbit, cancels every steady torque except those about one axis, the pitch axis normal to the orbit. In an orbit near the equator the field lies along that same axis, and a coil cannot push about the direction of the field.

How much each change moves the growth rate. The percentage difference in fσ₈ from growth in relativity inside a cosmological constant's expansion, against redshift, for three alternatives started from the same early amplitude. Changing the expansion to the evolving fit with DES-SN5YR and keeping relativity moves it by −1.2 per cent at redshift 0.5, and by at most 2.1 per cent anywhere below redshift 2. Keeping the constant's expansion and changing gravity to γ = 0.68 moves it by −8.6 per cent; doing both, −10.3. The shaded band is ±8 per cent, about the error of one of the better redshift-space measurements so far. The difference between a constant and the evolving fit is several times smaller than any single measurement; the difference between two laws of gravity is comparable to one. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is gravity.

Cosmology

The growth that tells a substance from a law

In relativity an expansion history leaves no freedom in how fast structure grows inside it, so every fitted history predicts its own growth. The evolving dark energy the surveys prefer changes that growth by one or two per cent; a change in the law of gravity, with the expansion held fixed, changes it by nine. Growth is a weak test of whether the dark energy is changing and a strong test of whether it is there at all.

$How far a sample of lenses can take the Hubble constant. The fractional uncertainty in H₀ from a sample of N lenses, both axes logarithmic: the independent errors of each lens — 2 per cent from its delay, 3 from the rest of its lens model, 2.5 from its line of sight and its own mass sheet as well as its kinematics fix it — fall as 1/√N; the shared errors — 1 per cent from the calibration of the line of sight, and the orbital bias of the previous figure — do not. With one aperture dispersion per lens the independent error is 10.1 per cent per lens and the floor 1.6 per cent, reached at about 42 lenses; with six resolved rings per lens, 5.9 and 1.1 per cent, reached at about 29. Dotted: the two floors. A sample that adds lenses without changing how each is measured stops improving at its floor; the resolved kinematics lower both the slope's starting point and the floor.$ Galaxies

A hundred lenses and one assumption

A time-delay lens gives the Hubble constant to five or ten per cent, and the hope is that a hundred give it to one. The errors that differ from lens to lens — its delay, its own line of sight, its own internal sheet — do average away. The errors every lens shares because every lens was analysed the same way do not, and they set a floor that a sample reaches after a few dozen lenses and never goes beneath.

The binaries in a fed core are spread evenly in energy. The steady distribution of binding energies among the hard binaries of a core where new binaries enter at x = 5 and leave by ejection from a cluster whose escape speed is 4 times its dispersion: the number per unit binding energy, against binding energy in units of mσ², from 20,000 simulated binaries weighted by the time each spends at each energy. Between twice the entry energy and a third of the typical ejection energy (about 120) it is flat to within 5 per cent, and it falls away smoothly near ejection because a single encounter can throw a binary out from anywhere above about 24. A flat distribution is what a population with a constant flux through energy and a constant drift must have: binaries pass every energy at the same rate, and spend the same time at each.

Gravitation

The binaries a core keeps are spread evenly in energy

A hard binary in a star cluster gains binding energy at the same average rate however hard it already is. So in a core that is fed new binaries and loses hardened ones, every binding energy is passed through at the same speed, and the steady population is flat in energy — which, counted per factor of energy, means most binaries are within a factor of ten of being thrown out. Where they were born hardly matters; where they leave decides everything.

Orbits

The jet that pushes a comet also turns it

The gas a comet sheds near the Sun pushes its orbit, and because the vents are not arranged symmetrically about its spin axis, it also twists the nucleus. In 2017 one small comet's day more than doubled in two months. The twist per orbit grows as the inverse square of the nucleus's radius while its supply of ice grows only as the radius, so below about a kilometre a comet is spun apart before it runs out of ice — and the shortage of small comet nuclei may be the result.

What DI Herculis's orbit should do, and what it does. The rate at which the periastron of the eclipsing binary DI Herculis advances, in degrees per century. Top: the prediction for stars whose spins are aligned with the orbit — 2.34 from general relativity, computed from the masses (5.17 and 4.52 solar masses), the 10.550164-day period and the eccentricity of 0.489; and 1.93 from the stars' tidal and rotational distortion, as published stellar models give it, of which the rotational flattening is 1.50 and the tides 0.43. Total 4.27. Bottom: the observed rate, between about 0.6 and 1.4 degrees per century in the published timing analyses — a quarter to a seventh of the prediction. For a quarter of a century after the discrepancy was established it was cited as possible evidence against general relativity, which contributes more than half the predicted total by itself.

Stars

A precession slowed by stars lying on their sides

The eclipsing binary DI Herculis turns the long axis of its orbit at a quarter to a seventh of the rate relativity and the stars' own distortion predict, and for a quarter of a century that gap was offered as evidence against general relativity. The answer was in the stars: both spin nearly on their sides, and a flattened star tipped past 54.7 degrees drags the orbit's periastron backwards instead of forwards. Explained, the slow precession becomes a measurement of something no spectrum shows — how far the spins lean towards the Earth.

$In a mixed field the torque follows the dipole and ignores the rest. The torque of a field that is part dipole and part quadrupole, with the total surface strength fixed, as a fraction of a pure dipole's, against the fraction of the field strength in the dipole, for wind magnetisations of 300, 3,000, 30,000, using the broken power law fitted to simulated mixed-field winds. Above a dipole fraction of 0.39, 0.24, 0.15 respectively, the torque is exactly what the dipole component alone would give: the quadrupole has fallen below the dipole long before the wind reaches its Alfvén radius, and contributes nothing to the lever. Below it, the quadrupole sets a floor. A star whose surface is mostly quadrupole by strength can therefore be braked almost entirely by its small dipole — and a measurement of the total field says little about the torque unless it says how much of the field is dipolar.$ Stars

A lever set by the dipole and not the field

A star's wind carries away its spin because the magnetic field holds the gas in step with the surface out to the Alfvén radius, and the torque goes as the square of that radius. How far out the field can hold depends less on how strong it is than on how it is arranged: a quadrupole of the same surface strength holds the Sun's wind to half the distance and brakes it at a fifth of the rate. In a mixed field only the dipole reaches far enough to matter — so the field that brakes a star is the part of it that is hardest to measure.

How far inside the door a day of compression reaches. The innermost drift shell at which phase-space density has halved after a day of compression, against where the compression put the last closed drift shell, for diffusion at Kp = 4, 5, 6, 7. With the door at L = 5 and Kp = 6 the loss reaches L = 3.8; at Kp = 4 it reaches only 4.3. The diagonal is the door itself: everything above the curves and below the diagonal was emptied without ever being opened. A strong storm both brings the door in and speeds the diffusion that drains the shells behind it, and the two together carry the loss a full Earth radius or more inside the shells that were opened.

Spaceflight

A belt drained through its outer door

When a storm's solar wind squeezes the magnetopause inward, the outermost drift shells of the radiation belt are cut open and their electrons leave within minutes. The shells inside are untouched by the boundary and by the atmosphere alike, and they empty anyway: the opened region is a sink, and the same diffusion that fills the belt from outside carries electrons out to it. A day of compression drains the belt a full Earth radius inside the shells that were actually opened.

A dropout that partly comes back. The flux of 1 MeV electrons at L = 4.5, relative to before the storm, through a storm whose ring current deepens to Dst = −150 nT over eight hours and decays with a one-day e-folding. Dashed: the adiabatic part alone, with phase-space density unchanged — the flux falls to 18 per cent at the storm's peak and returns completely. Solid: the same with a real loss of 60 per cent of the phase-space density during the main phase, as compression and outward diffusion would cause. The flux falls to 7 per cent and recovers only to 39. Seen in flux at fixed energy, the two causes look the same while the storm is on, and only the recovery — days later — tells how much of the dropout was electrons lost and how much was the field.

Spaceflight

A dropout that is only the field

During a storm the ring current weakens the field the radiation belt sits in, and every electron, keeping its adiabatic invariants, loses energy. A detector counting electrons at a fixed energy and place sees the flux fall by half or more — and not one electron has gone anywhere. When the ring current decays, the electrons regain their energy and the flux comes back. Only in the coordinates the electrons themselves conserve can a storm's real losses be told from the field's sleight of hand, and the bigger the storm, the larger the sleight.

$The number of lit dwarfs at a void's centre, against the date it was reionised. The number of haloes heavier than 3.2·10⁹ M☉/h that had collapsed from the matter of a 5 Mpc/h sphere by the redshift at which that sphere was reionised, on a logarithmic axis, against that redshift — for a sphere at the centre of a void with 20 per cent of the mean density today (solid) and an average sphere of the same size (dashed). If only those haloes could make stars, they are the sphere's dwarf galaxies. Reionised at 5.3: 4.3 in the void sphere, 195 in the average one; reionised at 6: 2.0 in the void sphere, 121 in the average one; reionised at 7.7: 0.22 in the void sphere, 29 in the average one. Each unit of delay in the date multiplies the void's count by 3.2 and the average sphere's by 2.1: the void's dwarfs sit further out in the tail of the distribution, and a tail is steep. A count of a few galaxies at a void centre is a statement about when that void's gas was heated, to within a fraction of a unit of redshift — provided the threshold is known.$ Cosmology

A void dated by the dwarfs that lit in time

Reionisation did not heat the universe all at once, and the centres of the large voids were among the last places it reached. A dwarf halo there could make stars only if it had grown heavy enough before its own gas was heated — so the number that did is a clock for one void, and a clock that runs faster the emptier the void.

The heaviest atom a hydrogen wind can carry, against the strength of the wind. The crossover mass — the heaviest atom an escaping hydrogen flow drags out with it — against the hydrogen flux at the base of the flow, on a logarithmic axis, for the Earth's gravity and gas at 1000 K. Atoms lighter than the crossover leave with the hydrogen; heavier ones stay. The crossover rises in proportion to the flux: lifting neon takes 9.7·10¹² hydrogen atoms per square centimetre per second, argon 1.8·10¹³, krypton 4.3·10¹³ and xenon 6.6·10¹³. The Earth loses about 3·10⁸ today, which lifts nothing heavier than hydrogen. Driven by a young Sun at its saturated X-ray and ultraviolet output, an energy-limited flow would reach 2.2·10¹³ on Mars, a crossover of 114; 2.3·10¹³ on Venus, a crossover of 50; 10¹³ on the Earth, a crossover of 21 — the smaller planet's weaker gravity both raises the flux per unit area and lowers the force holding the heavy atoms back.

Exoplanets

The heaviest atom a hydrogen wind can lift

A planet losing hydrogen fast enough to blow off as a wind drags other gases with it, but only those lighter than a crossover mass set by how strong the wind is. Below it every gas leaves in proportion to its distance from the limit and its isotopes are sorted by the same amount per mass unit; above it nothing goes. The Earth's xenon is sorted and its krypton is not, which no wind of neutral atoms can arrange.

Galaxies

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

$How long a chaotic orbit stays away, with and without islands. The fraction of recurrence times to a small square in the chaotic sea of the standard map that exceed a given number of iterations, on logarithmic axes, counted along one orbit of 6·10⁷ iterations at K = 1.5 and K = 7 (772,165 recurrences, mean 78; 548,043 recurrences, mean 109). Dashed: exponentials with the same means. At K = 7, where the islands have all but vanished, the recurrences are exponential: the orbit forgets where it was in a few iterations. At K = 1.5 they part from the exponential after a few hundred iterations and fall instead as a power of time, with exponent 1.17 between a thousand and a hundred thousand iterations; the step at the left is the share of visits that simply stay in the square for a second iteration. A recurrence a thousand times longer than the mean is found once in 257,388 recurrences, where an exponential would give one in e to the thousand. Each mean is the area of the chaotic sea divided by the area of the square — 110 when the sea fills the torus, 78 at K = 1.5, where the islands take 29 per cent of it. The islands shorten the average absence and lengthen its tail.$ Gravitation

A chaotic orbit clings to the edge of order

In a chaotic sea with no islands, an orbit forgets where it was within a few steps, and the time it stays away from any region falls off exponentially. Put islands of regular motion in the sea and the long absences multiply by factors of millions: the orbit reaches an island's edge, is caught in the hierarchy of smaller islands round it, and behaves as if it were regular for thousands of steps. The tail becomes a power law, and a rare few absences carry a large share of the orbit's whole history.

Orbits

A kick that writes over the record of the pulses

A star that becomes a white dwarf loses half its mass in pulses, and a companion far enough away feels each pulse as a sudden jolt to its orbit's shape. The wide binaries of white dwarfs should therefore carry the pulses' record in their eccentricities. They would, except that the white dwarf itself is born moving at a few tenths of a kilometre a second, and a kick that small writes more eccentricity than all the pulses together.

The summer sunshine at Mars's north pole over the last million years. The sunshine reaching Mars's north pole at the summer solstice, relative to its average, over the last 1,000 thousand years (blue), from a forcing built from the main periods of Mars's recent orbit: the tilt varying by about 3.5 degrees in 120,000 years and 1 degree in 99,000, the eccentricity by 0.03 in 95,000 years and 0.02 in 2.4 million, and the perihelion going round the seasons once in 51,000 years. The part due to the tilt alone is dashed. The sunshine swings by a factor of 2.06 between its extremes: the tilt sets a slow swell with a period of 120,000 years, and the precession, whose strength the eccentricity sets, cuts a faster ripple of 51,000 years into it, strongest when the orbit is most eccentric. These two periods, and their ratio of 2.35, are what a record of the sunshine should carry.

The observed sky

Layers that keep the rhythm and lose the dates

Mars's polar caps are stacks of ice and dust laid down while the planet's tilt and orbit changed the summer sunshine, and the question is whether they can be read as a record of that sunshine. They can — for the rhythm. Even when the ice accumulates faster under weak sunshine than under strong, the cycles of the tilt and the precession survive in the depth of the stack in their correct proportion. What does not survive is the date of any one layer.

The propellant a long departure burn wastes, under three ways of pointing it. The gravity loss of a departure from a 300-km orbit to a hyperbolic excess speed of 3 km/s with an exhaust speed of 4.41 km/s, against the stage's initial thrust-to-weight ratio, both logarithmic, for thrust held along the velocity (orange), held fixed in space at the best direction (gold) and turned at the best constant rate (blue). At a thrust-to-weight of 0.05: 1264 m/s, 3597 m/s and 1521 m/s. At 0.2: 166 m/s, 328 m/s and 154 m/s. At 0.8: 12 m/s, 20 m/s and 10 m/s. No one of the three is best everywhere. For burns of a few minutes to half an hour the constant-rate turn beats following the velocity by up to a tenth; for the longest burns, which sweep most of an orbit, following the velocity wins, because the direction that does most good turns faster as the vehicle gains speed and no constant rate keeps up. A fixed attitude, the simplest to fly, costs 88 per cent more than the best turn even for the strongest stage drawn, and several times more for weak ones. The true optimum, which no simple family contains, lies below all three and is computed numerically for each mission.

Spaceflight

No single way to point a long burn

A departure burn that lasts tens of minutes wastes propellant because it cannot all happen at the lowest, fastest point of the orbit, and how much it wastes depends on where the engine points while it runs. Three simple rules — follow the velocity, hold one direction in space, turn at a steady rate — are each best somewhere and none everywhere. A steady turn saves up to a tenth for burns of minutes; following the velocity wins for burns that last most of an orbit; holding still costs twice as much as either.

The cost of turning an orbit, against how much of each revolution the engine runs. The velocity spent per unit of inclination change, relative to an instantaneous turn at a node, when a small out-of-plane thrust is applied only in arcs centred on the two nodes, against the half-width of each arc (blue). Thrusting all the time costs π/2, 1.571; arcs of ±60° cost 1.209; ±30°, 1.047; ±10°, 1.0051. The price is time: the engine is off for the rest of each orbit, and the turn takes longer by the inverse of the duty times the efficiency (orange, right-hand scale) — relative to a push that was always fully effective, 1.57 times as long for continuous thrust, 3.14 for ±30°, 9.0 for ±10°. Most of Edelbaum's penalty is spent in the quarter-orbits near the points where a push turns the node instead of the plane.

Spaceflight

A plane turned only where turning is cheap

A low-thrust engine that turns an orbit's plane continuously pays π/2 times what an impulse at the node would cost, because for half of every revolution it is pushing where a push mostly swings the node instead of tilting the plane. Thrusting only in arcs about the nodes recovers almost all of that: arcs of ±30 degrees cost five per cent more than an impulse, at the price of taking twice as long. It is how geostationary satellites hold their planes, and it stops working for large turns, where a chord is shorter than any arc.

A conversion read galaxy by galaxy, with or without an alignment. The mean ellipticity of disc galaxies along a tidal template, in bins of each galaxy's own spin estimator — the quantity whose population average is the Lee–Pen alignment — for 60,000 discs a fifth as thick as they are wide, in three samples: spins set by the torque (net alignment 0.36), 85% of them randomised (0.054), and all of them random (0.002). The points trace the same curve through the origin in every sample — not quite straight, flattening at both ends, because a disc's shape saturates when its spin lies in the sky's plane — and a straight line fitted through each sample (dashed) has slope 0.279, 0.290, 0.292. That slope is the projection factor, what a disc's shape shows of its spin, and it does not care whether the population is aligned. A sample with no alignment at all measures the conversion as well as one with a strong one; only the population's mean, where all its galaxies pile up, moves.

Cosmology

A conversion measured without an alignment

The factor that turns a disc galaxy's spin alignment into the shape alignment a lensing survey fits looked like something only a sample with a detected alignment could test. It is not. It is a relation each galaxy obeys on its own, so it can be read off a few hundred discs with velocity maps whether or not their spins point anywhere in particular — and what is left for the hard measurement to test is only whether a disc's shape is anything more than its projected spin.

Exoplanets

A fraction read in each channel's own units

Hot Jupiters arrive by two routes, one that ends at twice the planet's Roche distance and one that ends where the disc stopped. Fitting the mixture in Roche distances alone, as the edge invites, gives a fraction that drifts by almost half if the planets weigh more than assumed. Each route has to be read in the units in which it does not move — and then the answer is limited not by how well the planets are measured but by how their radii have changed since they arrived.

$The angular momentum a halo has and the angular momentum a disc needs. The fraction of the mass with specific angular momentum below j, against j in units of the mean, for a halo's material — the universal profile fitted to simulated haloes, with shape parameter μ of 1.1, 1.25, 1.5 — and for an exponential disc with a flat rotation curve (black). Both have the same mean, as a disc that kept its halo's angular momentum would. They do not have the same shape. Below a tenth of the mean a disc holds 2% of its mass; a halo with μ = 1.25 holds 11%, and one with μ = 1.1, 15%. The halo also stops at a largest j, 4.0 times the mean for μ = 1.25, beyond which the disc still has a thin tail. A disc built from a halo's gas without any exchange of angular momentum would be too concentrated at its centre and cut off at its edge.$ Galaxies

A halo with too much slow gas for a disc

A disc's size follows from its halo's average spin, and the average is all the simplest account of disc formation uses. The whole distribution does not match. A halo holds five times as much of its material at low angular momentum as an exponential disc can accommodate, so gas that kept its angular momentum exactly would build a dense centre, a thin body and a hard edge — and the bulgeless discs that are common in the nearby universe say that something removed a fifth to a quarter of the slowest gas before it settled.

Galaxies

A stirring supernovae cannot pay for

The gas in a disc forming stars at redshift two moves at fifty kilometres a second in random directions, five times faster than in a spiral today, and turbulence that fast decays in a few tens of millions of years. Supernovae are the obvious source of the energy, and they run out at about thirteen kilometres a second however fast the disc forms stars. What holds the discs up is gravity — gas stirring itself by falling inward — and a disc stirred that way is also a disc pouring its gas into its own centre.

Sixty futures that began a trillionth apart. The momentum of the standard map at K = 3, a kicked rotor whose phase space is almost entirely chaotic, for 60 members started 10⁻¹² apart in angle, against iteration. For 41 iterations they are one trajectory; then the separation, growing by a factor of about 2.0 each step, reaches the size of the system and they become independent random walks. Three of the 60 cross the threshold at |p| = 90 within 400 iterations (dark), the stand-in for an orbit that goes unstable. Which members cross is decided by differences far below anything that could be measured; how many cross is a property of the dynamics, and it is the only thing the ensemble can be asked for.

Gravitation

A probability for a system with one future

The Solar System will do one thing, and an integration of it is wrong about which thing within a hundred million years. What long integrations report instead is a probability — one per cent that Mercury's orbit goes unstable before the Sun dies — and a probability of a single event is a strange object. It is a count over an ensemble of futures that differ by less than anything measurable, which makes sense only because chaos makes those futures independent; and how many of them are needed is a binomial question with an unforgiving answer for rare events.

The flattening a halo has, and the flattening an orbit feels. Three model haloes with flat rotation curves, each built from thin similar-shell layers whose axis ratio changes with radius (dashed), and the axis ratio a nearly circular orbit at each radius responds to (solid) — the a²-weighted mean over the shells inside the orbit, because a shell exerts no force on anything within it. A halo flattened to 0.85 everywhere is felt as 0.85 everywhere. A halo round inside and flattened to 0.8 beyond about 30 kpc is felt as round by every orbit within 20 kpc — 0.988 at 20 — and at 60 kpc as 0.83, where its own shells are 0.80. A halo flattened inside and round outside is felt as flattened well beyond where its shells have become round. An orbit reports a running average of everything inside it, weighted towards its own radius, and nothing at all about what lies outside.

Gravitation

A shell outside the stream turns nothing

A flattened halo turns the plane of every orbit inside it, and a stream of stars several orbits long records how far. But the turning is set only by the halo inside the orbit — a flattened shell enclosing it exerts no force on it at all — and within that, mostly by the shells just inside. So a stream measures a running average of the halo's shape out to its own radius, a halo round inside and flattened outside looks round to every stream in its core, and reading the shape of each layer back out of the streams costs more precision the further out the layer lies.

Which pairs of planets the angular momentum deficit can bring together. For each adjacent pair of planets, the system's angular momentum deficit divided by the critical deficit at which that pair's orbits could touch, on a logarithmic scale; below one the pair cannot collide under secular evolution, whatever happens. Blue: using only the deficit of the pair's own subsystem — the four terrestrial planets (7.8 × 10⁻⁹ in units of a solar mass times √AU) or the four giants (4.3 × 10⁻⁶). Orange: using the whole Solar System's deficit, which the giants hold almost entirely. On their own budget the inner pairs sit at 0.21, 0.12, 0.34 — safe. On the whole budget they sit at 112, 65, 187. The giants are safe either way, at 0.05, 0.18, 0.36. The terrestrial planets are stable because almost none of the giants' deficit reaches them, not because there is too little of it.

Orbits

A deficit the giants hold and Mercury could borrow

The angular momentum the planets' orbits lack of circular motion is conserved by every secular interaction, to every order, so it is a budget no planet can overdraw. Compared with what it would take to bring a pair of orbits into contact, it proves without an integration that the terrestrial planets cannot collide among themselves — and shows that they are safe only because the giants, who hold five hundred times as much, lend them almost none. A loan of half a per cent of Jupiter's would be enough to unsettle Mercury.

Where a spinning sphere of rubble is steepest. The slope of the surface of a spinning sphere of loose material against latitude — the angle between the local vertical and the combined pull of gravity and rotation — for spins of 30%, 60%, 80%, 90%, 97% of the critical rate, measured in its square, at which the equator would lift off: for a density of 1.19 g cm⁻³ the critical period is 3.03 hours and these are periods of 5.5, 3.9, 3.4, 3.2, 3.1 hours. The slope is zero at the pole and at the equator, where the pull and the vertical are aligned by symmetry, and steepest in between: at 30% it peaks at 10° at latitude 40°, at 97% at 70° at 10°. The band marks the angle of repose of loose rock, 30° to 40°. From 80% of the critical rate the steepest band is past it: the surface begins to slide well before the equator has lost its weight.

Orbits

A surface that slides before the body sheds

A rubble pile spun towards its limit does not wait for its equator to lose its weight. Long before that, its mid-latitudes grow steeper than loose rock can stand, and the surface flows — downhill, which on a spinning body means towards the equator. The ridge that so many small asteroids wear is built in that interval, at spins a sixth slower than the one that would fling material off, and whether it is built at all depends on a cohesion of less than a pascal.

$The Moon's Cassini states as it recedes. The angle between the Moon's spin axis and its orbit normal in each Cassini state, signed so that the present state is positive, against the Earth–Moon distance in Earth radii, for the Moon's present shape (β = (C − A)/B fixed) and orbital inclination. Today the Moon sits in Cassini state 2 at 6.68°. Closer in, two more states exist, a low-obliquity state 1 and an unstable state 4; they meet and vanish at 33.8 Earth radii, where the spin-axis precession, which weakens as the cube of the distance, falls to the critical fraction of the orbit's regression. A Moon that formed close to the Earth sat in state 1 at small obliquity, growing to 21° as it receded; at 33.8 Earth radii that state disappeared from under it, and it had to move to state 2 — which at that distance is at 46°. Since then, the obliquity has relaxed along state 2 to its present value.$ The observed sky

A state the Moon had to jump to

The Moon's spin axis sits a degree and a half from the ecliptic pole, locked in a geometry Cassini described in 1693. The lock is an equilibrium between the Earth's torque on the Moon's bulge and the slow regression of its orbit, and both depend on the Moon's distance. When the Moon was closer than about 34 Earth radii there were two more equilibria — and the one it almost certainly occupied vanished as it receded, leaving it to jump to the present one at an obliquity near 45°.

Whether a field helped, depending on what limits the escape. The ratio of the atmospheric loss of a magnetised Earth to that of the same Earth without a field, against the Sun's age, both on logarithmic scales, set at 2 today — the magnetised planet losing more — for two ways an unmagnetised planet's loss can scale. If both planets lose atmosphere in proportion to the solar-wind power they intercept (blue), the magnetosphere's cross-section shrinks as the young Sun's denser wind compresses it while the bare planet's does not, and the ratio falls to 0.46 at 0.7 billion years: the field protected the young Earth, on balance from before 1.9 billion years. If instead the bare planet's loss is set by the ions the Sun's ultraviolet makes above it (orange), the ultraviolet rises more slowly into the past than the wind's pressure does, and the ratio grows to 3.7: the field cost the young Earth more than it saves today. The answer to whether a magnetic field protects an atmosphere is not a property of fields; it is a statement about which process limits the escape, and the young Sun amplifies the difference.

The observed sky

A shield that depends on what limits the leak

Today the Earth, with its magnetic field, loses ions to space about as fast as Venus and Mars without one — a little faster, by most measures. Whether that was also true under the young Sun, whose wind pressed eighty times harder, depends on something no measurement of the present can settle. If escape is limited by the wind's power, the field protected the young Earth; if it is limited by the ions the Sun's ultraviolet makes, the field cost it more than it does now. The young Sun does not decide the question. It amplifies whichever answer is right.

Two arrivals with the same weak thruster. Two spacecraft entering Mars's sphere of influence (outer circle, 577 thousand km) from the right, each thrusting continuously against its velocity at 0.3 mm s⁻² — about what an electric thruster gives a spacecraft of a few tonnes — and aimed to pass 6 planetary radii from the centre. Arriving at 0.40 km/s (blue), the thrust removes the excess while the craft crosses the sphere: it is captured 10 days after entry, having spent 268 m/s, on an orbit whose apoapsis lies inside the sphere, where the planet rather than the Sun keeps it, and the drawn path follows the same thrust winding it inward until it is within one and a half planetary radii. Arriving at 0.80 km/s (orange), it passes through on a bent hyperbola and leaves again: the thrust had too little time, and too little leverage, to remove an excess twice as large.

Spaceflight

A thruster that must arrive slow

A rocket arriving at a planet stops by burning hard at the bottom of the gravity well, where every metre a second it spends removes the most energy. An electric thruster cannot burn hard anywhere. It can capture only arrivals slower than a few hundred metres a second, so it has to arrive with almost no excess at all — and then it pays the full circular speed of its final orbit to spiral down, two and a half times what a burn from the same start would cost, because a spiral spends its thrust at every speed on the way rather than at the highest one.

The last correction: accurate if late, cheap if early. For a spacecraft approaching a moon at 6 km/s with a 30 km aim error left by navigation from the Earth, which locates the moon by camera to 20 microradians and acts an hour after its last image: the error remaining at the flyby after one final correction (blue, km) and that correction's cost (orange, m/s), against how long before the flyby it is made, all on logarithmic scales. The error falls in proportion to the time-to-go, because the moon is seen from closer, until it meets a floor: 3.1 km from how well the limb of a Europa-sized moon can be fitted once it fills the camera's view, with the burn's own two-per-cent error, 0.6 km, beneath it. The cost rises as one over the time-to-go, because the same error has to be removed in less time. The floor is reached 6.2 hours out, and a correction later than that costs more and buys nothing. A correction a day out costs 0.35 m/s and delivers 10.8 km; one six hours out, 1.4 m/s for 3.2 km. With 1 m/s to spend, the best is 4.2 km, at 8.7 hours out.

Spaceflight

An aim bought with the last day of the approach

A spacecraft that photographs the moon it is approaching can aim its flyby far more precisely than the ground can tell it to, because it sees the target from closer every hour. But each hour closer leaves less time for a correction to act, so every kilometre of aim is bought with propellant — and below a few kilometres the camera stops helping at all, because a moon that fills the view can be located only as well as its limb can be fitted. For a small craft with a metre a second to spend, the aim comes out at four kilometres, set by geometry and not by the antenna.

$A second peak, seen through three precisions. The distribution of measured core fractions for a population in which 90 per cent of planets sit at their stars' prediction (0.32, with an intrinsic scatter of 0.03) and 10 per cent at 0.65 (scatter 0.08), each planet compared with its own star, observed with measurement errors of 0.04, 0.08, 0.15 in core fraction. At 0.04 the dense group is a separate hump; at 0.08 it is a shoulder; at 0.15 it has become a long upper tail on a single peak, indistinguishable by eye from a population that is simply broad. The current measurements sit between the second and third curves.$ Exoplanets

A second peak the errors have to let through

Whether the densest small planets are a population of their own or the far edge of the ordinary one is not a question about any single planet. It is a question about the shape of a histogram seen through measurement errors as wide as the gap it is looking for, and the number of planets it needs rises fifteen-fold for every factor of three in lost precision — which puts the answer a few dozen planets beyond the samples that first suggested it.

$A polarisation fraction that should change with frequency. The polarisation fraction of dust emission relative to its value at 353 GHz, against frequency on a logarithmic axis. A single population at a single temperature gives a fraction that does not change at all. If the carbon grains are unaligned and warmer than the silicates, they take a growing share of the emission toward high frequency and dilute the polarisation: at 1200 GHz the fraction falls to 0.90 of its 353 GHz value for a 3 K difference and 0.85 for 5 K. Half-aligned carbon halves the effect. The shaded band is flat to within ten per cent between 100 and 1200 GHz, the precision with which the satellite and balloon measurements across this range agree with no change.$ Galaxies

The grains that glow and the grains that align

Interstellar dust polarises starlight and polarises its own glow, and for decades the standard account gave the two jobs to different grains — aligned silicates, unaligned carbon. Two populations at two temperatures predict a polarisation fraction that falls toward high frequency; what satellites and balloons measured was a fraction that does not change. That flatness, which no grain shape can fake, says the grains that glow are the grains that align.

Gravitation

A Love number measured at nuclear density

The number that told Cassini there was water under Titan's ice is defined the same way for a neutron star, and in the last tenth of a second before two of them merge it changes the shape of the gravitational wave they emit. What the wave measures is mostly not the Love number itself but the star's size, raised to the sixth power — which makes a faint tidal signal into one of the few direct measurements of how large matter at nuclear density lets a star be.

Six moons, and how long each takes to move. The time each of Saturn's six largest moons would take to move outward by its own orbital radius, a divided by its rate of change, on a logarithmic scale. Bars: a constant tidal quality factor tuned to the inner moons' measured motion, under which the time grows steeply with distance and falls with mass — 6 billion years for Enceladus, 210 billion for Titan. Dashed: Titan's measured migration, 11 centimetres a year, which is 11 billion years — 19 times faster than the constant-Q prediction and within a factor of two of the inner moons, where the constant-Q picture puts them a factor of 33 apart. Resonance locking predicts exactly that: one time scale for every locked moon, set by the planet rather than by the moon.

Gravitation

A migration timed by the planet and not the moon

Titan is Saturn's most distant large moon, and by the standard account of tides it should be the slowest to move — a two-hundred-billion-year journey across its own orbit. It is moving eleven centimetres a year, twenty times faster, on the same time scale as the small moons close to the planet. A quality factor cannot do that. An oscillation inside Saturn, tuned to each moon in turn and dragged along by the planet's own slow evolution, can.

A two-body arc flown in the real field. How far from its target a spacecraft arrives when it flies the velocity Lambert's two-body problem gives, through the Earth's real field with its equatorial bulge, against the angle of the transfer, from a 400 km orbit to a point on a 420 km orbit, on a logarithmic scale. In a pure two-body field every solution arrives within a metre of the target; exactly 180° is left out, where the two points and the Earth's centre are in line and the transfer's plane is not determined. In the real field the miss grows with the transfer: at 51.6° inclination, 4.2 km after 60° and 43.0 km after 300°. An equatorial orbit misses most, because the bulge's extra pull is strongest over the equator and acts on it all the way round; an inclined orbit feels it only part of the time but also has its plane turned.

Orbits

An answer from two bodies, flown through a flattened Earth

Lambert's problem gives the exact velocity that carries a spacecraft between two points in a given time — exactly, for a spherical Earth. The real Earth bulges at the equator, by a thousandth of its central pull, and a spacecraft flying the exact two-body answer between two points in low orbit arrives tens of kilometres from where it was sent. The repair is to keep the two-body solver and lie to it about the target, and the lie converges by a factor of a hundred each time it is told.

Too slow to shatter. The kinetic energy a projectile brings per gram of a 2,000 kg satellite, against the impact speed on logarithmic axes, for projectiles of 1, 10, 100 kg. Above 40 joules per gram (dashed) the standard breakup model counts the collision as catastrophic — the whole target is shattered; below it, only a crater is made. The speeds needed are 12.6, 4.0, 1.3 km/s respectively. In the geostationary ring, where nothing crosses faster than about 0.8 km/s (shaded), only a projectile of hundreds of kilograms — another satellite — can shatter a satellite; in low orbit, at ten kilometres a second, a kilogram can.

Spaceflight

A ring where nothing leaves and little shatters

Every argument about orbital debris in low orbit ends in the atmosphere, which pulls fragments down in years or centuries. The geostationary ring has no atmosphere, so its removal term is exactly zero and anything left there stays for millions of years. It is saved by a second fact of the same geometry. Everything in the ring moves at nearly the same speed in nearly the same direction, so collisions there are slow, and a slow collision usually leaves a crater where a fast one leaves a cloud.

How fast to switch. The rms false polarisation left by a 1 per cent rms brightness fluctuation, against the modulation frequency, both on logarithmic scales, for fluctuations whose spectrum turns down at 5, 20, 100 Hz. Below the turnover the false signal is the whole fluctuation, about 0.7 per cent; above it, it falls steeply as the pairs are taken faster than the air changes. To reach a part in ten thousand (dashed) — the precision at which stellar and planetary polarisation becomes interesting — the switching must run at 3,311 Hz, 13,183 Hz, 66,069 Hz respectively: switching at hertz is useless, and a single beam needs kilohertz or more.

Starlight

A polarisation measured faster than the air changes

Polarisation is a difference between two brightnesses, and the brightness of a star seen through the air changes by a per cent from one millisecond to the next. Measure the two one after the other and the air's flicker becomes a false polarisation a hundred times larger than the signals worth having. The cure is to switch between them faster than the air can change, or to measure both at once and then swap them — turning a calibration problem into a differential one.

Starlight

A profile that averages a decade of depth

Each point on a star's disc reports the temperature at a different depth, so a limb-darkening profile looks like a depth sounding of the atmosphere waiting to be read. It is one, but a blurred one. Every angle averages a whole decade of optical depth, the averages overlap, and recovering the temperature structure from them is the inversion of a Laplace transform — a problem whose resolution grows only as the logarithm of the data's precision.

$A rare crossing reached in stages. Multilevel splitting on a kicked rotor (the standard map at K = 10), aimed at the event |p| > 450 within 400 iterations. 16 members are run until each reaches the first level or runs out of time; those that reach it are copied back up to 16, each copy restarted from its parent's crossing point with its angle nudged by a thousandth of a radian, and run on toward the next. The levels (dashed) are not fixed values of |p| but curves that rise as time runs out, so a member counts as closer to the event only if it has the time left to finish. 4 levels; the fractions passing them are 0.13, 0.06, 0.63, 0.88, and their product, 0.0043, is the estimate. Each stage is a common event; the rare one is never waited for.$ Gravitation

A rare event reached by cloning the near misses

Counting how often an ensemble of integrations goes unstable works for an event of one in a hundred and becomes hopeless for one in a million, because a ten per cent answer needs a hundred events and so a hundred million futures. Splitting gets there by a different route: wait for the members that come closest, copy them, and run the copies on. Each stage measures a common event and the rare one is a product of common ones — provided the copies forget their parents and the stages know how much time is left.

An iteration that slows toward the straight-through transfer. The contraction ratio of the corrected-target iteration — how much each pass shrinks the miss — against the transfer angle, for a transfer between orbits of 51.6° inclination at 400 and 420 km, with the Earth's bulge at its real strength and at 10 and 30 times it, on a logarithmic scale. The iteration converges where the ratio is below 1 (dashed). At the real strength it shrinks the miss 1857-fold per pass at 150°, 83-fold at 178° and 8-fold at 181°: always convergent, ever slower. At 10 times the strength it fails between 181° and 184°; at 30 times it fails between 182.5° and 192°. The slowing is geometric: near 180° the transfer's plane is set by a sideways shift of the aim point, so correcting the bulge's sideways miss tilts the whole arc.

Orbits

A correction that tilts the arc it meant to aim

A transfer designed with Lambert's two-body solution and flown through the Earth's flattened field misses its target, and the standard repair is to aim at the target minus the miss and solve again. That converges fast — until the transfer angle approaches 180°. There the two end points and the Earth's centre are nearly in line, the arc's plane is decided by a sideways shift of the aim, and correcting the bulge's sideways miss rotates the whole arc. The iteration slows by a factor of a hundred, and with a larger perturbation it runs away. Correcting only within the plane fixes it, at the cost of a plane change paid at arrival.

The observed sky

A year that depends on the season counting it

The tropical year is quoted to eight decimal places, and it is not one number. Counted from the June solstice it is fifty seconds shorter than counted from the December solstice, because the Earth's perihelion moves through the seasons and the planet covers the last of each year's shortfall at whatever speed it happens to be moving there. The Gregorian calendar was fitted to the year counted from the March equinox, which is why its average year is the length it is — and that year is itself drifting. On Mars the spread is fourteen minutes, and a calendar's leap rule can be more precise than the year it is meant to hold.

Spaceflight

A turn spread where the climb leaves room

An electric spacecraft spiralling from low orbit to geostationary has to remove its launch inclination on the way, and the classic steering law tilts its thrust out of the plane by the same angle all the way round each orbit. But out-of-plane thrust turns the plane only near the nodes, while in-plane thrust raises the orbit equally well everywhere. Letting the tilt follow the nodes — leaning hard where turning works and lying flat where it does not — saves 150 metres a second on a transfer from Cape Canaveral, with no coasting, because the climb fills exactly the parts of each orbit where the turn would be wasted.

Gravitation

A ring that is thinner from one side

Saturn's A ring is not a sheet of particles but a field of tilted slabs, each a few dozen metres across, pulled together by the ring's own gravity and sheared apart again within an orbit. Starlight passing through it threads the gaps or meets the slabs depending on its direction, so one patch of ring has a different optical depth for every observer — and a uniform model of it hides most of the particles it contains.

A cluster sinking by Chandrasekhar's formula alone. The orbital radius of a 2.0 × 10⁵ solar-mass globular cluster released on a circular orbit at 1 kpc in the two Fornax halo models, decaying under Chandrasekhar's dynamical friction with a velocity dispersion of 11.7 km/s and a Coulomb logarithm of 3. In the cusp it reaches the centre in 6.7 billion years. In the core the formula gives no arrival within 12 billion years, crossing into the core at 2.8. The formula already slows the cluster inside the core, because the circular speed there drops below the stars' velocity dispersion and few stars move slower than the cluster; but it still lets the cluster creep inward without limit. N-body experiments of clusters sinking into cored haloes find something stronger: the decay stops altogether near the core, which the local formula cannot produce at any setting of its logarithm.

Gravitation

A drag that stops where every orbit keeps time

Chandrasekhar's formula reads the density and the velocities where a sinking body is, and nothing about the orbits those stars are on. Inside a constant-density core every orbit has the same frequency, the differential motion that drags a body inward disappears, and the sinking stops — which is why the five globular clusters of the Fornax dwarf galaxy are still where they are.

Orbits

A wall that leans towards the body that damps

A hot Jupiter's eccentric orbit is rounded off by two tides at once, one dissipated in the planet and one in its star, and they depend on the planet's mass in opposite ways. So the period inside which planets are circular should fall with planet mass if the planet does the work and rise if the star does — and the heavy hot Jupiters that keep eccentric orbits where light ones never do say which.

$What colour removes, and what it can only estimate. The share of a synthetic family's linked membership that is not part of the family, against the clustering cutoff, with the parts a colour test can and cannot see. The true interloper fraction rises with the cutoff, from 0 per cent at 20 m/s to 84 per cent at 110. A colour cut removes the interlopers of the other type — 12 per cent of the list at 60 m/s — and leaves those of the family's own type, 6 per cent, which can only be estimated by scaling the detected ones by the background's mix. That estimate (dashed) tracks the truth on average and scatters about it, because a few dozen interlopers are a few dozen random draws.$ Orbits

A family checked by a colour it was not chosen by

Asteroid families are found by clustering orbits, and the clustering absorbs background asteroids that happen to orbit nearby. Fragments of one body share its composition, and the survey that found them measured their colours for free — so colour is a second membership test the clustering never used. It removes the interlopers that look different, and it can only estimate the ones that look the same.

The rhythms in the rock, set by the Moon's distance. The periods of the climatic precession cycles — the precession constant plus the perihelion frequency of each of the planets that dominate the Earth's eccentricity — and of the obliquity cycle, the precession constant plus the node frequency s₃, against the Earth–Moon distance. Today they are 23.8, 22.4, 19.0 thousand years for precession and 41 for obliquity. At 341,000 km they were 14.7, 14.2, 12.8 and 20.0. The planetary frequencies are held at today's values, which the chaos of the inner solar system makes uncertain beyond a few tens of millions of years; the precession constant is the term that moves.

The observed sky

A rhythm in the rock that measures the Moon

The Earth's axis precesses because the Sun and Moon pull on its equatorial bulge, and two-thirds of the pull is the Moon's. When the Moon was nearer and the day shorter, the axis turned faster, and the climate cycles that precession drives were shorter. Sediments laid down 1.4 billion years ago record those cycles — and with nothing in the rock but layer thicknesses, the ratio of two rhythms gives the Moon's distance at the time.

Two satellites moved apart by the attitude they hold. The along-track separation of two identical 5 kg satellites at 475 km, starting together, when one flies broadside to the airflow and the other edge-on for 15 days and then they swap, so that their relative drift ends at zero. The separation grows quadratically and then levels off. At solar minimum the manoeuvre moves the pair 2,797 km apart; mid-cycle 8,299; at solar maximum 24,620 — 206° of the orbit's 43,060 km circumference. No propellant is spent; the same command produces a tenfold different result depending on the state of the Sun.

Spaceflight

A rendezvous paid for in air

Two satellites in the same low orbit can be moved along it relative to each other with no propellant at all, by turning one broadside to the thin air and the other edge-on. The draggier one sinks and speeds up, the gap grows as the square of the time, and swapping attitudes halfway brings the drift back to zero. The cost moves from fuel to time, to altitude, and to knowing an atmosphere that changes tenfold with the Sun.

An arrival held without a burn, and released. A body approaching a planet from a heliocentric orbit just outside the planet's, drawn in the planet's rotating frame (the Sun far to the left, the planet's orbital motion upward), in units in which the Hill radius is 0.69. It drifts in along the shear of the differential rotation, passes through the neck at the Lagrange point L2 into the region the planet controls, loops round the planet 3.3 times and leaves through a neck again. No burn is drawn: the capture and the release are both the Sun's tide working against the planet's pull. For Mars one unit of distance is 1.56 million km and an orbit round the planet near the Hill radius takes about a year.

Spaceflight

An arrival the Sun hands over

A spacecraft that reaches a planet with no speed to spare can be captured with no burn at all, passing through the neck between the planet's gravity and the Sun's into a loose, looping orbit. The capture is temporary and chaotic — a few revolutions, and arrivals a few hundred kilometres apart end differently — but while it lasts the spacecraft is already bound, and the burn that keeps it is taken from there.

One likelihood, read with three priors. A Gaussian likelihood in the effective neutrino mass sum, peaking at −0.08 eV with a width of 0.05 eV — of the shape that combinations of microwave-background and baryon-acoustic data have been reported to give. Read with no prior it says the data prefer a negative sum. Read with the physical prior, sum ≥ 0 (shaded right of zero), its 95 per cent upper limit is 0.059 eV, at the normal-ordering floor. Read with the oscillation prior, sum ≥ 0.059 eV, its limit is 0.102 eV — but only 0.3 per cent of the likelihood lies above the floor at all, which is the more informative number: the data and the oscillation experiments disagree at about 2.8 standard deviations.

Cosmology

A bound below the floor is not a mass

Oscillation experiments prove that neutrinos weigh at least 0.059 eV between them. Cosmology's latest upper bound on their total is 0.064 eV, and a bound computed differently would sit below the floor. The usual reading is that cosmology has nearly weighed the neutrino. The better one is that the fit's likelihood peaks at a negative mass: the data see more clustering than any real neutrino allows, and the bound is a symptom of that tension rather than a measurement of anything.

Velocities that stop pointing along the stream. The angle between the stream's direction on the sky, as seen from the Galactic centre, and the mean motion of its stars, along the stream, for Large Magellanic Cloud masses of 0.0, 0.5, 1.5, 2.5 × 10¹¹ solar masses. In a static halo the misalignment stays at 1° or below — the small offset any stream has from its progenitor's orbit. With the Cloud it grows, peaking at 4°, 13°, 32° for the three masses, on the stretch of stream that passed nearest the Cloud. The misalignment is a measurement: with the halo's own potential known, its size along the stream gives the Cloud's mass.

Galaxies

A stream that points the wrong way

In a halo that holds still, the stars of a tidal stream move along it. The Large Magellanic Cloud, falling into the Milky Way for the first time, pulls on a passing stream directly and drags the Galaxy's own centre towards itself, and the stream is bent sideways faster than its stars' motions can follow. Their velocities stop pointing along the track — and how far they stop is a measurement of the Cloud's mass.

Exoplanets

An envelope that boils down before the starlight strips it

The usual story of how small planets lose their hydrogen starts with the star's ultraviolet light and runs for a hundred million years. It skips the first million. When the gas disc that fed a young planet disappears, an envelope that was swollen and held in by the disc's pressure is suddenly exposed, and it pours off as a wind powered by ordinary heat — until it has shrunk to the size at which the same number near thirty that governs every escaping atmosphere shuts the flow off.

Galaxies

A molecule that forms too slowly to wait for

The atomic hydrogen in a galaxy's disc turns molecular wherever a cloud is thick enough to shield itself from starlight, and stars form where it does. That is an equilibrium statement, and equilibrium takes time. Molecular hydrogen forms on dust grains, so in gas with a tenth of the Sun's metals it forms ten times more slowly — slowly enough that a cloud can finish collapsing before most of its hydrogen has become molecular, and form its stars from atoms.

Gravitation

A population that cannot share the cluster's temperature

Encounters between stars push a cluster toward equal kinetic energies, so heavy bodies slow down and sink. Spitzer showed in 1969 that past a certain point they cannot slow down enough: the heavy population falls away from the rest and becomes a cluster of its own. Stellar black holes are past that point by a factor of ten, in every globular cluster that kept them — and the subsystem they form, not the stars' binaries, is what holds most clusters' cores open until the black holes have been spent.

$A tilt that wanders up, one passing body at a time. The inclination of the Moon's orbit to the plane it started in, against the number of lunar-mass bodies that have passed the Earth–Moon pair at 5 km/s within 300 Earth radii, with the Moon at 30 Earth radii, for four different sequences of passages. Each passage tilts the orbit by a fraction of a degree in a random direction, so the inclination performs a random walk and grows as the square root of the number of passages: after 1,000 the four runs stand at 3.8, 1.8, 3.9, 0.2 degrees. The dashed line is the Moon's present inclination, 5.145 degrees. Whether a given history reaches it is a matter of chance; how far a typical one gets is set by the passages' number, mass and speed.$ Orbits

A tilt built from a thousand near misses

The Moon's orbit is tilted five degrees to the ecliptic, and slow tidal recession cannot have done it. One candidate is the leftover debris of planet formation: bodies the size of the Moon, passing the young Earth–Moon pair, each giving its orbit a kick of a fifth of a degree in a random direction. A random walk of a few thousand such kicks reaches five degrees — and the bodies that would supply them are the same population whose impacts left the Earth's mantle its gold.

The observed sky

Two exponents a solar cycle cannot pull apart

Whether a magnetic field protected the early Earth's air depends on what limits the escape of ions from a planet without one — the ultraviolet that makes them or the solar wind that carries them off. Measured at Mars through a solar cycle, the two causes rise and fall together, and the fit that tries to separate them leaves each exponent uncertain by half its value. Remarkably, the young Sun's loss is predicted well anyway. What spoils the prediction is not the confusion but a switch between the two limits that the present-day data cannot see coming.

Where on the orbit the turning is done, and where the shadow falls. How effectively out-of-plane thrust turns a circular orbit's plane at each point round the orbit, measured from the ascending node: it goes as the absolute value of the cosine of that angle, greatest at the two nodes and zero halfway between them. The shaded bands are the Earth's shadow on an orbit 6,378 km up with the Sun in the orbit's plane, 60 degrees of the circle, placed once over a node and once between the nodes. Over the node the shadow switches the engine off where it would have done the most turning, and the orbit loses 25% of its turning; between the nodes it switches the engine off where it was doing little, and the loss is 7%.

Spaceflight

A turn that costs more when the shadow sits on the node

An electric spacecraft turning its orbit's plane does most of the turning near the two nodes, where a push out of the plane tilts it most. Its engine runs on sunlight and stops in the Earth's shadow, and the shadow covers a sixth of the orbit wherever it happens to fall. When it falls on a node it removes a quarter of each orbit's turning, and a law that concentrates its thrust at the nodes loses half. The node drifts one way and the Sun the other, so the shadow slides across the nodes over weeks — and a transfer's length depends on the day it starts.

Starlight

A hundred kelvin under every radius

The temperatures of cool stars are quoted to tens of kelvin, and the scale they are all measured on is uncertain by about a hundred. That hundred is under almost everything else: a star's radius moves by twice its fractional error, a transiting planet's density by six times, a main-sequence age by ten per cent and an iron abundance by a fifth. And because a scale error moves every star of a sample the same way, no amount of averaging removes it.

Where the tide's energy goes, and what the deep ocean needs. Power in terawatts. The Moon and Sun do work on the oceans at about 3.7 TW, measured from the slowing of the Earth's spin and the Moon's recession and, since the 1990s, located by satellite altimetry: about 2.6 TW is lost in shallow seas, about 1 TW in the deep ocean where the tide flows over ridges and seamounts and makes internal waves, and about 0.1 TW in the solid Earth. The open bar is what Munk and Wunsch estimated the deep ocean needs to keep its stratification against the upwelling of cold water, about 2 TW; the winds supply part of it, the tides — on these estimates — about half.

Gravitation

The tide that holds up the deep ocean

Cold water sinks near the poles and fills the deep ocean, and something has to stir heat back down into it or the whole ocean below the surface would become one cold pool. The stirring costs about two terawatts. The winds supply some of it. The rest comes from the Moon: about a terawatt of the tide's energy is lost not on continental shelves but in the deep ocean, where the tide flowing over ridges and seamounts makes waves on the ocean's internal layers that break far below the surface.

The observed sky

A Moon measured in banded iron

Layers of iron-rich sediment laid down 2.46 billion years ago in what is now Western Australia keep time with the Earth's precession, and the precession's period says how far away the Moon was: about 321,800 kilometres, with a day of seventeen hours. Together with an older-is-closer point from 1.4 billion years ago, it measures how hard the tides worked between them — about a fifth as hard as today. Held further back, that efficiency brings the Moon out from the Earth a few hundred million years too early, which says the young tides worked harder than the middle-aged ones.

How far a change of attitude moves a satellite before the encounter. The along-track shift a 5-kg satellite achieves by switching from 0.01 to 0.1 m² of frontal area when warned of a conjunction, against the warning time, at 400, 500, 600 km, between solar minimum (lower edge) and maximum (upper edge), logarithmic. The shift grows as the square of the time. In a day it is 26.1 to 156 km at 400 km; 3.9 to 39 km at 500 km; 0.6 to 10 km at 600 km. The dashed line is the 556 m needed to bring a 300-m position uncertainty's collision risk below one in ten thousand.

Spaceflight

A dodge made of air

A small satellite with no thruster can still move along its orbit by turning broadside to the thin air, and the question is whether that is enough to get out of the way of debris. Below six hundred kilometres the answer, measured in distance, is easily yes: a day's warning buys kilometres when a few hundred metres would do. Measured in predictability it is no. The density of the air is forecast to about thirty per cent, so the dodge goes somewhere nobody, including the satellite's own operator, can say in advance to the precision the dodge was meant to achieve.

Every orbit turns by about the same angle, whatever its size. How far the Earth's equatorial bulge turns an ellipse with a 300-km perigee in one revolution, in degrees, against its semi-major axis, logarithmic: the line of apsides (solid) and the node (dashed), for inclinations of 28.5, 51.6, 98°. The turn per orbit depends on the semi-latus rectum, which for an elliptical orbit lies between one and two perigee radii, so it changes by less than a factor of four from a near-circular orbit to one reaching the Moon's distance: at 28.5°, 0.56° of apsides per orbit at 8,000 km and 0.20° at 300,000 km. Per day the large orbit turns far less, because it completes so few revolutions; per revolution, almost the same.

Spaceflight

An orbit lost to a burn that did not fire

A weak stage leaving the Earth splits its departure into several burns at perigee, coasting round a staircase of ever larger ellipses between them. If one burn does not fire, the stage goes round once more. That costs a revolution of time — an hour and a half early in the staircase, more than a day at the top — and a revolution of turning by the Earth's bulge, which comes out at nearly the same angle for every step. The angle is cheap to undo at the slow apogee of a large orbit. The time is not, when the target is the Moon.

$How far a wrong metallicity term moves the Hubble constant. The fractional change in the Hubble constant against the error in the adopted metallicity term, in magnitudes per dex, for supernova hosts whose Cepheids are 0.1, 0.2, 0.4 dex richer in metals than the geometric calibrators'. A term too weak (too close to zero) by 0.2 mag/dex with hosts 0.2 dex richer moves H₀ by 1.9%. The shaded band is the ±8% disagreement between the local measurement and the microwave background's prediction: reaching it with a 0.2-dex difference needs the term wrong by 0.9 mag/dex.$ Stars

A metallicity term too small to close a gap

Cepheids of the same period are slightly brighter when they are richer in metals, by about a fifth of a magnitude per factor of ten in metal content. The galaxies whose supernovae calibrate the Hubble constant hold Cepheids a little richer than the galaxies whose distances are known geometrically, so the term enters the answer. How much it can move it is simple to compute: a few per cent at most for any plausible error — and the eight per cent separating the local Hubble constant from the early universe's would need the term to be wrong by four times its own size.

A loop round the Earth that the Earth does not make. The path over one year, in the frame that turns with the Earth (Earth at the centre, the Sun far to the left, the Earth's motion upward), of bodies on heliocentric orbits with the Earth's period and eccentricities of 0.05, 0.1, 0.15, started at the Earth's mean longitude. Each traces a retrograde loop round the Earth, about 2ae long in the direction of motion and ae wide towards the Sun: for e = 0.1, 0.395 by 0.199 AU. The small circle is the Earth's Hill sphere, 0.010 AU in radius, inside which the Earth's gravity dominates the Sun's tide. Every loop lies far outside it. The loop is a difference between two ellipses with one period, and would be traced the same way if the Earth had no mass at all.

Gravitation

A moon the Earth does not hold

Seen from the Earth, the small asteroid Kamoʻoalewa circles the planet once a year, backwards, between forty and a hundred times as far away as the Moon. It is not a moon. It orbits the Sun, with the Earth's period and a slightly different eccentricity, and the loop is the difference between two ellipses — one the Earth would trace even with no mass at all. What the Earth's gravity does is subtler: it keeps the loop from drifting away.

How long a wide pair survives a halo of heavy objects. The time for a binary of one solar mass to be dissolved by passing compact objects, against its separation, both logarithmic, if the whole local dark-matter halo — 0.01 M☉ per cubic parsec, moving past at 300 km/s — is made of objects of 10, 100, 1000 M☉. A pair 0.1 pc apart (about 20,000 AU) lasts 477, 48 and 4.8 billion years respectively; at 1 pc, 47.7, 4.8 and 0.48. The heavier the objects at fixed total density, the fewer and harder the kicks, and the sooner the widest pairs go. The dashed line is 10 billion years, the age of the halo's oldest binaries.

Gravitation

Pairs of stars too loosely bound to survive heavy dark matter

Two stars a tenth of a parsec apart are held together by a force so weak that a passing mass of a few hundred suns, at a distance of a parsec, can loosen them. If the dark matter round the Milky Way were made of heavy compact objects, such pairs would have been unbound long ago. Some have survived ten billion years in the halo. The widest of them sets a ceiling on how heavy the dark matter's pieces can be — and the ceiling has moved twice, both times because of one pair.

The observed sky

A year kept by a star seen for one morning

For some seventy days each year Sirius is lost in the Sun's glare, and then one morning it rises just before dawn, for a few minutes, low in the east. The Egyptians kept a calendar of exactly 365 days, and against it that first morning slipped a day every four years, going round the whole calendar in about fourteen and a half centuries. The star's own year turned out to be the Julian year to within minutes — not because it had to be, but because two effects of precession cancel for Sirius seen from Egypt.

The Moon's reach north and south, over two cycles of its node. The greatest declination the Moon reaches in a month, north or south, from 2010 to 2045: about 23.44° plus the orbit's 5.145° tilt times the cosine of the node's longitude, which regresses once in 18.61 years. At the major standstill (2025.1, and 2006.5, 2043.7) the Moon reaches 28.6°, five degrees beyond the Sun's solstice declination (dashed); at the minor standstill, 18.3°, five degrees short of it. Near each extreme the reach changes slowly for two or three years, which is the standstill.

The observed sky

A Moon that rises further north than the Sun

The Sun's rising point swings north and south with the seasons between limits that do not change from one year to the next. The Moon's swings with the month, and its limits themselves open and close over 18.6 years as the plane of its orbit turns. At the widest, in 2024 and 2025, the Moon rose five degrees of declination further north than the midsummer Sun and skimmed the southern horizon of northern Scotland; at the narrowest, nine years later, it stays well inside the Sun's range.

Spaceflight

A tow with no rope

A spacecraft parked beside an asteroid pulls it by gravity alone, with a force set by the spacecraft's mass and its distance and nothing else. Twenty tonnes hovering a hundred and fifty metres from the centre of a two-hundred-metre asteroid moves it by two millimetres a second in a year. That is the gentlest engine there is, and its limit is not the pull but the exhaust: the spacecraft must thrust away from the rock to stay put without blasting it, and the angle that takes decides how close it can hover.

Elements sorted by how easily they lose an electron. The abundance of each element in the solar wind relative to the photosphere, normalised to oxygen, against its first ionisation potential, logarithmic: in the slow wind (filled) and the fast wind from coronal holes (open). Representative values. Elements easy to ionise — K, Na, Ca, Mg, Fe, Si, all below about 9 eV — are about three times over-abundant in the slow wind and about one and a half in the fast; those harder to ionise sit near one. The step comes at about 10 eV, which is the energy of a photon from the Lyman-α line of hydrogen: the sorting happens where hydrogen's own light sets the ionisation.

Stars

A wind that carries the ledger of its loops

The slow solar wind is richer in iron, magnesium and silicon than the Sun's surface is — about three times richer, relative to oxygen — and the elements that are enriched are exactly the ones that lose an electron easily. That sorting happens low in the chromosphere, inside closed magnetic loops, and it takes days. The fast wind from coronal holes barely shows it. So the composition of the wind arriving at the Earth is a record of where it was held, and for how long, before the field let it go.

The static universe is a saddle in the plane of size and speed. Every history of a closed universe of dust and an exactly balancing cosmological constant, as curves of constant energy in the plane of size against rate of change of size, in units of the static radius and 1/√(4πGρ). The two thick curves cross at the static point: they are the histories that approach it for an infinite time or leave it after an infinite time. Above them a universe expands from a big bang through forever or contracts from forever; to the left it rises from a big bang and falls back; to the right it contracts, bounces, and expands. No history other than the point itself stays near it.

Cosmology

A balance that is a saddle

In 1917 Einstein added a constant to his equations to hold the universe still, and the constant does exactly that: there is one size at which the attraction of matter and the repulsion of the constant cancel. Eddington showed in 1930 that the balance is a pencil on its point. A thousandth out and it is gone in seven units of its own free-fall time. The instability is not a flaw of that one model. It is why the real universe could cross the same balance without stopping.

Gravitation

A particle on a line that can spell any word

Two equal stars orbit each other, and a speck of dust moves up and down on the line through their centre, perpendicular to their orbit. It has one coordinate and cannot leave that line. In 1973 Jürgen Moser proved that for almost any list of whole numbers, as long as they are large enough, there is an orbit whose successive excursions last exactly that many periods of the stars. The list can be chosen at random, so the orbit can do anything a coin can.

Jupiter and Saturn trading a degree of longitude over nine centuries. Mean longitudes of Jupiter and Saturn in arcminutes, with their straight trends taken out and a 40-year smoothing applied, from a planar integration of the Sun, Jupiter and Saturn from their elements at the year 2000. Jupiter swings ±20′ and Saturn ±50′, in opposite senses, with a period of about 870 years. Comparing Ptolemy's positions with those of his own day, Kepler could not fit one mean motion to both, and by 1676 Halley reported Jupiter gaining and Saturn losing. Before 1785 that looked as if the Solar System were unstable; the curve shows that it is a very long period.

Orbits

A term too small to keep that a denominator kept

Every planetary theory is an expansion, and the order of its terms is meant to say which can be dropped. In the 1770s the largest unexplained motion in the Solar System, Jupiter speeding up and Saturn slowing down by a degree over centuries, came from a term that the ordering called negligible. Five of Saturn's orbits nearly match two of Jupiter's, and that one near-match divides a tiny third-order coefficient twice by a tiny frequency. An integration of the three bodies shows the swap directly, and shows what Laplace saw: it reverses every nine centuries.

Orbits

A perigee that moved twice as fast as Newton

The Sun's tide on the Moon should turn the line of its perigee once in eighteen years. It takes less than nine. For sixty years that was the outstanding failure of the law of gravitation, and in 1747 Clairaut told the Paris Academy that the inverse-square law would have to be amended. Two years later he withdrew the remark. The law was right and the calculation was unfinished: the series for the perigee has a small parameter that is not small, and its second term is seventy per cent of its first.

$A reservoir so large that it does not notice. The fractional increase in the heat of the Sun's convective envelope if a fraction 0.001, 0.003, 0.01 of the surface is held under spots, against time. The envelope holds about 7×10⁴⁵ erg, 58 thousand years of the Sun's light. Spots covering 0.3% of the surface for an eleven-year cycle add 4.2e-7 of that, a temperature change of 0.4 parts per million at most, spread over the whole envelope. So the blocked heat simply remains in the star, and the surface flux shows the deficit. It levels off only after the envelope's thermal time, when the star has adjusted.$ Stars

A spot whose missing heat goes nowhere

A sunspot blocks most of the energy trying to come up beneath it, and a large group visibly dims the whole Sun by a thousandth. Energy is conserved, so the blocked flux ought to come out somewhere else, and for a century observers looked for the bright ring around every spot. There is none to speak of. The heat has gone into the rest of the convection zone, spread so fast and so widely that the surface cannot notice, and it will stay there for longer than human history. For the time being the Sun simply is fainter, without any contradiction.

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