Astronomy & Celestial Mechanics
the sky, computed
Theme

Two bodies, and no more

The two-body problem is solved exactly and the three-body problem is not solved at all. Almost everything since has been about living with that.
Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Orbits

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

 A vector that does not change, drawn five times. The Laplace–Runge–Lenz vector A = v × L − GM r̂, constructed at five points of one orbit at e = 0.45 under a force ∝ r^−2, each drawn from the body rather than from the focus so that its length and direction can be compared point by point. Under the inverse square every one of them is the same vector: identical to machine precision in length and in direction. Its length is 0.4500, which is the eccentricity, and it points at pericentre — so the orbit's orientation is a conserved quantity and not a constant of integration, which is why the ellipse does not turn. Orbits

The third thing that is conserved

Energy fixes an orbit's size and angular momentum fixes its shape. Neither fixes which way it points — so a curve that closes needs a third conserved quantity, and only two force laws in the universe supply one.

 The three anomalies at E = 1.15 rad. The auxiliary circle construction. The eccentric anomaly E is measured at the centre, the true anomaly ν at the focus, and the mean anomaly M is time expressed as an angle. Here E = 1.150, ν = 1.827 and M = 0.602 radians, related by M = E − e sin E. Orbits

The position that has no formula, and is computed anyway

Kepler's equation relates where a body is to when it is there. It cannot be solved in elementary functions, Kepler said so, and nobody has managed it since — which has stopped nothing.

 The effective potential, for three angular momenta. The radial motion of an orbiting body is one-dimensional motion in an effective potential: the attraction −GM/r plus the centrifugal term L²/2r² that the angular momentum contributes. The barrier at small radius is what stops a body with any angular momentum at all from reaching the centre, and the bottom of each well is the circular orbit. Orbits

The wall that angular momentum builds

A body falling toward a star almost never arrives. Sideways motion, which looks like a detail of the initial conditions, turns the attraction into a well with a wall around the middle of it — and the wall is why it costs more fuel to hit the Sun than to leave the solar system.

 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.

 The mass correction against the mass. Each planet's departure from the massless harmonic law, against its own mass in solar units, on logarithmic axes. The exact law puts every point on the diagonal. Jupiter and Saturn are the only planets whose mass correction is larger than the perturbations from everything else, and Saturn's measured departure has the opposite sign. Orbits

The third law is wrong by the mass of the planet

Kepler's harmonic law says the square of the period goes as the cube of the size. Newton's version has one more term in it, and the term is the orbiting body's own mass — negligible for a planet, decisive for a binary star, and the reason a period can be converted into a mass at all.

 1I/ʻOumuamua: an orbit at e = 1.201, and the angle it turned through. The open branch of a conic at eccentricity 1.201 and periapsis 0.2559 AU, with the Sun at the occupied focus. Two numbers fix everything else: |a| = q/(e−1) = 1.273 AU, an impact parameter b = |a|√(e²−1) = 0.847 AU, an asymptote at ν∞ = arccos(−1/e) = 146.37° from periapsis, and a deflection of δ = 2ν∞ − 180° = 112.74° — which is the same statement as sin(δ/2) = 1/e. The asymptotes cross at |a|e = 1.529 AU from the Sun on the periapsis side, where a bound orbit's centre would be on the other. The speed left over at infinity is √(μ/|a|) = 26.40 km/s. Schematic in one respect: the drawing runs at about 119 px to the AU, so the Sun's own disc would be far smaller than its marker. Orbits

The orbit that has no period

Above an eccentricity of 1 the conic is open, the energy is positive and the semi-major axis is negative — and the vis-viva relation survives the sign change without a single alteration. What replaces the period is a speed, and that speed is what says where a visitor came from.

 The aim point and the miss distance are the same number far out and nothing like it close in. Periapsis distance against aim point for a hyperbolic approach to Jupiter at v∞ = 5.6 km/s, both in planet radii. The diagonal is where the two would be equal — where gravity did nothing — and the curve falls below it everywhere, by more the closer in the aim is. A trajectory aimed at 10.68 radii grazes the surface, because gravitational focusing means the planet's effective size is √(1 + v_esc²/v∞²) times its radius. The slope of this curve is what a navigation team cares about: it is 0.208 at an aim of 12 radii and 0.936 at 150, so the same correction manoeuvre changes the periapsis distance by 4.5 times as much at one end of the range as at the other, while changing B by exactly the same amount at both; far outside this plot, where focusing has run out, it reaches 1.000 and the two numbers become the same one. That is why the aim point is the coordinate a manoeuvre is quoted in, why an error ellipse is published in the B-plane, and why the turn angle — 2 arctan(μ/Bv∞²), 156.0° at 12 radii and 77.8° at 70 — is thought of as a function of B rather than of anything the spacecraft does. 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 conics through one periapsis, drawn by one solve. Distance from the Sun against time for four orbits sharing a periapsis of 0.5 AU, at e = 0.6, e = 1, e = 1.0001, e = 1.4, over 900 days. Every point on every curve came from the same universal Kepler solve — no branch on the conic class anywhere in it — and each curve was then checked against the classical solution of its own kind at the midpoint: Kepler's equation at e = 0.6 agrees to machine precision; Barker's cubic at e = 1 agrees to machine precision; e sinh H − H at e = 1.0001 agrees to machine precision; e sinh H − H at e = 1.4 agrees to machine precision. The curve to read twice is e = 1.0001: over this arc it lies within 0.02% of the parabola and is indistinguishable from it, and it is the only one of the four whose fate the drawing cannot show. What the classical parameterisation costs there is arithmetic rather than impossibility: at the midpoint of this arc, e sinh H − H throws away 3.3 of its sixteen digits to cancellation, against 0.2 at e = 1.4 — enough to matter to an ephemeris and not enough to stop a plot. Orbits

The one solve that does not ask which conic it is

Kepler's equation is for ellipses, Barker's cubic for parabolas, and a hyperbolic sine for the rest — three parameterisations of one motion, each worst exactly where its neighbour takes over. The universal variable removes the question, and the removal is not a convenience.

 One squaring, and the singularity is gone. Left, a harmonic oscillator: an ellipse centred on the origin, marked at 24 equal steps of its own phase. Right, the same points after squaring as complex numbers, u ↦ u². The image is an ellipse with the origin at a focus — checked here by the focal property, r₁ + r₂ = 2a to nine figures at four points — with semi-axes (A²+B²)/2 = 2.5000 and AB = 1.5000 and the focus at (A²−B²)/2 = 2.0000, which is ae exactly. Three things follow at once. The angle doubles, so one turn of the oscillator is one whole orbit traversed twice as fast in phase; the equal phase steps on the left arrive as the eccentric anomaly on the right, which is why that anomaly and not the true one is what the equations want; and the collision at r = 0, where the inverse square is infinite, is the point u = 0, where the oscillator has a perfectly ordinary velocity. The singularity was a property of the coordinates. Orbits

The singularity that is a change of variable

The Kepler problem blows up at zero separation, and a fixed-step integrator falls apart long before it gets there. Divide time by the radius and the equations become a harmonic oscillator — exactly, for every conic at once.

 An eccentricity and an inclination trading, at 65° of mutual tilt. The secular equations integrated from a nearly circular orbit (e = 0.02) inclined at 65° to a distant perturber's plane, over three oscillations. Above, the eccentricity; below, the inclination, with the constant √(1−e²)cos i drawn as the flat line it is. The eccentricity climbs to 0.8380 and the inclination falls to 39.25° at the same instant, and neither is a coincidence: the product is fixed, so one can only rise as the other falls. That floor is the same for every starting tilt — at maximum eccentricity j = √(5/3)Θ, so cos i = √(3/5) and the inclination arrives at 39.23° whether the orbit began at 50° or at 89°. The closed form for a circular start is e_max = √(1 − (5/3)cos²i₀) = 0.8380, which contains nothing about the perturber — not its mass, not its distance. Those set the clock and not the amplitude, and the period here is 4.83 Kozai times. What the figure cannot show is what happens at the top of the cycle in a real system: at e = 0.838 the pericentre is 0.1620 of the semi-major axis, where tides, general relativity or a stellar surface all intervene, and the quadrupole picture ends. 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.

 5 transfers through the same two points in the same 1400 days. Time of flight against semi-major axis for every transfer through two points 135° apart at 1 and 1.524 AU, with the revolution count running from 0 to 2. Each count contributes two branches, and for N ≥ 1 the pair folds: the time has a minimum at a = 1.2426 AU for one revolution — only 2.2 per cent above the minimum-energy value of 1.2161, which is why the horizontal axis is the excess over that value and logarithmic — so a flight time above it is met twice and below it not at all. Reading the crossings of the 1400-day line off the drawn curves gives 5 of them — 0 revs high, 1 rev low, 1 rev high, 2 revs low, 2 revs high — which is 2N + 1 with N = 2, and the count is a property of the time rather than of the geometry. That is the practical content: a root-finder started from a single guess returns one of these 5 and gives no sign that the other 4 exist, and the cheapest of them is often not the one nearest the guess. Orbits

One time of flight and five ways round

Lambert's theorem says two positions and an interval fix the transfer. Allow the transfer to complete whole revolutions and that stops being true: the flight time folds, one number admits five arcs, and the cheapest of them is usually not the one a solver started nearest to.

 After one revolution the error is 3π times longer than it is wide, and after 300 it is 2827. The two semi-axes of a fitted orbit's position uncertainty, against elapsed revolutions, for a solution whose semi-major axis is uncertain by 12 kilometres. The radial extent does not grow at all: a body on a slightly larger orbit is slightly further out and stays so. The along-track extent grows linearly, because δn/n = −(3/2)δa/a makes a semi-major-axis error into a mean-motion error and a mean-motion error into a phase that runs away — a·δM = 3πN·δa after N revolutions. The ratio is 3π ≈ 9.42 after a single revolution and 2827 after 300, which is why an asteroid recovered after one apparition is found within a few arcseconds of its predicted place along its own track and could be a long way from it in time. Every consequence of this in practice — that an impact probability is a one-dimensional integral rather than a volume, that a keyhole is an interval, that the next observation worth taking is the one across the track rather than the one that fits best — is a restatement of these two lines diverging. Orbits

An error that is nearly all in one direction

A fitted orbit's uncertainty is not a ball. Within a few revolutions it has collapsed onto a line along the track, because an error in the size of an orbit is an error in its period and an error in period is a phase that runs away — which is why an impact probability is an integral along a curve rather than over a volume.

 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.

 Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller. Gravitation

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

 Why a shell pulls a point inside it not at all. A double cone from a point inside a uniform shell. In the narrow-cone limit the far patch is 2.80 times further away and 2.80 times wider — the same number, checked to a part in a thousand before this figure is drawn. Its mass is greater by the square of that ratio and its pull weaker by the same square, so the two cancel in every direction. Gravitation

A sphere pulls exactly like a point, and the proof is a pair of cones

Every orbit ever computed treats the Sun as a dot. That is not an approximation — for a spherical body it is exact, and the reason is a cancellation between two patches of a shell.

 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 two-body problem, and the one-body problem it is. Left: two bodies with mass ratio 0.4 on ellipses of eccentricity 0.5 about their common barycentre, the heavier one on the smaller orbit. Right: the same system as one body of the reduced mass on a single ellipse of the same eccentricity about a fixed centre, at the separation of the two. The right-hand curve is the point-by-point difference of the two left-hand curves, so the substitution is drawn, not merely stated. Gravitation

Two bodies replaced by one that does not exist

The two-body problem is solved by turning it into a one-body problem about a fixed centre. The substitution is not an approximation — it is exact, and the body it invents has a mass no object in the system has.

 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.

 Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system. Gravitation

The curve that says where a body cannot go

The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.

 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.

 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.

 The Hill radius against the mass ratio, and where four planets keep their outermost moon. The Hill radius R_H = a(μ/3)^⅓ and the exact L₁ distance from the same quintic, both in units of the planet's own semi-major axis, against the mass ratio μ = m/(M+m) from 10⁻⁸ to 0.12. The two are the same line to the width of the stroke across the whole planetary range — the cube-root formula runs 0.33% short of the exact L₁ distance at the Earth's mass ratio and 10.0% short at the right-hand edge — so the departure between them is drawn on its own percentage scale at the right, which is the only place it can be seen. Below the curves are the numerically determined stability limits, 0.5 R_H for a prograde satellite and 0.7 R_H for a retrograde one, and the outermost known satellite of each of four planets, each at a Hill fraction computed from that planet's own mass and orbit: Moon 0.257, Sinope 0.451, Phoebe 0.198, Neso 0.424. Of the four, three are retrograde — Sinope, Phoebe, Neso — and they reach 0.451 of their planet's Hill radius against 0.257 for the one prograde satellite: 1.8 times as far out, in a diagram whose two stability limits stand in the ratio 0.7 to 0.5. No equilibrium argument predicts that, and it is what the drawing makes plain. Gravitation

The region a planet may keep a moon in

A satellite is not held by orbiting but by orbiting inside the radius at which the Sun's tidal field would take it away. That radius is the same distance the innermost Lagrange point sits at, asked a different question — and the boundary the sky actually respects is smaller, by a factor that depends on which way the moon is going round.

 Forty-two minutes, from anywhere to anywhere. Left, the gravitational field inside the Earth: a straight line for a uniform sphere, because the enclosed mass grows as r³ and the field as r, and the PREM curve for the real one, which is nearly flat through the whole mantle at about 9.94 m/s² because the dense core is already all below. Right, what falls through it. A body dropped down a diametric tunnel through a uniform Earth executes simple harmonic motion with ω = √(g/R), reaching the far side in 42.2 minutes; a body dropped down a chord at 0.6 of the radius feels only the component along the tunnel, which is the same ω times a smaller distance, so it arrives in the same time from a shorter trip. The period does not contain the length of the tunnel, its direction, or where the body starts. And 2π/ω is also the period of a circular orbit grazing the surface, 84.3 minutes — the tunnel and the orbit are one ellipse, seen twice. The real Earth is not uniform and gets there in 38.2 minutes instead, which is the honest number and is 9% quicker. Gravitation

The tunnel that takes the same time from anywhere

Inside a uniform sphere the field grows in proportion to the distance from the centre, which is Hooke's law. A body dropped down any straight tunnel arrives in the same time — and that time is the period of an orbit skimming the surface.

 Energy error over 240 revolutions, at one step size. The relative error in total energy against revolution number, for three integrators run on the same Kepler orbit at e = 0.5 with the same step of 200 per revolution. The exact energy is a constant, so every curve here is the method rather than the problem. Euler climbs steadily: its energy at the end is 106.5% wrong, and the orbit it draws has spiralled outwards. Runge–Kutta 4 begins 4.7e+2 times more accurate than leapfrog and ends at 2.22e-4, having grown by a factor of 10 across the run: the error is SECULAR. Leapfrog oscillates inside a band and stays there — worst error 2.62e-3, and the second half of the run is no worse than the first, which is measured here rather than claimed. That is the property that decides whether a five-billion-year integration means anything, and it is not accuracy: a symplectic method is the exact solution of a Hamiltonian a step-size away from the intended one, so its energy cannot wander, while a more accurate non-symplectic method has no such constraint and eventually wanders further. Gravitation

Wrong about where, and right about how much

Runge–Kutta is the more accurate method and loses energy steadily; leapfrog is cruder and its energy error never leaves a band. Over five billion years only one of those properties survives — and neither method knows where the planet is.

 A body that heats up because it is losing energy. A uniform self-gravitating sphere of 1.0 solar masses radiating at 1.0 solar luminosities, with no nuclear source at all, from 3.0 solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls, and the temperature rises. That is not a paradox and it is not a special case. The virial theorem makes 2K = −U for any self-gravitating gas in equilibrium, so E = U + K = −K, and −dE/dt = +dK/dt: energy leaving as light is energy arriving as heat. The bookkeeping is exact and is measured here rather than quoted — over the run 5.465·10⁴⁰ J of gravitational energy is released and 2.733·10⁴⁰ J is radiated, a ratio of 2.0000. Half the release is spent on the star's own heat and only half escapes. The consequence is a body with a negative heat capacity, which is why a contracting protostar gets hotter until it ignites, why a globular cluster's core runs away instead of settling, and why nothing self-gravitating ever comes to thermal equilibrium. Gravitation

The system that gets hotter as it loses energy

The virial theorem makes a self-gravitating body's total energy equal to minus its kinetic energy. Radiating heat away therefore raises the temperature, there is no equilibrium to settle into, and every star and every cluster is running away from 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.

 An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation. Gravitation

An average that weighs what cannot be watched

A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.

 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.

 Three orders of magnitude in astronomical units, and a factor of 9.6 in mutual Hill radii. Every adjacent pair of planets in three systems, plotted by the separation between them measured in their own mutual Hill radius, ((m₁+m₂)/3M⋆)^⅓ · (a₁+a₂)/2. In astronomical units the same 17 separations span a factor of 2554 — from 0.0043 AU between two TRAPPIST-1 planets to 10.9 between Uranus and Neptune — and carry no visible structure at all. In this unit they span 9.6, with a median of 11.8. The solid line at 2√3 = 3.46 is a theorem: two planets on circular coplanar orbits wider apart than that can never have a close encounter, whatever else happens, and every pair here is clear of it. The dashed line at 10 is a fit, to integrations of systems of several planets over billions of years, and it is the one that bites — a system packed tighter than about ten does not survive, which is why the observed distribution has a floor there and not at the theorem. A planetary system's spacing is not measured in kilometres. It is measured in a unit the planets define. 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.

 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

How long a system takes to forget

A star crossing a galaxy is deflected by every other star, and the deflections add as a random walk. The time for that walk to change a star's energy by its own amount is a hundred million billion years for a galaxy and a billion for a globular cluster, and everything about how the two are modelled follows from which side of a Hubble time that falls on.

 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.

 GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst. Gravitation

A distance with no ladder under it

The frequency sweep of an inspiral fixes the chirp mass with no distance in it, and the amplitude then gives the luminosity distance directly, because one expression fixes both. That is a distance measured with nothing calibrated beneath it — and its error budget is one angle.

 Nothing visible in any pulsar, and a quadrupole in the angle between them. Above: 4 millisecond pulsars' timing residuals over 15 years, at the few hundred nanoseconds a good one reaches. Each wanders, and none of them shows anything a reader could call a signal; a gravitational-wave background of amplitude 2.4·10⁻¹⁵ at one cycle per year contributes a common part to all of them that is smaller than each pulsar's own red noise. Below: the correlation between pairs, against the angle on the sky between them. 2211 pairs out of 67 pulsars, binned into 15 angles, against three curves with no free parameters between them. A quadrupolar background gives the Hellings–Downs shape — positive for nearby pulsars, negative near 83°, and back up to exactly half its zero-separation value at 180° because a background looks the same in opposite directions. An error in the observatory clock would give a flat line, because it shifts every pulsar identically. An error in the solar-system ephemeris would give a cosine, because it moves the barycentre in one direction. The drawn points prefer the quadrupole over the flat line by Δχ² = 358. That is the detection: not a waveform, not an event, not a moment — a shape in an angle, accumulated over fifteen years, on data taken for another purpose entirely. Gravitation

A detector the size of the galaxy

At a nanohertz no instrument can be built, so the clocks already in the sky are used instead. The signal is in no single pulsar's data — it is in the correlation between pairs as a function of the angle between them, and that curve has no free parameters at all.

 The adaptive step starts 5.7× more accurate and ends 7.5× worse. The envelope of the relative energy error against elapsed revolutions — the worst error within each plotted interval rather than the error at one sample in it — for the same second-order symplectic integrator run two ways on a Kepler orbit at e = 0.5. The flat band is a fixed step of one 200th of a period: its error oscillates once per revolution and the envelope does not grow, because a symplectic method at constant h is the exact solution of a nearby Hamiltonian and is conserving that one. The rising curve is the same method with the step refined where the orbit is fast — h ∝ r^3/2, the free-fall time, varying by a factor of 5 around the orbit — which is the first thing anybody reaches for at a close encounter and which is strictly more accurate step for step: over its first three revolutions it stays a factor of 5.7 below the fixed run. By 1500 revolutions it is a factor of 7.5 above it and still climbing. Changing the step changes which Hamiltonian is being conserved, the errors from successive steps stop cancelling, and what is left is a random walk with no bound at all. The practical consequence is that a solar-system integration cannot adapt its step: it either keeps a step short enough for the closest encounter it will ever meet, or it detects the encounter and hands that piece of the trajectory to an entirely different, non-symplectic method for the duration — which is what every long-term integration of the planets actually does. 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.

 A wave whose crests count out 35 grams per square centimetre of ring. Optical depth across a spiral density wave in Saturn's A ring, drawn outwards from the 5:3 inner Lindblad resonance with Mimas at 131,988 km. The wave is launched at the resonance on the left and damps away to the right. Its wavelength is not constant: it starts at about 17.6 km and has shortened to 3.58 km by the last crest drawn, because the wavenumber grows in proportion to the distance from resonance and the ring's own self-gravity is the only restoring force in the dispersion relation. That makes the pattern a chirp with exactly one unknown in it. Fitting the 24 crest positions actually drawn here — the square of each one's distance from resonance against its number, which is a straight line — returns a surface density of 35.0 grams per square centimetre against the 35 the profile was built from. The rings have been weighed this way rather than by anything touching them: the mass per unit area follows from counting bright bands in a light curve as a star sets behind the ring. 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 sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats. The observed sky

The face that is not quite fixed

The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground.

 Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses. Stars

The only stars whose masses are known

A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else.

 The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round. Stars

The disc that has to throw angular momentum away

Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for.

 A distance of 52.0 parsecs with nothing underneath it. Two ways to a distance for the same pair. The orbital parallax needs no iteration and no assumption: a double-lined spectroscopic orbit gives the relative orbit's linear size as (K₁+K₂)P√(1−e²)/2π sin i = 0.2268 AU, an astrometric orbit gives its angular size as 4.36 milliarcseconds, and the ratio is 52.0 parsecs — a length divided by an angle, with no step of the distance scale below it and no property of the stars assumed. The curves show the dynamical parallax, the version available when only one spectrum can be measured: guess the mass sum, take the linear size from the harmonic law, divide by the angular size, convert the apparent magnitude to an absolute one and read a new mass sum off a mass–luminosity relation. Three starting guesses spanning a factor of 10 in mass converge to the same distance in 8 passes and agree to 0.001 per cent. It converges because the distance depends on the assumed mass only as its cube root — the measured exponent here is 0.3333 — so a factor of two in the mass is 26 per cent in the distance, and one pass removes most of that. What it converges to is not the orbital parallax: the iteration settles at 54.2 pc against 52.0, 4.2 per cent away, because the fixed point is set by the mass–luminosity relation and the apparent magnitude rather than by anything measured about this orbit. The same insensitivity that makes it converge is why it is never better than the relation it leans on. Stars

Two orbits of one pair, and a distance falls out

Measure the same binary spectroscopically and astrometrically and the orbit comes back twice — once as a length in kilometres and once as an angle on the sky. The ratio is a distance that owes nothing to parallax, nothing to a standard candle, and nothing to any assumption about the stars.

 The light curve of AI Phoenicis, computed from its elements. Total light against orbital phase, computed by overlapping two discs of radius 1.805 and 2.9303 solar radii at an inclination of 88.5°, each weighted by its own surface brightness. The two eclipses hide the same area of sky and have different depths — 48.0 and 19.1 per cent — because what is lost is the light of whichever star is behind, and the ratio of the depths is therefore the ratio of the two surface brightnesses. Two things are left out and both matter to a real solution: this is the bolometric light rather than the light in a filter, and the discs are uniform, where a real one is limb-darkened and so has a deeper, rounder eclipse than the flat-bottomed one drawn here. Stars

Two radii, from a light curve alone

The radius of a star is not measured. It is inferred, from a temperature and a luminosity, through a model. There is one exception — a pair of stars that eclipse each other, whose light curve and velocity curves between them give both radii, both masses and the ratio of temperatures with no model of a stellar interior anywhere in the chain.

 A spectrum belonging to no temperature at all. The disc's summed emission, with the individual annuli drawn faintly beneath it. Each ring is a blackbody at its own temperature, and each is drawn at the area it actually has — the outer rings are cool and enormous, the inner ones hot and small. The sum has three parts and only the two ends belong to a temperature: a Rayleigh–Jeans rise of slope 2 from the outermost ring, a Wien cutoff at the hottest, and between them a stretch of slope 0.316, against the 1/3 that comes out of integrating ν²T(r)r dr with T ∝ r^−3/4. That middle section is the observational signature of a disc: no single blackbody produces it, no photosphere produces it, and its width rather than its peak is what says how far in the disc goes. What the figure cannot show is that a real disc's innermost rings are neither thin nor blackbodies, which is where the model's clean edges stop. Stars

A spectrum that is a stack of temperatures

An accretion disc is not hot. Its inner edge is, its outer edge is not, and the temperature runs continuously between them as a power of radius — so what leaves the disc is the sum of a great many blackbodies at different temperatures, which is a spectrum with no temperature in it and a slope no single body can produce.

 A measured mass of 2.08 deletes an equation of state. Mass against radius for three neutron-star equations of state, each a polytrope P = Kρ² integrated through the Tolman–Oppenheimer–Volkoff equation from the centre outwards until the pressure reaches zero. Every sequence rises, turns over and falls; only the rising part is stable, because past the maximum adding mass makes the star smaller and the smaller star cannot hold itself up. The maxima here are 1.60, 2.10, 2.57 solar masses, in the order of increasing stiffness — the same nuclear matter with a slightly harder response to compression supports a heavier star, and nothing else in the calculation changes. The horizontal band is PSR J0740+6620, whose mass of 2.08 ± 0.07 solar masses comes from the Shapiro delay of its own pulses passing its companion, which is a timing measurement and involves no model of the star at all. It sits above the maximum of one of the three, and those are not disfavoured but excluded: an equation of state that cannot hold up two solar masses is wrong, whatever else recommends it. The two lines at the left are exact and no star may cross them — the Schwarzschild radius, and the bound above it inside which the speed of sound in the matter would exceed the speed of light. A caution about the curves themselves: a Γ = 2 polytrope is a stand-in for nuclear matter and runs a kilometre or two large in radius at fixed mass, so read the ordering and the maxima rather than the radii. Stars

A radius that decides what matter can be

Nobody can make matter at four times the density of an atomic nucleus, and no calculation settles what it does there. What can be done is to weigh a neutron star — every candidate description of that matter predicts a heaviest star it could hold up, and a single measured mass above that value deletes the description permanently.

 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.

 Catching a target 40° ahead. A phasing manoeuvre. Dropping into an orbit 6% lower shortens the period to 0.9553 of the target's, so the chaser gains 16.1° each lap and closes 40° in 3 revolutions. Speeding up would have lost ground instead. Spaceflight

Catching up by slowing down, which cost Gemini 4 its fuel

To reach something ahead in the same orbit, a spacecraft must fire backwards. Pointing at the target and thrusting makes the gap grow, and a crew found that out in orbit before anyone had flown the correct manoeuvre.

 Total Δv against the radius ratio. Total transfer Δv, in units of the starting circular speed, against the ratio of the two circular radii. The Hohmann transfer is cheapest at small ratios; the bi-elliptic transfers overtake it, and the limiting one — an intermediate apoapsis taken to infinity — crosses at a ratio of 11.94. Above about 15.6 every bi-elliptic transfer beats the Hohmann. Spaceflight

Going too far in order to arrive cheaply

The Hohmann transfer is the cheapest two-burn route between circular orbits. Past a radius ratio of 11.94 the cheapest route is three burns, and it goes far beyond the destination first.

 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.

 Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between. Spaceflight

The same planet, three times

A flyby cannot change the encounter speed, only its direction, so a tour has to be designed in the space of what is conserved. One pass moves a spacecraft along a single curve and no further than the planet can bend it — and reaching a distant target means walking that curve, returning to the same planet again and again.

 What a 1 km/s burn is worth, against where it is spent. A vehicle arriving at Jupiter with an excess speed of 5.6 km/s, burning 1 km/s along its velocity at one point of the hyperbola. The vertical axis is the excess speed it leaves with. Spent at the surface the burn is worth 12.33 km/s of departure speed; spent far away it is worth 7.20. The energy bought is v·Δv, so the same propellant is worth 5.9 times as much at the bottom of the well — and nothing about the rocket has changed. Spaceflight

The same burn is worth more when moving fast

A rocket firing for ten seconds delivers the same change of speed wherever it is. It does not deliver the same change of energy, because energy is quadratic in speed — so the identical burn buys six times as much at the bottom of a gravity well as at the top, and every escape manoeuvre ever flown is arranged around that fact.

 What 28.5° of plane change costs, at three orbital speeds. The cost of rotating an orbital plane, 2v sin(Δi/2), against the angle turned, for the three speeds a transfer to geostationary orbit passes through: 7.669 km/s in a 400 km circular orbit, 3.075 km/s once circular at 42,164 km, and 1.618 km/s at the apogee of the ellipse between them. All three are √(μ/r) or vis-viva at μ⊕ = 398,600 km³/s². A 28.5° turn therefore costs 3.775, 1.514 or 0.797 km/s depending only on where it is done — the apogee figure is 21% of the low one, and that fraction is the speed ratio, so it is the same at every angle. Past 23.9° a turn in the low orbit costs more than the 3.176 km/s that leaves Earth from it, and a full reversal costs 2v = 15.34 km/s, 4.8 times the escape burn. Spaceflight

The cheapest place to turn

Rotating an orbital plane costs 2v sin(Δi/2), and the only quantity in that expression that a mission controls is the speed. Where the turn is made therefore matters more than how far it turns.

 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.

 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.

 Along a contour is free; across one costs, and 6 resonances sit on this one. Perihelion against aphelion, in units of Jupiter's orbit, with contours of constant v∞ — which is the Tisserand parameter through v∞² = (3 − T)v_p². Every contour crosses the line r_p = r_a = 1, because a spacecraft has to be at the planet's orbit to have an encounter there at all, and every point on one contour is reachable from every other point on it with no propellant: a flyby rotates the v∞ vector without changing its length, which moves the pump angle and slides the spacecraft along the curve. A burn is the only thing that moves it between curves, and that is the whole economy of a gravity-assist tour. The diagonals are resonant orbits — 1:1, 3:2, 2:1, 5:2, 3:1, 4:1 with the planet — and each is a straight line of slope −1 because a resonance fixes the semi-major axis and r_p + r_a = 2a. They matter because a spacecraft on one comes back to the same place at the same time as the planet, which is what makes a second encounter possible without waiting for a chance alignment. The marked points are where the v∞ = 0.3 contour meets each: a tour is a walk along the highlighted curve from one to the next, and the arithmetic that decides whether it can be walked is the turn one flyby delivers. At 1.35 Jupiter radii and 3.9 km s⁻¹ that turn is 163°, so the longest step drawn here needs 0.1 encounters — which is why a real tour has dozens of flybys and why the ones with the largest steps are the ones that need a deep-space manoeuvre in between. Spaceflight

A map of the transfers that are free

Drawn as contours of perihelion against aphelion, the invariant that survives an encounter becomes a map. A flyby slides a spacecraft along its own contour and costs nothing; a burn is the only thing that moves it between contours — so tour design is reading a graph.

 An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit. Spaceflight

A position measured from a frequency

A spacecraft is unresolvable and unreachable, and everything known about where it is comes from two scalars — a round-trip light time and a Doppler shift. Neither is an angle. The orbit solution returns two angles anyway, because the antenna is bolted to a rotating planet.

 A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky. Spaceflight

An angle measured against a quasar

A tracking station measures how fast a spacecraft is receding, which is one number where three are wanted. The two missing angles come from the Earth's rotation, slowly, and near a planetary encounter there is no time for slowly — so the position is instead measured directly, as a difference of arrival times between two antennas, referred to a quasar a few degrees away.

 A fold 0.06 Einstein times wide, and two configurations that make the same one. A binary lens of mass ratio 0.003 at a projected separation of 1.5 Einstein radii. Top left: the source plane, with the caustic — the set of source positions at which the magnification is formally infinite — and the track of a background star across it. A single lens has no such curve; it magnifies smoothly and diverges only at one point. A second mass makes the lens mapping fold, and the fold has edges: crossing one, the number of images changes from 3 to 5, because a pair is created out of nothing on the critical curve. The small closed curve near the origin is the central caustic, always there; the larger one at 0.83 Einstein radii is the planetary caustic, and its distance from the origin is s − 1/s, which is where the planet's own image lies. Top right: the central caustic drawn twice, once for s = 1.5 and once for s = 0.667. They are 0.0184 and 0.0169 Einstein radii across and they lie on top of each other. That is not a coincidence of these numbers: to the order that a central-caustic anomaly is measured, a close binary and a wide one with the reciprocal separation produce the same perturbation, so an event with only a central anomaly returns two separations and no way to choose. Below: the light curve along the track. The smooth part is what a single lens of the same total mass would do; the spikes are the two crossings, 0.06 Einstein times apart, so a few hours inside an event lasting a month. The two curves differ in one thing only — the size of the source. A point source diverges at each fold and reaches 27; a source of angular radius 0.006 Einstein radii averages over its own disc and reaches 9 — an eighth of a source radius inside the fold the two are 8 and 4, with the divergence replaced by a rounded shoulder whose width is the source's own diameter. Everywhere else in this collection the finite size of a star is a nuisance that degrades a measurement. Here it is the ruler: the fold is a straight edge of known sharpness sweeping across a disc, so the shape of that shoulder gives the source's angular radius, and dividing it by the crossing time gives the angular Einstein radius — which is the one quantity a light curve otherwise cannot supply. Exoplanets

A light curve with a fold in it

A single lens magnifies smoothly. A second mass makes the lens mapping fold, and a fold has an edge — a curve across which two images appear out of nothing and the magnification formally diverges. Crossing it turns the finite size of the source star from a nuisance into a ruler.

 The wobble of a star with a circular companion. The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity. Exoplanets

The star moves, and the mass is a lower bound

A planet is found by watching its star fall towards it. The wobble gives a mass multiplied by the sine of an angle nobody has measured — and the shortfall is not a rounding error.

 Mass against radius, and what lies between the curves. Planetary radius against mass on logarithmic axes, in Earth units, with composition curves computed from interior models rather than drawn through the points. The solid-planet curves are R ∝ M^(1/3.7) — flatter than a constant density because a heavier planet compresses itself — and the hydrogen curve turns over near three Jupiter masses, where degeneracy pressure takes over and adding mass makes the planet smaller. A mass alone or a radius alone places a planet on a line; only both together place it between two curves, and that is the whole argument for measuring a planet twice. Exoplanets

Two methods, and one density

A transit gives a radius. A wobble gives a mass. Neither says what a planet is made of, and the two together say it in one number — which is the only reason both are worth doing on the same object.

 A 6.2 m s⁻¹ signal at the rotation period, made by no planet at all. Above: the apparent radial velocity of a star carrying one dark spot over 0.4 per cent of its disc, rotating with an equatorial velocity of 3.2 km s⁻¹, over 3 rotations — and beside it a circular-orbit planet of the same period fitted to the same amplitude, 6.2 metres a second. That amplitude is several times the precision of a modern spectrograph and squarely inside the range in which warm sub-Neptunes are claimed, so the two are competing on equal terms. The spot signal is not a sinusoid: the spot is in view for half the rotation and hidden for the other half, so the curve is truncated, and its power at half the period is 1.16 of its power at the period. A Keplerian orbit at the same period has none there at all. Below: the diagnostic that actually settles it. A planet moves the whole spectrum bodily, so every line keeps its shape and the bisector — the locus of midpoints up a line profile — does not move; a spot removes light from one side of the profile, so the bisector tilts in step with the velocity and against it. The two clouds correlate at -0.92 and 0.14. What the picture cannot show is why this took so long to become routine: measuring a bisector to a few metres a second needs a signal-to-noise ratio of several hundred per spectrum, so for two decades the diagnostic existed and could not be applied to the faint stars the interesting claims were about. 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.

 Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 3 billion years. It falls at 4.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed. 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.

 S2's orbit, and the mass it implies. The orbit of S2 about the centre of the Milky Way, drawn from its measured elements: a semi-major axis of 0.1251 arcseconds, which at 8.28 kpc is 1036 AU, an eccentricity of 0.8843, and a period of 16.05 years watched round twice. Kepler's third law in solar units — a³/P² — gives 4.31 million solar masses, by exactly the calculation that weighs a planetary system. At periapsis the star is 120 AU from the focus, 1407 Schwarzschild radii, and the mean density inside that radius is 5.3e+15 M☉ per cubic parsec — a million times the densest star cluster known. That density, not the mass, is the argument: there is no configuration of stars that would fit. Galaxies

The mass at the centre that is not stars

One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole.

 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.

 Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through. 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.

 Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium. Gravitation

A pair that heats what is trying to cool

A star cluster has a negative heat capacity, so it cannot reach equilibrium — its core contracts and gets hotter without limit. What stops it is a binary, and which way a binary exchanges energy with the stars around it is settled by one comparison of two energies.

 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.

 Accretion from a medium a body is moving through, against how fast it moves. The rate at which a gravitating body captures gas out of a medium it is ploughing through, divided by the rate it would capture at rest, against its Mach number, both axes logarithmic. The accretion radius is set by where the body's escape speed matches the speed of the gas relative to it, and that relative speed combines the sound speed with the motion in quadrature; the rate carries the square of that radius times the speed, which leaves one plus the square of the Mach number to the power minus three halves. Subsonic motion therefore costs almost nothing — the curve is flat below Mach one third — while supersonic motion costs the inverse cube of the speed, drawn here with a measured logarithmic slope of -2.99. The same focusing produces the wake and the drag: the body pulls a denser column behind it, that column pulls back, and the material closest to the axis is captured. Accretion and dynamical friction are not two processes but two accounts of one, which is why a body that grows by this mechanism is also being slowed by it. Stars

The wake and the meal are one calculation

A body moving through gas gathers what passes inside the radius at which its escape speed matches the flow. That single radius sets both the drag it feels and the rate at which it grows, so accretion and friction are not two processes but two readings of one — and the rate falls as the inverse cube of the speed.

 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.

 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.

 An invariant that moves by 8.0e-3 once the planet's orbit is real. The Tisserand parameter of a comet on an orbit of semi-major axis 5 and eccentricity 0.8, followed through a close passage of Jupiter, integrated twice. The lower trace has Jupiter on a perfect circle, which is the problem the parameter is an exact constant of: it survives the encounter having moved by 1.0e-4, which is the integrator's own error and not a physical change, and the spike at the moment of closest approach is the osculating elements being briefly meaningless while the comet is inside Jupiter's sphere of influence rather than the constant failing. The upper trace is the same encounter with Jupiter on its real orbit, eccentricity 0.0489. The parameter comes out changed by 8.0e-3, 79 times as much, because the Jacobi constant exists only when the rotating frame is uniformly rotating and a planet on an ellipse does not provide one. That number is small and it is not negligible: comet families are separated by boundaries in this parameter placed to two decimal places, and a comet that drifts across one over several encounters has changed class without anything having happened to it that a single encounter could account for. Orbits

An invariant that is only almost one

The Tisserand parameter survives a close encounter with Jupiter exactly, and comet families are separated by boundaries in it drawn to two decimal places. The exactness holds for a Jupiter on a circle. Jupiter's eccentricity is 0.0489, and integrating the same encounter twice shows what that costs.

 A line nothing spun up by accretion can lie above, and the millisecond pulsars beneath it. The period–period-derivative diagram with the spin-up line drawn on it. An accreting neutron star is torqued by the disc until its magnetosphere turns at the same rate as the material arriving there, which fixes an equilibrium period as a function of the magnetic field and the accretion rate. Eliminating the field between that relation and the dipole formula that every point in this diagram is already read through leaves a straight line of slope 1.33, drawn here for accretion at the Eddington rate — the fastest a star can be pushed. The 7 recycled pulsars all sit below it, which is what the figure is for: none of them was spun up faster than the limit allows, and their positions are a record of how much mass each one received rather than of how old it is. The young pulsars are in the opposite corner, above the line and to the right, spinning down from birth. The two populations are not two stages of one life. A star that reaches the bottom left has been fed by a companion for a hundred million years, which is why almost every millisecond pulsar has one and almost no young pulsar does. Stars

A corner of the diagram that has to be earned

A pulsar spinning a thousand times a second cannot have been born that way and stayed that way, because its own radiation would have slowed it in a few million years. It got there by being fed, and the line it cannot lie above is where the accretion torque balances the magnetic one.

 Prograde and retrograde, the same encounter. The same encounter twice: a companion of 1 times the primary's mass passing at 1.5 disc radii, with the disc spinning the same way the companion orbits and then the other way. Nothing else differs. The prograde disc grows a bridge towards the companion and a tail away from it; the retrograde one is barely disturbed. The reason is a resonance rather than a force: in the prograde case the outer particles keep pace with the companion for a substantial part of the encounter and are pulled the whole time, and in the retrograde case they sweep past it and the impulses cancel. Galaxies

Whether it merges is a ratio of two times

Two galaxies passing each other either merge or do not, and what decides it is not how close they come. It is whether the encounter lasts long enough for the internal motions to respond — a slow, prograde passage transfers orbital energy into stellar orbits and the pair is bound; a fast one leaves both galaxies heated and still moving.

 A gravity assist with a 68° 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

The planet pays, and it shows

A gravity assist takes energy from a planet and gives it to a spacecraft, and the planet's loss is exactly the spacecraft's gain. For a two-tonne probe past Jupiter that loss is unmeasurable. Do it with a hundred Earth masses of icy debris and the same bookkeeping moves Neptune outward by several astronomical units.

 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. 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.

 Two bodies at a mass ratio of 2.6 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 2.6 to 1, so the heavier body's path is 2.6 times smaller. Stars

The companion survives, and it is moving

When one star of a close pair explodes, the other is left holding an orbital velocity and no orbit. Whether the pair stays bound turns on a single question — did more than half the total mass leave? — and the stars that answer no are running across the Galaxy at a hundred kilometres a second with nothing visible behind them.

 Central pressure bracketed without a model: 6 bodies, 23 decades apart. What can be said about the middle of a body from its mass and its radius alone. The lower end of each bar is GM²/8πR⁴, which follows from hydrostatic equilibrium and nothing else — no equation of state, no composition, no temperature, no assumption whatever about how the density is arranged inside. The upper end costs one more assumption, that the density does not increase outward, and it needs a central density, which is a model output rather than an observation and is why that edge is drawn as the softer one. The dot is what a full structural model gives. For the first five bodies every dot lies inside its bar, and what is worth noticing is how wide the bar is: Sun's rigorous floor is 4.48e+13 pascals against a modelled 2.34e+16, a factor of 522. The bound is true and nearly useless there, because most of a centrally condensed body's pressure comes from the concentration and the derivation deliberately knows nothing about it. The relativistic entry is the exception, and the reason to draw the figure at all. neutron star's modelled central pressure is 8.5 times the Newtonian ceiling — a body no Newtonian arrangement of matter with density falling outward can produce. The floor still holds, and holds for a statable reason: relativity makes the pressure gradient steeper than Newtonian gravity does, so the true central pressure can only exceed what the Newtonian derivation demands. The bracket therefore does more than constrain an interior. Applied at a small enough radius it breaks, and where it breaks is where Newtonian hydrostatics has stopped being the right equation. Stars

A floor under the centre that assumes nothing

There is a lower bound on the pressure at the centre of any body in hydrostatic equilibrium, and it needs no equation of state, no composition and no temperature — only a mass and a radius. For the Sun it is nearly useless. For a neutron star it says which theory of gravity the interior needs.

 Two interiors, 2.59 mm/s apart, against a floor of 0.02. What a radio link measures when a spacecraft flies past a moon. The horizontal axis is time from closest approach in minutes and the vertical axis is the accumulated change in the line-of-sight velocity in millimetres per second, after the pull of the moon as a point mass has been fitted and removed. What is left is the part of the field that is not spherically symmetric, and the two curves are what two interiors predict for it. The upper one is a body whose tidal Love number is 0.616 — a shell floating on a global liquid layer, free to deform almost as a fluid would. The lower one is a body solid throughout, at 0.03. They differ by 2.59 millimetres per second in the accumulated deflection, against a floor of 0.02 for a coherent two-way X-band link integrated over tens of seconds: 130 standard deviations in a single pass. That ratio is the whole reason the measurement is possible, and it is worth stating what is being compared. The point-mass deflection itself is 837 metres per second — five orders of magnitude larger than the signature of interest — so the interior is not read off the Doppler curve but off the residual left after a model of everything larger has been subtracted: the moon's mass, the planet's, the spacecraft's own thrusting and outgassing, the plasma along the path, the station's motion, and relativity. Every one of those has to be right to a part in a hundred thousand before the last curve here means anything, which is why gravity science needs many passes and a global fit rather than one flyby and a subtraction. The published uncertainty on Titan's k₂ is eleven per cent rather than the fraction of a per cent this signal-to-noise would suggest, and the difference is entirely correlations with the other parameters in that fit — a reminder that a formal error on a curve is not the error on the number extracted from it. The straight-line path assumed here is exact only in the limit of a fast flyby; a slow one bends, and the bending is solved for rather than approximated. Spaceflight

An ocean found in a Doppler residual

A spacecraft flying past a moon is deflected by hundreds of metres a second, and the part of that deflection which says whether the moon has an ocean is two millimetres a second. Everything larger has to be modelled and removed first, exactly, and what is left over is an interior.

 Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left. Orbits

Ninety-nine per cent of the mass and none of the spin

The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

 Every pair arrives circular. Eccentricity against gravitational-wave frequency for four binaries of 30 and 30 solar masses, each starting at 0.2 astronomical units with an eccentricity of 0.3, 0.7, 0.9, 0.99. Both axes are logarithmic and the tracks run left to right as the orbit shrinks. The curves are Peters' closed solution a(e), and they are checked against Peters' differential equation at three eccentricities on each track rather than against the integral they came from. The ordering is preserved — a pair that starts rounder stays rounder — but by the time the orbit is radiating at 10 hertz, where a ground-based detector begins to hear it, the four eccentricities are 8.8·10⁻⁸, 5.6·10⁻⁷, 3.6·10⁻⁶, 1.5·10⁻⁴. All four are far below anything a detector could measure. That is the figure's whole content and it is a strong statement: radiation reaction removes angular momentum faster, relative to energy, than a circular orbit would need, so eccentricity is destroyed on the way in. A binary observed to be eccentric in band therefore cannot have spent long shrinking quietly, and must have been put on that orbit recently — by a third body, or in the crowded centre of a cluster. The figure assumes the two bodies are points and nothing else acts on them, which is exactly the assumption an eccentric detection would refute. Gravitation

Every pair arrives circular

Gravitational radiation drains a binary's energy and its angular momentum at rates that do not keep step, and the mismatch destroys eccentricity far faster than it shrinks the orbit. A pair that starts at 0.99 and spirals in from a fifth of an astronomical unit is round to better than a part in a hundred million by the time a detector can hear it.

 The last parsec, priced in stars. The two timescales that shrink a pair of 10⁸-solar-mass black holes at the centre of a merged galaxy, against their separation in parsecs, both axes logarithmic, for a stellar velocity dispersion of 200 km/s and a central density of 500 solar masses per cubic parsec. Ejecting stars hardens the binary at a rate proportional to the separation, so that timescale grows as the orbit shrinks; gravitational radiation goes as the fourth power of the separation, so its timescale collapses. Neither alone finishes the job and the crossing of the two is where the answer is set. With the loss cone kept full, the crossing is at 2.9e-2 parsecs and the total is 4.2·10⁸ years, inside a Hubble time. With the supply of low-angular-momentum orbits emptied by a factor of 100 — which is what happens in a smooth spherical nucleus, because the stars that could interact have already been thrown out and two-body relaxation refills the orbits far too slowly — the crossing moves outward to 7.3e-2 parsecs and the total becomes 1.7·10¹⁰ years. That is the final-parsec problem, and it is not a problem about gravity: it is a problem about supply. Real nuclei are not spherical, and the figure cannot show what a triaxial potential does, which is to keep feeding the binary orbits it has not already used. Galaxies

The last parsec, and the stars that are not there

Two galaxies merge and their central black holes sink towards each other, and then stop. From about a parsec apart, friction no longer works and gravitational radiation is not yet strong enough — and the only mechanism in between throws away the stars it depends on faster than they can be replaced.

 A circular orbit has one lock, and an eccentric one has several. The strength of each spin–orbit resonance against orbital eccentricity, as the Hansen coefficient H(p, e) that multiplies the restoring torque on a permanently non-spherical body. At zero eccentricity every curve but the synchronous one is exactly zero — the figure checks that rather than showing it — so a body on a circular orbit can lock only by turning once per orbit. Away from zero the others switch on: at Mercury's eccentricity of 0.2056 the 3:2 resonance has 73 per cent of the synchronous one's strength and more than twice the 2:1's. A planet spinning down through this family therefore meets the 3:2 before the 1:1 and has a real chance of being caught there, which is what happened — Mercury turns three times for every two orbits, a fact discovered by radar in 1965 after a century of assuming it was locked. The free libration of that locked state follows from the same coefficient and the measured 2.03e-4 for (B − A)/C: 12.1 years, against a measured period near twelve. What the figure cannot show is the capture probability itself, which depends on how the tide dissipates and ranges from a few per cent for a simple constant-lag tide to more than half once friction between a liquid core and the mantle is included. Gravitation

A rotation locked to the orbit, but not one to one

Mercury turns exactly three times for every two circuits of the Sun. That was not what anybody expected, and it is not an accident — on a circular orbit a tidally despun body has exactly one place to lock, and on an eccentric one it has several — with the strength of each set by a coefficient that vanishes when the eccentricity does.

 Where the disc stops, and the spin that follows from it. Two radii against accretion rate, for a neutron star with a 10⁸-gauss field. The falling curve is the magnetospheric radius, where the field's stress on the disc matches the rate at which the flow carries angular momentum inward; it goes as the accretion rate to the minus two sevenths, which is a weak enough dependence that the factor of 1000 in supply drawn here moves the boundary by a factor of 7.2. The horizontal lines are corotation radii for three spin periods — the radius at which the disc orbits as fast as the star turns. Above corotation the field is spinning the gas faster than it wants to go and flings it out; below, the gas is faster and spins the star up. So the crossing is an attractor, and a star accreting steadily walks to the period where the two coincide. That period goes as the field to the six sevenths and the rate to the minus three sevenths — which is why a neutron star that has swallowed a tenth of a solar mass from a companion comes out at a few milliseconds, and why the millisecond pulsars have fields ten thousand times weaker than the young ones. Gravitation

The period a star is pulled towards

A magnetised neutron star does not let a disc reach it. The disc stops where the field's stress wins, and if that radius lies inside the corotation radius the star is spun up while outside it the star is spun down — so there is one period at which nothing changes, and an accreting star walks to it.

 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.

 The best-fitting eccentricity of a circular orbit. What a fitted eccentricity comes out at when the orbit's true eccentricity is 0 and each component of the eccentricity vector carries an error of 0.03. The distribution is not centred on the truth and cannot be: an eccentricity is the length of the vector (e cos ϖ, e sin ϖ), lengths are not negative, and a quantity bounded below by zero whose components scatter symmetrically has a distribution pushed away from the bound. For a circular orbit the most likely fitted value is exactly one error bar, 0.0300 here, and the mean is 0.0376 — 1.2533 error bars, which is √(π/2) and comes from geometry rather than from any property of the data. The practical consequence is a catalogue of small eccentricities that are all measurements of their own error bars, and the fix is not a better fit but a different question: an upper limit rather than a value. Orbits

An eccentricity that cannot be zero

Fit an orbit to noisy data and the eccentricity that comes back is never zero, not even when the orbit is a perfect circle. The reason has nothing to do with the data and everything to do with the fact that a length cannot be negative.

 Three causes, three shapes, one residual. An eclipse-timing residual: the observed minus the computed time of each eclipse, against cycle number, in days. Three effects are superposed and each has its own functional form. A slow change in the orbital period — from mass transfer or from magnetic braking — integrates to a parabola. A third body in a wide orbit moves the whole binary towards and away from the observer, so its light-travel time adds a sinusoid at the third body's period. And an eccentric orbit whose apsides are precessing moves the two eclipses in opposite directions, which is a sinusoid that changes sign between primary and secondary minima. The last is why both eclipses have to be timed: a third body moves them together and apsidal motion moves them apart, and a series of primary minima alone cannot tell the two apart at all. Stars

Three causes with three shapes in one curve

The times of eclipse in a binary star are a clock, and the clock runs late and early. Three completely different things make it do so — a third body, a changing period, and a slowly turning orbit — and they are separable only because each imposes a different shape on the residual.

 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.

 A bump that crosses the line from -41 to 25 km/s. The residual of a rotationally broadened line profile during a transit, drawn at five epochs and offset vertically. The planet covers a strip of the stellar disc whose radial velocity is the projected rotation at that position, so it removes light from one velocity and leaves a bump in the residual there. As the planet crosses, the bump travels across the profile — and where it starts and ends is set by the geometry of the chord. An orbit aligned with the star's equator gives a track symmetric about the line centre; this one, tilted by 30 degrees, runs from -41 to 25 kilometres a second and is not. The measurement is of a path rather than of a centroid, which is why it works on rapidly rotating stars where the velocity anomaly is swamped by the line's own width. Exoplanets

A shadow crossing a rotating line

A transiting planet hides a strip of a rotating star, and that strip has a definite velocity. So the planet removes light from one place in the line profile and leaves a bump there — a bump that travels across the line as the transit proceeds, tracing the path the planet took across the disc.

 A virial mass inflated 23.2-fold by orbits nobody resolved. The factor by which a virial mass is overestimated when the velocity dispersion is measured from single-epoch spectra, against the system's true dispersion, for four numbers of observing epochs. Every star in a binary carries its own orbital velocity, which adds in quadrature to the system's own, and the orbital velocities of ordinary binaries are of order a kilometre a second. A system whose real dispersion is 0.3 km/s therefore measures 1.45, and since a virial mass goes as the square of the dispersion the mass comes out 23.2 times too large. Repeat epochs fix it: the orbital velocities are uncorrelated between visits while the system's own are not, so the binary variance falls as one over the number of epochs and the correction is measured rather than modelled. Galaxies

A dispersion inflated by orbits nobody resolved

A velocity dispersion measured from single spectra is not the dispersion of the system's centres of mass. Every star in a binary carries its own orbital velocity of a kilometre or so, and for a dwarf galaxy whose real dispersion is smaller than that, the measured value — and the dark-matter content computed from its square — is mostly binaries.

 The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one. 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.

 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.

 The same equation, run away at q = 0.8 and settled at q = 0.5. The donor's overfill of its own Roche lobe against time, integrated for 3 mass ratios at a donor adiabatic response of −0.33 — a star with a deep convective envelope, which expands as it loses mass. The overfill sets the transfer rate and the transfer rate changes the overfill, and the sign of that feedback is the stability criterion: where the lobe shrinks faster than the star does, ζ_L > ζ_ad, the overfill grows and the growth is exponential. At q = 0.8 the lobe responds at 0.03 and the overfill runs away; At q = 1.4 the lobe responds at 1.32 and the overfill runs away. At q = 0.5 it responds at -0.62 and the transfer throttles itself back. The critical ratio for this donor is 0.63 — below equal masses, which is the result the whole subject turns on: a giant transferring to a lighter companion is already unstable before the mass ratio has reversed, and what follows is not accretion but a common envelope. Nothing here is a fitted rate. The vertical scale is logarithmic and the runaway is a straight line on it, which is what an exponential is. 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.

 Two separations that draw the same caustic. The width of the central caustic against the separation, computed from the lens equation for a mass ratio of 0.003, for each separation s and for its reciprocal 1/s. The two curves lie nearly on top of each other. That is the close–wide degeneracy, and it is a theorem rather than a coincidence: expanding the binary lens equation near the primary shows that the central caustic depends on the separation only through s + 1/s to first order in the mass ratio, and that combination is invariant under s → 1/s. It is first order and not exact, which the figure shows rather than hides — the two agree to 0.7 per cent at s = 2.8 and only to 13.1 at s = 1.4, because a smaller separation is closer to the resonant regime where the central and planetary caustics have not yet separated. Reducing the mass ratio to 1.0e-3 brings the worst case to 4.8 per cent, which is the first-order statement being checked rather than quoted. What that means for a measurement is uncomfortable. An event whose planetary signal comes from the source passing near the central caustic — which is most of them, because the central caustic sits where the magnification is already high and the event is already being watched — cannot distinguish a companion at 2.8 Einstein radii from one at 0.357. For a typical lens that is the difference between a planet at four astronomical units and one at less than one. And the disagreement the figure measures is not the way out: at 2.8 Einstein radii the caustics differ in width by 0.7 per cent, which is far below what a light curve sampled through a night's seeing can separate, so the ambiguity is real in the data even where it is not exact in the mathematics. Exoplanets

Two systems that draw the same curve

A binary lens with separation s and one with separation 1/s have central caustics that agree to first order in the mass ratio. The same event is therefore a planet at four astronomical units or one at less than one, and no amount of photometric precision decides between them.

 Two galaxies that stopped moving apart, and the mass that did it. The separation of two galaxies on a radial orbit that began together at the big bang, against cosmic time, solved so that after 13.797 Gyr they are 770 kpc apart and approaching at 110 km/s — the present separation and approach speed of the Milky Way and the Andromeda galaxy. The curve is a cycloid, r = A(1 − cos θ) and t = B(θ − sin θ), and only one cycloid passes through that point with that slope. It rose to 1037 kpc, turned round when the universe was 8.5 Gyr old, and on this purely radial orbit the two meet 3.3 Gyr from now. Its period fixes the mass: A³/(GB²) = 4.2·10¹² solar masses. The dashed curve is the same calculation with the cosmological constant's outward push included, integrated rather than solved; to arrive at the same place at the same speed against that push it needs 4.76·10¹² solar masses, 13 per cent more. Far more than the stars of the two galaxies, it is the timing argument's measurement of the Local Group's dark matter. Cosmology

The age of the universe weighs the Local Group

The Andromeda galaxy is approaching the Milky Way, and in an expanding universe that means the two once moved apart, stopped and turned round. One radial orbit passes through their present separation with their present speed after exactly the age of the universe, and its period fixes the mass that turned them — four trillion suns, twenty-five times what their stars can account for.

 How far the returning debris swings round, against the hole's mass. The relativistic advance of pericentre, per orbit, of the most bound debris from a disrupted star of 1 solar radius and 1 solar mass, passing at the tidal radius, against the hole's mass. Δω = 6π GM / c² a(1 − e²), and for these nearly parabolic orbits a(1 − e²) is about twice the pericentre, so the angle is set by how many gravitational radii the pericentre is — and that falls as the mass to the minus two thirds, because the tidal radius grows as the cube root of the mass while the gravitational radius grows as the mass. At 10⁵ solar masses the pericentre is 219 gravitational radii and the stream swings round by 2.5°; at 10⁶ solar masses the pericentre is 47 gravitational radii and the stream swings round by 11.6°; at 10⁷ solar masses the pericentre is 10 gravitational radii and the stream swings round by 53.4°. A light hole barely bends its debris's orbit, and a heavy one bends it by tens of degrees. 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.

 A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has. Exoplanets

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

 Mars: a sundial 40 minutes ahead and 51 behind. The equation of time on Mars over one of its years, 668.6 sols long, against sols since the northern spring equinox, computed from Kepler's equation for an orbit of eccentricity 0.0934 and an axis tilted 25.19°, with perihelion at solar longitude 250.87°. The solid curve is the difference between true and mean solar time: it runs from −51.1 local minutes, 611 sols after the equinox, to +39.9, 385 sols after it — −52.5 to +41.0 in Earth minutes, since a local minute is a 1,440th of a sol. The dashed curves are its two parts. The eccentricity term, one cycle a year, swings by ±42.9 minutes; the obliquity term, two cycles, by ±11.4; the first is 3.74 times the second. Perihelion falls 485 sols after the equinox, marked, and the vertical lines are the equinoxes and solstices. The observed sky

On Mars the orbit outweighs the tilt

The equation of time is the sum of two terms, one from the shape of the orbit and one from the tilt of the axis, and on the Earth they are nearly the same size. On Mars the orbit's term is almost four times the tilt's, a sundial runs from fifty-one minutes behind the clock to forty ahead, and the figure-of-eight the Sun traces in the Earth's sky becomes a teardrop. Nothing about the two terms is different; only their ratio is.

 A Sun that runs backwards for 8.1 days. The rate at which the Sun moves across the sky of a planet with a 3:2 spin–orbit ratio and eccentricity 0.2056, in degrees of hour angle per Earth day, against days from perihelion, over one 87.97-day orbit. The rate is the spin rate minus the rate at which the Sun's direction turns because the planet moves along its orbit, and by Kepler's second law that orbital rate peaks at perihelion, at 1.551 times its mean. The spin is 1.5 times the mean orbital rate, so for 8.1 days around perihelion, from −4.0 to +4.0 days, the orbital rate wins, the rate is negative, and the Sun moves backwards across the sky by 1.11 degrees before resuming. At perihelion it is moving at 0.21 degrees a day in the wrong direction. Away from perihelion the Sun crosses the sky at up to 3.4 degrees a day, and a whole solar day, noon to noon, takes 175.9 days — two orbits. The observed sky

A Sun that stops and runs backwards

On Mercury the equation of time is not a correction but a reversal. The planet turns three times for every two orbits, and near perihelion its orbital motion briefly outruns its spin, so the Sun halts, backs up by a degree over eight days and resumes. From one longitude that is three noons in a week; from another, a sunrise, a sunset and a second sunrise — and a slightly rounder orbit would have stopped it happening at all.

 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.

 The two force laws whose orbits close. The apsidal angle — the angle swept from periapsis to the next apoapsis — against the exponent of the force law, for F ∝ r^p. The dashed curve is the near-circular limit π/√(3+p), which has a closed form; the solid curve is the same angle for an orbit of eccentricity 0.4, computed by quadrature of ∫(L/r²)dr/√(2(E−U)) between its two turning points, with U the effective potential. An orbit closes when the apsidal angle is a rational multiple of π, and an orbit closes at every eccentricity only where the two curves meet: p = −2 at exactly 180° and p = +1 at exactly 90°, which is Bertrand's theorem. The quadrature returns 180.0000° and 90.0000° at those two exponents and departs from the near-circular curve by 1.4° at p = 0. The angle diverges as p approaches −3, where the circular orbit stops being stable and there is no well left to oscillate in. 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.

 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.

 How many planets a star has is the hardest thing a catalogue measures. The multiplicity distribution a transit catalogue would contain, for systems that all truly hold 5 planets, at four mutual inclination dispersions. 40,000 systems are drawn per dispersion with an isotropic viewing direction and Rayleigh-distributed inclinations about a common plane, at semi-major axes of 12, 16, 21, 27, 34 stellar radii; the bars are conditioned on at least one planet transiting, which is what makes a system appear in a catalogue at all. At 0.5° of dispersion 33 per cent of the detected systems show all 5 planets and the mean apparent multiplicity is 3.13; at 10° it is 1.39, with 68 per cent of them showing exactly one. Every one of those systems has 5 planets. The entire difference between a catalogue of singles and a catalogue of compact multiples is one number that nothing in the light curve measures. And the two effects run in opposite directions: the fraction of stars showing any planet RISES with the dispersion — 8%, 9%, 12%, 19% across the four — because scattering the orbits gives more of them a chance to cross the line of sight, while the number seen per detected star falls by a factor of 2.2. A survey that scatters its systems finds more stars with planets and fewer planets per star, and neither number on its own says which has happened. What no figure here can show is the true dispersion, because the observable is the ratio of those two and a system with fewer planets and a tighter plane reproduces it exactly. Exoplanets

How many planets a star has is not a measurement

Draw five thousand identical five-planet systems, scatter their orbital planes by half a degree, and a third of the detections show all five. Scatter them by ten degrees and two thirds show exactly one. Every system has five.

 One square root that raises the orbit and turns it. The velocity budget from a 300 km circular orbit to geostationary, against the plane change carried out along the way. Edelbaum's closed form — √(v₁² + v₂² − 2v₁v₂cos(½πΔi)) with Δi in radians — puts the whole continuous manoeuvre in one square root, and at Δi = 0 it collapses to |v₁ − v₂| = 4.651 km/s, which is the spiral's cost with no plane change in it. The two-impulse curve puts its rotation into the circularisation burn at apoapsis, where the vehicle is moving at 1.608 km/s and a rotation is cheap. At 28.5° the continuous transfer costs 5.951 km/s against the impulsive 4.256 — the plane change adds 1.300 to one and 0.363 to the other. That is the opposite of the usual claim that low thrust turns for free. It turns continuously, which is not the same thing: the gain is that the propellant is not the budget, and the Δv is worse here as it is everywhere else. Spaceflight

One square root that raises the orbit and turns it

Edelbaum put the plane change inside the same radical as the raise, and the half-pi in its cosine is the whole result — a continuous turn costs π/2 times an impulsive one below 140° and less above it.

 A wall measures a ratio, and a ratio is a line. The tidal quality factor against the system's age, with each line the locus of one measured circularisation boundary. For a planet the eccentricity damping time τₑ goes as Q′P^(13/3), so the cut-off period goes as (age/Q′)^(3/13) and only the ratio of the two appears. A measured wall is therefore a straight line of slope exactly 1 in this plane and never a point on it: a 5-day boundary is consistent with Q′ = 2·10⁵ in a one-billion-year-old system and with Q′ = 2·10⁶ in a ten-billion-year-old one, and nothing in the light curve chooses. The stellar version of this measurement escapes because the cluster supplies the age, from a main-sequence turn-off that owes the tide nothing — which is why a cut-off period read off four clusters is a dissipation measurement and the same wall in the hot-Jupiter plane is not. The shaded band is what a field star's age is actually worth: known to a factor of 3, it leaves Q′ known to a factor of 3 and no better, against a quantity whose published values for giant planets span 10⁴ to 10⁹. What would break the degeneracy is a second measurement with a different power of P in it — an orbital decay rate, which goes as Q′⁻¹ with no age in it at all, and which has now been measured for one planet. Orbits

A wall measures a ratio, and a ratio is a line

The same circularisation boundary drawn for planets probes the dissipation inside the planet rather than inside the star. But the boundary depends only on age over Q′, so with no cluster to date the system the measurement is a line in a plane and never a point on it.

 Two damping times, one crossing, and the slope that separates them. Circularisation timescale against orbital period for the two tidal mechanisms, both logarithmic, normalised to 1.217 Gyr at 10 days. On logarithmic axes a power law is a straight line and the index is its slope, so the figure's content is that the two lines have different slopes and one crossing. The equilibrium tide gives 5.33, the bulge raised on a convective envelope being dragged ahead by a viscosity that is turbulent convection itself; the dynamical tide gives 7, gravity waves launched at the convective boundary carrying angular momentum to wherever they break. The horizontal lines are the ages of populations a boundary can be read in: where each curve crosses one is the wall that population shows — at 0.125 Gyr, 6.5 days against 7.2; at 0.625 Gyr, 8.8 days against 9.1; at 4 Gyr, 12.5 days against 11.9; at 10 Gyr, 14.8 days against 13.5. One cluster measures one number and both theories have a free normalisation, so one cluster cannot choose. The separation across every age available is a factor of 1.11. The discriminator that does not depend on the normalisation is mass: the equilibrium tide needs a convective envelope, and above about 1.3 solar masses there is not one, so the two predict different behaviour on either side of a boundary the theory names in advance. That is a measurement about where the wall stops behaving, not about where it is — and it is the reason the samples had to grow from tens of binaries per cluster to hundreds. 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.

 An eccentricity of 0.4 swings the surface by 126 K or by 0.1, depending on the length of the year. The peak-to-trough swing in surface temperature over one orbit, against orbital period, for a planet with eccentricity 0.4 receiving on average the flux the Earth does, for surface layers of 1, 10, 50 metres of water. The dashed line is the 126 K the surface would swing through with no heat capacity. Every curve rises from near zero at short periods, where the orbit is over before the layer can respond and the planet feels only the average flux, towards the full swing at long periods, where every part of the orbit lasts long enough to be felt in full. The crossover is where the orbital period is comparable with the layer's thermal time. The vertical marks are the orbital periods of the Earth-flux orbit round stars of 0.1 M☉ (7 days), 0.5 M☉ (0.2 years), 1 M☉ (0.9 years). At a fixed eccentricity a planet in the habitable zone of a small star is thermally averaging almost regardless of how much water it has, and one round a Sun-like star is not unless it has an ocean — the ordering is set by the star through the period, which is the one quantity the flux-averaged habitable zone discards. 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.

 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.

 Two assembly histories, and the sign that separates them. The distribution of effective spin produced by two ways of making a black-hole binary, from 40,000 draws each. A pair that evolved together as two stars keeps its spins near the orbital axis — drawn here aligned to within 25 degrees — and cannot produce a negative projection at all: 0.0 per cent of its mergers fall below zero, and its mean is 0.45. A pair assembled by chance encounters in a dense cluster has no memory of any plane, so its tilts are isotropic, its distribution is symmetric, and exactly half of its mergers have an effective spin below zero. The sign of one number distinguishes two histories — and it does so without any of the individual spins being measured, which is the only reason the question is answerable at all from a catalogue in which every event's spins are individually uncertain. The third curve is a mixture with 50 per cent of its mergers drawn isotropically: its mean is 0.22 and 25 per cent of it lies below zero, which is the shape an observed distribution with a small positive mean and a real negative tail requires. Neither channel alone produces it. Gravitation

A spin that is measured as an angular momentum

Every other method of measuring a black hole's spin reads it from what the hole does to matter. A merging pair's waveform reads it from the orbit instead — and what it recovers is one mass-weighted projection onto the orbital axis, which is a number that two non-rotating holes and two extremal ones can share.

 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.

 Two ranges that look identical until something is spent. The bearing to a target, as a camera measures it, for two relative orbits that differ only in size — one 2.5 times further away than the other in every coordinate at every instant. Before the manoeuvre the two histories are identical to machine precision, and that is not a numerical accident but a property of the equations: the linearised relative dynamics are homogeneous, so scaling a whole trajectory gives another trajectory, and a direction is unchanged by scaling. An angles-only tracker is therefore exactly blind to range, and no amount of observing improves it. At 42 per cent of the way through, both vehicles fire the same 0.05 m/s impulse — and the histories separate, by up to 2.3 degrees, because an impulse measured in metres per second is the one quantity in the problem that does not scale with the trajectory. The manoeuvre is what makes range observable, which is why an autonomous vehicle navigating on a camera alone flies a deliberate zig-zag rather than a straight approach, and why the manoeuvre is chosen for what it reveals as much as for where it goes. Spaceflight

A camera that cannot see how far

The equations that make a close approach tractable are linear and homogeneous, so scaling an entire relative trajectory gives another valid one with every bearing identical. A vehicle navigating on angles alone is therefore not merely imprecise about range — it is exactly blind to it, and the only thing that restores the scale is spending propellant.

 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.

 A ceiling the outer orbit imposes. The greatest eccentricity a cycle can drive an inner orbit to, against the mutual inclination it started at, for four ratios of the inner orbit's angular momentum to the outer one's. The bottom curve is the test-particle case and it is the familiar closed form, √(1 − (5/3)cos²i₀), reaching one at ninety degrees — recovered here by a solver that knows nothing about it, which is the check. Every other curve is lower. The reason is conservation: the cycle works by moving angular momentum out of the inner orbit, and it has nowhere to put it but the outer one. A test particle can give up all of its angular momentum because the outer orbit is infinite by comparison; an inner binary carrying 1 times as much as its companion cannot, because tilting the outer orbit by the required amount changes the mutual inclination and shuts the cycle down. At ninety degrees the ceiling falls from 1.000 to 0.866. The pericentre a mechanism can deliver is bounded by what the perturber can absorb, and the test-particle formula is an upper bound rather than a prediction. 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.

 Where a disc pushes a planet outward. The sign of the total torque on a low-mass planet, as a function of the power-law indices of the disc's surface density (Σ ∝ r^−α, horizontal) and temperature (T ∝ r^−β, vertical), from the fitting formulae of Paardekooper and collaborators for an adiabatic disc with γ = 1.4 and the horseshoe torque unsaturated. Above the line the total is positive and the planet migrates outward; below it the planet migrates inward. The line is straight because every part of the torque is linear in the two indices, and it is steep in α because a shallow density profile strengthens the vortensity part and weakens nothing. The minimum-mass nebula (α = 1.50, β = 0.50) sits at a total of −3.76 in units of Γ₀/γ; The viscously heated inner disc (α = 0.60, β = 0.90) sits at a total of +0.74 in units of Γ₀/γ; The irradiated outer disc (α = 1.07, β = 0.43) sits at a total of −2.65 in units of Γ₀/γ. The minimum-mass nebula every textbook starts from is on the inward side by a wide margin, and a disc heated by its own accretion is on the outward side: the same planet can migrate either way depending on how the gas is heated, which the Lindblad torque alone could never say. 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.

 Circularised at twice the closest approach. Eccentricity against semi-major axis, measured in units of the planet's Roche distance 2.16 Rₚ(M⋆/Mₚ)^⅓, for planets delivered on orbits of eccentricity 0.97 with periastra of 1.2, 1.6, 2.2, 3 Roche distances. Tides raised on the planet dissipate orbital energy at periastron and exert almost no torque over the rest of the orbit, so the orbit's angular momentum is nearly conserved while its energy is removed: a(1 − e²) stays fixed and each track runs down and to the left along it. Each ends on a circle of radius q(1 + e₀), which for e₀ near one is twice the periastron — 1.2 → 2.36, 1.6 → 3.15, 2.2 → 4.33, 3 → 5.91. The shaded region above the curve is where the periastron is inside the Roche distance and the planet is stripped on its first close passage. So the channel cannot deliver a circular orbit inside twice the Roche distance: the inner edge of the hot Jupiters is a prediction, stated in units that depend on each planet's own radius and mass rather than on its period. 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.

 One eccentric planet, drawn as two circular ones. Top: the velocity of a star with one companion on an orbit of eccentricity 0.25 (argument of periastron 0.9 radians), in units of its semi-amplitude K, over two orbits — and on top of it the sum of two circular orbits, one at the same period and one at half of it, fitted by taking the first two terms of the exact curve's Fourier series. The second orbit's amplitude comes out at 0.246 of the first, which is the eccentricity to first order. Bottom: the difference between the two, on a scale 12 times magnified; its largest excursion is 0.0889 K and its rms 0.0473 K, second order in the eccentricity. For a planet with K = 30 m/s that is 1.42 m/s rms — below the noise of most instruments — so an eccentric single planet and a pair of planets in a 2:1 period ratio are the same measurement unless the data are good enough to see a term of order e². Exoplanets

One eccentric planet, or two circular ones

The velocity curve of a star with one planet on an eccentric orbit is, to first order in the eccentricity, exactly the curve of two planets on circular orbits with periods in the ratio two to one. The difference between the two readings is a term in the eccentricity squared, and for the modest eccentricities most planets have, it is smaller than the noise.

 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.

 A hotter summer that is also a shorter one. Daily-mean sunlight at 65°N through one year beginning at the March equinox, for an orbit of eccentricity 0.0167 and obliquity 23.44°, drawn twice: with perihelion at the June solstice, when the northern summer is closest to the Sun, and with perihelion at the December solstice, half a precession cycle later. The first summer peaks 6.9% higher, and it is also shorter, because the Earth moves fastest near perihelion: days above 275 W m⁻² number 139 against 144. The energy delivered on those days — the shaded area above the line — is 5.055 against 4.989 GJ m⁻², a difference of 1.3%. The second law makes the intensity and the duration trade against each other almost exactly, so a summer measured by how much energy it delivers barely feels precession at all. 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.

 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 cheapest circular orbit to stop in at Mars is not the lowest. The speed change needed to capture into a circular orbit round Mars, in km/s, against the orbit's radius in Mars radii, for arrival excesses of 2.65, 4, 6 km/s — the first being a minimum-energy transfer from the Earth. One burn at periapsis, where the spacecraft is fastest, turns the hyperbola into the circle. The cost has a minimum, at a radius of 2μ/v∞² from the planet's centre, where it is exactly v∞/√2: 3.6 radii and 1.87 km/s for 2.65 km/s; 1.6 radii and 2.83 km/s for 4 km/s; 0.7 radii and 4.24 km/s for 6 km/s. Lower orbits cost more because the circular speed there is higher than the gain from arriving deeper in the well; higher orbits cost more because the burn is made where the spacecraft is moving slowly and the Oberth effect is lost. A faster arrival moves the optimum inward, and at a large enough excess it moves below the planet's surface, where every circular orbit is on the expensive side of the minimum. Spaceflight

The cheapest circle to stop in

A spacecraft arriving at a planet has to lose speed to stay, and the cheapest circular orbit to lose it into is not the lowest. It sits at a radius set only by the arrival speed — twice the planet's gravitational parameter divided by the square of the excess — where the braking costs exactly the excess divided by the square root of two. Most orbiters want something else, and the arithmetic of what they do instead decides which planets are cheap to stop at and which are not.

 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.

 A bead on a rotating wire: when gas is flung off a disc. The effective potential — gravity plus the centrifugal potential in the frame rotating with the footpoint — for gas sliding along a rigid magnetic field line that leaves a Keplerian disc at 20°, 30°, 40° from the vertical, in units of GM over the footpoint radius, against distance along the line in units of that radius. The field line corotates with the disc where it is anchored, like a wire whipped round, and gas on it behaves like a bead. Close to vertical the potential rises away from the disc and the gas is bound to its footpoint. Inclined by more than 30° it falls, and gas is flung outward along the line with nothing but the rotation to drive it. The 30° threshold is exact for a Keplerian disc — the second derivative of the potential at the footpoint changes sign there — and it was the result that made magnetised disc winds a mechanism rather than a speculation. 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.

 How planet occurrence rises with the host star's iron, by kind of planet. Planet occurrence relative to its value at solar metallicity, on a logarithmic axis, against the host star's iron abundance [Fe/H] (zero is the Sun; one unit is a factor of ten). Each class is a power law, occurrence ∝ 10^(β[Fe/H]). Giant planets within a few astronomical units follow β = 2, the relation found in the first large radial-velocity survey to measure it: 12 per cent of stars at [Fe/H] = +0.3 have one, against 3 per cent at solar and under 0.3 per cent at −0.5 — a factor of 158 across the range drawn. The small planets found by transits rise far more gently; the slopes drawn, β = 0.5 for sub-Neptunes and 0.2 for super-Earths, are representative of published fits, which range from nearly flat to about a doubling across this interval depending on orbital period. Exoplanets

A threshold that makes iron matter only to giants

A star with twice the Sun's iron is four times as likely to have a giant planet, and barely more likely to have a small one. The steep law and the flat one look like two mechanisms. A core racing its disc, run over a spread of discs, produces both — steep where winning is rare, flat where it is common.

 The true obliquity a projected angle allows, against the tilt of the star. The true angle ψ between a transiting planet's orbit and its star's spin axis, against the inclination of that spin axis to the line of sight, for projected angles λ of 0°, 30°, 60°, 90°, 130° — from cos ψ = sin i★ cos λ for an orbit seen edge-on. At i★ = 90°, a star seen equator-on, the projection loses nothing and ψ = λ. As the star tips towards pole-on the true angle moves towards 90° from whichever side it started, so every curve ends at 90° when i★ = 0. Two things follow exactly. The projected angle is a lower bound for an aligned-looking system and an upper bound for a retrograde-looking one — ψ always lies between λ and 90°. And λ = 90° needs no inclination at all: a planet whose projected orbit crosses the stellar equator at right angles is on a polar orbit whatever the tilt, which is why the pile-up of polar orbits could be seen in projected angles alone. Exoplanets

An angle hidden by the tilt of the star

The line profile of a transited star gives the angle between orbit and spin as projected on the sky. The true angle needs the tilt of the star's spin axis, and every way of measuring that tilt is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits.

 A cold sphere collapsing onto the virial theorem. The virial ratio 2K/|W| — twice the kinetic energy over the magnitude of the potential energy — counted over the bound shells, of 400 concentric shells released almost at rest (2K/|W| = 0.1) from a sphere whose density falls as one over the radius, against time in units of the sphere's free-fall time. It starts near zero; overshoots to 1.7 as the collapse reaches the centre, the inner shells first; rebounds and oscillates; and settles to 1.03, the virial value, staying within ten per cent of it from about 4.9 free-fall times on. The shells never collide with anything; each feels only the mass inside it. What damps the oscillation is phase mixing — shells of different energy oscillate with different periods and drift out of step — and what redistributes the energy is the potential changing while they move through it. A system of stars reaches equilibrium this way in a few dynamical times, where encounters between the stars would take far longer than the age of the universe. Gravitation

An equilibrium reached without a single encounter

Every dynamical mass of a galaxy assumes its stars obey the virial theorem, and no two of those stars have ever exchanged energy. A collapse computed shell by shell shows how a system gets there anyway — in a few free-fall times, at half the size it fell from, by a potential that changes while its orbits are still moving through it.

 The eccentricity the planets force on an asteroid, and where it diverges. The forced eccentricity of a body in the inner Solar System against its semi-major axis, from the linear secular theory with all eight planets — its two giant-planet frequencies set to the measured 4.257 and 28.245 arcseconds a year — on a logarithmic axis: the largest value it reaches over the planetary cycles (solid) and its present value (dashed). Across most of the asteroid belt it is a few hundredths — 0.048 at 2.8 AU — the part of every asteroid's eccentricity that belongs to Jupiter and Saturn rather than to the asteroid. Near 2.04 AU it diverges: there the asteroid's own apsidal precession rate equals the frequency of the planetary mode dominated by Saturn, the denominator A − g₆ vanishes, and the theory's forced eccentricity goes to infinity. That is the ν₆ secular resonance. Real asteroids near it do not reach infinite eccentricity; they reach eccentricities large enough to cross the orbits of Mars and the Earth, and are removed. The location is for a body of low eccentricity and inclination; the full theory curves the resonance to larger distances at higher inclination, which is why the belt's inner edge is not a single number. Orbits

A denominator that emptied the inner belt

The eccentricity Jupiter and Saturn force on an asteroid is a fraction, and its denominator is the difference between the asteroid's own precession rate and one of the planets'. At two astronomical units that difference is zero. Where the zero sits depends on where Saturn is — and the belt's quiet orbits say Saturn did not take long to get there.

 The pull of a ring, a shell and a point, in the plane. Radial force in the plane of a thin ring of radius 1, against distance from its centre, in units where the same mass at a point would pull with one over the distance squared; positive is outward. Inside a spherical shell of the same radius and mass the force is exactly zero. Inside the ring it is not: it points outward, toward the nearer part of the ring, at 0.34 at half the radius and 2.75 at nine tenths, growing without limit toward the rim. Outside, the ring pulls inward harder than a point would, by 25 per cent at twice its radius and 9 per cent at three times, because the near side of the ring is closer than the centre. The shell is indistinguishable from the point everywhere outside it. Gravitation

Inside a ring the pull points outward

A hollow sphere exerts no force on anything inside it, and the proof is a pair of cones whose areas grow exactly as fast as gravity weakens. Cut the sphere down to a ring and the arcs grow only as the distance, the nearer side wins, and a planet set in the middle of a rigid ring falls into it within a handful of orbits — which is why Saturn's rings cannot be solid, and why a flat galaxy spins faster than its own enclosed mass allows.

 The same energy given to a cluster two ways. The distribution of the specific energies of 20,000 stars in a Plummer cluster (units G = M = a = 1), before and after an impulsive compressive shock of strength k = 0.06, which changes each star's vertical velocity by k times its height. Stars whose energy ends above the potential at the tidal radius, −0.196, marked by the vertical line, are lost. The thin curve is the cluster before. The dashed curve adds only the half-square term, the heating the textbook integral computes: every star gains a little, and 0.8 per cent of the stars cross the line. The solid curve adds the whole change, including the cross term between each star's own velocity and its kick, which averages to zero — the mean energy input is 1.59e-3 against 1.52e-3 — but broadens the distribution: 1.6 per cent of the stars cross. The same energy, 2.2 times as many escapers. 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.

 The separation of the Milky Way and Andromeda, run forward. The distance between the Milky Way and Andromeda against time from now, in billions of years, integrated from today's separation of 770 kpc and radial approach of 110 km/s, for four values of Andromeda's sideways (tangential) velocity: 0, 17, 57, 100 km/s. The haloes are extended Hernquist spheres of 1.15 and 1.5 trillion solar masses, and dynamical friction drains the orbit at each passage. At 0 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.0 Gyr. At 17 km/s the first approach is so nearly head on that the centres pass within 10 kpc and the pair merges at 4.8 Gyr. At 57 km/s the first pericentre is 117 kpc at 4.5 Gyr, and the pair merges at 9.2 Gyr. At 100 km/s the first pericentre is 321 kpc at 4.8 Gyr, and the pair has not merged by 14 Gyr. The measured sideways speed runs from about 17 km/s (Hubble) to about 57 km/s (Gaia), and the outcomes across that range differ by billions of years. Galaxies

A collision decided by a speed across the sky

Andromeda is coming toward the Milky Way at a hundred and ten kilometres a second, and that part is measured to a few per cent. Whether the two galaxies merge in the next ten billion years depends on how fast Andromeda is also moving sideways — a drift of fifteen millionths of an arcsecond a year, measured twice, with answers of seventeen and fifty-seven kilometres a second. One is a near head-on collision in under five billion years, the other a coin toss.

 An inner orbit's inclination, with and without the octupole. The inclination of an inner orbit to the plane of a distant perturber against time, in units of the quadrupole timescale, starting at 65 degrees with a nearly circular orbit, for octupole strengths ε = 0 and 0.03. At ε = 0 — a circular outer orbit — the inclination oscillates in the regular Kozai–Lidov cycle and never crosses ninety degrees, because the angular momentum along the outer orbit's axis is conserved and cannot change sign. With the outer orbit eccentric, the inclination drifts from cycle to cycle and crosses ninety degrees, the orbit turning from prograde to retrograde, first at t ≈ 120, 2 times in all within the time shown. The integration conserves the secular energy to 7·10⁻⁶. 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.

 Propellant a moon saves, against how close the flyby comes. The reduction in the Jupiter orbit-insertion burn bought by one flyby of a Galilean moon on the way in, in metres per second, against the flyby altitude, for an arrival at 5.6 km/s, a periapsis of five Jupiter radii and a capture orbit of 200 days, whose insertion alone costs 825 m/s. At 300 km: Io 268 m/s, Europa 122 m/s, Ganymede 191 m/s, Callisto 124 m/s. The saving falls with altitude because the turn angle does, roughly as one over the flyby distance once that is well above the moon's surface. Io saves most: it is deep in Jupiter's well, where it moves fastest, so a given turn of the relative velocity takes more energy from the spacecraft. Ganymede, the most massive, is second. 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.

 Small planets and giant planets per star, against the mass of the star. Planets per star against the host star's mass, both on logarithmic axes, for small planets (one to four Earth radii, orbital periods under 200 days) and giant planets (above a third of Jupiter's mass, within 2.5 astronomical units). The small-planet rate is drawn as 1.1 per star at one solar mass rising as M^−0.6; the giant rate as 7 per cent at one solar mass falling towards small stars as M^1. Both are representative of published fits, not fits of their own. At 0.3 solar masses a star has 2.3 small planets and a 2.1 per cent chance of a giant; at one solar mass 1.1 and 7 per cent. The ratio of small planets to giants changes by a factor of 7 between the two, so a statement about how common planets are depends on which star it is about as much as on which planet. Exoplanets

More small planets round smaller stars

A red dwarf has about twice as many small planets as the Sun's kind of star and a third as many giants. The two populations move in opposite directions along the same axis, and the small planets were built from discs several times poorer.

 Giant planets per logarithmic interval of orbital distance, and who can see where. Giant planets (a third of Jupiter's mass to thirteen Jupiters) per star per unit of the natural logarithm of orbital distance, against distance, round Sun-like stars. The curve is a broken power law representative of the long radial-velocity surveys: rising as a^0.8 to a peak at 3 AU and falling beyond it as a^−1, which integrates to 2.7 per cent of stars inside 1 AU and 12 per cent in all. The shaded bands are where each method has the sensitivity to measure it: radial velocities out to about 7 AU, where two decades of monitoring cover one orbit, and direct imaging beyond about 10 AU, where a young giant can be separated from its star. The peak sits near the outer edge of the radial-velocity window, where a planet has completed one or two orbits in the data, and the fall beyond it lies entirely outside that window — so the peak's position is where the two methods have to be joined rather than a place either can measure alone. Exoplanets

A peak that neither survey can see alone

Giant planets become commoner with distance from their star out to a few astronomical units, and then rarer. The rise is measured by radial velocities and the fall by direct imaging, and the peak between them sits where neither method is comfortable.

 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.

 Which of three bodies is thrown out. Of 60 integrations of three bodies of mass 1, 2, 3 released from rest in random triangles, 57 broke up within the time integrated, each leaving one body unbound and a binary behind. The escaper was the body of mass 1 in 46 runs (81 per cent), 2 in 6 runs (11 per cent), 3 in 5 runs (9 per cent). The bars beside them are the shares the statistical theory of break-up predicts to leading order, in proportion to the escaper's mass to the power −3: 86, 11, 3 per cent. Every run is deterministic and none of them could be predicted without integrating it, but the ensemble is: the lightest body leaves most often when the masses differ, because the energy of the binary left behind is shared between the escaper and the binary's recoil in inverse proportion to their masses, and the lightest body gets the most speed. Energy is conserved in every run to better than 3·10⁻⁴. Gravitation

The lightest body leaves

Three stars close together, with no hierarchy between them, cannot stay together. One of them is thrown out, which one is unpredictable in any single case, and across many cases it is almost always the lightest — a statistical law emerging from a deterministic chaos that cannot be integrated far enough to forecast.

 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.

 Lambert's time of flight in the semi-major axis and in Lancaster and Blanchard's variable. The non-dimensional time of flight T = t√(2μ/s³) of the transfer between radii 1 and 1.524 through 135 degrees, against two variables that label the same family of orbits. Left: against the semi-major axis in units of the minimum-energy ellipse's. The family is two curves — the fast branch, which passes the empty focus on one side, and the slow branch, on the other — joined at a = aₘ with T = 1.566, where both have a vertical tangent: the time changes infinitely fast with the semi-major axis there, and every a above aₘ corresponds to two times. Right: against x, the variable of Lancaster and Blanchard, in which a = aₘ/(1 − x²). The same family is one smooth, decreasing curve through T = 1.566 at x = 0 to the parabolic time 0.662 at x = 1, with no tangent vertical and every time given by exactly one x. Nothing about the orbits differs between the panels; only the label on them does. Orbits

The variable that makes Lambert's problem easy

Written in the semi-major axis, the time-of-flight equation has two branches joined by a vertical tangent, and a solver near the join struggles. Written in a variable Lancaster and Blanchard chose in 1969, it is one smooth curve — and every trajectory library now solves it that way.

 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.

 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.

 Heliocentric latitude bought three ways, for the same launch. The highest inclination to the ecliptic a spacecraft can reach, against the hyperbolic excess speed it leaves the Earth with. Launched directly into an inclined orbit at 1 AU (dashed), it pays 2v⊕ sin(i/2) — 15.4° for 8 km/s and 23.2° for 12. Launched on a transfer to Mars whose plane is tilted — the wall across the porkchop plot, where transfers near half a turn are forced steeply out of the plane — it reaches latitude i half-way, for 13.7° at 8 km/s and 21.5° at 12: the same list price, starting from the cost of getting to Mars at all. Launched to Jupiter, it can turn there: a flyby rotates its Jupiter-relative velocity, and the steepest heliocentric orbit a flyby at 6 Jupiter radii can leave it on climbs to 90° for a 12 km/s launch, reaching the 79° of Ulysses's orbit (dotted) from about 10.6 km/s — close to the launch speed that mission actually used. Inclination bought near the Earth costs the Earth's orbital speed; bought at Jupiter it costs Jupiter's, less than half as much, and the flyby pays. Spaceflight

The wall sells inclination at the dearest price

Transfers to Mars that go almost exactly half-way round the Sun are forced into planes tilted steeply out of the ecliptic, which is why they form a wall of expensive dates across every launch chart. A mission that wanted to leave the ecliptic could use the wall on purpose. It would be buying inclination at the Earth's orbital speed — the dearest place to buy it — and the routes that reach the Sun's poles have always gone somewhere slower to make the turn.

 A flattened shell that pulls nothing inside it. The strength of the gravitational field of a thin homoeoid — the shell between two similar concentric ellipsoids, here with axis ratio 0.5 — along its long axis (solid) and its short axis (dashed), in units of GM/a² against distance from the centre in units of the long semi-axis a, from 20,000 equal masses placed on it. The dotted curve is a point mass. Inside the shell the field is zero along both axes to within 0.0001 — the numerical residue of a finite sampling — however flattened the shell is: this is Newton's theorem for a homoeoid, the flattened generalisation of the spherical shell's empty interior. Outside, the shell does not pull like a point: at 1.2 a along the long axis the field is 1.003 against the point's 0.694, and at 0.60 a along the short axis, just outside the shell there, 0.901. The exterior field remembers the shape; the interior does not know there is one. Gravitation

The shell can be flattened if it thins at the poles

A spherical shell exerts no force anywhere inside it, and a ring exerts a force that points outward. Between them lies a shape Newton found that keeps the sphere's empty interior however far it is flattened: the homoeoid, a shell bounded by two similar ellipsoids, thick at the equator and thin at the poles in exactly the right proportion. It is why the gravity inside any flattened body can be computed at all — and why the planes of stellar streams measure the shapes of dark haloes.

 Four transiting planets, one star, and a density that must agree. The stellar density implied by each of four transiting planets of one simulated star, each fitted as though its orbit were circular, relative to the star's true density, against orbital period, both logarithmic, with one-sigma errors of 8, 10, 14, 20 per cent from the transit fits. Three planets have eccentricities of a few hundredths and agree with one another and with the truth (dashed line: their weighted mean, 0.950). The planet at 15.9 days has e = 0.2 with its periastron 70° from the line of sight, which makes it cross the star faster than a circular orbit would, and its implied density is 2.10 times the star's — 5.2 standard deviations from the others. A single planet's density can be reconciled with any star by choosing its eccentricity; several planets must share one star, and the odd one out is caught. Exoplanets

Every planet of a star must agree on its density

A single transiting planet gives the density of its star only if its orbit is known to be circular, and an eccentric orbit can disguise almost any star as almost any other. A star with four transiting planets has four such measurements of one number. They must agree — and the way they disagree measures how eccentric the orbits are, catches the candidate that orbits some other star, and pins down the star better than its own spectrum does.

 The least a star can average, whatever it is made of. The lower bound on the mass-averaged temperature of a main-sequence star of ideal gas, μmₕGM/6kR, against its mass, both logarithmic, with the radius taken from the main-sequence mass–radius relation, for mean molecular weights of 0.5 and 0.62 — 0.5 is fully ionised hydrogen, the lightest possible gas and so the lowest floor; 0.62 is the solar mixture. For a star of the Sun's mass and radius the floor is 1.9 million kelvin whatever the star is made of. The dashed curve is the average a centrally condensed star actually has, for the index-three polytrope that approximates the Sun: three times the floor, 7.2 million kelvin for the solar mixture. The floor rises slowly with mass, because radius grows almost as fast. Nothing in it refers to nuclear reactions: mechanics alone requires a star to be hot enough inside that hydrogen is ionised throughout and fusion is within reach. Stars

Mechanics sets a star's temperature before fusion does

A ball of gas with the Sun's mass and radius must average at least two million kelvin inside, whatever it is made of and whatever powers it, because nothing else can hold it up. The bound comes from the virial theorem and one inequality about gravitational binding. Where it is met, a body can be a star; where it demands temperatures a body plainly does not have — the Earth, Jupiter, a white dwarf — it proves that something other than hot gas is holding that body up.

 Curvature turns a curve into an island. The companions — separation across, mass up, on edge-on circular orbits — that fit simulated velocities of a star with a 12 Jupiter-mass companion at 22 AU (period 103 years), within a Δχ² of 6.18 of the best fit, for observing spans of 6, 12, 24 years with 60 velocities of 2 m/s noise each. Over 6 years the data show a slope and the allowed region is a long band along M ∝ a², spanning 3.3 decades in mass. As the span grows the curvature of the orbit appears, and the band contracts — to 1.0 decades at 24 years — towards the true companion. Inclined orbits, which the fit leaves out, would extend every region to higher masses. 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.

 How often a merged black hole is kicked faster than a given speed. The fraction of black-hole mergers whose remnant recoils faster than the speed on the horizontal axis, for mass ratios spread evenly in logarithm between 0.1 and 1 and spins of 0.9 of the maximum, spins within 10° of the orbit; spins within 30°; spins in random directions, from a fitting formula to numerical relativity over 4000 random mergers each. The median kicks are 113, 234, 534 km/s. Faster than 100 km/s: 62%, 85%, 96%. Faster than 300 km/s: 5%, 40%, 74%. Faster than 1000 km/s: 0%, 2%, 25%. Gas accreted before the merger is expected to drag both spins into line with the orbit, which is why a gas-rich merger keeps its black hole far more often than a gas-poor one. Galaxies

A black hole thrown out by its own merger

When two black holes merge, the gravitational waves they radiate carry momentum as well as energy, and unless the two are identical and spinning in line the waves leave more in one direction than the other. The remnant recoils, at anything from a few tens of kilometres a second to several thousand — and whether a galaxy keeps its central black hole comes down to how its spins were pointing in the last orbit before the merger.

 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.

 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.

 A fall into the Sun from a standing start. The distance of a body released at rest 1 AU from the Sun, as a fraction of its starting distance, against time as a fraction of the fall's total duration, 64.6 days: the exact radial orbit (solid) and a fall at the constant acceleration felt at the start (dashed). The exact fall begins exactly like the constant one and finishes far faster, because the pull grows as the inverse square of the shrinking distance: it reaches the Sun at 79 per cent of the time the constant pull would take. The curve is a cycloid, and the total time is a fifth of the circular orbital period at the starting distance — exactly 1/(4√2) of it — because a straight fall is half of an ellipse whose semi-major axis is half the starting distance. Orbits

A fall is half an orbit with no width

Stop a planet dead and it falls into the Sun, in a time that is not a quarter of its year or a half but exactly one over four times the square root of two. The fall is an ellipse squeezed to a line, it spends four-fifths of its duration in the outer half of its path, and in a uniform cloud every shell of it arrives at the centre at the same moment — which is where the free-fall time of star formation comes from.

 The plane change wants a gentle flyby. The lowest inclination to the Moon's orbit that one flyby can leave a spacecraft in, against the flyby's height above the Moon, for a spacecraft arriving from an orbit of perigee 6678 km inclined 51.6°: with no constraint on where the new orbit goes (dashed), and requiring the new perigee to lie between the atmosphere and 20,000 km, so that the spacecraft comes back to the Earth (solid). Unconstrained, any flyby closer than tens of thousands of kilometres can bring the inclination to nearly zero. But a close flyby turns the velocity so far that it also flings the spacecraft outward, and no aim point brings it back near the Earth; only from about 9,500 km up does a returning orbit become possible. Between that height and about 18,000 km the flyby can both bring the spacecraft back and leave its orbit nearly in the Moon's plane; further out still, the turn is too small to remove the inclination. The useful flyby is a gentle one, in a window. Spaceflight

A plane change borrowed from the Moon

Turning an orbit's plane by the engine costs a fixed fraction of the spacecraft's speed, which is why it is done as far from the Earth as possible. Farther still, at the Moon's distance, a spacecraft is moving so slowly that the Moon can turn its velocity through tens of degrees for nothing — but a flyby close enough to turn it hard also flings it outward, and the useful flyby is a gentle one, in a window of heights the arithmetic picks out.

 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.

 A period set by the donor's density, running in two directions. The orbital period at which a donor exactly fills its Roche lobe, against the donor's mass, onto a 0.7 M☉ white dwarf. A lobe-filling star's period is fixed by its mean density alone, so the period follows the donor's mass–radius relation. A main-sequence donor shrinks as it is stripped and grows denser, and the period falls: from 6.0 hours at 0.6 M☉ to 1.5 hours at the hydrogen-burning limit. A white-dwarf donor swells and grows less dense, and the period rises — 3.5 minutes at 0.2 M☉, 13 at 0.05, 47 at 0.01. The binaries with helium-rich spectra and periods between five minutes and an hour lie along the second curve and nowhere near the first, and their period is a mass gauge for a star that cannot be seen. Stars

A donor that grows as it is eaten

A white dwarf that loses mass gets bigger, so a binary in which one white dwarf feeds another runs backwards — its period lengthens as the donor shrinks. Whether the pair survives first contact at all depends less on the donor than on whether the stream finds room to form a disc, and for most pairs it does not.

 Orbital energy spent against the envelope it has to lift. The orbital energy a 0.6 M☉ companion releases by spiralling in from the onset of a common envelope down to a final separation, against that separation (curve), and the binding energy of the 0.85 M☉ envelope of a 1.2 M☉ giant with a 0.35 M☉ core, divided by the efficiency, for αλ of 0.25, 0.5, 1 (horizontal lines). Energies are in units of GM☉²/R☉. The giant fills its lobe at a separation of 118 R☉, and the companion releases nothing net until the orbit has shrunk below 34 R☉, because the envelope's mass leaves with the envelope; from there it falls until the energy released pays for the envelope, so each crossing is a final orbit — 1.29, 2.48, 4.63 R☉. The released energy grows only as one over the separation, so halving the efficiency halves the final orbit: a factor of four in αλ is a factor of four in the outcome, and there is no measurement of αλ that does not run through outcomes like these. Stars

The least known number in binary evolution

When a giant swallows its companion, the companion spirals inward until the orbital energy it has released is enough to throw the envelope off. The final orbit is set by one product of two unknowns, it moves in proportion to that product, and the merger time of the white dwarfs it leaves moves as its fourth power.

 The plane a moon's orbit turns about, between equator and ecliptic. The inclination of the Laplace plane — the plane about which a satellite's orbit precesses — to the Earth's equator, against distance from the Earth in Earth radii, for an obliquity of 23.44°. Close in, the Earth's equatorial bulge dominates and the plane is the equator; far out, the Sun dominates and it is the ecliptic, 23.44° away. The dashed curve uses today's bulge, which makes the transition at 8.4 Earth radii. The solid curve lets the bulge grow with the faster spin the Earth had when the Moon was closer — a 5.3-hour day at 5 Earth radii, from the system's angular momentum alone — which moves the transition out to about 14. The Moon today, at 60.3 Earth radii, has a Laplace plane within 0.0022° of the ecliptic, so its 5.15° inclination is entirely free: nothing forces it. 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.

 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.

 A nearly aligned population, seen in projection and in three dimensions. 60,000 transiting systems whose stars are tilted from their planets' orbits by a small random amount: each of the tilt's two components is Gaussian with a width of 5°, and its direction around the orbit normal is random. Orange: the true obliquity ψ, which is never negative and is almost never near zero — its median is 5.9°, because a tilt with two components is small only when both of them are small at once. Blue: the projected angle λ the line profile measures, which is symmetric about zero and peaked there. The projected angles look better aligned than the true ones, and they are not hiding anything: for small tilts λ is one component of a two-component tilt, so the rms true angle, 7.04°, is √2 times the rms projected angle, 5.01°, with no knowledge of any star's inclination. 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.

 A star brighter at its poles, and three paths a planet might take across it. A star rotating at 0.6 of its critical angular velocity, seen with its spin axis tipped 30° towards the observer from the plane of the sky, drawn with that axis vertical. Its outline is the Roche surface, 1.05 times as wide as it is tall in projection. The contours are the surface brightness, from von Zeipel's law with exponent 0.2 and a linear limb darkening: gravity at the equator is weaker, so the equator is cooler and its intensity there is 81 per cent of the pole's before limb darkening. Three transit chords at the same impact parameter, 0.3 polar radii, cross at projected spin–orbit angles of 0°, 45°, 90°. Only the first is symmetric about the star's axis; the others run from bright regions to dark ones or the reverse, and the light curve records which. Exoplanets

A transit that runs differently backwards

A star spinning near break-up is brighter at its poles than its equator, so a planet crossing it at an angle covers bright surface on the way in and dim surface on the way out. The light curve stops being symmetric in time, and the part that differs forwards and backwards measures the true spin–orbit angle from photometry alone — on exactly the stars where the line-profile method struggles.

 Five clocks around the black hole at the centre of the Galaxy. Timescales against distance from a black hole of 4.3·10⁶ solar masses at the centre of a cluster whose stellar mass inside r is 10⁶ (r/1 pc)^1.5 solar masses, made of stars of 1 solar mass. Ordinary two-body relaxation (orange) is slow everywhere, 8.4·10¹⁰ years at 0.01 pc. Orbits that close are nearly Keplerian and keep their orientation until they precess — through the stars' own mass farther out, through relativity close in — and over that coherence time the torques between them add rather than random-walk. That makes scalar resonant relaxation (blue) faster than two-body relaxation inside about 0.57 pc, and vector resonant relaxation (green), which turns orbital planes and is not stopped by precession within a plane, faster still: 3.5·10⁷ years at 0.1 pc, against the 6·10⁶-year age of the young stars there. Coefficients of order one are set to one; the slopes are the content. 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.

 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.

 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 companion's mass against the visible star's, for f = 6.08 M☉: never below 6.08, and 7.3 M☉ for a 0.7 M☉ star seen edge-on. The mass of the unseen companion against the mass of the visible star, both in solar masses, for a mass function of 6.08 M☉, at inclinations of 90°, 70°, 55°. Every curve starts at or above 6.08 when the visible star weighs nothing and rises as it gets heavier, because a heavier visible star moves less for the same companion. For a visible star of 0.7 solar masses — V404 Cygni's donor is a cool subgiant of about that mass — the companion is 7.3 M☉ at 90°, 8.6 M☉ at 70°, 12.4 M☉ at 55°. The shaded band is the range of neutron-star masses, up to about 2.3 M☉; no curve comes near it. A mass function above that limit is a proof that the companion is too heavy to be a neutron star, without knowing the inclination or the visible star's mass at all. Gravitation

A floor under the unseen companion

A star that swings back and forth in velocity with nothing visible beside it is orbiting something. One velocity curve gives one number about the something — the mass function — and that number is not an estimate of its mass but a floor under it, reached only if the visible star weighed nothing and the orbit were seen exactly edge-on. When the floor stands higher than any neutron star can, the companion is a black hole, whatever the inclination turns out to be.

 The path of a star with a dark companion: a 2.66 mas orbit riding on its parallax and proper motion. The track on the sky of a star like the visible member of Gaia BH1 over 3 years, in milliarcseconds, east to the left: proper motion of about 27 mas a year, a yearly parallax loop of 2.09 mas, and on top of both the star's own orbit about the barycentre it shares with an unseen companion, a 185.6-day ellipse of semi-major axis 2.66 mas, eccentricity 0.45, inclined at 126.6°. The dashed line is the path the star would follow alone. The difference between the two, a loop more than 1.3 times the parallax's size, is the orbit — a semi-major axis of 1.27 AU at the parallax's distance, which with the visible star's mass gives the companion's. Gravitation

A black hole weighed by a wobble on the sky

A velocity curve weighs an unseen companion only up to the tilt of its orbit. The same orbit followed on the sky, as a loop a few thousandths of an arcsecond across riding on a star's parallax, shows the tilt as the shape of the ellipse — and with the parallax turning the angle into astronomical units, Kepler's third law gives the companion's mass outright. That is how the nearest black holes were found: silent, unfed, and on orbits no X-ray survey could have reached.

 The pitch wheel, dumped into the field at four inclinations. Momentum stored in the pitch wheel of an Earth-pointing satellite at 500 km, over 3 days, under a steady pitch torque of 0.2 μN m, with coils of at most 0.2 A m² dumping it into the Earth's field under a law of gain 0.0005 per second. Without coils the store grows in a straight line to 51.8 mN m s. In orbits inclined at 0°, 28.5°, 51.6°, 97.4° it ends at 7.51, 1.18, 0.54, 0.40 mN m s. In the equatorial orbit the field lies close to the pitch axis, so only a small share of the stored momentum is across the field where the coils can remove it, and the store climbs until that share, times the gain, matches the torque. 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.

 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.

 Ejected before touching, or touching before ejected. The separation at which a hardening pair of 0.7-solar-mass main-sequence stars is typically ejected — where one encounter's recoil exceeds an escape speed 4 times the dispersion, at binding energy about 120 mσ² — against the cluster's velocity dispersion, both axes logarithmic, beside the separation at which the two stars fill their Roche lobes and begin to merge, 0.0082 au (1.8 solar radii). The ejection separation falls as the inverse square of the dispersion, because a hotter cluster needs a harder binary to throw it out. The lines cross at a dispersion of about 18 km/s. In clusters slower than that — open clusters and nearly all globulars — a binary is ejected while its stars are still apart. In faster ones — the most massive globulars and the nuclear star clusters at galaxies' centres — it is driven into contact first, and its two stars become one. Gravitation

Ejected from a globular, merged in a nucleus

A hard binary in a star cluster hardens until something ends it: an encounter that throws it out, or its own stars touching. Which comes first is decided by one comparison — the separation at which the cluster's escape speed is exceeded against the separation at which the stars meet — and the first shrinks as the square of the cluster's velocity dispersion. In a globular cluster binaries leave while their stars are still apart; in the nucleus of a galaxy they merge first, and the heating that holds the cluster up becomes a factory of stars more massive than they should be.

 The velocities that decide whether Jupiter keeps a visitor. A velocity diagram in Jupiter's orbital plane, in km/s, with the Sun's frame at the origin. Jupiter moves at 13.06 km/s (the arrow along the axis). The dashed circle is the escape speed from the Sun at Jupiter's distance, 18.47 km/s: a body whose heliocentric velocity ends inside it is bound. A visitor arriving with 3 km/s to spare reaches Jupiter's distance at 18.71 km/s, just outside the circle; here its velocity makes 150° with Jupiter's, so it meets Jupiter at 30.7 km/s. A flyby keeps that relative speed and only turns it, by at most 81° for a pass grazing the cloud tops — so the possible outcomes lie on the drawn arc of a circle of radius 30.7 km/s about Jupiter's velocity. The part of the arc inside the escape circle (darker) is the set of flybys that leave the visitor bound to the Sun; the slowest outcome is 17.7 km/s. Capture needs the visitor to arrive barely faster than escape, so that the turn of its velocity relative to Jupiter can take the small excess away. Orbits

Jupiter keeps only the visitors already nearly bound

An interstellar object passing Jupiter can lose enough speed to stay, but one flyby can only turn the visitor's velocity relative to the planet, never shorten it, and the turn removes at most a few kilometres a second from its speed around the Sun. So capture is common for visitors that arrive with less than a kilometre a second to spare and almost impossible for the speeds the three known visitors had. A census of the slow tail of the interstellar population, and the orbits a capture leaves — more than a third of them backwards — follow from the same arithmetic.

 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.

 Two eccentricity distributions for the planets that transit alone. Two hypotheses for the orbital eccentricities of planets that transit their stars alone, as probability densities on a logarithmic axis. Solid: a mixture of two populations, 70 per cent at σₑ = 0.04 and 30 per cent at σₑ = 0.25 — a cold majority like the planets in multi-planet systems and a hot minority. Dashed: a continuum, each planet's Rayleigh scale drawn evenly in its logarithm between 0.015 and 0.34, chosen as the continuum whose densities most resemble the mixture's. The two differ most between eccentricities of 0.1 and 0.15, where the mixture dips between its two populations and the continuum does not: 5.6 and 12.0 per cent of planets respectively. A trough there is what a separate hot population would leave, and it is the feature a measurement has to find. Exoplanets

A gap in eccentricity a density can only just see

Planets that transit alone have more eccentric orbits than planets with transiting siblings, and the excess can be fitted as a separate hot population or as a smooth spread with no gap. The two eccentricity distributions look plainly different. The stellar densities through which they are measured look almost the same, and how many planets it takes to tell them apart is set less by the star than by the transit.

 Dark companions' masses with and without a gap, as an astrometric survey would see them. The distribution of the masses of dark companions in a sample found by their stars' astrometric orbits, per unit of the logarithm of mass, for two hypotheses: neutron stars near 1.35 solar masses and black holes near 8 in equal numbers with nothing between 2 and 5 (solid), or the same with the gap filled at the density the black holes have at 5 (dashed; 16 per cent of the sample in it). Each is drawn as measured with mass errors of 10 per cent (blue) and 30 per cent (orange). With 10 per cent errors the empty gap stays empty: 0.2 per cent of the sample lands between 2.5 and 4.5 solar masses, against 10 per cent if it is filled. With 30 per cent errors the edges spill in — 4.3 per cent against 12 per cent — and the gap becomes a shallow dip that a modest sample cannot distinguish from a filled one. Gravitation

A gap that counting can test and impostors can fill

The black holes in X-ray binaries leave a gap between the heaviest neutron stars and about five solar masses, and nobody knows whether it is real or an artefact of how those masses are measured. Astrometric orbits give masses that need no light curve, and with errors of ten per cent a few dozen of them would settle it. The difficulty is not the counting. It is that pairs of white dwarfs and neutron stars, dark and close, can weigh exactly what a gap object would.

 How much a dying star stirs its companion's orbit, against their separation. The rms change in orbital eccentricity of a 0.8 M☉ companion when a 2 M☉ star becomes a 0.6 M☉ white dwarf, against the binary's separation before the loss, both logarithmic. Blue: the mass loss alone — twelve thermal-pulse episodes 500 years wide and a superwind 10,000 years wide, each adding a kick of random sign whose size falls off exponentially once it lasts longer than about a sixth of an orbit. Below 100 au the orbit is slow to every episode and the eccentricity is untouched. The pulses switch on near 229 au, and by 3,000 au the change is 0.26 and the superwind too has become sudden. Dashed: the change a velocity kick to the white dwarf of 0.05 or 0.2 km/s would make on its own, which grows as the square root of the separation. A 0.05 km/s kick outweighs the pulses out to 251 au; a 0.2 km/s kick outweighs the pulses at every separation drawn. 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 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 disc weighed by how far its stars rise. The share of the circular speed at 2.2 disc scale lengths supplied by the disc, against the vertical velocity dispersion of its stars measured at 1.5 scale lengths, for a disc of scale length 3 kpc whose rotation curve reaches 200 km/s, with scale heights of 0.30, 0.41, 0.60 kpc. The share grows linearly with the dispersion, because the surface density goes as its square and the disc's speed as the root of the surface density. A maximal disc, supplying about 85 per cent of the speed (dashed), needs a dispersion of 44 km/s at the middle thickness. A measured 24 ± 3 km/s (shaded) gives 0.41–0.52: a sub-maximal disc, with the halo supplying most of the speed even at the disc's own peak. Galaxies

A disc weighed by how far its stars climb

The rotation curve cannot say how much of a galaxy's speed belongs to its disc, because any split between disc and halo fits it equally well. The stars' motions perpendicular to the disc can, with no halo in the argument at all — and when that was done for thirty face-on spirals the discs came out light. But the vertical weighing needs the disc's thickness, which a face-on galaxy hides, and it needs the dispersion of the stars that hold the disc up, which the light measured may not belong to.

 What a deformability is made of. The two factors of the tidal deformability Λ = ⅔ k₂ C⁻⁵ for a 1.4 solar-mass neutron star, each relative to its value at 12 km, on a logarithmic scale, against the star's radius. The Love number k₂ — the same number that measures a moon's response to a planet's tide — rises by a factor of 1.6 between 10 and 14 km; the inverse fifth power of the compactness rises by 5.4. Of the change in Λ across that range, 78 per cent, on a logarithmic scale, comes from the compactness: most of what a deformability measures is how large the star is for its mass. The Love number carries the rest: how centrally concentrated the matter is, which is where the equation of state enters beyond the radius. 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.

 One orbit seen from three angles. An orbit of eccentricity 0.5 with its periastron 40° from the line of nodes, seen face on and tilted by 60° and 75°, each panel centred on its apparent ellipse. Every projection is again an ellipse — of apparent eccentricity 0.88 and 0.97 — but only the face-on view has the star (red) at one of its foci (open circles). Tilted, the star sits 0.48 and 0.57 apparent semi-major axes from the nearer focus. The displacement is what measures the tilt. Orbits

An orbit seen at an angle keeps everything but its focus

No orbit outside the solar system is seen face on. It is seen as a shadow on the sky, and the shadow of an ellipse is an ellipse, swept at equal areas in equal times, indistinguishable in shape and timing from a real orbit. Only one thing gives the tilt away. The star the companion circles projects to where the true focus projects, and that is not where the apparent ellipse has its focus.

 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.

 Two places one tidal dwarf could sit. The radial acceleration relation — observed acceleration against the acceleration the baryons alone produce, logarithmic, in m/s² — with the fitted relation for ordinary disc galaxies (solid) and the line of equality (dashed). The marks are a tidal dwarf galaxy of 10⁹ solar masses of gas and stars in an exponential disc of scale length 1.5 kpc, sampled every half kiloparsec to 6 kpc. If it carries no dark matter, as debris pulled from a disc should not, its points lie on the dashed line (open). If the relation is a law that every galaxy obeys, they lie on the curve (filled), and at the outermost radius the acceleration is 5.6 times the baryons' — a rotation speed 2.37 times higher. Every point is below a₀, where the two predictions are furthest apart. Galaxies

A galaxy born without a halo

When two galaxies collide, some of the gas flung out along their tidal tails condenses into new dwarf galaxies. Debris launched from a disc cannot take the dark halo with it, so these dwarfs should rotate at the speed their own gas and stars allow and no faster. The relation that predicts every ordinary galaxy's rotation from its baryons says they should rotate twice as fast. One rotation curve decides between a halo and a law — except that the inclination of a lumpy young disc can move the answer by more than the difference.

 A cold slab collapsing into a spiral. 800 self-gravitating sheets in phase space — position across, velocity up — at times 0, 3, 8, 120 in units where a sheet released from the edge reaches the centre in about 1.4. They start as a cold rectangle, uniform in position, with 10 per cent of the kinetic energy equilibrium needs. The slab falls in, overshoots, and the sheets, each orbiting at a period set by its own energy, wind into a spiral whose arms grow thinner and closer until they can no longer be told apart. By the last panel the system is in a steady state: a dense core and a halo of sheets that were flung out in the first collapse. Nothing in the dynamics is an encounter; every change of a sheet's energy comes from the potential changing in time. Gravitation

A relaxation that stops before it is finished

A cold, collapsing system of stars reaches virial equilibrium in a few crossing times without any star meeting another, because the potential changing in time reshuffles their energies. Lynden-Bell's theory predicted the state it should reach — the most probable one. Computed exactly, in a slab of self-gravitating sheets where every force is known, the collapse reaches equilibrium and not that state: the fluctuations that drive the mixing are what the mixing destroys, and they die before the job is done, leaving a profile and an ordering of energies that remember how the collapse began.

 A wall that leans the way the damping body says. The orbital period inside which a hot Jupiter's eccentricity is damped within 3 billion years, against the planet's mass, around a Sun-like star with planet Q′ = 10⁶, star Q′ = 10⁷ and a planetary radius of 1.1 Jupiter radii. If the planet's own tide did all the damping, the wall would fall with mass, from 5.5 days at one Jupiter mass to 3.2 at ten. If the star's tide did, it would rise, from 1.5 to 2.5 days. Together they give 5.5 and 3.4. Points are measured planets: filled where the orbit is eccentric, open where it is circular. The massive planets HAT-P-2 b, XO-3 b and WASP-14 b keep eccentric orbits at periods where lighter planets are all circular, which is the lean the planet's tide predicts and the opposite of the star's. 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.

 One parabola, two causes. Transit times of a planet on a 1.0914-day orbit over 12 years, minus the best straight-line ephemeris, with timing errors of 30 seconds — the shape a period shrinking by 29 milliseconds a year produces, a parabola 116 seconds deep. The curve is drawn once because it is the same curve for both explanations: an orbit decaying at that rate, or a star accelerating towards the observer at 2.9e-6 m/s², which shortens the light-travel time to each successive transit by the same steadily growing amount. Nothing in the transit times can tell them apart. Exoplanets

A clock that slows, or a star that falls

Transit times that curve away from a straight line can mean a planet's orbit is shrinking — the tides draining it into its star — or that the star itself is accelerating along the line of sight under a distant companion's pull. Both draw the same parabola, and the transit times cannot choose. The star's velocities can, but only for the shortest orbits: the acceleration that would fake a given decay grows as one over the period, and beyond a day or two an unseen Saturn can mimic more decay than the tides produce.

 The ground an eccentric Jupiter sweeps on its way in. The periastron and apoastron of a Jupiter delivered on an eccentric orbit, against its semi-major axis as tides shrink it at fixed angular momentum from 5 au to 0.060 au, both on logarithmic scales. The periastron stays at 0.03 au throughout; the apoastron falls from 10.0 au toward the final circular orbit, a 5.3-day hot Jupiter. Every radius between the two curves is crossed twice an orbit for the whole of the delivery. The shaded strip below the dashed line is where a small companion would have to orbit to be found transiting with the hot Jupiter — periods under 50 days, inside 0.27 au — and the migrator's orbit covers it until the semi-major axis has fallen to 0.15 au, after which the periastron alone keeps passing through its inner edge. A planet there is scattered out, thrown into the star, or swallowed; none survives a delivery that takes thousands of orbits. Exoplanets

A neighbour an eccentric arrival cannot leave

A hot Jupiter that arrived on an orbit of eccentricity near one swept every distance between its closest and farthest point, thousands of times, while tides shrank its orbit — and anything small in that range was scattered, swallowed or thrown into the star. A hot Jupiter carried in by the disc can keep its neighbours. So a hot Jupiter with a close companion came through the disc, and the share of them that have one puts a floor under how many did — a floor whose height is set less by the count than by three chances nobody has measured.

 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.

 Where to stage, if the first stage is coming home. The payload a two-stage rocket puts into low orbit, as a share of its lift-off mass, against how much of the 9.4 km/s it needs is supplied by the first stage, with exhaust speeds of 3 and 3.4 km/s and structural fractions of 0.05 and 0.06. Thrown away, the first stage is best asked for 4.1 km/s and the rocket carries 3.5%. Landed downrange — 0.02 more of its mass in legs and fins, and propellant kept for an entry and a landing burn — the best split moves down to 3.1 km/s and the payload falls to 3.0%. Flown back to the launch site, which first cancels the stage's downrange speed, it moves to 2.5 km/s and 2.7%. A recoverable first stage is asked to do less, and the upper stage does the rest. Spaceflight

A booster that flies home pays with the stage above it

A first stage that lands carries legs, fins and the propellant to bring itself back, all of which the rocket equation charges at the exponential rate it charges everything else. Recovered from a rocket designed to be thrown away, it costs a fifth of the payload. Recovered from a rocket designed around it, the toll falls by a third — because the best place to separate a stage that is coming home is earlier and slower, and the stage above it does the rest.

 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.

 A moonlet's period measures its primary's density. The orbital period of a small satellite against the radius of its orbit in units of the primary's radius, both logarithmic, for primaries of bulk density 1, 2, 3.5 g/cm³. The size of the asteroid cancels: a moonlet at five radii from a body of 2 g/cm³ takes 26 hours whether the body is a kilometre or a hundred across. At one radius the curves give the spin period at which a strengthless rubble pile flies apart — 3.3, 2.3 and 1.8 hours. Didymos–Dimorphos, at 3.1 radii and 11.92 hours, gives 2.4 g/cm³; 1999 KW4, at 3.9 radii and 17.42 hours, gives 2.1 g/cm³. Orbits

A moonlet that weighs a pile of rubble

Put Kepler's third law round an asteroid and its size cancels: a moonlet's period depends only on how many of the asteroid's radii away it orbits and on the asteroid's density. So the first asteroid moon ever seen, a mile-wide speck beside Ida, turned a picture into a density — and the density came out too low for solid rock. A fifth to a half of the asteroids weighed this way is empty space.

 How hard a hovering spacecraft pulls an asteroid. The velocity change a spacecraft hovering beside a 200-m asteroid gives it in a year by gravity alone, in millimetres per second, against the hovering distance from the asteroid's centre in asteroid radii, for spacecraft of 5 t, 20 t, 50 t. The pull depends on the spacecraft's mass and the distance, not on the asteroid's mass. A 20-tonne tractor at 1.5 radii gives 1.9 mm/s a year; at 3 radii, a quarter of that. Closer is better, and the limit on how close is set by the exhaust: inside the shaded region the engines would have to point so steeply outward to keep their 20° plumes off the surface that holding station costs more than the pull is worth. 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.

 A shift in the time and a change in the length, a quarter-cycle apart. The transit timing shift and duration change of a planet of 0.5 Jupiter masses in a 287-day orbit, carrying a moon of 10 Earth masses at 20 planetary radii, against the moon's orbital phase at the moment of transit, for two cycles. The timing swings by ±42 minutes and the duration by ±10.1 minutes. The moon's period is 12.9 days. Timing is largest when the moon is farthest along the track, and duration when the moon is moving fastest along it, so they are in quadrature; a perturbing planet would produce nothing of that kind in the duration. Exoplanets

A moon read from how late and how long

A moon pulls its planet round their common centre, and at each transit the planet is somewhere slightly different along that small orbit, moving slightly differently. So the transit arrives early or late, and it lasts longer or shorter. The two effects depend on the moon in different ways and they run a quarter of a cycle apart. Neither alone says anything about the moon. Their ratio gives its period, which a series of transits sampled once an orbit cannot otherwise reach.

 The same added mass, a different size. A galaxy's half-light radius against its mass, relative to its starting values and both on doubling axes, as it grows by repeated dry mergers with companions of mass ratio 1:1, 1:3, 1:10, 1:30, each companion's squared dispersion scaling as the square root of its mass. Equal mergers keep radius proportional to mass; small mergers approach the limit of radius as mass squared. To reach twice the mass, 1:1 gives ×2.0, 1:10 gives ×3.6. The straight dashed line is the mass-squared limit. Galaxies

A galaxy that grows faster by eating small

Two equal elliptical galaxies merge and the result has twice the mass and twice the size. A galaxy that takes in the same mass as thirty small companions ends up nearly four times the size. The difference is one line of the virial theorem, in which companions bring mass without much energy, and it may answer a puzzle thrown up by the first deep infrared surveys: the most massive galaxies at redshift two are a fifth of the size of anything like them today.

 One orbit, chosen to spell 1, 3, 1, 4, 1, 5. The height of the particle on the axis of an equal-mass binary of eccentricity 0.3, from launch at phase 210° and speed 1.3105, found by a seeded nested search for the sequence of excursions 1, 3, 1, 4, 1, 5, in whole binary periods. Between each pair of crossings the particle's excursion lasts the listed number of periods, to the nearest whole one. Moser's theorem is about every sequence of large enough integers at small enough eccentricity and is exact; this is one finite word at one finite eccentricity, found numerically to show the kind of thing the theorem says exists. 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.

 The first-order perigee rate is right for a slower Sun, and half wrong for ours. The rate at which an orbit's perigee turns, in turns per orbit, against the ratio m of the Sun's motion to the orbit's, from separate integrations of a satellite with the Moon's eccentricity under a solar tide on a circular path. The first-order rate (3/4)m² gives 0.0042 at the Moon's m of 0.0748, a perigee period of 18 years. The integrations give 0.0085, 8.9 years; the Moon's perigee takes 8.85. At m = 0.01 the integration gives 8.3e-5 against a first-order 7.5e-5. So the disagreement is not a matter of neglected forces: it depends on how large m is. 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.

 An outward path and a return path that cross, seen from a frame turning with the Moon. A trajectory in the Earth–Moon restricted three-body problem, in the frame rotating with the Moon. It leaves a 185 km parking orbit at 10.95 km/s, passes 716 km behind the Moon 3.0 days later, and returns to a perigee of -9 km without an engine firing, 6.0 days after departure. The crossing of the two legs is why it is called a figure of eight. The Moon's pull turns the direction of travel, and in a stationary frame the Earth is a much weaker influence on the far side; neither fact alone makes the geometry close. The Earth and Moon are at true scale. Spaceflight

A way home that needs no engine

A spacecraft can leave the Earth so that the Moon's gravity turns it round and sends it home with no further burn. Drawn in the frame that turns with the Moon the path is a figure of eight. It was flown to the Moon by every early Apollo mission for one reason: if the engine fails, the crew comes back. That safety is exquisitely narrow. A tenth of a metre a second at departure moves the perilune by tens of kilometres and the return by thousands, and the perilune it allows is not one that most landing sites can use.

 A binary that goes silent for part of its shrinking. The orbital period of a white dwarf of 0.8 M☉ and a donor from 0.5 M☉, in a one-zone evolution, against time. Mass transfer drives the period down fast while magnetic braking acts. At 3.17 hours the donor reaches 0.2 M☉ and loses its radiative core and with it the brake. Its radius, swollen by the fast transfer, relaxes and it falls inside its Roche lobe; transfer stops (dashed, no accretion, nothing to see). Gravitational waves alone close the orbit, until contact returns at 2.30 hours, 0.60 billion years later. The observed gap runs from 2.15 to 3.18 hours. The donor's thermal time at 0.2 M☉, set here to 3×10⁸ years, is chosen to place the upper edge, so that edge is fitted and the lower one is not. The period reaches a minimum of 68 minutes against an observed 80, a shortfall every model of this simplicity shares. Stars

A binary that goes silent for an hour of period

Binaries of a white dwarf feeding off a small star can be found at almost any orbital period down to about eighty minutes, except between about two and three hours, where they are strikingly rare. They have not stopped existing there. They have stopped shining. The moment the donor loses its radiative core, the wind's magnetic brake appears to switch off. The star, swollen by the transfer it was being driven through, shrinks back inside its own Roche lobe, and the feeding stops. Only gravitational waves remain to close the orbit, and it takes them about half a billion years.

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