The collection

Every essay — page 14

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence. Essays 261–280 of 514.

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

Red, gas-poor, and still spiral-shaped

A galaxy falling into a cluster meets a headwind of hot gas at two thousand kilometres a second. Where that ram pressure exceeds the disc's own grip on its gas, the gas goes; the stars, which are not a fluid and feel no pressure, do not. The result is a spiral with no fuel, and it is the commonest kind of galaxy in a cluster.

8 figures · Clusters

Cosmology

One object, seen once, from inside — and every number in it the output of a model.

$Bubbles that meet at z = 5.3, and a scattering depth of 0.047. The fraction of the volume of the universe filled by ionised bubbles, integrated from redshift 20 down to 4.5. The equation has two terms and no others: photons escaping from young galaxies open new volume, at a rate taken from the measured cosmic star formation history with an escape fraction of 0.2; recombinations inside the bubbles close it again, on a timescale that is one over the density times the recombination coefficient times a clumping factor of 3. Early on the density is high and recombination wins almost everything; the curve is nearly flat. As the universe expands the recombination time lengthens as the cube of one plus the redshift while the star formation rate is still rising, the balance tips, and the filling factor runs to one in under half a billion years. It reaches unity at redshift 5.31, which is overlap — the moment the bubbles meet and the last neutral walls between them disappear. The same integration gives an electron-scattering optical depth of 0.0465 for the microwave background, against the 0.054 that is measured, and that agreement is the check: the two observations constrain the same history from opposite ends, one fixing when it finished and the other how long it took.$

It ends when the walls meet

Reionisation is usually described as the universe becoming transparent, which makes it sound like a change of state. It is not. Each young galaxy opens a bubble of ionised gas around itself and recombination closes it again, and what ends the epoch is geometric — the bubbles meet, and the last neutral walls between them disappear.

8 figures · Reionisation

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

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.

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.

8 figures · Galaxy interactions

Spaceflight

Celestial mechanics used forwards — where to burn, and what it costs.

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.

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.

8 figures · Gravity assist

Exoplanets

Planets nobody has seen, weighed and measured from a dip, a wobble and a delay.

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.

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.

8 figures · Planet migration

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

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.

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.

10 figures · Supernovae

An equation of state is already a star

Write down how a gas's pressure depends on its density, insist that the pressure hold the weight up, and everything else follows — the run of density, the fraction of the mass inside each radius, and, at one particular index, a mass that does not care what the radius is.

8 figures · Polytropes

A clock with no fuel in it

A white dwarf has nothing left to burn, so its brightness is a record of how long it has been cooling. The faintest ones in the Galaxy are not faint because they are small — they are as faint as anything has had time to become, and the place where they stop is a date.

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

A fluid that turns as one piece

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

9 figures · Internal rotation

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

How much a world gives

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

8 figures · Love numbers

Whether the heavy material sank

A moment of inertia is 0.4 of MR² for a uniform sphere and less for everything that has differentiated, and it is measured by watching a body wobble. Mercury's 0.346 says most of the planet is iron core; the Moon's 0.393 says almost none of it is.

8 figures · Moment of inertia

Orbits

Kepler's three laws, and the family of curves a single force allows.

The 2.3-hour spin barrier, and the small bodies that are allowed through it. Rotation period against diameter for a synthetic asteroid population, both axes logarithmic, with the horizontal lines marking where a body held together by nothing but its own gravity would fly apart. That limit is P = √(3π/Gρ) and it contains only the density: 3.30 hours at 1 gram per cubic centimetre, 2.33 hours at 2 gram per cubic centimetre, 1.91 hours at 3 gram per cubic centimetre. Size does not appear in it, which is what makes the figure's shape informative rather than obvious — a barrier that depended on size would be drawn as a slope, and a horizontal line crossing five decades of diameter is a much stronger statement. The observed population respects it. Above about two hundred metres nothing rotates faster than the 2.3-hour line, and the crowding just below that line is real: bodies pile up against a limit they cannot cross. Below two hundred metres the wall stops applying and the fast rotators appear, some of them turning in minutes. Nothing about gravity changes at that size. What changes is that a body small enough is a single coherent rock with tensile strength, while a body large enough is a pile of fragments with almost none, and the barrier is a measurement of which is which. The picture is a synthetic population drawn at random from a fixed seed rather than a catalogue, so the individual points are not asteroids; what is real is the barrier, its value, and the fact that only the smallest bodies are found beyond it.

A wall with no size in it

Spin a body held together by nothing but its own gravity, and past a certain rate it comes apart. The rate depends on density alone — 2.3 hours for the stuff asteroids are made of — and the observed population respects the limit exactly, for every body larger than a couple of hundred metres.

8 figures · Rubble piles

Galaxies

Where the unknown stops being a number and becomes a profile — and most of it is not light.

An argument about the innermost kiloparsec

Simulations of cold dark matter have produced the same halo for thirty years — a density rising inward without limit. The rotation curves of the smallest galaxies say the density flattens off. The disagreement is confined to a region a thousandth of the halo's size, and it has not been settled.

8 figures · Halo profiles

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

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.

8 figures · Hydrostatic equilibrium $Where each mode turns back: ℓ = 0 through the centre, ℓ = 300 in the outer 2 per cent. Why a set of frequencies is a depth profile and a single frequency is not. An acoustic wave travelling into a star meets a rising sound speed and is refracted back; it turns where its horizontal phase speed matches the local sound speed, which happens at c(r)/r = 2πν/√(ℓ(ℓ+1)). The horizontal axis is the angular degree on a logarithmic scale and the vertical axis is the fractional radius of that turning point, drawn at 2000, 3090, 4000 microhertz. The ordering is the content. A radial mode, ℓ = 0, has no horizontal phase speed at all and passes straight through the centre. Degrees one and two turn deep in the core. By ℓ = 300 the mode is trapped in the outer 2 per cent and knows nothing about anything below. So a frequency measured to a part in ten thousand constrains an average of the interior weighted in a way the mode itself decides, and measuring thousands of modes of different degree gives thousands of differently weighted averages — which is a solvable inverse problem, and is how the base of the convection zone was located at 0.713 of the radius rather than assumed. The sound speed here is a polytrope's rather than a tabulated solar model's, so the curve is the right shape and the wrong star in its outer tenth, where the real Sun is convective and this one is not. What the picture cannot show is the frequency dependence at fixed degree, which is weaker but not negligible: a higher-frequency mode of the same degree turns slightly deeper, and the three curves separating toward the right is that effect.$

Every note turns back at its own depth

A sound wave heading into a star is refracted by the rising sound speed and turns around before it reaches the middle. Where it turns depends on which mode it is — so a list of frequencies is not one average of the interior but thousands of differently weighted ones, and that is a solvable problem.

8 figures · Asteroseismology

Orbits

Kepler's three laws, and the family of curves a single force allows.

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

A drift rate that says what the surface is made of

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

9 figures · Non-gravitational forces

Stars

The diagram that sorted them, and the one quantity that decides a star's whole life.

$14 glitches, and the 1.5 per cent of Vela that is not slowing down. Accumulated fractional spin-up against time for a Vela-like pulsar over 40 years, in parts per million. The underlying spin-down has been removed, so a perfectly braking pulsar would be a flat line at zero. What is drawn instead is a staircase: 14 sudden jumps of a few parts per million, each rising in less than a minute and then relaxing partway back over a couple of months, leaving a permanent step behind. Nothing outside the star can deliver angular momentum on a timescale of seconds, so the source is internal, and the only internal component that could have any to give is one that has not been slowing down with the rest. That is a neutron superfluid: it carries its rotation in quantised vortices, the vortices pin to the crustal lattice and cannot migrate outward, and so the superfluid keeps the spin it had while the crust brakes past it. The reservoir grows until the pinning fails somewhere, and a glitch is the unpinning. The straight line through the staircase is the glitch activity, 0.68 parts per million per year, and it converts directly into an interior measurement: the crust cannot on average take more than the superfluid stores, so the decoupled component must hold at least 2τ_c times the activity of the star's moment of inertia, which here is 1.5 per cent. It is a lower bound rather than a value, and it is one of the very few quantitative statements about the inside of a neutron star that needs no equation of state at all. The individual glitch times and sizes here are drawn from a seeded generator rather than a catalogue; what is real is the staircase's shape, the partial healing, and the arithmetic that turns a slope into a fraction.$

The part of a star that never slowed down

A pulsar's spin decays smoothly for years and then jumps upward inside a minute. Nothing outside can deliver angular momentum that fast, so something inside has been storing it — and the rate at which the jumps accumulate is a lower bound on how much of the star is not braking with the rest.

9 figures · Pulsars

Starlight

Distance, brightness and colour: what can be measured when nothing can be visited.

$A better measurement that made the model worse: 0.9 per cent in the sound speed. The fractional difference between the Sun's sound speed as its own oscillations measure it and as a structural model predicts it, against fractional radius. Zero would be agreement. The lower curve is the model built on the solar abundances used until the mid-2000s, and it hugs the axis: a part in a thousand across most of the interior, which was for a long time the best-tested piece of stellar physics anybody had. The upper curve is the same model with the abundances re-measured using three-dimensional atmospheres and without assuming local thermodynamic equilibrium — better measurements by every methodological standard, which lowered carbon, nitrogen and oxygen by around thirty per cent. The disagreement grows to 0.9 per cent, and it is not spread through the star: it peaks at 0.683 of the radius, just beneath the base of the convection zone at 0.713. The same substitution moves the model's own convection-zone base from 0.715 to 0.729, against a seismic value known to about a thousandth. What is being tested here is not really the abundances but what converts a composition into a structure, which is the opacity: the metals whose abundances fell are exactly the ones whose bound–free absorption dominates at those temperatures, and an opacity larger by some fifteen per cent near that boundary would restore the agreement. Laboratory measurements of iron at those conditions have since come in high by about that much, which is a satisfying result to have arrived at by way of a discrepancy in the sound speed of the Sun. The curves are published inversions and model differences rather than anything computed here; what the figure adds is where they peak and by how much.$

A better measurement that made the model worse

The Sun's composition was re-measured with better atmospheres and better physics, and the carbon, nitrogen and oxygen abundances fell by about thirty per cent. The improved model then disagreed with the Sun's own oscillations by ten times as much as the model it replaced, and it still does.

9 figures · Opacity

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

9 figures · Limb darkening

Gravitation

Two bodies pulling on each other, and everything that goes wrong at three.

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

A core weighed by something that never went in

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

8 figures · Oblateness

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