Astronomy & Celestial Mechanics
the sky, computed
Concept

Virial theorem — where it appears

The relation making a bound system's time-averaged kinetic energy half the magnitude of its potential energy, from which a mass follows with no orbit solved. It is an average over time that a single snapshot is not, and applying it to a system caught mid-collapse gives a mass that is wrong in a predictable direction.

Named by 16 essays across 5 fields — each of them below, with the objects they name alongside it.

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.

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.

gravitation · Virial theorem 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.

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.

gravitation · Virial theorem Mass from a velocity dispersion, across nine decades. The virial estimate M = 5 σ²R_e/G, drawn for half-light radii from 0.2 to 20 kpc, with four systems placed on it: M32 at σ = 75 km/s and R_e = 0.1 kpc, giving 6.5e+8 M☉; the Milky Way's bulge at σ = 105 km/s and R_e = 1 kpc, giving 1.3e+10 M☉; M87 at σ = 320 km/s and R_e = 7 kpc, giving 8.3e+11 M☉; the Coma cluster at σ = 1000 km/s and R_e = 1500 kpc, giving 1.7e+15 M☉. The same expression spans from a dwarf elliptical to a cluster of galaxies — nine decades of mass — because the virial theorem does not care what the moving objects are. In the last case they are whole galaxies, and the estimate exceeds everything visible in the cluster by a factor of about fifty, which is where the problem was first noticed.

A galaxy held up by disorder

An elliptical galaxy barely rotates. What holds it up against its own gravity is the randomness of its stars' motions, and the virial theorem turns that randomness into a mass by exactly the reasoning that turns a rotation speed into one.

galaxies · Velocity dispersion 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.

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.

gravitation · Binary heating Gas kept inside 3.6 to 6.8 kiloparsecs, depending only on the speed of the fall. The pressure holding a disc galaxy's gas down, against radius, with the ram pressure of the cluster gas it is falling through drawn as three levels. The restoring pressure is two pi times the gravitational constant times the product of the stellar and gaseous surface densities, both exponential with scale lengths of 3 and 5 kiloparsecs, so it falls steeply outward; the ram pressure is the intracluster density times the square of the infall speed and does not depend on radius at all. Where the level crosses the curve, the gas goes. A galaxy entering at 700 kilometres a second keeps everything inside 6.8 kiloparsecs; at 1600 it keeps only 3.6. Nothing in this touches the stars, which are not a fluid and feel no pressure at all, so the galaxy emerges with its stellar disc and its rotation curve intact and its star formation stopped from the outside in. That is the mechanism behind the most conspicuous fact about clusters: their spirals are red, gas-poor and still spiral-shaped, which no process acting on the stars could produce.

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.

galaxies · Clusters 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.

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.

stars · Hydrostatic equilibrium A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

galaxies · Galaxy spin One dimensionless number decides whether a cloud may collapse. Field strength against hydrogen column density, with three loci of constant mass-to-flux ratio. The ratio of mass to magnetic flux is conserved under flux freezing, so it cannot be changed by anything that happens during a collapse — which is what makes it a criterion rather than a description. Its critical value is the pure constant 1/(2π√G), and dividing by that gives the dimensionless λ drawn here: below λ = 1 the field can hold the cloud up for ever, however cold it gets, because gravity and the magnetic force scale the same way with radius; above it no field strength suffices. Each locus is a straight line of slope exactly one, because at fixed λ the required field is exactly proportional to the column. Zeeman measurements of dense cores put them a little above the line and their envelopes a little below it, which is the arrangement a slow leak of flux out of the centre would produce and is the observational case for ambipolar diffusion.

A threshold with no free parameter in it

Divide a cloud's mass by the magnetic flux threading it. Gravity and the magnetic force both fall as the inverse square of the radius, so the ratio cannot change during a collapse — and the critical value that separates a cloud which must collapse from one that never can is one over two pi root G, a pure constant.

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

The epoch nobody saw moves the tilt

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

cosmology · Inflation How the scatter depends on how much dispersion is added to rotation. The scatter of the baryonic Tully–Fisher relation, in dex of mass, when the same 80 turbulent model discs are measured with S = √(K Vᵣₒₜ² + σ²), against the weight K given to the rotation, for rotation measured at 1 and 2.2 disc scale lengths. Measured at 1 scale length the scatter is smallest, 0.076 dex, at K = 0.55, and is 0.076 dex at K = 0.5; measured at 2.2 scale lengths the scatter is smallest, 0.077 dex, at K = 0.23, and is 0.134 dex at K = 0.5. For an exponential disc with constant dispersion the pressure correction is exactly K = one over twice the radius in scale lengths, so the best weight depends on where the rotation is measured. The widely used K = 0.5 is the value that makes a pure rotator and a pure isothermal sphere of the same mass agree; it is also the pressure correction for rotation measured at one scale length, and not at any other.

A disc that turns slower than its mass requires

The star-forming discs of ten billion years ago were not the thin, cold, orderly discs of today. Their gas moved randomly at tens of kilometres a second, and that motion is a pressure that holds up part of each disc, so it turns more slowly than its mass alone would require. Read those rotation speeds as if rotation did all the work and the Tully–Fisher relation tilts and scatters as if galaxies had evolved, when what differs is how much of their weight is carried by disorder.

galaxies · Tully–Fisher A planet in the middle of a 0.08 M☉ star's zone spends 94 Myr too hot to keep an ocean. How long a planet spends receiving more flux than the runaway-greenhouse limit while its star contracts onto the main sequence, against stellar mass, for planets at the inner edge, in the middle, at the outer edge of the zone the star will have once it settles. The luminosity is the contraction law of a fully convective star, falling as t^(−2/3) from an age of 1 Myr until arrival; the limit is the runaway flux for the star's temperature. A planet at the inner edge is too hot for 313 Myr round a 0.08 M☉ star and 17.8 Myr round a 0.6 M☉ one; a planet in the middle is too hot for 94 Myr round a 0.08 M☉ star and 5.5 Myr round a 0.6 M☉ one; a planet at the outer edge is too hot for 39 Myr round a 0.08 M☉ star and 2.0 Myr round a 0.6 M☉ one. A runaway greenhouse is not a hot climate; it is a state in which the ocean is entirely in the atmosphere as steam, where ultraviolet light splits it and hydrogen escapes. A planet in the eventual zone of the commonest stars in the galaxy begins its life in that state for tens to hundreds of millions of years — a span comparable with the whole assembly of the Earth. The durations are measured from 1 Myr; a rocky planet may take tens of millions of years to finish forming, and one that formed later misses the start of its exposure, while the smallest stars' arrival times are somewhat short in this model, which lengthens the end of it.

Steam before the zone existed

The smallest stars take hundreds of millions of years to contract onto the main sequence, shining at many times the luminosity they will settle at. A planet in the habitable zone such a star will eventually have spends that time with its ocean in the air as steam, while starlight splits the water and the hydrogen leaves — so the zone of the commonest star in the galaxy is a place that had to survive being too hot first.

exoplanets · Habitable zone 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.

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.

gravitation · Virial theorem 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.

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.

gravitation · Impulse approximation 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.

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.

stars · Hydrostatic equilibrium 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.

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.

gravitation · Virial theorem 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.

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.

galaxies · Galaxy interactions

Named alongside it

The objects these essays reach for when they reach for this one.

Velocity dispersionCore collapseDark matter haloRelaxationBinary heatingDark matterDynamical massEquation of stateGlobular clusterHydrostatic equilibriumKinetic energyNegative heat capacity

All concepts