Astronomy & Celestial Mechanics
the sky, computed
Concept

Polytrope — where it appears

A body whose pressure is a fixed power of its density, so that hydrostatic equilibrium reduces to one differential equation. The single exponent, written as 1 + 1/n, fixes the whole run of density: n = 1.5 is a non-relativistic degenerate gas or a convective envelope, and n = 3 is a radiative star or a relativistic one.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

From 1.667 to 1.343, and the limit is in the second number. The local slope Γ = d ln P / d ln ρ of the exact degenerate electron equation of state, against density, differentiated numerically from the drawn pressure rather than quoted. It is 1.6666 at 10⁴ kg/m³ — the 5/3 of a non-relativistic Fermi gas — and 1.3430 at 10¹² — the 4/3 of a relativistic one — falling monotonically between, and passing 1.5 at 3.63·10⁹ kg/m³. Nothing gives way at either end; the electrons simply run out of room to go faster, because they are already near c. The mass limit is in the exponent and not in the strength. Hydrostatic balance for a polytrope of index Γ gives M ∝ ρ_c^(3Γ−4)/2, so at Γ = 4/3 the exponent is zero: the mass no longer depends on the central density at all, and there is exactly one mass a fully relativistic degenerate star can have — 1.459 M☉ at μ_e = 2. Above it, squeezing harder buys pressure that rises more slowly than gravity does, and there is nothing left to stop at.

The mass a cold star cannot exceed

A degenerate star's radius falls as its mass rises, and nothing in that relation suggests a limit. Making the electrons relativistic softens the pressure law until pressure and gravity scale the same way with radius — and the radius drops out of the balance, leaving one mass and no room to argue.

stars · Degeneracy Lane–Emden solutions for n = 0, 1, 1.5, 3, 4.5, 5, and the one that has no surface. The dimensionless density θ against the dimensionless radius ξ, for polytropic indices 0, 1, 1.5, 3, 4.5, 5. Each curve is the whole structure of a star whose pressure is K times its density to the power 1 + 1/n: the equation of state and hydrostatic equilibrium leave one second-order differential equation, and this is its solution. Every curve starts at θ = 1 with zero slope, because the density is greatest at the centre and has no cusp there. What separates them is where they end. At n = 0 the density is uniform and the surface is at ξ₁ = 2.4495; by n = 3 it has moved out to 6.8968 and the central density is 54.2 times the mean. At n = 5 the curve reaches zero only at infinity — a configuration of infinite radius and, remarkably, finite mass — and every index above it has neither. The three curves that have closed forms, n = 0, 1 and 5, are drawn from the same numerical integration as the rest and agree with those forms to better than two parts in a million, which is what licenses reading the others off the picture. What the figure cannot show is the scale: ξ is radius divided by a length that depends on the central density and on K, so two stars of the same index and wildly different sizes have the same curve here.

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.

stars · Polytropes The Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly.

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.

gravitation · Love numbers 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 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.

gravitation · Oblateness The inflation threshold at 2·10⁵ W m⁻², and the 0.69 R_J above it. Radius against the starlight received, for a population of Jupiter-mass planets. The horizontal line is what a structural model gives for a cold, old, Jupiter-mass ball of hydrogen and helium: 1.06 Jupiter radii, and it hardly depends on mass at all in this range, because degeneracy is beginning to set in and the mass–radius relation is flattening toward its turnover. Planets receiving less than about 2·10⁵ watts per square metre sit on that line, with a median of 1.06, which is the control the rest of the figure depends on: the models are not wrong in general. Above the threshold the radii climb, reaching a median of 1.75 — half again the size a cold planet of the same mass can be — and the onset is sharp enough to be called a threshold rather than a trend. Starlight by itself will not do this. Irradiation is absorbed high in the atmosphere and re-emitted from there; it slows the escape of heat from below, which delays contraction, but it cannot deposit energy beneath the radiative–convective boundary, and it is the interior entropy that sets the radius. So the excess is evidence for a mechanism that carries roughly half a per cent of the incident flux down to pressures of tens of bars — ohmic dissipation of currents driven through a partly ionised atmosphere, breaking gravity waves, and tidally forced turbulence are the candidates, and the threshold is the number each of them has to reproduce. The points are a synthetic population from a seeded generator, not a catalogue; what is real is the threshold, the size of the excess, and the fact that the un-irradiated planets sit exactly where they should.

A radius no cold planet is allowed

A Jupiter-mass ball of hydrogen has a maximum size, and it is about 1.06 Jupiter radii however old or young it is. Hundreds of hot Jupiters are half again that, and the excess switches on sharply above a threshold in the starlight they receive — which means something is putting energy in deep.

exoplanets · Planet composition The most pressure a core can supply, and what its envelope asks for. The pressure at the boundary of a star's isothermal core, on a logarithmic scale, against the fraction of the star's mass the core holds. The coloured curves are the most pressure a core of that mass can supply at its edge — the maximum of the isothermal sphere, 1.40 c⁸/(G³M²) for a core of mass M — for cores of mean molecular weight 0.6, 1.0, 4/3; the flat grey line is the pressure the envelope, of molecular weight 0.6, exerts on the core at the same temperature. The core's ceiling falls as the square of its mass, and it can hold the envelope up only while it lies above the line. The curves cross it at core fractions of 0.370, 0.133, 0.075: a core of the same molecular weight as its envelope could hold 37 per cent of the star, and a fully ionised helium core, μ = 4/3, under this envelope only 7.5 per cent. The limit exists because the core is heavier per particle than the gas it has to hold up.

A core too heavy for the gas around it

When a star's centre runs out of hydrogen, the helium left behind stops making energy, settles to one temperature, and becomes the one member of the polytrope family with no surface of its own. It can hold up the envelope around it only while it is less than about a tenth of the star. Past that there is no balance to be had, and the star crosses to the giant branch in a few per cent of its lifetime — which is why that part of the diagram is empty.

stars · Polytropes 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 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.

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.

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

A star tipped over at a thousand times its horizon

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

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

A soft centre the rest of the core outvotes

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

stars · Polytropes The gradient radiation would need, as a young star shrinks. The radiative temperature gradient at the centre of a contracting, fully convective pre-main-sequence star — the gradient radiation alone would need to carry the luminosity — against the star's radius, which shrinks to the right, for 0.3, 0.5, 1, 2 solar masses, logarithmic. The centre stays convective while the gradient exceeds the adiabatic 0.4 (dashed). As the star shrinks its centre gets hotter and denser, Kramers opacity, ∝ ρT⁻³·⁵, falls, and the gradient falls as the radius to the power 2.5; where it crosses 0.4 a radiative core appears. 0.3 M☉: not before the main sequence; 0.5 M☉: not before the main sequence; 1 M☉: at 1.94 R☉; 2 M☉: at 7.62 R☉. Dots: each mass's main-sequence radius, where contraction stops. A Sun-like star leaves its convective line at 1.9 solar radii; the lightest never do.

Where a shrinking star stops boiling at its centre

A young star descends the Hertzsprung–Russell diagram at almost fixed temperature, convective from centre to surface, until something at its centre changes and it turns sharply left toward the main sequence. The something is the opacity. As the star shrinks its centre grows hotter and denser, the Kramers opacity there falls, and radiation becomes able to carry the luminosity without the steep gradient that drives convection. A Sun-like star crosses that line at about twice its present radius, two million years old; a star of a third of the Sun's mass never crosses it at all.

starlight · Opacity

Named alongside it

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

Degeneracy pressureHydrostatic equilibriumEquation of stateRadiation pressureCentral condensationChandrasekhar limitKelvin helmholtz timescaleMass radius relationAdiabatic indexDynamical instabilityEddington limitGeneral relativity

All concepts