Astronomy & Celestial Mechanics
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Concept

Chandrasekhar limit — where it appears

The greatest mass a cold degenerate star can support, about 1.44 solar masses, set by the exponent of the pressure law rather than by any strength. It is why a Type Ia supernova has a repeatable brightness, since the exploding object always has very nearly the same mass.

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

Radius against mass for a degenerate star. The mass–radius relation for electron-degenerate matter. More mass gives a smaller star, and the radius reaches zero at 1.46 solar masses — the Chandrasekhar limit, solved from the same expression that draws the curve rather than quoted alongside it.

A star held up by a rule about counting

A white dwarf makes no energy and does not collapse. What holds it up is not heat or pressure in any ordinary sense — it is a quantum rule forbidding two electrons from occupying the same state.

stars · Degeneracy 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 A light curve that is a decay chain. Bolometric luminosity against days since explosion, for three Type Ia models synthesising 0.3 M☉, 0.6 M☉, 0.9 M☉ of ⁵⁶Ni, from Arnett's one-zone diffusion model. Nothing in the shape is fitted: the two timescales are the laboratory half-lives of ⁵⁶Ni and ⁵⁶Co, 8.8 and 111.3 days, and the rise is those decays seen through an envelope that takes 11.3, 15.5, 18.6 days to leak. Each peaks at 14, 18, 21 days, and at maximum the luminosity equals the instantaneous deposited power to within 0.6% — Arnett's rule, and the only reason a peak brightness can be read as a mass of nickel. After maximum the curves settle to nearly straight lines on this logarithmic axis, declining at 0.0360 magnitudes a day against ⁵⁶Co's own 0.0098. The difference is gamma-ray escape: the ejecta become transparent to the very photons that are supposed to be heating them, and the tail is therefore steeper than the isotope. The one thing the tail is not is a property of the star — it is a half-life, plus a column density that is falling as t⁻².

The candle that has to be standardised

A Type Ia's light is the decay of half a solar mass of nickel-56 seen through an expanding envelope. Their peak brightnesses span six-tenths of a magnitude — and it is the width of the light curve, not anything else, that says which.

stars · Supernovae 7 solar masses of progenitor become 0.56 of white dwarf. Above: the initial–final mass relation. The steep line is what a star would leave if it kept everything; the shallow one is what it actually leaves, M_f = 0.08M_i + 0.489, fitted to white dwarfs in open clusters. Everything between 1 and 8 solar masses ends up between 0.57 and 1.13 — a range of 7 compressed by a factor of 12 — and the fraction thrown away rises from 43% at the bottom to 86% at the top. A star spends most of its mass in the last per cent of its life, at rates no theory predicts from first principles, and this relation is measured by running stellar evolution backwards: a cluster's turn-off gives the age, a white dwarf's temperature gives its cooling time, the difference gives how long its progenitor lived, and a model turns that into the mass it was born with. Below: the white dwarf mass distribution that follows, from a Salpeter initial mass function pushed through the relation above. It peaks at 0.61 solar masses, which is what every survey of white dwarfs measures. Two things make that peak and the flatness of the relation is only one of them: the unsmoothed distribution (the stepped curve) falls monotonically and is largest at its low-mass edge, because the initial mass function is steep. The edge is where it is because no star lighter than about 1.0 solar masses has had time to die yet, and the scatter of the measurement rounds that cliff into a maximum a little above it. A flat relation produces a narrow range; it takes the age of the Galaxy to produce a peak.

The mass a star does not keep

The initial–final mass relation is nearly flat, so everything from one to eight solar masses ends as a white dwarf between 0.57 and 1.13 — and the observed distribution piles up at 0.6 almost regardless of what went in. The relation is measured by running stellar evolution backwards, because nothing predicts the loss.

stars · Stellar evolution 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

Named alongside it

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

White dwarfDegeneracy pressureEquation of stateHydrostatic equilibriumMass radius relationPolytropeStandard candleArnetts ruleAsymptotic giant branchBlack holeCentral condensationCooling age

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