The mass no cold matter can hold up

A gas of fermions has a pressure that does not go away when the temperature does, because the exclusion principle forces the particles into states with momentum whether they are hot or not. That pressure is what holds up a star that has stopped burning, and following it to its limit produces one of the sharpest numbers in astrophysics.

Why heavier means smaller #

For a non-relativistic degenerate gas the pressure goes as the five-thirds power of the density. Put that into hydrostatic balance against gravity and the mass–radius relation comes out as

$$R \propto M^{-1/3}$$

which is not a small departure from ordinary behaviour but a reversal of it. Adding mass to a white dwarf makes it smaller.

The reason is a competition of powers. Compressing the star raises the density, and the pressure gained goes as $\rho^{5/3}$ while the gravitational weight to be supported goes as $\rho^{4/3}$ — so at low density the pressure wins the competition easily and the star settles at a large radius, and as the mass rises the balance is struck at ever smaller radii.

That single relation already produces a striking number. A white dwarf of the Sun’s mass has a radius near the Earth’s, and the figure’s curve crosses that line just under one solar mass. A body containing a Sun’s worth of matter in a planet’s volume has a mean density of a tonne per cubic centimetre, which is what makes the first observation of one — Sirius B, in 1915 — such an awkward result.

Where the limit comes from #

The five-thirds law assumes the electrons are slow. As the star is compressed they are not: the Fermi momentum rises as $\rho^{1/3}$, and once it approaches $m_e c$ the energy of an electron is $pc$ rather than $p^2/2m$, and the pressure law changes.

For an ultra-relativistic degenerate gas $P \propto \rho^{4/3}$, and that exponent is special. A self-gravitating ball with $P = K\rho^{4/3}$ is an $n = 3$ polytrope, and for $n = 3$ the mass that hydrostatic balance permits is

$$M = 4\pi\left(\frac{K}{\pi G}\right)^{3/2}\times 2.01824$$

with no radius in it. The radius has cancelled out entirely. There is one mass such a star can have, and it does not depend on how big the star is.

The number 2.01824 is the Lane–Emden constant $\xi_1^2|\theta'(\xi_1)|$ for $n=3$, and the figure integrates the Lane–Emden equation to get it rather than quoting it, obtaining 2.0182. With $K$ computed from the fundamental constants and two nucleons per electron, the limiting mass is 1.456 solar masses, and the full structure integration approaches 1.452.

That is why the limit exists and why it is a limit rather than a crossing. Below the limiting mass the electrons are only partly relativistic, the effective exponent is between the two, and there is a finite radius at which balance holds. As the mass rises the star shrinks, the electrons become more relativistic, the exponent falls toward four-thirds, and the radius at which balance holds runs to zero. Above the limit there is no radius at all.

The powers, and why four-thirds is the boundary #

The general statement is worth extracting because it applies far beyond stars.

For a polytrope $P = K\rho^\gamma$, dimensional analysis of hydrostatic balance gives $M \propto R^{(3\gamma-4)/(\gamma-1)}$ up to constants. The exponent vanishes when $\gamma = 4/3$: the mass becomes independent of the radius, which is the case above.

For $\gamma > 4/3$ the mass rises with radius, and a star squeezed slightly gains more pressure than it needs — it is stable. For $\gamma < 4/3$ it loses more than it gains, and any compression runs away. So four-thirds is the boundary of stability for a self-gravitating ball, and a star whose equation of state softens to that exponent is exactly marginal.

That marginality is the reason the limit is so sharp and the reason small corrections matter enormously near it. Anything that softens the equation of state slightly below four-thirds — electron capture onto nuclei, general-relativistic corrections, the finite temperature of a real star — destabilises it, and the collapse begins somewhat below 1.456 rather than at it.

A self-gravitating gas has a negative heat capacity: losing energy raises its temperature, so a star that radiates gets hotter and radiates faster. The white dwarf is the state reached when that runaway is stopped — degeneracy pressure does not depend on temperature, so the star can cool without shrinking further, and the runaway ends. That is why a white dwarf is stable in a way an ordinary star is not.

What the star is, physically #

It is worth stating what has happened to the matter, because “degenerate” is doing a lot of work.

The electrons are not bound to particular nuclei. Their wavefunctions overlap across the whole star, the exclusion principle fills the momentum states from the bottom up, and the Fermi energy near the limit is several times $m_ec^2$ — a few million electron-volts, or tens of billions of kelvin in temperature units. Against that, the star’s actual thermal energy is negligible even at ten million kelvin.

That is the sense in which the star is cold: not that it is cool to the touch, but that its thermal energy is irrelevant to its structure. Which is what makes the calculation possible at all — the structure has no temperature in it, so it can be integrated once and for all rather than solved along with an energy-transport equation.

It also explains the star’s most peculiar property. A normal star that loses energy contracts and heats up, because gravity gives back twice what radiation removes. A white dwarf loses energy and does not contract, because its pressure does not care about its temperature. It simply cools, at fixed radius, for ever — a cooling curve that is now used as a clock, since the oldest and coolest white dwarfs in a cluster date it.

How the curve was obtained #

The mass–radius curve is not a formula evaluated; it is a set of stars integrated, and the distinction is what makes the crossover between the two exponents come out rather than being put in.

The exact equation of state of a degenerate electron gas is parameterised by $x = p_F/m_ec$, with

$$\rho = Bx^3, \qquad P = A\left[x(2x^2-3)\sqrt{1+x^2} + 3\,\mathrm{arcsinh}\,x\right]$$

which is $\propto x^5$ for small $x$ and $\propto x^4$ for large — the five-thirds and four-thirds laws, in one expression, with no case analysis.

Given a central $x$, the pair $dP/dr = -Gm\rho/r^2$ and $dm/dr = 4\pi r^2\rho$ is integrated outward until the density falls to nothing. That radius is the star’s, and that mass is the star’s. Doing it for eleven central densities spanning several decades gives the eleven points on the curve, and the asymptote is where they pile up.

Nothing about the limit is in the integration. What comes out is a sequence of stars, each of which was found by solving a differential equation, and the fact that their masses converge on 1.452 is a result rather than an input — checked here against 1.456 from the polytrope formula, which is a completely separate calculation.

One numerical detail is worth recording because it produced a spectacularly wrong answer first. The step size is chosen so that no step changes $x$ by more than half a per cent, which is the right rule everywhere except at the very start: there $m = 0$, so the density gradient is zero, and a rule based on the gradient permits an arbitrarily large step. A first attempt with no radial cap took a first step of a megametre at the central density and produced a star of 289 solar masses. A step rule that is unconstrained wherever the derivative vanishes is unconstrained exactly where the integration begins, which is a trap worth knowing about in any shooting problem.

The number and the composition #

The limiting mass depends on one property of the matter besides the constants: the number of nucleons per electron, $\mu_e$.

Carbon and oxygen, which is what a white dwarf from an ordinary star is made of, have $\mu_e = 2$, giving 1.456 solar masses. Hydrogen would have $\mu_e = 1$ and a limit four times larger, but a hydrogen white dwarf would ignite long before reaching it. Iron, with $\mu_e = 2.15$, gives 1.26.

The dependence is $\mu_e^{-2}$, which is steep, and it means the limit is a property of the composition rather than a universal number. Quoting “1.4 solar masses” without saying what it is made of is the standard shorthand and is correct for the case that occurs.

That sharpness has an observational payoff. A white dwarf accreting from a companion approaches the same limiting mass whatever the details, ignites its carbon, and detonates — and because the mass at detonation is always about the same, the explosion always releases about the same energy. That is why type Ia supernovae are standard candles, why they can be used to measure distances across the universe, and why the accelerating expansion was found with them.

A star’s fusion can reach iron and no further, because the binding-energy curve peaks there. A white dwarf is what is left when that fuel is gone and no further release is available — so its composition is carbon and oxygen for most stars, and the limit computed here depends on that through the number of electrons per nucleon.

The argument that took twenty years to be accepted #

The physics above was done by Chandrasekhar in 1930, on a boat from Madras to England, when he was nineteen. The reception it got is worth recording, because the objection is instructive rather than merely unfortunate.

Eddington — the most authoritative astrophysicist alive, and the man who had established that starlight bends by twice the Newtonian amount — attacked the result publicly and repeatedly. His objection was not to the arithmetic, which he did not dispute, but to the conclusion: he held that a star above the limit must find some way to avoid collapsing, and that a theory predicting otherwise was reducing itself to absurdity.

The specific technical complaint was that combining special relativity with non-relativistic quantum statistics was illegitimate. That is a real question and it has since been answered: the relativistic treatment gives the same $\rho^{4/3}$ law.

What is worth taking from the episode is not that a great physicist was wrong but why the argument was so hard to settle. Both sides agreed on the equations and disagreed about whether a mathematical result with an unacceptable physical consequence should be believed. There was no experiment available; the first neutron star was found in 1967 and the first stellar-mass black hole candidate in 1971. Chandrasekhar received the Nobel Prize in 1983, fifty-three years after the calculation.

What is above the limit #

The star does not stop being a star; it stops being a white dwarf.

At densities above about $10^{10}$ kilograms per cubic metre it becomes energetically favourable for electrons to be captured onto protons, producing neutrons and neutrinos. That removes electrons, which removes the pressure, which accelerates the collapse — a positive feedback that makes the transition abrupt.

What halts it is neutron degeneracy at nuclear density, some $10^{17}$ kilograms per cubic metre, and the same argument applies with neutrons in place of electrons. The neutron star that results has its own limiting mass, around two solar masses, though the number is much less certain because it depends on the equation of state of matter at nuclear density, which is not known to better than a factor.

Above that, nothing known holds. The general-relativistic version of the stability argument — the Tolman–Oppenheimer–Volkoff equation — has no solution, and the collapse continues.

The white dwarf sits far above the line at which an object would have a horizon, and the neutron star not far above it. That is worth knowing because it says which limit binds: a white dwarf exceeding its mass limit collapses to a neutron star rather than to a hole, and a neutron star exceeding its own has nowhere left to go — the horizon is the next thing down.

The low-mass end of the same sequence is the radius above which a body cannot help being round, where its own gravity beats the strength of its material. That and the Chandrasekhar limit are the two ends of one question — what holds an object up against its own weight — with rock at the bottom and degenerate electrons at the top, and nothing else in between that matters.

The same argument at other scales #

The competition of powers that produces the limit is not confined to stars, and recognising it elsewhere is the best test of whether it has been understood.

A planet’s size is set by the same balance one exponent up. Cold matter at low pressure is held apart by electrostatic forces rather than by degeneracy, giving a nearly constant density, so a planet’s radius grows as the cube root of its mass. As the mass rises, degeneracy takes over, the exponent changes, and the radius passes through a maximum — at about the mass of Jupiter, which is why Jupiter and a brown dwarf several times more massive are much the same size. The size at which a body becomes round is the bottom end of the same sequence.

A metal’s electrons supply a pressure of the same kind and nothing gravitational balances it. The Fermi pressure in copper is about $4\times10^{10}$ pascals, which is what the lattice’s electrostatic attraction has to contain, and the balance sets the metal’s density. It is the same arithmetic with a Coulomb term where the gravitational one is.

And a nucleus is the same problem again, with the strong force supplying the attraction and nucleon degeneracy the pressure — which is why the binding energy per nucleon has the shape it does and why nuclei have a nearly constant density regardless of size.

What differs between the four cases is what supplies the attraction and over what range. What is the same is that a cold fermion gas has a pressure fixed by its density alone, that the pressure’s exponent decides stability, and that four-thirds is the boundary in every case where the attraction is gravity.

The Chandrasekhar limit is a much less exotic version of an equation of state running out. It has been reliable across a wide range and stops being so where the electrons become relativistic, and the failure is complete rather than gradual — the exponent changes from 5/3 to 4/3 and the pressure stops rising fast enough to win. Every such limit in physics has that shape.

A stellar mass written in four constants #

The limit has a property that is easy to walk past: no astronomy went into it. The derivation used the exclusion principle, special relativity, Newtonian gravity and the mass of a nucleon, and every one of those is available without ever looking at a star.

Follow the dimensions through and the mass that comes out is

$$M \sim \frac{1}{m_H^2}\left(\frac{\hbar c}{G}\right)^{3/2},$$

which is the Planck mass cubed divided by the square of the proton mass. Evaluate it and the answer is $3.7\times10^{30}$ kilograms — 1.85 times the mass of the Sun, before any of the dimensionless factors that the careful integration supplies.

Read the other way it is a count: the number of nucleons such a star can hold is about $(M_{\text{Pl}}/m_p)^3 \approx 2\times10^{57}$, a pure number built out of how weak gravity is compared with the strength that binds a nucleus. The largest cold object the universe permits contains that many particles because gravity is $10^{38}$ times weaker than the strong force, and for no other reason.

That is an unusually direct instance of something this collection keeps meeting. A quantity that seems to belong to a particular kind of object — the heaviest possible dead star — turns out to be a combination of constants that were measured in laboratories, and the object is where the combination happens to be displayed.

Almost no white dwarf is anywhere near it #

The number 1.4 is quoted so often that it is worth saying plainly: the overwhelming majority of white dwarfs are nowhere close to it, and the reason is not the physics on this page.

The observed distribution of white-dwarf masses peaks sharply near 0.6 solar masses, and it is narrow. Sirius B, one of the heaviest well-measured examples, is about 1.0. What sets those numbers is not degeneracy at all but mass loss: a star of several solar masses spends its last phase shedding its envelope in a wind, and what is left as a white dwarf is the core, which is a small and weakly-varying fraction of what went in. A star would have to begin at around eight solar masses to leave a remnant approaching the limit, and above that it does not make a white dwarf at all — it burns further and ends another way.

So the ceiling is real and mostly untouched. A single star cannot walk up to it. The white dwarfs that do reach it are the ones with a companion to steal from, which is what makes the limit an event rather than a boundary on a graph: the mass rises slowly, from outside, until an object that has been stable for a billion years stops being able to be.

What the picture cannot show #

The star is treated as non-rotating and unmagnetised. Rotation supports additional mass, and rapidly rotating white dwarfs above the limit are possible in principle; magnetic fields do the same to a lesser degree. Both are neglected in the classic result and both are argued to matter in some supernova progenitors.

General relativity is absent from the integration. Newtonian hydrostatic balance is used, and near the limit the star’s compactness is enough that the relativistic correction is not negligible. It lowers the limiting mass slightly and, more importantly, makes the marginal case unstable rather than neutrally stable.

The equation of state is that of a perfect degenerate gas. Real matter at those densities has Coulomb corrections between the ions, which lower the pressure by a few per cent, and at lower densities the ions crystallise — a white dwarf’s interior is expected to be a lattice, with a latent heat that shows up in the cooling curve.

The exact equation of state is for a free gas. The electrons in a real white dwarf sit in the field of the ions, and at the lower densities the interaction is not negligible: it lowers the pressure, which lowers the radius at a given mass. The correction is a few per cent at a solar mass and smaller near the limit, where the electron energies dwarf everything.

And the limit is approached and not reached. The curve’s asymptote is where a model runs out rather than where a star does, and the curve’s asymptote is a mathematical feature of an idealised equation of state. Real stars begin to collapse a little below it for the reasons above, and the observed masses of white dwarfs cluster around 0.6 solar masses with very few near the limit at all — the limit governs what happens to the few that get there rather than describing the population.

The ladder from here #

Later rungs on this anchor: the Lane–Emden equation and the polytrope family, of which the two cases here are members; the Tolman–Oppenheimer–Volkoff equation and the neutron star’s own limit; the cooling curve of a white dwarf and its use as a clock for the age of the galaxy’s disc; electron capture and the physics of the collapse itself; and the type Ia supernova, where the sharpness of the limit becomes a distance ladder.

The neighbouring ladders are degeneracy pressure, which is the support this essay pushes to its limit, and the negative heat capacity of a gas ball, which is the runaway a degenerate star is immune to. The exclusion principle is where the pressure comes from in the first place.