The mass a cold star cannot exceed

A white dwarf is held up by a counting rule: the exclusion principle forbids two electrons from occupying the same state, so squeezing the gas forces electrons into higher-momentum states and the resulting pressure has nothing to do with the temperature. A cold star can be perfectly stable.

The relation that follows is already strange. Non-relativistic degenerate pressure goes as $\rho^{5/3}$, and combining that with hydrostatic balance gives

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

More mass makes a smaller star. Nothing in ordinary experience behaves that way. And nothing in that expression suggests a limit: it is a smooth power law, defined for every positive mass, and it says a sufficiently heavy white dwarf is merely a very small one.

The exponent that removes the radius #

Degeneracy pressure comes from momentum, and there are two regimes.

At low density the electrons are non-relativistic, so the momentum $p$ of an electron corresponds to kinetic energy $p^2/2m$ and speed $p/m$. The pressure is roughly (number density) × (momentum) × (speed), which gives $P \propto n \cdot p \cdot p/m \propto n^{5/3}$, since the Fermi momentum goes as $n^{1/3}$.

At high density the electrons are relativistic, and their speed can no longer rise: it is $c$, whatever their momentum. So the pressure becomes (number density) × (momentum) × $c \propto n^{4/3}$.

That change of exponent is everything. Consider a self-gravitating sphere of mass $M$ and radius $R$. Gravity’s characteristic pressure scales as $GM^2/R^4$. The gas’s pressure at mean density $\rho \sim M/R^3$ scales as $\rho^\Gamma \sim M^\Gamma/R^{3\Gamma}$. Balance requires

$$\frac{M^\Gamma}{R^{3\Gamma}} \sim \frac{M^2}{R^4}.$$

At $\Gamma = 5/3$: $M^{5/3}/R^5 \sim M^2/R^4$, giving $R \sim M^{-1/3}$. A relation between mass and radius — pick a mass, get a radius.

At $\Gamma = 4/3$: $M^{4/3}/R^4 \sim M^2/R^4$, and the radius cancels from both sides. What is left is $M^{4/3} \sim M^2$, which is not a relation between mass and radius at all. It is satisfied by exactly one value of $M$ and by no other, at any radius whatever.

That one value is the Chandrasekhar limit. For a mean molecular weight per electron $\mu_e = 2$ — carbon, oxygen, anything with equal numbers of protons and neutrons — it is $1.44\ M_\odot$.

Not a strength, a scaling #

The distinction the essay title turns on is worth stating flatly, because the usual summary gets it wrong.

A bridge fails when the load exceeds the strength of the steel. That is a limit of the first kind: a material property is exceeded, and using stronger steel raises the limit.

Degeneracy pressure has no such property. There is no maximum pressure a degenerate gas can supply — the pressure rises without bound as the density rises, for ever. Squeezing a $1.5\ M_\odot$ white dwarf produces more pressure at every stage, and it still collapses.

What fails is not the magnitude of the pressure but its rate of increase. Compress the star by a factor $x$ in radius. Gravity’s demand rises as $x^4$. At $\Gamma = 5/3$ the pressure supplied rises as $x^5$, which outruns the demand, so the compression is resisted and the star finds a new equilibrium. At $\Gamma = 4/3$ the supply rises as $x^4$ — exactly the same power — so the two track each other and the star is indifferent to its own size. Any further softening, from any cause, and the demand wins at every radius.

A limit of that kind cannot be raised by any material being stronger. It is set by the exponent, and the exponent is set by the fact that electrons cannot exceed the speed of light.

The radius that falls as the mass rises #

The inverted mass–radius relation deserves a moment on its own, because it is the first strange thing in the subject and it is strange for a reason that is not the limit.

An ordinary star is supported by thermal pressure, which depends on temperature, which depends on the rate at which energy is being produced. Adding mass to it adds fuel, raises the central temperature, raises the pressure, and the star gets bigger. Mass and radius rise together, which is what everything in ordinary experience does.

A degenerate star has no such loop. Its pressure comes from the density alone, and adding mass adds weight without adding anything to push back with except the compression that weight itself produces. The star must therefore shrink until the higher density supplies the higher pressure — and the arithmetic of how much it must shrink gives $R \propto M^{-1/3}$.

The numbers are worth stating because they are outside experience. Sirius B is 1.02 solar masses in a sphere 84 per cent of the Earth’s radius, so its mean density is about $2\times10^9$ kg per cubic metre: a teaspoon of it weighs ten tonnes. A dwarf twice its mass would be smaller still, by a factor of $2^{1/3}$ in radius and a factor of four in density — and the extrapolation is exactly what the essay is about, because it runs into the limit long before the relation stops being interesting.

The relation is also a measurement. Its coefficient depends on $\mu_e$, and $\mu_e$ depends on composition, so a white dwarf whose mass and radius are both known independently — from a binary orbit and a gravitational redshift, say — constrains what it is made of. That is one of very few ways to say anything at all about the interior of an object with no observable surface features and a spectrum consisting almost entirely of pressure-broadened hydrogen.

What was actually measured #

Three things, and they are unusually well separated from the theory.

White dwarf masses. Measured by gravitational redshift for the brightest examples — Sirius B’s spectral lines are shifted by 80 km/s equivalent, which is a direct read of $GM/Rc^2$ — and by orbital dynamics where a dwarf is in a binary. The Sirius B measurement gives $1.018\ M_\odot$ at a radius of $0.0084\ R_\odot$, which is 84 per cent of the Earth’s radius at slightly more than the Sun’s mass. Masses like that are known directly only in binaries, and Sirius B is one.

The mass distribution. Some tens of thousands of white dwarf masses are now known from spectroscopic fitting of the Balmer line profiles, whose widths depend on the surface gravity. The distribution peaks sharply at $0.6\ M_\odot$ and has a tail extending to about 1.3, and there is nothing above 1.4. That empty region is the observational statement of the limit, and it is an absence rather than a measurement, which is the kind of evidence this subject usually has to make do with.

The luminosity of type Ia supernovae. This is the third measurement and it is the consequential one. A carbon–oxygen white dwarf accreting from a companion approaches the limit; at about $1.38\ M_\odot$ the core reaches carbon ignition, and because the star is degenerate there is no thermostat — the same absence that makes the helium flash a flash — so the burning runs away and unbinds the star entirely.

Every such explosion therefore involves nearly the same mass of fuel, and produces nearly the same amount of ⁵⁶Ni, whose decay powers the light curve — a light curve shaped by radioactivity rather than by pulsation, but standardised by the same trick of relating a shape to a luminosity. The peak luminosities cluster within a factor of about three, and a further correlation between peak brightness and the rate of decline tightens them to about 10 per cent.

The number, built out of constants #

The scaling argument says a unique mass exists and does not say what it is. Carrying the constants through is worth doing once, because the answer contains no astronomy whatever.

Solving the balance properly means solving the Lane–Emden equation for the polytrope with $\Gamma = 4/3$, which is index $n = 3$, and its dimensionless mass integral is a pure number: 2.018. What multiplies that number is

$$M_{\rm Ch} \;\simeq\; \frac{2.018}{4\pi}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2},$$

and every symbol in it is a constant of nature. Planck’s constant and the speed of light come from the electrons being relativistic and quantum-mechanical; the gravitational constant comes from the weight they are holding up; the proton mass comes from the nucleons whose weight it is. There is no stellar physics in the expression — no opacity, no nuclear cross-section, no composition beyond the ratio of nucleons to electrons.

Grouping it differently makes the point sharper. $(\hbar c/G)^{1/2}$ is the Planck mass, $2.18\times10^{-8}$ kg, so

$$M_{\rm Ch} \sim \frac{M_{\rm Pl}^3}{m_H^2}.$$

A limit on the mass of a star is the cube of the Planck mass divided by the square of the proton mass, times a number of order one. That combination is about $1.8\times10^{30}$ kg, and the Sun is $2.0\times10^{30}$.

The two exponents that bracket the argument are worth seeing on one axis, over the full range of densities a white dwarf spans.

That is the reason the limiting mass is approached asymptotically rather than crossed. A star at 1.3 solar masses has a slope somewhere near 1.4 and a finite radius; adding mass steepens the compression, lowers the slope further, and shrinks the radius faster than the added mass would suggest. The radius reaches zero only in the limit, and in practice something else happens first — electron capture, or carbon ignition, or the onset of general-relativistic instability — at a mass a per cent or two below the formal limit.

Which of those intervenes depends on the composition, and the difference matters observationally. A carbon–oxygen white dwarf approaching the limit ignites its carbon and detonates; an oxygen–neon–magnesium one loses its electrons to capture on neon and collapses to a neutron star instead. Two objects at the same mass, made of matter with the same $\mu_e$ and therefore the same formal limit, end in a type Ia supernova and in a core-collapse event respectively. The limit says when something must happen and says nothing at all about what. That is worth holding onto, because the limit is usually introduced as the explanation of the type Ia supernova and it is nothing of the kind. It explains why there is a characteristic mass at which something happens, which is why type Ia supernovae are standardisable at all; the explosion mechanism itself is a separate and much less settled problem.

Why an ordinary star is near the same number #

That near-coincidence is not one, and it is the most useful thing the previous section produces.

The same combination of constants sets the mass scale of stars in general, and the reason is that the argument above is not specific to degeneracy. Any object supported by a pressure that goes as the four-thirds power of density has this mass and no other, and a gas of relativistic particles of any kind — photons included — has exactly that exponent. A star massive enough to be supported mainly by radiation pressure is therefore near the same mass, for the same reason, with the same constants.

So the upper end of the stellar mass range is set by the same expression as the white-dwarf limit, differing only in the numerical factor. And the lower end, the hydrogen-burning threshold at 0.08 solar masses, is a fixed fraction of it, because that threshold is where degeneracy halts contraction before ignition — the same physics again, evaluated at a different place.

Stars span a factor of a thousand in mass, and the whole range sits within a couple of orders of magnitude of $M_{\rm Pl}^3/m_H^2$. The reason a star is the size it is, rather than a thousand times larger or smaller, is a statement about the ratio of gravitational to nuclear strength, and it is the same statement that decides how heavy a white dwarf may be.

That also explains why the limit is so nearly the same for every white dwarf ever measured. Its one free parameter, $\mu_e$, ranges over a factor of about 1.08 across every plausible composition — from 2 for anything with equal nucleons and electrons to 2.15 for iron — so the limit itself varies by 14 per cent at the very most, and by nothing at all across the carbon-oxygen interiors that make up almost all of them. A quantity assembled from four fundamental constants and one ratio that barely moves is as close to a universal number as astronomy has, and that is the property the standard candle is built on.

Where the model stops #

The limit is not the ignition mass. Carbon ignition happens at about $1.38\ M_\odot$, before the star reaches 1.44, because the core heats as it contracts. So no white dwarf ever actually reaches the Chandrasekhar mass by accretion — it explodes first. The limit is the asymptote the physics points at, and the observable event happens just short of it.

Rotation raises it. A rapidly rotating white dwarf is supported partly by centrifugal force, and differentially rotating models are stable well above 1.44 — up to about 2 solar masses for extreme cases. The “super-Chandrasekhar” supernovae, of which a handful are known with inferred ejected masses near 2 $M_\odot$, are usually explained this way or as mergers of two dwarfs.

Electron capture intervenes. At densities above about $10^{13}$ kg/m³ it becomes energetically favourable for electrons to be absorbed by nuclei, which removes the very particles supplying the pressure. That is a second softening on top of the relativistic one, and for oxygen–neon–magnesium cores it triggers collapse rather than explosion — a different fate for a star that reached the same mass by a different route.

General relativity subtracts a little. The Newtonian analysis above ignores the fact that pressure itself gravitates. Including it destabilises the star slightly earlier, lowering the limit by a fraction of a per cent for a carbon dwarf and by considerably more for a neutron star, where the analogous limit is not calculable at all without knowing the equation of state of matter at nuclear density — the same object whose orbit around another neutron star is measured to be shrinking.

And the limit is not unique to white dwarfs. The same argument, with neutrons in place of electrons, gives the Tolman–Oppenheimer–Volkoff limit for neutron stars — around 2.2 solar masses, though the number depends on nuclear physics nobody can measure. The structure of the argument survives entirely; only the particle changes.

What the picture cannot show #

The collapse. Every figure here is an equilibrium: a curve of states that balance. The interesting thing about the limit is what happens when no state balances, and that is a dynamical process with no point on any of these curves.

Time. A white dwarf approaching the limit does so over millions of years of accretion, and the ignition and disruption take seconds. No axis on this page carries either.

The interior. The mass–radius relation is a relation between two global quantities, and the star it describes has a density running over eight orders of magnitude from centre to surface. The exponent $\Gamma$ drawn in the first figure is a function of density, so it is a different number at every depth in a real star, and the “the star has $\Gamma = 4/3$” of the scaling argument is an average that the exact calculation replaces with an integral.

A result that was refused in public #

Chandrasekhar computed the limit in 1930, at nineteen, on the voyage from Madras to Cambridge, and published the essential result in 1931. He presented the full theory to the Royal Astronomical Society in January 1935.

Arthur Eddington — the most eminent astrophysicist alive, and Chandrasekhar’s own mentor — rose immediately afterwards and said that the result was a reductio ad absurdum: since a star collapsing indefinitely was manifestly impossible, there must be a law of nature preventing it, and the relativistic degeneracy formula must therefore be wrong. He maintained that position for the rest of his life.

The physics community largely sided with Eddington on authority and largely knew he was wrong on the merits; Dirac, Bohr and Rosenfeld all confirmed to Chandrasekhar that the degenerate equation of state was correct. Chandrasekhar left the field, worked on stellar dynamics, radiative transfer and half a dozen other subjects, and received the Nobel Prize for this work in 1983 — fifty-three years after doing it.

What Eddington found absurd is the part that turned out to matter. A star with no stable endpoint does collapse, and the objects it collapses into — neutron stars and black holes — were not merely permitted by the limit but implied by it. The conclusion he refused as impossible is now the standard account of where most of the heavy elements come from, and of where the sharpest measurement in cosmology gets its calibration — one number that is an age, a size and a density included.

The composition enters the limit through one number, and it is worth drawing a case where that number is different.

That is why the limit is quoted as 1.4 solar masses and not as a constant of nature. It is a constant of nature multiplied by $\mu_e^{-2}$, and $\mu_e$ is a fact about what the star is made of. A white dwarf of pure hydrogen would have a limit near five and a half solar masses and does not exist, because hydrogen at those densities ignites long before it becomes degenerate.

So the number that is used is the number for matter that has finished burning hydrogen — helium, carbon, oxygen, all at two nucleons per electron — and the coincidence is that every one of those gives the same answer. The limit is insensitive to composition across the entire range of compositions a white dwarf can actually have, and sensitive to it in a range that is physically unreachable. That is a stronger statement than the usual one, and it is the reason a single number does so much work.

Where the ladder goes next #

Later rungs on this anchor: the Lane–Emden equation for $n = 3$, and the exact numerical coefficient in front of the $\mu_e^{-2}$. White dwarf cooling as a clock, and the age of the Galactic disc read off the faint end of the luminosity function. Crystallisation of the carbon–oxygen interior, and the release of latent heat that pauses the cooling. Electron capture and the accretion-induced collapse channel. The Tolman–Oppenheimer–Volkoff limit and what the neutron-star maximum mass says about matter at nuclear density. And the double-degenerate channel for type Ia supernovae, in which two white dwarfs merge and the exploding mass is not the Chandrasekhar mass at all — which is the leading current worry about the standard candle everything above rests on.