A core too heavy for the gas around it

A polytrope is a star built from one number. Choose how pressure rises with density, demand that the pressure hold the weight up, and the whole structure follows — the run of density, the central condensation, the relation between mass and radius. The one number is the index, and a main-sequence star is described surprisingly well by a single value of it: about 3 for a star whose energy leaks out by radiation, 1.5 for one stirred by convection.

That description ends at a definite moment. When the hydrogen at the centre runs out, the star is no longer one body with one equation of state. It is a core of helium, with no fuel of its own, wrapped in an envelope of hydrogen that is still burning at its base. The two halves obey different laws and have different compositions, and the question of whether the star can stay in equilibrium becomes a question about whether the two halves can agree on the pressure at the surface where they meet. A single index cannot answer that. Two indices — one for the core, one for the envelope — answer it exactly, and the answer is a limit on how large the core can be.

A core that makes no energy settles to one temperature #

The first step is to see what the core becomes. Energy flows outward through a star at a rate set by the temperature gradient: where there is a gradient, heat moves down it. In a region that generates no energy, the luminosity passing through each sphere is whatever enters from below — and at the centre of an exhausted core, nothing enters from below. The luminosity is zero throughout, so the temperature gradient is zero throughout, and the core comes to a single temperature, set by the hydrogen-burning shell that sits on its surface.

A gas at a single temperature has pressure proportional to density, and in the polytrope family that is the limit of infinite index. It is the one member with no edge. Every finite-index polytrope reaches zero density at some radius and stops there; the isothermal sphere never does. Its density falls, eventually as the inverse square of radius, forever, and the enclosed mass grows without limit. A finite isothermal core therefore exists only because something outside it stops it: a pressure applied at its surface, which in a star is the weight of the envelope pressing down.

That changes the question. A self-contained star asks what radius its equation of state gives it. A bounded isothermal core asks something else: given a mass and a temperature, what pressure can it push back with at its edge? And the answer, which is the whole of what follows, is that there is a most it can manage.

An isothermal sphere can push back only so hard #

Take an isothermal sphere of fixed mass and temperature and imagine choosing where its edge lies. A sphere cut off very far out is large and diffuse, and the pressure at its edge is small. Pull the edge inward — hold the same mass in a smaller volume — and the pressure at the edge rises, because the gas is denser. That is the behaviour any gas shows when compressed, and it is what makes a gas stable: squeeze it and it pushes back harder.

But as the sphere is compressed its own gravity grows too, and it grows faster. The gas at the centre is pulled inward more strongly, the density profile steepens, and the ratio of central density to edge density climbs. The pressure at the edge rises more and more slowly and then, at a definite contrast, stops rising at all.

The maximum sits at a density contrast of 14.1, where the pressure at the edge is $1.40\,c^8/(G^3M^2)$, with c the isothermal sound speed, which carries the temperature and the molecular weight. These two numbers come straight out of integrating the isothermal equation, and they are the same numbers that describe a molecular cloud core on the edge of collapse. In a cloud the outside pressure comes from the warmer gas around it and the result is called the Bonnor–Ebert mass: the most a cloud at a given temperature can hold together under a given pressure. In a star the mathematics is identical and the application is inverted. The outside pressure is not given by the surroundings; it is demanded by an envelope that has to be held up. A core past the peak cannot supply it.

The dependence on mass is the part to keep. At fixed temperature the ceiling falls as the inverse square of the core’s mass. A heavier core can push back less hard at its surface, not more, because the extra mass deepens its own gravity well faster than it adds thermal support. That inversion is the root of the limit: the core grows as the shell deposits helium on it, and every gram it gains lowers the ceiling on what it can support.

Where the ceiling meets the weight of the envelope #

The envelope’s side of the balance is fixed by the same kind of estimate that sets any star’s central pressure. Hydrostatic equilibrium makes the pressure at the base of the envelope scale as $GM^2/R^4$, with M the star’s mass and R its radius. The temperature there is set by the virial balance — the same balance that makes a self-gravitating system hotter as it loses energy — and scales as $\mu GM/R$, with μ the envelope’s mean molecular weight. Eliminating the radius between those two gives the envelope’s demand in the same currency as the core’s ceiling: four powers of temperature over the molecular weight, divided by $G^3$ and by the square of a mass. The difference is that the envelope’s demand carries the star’s mass and the envelope’s molecular weight, and the core’s ceiling carries the core’s mass and the core’s molecular weight.

So the ratio of the two depends on nothing but the fraction of the star that is core and the ratio of the two molecular weights. The temperature, the radius, even the star’s total mass all cancel. The only constant the scaling cannot supply is the numerical factor in the envelope’s demand, which depends on the detailed structure of the envelope. Schönberg and Chandrasekhar found it in 1942 by integrating composite models — an isothermal core fitted to a polytropic envelope — and the result is that the core can hold the envelope up only while its mass fraction satisfies $q < 0.37\,(\mu_{\rm env}/\mu_{\rm core})^2$.

The molecular weights are what make the number small. A fully ionised helium core has μ = 4/3: each helium nucleus and its two electrons are three particles sharing four mass units. The envelope, mostly hydrogen, has μ close to 0.6. At the same temperature and density the helium supplies less than half the pressure, because pressure counts particles and the helium has fewer of them. The ceiling scales as the fourth power of the inverse molecular weight, so the helium core’s ceiling is lower by a factor of about twenty-four, and the limit falls from 37 per cent of the star to 7.5. A core that had somehow kept the envelope’s composition could hold up a third of the star; a helium core holds up less than a tenth.

The figure also shows why this is a limit and not a gradual change. The ceiling curve crosses the demand line once and only once. On one side the core supplies more than the envelope asks, and the two settle into a balance; on the other the core cannot supply what is asked at any radius. There is no intermediate state with a partly satisfied envelope. There is equilibrium, and then there is not.

Two cores that balance the same envelope, and one that survives #

Below the limit the balance has a feature that the pressure-versus-contrast diagram hides, and it matters for what happens when the limit is crossed. It is clearest in a rougher picture: a core of fixed mass, treated as uniform, with the pressure at its edge estimated from the virial theorem as a thermal term minus a gravitational one.

The curve is a hump, and a demand below the top crosses it twice. On the large side of the hump, compressing the core raises the pressure it pushes back with, so a small disturbance is corrected: that solution is stable, and it is where a real core sits. On the small side, compressing the core lowers the pressure, so a small inward push is not resisted but rewarded, and the core falls in. As the shell adds helium the hump sinks, the two solutions slide toward each other, and at the Schönberg–Chandrasekhar mass they meet at the top and annihilate. There is then no radius at which the core can sit.

What the core does next is contract, and it does so on the timescale set by how fast the star can radiate away the gravitational energy the contraction releases — the thermal, or Kelvin–Helmholtz, timescale. That is far shorter than the nuclear timescale on which the main sequence was lived. The core shrinks and heats; the energy it releases, together with the shell’s output, pushes the envelope outward; and by the mirror principle the contraction inside is matched by an expansion outside. The star swells toward the red giant branch until the core is hot enough to ignite helium, or dense enough for a different kind of pressure to take over.

A degenerate core has no ceiling #

That different pressure is the way out, and it decides which stars ever meet this limit at all.

The ceiling exists because an isothermal gas’s pressure is proportional to its density: compress it by a factor and its pressure rises by the same factor, which in a core of fixed mass means the edge pressure rises only as the inverse cube of radius while gravity’s opposing term rises as the inverse fourth power. Gravity wins at small radii. A degenerate gas — one dense enough that its electrons are packed by the exclusion principle rather than by temperature — has pressure rising as the five-thirds power of density, so its thermal term grows as the inverse fifth power of radius. Now the support grows faster than gravity, and compressing the core always buys more pressure.

So a core that is already degenerate when it grows past a tenth of the star simply settles into a smaller radius and carries on. This is what happens in stars of less than about two solar masses. Their exhausted cores are dense and cool enough for degeneracy to set in before the core reaches the limit, and the core then grows slowly and steadily, at the rate the shell delivers helium, without any thermal-timescale contraction. The star moves off the main sequence gradually, along the subgiant branch, on a nuclear timescale.

Stars above about two solar masses have hotter, less dense cores that stay far from degeneracy. For them the limit is a real wall, and it arrives almost at once. Such a star burns hydrogen in a convective core, which stirs its contents evenly, so hydrogen runs out through the whole convective region at the same moment — and the helium core it leaves behind already holds about a tenth of the star. It is born at the limit. Anything that enlarges the convective core, like convective overshoot mixing fresh fuel in from above, leaves a heavier core behind and puts the star further past the limit on the day hydrogen runs out.

The envelope sets the limit too #

Because the limit depends on the ratio of two molecular weights, it depends on the envelope’s composition as well as the core’s. An envelope enriched in helium — by mixing that drags processed material up from the core, or in a star born helium-rich — is heavier per particle, pushes down harder at the same temperature, and moves the balance point.

The direction of the effect runs against the obvious guess. A heavier envelope, with fewer particles per unit mass, exerts less pressure at a given temperature, not more, and the scaling places the molecular weight to the fourth power in the demand’s denominator. So a helium-enriched envelope raises the limit: with $\mu_{\rm env} = 0.8$ a helium core can hold 13.3 per cent of the star before it runs out of support. Any process that mixes helium outward during the main sequence therefore lets the star keep a larger core in equilibrium and delays the collapse. That is one of the reasons the limit, though exact as a mathematical statement about composite polytropes, is always quoted in real stars as a range — somewhere around eight to ten per cent for solar composition, moved by the details of mixing that no single number carries.

Why the gap in the diagram is empty #

The limit is not only a statement about stellar interiors. It predicts something visible, and the prediction was the first thing it explained.

In the colour–magnitude diagram of a star cluster, the main sequence is densely populated and so is the giant branch, and for stars a little more massive than the Sun the region between them is almost empty. That emptiness, the Hertzsprung gap, is a statement about time. The number of stars in any region of the diagram is proportional to how long a star spends there, so a nearly empty region is one that stars cross quickly. The limit supplies the reason for the speed.

For a star above the band, the crossing proceeds on the thermal timescale — a few tenths of a per cent of the main-sequence lifetime on these rough scalings, and a few per cent in detailed models once the time to rearrange the whole envelope is counted. A cluster with a hundred stars on its upper main sequence should therefore have one or a few caught in the gap. That is what is seen, and a deficit read as a speed is one of the most reliable arguments in stellar astrophysics precisely because it needs no model of the stars themselves, only the assumption that they are arriving at a steady rate.

For a star below the band the prediction reverses. The degenerate core never meets the limit; the crossing is nuclear-timed; and the region is filled by a well-populated subgiant branch. The transition between the two behaviours — where the gap opens as the cluster’s turn-off mass rises — is itself a measurement of the mass at which cores stop being degenerate at hydrogen exhaustion, which is one of the calibrations stellar models are tested against.

What the limit is and is not #

The limit is exact in the sense that matters for a polytrope: given an isothermal core and a polytropic envelope with a jump in molecular weight at the boundary, the coefficient 0.37 and the square of the molecular-weight ratio follow from integrating two equations and matching them. It is approximate in every other sense. Real cores are not perfectly isothermal, because the shell’s heat leaks in; real envelopes are not polytropes; real boundaries between core and envelope are gradients in composition rather than steps. Detailed stellar models find the limit acting at somewhat different fractions and as a gradual steepening of the evolution rather than a cliff.

What survives all of those refinements is the mechanism. A core with no energy source settles to one temperature; a gas at one temperature can push back only so hard; the push available falls as the core grows and as its particles get heavier; and the envelope’s demand does not fall with it. That is a structural fact about self-gravitating gas, independent of any detail of nuclear physics, and it is why the same mathematics serves a cold molecular cloud and the centre of a massive star.

Still open: how isothermal an exhausted core really is #

The argument depends on the core being at one temperature, and that depends on the core generating nothing and receiving nothing. Neither is quite true. The shell above the core heats its surface; slow contraction releases gravitational energy inside the core even before the limit is reached; and neutrino losses cool the centre. Each of these puts a gradient back into a core the argument treats as flat, and a core with a gradient is a polytrope of large but finite index, with a ceiling that differs from Bonnor and Ebert’s.

How much that matters is measurable in principle. The size of the core at hydrogen exhaustion, and how quickly it contracts afterwards, leave signatures in the oscillation frequencies of stars just leaving the main sequence, where mixed modes probe the core directly. Whether those frequencies can be inverted precisely enough to measure the core’s temperature gradient, and so test whether the limit bites where the composite polytrope says it should or somewhere else, is a question the current generation of space photometry has begun to answer for a few dozen stars and not yet for the population.