The ball of gas that heats up as it cools

Every body anybody has ever put a thermometer in gets hotter when energy is added to it — the counting behind that is equipartition’s, and it presumes something other than the body itself is holding it together. That is what a positive heat capacity means, it is what makes a temperature useful, and it is the reason two bodies placed in contact end up agreeing. A ball of gas held together by its own gravity does the opposite.

The result is not an approximation and it does not depend on the gas. It follows from one theorem about bounded motion under an inverse-square force, and its consequences reach from why a protostar heats itself to fusion temperatures to why a star cluster cannot come to equilibrium.

Two energies, and the fixed ratio between them #

For any system of particles held together by mutual inverse-square attraction, bounded in space and averaged over long enough, the kinetic and potential energies satisfy

$$2K + U = 0.$$

That is the virial theorem. The derivation is short — differentiate the moment of inertia twice, average, and note that a bounded system’s moment of inertia cannot grow without limit, so its second derivative averages to zero — and the ratio it gives is exact.

The total energy is therefore

$$E = K + U = K - 2K = -K,$$

and since the kinetic energy of a monatomic gas is $\tfrac32 NkT$, the temperature is $T = -\tfrac{2E}{3Nk}$ and

$$C = \frac{dE}{dT} = -\tfrac32 Nk.$$

Negative, and equal in magnitude to the heat capacity the same gas would have in a box. The figures compute that number from the drawn points and it agrees to the digits printed.

For a solar mass of hydrogen at one solar radius the same arithmetic gives a mean temperature of 2.8 million kelvin — which is the right order for the interior of the Sun, obtained from nothing but a mass, a radius and a theorem.

What the sign means in practice #

A body with a negative heat capacity in contact with a reservoir cannot come to equilibrium with it. Suppose it is hotter than its surroundings, so it loses heat; losing heat makes it hotter, so it loses faster, so it gets hotter still. Suppose it is colder, so it gains heat; gaining heat makes it colder, so it gains faster. Either way the temperature difference grows.

Every measured heat capacity of a laboratory substance is positive, and the counting that produces them guarantees it: add energy to a box of gas and it gets hotter — and what is being counted there is mass in one of its other forms, since the binding energy of a self-gravitating body is a real part of what it weighs. A self-gravitating ball does the opposite, and the reason is that it is not in a box. Its size is a variable, the energy budget includes the gravitational term, and the virial relation ties the two together so tightly that removing energy leaves the remaining particles moving faster.

The mechanism of the flip is worth stating in words. In a box, adding energy raises the kinetic energy and nothing else, because the walls hold the volume. In a self-gravitating ball there are no walls: adding energy makes the ball expand, expansion does work against gravity, and the work done exceeds the energy added — the excess coming out of the kinetic energy, which falls. Expansion is the mechanism, and the factor of two in the virial theorem is what makes the excess win.

The nine million years #

The most consequential application of the theorem was made before anybody knew it was wrong, and the wrongness took fifty years to explain.

A cloud contracting under its own gravity releases gravitational energy. Half of that release goes into heating the cloud, by the theorem; the other half is available to be radiated. So a body that has contracted from a great distance to a radius $R$ has radiated an amount equal to its present total energy, and dividing that by its luminosity gives how long it could have been shining.

Kelvin made that estimate in the 1860s and defended it for forty years against geologists, who wanted hundreds of millions of years for sedimentation, and against Darwin, who wanted more still for natural selection. The dispute is often told as a story about a physicist’s arrogance, and it is a better story than that: Kelvin’s arithmetic was right, his data were right, and his conclusion followed. What was missing was a source of energy with no gravitational or chemical counterpart, and no amount of care with the two known sources could have revealed it.

What fraction of the mass each process converts is what makes the timescale argument decisive. Chemical burning manages about a part in $10^9$ and hydrogen fusion about seven parts in a thousand — a factor of seven million — and that factor is the difference between a Sun that could last twenty million years and one that has already lasted four and a half thousand million. Kelvin’s estimate was not a mistake in the arithmetic; it was the right arithmetic for the only energy source he knew about.

The Kelvin–Helmholtz mechanism is not wrong; it is merely not the main term for a star on the main sequence. It is the main term before fusion starts, and it is what heats a protostar: a cloud collapses, half the released energy stays as heat, the temperature climbs, and when it reaches about ten million kelvin fusion ignites and the contraction stops. The negative heat capacity is the reason a collapsing cloud does not merely fall together but heats itself on the way — the collapse is its own furnace.

The same sign, in a system nobody would call a star #

The negative heat capacity is not peculiar to gas balls. It belongs to any bounded system held by an inverse-square attraction, which includes a star cluster, a galaxy and — with the sign of the charge reversed — nothing at all, because electrostatic systems screen themselves and gravitational ones cannot.

The same sign turns up in a system nobody would call astrophysical: a cluster of stars, or any collection held together by its own gravity with nothing else acting. Such a system evaporates — the outer members carry energy away, the remainder contracts, and contracting makes the remainder hotter in the sense that its members move faster. There is no equilibrium to settle into, which is why a self-gravitating system has no proper thermodynamic limit and why the usual machinery declines to apply to it.

For a star cluster the consequence has a name. Two-body encounters redistribute energy between stars, and the ones that gain enough leave. A cluster losing its fastest members loses energy, so by the theorem it contracts and its remaining stars speed up, so more of them can leave. Evaporation accelerates itself. Over many crossing times the core of a cluster contracts without limit while its halo expands — the two halves running away from each other in temperature rather than approaching one another — and the process has to be stopped by something else entirely, usually the formation of a tight binary in the core that acts as an energy source.

The general lesson is that gravity organises rather than disorders — which sits awkwardly beside entropy as a count of arrangements, and is the reason the counting has to be redone when the interaction is long-ranged. Left alone, a gravitating system does not smooth out into a uniform warm mush the way a gas in a box does; it separates into a dense hot part and a diffuse cool one, and goes on separating. That is why the universe contains structure at all.

The sign, measured in a laboratory #

The negative heat capacity is usually presented as gravity’s peculiarity, and it is not confined to gravity. It has been measured in an object of a hundred and fifty atoms sitting in a vacuum chamber.

The condition for it is not long-range attraction but the difference between two ways of specifying a system’s energy. A body in contact with a large reservoir has a temperature imposed on it and its energy fluctuates; a body with a fixed energy has a temperature that follows from it. For a large system the two descriptions agree. For a small one they need not, and where they disagree the fixed-energy heat capacity can be negative while the fixed-temperature one — which is a variance and must be positive — is not.

Where it happens is near a first-order transition. A cluster of atoms melting has to pay a latent heat, and in a small cluster that cost is comparable with the total energy. Adding energy converts some of the cluster from solid to liquid, and the conversion absorbs more energy than the temperature rise requires — so over a range of energies the temperature falls as energy is added.

That was measured in 2001, on clusters of about 150 sodium atoms held as ions, whose energy was set by photon absorption and whose temperature was read from how they fragmented. The heat capacity came out negative over a window of a fraction of an electronvolt, exactly where the melting transition sits.

Which changes what the sign means. It is not a signature of gravity; it is a signature of a system whose parts rearrange when energy is added, in a way that consumes more than the addition. Gravity produces that by letting the body expand; a melting cluster produces it by converting phase. The two mechanisms have nothing in common except that neither has a rigid container.

Kelvin’s other estimate, wrong for another reason #

The nine million years is the famous failure and Kelvin made a second age estimate that failed independently, and the pair together are more instructive than either.

The other estimate is of the Earth rather than the Sun. Take a planet that began molten, let heat conduct outward, and measure the temperature gradient near the surface today: the gradient tells how far the cooling has got, and running it backwards gives an age. Kelvin’s answer, over several attempts, ranged from twenty to a hundred million years.

Two things were wrong with it and only one is the one usually named. The famous one is radioactivity, discovered in 1896: the Earth’s interior contains a heat source Kelvin had no way to know about, so it has cooled far more slowly than a body with no source.

The other was pointed out in 1895, before radioactivity, by John Perry — who had been Kelvin’s assistant. Kelvin’s calculation assumes heat leaves by conduction. If the deep interior is fluid enough to convect, heat is carried to the base of the crust far faster than conduction would move it, the surface gradient is maintained by a much larger reservoir, and the age inferred from it is far too small. Perry showed the correction could be a factor of ten or more.

Perry was right, convection is what the mantle does, and the argument was ignored. That is the more uncomfortable half of the story: the failure that is remembered was one nobody could have avoided, and the failure that was avoidable was pointed out in print and dismissed.

The general moral is worth keeping separate from the history. An estimate built on a transport mechanism is only as good as the assumption that the named mechanism is the operative one — and checking which mechanism dominates is a different kind of work from doing the calculation carefully.

Using it backwards #

One more use of the theorem is worth a sentence, because it is how most of the mass anybody has ever inferred was inferred.

Run the relation the other way. For a bound system in equilibrium the kinetic and potential energies are in a fixed ratio, so measuring the kinetic energy — from the speeds of the members — and the size gives the potential energy, and therefore the mass. No knowledge of what the mass is made of is required, and no light needs to come from it.

Applied to a cluster of galaxies in 1933, the method gave a mass far larger than the light suggested, and the discrepancy has never gone away. Applied to a globular cluster it gives a mass-to-light ratio; applied to a galaxy’s outer parts it gives a rotation curve’s implication; applied to a star-forming cloud it says whether the cloud is bound at all.

What the method inherits from the theorem is its assumptions, and they are the ones this essay has already listed: the system must be bound, it must be in equilibrium, and the average must be over long enough. A cluster still assembling, or one whose members are on their first pass, satisfies none of them, and a virial mass computed for such a system is a number with no meaning attached.

The thermostat, and the runaway when it fails #

Once fusion begins, the negative heat capacity becomes a regulator of remarkable stability. Suppose the fusion rate rises briefly. Extra energy is released, the core expands, expansion cools it, and — because the reaction rate depends on temperature through an enormous exponent — the fusion rate falls sharply. The star holds itself steady, and the mechanism is the sign of its heat capacity.

The thermostat works because the rate sits at the steep end of an exponential. Stellar fusion goes roughly as $T^4$ for the proton–proton chain and as $T^{17}$ for the CNO cycle, so a per cent of extra temperature is a large increase in output — and the star, having a negative heat capacity, responds to extra output by expanding and cooling. The regulation is tight, automatic, and depends on the sign this essay is about.

The regulator fails when the gas stops being able to expand. In a degenerate core the pressure comes from the exclusion principle rather than from the temperature, so heating it does not expand it, so nothing cools the reaction. That is a thermonuclear runaway: it is what a helium flash is in a red giant, and what a type Ia supernova is in a white dwarf that has been pushed over the limit. The presence or absence of a negative heat capacity is the difference between a star that burns steadily for ten billion years and one that detonates.

The runaway happens when that sign is removed. A gas whose pressure is set by degeneracy rather than by thermal motion does not expand when it is heated, so the thermostat is disconnected: the temperature rises, the rate rises far faster, and nothing pushes back. That is a helium flash in a red giant, and it is a nova on a white dwarf — the same disconnection twice, with the same cause.

The number that decides which mechanism a body runs on #

Comparing timescales is the way to tell which of a body’s possible energy sources is actually supplying it, and the comparison is a division.

The Kelvin–Helmholtz time is the total energy divided by the luminosity — how long the body could shine on gravity. The nuclear time is the available nuclear energy divided by the same luminosity. The free-fall time is how long the body would take to collapse if the pressure vanished, and it depends on nothing but the mean density.

For the Sun those three are 9.4 million years, 10 billion years, and 30 minutes. The enormous separation between them is what makes a star a simple object: it is in hydrostatic equilibrium because the free-fall time is negligible, it is in thermal equilibrium on a nuclear timescale because the Kelvin–Helmholtz time is short by comparison, and its structure at any moment is settled without reference to its history.

Where two of the three become comparable, the object is interesting and hard. A pulsating variable star has a period near its own free-fall time. A protostar has a Kelvin–Helmholtz time comparable with its accretion time. A supernova progenitor’s final burning stages have nuclear times shorter than the free-fall time, which is why the collapse is not stopped by anything. Every difficult problem in stellar structure is a place where a ratio that is usually enormous has come down to one.

Where the model stops #

A uniform sphere is not a star. The relations used here — $U = -\tfrac35 GM^2/R$ and a single mean temperature — are for constant density, and a real star is centrally condensed by four orders of magnitude. Every number moves by a factor of order one and no sign changes, because the sign comes from the theorem and not from the density profile.

The theorem is a time average and a snapshot is not. For a body in hydrostatic equilibrium the distinction is immaterial; for a system that is oscillating, collapsing or has recently been disturbed it is not, and applying the virial ratio to a single instant of such a system gives a wrong answer. Using the theorem the other way round — as a mass estimator averaged over a population of orbits — is a measurement technique with its own difficulties and belongs to the collection that owns the sky.

Radiation pressure is neglected. In a massive star radiation carries a large share of the pressure support, the effective ratio of specific heats falls toward 4/3, and the virial coefficient changes; at exactly 4/3 the total energy is zero and the star is marginally bound, which is where the upper mass limit comes from.

And there is no thermodynamic limit. Every argument in ordinary statistical mechanics assumes that doubling the size of a system doubles its energy and entropy, and gravity’s long range breaks that: the energy of a self-gravitating ball goes as the square of its mass rather than as its mass. Extensivity fails, the entropy has no maximum, and the standard machinery — including the second law’s usual statement — has to be applied with care or not at all.

What the pictures cannot show #

The hero figure draws a straight line with a negative slope and cannot draw what makes it possible, which is that the volume is free to respond. Every point on that line is a different radius, and a picture of the body at each would show a different object — so what is plotted is a family of equilibria rather than a process any single body undergoes.

Nor can any figure show the runaway that the negative sign implies. A body with a negative heat capacity in contact with an ordinary one has no steady state to draw, and drawing the two at some instant asserts an equilibrium the physics denies.

Where this ladder goes next #

The rung below asked when a body’s own gravity overwhelms the strength of what it is made of, and answered with a size. This rung asks what happens once gravity is the only thing holding a body together, and answers with a sign — the one property that makes gravitating matter thermodynamically unlike everything else.

The rungs above are the ones this essay pointed at. The Jeans criterion, which asks when a cloud’s self-gravity beats its own pressure and therefore when it collapses at all. The gravothermal catastrophe, in which a self-gravitating system with a hot core and a cool halo has no stable configuration and the core collapses indefinitely — the negative heat capacity applied to a system in contact with itself. And the maximum entropy that does not exist, which is the formal statement of why gravity has no equilibrium statistical mechanics.

The habit worth carrying away is about which quantities are safe to assume. A heat capacity is not required to be positive; it is positive whenever the confining agency is external, and gravitating systems are the case where it is not. The same caution applies to any system whose container responds to its contents — a balloon whose skin softens when warmed, a chemical reaction that changes the volume it occupies, a population whose resources grow with it. The stabilising sign that makes equilibrium reasoning work is an assumption about the boundary rather than a law.