Astronomy & Celestial Mechanics
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Gravitation

The system that gets hotter as it loses energy

The virial theorem makes a self-gravitating body's total energy equal to minus its kinetic energy. Radiating heat away therefore raises the temperature, there is no equilibrium to settle into, and every star and every cluster is running away from one.

Assumes Virial theorem and Hydrostatic equilibrium.

16 min read 8 figures Scales that defeat intuitionTwo bodies, and no more

Take a box of gas, put it in a cold room, and it cools. Every physical intuition about heat is built on that observation, and it holds for everything anybody handles. It does not hold for anything held together by its own gravity, and the reversal is not a subtlety or a limiting case — it is the ordinary behaviour of stars, clusters and galaxies, and it is why none of them ever settles down — including, on a long enough view, the clusters whose masses it is used to measure.

A body that heats up because it is losing energy. A uniform self-gravitating sphere of 1.0 solar masses radiating at 1.0 solar luminosities, with no nuclear source at all, from 3.0 solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls, and the temperature rises. That is not a paradox and it is not a special case. The virial theorem makes 2K = −U for any self-gravitating gas in equilibrium, so E = U + K = −K, and −dE/dt = +dK/dt: energy leaving as light is energy arriving as heat. The bookkeeping is exact and is measured here rather than quoted — over the run 5.465·10⁴⁰ J of gravitational energy is released and 2.733·10⁴⁰ J is radiated, a ratio of 2.0000. Half the release is spent on the star's own heat and only half escapes. The consequence is a body with a negative heat capacity, which is why a contracting protostar gets hotter until it ignites, why a globular cluster's core runs away instead of settling, and why nothing self-gravitating ever comes to thermal equilibrium.
Fig. 1 A self-gravitating sphere of one solar mass radiating at one solar luminosity, with no nuclear source at all, contracting from three solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls and the temperature rises. The bookkeeping is measured off the arrays rather than quoted — over the run 5.5×10405.5\times10^{40} J of gravitational energy is released and 2.7×10402.7\times10^{40} J radiated, a ratio of exactly 2. Half the release is spent on the body’s own heat and only half escapes. Nothing in the setup is unusual: it is an ideal gas obeying the virial theorem and losing energy at a steady rate.

Two lines of algebra

The virial theorem for a bound, self-gravitating system in equilibrium is 2⟨K⟩=−⟨U⟩2\langle K\rangle = -\langle U\rangle, and weighing a cluster with it is the base rung of this ladder. Its consequence for energy accounting takes one further line.

E=K+U=K−2K=−K.E = K + U = K - 2K = -K.

The total energy is minus the kinetic energy. So

dEdt=−dKdt,\frac{dE}{dt} = -\frac{dK}{dt},

and since the kinetic energy of an ideal gas is its temperature times a constant, losing energy is heating up. Written as a heat capacity, C=dE/dT<0C = dE/dT < 0: the system’s heat capacity is negative, and for a monatomic ideal gas held together by gravity it is exactly −32Nk-\tfrac32 Nk.

The factor of two in the figure is the same statement seen from the gravitational side. Contracting from R1R_1 to R2R_2 releases ∣ΔU∣|\Delta U|; of that, ∣ΔU∣/2|\Delta U|/2 becomes kinetic energy and stays, and ∣ΔU∣/2|\Delta U|/2 leaves as radiation. A contracting body can only ever radiate half of what it releases, and it is the other half that makes it hot.

What it explains, before it becomes strange

Two of the most important facts about stars follow immediately and comfortably, and they are worth taking first.

A protostar reaches fusion temperature with no source of heat. A cloud fragment collapsing under its own gravity has nothing burning in it, radiates freely, and gets hotter the whole way down — and it keeps getting hotter until the centre reaches about 10710^7 K and hydrogen fusion begins. The energy for the entire pre-main-sequence phase comes from the contraction, half of it appearing as the star’s own light. Gravity is the star’s first fuel, and it is spent before anything is ignited.

A star that loses its nuclear source contracts and heats. When the core hydrogen runs out, the core has no support and contracts; the virial theorem says its temperature rises; the temperature rise ignites a shell around it. That is the mirror that makes a red giant — the core shrinking and the envelope expanding are a single event, and the direction of each is set by this theorem.

The mirror: an envelope 62 times larger around a core 1.48 times smaller. Envelope radius and core radius against helium-core mass for a 1 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 2.0 to 122 solar radii — 0.57 astronomical units — while the degenerate core contracts from 13,848 to 9,368 kilometres, and the two are separated by a factor of 364 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5000 to 3700 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.
Fig. 2 The mirror itself. As the helium core contracts the envelope expands, and both are driven by the same balance: the shell source sits at a fixed fraction of the core’s surface, its luminosity climbs as the core shrinks and heats, and the envelope has no choice but to swell. A star does not get bigger because it is running out of fuel; it gets bigger because the part of it that has run out is getting smaller. The negative heat capacity is what makes the core’s shrinkage self-sustaining rather than self-limiting.

The Kelvin–Helmholtz problem, and the answer it forced

Before nuclear energy was known, the contraction was the theory of the Sun, and the negative heat capacity is what made it work — a Sun that shrank slowly would heat rather than cool, which is why it could shine steadily while shrinking.

The timescale is the one the opening figure runs on: tKH=∣E∣/L=GM2/2RLt_{\rm KH} = |E|/L = GM^2/2RL, which for the Sun is around thirty million years depending on the structure factor assumed. Kelvin and Helmholtz took that as the age of the Sun, and it was the strongest argument physics had against Darwin and against the geologists, both of whom needed hundreds of millions of years.

They were right about the physics and wrong about the source. What the argument really established, and what survives, is a lower bound on how fast a star can change: any star out of thermal balance readjusts on tKHt_{\rm KH}, and no star can vary its luminosity faster than that without something other than gravity being involved.

Where it stops being comfortable

A negative heat capacity has a consequence that no ordinary thermodynamic system has: two such systems in contact do not come to a common temperature. They run away from each other.

Put a hot self-gravitating region next to a cool one, in thermal contact. Heat flows from hot to cool, as it must. The hot region, having lost energy, gets hotter. The cool region, having gained it, gets cooler. The temperature difference is now larger than it was, so the flow speeds up. There is no equilibrium in that direction and the process accelerates without limit.

That is the gravothermal catastrophe, identified by Antonov in 1962 and Lynden-Bell and Wood in 1968, and it is the standing embarrassment of statistical mechanics applied to gravity: there is no maximum-entropy state for a self-gravitating system, so the usual apparatus of thermodynamics does not apply and cannot be made to.

The failure is worth stating precisely, because it is not that the entropy is hard to compute. It is that the entropy has no maximum: for a system of fixed energy confined to a fixed volume, the entropy can always be raised by moving a little more mass into the core and letting the rest expand, and the process has no end. Antonov’s contribution was to show exactly where the local maximum disappears — beyond a density contrast of about 709 between the centre and the boundary, the equilibrium that exists at lower contrasts ceases to be even locally stable. Below that number a self-gravitating isothermal sphere can sit still; above it, nothing can.

The centre that shrank: 1.59·10⁵ to 1.66·10⁶ g cm⁻³. Central density of the helium core against its mass, for a 1 solar-mass star, from the cold degenerate radius relation. Two factors are drawn and they are not the same factor. Along the giant branch the density rises by 10.4 — exactly the square of the core-mass ratio, because a core whose radius goes as M⁻¹ᐟ³ has a density going as M². The larger step happened before the curve starts: the Sun's centre today is about 150 grams a cubic centimetre and is an ordinary hot gas, so the base of the giant branch is already 1,061 times denser than that, and no part of this curve draws the transition. The relation used is the zero-temperature one, so every density here is an upper bound, and the bound is loosest at the left-hand end where a real core is only partly degenerate.
Fig. 3 What negative heat capacity looks like inside one star. As the core contracts it heats — losing energy makes it hotter, because half the released gravitational energy goes into heat and the other half is radiated — and the density contrast between core and envelope runs away. The same sign is why a star cannot reach thermal equilibrium by cooling and why a self-gravitating cluster evaporates rather than settling. One property of gravity, appearing at both ends of the mass scale.

What was actually measured

Three separate measurements bear on this, and none of them is a fit to the theory.

The core-collapse fraction. Of roughly 150 globular clusters in the Galaxy, about 20% have surface-brightness profiles that rise as a power law into the centre rather than flattening into a core. The theoretical collapse time is about fifteen relaxation times, and for a typical cluster that is a few billion years — so a fraction of order a fifth having got there in a Hubble time is what the theory predicts, and it is what is counted.

The velocity dispersions. A cluster can be weighed three ways — from its dispersion, from its light with an assumed mass-to-light ratio, and from the motions of individual stars — and the first of those is the virial theorem used as an instrument. The same measurement, taken as a function of radius, shows the dispersion rising towards the centre in collapsed clusters, which is the negative heat capacity’s signature read directly off the sky.

The binaries. What halts the collapse is a genuine surprise and it has been observed. As the core contracts, three-star encounters form binary pairs; a subsequent encounter with a third star tightens the binary and ejects the third star at speed. A hard binary is an energy source, converting orbital binding energy into kinetic energy of the cluster, and a handful of them can reverse the collapse of an entire core. Core-collapsed clusters are observed to have enhanced binary fractions and enhanced populations of the objects that binaries make — blue stragglers, millisecond pulsars, X-ray sources — concentrated in exactly the region the theory says the energy is coming from. The same argument at a different mass shows that nothing in it depends on the star being the Sun.

A body that heats up because it is losing energy. A uniform self-gravitating sphere of 5.0 solar masses radiating at 1.0 solar luminosities, with no nuclear source at all, from 3.0 solar radii. Everything is in units of the starting energy, and the two curves that matter run in opposite directions: the total energy falls, and the temperature rises. That is not a paradox and it is not a special case. The virial theorem makes 2K = −U for any self-gravitating gas in equilibrium, so E = U + K = −K, and −dE/dt = +dK/dt: energy leaving as light is energy arriving as heat. The bookkeeping is exact and is measured here rather than quoted — over the run 1.366·10⁴² J of gravitational energy is released and 6.831·10⁴¹ J is radiated, a ratio of 2.0000. Half the release is spent on the star's own heat and only half escapes. The consequence is a body with a negative heat capacity, which is why a contracting protostar gets hotter until it ignites, why a globular cluster's core runs away instead of settling, and why nothing self-gravitating ever comes to thermal equilibrium.
Fig. 4 A five-solar-mass sphere contracting at the same luminosity. Every curve keeps its shape and the timescale shortens, because the available gravitational energy goes as the square of the mass while the radiating area does not. The sign of the temperature change is unaltered; only how quickly it happens is.
The mirror: an envelope 27 times larger around a core 1.35 times smaller. Envelope radius and core radius against helium-core mass for a 1.5 solar-mass star, both in solar radii on one logarithmic axis. Two curves with opposite slopes at every point: the envelope grows from 3.8 to 103 solar radii — 0.48 astronomical units — while the degenerate core contracts from 12,655 to 9,368 kilometres, and the two are separated by a factor of 530 in the middle of the range. The figure is drawn from two stated relations rather than from a stellar model: a core-mass–luminosity law of the sixth power, calibrated to 2,500 solar luminosities at 0.48 solar masses of core, and a Hayashi temperature falling from 5422 to 4013 kelvin across the range. Everything else follows exactly — the radius by the Stefan–Boltzmann law, the core radius by the cold degenerate relation R ∝ M⁻¹ᐟ³. Note what the tip radius is and is not: it is the red-giant branch, and the later asymptotic-giant tip is a different and larger number.
Fig. 5 The mirror for a star half again as massive as the Sun. The envelope still expands by a factor of tens while the core shrinks, and the ratio between the two is if anything more extreme. The behaviour is a property of a shell source above a contracting core, not of any particular mass.

The same sign, three scales up

The argument used nothing about stars. It used a bound system, gravity, and pressure from random motion — so it applies wherever those three hold, and they hold on every scale the subject has.

Galaxies. A galaxy’s stars are a collisionless gas whose “temperature” is its velocity dispersion, and the virial theorem is how a galaxy held up by disorder is weighed. Its negative heat capacity shows up as a rule that sounds backwards and is not: a galaxy that loses energy to dynamical friction, to a merger, or to gas being driven out becomes more strongly bound and more concentrated, and the stars in it move faster afterwards.

Halos. The dark matter halo around a galaxy is a self-gravitating system with no dissipation at all, and that turns out to be the important fact about it. Ordinary gas can radiate, so it loses energy and sinks to the centre, which is why galaxies have discs; dark matter cannot, so it stays puffed up, which is why the rotation curve refuses to fall out to many disc radii. The two components share a potential and behave in opposite ways because only one of them has a way to shed heat.

The mechanism, schematic: 33% of the mass inside 0.045% of the radius. A schematic cross-section of a 1 solar-mass red giant carrying a 0.33 solar-mass helium core, beside the two proportions no single cross-section can hold. The inert degenerate core is 33% of the mass and 0.045% of the 34 solar-radius star, so drawing it at 16 per cent of the drawn radius exaggerates it 356 times over. The scale is therefore broken, the break is stated on the drawing, and the two bars carry the numbers the disc cannot: by radius the core is a line thinner than the one drawn for it, by mass it is a third of the star. Between core and envelope is the hydrogen-burning shell, where all 280 solar luminosities are made, in a layer thin enough that the drawing exaggerates it too. Outside it the envelope convects over nearly the whole radius, which is why a giant's surface can show material processed near the shell.
Fig. 6 And where the energy is actually being made while this happens. Hydrogen burns in a thin shell around the inert core, and the shell’s luminosity is set by the core’s mass and radius rather than by anything about the envelope — so as the core contracts and heats, the shell burns faster and the star gets brighter while the core gets smaller. The envelope’s expansion is a response to that, not a cause of it, which is the whole content of the mirror.

Why thermodynamics objects

A negative heat capacity is not merely unfamiliar; in the framework most of thermodynamics is taught in, it is impossible. It is worth seeing exactly which assumption it breaks.

In the canonical ensemble — a system in contact with a heat bath at fixed temperature — the heat capacity is proportional to the variance of the system’s energy, and a variance cannot be negative. That is a proof, and it is sound: no system in equilibrium with a bath can have a negative heat capacity, because such a system is unstable. Give it a little energy and it cools, so it takes more from the bath, and cools further; the runaway does not stop.

Self-gravitating systems are described in the microcanonical ensemble instead — fixed energy, no bath — and there the proof does not apply. The heat capacity is defined through the entropy’s curvature rather than through an energy variance, and nothing forbids the sign.

So the two ensembles, which agree for every ordinary substance, do not agree here. That failure has a cause: the usual equivalence assumes the system’s energy is extensive, meaning that doubling the amount of material doubles the energy, which requires that a particle interact only with its neighbours. Gravity is long-range and unshielded, so every particle interacts with every other, the potential energy grows faster than the volume, and there is no thermodynamic limit to take.

A self-gravitating system is therefore not a thermodynamic system in the textbook sense, and its strange behaviour is not a curiosity of astronomy but the generic behaviour of long-range interactions. The same non-extensivity, and the same ensemble inequivalence, shows up in plasmas, in some spin models and in two-dimensional fluid turbulence.

The sharpest case of all

The cleanest example of the sign is not a star or a cluster but the one system whose thermodynamics was written down last.

A black hole has a temperature, inversely proportional to its mass, and it radiates accordingly. Radiating removes energy, which is mass; less mass is a higher temperature; so it radiates faster still. The heat capacity is negative, and unlike a star’s there is nothing to arrest the runaway — the evaporation accelerates without limit until the hole is gone.

The numbers make the runaway academic for anything astrophysical. A hole of a solar mass has a temperature of about 60 nanokelvin, far below the 2.7 K of the microwave background, so it absorbs vastly more than it emits and grows rather than shrinks. Only when the background has cooled below that, in some 102010^{20} years, can any stellar-mass hole begin to evaporate at all — and the evaporation then takes 106710^{67} years.

The consequence for the argument is the same as in the previous section, at its most extreme. A black hole cannot be in stable equilibrium with an infinite heat bath: if it is hotter it evaporates away, if it is colder it grows without bound. It can be in equilibrium with a finite one, in a box small enough that emitting radiation raises the box’s temperature faster than it raises the hole’s — which is the standard setting in which black hole thermodynamics is made consistent, and which is the same trick as noting that a star cluster is stable only because it is not in contact with anything.

The system that gets hotter as it loses energy is therefore not an oddity of gravitating gas. It is what gravity does to thermodynamics at every scale where gravity is the dominant interaction, and it is the one place where the textbook proof of a positive heat capacity has to be read for its assumptions rather than its conclusion.

There is a corollary about how the sign should be reported, since “negative heat capacity” invites the wrong picture. Nothing is absorbing heat and getting colder in the ordinary sense. What happens is that the system converts potential energy into kinetic energy faster than it loses energy at the boundary, and the temperature is a measure of the kinetic part alone. Written in terms of the total energy the behaviour is strange; written in terms of the two components separately it is arithmetic, and the strangeness is entirely in the choice of what to call the system’s energy.

It also explains why the effect disappears the moment the system stops being self-gravitating. A gas in a rigid container has a fixed volume, so releasing energy cannot convert potential into kinetic and the heat capacity is positive in the ordinary way. Gravity’s peculiarity is that it supplies the container as well as the contents, and a container that shrinks when the gas cools is the whole of the mechanism.

The distinction also settles a question that comes up whenever the sign is first met: whether a star could be used to build a perpetual motion machine of the second kind, running between it and a colder body. It could not. The star’s temperature rises as it radiates, but the energy it radiates came from its own gravitational field, and the field is a finite reservoir being drained. What looks like a violation is a system in the middle of a one-way process, and the process ends — in degeneracy, in fusion, or in collapse.

The same reasoning disposes of the related worry about the second law: entropy increases throughout, because the energy radiated away is spread into a vastly larger number of low-energy photons than the states the contracting system gives up. Gravitational contraction is an entropy-increasing process, and the temperature rising as it proceeds is not in tension with that.

Where the model stops

The theorem needs three things, and each failure is a real astronomical case.

It needs the system to be bound and in equilibrium. A cluster still forming, or one being tidally stripped by the Galaxy, satisfies neither, and its virial mass is wrong in a direction that depends on which.

It needs the pressure to come from thermal motion of an ideal gas. In a degenerate core the pressure does not depend on the temperature at all, so the feedback loop is cut: the core can lose energy without contracting and can gain it without expanding, which is why a degenerate helium core ignites explosively instead of settling. Degeneracy is the one thing that gives a self-gravitating body a positive heat capacity, and it is the reason white dwarfs simply cool.

And it needs the total energy to be negative and the system finite. In an infinite medium the potential energy is not defined and the argument does not start; that is why the same reasoning cannot be applied naively to the expanding Universe, where the resolution is a different one entirely. Two more readings of the same star make the separation of scales explicit.

The centre that shrank: 4·10⁵ to 1.66·10⁶ g cm⁻³. Central density of the helium core against its mass, for a 2 solar-mass star, from the cold degenerate radius relation. Two factors are drawn and they are not the same factor. Along the giant branch the density rises by 4.2 — exactly the square of the core-mass ratio, because a core whose radius goes as M⁻¹ᐟ³ has a density going as M². The larger step happened before the curve starts: the Sun's centre today is about 150 grams a cubic centimetre and is an ordinary hot gas, so the base of the giant branch is already 2,665 times denser than that, and no part of this curve draws the transition. The relation used is the zero-temperature one, so every density here is an upper bound, and the bound is loosest at the left-hand end where a real core is only partly degenerate.
Fig. 7 The central density of a two-solar-mass star’s helium core through the same evolution. It rises by more than an order of magnitude while the star’s surface is moving outward by two, which is the whole of the mirror stated as one number rather than two curves.
The mechanism, schematic: 33% of the mass inside 0.004% of the radius. A schematic cross-section of a 2 solar-mass red giant carrying a 0.67 solar-mass helium core, beside the two proportions no single cross-section can hold. The inert degenerate core is 33% of the mass and 0.004% of the 326 solar-radius star, so drawing it at 16 per cent of the drawn radius exaggerates it 4,324 times over. The scale is therefore broken, the break is stated on the drawing, and the two bars carry the numbers the disc cannot: by radius the core is a line thinner than the one drawn for it, by mass it is a third of the star. Between core and envelope is the hydrogen-burning shell, where all 17,945 solar luminosities are made, in a layer thin enough that the drawing exaggerates it too. Outside it the envelope convects over nearly the whole radius, which is why a giant's surface can show material processed near the shell.
Fig. 8 The same structure drawn to scale. A third of the mass sits inside four thousandths of the radius, and the burning shell is the boundary between the two regions. Nothing in the picture is a matter of degree: the core and the envelope are separated by four orders of magnitude in radius and behave as two nearly independent objects.

Where this ladder goes next

The base rung used the virial theorem as a weighing instrument; this rung has used it as a thermodynamic statement. The next asks what the theorem’s failure looks like — what a system does while it is still on its way to satisfying it, which is the violent relaxation that turns a collapsing cloud into a galaxy in a few dynamical times rather than the many relaxation times a two-body process would need.

Further along sits the question that the gravothermal catastrophe leaves open and that is still open. Statistical mechanics assumes an entropy maximum exists; gravity provides a system for which it does not, and every attempt to write down a thermodynamics of self-gravitating matter has to decide what to do about that. The astrophysical answer has been to stop looking for an equilibrium and to compute the evolution instead, which works and is not the same as understanding it.

Where this goes next

An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation. An average that weighs what cannot be watched A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not. Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is. A star is held up by its own weight A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it. A cluster weighed three ways, and its stars weighed once. A cluster of galaxies with a velocity dispersion of 1000 km/s, gas at 8 keV and a strong-lensing Einstein radius of 25 arcseconds, each turned into a mass inside 1.5 Mpc by its own relation and nothing else: 7.0, 8.9 and 15.2 × 10¹⁴ M☉. The three assume, respectively, that the galaxies are in equilibrium, that the gas is, and nothing whatever — so their agreement to within a factor of 2.2 is not three restatements of one assumption. The lensing bar is the loosest of the three and is drawn that way deliberately: it measures the mass inside a cylinder of radius 109 kpc, 1.11×10¹⁴ M☉, and carrying that out to 1.5 Mpc as though it were a sphere overstates it. The stars are 2.9 per cent of it. A cluster weighed three ways The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Binary heatingCore-collapseGravothermal catastropheKelvin helmholtz timescaleNegative heat capacityRelaxationSelf-gravitating systemThermal equilibriumThermodynamic limitVirial theorem