Mechanics sets a star's temperature before fusion does

A body in hydrostatic equilibrium is one in which the pressure at every depth supports the weight of everything above it. That condition alone sets a floor under the central pressure: any body of mass $M$ and radius $R$, whatever it is made of, must have a central pressure of at least $GM^2/8\pi R^4$. The floor is rigorous and weak, and its usefulness lies exactly in its assuming nothing about the material.

A pressure is not yet a temperature. To go from one to the other needs an equation of state, and the most general statement that can be made without choosing one in detail is that the material is an ideal gas — atoms or ions and electrons moving freely, with pressure proportional to density times temperature. For a star that is very nearly true; for many other bodies it is not. Given that one assumption, and nothing about the star’s composition, energy source or structure, hydrostatic equilibrium sets a floor under the star’s mean temperature too. The floor says a star of the Sun’s mass and radius must average at least two million kelvin inside. Before anyone knew where the Sun’s energy came from, a calculation of this kind was the first evidence that the inside of a star is hot enough for nuclear reactions.

The virial theorem, and the least bound arrangement #

Multiply the equation of hydrostatic equilibrium by the radius and integrate over the whole star, and it becomes the virial theorem: the integral of three times the pressure over the star’s volume equals minus its gravitational binding energy,

$$\int_0^M \frac{3P}{\rho}\,dm = -W = \int_0^M \frac{Gm}{r}\,dm.$$

For an ideal gas $P/\rho = kT/\mu m_H$, where $\mu$ is the mean mass of a particle in units of the hydrogen atom’s, so the left-hand side is $3k/\mu m_H$ times the mass-averaged temperature, times the mass. The right-hand side is the binding energy, and it has a lower limit. Every shell of mass $dm$ at radius $r$ contributes $Gm\,dm/r$, and $r$ is never more than $R$; so the binding energy is at least $\int Gm\,dm/R = GM^2/2R$, the value it would have if all the mass were in a thin shell at the surface. Putting the two together,

$$\langle T\rangle \ \ge\ \frac{\mu m_H}{6k}\,\frac{GM}{R}.$$

The inequality is sharp in one direction only. The thin shell is the least bound way to arrange a given mass within a given radius, and every real body is more concentrated and more tightly bound: a uniform sphere by a fifth more, a fully convective star by three-quarters more, a star with the Sun’s degree of central concentration by three times. Since the mean temperature of an ideal-gas star is proportional to its binding energy, real stars are hotter on average than the floor by the same factors. The floor is weak because it assumes the least concentration possible, and it is useful because it assumes nothing else.

Two million kelvin, whatever the Sun is made of #

The floor depends on composition only through the mean molecular weight, and the lightest possible gas — fully ionised hydrogen, a proton and an electron sharing the mass of one hydrogen atom — has $\mu = 1/2$. Any other composition has a larger $\mu$ and a higher floor. So the smallest floor a body of the Sun’s mass and radius can have, whatever it is made of, is 1.9 million kelvin, and for the Sun’s actual mixture it is 2.4 million. The Sun’s real mass-averaged temperature, from models, is about three times higher, as the index-three polytrope that approximates it predicts, and its central temperature is fifteen million kelvin.

Along the main sequence the floor changes slowly, because a star’s radius grows almost in proportion to its mass: every star from a tenth of the Sun’s mass to fifty times it must average at least a million or so kelvin inside. That settles two things about stars at once, without any nuclear physics. Their hydrogen is ionised throughout their interiors, because hydrogen is fully ionised well below a million kelvin at stellar densities. And their interiors are within the range in which nuclear reactions between hydrogen nuclei proceed at a meaningful rate, which requires temperatures of millions of kelvin because of the electrical repulsion between the nuclei.

The argument that came before fusion #

The historical order was exactly this. In the 1920s Eddington worked out the interior temperatures of stars from their masses, radii and the equations of equilibrium, and concluded that the centre of the Sun is at tens of millions of kelvin — a figure he could derive without knowing what the Sun burns. Physicists of the time calculated that nuclei would need even higher temperatures to overcome their mutual repulsion, and objected that the stars were not hot enough for the process Eddington suspected. His reply was that the stars were as hot as mechanics required, and that if the physics said they were too cool for fusion, the physics would have to find a way; it did, when quantum tunnelling was understood a few years later. The temperature was established first, by the argument drawn here and its refinements, and the nuclear physics had to fit it.

The episode is a useful model of how a bound earns its keep. Eddington’s temperatures depended on assumptions about the opacity and composition of stellar interiors that turned out to be partly wrong — he assumed far more heavy elements than the Sun contains — and his detailed numbers moved as those were corrected. The order of magnitude did not move, because the virial argument underneath it needs neither: any interior that holds up a solar mass within a solar radius with the pressure of a gas is at millions of kelvin. The refinements changed the factor of three between the floor and the truth, never the fact that the floor was millions of kelvin.

The same logic runs every time a star contracts. A self-gravitating ball of gas gets hotter as it loses energy: radiating heat away lets it contract, contraction deepens its binding, and the virial theorem then demands a higher mean temperature. The floor rises as the inverse of the radius. A contracting protostar heats up until its interior reaches the temperatures at which hydrogen burns, and then stops contracting, because the energy radiated is replaced. The floor is the mechanical half of why stars ignite.

Where the floor is impossible #

The floor assumes ideal-gas pressure, and applied to bodies that are not held up by ideal-gas pressure it gives answers they cannot meet. That is its second use.

The Earth, treated as an ideal gas of its own composition, would have to average nearly forty thousand kelvin inside to hold itself up; its interior is a few thousand. Jupiter, as an ideal gas of hydrogen and helium, would have to average over eighty thousand; it is nearer fifteen or twenty. A white dwarf of 0.6 solar masses, as an ideal gas of carbon and oxygen nuclei and electrons, would have to average three hundred million kelvin; its interior is perhaps ten million. Each of these bodies misses the floor by factors of several to forty. Since the floor follows from hydrostatic equilibrium and the ideal-gas law alone, and the bodies are plainly in equilibrium, the ideal-gas law must be what fails.

What holds them up instead is known in each case. The Earth is held up by the stiffness of rock and iron, whose pressure comes from the resistance of atoms’ electron shells to being squeezed together and hardly depends on temperature. Jupiter’s interior is hydrogen compressed until its electrons are partly degenerate and the electrostatic interactions between particles matter as much as their motion. The white dwarf is held up almost entirely by degenerate electrons, whose pressure comes from the exclusion principle rather than from heat. The floor did not know any of that. It showed that each body needs some pressure that is not thermal, and by how much, from its mass and radius alone.

The brown dwarf sits in between, missing the floor by a factor of two — partly thermal, partly degenerate. That borderline position is not an accident, and it is where the last figure leads.

When light carries part of the weight #

In massive stars the pressure is not all gas pressure. The radiation in a hot interior exerts a pressure of its own, proportional to the fourth power of the temperature, and in the most massive stars it supports a large part of the weight. Eddington’s standard model gives the share in closed form: negligible for the Sun, a few per cent at ten solar masses, a third at sixty. The virial integral then splits into a gas part and a radiation part, and the floor on the gas temperature is lowered in proportion to the gas’s share.

The split matters because radiation is a poor supporter. A gas’s contribution to the virial balance grows as it is compressed, which is why a contracting gas star is stable; radiation’s contribution, per unit of internal energy, is only half as large, and a star supported mostly by radiation is barely bound — a small disturbance can unbind its outer layers. That is one form of the argument that stars cannot be arbitrarily massive: as the mass rises, radiation’s share approaches one, the star’s binding approaches zero, and above a hundred or so solar masses stars lose mass in eruptions and winds faster than they can hold it. A brightness that would blow the star apart is the same limit approached from the surface.

The least mass of a star #

The floor rises as a ball of gas contracts, but a ball cannot contract forever. As it shrinks, the electrons in it are squeezed together, and when their spacing becomes comparable to their quantum wavelength — when the Fermi energy at the ball’s mean density rises to the thermal energy — they become degenerate and resist further compression with a pressure that does not need heat. From then on the ball contracts only slowly, and as it radiates it cools rather than heating. The largest temperature it ever reaches is the one it had when degeneracy took over.

The two quantities scale differently. The virial temperature rises as the inverse of the radius; the Fermi energy at the mean density rises as the inverse square. They meet at a radius that falls with mass, and the temperature at the meeting rises as the four-thirds power of the mass. A massive ball reaches the temperatures of hydrogen burning long before degeneracy intervenes, and becomes a star. A light one is caught by degeneracy first and never gets hot enough: it becomes a brown dwarf, glowing from its contraction and then cooling for the rest of time. The crossing in this order-of-magnitude argument falls at a twentieth of a solar mass; detailed models, which follow the centre rather than the average and use the full equation of state, put it at 0.075 solar masses. The argument gets the scale from nothing but the virial floor and the exclusion principle, and the brown dwarf that misses the ideal-gas floor by a factor of two is a body sitting right at that boundary.

A floor that rises as a star ages #

The molecular weight in the floor is not fixed over a star’s life. Every four hydrogen nuclei fused into one helium nucleus reduce the number of particles in the core — eight particles, four protons and four electrons, become three, a helium nucleus and two electrons — and so raise the mean molecular weight from about 0.6 towards 1.3. At fixed mass and radius, the virial floor rises in proportion. The core cannot simply stay as it was: with fewer particles to share the thermal energy the virial balance requires, it must either heat up or contract, and in practice it does both, slowly, over the main-sequence lifetime. A hotter, denser core burns hydrogen faster, and the star brightens.

That is why the Sun is about thirty per cent brighter now than when it formed, and why every main-sequence star brightens steadily as it ages, with no change in its mass. The effect enters the steep dependence of a star’s luminosity on its molecular weight — the fourth power in the simplest scaling — and it begins with the virial floor: the temperature mechanics demands rises as the gas becomes heavier per particle, and the star has to comply.

The same floor for a cluster of galaxies #

The argument does not care what the particles are or how many there are, only that they are in equilibrium in their own gravity. A cluster of galaxies is the largest example. Its mass, most of it dark matter, holds a diffuse atmosphere of hot gas in equilibrium, and the virial theorem applied to that gas gives a temperature of order $\mu m_H GM/kR$ for the cluster’s mass and radius — tens of millions of kelvin for a cluster of a thousand trillion solar masses spread over a couple of megaparsecs. The gas is observed at exactly those temperatures, glowing in X-rays, and its temperature is one of the standard ways a cluster is weighed: measure the temperature and the size, and the virial relation gives the mass.

The logic is the same in both directions. For a star, a known mass and radius give a minimum temperature; for a cluster, a measured temperature and radius give a mass. And the same caveat applies: the relation holds for gas in equilibrium supported by its own thermal pressure, and a cluster whose gas is partly supported by turbulence or magnetic fields — as most are, by some tens of per cent — has a temperature that underestimates its mass, which is one of the main uncertainties in using clusters to measure cosmology.

What the floor does not say #

The floor is a statement about the mass-averaged temperature, not about the temperature anywhere in particular: a star could in principle be very hot in a small region and cool elsewhere and satisfy it. It assumes the gas is fully ideal and ignores the energy of ionisation, which in the outer layers of cool stars absorbs some of the heat and changes the relation between pressure and temperature. It uses the least bound arrangement of mass, which no real star has, so it is always a factor of a few below the truth. And for rotating or magnetised bodies, the virial theorem acquires extra terms — rotational kinetic energy, magnetic energy — that share the support with pressure and lower the temperature the gas needs; for most stars they are small, and for rapidly rotating or strongly magnetised ones they are not.

A bound that was a prediction #

It is worth being clear about what kind of result this is. The floor is not a model of a star. It is what can be said about any star before a model is built, from its mass and radius and the assumption that it is a gas — and what it says is enough to establish that stars are hot enough inside for hydrogen to be ionised and for nuclear reactions to be possible, that the rocky and giant planets and the white dwarfs are not held up by heat, and roughly where the boundary between stars and brown dwarfs must lie. Supplying an equation of state turns the inequality into an equality and the floor into a temperature profile. The inequality is how much of that profile is fixed by mechanics alone: for the Sun, the average to within a factor of three.

Still open: how hot the smallest stars really are inside #

The boundary between stars and brown dwarfs is set by a competition between thermal and degenerate pressure at temperatures where both matter and the equation of state is at its least certain: hydrogen partially degenerate, strongly coupled by electrostatic forces, near the densities at which it may become metallic. The models that put the limit at 0.075 solar masses depend on that equation of state and on the opacities of the cool atmospheres that regulate how fast the objects lose heat, and the measured masses of the lowest-mass stars and most massive brown dwarfs in binaries have sometimes disagreed with the models by more than their errors. Whether the models’ interiors are as hot as they say — whether the smallest stars reach the temperatures the floor and the models require, or burn hydrogen in a way the models do not capture — is being tested by the growing sample of such binaries with dynamical masses, one pair at a time.