The number of protons that makes a star

Where every model runs out at once built the Planck scale out of three constants, and found a mass of about 22 micrograms at which gravity between two quantum particles stops being negligible. A proton is about $10^{19}$ times lighter than that. So the gravitational attraction between two protons, measured against the natural unit of force that $\hbar$ and $c$ provide, is the square of that ratio:

$$\alpha_G = \frac{G m_p^2}{\hbar c} \approx 5.9 \times 10^{-39}.$$

It is the weakest number in physics that anyone commonly uses. The electric equivalent, the fine-structure constant, is about $1/137$. Compared with that, gravity between two protons is $10^{36}$ times feebler. It would seem that a number like that has no consequences at all, that it can be dropped from any calculation of anything. It can, for atoms, for molecules, for the things on a kitchen table. Yet raise it to the power minus three halves, and it gives $2.2 \times 10^{57}$, which is very close to the number of protons in the Sun.

That is not a coincidence, and the argument that shows why is short. It is an argument about orders of magnitude, in which the numerical factors of a few are not to be taken seriously, and it needs no theory of how a star is built. It takes a ball of gas and asks two questions. How few protons can such a ball have and still reach fusion? And how many can it have before light takes over? The answers turn out to be, up to modest factors, the same number.

Why gravity needs so many #

The first thing to get straight is why the number has to be large at all. The gravitational binding energy per proton in a ball of $N$ protons with spacing $d$ between them goes roughly as $G m_p^2 N^{2/3}/d$. The factor $N^{2/3}$ is there because the ball’s radius is about $N^{1/3} d$, while its mass is $N m_p$. A proton in the ball feels a mass of $N m_p$ at a radius $N^{1/3} d$, so its potential energy has $N$ on top and $N^{1/3}$ below.

Compare that with what the same protons could do through any other force. At the same spacing the electric energy of a pair is about $\alpha\hbar c/d$. For gravity to compete with that, the factor $\alpha_G N^{2/3}$ would have to reach $\alpha$, which it does near $N \sim (\alpha/\alpha_G)^{3/2} \approx 10^{54}$. That is the scale of Jupiter. It is why the size at which a body becomes round is a scale for rocks and why a giant planet is the largest cold object that grows as more is added. Gravity is hopeless on small scales, and the only way it can do anything is to involve very many particles. The factor of $2/3$ in the exponent is how the number gets its power of $3/2$.

The floor: a ball that is not hot enough #

Now take the ball and let it contract. As the ball of gas that heats up as it cools showed, a self-gravitating ball heats as it shrinks. The virial theorem sets its thermal energy per particle at about the gravitational binding per particle, which rises as spacing falls. Left to itself, the ball would get hotter without limit. What stops it, in a ball with no fusion, is that the electrons are being squeezed. The exclusion principle gives each electron a kinetic energy of about $\hbar^2/(2 m_e d^2)$ just from being confined to a volume $d^3$, whatever its temperature, as the pressure that is not a temperature explained. Energy spent on that crowding does not appear as heat. So the thermal energy per particle is about

$$kT \approx \frac{G m_p^2 N^{2/3}}{d} - \frac{\hbar^2}{2 m_e d^2}.$$

The first term grows as $d$ falls, the second grows faster, and the difference has a peak.

Setting the derivative to zero gives the hottest such a ball can get:

$$kT_{\max} \approx \tfrac{1}{2}\,\alpha_G^2\, N^{4/3}\, m_e c^2.$$

It has no $G$ or $\hbar$ apart from what sits in $\alpha_G$. The electron’s rest energy is the only other scale, and that is not because electrons are moving at any large speed; it is a way of writing $\hbar^2/(m_e d^2)$ with $d$ eliminated. A direct search over spacing confirms that the peak of the curve and the formula agree to about a per cent.

Protons fuse, at a rate that matters to a star, near a temperature whose thermal energy is about a kiloelectronvolt, ten million kelvin. That temperature is itself interesting. It is much lower than the Coulomb barrier between two protons would suggest, because of the wall that is not quite a wall, and a good deal of the physics of quantum tunnelling and of nuclear rates goes into its value. Fortunately the answer depends on it only weakly. Put the kiloelectronvolt in the formula and solve for $N$.

The calculation has no right to be that close. It has factors of a few at every stage: nothing was said of the star’s density profile, or of the fact that deuterium burns at a lower temperature, and a real calculation of the hydrogen-burning limit takes a stellar interior code. Measured against that, a factor of about three is a pleasant surprise and should not be over-read. What the sum establishes is the structure. A ball below this count contracts until electrons stop it, and never fuses protons. Those exist; they are brown dwarfs, cooling slowly from the heat of their contraction, and the transition is observed at about 0.075 solar masses. A body like that is a failed star for exactly this reason, and it fails at its peak temperature, without any nuclear burning to explain.

The ceiling: a ball held up by light #

At the other end, no fusion is involved at all. The pressure that weighs down what it holds up and the brightness a mass cannot exceed both approached this limit from the point of view of what holds a star up. Here the question is simply how the shares of pressure divide. The gas pressure in a star is about $nkT$, and the radiation pressure is about $aT^4/3$. For a star in hydrostatic balance, a star’s central pressure scales as $G M^2/R^4$. With the gas taken as ideal, those relations can be combined so that the radius falls out and the ratio of the two pressures depends only on the mass.

Eddington worked this out in 1926, using a star built as a polytrope of index three. If $\beta$ is the fraction of the pressure from gas, then

$$\frac{1 - \beta}{\beta^4} = 0.003 \left(\frac{M}{M_\odot}\right)^2 \mu^4,$$

where $\mu$ is the mean mass per particle in units of the proton’s, about 0.6 for an ionised star of cosmic composition. Eddington called it the most important relation in his book. The coefficient is not arbitrary: written out, it is a pure number times the combination $(N/N_\star)^2$, where $N_\star = \alpha_G^{-3/2}$ is the same number, and the only constants that survive the cancelling are $G$, $\hbar$, $c$ and the proton’s mass. Radiation’s constant $a$ has $\hbar$ and $c$ in it, and its Boltzmann factors cancel those of the gas.

The picture is the thing to notice. There is no sharp edge, only a smooth climb, from a negligible share at a solar mass to half at about 142. But a star dominated by radiation is close to the edge of instability, for a reason the mass no cold matter can hold up gave for white dwarfs: a gas whose particles move near the speed of light has pressure proportional to the four-thirds power of density, and four-thirds is exactly the exponent at which gravity and pressure scale alike. Nothing then resists a change of radius. The Chandrasekhar mass, about 1.4 solar masses for a white dwarf, comes out proportional to the same $\alpha_G^{-3/2}$, for the same reason, since in that case the relativistic particles are electrons held by exclusion, and here they are photons.

So a very massive star has a ceiling. The observed upper limit is near 150 solar masses, and some reported stars may reach 200 or more with some difficulty about how to count them, and they shed mass furiously in stellar winds. It is not a law as sharp as the Chandrasekhar limit. The word ceiling here is a shorthand for a steep and growing difficulty, and nothing in the quartic forbids a star of 300 Suns from existing for a while.

A window that slides without changing shape #

Now the point of the exercise. The floor depends on $\alpha_G$ through $\alpha_G^2 N^{4/3}$ being held constant, so $N_{\min} \propto \alpha_G^{-3/2}$. The ceiling depends on $(N/N_\star)^2$ being held constant, so $N_{\max} \propto \alpha_G^{-3/2}$ as well. They came from completely different physics, one from electron crowding and fusion and the other from the balance of light and gas, and both have the same exponent.

Change the strength of gravity and the window moves as a unit. With gravity a thousand times stronger, stars would be $3 \times 10^4$ times less massive, and they would still come in a range of the same width. A proton count of a star is therefore the reciprocal of the weakness of gravity, to the three-halves power, multiplied by factors that depend on the fine-structure constant, on the mass ratio of electron and proton and on what happens at the fusion temperature, all of which enter at low power or logarithmically.

That is the reason stars are all roughly the same. They have the same temperatures, within a factor of ten, and the same central densities in a similar range. The question of why the visible mass of the universe comes in such uniform lumps has an answer with very little astrophysics in it. Clouds fragment in complicated ways, and a great deal is not understood about why stars form with the distribution of masses they do. But the range within which they can do it is set before any of those processes start.

Bigger numbers from the same source #

The same number keeps turning up. A star’s lifetime on the main sequence, estimated by dividing its energy reserve by its luminosity, comes out near $\alpha_G^{-1}$ times the fundamental unit of time $\hbar/(m_p c^2)$, multiplied by factors of a few powers of $\alpha$. That works out to around $10^{10}$ years. Brandon Carter and later Bernard Carr and Martin Rees noted that much of the macroscopic world follows in this way. The mass of a planet, the height of a mountain and the size of a person have been estimated from $\alpha$ and $\alpha_G$ alone, in papers that have an irresistible quality of telling the reader how tall they should be.

A word of caution belongs with those estimates. They work in the sense that they get the right power of ten. They do not derive the fine structure of what they describe, and their success is partly due to the freedom in choosing which factors of a few to leave out. A prediction that may be wrong by a factor of three on a logarithmic plot looks exact, and the eye forgives what an experimenter would not.

And there is a question that the calculation leaves untouched. Why is $\alpha_G$ so small? In the language of the nineteen decades a slow coupling buys, the proton’s mass comes from the running of the strong force and has no direct connection to gravity, and so the question is why the scale at which the strong coupling grows strong sits nineteen decades below the Planck mass. The logarithm there is what makes a large ratio from a modest one. So this smallest number in physics is itself explained as a smooth consequence of a coupling of about 1/40 at high energy. The ratio is then set, and stars inherit it.

A related check comes from an unexpected direction. The Chandrasekhar mass and the ceiling of the window both have the same dependence on $\alpha_G$, and both are about relativistic particles holding up a ball. The hole that outlives everything and then does not has the Planck scale entering by another route, at the other end of a star’s life. Across the whole of stellar astrophysics, only the numerical coefficients change from one case to another. The power of the coupling stays the same, and that is the sign that a single scale is at work.

What the arithmetic is not saying #

The sums here all stand for estimates. No density profile was solved for. The kinetic term assumed a non-relativistic gas of electrons throughout. The fusion temperature was a round number, a kiloelectronvolt, and the true threshold depends on which reaction is in question and on the composition. The real floor at 0.075 solar masses comes from detailed calculations of opacity and energy transport, and it depends on metallicity. The figure gives only the structure of the dependence.

The quartic relation assumes that a star is a polytrope with index three throughout and that its mean composition stays fixed. Real massive stars have convective cores, strong winds and a distribution of composition that changes as they burn. Eddington’s number should be read as a good guide to where radiation becomes dominant, and not as the actual mass of the largest possible star, which is uncertain. Reports of stars of several hundred solar masses have been revised downward after better measurement, and the question of the true upper limit has not been closed.

The figure of varying gravity is a calculation within this simple model, holding all other constants fixed. It says nothing about whether the coupling can vary or whether anything else would stay the same if it did. A universe with stronger gravity would also collapse and form structure differently, and any actual argument about other universes would have to take all of that on. The window’s sliding is a statement about arithmetic.

Nor does any of it say how many stars of each mass a cloud will make. The observed mass function peaks at a few tenths of a solar mass and falls steeply above, for reasons of turbulence, fragmentation and heating that have nothing to do with these limits. A window says what is possible and has no opinion about what is common.

Still open: whether the first stars used the same window #

The first stars, with no heavy elements to help them cool, are thought to have been unusually massive, perhaps ten to several hundred times the Sun, a range calculated by simulation and not yet observed directly. Their window is the same one, because neither limit depends on composition except through modest factors. But the part of the window in which they formed may have been its upper end, because the gas could not fragment into small clumps. Whether the very largest of them passed the pair-instability boundary, where radiation so hot that it creates electron–positron pairs saps the pressure that holds the star up, and exploded leaving no remnant, is a matter of active searching. Telescopes looking for the chemical signature of such an explosion in old stars have found some candidates, though nothing is agreed.

There is also the top end of the observational question. If a star was truly seen above the point where the quartic and instability make it hard to exist, then the ceiling would need a mechanism to delay the loss of mass, or the star would turn out to be a merger or an unresolved pair. High-resolution imaging has already split some of the most famous massive stars into several, and that work goes on.

Gravity between two protons is $5.9 \times 10^{-39}$ of the unit that $\hbar$ and $c$ set, and its power of minus three halves, $2.2 \times 10^{57}$, sits between a floor at about $3 \times 10^{55}$ from a peak temperature and a ceiling near 142 solar masses from the rising share of radiation pressure; change the coupling, and floor and ceiling slide together, keeping the same width. A star’s mass looks like an accident of a cloud. Its range is arithmetic. The habit worth keeping is to look, when something in nature comes in a suspiciously narrow range, for the dimensionless number that would set it, since an honest order of magnitude can be worth more than a detailed model that hides why the answer came out that way.