A star that has to make its own catalyst

Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, radiation leaks out faster, and the star settles brighter and hotter at its surface. That argument concerned the metals’ effect on how light gets out of a star. There is a second effect, which concerns how the light is made in the first place, and it becomes decisive at the limit the first argument approached: a star with no metals at all.

The first stars formed from gas made in the first few minutes of the universe — hydrogen, helium and a trace of lithium, with no carbon, nitrogen or oxygen whatever. A low-mass star of that composition could burn hydrogen as the Sun does. A massive one could not, at least not at first, because the way massive stars burn hydrogen depends entirely on the elements this gas lacks.

The other half of what the metals do #

The first effect of removing metals, the one already established, is on how light escapes. In a star like the Sun about half the opacity of the interior comes from metals — bound electrons of iron, oxygen and the rest absorbing radiation — and removing them lets the radiation diffuse out faster.

That effect is real and continuous: a star with a hundredth of the Sun’s metals is noticeably brighter at the same mass, and the trend continues toward zero. But it changes the luminosity by tens of per cent, not by factors, and it is the same in kind for stars of every mass. The effect this essay is about is different in kind. It does not modify how the light is carried but whether the star can make it in the usual way at all, and it applies only to stars massive enough to depend on the CNO cycle — where it changes the central temperature by a factor of three.

Two ways to burn hydrogen #

Hydrogen fuses into helium by two routes, and they divide the stars between them. In the Sun the proton–proton chain dominates: protons combine directly, through a slow first step that needs one of them to turn into a neutron, and the rate rises roughly as the fourth power of the temperature near fifteen million kelvin. In stars more than about one and a half times the Sun’s mass the CNO cycle dominates: a proton is captured by a carbon nucleus, then by the nitrogen it becomes, then by oxygen, and the cycle ends by emitting a helium nucleus and returning the original carbon. The carbon, nitrogen and oxygen are catalysts — not consumed, only borrowed — and the cycle’s rate is proportional to how much of them there is.

The CNO cycle has a far steeper dependence on temperature, because its reactions must overcome the larger electric charges of carbon and nitrogen. Near twenty-five million kelvin its rate rises as the temperature to the sixteenth or eighteenth power. That steepness is why it takes over in massive stars: their cores are hotter, and above a crossover temperature the CNO cycle outruns the proton–proton chain by orders of magnitude for the same fuel. It is also why massive stars have convective cores — energy generated that steeply is concentrated at the very centre.

A star that cannot settle #

A massive star made of hydrogen and helium alone begins life the way any star does: a contracting cloud whose centre heats as gravity squeezes it, until fusion supplies the energy the star radiates and the contraction stops. With no catalyst the only route available is the proton–proton chain, and the chain’s shallow temperature dependence makes it a poor thermostat for a luminous star. A thirty solar-mass star radiates a hundred thousand times the Sun’s luminosity, and the proton–proton chain can supply that only at a core temperature near two hundred million kelvin.

The star never gets there, because something else happens first. As the core passes a hundred million kelvin the helium begins to fuse. Three helium nuclei combine into carbon through the triple-alpha process — two of them first forming an unstable beryllium nucleus that lasts long enough to catch a third, a sequence that works only because carbon has an excited state at just the right energy. The triple-alpha rate rises with temperature even more steeply than the CNO cycle’s, roughly as the fortieth power near a hundred million kelvin, so a core heating through that temperature begins to make carbon quickly.

The carbon it makes is the catalyst it lacks. The CNO cycle switches on, its rate proportional to the tiny carbon fraction, and its steep temperature dependence lets even a minute fraction carry the star’s luminosity at a high enough temperature. The contraction stops at the point where the carbon made is just enough.

How quickly the core gets there #

A massive star contracts quickly. Its contraction before fusion takes over is governed by the Kelvin–Helmholtz time, the gravitational energy it can release divided by the luminosity it radiates, and a star of thirty solar masses that is a hundred thousand times as luminous as the Sun has a Kelvin–Helmholtz time of tens of thousands of years — against tens of millions for the Sun. In that time a metal-free core heats past the temperatures at which a present-day star of the same mass would already have stopped contracting, on its way to the hundred million kelvin at which helium begins to fuse.

That is why the carbon in the first figure is counted over a hundred thousand years. It is the order of the time the core spends near the crossing before settling, and the crossing barely depends on it: ten times longer or shorter moves it by about five per cent in temperature, because the triple-alpha rate changes by orders of magnitude over that range. The answer is pinned by the steepness of the reactions, not by the details of the contraction.

The luminosity of a massive star rises less steeply with its mass than the familiar three-and-a-half power, as radiation pressure takes over from gas pressure in supporting it. The figures use a single power of 2.2 above ten solar masses. A metal-free star’s lower opacity raises its luminosity a little at each mass, which raises the energy the core must supply and moves the crossing a little hotter; the shift is small for the same reason as everything else here.

Where it settles #

The first figure finds that point. The carbon needed falls steeply with temperature: at a hotter core, a smaller fraction of catalyst suffices. The carbon made rises more steeply still: a hotter core runs the triple-alpha process exponentially faster. The two curves cross at about a hundred and eight million kelvin, and the carbon fraction there is under two parts in ten billion — a carbon nucleus for every few billion hydrogen nuclei.

The crossing is not sensitive to the details. The time allowed for making carbon, the fraction of the star’s mass that generates the energy, and the luminosity assumed each move the crossing by a few per cent in temperature, because both curves are so steep that a small shift in temperature changes either by orders of magnitude. Detailed calculations of metal-free massive stars, with full stellar structure in place of the scalings used here, find the same answer: central temperatures near a hundred million kelvin and carbon fractions of $10^{-10}$ to $10^{-9}$, reached in the first fraction of a per cent of the star’s life and held thereafter.

That is the surprising connection. A metal-free massive star runs on the same CNO cycle as every massive star of the present day; it simply makes its own catalyst first. The cycle that the present generation inherits as pollution from earlier stars, the first generation manufactures in its own core in the first hundred thousand years, by the reaction that will later make all the carbon in the universe. The first stars burned hydrogen by the light of carbon they had just made, at a temperature most stars reach only when they begin to burn helium near the end of their lives.

Three times hotter at every mass #

The ratio is set by the catalyst. A present-day massive star carries about one per cent of its mass in carbon, nitrogen and oxygen; a metal-free one makes a few parts in ten billion. The CNO cycle’s rate is proportional to the catalyst and steeply dependent on temperature, so a factor of a hundred million in catalyst is made up by a factor of about three in temperature — the sixteenth-power dependence turning a factor of $10^8$ into a factor of a few. At every mass the same arithmetic applies, and the ratio stays near three.

A core three times hotter cannot be the same size. A star is held up by its gas pressure against its gravity, and in a star of a given mass the central temperature measures how compressed it is: halving the radius doubles the temperature. A metal-free star of thirty solar masses with a core three times hotter is about a third the radius of its present-day counterpart.

A smaller, bluer star #

The luminosity of a massive star is set mainly by its mass, and a metal-free star of a given mass is about as luminous as a present-day one — slightly more, because its lower opacity lets radiation out more easily. The same luminosity from a third of the radius means a surface temperature higher by the square root of three, and the metal-free stars of thirty to a hundred solar masses come out at sixty to ninety thousand kelvin at their surfaces in this estimate, approaching a hundred thousand kelvin for the most massive.

A blackbody at those temperatures emits most of its energy beyond 13.6 electronvolts — the energy that ionises hydrogen — and a substantial tail beyond 24.6 electronvolts, which ionises helium. The most massive add photons beyond 54.4 electronvolts, enough to strip helium’s second electron, which present-day stars barely produce. A population of such stars would produce several times more hydrogen-ionising photons per unit of mass than a present-day population, and a distinctive signature: the recombination line of singly ionised helium at 1640 ångströms, emitted by gas around them as the helium recaptures its second electron.

What the first stars did to the universe #

That ionising output is why the first stars matter beyond themselves. They formed in the first few hundred million years, in gas that had cooled after the Big Bang into a neutral state, and their ultraviolet light began the ionisation of the gas between the galaxies that was complete by about a billion years. Every estimate of how quickly that happened depends on how many ionising photons each star made, which depends on how hot its surface was — and the hot surface is a direct consequence of the missing catalyst.

They also ended the era they began. Their short lives — a few million years for stars of tens of solar masses — ended in supernovae that spread the first carbon, oxygen and iron into the gas around them. The very next generation of stars formed from gas that contained some of the catalyst the first lacked, and as soon as the carbon fraction exceeded a few parts in ten billion, a new massive star no longer needed to make its own. A metal content a ten-thousandth of the Sun’s — a catalyst fraction of a millionth — already brings a thirty-solar-mass star’s core down from about a hundred million kelvin to about sixty, and a hundredth of the Sun’s brings it to about forty-four million, ten above a present-day star’s. The regime described here belongs only to the very first stars, and it ended as soon as their debris had been stirred into the gas around them.

Why none has been seen #

No metal-free star has been observed. The most metal-poor stars known, in the halo of the Milky Way, have iron abundances below a ten-millionth of the Sun’s, but they contain carbon, and their compositions look like the debris of one or a few earlier supernovae rather than primordial gas. They are low-mass stars that have lived for thirteen billion years — and a metal-free star of that mass would also have lived that long, so their absence suggests that the first generation contained few or no low-mass stars, or that those few have been hidden by accreting metals from the interstellar gas they have passed through since. It is a sharper version of a long-standing puzzle: there are also far fewer low-metal stars of the Sun’s type than a closed-box history of the Galaxy predicts, and both deficits point to the earliest gas being enriched, or diluted, faster than a simple accounting allows.

The massive metal-free stars are gone, by definition: their lives were millions of years long, thirteen billion years ago. They can be seen only at the distances where their light is still arriving — in galaxies at redshifts of ten or more — and only as populations, not as individual stars. The helium line at 1640 ångströms, the bluest colours, and an unusual ratio of ionising photons to the visible light are the signatures searched for in the spectra of the most distant galaxies, and several tentative detections have been reported in galaxies observed by the latest infrared telescopes, none yet beyond question.

A thermostat with a different setting #

A star on the main sequence is a thermostat: if its core produces too much energy it expands, cools and produces less; if too little, it contracts, heats and produces more. The setting of the thermostat — the temperature at which production matches the luminosity — depends on how steeply the energy generation rises with temperature and on how much fuel and catalyst it has to work with.

A metal-free massive star has the same thermostat with a different setting. Its catalyst fraction is set by its own past, not inherited, and once made it changes little: the triple-alpha process keeps adding carbon at a rate that depends steeply on temperature, so if the core heats slightly more carbon is made, the CNO cycle speeds up, and the core expands and cools. The carbon fraction is itself regulated. The star burns most of its hydrogen at a nearly constant central temperature, as present-day stars do, only three times higher — and its main-sequence lifetime, set by its fuel and its luminosity rather than by its temperature, is nearly the same as a present-day star’s of the same mass.

What the figures leave out #

The stellar structure is homology, not a model. The core density is scaled from the Sun’s as the cube of the central temperature divided by the square of the mass, the luminosity is a single power of the mass, and a fixed fifth of the mass generates the energy. Those are the scalings that make the arithmetic visible; a stellar evolution code gives central temperatures within ten or twenty per cent of these.

The reaction rates are textbook fits. The analytic forms for the three energy-generation rates are accurate to tens of per cent over the temperatures drawn, which moves each crossing by a few per cent in temperature, because the rates are so steep.

The carbon is made over a fixed time. The time allowed for the triple-alpha process to accumulate carbon, a hundred thousand years, is the order of the time a massive star takes to contract; a factor of ten either way moves the crossing temperature by about five per cent and the carbon fraction by a factor of a few.

The surface follows the core. The radius of the metal-free star is scaled from the present-day star’s by the ratio of central temperatures, and its surface temperature from that radius at the same luminosity. Real metal-free stars have different envelope structures, and their surface temperatures in detailed models are similar but not identical.

Still open: the mass the first stars were born with #

The scalings here apply to any metal-free star massive enough to need the CNO cycle. How massive the first stars actually were is the larger open question, and it decides everything about their effect. Gas with no metals cools only through molecular hydrogen, which cannot bring it below a few hundred kelvin; warmer gas fragments into larger clumps, so the first stars are expected to have been born massive. Simulations that follow the collapse further find that the discs around the first protostars fragment into several stars, some of them much smaller, and the resulting mass function may have extended from a fraction of a solar mass to hundreds.

Which part of that range dominated determines how much ionising light the first stars produced, how much of their carbon and oxygen they returned to the gas, and whether any of the smallest survive today. The core temperature of a hundred million kelvin is a consequence of the chemistry, and it is not in doubt. The number of stars that ran at it, and how large they were, is what the searches for the 1640-ångström line in the earliest galaxies are trying to measure.