Why the biggest stars die first, and take the galaxy with them

More fuel ought to mean longer life. It is the intuition behind every fuel tank ever built, and for stars it is wrong by a factor of tens of thousands.

The reason is that a star’s mass sets both the size of the tank and the rate of consumption, and it sets the second far more steeply than the first. Fuel goes as $M$. Consumption goes as $M^{3.5}$. The ratio — the lifetime — goes as $M^{-2.5}$, and the minus sign is the whole story.

The arithmetic #

A star’s main-sequence lifetime is the fuel divided by the burn rate:

$$t \approx \frac{fMc^2\epsilon}{L} \propto \frac{M}{M^{3.5}} = M^{-2.5}.$$

Only the core’s hydrogen is available — about 10% of the star for most masses — and fusion converts 0.7% of that mass to energy. The Sun’s figure comes out at about ten billion years, of which four and a half have passed.

Scaling from there:

A thirty-solar-mass star has thirty times the fuel and gets through it in five million years — a two-thousandth of the Sun’s span. Human civilisation is about a two-thousandth of that. A star can be born, live and explode in a time that would be a geological eyeblink.

At the other end the numbers stop being meaningful in a different way. A 0.2 solar-mass red dwarf has a computed lifetime of 400 billion years, which is thirty times the current age of the universe. Not one has ever finished. Every red dwarf ever formed is still on the main sequence, and the statement about their lifetimes is a prediction that will not be tested.

The clock in a cluster #

The steepness makes stellar lifetimes into a dating method, and it is the main one available for objects older than any radioactive clock can reach.

Stars in a cluster all formed at about the same time. As the cluster ages, the most massive stars leave the main sequence first, then the next most massive, and the top of the sequence erodes downward. The mass at the top edge — the turn-off — is the mass whose lifetime equals the cluster’s age. Because the lifetime relation is steep, the method is sensitive. A modest change in turn-off mass corresponds to a large change in age, and the turn-off is a sharp feature in a colour–magnitude diagram rather than a gradual one. Young open clusters like the Pleiades still have their B stars — hot, blue and obvious on any colour measurement — and come out around 100 million years. Globular clusters have turn-offs near 0.85 solar masses and come out at 12–13 billion years.

That last number has been consequential. For much of the 1990s the best globular ages exceeded the best estimates of the age of the universe, which is a straightforward contradiction. It was resolved from both ends — Hipparchus’s revision of the distance scale lowered the ages, and the discovery of accelerated expansion raised the universe’s age — and the current numbers agree with about a billion years to spare. The distance scale was the larger part of the correction, which is a reminder of how far an error at the bottom of the ladder travels.

Rare, brief, and in charge #

Massive stars are scarce. The initial mass function falls steeply — roughly as $M^{-2.35}$ — so for every star above 20 solar masses there are several thousand below one. They are also short-lived, so at any moment there are very few of them.

They nevertheless dominate almost everything about a galaxy.

Light. A single 30-solar-mass star outshines 65,000 Suns, and its output peaks in the ultraviolet. A handful of them outshines a million ordinary stars, so the visible light of a star-forming galaxy comes overwhelmingly from a tiny minority. What a distant galaxy looks like is a statement about its massive stars and almost nothing else.

Chemistry. The universe began with hydrogen, helium and a trace of lithium. Every other element was made in stars, and the heavy ones were made in massive stars and their supernovae. Carbon, oxygen, silicon, iron — the whole of the periodic table past helium — was assembled in objects that lasted a few million years and then dispersed their contents.

Structure. Their winds and explosions stir the interstellar medium, compress clouds into new stars, and drive gas out of small galaxies entirely. Star formation regulates itself through the deaths of the stars it makes, and the regulator is the massive minority.

The Sun contains about 1.4% of elements heavier than helium, which means about 1.4% of the Sun was once inside a star that had already died. The same is true of the Earth, and of everything on it. The carbon in a body was made in a star that lasted a few million years and exploded before the solar system existed.

The division of labour between mass ranges is sharper than that summary suggests, and it is worth having. Oxygen, neon, magnesium and silicon come almost entirely from core-collapse supernovae — massive stars, so they appeared early and fast. Iron comes about half from those and half from type Ia supernovae, which involve white dwarfs and therefore lag by a billion years or more. Carbon and nitrogen come substantially from intermediate-mass stars shedding envelopes on the asymptotic giant branch, on timescales of hundreds of millions of years. The elements heavier than iron split again, between slow neutron capture in those same giants and rapid capture in neutron-star mergers.

The consequence is that the ratio of one element to another is a clock. A star whose iron is low relative to its oxygen formed early, before the type Ia supernovae had contributed; measuring that ratio in stars across the galaxy reconstructs the order in which its parts were assembled. Chemistry is used as chronology, and the reason it works is entirely the spread in stellar lifetimes described above.

What was actually measured #

None of these lifetimes has been watched. The longest astronomical record is four thousand years; the shortest stellar lifetime is a million.

What is measured is a population, and measuring one requires a distance, a brightness and a colour for every star in it. A cluster’s colour–magnitude diagram shows the turn-off; the models supply the mapping from turn-off mass to age; the age follows. That chain has two inputs that are measurements — the turn-off position and the cluster’s distance — and one that is theory.

The theory is checked where it can be. Stellar structure models predict the Sun’s present luminosity, radius and neutrino output; all three are measured, and the neutrino count was famously a third of the prediction for thirty years. That deficit turned out to be a property of neutrinos rather than of the Sun, and its resolution — neutrino oscillation — was a discovery in particle physics obtained by counting the products of nuclear reactions in a star’s core.

There is also direct evidence at the fast end. SN 1987A exploded in the Large Magellanic Cloud, and the progenitor had been photographed: a blue supergiant of about 20 solar masses. A star of that mass should live around 10 million years, and the surrounding stellar population is about that age. It is not a measurement of a lifetime, but it is a consistency check on one, done with the star’s own corpse.

Three clocks that do not depend on each other #

The turn-off is one clock and it has a theory inside it. What makes stellar ages believable is that two other clocks, built on entirely different physics, give the same answers.

The first is the star’s own ringing. A star is a resonant cavity. Convection near the surface excites sound waves that travel inward, refract, and return, and the modes that fit produce brightness variations of a few parts per million with periods of minutes. Measuring their frequencies is asteroseismology, and it gives two quantities almost model-independently: the large frequency separation, which scales as the square root of the mean density, and the frequency of maximum power, which scales with surface gravity and temperature. Density and surface gravity together give a mass and a radius.

That is the important part. An age follows from a mass and a radius, because a star’s radius grows measurably as its core hydrogen is consumed — and unlike the turn-off method, it works on a single field star with no cluster required. The Kepler mission observed several thousand stars precisely enough to do it, and the resulting ages for individual stars in the galactic disc have uncertainties of 10–20%. More finely, the small frequency separation is sensitive to the composition gradient in the core, which is a direct probe of how much hydrogen is left. The clock is being read off the interior rather than inferred from the surface.

The second is a cooling body. A white dwarf produces no energy. It is a hot object of known mass and known radius radiating into space, and the physics of how fast it cools is ordinary thermodynamics with a well-understood equation of state. The faintest white dwarfs in a cluster are therefore the oldest, and the cut-off at the faint end of the white-dwarf sequence gives an age with no stellar evolution model in it at all — only cooling. Applied to globular clusters, white-dwarf cooling ages agree with turn-off ages to within about a billion years, and the two methods share almost no assumptions beyond the distance.

The third is radioactive. A handful of extremely metal-poor stars show absorption lines of thorium and uranium, elements with half-lives of 14 and 4.5 billion years. Comparing their abundance with that of a stable element produced by the same process gives an age for the material directly, in exactly the way a rock is dated. The measurement is very hard — the uranium line is faint and sits in a crowded region of the spectrum — and it has been made for a few stars. It gives ages of 12 to 14 billion years, consistent with the clusters.

Three clocks, resting on stellar structure, on thermodynamics, and on nuclear decay respectively. That they agree is the reason a number like “13 billion years” is quoted without hedging, and it is a better argument than any one of them makes alone.

What the short lives leave behind #

The stars that finish quickly are the ones that change everything else, and their output is visible in the light of the galaxies they sit in. That is why the colour of a distant galaxy dates its star formation. A galaxy still forming stars is blue, because its massive stars are present; one that stopped is red, because they are gone and only the long-lived remain. Colour as a thermometer becomes colour as a clock, on a scale of galaxies. There is a pleasing circularity in that. The ladder is climbed on stellar deaths, and the calibration of those deaths depends on cluster ages, which depend on the turn-off, which depends on the lifetime relation above. Astronomy’s distance scale and its stellar physics hold each other up.

The masses that leave nothing #

At the very top of the mass range the story changes qualitatively, and the change produces a gap in the population that has been observed.

Above about 130 solar masses, a star’s core becomes hot enough that photons begin producing electron–positron pairs. Making a pair consumes energy that was supporting the star, so the pressure drops just when it is most needed; the core contracts, oxygen ignites explosively, and the star is destroyed completely. A pair-instability supernova leaves no remnant at all — no neutron star, no black hole, just the dispersed products.

Between roughly 65 and 130 solar masses the instability is partial: the star pulses violently, sheds shells and survives, arriving at core collapse with much less mass than it started with.

The consequence is a black hole mass gap. Stars in that range cannot leave black holes between about 65 and 130 solar masses, because either the star is blown apart entirely or it has shed the mass first. Gravitational-wave observations of merging black holes now number in the hundreds, and the mass distribution shows a deficit in roughly that range — a prediction from stellar structure, confirmed by an instrument that detects nothing electromagnetic.

The same curve read with different masses marked is the whole of what the relation is used for.

Two clocks from the same population #

The steepness of the lifetime relation has a use that has nothing to do with dating anything: it makes different parts of a galaxy’s spectrum sensitive to star formation over different intervals.

Only the most massive stars produce enough ultraviolet to ionise hydrogen, and they live about ten million years. So the strength of the hydrogen recombination lines coming out of a galaxy measures how many such stars exist now, which measures the star formation rate averaged over the last ten million years and no longer.

The ultraviolet continuum comes from a broader range of masses, down to a few solar masses, whose lifetimes reach a hundred million years or more. The same galaxy’s ultraviolet brightness therefore reports an average over that much longer interval.

Comparing the two is a measurement of whether the rate has been steady. A galaxy whose line emission is weak relative to its ultraviolet has been forming stars and recently stopped; one where the ratio runs the other way has just started. That comparison is a standard diagnostic, and it is entirely a consequence of the relation in this essay: two tracers, two mass ranges, two lifetimes, and therefore two clocks running at different speeds over the same population.

The same pair of tracers is what makes a starburst identifiable as a burst rather than as a high steady rate, which is a distinction no single measurement can draw.

Where the model stops #

Constant luminosity. The estimate treats $L$ as fixed, and it is not — the Sun has brightened about 30% since it formed, and will brighten further. Full models integrate the changing output and give somewhat shorter lives than the simple scaling.

A fixed fuel fraction. The 10% figure varies with mass. Massive stars have convective cores that stir in fresh hydrogen, extending the burn; low-mass stars are fully convective and can eventually use nearly all their hydrogen, which is why red-dwarf lifetimes are even longer than the scaling suggests.

Mass loss. Massive stars shed substantial mass in winds, so a star does not keep the mass it started with — and mass is the parameter everything else depends on. Above about 40 solar masses the loss is large enough to change the outcome.

Single stars. Binary interaction rewrites the whole calculation for a substantial fraction of massive stars, and mass transfer can rejuvenate a star that ought to have finished.

The figures also flatter the situation, and it is the usual logarithmic flattery. Six decades in lifetime on one plot makes the range look surveyable. It is not: the difference between the top and bottom of that axis is the difference between a few million years and a few hundred billion, which is the difference between something shorter than the age of the Alps and something longer than the universe has existed by a factor of thirty.

Two more readings mark the boundaries the relation is used to place.

The ladder from here #

Later rungs: the fuel fraction and core convection. The Sun’s evolving luminosity and the faint young Sun problem, which the geological record makes into a real puzzle. The turn-off as an age indicator, done properly. Isochrones. The initial mass function and its measurement. Nucleosynthesis, and which elements come from which mass range. Core-collapse supernovae. Mass loss in massive stars. Binary interaction and its statistics. And the first stars, which had no metals at all, may have been far more massive than anything since, and have never been observed.

Red dwarfs will still be fusing hydrogen when the universe is a hundred times its present age, and none has ever been seen to do anything else. The most common star in the galaxy has a life history that is entirely a prediction.