Astronomy & Celestial Mechanics
the sky, computed
Stars

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

A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it.
11 min read 6 figures Scales that defeat intuitionThe sky as a clock

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 MM. Consumption goes as M3.5M^{3.5}. The ratio — the lifetime — goes as M−2.5M^{-2.5}, and the minus sign is the whole story.

How long a star lasts, against its massMain-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.0.321.03.2103210^610^710^810^910^1010^1110^12mass (solar units)the age of the universe0.5M☉ — 80 Gyr1M☉ — 10 Gyr3M☉ — 458 Myr10M☉ — 23 Myr30M☉ — 1 Myrslope ≈ −2.5
Fig. 1 Main-sequence lifetime against mass, on logarithmic axes. The dashed horizontal line is the age of the universe; every star above it has not had time to finish, and every star to the right of where the curve crosses it has been dead for a long time.

The arithmetic

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

t≈fMc2ϵL∝MM3.5=M−2.5.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:

Mass Luminosity Lifetime
0.2 M☉ 0.005 L☉ 4 × 10¹¹ yr
1 M☉ 1 L☉ 1 × 10¹⁰ yr
3 M☉ 55 L☉ 5 × 10⁸ yr
10 M☉ 4,400 L☉ 2 × 10⁷ yr
30 M☉ 65,000 L☉ 5 × 10⁶ yr

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.

Luminosity against massMain-sequence luminosity against mass, both in solar units, on logarithmic axes. The slope is between three and four across most of the range, so a small spread in mass becomes an enormous spread in output.0.100.321.03.210320.011.0100100001000000mass (solar units)0.2M☉ → 0.01L☉1M☉ → 1L☉5M☉ → 391L☉20M☉ → 50,088L☉dashed: a pure slope of 3.5
Fig. 2 The relation that does the damage. Luminosity climbing as the three-and-a-half power of mass is what makes the fuel supply irrelevant — consumption outruns capacity by two and a half powers.

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.

The Hertzsprung–Russell diagramLuminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units
Fig. 3 The main sequence, with the mass markers that set the turn-off. A cluster’s diagram is this one with the upper part removed, and how much has been removed is how old the cluster is.

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−2.35M^{-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.

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.

How long a star lasts, against its massMain-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.0.100.321.03.2103210^610^710^810^910^1010^1110^12mass (solar units)the age of the universe0.15M☉ — 512 Gyr1M☉ — 10 Gyr8M☉ — 39 Myr60M☉ — 313 kyrslope ≈ −2.5
Fig. 4 The same curve with different reference masses. The 8-solar-mass mark is the boundary between ending as a white dwarf and ending as a supernova, and it corresponds to a lifetime of about 40 million years.

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.

Blackbody curves at 3200, 30000 KThermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.visible050010001500200000.20.40.60.811.2wavelength (nm)3200 K, peak 906 nm30000 K, peak 97 nmeach curve scaled to its own peak
Fig. 5 A red dwarf and a massive star. The hot star’s output peaks in the ultraviolet and is enormously larger, so the visible appearance of a star-forming region is set almost entirely by objects that will not exist in ten million years.

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.

The distance ladder, and its overlapsThe reach of each distance technique on a logarithmic scale in parsecs. Each rung is calibrated where it overlaps the one below it, so an error low on the ladder propagates all the way to the top.10^-610^-410^-210^010^210^410^610^810^10radar rangingdirect: a timed echoparallaxgeometry, and nothing assumedspectroscopic parallaxassumes a star like the calibratorsCepheid variablesassumes the period–luminosity relation holdsTully–Fisherassumes rotation tracks luminositytype Ia supernovaeassumes a standard explosionHubble's lawassumes the expansion rate is knowndistance (parsecs)each rung is calibrated on the one belowan error at the bottom moves everything above it
Fig. 6 The distance techniques. Type Ia supernovae are the top rung, and they are the deaths of stars — the endpoint of the mass sequence used as a measuring instrument at scales nothing else reaches.

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.

Where the model stops

Constant luminosity. The estimate treats LL 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.

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.

Where this goes next

0.100.321.03.210320.011.0100100001000000mass (solar units)0.2M☉ → 0.01L☉1M☉ → 1L☉5M☉ → 391L☉20M☉ → 50,088L☉dashed: a pure slope of 3.5 Mass decides everything, by a power of three and a half Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars. R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units The diagram that sorted the stars, by plotting two things against each other Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy. visible050010001500200000.20.40.60.811.2wavelength (nm)3000 K, peak 966 nm5800 K, peak 500 nm10000 K, peak 290 nmeach curve scaled to its own peak Colour is a thermometer, and it reads across the galaxy A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy.