A lifetime that stops getting shorter

A star’s main-sequence life is a budget: the hydrogen it can burn, divided by the rate at which it burns it. The fuel grows in proportion to the star’s mass. The rate grows much faster, because a heavier star must be hotter inside to hold itself up and burns its fuel far more fiercely — mass decides everything, by a power of three and a half for stars like the Sun. Divide one by the other and the lifetime falls as the mass to a power of about two and a half. The Sun lives ten billion years; a star of ten solar masses, about thirty million; the biggest stars die first, and their deaths enrich the galaxy long before the small stars have done anything.

The same arithmetic carried to the top of the mass range gives absurd answers. A star of a hundred and fifty solar masses, burning at a hundred and fifty to the power three and a half times the Sun’s rate, would last a few tens of thousands of years. Stars that massive exist — the most massive known, in the cluster R136 in the Large Magellanic Cloud, are estimated at a hundred and fifty to two hundred solar masses — and they live for millions of years, not tens of thousands. Something stops the steep law. It is the same thing that stops stars from being arbitrarily massive in the first place: their own light.

A lifetime with a floor #

The luminosity of a star cannot exceed the point at which the outward push of its radiation on the electrons in its outer layers balances the inward pull of gravity. That is a brightness that would blow the star apart: the Eddington luminosity, $L_\mathrm{Edd} = 4\pi G M c/\kappa$, about thirty-two thousand times the Sun’s luminosity per solar mass for electron scattering. It grows only in proportion to the mass. For stars like the Sun it is far above the actual luminosity and irrelevant. For the most massive stars it is approached: their luminosities are tens of per cent of the limit, and the steep mass–luminosity law bends over to follow it.

Once luminosity is proportional to mass, and fuel is too, the mass cancels and the lifetime is a constant. The figure’s simple model, with a steep law below and a cap at forty per cent of the Eddington luminosity above, shows the curve bending from a steep fall onto a floor of two and a half to three million years — a little higher at the very top, because the most massive stars burn a larger fraction of their hydrogen in their larger convective cores. Detailed stellar models agree with the shape: their lifetimes level off at about three million years above sixty solar masses and change by only a few tenths of a million years between a hundred and three hundred.

The argument has a familiar shape elsewhere in astrophysics. A quasar growing at the Eddington limit doubles its mass on a timescale set by exactly the same combination of constants — the efficiency of turning accreted mass into light, the electron opacity, the speed of light and $G$ — and that timescale, the Salpeter time of about forty-five million years for an efficiency of ten per cent, is independent of the black hole’s mass for the same reason the stellar floor is: both the output and the reservoir scale with the mass. A ceiling on the light, and not on the mass, turns into a ceiling on the rate at which anything gravitationally bound can process its own mass through radiation, whether the processing is fusion in a star or accretion onto a hole. The stellar floor is about twenty times shorter than the Salpeter time because hydrogen fusion releases fourteen times less energy per kilogram than accretion and the star burns only part of its fuel.

How much the cap matters #

The level of the floor depends on how close to the limit the most massive stars actually shine, which is not a fixed number.

A cap at a fifth of the limit puts the floor near five million years; one at four-fifths, near one and a half. Where real stars sit depends on how their luminosity responds to their mass at the top of the range, and stellar models put the Eddington factor of a hundred-solar-mass star on the main sequence at a third to a half, rising as the star evolves and loses mass. The floor’s existence does not depend on the value: any cap proportional to mass gives a lifetime that stops falling. Its level does, by the same factor, and that is the uncertainty in the ages of the youngest clusters.

A floor written in constants #

The level of the floor follows from a short calculation. A star burning a fraction $f$ of its hydrogen, of mass fraction $X$, releases $\varepsilon = 0.007$ of the burned mass as energy; shining at a fraction $\Gamma$ of the Eddington luminosity, it lasts

$$t = \frac{\varepsilon X f\, \kappa c}{4 \pi G \Gamma},$$

and the star’s mass has gone.

What sets the floor is a handful of constants: the fraction of rest mass released when hydrogen fuses to helium, which is nuclear physics; the opacity of free electrons to light, which is a limit written for free electrons and fixed by the Thomson cross-section; the speed of light; and Newton’s constant. To these the star adds only how close to its limit it shines and how much of its hydrogen it can reach. A star at forty per cent of its limit, burning half its hydrogen, lives about two and a half million years. No star burning hydrogen can live much less than a million, whatever its mass, because no star can shine brighter per unit mass than the limit allows.

That gives the floor a curious status. It is not a property of any particular kind of star. It is the time it takes radiation, flowing out at the greatest rate gravity can contain, to carry away the nuclear energy of a fixed fraction of the mass. Any object fusing hydrogen and held by its own gravity against its own light obeys the same bound.

Why the brightest stars live as long as merely bright ones #

The floor reshapes how the most massive stars matter. A star of forty solar masses and one of a hundred and fifty live almost exactly as long, three million years or a little more. Their lifetimes differ by tens of per cent where their luminosities differ by a factor of five. In a cluster of stars born together, all the stars above about sixty solar masses leave the main sequence within a few hundred thousand years of each other.

The turn-off, the most massive star still on the main sequence, is the standard clock for a cluster’s age, and for most clusters it is a good one: each factor of ten in age moves it by a factor of about two and a half in mass. Near the floor it fails. A cluster of two million years has lost none of its massive stars, a cluster of three and a half has lost all those above about thirty-five solar masses, and between the two the turn-off mass runs from the very top of the mass range to tens of solar masses in a million years. Dating the youngest clusters by their turn-off is like dating a burning candle by its height when every candle in the box is the same length.

The youngest clusters are dated instead by what their massive stars are doing: the appearance of Wolf–Rayet stars, which have lost their outer hydrogen and show the products of burning at their surfaces, after about three million years; the first supernovae after three to four; the red supergiants after about seven. It is also why a single massive star in the field is so hard to date: a lone star is dated only as it leaves the main sequence, and for a very massive star that departure falls in a window the floor has squeezed to almost nothing.

Why the Sun is nowhere near its floor #

The same formula shows why ordinary stars are nowhere near the floor, and it is not only that they shine far below their limits.

The Sun shines at about a thirty-thousandth of its Eddington luminosity, and burns only the tenth of its hydrogen that its radiative core can reach. Put those numbers in the formula and the lifetime comes out at six billion years — the right order, and short of the true ten only because the Sun has been fainter for most of its life than it is now. The floor for the most massive stars and the Sun’s ten billion years are the same calculation with $\Gamma$ different by four orders of magnitude, and the steep mass–luminosity law is nothing but the statement of how $\Gamma$ climbs with mass — slowly enough at low mass that most of the light of a galaxy comes from stars that are not most of its stars, and then, near the top, so close to one that it can climb no further.

That is also why the floor is a statement about electrons rather than about nuclear physics. The fusion rate inside a massive star is extraordinarily sensitive to temperature, through the CNO cycle that such a star has to make its own catalyst for, but the rate at which the star can let its energy out is set by how the light pushes on the free electrons in its envelope. The core adjusts to supply whatever the envelope can carry, and at the top of the mass range the envelope carries all it can.

A burst that releases everything together #

The floor has a consequence for whole galaxies. The massive stars of a burst of star formation are the source of almost all its ultraviolet light, its winds and, at the end, its supernovae — the feedback that heats and disperses the gas from which they formed.

Per kilogram of star, the most massive stars do not radiate much more over their lives than the Sun does — a few times more, because they burn a larger fraction of their hydrogen. What makes them dominant is that they do it fast, and with the floor, all at the same time. The ultraviolet output of a young cluster stays nearly constant for its first three million years and then drops steeply as the most massive stars all leave the main sequence together; the first supernovae follow within a million years more. The gas round the cluster receives its energy in a single pulse a few million years long.

That timing is visible in the clouds where stars form. A lifetime read from how gas and stars sit apart finds that molecular clouds are dispersed within a few million years of forming their first massive stars — before any supernova has exploded, by the light and winds alone. The floor is why that dispersal time is so similar from one galaxy to another: the clock that runs it is set by the same constants everywhere.

Where the floor is seen #

The floor shows up most clearly in the light of whole star-forming regions, where the individual stars cannot be resolved. A young starburst’s spectrum is dominated by its most massive stars, and as it ages it passes through a sequence of features that the floor times. For its first two or three million years the spectrum shows the hot, blue continuum and the wind lines of O stars, with an ultraviolet output that is nearly constant. Then, within a few hundred thousand years, the broad emission lines of Wolf–Rayet stars appear — massive stars that have stripped their outer layers in their winds — and shortly after that the ultraviolet output begins to fall as the most massive stars leave the main sequence together. The Wolf–Rayet features in galaxies’ integrated spectra are seen only in a narrow window, between about three and five million years after a burst, and their presence dates the burst to within a million years.

The cluster R136, at the heart of the Tarantula Nebula in the Large Magellanic Cloud, is the nearest example of a region younger than the floor. Its brightest stars, among the most massive known, are still on the main sequence or just leaving it; the cluster’s age is estimated at one to two million years, and no supernova has yet exploded there. A few million years from now all of them will leave the main sequence within a short interval of one another, and the region will pass from a nebula lit by hot stars to one shaped by supernovae over the course of perhaps a million years — a transition timed by the same constants that time it in every galaxy.

A floor that the most massive stars approach and then leave #

The floor holds for stars burning hydrogen at a fraction of the Eddington limit, and the most massive stars complicate it in two ways. They lose mass rapidly in winds driven by their own radiation, so a star born at two hundred solar masses may be a hundred and fifty by the middle of its life and much less at the end; the lifetime depends on the mass it had when it burned its fuel rather than when it formed, and the mass a star does not keep can be most of it. And a star shining close to its Eddington limit is near the edge of stability: its outer layers can inflate, become loosely bound and erupt, as the luminous blue variables do, losing solar masses in a single episode.

Neither changes the floor much, because both act in the same direction: they keep the star’s luminosity below its limit and its burned fraction roughly constant. But they do mean that the masses of the most massive stars, inferred from their luminosities on the assumption that they sit on a mass–luminosity relation, are uncertain by tens of per cent, and the lifetime that the floor fixes is better known than the masses it applies to.

A toy model of the most massive stars #

The figures use a toy model: a single power law for the luminosity of stars below the cap, a cap at a fixed fraction of the Eddington limit, and a burned fraction rising as the logarithm of mass. Real stellar models compute the luminosity from the structure, which depends on composition, rotation and mixing, and their burned fractions rise and level off in ways a logarithm only imitates. The model’s lifetimes are within a few tens of per cent of detailed models over the whole range — enough to show the shape — but its turn-off ages and energies should be read as illustrations, not predictions. Metal-poor stars have lower opacity and shine closer to their limits at a given mass, and stars rotating rapidly mix fresh hydrogen into their cores and live longer; both move the floor by tens of per cent, neither removes it.

Still open: how massive a star can be born #

The floor answers how long the most massive stars live; it does not answer how massive they can be. The Eddington limit applies to a star in equilibrium, and a star forming by accretion faces a version of it during its growth: the light of the young star pushes on the infalling gas and dust, which have a much higher opacity than electrons, and should halt accretion at a few tens of solar masses. Stars of a hundred and fifty or more exist nonetheless, either because accretion is channelled through discs and flows that let the light escape along other paths, or because they are the products of mergers in dense young clusters. If the upper end of the mass range is set by mergers, its stars are younger than their clusters, rejuvenated by the collision, and the floor’s clock started late for them — one more way in which the most massive stars stand slightly apart from the arithmetic that otherwise governs every star, and a forecast is still all that can be made at the other end of the mass range for the opposite reason.