A lifetime that is still a forecast

A star thirty times the Sun’s mass has thirty times the fuel and burns it forty thousand times faster, and it is gone in a few million years. The same arithmetic run the other way says that a small star should live a very long time: less fuel, but burned far more slowly still, so that the lifetime rises steeply as the mass falls. The Sun lives ten billion years. A star of half its mass lives more than fifty.

That is the part of the calculation that is usually drawn. The interesting part is at the bottom, where two things happen at once. The lifetimes pass the age of the universe, so that the stars in question have never been seen to finish; and a change in the stars’ internal structure multiplies their fuel by nearly ten, so that the lifetimes are longer than the simple arithmetic says by the same factor. Most of the stars in the Galaxy sit in that part of the diagram, and the whole of their lives is a prediction.

Fuel, and the part of it a star can reach #

The lifetime is the hydrogen a star burns divided by the rate at which it burns it. The rate is the luminosity, which rises steeply with mass — as the fourth power near the Sun’s mass and more gently, closer to the square, among the smallest stars. The hydrogen burned is not the hydrogen the star contains.

In the Sun, fusion happens only in the core, the inner tenth of the mass, where the temperature exceeds about ten million kelvin. Above the core lies a thick radiative zone, where energy leaks outward as light, slowly and without stirring the gas. The helium made in the core stays there; fresh hydrogen from above never comes down to replace it. When the core’s hydrogen is gone — about a tenth of the star’s total — the Sun leaves the main sequence, with nine-tenths of its hydrogen still in its envelope, unused and unreachable. That tenth is where the ten-billion-year figure comes from.

A star of less than about a third of a solar mass is built differently. Its interior is too opaque and too cool for radiation to carry the energy out, and the gas boils instead: convection runs from the centre to the surface, and every parcel of gas is carried down through the core and back up again in a few years. The helium made in the core is mixed through the whole star, and the hydrogen of the whole star is brought down to the core. Such a star burns nearly all its hydrogen before it runs out.

The switch multiplies the fuel by nearly ten. The figure draws it as a smooth step between the two limits — the shape of the transition is an interpolation, since the partly convective stars between a third and three-quarters of a solar mass burn fractions that depend on the details of their mixing — and its consequence is the gap between the two curves in the first figure. The simple rule, a tenth burned at every mass, gives a tenth-solar-mass star a lifetime of about eight hundred billion years. The whole star burned gives about eight trillion.

A gap in the main sequence where the stirring begins #

Why the transition happens near a third of a solar mass is a question about opacity. Energy leaves a stellar interior by radiation if the temperature falls gently enough outward for light to carry the flux, and by convection if it would have to fall more steeply than a rising parcel of gas cools by expanding. In small stars the interior is cool enough that ions of hydrogen and heavier elements absorb light strongly, radiation cannot carry the flux, and the steeper gradient takes over. Below about 0.35 solar masses the radiative zone that separates the Sun’s core from its envelope disappears altogether.

The boundary is visible. When the space astrometry mission measured parallaxes for millions of nearby stars, their Hertzsprung–Russell diagram showed a narrow, slightly depleted stripe running across the main sequence at the luminosity of stars of about a third of a solar mass — a gap, a few hundredths of a magnitude wide, in what had always been drawn as a smooth band. The explanation offered is that stars just above the transition mass are unstable in a specific way: their small radiative zone opens and closes as helium-3, made in the core, builds up and alters the core’s structure, so that the star oscillates between partly and fully convective, and its luminosity and radius shift slightly as it does. Stars caught in that phase are displaced from their neighbours, leaving the stripe thin. The change in structure that multiplies a star’s fuel by ten is marked on the diagram by a hairline, found only when the diagram was drawn with enough stars and precise enough distances.

The stars that have never finished #

The universe is 13.8 billion years old. Setting the lifetime equal to that gives a mass, and the mass is the most consequential number on the diagram: 0.90 solar masses. A star lighter than that formed at the very beginning of the Galaxy has not had time to exhaust its core. Every such star ever born is still on the main sequence.

Weight the stars by how many of each mass are born — far more small ones than large, with a standard initial mass function — and assume the Galaxy has formed stars at a steady rate over its whole history, and the fraction of all stars ever born that have died is 7.8 per cent. Every white dwarf, every neutron star, every stellar black hole and every supernova remnant comes from that one star in thirteen. Those few stars carried half of all the mass, because they are the heavy ones, and they made every element heavier than helium; the other twelve-thirteenths of the stellar population has contributed nothing yet but light, and a modest amount of that.

The fraction depends on the assumptions — on the mass function’s shape and on the Galaxy’s history of star formation, which was not steady — but no reasonable choice moves it above a few tenths. Stellar death is rare. It is the youth of the universe, not the physics of stars, that makes it so.

The part of the diagram no star has crossed #

On the Hertzsprung–Russell diagram, which plots luminosity against surface temperature, the main sequence is a band from hot, bright stars at the upper left to cool, faint ones at the lower right, and stars sit on it rather than travelling along it for most of their lives.

The bold part of the band, everything cooler than about 5,400 kelvin, has never produced a dead star. Its stars have arrived and stayed. When the tracks by which stars leave the main sequence are calibrated — the bend where a cluster’s main sequence turns off, the giant branches, the white dwarfs — every one of them comes from stars heavier than the edge. The oldest globular clusters in the Galaxy are turning off at about 0.8 solar masses, which is as low as any observed departure from the main sequence has ever been.

That has a consequence for the theory. The physics of stellar interiors is tested by stars that have lived their lives: their positions on the diagram at each stage, their ages from clusters, the masses of their remnants. For the red dwarfs, every one of those tests is missing. What is measured about them is their present state — masses from binary orbits, radii from eclipses and interferometry, luminosities, temperatures — and the present state is where the theory has had its most persistent trouble, in the stars’ sizes. A theory with a discrepancy in the part it can check is being used to forecast trillions of years it cannot.

Arriving takes a billion years #

The lower main sequence is slow at both ends of the life. A star forms by contracting from a cloud, releasing gravitational energy as light, and it reaches the main sequence when its core becomes hot enough for hydrogen fusion to supply that light instead. The contraction takes of the order of the Kelvin–Helmholtz time — the time the star’s gravitational binding energy would last at its main-sequence luminosity.

For the Sun that is thirty-one million years, a blink against its ten-billion-year life. For a star of a tenth of a solar mass it is about 1.7 billion years: the smallest stars take longer merely to settle onto the main sequence than most massive stars live. A red dwarf that formed with the Sun has only recently finished contracting. The two timescales never cross — arriving always takes a small fraction of staying — but for the smallest stars that small fraction is a large part of the history of the Galaxy, and a young red dwarf is larger and brighter than an old one of the same mass for longer than the Sun has been shining.

The slow arrival leaves a clock of its own. Lithium is destroyed at about two and a half million kelvin, well below the temperature of hydrogen burning, and in a fully convective star every parcel of gas is carried down to the hot centre, so the star’s whole supply of lithium is burned as soon as its contracting core reaches that temperature. The more massive a young red dwarf, the sooner that happens. In a young cluster, therefore, there is a sharp boundary in brightness: stars above it have lost their lithium, stars below it — lighter, and still contracting — have not yet. The position of that boundary moves to fainter stars as the cluster ages, and reading it gives the cluster’s age with less dependence on uncertain physics than the turn-off at the top of the main sequence. It is one of the few clocks the lower main sequence offers, and it works because arriving there is slow.

Below about 0.08 solar masses the core never gets hot enough, because the electrons become degenerate first and their pressure stops the contraction before fusion can begin. Those are the brown dwarfs, and they have no main sequence at all: they contract, glow faintly with the heat of contraction, and fade. The boundary between them and the lowest-mass stars is set by the same race between contraction and degeneracy that sets the arrival time above it.

What is measured about them now #

The present state of the red dwarfs is known with some precision, and every piece of it comes from a companion. Masses come from binary orbits, where two stars’ motions around each other give each one’s mass through Kepler’s third law with no model of either. Radii come from eclipses, where the duration of each eclipse and the shape of its ingress give both stars’ sizes as fractions of their separation. Together they give the mass–radius relation of the lower main sequence to a few per cent, from a few dozen eclipsing pairs of red dwarfs found in wide-field photometric surveys.

It is that relation which disagrees with the models. At a given mass, the measured radii are three to five per cent larger than stellar-evolution calculations predict, and the stars are correspondingly cooler, so that their luminosities come out about right. The leading explanation is magnetic: the eclipsing pairs are close binaries, spun up and kept rapidly rotating by their companions’ tides, and their strong magnetic fields suppress convection near the surface and cover it with spots, puffing the star up slightly. Single, slowly rotating red dwarfs, whose radii can be measured by interferometry, agree with the models better. The effect is small for the present and it is unknown for the future: whether a magnetically puffed star also burns its fuel differently over trillions of years is not a question present stars can answer.

What happens after, according to the models #

Stellar-evolution calculations carried to the end of a red dwarf’s life predict something no star in the present universe has done. A fully convective star of a tenth of a solar mass does not swell into a giant when its hydrogen runs low, as the Sun will. It stays compact, and as its helium fraction rises its interior grows hotter and denser, so its luminosity and its surface temperature both rise: over trillions of years it moves up and to the left on the diagram, becoming a small, hot, bluish star before its hydrogen is finally exhausted, and then contracts into a helium white dwarf. The models call these late stages blue dwarfs.

No blue dwarf exists and none will for trillions of years. The prediction rests on the same physics that the Sun’s future does — the equation of state, the opacities, the nuclear rates — at densities and compositions where the physics is reasonably well known; what it cannot rest on is a single observation of the process. The life history of the commonest kind of star is entirely a forecast, made with a theory that has never watched one finish.

A galaxy that will be lit by its smallest stars #

The long lives change what the Galaxy will look like. Today its light is dominated by the stars that will not be here long: a few per cent of the stars by number, the massive and middle-mass ones, supply most of the luminosity, while the red dwarfs that make up three-quarters of the stars supply a few per cent of it. As the gas from which stars form is used up — which in a galaxy like the Milky Way will take some tens of billions of years more — no new massive stars will replace the ones that die, and the Galaxy’s light will fall and redden, with the red dwarfs supplying an ever larger share of what remains.

Calculations carried forward on this basis give a Galaxy whose last stars are red dwarfs shining for ten to a hundred trillion years after the last star formation, after which only white dwarfs, neutron stars, brown dwarfs and black holes remain, cooling. The era of stars as a whole is therefore a thousand times longer than the time that has already passed, and almost all of it belongs to stars of less than half a solar mass. On that timescale the present universe — with its supernovae, its bright galaxies and its short-lived massive stars — is an unrepresentative first flash, and the Sun, which will be dead within eight billion years, is one of the short-lived ones.

What the figures leave out #

The lifetimes use the family’s broken power-law relation between mass and luminosity, which is a fit to present-day main-sequence stars and does not include the slow brightening every star undergoes as its core’s composition changes; a real star’s lifetime is shorter than its present fuel divided by its present luminosity by a modest factor for that reason. The fraction burned is a smooth interpolation between two limits, a tenth and nearly all, not the output of stellar models, whose transition depends on convective overshoot and on how the partly convective stars mix. The census of dead stars assumes a steady rate of star formation over 13.8 billion years and one standard mass function; the Galaxy formed stars faster in the past, which raises the fraction dead slightly, and the mass function at the lowest masses is uncertain. The pre-main-sequence times are the Kelvin–Helmholtz time evaluated at arrival, an order of magnitude rather than a computed track. And every star is treated as isolated: close binaries exchange mass and merge, and some low-mass stars have been made to leave the main sequence early by a companion’s interference.

Still open: whether the models of the smallest stars are right #

The red dwarfs are the most common stars, the longest-lived, and the ones around which most small planets orbit, inside habitable zones that move as their stars evolve. Their present properties are measured to a few per cent and disagree with the models by about that much, in the direction that suggests magnetic fields are doing something to their surfaces that the standard physics leaves out. Whether the disagreement also changes their futures — their lifetimes, their late brightening, the fraction of their hydrogen they actually burn — is not known, and will not be observed directly for as long as there are observers. What can be tested is the calibration: the radii and luminosities of red dwarfs of known age, in clusters and in binaries with white dwarfs whose cooling clocks date them. Every such measurement constrains a forecast of trillions of years with a few per cent of evidence from the first thirteen billion.