The mass a star does not keep

The rungs below followed a star’s interior: a core that contracts while the envelope swells, and a degenerate core that eventually cannot be held up at all. Both are accounts of what happens to the material that stays.

Most of it does not stay. A star of five solar masses ends as a white dwarf of about nine tenths of one, which means it has thrown away four fifths of itself, and nearly all of that in the last per cent of its life. What is left is not a residue of what went in; it is a quantity that turns out to be almost independent of what went in.

What the relation is #

Fitted to white dwarfs in open clusters, the relation is close to a straight line,

$$M_{\text{final}} \approx 0.08\,M_{\text{initial}} + 0.49,$$

over the range from one solar mass — below which nothing has had time to die — to about eight, above which the core reaches carbon ignition and the star does something else entirely.

Two features of that line are the whole content of this rung.

The slope is small. A tenth of a solar mass of extra final mass costs one and a quarter solar masses of extra initial mass. Everything else goes.

The intercept is close to the minimum white dwarf mass. Half a solar mass is roughly the smallest core a star of any mass develops before its envelope leaves, which is why the line starts there.

The fraction thrown away therefore rises from about 43 per cent at one solar mass to 86 per cent at eight. A star’s death is mostly an act of disposal.

Why the peak is not just the flatness #

The usual short account is that a flat relation compresses everything into a narrow range, which is why white dwarfs are all about 0.6 solar masses. Half of that is right and the half that is missing is the interesting half.

Push a Salpeter initial mass function through the relation. The number of white dwarfs per unit final mass is the number per unit initial mass times the Jacobian $dM_i/dM_f$, and the Jacobian is one over the slope — a factor of twelve and a half. So the density is multiplied by twelve everywhere, and the shape is the initial mass function’s shape, stretched.

The initial mass function is steep and monotonically falling. So the derived distribution is monotonically falling too. It has no interior maximum at all; its largest value is at its low-mass edge.

The peak is the edge. And the edge is where it is for a reason that has nothing to do with stellar structure: no star lighter than about one solar mass has had time to die yet, because the Galaxy is thirteen billion years old and a 0.9-solar-mass star lives longer than that. The distribution is truncated there, and the scatter of the measurements — a few hundredths of a solar mass, from the surface gravities and from the intrinsic scatter of the relation itself — rounds the cliff into a maximum a little above it.

A flat relation produces a narrow range. It takes the age of the Galaxy to produce a peak.

How the relation is measured #

Nothing about the mass-loss rate is computed from first principles, so the relation is measured, and the measurement runs the theory backwards through four steps. It is worth setting out because every step is an inference and one of them is a model.

One: the cluster’s age. A star cluster’s colour–magnitude diagram bends where stars are leaving the main sequence, which is a place rather than a track, and the mass at that bend is fixed by the cluster’s age. That is an age read off a bend, and it is the only step that involves no white dwarf.

Two: the white dwarf’s cooling age. A white dwarf has no fusion. It radiates its stored heat and cools along a track fixed by its mass and composition, so its temperature and radius give the time since it became a white dwarf.

Three: subtract. The cluster’s age minus the cooling age is how long the progenitor lived before dying — its nuclear lifetime.

Four: invert a model. A nuclear lifetime is turned into an initial mass by a stellar-evolution model, because lifetime and mass are related by a steep and well-understood function. This is the step that is a model rather than an observation, and it is where the relation’s systematic uncertainty lives: a change in the assumed convective overshooting or the assumed metallicity moves every inferred initial mass.

Only then is the final mass plotted against it — and the final mass comes from the white dwarf’s own spectrum, through the pressure broadening of its hydrogen lines, which gives the surface gravity, which with a mass–radius relation gives the mass.

What a cooling age actually is #

Step two above deserves more than a sentence, because it is the clock the whole relation is timed by and it is a peculiar one.

A white dwarf is a degenerate carbon–oxygen sphere with a thin envelope. It generates no energy at all: what it radiates is the residual thermal energy of its ions, since the electrons are degenerate and their energy is not available. That reservoir is finite and its rate of loss is set by how fast heat crosses the non-degenerate envelope, which is an opacity problem.

The result is a cooling law close to $L \propto t^{-7/5}$, so a white dwarf spends most of its life faint. That has two consequences the measurement depends on. The cooling age is a steep function of luminosity, which makes it well determined for a hot young remnant and poorly determined for a cold one — so the useful cluster white dwarfs are the recently formed ones in young clusters. And the very coldest white dwarfs in the Galactic disc give a lower bound on the disc’s age that is independent of everything nuclear.

What actually removes the mass #

Two mechanisms, operating at different times and with very different reliability.

A slow wind through the giant phases. A red giant loses perhaps $10^{-8}$ to $10^{-7}$ solar masses a year, driven by radiation pressure on dust that condenses in the cool outer atmosphere — the same grains that redden everything behind them and drags the gas with it. Over a hundred million years on the giant branch that is a few hundredths of a solar mass — significant for a low-mass star, negligible for a high-mass one.

A superwind at the end of the asymptotic giant branch. The rate climbs to $10^{-5}$ or $10^{-4}$ solar masses a year, and the entire remaining envelope leaves in a few tens of thousands of years. This is where nearly all of the mass goes, and it is the phase that no theory predicts: the rate depends on the coupling between pulsation, dust formation and radiation pressure in a cool, extended, unstable atmosphere, and the prescriptions used in stellar-evolution codes are fitting formulae calibrated against observations rather than derived quantities. The visible remnant of the superwind is a planetary nebula: the ejected envelope, ionised by the exposed core, glowing for perhaps twenty thousand years before it disperses. Their number is a check on the arithmetic — the rate at which stars are dying in the Galaxy, multiplied by twenty thousand years, gives roughly the number that should be visible, and it does.

What the relation is worth #

Three things it settles, and they are worth setting out because a nearly flat line looks like an uninteresting result.

It explains why type Ia supernovae exist and are standard. Everything below eight solar masses leaves a carbon–oxygen white dwarf far below the Chandrasekhar limit, so accretion is the only way to reach that limit — and reaching a fixed limit is what makes the explosion standardisable.

It fixes the mass returned to the interstellar medium. Integrating the mass lost over an initial mass function gives the fraction of a stellar population’s mass that goes back into gas, which is around a third for a single generation and which every model of galactic chemical evolution needs.

And it puts a ceiling on the white dwarf mass that is nothing to do with degeneracy. The Chandrasekhar limit is 1.4 solar masses. The heaviest white dwarf a single star can make is about 1.1 to 1.3, because a heavier core would have ignited carbon. So a white dwarf found above about 1.25 solar masses is not the product of a single star’s evolution, and is evidence of a merger.

Where the mass goes, seen directly #

There is a check on all of this that does not involve white dwarfs at all: weigh the ejecta.

A planetary nebula’s mass can be estimated from its emission measure — the recombination line flux fixes the product of density squared and volume, and the angular size and distance fix the volume. The answers come out between about 0.1 and 1 solar mass, with a distribution consistent with what the initial–final mass relation requires, and with the same distance problem that afflicts everything in the Galaxy: the individual masses are uncertain by factors because the distances are.

The rate is a second check. Multiply the number of stars dying per year in the Galaxy — around one per year, from the initial mass function and the star-formation history — by the twenty thousand years a nebula stays visible, and the expected population is around twenty thousand. The catalogued number is a few thousand, and the shortfall is understood as extinction and confusion in the plane rather than as a discrepancy.

Weighing the wind while it leaves #

The chain above infers the lost mass by subtracting the remnant from the progenitor, which is an accounting exercise on two endpoints. There is a second measurement that watches the loss happen, and it is worth having because it is independent of every stellar-evolution model in the chain.

An asymptotic giant’s wind is molecular. Carbon monoxide is abundant in it, resistant to dissociation, and radiates through a ladder of rotational lines in the millimetre band, so the envelope is observable as an expanding CO shell around the star. The line profile gives the expansion speed directly — it is doubly peaked, with the two horns being the approaching and receding faces of a shell, separated by twice the outflow velocity — and the integrated flux, with an assumed abundance, gives the mass.

The numbers that come out are consistent with what the initial–final mass relation requires. Outflow speeds cluster between 5 and 25 kilometres per second, which is a few times the escape speed from an extended giant and nothing like the speed of a hot star’s wind. Rates for the extreme objects reach $10^{-4}$ solar masses a year: the nearest carbon star, IRC+10216, is losing about $2\times10^{-5}$ and is surrounded by a molecular envelope tens of thousands of astronomical units across, containing more mass than it will end up keeping.

The measurement’s weakness is the abundance. What is observed is CO, and what is wanted is total mass, so every rate carries the assumed carbon-to-hydrogen and oxygen-to-hydrogen ratios — which is exactly the quantity that the dredge-up alters during the phase being measured. The rates are good to a factor of two or three, and that is why the accounting chain, for all its model dependence, remains the primary route to the relation.

The loss is not steady #

One further complication, and it is visible in the images rather than inferred.

Several of these envelopes are not smooth. They contain detached shells — thin, nearly spherical arcs of enhanced density at several thousand astronomical units, separated by regions of much lower density, so the star was losing mass fast, then slowly, then fast again. The spacing gives the interval between episodes: some tens of thousands of years.

That interval is a recognisable number. An asymptotic giant burns hydrogen in a shell for most of the time, and periodically the helium accumulating beneath it ignites in a flash — a thermal pulse — which briefly restructures the star, alters its radius and luminosity, and mixes processed material to the surface. The interpulse period for a solar-mass star is a few tens of thousands of years, and the shells are spaced accordingly.

So the superwind is not one event but the envelope of a series, modulated by a nuclear instability deep inside the star. Two things follow. The mass-loss prescriptions used in evolution codes are averages over a process that is not smooth, which is part of why they are fitting formulae. And the surface composition changes as the loss proceeds, because each pulse dredges up carbon — so a star can leave the asymptotic giant branch as a carbon star having entered it as an oxygen star, with the wind’s dust chemistry, and therefore its opacity and its driving, changing partway through.

The quantity being calibrated is an average over an episodic process whose amplitude nobody can compute, which is a fair summary of why a relation this consequential is still fitted to twenty-odd clusters rather than derived.

A last consequence of that episodic structure is observational rather than theoretical. Because the loss is concentrated into brief intervals, the number of stars caught in the act is small, so the phase that removes most of a star’s mass is represented in any survey by a handful of objects — and the nearest of them, at a few hundred parsecs, are the ones every prescription is fitted to.

The relation’s two ends behave differently, and the lifetime curve underneath it is what says why, so both are worth reading at other markings.

What is not settled #

The scatter is real and unexplained. Cluster white dwarfs scatter about the fitted line by more than their individual error bars, by roughly a tenth of a solar mass. Rotation, metallicity, binarity and magnetic fields are all candidates, and none of them has been shown to account for it.

The metallicity dependence is expected and barely measured. Mass loss is driven by dust, dust needs metals, and a metal-poor star should therefore lose less and leave a heavier remnant — a population effect that colour, read as a thermometer, is the usual handle on. Globular-cluster white dwarfs — the only metal-poor sample available — sit at about 0.53 solar masses, lighter rather than heavier, which is the opposite of the naive prediction and is usually attributed to their progenitors being lighter still.

And the top of the range is uncertain by a factor. Whether the boundary between white dwarf and neutron star is at seven solar masses or nine depends on the treatment of convective overshooting, and shifting it changes the supernova rate of a stellar population by tens of per cent — which is a large fraction of the disagreement between predicted and observed supernova rates in galaxies.

One more marking covers the mass range in which the relation’s slope is steepest.

Where this ladder goes next #

This rung has taken the fate of a star’s envelope and found that the number that matters is not predicted anywhere: the relation between what a star was and what it leaves is measured by dating two clocks in the same cluster and subtracting.

The rung above is the returned material rather than the remnant. What comes off an asymptotic giant is not the composition that went in — the star has dredged carbon and s-process elements to its surface — so the wind is the main source of carbon and of about half the elements heavier than iron, and the enrichment history of a galaxy is written by this phase.

Beside it lies the same accounting for the stars this relation excludes: what an eight-to-twenty-solar-mass star leaves, where the answer is a neutron star and the mass loss is driven by line opacity rather than dust, and where the relation is far less flat.

And below it, the habit: when nothing predicts a quantity, calibrate it against something the theory does get right. The mass-loss rate is not computable. The nuclear lifetime is, to a few per cent — so the relation is built out of lifetimes and ages, and the uncomputable quantity is never used at all.