A lone star is dated only as it leaves

A cluster has an age because it is a population: its stars were born together with a range of masses, and the point at which they leave the main sequence is a clock that every member reads together. A star in the field has no such population. It has one temperature, one luminosity, one composition, and whatever age it has must be read from where that single point sits on the Hertzsprung–Russell diagram.

The method is the obvious one. Compute isochrones — the positions on the diagram of stars of every mass at a single age — for a grid of ages and compositions, and ask which of them pass through the measured point within its errors. With a parallax from the Gaia satellite and a good spectrum, a nearby star’s luminosity is known to a few per cent and its temperature to about seventy kelvin, and modern surveys have produced isochrone ages for millions of stars this way. This essay asks what those ages are worth, and the answer depends almost entirely on where on the diagram the star is.

Why the main sequence is a crowd #

A star on the main sequence does not travel along it. It sits near the place its mass puts it, slowly brightening as helium accumulates in its core and the core contracts and heats. The Sun has brightened by about forty per cent since it formed and will roughly double its starting luminosity before it leaves the main sequence — enough to carry the band where water can be liquid half as far again outwards — while its temperature changes by only a hundred or two kelvin. On the diagram, a star’s whole main-sequence life is a short step, mostly upwards.

An isochrone is a line through all those steps at once — every mass, each at the point its own step has reached by a given age. Because the step is short and points almost along the main sequence itself, the isochrones for different ages lie very close together on it. The figure’s error box around the Sun-like star covers isochrones from 3.4 to 5.6 billion years, and the lines it does not cover are not far away. A one-solar-mass star at 4.6 billion years and a slightly more massive star at three are, within these errors, in the same place.

The isochrones separate only where stars leave the main sequence. Once the hydrogen in the core runs out, a star of about the Sun’s mass crosses rightwards towards the giant branch in a tenth of its main-sequence lifetime, its temperature falling by about a thousand kelvin over a billion years. That is the subgiant branch, and on it the isochrones fan out: positions that differ by a measurable amount correspond to ages that differ by a few per cent. The error box around the subgiant in the figure, no larger than the dwarf’s, contains a range of ages of only 0.4 billion years.

The model behind the figures is parametric: luminosity and temperature on the zero-age main sequence as powers of mass with a metallicity correction, a brightening over the main-sequence lifetime fitted to the Sun’s, a lifetime proportional to the mass to the minus two and a half, and a straight crossing of the subgiant branch. It reproduces the Sun’s present place to within a few kelvin and a few per cent in luminosity. It is not a stellar evolution code, and the essay’s numbers are not the ones a survey quotes, but the geometry — where isochrones crowd and where they spread — is what every model agrees on, and it is that geometry the conclusions rest on.

Four stars, four precisions #

The age of a single star is a probability distribution: every combination of mass, age and composition that predicts the observed position is weighed by how well it fits and by how common such stars are. The figure shows it for one-solar-mass stars at four ages. A two-billion-year-old dwarf is dated to within a factor of two either way, its distribution running down to the youngest ages the grid allows. A Sun-like star at 4.6 billion years is dated to a quarter. At eight billion years, when a solar-mass star has used four-fifths of its hydrogen, the precision is eight per cent. On the subgiant branch it is two.

The one thing that changes between them is how fast the star is moving on the diagram. The two-billion-year-old star has brightened by fifteen per cent since it formed; a measurement that fixes its luminosity to three per cent fixes that fraction only to about a fifth, and the age with it. The subgiant is cooling by about ninety kelvin every hundred million years, so a measurement good to seventy kelvin fixes its place on the branch, and its age, to within about a hundred million years.

What the probability is weighed against #

A distribution that wide depends on what is assumed before the measurement. Every combination of mass, age and composition that fits the position is weighted by how many such stars are expected: lighter stars are commoner, so the fit leans towards the lighter, older end of the ridge; ages are taken as equally likely anywhere from zero to fourteen billion years, so a star whose position allows every age from one to six is given a median near the middle of that range. The two-billion-year-old dwarf’s distribution is as broad as it is partly because the prior has nothing to stop it, and runs into the hard edge at zero age.

For a subgiant none of this matters. Its measured position allows only a narrow range of ages, and any reasonable prior is nearly flat across a range that narrow, so the answer is the same whatever is assumed. For a dwarf the prior is a large part of the answer, and two surveys that differ only in their priors can report ages for the same star that differ by more than their quoted errors. The test of whether an isochrone age is a measurement is whether it survives a change of prior, and on the main sequence it often does not.

The stage, not the star #

Plotted against the fraction of its main-sequence life a star has used, the precision falls steadily from tens of per cent to a few, and the curve does not depend on the star’s mass. A 1.2-solar-mass star half-way through its life is dated exactly as well as a 0.9-solar-mass star half-way through, although one is four billion years old and the other eight. The isochrone method is a measurement of how far a star has evolved, and an age only through the conversion from that fraction to years.

That has a direct consequence for which stars can be dated. A star of less than about 0.7 solar masses has not used half its main-sequence life even if it is as old as the Galaxy, and no isochrone age for it will be better than about twenty per cent however well its position is measured; for the red dwarfs, which are most of the stars in the Galaxy, the method gives essentially nothing. The stars that can be dated precisely by their position are the ones near the end: late main-sequence stars somewhat more massive than the Sun, and subgiants. A survey that quotes isochrone ages for millions of stars is quoting precise ages for a minority of them, and little more than bounds for the rest.

An age that is a mass #

The reason a dwarf is poorly dated is visible when mass and age are drawn together. The measured position allows a narrow ridge: a slightly heavier star younger, a slightly lighter star older, each predicting the same luminosity and temperature. A main-sequence lifetime falls as the mass to the minus two and a half, but the ridge is steeper than that, because a heavier star also starts brighter and needs less brightening to reach the observed luminosity — so it must be younger still. Along the ridge, one per cent more mass means almost ten per cent less age.

The position therefore measures a combination of mass and age well and each of them poorly, and the age uncertainty is the mass uncertainty multiplied by about ten. A star’s mass can be known to a few per cent from its position — which is what makes a dwarf’s mass a useful quantity — and that same few per cent becomes a few tens of per cent in age. On the subgiant branch the ridge rotates: the star’s temperature depends on how far across the branch it has got, which is a strong function of age and a weak one of mass, and the two quantities separate.

Any independent measurement of mass breaks the degeneracy. A star in a binary with a measured orbit has its mass to a per cent or two, and its isochrone age improves accordingly. So does a star whose oscillations have been measured: the comb of frequencies in its light gives the mean density and, with the radius, the mass, and the frequencies of its modes that penetrate the core measure how much hydrogen has been used — the fraction of life itself, without passing through the diagram at all.

The cluster’s clock, read off one star #

A subgiant’s position does for one star what a cluster’s turn-off does for many. In a cluster, the stars just leaving the main sequence are those whose lifetime equals the cluster’s age, and the bend in the cluster’s sequence is where that happens; reading the age means finding the mass at the bend. A single subgiant is one of those stars caught in the act: it has just exhausted its core hydrogen, so its age is its main-sequence lifetime plus the short time since, and its temperature says how long that has been. The mass that a cluster’s population supplies is supplied for the subgiant by its luminosity, because on the subgiant branch luminosity depends on mass and hardly on age.

So the isochrone method does not fail for field stars. It fails for field stars that are not at the turn-off, and the turn-off is where it was always going to work — for a population or for one star. A dwarf in the field is like a cluster that has no stars yet leaving its main sequence: young, with a turn-off off the top of the diagram, and an age that can only be bounded.

What a systematic error costs #

Statistical errors average away over a sample; systematic ones do not, and the ages of field stars are dominated by systematics: the temperature scale, which for spectroscopic and colour-based methods disagrees at the level of fifty to a hundred kelvin; the metallicity scale; and the bolometric correction and extinction that turn a parallax into a luminosity, a step that is itself not a simple inversion.

The temperature scale matters most for the dwarf. Fifty kelvin hotter moves the Sun-like star along the narrow mass–age ridge to a higher mass, and the age falls by seventeen per cent. A luminosity three per cent too high moves it up the main sequence, older by seven. The subgiant hardly notices either: its age is set by where it is across the branch, and fifty kelvin is a small fraction of the several hundred it crosses. The metallicity error barely moves either star in this model, because a metal-richer star at the same position is both fainter for its mass and longer-lived, and the two effects nearly cancel; real stellar models do not cancel quite so neatly, and the metallicity scale is a significant systematic for dwarfs in practice.

The asymmetry is the same as for the random errors, and for the same reason. A dwarf’s age is read from a short step on a crowded sequence, so a small shift in where the star is measured reads as a large shift in age. A subgiant’s is read from a long excursion across an open part of the diagram.

What a sample sees #

The consequences for a study of the Galaxy are visible when a population is dated. The subgiants recover their ages almost exactly. The dwarfs scatter widely, and the scatter is not symmetric: a star near the zero-age main sequence can be scattered by its measurement errors only towards older ages, because there are no younger ones, so the youngest dwarfs are recovered too old on average. Any relation between age and another quantity — metallicity, motion, height above the Galactic plane — built from dwarfs is blurred by the scatter and lifted at its young end.

That is why the most precise maps of the Galaxy’s history built from isochrone ages have used subgiants alone. Their rarity is the price. A star spends only about a tenth as long on the subgiant branch as on the main sequence, so among all the stars in a survey, which are mostly dwarfs well short of their turn-off, subgiants are a few per cent. The Sun will not become one for another five billion years, and every star in the Galaxy lighter than about 0.8 solar masses, which is most of them, has never been one. They are a small fraction of all stars, because the subgiant phase is short, but in a survey of millions there are hundreds of thousands, and each is dated to a few per cent. The age–metallicity relation of the solar neighbourhood, the timing of the thick disc’s formation and the epoch of the merger that built the inner halo have all been read from such samples, and the dwarfs that make up most of the same surveys contributed little to the clocks.

Other clocks for the stars this one cannot reach #

For the dwarfs, other clocks exist, each calibrated on clusters and each working where the diagram does not. A cool star’s rotation slows as its wind carries away its angular momentum, and after the first few hundred million years the spin rate depends on mass and age in a way that can be inverted: gyrochronology, precise for young and middle-aged dwarfs exactly where the isochrones crowd, and failing in older stars, whose spin-down appears to stall. Oscillations give the fraction of hydrogen burned directly, for the stars bright and active enough to show them. A star with a white-dwarf companion can be dated by the companion’s cooling, plus the lifetime of the companion’s progenitor. Lithium depletion dates the youngest. Chemical clocks — ratios of elements made on different time-scales, such as yttrium to magnesium — are calibrated on stars dated by the other methods and extend their reach.

Each of these is weakest where another is strongest, and the isochrone method’s place among them is the end of the main sequence and the branch beyond it. It is the only one that needs nothing but the position, which is why it is applied to every star in a survey, and the figures say which of those ages to believe.

What the model leaves out #

The evolutionary model is parametric and smooth; real isochrones have a hook at the turn-off for stars above about 1.1 solar masses, where a convective core contracts briefly at the end of the main sequence, and that hook makes ages ambiguous over a narrow range. The models’ own uncertainties — the mixing length that sets a convective star’s radius, convective overshoot, the diffusion of helium and metals, the solar abundance scale — shift isochrones by amounts comparable with the measurement errors, and they are systematic in the same way the temperature scale is. Subgiant ages from real surveys carry about seven per cent uncertainty rather than two for this reason. The mass prior is Salpeter’s; a flat prior on age is assumed, and a different prior moves the poorly constrained dwarf ages considerably and the subgiants hardly at all. Binaries, which are common and appear brighter than single stars, are ignored. And the metallicity dependence is a single linear coefficient for each quantity, which is the reason its systematic cancels more neatly here than in real grids.

Still open: whether a dwarf can be dated without its neighbours #

Most stars in the Galaxy are dwarfs, and most of what is known about the history of the disc is known from the few stars the isochrones can date. The question is whether a dwarf’s age can be measured to ten per cent from the star itself. Each candidate is calibrated on clusters and limited by them: gyrochronology by its stall at old ages, oscillations by their amplitude in cool dwarfs, chemical clocks by their dependence on where in the Galaxy a star was born. A space photometry mission designed to measure oscillations in Sun-like dwarfs, and spectroscopic surveys large enough to calibrate chemical clocks on subgiants, are the two routes being taken. If either works, the isochrone method’s limit will be the one already drawn here — precise only for stars that are leaving — and the rest of the diagram will be dated by clocks that do not read the diagram at all.