Four clocks and the stars none of them reads

A star in a cluster is dated by the cluster: all its members share an age, and the place where the more massive ones are leaving the main sequence reads it off a bend. A star alone has no such help. Its position on the Hertzsprung–Russell diagram dates it only as it leaves the main sequence, because for most of a dwarf’s life it sits almost still while its age runs through billions of years. Other clocks exist — rotation, oscillations, chemistry — and each is said to work where the diagram does not. The question here is whether, between them, they cover the stars that need dating.

The hero figure lays the four side by side for a star of the Sun’s mass, with representative precisions from the literature stated in its description, as the fractional age uncertainty against the age. Rotation is good early, once the star’s spin has forgotten how it started, and stays at about fifteen per cent until its magnetic braking weakens, near the Sun’s present age; after that it is useless. The isochrones, the oscillations and the chemical clock all improve with age, because their uncertainty in years is roughly constant and the age grows. The combination, heavy, is fourteen per cent at a billion years, nine at three billion and four at eight. A Sun-like star is datable to better than ten per cent from about two and a half billion years until it leaves the main sequence.

Four clocks, four kinds of error #

The clocks differ in how their errors scale, and the scaling is what decides where each works.

Isochrones read an age from where a star sits on the diagram relative to model tracks. On the main sequence a star moves slowly — a Sun-like star brightens by about forty per cent over its first five billion years — and errors of a few per cent in temperature and of a tenth of a decade in metal content blur its position across a large part of that motion. The age uncertainty in years is therefore a roughly fixed fraction of the main-sequence lifetime, of order a tenth of it. For a Sun-like star that is about eight hundred million years at any age; as a fraction of the age it is large early and small late.

Rotation reads an age from a spin that slows predictably: after the first few hundred million years a star’s wind has erased the memory of its initial spin, and the period grows as roughly the square root of age. A period measured to a few per cent, with a scatter at fixed age of several per cent, gives the age to about fifteen per cent, whatever the age, so long as the braking law holds. It does not hold forever. Old stars appear to stop spinning down once their rotation is slow enough that their dynamos change character, and from then on the period no longer changes with age.

Oscillations read an age from the frequencies of sound waves trapped inside the star, which measure its interior structure — in particular how much of the hydrogen in its core has turned to helium. The precision, in years, is again a fraction of the main-sequence lifetime, somewhat better than the isochrones’, but it applies only where the oscillations can be seen. Their amplitude grows with the star’s luminosity divided by its mass, and below about nine-tenths of a solar mass it is too small to detect with present photometry.

Chemistry reads an age from the ratio of two elements the Galaxy has made on different timescales — yttrium, produced mostly in old low-mass stars, against magnesium, produced promptly in supernovae. In stars like the Sun, born from gas enriched progressively over time, the ratio has climbed by about four hundredths of a decade per billion years, and a precise abundance analysis measures it to about two hundredths, giving the age to about half a billion years. But the calibration holds only for stars very like the Sun in mass and metal content — solar twins — because the ratio also depends on where and when in the Galaxy a star formed.

A slightly heavier star #

The more massive dwarfs are easier, for a reason that is also a limitation.

A star of 1.2 solar masses moves across the diagram quickly, burning its hydrogen in about six billion years, so the isochrones and oscillations reach good precision soon. Its rotation clock is short-lived: such a star’s thin convective envelope brakes weakly, its spin converges early and stalls early, and by two and a half billion years its period no longer tracks its age. The clocks hand over from one to another smoothly in the figure, and the combination stays near ten per cent through most of the star’s life. The heavier dwarfs are well dated because they age conspicuously.

A K dwarf has one clock #

Move from the Sun to a star of eight-tenths of its mass, an early K dwarf, and two of the four clocks disappear.

The oscillations are gone because the star is too faint for its mass. The chemical clock is gone because it is not a solar twin. The isochrones remain, but the star’s main-sequence lifetime is about seventeen billion years, longer than the age of the universe, and it barely moves on the diagram in the time available: its isochrone age uncertainty is more than a billion years at any age, larger than its age for most of its life. What is left is rotation, at about fifteen per cent, and the combination of the two rarely does better.

That fifteen per cent rests on the rotation clock’s calibration holding for K dwarfs through most of their lives, which is less secure than it sounds. The calibration comes from open clusters, and old open clusters are rare: the oldest with well-measured rotation periods for K dwarfs are a few billion years old. Within that range the K dwarfs have shown surprises — a temporary stall in their spin-down at around a billion years, during which their periods barely change for several hundred million years, attributed to angular momentum being exchanged between the star’s core and envelope. A clock that stalls is a clock that cannot date the stall’s interval, and its calibration beyond the oldest clusters is an extrapolation.

The isochrone figure for this star is worth reading carefully, because it says something about the diagram that is easy to miss. The isochrone curve and the combination meet only near the end of the drawn range, at eleven billion years, where the K dwarf has finally moved far enough from where it began to be placed on its track. That is not the end of its main sequence; at seventeen billion years it is barely a third of the way through. Every K dwarf in the Galaxy is younger than its own main-sequence lifetime by a large margin, so every K dwarf is still in the part of its life where the diagram cannot date it. Isochrones are a clock that reads only near its end, and the universe is not old enough for the K dwarfs’ clocks to have reached theirs.

Where some clock works #

The map makes the pattern plain. Above about nine-tenths of a solar mass, where the oscillations become visible and the stars move appreciably on the diagram in their lifetimes, some combination of clocks reaches ten per cent over most of the main sequence after its first two billion years. Below it, the shaded region ends. The threshold is not a precise number — it depends on the precisions assumed, and better photometry or better rotation calibrations would move it — but its existence does not: the clocks that improve with age all need a star that changes visibly in its lifetime or rings loudly enough to hear, and the K and M dwarfs do neither.

Those are the commonest stars, and the most common star is the one whose life history is a forecast rather than an observation. Below 0.85 solar masses lie roughly three-quarters of the main-sequence stars in the Galaxy’s disc, and most of the stars around which small planets have been found. Most stars are small and most of the light is not, and the same imbalance runs through dating: the stars that are easiest to date are a minority of the stars there are.

The threshold matters to the shape of that conclusion, and it is worth seeing what a looser one does.

At twenty per cent the gap closes, and that is the honest summary of where the commonest stars stand: datable to about fifteen per cent by one clock, if that clock’s calibration can be trusted out to ages it has not been tested at. The difference between the two maps is not a difference of measurement but of trust. A ten-per-cent requirement exposes that only one method reaches the K dwarfs; a twenty-per-cent one hides it behind that method’s nominal precision.

The rotation clock’s window #

The rotation clock is the only one that reaches the low-mass stars, so its window decides what can be dated among them.

The window has two edges, and both move with mass. Its opening — the time by which stars of a given mass have converged onto a single spin–age relation — comes later for lower masses, because their weaker winds take longer to erase the starting spread. Its closing — the onset of weakened braking — comes later too, because the slower-rotating, less massive stars reach the critical rotation later in their lives. For a heavier dwarf the window closes within a few billion years; for a Sun-like star at about its present age, which is why the Sun’s own period is a weak constraint on its age; for a K dwarf, perhaps not before eight billion years, if the braking law holds that long.

The closing edge for K dwarfs has not been observed, because no cluster old enough with K-dwarf rotation periods exists close enough to measure. The weakened-braking picture rests on stars somewhat more massive than the Sun, measured by oscillations and rotation together, and its extension to K dwarfs is a prediction. If the K dwarfs’ braking weakens earlier than the extrapolation says, their window closes sooner and the region of the map no clock reaches grows.

Clocks calibrated by clocks #

None of the four clocks is fundamental. Each is calibrated against stars whose ages come from another, and the chain has a direction. Isochrones rest on stellar evolution theory and are tested on clusters, where the turn-off dates the whole population. Rotation is calibrated on those same clusters: the periods of their K and G dwarfs, at ages set by their turn-offs, define the spin–age relation. The chemical clock is calibrated on solar twins dated by isochrones and oscillations. Oscillation ages are the most direct, depending on the physics of the interior rather than on any empirical relation, but they too assume a model of the star’s structure.

So the combined precision in the figures is a precision relative to the isochrone scale at the root of the chain, and any systematic error there propagates down. An error of ten per cent in the turn-off ages of the calibrating clusters — the size of the differences between model sets — shifts every rotation age by the same ten per cent, and no combination of clocks calibrated on it can detect it. The independent check is the one star whose age is known without any of them.

The Sun as a test #

The Sun’s age is known from the decay of radioactive isotopes in the oldest meteorites, which formed with it: 4.567 billion years, to a fraction of a per cent. It is the one star against which every clock can be scored absolutely.

The clocks do well, with caveats that illustrate the figures. Helioseismology, with millions of oscillation modes measured over decades, gives an age within a few per cent of the meteoritic one — far better than the stellar precisions drawn, because no other star is so close. Isochrones fitted to the Sun’s luminosity, temperature and composition give a few billion years with an uncertainty that depends mostly on the composition assumed, illustrating why isochrones are weak on the main sequence. Rotation gives an age close to the right one if the Sun’s period of about twenty-five days is read with the spin–age relation from younger clusters — but whether the Sun is just before, at or just past the onset of weakened braking is itself argued about, and its period alone cannot settle that. The Sun is a test passed, of a clock whose window may be closing at exactly the Sun’s age.

What the gap costs #

The ages of individual stars are wanted for two large programmes, and the gap affects both.

The first is the history of the Galaxy’s disc: when its stars formed, how their orbits have been heated over time, how its chemistry evolved. Those questions are asked of large samples, and statistical ages — a population of stars dated in bulk by isochrones, with the errors averaging down — answer some of them well. But questions about the oldest stars in the thin disc, or about how a population’s kinematics depend on age at the level of a billion years, need individual ages to better than about twenty per cent, and for most of the disc’s stars that is not available.

The second is the evolution of planetary systems: whether planets migrate or lose atmospheres over billions of years, whether the radius valley shifts as systems age, whether close-in planets are destroyed by tides. Each needs ages for the host stars, most of which are K and early M dwarfs, and each is limited at present by the precision of those ages more than by the precision of the planets’ measurements.

What the figures assume #

The precisions are representative, not measured for any one star. Isochrone precision depends on how well the temperature, luminosity and metal content are known; rotation precision on the scatter of the spin–age relation, which is itself a subject of measurement; oscillation precision on the quality of the photometry and the star’s brightness. The figures use values typical of the best current work, and the map’s boundary moves if they change.

The clocks are independent. In practice they share inputs — all depend on the star’s mass and composition, and the rotation and chemical clocks are calibrated against ages from the others. Combining them as independent overstates the combined precision somewhat.

The rotation model is simple. Convergence, square-root spin-down and weakened braking are idealisations of a spin history that has stalls, mass-dependent exponents and a dependence on metal content; the window’s edges are approximate.

A white-dwarf companion is not used. A dwarf with a white-dwarf companion can be dated by the companion’s cooling plus its progenitor’s lifetime, often to ten or twenty per cent at any age — a fifth clock available for a small fraction of stars, which is how several of the rotation calibrations for old K dwarfs have been extended.

Still open: a clock for the commonest stars #

The missing clock would be one that works for a single low-mass dwarf at a few billion years. Candidates are being pursued. The magnetic activity of a star — the strength of the emission from its chromosphere — declines with age, more steeply than rotation in some ranges, and can be measured for faint stars; but it varies through a star’s activity cycle by as much as it changes over a billion years. The lithium in K dwarfs is depleted slowly over billions of years, at rates that depend on mass in ways not yet calibrated. Kinematics give statistical ages for populations but not for individual stars. And wide binaries — a K dwarf with a more massive companion whose age can be read by isochrones or oscillations — let the easier clock date the harder star, one pair at a time.

Each of these is being calibrated against the old open clusters and the wide binaries that the space astrometry catalogues have found in large numbers. Whether any of them reaches ten per cent for a single K dwarf at five billion years is the question that decides whether the commonest stars in the Galaxy can ever be dated as individuals — or whether their ages will remain properties of the populations they belong to.