An age read off a bend

A star’s age is not observable. Its mass is, sometimes; its temperature and luminosity are, given a distance; its composition is, from a spectrum. Its age is not among the things a photon carries — which puts it in the same category as a distance before something is known, a quantity that has to be inferred rather than recorded.

For a cluster, it is. Not because a cluster contains any information a single star does not, but because it contains many stars of one age — and one age applied to a range of masses produces a feature in a diagram that a single star cannot produce.

Why the bend exists #

Two facts, and the whole method is their product.

The first is that a main-sequence star’s lifetime falls steeply with its mass. Fuel is proportional to mass and consumption is proportional to luminosity, which goes as roughly $M^{3.5}$, so the lifetime goes as $M^{-2.5}$ — the biggest stars die first, by a very large factor. The second is that a star on the main sequence stays put. The main sequence is a place stars sit rather than a track they travel: a star of a given mass arrives at a particular luminosity and temperature, brightens by perhaps forty per cent over its whole main-sequence life, and then leaves. If stars moved along the main sequence, the top of it would be populated by stars that had arrived from below and there would be no bend at all.

Put those together. In a cluster of age $t$, every star with lifetime less than $t$ has gone, and every star with lifetime greater than $t$ is exactly where it started. The main sequence is therefore truncated, and the truncation point is the mass whose lifetime is $t$.

Isochrones, and the word “track” #

The curves in the opening figure are isochrones — loci of constant age across a range of masses — and the distinction from evolutionary tracks is the one thing about this subject that is worth being pedantic over.

An evolutionary track is one star’s path over time: fixed mass, varying age. An isochrone is one instant across a population: fixed age, varying mass. They are orthogonal slices of the same two-parameter surface, and they look sufficiently alike near the turnoff that they are constantly confused.

The practical consequence is that the giant branch of an isochrone is a narrow feature, populated by a small mass range, and it is therefore sparsely populated: an old cluster has thousands of main-sequence stars and a few dozen giants. The turnoff is where the statistics are, and it is also the most sensitive part of the diagram to the age, which is a fortunate coincidence rather than a designed one.

What is actually fitted #

In practice nobody reads a turnoff off a diagram by eye. What is done is to fit a family of isochrones to the observed colour–magnitude diagram, and the fit has more free parameters than the age.

The colour–magnitude diagram is not the theoretical diagram: the horizontal axis is a colour index, not a temperature, and the vertical is an apparent magnitude, not a luminosity. Converting between them requires a distance, a reddening, and a transformation from the theoretical temperature and gravity to observed colours — and each of those is uncertain.

Reddening shifts the diagram horizontally and, because extinction is greater in the blue, also vertically; the two effects are correlated and the correlation runs roughly along the isochrone, which is unhelpful.

And composition matters more than it sounds. A metal-poor star of a given mass is hotter and bluer than a metal-rich one, because the opacity is lower, so the whole isochrone slides. The age–metallicity degeneracy is not exact but it is close enough to be the second-largest source of uncertainty in most old-cluster ages after the distance.

How sensitive the bend actually is #

The method’s usefulness rests on a quantitative claim that is easy to make and worth checking: that the turnoff moves enough with age to be readable.

At 12 billion years the turnoff mass is 0.94 solar masses. At 10 billion it is 1.00, and at 14 it is 0.89. So a twenty per cent change in age is a six per cent change in turnoff mass — which, through the mass–luminosity relation’s slope of about four, is a twenty-five per cent change in turnoff luminosity, or a quarter of a magnitude.

A quarter of a magnitude is comfortably measurable. It is also, and this is the difficulty, the same size as the distance and reddening uncertainties, so the age is not limited by how well the bend can be located but by how well the vertical axis is calibrated.

A second sensitivity runs the other way and is exploited constantly. Because the turnoff luminosity is a steep function of age, the vertical position of the turnoff can be used as a standard candle once the age is assumed — which is main-sequence fitting, the technique that puts clusters on the distance ladder in the first place. The circularity is real and is broken by fitting the unevolved part of the main sequence, well below the turnoff, where the position depends on composition and hardly at all on age.

The clusters that were older than the universe #

In the early 1990s the best globular-cluster ages were 16 to 18 billion years. The best value of the expansion rate implied an age for the universe of 10 to 13. The clusters were older than the thing containing them, by a margin of several standard deviations, and the discrepancy was serious enough to have a name.

It resolved from the distance side. Hipparcos measured trigonometric parallaxes for a set of nearby metal-poor subdwarfs — the field equivalents of globular-cluster main-sequence stars — and the resulting main-sequence-fitting distances to the clusters came out larger than the previous ones by about 0.1 to 0.2 magnitudes in modulus. Larger distance means brighter absolute magnitude at the turnoff, which means a more massive turnoff star, which means a younger cluster. The ages fell to 11 to 13 billion years.

The point worth taking is which side moved. The cosmological age also changed in the same decade, upward, when the expansion turned out to be accelerating — but the larger correction was to the cluster ages, and it came from a parallax measurement of stars that were not in any cluster. The oldest ages in astronomy rest on the shortest distance measurement there is.

The clusters themselves, and what they are for #

Two kinds of object get called clusters and they date differently.

Open clusters are young, sparse, in the disc, and dissolving. A few hundred to a few thousand stars, ages from a million years to a few billion, and turnoffs high on the main sequence where the isochrones are widely spaced. They are the best-dated objects in the Galaxy and are what the disc’s chemical history is reconstructed from — each is a sample of the interstellar medium at the moment it formed, with a composition and an age.

Globular clusters are old, dense, in the halo, and gravitationally bound for the age of the universe. A hundred thousand to a million stars, ages of 11 to 13 billion years, and turnoffs near one solar mass where the isochrones crowd together. They are dated much less precisely in relative terms and are far more consequential, because their ages are a lower bound on the age of the universe that owes nothing to cosmology.

That lower bound is the reason the subject has been fought over. A globular cluster age is a measurement made entirely inside the Milky Way, from stellar physics and a parallax, with no expansion, no redshift and no model of anything larger than a star. When it agrees with the cosmological age it is a genuine consistency check between two disciplines that share no equipment.

The agreement is now good. Cosmology gives 13.8 billion years; the oldest globulars give 12.5 to 13 with uncertainties of about a billion; and the difference — half a billion to a billion years — is a plausible interval between the Big Bang and the first star clusters. That the residual comes out positive and small is not something either side arranged, and it is the strongest single argument that both are approximately right.

What else the diagram gives away #

Once a cluster is a single-age, single-composition population, several things become measurable that are not measurable anywhere else.

The helium abundance shifts the main sequence and the turnoff differently, so a well-observed cluster constrains it — which matters because helium is otherwise almost unmeasurable in a cool star, having no lines at those temperatures.

The white dwarf cooling sequence gives a second, independent age. White dwarfs have no fusion, so they simply cool, and the faintest one in a cluster is the first one formed. The two ages agree in the clusters where both have been measured, and they share almost no physics. And blue stragglers: stars above the turnoff that ought not to exist, since a star of that mass should have left long ago. They are the direct evidence that a cluster is not a passive collection — they are made by collisions and by mass transfer across a Roche lobe, and their number per cluster is a measurement of the dynamics rather than of the age.

What the method assumes #

Three assumptions, all approximately true and none exactly.

One age. Globular clusters were long taken to be strictly coeval, and are not: high-precision photometry from space has resolved multiple main sequences in several of them, implying two or more populations differing in helium and in light-element abundances, formed within a few hundred million years of one another. That spread is small compared with 12 billion years, so the ages survive; the story of how clusters formed does not.

One composition. The same observations killed this one too, and more decisively — the abundance spreads in sodium, oxygen and aluminium within a single cluster are large, even where iron is uniform.

One distance. Safe, and the least of the problems.

None of these upsets the measurement, and it is worth being clear why: the turnoff is set overwhelmingly by the dominant population, and second-order structure moves it by far less than the distance uncertainty does. The method’s error budget is still dominated by the same thing it was dominated by in 1950, which is how far away the cluster is.

The remaining assumptions are about the population rather than the stars, and they are the ones a diagram cannot check for itself.

The clock that depends on how far the mixing goes #

There is a term in the lifetime calculation that is not physics in the sense the rest of it is, and it sets the age scale of every intermediate-mass cluster.

A star above about 1.2 solar masses burns hydrogen in a convective core. The convection mixes fresh fuel inward, so the amount of hydrogen available is not what sits inside the nuclear-burning region but what sits inside the mixed region — and the boundary between them is not sharp. A convective element arriving at the formal boundary, where the acceleration reverses, is still moving, and it overshoots into the stable layer beyond before it stops.

How far it overshoots is not calculable from first principles. It is parameterised as some fraction of a pressure scale height and the fraction is fitted, and the fit changes the main-sequence lifetime by ten to twenty per cent at a given mass — which is ten to twenty per cent of every age derived from a turnoff in that mass range.

The observational handle is the shape of the turnoff rather than its position. Overshooting extends the main sequence to higher luminosities and makes the turnoff hook over more sharply, so a cluster with a well-populated turnoff constrains the parameter directly. That is how it is calibrated, on clusters of a few hundred million years where the turnoff sits at two or three solar masses and there are enough stars in it to see the shape.

The awkwardness is that space photometry has since resolved intermediate-age cluster turnoffs into features that are extended rather than sharp, spread over a range that would correspond to a few hundred million years of age spread. Two explanations are live — a genuine spread in formation times, or a spread in rotation rates, since a rotating star is mixed differently and lives longer. The second would mean the turnoff is measuring rotation as well as age, and the two are not separable from the diagram alone.

None of this moves the globular-cluster ages much, because at 0.9 solar masses the core is radiative and there is no convective boundary to overshoot. It moves the open clusters, which are the objects the disc’s history is reconstructed from, and it moves them systematically rather than randomly.

There is a second-order consequence worth naming. Because the fitted overshooting parameter is calibrated at two or three solar masses and then applied across the whole grid, any error in it propagates to masses where nothing constrains it — and the models are used well outside the range the calibration covers, in both directions.

There is a second clock in the same clusters, and comparing the two is the only external check the method has. The faintest white dwarfs in a cluster have been cooling since they formed, and their luminosity is a age that shares nothing with the turnoff: it depends on the heat capacity of a degenerate interior and on an envelope opacity, not on nuclear timescales or convective mixing. Where both have been measured — a handful of open clusters and, with difficulty, two globulars — the two ages agree to about a billion years, and they disagree systematically in the sense that the white-dwarf ages run slightly younger. That is a real result rather than a reassurance: two clocks built out of unrelated physics, agreeing to ten per cent, bound the systematic error of each in a way that no amount of work on either alone could.

Ages that are only ever relative #

The most reliable thing a turnoff delivers is not an age at all but an age difference, and the distinction is worth being explicit about because the two are quoted in the same units.

An absolute age needs a distance, a reddening, a composition and a set of models, and its uncertainty is dominated by the first. Comparing two clusters of similar composition, most of that error is common: the models are the same, the transformations are the same, and if the distances are obtained by the same technique their errors are correlated. What survives the comparison is much smaller than what survives either measurement alone.

So relative ages of globular clusters are good to a few hundred million years while their absolute ages are uncertain by a billion, and every result about the order in which the halo assembled rests on the first number rather than the second. The clusters divide into an old group with essentially no age spread and a younger group that is spread over several billion years and is preferentially at larger radii — which is read as the second group having been accreted with dwarf galaxies rather than formed in place.

The technique that gives those relative ages avoids the distance entirely. Measure the magnitude difference between the turnoff and the horizontal branch, both features of the same cluster, and the distance modulus cancels in the subtraction. What it does not cancel is the horizontal branch’s own dependence on composition and on whatever else sets it, and the second-parameter problem — two clusters of the same metallicity with visibly different horizontal branches — has been open since the 1960s.

A differential measurement is better than its absolute counterpart exactly in proportion to how much the two share, and here they share everything except the feature being used as the reference.

Everything above treats the turnoff as a point on an isochrone. It is also the end of a track, and the two masses that bracket a cluster’s turnoff behave differently enough to be worth drawing.

That distinction matters because it puts a discontinuity into the middle of the age scale. Below about 1.8 solar masses the core is degenerate at helium ignition and the star flashes; above it, the core is not and the star does not. The luminosity of the giant branch’s tip is nearly independent of mass on the first side and rises with mass on the second, so the same observable — the brightest red giant in a cluster — means different things for clusters older and younger than about a billion years.

Isochrone fitting handles that by construction, since the isochrone is built from tracks that already contain it. What it means in practice is that the age uncertainties are not smooth in age: they are small either side of the transition and larger across it, and a cluster whose turnoff mass sits near 1.8 is dated worse than one either side of it for reasons that have nothing to do with its photometry. That is a general feature of ages derived from stellar structure rather than from a decaying nucleus: the uncertainty is inherited from wherever the models change character, and the models change character at masses set by physics that has nothing to do with the cluster being dated. A radioactive clock has no such feature, which is one reason the two kinds of age are compared whenever both are available. The comparison is not often possible — very few clusters contain a star with a measurable radioactive abundance — but where it has been done the two agree to within their uncertainties, which is the only external check the isochrone scale has ever had.

Where the ladder goes next #

The rung above is the field: a single star with no population around it, whose age has to be got some other way. Three routes exist — the white-dwarf cooling age if it has a degenerate companion, the rotation period through gyrochronology, and asteroseismology, which measures the mean density and therefore the evolutionary state directly. All three are calibrated on clusters, which is to say on this rung. The one that has changed the subject is the third, because an interior read from a comb of frequencies gives an age for a star that belongs to nothing at all.