An expansion clock read from the stars' motions

The bottom of the stellar mass function is measured in young clusters, because that is the only place brown dwarfs are bright enough to count. A brown dwarf has no nuclear fuel to hold its luminosity steady; it shines by contracting, and it fades as it does, so a brown dwarf of forty Jupiter masses at a million years old is as bright as a low-mass star, and at a billion years is too faint to find. In a cluster a few million years old they are still luminous, and a census of the cluster’s members by brightness becomes a census by mass — once every brightness has been converted into a mass at the cluster’s age.

That conversion is where the turnover of the mass function became a statement about the age. A brown dwarf’s brightness is roughly its mass raised to a power, divided by its age to another, and the mass function’s turnover, its brown-dwarf fraction and its lowest-mass end all slide when the assumed age changes. The ages used are model ages, read from where the cluster’s stars sit on evolutionary tracks, and pre-main-sequence tracks disagree among themselves by a factor of two for the youngest clusters.

There is another clock, and it needs no stellar model at all. Young associations are expanding. Their stars formed together in a cloud that was then dispersed by the stars’ own winds and radiation, and with the gas gone the group is unbound and its members coast apart. Measure how far apart and how fast, and the ratio is an age.

A slope that is an inverse time #

The idea is as old as the expansion of the universe and identical to it. Stars that left a small region at the same time with a range of speeds are, after a while, sorted by speed: the fast ones furthest out, the slow ones near the middle. Plot each star’s velocity in some direction against its position in that direction and the points lie along a line whose slope is the inverse of the time they have been moving. A star ten parsecs from the centre moving outward at a kilometre a second has been travelling for about ten million years.

The method was proposed in the 1940s for the O and B associations, using the few proper motions then available, and it gave ages that disagreed with the stellar ones in ways nobody could resolve, because the measurements were poor. It became precise only with space astrometry, which now gives five numbers from one wiggle — position, parallax and proper motion — for every member of the nearby associations, accurate enough that the positions in three dimensions and the motions across the sky are known to a fraction of a parsec and a fraction of a kilometre a second. The opening figure shows a model association of the kind now measured routinely: two hundred members, one direction of the three, and a fitted age within a few per cent of the true one.

A clock that runs fast in the youngest groups #

The model behind the slope assumes the stars started from a point. They did not. They formed throughout a cloud a parsec or more across, and a star’s present position is where it started plus how far it has gone.

The starting spread adds scatter to the positions that has nothing to do with the velocities, and scatter in the independent variable weakens a fitted slope. A weaker slope is a longer age, so the clock runs fast, and the effect falls as the inverse square of the age: when the stars have travelled much further than the size they started in, where they started stops mattering. At twenty million years the error is a per cent or two. At three million, for an association born three parsecs across, it nearly doubles the age.

That is the wrong way round for the brown dwarfs. The associations whose brown dwarfs are brightest, and where the bottom of the mass function is measured best, are the youngest — exactly where the expansion clock is least reliable. The bias can be corrected if the birth region’s size is known, which it is not, except by inference from the densest groups still embedded in their clouds.

A clock that starts when the gas leaves #

The second bias goes the other way, and it is not a matter of measurement at all.

The stars did not begin to coast apart when they formed. While the cloud’s gas was still there, its gravity held the group together, and the expansion began only when the gas was driven out — by the winds and radiation of the most massive stars, or by their first supernovae — a delay of perhaps one to a few million years. The expansion clock measures the coasting, not the stars’ lives. For an association five million years old that began expanding three million years after its stars formed, the clock reads two million, half the true age.

The measurements show it. Associations whose expansion ages have been measured from space astrometry, and whose stellar ages are known from their positions on evolutionary tracks or from the lithium their low-mass stars have burned, come out with expansion ages that are younger, typically by a few million years. That offset is now read as a measurement of the delay itself — how long a forming cluster keeps its gas before it loses it, which is one of the central unknowns of star formation, tied to how much of a cloud that cannot hold itself up turns into stars before the rest is blown away — rather than as an error in either clock.

Hubble’s law in a cluster #

The expansion clock is the same argument that gave the first age of the universe, and it shares its biases. A velocity proportional to distance means a common starting moment, and the inverse of the constant of proportionality is an age, a size and a density at once. For the universe, the simple inverse is wrong because the expansion has not been uniform — gravity slowed it and dark energy has since sped it up — and the true age is the inverse slope times a factor that depends on the history. For an association the history is simpler but not trivial: a coasting phase preceded by a held-together phase, and the inverse slope measures only the coasting.

The universe has a second bias that an association shares exactly. Galaxies do not start at a point; they have peculiar velocities on top of the expansion, which add scatter to the velocity–distance relation and must be averaged out over large distances before the slope means anything. Stars in an association have starting positions on top of their expansion, which add scatter to the positions and must be outrun by the expansion before the slope means anything. In both cases the clock is cleanest for the objects that have travelled furthest, and least reliable for the nearest and the youngest.

Statistics are not the limit #

It would be natural to think that the expansion age’s weakness is the number of stars, and that more members would cure it. The arithmetic says otherwise.

The statistical uncertainty of a fitted slope falls as the inverse square root of the number of stars, and its scale is set by the velocity error per star and by the scatter the starting positions add. With velocities known to a few tenths of a kilometre a second, a few dozen members give an age to a few per cent; the nearby associations have hundreds. Counting more stars sharpens a clock whose accuracy it cannot improve, because the starting spread and the delay are biases, not noise, and they are the same for every member.

The velocities that matter most are the hardest. Motions across the sky come from proper motions and distances, and are known well once a distance is not taken as one over the parallax for the fainter members; motions along the line of sight need spectra, are measured for fewer members, and are affected by orbital motion in unresolved binaries, which adds a few kilometres a second of scatter to the stars that have companions — a dispersion inflated by orbits nobody resolved. Fits that use only the motions across the sky avoid that, at the cost of one of the three dimensions.

There is also the question of membership. An association’s outskirts mix with the field population of the Galaxy, which has its own motions, and a few interlopers at large distances from the centre with velocities pointing outward by chance steepen the fitted slope and shorten the age. The members at the edge are the ones that carry most of the leverage in a fit of velocity against position, so the clock is most sensitive to exactly the stars whose membership is least certain. Analyses guard against this by fitting the expansion and the membership together, with a model of the field’s velocities as a background — another place where the clock’s accuracy is set by something other than the count.

A third clock, in the lithium #

Young clusters have a second model-light clock, and it constrains the delay from the other side. Lithium is destroyed at temperatures of a few million kelvin, and a contracting low-mass star’s centre reaches that temperature at an age that depends steeply on its mass, because mechanics sets a star’s central temperature before fusion does. In a cluster of a given age, the stars above a certain mass have burned their lithium and those below it have not, and the boundary between them — a sharp jump in the lithium line’s strength along the cluster’s sequence — moves to fainter stars as the cluster ages. The physics is the contraction of a fully convective ball of gas and one nuclear reaction, much simpler than the full evolutionary tracks, and the ages it gives for clusters of twenty to a hundred million years are among the most reliable known.

Where both the lithium clock and the expansion clock can be read in the same association, their difference measures the delay directly, with neither clock depending on the evolutionary tracks of the hot stars. The comparisons made so far give expansion ages a few million years younger than lithium ages, in the same direction and of the same size as the comparisons with the tracks. Three clocks, two differences, one delay: the motions measure the coasting, the lithium the stars’ contraction, and the gap between them is how long the gas stayed.

What the age does to a brown dwarf’s mass #

The reason to want an age at all is the conversion of brightness to mass, and the conversion is steep.

A contracting brown dwarf’s luminosity falls roughly as its mass to a power near two and a half divided by its age to a power near one, so at fixed brightness the inferred mass rises as the age to a power near a half. An age off by a factor of two moves every inferred mass by forty per cent, and moves them all the same way. Since the mass function is a count of objects in bins of mass, shifting every mass by forty per cent slides the whole bottom of the function — its turnover, its brown-dwarf fraction — sideways by that amount, without changing its shape. The shape is a property of the cluster; the position is a property of the age.

Why the bottom is where the age matters #

The age matters far more for brown dwarfs than for stars, and the reason is the difference between a thermostat and a cooling body. A star on the main sequence has a luminosity set by its mass and held steady by fusion; once it has settled, its age barely changes its brightness, and a census of stars by brightness is a census by mass whatever the cluster’s age. That is why most of a population’s mass sits in small stars and can be counted reliably: the conversion from light to mass is fixed.

A brown dwarf has no thermostat. It is cooling, and its luminosity is a record of how long it has been cooling as much as of how much it weighs. The same is true of the young low-mass stars that have not yet reached the main sequence, which are still contracting toward it; in a cluster a few million years old, every object below about half a solar mass is in that state. So in the youngest clusters the age enters the conversion for most of the members, and an age error shifts most of the mass function. In a cluster a hundred million years old it enters only below the hydrogen-burning limit, and shifts only the brown dwarfs — which by then are too faint to count anyway.

Masses read on an expansion clock #

The last figure puts the two halves together: brown dwarfs whose masses are read using the expansion age, with its two biases.

With no delay the starting spread makes the clock run fast in the youngest clusters and the masses come out slightly heavy. With a delay of a million or two years, the clock reads young and the masses come out light, by more as the cluster is younger. The two biases have opposite signs and different dependences on age, so somewhere they cancel — for some cluster, at some age, the expansion clock is right by accident — and nothing in the motions alone says which cluster or which age.

That is not a failure of the method so much as a statement of what it measures. An expansion age measures the coasting time. A stellar age measures the time since the stars formed. Their difference is the delay, and if the delay is known — from the clusters where both clocks have been read precisely — the expansion age becomes an independent stellar age for the clusters where the tracks are least reliable, which are the youngest, and which are where the brown dwarfs are counted.

What the model leaves out #

The model association coasts in a straight line from a Gaussian blob with an isotropic spread of velocities. Real associations are not so tidy. Many formed in several subgroups, each with its own centre and its own moment of gas expulsion, so that a single slope through the whole association averages several clocks. Some expand faster in one direction than another, because the cloud they came from was elongated, and the slope then depends on the direction measured. And the Galaxy’s tidal field and the residual gravity of the group itself bend the paths slightly over tens of millions of years, which matters little for the young groups but distorts the oldest expansion ages.

The brown-dwarf conversion is a simple contraction law with a fixed effective temperature at each mass, which captures the steepness of the dependence on age and not its details. Real cooling tracks bend where deuterium burning briefly holds a young brown dwarf’s luminosity up, at a few million years, and the age dependence is shallower there and steeper either side.

Still open: the age of a cluster as a parameter #

The expansion clock turns the age of a young cluster from a property of stellar models into a measurement of motions, and in doing so it reveals that the cluster has two ages: the time its stars have existed and the time they have been flying apart. For the mass function the first is what matters, since it is what a brown dwarf has been cooling for. The expansion clock supplies the second, and the difference between them is a delay that is itself a quantity star formation has to explain.

The way forward is the one the turnover essay proposed and the motions now make possible: treat the age as a parameter fitted alongside the mass function, constrained by every clock available — the expansion, the lithium in the low-mass stars, the tracks of the higher-mass ones — with the delay as a further parameter linking two of them. A mass function reported that way carries its age’s uncertainty in its turnover instead of hiding it, and the comparison of clusters, which is how the question of whether the mass function is the same everywhere is asked, becomes a comparison of clusters whose ages have been read on clocks with known biases rather than on one model’s word.