A turnover whose position is an age

The initial mass function — the distribution of masses with which stars are born — is the single most used function in astrophysics that nobody measures directly. It is inferred, and the inference passes through a mismatch at every step: stars are counted by their light and wanted by their mass, and the light of a star depends on its mass far more steeply than its number does. For massive stars the inference needs a correction for those already dead, a lifetime divided by an age. For the smallest objects, at the other end of the function, it needs an age for a different reason.

Below about 0.075 solar masses, an object never reaches a central temperature high enough to sustain hydrogen fusion; mechanics sets its central temperature before fusion can, and degeneracy pressure halts its contraction first. Such an object is a brown dwarf. It shines by contracting and then by cooling, and at no point does its brightness hold steady the way a star’s does on the main sequence. Even objects above the limit, in a cluster a few million years old, have not yet arrived on the main sequence and are still contracting. In a young cluster, every object below about half a solar mass is fading.

So the brightness of a young low-mass object is a function of two things, its mass and its age. The hero figure draws it. Each line is one mass, from ten Jupiter masses to half a solar mass, fading as it ages; the dashed horizontal line is one observed luminosity, about a hundredth of the Sun’s. That luminosity belongs to an object of 38 Jupiter masses if the cluster is a million years old, 52 if it is two, 80 if five and 111 if ten. Across a factor of ten in age the inferred mass changes by a factor of three — and crosses the hydrogen-burning limit on the way.

Why young objects fade as they do #

A young low-mass object is fully convective, and its surface temperature is pinned near a value set by the opacity of its outer layers, about 2,500 to 3,500 kelvin depending on mass — the Hayashi limit. At that fixed temperature it contracts, radiating the gravitational energy it releases, and the virial theorem fixes how its radius shrinks: for a fully convective sphere, the radius falls as the age to the power $-1/3$, so the luminosity, proportional to the radius squared at fixed temperature, falls as the age to the power $-2/3$.

That law has a mass dependence too. At a given age a heavier object has more gravitational energy to release and is larger and brighter, roughly as the mass to the power 4/3, modified by the slow rise of the Hayashi temperature with mass. Inverting for the mass at fixed luminosity and temperature gives

$$M \propto L^{3/4}\, t^{1/2}.$$

The mass inferred from a given luminosity goes as the square root of the age assumed. An age wrong by a factor of two gives masses wrong by forty per cent; an age wrong by a factor of four, by a factor of two.

That is a large lever, and the ages of young clusters are not well known. They are themselves read from the same contraction tracks — from where a cluster’s stars sit on the Hertzsprung–Russell diagram relative to models — and different model sets give ages that differ by factors of two for the same cluster. Independent clocks, such as the lithium depletion boundary, work only for clusters older than about twenty million years. For the youngest regions, where brown dwarfs are brightest and easiest to count, the age is the weakest number in the analysis.

The same object, star or not #

The boundary between stars and brown dwarfs is a fixed mass, and an object near it can be put on either side by the age alone.

The limit itself is a theoretical quantity. Shiv Kumar computed in 1963 that below a certain mass a contracting object becomes degenerate before its centre is hot enough to burn hydrogen steadily, held up by a counting rule rather than by heat; modern models put the limit at about 0.075 solar masses for solar composition. Whether a given object in a young cluster lies above or below it cannot be read from its brightness without its age. In a cluster of uncertain age the census of brown dwarfs is a census of objects fainter than a luminosity that itself depends on the age assumed.

There is one observational test that does not need the age, for clusters old enough. Lithium is destroyed at a central temperature of about two and a half million kelvin, which objects below about 0.065 solar masses never reach. A cluster member that still shows lithium in its spectrum, and is fainter than the faintest stars that have destroyed theirs, is a brown dwarf by a criterion owing nothing to the luminosity–age relation. The first brown dwarfs confirmed in a cluster, in the Pleiades in 1995, were confirmed this way. But the test works only once the cluster is old enough for the low-mass stars to have burned their lithium — tens of millions of years at least — and the youngest regions, where brown dwarfs are brightest and most numerous in surveys, are too young for it.

A turnover that moves #

The mass function of a young cluster, built object by object from luminosities, inherits the age’s error as a shift.

The simulation draws twenty thousand objects from a log-normal mass function — the form most measurements find, with a broad peak near a fifth of a solar mass and a width of about half a decade — gives them all the same age, computes their luminosities, and then reads their masses back with an assumed age. With the right age the distribution is recovered. With an age too young by a factor of two, every object is read as lighter by about a quarter, and the whole distribution slides down the mass axis; too old by a factor of 2.5, it slides up by half.

The shape survives the error; its position does not. Because the inferred mass is proportional to the square root of the assumed age for every object alike, a wrong age multiplies every mass by the same factor and moves the distribution bodily in log mass without changing its width or its skew. The turnover — the peak, which is the feature most often compared between regions — moves with it, and so does the hydrogen-burning limit’s place in the distribution.

For an older cluster the same fractional error has nearly the same effect, because the dependence on age is a power law. What changes is how likely the error is. Ages of a few million years are usually quoted to a factor of two; the uncertainty in the turnover’s position that follows is about forty per cent, comparable to the differences between regions that comparisons of the turnover have been used to find or rule out.

A ratio that is an age #

The quantity most often quoted for the substellar end of the mass function is the ratio of brown dwarfs to stars, a single number that can be compared between regions with modest data.

The ratio is sensitive to the age because the boundary between its two bins is a fixed mass, while the inferred masses slide. A cluster read as younger than it is has its faint stars read as brown dwarfs, and the ratio rises; one read as older loses brown dwarfs to the stellar side. A factor of two in age moves the ratio by more than half its value.

For an older cluster read as one million years old — the age often assumed for embedded regions by default — the inferred brown-dwarf ratio is more than two and a half times the true one. That is the size of difference that has, in the past, been reported between star-forming regions and read as evidence that brown dwarfs form more efficiently in some environments than others: denser clusters, or regions bathed in ultraviolet light from massive stars, or regions where gas fragments differently. Some such differences have later shrunk when the regions’ ages and distances were revised.

Universality, tested through an age #

The question these comparisons address is one of the central questions about star formation: is the initial mass function universal? The function measured in the solar neighbourhood, in young clusters and in old globular clusters looks remarkably similar, with a peak near a fifth of a solar mass and a high-mass slope close to Salpeter’s. If it is universal, star formation is governed by something that sets the same characteristic mass everywhere — the thermal physics of collapsing gas, perhaps, or the turbulence that fragments it. If it varies, the variation is a measurement of what controls it.

At the substellar end the test runs straight through the age. A turnover that sits at a different mass in two regions, or a brown-dwarf ratio that differs by a factor of two, is evidence for a non-universal mass function only if the regions’ ages are known well enough to rule out the shift the figures show. For most young regions they are not. The honest conclusion from the present data is that the substellar mass function is consistent with being universal within the uncertainty that ages contribute — which is a statement about how large that uncertainty is as much as about star formation.

The present-day mass function and the initial one differ at the high-mass end by the stars that have died; at the low-mass end they differ by nothing, but the present-day masses are read through the present-day age. The two ends of the function share the same structure: what is counted is a luminosity, what is wanted is a mass, and the age converts one into the other.

What the turnover was supposed to tell #

The interest in the turnover’s position is not merely bookkeeping. If the characteristic mass of star formation — the mass at which the function peaks — is set by the thermal physics of collapsing gas, it should depend on the gas’s temperature and density, and so differ between a cold, dense cloud and a warm, diffuse one, and between today’s metal-rich clouds and the metal-poor gas of the early universe. The Jeans mass, the scale at which a cloud cannot hold itself up, goes as the temperature to the 3/2 power and the inverse square root of the density; a factor of two in temperature is a factor of nearly three in the Jeans mass.

Measured turnovers in nearby star-forming regions, which span a range of temperatures and densities, differ much less than that — and that near-constancy is itself one of the puzzles of star formation, which theories of fragmentation have tried to explain by feedback from the first stars to form, or by the way a collapsing cloud’s temperature tracks its density. The puzzle is only as sharp as the measurements. If each region’s turnover is uncertain by forty per cent from its age alone, a difference of a factor of two between regions is hard to establish and a constancy to twenty per cent is impossible to.

Masses that do not need an age #

The degeneracy breaks wherever a mass can be measured directly, and for brown dwarfs that means binaries. A pair of brown dwarfs orbiting each other with a measurable period and separation gives their total mass by Kepler’s third law, independent of any model; their luminosities then calibrate the luminosity–mass–age relation at that point. A few dozen such dynamical masses now exist, most for brown dwarfs in the field, a few hundred million to a few billion years old, and they have found the evolutionary models good to roughly ten to twenty per cent in mass at a given luminosity and age — with discrepancies in both directions that are not yet understood.

For young clusters the dynamical masses are rarer, because young brown-dwarf binaries are far away and their orbits long. Eclipsing young binaries are the most valuable: their masses and radii come directly from the light curve and velocities, and their luminosities and temperatures then test the contraction tracks at a known mass. The handful known show that the youngest objects’ radii are sometimes larger than models predict for their ages — a sign that either the models or the ages are off, in the way the hero figure says they must trade against each other.

The same problem, a thousand times older #

Brown dwarfs in the field around the Sun are not young; they have ages anywhere from a few hundred million years to ten billion, and they have long since stopped contracting and are simply cooling, like a planet. Their luminosity falls with age as roughly the 1.3 power and rises with mass as about the 2.6 power, so the mass inferred from a luminosity again goes as about the square root of the age assumed. For a single field brown dwarf the age is unknown within a factor of ten, and its mass within a factor of three.

The field’s substellar mass function is therefore derived differently: not object by object, but by simulating a population with an assumed star-formation history and mass function, letting every object cool for its age, and comparing the predicted distribution of luminosities and temperatures with the one observed. The age enters as a distribution rather than a number, and the mass function that comes out is constrained to be consistent with it. It is the approach the young clusters, with their single uncertain ages, are now beginning to adopt in the form of fitting the age and the mass function together.

What the simulation leaves out #

The contraction law is the simplest one. Real tracks include deuterium burning, which stalls the contraction of objects above about thirteen Jupiter masses for a few million years and changes the luminosity–age relation at the youngest ages; and the surface temperature is not fixed but drifts with age and mass. The square-root dependence of inferred mass on assumed age is close to what detailed models give for ages of one to ten million years.

Every object in a cluster has one age. Real star-forming regions show spreads of ages, or apparent ages, of a few million years, and an age spread adds a corresponding spread to the inferred masses on top of the shift drawn here.

The mass function is a single log-normal. The conclusion that a wrong age shifts it without reshaping it holds for any shape, because the shift is a constant factor in mass.

Distances, extinction and unresolved binaries are perfect. Each enters the inferred luminosity and therefore the inferred mass, with its own errors, and the age error is in addition to them. A distance wrong by twenty per cent moves luminosities by forty and masses by about thirty, comparable to the age effect.

Still open: the age of a cluster as a mass-function parameter #

The figures suggest a reframing that the field has begun to adopt: treat the age not as a fixed input but as a parameter fitted alongside the mass function, with whatever independent constraints exist — kinematic ages from the cluster’s expansion, lithium for the older clusters, the positions of the higher-mass stars on their tracks. A mass function reported with the age marginalised over carries an honest uncertainty in its turnover and its brown-dwarf ratio, and comparisons between regions then test universality with the age uncertainty included rather than hidden.

The kinematic ages are the most promising independent clock. The space astrometry catalogues measure the motions of young stars precisely enough to trace a cluster’s expansion back to its compact origin, and the time since then is an age that owes nothing to stellar models. Where such ages exist, they have sometimes agreed with the model ages and sometimes differed from them by a factor of two. Every such comparison is also a measurement of how far a substellar mass function from that cluster could have been misplaced — and of how much of the variation once reported between regions was the age.