A mass function corrected by an age

The initial mass function is one of the most heavily used quantities in astrophysics. It converts a mass of gas into a population of stars, a population of stars into a luminosity, a luminosity into a star-formation rate, and a star-formation rate into a chemical evolution. Nearly every extragalactic result rests on it.

It is also not observable. What a star count measures is the stars that are there now, and the ones born massive are not.

The gap between the two is not a subtlety. The biggest stars die first, and they die so much faster than everything else that in an old population essentially none of them remains. Counting what is there and calling it the mass function is therefore not an approximation with a small error; it is a different quantity.

It is worth naming the two quantities carefully, because the literature is not always careful. The initial mass function is the distribution of masses at birth. The present-day mass function is the distribution among the stars now alive. Nearly every theoretical use wants the first and nearly every observation gives the second, and the phrase “the mass function” is used for both. Where a paper’s conclusion turns on the high-mass end, which of the two is meant decides the answer by orders of magnitude.

The correction is an exponent, not a factor #

The arithmetic is unusually clean and it is worth doing.

A star’s main-sequence lifetime falls steeply with mass — as roughly $M^{-2.5}$, because the luminosity rises as $M^{3.5}$ and the fuel supply only as $M$. In a population that has been forming stars at a constant rate for a time $T$, the fraction of stars of mass $m$ still alive is their lifetime divided by $T$, provided that is less than one.

So the present-day mass function is the initial one multiplied by $\tau(m)/T$ above the turn-off, and multiplied by one below it. A multiplication by a power law is an addition of exponents: if the initial function has slope $-\alpha$, the counted one has slope $-(\alpha + 2.5)$ above the turn-off.

That is a large correction. A Salpeter slope of $-2.35$ becomes $-4.85$ in the counts, which means that between one and ten solar masses the observed function under-represents the massive stars by a factor of three hundred.

Notice which part of that arithmetic is secure. The exponent 2.5 comes from the mass–luminosity relation, and mass decides everything by a power of three and a half is one of the better-established relations in stellar astrophysics — measured on eclipsing binaries, checked against models, and stable across the mass range that matters here. So the form of the correction is not in doubt. What is in doubt is the normalisation, which requires the age, and the assumption of a constant birth rate, which requires the history.

What has to be known to apply it #

The correction requires the star-formation history, and that is where the difficulty lives.

For a single burst — a star cluster, where everything formed at once — the correction is a step: everything below the turn-off is present, everything above it is gone. That is the cleanest case and it is why clusters are the standard laboratory. Even there the correction is not trivial, because the massive stars have not merely died, they have removed themselves from the count in a way that depends on when.

For constant star formation the correction is the smooth power law above. For anything in between it is a convolution of the birth rate with the lifetime, and the answer depends on the shape of the history rather than only on its duration.

That last observation is the useful one. The same data constrain both, and separating them requires either an independent age — from an isochrone fit, from a white-dwarf cooling sequence, from asteroseismology — or a population simple enough that one of the two is known.

There is a case in between that is common and awkward: a galaxy, or a region of one, which is neither a single burst nor a steady state. Real star-formation histories are bursty on scales of tens of millions of years, and a burst that ended twenty million years ago leaves a population whose most massive stars are gone while the ones just below them are not. The counted function then has a bump rather than a smooth steepening, at a mass set by the time since the burst, and fitting a power law through it returns a slope that is a property of the timing rather than of the function. That is the leading explanation for several reported detections of a non-standard mass function in nearby star-forming regions.

The other end, which needs no correction and has its own problem #

Below the turn-off nothing has died, so the counted function is the initial one exactly. That is the good news, and the low-mass end has a different difficulty of its own.

Low-mass stars are faint. A magnitude-limited survey detects them only nearby, so the sample is small, and the selection is on flux while the quantity wanted is a mass. Converting between the two requires a mass–luminosity relation that is itself steep, so a small error in the relation is a large error in the inferred mass function.

And the very low-mass end runs into the hydrogen-burning limit, below which objects are brown dwarfs that cool and fade continuously. There the concept of a main-sequence lifetime does not apply, the luminosity depends on age as well as mass, and the counted function is not the initial one for a completely different reason.

One practical consequence of all of this is worth stating on its own. The three quantities — the mass function, the star-formation history and the age — are determined by the same counts, and any two of them fix the third. So a paper that measures a mass function has assumed a history, one that measures a history has assumed a function, and one that measures an age has assumed both. That is not a criticism; it is the structure of the problem, and the honest presentations say which two were assumed. What is not defensible, and is common, is measuring one of the three from a population and then using the result to argue about a second in the same population, since the second was an input to the first.

What was actually measured #

Three results stand out, and the third is the one that changed how the quantity is used.

Salpeter’s original slope. Measured in 1955 from the solar neighbourhood, by exactly the procedure above: count the stars, correct for the ones that have died using a lifetime and an assumed constant birth rate, and fit a power law. The answer, $-2.35$, has survived seventy years of remeasurement for stars above about half a solar mass.

The flattening below a solar mass. Later work established that the function is not a single power law: it flattens below about half a solar mass and turns over near the hydrogen-burning limit. That part needs no lifetime correction at all, which is why it was established more securely and earlier than the high-mass slope despite being harder to observe.

The universality question. Whether the function is the same everywhere is the question that matters for extragalactic work, and it is answered by comparing populations with different ages, metallicities and densities. The comparisons are consistent with universality within their uncertainties — and the uncertainties are dominated by the corrections in this essay, so the statement is weaker than it looks. Claims of a non-universal function in extreme environments recur, and each has to be defended against the possibility that the star-formation history rather than the mass function is what differs.

And there is a fourth measurement that deserves mention because it bypasses the correction entirely. In a resolved cluster young enough that no star has died, the counted function is the initial one, with no lifetime correction of any kind. Such clusters are rare, small and dust-obscured, and the mass functions measured in them are the cleanest available — and they are measured on a few hundred stars each, so their statistical precision is poor. The best-determined and the least-corrected measurements are different measurements, which is an uncomfortable and common situation.

The universality question, restated as an observation. The evidence for universality is that populations of very different ages and metallicities, corrected as described here, give consistent slopes. The corrections applied to those populations are different from each other and are the dominant uncertainty in each. So the statement “the function is universal” is inseparable from the statement “the corrections are right”, and a systematic error in the corrections that scaled with age would produce exactly the appearance of universality it is being used to establish. Nobody thinks that is what is happening, and nobody has a test that would show it if it were.

Where the picture stops #

Three of them, and the second is the one that biases the answer.

The count is over a volume, and the volume is not the population. A survey of the solar neighbourhood counts stars in a small cylinder around the Sun, and the massive stars in it were born in the disc’s midplane while the low-mass ones have had time to be scattered to larger heights. So the vertical distribution depends on mass, and the correction from a local count to a column density depends on a scale height that is itself a function of mass and age. That correction is comparable to the evolutionary one for the intermediate masses and it is routinely folded in without comment.

Binaries are counted as one star. A substantial fraction of stars are in pairs, and an unresolved pair is counted once, at a mass inferred from the combined light — which is too large. That flattens the measured function at the low-mass end and depletes the counts, and correcting for it requires knowing the binary fraction as a function of mass, which is itself measured from surveys with their own selection.

Massive stars leave before they die. A cluster loses its stars dynamically as well as by evolution, and the loss is mass-dependent: two-body relaxation drives the light stars outward and the heavy ones inward, so a mass function measured inside a cluster’s half-mass radius is steeper than the cluster’s own. Correcting for that requires a dynamical model and the correction is comparable to the evolutionary one.

And the massive end is measured on very few stars. Above ten solar masses the counted numbers in any single population are tens rather than thousands, so the Poisson errors are large and the fitted slope is unstable. Combining populations to improve the statistics requires assuming they share a mass function, which is the question being asked.

There is a fourth, and it concerns what “mass” means in the measurement. Nothing counts stars by mass. What is counted is stars by luminosity, or by colour and magnitude, and the conversion to mass runs through a mass–luminosity relation that depends on age, metallicity and evolutionary state. For main-sequence stars that conversion is reliable; for anything evolved it is not, and evolved stars are the luminous ones that dominate a magnitude-limited count. So the correction discussed in this essay sits on top of a conversion whose own uncertainty is comparable to it, and the two are correlated because both depend on the age.

Why this is a rung on the mass-function ladder #

The recurring shape here is worth naming, because it is the same one that runs through the selection essays of this collection.

What is counted is what survives, and survival is a function of the quantity being measured. For a magnitude-limited survey, survival is a function of luminosity. For a mass function, survival is a function of mass through the lifetime. In both cases the correction is calculable, in both cases it depends on something else that has to be assumed, and in both cases the assumption is correlated with the answer.

The response is the same too, and it is the one this collection keeps arriving at: model the observation rather than correcting it. Predict what a survey of a population with a given mass function and star-formation history would have counted, apply the selection, and compare against what was counted. That does not remove the degeneracy between the mass function and the history — nothing does, from counts alone — but it puts the degeneracy in the answer where it belongs, rather than in a correction applied silently beforehand.

Most stars are small and most of the light is not, and the gap between those two statements is where every use of the mass function lives. A quantity that converts between mass and light, measured through a survey that selects on light, to be applied to systems whose mass is what is wanted: the circularity is structural, and managing it rather than escaping it is the whole of the practice.

One further observation about the age at which the correction bites. The turn-off mass falls as the age to the power minus two fifths, so it is 2.5 solar masses at a billion years, 1.4 at five billion, and 1.0 at ten. That means the correction is negligible for the mass range that dominates a galaxy’s mass and enormous for the range that dominates its light. A measurement of stellar mass from counts is therefore nearly correction-free and a measurement of star-formation rate from light is nearly all correction — which is why an age read off a bend and a mass read off a count are such different kinds of measurement despite using the same data.

It is worth stating what changes if the function is not universal, because that is what the effort is for. A galaxy’s stellar mass is measured from its light, and the conversion factor is an integral over the mass function — dominated by low-mass stars that contribute mass and no light. Changing the low-mass slope changes every galaxy mass in the universe by a factor; changing the high-mass slope changes every star-formation rate. Neither change is detectable in the galaxies themselves, because both are degenerate with the quantity being measured. The whole census of the universe’s stars rests on a function measured in the solar neighbourhood and assumed to hold at redshift two, and the assumption is defended by the absence of evidence against it rather than by evidence for it. The gas runs out before the galaxy does is one of the arguments that depends on it most directly.

One further caution belongs with the arithmetic, because it decides where the correction is safe. The factor being divided out is a lifetime over an age, and it goes to zero for masses whose lifetime is shorter than the population’s age — so dividing by it amplifies whatever is in the numerator without bound. For a ten-billion-year-old population the correction at three solar masses is a factor of thirty and at five it is a factor of hundreds, which means a handful of stars, or a handful of contaminating interlopers, becomes a large inferred population. The correction is therefore trustworthy only over the mass range where a substantial fraction of the stars born are still present, and every statement about the high-mass end of an old population’s birth function is an extrapolation rather than a measurement.

Where the ladder goes next #

The next rung is the dynamical correction: how a cluster’s own evolution reshapes its mass function, and why an observed function is a property of a radius as well as of a population. Further up sits the extragalactic use — what happens to a galaxy’s inferred stellar mass when the assumed function is changed, and why that single assumption is one of the largest systematics in the census of the universe’s stars.