Concept

Turn-off mass — where it appears

The stellar mass whose main-sequence lifetime equals a population's age, so that stars heavier than it have died. It converts a present-day mass function into a birth function, and dividing by the surviving fraction near it amplifies small numbers without bound.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

$7 solar masses of progenitor become 0.56 of white dwarf. Above: the initial–final mass relation. The steep line is what a star would leave if it kept everything; the shallow one is what it actually leaves, M_f = 0.08M_i + 0.489, fitted to white dwarfs in open clusters. Everything between 1 and 8 solar masses ends up between 0.57 and 1.13 — a range of 7 compressed by a factor of 12 — and the fraction thrown away rises from 43% at the bottom to 86% at the top. A star spends most of its mass in the last per cent of its life, at rates no theory predicts from first principles, and this relation is measured by running stellar evolution backwards: a cluster's turn-off gives the age, a white dwarf's temperature gives its cooling time, the difference gives how long its progenitor lived, and a model turns that into the mass it was born with. Below: the white dwarf mass distribution that follows, from a Salpeter initial mass function pushed through the relation above. It peaks at 0.61 solar masses, which is what every survey of white dwarfs measures. Two things make that peak and the flatness of the relation is only one of them: the unsmoothed distribution (the stepped curve) falls monotonically and is largest at its low-mass edge, because the initial mass function is steep. The edge is where it is because no star lighter than about 1.0 solar masses has had time to die yet, and the scatter of the measurement rounds that cliff into a maximum a little above it. A flat relation produces a narrow range; it takes the age of the Galaxy to produce a peak.$

The mass a star does not keep

The initial–final mass relation is nearly flat, so everything from one to eight solar masses ends as a white dwarf between 0.57 and 1.13 — and the observed distribution piles up at 0.6 almost regardless of what went in. The relation is measured by running stellar evolution backwards, because nothing predicts the loss.

stars · Stellar evolution $A high end that is 2.5 in slope steeper than the one stars were born with. The mass function a star count measures against the one stars were born with, for a population that has been forming stars steadily for 10 billion years. Below about a solar mass nothing has had time to die, so the two coincide exactly. Above it the fraction still alive is the main-sequence lifetime divided by the age, and since the lifetime falls as the two-and-a-half power of the mass, the present-day function is steeper than the initial one by exactly that exponent. A count of massive stars in an old population therefore under-represents them by orders of magnitude, and reading it as an initial mass function gives a slope far too steep. The correction is large, it is calculable, and it depends on the star-formation history — which is usually the thing the mass function was going to be used to constrain.$

A mass function corrected by an age

Counting stars by mass gives the stars that are alive. What every argument needs is the stars that were born, and the two differ by the fraction of each mass still on the main sequence — which is a lifetime divided by an age, and which for massive stars in an old population is a very small number.

stars · Initial mass function

Named alongside it

The objects these essays reach for when they reach for this one.

Asymptotic giant branch Chandrasekhar limit Completeness Cooling age Dredge up Initial final mass relation Initial mass function Main sequence lifetime Mass loss Mass loss rate Mass-to-light ratio Planetary nebula

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