Astronomy & Celestial Mechanics
the sky, computed
Generator

The stellar-lifetime generator

How long a star lasts, against its mass
How long a star lasts, against its mass. Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.

Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.

6 essays call stellar-lifetime. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

How long a star lasts, against its mass. Main-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years. Stars

Why the biggest stars die first, and take the galaxy with them

A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it.

 The main sequence against its own ceiling. Luminosity against mass, in solar units, with the Eddington limit drawn on the same axes. The limit is exactly linear in mass. The main sequence is not — until it is: above about 55 solar masses the relation flattens to a straight proportionality as well, and the two lines then run parallel at a fixed ratio of 83 per cent. They never cross on this range, which is the more interesting outcome: the most massive stars do not approach their limit gradually, they are built at a fixed fraction of it. The Sun, by contrast, radiates 0.0026 per cent of its own ceiling. Stars

A brightness that would blow the star apart

Light pushes. For a star of a given mass there is a luminosity at which the outward push on its own outer layers equals the inward pull of gravity, and it depends on nothing but the mass and the opacity — not on the star's structure, its composition or its age.

 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. Stars

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.

 The same well after the primary has lost 45 per cent of its mass. Two effective potentials for one body: the solid curve before the primary loses mass and the faint one after, both at the same angular momentum, because a central force of any strength exerts no torque. The well shallows and its floor moves out from r = 1.00 to 1.82. The body's own level moves with it — from E = -0.420 to -0.127 — and the two horizontal lines are drawn where the radial action is conserved, which puts the turning points at 0.71–1.67 before and 1.30–3.03 after. The ratio between them is 2.3333 in both, so the orbit is the same shape at a larger size: everything about the body's path has scaled and its eccentricity of 0.4 has not moved. That is what a slow change leaves behind, and it is not what a sudden one leaves. Orbits

The well moves, and the body keeps its share of it

When the Sun becomes a white dwarf it will throw away half its mass, and every planet's orbit will swell by the same factor. Their eccentricities will not change at all — provided the loss is slow, and the only meaning "slow" has here is slow compared with one orbital period.

 How long a star lives on the main sequence, down to the smallest stars. Main-sequence lifetime against mass, on logarithmic axes, as the hydrogen a star burns divided by its luminosity, scaled so that the Sun lives ten billion years. The dashed line assumes every star burns a tenth of its hydrogen, as the Sun does; the solid line lets stars below about a third of a solar mass, which are convective throughout, burn nearly all of it. The horizontal line is the age of the universe, 13.8 billion years: every star lighter than 0.90 solar masses that has ever formed is still on the main sequence. A star of a tenth of a solar mass lives about 8 trillion years — 597 times the present age of the universe — and its life so far is a few thousandths of one per cent of it. Stars

A lifetime that is still a forecast

No star lighter than nine-tenths of the Sun has ever died, because none has had time. The smallest are convective to their centres, burn almost all their hydrogen rather than a tenth of it, and will shine for trillions of years. Most of the stars in the Galaxy therefore have a life history no observation has tested, and only one in thirteen stars ever born has finished.

 A lifetime that stops getting shorter. Main-sequence lifetime against stellar mass, both logarithmic, from a simple model in which luminosity rises as the 3.8th power of mass until radiation pressure caps it at 40% of the Eddington limit (solid), and without the cap (dashed). The Sun lives 10 billion years, a 10 M☉ star 36 million, a 40 M☉ star 3.2 million, and above about 60 M☉ the lifetime stops falling: 2.7 million at 100 M☉, 3.0 million at 300. Without the cap the steep law would give the heaviest stars lifetimes of tens of thousands of years; with it, every very massive star lives a few million years whatever its mass. Stars

A lifetime that stops getting shorter

A star twice the Sun's mass lives a fifth as long, ten times the mass a few hundredths. Extrapolate the steep law that says so to the most massive stars known, a hundred and fifty or two hundred suns, and they should burn out in a few tens of thousands of years. They do not: every star above about sixty solar masses lives between two and a half and three million years, whatever its mass. Radiation pressure caps how bright a star can be per unit of its mass, and once luminosity and fuel both grow in proportion to mass, the mass cancels out of the lifetime.

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