Astronomy & Celestial Mechanics
the sky, computed
Concept

Mass loss — where it appears

Material a star sheds during its life rather than in its final explosion. For massive stars it is driven by radiation pressure on spectral lines and can remove most of the star's mass, which makes the remnant it leaves depend on its metallicity as well as its birth mass.

Named by 8 essays across 3 fields — 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 Two responses to the same transfer, turning round at q = 0.79 and q = 1. What conservative mass transfer does to the orbit and to the lobe, plotted against the mass ratio of donor to accretor on a logarithmic axis. Both curves are logarithmic derivatives with respect to the donor's mass, so a positive value means the quantity shrinks as the donor loses mass and a negative one means it grows. The orbit's response is exactly twice the mass ratio less one, which follows from holding the total mass and the total angular momentum fixed and nothing else, and it crosses zero at equal masses: transfer from the heavier star draws the orbit in, transfer from the lighter one pushes it out. The lobe's response adds to that the change in the lobe's shape, and it crosses zero earlier, at a mass ratio of 0.788. Between those two crossings the orbit is still widening while the lobe is already closing. To the right of both, a donor that loses mass finds its lobe shrinking around it, which is the runaway the essay is about: the transfer narrows the valve it is flowing through.

The flow that narrows its own channel

Two stars close enough share a surface, and the point where that surface pinches is a valve. What comes through it changes the orbit, and the orbit changes the valve — with a sign that reverses at equal masses, which is why some binaries transfer quietly for a hundred million years and others tear themselves apart in a thousand.

stars · Mass transfer A wind that writes its own terminal speed on the blue edge, at 2017 km/s. A P Cygni profile, computed by integrating a spherical wind on a grid rather than drawn. The wind accelerates outward as one minus the inverse radius to the power 0.8, reaching 2000 kilometres a second, and its density falls as the inverse square of the radius over its own speed; the horizontal axis is the Doppler shift in units of that terminal speed, with blue to the left. The column directly in front of the stellar disc is moving toward the observer at every speed from nearly nothing near the surface up to the terminal value far out, so it takes light out of the beam across that whole range and the absorption trough runs from the rest wavelength to the blue edge at -1.01 of the terminal speed. That edge is the measurement: no model of the star, its distance or its mass enters it, only the geometry of a column seen end-on. The emission comes from everywhere else, where the wind moves across the line of sight and can only add photons; it is Infinity times stronger to the red than to the blue, because the part of the shell receding directly away is hidden behind the star and its blueshifted counterpart is not. Two features, one shell, and between them a velocity and a mass-loss rate.

A speed read off an edge

A hot star drives material off itself at thousands of kilometres a second, and the profile that wind prints on the star's own spectrum has a sharp blue edge. That edge is the terminal velocity, measured with no model of the star, no distance, and no calibration — one of the few numbers in stellar astrophysics obtained from geometry alone.

starlight · Stellar winds Two bodies at a mass ratio of 2.6 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 2.6 to 1, so the heavier body's path is 2.6 times smaller.

The companion survives, and it is moving

When one star of a close pair explodes, the other is left holding an orbital velocity and no orbit. Whether the pair stays bound turns on a single question — did more than half the total mass leave? — and the stars that answer no are running across the Galaxy at a hundred kilometres a second with nothing visible behind them.

stars · Supernovae 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.

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.

orbits · Effective potential A push that depends on how fast the wind is already going. The force multiplier — the radiative push on a line-driven wind as a multiple of the push on its free electrons — against t, the electron-scattering optical depth across one thermal Doppler width, which is small where the wind accelerates steeply. The solid curve is an ensemble of spectral lines whose strengths follow a power law, the number of lines stronger than a given strength falling as that strength to the power 1 − α, with α = 0.6. If every line were optically thin the push would be the flat line at 2000, independent of the flow; if every line were thick it would fall as 1/t, set only by the velocity gradient. The ensemble lies between: over most of the range it falls as t to the power −0.59, because as t rises lines cross from thin to thick one after another. That single exponent is the whole of the theory — it fixes the terminal speed, the mass-loss rate and how both scale with the star — and it is a property of the atomic physics of the ions in the wind, not of the star.

A rate the wind has no choice about

A hot star's wind is driven by starlight caught in hundreds of thousands of spectral lines, and the way those lines are distributed in strength fixes everything else. The mass-loss rate turns out not to be a free parameter at all but the one value for which a smooth flow can leave the surface — and multiplying it by the wind's speed and the square root of the star's radius cancels the star's mass, leaving a relation with the luminosity alone.

starlight · Stellar winds One history of mass loss, three orbits, three outcomes. The eccentricity of three planets' orbits against the fraction of the star's mass lost so far, as the star sheds 45 per cent of its mass in 6 equal pulses at irregular intervals. Each planet starts with e = 0.05. The pulses are the same events in the star's own time, and the planets differ only in their periods, so each pulse lasts 3, 0.3, 0.03 orbital periods respectively. For the innermost planet, to which every pulse is slow, the eccentricity ends at 0.050: the orbit swells and keeps its shape. For the outermost, to which every pulse is sudden, each pulse kicks the eccentricity by an amount set by where the planet happened to be, and it ends at 0.298. The middle planet ends at 0.090. The same total loss leaves three different shapes, and only one of them is predicted by either limit alone.

The size forgets how the mass was lost, and the shape does not

A dying star loses half its mass, and a planet round it keeps its orbit's shape if the loss is slow and has it scrambled if the loss is sudden. Real stars lose mass in pulses — each one sudden or slow depending on the planet — and what survives a train of pulses is neither answer: the orbit's size settles to the slow value quickly, while its eccentricity random-walks and keeps a record of the history.

orbits · Effective potential How much a dying star stirs its companion's orbit, against their separation. The rms change in orbital eccentricity of a 0.8 M☉ companion when a 2 M☉ star becomes a 0.6 M☉ white dwarf, against the binary's separation before the loss, both logarithmic. Blue: the mass loss alone — twelve thermal-pulse episodes 500 years wide and a superwind 10,000 years wide, each adding a kick of random sign whose size falls off exponentially once it lasts longer than about a sixth of an orbit. Below 100 au the orbit is slow to every episode and the eccentricity is untouched. The pulses switch on near 229 au, and by 3,000 au the change is 0.26 and the superwind too has become sudden. Dashed: the change a velocity kick to the white dwarf of 0.05 or 0.2 km/s would make on its own, which grows as the square root of the separation. A 0.05 km/s kick outweighs the pulses out to 251 au; a 0.2 km/s kick outweighs the pulses at every separation drawn.

A kick that writes over the record of the pulses

A star that becomes a white dwarf loses half its mass in pulses, and a companion far enough away feels each pulse as a sudden jolt to its orbit's shape. The wide binaries of white dwarfs should therefore carry the pulses' record in their eccentricities. They would, except that the white dwarf itself is born moving at a few tenths of a kilometre a second, and a kick that small writes more eccentricity than all the pulses together.

orbits · Effective potential

Named alongside it

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

Adiabatic invariantAsymptotic giant branchImpulsive approximationStellar windWhite dwarfAngular momentumEccentricityEffective potentialEscape velocityMass loss rateNatal kickRandom walk

All concepts