Astronomy & Celestial Mechanics
the sky, computed
Concept

Stellar wind — where it appears

The continuous outflow of material from a star's surface, driven by radiation pressure on dust in cool giants and on spectral lines in hot ones. Its rate is not predicted from first principles, and for a dying star it removes most of the star.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside 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.

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.

stars · Eddington limit 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 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 A lever 12 radii long, and the spin it removes. A magnetised stellar wind, drawn with the Alfvén surface at 12 stellar radii — the schematic distance at which the wind's inertia finally beats the field. Inside it the gas is forced to turn with the star, so every gram that leaves carries the specific angular momentum of the radius at which it broke free rather than of the surface it came from, and the lever squares: J̇ = (2/3) Ṁ Ω r_A². Beyond the surface the streamlines curve backwards, because angular momentum conservation makes the azimuthal speed fall as 1/r while the radial speed does not. With a moment of inertia coefficient of 0.073 and a mass loss of 2.3·10⁻¹⁴ solar masses a year, the star loses a fraction 2.3·10⁻¹⁴ of its mass and a fraction 3·10⁻¹¹ of its angular momentum in the same year — a ratio of 1,315, which is (2/3)(r_A/R)²/k² and nothing else. The e-folding time for the spin is 3.3·10¹⁰ years against 4.3·10¹³ years for the mass. Nothing here is to scale in one respect that matters: the wind's density falls by more than ten orders of magnitude across the drawn region, so the streamlines are drawn as though the flow were visible when almost none of it is.

A wind that takes no mass and all the spin

The Sun loses about a ten-thousandth of itself to its own wind over its whole life. It loses most of its rotation to the same wind, and the whole of that asymmetry is one geometric factor — the gas is forced to keep turning with the star until it is a dozen or more radii out from the surface it left.

stars · Magnetic braking The shield that does not shield. Above: measured ion escape rates for three planets against their surface magnetic field. Venus and Mars have no dynamo at all and Earth has one, and the three rates lie within a factor of 9 — with the magnetised planet losing the most. The intuition that a magnetosphere protects an atmosphere is not a small correction away from being right; the measurement does not support it. Below: why. A dipole's field lines are not all closed. Those emerging within a polar cap reconnect with the wind's own field and lead straight to space, and the cap's area is set by how far the magnetosphere reaches — a boundary at ten planetary radii still leaves 5.1 per cent of the surface open. So a magnetosphere is both a shield and a funnel: it deflects the wind from most of the planet and collects ions from the whole ionosphere into the polar wind, which is exactly what an instrument above the poles measures leaving. Whether the net is protection depends on quantities nobody can compute from the field strength alone, and the three points above are the state of the evidence.

The shield that is also a funnel

A magnetic field is supposed to protect an atmosphere from the stellar wind. Venus and Mars have no dynamo and Earth has one, and their measured ion escape rates lie within a factor of a few — with the magnetised planet losing the most, because a dipole's polar field lines are open and lead straight to space.

exoplanets · Atmospheric escape 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 Thin winds cannot cool their own shocks. The ratio of the cooling time of gas behind a shock to the time the wind takes to carry it away, against the wind's mass-loss rate, for shocks of 300, 500, 800 km/s at three stellar radii from a star of 10 solar radii. The ratio goes as the fourth power of the shock speed and inversely as the density, and so as one over the mass-loss rate. Above the dashed line the shocked gas leaves the star before it can radiate its heat and stays at millions of kelvin. The crossing falls at 2·10⁻⁹, 10⁻⁸, 9·10⁻⁸ solar masses a year. Winds thinner than that stay hot behind their shocks: the measured weak winds, 10⁻¹⁰ to 10⁻⁸, are hot behind any shock faster than about 500 km/s and the thinnest of them behind any shock at all, while the dense winds of the most luminous stars, above 10⁻⁷, cool their shocks except the fastest. The weak-wind problem begins about where the shocks stop cooling.

A wind that may be too hot to see

Below a certain luminosity the winds of hot stars are measured at a tenth to a hundredth of what the theory of line-driven winds predicts. Either the theory fails for thin winds, or the winds are there and hot — heated by their own shocks and unable to cool, invisible to the ultraviolet lines that measure them. X-ray line shapes, which weigh the whole wind regardless of its ions, have begun to say which.

starlight · Stellar winds In a mixed field the torque follows the dipole and ignores the rest. The torque of a field that is part dipole and part quadrupole, with the total surface strength fixed, as a fraction of a pure dipole's, against the fraction of the field strength in the dipole, for wind magnetisations of 300, 3,000, 30,000, using the broken power law fitted to simulated mixed-field winds. Above a dipole fraction of 0.39, 0.24, 0.15 respectively, the torque is exactly what the dipole component alone would give: the quadrupole has fallen below the dipole long before the wind reaches its Alfvén radius, and contributes nothing to the lever. Below it, the quadrupole sets a floor. A star whose surface is mostly quadrupole by strength can therefore be braked almost entirely by its small dipole — and a measurement of the total field says little about the torque unless it says how much of the field is dipolar.

A lever set by the dipole and not the field

A star's wind carries away its spin because the magnetic field holds the gas in step with the surface out to the Alfvén radius, and the torque goes as the square of that radius. How far out the field can hold depends less on how strong it is than on how it is arranged: a quadrupole of the same surface strength holds the Sun's wind to half the distance and brakes it at a fifth of the rate. In a mixed field only the dipole reaches far enough to matter — so the field that brakes a star is the part of it that is hardest to measure.

stars · Magnetic braking

Named alongside it

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

Mass loss rateMass lossAlfven radiusAngular momentumEddington limitMagnetic brakingRadiation pressureTerminal velocityAccretionAngular momentum transportAsymptotic giant branchAtmospheric escape

All concepts