A speed read off an edge

Almost every number in stellar astrophysics is the end of a chain. A luminosity needs a distance, a distance needs a calibration, a calibration needs a parallax; a mass needs an orbit of two stars about each other; a radius needs an eclipse or an interferometer. The wind speed of a hot star needs none of those. It is read off the edge of a line, and the only thing standing between the measurement and the number is a Doppler shift.

The name comes from the star P Cygni, which brightened by several magnitudes in 1600, was described by Kepler’s contemporaries, and has been shedding mass ever since.

The geometry, and why it produces two features #

Put an observer far away along one axis and a spherically expanding shell around a star. Every parcel of that shell is moving radially outward at a speed set only by its distance from the star. The Doppler shift a parcel imposes on the light it interacts with is the component of that motion along the line of sight — a projection, and therefore a lower bound on the speed unless the geometry is known, which here it is.

Divide the shell into three regions by what the observer sees behind them.

The column in front of the stellar disc has the star behind it. Its material is moving almost directly toward the observer, so it is blueshifted, and it can remove photons from the beam. The blueshift runs from nearly zero for parcels just above the photosphere — where the wind has barely started — to the full terminal speed far out. That is one continuous absorption trough covering the whole blue side.

The material off to the sides has nothing behind it. It is moving across the line of sight, so its shift is small, and there is nothing for it to absorb. It can only scatter photons into the beam: emission, centred near the rest wavelength.

The material directly behind the star is hidden. That region would have been receding fastest and would have produced the reddest emission, and its absence is why the emission peak is displaced slightly to the red rather than being symmetric.

The edge is the measurement #

The blue edge is the sharpest feature in the profile, and its sharpness is the whole reason the measurement is good.

At the edge the absorbing material is at the terminal velocity, which means it is far from the star, where the wind has stopped accelerating. Far from the star the velocity gradient is small, so a large range of radii all contribute at very nearly the same Doppler shift. The absorption therefore piles up at one wavelength and then stops, because there is no material moving faster than the terminal speed at all.

What blurs the edge is the intrinsic width of the line — thermal motions and turbulence in the wind — and that is a few tens of kilometres a second against a terminal speed of thousands. So the edge is defined to about one per cent.

This is a rare thing in the subject: a spectroscopic measurement that does not need a flux calibration, an absolute wavelength scale beyond the line’s own rest wavelength, or a distance. Compare it with the ordinary use of a line shift as a speedometer, which measures one velocity because the absorbing gas has one velocity. Here the gas has every velocity between zero and the maximum, and the profile is a map of how much material there is at each.

The velocities involved #

The numbers are worth stating because they are outside ordinary stellar experience. An O star’s wind reaches two to three thousand kilometres a second. The escape speed from its surface is about the same, which is not a coincidence: the wind is driven from near the surface, so whatever accelerates it has to overcome roughly the escape speed and then has the whole rest of the star’s neighbourhood to add the surplus. By contrast the Sun’s wind reaches four hundred kilometres a second, and it is driven differently — by gas pressure in a hot corona rather than by radiation on spectral lines. That corona is itself heated by the magnetic field the Sun’s rotation and convection between them wind up, which is why solar-type winds and hot-star winds have almost nothing in common except the word.

What drives it #

The mechanism for hot stars is momentum transfer from starlight to spectral lines, and the reason it works so well is a resonance effect that has nothing to do with brute force.

A photon absorbed by an ion transfers its momentum, and which ions are present at all is fixed by the temperature through the ionisation balance. If the ion could absorb only at one exact wavelength, it would very quickly Doppler-shift itself out of that wavelength and stop absorbing. But the wind is accelerating, so as an ion speeds up it moves into resonance with photons it had not been absorbing, and a new supply becomes available continuously. The wind is therefore driven by a sweep through the star’s spectrum rather than by a fixed slice of it. That metallicity dependence is one of the most consequential facts in the subject. It means the first generations of stars, formed from gas with no metals in it, could not have lost mass this way — so they kept their mass, ended differently, and left different remnants. The metals that make the forest were themselves made by earlier stars, so a galaxy’s winds get stronger as it ages, and a star’s fate depends on when it was born.

Getting a rate out of it #

The terminal velocity is easy. The mass-loss rate is not, and the reason is a square.

The absorption in a resonance line is proportional to the density of absorbers, so a saturated line gives only a lower limit and an unsaturated one gives a column density. Emission, though, comes mostly from recombination, which is a two-body process and therefore proportional to the square of the density. Two diagnostics with different powers of density do not measure the same average unless the wind is smooth.

It is not smooth. Winds are clumped — the line-driving mechanism is itself unstable, because a parcel that speeds up absorbs more and speeds up further — and a clumped wind of the same total mass emits more than a smooth one, in proportion to the mean of the square over the square of the mean.

The consequence is a long-running disagreement. Rates derived from emission diagnostics exceeded rates derived from absorption diagnostics by factors of three to ten, until the clumping was taken into account; the current view is that the emission-based rates were too high by roughly the square root of the clumping factor, and the absorption ones were closer. The correction is a factor of about three, and it changes the predicted final masses of massive stars substantially.

What the star loses #

Over its life a massive star can shed a large fraction of itself this way, and that is the reason the subject matters beyond spectroscopy. The mass a star does not keep is therefore not a footnote to its evolution but a controlling parameter of it, and the wind is the largest term in it for anything above about ten solar masses.

There is also a limit that the whole subject sits underneath. A star cannot radiate more than the luminosity at which radiation pressure on free electrons balances gravity, and the most luminous stars sit close enough to it that small changes in opacity produce very large changes in mass loss.

The wind seen in X-rays #

There is a second observable produced by the same wind, from a completely different part of the spectrum, and it supplies a mass-loss rate by a route that does not go through the density squared.

A line-driven wind is unstable. A parcel that speeds up slightly absorbs photons it was not absorbing, accelerates further, and runs into the slower material ahead of it. The result is a wind full of shocks, and shocks at a thousand kilometres a second heat gas to a few million kelvin — which radiates in X-rays.

The X-ray lines from that shocked gas are themselves broad, because the emitting material shares the wind’s expansion, and their profiles carry a measurement. A photon emitted on the far side of the star has to cross the whole wind to reach the observer and may be absorbed on the way; one emitted on the near side does not. So the red half of each line, which comes from the receding far side, is suppressed relative to the blue half, and the degree of suppression measures the wind’s optical depth along the line of sight.

That optical depth is linear in the density rather than quadratic, so it is insensitive to clumping — which is exactly the property the emission-based rates lack. Two diagnostics with different powers of density, one of them clumping-blind, is precisely the pair needed to measure the clumping factor rather than to argue about it.

The rates that came out were lower than the emission-based ones by factors of several, consistent with the absorption-based ones, and the agreement is what settled the argument in the direction the previous section describes.

The measurement is expensive. It requires a high-resolution X-ray spectrometer above the atmosphere, and it has been done well for a handful of the brightest hot stars. The technique that settled the systematic is available for perhaps a dozen objects, and the correction it established is applied to thousands.

What was actually measured #

The chain for a terminal velocity is short and worth stating because it is so unusually short.

An ultraviolet spectrum is taken, since the strong resonance lines of the abundant ions in a hot wind — carbon, nitrogen, silicon — lie between 1200 and 1600 ångströms and are inaccessible from the ground. The blue edge of a saturated line’s absorption trough is located, in wavelength. The rest wavelength of the transition is known from the laboratory to many figures. The ratio gives a velocity through the Doppler formula, and the star’s own systemic velocity is subtracted.

That is the whole of it. No distance, no photometric calibration, no model atmosphere. The dominant uncertainty is deciding exactly where the edge is when it is blurred by turbulence, and that is a few per cent.

Everything else about a wind is much harder. The rate requires a model, the clumping requires a model of the model, and the acceleration law requires fitting the shape of a trough whose shape also depends on the ionisation balance through the wind.

The profile has three parameters and each of them changes it in a way the others cannot imitate.

The other place the same profile appears #

The shape is not confined to hot stars, and the places it turns up are a useful check that the argument is about geometry rather than about any particular physics.

Novae show it, from the shell ejected in the outburst; supernovae show it, at speeds ten times higher, in lines whose blue edges give ejecta velocities of ten thousand kilometres a second and more; and quasar absorption systems show it, from outflows driven off an accretion disc rather than off a star. In every case the reading is the same: a sharp blue edge is a speed, obtained from geometry, and everything else about the profile is a model. The consistency across four decades in velocity and twenty in luminosity is the best argument that the interpretation is right, because the alternative explanations for the shape are all specific to one kind of object.

The winds that are too weak #

The line-driving theory predicts a mass-loss rate as a function of luminosity, metallicity and effective temperature, and across most of the range it works: the predicted and measured rates agree within the uncertainties for the luminous O stars where the diagnostics are best.

Below a threshold it stops working, and the failure is large.

For O stars fainter than about a hundred thousand solar luminosities, the measured rates fall an order of magnitude or more below the prediction — and the discrepancy grows as the luminosity falls, so it is a systematic trend rather than a scatter. The effect is called the weak-wind problem and it has been unresolved since it was identified two decades ago.

Two families of explanation are on offer. The first is that the winds really are that weak, because something in the driving fails at low luminosity — the wind becomes optically thin in the lines that drive it, or the ions responsible recombine to states that do not, so the coupling between the radiation and the gas breaks down. The second is that the winds are not weak and the diagnostics are failing: if most of the wind is at a temperature where the observed ions do not exist, the tracer used is measuring a small fraction of the material and the inferred rate is correspondingly low.

The second explanation has direct support. X-ray observations of some weak-wind stars find more material than the ultraviolet diagnostics do, which is what a hot, largely invisible wind component would produce. It does not account for the whole discrepancy.

Whichever it is, the consequence is the same for anything downstream. The mass a star of moderate luminosity loses over its life is uncertain by an order of magnitude, and that propagates into its final mass, its remnant, and the amount of processed material it returns. A theory that works for the brightest objects and fails for the common ones is the wrong way round for population synthesis, which needs the common ones.

The wind that is not steady #

Everything above treats the wind as a steady flow, and for the terminal velocity that is adequate. For the rate it is not, and the departures are large enough to have their own literature.

The instability that produces the shocks also produces structure on every scale down to the resolution of the calculations, so the wind is not a smooth outflow but a spray of dense shells separated by rarefied gas. Time-resolved spectroscopy sees it directly: the absorption troughs of the strong ultraviolet lines contain narrow features that appear near the rest wavelength and migrate blueward over a day or two, accelerating outwards as the material they belong to does.

Those are discrete absorption components, they recur on the star’s rotation period rather than randomly, and they are read as large-scale structures — spiral streams anchored to something on the surface, sweeping past the line of sight as the star turns.

The consequence for the measurement is a caution rather than a correction. A single spectrum measures the wind along one line of sight at one moment, and a wind with rotating structure gives different answers at different phases. The terminal velocity is safe, because the edge is set by the fastest material and there is fastest material in every direction. The rate is not, and the scatter between epochs is comparable with the systematic uncertainties the previous sections discussed.

There is also an episodic component that no steady model contains at all. The most luminous stars undergo eruptions in which they shed a solar mass or more over years — far more mass than the steady wind removes over the same interval — and whether those events dominate the total mass lost over a star’s life is unresolved. A rate averaged over a century may be a poor guide to a total integrated over a million years, and the star that supplies this essay’s name is one of the objects for which that is demonstrably true.

And the two extremes of the third parameter, which is the one that carries the mass-loss rate.

Where the ladder goes #

The obvious next rung is the driving law itself: why the terminal velocity comes out at a couple of times the escape speed, what sets the exponent in the velocity law, and how the rate scales with luminosity and metallicity. That is the theory of line-driven winds, and it produces a relation between mass-loss rate and luminosity that is testable against the profiles this essay reads.

The other direction is what the wind does to its surroundings rather than to its star. A wind at two thousand kilometres a second carrying ten to the minus six solar masses a year deposits as much energy into the interstellar medium over a star’s life as the supernova that ends it — which makes hot stars a source of mechanical energy in a galaxy, not only of light.