Astronomy & Celestial Mechanics
the sky, computed
Starlight

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.

Assumes Stellar winds, Line formation and Eddington limit.

14 min read 5 figures Light is the only messengerMeasuring the unreachable

The theory of line-driven winds makes the mass-loss rate of a hot star an eigenvalue: the one rate at which a smooth flow can leave the surface, fixed by the distribution of line strengths in the star’s spectrum. For the most luminous stars it works. Their measured rates, from the strength of their ultraviolet and optical lines, agree with the prediction to within the uncertainty that clumping in the wind introduces. Below a luminosity of about 105.210^{5.2} times the Sun’s the agreement fails, and it fails by more than any adjustment of the theory’s parameters can fix: the measured rates of the fainter O stars sit ten to a hundred times below the prediction. That is the weak-wind problem.

The essay on the theory ended with two readings of it. Either the winds really are that weak — the line-driving mechanism has a failure mode in thin winds that the steady theory leaves out — or the winds are there and the measurement misses them. This essay is about the second reading, because it makes a definite prediction that can be tested, and the test has begun.

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.
Fig. 1 The ratio of the cooling time of gas behind a shock to the time the wind carries it away, against the mass-loss rate, for shocks of 300, 500 and 800 km/s at three stellar radii from a star of ten solar radii. Above the dashed line the shocked gas stays at millions of kelvin. The crossings fall at 2 × 10⁻⁹, 10⁻⁸ and 9 × 10⁻⁸ solar masses a year; the measured weak winds, 10⁻¹⁰ to 10⁻⁸, are hot behind any shock faster than about 500 km/s.

Winds full of shocks

A line-driven wind is not smooth. The driving force on a parcel of gas depends on how much of the star’s light its lines can absorb, and a parcel moving slightly faster than its neighbours has its lines shifted out of the shadow of the gas below it, absorbs more, and accelerates further. The instability grows quickly, and simulations of line-driven winds find them broken into dense shells and fast, thin streams that collide. The collisions are shocks at a few hundred kilometres a second, and gas passing through such a shock is heated to millions of kelvin.

That hot gas is seen. Hot stars are X-ray sources, at about a ten-millionth of their total luminosity, and the X-ray spectra show emission lines from highly ionised oxygen, neon and silicon at the temperatures shocks of that speed produce, broadened by the wind’s motion — which places the hot gas in the wind rather than near the surface. In the luminous stars with dense winds, the hot gas is a small fraction of the whole.

Thin winds cannot cool

Whether shocked gas stays hot depends on a comparison of two times. Gas cools by radiating, and the radiation rate per particle rises with density, because cooling needs collisions between particles. The flow carries the gas away from the star on the time it takes to travel a stellar radius or so. In a dense wind the cooling time is far shorter than the flow time, the shocked gas cools almost as soon as it is heated, and only a thin layer behind each shock is hot at any moment. In a thin wind the cooling time is longer, the shocked gas leaves before it can cool, and it stays hot all the way out.

The ratio of the two times goes as the fourth power of the shock speed and inversely as the mass-loss rate. The first figure puts the crossing — where the shocks stop cooling — at between a few times 10−910^{-9} and 10−710^{-7} solar masses a year, depending on the shock speed. The measured weak winds lie below it. In other words, the luminosity below which the theory appears to fail is close to the luminosity below which the winds’ shocks should stop cooling, and a wind that cannot cool its shocks will have much of its gas hot.

A rate that is a power of the luminosity, and a power set by the ions. The mass-loss rate of a line-driven wind, in solar masses a year on a logarithmic axis, against luminosity, for three values of the ensemble exponent α, with the mass rising along the main sequence as the 1/2.3 power of the luminosity and all three passing through the calibration point, 10⁻⁶ solar masses a year at 3.16·10⁵ solar luminosities. The theory gives the rate as the luminosity to the power 1/α times the effective mass to the power 1 − 1/α, so the slope is 1.57 for α = 0.5, 1.38 for α = 0.6, 1.24 for α = 0.7: a modest change in the atomic physics changes how steeply mass loss rises with brightness, and at the faint end of the range drawn the three differ by a factor of 2.1. The shaded region, below 1.58·10⁵ solar luminosities, is where measured rates fall one to two orders of magnitude below the prediction — the weak-wind problem — a gap no reasonable value of α closes.
Fig. 2 The mass-loss rate of a line-driven wind against luminosity, for three values of the line-ensemble exponent, all passing through 10⁻⁶ solar masses a year at 3.16 × 10⁵ solar luminosities. The theory’s slope is set by the exponent; below about 10^5.2 solar luminosities the measured rates fall one to two orders of magnitude beneath every curve.

A blind spot in the ultraviolet

The rates are measured from lines that only see cool gas. The strongest wind lines of an O star are ultraviolet resonance lines of carbon, nitrogen, silicon and phosphorus in particular ionisation stages — four-times-ionised phosphorus, three-times-ionised carbon — and the strength of each line measures the column of that one ion. Converting the column to a mass-loss rate requires knowing what fraction of the element is in that stage, which is computed from a model of the wind’s ionisation by the star’s radiation.

The rate an ultraviolet line reports, if part of the wind is hot. The mass-loss rate an ultraviolet resonance line would report, as a fraction of the true rate, against the fraction of the wind that is too hot to hold the line's ion. A resonance line measures the column of one ion, which is the mass-loss rate times the fraction of the element in that ionisation stage; gas at millions of kelvin holds the element in far higher stages, so it is invisible to the line. If 90 per cent of the wind is hot, the line under-reads by a factor of ten; if 99 per cent, by a hundred — the size of the deficit measured for the weak-wind stars. On this reading the weak winds are not weak but hot: most of their mass passes through the ultraviolet's blind spot.
Fig. 3 The mass-loss rate an ultraviolet line would report, as a fraction of the true rate, against the fraction of the wind too hot to hold the line’s ion. If 90 per cent of the wind is hot the line under-reads by ten; if 99 per cent, by a hundred — the size of the measured deficits.

Gas at millions of kelvin is ionised far beyond those stages, and so holds none of the ions the lines measure. If most of a weak wind is hot, its ultraviolet lines see only the cool remainder, and the rate inferred from them is too small by the inverse of the cool fraction. A wind ninety-nine per cent hot reads a hundred times weaker than it is. On this reading the weak winds are not weak; their mass passes through a blind spot of the instrument used to weigh them.

There is already evidence that hot gas reaches the measured ions. The ultraviolet spectra of O stars show lines of five-times-ionised oxygen and four-times-ionised nitrogen that the star’s own radiation is too soft to produce — superionisation, a puzzle for decades until it was traced to the X-rays from the wind’s shocks, which strip inner electrons from ions and leave them more highly ionised than the balance set by the star’s light would allow. The ion fractions the rate measurements depend on are therefore set partly by the hot gas even where it is a minority, and the models that convert lines to rates now include it. In a wind that is mostly hot, the correction is no longer a correction.

The optical measurement does not help. The hydrogen line Hα, which measures the rates of dense winds through recombination, is sensitive to the density squared and becomes too faint to measure at all below about 10−710^{-7} solar masses a year — just where the problem begins.

The correction that came before

This is not the first time the ultraviolet and the optical rates have disagreed. In the 2000s, rates measured from the phosphorus lines of luminous O stars came out ten times lower than the rates from Hα for the same stars. The resolution was clumping: Hα, sensitive to the density squared, overestimates the rate of a wind whose gas is concentrated in dense clumps, while phosphorus, sensitive to the density, does not — but the ultraviolet lines are themselves affected by clumps that are optically thick, which hide part of the gas from the line. The rates in use now are the result of fitting both kinds of line with a clumping model, and they came down by a factor of two to three from the older values.

The weak-wind problem is a larger discrepancy, in fainter stars, and clumping does not explain it: the deficit is in the ultraviolet lines that clumping affects least, and Hα is too faint to compare. But the episode is a warning about what kind of problem it may be. Each time a wind’s rate has been measured in a new way, the answer has depended on an assumption about the wind’s structure that the measurement could not test, and each time the structure turned out to be more complicated than assumed.

A trough that saturates

There is a second limit on the ultraviolet lines, and it explains why the weak-wind measurements rely on particular, rarer ions.

An absorption trough that stops deepening. The blue-shifted absorption of a P Cygni line formed in a wind accelerating as a β = 1 law, against velocity in units of the terminal speed, for line optical depths whose scale is 0.1, 1, 10, 100 — proportional to the mass-loss rate times the ion fraction. A thin wind gives a shallow trough whose depth grows in proportion to the rate; a dense one gives a black, saturated trough that cannot get any deeper. The absorbed fraction of the continuum across the trough is 0.07, 0.41, 0.94, 0.98: a factor of ten in rate changes it by 0.34 at the thin end and by 0.04 at the dense end. So each ultraviolet line measures rates over about two decades and gives only a bound outside them, which is why weak-wind measurements lean on the unsaturated lines of rarer ions — and why their ion fractions, which a hot component would change, matter so much.
Fig. 4 The blue-shifted absorption of a P Cygni line formed in a wind with a β = 1 velocity law, for line optical-depth scales of 0.1, 1, 10 and 100. The absorbed fraction of the continuum across the trough is 0.07, 0.41, 0.94 and 0.98: a factor of ten in rate changes it by 0.34 at the thin end and 0.04 at the dense end.

A wind’s absorption trough reads its terminal speed at its blue edge, but its depth reads the rate only while the line is unsaturated. A strong line in a dense wind is black across its whole width and does not get deeper as the rate rises; a line in a thin wind is shallow and its depth tracks the rate. Each line therefore measures rates over about two decades and gives only a bound outside them. For the weak winds the strong carbon and nitrogen lines are already unsaturated and the measurements are made from them, but their ion fractions are the most sensitive to the wind’s temperature structure — so the lines that can measure a weak wind are exactly the ones a hot component would distort most.

A measurement that does not care about ions

The test needs a measurement of the wind’s total mass that does not depend on any particular ion. X-ray line shapes provide it.

An X-ray line whose shape weighs the wind it shines through. Profiles of an X-ray emission line from hot gas spread through a hot star's wind outside 1.5 stellar radii, against wavelength shift in units of the terminal speed, for wind optical depths τ⋆ of 0.1, 1, 4. Emission from the receding far side has to cross more of the cool wind than emission from the approaching near side, so as the wind thickens the red half of the line is absorbed and the profile becomes skewed and blue-shifted: its centroid moves from −0.06 to −0.32 of the terminal speed. The optical depth is the mass-loss rate times the continuum opacity, which does not depend on which ions are present, so the line's shape measures the whole wind — hot and cool together. For the weak-wind star μ Columbae, X-ray profiles measured this way gave a rate roughly an order of magnitude above the ultraviolet one, with most of the wind hot.
Fig. 5 Profiles of an X-ray emission line from hot gas spread through a wind outside 1.5 stellar radii, for wind optical depths τ⋆ of 0.1, 1 and 4. Emission from the receding far side crosses more of the cool wind and is absorbed, so the line becomes skewed and blue-shifted as the wind thickens; its centroid moves from −0.06 to −0.32 of the terminal speed.

The X-ray lines are emitted by the hot gas throughout the wind and absorbed on their way out by the cooler gas. The absorption is by the continuum opacity of the cool wind’s metals — photoelectric absorption of X-rays by inner-shell electrons — which depends on how much material is in the way, not on which ionisation stage it is in. Emission from the side of the wind moving away from the observer has to cross more of the wind than emission from the approaching side, so a thick wind absorbs the red half of each line and the profile becomes skewed and shifted towards the blue. The degree of skew measures the wind’s optical depth, and so its mass-loss rate, independently of the ultraviolet ion fractions. Owocki and Cohen worked out the profile shapes in 2001, and they have since been fitted to the X-ray spectra of a dozen hot stars.

For the dense winds of the luminous stars the X-ray rates agree with the ultraviolet and Hα rates once clumping is allowed for, which validates the method. For the weak-wind star μ Columbae, Huenemoerder and collaborators found in 2012 that the X-ray profiles imply a mass-loss rate roughly an order of magnitude above the ultraviolet value — with most of the wind at X-ray-emitting temperatures. That is what the hot-wind reading predicts, and it is the clearest evidence that at least one weak wind is a hot wind.

The other explanation

The alternative is not dead, and it makes a different prediction. In a thin wind the radiation pushes on a minority of ions — the ones whose lines absorb starlight — and they share the push with the rest of the gas through collisions. If the gas is thin enough, the collisions are too rare to share it: the driving ions slip through the bulk of the gas, accelerating alone, and the bulk is left behind. Ion decoupling would truly reduce the mass that leaves, and the weak winds would really be weak. It would also leave a characteristic signature: the driving ions moving faster than the bulk, heating it by friction, and the wind reaching a lower terminal speed than its edge suggests.

The two readings predict opposite things for the X-ray profiles. A hot wind has a large total mass, so its X-ray lines are skewed by its own opacity; a truly weak wind is thin, and its X-ray lines are nearly symmetric. Of the few weak-wind stars observed well enough, μ Columbae favours the hot wind. Others are too faint in X-rays for their profiles to be measured precisely, and it is possible that both effects operate in different stars, or at different luminosities.

A wind between two kinds

There is a sense in which the weak-wind stars sit on a boundary between two kinds of wind. The Sun’s wind is not driven by lines at all: it is driven by the pressure of a corona heated to a million kelvin by the dissipation of magnetic energy, and its field holds the gas in some places and opens it in others. It is a hot wind from the start, thin, carrying away a ten-trillionth of the Sun’s mass a year — and nearly all its angular momentum. The winds of the most luminous hot stars are the opposite: cool, dense, driven by the momentum of starlight absorbed in lines.

A weak-wind O star, if the hot reading is right, has a wind launched by lines but carried for most of its length as hot gas, at temperatures near a corona’s. It is not clear that the line-driving theory’s steady solution still describes the outer flow of such a wind, since a hot gas has a pressure of its own and loses the ions the lines need. The hot component may change the terminal speed, the shape of the velocity law and the coupling of the wind to the star’s magnetic field, in ways the measured blue edges are just beginning to test.

The floor under the force

What sets the lowest luminosity at which line driving works at all is the comparison the Eddington limit makes. Light pushes on the free electrons of a hot star’s atmosphere with a force that, for an O star, is a few tenths of gravity on its own; lines multiply it by the force multiplier, which in a dense wind is a thousand or more. In a thin wind the strongest lines are unsaturated and the multiplier falls, and in the thinnest the lines that could drive the wind are too few to lift anything. Line-driven winds therefore have a floor, and stars much fainter than the weak-wind threshold — early B stars on the main sequence — have winds so thin they are barely detected in any band. Whether the weak-wind problem is the approach to that floor, or a hot wind hiding above it, is the question the X-ray profiles are answering one star at a time.

Why it matters beyond the stars

The rates matter because of what the wind takes. Over a massive star’s life its wind removes a large fraction of its mass, and how much is removed decides how massive the star is when it dies, whether it explodes, and how heavy a black hole it leaves. The most common massive stars are the ones below the weak-wind threshold — luminosity falls steeply with mass, and there are far more stars of twenty solar masses than of sixty — so the uncertainty sits in the part of the population that dominates the numbers. A factor of ten in the rates of those stars changes the predicted masses of black holes from their deaths by solar masses, which is the level at which the black holes detected by their gravitational waves are now being compared with the models.

The rates also set how the winds of massive stars stir the gas around them, and how much momentum young clusters put into their natal clouds. Most of that feedback comes from the most luminous stars, whose rates are secure; the weak-wind problem matters there less. It matters most for the stars’ own evolution, where every star’s wind counts.

What the figures simplify

The cooling comparison uses a single shock speed and a fixed radius, where real winds have a distribution of shock strengths at every radius. The hidden-fraction figure assumes the hot gas holds none of the measured ion and the cool gas all of it, where real ionisation is a continuum. The P Cygni profiles use the Sobolev approximation and a single velocity law, and the X-ray profiles a single onset radius and a smooth wind, where clumping in the cool wind reduces its effective opacity — porosity — and makes the X-ray rates somewhat uncertain in the other direction. Each of these is handled in the published analyses, and none removes the central fact that a wind can be hot enough to escape the ultraviolet’s notice, and that X-ray profiles are the measurement that can see it.

Still open: whether every weak wind is hot

One star with well-measured X-ray profiles is evidence and not a demonstration. The next generation of X-ray spectrographs, with the sensitivity to measure line profiles in fainter stars, will give the X-ray rates of a sample of weak-wind stars rather than one. If they all come out high, the weak-wind problem was an ionisation problem, the line-driving theory is right down to lower luminosities than it appeared, and the mass-loss rates used in stellar evolution need revising upward for the most common massive stars. If they come out low, the winds are truly weak and the theory needs a decoupling mechanism. If they scatter, both are at work. In each case the answer will come from the shapes of a few X-ray lines, which see a wind’s mass without asking what its ions are.

Where this goes next

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. 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. 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.

The objects this essay names

Each one links to every other essay that touches it.

IonisationLine driven windMass loss rateO type starP cygni profileShock heatingStellar windX-ray emission