A rate the wind has no choice about

The terminal speed of a hot star’s wind is read off the blue edge of a P Cygni profile, and for the hottest stars it comes out at two or three thousand kilometres a second. The mass-loss rate is harder, but it is measured too, from the strength of emission lines and radio free–free emission, at around a millionth of a solar mass a year for a luminous O star. The mechanism is not in doubt either: starlight is absorbed in the ultraviolet lines of metal ions, each absorption hands the ion the photon’s momentum, and because the wind accelerates, each ion keeps Doppler-shifting into fresh, unabsorbed light.

What that description does not say is why the numbers are what they are. Why should the terminal speed be close to two and a half times the escape speed from the surface, for almost every hot star? Why should the rate be a millionth of a solar mass a year rather than a thousandth or a billionth? The answers turn out to come from a single exponent that describes how the driving lines are distributed in strength — an exponent that belongs to the atomic physics of iron, carbon, nitrogen and oxygen, and not to any particular star.

One line thin, one line thick, and a hundred thousand of them #

A single spectral line in a moving wind can be in one of two states, and they push in completely different ways.

If the line is optically thin — weak enough that it absorbs only a small fraction of the light at its own wavelength — every ion in the line absorbs independently, and the total push is proportional to how many ions there are. It does not matter how fast the wind is accelerating. If the line is optically thick, it absorbs essentially all the light in a narrow slice of the spectrum, the slice whose width is the thermal Doppler width of the ion. The number of ions no longer matters; what matters is how fast the wind accelerates through that slice, because a steeper acceleration sweeps the line across more of the stellar spectrum per unit distance. A thick line’s push is proportional to the velocity gradient divided by the density, and independent of the line’s own strength.

The electron-scattering optical depth across one Doppler width — the quantity $t$ in the figure — measures which regime a line is in. Where the wind is dense or accelerates slowly, $t$ is large, and more lines are thick; where it is thin or accelerates quickly, $t$ is small, and more lines are thin. A line of strength $\eta$, measured relative to electron scattering, is thick when $\eta t$ exceeds about one. It is the same transition that makes a saturated absorption line stop measuring how much of its element is there, turned from a problem of measurement into a law of force.

A real hot star has hundreds of thousands of lines, from a few very strong resonance lines of carbon and nitrogen to a vast number of weak lines of iron-group ions. Castor, Abbott and Klein’s step in 1975 was to describe them statistically: the number of lines stronger than $\eta$ falls as a power of $\eta$, and the exponent is written $1 - \alpha$. For such an ensemble the total push — the force multiplier — is a sum over lines each either thin or thick, and as $t$ rises, lines cross from the first state to the second one after another. The result is a power law between the two extremes, falling as $t^{-\alpha}$.

The figure computes that sum over an ensemble with $\alpha = 0.6$, and the slope read off the drawn curve is $-0.59$. At very small $t$, every line is thin, and the push saturates at a value set by the total strength of the line list — about two thousand times the push on the electrons alone. That saturation value matters for what follows: it means line driving can, in principle, exceed gravity by a large factor even for a star far below its electron-scattering Eddington limit. In practice the wind never reaches that regime, because the flow adjusts its own velocity gradient until the push and gravity balance, and $\alpha$ sets how. (A disc fed far above its limit meets the same balance from the other side: there the supply is fixed and the light adjusts.)

The exponent is measured from atomic line lists — hundreds of millions of transitions computed for ions of every element — rather than from stars, and it comes out between 0.5 and 0.7 for the temperatures of O and early B stars. Where it sits in that range depends on which ionisation stages are present, and the ionisation balance is fixed by the temperature.

A rate that is an eigenvalue #

The equation of motion of a line-driven wind is short, once the force multiplier is written as a power law. Gravity pulls inward as $1/r^2$; the line force pushes outward in proportion to $L\, t^{-\alpha}$, and since $t$ is inversely proportional to the velocity gradient, that makes the push proportional to the gradient raised to the power $\alpha$. Writing the square of the wind speed in units of the escape speed as $w$, and measuring distance by $x = 1 - R/r$, which runs from 0 at the surface to 1 at infinity, the whole balance reduces to

$$w' + 1 = C\, (w')^{\alpha},$$

where $w'$ is the acceleration, the $1$ is gravity and the right-hand side is the line force. The constant $C$ carries the luminosity, the line list and — crucially — the mass-loss rate, which enters with a negative power: $C \propto \dot M^{-\alpha}$. A heavier wind has more material to be pushed by the same light, and every ion receives less of it.

The equation says that $C$ must equal $(w' + 1)/(w')^{\alpha}$, and that function has a minimum. If $C$ is below it — if the mass-loss rate is too high — there is no acceleration at which the push can balance gravity, and the wind cannot exist; the radiation simply cannot lift that much mass. If $C$ is above it, there are two accelerations that work, a shallow one and a steep one, and neither alone connects a flow starting slowly at the stellar surface to one coasting at infinity. The only value of $C$ that gives a single, smooth solution from the surface outward is the minimum itself, where the line is tangent to the curve.

The mass-loss rate is therefore not a free parameter of the wind. It is the eigenvalue of the flow, the largest rate the radiation can lift, and the wind takes it because any smaller rate leaves a push over and any larger one fails. That is the same structure as the Parker solution for the solar wind, where the sonic point selects a unique flow from a family, and it is why a hot star does not need to be told how much mass to lose. Its luminosity and its line list decide.

At the tangent point the acceleration is $w' = \alpha/(1-\alpha)$, which is 1.5 for $\alpha = 0.6$. Since $w'$ is constant, $w$ grows linearly in $x$, and the velocity follows $v = v_\infty \sqrt{1 - R/r}$ with

$$v_\infty = v_{\rm esc}\sqrt{\frac{\alpha}{1-\alpha}}.$$

The terminal speed is a fixed multiple of the escape speed, and the multiple is set by the line-strength exponent alone. That explains why the ratio is nearly the same for so many stars. It does not yet give the right number.

Half the measured speed #

For $\alpha = 0.6$ the formula gives a terminal speed of 1.22 times the escape speed. The measured ratio, for stars hotter than about 21,000 kelvin, is 2.65.

The missing factor of two is the star’s size. The derivation above treats the star as a point, so that every photon arriving at a parcel of wind comes from exactly behind it. Close to the surface that is badly wrong: a parcel just above the photosphere sees the star as a hemisphere filling half its sky, and light arriving at an angle is Doppler-shifted differently and pushes less effectively along the radial direction. The correction weakens the push at the base of the wind, where the mass-loss rate is decided, and strengthens it relatively further out, where the speed is decided. A smaller rate with the same light at large radii means each gram is driven faster, and the finite-disc calculations of the 1980s raised the predicted ratio to about 2.5 to 3, in agreement with the measurements. A number that was wrong by a factor of two turned out to be measuring the angular size of the star as seen from its own wind.

The steps in the measured ratio are the more striking feature, because they are not in the point-star formula at all. Between 21,000 and about 18,000 kelvin the ratio halves, from 2.65 to 1.3. The cause is iron. In the hotter stars, iron in the inner wind is mostly Fe IV; below the jump it recombines to Fe III, which has a far richer set of lines in the part of the ultraviolet where these cooler stars emit most of their light. The line list changes, $\alpha$ drops, and the flow responds exactly as the eigenvalue argument says it should: more lines at the base lift more mass, the rate rises by a factor of a few, and with more mass to push the terminal speed falls. The effect is called the bi-stability jump, and its temperature, its sign and its size were predicted from the ionisation of iron before being confirmed in the measured speeds.

The jump matters beyond the wind. Stars evolving to cooler temperatures after the main sequence cross it, and the higher rate below it strips mass faster. Some models of luminous blue variables — the unstable, eruptive massive stars — have their outbursts triggered by a star moving back and forth across the jump as its radius changes.

A power of the luminosity set by the ions #

The eigenvalue condition also gives the rate itself. Setting $C$ at its minimum and unpacking what it contains, the mass-loss rate goes as

$$\dot M \propto L^{1/\alpha}\, M_{\rm eff}^{\,1 - 1/\alpha},$$

where $M_{\rm eff}$ is the stellar mass reduced by the part of gravity that electron scattering cancels. For $\alpha = 0.6$, doubling the luminosity multiplies the rate by more than three.

Along a real sequence of O stars the mass rises with the luminosity, and the mass factor partly offsets the luminosity factor, so the rates in the figure rise as the 1.2 to 1.6 power of the luminosity rather than the 1.4 to 2 power of $L^{1/\alpha}$ alone. The spread between the three exponents is a factor of two at the faint end of the range drawn — a real difference, but not an enormous one. Across the range where the diagnostics are best, the luminous O stars and supergiants, the predicted rates and the measured ones agree to within the factor of two to three that clumping in the wind introduces into the measurements.

Below about $1.6\times10^5$ solar luminosities that agreement fails, and it fails by more than any value of $\alpha$ can fix. The measured rates of fainter O stars sit one to two orders of magnitude below the prediction — the weak-wind problem. The line-driving theory has an internal reason to fail there: at low density the wind’s lines become optically thin to their own driving radiation, the gas that absorbs the light and the gas that carries the mass can decouple, and the ions doing the driving can leave the rest behind. Whether that happens, or whether the weak winds are ordinary winds that the ultraviolet diagnostics under-count, has not been settled.

Iron, and the stars that had none #

The line list is a list of metal lines, so the rate depends on how much metal the star has. Detailed calculations give $\dot M \propto Z^{0.7}$ to $Z^{0.85}$ at fixed luminosity, with iron supplying most of the driving at the base of the wind where the rate is set, and carbon, nitrogen and oxygen supplying most of it further out where the speed is set.

That dependence reaches a long way. A forty-solar-mass star in the Milky Way loses a substantial fraction of its mass to its wind over a life of a few million years; one in the Small Magellanic Cloud, at a fifth of solar metallicity, loses much less and ends heavier. The heaviest stellar-mass black holes found by gravitational-wave detectors, at thirty to eighty solar masses, are heavier than any in the Milky Way, and the most natural explanation is that they formed from metal-poor stars whose line-driven winds were too weak to strip them. The first stars, with no metals at all, would have had no line-driven winds whatever. Their only mass loss would have come from pulsations, rotation or eruptions, and they would have died at close to the mass they were born with.

A momentum from which the mass cancels #

The most useful consequence of the theory is an accident of its exponents. The terminal speed goes as the escape speed, $\sqrt{2GM/R}$. Multiply the rate by the terminal speed and by the square root of the radius, and the radius cancels:

$$\dot M\, v_\infty \sqrt{R} \;\propto\; L^{1/\alpha}\, M^{3/2 - 1/\alpha}.$$

For $\alpha = 2/3$ the mass drops out exactly. For $\alpha = 0.6$ it is left with a power of $-0.17$, small enough to be nearly invisible.

The result is the wind-momentum–luminosity relation. The model stars in the figure span a factor of three in mass at each luminosity, and they lie on a single line with a scatter of 0.02 dex — the mass has cancelled to that precision. Real O stars and supergiants follow such a relation, with a scatter set largely by the measurement of the rates. Since the wind momentum can be measured from a spectrum — the terminal speed from the P Cygni edge, the rate from the Hα emission, the radius from the continuum — and the relation then gives the luminosity, the combination is a distance indicator. It was proposed in the 1990s for the brightest blue supergiants in galaxies out to ten or twenty megaparsecs — where every method is checked against the others — beyond the reach of individual Cepheids in crowded fields, as one more step on the distance scale that shares no systematic error with the others.

Its limitations are the ones the theory predicts. The relation shifts with metallicity, as the figure shows, and its calibration has to be carried to each galaxy’s own composition. The clumping correction to the rate enters the zero point directly. And the relation holds only where line driving works as described — for the supergiants that are bright enough to be seen at those distances, which is where it works best.

What was actually measured, and what was calculated #

It is worth being explicit about which parts of this are observation and which are inference, because a theory as neat as this one invites more confidence than it has earned.

The terminal speeds are measured directly and precisely, from the edges of saturated ultraviolet lines, for hundreds of O and B stars. The escape speeds require masses, which come from surface gravities fitted to the pressure-broadened wings of lines, and those carry errors of 30 per cent or more. The ratio of the two, and the jump in it, are therefore measured with the precision of the masses. The mass-loss rates are measured by at least three methods — ultraviolet absorption, Hα and infrared emission, radio free–free emission — that respond to the density to different powers, so they disagree unless the wind’s clumping is modelled; the corrections are factors of two to three and still under discussion. The line-strength exponent is not measured in any star at all; it is computed from atomic data, and the predictions made with it are then compared with the rates.

What the comparison shows is that the theory reproduces the terminal speeds once the star’s finite size is included, predicts the bi-stability jump in sign, temperature and size, and reproduces the rates of luminous stars within the uncertainty of the clumping correction. It fails for the weak winds, and it describes only the steady part of a wind that is intrinsically unstable: a parcel that speeds up sees more unabsorbed light and speeds up further, so the smooth solution breaks into shocks and clumps that emit the X-rays hot stars are seen to produce. The eigenvalue argument gives the mean rate of a flow whose structure it cannot describe.

Still open: whether the faint stars’ winds are there #

The theory’s failure is in the least dramatic stars, and that is the wrong place for it to fail. Population synthesis, stellar evolution models and predictions of black-hole masses all need mass-loss rates across the whole range of luminosity, and the most common massive stars are the ones below the weak-wind threshold. If their winds really are ten or a hundred times weaker than predicted, the line-driving mechanism has a failure mode that the steady theory does not include, and something about the decoupling of absorbing ions from the bulk of the gas has to be added. If the winds are there and the ultraviolet lines are failing to trace them, the correction is to the diagnostics, and the X-ray observations that find more material than the ultraviolet does are already pointing that way. The measurement that would decide it is a rate from a diagnostic that does not depend on the ionisation of the tracer — which, for these stars, has not yet been made with the precision needed.