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EAS Publications Series, Vol. ?, 2011

STELLAR WINDS

Stan Owocki1

Abstract. A “stellar wind” is the continuous, supersonic outflow ofmatter from the surface layers of a star. Our sun has a solar wind,driven by the gas-pressure expansion of the hot (T > 106 K) solarcorona. It can be studied through direct in situ measurement by in-terplanetary spacecraft; but analogous coronal winds in more distantsolar-type stars are so tenuous and transparent that that they are diffi-cult to detect directly. Many more luminous stars have winds that aredense enough to be opaque at certain wavelengths of the star’s radia-tion, making it possible to study their wind outflows remotely throughcareful interpretation of the observed stellar spectra. Red giant starsshow slow, dense winds that may be driven by the pressure from mag-netohydrodyanmic waves. As stars with initial mass up to 8 M⊙ evolvetoward the Asymptotic Giant Branch (AGB), a combination of stellarpulsations and radiative scattering off dust can culminate in “super-winds” that strip away the entire stellar envelope, leaving behind ahot white dwarf stellar core with less than the Chandrasekhar mass of∼ 1.4M⊙. The winds of hot, luminous, massive stars are driven by line-

scattering of stellar radiation, but such massive stars can also exhibitsuperwind episodes, either as Red Supergiants or Luminous Blue Vari-able stars. The combined wind and superwind mass loss can strip thestar’s hydrogen envelope, leaving behind a Wolf-Rayet star composedof the products of earlier nuclear burning via the CNO cycle. In addi-tion to such direct effects on a star’s own evolution, stellar winds canbe a substantial source of mass, momentum, and energy to the inter-stellar medium, blowing open large cavities or “bubbles” in this ISM,seeding it with nuclear processed material, and even helping triggerthe formation of new stars, and influencing their eventual fate as whitedwarves or core-collapse supernovae. This chapter reviews the proper-ties of such stellar winds, with an emphasis on the various dynamicaldriving processes and what they imply for key wind parameters likethe wind flow speed and mass loss rate.

1 Bartol Research Institute, Department of Physics & Astronomy, University of Delaware,Newark, DE 19716 USA

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Contents

1 Introduction and Background 4

2 Observational Diagnostics and Inferred Properties 7

2.1 Solar Corona and Wind . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Spectral Signatures of Dense Winds from Hot and Cool Stars . . . 9

2.2.1 Opacity and Optical Depth . . . . . . . . . . . . . . . . . . 92.2.2 Doppler-Shifted Line Absorption . . . . . . . . . . . . . . . 112.2.3 Asymmetric P-Cygni Profiles from Scattering Lines . . . . 132.2.4 Wind Emission Lines . . . . . . . . . . . . . . . . . . . . . . 162.2.5 Continuum Emission in Radio and Infrared . . . . . . . . . 18

3 General Equations and Formalism for Stellar Wind Mass Loss 19

3.1 Hydrostatic Equilibrium in the Atmospheric Base of any Wind . . 193.2 General Flow Conservation Equations . . . . . . . . . . . . . . . . 203.3 Steady, Spherically Symmetric Wind Expansion . . . . . . . . . . . 213.4 Energy Requirements of a Spherical Wind Outflow . . . . . . . . . 22

4 Coronal Expansion and Solar Wind 23

4.1 Reasons for Hot, Extended Corona . . . . . . . . . . . . . . . . . . 234.1.1 Thermal Runaway from Density and Temperature Decline

of Line-Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.2 Coronal Heating with a Conductive Thermostat . . . . . . 244.1.3 Incompatibility of Hot Extended Hydrostatic Corona with

ISM Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Isothermal Solar Wind Model . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Temperature Sensitivity of Mass Loss Rate . . . . . . . . . 284.2.2 The Solar Wind as a Thermostat for Coronal Heating . . . 29

4.3 Driving High-Speed Streams . . . . . . . . . . . . . . . . . . . . . . 304.4 Relation to Winds from other Cool Stars . . . . . . . . . . . . . . . 31

4.4.1 Coronal Winds of Cool Main Sequence Stars . . . . . . . . 314.4.2 Alfven Wave Pressure Driving of Cool Giant Winds . . . . 314.4.3 Superwind Mass Loss from Asymptotic Giant Branch Stars 33

4.5 Summary for the Solar Wind . . . . . . . . . . . . . . . . . . . . . 34

5 Radiatively Driven Winds from Hot, Massive Stars 35

5.1 Radiative Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 365.1.1 Electron Scattering and the Eddington Limit . . . . . . . . 365.1.2 Driving by Doppler-Shifted Resonant Line Scattering . . . . 365.1.3 Radiative Acceleration from a Single, Isolated Line . . . . . 375.1.4 Sobolev Localization of Line-Force Integrals for a Point Star 38

5.2 The CAK Model for Line-Driven Winds . . . . . . . . . . . . . . . 395.2.1 The CAK Line-Ensemble Force . . . . . . . . . . . . . . . . 395.2.2 CAK Dynamical Solution for Mass Loss Rate and Terminal

Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Stan Owocki: Stellar Winds 3

5.3 Extensions of Idealized CAK Model . . . . . . . . . . . . . . . . . 425.3.1 Finite-Size Star . . . . . . . . . . . . . . . . . . . . . . . . . 425.3.2 Radial Variations in Ionization . . . . . . . . . . . . . . . . 435.3.3 Finite Gas Pressure and Sound Speed . . . . . . . . . . . . 43

5.4 Wind Instability and Variability . . . . . . . . . . . . . . . . . . . 435.5 Effect of Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.6 Summary for Radiatively Driven Massive-Star Winds . . . . . . . . 48

6 Wolf-Rayet Winds and Multi-line Scattering 50

6.1 Example of Multiple Momentum Deposition in a Static Gray Envelope 516.2 Multi-Line Transfer in an Expanding Wind . . . . . . . . . . . . . 536.3 Wind-Momentum-Luminosity Relation for WR Stars . . . . . . . . 546.4 Cumulative Co-Moving-Frame Redshift from Multi-line Scattering 556.5 Role of Line Bunches, Gaps, and Core Thermalization . . . . . . . 566.6 Summary for Wolf-Rayet Winds . . . . . . . . . . . . . . . . . . . 59


1 Introduction and Background

The Sun and other stars are commonly characterized by the radiation they emit.But one of the great astronomical discoveries of the latter half of the past centurywas the realization that nearly all stars lose mass through a more or less continuoussurface outflow called a “stellar wind”. While it was long apparent that starscould eject material in dramatic outburts like novae or supernovae, the concept ofcontinuous mass loss in the relatively quiescent phases of a star’s evolution stems inlarge part from the direct in situ detection by interplanetary spacecraft of a high-speed, supersonic outflow from the sun. For this solar wind, the overall rate ofmass loss is quite modest, roughly 10−14M⊙/yr, which, even if maintained over thesun’s entire main sequence lifetime of ca. 1010 year, would imply a cumulative lossof only a quite negligible 0.01% of its initial mass. By contrast, other stars – andindeed even the sun in its later evolutionary stages as a cool giant – can have windsthat over time substantially reduce the star’s mass, with important consequencesfor its evolution and ultimate fate (de Koter et al. 2008; Maeder & Meynet 2008;Vink 2008; Willson 2008). Moreover, the associated input of mass, momentum,and energy into the interstellar medium can have significant consequences, formingvisually striking nebulae and wind blown “bubbles” (Castor et al. 1975), and evenplaying a feedback role in bursts of new star formation that can influence theoverall structure and evolution of the parent galaxy (Oey & Clark 2007). Inrecent years the general concept of a continuous “wind” has even been extendedto describe outflows with a diverse range of conditions and scales, ranging fromstellar accretion disks (Calvert 2004), to Active Galactic Nuclei (Proga 2007), toeven whole galaxies (Breitschwerdt & Komossa 2000).

In the half-century since this concept of a wind from the sun or stars took hold,there has amassed a vast literature – consisting of thousands of journal papers,roughly a hundred conference proceedings, dozens of review articles, and even ahandful of books – on various aspects of solar and stellar wind mass loss. It isneither feasible nor desirable to attempt any comprehensive survey of this litera-ture, and interested readers looking for entry points beyond the topical coveragehere are encouraged to begin with a few key complementary reviews and books.For the solar wind, the original monograph Interplanetary Dynamics by Parker(1962) provides a still relevant and fundamental basis, while the recent Basics

of the Solar Wind by Meyer-Vernet (2007) gives an accessible modern summary.Particularly insightful reviews include those by Leer et al. (1982), Parker (1991),and Cranmer (2009). Proceedings of the regular series of dedicated solar windconferences, the most recent being Solar Wind 12 (Maksimovic et al. 2010), alsomake a good entry point into the evolving solar wind literature. For stellar winds,the text by Lamers & Cassinelli (1999) provides a good general introduction, whilemore targeted reviews tend to split between those focused on hot, massive stars(Kudritzki 2000; Owocki 2000, 2004; Puls et al. 2009) vs. cool, low-mass giants(Holzer & MacGregor 1985; Holzer 1987; Willson 2000; Dupree 2004).

This chapter aims to give a broad physical overview of the properties and pro-cesses involved in solar and stellar wind mass loss. A general theme is to identify


the forces and energies that can overcome the gravitational binding that holds ma-terial onto a nearly hydrostatic stellar surface, and thereby lift and accelerate theoutermost layers into a sustained outflow through which material ultimately es-capes entirely the star’s gravitational potential. This competition between outwarddriving and inward gravity is key to determining two basic wind characteristics,namely the mass loss rate M and terminal flow speed v∞.

Stellar wind mass loss rates span many decades, reaching more than a billiontimes that of the solar wind, M > 10−5M⊙/yr, for luminous giant stars; theexact values depend crucially on the specific wind driving mechanism. But for awide range of driving mechanisms, a quite general rule of thumb is that the windterminal speed v∞ scales in proportion to the escape speed ve from the stellarsurface. One intuitive way to think of the process is that in general the outwarddriving mechanism tends to tune itself to siphon off just enough mass from thestar to keep the outward force an order-unity factor above gravity; the net effective“anti-gravity” means wind material effectively “falls off” the star, reaching a finalspeed comparable to ve.

In practice, the proportionality factor can vary from about a third (for redgiants) to about factor three (for blue supergiants). Note that in the former case90% of the energy to drive the wind is expended in lifting the material out of thestar’s gravitational potential, with the remaining 10% expended in accelerationto the terminal flow speed; in the latter case, this relative allotment of energy isreversed. The total wind energy loss rate, Lwind ≡ M(v2

e +v2∞)/2, is typically only

a tiny fraction of the star’s radiative luminosity, about 10−6 for the solar wind,and ranging up to a few percent for winds from luminous, massive stars. Duringbrief “superwind” episodes of massive “Luminous Blue Variable” (LBV) stars, theradiative and wind luminosity can even become comparable.

The specific driving mechanisms vary with the various types of stellar wind,which can be broadly organized into three general classes:

1. Coronal Winds from the sun and other cool, main-sequence stars. Thesecan be characterized as thermally or gas-pressure driven, with the main out-ward force to overcome gravity stemming from the gas pressure gradientassociated with the high-temperature corona. The key issues thus lie in un-derstanding how mechanical energy generated in the near-surface convectionzone is transmitted upward (e.g. via wave oscillations in the magnetic field)to heat and drive a high-temperature coronal expansion. While importantas a prototype, and because the earth itself is embedded in the solar coronalwind, the low M means such coronal winds have negligible direct effect onthe star’s evolution. (On the other hand the loss of angular momentum inmagnetized coronal winds does seem sufficient to cause an evolutionary de-cline of rotation rates in the sun and other cool stars; Weber & Davis 1967;Sholz 2009).

2. Winds from Cool Giants and Supergiants. The lower surface gravity of giantstars facilitates much stronger, but slower wind mass loss. For example, theslow (v∞ ∼ 10 − 50 km/s), moderately strong (M ∼ 10−8M⊙/yr) winds of


Red Giants may be driven by the direct momentum (vs. energy) addition ofmagnetic Alfven waves. But in later evolutionary stages, stellar pulsations,perhaps augmented by radiative scattering on dust, can apparently induce amuch stronger, even runaway mass loss. Over the final few thousand years,this can reduce stars with initial mass as high as 8 M⊙ to a remnant whitedwarf below the Chandrasekhar limit of ∼ 1.4 M⊙! Similarly, initially hotstars with M ≈ 8− 40M⊙ can evolve through a Red Supergiant phase, witha strong mass loss that has importance consequences for their subsequentevolution and eventual demise as core-collapse supernovae (SN).

3. Winds from Hot, Massive, Luminous Stars. The high luminosity of hot, mas-sive stars means that the momentum of the light scattering off the electronsand ions in the atmosphere is by itself able to overcome the gravity and drivewinds with v∞ ∼ (1−3)ve ∼ 1000−3000 km/s and M ∼ 10−10−10−5M⊙/yr.As quantified in the so-called CAK (after Castor, Abbott & Klein 1975) for-malism, for winds from OB-type stars the essential coupling between radia-tive momentum and gas is via line scattering by heavy, minor ions; regulationof the driving by line saturation leads to relatively simple, analytic scalinglaws for the dependence of mass loss rate and flow speed on stellar parame-ters like luminosity, mass, and radius. In the subclass known as Wolf-Rayet(WR) stars, the mass loss is so high that the stellar “photosphere” lies inthe wind itself, characterized now by strong wind-broadened emission linesof helium and abundant heavy elements like Carbon, Nitrogen, or Oxygen(CNO); in evolved WR stars the depletion or absence of hydrogen indicatesthe mass loss has actually stripped away the star’s original hydrogen en-velope, exposing a core of material processed by various stages of nuclearburning. This mass loss may also be augmented by Luminous Blue Variable(LBV) phases characterized by eruptions or “superwinds” with mass lossrates up to 0.01 − 1 M⊙/yr lasting several years or decades. The 1840-60giant eruption of η Carinae provides a key prototype.

Of these, the first and third have the more extensively developed theory, and assuch will constitute the major focus of detailed discussion below. For the secondclass, see previous reviews (Holzer & MacGregor 1985; Holzer 1987; Willson 2000;Dupree 2004) and references therein. (See, however, §4.4 below.)

The next section (§2) summarizes the diagnostic methods and resulting in-ferred general properties for these various classes of stellar wind. The followingsection (§3) introduces the basic dynamical conservation equations governing windoutflow, which are then applied (§4) to gas-pressure-driven solar wind. The lasttwo sections give a quite extensive discussion of radiatively driven mass loss fromhot, massive, luminous stars, including the CAK line-driving of OB winds (§5),and the multi-line scattered winds of Wolf-Rayet stars (§6).


Fig. 1. Two telescopic clues to the existence of the solar wind. Left: The dual tails

of Comet Hale-Bopp. The upper one here consists mostly of dust slowly driven away

from the comet by solar radiation; it is tilted from the anti-sun (radial) direction by the

comet’s own orbital motion. The lower plasma tail comes from cometary ions picked

up by the solar wind; its more radial orientation implies that the radial outflow of the

solar wind must be substantially faster than the comet’s orbital speed. Right: White-

light photograph of 1980 solar eclipse, showing how the million degree solar corona is

structured by the solar magnetic field. The closed magnetic loops that trap gas in the

inner corona become tapered into pointed “helmet” streamers by the outward expansion

of the solar wind. Eclipse image courtesy Rhodes College, Memphis, Tennessee, and

High Altitude Observatory (HAO), University Corporation for Atmospheric Research

(UCAR), Boulder, Colorado. UCAR is sponsored by the National Science Foundation.

2 Observational Diagnostics and Inferred Properties

2.1 Solar Corona and Wind

The close proximity of the sun makes it possible to obtain high-resolution observa-tions of the complex magnetic structure in the solar atmosphere and corona, whilealso flying interplanetary spacecraft to measure in situ the detailed plasma andmagnetic field properties of the resulting solar wind. Even before the spacecraftera, early visual evidence that the sun has an outflowing wind came from anti-solardeflection of comet tails (Biermann 1951; figure 1a), and from the radial striationsseen in eclipse photographs of the solar corona (Newkirk 1967; figure 1b). Nowa-days modern coronagraphs use an occulting disk to artificially eclipse the brightsolar surface, allowing ground- and space-based telescopes to make routine obser-vations of the solar corona.

Figure 2a (from McComas et al. 1998) combines such coronagraphic imagesof the inner and outer corona (taken respectively from the SOHO spacecraft, andfrom the solar observatory on Mauna Loa, Hawaii) with an extreme ultravioletimage of the solar disk (also from SOHO), and a polar line plot of the latitudinalvariation of solar wind speed (as measured by the Ulysses spacecraft during itsinitial polar orbit of the sun). The equatorial concentration of coronal brightnessstems from confinement of coronal plasma by closed loops of magnetic field, with


Fig. 2. Left: Polar plot of the latitudinal variation of solar wind speed (jagged lines)

and magnetic polarity (indicated by red vs. blue line color), as measured by the Ulysses

spacecraft during its inital polar orbit of the sun. This is overlayed on a composite of

coronagraphic images of the inner and outer corona, and an extreme ultraviolet image

of the solar disk, all made on 17 August 1996, near the minimum of the solar sunspot

cycle. Right: X-ray images of the sun made by the Yokhoh satellite from the 1991 solar

maximum (left foreground) to the 1996 solar minimum (right background).

the outer loops pulled into the radial “helmet” streamers by the outflow of thesolar wind. By contrast, the open nature of magnetic field lines over the polesallow material to flow freely outward, making for reduced density coronal holes

(Zirker 1976). A key point from figure 2a is that the solar wind arising fromthese polar coronal holes is much steadier and faster (about 750 km/s) than theequatorial, streamer-belt wind measured near the ecliptic (with speeds from 400to 750 km/s). This provides a vivid illustration of how the magnetic structure ofthe corona can have a profound influence on the resulting solar wind.

Early direct evidence for the very high coronal temperature came from eclipseobservations of emission lines from highly stripped ions of calcium and iron, themost prominent being the Fe+13 “green” line at 5303A. This stripping of 13electrons from iron arises from high-energy collisions with free electrons, implyinga characteristic electron temperature around one and half million Kelvin (1.5 MK)in the dense, coronal loops that dominate the line emission. Nowadays the hightemperature of these coronal loops is most vividly illustrated by soft X-ray images,such as those shown in figure 2b from the observations by the Yokhoh satellite madeover the 1991-1996 declining phase of the sun’s 11-year magnetic activity cycle.Unfortunately, the relative darkness of coronal holes means it is much more difficultto use emission lines and X-rays to infer the temperature or other properties ofthese important source regions of the solar wind.

Complementary studies with space-borne coronal spectrographs (such as theUVCS intrument on SOHO; Kohl et al. 1995) use the strength and width ofultraviolet scattering lines to infer the flow speed and kinetic temperatures of


various ion species throughout the corona. Results show the outflow speed incoronal holes already exceeds 100 km/s within a solar radius from the surface;morever, ion temperatures are even higher than for electrons, up to 4 MK forprotons, and 100 MK for oxygen! The recent, very comprehensive review of coronalholes by Cranmer (2009) provides an extensive discussion of how these differenttemperatures may be linked to the detailed mechanisms for heating the corona,e.g. by magnetic waves excited by the complex convective motions of the solaratmosphere.

In situ measurements by Ulysses and many other interplanetary spacecraftshow that the solar wind mass flux (given by ρvr2, the product of density, flowspeed, and the square of the local heliocentric radius) is remarkably constant,varying less than a factor two in space and time (Withbroe 1989). If extendedover the full sphere around the sun, the corresponding global mass loss rate worksout to M ≡ 4πρvr2 ≈ 2 − 3 × 10−14M⊙/yr. In the ecliptic plane, the typicalterminal flow speed is v∞ ≈ 400 km/s, but it extends up to v∞ ≈ 750 km/sin both ecliptic and polar high-speed streams that are thought to originate fromcoronal holes.

Finally, while many other cool, solar-type stars show clear signatures of mag-netic activity and hot, X-ray emitting coronae, any associated coronal type windsare simply too tenuous and transparent to be detected directly by any aborptionsignatures in the spatially unresolved stellar flux. In a handful of stars, there isevidence for an “astrosphere” (Wood 2004), the stellar analog of the bullet-shapedheliospheric cavity carved out by the solar wind as the sun moves through thelocal interstellar medium. A wall of neutral hydrogen that forms around the noseof this bullet can lead to a detectable hydrogen line absorption in the observedspectrum of stars positioned so that their light passes through this wall on its wayto earth. But apart from such indirect evidence, most of what is inferred aboutcoronal winds from other stars is based on analogy or extrapolation of conditionsfrom the much more detailed measurements of the sun’s coronal wind.

2.2 Spectral Signatures of Dense Winds from Hot and Cool Stars

2.2.1 Opacity and Optical Depth

The much higher mass loss rates in stellar winds from both hot and cool giantsmakes them dense enough to become opaque in certain spectral bands, impartingdetectable observational signatures in the star’s flux spectrum that can be used toinfer key wind conditions. The coupling of wind material to stellar light can takevarious forms (Mihalas 1978), including:

• free-electron (Thomson) scattering;

• free-free absorption and emission by electrons near ions;

• bound-free (ionization) absorption and free-bound (recombination) emissionby atoms or molecules;


• bound-bound (line) scattering, absorption, and/or emission by atoms ormolecules;

• scattering, absorption, and emission by dust grains.

In parsing this list, it helpful to focus on some key distinctions. Scattering

involves a change in direction of light without true absorption of its energy, whileemission creates new light, e.g. from the energy within the gas. Bound-bound

processes lead to narrow wavelength features in the spectrum, known as (absorp-tion or emission) lines. Most other processes involve the broad spectral continuum

of stellar radiation, albeit with bound-free or free-bound edges at the wavelengthassociated with the energies for ionization/recombination of the ion, or broad vari-ations due to wavelength dependence of the cross section, e.g. for dust.

In the context of determining stellar wind mass loss rates, a particularly keydistinction lies between single-body processes (e.g., electron scattering, bound-freeabsorption, or dust scattering and absorption), and two-body processes involvingboth an ion and a free electron (e.g., free-free absorption, and free-free or free-bound emission). The strength of the former scales in proportion to the singledensity of the species involved, whereas the latter depend on the product of theelectron and ion density, and so have an overall density-squared dependence. Ina wind with a high degree of spatial clumping, such density-squared processeswill be enhanced over what would occur in a smooth flow by a clumping factorCf ≡

√

〈ρ2〉/ 〈ρ〉, where the square brackets represent volume averaging on a scalelarge comparized to the clump size. For a wind in which material is mostly inclumps filling only a fraction fv of the total volume, any mass loss rate diagnosticthat scales with ρ2 will tend to overestimate the true mass loss rate by about afactor Cf = 1/

√fv. Further details can be found in the proceedings of a recent

conference on wind clumping (Hamann et al. 2008).Single-density continuum processes (e.g., electron scattering, bound-free ab-

sorption, or dust scattering and absorption) can be characterized by the opacity,κ, which is effectively a cross section per unit mass, with CGS units cm2/g. Theoverall effectiveness of continuum absorption or scattering by the wind dependson the associated optical depth from some surface radius R to the observer. Alongthe radial direction r, this is given by the integral of the wind opacity κ timesdensity ρ(r),

τ∗ =

∫ ∞

R

κρ(r)dr ≈ κM

4πv∞R= 2.5

κR⊙

κe R

M−5

v1000. (2.1)

The second, approximate equality here applies to the simple case of a steady,spherically symmetric wind with mass loss rate M , and approximated for sim-plicity to flow at a constant speed v∞. The last equality uses the shorthandM−5 = M/(10−5M⊙/yr) and v1000 = v∞/(1000 km/s) to give numerical scalingsin terms of the opacity for electron scattering, which for a fully ionized mediumwith standard hydrogen mass fraction, X = 0.7, has a CGS value κe = 0.2(1+X) =0.34cm2/g. For example, electron scattering in the solar wind has τ∗ ≈ 10−8, con-firming it is very transparent. But the winds from Wolf-Rayet stars, with a billiontimes higher M , can become optically thick, even to electron scattering.


For the strong, slow winds from cool giants, the fraction of free electrons be-comes small, but the cool, dense conditions often allow molecules and even dustto form. Spherical dust of radius a, mass md, density ρd has a physical cross sec-tion σ = 3md/4aρd; conversion of a mass fraction Xd of wind material into dustthus implies an overall opacity κd = 3Xd/4aρd. For standard (solar) abundances,Xd ≈ 2 × 10−3, and since ρd ∼ 1 g/cm3 (most dust would almost float in water),note that for dust grains with typical size a ≈ 0.1µ, the corresponding opacity isof order κd ≈ 100 cm2/g, several hundreds times greater than for free electronscattering.

This geometric cross section only applies to wavelengths much smaller thanthe dust size, λ ≪ a; for λ ≥ a, the associated dust opacity decreases as κd(λ) ∼(λ/a)−β , where β = 4 for simple Rayleigh scattering from smooth spheres of fixedsize a. In practice, the complex mixtures in sizes and shapes of dust typically leadto a smaller effective power index, β ≈ 1− 2. Still, the overall inverse dependenceon wavelength means that winds that are optically thick to dust absorption andscattering can show a substantially reddened spectrum.

Moreover, the energy from dust-absorbed optical light is generally reemitted inthe infrared (IR), at wavelengths set by the dust temperature through roughly thestandard Wien’s law for peak emission of a blackbody, λ = 2.90µm/(T/1000K).For typical dust temperatures of several hundreds to a thousand degrees, this givesa strong IR excess in emission peaking in the 3-10 micron range. Such IR excesssignatures of wind dust are seen in very cool giant stars, particularly the so-calledCarbon stars, but also in the much hotter Carbon-type Wolf-Rayet stars, knownas WC stars. A good overview of dust opacity and emission is given by Li (2005).

2.2.2 Doppler-Shifted Line Absorption

It is much more common for winds to become optically thick to line absorptionor scattering that results from transitions between two bound levels of an atom(including ions), or molecule. Because this is a resonant process, the associatedcross sections and opacities can be much larger than from free electron scattering,enhanced by factor that depends on the “quality” of the resonance1. For atomictransitions, classical calculations (i.e. without quantum mechanics) show the res-onance enhancement factor is given roughly by q = Aλo/re, where A is the massabundance fraction of the atom, λo is the photon wavelength for the line transi-tion, and re = e2/mec

2 = 2.6× 10−13 cm is the classical radius for electron chargee and mass me (Gayley 1995). For ultraviolet line transitions with λo ≈ 250 nmand from abundant ions with A ≈ 10−5, one finds q ≈ 103.

As measured in the rest frame of the ion, this line opacity is confined to a verynarrow range around the resonant wavelength λo. But the Doppler shift from theion’s random thermal motion gives lines a finite Doppler width ∆λD = λovth/c,

1The effect is somewhat analogous to blowing into a whistle vs. just into open air. Like thesound of whistle, the response occurs at a well-tuned frequency, and has a greatly enhancedstrength.


Velocity

RadiusWavelength

vth

lSob =vth/(dv/dr)

λline

Fig. 3. Two perspectives for the Doppler-shifted line-resonance in an accelerating flow.

Right: Photons with a wavelength just shortward of a line propagate freely from the

stellar surface up to a layer where the wind outflow Doppler shifts the line into a resonance

over a narrow width (represented here by the shading) equal to the Sobolev length, set

by the ratio of thermal speed to velocity gradient, lSob ≡ vth/(dv/dr). Left: Seen

from successively larger radii within the accelerating wind, the Doppler-shift sweeps out

an increasingly broadened line absorption trough blueward of line-center in the stellar

spectrum.

where for ions with atomic mass mi and temperature T , the thermal speed vth =√

kT/mi. For typical observed ions of carbon, nitrogen, and oxygen (CNO) in ahot-star wind with temperatures a few times 104 K, one finds vth ≈ 10 km/s.

In a stellar wind with flow speed v ≈ 1000 km/s ≫ vth, the Doppler shift fromthe wind flow completely dominates this thermal broadening. As illustrated infigure 3, a stellar photon of wavelength λ < λo − ∆λD emitted radially from thestar thus propagates freely through the wind until reaching a location where thewind speed v has shifted the line into resonance with the photon,

λ = λ′o ≡ λo(1 − v/c) . (2.2)

The width of this resonance is given by lSob = vth/(dv/dr), known as the Sobolevlength, after the Russian astrophysicist V. V. Sobolev, who first developed thetheory for line radiation transport in such high-speed outflows (Sobolev 1957,1960). For a wind of terminal speed v∞ from a star of radius R, a characteristicvelocity gradient is dv/dr ≈ v∞/R, implying lSob/R ≈ vth/v∞ ≈ 0.01 ≪ 1, andso showing this Sobolev line resonance is indeed quite sharp and narrow.

The Sobolev approximation assumes a narrow line limit vth → 0 to derivelocalized, analytic solutions for the line transport, using the fact that the opacitythen becomes nearly a δ-function about the local resonant wavelength, κ(λ) =qκeδ(λ/λo − 1 + v/c). For wavelengths from line center λo to a blue-edge λ∞ ≡


λo(1 − v∞/c), the associated wind optical depth then becomes

τ(λ) =

∫ ∞

R

κ(λ)ρdr =

[

qκeρc

dv/dr

]

res

≡ τSob , (2.3)

where in the second equality the wind quantities are to be evaluated at the relevantresonance radius where v(rres) ≡ c(1 − λ/λo).

As a simple explicit example, for a steady wind with a canonical wind velocitylaw v(r) = v∞(1 − R/r), one finds

τ(λ) =τo

1 − R/rres=

τov∞c(1 − λ/λo)

, (2.4)

where

τo ≡ qκeMc

4πRv2∞

=qc

v∞τ∗,e . (2.5)

The latter equality in eqn. (2.5) emphasizes that this line optical depth is a factorqc/v∞ larger than the continuum optical depth for electron scattering, as givenby eqn. (2.1). For strong lines (q ≈ 1000), and even for moderately fast windswith v∞ ≈ 1000 km/s, this enhancement factor can be several million. For windswith v∞ ≈ 1000 km/s and R ≈ 1012 cm, such strong lines would be opticallythick for mass loss rates as low as M ∼ 10−10 M⊙/yr; for dense winds with M ∼10−6 M⊙/yr, strong lines can be very optically thick, with τo ∼ 104!

Defining a wind-scaled wavelength displacement from line center, x ≡ (c/v∞)(λ/λo − 1), eqn. (2.4) gives the very simple wavelength scaling, τ(x) = τo/(−x).For a simple “point-source” model (which ignores the finite size of the star and soassumes that all radiation is radially streaming), this predicts the observed stellarflux spectrum should show a blueward line-absorption trough of the form,

F (x) = Fce−τo/|x| ; − 1 < x < 0 , (2.6)

where Fc is the stellar continuum flux far away from the line. (For the somewhatmore general velocity law v(r) = v∞(1−R/r)β , the optical depth scaling becomesτ(x) = τo/β|x|2−1/β .)

2.2.3 Asymmetric P-Cygni Profiles from Scattering Lines

Strong spectral lines often tend to scatter rather than fully absorb stellar radia-tion, and this leads to the emergent line-profile developing a very distinctive formknown as a “P-Cygni profile”, after the star P Cygni for which its significance asa signature of mass outflow was first broadly recognized.

Figure 4 illustrates the basic principles for formation of such a P-Cygni line-profile. Wind material approaching an observer within a column in front of the starhas its line-resonance blue-shifted by the Doppler effect. Thus scattering of stellarradiation out of this direction causes an absorption trough on the blue side of theline profile, much as described in the simple, pure-absorption analysis given above.However, from the lobes on either side of this absorption column, wind material


+=

Velocity ; Wavelength

Flu

x

P-Cygni Line Profile

Blue-ShiftedAbsorption

Symmetric Emission

V=0V∞V=0V∞

Observer

Fig. 4. Schematic illustration of the formation of a P-Cygni type line-profile in an ex-

panding wind outflow.

can also scatter radiation toward the observer. Since this can occur from either theapproaching or receding hemisphere, this scattered radiation can either be blue-shifted or red-shifted. The associated extra flux seen by the observer thus occursas a symmetric emission component on both sides of the line center. Combinedwith the reduced blue-side flux, the overall profile has a distinctly asymmetricform, with apparent net blueward absorption and redward emission.

Derivations of the full equations are given, e.g. in Mihalas (1977) and Lamers& Cassinelli (1999), but a brief outline of the quantitative derivation is as follows.

First, the above point-star computation of the absorption trough from stellarlight scattered out of the line of sight to the observer must now be generalized toaccount for rays parallel to the radial ray to the observer, but offset by a distancep, with 0 ≤ p ≤ R. Denoting z as the coordinate along such rays, with z = 0 atthe midplane where the rays come closest the star’s center, then the optical depthfrom the star (at z∗ =

√

R2 − p2) to the observer (at z → +∞) is (cf. eqn. (2.3))

τ(p, λ) =

∫ ∞

z∗

κ(λ)ρdz =

[

qκeρc

dvz/dz

]

res

. (2.7)


The resonance condition is now vz(zres) ≡ c(1−λ/λo) = −v∞x, with the ray pro-

jection of the radial wind speed given by vz = µv, where µ = z/√

z2 + p2. Writtenin terms of the wind-scaled wavelength x (≡ (c/v∞)(λ/λo−1)), the observed direct

intensity is thus attentuated from the stellar surface value,

Idir(p, x) = Ice−τ(p,x) , (2.8)

which upon integration over the stellar core gives for the observed direct flux (cf.eqn. (2.6))

Fdir(x) = Fc

∫ R

0

e−τ(p,x) 2p dp

R2. (2.9)

For the scattered, emission component, the formal solution of radiative transfer(Mihalas 1977) takes a purely local form within the Sobolev approximation,

Iscat(p, x) = S(

1 − e−τ(p,x))

, (2.10)

where the source function S for pure-scattering is given by the wavelength- andangle-averaged mean intensity, customarily denoted J . In static atmospheres,computing this source function requires a global solution of the line-scattering,but in an expanding wind, methods developed by Sobolev (1960) allow it to becomputed in terms of purely local “escape probabilities”. Overall, the radial scalingfollows roughly the optically thin form for diluted core radiation Sthin = WIc,where the dilution factor W ≡ (1 − µ∗)/2, with µ∗ ≡

√

1 − R2/r2. The observedscattered flux is then computed from integration over all suitable rays,

Fscat(x) =2π

r2

∫ ∞

pmin

Iscat(p, x) p dp , (2.11)

where pmin = R for red-side wavelengths (λ > λo) that scatter from the backhemisphere that is occulted by the star, and pmin = 0 otherwise.

The total observed flux, F (x) = Fdir(x) + Fscat(x), combines the blue-shiftedabsorption of the direct component plus the symmetric emission of the scatteredcomponent, yielding then the net asymmetry of the P-Cygni profile shown infigures 4 and 5.

The quantitative fitting of observed line-profiles often uses the so-called SEI(Sobolev with Exact Integration) method (Lamers et al. 1987; Groenewegen &Lamers 1989), in which the Sobolev approximation is only used for computation ofthe scattering source function, which as a wavelength- and angle-averaged quan-tity tends to cancel errors associated with the finite width of the line resonance.However, because microturbulence in a stellar wind can effectively broaden thatresonance to many (factor ten or more) times the thermal value vth ≈ 10 km/s,an exact integration over depth is used for the formal solution of line-transfer.

Measurements of the blue-edge of a strong absorption trough immediately givesthe wind terminal speed. Moreover, if the line is detectable but not saturated (i.e.,0.1 < τo < 3), then the observed depth of the trough can in principal be used to


Flux

Wavelengthτο=10

το=1

το=0.1

1.0 0.5 0.0 0.5 1.0

0.5

1.0

1.5

2.0

Fig. 5. Left: Observed PCygni line profile for C+3 from NGC 6543, the central star in

the “Cat’s Eye” nebula. Right: Theoretical P-Cygni profile for a wind with velocity law

v(r) = v∞(1 − R/r), plotted as flux vs. wind-scaled wavelength x = (c/v∞)(λ/λo − 1);

the three curves show results for a weak, moderate, and strong line, with characteristic

line strengths (see eqn. 2.5) τo = 0.1, 1, and 10.

infer M , assuming one knows the line-opacity. In practice, uncertainties in theionization fraction of prominent ions, which feed directly into the opacity strengthfactor q, often limit the utility of this method.

But recent FUSE observations of UV lines from P+4 (a.k.a. PV) have provenparticularly useful for inferring mass loss rates of O stars (Fullerton et al. 2006).Because PV is often nearly the dominant ionization stage, the ionization cor-rection is less uncertain. Moreover, the low elemental abundance of phosphorusmeans that the lines, unlike those from more abundant elements, are not generallysaturated. Finally, in contrast to density-squared diagnostics, line-scattering isa single-density process, and so is not affected by the degree of wind clumping.Indeed, mass loss rates inferred by this method tend to be substantially lower, bya factor of ten more, compared to those inferred from the density-squared diag-nostics discussed in the next subsections (Fullerton et al. 2006). The resolution ofthis mass loss discrepancy is matter of much current research and discussion (see,e.g., Puls et al. 2006 and Hamann et al. 2008).

2.2.4 Wind Emission Lines

In WR stars with very dense winds, the emergent spectrum is characterized byvery broad (> 1000 km/s Doppler width) emission lines from various ionizationstages of abundant elments like He, N, C, and O. The recombination of an electronand ion generally occurs through a downward cascade through the many boundlevels of the ion, giving the bound-bound line emission. The involvement of an ionplus free electron means this is one of the processes that scale with the square ofthe gas density. In the narrow-line (Sobolev) approximation, the line emissivity

(energy per unit time, volume, and solid angle) at wind-scaled wavelength x andfrom some wind radius with velocity v(r) is proportional to

η(x, r, µ) ∝ ρ2δ(x + µv(r)/v∞) , (2.12)


-1.0 -0.5 0.0 0.5 1.0

1.0

1.5

2.0

2.5

3.0

Fig. 6. Left: Observed flat-top line profiles of twice ionized carbon CIII (λ5696A) from

the Wolf-Rayet star WR 137 (from Lepine and Moffat 1999), plotted vs. velocity-unit

wavelength shift from line center. Right: Simple analytic flat-top emission line profile

plotted vs. wind-scaled wavelength x = (c/v∞)(λ/λo − 1); the model assumes a velocity

law v(r) = v∞(1 − R/r) with emission starting at a minimum speed vmin = v∞/2.

where µ is the direction cosine for the emitted radiation relative to the local radialdirection. Integration over all directions µ then gives for the emission luminositycontribution from a narrow shell between r and r + dr,

dL(x)

dr∝ r2ρ2

∫ +1

−1

dµ δ(x + µv(r)/v∞) (2.13)

∝ r2ρ2 v∞v(r)

if |x| ≤ v(r)

v∞(2.14)

= 0 otherwise. (2.15)

Thus each such shell produces a flat-top profile extending to x = ±v/v∞ aboutline center. If the ionization stage to produce the emission doesn’t form until nearthe wind terminal velocity, then the observed profile is indeed flat-topped, with ahalf-width given by v∞.

But if the ion already forms at a lower radius with a velocity vmin < v∞, thenthe profile exhibits a flat-topped center for |x| < vmin/v∞, with tapered wingsextending to x = ±1 (see, e.g., Dessart & Owocki 2002, 2005). For example,for the case of the standard velocity law v(r) = v∞(1 − R/r), the emission at|x| > vmin/v∞ depends on the integral,

L(x) ∝∫

ρ2r2

vdr ∝

∫

dr

v3r2∝∫

dx

x3∝ 1

2x2, (2.16)

where the second integrand follows from mass continuity, and the third uses thefact that Rdr/r2 = dv/v∞ = −dx. One thus obtains the simple analytic resultthat the line wing tapers off as 1/x2 from each side of the inner flat top over|x| < vmin/v∞.

Using this simple model, the right panel of figure 6 plots a sample flat-topprofile for vmin/v∞ = 0.5 . The left panel of figure 6 (taken from Lepine and


Moffat 1999) shows a corresponding observed proile, for the CIII line from thestar WR 134. The observed half-width of this emission emission line directlyimplies a wind speed of nearly 2000 km/s for this star. Other WR stars showhalf-widths that imply wind speeds in the range 1000-3000 km/s.

Since the emission strength is proportional to ρ2 ∼ M2, careful modeling of theexpanding wind and atmosphere can in principal enable one to use the observedstrength of emission lines to also estimate the stellar mass loss rate. But suchestimates depend on the wind velocity law, and, as noted above, on the degree ofwind clumping (Hamann 1996; Hillier & Miller 1999; Hamman et al. 2003; Hillier2003; Puls et al. 2006).

For O-stars, a key mass loss rate diagnostic is the Balmer (Hα) line emissionfrom the n = 3 to n = 2 transition of recombining Hydrogen (Puls et al. 1996). Ittends to be dominated by emission very near the star, and so is not as sensitive tothe wind terminal speed. As a ρ2 diagnostic, it is also senstive to wind clumping,but in this case, in the relatively low-speed flow near the stellar surface.

2.2.5 Continuum Emission in Radio and Infrared

The dense, warm stellar winds from hot supergiants can also produce substantialcontinuum emission at radio and infrared (IR) wavelengths. Those with spectrashowing a flux that decreases with frequency are identified as “non-thermal”, andare nowadays associated with binary sytems, wherein electrons accelerated in theshock collision between the two winds emit radio waves through gyration aboutthe local magnetic field (van Loo et al. 2006).

More useful as a wind diagnostic are the thermal emitters, wherein the radio/IRarises from free-free emission by electrons with thermal energies characteristic ofthe wind temperature. A landmark analysis by Wright & Barlow (1976) showedhow the observed flux of thermal radio/IR emission could be used as a quite directdiagnotic of the wind mass loss rate.

Both free-free emission and aborption are density-squared processes. For ab-sorption in a wind moving at constant speed v∞, the radial optical depth from aradius r to the observer is thus given by

τν(r) = K(ν, T )

∫ ∞

r

ρ2(r′) dr′ = K(ν, T )M2

48π2v2∞r3

, (2.17)

where the dependence on frequency ν and temperature T is set by K(ν, T ) ∼ν−2T−3/2 (Allen 1973; Wright & Barlow 1975). Solving τν(Rν) = 1 allows us todefine a characteristic radius for the “radio/IR photosphere” at frequency ν,

Rν =

[

K(ν, T )M2

48π2v2∞

]1/3

. (2.18)

For hot supergiants, the wind density is so high that this radio/IR photospherecan be at hundreds of stellar radii, Rν > 100R.


One can then estimate the radio/IR luminosity by a simple model that accumu-lates the total outward emission from a sphere with this photospheric radius; sincefree-free emission is a thermal process, the surface brightness of this photosphereis set by the Planck function Bν(T ), giving

Lν = 2πBν4πR2ν = 8π2Bν

[

K(ν, T )M2

48π2v2∞

]2/3

(2.19)

For radio/IR frequencies and typical stellar wind temperatures, the Planck func-tion is well approximated by its Rayleigh-Jeans form, Bν ≈ 2ν2kT/c2. Usingthe above scaling for K(ν, T ) ∼ ν−2T−3/2, the radio/IR luminosity scales withfrequency as Lν ∼ ν2/3, but becomes independent of temperature.

The luminosity scales with wind parameters as Lν ∼ (M/v∞)4/3, but other-wise depends just on known atomic constants. As such, if the wind terminal speedis known from other diagnostics, such as P-Cygni line profiles, measurements ofthe radio/IR flux can provide (for known stellar distance) a relatively robust di-agnostic of the wind mass loss rate M . But again, because of the scaling withdensity-squared, in a clumped wind such radio/IR inferred mass loss rates areoverestimated by a factor Cf = 1/

√fv, where now fv is the clump volume filling

factor in the distant, outer wind. Comparison with mass loss rates derived fromHα, which also scales with density-squared but is formed near the star, can thusgive a clue to the radial evolution of wind clumping (Puls et al. 2006).

3 General Equations and Formalism for Stellar Wind Mass Loss

With this background on wind diagnostics, let us now turn to theoretical modellingof stellar winds. This section lays the groundwork by defining the basic dynamicalequations for consevation of mass, momentum, and energy. The following sectionsreview the mechanisms for wind driving for the different type of stellar winds.

3.1 Hydrostatic Equilibrium in the Atmospheric Base of any Wind

A stellar wind outflow draws mass from the large reservoir of the star at its base.In the star’s atmosphere the mass density ρ becomes so high that the net mass fluxdensity ρv for the overlying steady wind requires only a very slow net drift speedv, much below the local sound speed. In this nearly static region, the gravitationalacceleration ggrav acting on the mass density ρ is closely balanced by a pressuregradient ∇P ,

∇P = ρggrav , (3.1)

a condition known as hydrostatic equilibrium. In luminous stars, this pressurecan include significant contributions from the stellar radiation, but for now let usassume (as is applicable for the sun and other cool stars) it is set by the ideal gaslaw

P = ρkT/µ = ρa2 . (3.2)


The latter equality introduces the isothermal sound speed, defined by a ≡√

kT/µ,with T the temperature, µ the mean atomic weight, and k Boltmann’s constant.The ratio of (3.2) to (3.1) defines a characteristic pressure scale height,

H ≡ P

|∇P | =a2

|ggrav|. (3.3)

In the simple ideal case of an isothermal atmosphere with constant sound speeda, this represents the scale for exponential stratification of density and pressurewith height z

P (z)

P∗=

ρ(z)

ρ∗= e−z/H , (3.4)

where the asterisk subscripts denote values at some some surface layer where z ≡ 0.In practice the temperature variations in an atmosphere are gradual enough thatquite generally both pressure and density very nearly follow such an exponentialstratification.

As a typical example, in the solar photosphere T ≈ 6000 K and µ ≈ 10−24 g,yielding a sound speed a ≈ 9 km/s. For the solar surface gravity ggrav =GM⊙/R2

⊙ ≈ 2.7 × 104 cm/s2, this gives a pressure scale height of H ≈ 300 km,which is very much less than the solar radius R⊙ ≈ 700, 000 km. This impliesa sharp edge to the visible solar photosphere, with the emergent spectrum welldescribed by a planar atmospheric model fixed by just two parameters – typicallyeffective temperature and gravity– and not dependent on the actual solar radius.

This relative smallness of the atmospheric scale height is a key general char-acteristic of static stellar atmospheres, common to all but the most extremelyextended giant stars. In general, for stars with mass M, radius R, and surfacetemperature T, the ratio of scale height to radius can be written in terms of theratio of the associated sound speed a∗ to surface escape speed ve ≡

√

2GM/R,

H∗

R=

2a2∗

v2e

≡ 2s . (3.5)

One recurring theme of this review is that the value of this ratio is also of directrelevance to stellar winds. The parameter s ≡ (a∗/ve)

2 characterizes roughly theratio between the gas internal energy and the gravitational escape energy. Forthe solar photosphere, s ≈ 2.2 × 10−4, and even for very hot stars with an orderof magnitude higher photospheric temperature, this parameter is still quite small,s ∼ 10−3. However, as discussed further in the next section, for the multi-million-degree temperature of the solar corona, this parameter is much closer to unity,and that is a key factor in the capacity for the thermal gas pressure to drive theoutward coronal expansion that is the solar wind.

3.2 General Flow Conservation Equations

Let us now consider a case wherein a non-zero net force leads to a net acceleration,

dv

dt=

∂v

∂t+ v · ∇v = −∇P

ρ− GM

r2r + gx . (3.6)


Here v is the flow velocity, gx is some yet-unspecified force-per-unit-mass, and thegravity is now cast in terms of its standard dependence on gravitation constantG, stellar mass M, and local radius r, with r a unit radial vector.

Conservation of mass requires that any temporal change in the local density ρmust arise from a net divergence of the mass flux density ρv,

∂ρ

∂t+ ∇ · ρv = 0 . (3.7)

Conservation of internal energy takes a similar form, but now accounting also forany local sources or sinks of energy,

∂e

∂t+ ∇ · ev = −P ∇ · v −∇ · Fc + Qx . (3.8)

For an ideal gas with ratio of specific heats γ, the internal energy density e isrelated to the pressure through

P = ρa2 = (γ − 1)e . (3.9)

On the right-hand-side of eqn. (3.8), Qx represents some still unspecified, netvolumetric heating or cooling, while Fc is the conductive heat flux density, takenclassically to depend on the temperature as

Fc = KoT5/2 ∇T , (3.10)

where for electron conduction the coefficient Ko = 5.6 × 10−7 erg/s/cm/K7/2

(Spitzer 1962).Collectively, eqns. (3.6)-(3.9) represent the general equations for a potentially

time-dependent, multi-dimensional flow.

3.3 Steady, Spherically Symmetric Wind Expansion

In application to stellar winds, first-order models are commonly based on thesimplifying approximations of steady-state (∂/∂t = 0), spherically symmetric,radial outflow (v = vr). The mass conservation requirement (3.7) can then beused to define a constant overall mass loss rate,

M ≡ 4πρvr2 . (3.11)

Using this and the ideal gas law to eliminate the density in the pressure gradientterm then gives for the radial equation of motion

(

1 − a2

v2

)

vdv

dr= −GM

r2+

2a2

r− da2

dr+ gx . (3.12)

In general, evaluation of the sound speed terms requires simultaneous solution ofthe corresponding, steady-state form for the flow energy, eqn. (3.8), using also theideal gas eqn. (3.9). But in practice, this is only of central importance for proper


modelling of the pressure-driven expansion of a high-temperature (million degree)corona, as discussed in the next section for the solar wind.

For stars without such a corona, any wind typically remains near or below thestellar photospheric temperature T, which as noted above implies a sound speeda∗ that is much less than the surface escape speed ve. This in turn implies thatthe sound-speed terms on the right-hand-side of eqn. (3.12) are quite negligible,of order s ≡ a2

∗/v2e ∼ 10−3 smaller than the gravity near the stellar surface. These

terms are thus of little dynamical importance in determining the overall windproperties, like the mass loss rate or velocity law (see §4). However, it is still oftenconvenient to retain the sound-speed term on the left-hand-side, since this allowsfor a smooth mapping of the wind model onto a hydrostatic atmosphere througha subsonic wind base.

In general, to achieve a supersonic flow with a net outward acceleration, eqn.(3.12) shows that the net forces on the right-hand-side must be positive. Forcoronal winds, this occurs by the sound-speed terms becoming bigger than gravity.For other stellar winds, overcoming gravity requires the additional body forcerepresented by gx, for example from radiation (§5).

3.4 Energy Requirements of a Spherical Wind Outflow

In addition to these force or momentum conditions for a wind outflow, it is in-structive to identify explicitly the general energy requirements. Combining themomentum and internal energy equations for steady, spherical expansion, integra-tion yields the total energy change from the base stellar radius R to some givenradius r

M

[

v2

2− GM

r+

γ

γ − 1

P

ρ

]r

R

=

∫ r

R

(

Mgx + 4πr′2Qx

)

dr′ − 4π[

r′2Fc

]r

R. (3.13)

On the left-hand-side, the terms represent the kinetic energy, gravitational po-tential energy, and internal enthalpy. To balance this, on the right-hand-side arethe work from the force gx, the net volumetric heating Qx, and the change inthe conductive flux Fc. Even for the solar wind, the enthalpy term is generallymuch smaller than the larger of the kinetic or potential energy. Neglecting thisterm, evaluation at arbitrarily large radius r → ∞ thus yields the approximaterequirement for the total energy-per-unit-mass,

v2∞

2+

v2e

2≈∫ ∞

R

(

gx + 4πr′2Qx/M)

dr′ +4πR2Fc∗

M, (3.14)

where Fc∗ is the conductive heat flux density at the coronal base, and and the outerheat flux is assumed to vanish asymptotically far from the star. This equationemphasizes that a key general requirement for a wind is to supply the combinedkinetic plus potential energy. For stellar winds, this typically occurs through thedirect work from the force gx. For the solar wind, it occurs through a combinationof the volume heating and thermal conduction, as discussed next.


4 Coronal Expansion and Solar Wind

4.1 Reasons for Hot, Extended Corona

4.1.1 Thermal Runaway from Density and Temperature Decline of Line-Cooling

In the sun and other relatively cool stars, the existence of a strong near-surface con-vection zone provides a source of mechanical energy to heat the upper atmosphereand thereby reverse the vertical decrease of temperature of the photosphere. Fromphotospheric values characterized by the effective temperature Teff ≈ 6000 K, thetemperature declines to a minimum Tmin ≈ 4000 K, but then rises through alayer (the chromosphere) extending over many scale heights to about 10, 000 K.Above this, it then jumps sharply across a narrow transition region of less thanscale height, to values in the corona of order a million degrees! The high pres-sure associated with this high temperature eventually leads to the outward coronalexpansion that is the solar wind.

This sharp jump in temperature is a direct result of the inability of the increas-ingly rarefied material in the upper atmosphere to radiate away any mechanicalheating, for example associated with dissipation of some generic wave energy fluxFE over some damping length λd. To maintain a steady state with no net heating,this energy deposition must be balanced by radiative cooling, which can be takento follow the scaling (e.g., Cox and Tucker 1969; Raymond, Cox, and Smith 1976),

FE

λd= nHneΛ(T ) . (4.1)

Here Λ(T ) is known as the the optically thin cooling function, which can be readilytabulated from a general atom physics calculation for any assumed abundanceof elements, with the standard result for solar (or ‘cosmic’) abundances plottedschematically in figure 7. The overall scaling with the product of the hydrogen andelectron number densities, nH and ne, reflects the fact that the radiative coolingarises from collional excitation of ions by electrons, for the assumed abundance ofion species per hydrogen atom.

Consider then the nature of this energy balance within a hydrostatically strat-ified atmosphere, for which all the densities are declining exponentially in heightz, with a scale height H ≪ R. If the damping is linked to material absorption,it might scale inversely with density, but even so cooling would still scale withone higher power of density. Using eqn. (4.1), one thus finds the radiative coolingfunction needs to increase exponentially with height

Λ[T (z)] ∼ FE

λd ρ2∼ FE e+z/H . (4.2)

As illustrated in figure 7, for any finite wave flux this required increase should leadto a steadily higher temperture until, upon reaching the local maximum of thecooling function at a temperature of ca. 105 K, a radiative balance can no longerbe maintained without a drastic jump to very much higher temperature, above107 K.


10-21

10-22

10-23

Λ(T) [erg cm3/s]

Temperature(K)

105104 106 107

<-- Thermal Instability -->

~FE/λdρ2

Fig. 7. Radiative cooling function Λ(T ) plotted vs. temperature T on a log-log scale.

The arrows from the left represent the density-scaled rate of energy deposition, which

increases with the exponential decrease in density with height. At the point where this

scaled heating exceeds the maximum of the cooling function, the radiative equilibrium

temperature jumps to a much higher value, in excess of 107 K.

This thermal instability is the direct consequence of the decline in cooling ef-ficiency above 105 K, which itself is an intrinsic property of radiative cooling,resulting from the progressive ionization of those ion stages that have the boundelectrons needed for line emission. The characteristic number density at whichrunaway occurs can be estimated as

nrun ≈√

FE

λdΛmax≈ 3.8 × 107 cm−3

√

F5R⊙

λh(4.3)

where Λmax ≈ 10−21 erg cm3/s is the maximum of the cooling function [see fig-ure 7]. The latter equality evaluates the scaling in terms of typical solar values fordamping length λd ≈ R⊙ and wave energy flux density F5 = FE/105erg/cm2/s.This is roughly comparable to the inferred densities at the base of the solar corona,i.e. just above the top of the transition region from the chromosphere.

4.1.2 Coronal Heating with a Conductive Thermostat

In practice, the outcome of this temperature runaway of radiative cooling tendsto be tempered by conduction of heat back into the cooler, denser atmosphere.Instead of the tens of million degrees needed for purely radiative restabilization,the resulting characteristic coronal temperature is “only” a few million degrees. Tosee this temperature scaling, consider a simple model in which the upward energyflux FE through a base radius R is now balanced at each coronal radius r purely


by downward conduction,

4πR2FE = 4πr2KoT5/2 dT

dr. (4.4)

Integration between the base radius R and an assumed energy deposition radiusRd yields a characteristic peak coronal temperature

T ≈[

7

2

FE

Ko

Rd − R

Rd/R

]2/7

≈ 2 × 106 K F2/75 , (4.5)

where the latter scaling applies for a solar coronal case with R = R⊙ and Rd =2R⊙. This is in good general agreement with observational diagnostics of coronalelectron temperature, which typically give values near 2 MK.

Actually, observations (Kohl et al. 1999; Cranmer et al. 1999) of the “coronalhole” regions thought to be the source of high-speed solar wind suggest that thetemperature of protons can be significantly higher, about 4−5 MK. Coronal holesare very lower density regions wherein the collisional energy coupling between elec-trons and protons can be insufficient to maintain a common temperature. For com-plete decoupling, an analogous conductive model would then require that energyadded to the proton component must be balanced by its own thermal conduction.But because of the higher mass and thus lower thermal speed, proton conductivityis reduced by the root of the electron/proton mass ratio,

√

me/mp ≈ 43, relativeto the standard electron value used above. Application of this reduced protonconductivity in eqn. (4.5) thus yields a proton temperature scaling

Tp ≈ 5.8 × 106 K F2/75 , (4.6)

where now F5 = FEp/105 erg/cm2/s, with FEp the base energy flux associatedwith proton heating. Eqn. (4.6) matches better with the higher inferred protontemperature in coronal holes, but more realistically, modelling the proton energybalance in such regions must also account for the energy losses associated withcoronal expansion into the solar wind (see §3.3).

4.1.3 Incompatibility of Hot Extended Hydrostatic Corona with ISM Pressure

Since the thermal conductivity increases with temperature as T 5/2, the high char-acteristic coronal temperature also implies a strong outward conduction flux Fc.For a conduction-dominated energy balance, this conductive heat flux has almostzero divergence,

∇ · Fc =1

r2

d(r2 KoT5/2(dT/dr))

dr≈ 0 . (4.7)

Upon double integration, this gives a temperature that declines only slowly out-ward from its coronal maximum, i.e. as T ∼ r−2/7. The overall point thus isthat, once a coronal base is heated to a very high temperature, thermal conduc-tion should tend to extend that high temperature outward to quite large radii(Chapman 1961).


This radially extended high temperature of a corona has important implicationsfor the dynamical viability of maintaining a hydrostatic stratification. First, forsuch a high temperature, eqn. (3.5) shows that the ratio of scale height to radiusis no longer very small. For example, for the typical solar coronal temperature of2 MK, the scale height is about 15% of the solar radius. In considering a possiblehydrostatic stratification for the solar corona, it is thus now important to takeexplicit account of the radial decline in gravity,

d lnP

dr= −GM

a2r2. (4.8)

Motivated by the conduction-dominated temperature scaling T ∼ r−2/7, let usconsider a slightly more general model for which the temperature has a power-lawradial decline, T/T = a2/a2

∗ = (r/R)−q. Integration of eqn. (4.8) then yields

P (r)

P∗= exp

(

R

H∗(1 − q)

[

(

R

r

)1−q

− 1

])

, (4.9)

where H∗ ≡ a2∗R

2/GM. A key difference from the exponential stratification ofa nearly planar photosphere (cf. eqn. 3.4) is that the pressure now approaches afinite value at large radii r → ∞,

P∞

P∗= e−R/H∗(1−q) = e−14/T6(1−q) , (4.10)

where the latter equality applies for solar parameters, with T6 the coronal basetemperature in units of 106 K. This gives log(P∗/P∞) ≈ 6/T6/(1 − q).

To place this in context, note that a combination of observational diagnosticsgive log(PTR/PISM ) ≈ 12 for the ratio between the pressure in the transitionregion base of the solar corona and that in the interstellar medium. This impliesthat a hydrostatic corona could only be contained by the interstellar medium if(1 − q)T6 < 0.5. Specifically, for the conduction-dominated temperature indexq = 2/7, this requires T6 < 0.7. Since this is well below the observational rangeT6 ≈ 1.5−3, the implication is that a conduction-dominated corona cannot remainhydrostatic, but must undergo a continuous expansion, known of course as the solarwind.

But it is important to emphasize here that this classical and commonly citedargument for the “inevitability” of the solar coronal expansion depends cruciallyon extending a high temperature at the coronal base far outward. For example, abase temperature in the observed range T6 = 1.5−3 would still allow a hydrostaticmatch to the interstellar medium pressure if the temperature were to decline withjust a somewhat bigger power index q = 2/3 − 5/6.

4.2 Isothermal Solar Wind Model

These problems with maintaining a hydrostatic corona motivate consideration ofdynamical wind solutions (Parker 1962). A particularly simple example is that of


0 0.5 1 1.5 20

0.5

1

1.5

2

v/a

r/rc

Fig. 8. Solution topolgy for an isothermal coronal wind, plotted via contours of the

integral solution (4.13) with various integration constants C, as a function of the ratio of

flow speed to sound speed v/a, and the radius over critical radius r/rs. The heavy curve

drawn for the contour with C = −3 represents the transonic solar wind solution.

an isothermal, steady-state, spherical wind, for which the equation of motion [fromeqn. (3.12), without external driving (gx = 0) or sound-speed gradient (da2/dr =0)] becomes

(

1 − a2

v2

)

vdv

dr=

2a2

r− GM

r2, (4.11)

Recall that this uses the ideal gas law for the pressure P = ρa2, and eliminatesthe density through the steady-state mass continuity; it thus leaves unspecified theconstant overall mass loss rate M ≡ 4πρvr2.

The right-hand-side of eqn. (4.11) has a zero at the critical radius

rc =GM

2a2. (4.12)

At this radius, the left-hand-side must likewise vanish, either through a zero ve-locity gradient dv/dr = 0, or through a sonic flow speed v = a. Direct integrationof eqn. (4.11) yields the general solution

F (r, v) ≡ v2

a2− ln

v2

a2− 4 ln

r

rc− 4rc

r= C , (4.13)


where C is an integration constant. Using a simple contour plot of F (r, v) inthe velocity-radius plane, figure 8 illustrates the full “solution topology” for anisothermal wind. Note for C = −3, two contours cross at the critical radius(r = rc) with a sonic flow speed (v = a). The positive slope of these representsthe standard solar wind solution, which is the only one that takes a subsonic flownear the surface into a supersonic flow at large radii.

Of the other initially subsonic solutions, those lying above the critical solutionfold back and terminate with an infinite slope below the critical radius. Thoselying below remain subsonic everywhere, peaking at the critical radius, but thendeclining to arbitrarily slow, very subsonic speeds at large radii. Because suchsubsonic “breeze” solutions follow a nearly hydrostatic stratification (Chamberlain1961) , they again have a large, finite asymptotic pressure that doesn’t match therequired interstellar boundary condition.

In contrast, for the solar wind solution the supersonic asymptotic speed meansthat, for any finite mass flux, the density, and thus the pressure, asymtoticallyapproaches zero. To match a small, but finite interstellar medium pressure, thewind can undergo a shock jump transition onto one of the declining subsonicsolutions lying above the decelerating critical solution.

Note that, since the density has scaled out of the controlling equation of motion(4.11), the wind mass loss rate M ≡ 4πρvr2 does not appear in this isothermalwind solution. An implicit assumption hidden in such an isothermal analysis isthat, no matter how large the mass loss rate, there is some source of heating thatcounters the tendency for the wind to cool with expansion. As discussed below,determining the overall mass loss rate requires a model that specifies the locationand overall level of this heating.

4.2.1 Temperature Sensitivity of Mass Loss Rate

This isothermal wind solution does nonetheless have some important implicationsfor the relative scaling of the wind mass loss rate. To see this, note again thatwithin the subsonic base region, the inertial term on the left side of eqn. (4.11) isrelatively small, implying the subsonic stratification is nearly hydrostatic. Thus,neglecting the inertial term v2/a2 in the isothermal solution (4.13) for the criticalwind case C = −3, one can solve approximately for the surface flow speed

v∗ ≡ v(R) ≈ a

(

rc

R

)2

e3/2−2rc/R . (4.14)

Here “surface” should really be interpreted to mean at the base the hot corona,i.e. just above the chromosphere-corona transition region.

In the case of the sun, observations of transition region emission lines provide aquite tight empirical constraint on the gas pressure P∗ at this near-surface coronalbase of the wind (Withbroe 1987). Using this and the ideal gas law to fix theassociated base density ρ∗ = P∗/a2, one finds that eqn. (4.14) implies a mass loss


scaling

M ≡ 4πρ∗v∗R2 ≈ 56

P∗

ar2c e−2rc/R ∝ P∗

T5/26

e−14/T6 . (4.15)

With T6 ≡ T/106 K, the last proportionality applies for the solar case, and isintended to emphasize the steep, exponential dependence on the inverse temper-ature. For example, assuming a fixed pressure, just doubling the coronal temper-ature from one to two million degrees implies nearly a factor 200 increase in themass loss rate!

Even more impressively, decreasing from such a 1 MK coronal temperature tothe photospheric temperature T ≈ 6000 K would decrease the mass loss rate bymore than a thousand orders of magnitude! This reiterates quite strongly thatthermally driven mass loss is completely untenable at photospheric temperatures.

The underlying reason for this temperature sensitivity stems from the expo-nential stratification of the subsonic coronal density between the base radius Rand the sonic/critical radius rc. From eqns. (3.5) and 4.12) it is clear that thiscritical radius is closely related to the ratio of the base scale height to stellar radius

rc

R=

R

2H=

v2e

4a2, (4.16)

where the latter equality also recalls the link with the ratio of sound speed tosurface escape speed. Application in eqn. (4.15) shows that the argument of theexponential factor simply represents the number of base scale heights within acritical radius.

As noted in §2.1, in situ measurements by interplanetary spacecraft show thatthe solar wind mass flux is actually quite constant, varying only by about a fac-tor of two or so. In conjunction with the predicted scalings like (4.15), and theassumption of fixed based pressure derived from observed transition region emis-sion, this relatively constant mass flux has been viewed as requiring a sensitivefine tuning of the coronal temperature (e.g., Withbroe 1989).

4.2.2 The Solar Wind as a Thermostat for Coronal Heating

But a more appropriate perspective is to view this temperature-sensitive mass lossas providing an effective way to regulate the temperature resulting from coronalheating. This can be seen by considering explicitly the energy requirements for athermally driven coronal wind. From eqn. (3.13) (with gx = 0), the total energychange from a base radius R to a given radius r is

M

[

v2

2− v2

e

2

R

r+

γa2

γ − 1

]r

R

= 4π

∫ r

R

r′2Qxdr′ + 4π[

R2Fc∗ − r2Fc

]

. (4.17)

The expression here of the gravitational and internal enthalpy in terms of theassociated escape and sound speed ve and a allows convenient comparison of therelative magnitudes of these with the kinetic energy term, v2/2. Even at typicalcoronal temperature of 2 MK, the enthalpy term is only about a third of the


gravitational escape energy from the solar surface, implying that the initially slow,subsonic flow at the wind base has negative total energy set by gravity. Far fromthe star, this gravitational term vanishes, and the kinetic energy associated withthe supersonic solar wind dominates. Since enthalpy is thus not important ineither limit, let us for convenience ignore its relatively minor role in the globalwind energy balance, giving then

Ew ≡ M

(

v2∞

2+

v2e

2

)

≈ 4π

∫ ∞

R

r′2Qxdr′ + 4πR2(Fc∗) . (4.18)

Here the outer heat flux is assumed to vanish asymptotically far from the star,while Fc∗ < 0 is the (inward) conductive heat flux density at the coronal base. Eqn.(4.18) shows then that the total energy of the solar wind is just given by the total,volume-integrated heating of the solar corona and wind, minus the lost energyconduction back into the solar atmosphere. The analysis in §4.1.3 discussed howthe heating of a strictly static corona can be balanced by the inward conductiveflux back into the sun. But eqn. (4.18) now shows that an outflowing corona allowsa net difference in heating minus conduction to be carried outward by the solarwind.

In this sense, the problem of “fine-tuning” the coronal temperature to givethe observed, relativey steady mass flux is resolved by a simple change of per-spective, recognizing instead that for a given level of coronal heating, the stronglytemperature-sensitive mass and energy loss of the solar wind provides a very ef-fective “thermostat” for the coronal temperature.

4.3 Driving High-Speed Streams

More quantitative analyses solve for the wind mass loss rate and velocity law interms of some model for both the level and spatial distribution of energy additioninto the corona and solar wind. The specific physical mechanisms for the heatingare still a matter of investigation, but one quite crucial question regards the relativefraction of the total base energy flux deposited in the subsonic vs. supersonicportion of the wind. Models with an explicit energy balance generally confirma close link between mass loss rate and energy addition to the subsonic base ofcoronal wind expansion. By contrast, in the supersonic region this mass flux isessentially fixed, and so any added energy there tends instead to increase theenergy-per-mass, as reflected in asymptotic flow speed v∞ (Leer & Holzer 1982).

A important early class of solar wind models assumed some localized deposi-tion of energy very near the coronal base, with conduction then spreading thatenergy both downward into the underlying atmosphere and upward into the ex-tended corona. As noted, the former can play a role in regulating the coronalbase temperature, while the latter can play a role in maintaining the high coronaltemperature needed for wind expansion. Overall, such conduction models of solarwind energy tranport were quite successful in reproducing interplanetary measure-ments of the speed and mass flux of the “quiet”, low-speed (v∞ ≈ 350−400 km/s)solar wind.


However, such models generally fail to explain the high-speed (v∞ ≈ 700 km/s)wind streams that are inferred to emanate from solar “corona holes”. Such coronalholes are regions where the solar magnetic field has an open configuration that,in constrast to the closed, nearly static coronal “loops”, allows outward, radialexpansion of the coronal gas. To explain the high speed streams, it seems thatsome substantial fraction of the mechanical energy propagating upward throughcoronal holes must not be dissipated as heat near the coronal base, but insteadmust reach upward into the supersonic wind, where it provides either a directacceleration (e.g. via a wave pressure that gives a net outward gx) or heating(Qx > 0) that powers extended gas pressure acceleration to high speed.

Observations of such coronal hole regions from the SOHO satellite (Kohl et al.1999; Cranmer et al. 1999) show temperatures of Tp ≈ 4 − 5 MK for the protons,and perhaps as high as 100 MK for minor ion species like oxygen. The fact thatsuch proton/ion temperatures are much higher than the ca. 2 MK inferred forelectrons shows clearly that electron heat conduction does not play much role inextending the effect of coronal heating outward in such regions. The fundamen-tal reasons for the differing temperature components are a topic of much currentresearch; one promising model invokes ion-cyclotron-resonance damping of mag-netohydrodynamics waves (Cranmer 2000, 2008).

4.4 Relation to Winds from other Cool Stars

4.4.1 Coronal Winds of Cool Main Sequence Stars

Like the sun, other cool stars with surface temperatures T ∼< 10, 000 low enoughfor Hydrogen recombination are expected to have vigorous subsurface convectionzones that generate a strong upward flux of mechanical energy. For main-sequence

stars in which gravitational stratification implies a sharp, exponential drop insurface density over small scale height H ≪ R, this should lead to the same kindof thermal runaway as in the sun (§4.1; figure 7), generating a hot corona, andso also an associated, pressure-driven coronal wind expansion. As noted in §2.1,observations of X-ray emission and high-ionization UV lines do indeed providestrong evidence for such stellar coronae, but the low density of the very low massloss rate means there are few direct diagnostics of actual outflow in stellar coronalwinds.

4.4.2 Alfven Wave Pressure Driving of Cool Giant Winds

By contrast, cool giants do show much direct evidence for quite dense, cool winds,without any indication of a hot corona that could sustain a gas-pressure drivenexpansion. However, the mechanical energy flux density needed to drive suchwinds is actually quite comparable to what is inferred for the solar surface, FE ≈105 erg/cm2/s, which likely could again be readily supplied by their H-recombinationconvection zones. But a key difference now is that, owing to the much weaker grav-itational stratification, this need no longer lead to a thermal runaway.


In particular, consider again the general scaling of the ratio of the gravitationalscale height to stellar radius (cf. eqn. 3.5),

H

R=

2a2

v2e

=kTR

GµM= 4 × 10−4 (T/T⊙) (R/R⊙)

M/M⊙. (4.19)

In cool, solar-mass giant stars, the much larger radius can increase the surfacevalue of this ratio by a factor hundred or more over its (very small) solar value,i.e. to several percent. Moreover, at the > 1 MK coronal temperature of a thermalrunaway, the ratio would now be well above unity, instead of ca. 10% found forthe solar corona. This would formally put the critical/sonic radius below thestellar radius, representing a kind of free expansion of the coronal base, at asupersonic speed well above the surface escape speed, instead of critical transitionfrom subsonic to supersonic outflow. It seems unlikely thermal runaway could besustained in such an expanding vs. static medium.

But in lieu of runaway heating, it seems instead that the upward mechanicalenergy flux could readily provide the direct momentum addition to drive a rela-tively slow wind outflow against the much weaker stellar gravity, operating nowvia a gradient in the pressure associated with waves instead of the gas. In general,the mechanical stress or pressure exerted by waves scales, like gas pressure, withthe gas density ρ; but now this is multiplied by the square of a wave velocityamplitude δv, instead of the sound speed a, i.e. Pw = ρδv2.

Although the details of how such a wave pressure is generated and evolvesdepends on the specific type of wave, a general energetic requirement for anywaves to be effective in driving a wind outflow is that the fluctuating velocityδv must become comparable to the local gravitational escape speed somewherenear the stellar surface. This already effectively rules out wind driving by soundwaves, because to avoid strong damping as shocks, their velocity amplitude mustremain subsonic, δv < a; this imples Pw < Pgas, and thus that sound waves wouldbe even less effective than gas pressure in driving such cool winds (Koninx andPijpers 1992).

In this respect, the prospects are much more favorable for magnetohydrody-namic waves, particularly the Alfven mode. For the simple case of a radiallyoriented stellar magnetic field B = Br, the mean field can impart no outwardforce to propel a radial wind. But Alfven wave fluctuations δB transverse to themean field B give the field a wave pressure Pw = δB2/8π, for which any radial gra-dient dPw/dr does have an associated radial force. Moreover, since the fractionalamplitude f = δB/B of Alfven waves can be up to order unity, the associatedvelocity amplitude δv of these transverse fluctuations can now extend up to theAlfven speed,

vA ≡ B√4πρ

. (4.20)

Unlike for sound waves, this can readily become comparable to the stellar escapespeed ve, implying a ready potential for Alfven waves to drive a strong stellarwind.


In fact, for Alfven waves with a fractional amplitude f = δv/vA = δB/B,requiring δv = ve = v at a wind critical point near the stellar surface radius Rgives a direct estimate for the potential wind mass loss rate,

M ≈ δB2R2

ve= 1.2 × 10−7 M⊙

yr

δB2R5/2100

√

M/M⊙

, (4.21)

where R100 ≡ R/(100R⊙) and δB = fB is in Gauss. Indeed, note that for the sun,taking f = 0.1 and a mean field of one Gauss gives M = 1.2×10−14M⊙/yr, whichis somewhat fortuitously consistent with the mass flux measured by interplanetaryspacecraft (§2.1). More significantly, for giant stars with fluctuating fields δB =fB of a few Gauss, the scaling in eqn. (4.21) matches quite well their much higher,observationally inferred mass loss rates. For a wind with this mass loss scaling andterminal speed comparable to the surface escape v∞ ≈ ve, the energy flux densityrequired is

FE =Mv2

e

4πR2≈ 5 × 104 erg

cm2 sf2B2

√

M/M⊙

R100, (4.22)

which, as noted above, for typical cool-giant parameters is quite close to solarenergy flux density.

Actually, observations indicate such cool-giant winds typically have terminalspeeds of only 10-20 km/s, significantly below even their relatively low surfaceescape speeds ve ≈ 50 km/s, and implying that most all (∼ 90%) of the inputenergy goes toward lifting the wind material out of the star’s gravitational poten-tial. Explaining this together with the large mass loss rate demands a delicate finetuning in any wind model, for example that the Alfven waves are damped over alength scale very near a stellar radius (Hartmann and MacGregor 1980; Holzer,Fla and Leer 1983). For further details, the reader is referred to the very lucidreviews by Holzer (1987) and Holzer & MacGregor (1985).

4.4.3 Superwind Mass Loss from Asymptotic Giant Branch Stars

Stars with initial mass M < 8M⊙ are all thought to end up as white dwarfs withmass below the Chandrasekhar limit M < 1.4M⊙. This requires copius mass loss(M > 10−4M⊙/yr) to occur in a relatively brief (∼ 104 yr) “superwind” phase atthe end of a star’s evolution on the Asymptotic Giant Branch (AGB). It seemsunlikely that mere wave pressure generated in near-surface convection zones coulddrive such enourmous mass loss, and so instead, models have emphasized tappingthe more deep-seated driving associated with global stellar pulsations. The mosteffective seem to be the radial pressure or p-modes thought to be excited by theso-called “kappa” mechanism (Hansen, Kawaler and Trimble 2004) within interiorionizations zones (e.g. of Helium). Because of the much higher interior temper-ature and sound speed, such pulsations can have much higher velocity amplitudethan surface sound waves. Indeed, as the pulsations propagate upward throughthe declining density, their velocity amplitude can increase, effectively lifting gasparcels near the surface a substantial fraction of the stellar radius.


If the subsequent infall phase of these near-surface parcels meets with the nextoutward pulsation before returning to its previous mimimum radius, the net effectis a gradual lifting of material out of the much weakened gravitational potential.Moreover, as the gas cools through the expansion into ever larger radii and awayfrom the stellar photophere, dust can form, at which point the additional drivingassociated with scattering and absorption of the bright stellar radiation can propelthe coupled medium of gas and dust to full escape from the star. But such dust-driving is only possible when the pulsations are able to effectively “levitate” stellarmaterial in a much weakened gravity.

As reviewed by Willson (2000), models suggest the net mass loss can eventuallybecome so effective that the associated mass loss timescale τM = M/M becomesshorter than the timescale for interior evolution, representing then a kind of “deathline” at which the entire stellar envelope is effectively lost, leaving behind only thehot, dense, degenerate stellar core to become a white dwarf remnant. Overall, thisprocess seems arguably a largely interior evolution issue, more akin to a “criticaloverflow” (e.g., like the “Roche lobe” overflow in mass exchange binaries) than asteady, supersonic outflow from a fixed stellar surface. As such, the kind of simpli-fied steady-state wind formalism outlined here is perhaps of limited relevance. Seethe review by Willson (2000) and references therein for a more detailed discussionof this extreme final stage of cool-star mass loss.

4.5 Summary for the Solar Wind

• The solar wind is driven by the gas pressure gradient of the high-temperaturesolar corona.

• The hot corona is the natural consequence of heating from mechanical energygenerated in the solar convection zone. At low density radiative cooling can-not balance this heating, leading to a thermal runaway up to temperaturesin excess of a million degrees, at which inward thermal conduction back intothe atmosphere can again balance the heating.

• Outward thermal conduction can extend this high temperature well awayfrom the coronal base.

• For such a hot, extended corona, hydrostatic stratification would lead to anasymptotic pressure that exceeds the interstellar value, thus requiring a netoutward coronal expansion that becomes the supersonic solar wind.

• Because the wind mass flux is a sensitive function of the coronal temperature,the energy loss from wind expansion acts as an effective coronal thermostat.

• While the mass loss rate is thus set by energy deposition in the subsonicwind base, energy added to the supersonic region increases the flow speed.

• The high-speed wind streams that emanate from coronal holes require ex-tended energy deposition. Coronal-hole observations showing the proton


temperature significantly exceeds the electron temperature rule out electronheat conduction as the mechanism for providing the extended energy.

• Magnetic fields lead to extensive structure and variablity in the solar coronaand wind.

• Other cool main sequence stars likely also have hot, pressure-driven coronalwinds, but the dense, cool winds of cool giants could instead be driven bythe pressure from Alfven waves.

5 Radiatively Driven Winds from Hot, Massive Stars

Among the most massive stars – which tend also to be the hottest and mostluminous – stellar winds can be very strong, with important consequences for boththe star’s own evolution, as well as for the surrounding interstellar medium. Incontrast to the gas-pressure-driven solar wind, such hot-star winds are understoodto be driven by the pressure gradient of the star’s emitted radiation.

The sun is a relatively low-mass, cool star with a surface temperature about6000 K; but as discussed above, its wind arises from pressure-expansion of the veryhot, million-degree solar corona, which is somehow superheated by the mechanicalenergy generated from convection in the sun’s subsurface layers. By contrast, high-mass stars with much hotter surface temperatures (10,000-100,000 K) are thoughtto lack the strong convection zone needed to heat a circumstellar corona; theirstellar winds thus remain at temperatures comparable to the star’s surface, andso lack the very high gas-pressure needed to drive an outward expansion againstthe stellar gravity.

However, such hot stars have a quite high radiative flux, since by the Stefan-Boltzmann law this scales as the fourth power of the surface temperature. Becauselight carries momentum as well as energy, this radiative flux imparts a force to theatoms that scatter the light. At the level of the star’s atmosphere where this forceexceeds the inward force of the stellar gravity, material is accelerated upward andbecomes the stellar wind.

An important aspect of this radiative driving process is that it often stemsmostly from line scattering. As discussed in section 2.2, in a static medium suchline interactions are confined to radiation wavelengths within a narrow thermalwidth of line-center. However, in an accelerating stellar wind flow, the Dopplereffect shifts the resonance to increasingly longer wavelengths, allowing the linescattering to sweep gradually through a much broader portion of the stellar spec-trum (see figure 3). This gives the dynamics of such winds an intricate feedbackcharacter, in which the radiative driving force that accelerates the stellar winddepends itself on that acceleration.


5.1 Radiative Acceleration

5.1.1 Electron Scattering and the Eddington Limit

Before analyzing the radiative acceleration from line-scattering, let us first considerthe simpler case of scattering by free electrons, which is a “gray”, or frequency-independent, process. Since gray scattering cannot alter the star’s total luminosityL, the radiative energy flux at any radius r is simply given by F = L/4πr2,corresponding to a radiative momentum flux of F/c = L/4πr2c. As noted insection 2.2.1, the opacity for electron scattering is κe = 0.2(1 + X) = 0.34 cm2/g,where the latter result applies for the standard (solar) hydrogen mass fractionX = 0.72. The product of this opacity and the radiative momentum flux yieldsthe radiative acceleration (force-per-unit-mass) from free-electron scattering,

ge(r) =κeL

4πr2c. (5.1)

It is of interest to compare this with the star’s gravitational acceleration,GM/r2. Since both accelerations have the same 1/r2 dependence on radius, theirratio is spatially constant, fixed by the ratio of luminosity to mass,

Γe =κeL

4πGMc. (5.2)

This ratio, sometimes called the Eddington parameter, thus has a characteris-tic value for each star. For the sun it is very small, of order 2 × 10−5, but forhot, massive stars it can approach order unity. As noted by Eddington, electronscattering represents a basal radiative acceleration that effectively counteracts thestellar gravity. The limit Γe → 1 is known as the Eddington limit, for which thestar would become gravitationally unbound.

It is certainly significant that hot stars with strong stellar winds have Γe onlya factor two or so below this limit, since it suggests that only a modest additionalopacity could succeed in fully overcoming gravity to drive an outflow. But it isimportant to realize that a stellar wind represents the outer envelope outflow froma nearly static, gravitationally bound base, and as such is not consistent withan entire star exceeding the Eddington limit. Rather the key requirement for awind is that the driving force increase naturally from being smaller to larger thangravity at some radius near the stellar surface. The next subsection describes howthe force from line-scattering is ideally suited for just such a spatial modulation.

5.1.2 Driving by Doppler-Shifted Resonant Line Scattering

As discussed in section 2.2, the resonant strength and sensitive wavelength depen-

dence of bound-bound line opacity leads to strong, characteristic interactions ofstellar radiation with the expanding wind material, giving rise to distinct featuresin the emergent stellar spectrum, like asymmetric P-Cygni line-profiles, whichprovide key signatures of the wind expansion. These same basic properties – res-onance strength and Doppler-shift of the line-resonance in the expanding outflow


– also make the radiative force from line-scattering the key factor in driving mostmassive-star winds. Relative to free-electron scattering, the overall amplificationfactor for a broad-band, untuned radiation source is set by the quality of the res-onance, Q ≈ νo/A, where νo is the line frequency and A is decay rate of theexcited state. For quantum mechanically allowed atomic transitions, this can bevery large, of order 107. Thus, even though only a very small fraction (∼ 10−4)of electrons in a hot-star atmosphere are bound into atoms, illumination of theseatoms by an unattenuated (i.e., optically thin), broad-band radiation source wouldyield a collective line-force that exceeds that from free electrons by about a factorQ ≈ 107 × 10−4 = 1000 (Gayley 1995). For stars within a factor two of the free-electron Eddington limit, this implies that line-scattering is capable, in principle,of driving material outward with an acceleration on order a thousand times theinward acceleration of gravity!

In practice, of course, this does not normally occur, since any sufficiently largecollection of atoms scattering in this way would quickly block the limited fluxavailable within just the narrow frequency bands tuned to the lines. Indeed, in thestatic portion of the atmosphere, the flux is greatly reduced at the line frequencies.Such line “saturation” keeps the overall line force quite small, in fact well belowthe gravitational force, which thus allows the inner parts of the atmosphere toremain gravitationally bound.

This, however, is where the second key factor, the Doppler effect, comes intoplay. As illustrated in figure 3, in the outward-moving portions of the outer at-mosphere, the Doppler effect red-shifts the local line resonance, effectively desat-urating the lines by allowing the atoms to resonate with relatively unattenuatedstellar flux that was initially at slightly higher frequencies. By effectively sweepinga broader range of the stellar flux spectrum, this makes it possible for the line forceto overcome gravity and accelerate the very outflow it itself requires. As quantifiedwithin the CAK wind theory described below, the amount of mass accelerated ad-justs such that the self-absorption of the radiation reduces the overall line-drivingto being just somewhat (not a factor thousand) above what’s needed to overcomegravity.

5.1.3 Radiative Acceleration from a Single, Isolated Line

For weak lines that are optically thin in the wind (i.e. with a characteristic opticaldepth from eqn. (2.5), τo ≪ 1), the line-acceleration takes a form similar to thesimple electron scattering case (cf. eqn. 5.1),

gthin ≡ κvthνoLν

4πr2c2= wνo

q ge , (5.3)

where κ characterizes the opacity near line center, q ≡ κvth/κec is a dimensionlessmeasure of the frequency-integrated line-opacity (cf. section 2.2.2), and wνo

≡νoLν/L weights the placement of line within the luminosity spectrum Lν .

For stronger lines, the absorption and scattering of radiation can be treatedwithin the Sobolev approximation in terms of the optical thickness of the local


Sobolev resonance zone, as given in section 2.2 for non-radial rays integrated overthe finite-size star (see eqns. (2.7 - 2.9). In the further approximation of purelyradial streaming radiation from a point-like star, the radial acceleration takes theform,

gline ≈ gthin1 − e−τSob

τSob. (5.4)

wherein the reduction from the optically thin case depends only on the local radialSobolev optical depth given in eqn. (2.3), which can be rewritten here as

τSob = ρκlSob =ρκvth

dv/dr=

ρqκec

dv/dr≡ qt . (5.5)

In the final equality, t ≡ κeρc/(dv/dr) is the Sobolev optical depth for a line withintegrated strength equal to free electron scattering, i.e. with q = 1.

In the limit of a very optically thick line with τs ≫ 1, the acceleration becomes,

gthick ≈ gthin

τs= wνo

L

4πr2ρc2

dv

dr= wνo

L

Mc2vdv

dr, (5.6)

where the last equality uses the definition of the wind mass loss rate, M ≡ 4πρvr2.A key result here is that the optically thick line force is independent of the line-strength q, and instead varies in proportion to the velocity gradient dv/dr. Thebasis of this is illustrated by figure 3, which shows that the local rate at whichstellar radiation is red-shifted into a line-resonance depends on the slope of thevelocity. By Newton’s famous equation of motion, a force is normally understoodto cause an acceleration. The above shows that an optically thick line-force alsodepends on the wind’s advective rate of acceleration, v dv/dr.

5.1.4 Sobolev Localization of Line-Force Integrals for a Point Star

To provide a more quantitative illustration of this Sobolev approximation, letus now derive these key properties of line-driving through the localization of thespatial optical depth integral. Under the simplifying approximation that the stellarradiation flux is purely radial (as from a central point source), the force-per-unit-mass associated with direct absorption by a single line at a radius r is given by

gline(r) = gthin

∫ ∞

−∞

dx φ(x − u(r))e−τ(x,r). (5.7)

The integration is over a scaled frequency x ≡ (ν/νo − 1)(c/vth), defined from linecenter in units of the frequency broadening associated with the ion thermal speedvth, and u(r) ≡ v(r)/vth is the radial flow speed in thermal-speed units. Theintegrand is weighted by the line-profile function φ(x), which for thermal broad-

ening typically has the Gaussian form φ(x) ∼ e−x2

. The exponential reductiontakes account of absorption, as set by the frequency-dependent optical depth tothe stellar surface radius R,

τ(x, r) ≡∫ r

R

dr′κρ(r′)φ (x − u(r′)) . (5.8)


A crucial point in evaluating this integral is that in a supersonic wind, the variationof the integrand is dominated by the velocity variation within the line-profile.As noted above, this variation has a scale given by the Sobolev length lSob ≡vth/(dv/dr), which is smaller by a factor vth/v than the competing density/velocityscale, H ≡ |ρ/(dρ/dr)| ≈ v/(dv/dr). A key step in the Sobolev approximation isthus to recast this spatial integration as an integration over the comoving-framefrequency x′ ≡ x − u(r′),

τ(x, r) = −∫ x′(r)

x′(R)

dx′ vth

dv/dr′κρ(x′)φ(x′) ≈ τSobΦ(x − u(r)) , (5.9)

where the integrated profile

Φ(x) ≡∫ ∞

x

dx′φ(x′) , (5.10)

and the latter approximation in (5.9) uses the assumption that v(r) ≫ vth to for-mally extend the surface frequency x′(R) to infinity relative to the local resonancex′(r). As defined in eqn. (5.5), the Sobolev optical thickness τs = qt arises hereas a collection of spatial variables that are assumed to be nearly constant over theSobolev resonance zone, and thus can be extracted outside the integral.

Finally, a remarkable, extra bonus from this approximation is that the resultingoptical depth (5.9) now has precisely the form needed to allow analytic evaluationof the line-force integral (5.7), yielding directly the general Sobolev force expressiongiven in eqn. (5.4).

5.2 The CAK Model for Line-Driven Winds

5.2.1 The CAK Line-Ensemble Force

In practice, a large number of lines with a range of frequencies and strengths cancontribute to the wind driving. Castor, Abbott, and Klein (1975; hereafter CAK)introduced a practical formalism for computing the cumulative line-accelerationfrom a parameterized line-ensemble, under the simplifying assumption that thespectral distribution keeps the individual lines nearly independent. For the point-star model used above, direct summation of the individual line-accelerations asgiven by eqn. (5.4) yields

gtot = ge

∑

wνoq1 − e−qt

qt≈ ge

∫ ∞

0

qdN

dq

1 − e−qt

qtdq , (5.11)

where the latter equality approximates the discrete sum as a continuous integralover the flux-weighted number distribution dN/dq. Following CAK, a key furthersimplification is to approximate this number distribution as a simple power law inthe line strength q,

qdN

dq=

1

Γ(α)

(

q

Q

)α−1

, (5.12)


where Γ(α) is the complete Gamma function, the CAK power-law index satisifies0 < α < 1, and the above-mentioned cumulative line-strength Q provides a conve-nient overall normalization (cf. §4.2.2). Application of eqn. (5.12) in (5.11) thenyields the CAK, point-star, radiative acceleration from this line-ensemble,

gCAK =Q1−α

(1 − α)

ge

tα

=1

(1 − α)

κeLQ

4πr2c

(

dv/dr

ρcQκe

)α

. (5.13)

Note that this represents a kind of “geometric mean” between the optically thinand thick forms (eqns. 5.3 and 5.6) for a single line2.

5.2.2 CAK Dynamical Solution for Mass Loss Rate and Terminal Speed

The simple, local CAK/Sobolev expression (5.13) for the cumulative line-drivingforce in the point-star approximation provides a convenient basis for deriving thebasic scalings for a line-driven stellar wind. Since, as noted above, the gas pressureis not of much importance in the overall wind driving, let us just consider thesteady-state equation of motion (3.12) in the limit of zero sound-speed, a = 0.This simply requires that the wind acceleration must equal the line-accelerationminus the inward acceleration of gravity,

vdv

dr= gCAK − GM(1 − Γe)

r2, (5.14)

wherein we’ve also taken into account the effective reduction of gravity by thefree-electron scattering factor Γe. Note from eqn. (5.13) that the CAK line-forcegCAK depends itself on the flow acceleration it drives.

Since this feedback between line-driving and flow acceleration is moderated bygravity, let us define the gravitationally scaled inertial acceleration,

w′ ≡ r2vv′

GM(1 − Γe). (5.15)

In terms of an inverse-radius coordinate x ≡ 1−R/r, note that w′ = dw/dx, wherew ≡ v2/v2

esc(1 − Γe) represents the ratio of wind kinetic energy to the (electron-force-reduced) effective gravitational binding from the surface. Using eqn. (5.13),

2CAK dubbed the ratio gCAK/ge the “force multiplier”, written as M(t) = kt−α, with ka normalization constant. Note however that their original formulation used a fiducial thermalspeed vth in the definition of the optical depth parameter t = κeρvth/(dv/dr), which then givesan artificial thermal-speed dependence to the CAK normalization, k = (vth/c)α Q1−α/(1 − α).In the Sobolev approximation, the line-force has no physical dependence on the thermal-speed,and so the formal inclusion of a fiducial vth in the CAK parameterization has sometimes led toconfusion, e.g., in applying tabulations of the multiplier constant k. The formulation here avoidsthis problem, since the parameters q, Q, and t = κeρc/(dv/dr) are all independent of vth. Afurther advantage is that, for a wide range of hot-star parameters, the line normalization has arelatively constant value Q ≈ 2000Z, where Z is the metalicity relative to the standard solarvalue (Gayley 1995).


M > MCAK

M = MCAK

M < MCAK

Cw’α

w’

1+w’

1

O Solution

s

2 Soluti

ons

1 Solution

..

..

. .

Fig. 9. Graphical solution of the dimensionless equation of motion (5.16) representing a

1D, point-star, zero-sound-speed CAK wind, as controlled by the constant C ∼ 1/Mα.

If M is too big, there are no solutions; if M is small there are two solutions. A maximal

value M = MCAK defines a single, “critical” solution.

the equation of motion can then be rewritten in the simple, dimensionless form,

w′ = C w′α − 1, (5.16)

where the constant is given by

C =1

1 − α

(

QΓe

1 − Γe

)1−α (L

Mc2

)α

. (5.17)

Figure 9 illustrates the graphical solution of this dimensionless equation of mo-tion for various values of the constant C. For fixed stellar and opacity-distributionparameters, this corresponds to assuming various values of the mass loss rate M ,


since C ∝ 1/Mα. For high M (low C), there are no solutions, while for low M (highC), there are two solutions. The two limits are separated by a critical case withone solution – corresponding to the maximal mass loss rate – for which the functionCwα intersects the line 1 + w at a tangent. For this critical case, the tangency re-quirement implies αCcw

′α−1c = 1, which together with the original equation (5.16)

yields the critical conditions w′c = α/1 − α and Cc = 1/αα(1 − α)1−α.

From eqn. (5.17), this critical value of Cc defines the maximal CAK mass lossrate

MCAK =L

c2

α

1 − α

[

QΓe

1 − Γe

](1−α)/α

. (5.18)

Moreover, since eqn. (5.16) has no explicit spatial dependence, these critical con-ditions hold at all radii. Spatial integration of the critical acceleration w′

c fromthe surface radius R yields a specific case of the general “beta”-velocity-law,

v(r) = v∞

(

1 − R∗

r

)β

, (5.19)

where here β = 1/2, and the wind terminal speed v∞ = vesc

√

α(1 − Γe)/(1 − α).An important success of these CAK scaling laws is the theoretical rationale

they provide for an empirically observed “Wind-Momentum-Luminosity” (WML)relation (Kudritzki et al. 1995). Combining the CAK mass-loss law (5.18) togetherwith the scaling of the terminal speed with the effective escape yields a WMLrelation of the form,

Mv∞√

R∗ ∼ L1/α′

Q1/α′−1

(5.20)

which neglects a residual dependence on M(1 − Γe) that is generally very weakfor the usual case that α′ is near 2/3. Fits for galactic OB supergiants (Puls etal. 1996) give a luminosity slope consistent with α ≈ 0.6, with a normalizationconsistent with Q ≈ 103. Note that the direct dependence Q ∼ Z provides thescaling of the WML with metalicity Z.

5.3 Extensions of Idealized CAK Model

5.3.1 Finite-Size Star

These CAK results strictly apply only under the idealized assumption that thestellar radiation is radially streaming from a point-source. If one takes into ac-count the finite angular extent of the stellar disk, then near the stellar surface theradiative force is reduced by a factor fd∗ ≈ 1/(1 + α), leading to a reduced massloss rate (Friend & Abbott 1986; Pauldrach et al. 1986),

Mfd = f1/αd∗ MCAK =

MCAK

(1 + α)1/α≈ MCAK/2 . (5.21)

Away from the star, the correction factor increases back toward unity, which forthe reduced base mass flux implies a stronger, more extended acceleration, giv-ing a somewhat higher terminal speed, v∞ ≈ 3vesc, and a flatter velocity law,approximated by replacing the exponent in eqn. (5.19) by β ≈ 0.8.


5.3.2 Radial Variations in Ionization

The effect of a radial change in ionization can be approximately taken into accountby correcting the CAK force (5.13) by a factor of the form (ne/W )

δ, where ne is the

electron density, W ≡ 0.5(1 −√

1 − R∗/r) is the radiation “dilution factor”, andthe exponent has a typical value δ ≈ 0.1 (Abbott 1982). This factor introduces anadditional density dependence to that already implied by the optical depth factor1/tα given in eqn. (5.13). Its overall effect can be roughly taken into account withthe simple substition α → α′ ≡ α − δ in the power exponents of the CAK massloss scaling law (5.18). The general tendency is to moderately increase M , andaccordingly to somewhat decrease the wind speed.

5.3.3 Finite Gas Pressure and Sound Speed

The above scalings also ignore the finite gas pressure associated with a smallbut non-zero sound-speed a. Applying the full gas-pressure equation of motion(3.12) to the case with line-driving (gx = gCAK), then through a perturbationexpansion in the small parameter a/vesc, it is possible to derive simple scalings forthe fractional corrections to the mass loss rate and terminal speed (Owocki andud-Doula 2004),

δm ≈ 4√

1 − α

α

a

vesc; δv∞ ≈ −αδm

2(1 − α)≈ −2√

1 − α

a

vesc. (5.22)

For a typical case with α ≈ 2/3 and a ≈ 20 km/s ≈ vesc/30, the net effect is toincrease the mass loss rate and decrease the wind terminal speed, both by about10%.

5.4 Wind Instability and Variability

The steady, spherically symmetric, line-driven-wind models decribed above havehad considerable success in explaining the inferred general properties of OB-starwinds, like the total mass loss rate and mean velocity law. But when viewed morecarefully there is substantial evidence that such winds are actually quite highlystructured and variable on a broad range of spatial and temporal scales.

Large-scale variations give rise to Discrete Absorption Components (DACs)seen in UV spectra lines (Howarth and Prinja 1989). These begin as broad ab-sorption enhancement in the inner part of the blue absorption trough of an un-saturated P-Cygni line, which then gradually narrow as they drift, over a periodof days, toward the blue-edge of the profile. These DACs likely result from dis-turbances (e.g., localized magnetic fields or non-radial pulsations) at the photo-spheric base of the stellar wind, which through stellar rotation evolve in the windinto semi-regular “Co-rotating Interaction Regions” (CIRs: Mullan 1984; Cran-mer and Owocki 1996; much as detected directly in the solar wind at boundariesbetween fast and slow wind streams (see §2.1).

But there is also strong, though indirect evidence that OB winds have an ex-tensive, small-scale, turbulent structure. Saturated P-Cygni lines have extended


-2 2+ u0

gdir~+ u

gdir

-1 1

I*e-τ(x-u)

φ(x-u)

x-u

Fig. 10. Illustration of the physical origin of the line-driven instability. At an arbitrary

point in the expanding wind, it shows the line profile φ and direct intensity from the

star plotted vs. comoving frame frequency x − u. The light shaded overlap area is

proportional to the component line-acceleration gdir. The dashed profile shows the effect

of the Doppler shift from a perturbed velocity δu (in units of thermal speed vth), with

the resulting extra area in the overlap with the blue-edge intensity giving a perturbed

line-force δg that scales in proportion to this perturbed velocity.

black troughs thought to be a signature that the wind velocity is highly nonmono-tonic, and such stars commonly exhibit soft X-ray emission thought to originatefrom embedded wind shocks.

This small-scale flow structure likely results from the strong Line-Deshadowing

Instability (LDI) that is intrinsic to the radiative driving by line-scattering (Mac-Gregor et al. 1980; Owocki and Rybicki 1984, 1985). As noted above, there isa strong hidden potential in line-scattering to drive wind material with accelera-tions that greatly exceed the mean outward acceleration. For any small velocityperturbations with a length scale near or below the Sobolev length lSob, linear sta-bility analyses show that the perturbed radiative acceleration δg no longer scaleswith the velocity gradient δv′, as expected from the above Sobolev analysis, butrather in direct proportion to the perturbed velocity itself, δg ∼ δv (MacGregoret al. 1980; Owocki and Rybick 1984). The leads to a strong instabililty, in whichthe stronger acceleration leads to a faster flow, which in turns leads to an evenstronger acceleration (see figure 10).

The growth rate of this instability scales with the ratio of the mean drivingforce to the ion thermal speed, ωg ≈ gCAK/vth. Since the CAK line driving alsosets the flow acceleration, i.e. gCAK ≈ v dv/dr, the growth is approximately atthe flow rate through the Sobolev length, ωg ≈ v/lSob. This is a large factorv/vth ≈ 100 bigger than the typical wind expansion rate dv/dr ≈ v/R, implying


Height (R*)

log(

ρ)(g/cm3)

time(hr)

time(hr)

v(km/s)

v(km/s)

log(

ρ)(g/cm3)

Fig. 11. Results of 1-D, time-dependent numerical hydrodynamics simulation of the

nonlinear growth of the instability. The line plots show the spatial variation of velocity

(upper) and density (lower) at a fixed, arbitrary time snapshot. The corresponding grey

scales show both the time (vertical axis) and height (horizontal axis) evolution. The

dashed curve shows the corresponding smooth, steady CAK model. from the vertical in

intervals of 4000 sec from the CAK initial condition.

then that a small perturbation at the wind base would, within this lineary theory,be amplified by an enormous factor, of order ev/vth ≈ e100!

In practice, a kind of “line-drag” effect of the diffuse component of radiationscattered within the lines reduces this stability (Lucy 1984; Owocki and Rybicki1985), giving a net growth rate that scales roughly as

ωg(r) ≈gCAK

vth

µ∗(r)

1 + µ∗(r). (5.23)

where µ∗ ≡√

1 − R2/r2. This net growth rate vanishes near the stellar surface,where µ∗ = 0, but it approaches half the pure-absorption rate far from the star,where µ∗ → 1. This implies that the outer wind is still very unstable, withcumulative growth of ca. v∞/2vth ≈ 50 e-folds.

Because the instability occurs at spatial scales near and below the Sobolevlength, hydrodynamical simulations of its nonlinear evolution cannot use the Sobolev


approximation, but must instead carry out non-local integrals for both direct anddiffuse components of the line radiation transport (Owocki, Castor and Rybicki1988; Feldmeier 1995; Owocki and Puls 1996, 1999). The right panel of figure 11illustrates typical results of a 1-D time-dependent simulation. Because of the stabi-lization from line-drag, the flow near the photospheric wind base follows the steady,smooth CAK form (dashed curve), but beginning about r ≈ 1.5R, the strong in-strinsic instability lead to extensive wind structure, characterized by high-speedrarefactions in between slower dense, compressed shells.

Because of the computational expense of carrying out the nonlocal integrals forthe line-acceleration, 2-D simulations of this instability have so far been limited,either ignoring or using simplified approximations for the lateral components ofthe diffuse line-drag (Owocki 1999; Dessart and Owocki 2003, 2005). But resultsso far indicate the compressed shells of 1-D simulations should break up into small-scale clumps, with a characteristic size of a few percent of the local radius, and avolume filling factor fv ≈ 0.1. As noted in §2.2.1, this has important implicationsfor interpreting diagnostics that scale with the square of wind density, suggestingthat mass loss rates based on such diagnostics may need to be reduced by factorsof a few, i.e. 1/

√fv ≈

√10.

5.5 Effect of Rotation

The hot, luminous stars that give rise to line-driven stellar winds tend generallyto have quite rapid stellar rotation. This is most directly evident through the ex-tensive broadening of their photospheric absorption lines, which suggest projectedequatorial rotation speeds Vrot sin i of hundreds of km/s, where sin i ≤ 1 accountsfor the inclination angle i of the observer line-of-sight to the stellar rotation axis. Insome of the most rapid rotators, e.g. the Be stars, the surface rotation speed Vrot

is inferred to be a substantial fraction, perhaps 70-80% or more, of the so-calledcritical rotation speed, Vcrit ≡

√

2GM/R, for which material at the equatorialsurface would be in Keplerian orbit (Townsend et al. 2004).

Initial investigations (Friend & Abbott 1986; Pauldrach et al. 1986) of the effectof rotation on radiatively driven winds derived 1-D models based on the standardCAK line-driving formalism, but now adding the effect of a centrifugally reduced,effective surface gravity as a function of colatitude θ,

geff (θ) =GM

R2

(

1 − Ω2 sin2 θ)

, (5.24)

where Ω2 ≡ V 2rotR/GM , ignoring for simplicity any rotational distorton of the

surface radius R. This allows one to write the standard CAK mass loss ratescaling law (cf. eqn. 5.18) in terms of surface values of the mass flux m = ρv,radiative flux F , and effective gravity geff , relative to corresponding polar (θ = 0)values mo, Fo, and go = GM/R2,

m(θ)

mo=

[

F (θ)

Fo

]1/α [geff (θ)

go

]1−1/α

. (5.25)


c.b.a.0

90

180

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

Radius (R ) *

Co

-La

titu

de

Fig. 12. Contours of density in stellar wind from a star rotating at 75% of critical

rate, plotted vs. colatitude θ and radius r, spaced logarithmically with two contours per

decade. The superposed vectors represent the latitudinal velocity, with the maximum

length corresponding to a magnitude of vθ =100 km/s. The three panels show the cases

(a) without nonradial forces or gravity darkening, (b) with nonradial forces but no gravity

darkening, and (c) with both nonradial forces and gravity darkening.

Taking, as done initially, the surface to be constant in latitude, F (θ) = Fo, yields

m(θ)

mo=[

1 − Ω2 sin2 θ]1−1/α

; F (θ) = Fo . (5.26)

Since α < 1, the exponent 1−1/α is negative, implying that the mass flux increasesmonotonically from pole (sin θ → 0) toward the equator (sin θ → 1).

However, as first demonstrated by von Zeipel (1924), for a radiative stellar en-velope undergoing solid body rotation, the emergent radiative flux at any latitudeis predicted to vary in proportion to the centrifugally reduced effective gravity,F (θ) ∼ geff . Applying this equatorial “gravity darkening” for the surface fluxyields

m(θ)

mo= 1 − Ω2 sin2 θ ; F (θ) ∼ geff (θ) (5.27)

so that the mass flux now decreases towards the equator, with a maximum at thepole!

Recalling that the wind terminal speed tends to scale with the surface gravitythrough the escape speed, this 1-D analysis also predicts a latitudinally varyingwind speed that is proportional to a centrifugally reduced, effective escape speed,

v∞(θ) ∼ vesc

√

1 − Ω2 sin2 θ. (5.28)

The latitudinal variation of wind density is then obtained from ρ(θ) ∼ m(θ)/v∞(θ).More generally, the wind from a rotating star can also flow in latitude as

well as radius, requiring then a 2-D model. In this regard, a major conceptual


advance was the development of the elegantly simple “Wind Compressed Disk”(WCD) paradigm by Bjorkman & Cassinelli (1993). They noted that, like satel-lites launched into earth orbit, parcels of gas gradually driven radially outwardfrom a rapidly rotating star should remain in a tilted ‘orbital plane’ that bringsthem over the equator, where they collide to form compressed disk. Initial 2-D hydrodynamical simulations (Owocki, Cranmer, and Blondin 1994) generallyconfirmed the basic tenets of the WCD model (figure 12a), with certain detailedmodifications (e.g., infall of inner disk material). But later simulations (Owocki,Cranmer, and Gayley 1996) that account for a net poleward component of theline-driving force (figure 13) showed that this can effectively reverse the equator-ward drift, and so completely inhibit formation of any equatorial compressed disk(figure 12b). Indeed, when equatorial gravity darkening is taken into account, thelower mass flux from the equator makes the equatorial wind have a reduced, ratherthan enhanced, density (figure 12c).

The net upshot then is that a radiatvely driven wind from a rapidly rotatingstar is predicted to be both faster and denser over the poles, instead of the equator.This may help explain spectroscopic and interferometric evidence that the current-day wind of the extreme massive star η Carinae is faster and denser over the poles,leading to a prolate shape for its dense, optially thick ‘wind photosphere’ (Smithet al. 2003; van Boekel et al. 2003; Groh et al. 2010). Extensions of the rotationalscalings to the case of continuum-driven mass loss could also explain the bipolarshape of the Homunculus nebula (Owocki, Shaviv and Gayley 2004).

5.6 Summary for Radiatively Driven Massive-Star Winds

• The large ratio of luminosity to mass means OB stars are near the “Edding-ton limit”, for which the radiative force from just scattering by free electronsnearly cancels the stellar gravity.

• The resonance nature of line absorption by bound electrons makes theircumulative effective opacity of order a thousand times larger than free elec-trons, but the discrete energies of bound states means the opacity is tunedto very specific photon wavelengths.

• In a static atmosphere, saturation of line absorption keeps the associatedforce smaller than gravity; but in an accelerating wind, the associated Dopplershift of the line-frequency exposes it to fresh continuum radiation, allowingthe line-force to become strong enough to overcome gravity and drive theoutflow.

• Because lines have a thermally broadened velocity width (vth ∼< a) muchless than the wind flow speed (v ≈ vesc), the absorption or scattering ofphotons in such an accelerating wind occurs over a narrow resonance layer(of width lSob = vth/(dv/dr) ≈ (vth/vesc)R); following methods introducedby V. V. Sobolev, this allows nearly local solution of the line-transfer in suchaccelerating flow.


faster

pola r

wind

slower

equatorial

wind

rrN

(1) Stellar Oblateness =>poleward tilt in flux

(2) Pole-equator asymmetry in velocity gradient

Max[dv n/dn]

Flux

Fig. 13. Illustration of the origin of poleward component of the radiative force from a

rotating star. Left: The poleward tilt of the radiative flux arising from the oblateness

of the stellar surface contributes to a poleward component of the driving force. Right:

Since the wind speed scales with surface escape speed, the lower effective gravity of the

equator leads to a slower equatorial speed. The associated poleward increase in speed

leads to a poleward tilt in the velocity gradient, and this again contributes to a poleward

component of the line force.

• Within the CAK model for a power-law distribution of line strengths, thecumulative line-force scales with a power of the local velocity gradient dividedby the density.

• For the idealized case of negligible gas pressure and a radially streamingpoint-source of stellar radiation, application of this CAK line-force within asteady-state radial equation of motion yields analytic scalings for the massloss rate and wind velocity law.

• Accounting for the finite cone-angle of the stellar disk and radial ionizationbalance variations in the driving opacity yield order unity corrections to thepoint-star scalings. Corrections for a finite gas pressure are smaller, of order10%.

• Overall, the predictions of the CAK scalings agree well with an observation-ally inferred wind-momentum-luminosity relation for OB supergiants.

• Careful analysis indicates that hot-star winds have extensive structure andvariable, with small-scale turbulent structure likely arising from intrinsic in-stability of line-driving. The associated clumping has important implicationfor interpreting mass-loss rate diagnostics that scale with density squared.


• Rotation can also significantly affect the mass loss and wind speed, withthe gravity darkening from rapid rotation leading to a somewhat surprisingpolar enhancement of wind density and flow speed.

6 Wolf-Rayet Winds and Multi-line Scattering

The above CAK model is based on a simplified picture of absorption of the star’sradiative momentum by many independent lines, effectively ignoring overlap effectsamong optically thick lines. Since each such thick line sweeps out a fraction v∞/cof the star’s radiative momentum L/c, the ratio of wind to radiative momentumis

η ≡ Mv∞L/c

= Nthickv∞c

, (6.1)

which is sometimes termed the wind “momentum” or “performance” number. Toavoid overlap requires Nthick < c/v∞, which in turn implies the so-called “single-scattering” limit for the wind momentum, η < 1. Since most OB winds are inferredto be below this limit, the basic CAK model ignoring overlap still provides areasonably good model for explaining their overall properties.

However, such a single-scattering formalism seems quite inadequate for themuch stronger winds of Wolf-Rayet (WR) stars. Wolf-Rayet stars are evolved,massive, hot stars for which the cumulative mass loss has led to depletion of theoriginal hydrogen envelope. They typically show broad wind-emission lines ofelements like carbon, nitrogen, and/or oxygen that are the products of core nu-cleosynthesis. Overall, observations indicate that WR winds are especially strong,and even optically thick to continuum scattering by electrons. Notably, inferredWR wind momenta Mv∞ are generally substantially higher than for OB starsof comparable luminosity, placing them well above the OB-star line in the aboveWind-Momentum-Luminosity relation. In fact, in WR winds the inferred momen-tum numbers are typically well above the single-scattering value η = 1, sometimesas high as η = 10 − 50!

This last property has often been cast as posing a WR wind “momentumproblem”, sometimes with the implication that it means WR winds cannot beradiatively driven. In fact, it merely means that, unlike for OB stars, WR windscannot be treated in the standard, single-scattering formalism. However, momen-tum ratios above unity can, in principle, be achieved by multiple scattering betweenoverlapping thick lines with a velocity-unit frequency separation ∆v < v∞. Fig-ure 14 illustrates this for the simple case of two overlapping lines. But, as discussedbelow, the large momentum of WR winds requires much more extensive overlap,with thick lines spread densely throughout the spectrum without substantial gapsthat can allow radiation to leak out. An overall theme here is that understandingWR mass loss represents more of an “opacity” than a “momentum” problem.


λ

Radius r

λ =

Photon Wavelength in Co-Moving Frame

R*

λblue

red

λ*

WindVelocity

Fig. 14. The Doppler-shifted line-resonance in an accelerating flow, for two lines with

relative wavelength separation ∆λ/λ < v∞/c close enough to allow multi-line scattering

within the wind. Photons scattered by the bluer line are reemitted with nearly equal

probability in the forward or backward directions, but then further red-shifted by the

wind expansion into resonance with the redder line. Because of the gain in radial direction

between the line resonances, the red-line scattering imparts an additional component of

outward radial momentum.

6.1 Example of Multiple Momentum Deposition in a Static Gray Envelope

To provide the basis for understanding such multi-line scattering, it is helpfulfirst to review the momentum deposition for continuum scattering. The totalradial momentum imparted by radiation on a spherically symmetric circumstellarenvelope can be expressed in terms of the radiative force density ρgrad integratedover volume, outward from the wind base at radius R,

prad =

∫ ∞

R

4πr2ρgraddr, (6.2)


* * *

(a) (b) (c)

Fig. 15. Schematic photon trajectories in (a) hollow and (b) filled gray spheres, with

part (c) illustrating the net “winding” that makes even the filled case have a persistent

net outward push that implies multiple radial momentum deposition. See text for details.

where the radiative acceleration is given by a frequency integral of the opacity κν

over the stellar flux spectrum, Fν ,

grad =

∫ ∞

0

κνFν

cdν (6.3)

A particularly simple way to illustrate the requirements of multiple momentumdeposition is in terms of an envelope with a gray opacity κ. In this case, the fluxis just a constant in frequency, set at each radius by the bolometric luminositythrough F = L/4πr2. This implies grad = κL/4πr2c, and so yields

prad =L

c

∫ ∞

R

κρdr =L

cτ, (6.4)

where τ is the total wind optical depth. Thus the requirement for exceeding thesingle scattering limit for radiative momentum deposition, prad > L/c, is simplythat the envelope be optically thick, τ > 1.

Figure 15 provides a geometric illustration of how multiple momentum depo-sition occurs in an optically thick envelope. Figure 15a shows the case of a hollowshell with optical thickness τ = 5, wherein a photon is backscattered within thehollow sphere roughly τ times before escaping, having thus cumulatively impartedτ times the single photon momentum, as given by eqn. (6.4).

However, the same momentum deposition also occurs in a solid sphere withthe same total optical depth, even though, as shown in figure 15b, photons in thiscase undergo a much more localized diffusion without hemispheric crossing.

Figure 15c illustrates how these diffusive vs. direct-flight pictures of multi-ple momentum deposition can be reconciled by thinking in terms of an effective“winding”around the envelope. For each scattering within a spherical envelope theradial momentum deposition is unchanged by arbitrary rotations about a radiusthrough the scattering point. For figure 15c the rotations are chosen to bring all


the scattered trajectories into a single plane, with the azimuthal component alwaysof the same sense, here clockwise viewed from above the plane. In this artificialconstruction, scattering thus leads to a systematic (vs. random walk) drift of thephoton along one azimuthal direction, implying a cumulative winding trajectoryfor which the systematic outward push of the scattered photon is now apparent.

6.2 Multi-Line Transfer in an Expanding Wind

The multiple scattering by a dense ensemble of lines can actually be related to theabove gray envelope case, if the spectral distribution of lines is Poisson, and so isspread throughout the spectrum without extended bunches or gaps. As first notedby Friend and Castor (1983), and later expanded on by Gayley et al. (1995), awind driven by such an effectively gray line-distribution can be analyzed throughan extension of the above standard CAK formalism traditionally applied to themore moderate winds of OB stars. In the case of dense overlap, wherein opticallythick lines in the wind have a frequency separation characterized by a velocity ∆vmuch less than the wind terminal speed v∞, the mean-free-path between photoninteractions with separate lines is given by

1

ρκeff=

∆v

dvn/dn, (6.5)

where dvn/dn ≡ n · ∇(n · v) is the projected wind velocity gradient along a givenphoton direction n (see §4.2.6). The directional dependence of this velocity gradi-ent implies an inherent anisotropy to the associated effectively gray line-ensembleopacity κeff . However, a full non-isotropic diffusion analysis (Gayley et al. 1995)indicates that the overall wind dynamics is not too sensitive to this anisotropy.In particular, the total wind momentum can still be characterized by the effectiveradial optical depth, τeff

r , which in this case yields,

prad ≈ L

cτeffr =

L

c

∫ ∞

R

ρκeffr dr =

L

c

∫ ∞

R

1

∆v

dvr

drdr =

L

c

v∞∆v

, (6.6)

which implies τeffr = v∞/∆v. Neglecting a modest correction for gravitational

escape, global momentum balance requires prad ≈ Mv∞, thus implying

η ≈ v∞∆v

. (6.7)

For winds driven by a gray ensemble of lines, large momentum factors η ≫ 1simply require that there be a large number of optically thick lines overlappingwithin the wind, v∞ ≫ ∆v.

Note that eqn. (6.7) implies that the mass loss scales as

M ≈ L

c2

c

∆v, (6.8)

wherein c/∆v just represents the total, spectrum-integrated number of thick lines,Nthick. It is important to realize, however, that this number of thick lines is not


fixed a priori, but is itself dependent on wind properties like the mass loss rateand velocity law.

Self-consistent solution is again possible though through an extension of thestandard CAK formalism. The key is to account for the fact that radiation enteringinto resonance of each line does not in general come directly from the stellarcore, but instead has been previously scattered by the next blueward overlappingline. In the limit of strong overlap, the transfer between lines can be treatedas local diffusion, using however a non-isotropic diffusion coefficient to acccountfor the directional variation of the velocity gradient. In analogy to the finite-disk correction for point-star CAK model, one can then derive a “non-isotropicdiffusion” correction factor, fnid, to account for the diffuse angle distribution ofthe previously line-scattered radiation as it enters into resonance with then nextline. With this factor, one again finds that the mass loss follows a standard CAKscaling relation (5.18)

M = f1/αnid Mcak = f

1/αnid

L

c2

α

1 − α

[

QΓ

1 − Γ

](1−α)/α

. (6.9)

where f1/αnid is typically about one half. (See fig. 6 of Gayley et al. 1995.)

The wind velocity law in such multi-scattering models is found to have a some-what more extended acceleration than in standard finite-disk CAK models, withvelocity law indexes of roughly β = 1.5 − 2. The terminal speed is again scaleswith the effective escape speed, v∞ ≈ 3vesc.

This analysis shows that, within such “effectively gray” distribution of overlap-ping lines, multiline scattering can, in principle, yield momentum numbers η ≫ 1well in excess of the single-scattering limit. In this sense, there is thus no funda-mental “momentum” problem for understaneding WR winds.

6.3 Wind-Momentum-Luminosity Relation for WR Stars

As noted above (§4.3.5), this CAK mass-loss law together with the tendency forthe terminal speed v∞ to scale with the effective escape speed ve implies theWind-Momentum-Luminosity relation (5.20). For WR stars, such comparisons ofwind momentum vs. luminosity show a much greater scatter, but with momentaconsistently above those inferred for OB stars, typically reflecting more than afactor 10 higher mass loss rate for the same luminosity (Hamman et al. 1995).Assuming the same α ≈ 0.6 that characterizes OB winds, eqn. (5.20) suggeststhat WR winds must have a line opacity normalization Q that is more than afactor ≈ 10α/(1−α) ≈ 30 higher! Alternatively, this enhanced mass loss could alsobe obtained through a slightly lower α, representing a somewhat flatter numberdistribution in line opacity. For example, for stars with Eddington parameterΓ ≈ 1/2 and O-star value for Q ≈ 103, simply decreasing from α = 0.6 to α = 0.5

yields the required factor Q1/0.5−1/0.6 ≈ 10 increase in M .

While such a modest reduction in α may seem more plausible than a largeincrease in Q, it is generally not clear what basic properties of WR winds could


lead to either type of change in the line opacity distribution. In this context, it thusseems useful here to distinguish the classical momentum problem of achieving thelarge inferred WR momentum numbers, from an opacity problem of understandingthe underlying sources of the enhanced line opacity needed to drive the enhancedmass loss of WR winds. The wind-momentum vs. luminosity relation given ineqn. (5.20) yields a scaling of wind momentum numbers with η ∼ L1/α−1 ∼L2/3, implying that even OB winds should have large momentum numbers, evenexceeding the single scattering limit, for a sufficently large luminosity. Viewed inthis way, the fundamental distinguishing characteristic of WR winds is not so muchtheir large momentum number, but rather their enhanced mass loss compared toOB stars with similar luminosity. Identifying the sources of the enhanced opacityrequired to drive this enhanced mass loss thus represents a fundamental, unsolved“opacity problem”.

6.4 Cumulative Co-Moving-Frame Redshift from Multi-line Scattering

One can also view this multi-line scattering that occurs in a WR wind as a randomwalk over wind velocity. To achieve the rms target velocity v∞ in random incre-ments of ∆v requires stepping through (v∞/∆v)2 lines. Since the expected netredshift from each line interaction is of order ∆v/c, photons undergo a cumulativeredshift ∆E/E ≈ (v2

∞/∆v2)∆v/c over the course of their escape. The associatedradial momentum deposition factor is η ≈ (∆E/E)(c/v∞) ≈ v∞/∆v, as foundabove.

Figure 16 illustrates the comoving-frame redshift for a typical photon as itdiffuses in radius due to multi-line scattering. The photon track shown is acharacteristic result of a simple Monte Carlo calculation for a wind velocity lawv(r) = v∞(1 − R/r) and a constant line spacing ∆v = v∞/10. Each of the nodesshown represents scattering in one line. In this specific case, the photon escapesonly after interacting with nearly 90 lines, resulting in cumulative redshift of nearly9v∞/c. These are near the statistically expected values of (v∞/∆v)2 = 100 linescatterings resulting in a total redshift v2

∞/c∆v = 10v∞/c, as given by the aboverandom walk arguments.

This systematic photon redshift can also be related to the energy loss – orphoton “tiring” – that results from the work the radiation does to accelerate thewind to its terminal speed v∞. The ratio of wind kinetic energy to the radiativeenergy represents a “kinetic tiring number”

mkin =Mv2

∞

2L= η

v∞2c

. (6.10)

Since typically v∞/c < 0.01, for WR winds with momentum numbers of orderη ≈ 10, photon tiring is only a ∼5% effect. Although thus not of much significancefor either OB or WR winds, photon tiring does represent a fundamental limit tomass loss, with implications for understanding the giant eruptions from LuminousBlue Variable (LBV) stars.


1 2 3 4 5 6 7

2

4

6

8

10

Co

-Mo

vin

g R

edsh

ift

(v

un

its)

r/R*

∞

8

Fig. 16. Simulated photon redshift in the comoving frame as a photon executes a random

walk through the wind. The adopted line density is v∞/∆v = 10, which because of the

random walk character, implies the photon will interact with roughly 102 different lines

during escape. Since the comoving redshift between scatterings is v∞/10, this implies a

total comoving redshift of 10 v∞.

6.5 Role of Line Bunches, Gaps, and Core Thermalization

An essential complication for developing realistic models of WR winds stems fromthe inherently non-Poisson character of the spectral line distributions derived fromatomic databases of line lists. At any given wind radius, the dominant contribu-tion to the line opacity stems from a surprisingly small number of specific ion-ization stages of abundant heavy metals, primarly iron and iron-group elements.Moreover, for any given ion stage, the term structure is such that the lines arenotably “bunched” into relatively restricted ranges of wavelength. With just asmall number of distinct ion stages, the cumulative line spectrum thus likewiseexhibits extensive wavelength bunching. Within the gaps between these bunches,the radiation can propagate relatively unimpeded by line-scattering, thus repre-senting a potential preferential “leakage” that can significantly reduce the globalradiative momentum deposition.

To provide a physical perspective it is helpful to return to the above conceptthat the line-ensemble constitutes an effective continuum opacity, but now allowingthis to be frequency dependent to account for the relative bunches and gaps in thespectral distribution of lines. As seen from eqn. (6.3), the radiative acceleration


c. d.

a. b.

v

0

v 8

λ

v

0

v 8

Core Thermalization

Wind Redistribution

Fig. 17. Schematic diagram illustrating the role of line gaps, line bunches, and photon

thermalization for WR wind momentum deposition. The horizontal lines represent the

velocity/frequency spacing of optically thick lines in the wind. The four parts represent:

(a.) an effectively gray model (b.) a wind with fixed ionization and extensive gaps (c.) a

wind with ionization stratification that fills gaps (d.) the importance of limiting energy

redistribution in the wind.

for such a non-gray opacity depends on the spectral integral of the opacity timesthe frequency-dependent radiation flux. In general the flux spectrum at any givenlocation in an atmosphere or wind depends on self-consistent solution of a globalradiation transport problem, with overall characteristics depending critically onthe thermalization and frequency redistribution properties of the medium.

Figures 17a-d illustrate this key role of photon thermalization and redistribu-tion for simple non-gray line-distributions that are divided locally into two distinctspectral regions, representing either a line “bunch” or “gap”. Figure 17a first re-caps the effectively gray case in which the entire spectrum is covered by lines


at fixed velocity separation ∆v = v∞/10. Since this represents a total effectiveoptical depth τ = 10 for photons to escape from the surface (v = 0) to infinity(v = v∞), the global radiative momentum is prad = 10L/c, as follows from eqn.(6.4).

Figure 17b represents the case when lines have the same concentration ∆v =v∞/10 in half the spectrum (the bunch), with the other half completely line-free(the gap). Photons blocked in the bunched region are then rethermalized in thestellar core, and so tend to escape though the gap. Simple statistical argumentsshow that only about 0.5/(1 + τ/2) = 1/12 of the total flux now makes it outthrough the half of the spectrum covered by the bunch, implying from eqn. (6.4)that the total radiative momentum is now only prad/(L/c) ≈ 10×1/12+0×11/12 =5/6 < 1! This roughly represents the circumstance applicable to OB star winds,which have radially constant ionization, and so a radially fixed line-spectrum. Forsuch winds, any significant spectral gaps keep the momentum number to near thesingle scattering limit prad ≈ L/c, even if there is very extensive line overlap inspectral line bunches.

Figure 17c represents the case wherein there are again τ = 10 lines covering thefull spectrum, but now over a limited spatial range, occuring very close to the starin one spectral region, and very far away in the other. In the region where line-blocking occurs near the star, core thermalization again channels photons to theother spectral region. But in this second spectral region, the blocking occurs faraway from the star, with greatly reduced probability of backscattering to thermalredistribution in the stellar core. Again assuming purely coherent line-scatteringin the wind, this means the flux is nearly independent of the outer wind blocking.Since all the stellar flux must diffuse through a layer somewhere with τ = 10 lines,the global momentum deposition is now again, as in case a, simply prad = 10L/c.However, note that the flux distribution is still similar to case b, i.e. in proportion1/12 and 11/12 to the spectral regions corresponding respectively to the inner andouter wind blocking. This thus implies the same ratio for the relative depositionof radiative momentum. Overall this example, intended to represent the case ofan optically thick WR wind with ionization stratification, does indeed illustratehow it is possible to get a large global momentum deposition even when the line-opacity-spectrum is locally divided into gaps and bunches. However, this radiativemomentum tends to be deposited more in the outer wind, leaving a net deficit indriving needed to initiate the outflow in the lower wind. This may play a factorin inducing the inferred structure and variability of WR winds.

The final example in figure 17d illustrates the crucial importance of the assump-tion that the radiative transfer within the wind itself is through pure, coherentscattering. If there is significant thermalization or any other type of spectral en-ergy redistribution within the wind itself, then radiation will always tend to bechanneled into local gaps, thus again limiting the momentum deposition to a levelroughly characterized by the single scattering limit.


6.6 Summary for Wolf-Rayet Winds

• The classical “momentum problem” (to explain the large inferred ratio ofwind to radiative momentum, η≡ Mv∞/(L/c) ≫ 1) is in principle readilysolved through multiple scattering of radiation by an opacity that is suffi-ciently “gray” in its spectral distribution. In this case, one simply obtainsη ≈ τ , where τ is the wind optical depth.

• Lines with a Poisson spectral distribution yield an “effectively gray” cumula-tive opacity, with multi-line scattering occuring when the velocity separationbetween thick lines ∆v is less than the wind terminal speed v∞. In this case,one obtains η ≈ v∞/∆v.

• However, realistic line lists are not gray, and leakage through gaps in the linespectral distribution tends to limit the effective scattering to η ∼< 1.

• In WR winds, ionization stratification helps spread line-bunches and so fillin gaps, allowing for more effective global trapping of radiation, and thusη > 1.

• However, photon thermalization can reduce the local effectiveness of line-driving near the stellar core, making it difficult for radiation alone to initiatethe wind.

• The relative complexity of WR wind initiation may be associated with theextensive turbulent structure inferred from observed variabililty in WR windemission lines.

• Overall, the understanding of WR winds is perhaps best viewed as an “opac-ity problem”, i.e. identifying the enhanced opacity that can adequately blockthe radiation flux throughout the wind, and thus drive a WR mass loss thatis much greater than from OB stars of comparable luminosity.

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