16.4: Extended Atmospheres and the Formation of Stellar Winds

So far we have limited the subject of this book to the interiors and atmospheres of normal stars. However, we cannot leave the subject of stellar astrophysics without offering some comments about the transition between the outer layers of the stars and the interstellar medium that surrounds the stars. This is the domain where all of the simplifying assumptions, which made the description the inner regions of the star possible, fail. As one moves outward from the photosphere, through the chromosphere and beyond, he or she will encounter regions that involve some of the most difficult physics in astrophysics. Here the regions become sufficiently large that the plane-parallel assumption which so simplified radiative transfer is no longer valid, for the transition region may extend for many stellar radii. The density of matter is so low that LTE can no longer be applied to any elements of the gas. Because of the low density, the radiative interactions, while remaining crucial to the understanding of the structure of the medium, receive competition from previously neglected modes of energy transport. From observation, we know that turbulence is present in most stars. In some instances, the mechanical energy of the turbulent motion can be systematically transmitted to the outer layers of the star in amounts that compete with the energy from the radiation field. For some stars, the coupling of the magnetic field of the underlying star may provide a mechanism for the transmission of the rotational energy and momentum of the star to the surrounding low-density plasma.

The influx of energy from all sources seems to be sufficient to cause much of the surrounding plasma to be driven into the interstellar medium. Thus, we must deal with a medium in which hydrostatic equilibrium no longer applies. Continuous absorption is no longer of great significance so that the primary coupling of the radiation field to the gas is through the bound-bound transitions that produce the line spectra of stars. However, as we have seen, the frequency dependence of the line absorption coefficient is quite large, so Doppler shifts supplied by the mass motions of the gas will yield an absorption coefficient that is strongly dependent on the location of the gas. Thus the coupling of the radiation field is further complicated.

The region surrounding the star where these processes take place is generally known as the extended atmosphere of the star while the material that is driven away is referred to as the stellar wind. It is likely that all stars possess such winds, but the magnitude of the mass loss produced by them may vary by 8 powers of 10 or more. The primary processes responsible for their origin differ greatly with the type of star. For example, the wind from the sun originates from the inability of the hot corona to come into equilibrium with the interstellar medium, so that the outer layers "boil" away. The details of the solar wind are complicated by coupling of the rotational energy of the sun to the corona through the solar magnetic field. Thus the outer regions of the solar atmosphere are accelerated outward at an average mass loss rate of the order of 10^-14M_⊙ per year.

The heating of the corona is likely to result from coupling with the convective envelope of the sun. Thus corona-like winds are unlikely to originate in the early-type stars that have radiative envelopes. For the hotter stars on the left-hand side of the H-R diagram, the acceleration mechanism is most likely radiative in origin. The coupling of the turbulent motions of the photosphere to the outer regions of the atmosphere is so poorly understood that its role is not at all clear. However, in the case of some giants and supergiants, this coupling may well be the dominant source of energy driving the stellar wind.

In the face of these added difficulties, it would be both beyond the scope of this book and presumptuous to attempt a thorough discussion of this region. For that the reader is directed to the supplemental reading at the end of the chapter. We will merely outline some approaches to some aspects of these problems that have yielded a small measure of understanding of this transition region between the star and the surrounding interstellar space.

a. Interaction of the Radiation Field with the Stellar Wind

For those stars where the stellar wind results from the star's own radiation, not only is the wind driven by the radiation, but also the motion of the material influences the radiative transport. Thus one has a highly nonlinear problem involving the formidable physics of hydrodynamics and radiative transfer. This interplay between the two is the source of the major problems in the theory of radiatively driven stellar winds.

Since it is clear that we must abandon most of the assumptions in describing this region, we will have to replace them with others. Thus I shall assume that these outer regions are spherically symmetric. Although this is clearly not the case for many stars, an understanding of the problems posed by a spherically symmetric star will serve as a foundation for dealing with the more difficult problems of distorted stars. Since we have already dealt with the problems of spherical radiative transport (see Section 10.4) and departures from LTE (see Chapter 15), we concentrate on the difficulties introduced by the velocity field of the outer atmosphere and wind.

One of the earliest indications that some stars were undergoing mass loss came from the observation of the line profiles of the star P-Cygni. The Balmer lines (and others) in the spectrum of this star showed strong emission shifted slightly to the red of the stellar rest wavelength and modest absorption centered slightly to the blue side of the rest wavelength. This was recognized as the composite line profile arising from an expanding envelope with the emission being formed in a relatively transparent out- flowing spherical envelope. The "blue" absorption arose from that material that was moving, more or less, toward the observer and was seen silhouetted against the hotter underlying star (see Figure 16.6). Such line profiles are generally known as P-Cygni profiles and are seen during novae outbursts and in many other types of stars.

To understand the formation of such profiles, one must deal with the radiative transport of spectral line radiation in a moving medium. If the envelope surrounding the star cannot be assumed to be optically thin to line radiation, then one is faced with a formidable radiative transfer problem. In 1958, V. V. Sobolev²¹ realized that the velocity gradient in such an expanding envelope could actually simplify the problem. If the gradient is sufficiently large, then the relative Doppler shifts implied by that gradient could effectively decouple line interaction in one part of the envelope from that in another part. Since the outflow velocities often reach several thousand kilometers per second, the region of the envelope where photons are not Doppler-shifted by more than the line width can actually be quite thin. The contribution to the observed line profile at any given frequency therefore arises from emission (or absorption) occurring on surfaces of constant radial (i.e., line-of-sight) velocity. The extent of that contribution will then depend on the probability that a photon emitted on one of these surfaces can escape to the observer.

Figure 16.6 depicts the expanding envelope of a star and the regions that give rise to the corresponding P-Cygni profile.

Consider an envelope which is optically thick to line radiation surrounding a stellar core of radius \(\mathrm{r_c}\) whose continuum radiation passes freely through the envelope. Let us further assume that the line is such that complete redistribution is an appropriate choice for the redistribution function. Finally, we define \(\beta\) to be the probability that an emitted photon will escape from the envelope. Clearly such a probability will depend on the line width and the velocity gradient. We can then write the local value for the average mean intensity in the line as \[\bar{J}=(1-\beta) S(r)+\beta_c I_c\label{16.3.1}\]

where \((1-\beta) S(r)\) is the contribution to the local radiation field from those photons created in the line that did not escape and \(\beta_\mathrm{c}\mathrm{I_c}\) represents those continuum photons that have penetrated to that part of the envelope.

An estimate of \(\beta\) and \(\beta_\mathrm{c}\) will then provide us with a relationship between \(\overline{\mathrm{J}}\) and S which we can substitute into equations for the non-LTE source function developed in Chapter 15 [see equations (15.2.19) and (15.2.25)]. The mean escape probabilities \(\beta\) and \(\beta_\mathrm{c}\) will depend on local conditions only if the velocity gradient is sufficiently large that the conditions determining the probabilities can be represented by the local values of temperature and density.

Castor²⁰ has developed this formalism for a two-level atom and it is nicely described in Mihalas²¹. Basically, the problem becomes one of geometry. Contributions to the local mean intensity will come largely from surfaces within the envelope which exhibit a relative Doppler shift that is less than the Doppler width of the line. Castor finds this to be \[\beta=\frac{1}{2} \int_{-1}^{+1} \frac{1-e^{-\tau_0(r) /\left[1+\mu^2 \Phi(\mathrm{v})\right]}}{\tau_0(r) /\left[1+\mu^2 \Phi(\mathrm{v})\right]} d \mu\label{16.3.2}\]

where \[\mathrm{v}=\frac{v(r)}{v_{\mathrm{th}}}\label{16.3.3}\]

is simply the velocity measured in units of the thermal velocity characterizing the line width and \[\Phi(\mathrm{v})=\frac{d \ln \mathrm{v}}{d \ln r}-1\label{16.3.4}\]

which occurs from expanding the effective optical path in a power series of the velocity field and retaining only the first-order term. The parameter \(\tau_0\) is effectively the optical depth in the line along a radius from the star and is defined by \[\tau_0 \equiv \frac{\kappa_{\text {line }}(r) \rho(r)}{\mathrm{v}(r) / r}\label{16.3.5}\]

This quantity is essentially the line extinction coefficient multiplied by some effective length that is obtained by dividing by the dimensionless velocity gradient.

We may approximate \(\beta_\mathrm{c}\) by the following argument. Since \(\beta_\mathrm{c}\) is essentially an escape probability of a continuum photon leaving the envelope by traveling in any direction, it is essentially the same as \(\beta\) except for the probability of the photon's hitting the stellar core. That probability is given simply by the fraction of \(4\pi\) occupied by the core itself and is commonly called the dilution factor since it is the amount by which the local radiation density is diluted as one recedes from the star. The dilution factor \(W\) is therefore \[W \equiv \frac{1}{2}\left[1-\left(1-\frac{r_c^2}{r^2}\right)^{1 / 2}\right] \approx \frac{1}{4}\left(\frac{r_c}{r}\right)^2 \quad r \gg r_c\label{16.3.6}\]

so that \[\beta_c \approx W \beta\label{16.3.7}\]

We now have \(\beta\) and \(\beta_\mathrm{c}\) defined in terms of the velocity field, position, and parameters of the gaseous envelope that determine the local effective optical depth in the line \(\tau_0\). Remembering that \[\bar{J} \equiv \int_0^{\infty} \phi(\nu) J_\nu d \nu\label{16.3.8}\]

we may substitute into either of the non-LTE source functions given by equation (15.2.19) or (15.2.25) and obtain the source function for the line. Once the line source function is known, the solution for the equation of transfer (see Section 10.4 for spherical geometry) can be solved and the line profile obtained.

Solution of this problem also yields the radiant energy deposited in the envelope as a result of the line interactions. By doing this for a number of lines, the local effects of radiation on the envelope can be determined. This is one of the primary factors in determining the dynamics of the envelope.

b. Flow of Radiation and the Stellar Wind

Most progress in understanding stellar winds has been made with the radiation-driven winds of the early-type stars. While coronal winds of the later-type stars are important, particularly for slowing down the rotational velocity of these stars by magnetic breaking, the uncertainties concerning the formation of their coronal source force us to leave their discussion another to elucidate. What is presently known about them is nicely summarized by Mihalas²¹ (pp.521-540) and Cassinelli²². Instead, we concentrate on some aspects of the large winds encountered in the early-type stars.

In Chapter 6 [equation (6.5.2)] we showed that if the radiation pressure became sufficiently high, a star would become unstable. While many stars on the upper main sequence approach this limit, they still fall short. Yet these same stars exhibit significant mass loss in the form of a substantial stellar wind. In 1970, Leon Lucy and Philip Solomon²³ showed that, under certain circumstances, the coupling of the radiation field to the matter in the upper atmosphere through the strong resonance lines of certain elements could provide sufficient momentum transport to the envelope to drive it away from the star. This idea was developed more fully by Castor, Abbot, and Klein²⁴ to provide the foundations for the theory of radiatively driven stellar winds. Castor, Abbot, and Klein found that by including the effects of the large number of weaker lines present in material of the high atmosphere, accelerations 100 times greater than those found by Lucy and Solomon would result. This implied that mass loss from radiatively driven winds could be expected in cooler stars with higher surface gravities than the supergiants considered by Lucy and Solomon. Thus, stellar winds should be a ubiquitous phenomenon throughout the early-type stars.

The basic approach to describing any phenomenon of this type is to write down the conservation laws that must apply. To make the description as simple as possible, we assume that the flow is steady. Again, we assume spherical symmetry and remind the reader that the relevant conservation laws all have their origin in the Boltzmann transport equation developed in Chapter 1. Mass will be continuously lost by the wind, but for a steady flow, the mass flux through a spherical shell of radius r will be constant and given by \[\frac{d M}{d t}=4 \pi r^2 \rho \nu=\mathrm{const}\label{16.3.9}\]

The conservation of momentum requires \[\rho v \frac{d v}{d r}+\frac{d P}{d r}+\rho \frac{G M}{r^2}-\rho g_r=0\label{16.3.10}\]

where \(g_\mathrm{r}\) is the radiative acceleration. The conservation of energy and thermodynamics can be expressed by \[v \frac{d \mathrm{U}}{d r}-\frac{P v}{\rho^2} \frac{d \rho}{d r}=\frac{\nabla \cdot \vec{F}_t}{\rho}=\frac{Q_A+Q_R+Q_c}{\rho}\label{16.3.11}\]

where \(\mathbf{U}\) is the internal energy of the gas and \(\vec{\mathrm{F}}_\mathrm{t}\) is the total energy flux whose divergence can be described in terms of the energies Q_A, Q_R, and Q_C deposited locally in the gas by acoustical, radiative, and conductive processes respectively. Since both the radiative and gravitational accelerations vary as r^-2, it is common to introduce the parameter \[\Gamma=\frac{g_r}{g}\label{16.3.12}\]

If the wind is assumed to be radiatively driven, then Q_A = 0. Similarly the material density is sufficiently low that Q_C = 0. If the total energy in the flow is significantly less than the stellar luminosity, we may assume that radiative equilibrium may be locally applied, so that \[Q_R=\int_0^{\infty} 4 \pi \kappa_\nu\left(J_\nu-B_\nu\right) d \nu \approx 0\label{16.3.13}\]

which greatly simplifies the energy equation. The energy and momentum equations can now be combined to yield \[\frac{d}{d r}\left(\frac{v^2}{2}+\mathbf{U}+\frac{P}{\rho}-\frac{G M}{r}\right)=g_r\label{16.3.14}\]

which, after the total energy of the gas is defined to be \[E \equiv \frac{v^2}{2}+\mathbf{U}+\frac{P}{\rho}-\frac{G M}{r}\label{16.3.15}\]

can be integrated to give \[E=E_0+(\dot{M})^{-1} \int_0^{\infty} 4 \pi r^2 v \rho g_r d r\label{16.3.16}\]

Thus even if the total energy of the gas near the star is negative, continual radiative acceleration can introduce sufficient energy to drive the energy positive and thus allow it to escape.

As an example, consider the case where the gas in the wind is isothermal so that we can introduce an equation of state of the form \[P=c_s^2 \rho\label{16.3.17}\]

where the speed of sound \(\mathrm{c_s}\) is constant. We can then differentiate equation 16.3.9 and eliminate dP/dr from equation (16.3.10) to get \[\frac{r}{v} \frac{d v}{d r}=\frac{2 c_s^2-G M(1-\Gamma) / r}{v^2-c_s^2}\label{16.3.18}\]

For supersonic flow to occur, the numerator of the right hand side of equation (16.3.18) must vanish at the point the speed of sound is reached (that is, \(\mathrm{v=c_s}\)). This places significant constraints on the value that \(\Gamma\) must have at the transonic point. Cassinelli²² shows that this requires additional radiative heating of the gas as it approaches the transonic point. He suggests that scattering by electrons and resonance lines can do the work required to meet the conditions at the transonic point. Beyond the transonic point, further acceleration is most likely driven by the line opacity.

Since it has been shown that it is possible to drive stellar winds radiatively for stars whose luminosity is well below the Eddington luminosity, it remains for a complete and fully consistent model to be made. The problem here is largely numerical for the basic physical constraints are understood. To be sure these problems are formidable. Correct representation of the radiative acceleration requires calculation of the effects due to the myriads of weak spectral lines that populate the ultraviolet region of the spectrum where these stars radiate most of their energy. The energy balance equations must be solved to properly incorporate the effects of heating near the transonic point and to determine the temperature structure of the flow. Then the hydrodynamic flow equations will have to be solved numerically, incorporating an appropriate equation of state. Of course, all this must be done in non-LTE so that the proper ionization and excitation structure can be obtained. Undoubtedly the Castor, Abbot, and Klein models have pointed the correct way to understanding radiatively driven winds, but much remains to be done before the models can be used as a probe of the physics of the outer layers of stars.

This brief look at the problems of extended atmospheres and radiatively driven winds points out that the transition region between the normal stellar atmosphere and the interstellar medium is a difficult region to understand. A closer look at the chromospheres and coronas of other stars would have shown the same, or possibly more difficult, problems. Certainly the description of the transition zone for nonspherical stars will include more difficulties and require greater cleverness on the part of astrophysicists to understand them. Certainly this will be necessary if we are to understand the evolution of close binary stars in sufficient detail to understand their ultimate fate and the manner in which they influence the interstellar medium. Without that understanding, the evolution of the galaxy will be difficult, if not impossible, to understand. And without a reliable knowledge of galactic evolution can we ever hope to delineate the evolution, of the universe as a whole?