Astron. Astrophys. 333, 379-384 (1998)

2. Lack of closure relations

The envelope density profile derived in LU97 (Eq. 3) results in a logarithmically diverging mass, so an artificial cutoff of the atmosphere was taken in LU97 (Eq. 8). A related problem is that the pressure close to the photosphere is not handled in a self-consistent manner. In this section, we address these problems and discuss their implications.

We operate in a Newtonian potential, assume that the envelope is radiative, and link the envelope model to the atmosphere using the Eddington approximation in which the surface temperature, , where is the effective temperature. At the photospheric radius, , where the optical depth is 2/3, . The Eddington approximation requires that the atmosphere be thin so that the surface is not much larger than . The hydrostatic equation is:

where M, is the mass of the black hole alone. We neglect the envelope mass since . The equation of radiative transfer is

where the luminosity, L, and absorption coefficient, , are assumed to be constant throughout the envelope. Dividing the previous two equations yields the familiar result:

where is the Eddington luminosity. The right side of Eq.  6is constant, because the mass is dominated by the mass of the black hole, the luminosity is wholly supplied by the accretion disk at the base of the envelope, and is dominated by electron scattering. Integrating, we find

where the surface pressure, , is, in the Eddington approximation, equal to . This surface term was omitted in LU97.

Because pressure goes as the fourth power of temperature, throughout most of the envelope the surface pressure term in Eq.  7is unimportant, and

where is a constant. It then follows from the equation of state of the gas () and the radiation () that everywhere in the envelope except at low optical depth,

This relation was assumed in LU97.

It is now possible to solve the hydrostatic equation (Eq. 4) for temperature or, equivalently, density by using the approximate relation between P and (Eq.  9):

where is a constant of integration which was neglected in LU97. The constant is determined by the requirement that at the photospheric radius, , .

Following LU97, we find a relationship between the envelope mass and . The envelope mass is found by integrating Eq.  11from the base radius,

to (The exact location of base radius is an unknown; it is the poorly understood interface between the disk and envelope. The base radius, as argued in LU97, should be close to the tidal radius, because that is where the dynamical effects of angular momentum are expected to become important.)

An envelope which contains no more than a solar mass must be extremely close to the Eddington limit with . Eq.  15shows that the photospheric radius is exponentially sensitive to , suggesting that very small changes in base luminosity, which is controlled by a poorly understood accretion in the disk, may create large changes in the photospheric radius. For example, in this Newtonian approximation, the steady state radius would have to change from to  cm when changes from to . This result illustrates the extreme fine tuning (to three parts in ) which was required in LU97 to reach the steady state solution. As we describe in Sect.  3, the fine tuning required in the Schwarzschild case is less, but the level is above the local critical luminosity.

We now estimate the maximum radius at which the Eddington approximation is valid. The approximation requires that the atmosphere be thin, or equivalently, that the gas pressure scale height be much less than the radius at the photosphere. This limit is also a physical dividing line, because for further extended atmospheres, one expects high mass loss as the atmosphere is nearly isothermal at , so the escape velocity falls faster than the thermal velocity.

The gas pressure scale height at the photosphere can be evaluated using Eqs.  4and 8as:

At , the gas pressure scale height must be much less than the photospheric radius:

where we have written in terms of the envelope mass (Eq.  15). Note that result is independent of the black hole mass. The effective temperature therefore has the same scaling as in LU97:

These limiting results are surprisingly close to those of LU97 who found and . The difference, of course, is that our results show that smaller photospheric radii could exist. The exact photospheric radius cannot be determined without knowledge of the inner luminosity source.

An additional way to see the wind constraint on the radius is to consider in Eq.  10, which enforces the condition that at the photosphere (which radiates at the Eddington limit). The constant is zero when , and becomes negative for larger values of . Negative values of give (see Eq.  10) regions at the top of the envelope which approach an isothermal state and which would likely drive winds.

Even before the radius expands so far that the pressure scale height becomes comparable with the radius, winds could begin to play an important role-either in altering the envelope structure or in removing much the envelope mass. An estimate of the importance of the winds may be found by connecting, at the photospheric radius, isothermal wind solutions to the envelope solutions. Because of the extremely low densities ( g/cm³), the envelope is ionized and the cross-section is largely provided by electron-scattering, so an isothermal wind powered by the continuum cross-section (rather than line transitions), may be most appropriate. In this case, the following equation describes the outflow (e.g. Kudritzki, 1988):

where is the sound speed (in our case, isothermal so that and the sonic point, . Dimensionless radius, , and dimensionless velocity, , allow Eq.  21to be neatly integrated to obtain:

Connecting such outflows to the Eddington envelopes allows for determination of mass loss as a function of envelope photospheric radius. Generally, the time for such a wind to significantly reduce the envelope mass is at least 1000 years-many times longer than the other relevant time scales for the problem (results are shown in Fig. 1). For photospheric radii larger than  cm, the sonic point occurs below the photosphere, showing, in agreement with the limiting radius found above, that the hydrostatic solution (Eqs.  10 and  11) is no longer valid.

Fig. 1. Time scales for envelopes as a function of photospheric radius: dynamical time, thermal time, and time for wind driven mass loss. The thermal time scale is calculated assuming a energy input of and energy output of and  cm. The envelopes can stay in hydrostatic equilibrium only as long as . The mass loss to winds produced by an isothermal outflow are small enough that they do not appear to affect the system.

© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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