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

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4. Evolution in time

Both the Newtonian and relativistic envelopes require fine tuning of the luminosity in order to produce static extended envelopes, because [FORMULA] depends sensitively on the luminosity. In the Schwarzschild case, unless the base of the envelope knows to shine locally at [FORMULA] above the local critical luminosity, and the luminosity is restricted to a narrow regime, it appears difficult to maintain the type of envelopes discussed in Sect.  2. The dependence on luminosity is strong; a fraction of a percent change in base luminosity could increase the steady-state radius from [FORMULA] to [FORMULA]  cm. Since the energy source (the accreting torus) is probably not extremely well-coupled to the envelope, it is unlikely that the base luminosity is so finely tuned. Therefore, we consider how the photosphere will change with time for different base luminosities.

We identify three cases. If the base luminosity is sub-critical, the envelope is not much bigger than the accreting torus. When the luminosity is super-critical, the envelope must expand. If the super-criticality is less than [FORMULA], the expansion is modest and the envelope may remain in hydrostatic and thermal equilibrium. If the base luminosity is super-critical by a significant factor, e.g. 2, then the envelope expands hydrostatically on a thermal time scale, until it expands so much that the dynamical time scale becomes shorter than the thermal time scale. After that, the outer parts of the envelope are no longer in hydrostatic equilibrium, but this does not imply that they are instantly lost. The envelope may continue to expand, perhaps even with a structure similar to the hydrostatic solution, until the scale height of atmosphere becomes comparable to the radius (as discussed in Sect.  2) at which point a wind will set in.

We now examine the expansion of the envelope. The relevant time scales are the dynamical time scale, the thermal time scale, and the radiation time scale, over which time all matter would be accreted. The first two time scales are shown in Fig. 1 as a function of photospheric radius.

The dynamical time scale grows with photospheric radius:


The thermal time scale, defined as the time scale to unbind the envelope assuming the energy in the envelope increases at a rate of [FORMULA] (e.g. if the base luminosity were [FORMULA] and the surface radiated at [FORMULA]). The total energy of the hydrostatic envelope is [FORMULA] where the internal energy is


Note that the energy is strongly dependent on the location of the inner boundary, [FORMULA]. The gravitational energy, [FORMULA], so that the thermal time scale is


with a complex dependence on photospheric radius and black hole mass. The thermal time scale depends both on the net energy injection rate and on the base radius of the envelope. As both parameters depend on the unknown physics of the torus, and as such, their exact values are uncertain. We believe the base radius to be near the tidal radius [FORMULA]  cm, where rotational support likely becomes important. Similarly, thick tori may produce luminosities up to a few times the Eddington limit, but the exact value cannot be predicted. These parameters enter into the thermal time scale multiplicatively, and together may lengthen or shorten the thermal time scale by a factor of 2 or 3.

Lastly, the radiation time is the time to radiate, at the Eddington limit, all of the accretion energy of the tidal debris:


for an accretion efficiency of [FORMULA] and an envelope mass of [FORMULA]. This time scale does not depend on the photospheric radius, and is generally longer than the other relevant time scales of interest (for a [FORMULA] black hole).

As discussed above, if the luminosity at the base is strongly super-critical, then the envelope will expand and the photosphere will move to larger radii (see Fig. 2). For an envelope with its photosphere below [FORMULA]  cm, the dynamical time is shorter than the thermal time, so the envelope should expand in hydrostatic equilibrium. When the photosphere grows larger than [FORMULA]  cm, the envelope can no longer be in hydrostatic equilibrium. However, the material at small radii has a much shorter dynamical time, so may be able to adjust to a near-equilibrium state quickly. The envelope could continue to expand, perhaps with the interior in a near equilibrium state and the outer parts further from equilibrium. If the photospheric radius reaches [FORMULA]  cm, the thermal time scale is so short and the pressure scale height becomes so large compared to the radius, that it seems most likely either a strong wind would form and carry the excess energy away, or the entire envelope would be unbound.

[FIGURE] Fig. 2. Radius and temperature for sequences of expanding envelopes labeled with three different energy input rates (each envelope radiates at the Eddington limit) and a base radius of [FORMULA]  cm. The solid lines are the phase in which the envelopes are in hydrostatic equilibrium. The dashed lines signify that the envelope is not in hydrostatic equilibrium, but for the purposes of this illustration, we assume that the models do stay in hydrostatic equilibrium and calculate the time between models at [FORMULA], where [FORMULA] is the energy difference between models. These calculations are described in more detail in Sect.  4.

Fig. 2 illustrates possible evolutionary sequences for different values of the base luminosity. For Fig. 2, we make the assumption that the envelope is in a hydrostatic state (described by eqn.  10), even after the thermal time scale falls below the dynamical time scale. The time to move between envelopes [FORMULA], where [FORMULA] is the energy difference between sequential envelopes (A similar method was applied by Ulmer (1998b) to investigate the evolution of thick accretion disks.) Evolutionary sequences show the evolution of both photospheric radius and temperature with time. The curves become nearly vertical as the envelopes evolve to large radii, because the binding energy becomes so small that they can be unbound by very little energy input.

It seems likely that the envelope will expand on a time scale of months to years, depending on the base radius and the base luminosity, to [FORMULA]  cm where [FORMULA]  K. Subsequently, the envelope will expand, but will not be able to maintain hydrostatic equilibrium throughout.

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© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998