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

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3. Effects in the Schwarzschild metric

The Schwarzschild geometry has a significant impact on envelopes which radiate near the Eddington limit. At first appearance, the Schwarzschild geometry seems to be an unimportant correction to the problem, because the base of the photosphere is at [FORMULA], and redshift effects are of order 2%, [FORMULA]. However, because the envelopes are very close to the Eddington limit, as is required to produce an extended photosphere, the envelope structure is very sensitive to [FORMULA]. In particular for envelopes of [FORMULA], [FORMULA] is [FORMULA], so a 2% reduction in luminosity will significantly alter the envelope structure.

We quote below a number of results which were obtained in the study of x-ray bursts and envelopes around neutron stars (Paczynski & Anderson 1986; Paczynski & Prószynski 1986). For our purposes, the most important feature of the relativistic stellar structure equations is that the local critical luminosity does not scale with radius in the same manner as the local luminosity with the consequence that the structure is convective rather than radiative. Specifically, the local luminosity and critical luminosity determined by the local gravitational forces (equivalent to the Eddington luminosity at large r) are

[EQUATION]

where [FORMULA] is the luminosity as measured far from the black hole. Because of the different scalings, a near critical luminosity at the surface requires a super-critical luminosity at the base. As a consequence the star is convective. Alternatively, the convective nature can be seen by comparing the radiative and adiabatic gradients:

[EQUATION]

Even for a near critical surface luminosity, [FORMULA] quickly becomes larger than [FORMULA] as the coordinate, r, decreases.

As a consequence, the envelope must be convective. Whether or not the convection is efficient is difficult to determine because in radiation dominated regimes, convection is not fully understood, although progress is being made (e.g., Arons 1992). If convection is efficient, then by definition, [FORMULA] is nearly a constant. The equation of the temperature gradient yields:

[EQUATION]

so [FORMULA] is very closely a constant (to [FORMULA] part in [FORMULA]), and we recover the polytropic equation of state [FORMULA], which was used in the previous section. Assuming that convection is efficient, we can determine the relationship between the photospheric radius, the envelope mass, and the surface luminosity in the same manner as in Sect.  2. Ignoring the relativistic terms in the hydrostatic stellar structure equations, which is a good approximation (to [FORMULA]) because the terms enter multiplicatively rather than as differences, we recover eqn.  15. In contrast to the Newtonian case, in the Schwarzschild case, the luminosity at the base must be locally super-critical in order to support an extended envelope with a near-Eddington surface luminosity.

If the luminosity is [FORMULA] super-critical, and the envelope is able to expand, then photospheric radius is still sensitive to the base luminosity, but not exponentially sensitive as is the case for a Newtonian atmosphere. The luminosity at the photosphere is nearly equal (to better than one part in [FORMULA]) to the critical luminosity (Eqs.  24), so

[EQUATION]

Using Eq.  23to write the luminosity at the base, [FORMULA] as a function of photospheric radius, yields the result:

[EQUATION]

In steady state the photospheric radius would increase from [FORMULA] to [FORMULA]  cm with a [FORMULA] change in base luminosity.

There may be additional effects which compete with the relativistic effects. For neutron stars, the temperatures at the base of the envelope are so high that relativistic corrections to the Thompson cross section become important (e.g. Paczynski & Anderson 1986), but the temperatures around the tidal disruption created envelopes never reach such high temperatures. In our case, slight rotation may serve to reduce the critical luminosity at small radii, in the plane of rotation. Along the rotation axis, the envelope would likely be convective for the reasons described above.

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© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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