## 3. Stability analysisThe ambiguity of the shock locations and was first removed by Chakrabarti & Molteni (1993) for adiabatic flows around a non-rotating black hole in the context of Paczynski & Wiita (1980) potential. By checking the variation of the total pressure of the post-shock flow as the shock is perturbed, they argued that the shock at is stable for accretion, and unstable for winds, but did not give a definite conclusion about the shock at . More recently, Yang & Kafatos (1995) made a fully relativistic analysis on the stability of shocks in isothermal accretion flows onto a Schwarzschild black hole, and found that the shock at is unstable, while the shock at is stable except when is very close to the point at which . Here we follow Yang & Kafatos' (1995) method, extending to the case of adiabatic accretion and wind flows in Kerr geometry. Across the shock, the momentum flux density, defined as (different from only by a constant factor) is conserved, resulting exactly in Eq.(2e). If due to some perturbation, the shock location is moved from to , the momentum flux density may not be in balance, the resulting difference across the shock is The stability of the shock depends on the sign of
If , for accretion flows
(i.e. along decreasing The present case is mathematically more complex than that studied
by Yang & Kafatos (1995), so that it is not possible to obtain an
analytic criterion of the sign of , and the only
way to estimate is that through numerical
calculations. It is seen from Eqs (5) and (2) that to evaluate
, the values of and
are needed, which can be solved out by
combining the differential form of Eqs (2a) and (2c), i.e.
and ; similarly, the
values of and , needed
to evaluate , can be solved out by combining
the differential form of Eqs (2b) and (2d), i.e.
and . In this way, we
have performed calculations for the Schwarzschild black hole and the
Kerr hole, as well as for prograde flows and retrograde ones, and have
been able to reach the following conclusion: © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |