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Astron. Astrophys. 321, 665-671 (1997)

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3. Stability analysis

The ambiguity of the shock locations [FORMULA] and [FORMULA] 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 [FORMULA] is stable for accretion, and unstable for winds, but did not give a definite conclusion about the shock at [FORMULA]. 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 [FORMULA] is unstable, while the shock at [FORMULA] is stable except when [FORMULA] is very close to the point at which [FORMULA]. 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 [FORMULA] only by a constant factor)


is conserved, resulting exactly in Eq.(2e). If due to some perturbation, the shock location is moved from [FORMULA] to [FORMULA], 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 [FORMULA] If [FORMULA], for accretion flows (i.e. along decreasing r), when [FORMULA] (or [FORMULA]), the momentum flux just after the shock is larger (or smaller) than that just before the shock, so the shock should be put to shift towards further increasing (or further decreasing) r, thus the shock is unstable; but for wind flows (i.e. along increasing r) the shock is stable, because the imbalance of the momentum flux due to [FORMULA] would always cause the shock to move back towards its unperturbed location. On the contrary, [FORMULA] implies that the shock in accretion flows is stable, and the shock in winds is unstable.

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 [FORMULA], and the only way to estimate [FORMULA] is that through numerical calculations. It is seen from Eqs (5) and (2) that to evaluate [FORMULA], the values of [FORMULA] and [FORMULA] are needed, which can be solved out by combining the differential form of Eqs (2a) and (2c), i.e. [FORMULA] and [FORMULA] ; similarly, the values of [FORMULA] and [FORMULA], needed to evaluate [FORMULA], can be solved out by combining the differential form of Eqs (2b) and (2d), i.e. [FORMULA] and [FORMULA]. 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: for accretion flows the shock at [FORMULA] is always stable, while the shock at [FORMULA] is unstable except when the value of l is very close to that of [FORMULA] ; for wind flows the shock at [FORMULA] is always unstable, while the shock at [FORMULA] is stable except when l is very close to [FORMULA] . It is seen from Fig. 2 that when l is close to [FORMULA] the values of [FORMULA] and [FORMULA] become close to each other. To show more precisely the exceptional case when l is very close to [FORMULA], we give a numerical example: for the black hole's specific angular momentum [FORMULA] and prograde accretion flows (cf. Fig. 1b), when [FORMULA] the corresponding [FORMULA] is slightly smaller than 2.0754, in the numerical calculation when the parameter l is taken to be 2.0754 both the shocks at [FORMULA] and [FORMULA] are stable (in fact [FORMULA] and [FORMULA] are so close to each other now, so that it is practically not possible to distinguish them), but when [FORMULA] the shock at [FORMULA] becomes unstable, and the value 2.08 is still not the lower limit of l to which the resulting [FORMULA] keeps being an unstable shock location.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998