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

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5. Discussion

In this paper, we have studied theoretically the nature of Rankine-Hugoniot shocks in thin, axisymmetric accretion and wind flows in Kerr geometry. We have presented the energy-angular momentum parameter space for which the flow may develop shocks. The ambiguity in regard of the shock location is removed by stability analysis: the stable shock location is practically unique. We have also shown that the frame-dragging effect of a rapid Kerr black hole causes shocks in prograde flows to locate closer to the hole, and to be stronger than those in retrograde flows as well as those in the Schwarzschild hole case. These fully relativistic results agree qualitatively, thus confirm those obtained in terms of a pseudo-Newtonian black hole model, e.g. Chakrabarti & Molteni (1993) for a non-rotating black hole, and Sponholz & Molteni (1994) for a rotating one, further suggesting that shocks are likely to occur in astrophysical disks and winds.

We now make some comments on the assumptions for the flow made in this paper, i.e. that the flow has a conical shape, and that the flow motion is adiabatic. Certainly, none of these is strictly valid in realistic situations: at the shock the flow is likely to expand in the vertical direction, and to loss energy into its surroundings, and a sophisticated theory of the shock process taking all these effects into account is definitely awaited. However, from the existing literature it seems that the major conclusion made about the shock in the flow ignoring these detailed effects is still meaningful. Fukue (1987) constructed the first example of the standing shock in the adiabatic, conical accretion flow. The above mentioned works of Chakrabarti & Molteni (1993) and Sponholz & Molteni (1994) studied, both analytically and through numerical simulation, the nature of Rankine-Hugoniot shocks in a flow with constant thickness. The assumption that the flow has a constant thickness should not be virtually different from that the flow keeps a conical shape, since in both cases the flow has a fixed rigid surface and the vertical equilibrium in the flow is not assumed. On the other hand, the Rankine-Hugoniot shock in a flow whose shape is not fixed but the vertical equilibrium holds has been studied by, e.g. Chakrabarti (1989), and the relevant result is similar with that obtained using the other model, i.e. the flow has a fixed shape and no vertical equilibrium. In fact, as Chakrabarti (1992) showed, if the vertical motion is ignored, then there is virtually no difference in the transonic properties of axisymmetric, polytropic disks whether they are of constant thickness, or of conical shape, or in vertical equilibrium. Extending further, when the assumption of adiabatic flow is replaced by that of isothermal flow, as done by Yang & Kafatos (1995), the major conclusion reached accordingly, i.e. the multiplicity of the formal shock location and the uniqueness of the stable shock location, remains qualitatively the same as that in the adiabatic case. Altogether, the efforts of all the above authors and others seem to have indicated that shocks can be common in inviscid flows, independent of the flow's shape, and whether the flow motion is adiabatic or it is isothermal.

One factor that affects seriously the shock formation is viscosity. As Chakrabarti & Molteni (1995) showed for isothermal accretion flows, viscosity is repulsive to shocks in the sense that a large enough viscosity can cease shocks, and that the flow remains subsonic throughout the accretion disk and becomes supersonic only at the inner sonic point located very close to the black hole; for flows with a small viscosity shocks can still form, but they are weaker and located farther away as compared with those in inviscid flows. These results are sound in the view that the viscous process is to convert gravitational potential energy to thermal energy of the accreting matter continuously and gradually, thus reducing kinetic energy (converted from gravitational energy too) that an inviscid flow could gather, and weakening the shock, or even ceasing it if the viscosity is large. We believe that viscosity has similar effects on the flow studied in the present paper, then our conclusions made here should hold for a small viscosity, but not for a large one.

Recently Narayan & Yi (1994) constructed self-similar, advection-dominated accretion flows in which most of the viscously dissipated energy is carried in by the accreting gas as entropy rather than being radiated. Shock-included accretion flows studied in the present paper have similar properties of advection: the entropy at the inner sonic point is higher than that at the outer sonic point, this means that the flow is advecting all the entropy generated at the shock without radiating it, and all the energy is also advected along with the flow downstream into the black hole. It is also interesting to note that our flows all have the relativistic Bernoulli constant [FORMULA], identical with the positive value of the non-relativistic Bernoulli constant for flows of Narayan & Yi (1994). Accordingly, our flows are expected to occur in physical situations similar to that for Narayan & Yi(1994) flows, i.e. when the optical depth is either very small (the radiative cooling is unable to keep up with the viscous energy generation), or very large (the cooling time of the disk is longer than the accretion time). Astrophysically, both the Narayan & Yi flows and ours have very sub-Eddington luminosities. However this does not necessarily mean that these flows are difficult to detect observationally, as adequate accretion rates may result in detectable luminosities even for a very low efficiency. For this reason Abramowicz & Lasota (1995) described advectively dominated disks as secret guzzlers, of which several astrophysical objects have been regarded as candidates (see references in Abramowicz & Lasota 1995).

Apart from these similarities, the self-similar solution of Narayan & Yi (1994)has a limitation that it cannot be applied to the inner region of the disk due to the restrictions of the boundary, and it is in the inner region where the accretion flow eventually becomes supersonic. More precisely, as Abramowicz & Lasota (1995) analyzed, it is the difficult enterprise of solving the problem of transonic flows that masked for a long time the real physical issues of advection dominated flows. However, a complete and self-consistent theory for the black hole accretion flow must take into account the flow's transonic nature, as done very recently by Igumenshchev, Chen & Abramowicz (1996) who incorporated both the advective cooling and the transonic motion in the flow. To sum up, there have been two kinds of advection-dominated solutions for black hole accretion, in both of which the flow passes the inner sonic point near the black hole. The difference between them is: in the Narayan & Yi solution the dissipation of gravitational energy to thermal energy in the accretion disk is by some continuous viscous processes, and the flow remains subsonic throughout the disk except in the innermost region, while in the solution of the present paper the dissipation is via shocks, the additional outer sonic point in the flow is required. A detailed comparison between the two kinds of solutions will be a subject of our future work.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998