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Astron. Astrophys. 364, 901-910 (2000)

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1. Introduction

Accretion onto neutron stars and black holes gives the main energy supply in galactic X-ray sources. If the angular momentum of the matter provided by the massive companion star is sufficiently large, accretion occurs through a disk formed by the centrifugal forces. In this case, there exist two substantially different mechanisms which allow accreting matter to loose its angular momentum in the vicinity of the inner boundary of the accretion disk and fall onto the star. One of them is based on the classical turbulent viscosity model by Shakura (1972) and Shakura & Sunyaev (1973). Another mechanism is due to Sawada et al. (1986) who found out that spiral shocks originating inside the accretion disk are able to diminish the angular momentum of the matter to such an extent that the centrifugal forces cannot prevent its accretion on the star.

Another limit in the variety of accretion flows relates to accretion from a high-speed wind. If this wind is supersonic, a bow shock is forming near the black hole or near the stellar magnetosphere. Depending on the velocity of the incoming flow at infinity and on the cooling phenomena in the vicinity of the accretor, various accretion patterns can be obtained (see three-dimensional calculations by Ishii et al. 1993; Ruffert 1994). The existence of a steady-state solution for the wind accretion flow is the subject of current discussions. Numerous unsteady patterns resulting in the formation of transient accretion disks (Shima et al. 1998) coexist with the self-similar (Bisnovatyi-Kogan et al. 1979) and numerical Newtonian (Pogorelov et al. 2000) and relativistic (Font & Ibañez 1998a , 1998b; Font et al. 1999) steady-state solutions.

Under high wind velocity conditions, the angular momentum of infalling matter is sometimes not sufficient for the accretion disk formation at distances of the Alfvén radius where the magnetic pressure of the star is in balance with the dynamic pressure of the gas, and approximately spherical free-fall onto the magnetosphere takes place, see the discussion by Illarionov & Sunyaev (1975), Cassen & Pettibone (1976), Shapiro & Lightman (1976), Arons & Lea (1980). This occurs in some binary X-ray sources with a massive companion star where long-periodic pulsars are observed (Nagase 1989; Börner et al. 1987; Bisnovatyi-Kogan 1991; Anzer & Börner 1995).

Without angular momentum, the flow pattern reduces to a purely spherically-symmetric one (Bondi 1952; Ruffert & Arnett 1994). Self-similar spherically-symmetric accretion in the gravitational field of a point mass was thoroughly investigated by Kazhdan & Murzina (1994) for the case of an adiabatic flow with a power-law density distribution far from the accreting center. They performed the analysis of various situations, starting from hydrostatic equilibrium to an arbitrary Mach number at infinity. One of the important results obtained by these authors is that the accretion pattern strongly depends on the boundary conditions at the center, namely, on the accretion rate at [FORMULA]. The solutions can be subdivided into two groups. One group contains continuous, supersonic infall solutions (the Bondi solution belongs to this class). The other one contains various types of solutions with a shock wave expanding as [FORMULA]. These solutions for conditions fixed at infinity are characterized by the arbitrary position of the shock wave at a given time. This position can be determined uniquely only if we specify the nature of the infall near the center. This means that the shock wave is formed due to the presence of imaginary impermeable or partially impermeable surfaces at the origin. They decelerate the falling gas to subsonic velocities.

The paper of Kazhdan & Murzina (1994) gives us the guideline to what we might expect if the flow is not self-similar. In this case we must specify conditions both at the outer ([FORMULA]) and at the inner ([FORMULA]) boundary. Let the flow be supersonic at [FORMULA]. In this case we must specify all quantities on this spherical surface. If [FORMULA], p, and U are dimensionless density, pressure, and radial velocity in terms of [FORMULA], [FORMULA], and [FORMULA], respectively, the conservation relations for a spherically-symmetric shockless flow of perfect polytropic gas are




Here we introduced the Mach number [FORMULA], [FORMULA], and [FORMULA] ([FORMULA] is the polytropic index, M is the mass of the star, and G is the gravitation constant). Steady shockless solutions do not exist for arbitrary conditions on the outer boundary, even if the inner boundary allows free penetration.

For example, if the inner boundary ensures the maximum accretion rate ([FORMULA] attainable for the shockless flow which is at rest at infinity (the case of hydrostatic equilibrium), then, according to Bondi (1952), it has the sonic point at [FORMULA] with


The conservation relations (1)-(3) at this point give


If we want to preset some supersonic flow at [FORMULA], it is necessary to assume that


One can hardly expect any shockless supersonic accretion pattern originating from hydrostatic equilibrium at infinity if the total enthalpy lies without the interval (6).

If, for some reason, shockless accretion is not realized, then according to the self-similar solution by Kazhdan & Murzina (1994) a shock wave will originate at the inner boundary and propagate outward until it reaches the external boundary. Note that calculations with a subsonic inflow at [FORMULA] are numerically much more difficult.

The situation becomes even more complicated if the inner boundary as a whole or some of its parts are not completely permeable. The problem in this case is at least two-dimensional and simple mathematical manipulations are no longer possible. For this reason we have to perform systematic numerical parametric studies. Bisnovatyi-Kogan & Pogorelov (1997) investigated the accretion of slowly rotating matter onto a gravitating center emphasizing the possibility of steady-state accretion patterns. For all quantities fixed at [FORMULA], the dependence on the dimensionless parameter [FORMULA] was studied. Here [FORMULA] is the angular velocity which could be attained at the distance [FORMULA] from the axis of rotation under the assumption of constant angular momentum distribution. The cases corresponding to [FORMULA] were considered.

Chen et al. (1997) dwelt on the accretion patterns with expanding shock waves. Those authors also performed calculations for [FORMULA]. Igumenshchev et al. (1993) presented their results on the quasispherical accretion of matter onto a relativistic object with anisotropic X-ray luminosity and obtained the phenomenon of a hot gas outflow. Toropin et al. (1999) have recently calculated a quasi-spherical Bondi-like accretion onto a magnetic dipole. The study was performed using resistive magnetohydrodynamic (MHD) simulations in the axisymmetric case. The matter was assumed to have no angular momentum and the main results implied no rotation of the star. The authors specified supersonic Bondi solution at [FORMULA]. Note that this solution refers to a rather large accretion rate at [FORMULA] and can be expected only at the early stages of accretion on the magnetic field of the star. This seems to be the reason why they obtained only one type of the accretion pattern which possesses an outward moving shock wave. Note also that the initial stage of the time-dependent accretion flow corresponds to small radiative cooling (large values of [FORMULA]), which later becomes more efficient for optically thin plasma.

The model of accretion implemented numerically by Toropin et al. (1999) and in our paper corresponds to the scenario of Arons & Lea (1976). According to it, the plasma flow is initially decelerated by the star magnetosphere, with cusps forming in the polar regions (see Lipunov 1992). Later on, owing to the Rayleigh-Taylor instabilities which act in the equatorial region, clumps of plasma penetrate beneath the magnetopause and, threaded by the magnetic field, fall freely along the magnetic field lines onto the poles under the action of gravity. In the approach adopted by Toropin et al. (1999) the matter penetrates through the stellar magnetosphere owing to magnetic diffusion resulting from the finite conductivity of the plasma. This mechanism can result in the formation of an accretion picture which is quite different from what we expect in reality.

In our paper we choose another, although still simplified, approach which allows us to stay within a purely gas dynamic approximation of the problem. We assume that, on entering inside the magnetopause, the clumps of plasma are homogenized within a layer adjacent to the inner side of the initial magnetospheric surface and soon after that we again can consider the gas flow in the continuum approximation. Due to its complexity, for now this intermediate stage of the scenario is omitted and we investigate only the resulting flow. This flow occurs around some modified shape of the magnetosphere which is characterized by the presence of polar holes. The surface of the obtained magnetopause is impermeable for the accreting flow, whereas near the magnetic poles the matter can fall towards the star. These parts of the boundary are modeled by freely penetrable circular holes of specified radius. With this choice of the boundary we intend to simulate certain effects of the magnetic interaction. Outside the boundary we treat the flow as a purely gas dynamic process. Despite obvious simplifications, this model seems to be a natural initial step in the detailed study of the accretion process. Note in this connection that magnetohydrodynamic effects are mainly essential in description of the penetration process itself. The models governing this process are to be developed.

The paper is structured as follows. In Sect. 2 we state the mathematical problem and describe the dimensionless parameters governing it. Sect. 3 describes some specific features of the applied numerical method. In Sect. 4 we present various numerical results, and Sect. 5 gives the discussion.

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Online publication: January 29, 2001