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

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2. Statement of the problem

The considerations given in the previous section allow us to perform the analysis on the basis of the numerical solution of the Euler gas dynamic equations. Suppose the flow pattern to be axisymmetric with the gravitating object placed at the origin of the coordinate system and with the rotation axis aligned with the z-axis (Fig. 1). The matter is assumed to be ideal and thermally perfect. By assuming the heat release/absorption processes to be polytropic we can write the caloric equation of state in the form

[FIGURE] Fig. 1. Numerical grid and the profile of accreting magnetosphere.

[EQUATION]

where [FORMULA] is the internal energy per unit mass and [FORMULA] is the polytropic index. The system of governing equations in the Cartesian coordinate system shown in Fig. 1 reads:

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The superscript T means transposing. In the above system [FORMULA] is the total energy per unit volume. We normalized the quantities of density, velocity, and pressure by [FORMULA], [FORMULA], [FORMULA], where [FORMULA] is the reference length defined by the formula approximating the solid portion of the contracted magnetosphere of the star (Elsner & Lamb 1976)

[EQUATION]

In the first quadrant one has [FORMULA]. The angle [FORMULA] is counted off the z-axis. It is clear that at [FORMULA], the polar distance of the magnetosphere is [FORMULA]. At the equator, [FORMULA]. We use this simplified formula in our calculations, since it adequately describes the shape of the magnetopause in a way sufficient for our qualitative studies. We have written out the dimensionless system for the general case in which the accreting matter possess some angular momentum at infinity. Modifications necessary in the case of the quasispherical accretion of matter without angular momentum will be given in Sect. 4.

We assume the inflow at [FORMULA] to be supersonic. Thus, we must specify all quantities on the outer boundary. Taking into account this fact and inspecting the structure of the system (7)-(11), we arrive at the following set of dimensionless parameters of the problem: [FORMULA], [FORMULA], [FORMULA], and S, where [FORMULA]. As previously noted, [FORMULA] is the radial inflow velocity at [FORMULA] and [FORMULA] is the Keplerian velocity. By introducing these dimensionless parameters and initially assuming the flow to be spherically symmetric, we fix the values on the outer boundary as

[EQUATION]

where [FORMULA] is the [FORMULA]-component of the velocity vector.

Neglecting for the moment the velocity components in the angular directions, we can calculate the entropy function and the total enthalpy as

[EQUATION]

We now choose the dimensionless [FORMULA] to be 100 and introduce the following distribution of the angular velocity along the outer boundary:

[EQUATION]

The first interval here corresponds to the constant angular momentum and the second to the constant angular velocity distribution, respectively. The y-component v (normal to the plane of Fig. 1) of the velocity vector [FORMULA] is then [FORMULA].

To modify the quantities of [FORMULA] and [FORMULA] at [FORMULA], we assume [FORMULA] and [FORMULA] and calculate the new values from the formulas

[EQUATION]

Because of the symmetry of the problem, it suffices to seek solution only in the first quadrant. The conditions on the inner boundary are the following: nonpenetration is adopted for [FORMULA], and absorbing boundary conditions are used for [FORMULA]. The essence of these absorbing boundary conditions is described in Pogorelov & Semenov (1997). We use the quantity extrapolation at the supersonic exit [FORMULA] and apply the relations in the rarefaction fan to accelerate the outflow velocity to the sonic value if [FORMULA]. These conditions give results nearly identical to those obtained with the "vacuum" boundary conditions by Hunt (1971) if the latter are used in the framework of the Godunov-type schemes (Toro 1997).

The initial conditions are believed to be not very important if we seek a steady-state solution. On the other hand, they are of crucial importance if we study transient phenomena.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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