## 4. Simulation resultsOur numerical results are subdivided into two groups. One of them
deals with accretion of nonrotating gas onto the model magnetosphere.
The other one concerns accretion of slowly rotating matter. In the
former case, the ## 4.1. Quasispherical accretion onto magnetosphereIn Fig. 2 we present the numerical results for the following parameters: , , , and (the last parameter specifies the radius of the polar hole). The value of the polytropic index was chosen sufficiently small to account for the radiative cooling effects.
Note that the size of the holes in the adopted model is fully
determined by the thickness of the penetration layer, that is, on the
details of the Rayleigh-Taylor (interchange) instability of the
magnetopause. Thus, the radius of the holes is a parameter of the
problem. The depth of penetration cannot be smaller than the size of
the plasma blobs occurring inside the layer. According to Arons &
Lea (1976), the blob size is
(0.1-0.2) , or 0.15-0.3 in our
dimensionless units. This gives us a rough estimate for the variation
range of Observational estimates of the polar hot spot cannot be done unambiguously since the observed beam is determined by the combination of a hot spot size and an angular size of the emission beam. Assuming the relation between the size of the hot spot and Alfvénic radius (Baan & Treves 1973), we obtain the lower boundary of the angular radius of the hot spot of the order of 0.1 radian for the magnetic field (dipole component) Gs (Sheffer et.al. 1992; Baushev & Bisnovatyi-Kogan 1999; Scott et al. 2000). The size of the polar holes was initially chosen fairly large, since we would like to be sure that the matter can be accreted at the rate prescribed by the external boundary conditions. Later we shall investigate the dependence of the accretion pattern on this size. We describe Fig. 2 in detail, since the subsequent figures have the same structure. In the first quadrant, we use arrows to show the velocity vectors. The size of each arrow is proportional to the magnitude of the velocity vector. In order to avoid misinterpreting, we limited the size of the vectors shown. The regions with small velocity manifest themselves as white zones free of arrows. The second quadrant shows the streamlines corresponding to the steady state obtained. The third quadrant contains the contours of constant density logarithms. In this plot we can distinguish the discontinuities which are likely to occur in the progress of flow deceleration by the impermeable surface of the magnetosphere. In the fourth quadrant, we show the contours of constant Mach numbers. Dotted lines represent the level . In the adopted model, the flow is essentially the combination of a supersonic blunt body and a nozzle flow. It is therefore likely that a bow shock will appear in front of the impermeable portion of the magnetosphere. This shock decelerates the spherically-symmetric focusing stream of accreting matter to subsonic velocities. Such deceleration is caused by the nonpenetration boundary conditions. Note that there exists a stagnation point at , . In addition, the surface of the magnetosphere has a shape such that the radially oriented gravitation force always has a component directed along this surface towards the poles. The bow shock is finely resolved by the numerical scheme on the grid adapted to the magnetosphere surface. The streamline and especially the velocity vector pattern show the gas deflection at the magnetopause under the action of gravity. We can also distinguish a low-velocity zone adjacent to the stagnation point at the equator. The supersonic stream of matter can initially be freely accreted through the polar holes. In the course of time, however, as we approach the steady state, the freely falling column of matter surrounding the polar axis meets the convergent flow formed in the shock layer around the impermeable part of the magnetosphere. Collision of these two streams decelerates the former one, thus resulting in the origin of another shock. These two shocks form a combined bow shock around the magnetosphere. The resulting shock cannot generically be smooth and the derivative of the shock surface is discontinuous. It is clear that there must appear a third shock at this point. That is why, this point is called a triple point. We can see all these shocks in the density and Mach number contour plots. Their presence is also confirmed by the distribution of the lines (the sonic lines). Note that the stream inside the shock layer, spreading over the magnetosphere surface in the equatorial region, soon becomes supersonic again. There exists another sonic line in the vicinity of the polar holes. On this line the gas in the blunt body shock layer decelerates to subsonic velocities again. This implies the existence of a shock wave. The last sonic line is not very well seen among the Mach lines in Fig. 2. One can easily distinguish it in Fig. 3 and Fig. 4, where we show the solution for , , , and and 0.1, respectively. In these figures, the slip line between the two portions of the flow passing through the different parts of the bow shock is also very well seen. It is apparent that this slip line originates at the triple point.
It is worth emphasizing, that for large values of the parameter
Note that the streamlines oriented along the polar axis can find their way to the star only through a very narrow hole near this axis. The main portion of accreting matter is supplied by the supersonic stream formed in the narrow shock layer around the impermeable part of the magnetosphere. This, in fact, means that accretion of initially almost spherically symmetric flow occurs in a way consistent with that predicted by Arons & Lea (1976). It is interesting that even a very narrow polar hole with (Fig. 4) permits the accretion rate prescribed on the outer boundary. We can conclude that a steady-state accretion pattern can be realized along with the highly unsteady results predicted by Kazhdan & Murzina (1994) and obtained numerically by Chen et al. (1997) and Toropin et al. (1999). It is also worth mentioning that smaller sizes of the holes result in larger bow shock stand-off distances. This is quite reasonable, since in the absence of any holes an atmosphere will inevitably form around the star and the thickness of the shock layer will infinitely grow with time. On the other hand, larger cooling (smaller ) decreases the bow shock stand-off distance. This is shown in Fig. 5 which presents the accretion pattern for , , , and . For smaller gravitation, the bow shock stand-off distance increases, as seen from Fig. 6.
For narrow holes and larger polytropic indices no steady accretion pattern can be obtained. ## 4.2. Accretion of rotating matterFirstly, we show in Fig. 7 the results of the numerical modeling of accretion onto a black hole. In this case, the whole sphere is fully permeable. We do this in order to compare the results obtained by our numerical technique with those previously presented by Bisnovatyi-Kogan & Pogorelov (1997). We use the following set of parameters: , , , and . The last parameter implies that the rotation chosen in this case remains slightly sub-Keplerian. The results clearly show that the centrifugal force strongly deflects matter from the axis of rotation. As a result, a high-density region is formed near the accretor in the equatorial region. The density near the equator is times larger than that near the poles and almost all the matter accretes in a narrow layer. This layer is a predecessor of an accretion disk which is expected to appear for . The accretion is everywhere supersonic. It also remains steady, in contrast to the results of Chen et al. (1997) obtained for conditions on the outer boundary inconsistent with the accretion ability of the inner spherical surface. Note that presetting an arbitrary distribution of quantities on the outer boundary is sometimes not very physical, in contrast with the case of accretion from a uniform supersonic wind, where matter can pass by the accretor and accumulation of matter near it is less probable. This also concerns the choice of the Bondi parameters, since a nonstationary shock wave will spread to infinity in such an inconsistent case. As a result, sooner or later, it will reach the sonic point in the Bondi solution, thus making this solution unsuitable for construction of boundary conditions at . This, in other words, means that in order to obtain transient unsteady solutions, one must prescribe at the outer boundary such mass inflow that cannot be accreted by the star.
Note that the poloidal component of the velocity vector must vanish at the equator. This means that a transverse shock must appear in the flow field. This shock is clearly seen in the density isolines. The investigation of accretion of slowly rotating matter onto the
stellar magnetosphere is performed by fixing
and
and varying the parameters
, ,
and
It is important to note that the velocity of its propagation can sometimes be so large that the radial component of the velocity vector behind the shock becomes positive. This can be interpreted as an explosion-like formation of a gas cloud around the star. © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |