4. Simulation results
Our 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 y-component of the velocity vector and the corresponding equation for its determination vanish in the system of (7)-(11). If we normalize velocity and pressure by and , respectively, the structure of the system remains unchanged if we substitute by S in the source term.
4.1. Quasispherical accretion onto magnetosphere
In 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 d. The shape of the inner boundary in our case corresponds to the definition of plasmasphere introduced by Arons & Lea (1980).
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 S, that is, for strong gravitation the influence of the Mach number on the flow pattern becomes rather weak, since the matter is rapidly accelerating at large distances from the star and acquires hypersonic values ahead of the bow shock.
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 matter
Firstly, 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 d. Larger implies smaller rotation in comparison with the Keplerian rotation. Let , , and (Fig. 8). Though rotation is rather slow in this case, we can see a noticeable difference with the similar results for the quasispherical accretion shown in Fig. 2. It is apparent that the centrifugal force acts to decelerate the stream formed in the shock layer and there appears a possibility of accretion straight along the rotation axis (see the velocity vector distribution). Note that the shape of the bow shock becomes more complicated. We can see an additional line near the axis of rotation in this case. Larger rotation increases the described effect, see Fig. 9 (, , and ). The case of a smaller hole is shown in Fig. 10 (, , and ). One can notice that the main stream of accreting matter becomes very narrow under the action of the centrifugal force which pushes the the stream to the impermeable portion of the magnetosphere. For this reason, rapid rotation can make accretion rather inefficient, thus leading to development of a growing magnetosphere accompanied by a bow shock propagation outward. A steady-state solution cannot be obtained in this case. In Fig. 11 we show one of the solutions on the boundary between the steady-state and nonstationary solutions. The flow presented corresponds to , , and . The pattern is substantially different in this case. The triple point moves farther from the rotation axis and a low-velocity circulation zone originates behind the portion of the bow shock closer to the rotation axis (see the streamline and velocity vector distributions). The bow shock stand-off distance starts increasing right above the triple point, attains a maximum, and decreases to a rather small value at the rotation axis. The flow restructuring allows a large portion of the radially falling gas to be accreted avoiding preliminary compression in the shock layer near the impermeable surface of the magnetosphere. This permits us to obtain a steady-state solution. Smaller values of increase the compressibility of the falling matter which is highly compressed in the shock layer, as seen from the streamline distribution in Fig. 12 which corresponds to , , and . For larger cooling effects, there appears a portion of matter around the symmetry axis which is accreted at a supersonic speed without any shock. Thus, the accretion pattern becomes qualitatively different in this case. It also turns out that a certain portion of the bow shock is intersected at rather acute angles. Decreasing the holes will lead to a solution with a divergent bow shock. We can see the dynamics of the accreting pattern in Fig. 13 and Fig. 14 which correspond to , , and and 0.1, respectively. The solution shown in Fig. 13 represents a steady accretion with a very thin shock layer, owing to high compressibility of the mater at . Narrow polar holes cannot ensure sufficiently high accretion rates and the matter starts accumulating around the magnetosphere. This finally results in an unsteady solution with the bow shock moving towards the outer boundary (Fig. 14).
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