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Astron. Astrophys. 337, 311-320 (1998)

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4. Summary and discussion

Two dimensional planar wind accretion flows of a supersonic isothermal gas were analyzed numerically. The computational efficiency was improved as compared to earlier calculations by using an accurate local time stepping and a new type of upwind scheme.

Comparing our new results with earlier calculations one realises that 2D flows with [FORMULA] close to unity have so far not been studied very systematically. Matsuda et al. (1992) presented one case with [FORMULA] and one with [FORMULA]. Their resolution near the central object was much coarser than our resolution. They had also obtained flip-flop configurations, but the instability took much longer to develop. Their amplitudes were smaller and the widths of the accretion columns considerably larger. Boffin and Anzer (1994) also calculated one model with [FORMULA] which has a narrow oscillating accretion column, but again a low amplitude. 3D flows with [FORMULA] have been systematically studied by Ruffert (1996). In these flows the accretion regions are always much wider and do not show the systematic oscillations of entire accretion region, although the accretion itself is non-steady. These differences between 3D and 2D flows are quite general and independent of the value of [FORMULA].

We also have developed a numerical scheme which exactly conserves angular momentum (AMC scheme) and compared its performance to the earlier linear momentum conserving (LMC) scheme. We found that the results obtained with our LMC scheme depend very strongly on the resolution of the grid used, whereas the AMC scheme gives almost no difference for the two types of resolution considered here. Therefore we feel that the AMC calculations are much more reliable and all the results presented in the previous section are based on this scheme. The fact that our AMC and LMC results differ by large amounts suggests to us that many of the earlier calculations which basically conserve linear momentum should be taken with caution. This will be particularly important for the temporal behavior of the accretion of angular momentum [FORMULA] (t).

Our calculations of supersonic flows show that in all cases the accretion of both mass and angular momentum is very erratic. These large fluctuations can be seen in the hight values of the RMS variations of [FORMULA] and [FORMULA]. But their presence is even more obvious when one considers the time history curves for [FORMULA] and [FORMULA]. From these curves one finds that the amplitudes of [FORMULA] can reach up to 40 and have sharp peaks typically between 10 and 20. The maximum peaks of [FORMULA] are around 4 and the typical values lie between 1 and 2. Such a spiky behaviour had also been obtained in earlier 2D computations, but their amplitudes were much smaller. This behavior can be explained by the formation of Keplerian disks near the inner boundary. If the specific angular momentum of the infalling material is larger than that of the Kepler orbit at the innermost radius then this material cannot accrete.

If such high angular momentum material is flowing in long enough, a disk will form which blocks further accretion very efficiently. Material with very low angular momentum or with opposite rotation falling in during a subsequent phase interacts with the disk and can destroy it. This will lead to a burst of the accreted mass and angular momentum. After such a burst the process can be repeated and a new disk will form. Our calculations indicate that reversals of the disk rotation are quite common.

For the modeling of X-ray binaries fed by wind accretion the fluctuations of [FORMULA] are of major importance. They can be brought into relation with the observed spin-up and spin-down of these X-ray pulsars; see Anzer & Börner (1995). In their investigation they showed that the random fluctuations calculated by Ruffert (1994a) for 3d models were by a factor [FORMULA] too low in order to explain the pulse period variations observed in the source Vela X-1. However our new calculations give variations of [FORMULA] which are substantially larger than those found by Ruffert. We have obtained typical values of RMS [FORMULA] of the order of 0.2 in our dimensionless units (see Table 2). This result can also be formulated as:

[EQUATION]

since [FORMULA] is typically of the order unity. On the other hand Ruffert (1996) gives RMS(j)=0.01[FORMULA] for [FORMULA] and RMS(j) = 0.03[FORMULA] for [FORMULA]. Taking into account that RMS([FORMULA])=RMS ([FORMULA]j) and [FORMULA] [FORMULA] we have RMS ([FORMULA]) = (0.01-0.03)[FORMULA]. Therefore our values for the fluctuations are a factor 3 to 10 larger than those of Ruffert's 3D calculations. Therefore on the basis of our calculations one might conclude that the observed period fluctuations could in principle, be caused by random fluctuations of [FORMULA]. But the amplitudes are only marginally large enough and any slight reduction of the efficency would rule out this interpretation. There is in particular the aspect that our calculations are two-dimensional whereas the real flows are three-dimensional and the difference in amplitudes between 2D and 3D flows could be sufficiently large to make the described interpretation invalid. To really answer this question requires full 3D computations, taking angular momentum conservation into account.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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