Astron. Astrophys. 317, 793-814 (1997)

5. Results of models with 20% velocity gradient

Models KM, KS, LM and LS break the axisymmetry very strongly since they have as boundary condition a 20% velocity gradient over one accretion radius. This asymmetry induces a strong clockwise vortex around the accretor as can be seen in Fig. 7. The sign of the velocity gradient has been chosen in such a way that upstream of the accretor the higher velocities are on the negative y -side of the xy -plane (lower half of the contour plots). However, the numeric simulations confirm the sign of the analytically estimated accreted angular momentum (Eq. (7)): the vector points into the plane of the plot, which corresponds to a clockwise rotation in these contour plots. A note of caution is necessary, however: recall that Eq. (7) was derived only to lowest order in  and  , thus assuming that  and  are small compared to unity. Obviously this is questionable with the choice for the models presented in this section. It is still interesting to see by how much the analytic estimates deviate from the numerical models is this extreme case.

Fig. 7a-d. Contour plots showing snapshots of the density together with the flow pattern in a plane containing the center of the accretor for all models with a velocity gradient of 20%. The contour lines are spaced logarithmically in intervals of 0.1 dex. The bold contour levels are labeled with their respective values (0.01 or 1.0). The dark shades of gray indicate a high density. The dashed contour delimits supersonic from subsonic regions. The time of the snapshot together with the velocity scale is given in the legend in the upper right hand corner of each panel.

5.1. Moderately supersonic accretors, Mach 3

The accretion rates of several quantities for models KS and KM can be found in Fig. 8. Contrary to model IM, model KM does not exhibit such strong fluctuations, thus it never even comes close (by a factor 3) to the analytically predicted value (-0.9) of Eq. (7), nor does it ever come close to the zero line. The large scale motion of the vortex around the accretor is fairly stable in time, which explains why the fluctuations relative to the mean of the specific angular momentum are smaller in model KM than in model IM.

Fig. 8a-d. The accretion rates of several quantities are plotted as a function of time for the moderately supersonic ( =3) models KM and KS with a velocity gradient of 20%. The left panels contain the mass and angular momentum accretion rates, the right panels the specific angular momentum of the matter that is accreted. In the left panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994a), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994a) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the left panels trace the x (dotted), y (thin solid) and z (bold solid) component of the angular momentum accretion rate. The same components apply to the right panels. The value (-0.9) of the specific angular momentum as given by Eq. (7) is outside the range of the plot for both models.

The fluctuations of the mass accretion rate (left panels of Fig. 8 of model KM seems to increase with time indicating that this model is still evolving in time and has not yet reached a steady mean state. These panels should be compared to the analogous panels (left in Fig. 4 for the models IM and IS. One notices that both the mean and the amplitude of the fluctuations is larger in model KM than in model IM Although the mean mass accretion rate of model KS is lower than the mean in model IS this might be a transient effect, since toward the end of model IS the mass accretion rate is of the same magnitude as in model KS. Thus for the larger accretors (M-models) the vortex allows more mass to be accreted, while for the small accretors (S-models) the difference is negligible. Both models KM and KS have one difference in common compared to model IM and IS respectively: the unstable flow, which manifests itself e.g. via a fluctuating mass accretion rate, begins much faster for the models with a large gradient, models KM and KS. Also the large-scale fluctuations of the specific angular momentum appear at roughly in models KM and KS, while it takes until in models IM and IS.

5.2. Highly supersonic accretors, Mach 10

The accretion rates of several quantities for the highly supersonic models LM and LS are shown in Fig. 9. The fluctuation amplitude of the mass accretion rate of model LM is larger than both the amplitudes of model JM (top left panel in Fig. 6) and of model FM (top left panel of Fig. 9 in Ruffert 1994), however the mean seems roughly equal. On the whole model LM looks more unstable and active than the models JM and FM with small or no gradients. The mass accretion rate of model LS does not seem to decline constantly during the first two time units as was the case for model JS.

Fig. 9a-d. The accretion rates of several quantities are plotted as a function of time for the highly supersonic ( =10) models LM and LS with a velocity gradient of 20%. The left panels contain the mass and angular momentum accretion rates, the right panels the specific angular momentum of the matter that is accreted. In the left panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994a), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994a) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the left panels trace the x (dotted), y (thin solid) and z (bold solid) component of the angular momentum accretion rate. The same components apply to the right panels. The value (-3.0) of the specific angular momentum as given by Eq. (7) is outside the range of the plot for both models.

The trend that the model with the smaller accretor (in this case model LS compared to model LM) has the z -component shifted closer to the zero line is repeated also for the highly supersonic models with large gradients. The fluctuations around the mean of the z -component has roughly the same amplitude and frequency as the fluctuations of the x and y -components around zero.

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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