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

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4. Results of models with 3% velocity gradient

4.1. Moderately supersonic accretors, Mach 3

The model denoted by IM in this work is identical to the simulation presented in Ruffert & Anzer (1995). Since for the parameters that I wanted to investigate in this paper a slightly different prescription of the velocity gradient was necessary (cf. Sect.  2.2), a comparison is called for to check how large the influence of the "tanh"-term is for small gradients [FORMULA]. Models IM and IM* are identical except for the "tanh"-term. Fig. 3a shows a contour plot of the density in one plane for model IM, Fig. 4 shows the accretion rates of several quantities for model IM, and Fig. 5 displays the same quantities for model IM*. One notices that both the mass accretion rate as well as the specific angular momentum accreted are similar in both models for the time over which both have been calculated. I stopped the simulation of model IM* at [FORMULA] because it looked so alike to model IM. Thus I conclude that for small gradients [FORMULA] the difference between the "tanh"-prescription and the simple linear dependence is indeed negligible.

[FIGURE] Fig. 3a-e. 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 3%. The contour lines are spaced logarithmically in intervals of 0.1 dex. The bold contour levels are sometimes labeled with their respective values (0.01 and 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.
[FIGURE] Fig. 4a-f. The accretion rates of several quantities are plotted as a function of time for the moderately supersonic ([FORMULA] =3) models IM, IS and IT with a velocity gradient of 3%. 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; Bondi 1952) 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 horizontal line in the right Panel of model IM shows the specific angular momentum value as given by Eq. (7). It is outside the range of the plot for models IS and IT.

[FIGURE] Fig. 5a and b. The accretion rates of several quantities are plotted as a function of time for model IM*. The left panel contains the mass and angular momentum accretion rates, the right panel the specific angular momentum of the matter that is accreted. In the left panel, 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 panel. The horizontal line in the right Panel shows the specific angular momentum value as given by Eq. (7).

Models IS and IT differed only in their numerical resolution: model IT was simulated with one grid level finer, while the accretor size was very small ([FORMULA]) compared to models IM and IM* (whose radii were 5 times larger). The density distribution of models IS and IT can be found in Fig. 3, while their accretion rates are shown in Fig. 4. Because model IT had one level of refinement more, the computational cost was larger, so the time the simulation was run is roughly half of the time of model IS. Until [FORMULA] both models show the same features: the mass accretion rate rises to roughly 6 units and starts fluctuating at about [FORMULA] time units. The specific angular momentum rises continuously until [FORMULA] then fluctuates around a value of roughly -0.03 while increasing the amplitudes of the fluctuations. Thus the use of 9 nested grids seems sufficient, since the calculation with 10 grids did not show any qualitative differences.

It is clear that model IT has not been evolved for long enough to obtain meaningful averages, since the fluctuation of the mass accretion rate has hardly begun when the simulation is stopped (Fig. 4). Although model IS was calculated for a longer time, the continuously decreasing mass accretion rate of model IS (Fig. 4) indicates that a stationary state has not yet been reached und so the simulation of this model should have been continued even further. However. the high computational cost precluded this. The fluctuations of the specific angular momentum, too, seem to increase with time corroborating the the statement. Thus the average quantities obtained from this model IS cannot be very exact.

Two uncertainties that could not be resolved by the single model IM presented in Ruffert & Anzer (1995), can now be answered. The first one pertained to the fact that the specific angular momentum (visible in the top right panel of Fig. 4) seemed to reach but not cross the zero-line. This seems to be a coincidence of the initial and boundary conditions, since in model IS (visible in the middle right panel of Fig. 4) the fluctuations are indeed large enough to change the sign of the specific angular momentum accreted. We will return to this point in Sect.  8.2. The second uncertainty was whether in the generic case the values of the specific angular momentum attained and exceeded the analytically estimated ones given by Eq. (7). Model IS clearly does not attain these values by a large margin, roughly a factor of 3 - the analytic value for model IS is the same as for model IM, since the accretor radius does not enter into Eq. (7). The smaller accretor radius seems to allow only smaller values of angular momentum to be accreted: if the lever arm (which is the radius of the accretor) is smaller the velocities have to be an appropriate amount larger (a factor of 5, roughly) to compensate. This is obviously not the case: the arrows in the left panels of Fig. 3 close to the surface of the accretor have roughly the same length. The smaller accretor sizes also have the effect that the time scale of the fluctuations of model IS are shorter than the ones of model IM (compare the right panels of Fig. 4).

The corresponding plot to the top left panel in Fig. 4 of model IM for the axisymmetric case can be found in the top left panel of Fig. 16 in Ruffert & Arnett (1994). For model IS the closest would be the top left panel of Fig. 22 in Ruffert & Arnett (1994). One can see, that while the amplitude of the fluctuations of the the z -component of the accreted angular momentum is comparable, the average of this component of the models with velocity gradients is clearly non-zero. Contrary to this, the x - and y -components fluctuate more strongly in the models without gradients, but in all models their temporal average is close to zero (see e.g. Table 1). The run, average and fluctuations of the mass accretion rate is similar in all models.

Due to the non-axisymmetric upstream boundary conditions it is not surprising that the shape of the bow shock is not symmetric either. There is an indication of this fact in the panels shown in Fig. 3 for the density distribution close to the accretor, but it is very prominent when inspecting the shock cone position further away from the accretor (see e.g. Fig. 1 in Ruffert & Anzer 1995). The temporal evolution shows the usual kinks and deformations of the shock cone that were described in the previous papers (Ruffert 1996, and references therein).

4.2. Highly supersonic accretors, Mach 10

The right panels of Fig. 3 display the density contours of models JM and JS, which are equivalent to models IM and IS, except for the different flow speed upsteam: Mach 10 for the J-models contrary to Mach 3 for the I-models. The corresponding accretion rates can be found in Fig. 6. The highly supersonic models JM and JS, too, do not converge to a quiescent steady state but show an unstable fluctuating flow pattern.

[FIGURE] Fig. 6a-d. The accretion rates of several quantities are plotted as a function of time for the highly supersonic ([FORMULA] =10) models JM and JS with a velocity gradient of 3%. 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 horizontal line in the right Panel of model JM shows the analytic specific angular momentum value as given by Eq. (7). It is outside the range of the plot for model JS.

In the same way as model IM, also model JM meets and exceeds the analytically given value (Eq. (7)) of the specific angular momentum (top right panel Fig. 6, horizontal line), however only in rare, short bursts. Thus, on average, the fraction of the specific angular momentum to the analytic value is smaller than the value of the fraction of model IM (cf. Table 1 colum [FORMULA]). However, similarly to the difference between models IM and IS, the specific angular momentum of model JS is smaller still. In fact, when looking at the bottom right panel of Fig. 6, the curve for the [FORMULA] component overlaps very strongly with the curves of the other components. There is, however, still clearly a systematic shift of [FORMULA] not visible in  [FORMULA] and  [FORMULA] which fluctuate around zero.

The left two panels in Fig. 6 (models JM and JS) can be compared to the two top panels of Fig. 9 in Ruffert (1994b) which show the results of the equivalent models without gradients (FM and FS, respectively). The magnitude and fluctuation amplitude of the mass accretion rate are similar which is also the case for the angular momentum accretion rates (note different scale of y -axis in figures). Since the mass accretion rate of model JS seems to be steadily declining (although also fluctuating, bottom left panel in Fig. 6), this model at the time we stopped the simulation is probably still influenced by transients generated through the initial conditions. Thus the average values of e.g. the specific angular momentum, have to be used with caution.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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