## 8. Analysis of results## 8.1. Mass accretion rateThe mass accretion rates obtained in this work for all models except IT, RL, and ST are collected in Fig. 12 together with the amplitude of their fluctuations (one standard deviation). This figure indicates that, to first order, the mass accretion rate is independent of , even for the large . Although the mean mass accretion rate does vary slightly with Mach number and accretor radius, even in units of , the variation of the rates across different velocity gradients remains within the fluctuations of the unstable flow. This is only in partial agreement with Fig. 1. Up to no variation is expected from Fig. 1 if all mass in the accretion cylinder is actually accreted (which is an assumption that enters when deriving Eq. (6), etc.). However, since the mass accretion rate seems to remain unchanged even for the models with (contrary to Fig. 1), we conclude that for these large velocity gradients not all matter in the accretion cylinder is accreted any longer.
The relative mass fluctuations, i.e. the standard deviation
Two general statements can be made. When, starting from axisymmetric models, slightly increasing in the velocity gradient to a few percent, the relative accretion rate fluctuation either remains unchanged or decreases. In the axisymmetric case, any eddy will produce a fluctuation of the accretion rates. As long as the velocity gradient is small, the vortex generated by the the incoming angular momentum around the accretor is of the same strength as the eddies and it might be able to stabilize the flow around the accretor. When further increasing the velocity gradient the relative fluctuations increase strongly. So the stabilizing effect is lost indicating that the vortex itself contributes to the eddies and the fluctuations. ## 8.2. Specific angular momentumAssuming a vortex flowing with Kepler velocity The term has erroneously been omitted in my
previous works (e.g. Ruffert, 1996). Although for short periods
of time the specific angular momentum can exceed
, it is difficult to imagine how accreted
matter can In Fig. 14, I plot for several models the numerically obtained quantities along with the amplitude of the fluctuations (one standard deviation, ), which can be found in Table 1. These are plotted in units of (Eq. (10)). Additionally, above the diamonds denoting the above mentioned ratio , I plot, using plus-signs, the values that one expects for the analytically estimated quantity from Eq. (7) and denoting by squares, from the semi-numerical estimate, Eq. (9).
Several trends can easily be noticed in Fig. 14. Model JM seems to be well below the general trend, indicating that the simulation was not evolved for long enough; I will not include this model in the following discussion. The four "K" and "L" models form a fairly homogeneous group accreting roughly 0.3 of the Kepler specific angular momentum. For the "I"- and "J"-models this fraction is roughly 0.1, thus confirming that for models with smaller gradients, the vortex around the accretor is less pronounced. These two groups vary less among themselves than the variation one would expect if the analytical estimates Eq. (7) (plus signs) or Eq. (9) (squares) were valid. Thus, when estimating the specific angular momentum one should be guided by the Kepler-values Eq. (10). When applying Eq. (6) one should bear in mind the allowable parameter range: small gradients, supersonic flow and small accretors. A good counter example is model RL, which exhibits a constant state and the sign of the accreted angular momentum is opposite to Eq. (7). The smaller lever arm acting in models with smaller accretors is included in Eq. (10). Still, the "S"-cases have a slightly smaller value of compared to the "M"-cases. The reduction is, however, not uniform for all models; the models with large gradients show reductions of at most a factor of 2, while the other models have a factor of 3. So small accretors impede high specific angular momentum accretion in some additional way than only via their smaller lever arm. At the distance of the surface (= radius of the accretor), matter seems to move in eddies at some fraction of the Kepler-speeds. This fraction is dependent more on the velocity gradient than on the size of the accretor or the Mach number. Recall, that from Fig. 1 one would expect the models with large gradients ("K" and "L") to accrete specific angular momenta a factor of 2 larger than the analogous models with small gradients ("I" and "J"). This is in contradiction to what is shown in Fig. 14, confirming again that the assumption of accreting everything from the accretion cylinder is not correct for large gradients. When the fluctuations of the specific angular momentum are larger
than its mean, then the accreted angular momentum can change sign,
indicating a reversal of the rotation direction of a disk around the
accretor. This will most easily be attained for models with a small
gradient, since the fluctuations need not be large in these cases.
These models have their "error bars" extending completely down to the
Shocks imply generation of entropy, thus if shocks appear more often when the direction of rotation of the disk is reversed one would expect that matter with higher specific entropy is accreted during phases when the specific angular momentum of the matter changes sign. To find such a possible correlation, I draw in Fig. 16 a dot connecting the two quantities for every second time step of the numerical simulation. This was done only for the four models JM, JS, IS, and SS which exhibited a change in sign of the specific angular momentum (see e.g. Fig. 14). The two models JM and IS (Fig. 16a and c) have an only small intrinsic scatter of the specific entropy accreted (roughly 10%) indicating that the fluctuations visible in Figs. 4 and 6 do not generate equally fluctuating shock structures. For model JM the entropy does not at all seem to correlate with the angular momentum, while for model IS a marginal indication exists that the specific entropy is slightly higher for larger (=more positive) specific angular momenta. On the other hand, the highest entropies appear at the most negative momenta. In contrast, models JS and SS exhibit a relatively large scatter of the specific entropy (roughly 40% and 25%, respectively) and fairly clear correlations. In model JS (Fig. 16b) high entropy material is accreted preferentially when the angular momentum is small (around zero). This result confirms what has been described at the beginning of this paragraph. The correlation in Model SS (Fig. 16d) is different again: no material with low specific entropy (less than say 4.9) is accreted when the specific angular momentum is positive. Thus the disk seems to be in constant turmoil (with many shocks) when the disk rotates in anti-clockwise direction (which is contrary to the "normal" direction, cf. beginning of this Sect. 8.2). However, when the disk rotates in clockwise direction both high and low entropy material (more or less than 4.9) is present.
In Fig. 17 the rate at which mass is accreted is plotted versus the specific angular momentum. Analogously to the differing correlations of the entropy, the four models JM, JS, IS, and SS display different behaviours. The correlation in model SS is clearest: the mass accretion rate is highest when the specific angular momentum is around zero and the rate decreases for more positive and more negative values of the specific angular momentum. A similar, but less clear, trend can be discerned for model JM (Fig. 17). Obviously, when the flow does not rotate around the accretor it falls down the potential to the surface of the accretor and can thus be absorbed more efficiently.
The maximum rate at which mass is accreted also decreases with increasing magnitude of specific angular momentum (independent of sign) in models JS and IS (Figs. 17b and c). However, the smallest mass accretion rates are scattered fairly uniformly along all momenta. ## 8.3. Comparison with previous worksTwo of the previously published The most important parameters of the Anzer et al. (1987) work are: the radius of the accretor is , the adiabatic index is , the velocity gradient is and the Mach number is . Models IM or SS from the present work most closely resemble these parameters. In Sect. 4 of Anzer et al. (1987) they report finding a ratio of 0.22 between the numerically obtained (SPH simulation with 7500 particles) specific angular momentum and the analytically estimated value. This value is within a factor of two of the results shown in Fig. 14. Of the five previously published © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |