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

8. Analysis of results

8.1. Mass accretion rate

The 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.

Fig. 12. Mass accretion rates (units: ) are shown as a function of the strength of the velocity gradient: 20% and 3% are the results from this work, while the values for models without gradient (at the x-axis position "-infty") are taken from Ruffert (1994a) and Ruffert & Arnett (1994). Diamonds () denote models in which the accretor has a radius of , triangles () models with . The large bold symbols belong to models with a speed of , while the smaller symbols belong to models with . The accretor radius and Mach number are also written near each set of points. All models have , except for the "star" which denotes model SS. The error bars extending from the symbols indicate one standard deviation from the mean (S in Table 1). Some points were slightly shifted horizontally to be able to discern the error bars.

The relative mass fluctuations, i.e.  the standard deviation S divided by the average mass accretion rate (cf. Table 1), have been collected in Fig. 13. Although some models cluster around a relative fluctuation of 5%-10% while others are around 15%-25%, it is not clear which combination of parameters is responsable for this division.

Fig. 13. The relative mass fluctuations, i.e.  the standard deviation S divided by the average mass accretion rate (cf. Table 1), is shown as a function of the strength of the velocity gradient: 20% and 3% are the results from this work, while the values for models without gradient (at the x-axis position "-infty") are taken from Ruffert (1994a) and Ruffert & Arnett (1994). Diamonds () denote models in which the accretor has a radius of , triangles () models with . The large bold symbols belong to models with a speed of , while the smaller symbols belong to models with . All models have , except for model SS (with ) denoted by a star ().

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 momentum

Assuming a vortex flowing with Kepler velocity V just above the accretor's surface with radius , the specific angular momentum of such a vortex is

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 on average (temporal) exceed this value. This implies that smaller objects (smaller ) can accrete only smaller specific angular momenta, which goes to zero like .

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).

Fig. 14. The average specific angular momentum (units: , Kepler velocity vortex at surface of accretor, as given by Eq. (10)) is shown for most models by diamond symbols ( in Table 1). The "error bars" extending from the symbols indicate one standard deviation from the mean ( in Table 1). The long error bars extending to the bottom axis are an indication that the fluctuations of the respective model are so large, that the specific angular momentum changes sign from time to time. The plus signs above the diamonds indicate the specific angular momentum according to the Shapiro & Lightman (1976) prescription, Eq. (7), while the squares denote the values taken from the semi-numerical estimate, Eq. (9) and Fig. 1. All models have , except for model SS (). The star () at the position of IM is the value taken from Ishii et al. (1993) (see text in Sect.  8.3)

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 x -axis in Fig. 14. For one of these models, model SS, Fig. 15 shows the reversal of the disk surrounding the accretor. The "normal" rotation direction is shown in Fig. 15e for model KS which has a large gradient . From the bottom right panel in Fig. 8 one can see that the specific angular momentum never changes sign indicating a strong regular flow around the accretor. This is visible in Fig. 15e: the disk (region of azimuthal flow) extends to a distance of at least 0.3  and is, however, slightly eccentric. In contrast, the disk around the models with small gradient, e.g. model SS, is smaller: roughly 0.1  . Figs. 15a to d show this small disk alternating between clockwise and anti-clockwise rotation. Shocks appear when matter originating outside the disk is accreted in the opposite direction of the current disk rotation, e.g. shortly after the disk has been counterrotating (in Fig. 15a), matter from downstream forces the disk to rotate in anti-clockwise direction (in Fig. 15b). In this figure a shock is visible at (,  ): note the velocity discontinuity, the change from supersonic to subsonic (dashed line) and the increase in density (darker shades) when going with the flow in anti-clockwise direction at (,  ).

Fig. 15a-e. A closeup view of the matter and velocity distribution around the accretor for two models with small accretor (). The density distribution is shown with contours and shades of gray: darker tones indicate higher density. Arrows are overlayed to show the instantaneous velocity of matter. The contour lines are spaced logarithmically in intervals of 0.5 dex for model SS and 0.2 dex for model KS. The bold contour levels are sometimes labeled with their respective values (0.01 and 1.0). 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.

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.

Fig. 16a-d. The specific entropy of accreted matter is plotted versus the specific angular momentum of this matter for models JM, JS, IS and SS. Each dot displays the two quantities at one moment in time. The times for which the dots are plotted are , , , and , respectively.

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.

Fig. 17a-d. The mass accretion rate is plotted versus the specific angular momentum of the accreted matter for models JM, JS, IS and SS. Each dot displays the two quantities at one moment in time. The times for which the dots are plotted are , , , and , respectively.

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 works

Two of the previously published two -dimensional simulations mentioned in the introductory chapter Sect.  1 investigate a velocity gradient (as was done here); Taam & Fryxell (1989) and Anzer et al. (1987). The most important parameters of the first work are: the radius of the accretor is , the adiabatic index is , the velocity gradient is or  , with a Mach number of or . Thus a model most similar to these conditions is model SS (cf. Table 1). When comparing the top two panels of Fig. 11 with the equivalent Figs. 10 and 11 in Taam & Fryxell (1989) one notices one main difference: model SS does not show the "flaring events" described by Taam & Fryxell (1989). These flaring events are due to the collapse of otherwise fairly stable disks around the accretor. In quasi-regular intervals the disk changes its direction of rotation and during these inversions the mass of the disk is accreted. This yields short episodes of very high accretion rates, termed flaring events by Taam & Fryxell (1989). In model SS the disk is much less stable than in Sequence 2 of Taam & Fryxell (1989), consequently the buildup and collapse of the disk is much more eratic, so no flaring events of the same magnitude as in Taam & Fryxell (1989) is seen in model SS. That the disk is so stable in the simulation by Taam & Fryxell (1989) is due to the fact that their calculation is two-dimensional, contrary to the three-dimensional models presented here. Once a disk is formed in a two-dimensional calculation hardly any matter can be accreted in radial direction. In three-dimensions, however, matter can still be accreted via the polar caps even if is disk is present in the equatorial plane. If this matter is focussed from above and below the disk, it can also act to disrupt the integrity of the disk, shortening the lifetime of the disk in three-dimensional calculations.

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 three -dimensional simulations mentioned in the introduction, we will concentrate our comparison to the results of Ishii et al. (1993). In the other cases the numerics is very coarse or questionable, e.g. few zones or particles, local time stepping not appropriate to the problem, etc. The parameters which were used in the "3D velocity inhomogeneous case" of Ishii et al. (1993) are the following. The radius of the accretor is , the adiabatic index is (isothermal), the velocity gradient is and their Mach number is . No model described in the present paper (cf. Table 1) has such a low adiabatic index, but the other parameters are most closely covered by model IM or KM. The value that Ishii et al. (1993) obtain numerically (24% is -0.11 devided by -0.45, these values are taken from Sect. 3.5 in Ishii et al. 1993) is represented by a star () in Fig. 14. Their value (24%) must be used with caution, since their model has an adiabatic index of and their numerical resolution of the accretor is only two zones. Extrapolating from the models presented in this paper, I would expect a higher value, since the average specific angular momentum of model SS (smaller ) is larger than of model IS (larger ), and the models with larger accretors ("M"-models) are larger still.

© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
helpdesk.link@springer.de