Astron. Astrophys. 346, 861-877 (1999)

4. Comparison of accretion rates

4.1. Mass accretion rates

I collected all the means of the mass accretion rates and their standard deviation into Fig. 13 together with the appropriate values for models without gradients. The standard deviation is shown as "error bar" in order to give an indication on how precisely one should interpret the mean values.

Fig. 13. Mass accretion rates (units: ) are shown as a function of the strength of the density gradient: the points to the right of lg gradient = -2 are the results from this work, while the values for models without gradient (at the x-axis position "-infty") are taken from Ruffert (1994) and Ruffert & Arnett (1994). Diamonds () denote models in which , triangles () models with , and squares () show The large bold symbols belong to models with a speed of , while the smaller symbols belong to models with . The adiabatic index and Mach number are also written near each set of points. 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.

(a) The mass accretion rates are fairly independent of the density gradient strength for small , but a decrease of the rates might be present when increasing the gradients: a slight trend toward smaller rates for larger gradients seems possible. (b) A clear increase of the rates is visible when decreasing the index . When going from 5/3 to 1.01 the accretion rate increases by over a factor of two. (c) Models with smaller Mach numbers have larger rates. This trend also applies to models with a velocity gradient as can be seen in Fig. 12 of R1.

Because the fluctuations of the mass accretion rate slightly decrease with decreasing , while at the same time the rate itself increases, it follows that the relative fluctuation decreases strongly with decreasing as can be seen in Fig. 14. No obvious trend seems visible on how the relative fluctuations change with gradient strength; a slight increase might be present when comparing the larger gradient fluctuation with the smaller gradient ones.

Fig. 14. 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 density 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 (1994) and Ruffert & Arnett (1994). Diamonds () denote models in which , triangles () models with , and squares () show The large bold symbols belong to models with a speed of , while the smaller symbols belong to models with .

The dependence of the relative fluctuations on velocity seems to invert: while they are clearly larger for smaller Mach numbers for the models, they are smaller for smaller Mach numbers in the models. And last, the simulations with produce relative fluctuations that vary more strongly with gradient strength than with Mach number.

4.2. Specific angular momentum

As has already been described in Sect. 3 the specific angular momentum reaches the analytic values given by Eq. (7) to within 10%, but only as long as the accretion flow is roughly stable and only for the models with small density gradients, . In Fig. 15 I show the specific angular momentum averaged over the whole time the models were evolved, excluding the initial transients. This gives a better picture of how the very unstable flow tends to decrease the average and produce a large fluctuation of the specific angular momentum. So I included as "error bars" the standard deviation of the fluctuations around the mean.

Fig. 15. The average accreted specific angular momentum (units: , Kepler velocity vortex at surface of accretor, as given by Eq. (10) in R1) 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 Eq. (7).

The unit I chose in Fig. 15 is the specific angular momentum that a vortex just at the surface of the accretor would have if it spun with the local Kepler velocity. Thus unity in these units indicates a Kepler orbit and matter that has more specific angular momentum than this would be flung off. One can see in Fig. 15 that the matter accreted is well below this value. For comparison, I also plot, using plus signs (+), the specific angular momentum as given by Eq. (7). These values tend to lie above the mean, especially for the models with large gradients (N,Q,U), and are well within one standard deviation from the mean for the small gradient models (M,P,T). For the latter models the fluctuations are so large that occasionally the sign of the accreted specific angular momentum reverses; this is indicated by the "error bars" extending completely down to the x-axis.

If one is only interested in the average specific angular momentum that is accreted, an inspection of Table 1 yields that the whole range of values between zero and about 70% is attained depending on the model parameters. Livio et al. (1986) reported values between 10% and 20% for their parameters.

If one reduces the size of the accretor, some point will be reached when the maximum specific angular momentum that can be accreted will become smaller than the amount present in the accretion cylinder. Setting Eq. (7) equal to Eq. (10) in R1 a relation is obtained between the gradients in the flow () and the radius of the accretor :

So for accretors smaller than this radius the angular momentum accretion should no longer be dominated by what is given in the accretion cylinder. For =0 and =0.03 and 0.2 we obtain and . Both these values are below what is currently possible to simulate numerically, but could be important in astrophysical objects. On the other hand for model VS, which has =0 and =1.0, an accretor radius of results, which is larger than the numerically used radius of . Thus in model VS not all the specific angular momentum offered in the incoming bulk flow can be accreted. Vice-versa, the maximum gradient that can be accommodated by an accretor of given radius is

The right panel of Fig. 9 compares the specific angular momentum accreted between the density gradient models presented here and the velocity gradient models shown in R1. The ratio (cf. legend of the x-axis of Fig. 9) is plotted of the specific angular momentum (values given in Table 1) for these two sets of models. If a pair of models experiences the same reduction in accretion of specific angular momentum, the ratio plotted would be one. If the density-gradient model of the pair suffers a greater decrease of accretion (due to e.g. stronger relative fluctuations of the unstable flow) this will be reflected by a ratio that is smaller than unity.

The pair MF/JS is less than zero, because the average angular momentum accreted by model MF is actually retrograde to the bulk flow momentum. For this pair, as well as for MS/IS, the very large `error' bar indicates that the fluctuations due to the unstable flow are very large compared to the average and so the latter value does not permit a strong statement. So although for four of the five models the ratios are above unity, which would indicate that the unstable flow actually increases the angular momentum accretion for the density gradient models as compared to the velocity gradient models, this reasoning is not a credible one. Additionally one has to keep in mind that the specific angular momentum of the incoming bulk flow of models KS and LS is actually larger than the maximum permitted by the Eq. 11. Thus the accreted momentum will be decreased due to this effect, too (angular momentum barrier), which explains why these model-pairs have the largest ratios.

4.3. Correlations

A correlation was found in R1 (Fig. 17) between the mass accretion rate and the specific angular momentum: the rate decreases when the magnitude of the momentum is largest. In Fig. 16 I draw a similar plot as in R1, each dot connecting the two quantities for every second timestep of the numerical simulation. A similar trend can be seen for model QS, which confirms that the dynamics is similar: when the flow does not rotate (low specific angular momentum) if falls down the potential to the surface of the accretor and thus produces a higher mass accretion rate. For all other models a correlation is not obvious.

Fig. 16. The mass accretion rate is plotted versus the specific angular momentum of the accreted matter for model QS. Each dot displays the two quantities at one moment in time. Only dots at time later than are plotted to avoid the initial transients.

© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999
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