## 3. Dynamics and accretion rates## 3.1. Results of models with =5/3 and 4/3I will describe the results of the models for which a ratio of specific heats of =5/3 was chosen together with those models of =4/3 because the evolution is very similar. The only exception is model MV: its slow relative bulk velocity of Mach 1.4 produces a significantly more stable flow. Figs. 2 and 6 show snapshots of the flow velocities and density distribution in the x-y-plane containing the accretor. The velocity pattern as well as the density contours within the shock cone indicate a strongly unstable flow. Also the shock cone itself has many bumps and kinks. Note how the density contours strongly bend over upstream of the
shock (at ) for models N
and Q indicating the large gradient as compared to models M
and P, where only the contour of
is seen to be detached from the
shock. The higher densities on the lower side of In contrast to what has been said, model MV with a slow (but
still supersonic relative velocity), shows a very regular flow
pattern: the stagnation point is about
0.3 downstream from the
accretor. Matter that comes within this point gets accreted while
matter that stays outside just passes the accretor. The mass accretion
rate of model MV (cf. Fig. 3) rises slowly and nearly
monotonically to saturate close to the Bondi-Hoyle formula value. Only
towards the end is there an indication that the flow might become
unstable for this model too. The component of interest (
The other 8 models (MS, MF, NS, NF, PS, PF, QS, QF) show very strong fluctuations of the accretion rates of mass and all angular momentum components (Figs. 4, 5, 7, and 8). A variation of factors of two is not uncommon, so the averages stated in Table 1 should be used with care and where possible the standard deviations taken into account. I include the accretion rate plots for all models in order to facilitate the judgement of how representative the average values are for the whole temporal evolution. A few trends can be discerned. All four models MS, MF, PS, PF,
i.e. the ones with small
display a fairly quiet initial transient phase. The Note that model PF is one for which the simulation was run fairly long compared to the timescale of fluctuations. This increases the confidence that the average mass accretion rate is a representative value and not a random one of a transient state. The angular momenta, however, still do not display the marginal positive shift that the z-component should have compared to the other two. No significant difference is observed when comparing two models
that only differ in Mach number. However the mass accretion rates are
significantly larger in the models with smaller
, again
## 3.2. Results of models with =1.01The first main obvious difference between the nearly isothermal models and the more adiabatic ones is that the shock cone is attached to the accretor, as can be seen in Fig. 10. The pressure around the accretor in this case is not sufficient to push away and support the shock cone. Not even the large density gradient is able to dislodge the shock from touching the surface of the accretor.
Both models with high Mach number (TF and UF) hardly show any activity of unstable flow within the shock cone, contrary to the moderately supersonic cases (TS and US). Again (as in model MV) a clear and stable stagnation point is present downstream of the accretor for these quiet models. It is a common feature of practically all simulations (including the ones by other authors) that when the shock cone is attached to the accretor no (or hardly any) instability is observed. Whether or not the shock cone is attached depends on many physical attributes of the models (e.g. stiffness of the equation of state, etc.) and numerical parameters (not least resolution). However, when these conditions collude to produce an attached shock, invariably the flow remains stable. The very active flow of the slower models (TS and US) reflects
itself in a higher variability of the mass accretion rate as compared
to models TF and UF (Figs. 11 and 12). However the
fluctuation of the mass accretion rates of all these
models is much smaller than the
fluctuations shown by the more adiabatic models described further
above. The average mass accretion rates of the slow models slightly
exceed the values predicted by the Hoyle-Lyttleton theory, while the
faster models only reach 80% of .
The
These models can be compared to models without gradients
(Ruffert, 1996) but with the same remaining parameters:
models TF and UF should be compared to model HS in Fig.
7e, while models TS and US can be compared to model GS
in Fig. 5e of Ruffert (1996). Both the different behaviour of the mass
accretion rates as well as the amplitudes of fluctuation of the
angular momentum accretion rates are comparable between the models,
indicating that the presence of a density gradient does not
significantly alter the accretion properties of such a strongly
unstable flow. Of course, the mean of the ## 3.3. Results of model VSThe density gradient in the previous set of models of and was chosen in order to facilitate the direct comparison between the results presented in this paper and the ones with velocity gradients shown in R1. However, Eq. 7 then predicts that the specific angular momentum accreted will be six times smaller for the density gradients as compared to accretion with velocity gradients. That this is true becomes clear when comparing the plots for models MS, MF, PS, NS and NF with equivalent models from paper R1: IS, JS, SS, KS and LS, respectively: In the models presented in this paper, the z-component of the angular momentum, which is the one influenced by the gradient, hardly rises above the random fluctuations of the other two components. In order to check the correct separation of the effect of the density gradient from the unstable nature of the flow, a model with larger gradient is helpful and will be presented in this subsection. The particular value of is suggested because for this case, Taam & Fryxell (1989) have found that a quasi-steady disk forms which does not change its sense of direction. Fig. 9 (left panels) shows the temporal evolution of the mass and angular momentum accretion for this large density model. The mass accretion rate remains very low over the whole simulated time on average being less than half the value of model QS (which is similar to VS in all parameters except ). Note that these average values listed in Table 1 are normalised to Eq. 7 and not to Eq. 9. Thus a big part of the reduction of specific angular momentum between model QS (0.55) and model VS (0.23) is probably due to the "tanh"-term represented by the factor (Fig. 1): for model QS while for model VS. Also note that model VS does not display as drastic a reduction in mass accretion rate as shown in Fig. 22 of Taam & Fryxell (1988) [note the different choice of time units between this paper and the one in Taam & Fryxell, 1988]. The reason probably being that in their 2D models accretion gets effectively shut off as soon as a stable disk structure forms, while in my 3D simulations accretion can still proceed practically unimpeded via the poles. The lower left panel of Fig. 9 shows the angular momentum accretion. In this large gradient model, the z-component clearly dominates compared to the other two components, i.e. the fluctuating flow cannot compete with the angular momentum available in the bulk flow gradient. Note also that the z-component never crosses the line which indicates the formation of a disk-like flow structure with rotation in an unchanging sense. However, I do not expect the specific angular momentum to reach a quasi-steady state because of the three-dimensional nature of my simulations: angular momentum can continue to be accreted from directions outside the plane of the disk and thus probably disturbing the disk, too. © European Southern Observatory (ESO) 1999 Online publication: June 17, 1999 |