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

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3. Dynamics and accretion rates

3.1. Results of models with [FORMULA]=5/3 and 4/3

I will describe the results of the models for which a ratio of specific heats of [FORMULA]=5/3 was chosen together with those models of [FORMULA]=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 [FORMULA]) for models N and Q indicating the large gradient as compared to models M and P, where only the contour of [FORMULA] is seen to be detached from the shock. The higher densities on the lower side of y (where it is negative) seem to produce an asymmetric shock cone: the lower part of the cone subtends a smaller angle with the [FORMULA]-axis than the upper half. However, the quickly varying accretion flow pattern tends to impinge on the shock front and dislodge it. This asymetry has been observed and commented on in Ishii et al. (1993) and Soker & Livio (1984). In the models with smaller [FORMULA] (models P and Q) the shock cone tends to be narrower and its minimum distance from the accretor smaller than for the models M and N.

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 [FORMULA] 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 (z) for the specific angular momentum saturates at about 75% of the value estimated analytically by Eq. (7). This is probably due to the fact that for bulk velocities close to a Mach number of unity the Hoyle-Lyttleton approximation brakes down because a certain fraction of mass is accreted practically spherically symmetrically, i.e. as Bondi tried to describe it with the approximative formula (Bondi, 1952). The other two components (x and y) fluctuate around zero (which is given by the symmetry of the boundary conditions), but indicate at a very early stage that the flow is mildly unstable.

[FIGURE] Fig. 2. Contour plots showing snapshots of the density together with the flow pattern for all models with an adiabatic index of 5/3. The contour lines are spaced logarithmically in intervals of 0.1 dex. The bold contour levels are sometimes labeled with their respective values (0.0, 0.2, and 0.4). Darker shades of gray indicate higher densities. 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. 3. The accretion rates of several quantities are plotted as a function of time for model MV ([FORMULA]=1.4, gradient of 3%, adiabatic index of [FORMULA]). 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: the top solid line is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952) and the lower one half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the left panel 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).

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 [FORMULA] display a fairly quiet initial transient phase. The x and y-components of the angular momentum fluctuate mildly around zero and the z-component reaches fairly precisely (to within 10%) the analytic estimate Eq. (7). This shows the applicability of the analytic result only to the initial quasi-stationary state. The models with large [FORMULA] do not have this quiet phase but become chaotic much quicker. This decreased stability also produces lower accretion rates for mass as well as lower specific angular momenta. The initial quiet transient phase has already been seen in R1: compare the present models to the ones from R1, IS, JS, KS, LS in Figs. 4, 6, 8, and 9 in R1, respectively.

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 [FORMULA], again ceteris paribus . The same applied to the velocity gradient models of R1, cf. model IS and SS in Figs. 4 and 11 respectively. This dependence of the accretion rate on the adiabatic index [FORMULA] is well known for stationary flows from analytic calculations, e.g. Bondi (1952) for spherically symmetric flows and Sects. 5 and 6 and Figs. 9 and 10 in Foglizzo & Ruffert (1997).

[FIGURE] Fig. 4. The accretion rates of several quantities are plotted as a function of time for the moderately supersonic ([FORMULA]=3) models MS and NS with an index [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: half the value of the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952). The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels. The horizontal line in the bottom panels show the specific angular momentum value as given by Eq. (7).

[FIGURE] Fig. 5. The accretion rates of several quantities are plotted as a function of time for the highly supersonic ([FORMULA]=10) models MF and NF with an adiabatic index of [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: half the value of the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952). The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels; the horizontal line shows the specific angular momentum value as given by Eq. (7).

[FIGURE] Fig. 6. Contour plots showing snapshots of the density together with the flow pattern in a plane containing the centre of the accretor for all models with an adiabatic index of 4/3. The contour lines are spaced logarithmically in intervals of 0.1 dex. The bold contour levels are labeled with their respective values (0.0 or 1.0). Darker shades of gray indicate higher densities. 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. 7. The accretion rates of several quantities are plotted as a function of time for the moderately supersonic ([FORMULA]=3) models PS and QS with an adiabatic index of [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels; the horizontal line shows the specific angular momentum value as given by Eq. (7).

[FIGURE] Fig. 8. The accretion rates of several quantities are plotted as a function of time for the highly supersonic ([FORMULA]=10) models PF and QF with an adiabatic index of [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels; the horizontal line shows the specific angular momentum value as given by Eq. (7).

[FIGURE] Fig. 9. Left panels: The accretion rates of several quantities are plotted as a function of time for the model with largest gradient ([FORMULA]). The top panel contains the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panel, the straight horizontal line shows the analytical half the Bondi-Hoyle value from the approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952). The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top panel trace the x (dotted), y (thin solid) and z (bold solid) component of the angular momentum accretion rate. The same components apply to the bottom panel. Right panel: Comparison of the ratio of specific angular momenta accreted for models with density gradients (MS, MF, PS, NS, NF) to models with velocity gradients (IS, JS, SS, KS, LS taken from paper R1), keeping all other parameters equal.

3.2. Results of models with [FORMULA]=1.01

The 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.

[FIGURE] Fig. 10. Contour plots showing snapshots of the density together with the flow pattern in a plane containing the centre of the accretor for all models with an adiabatic index of 1.01 The contour lines are spaced logarithmically in intervals of 0.1 dex. The bold contour levels are labeled with their respective values (0.0 or 1.0). Darker shades of gray indicate higher densities. 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.

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 [FORMULA] 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 [FORMULA]. The z-component of the specific angular momentum in the models with large gradient (US, UF) practically never reach the analytical estimate Eq. (7) while the models with small gradient (TS, TF) fluctuate strongly about this value, exceeding it by a factor of three or even reversing sign.

[FIGURE] Fig. 11. The accretion rates of several quantities are plotted as a function of time for the moderately supersonic ([FORMULA]=3) models TS and US with an adiabatic index of [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels; the horizontal line shows the specific angular momentum value as given by Eq. (7).

[FIGURE] Fig. 12. The accretion rates of several quantities are plotted as a function of time for the highly supersonic ([FORMULA]=10) models TF and UF with an adiabatic index of [FORMULA]. The top panels contain the mass and angular momentum accretion rates, the bottom panels the specific angular momentum of the matter that is accreted. In the top panels, the straight horizontal lines show the analytical mass accretion rates: dotted is the Hoyle-Lyttleton rate (Eq. (1) in Ruffert 1994), solid is the Bondi-Hoyle approximation formula (Eq. (3) in Ruffert 1994; Bondi 1952) and half that value. The upper solid bold curve represents the numerically calculated mass accretion rate. The lower three curves of the top 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 bottom panels. The horizontal line in the bottom panels show the specific angular momentum value as given by Eq. (7).

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 z-component is not zero in flows with gradients. Fig. 11 can also directly be compared with Fig. 11 from Ishii et al. (1993). Both the mass and angular momentum accretion rates show very similar magnitudes and evolution.

3.3. Results of model VS

The density gradient in the previous set of models of [FORMULA] and [FORMULA] 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 [FORMULA] 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 [FORMULA]). 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 [FORMULA] (Fig. 1): [FORMULA] for model QS while [FORMULA] 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 [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999
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