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

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2. Numerical procedure and initial conditions

Since the numerical procedures and initial conditions are mostly identical to what has already been described and used in previous papers (cf. Ruffert 1997, R1, and references therein) I will refrain from repeating every detail, but only give a brief summary.

2.1. Numerical procedure

The distribution of matter is discretised on multiply nested equidistant Cartesian grids (e.g. Berger & Colella, 1989) with zone size [FORMULA] and is evolved using the "Piecewise Parabolic Method" (PPM) of Colella & Woodward (1984). The equation of state is that of a perfect gas with a specific heat ratio [FORMULA] (see Table 1). The model of the maximally accreting, vacuum sphere in a softened gravitational potential is summarised in Ruffert & Arnett (1994) and Ruffert & Anzer (1995).


Table 1. Parameters and some computed quantities for all models. [FORMULA] is the Mach number of the unperturbed flow, [FORMULA] the parameter specifying the magnitude of the density gradient, [FORMULA] the ratio of specific heats, [FORMULA] the total time of the run (units: [FORMULA]), [FORMULA] the integral average of the mass accretion rate, S one standard deviation around the mean [FORMULA] of the mass accretion rate fluctuations, [FORMULA] the maximum mass accretion rate, [FORMULA] is defined in Eq. (3) of Ruffert & Arnett (1994), [FORMULA], [FORMULA], [FORMULA], are the averages of specific angular momentum components together with their respective standard deviations [FORMULA], [FORMULA], [FORMULA], s is the entropy (Eq. (4) in Ruffert & Arnett 1994), the radius of the accretor always is [FORMULA], the number of grid nesting depth levels [FORMULA], the size of one zone on the finest grid [FORMULA], the softening parameter for the potential of the accretor (see Ruffert 1994) [FORMULA] zones, the number N of zones per grid dimension is 32, and the size of the largest grid is [FORMULA].

A gravitating, totally absorbing "sphere" moves relative to a medium that has a distribution of density and velocity far upstream (at [FORMULA]) given by



with the redefined accretion radius (note slight difference to Eq. 1)


In this paper I only investigate models with gradients of the density distribution, thus for all models I set [FORMULA]. The values of [FORMULA] can be found in Table 1. In order to keep the pressure constant throughout the upstream boundary, I changed the internal energy or equivalently the sound speed c accordingly ([FORMULA]). Thus the Mach number [FORMULA] of the incoming flow varies too, since the velocity is kept constant. The Mach numbers given in the Table 1 refer to the values along the axis (y=0) upstream of the accretor.

The function "[FORMULA]" is introduced in Eq. (3) and (4) to serve as a cutoff at large distances y for large gradients [FORMULA]. The units I use in this paper are (1) the on-axis sound speed [FORMULA] as velocity unit; (2) the accretion radius (Eq. (5)) as unit of length, and (3) the on-axis density [FORMULA]. Thus the unit of time is [FORMULA].

I'll assume both [FORMULA] and [FORMULA] for the following estimates. Accreting all mass within the accretion cylinder, and taking the matter to have the density and velocity distributions as given by Eqs. (3) and (4) one obtains the mass accretion rate to lowest order in [FORMULA] and [FORMULA] to be


an equation very similar to Eq. (2). Further assuming that all angular momentum within the deformed accretion cylinder is accreted too, the specific angular momentum of the accreted matter follows to be (Ruffert & Anzer 1995; Shapiro & Lightman 1976; again to lowest order in [FORMULA])


For positive [FORMULA] the density is lower on the positive side of the y-axis, then the vortex formed around the accretor is in the anticlockwise direction, i.e. the angular momentum component in z-direction is positive.

The values of the specific angular momentum obtained from the numerical simulations will be compared to the values that follow from this Eq. (7) to conclude which of the above mentioned views - low or high specific angular momentum of the accreted material - is most more appropriate. As was stated further above, R1 found that a sizable amount (between 7% and 70%) is accreted, when velocity gradients are present. I will implicitly assume the component [FORMULA] when discussing properties like fluctuations, magnitudes, etc. From the symmetry of the boundary conditions the average of the x and y components of the angular momentum should be zero, although their fluctuations can be quite large. Taking the numerically obtained accretion rates of mass [FORMULA] and angular momentum [FORMULA], I calculate the instantaneous specific angular momentum [FORMULA], the temporal mean l of which is listed in Table 1, too.

Apart from serving as cut-off, the tanh-dependencies in Eqs. (3) and (4) have a gradient that is slightly less steep than simply linear. Thus less specific angular momentum is present at a given radius from the accretor and smaller values in the magnitude of the accreted quantity result.

One can numerically approximate the integrals of the mass flux and angular momentum over the cross section of the accretion cylinder, to obtain the coefficients f in the relations Eq. (6) and Eq. (7):



Here, I will only consider the effect of a density gradient. The unitless functions f are a function of [FORMULA] and the functional relation of [FORMULA], i.e. whether [FORMULA] depends purely linearly on [FORMULA] or as in Eq. (3) via the "tanh"-term. Fig. 1 shows the values of the functions f for the mass and specific angular momentum and for both the linear and "tanh" case. The only minimal deviation of the "tanh" curves from the linear ones indicates that the tanh-cutoff hardly acts within the accretion cylinder. Since [FORMULA] and is practically constant for [FORMULA], Eq. (6) is a good approximation in this range. If the prescription is correct that everything in the accretion cylinder is accreted, no difference in the accretion rate should be observable in the models with differing magnitude of gradient [FORMULA] (cf. Table 1). The same constancy applies to the accretion of specific angular momentum: its coefficient [FORMULA] remains relatively constant [FORMULA] in the range [FORMULA], which includes both [FORMULA] for which models were simulated. So although I used the same magnitudes for the gradients, the effects I expect in the accretion rates are markedly different from what was presented in paper R1. Only for model VS with [FORMULA], does [FORMULA] deviate appreciably from 0.25: it is [FORMULA].

[FIGURE] Fig. 1. The coefficient for the mass accretion rate [FORMULA] (dotted), defined by Eq. (8), and for the specific angular momentum [FORMULA] (solid), defined by Eq. (9), as a function of [FORMULA]. The thin curves show the values of f for a simple linear relation between [FORMULA] and [FORMULA], while the bold curves apply to a relation including the "tanh"-term as given in Eq. (4). The two vertical straight lines indicate the gradients that were used in the numerical models (cf. Table 1), [FORMULA] and [FORMULA]. Compare this figure with Fig. 1 in paper R1 and note the differences.

2.2. Models

The combination of parameters that I varied, together with some results are summarised in Table 1. The first letter in the model designation indicates the strength of the gradient: M, P, and T have [FORMULA], while N, Q, and U have [FORMULA] and V is [FORMULA]. The second letter specifies the relative wind flow speeds, F (fast), S (slow) and V stand for Mach numbers of 10, 3 and 1.4, respectively. I basically simulated models with all possible combinations of the two higher flow speeds (Mach numbers of 3 and 10), the two gradients (3% and 20%) and varying the adiabatic index between 5/3, 4/3 and 1.01. Model VS with the largest gradient of 100% facilitates a comparison to a previous two-dimensional simulation by Fryxell & Taam (1988). The grids are nested to a depth [FORMULA] such that the radius of the accretor [FORMULA] spans several zones on the finest grid.

As far as computer resources permitted, I aimed at evolving the models for at least as long as it takes the flow to move from the boundary to the position of the accretor which is at the centre (crossing time scale). This time is given by [FORMULA] and ranges from about 1 to about 10 time units. The actual time [FORMULA] that the model is run can be found in Table 1, as well as the parameters of the grids (L, g, etc.).

When modeling a BHL flow with a density gradient, one has to pay attention to the fact that matter parcels with possibly very different densities (which initially are separated upstream) will be focussed to find themselves close to each other along the accretion axis (U. Anzer, personal communication). This might render the flow additionally unstable. Although true in principle, this problem does not affect, in practice, the models I will present: a closer inspection of Eq. (3) reveals that the density jump from one end of the accretion radius to the other is only a factor 1.5 for the case with large gradient ([FORMULA]). This does not seem to have an additional influence as compared to models with constant density presented in paper R1.

The calculations are performed on a Cray-YMP 4/64 and a Cray J90 8/512. They need about 12-16 MWords of main memory and take approximately 160 CPU-hours per simulated time unit.

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© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999