## 2. Numerical procedure and initial conditionsSince the numerical procedures and initial conditions are mostly identical to what has already been described and used in previous papers (e.g. Ruffert, 1996; Ruffert & Anzer, 1994) I will refrain from repeating every detail, but only give a brief summary. ## 2.1. Numerical procedureThe distribution of matter is discretised on multiply nested equidistant Cartesian grids (e.g. Berger & Colella, 1989) with zone size 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 of or (see Table 1). The model of the maximally accreting, vacuum sphere in a softened gravitational potential is summarized in Ruffert & Arnett (1994) and Ruffert & Anzer (1995).
A gravitating, totally absorbing "sphere" moves relative to a medium that far upstream has a distribution of density and velocity given by with the redefined accretion radius In this paper I only investigate models with gradients of the velocity distribution; the values of can be found in Table 1. Thus for all models I set . Additionally, if only a density gradient is introduced without varying some other thermodynamic variable (e.g. temperature, entropy, etc.) at the same rate, pressure will not be in equilibrium (cf. e.g. Ho et al. 1989), so an additional thermodynamic variable should be varied, which complicates matters. The function " " is introduced in
Eq. (4) to serve as a cutoff at large distances When using the density and velocity distributions (Eqs. (3) and (4)) to calculate the mass accretion rate, assuming that all mass within the accretion cylinder is accreted, one obtains to lowest order in 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 1994; Shapiro & Lightman 1976; again to lowest order in ) Note the different signs with which the two
enter this equation. The density gradient acts
in the "expected" direction: if the density is higher on the positive
side of the The values obtained from the numerical simulations of the specific
angular momentum should 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
probably correct. Ruffert & Anzer (1995) find that the numerical
value is 0.72 times
One can numerically approximate the integrals (U. Anzer,
personal communication) of the mass flux and angular momentum over the
deformed cross section of the accretion cylinder, to obtain the
coefficients Here, I will only consider the effect of a velocity gradient. The
unitless functions
## 2.2. ModelsThe combination of parameters that I varied, together with some
results are summarised in Table 1. The first letter in the model
designation indicates the Mach number and the strength of the
gradient: I, J, and S have , while K, L, and R
have . The second letter specifies the size of
the accretor: M (medium) and S (small) stand for accretor radii of
0.1 , and 0.02
, respectively. I basically simulated models
with all possible combinations of two relative wind flow speeds (Mach
numbers of 3 and 10), two gradient strengths (3%
and 20%) and two different accretor sizes (0.02 and
0.1 accretion radii), all with an adiabtic index of 5/3. The
exeptional models are ST and SS - in which I used an
adiabatic index of 4/3 - and model RL which has a very large
accretor radius and a very slow relative flow velocity. The grids are
nested to a depth 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 center (crossing time scale). This time is given by and ranges from about 1 to about 10 time units. The actual time that the model is run can be found in Table 1. The velocity distribution following the "tanh"-prescription of Eq. (4) has an inflection point and thus is Kelvin-Helmholz unstable with an amplification time constant of roughly (Drazin & Reid 1981). During the time it takes matter to move from the boundary to the accretor, , random perturbations can grow by about which is smaller than unity even for the large gradients () used in the simulations and listed in Table 1. 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 40 CPU-hours per simulated time unit (for the models and Mach 10; the models take four times as long, etc.; is the size of a zone on the finest grid, see Table 1). © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |