Dynamical friction in flattened systems:: a numerical test of Binney's approach

We carry out a set of self-consistent N-body calculations to investigate how important the velocity anisotropy in non-spherical dark matter haloes is for dynamical friction. For this purpose, we allow satellite galaxies to orbit within flattened and live dark matter haloes (DMHs) and compare the resulting orbit evolution with a semi-analytic code. This code solves the equation of motion of the same satellite orbits with mass loss and assumes the same DMH, but either employs Chandrasekhar's dynamical friction formula, which does not incorporate the velocity anisotropy, or Binney's description of dynamical friction in anisotropic systems. In the numerical and the two semi-analytic models, the satellites are given different initial orbital inclinations and orbital eccentricities, whereas the parent galaxy is composed of a DMH with aspect ratio q(h) = 0.6. We find that Binney's approach successfully describes the overall satellite decay and orbital inclination decrease for the whole set of orbits, with an averaged discrepancy of less than 4 per cent in orbital radius during the first three orbits. If Chandrasekhar's expression is used instead, the discrepancy increases to 20 per cent. Binney's treatment therefore appears to provide a significantly improved treatment of dynamical friction in anisotropic systems. The velocity anisotropy of the DMH velocity distribution function leads to a significant decrease with time of the inclination of non-polar satellite orbits. But, at the same time, it reduces the difference in decay times between polar and coplanar orbits evident in a flattened DMH when the anisotropic DMH velocity distribution function is not taken into account explicitly. Our N-body calculations furthermore indicate that polar orbits survive about 1.6 times longer than coplanar orbits and that the orbital eccentricity e remains close to its initial value if satellites decay slowly towards the galaxy centre. However, orbits of rapidly decaying satellites modelled with the semi-analytic code show a strong orbital circularization (e > 0) not present in the N-body computations.