On the orbital periods for a particular case of spherical symmetry

Abstract

A particular case of steady state and spherical symmetry - the so-called logarithmic potential introduced as the first approximation for dark coronae of galaxies - is studied. Both time and angle dependence of the distance to the centre for the orbit of a bound test particle with arbitrary initial conditions are calculated numerically. The main attention is paid to the ratio of the sidereal period to the anomalistic one. It is found that this ratio is only slightly variable for a given mean distance to the centre and to increase with increasing orbital eccentricity. This quantitative result may be explained by the fact that the cumulative mass dependence on the distance corresponding to the logarithmic potential obeys a power law, the case where the ratio of the second derivative of the potential to the square of angular velocity for the same distance is constant. On the other hand, compared to the period of circular motion both periods increase with increasing eccentricity.