Distortion of Orbits Under Fluctuating Gravitational Forces

The motion of a celestial body under the influence of a central force is considered. The influence of other bodies on the gravitational attraction force of the two-body problem is modeled as fluctuating force perturbations about a mean value. The ordinary differential equation determining the orbits is derived in polar coordinates. The exact analytical solution is given. Three cases, namely the elliptic, parabolic and hyperbolic cases are investigated. The distortion of the orbits under the influence of small perturbational forces are depicted. The equation determining the escape angle is solved numerically by the Newton-Raphson method. Effects of fluctuation frequencies on the distortion of orbits as well as escape angles are studied in detail. It is found that the force fluctuations may alter the nature of the orbits and destroy their symmetries. When the fluctuation frequency is an integer number, the orbits are distorted but the escape angles remain unaffected for the parabolic case. For non-integer frequencies, the escape angles may decrease in magnitude.