Statistical Distribution of Keplerian Velocities

Abstract

Introduction T HE use of generalized functions, in particular of Dirac’s delta functions, has already been proposed1−5 as a useful tool in simplifying the derivations of probability distribution functions related to the statistical description of Keplerian orbits. The formalism that derives from an extensive use of these functions permits an easy conversion from classical astrodynamic formulas to probability distributions. The methodology was independently discovered in different forms by Au and Tam,6 Meshcheryakov,4 and Izzo,1 and it has been used in various applications1−3,5 mainly related to the description of collision probabilities and to the modeling of the space debris environment. The limitations in the description of the Molnyia and geostationary orbits of the orbital debris environment model proposed by Kessler et al.7 were alleviated by using results derived with this methodology.3 These results also generalized previous expressions8−10 commonly used in space flight mechanics problems dealing with the statistics of a large number of orbits and objects. In the evaluation of collision risks as well as in the problem of inferring the orbital motion of space debris from a measurement campaign such as the one carried by the Long Duration Exposure Facility or by the more recent Space Dust11 instrument, it is often necessary to have statistical models also on the velocities of orbiting objects. In this Note Dirac’s delta formalism is applied to derive the absolute velocity distribution of an orbiting object having uncertain orbital parameters.