A method of calculating fragment velocity additions for the analysis of the breakup of a launch vehicle upper stage

Abstract The estimation of velocity changes in an orbital breakup event of any space object is an important activity in the field of space debris. There are various methods described in the literature to estimate velocity additions. In one method, the normal, radial and tangential components of fragment velocity additions are estimated from the differences in semi-major axis, eccentricity and inclination with respect to those of parent object. Unfortunately, when the parent orbit has small eccentricity or the breakup occurs either near apogee or near perigee, the computation of radial component of velocity increment encounters a singularity and hence is prone to errors because of imperfect orbital element data. In another method, velocity increments are estimated from semi-major axis, eccentricity, inclination, right ascension of the ascending node and true anomaly of fragments as well as parent at the time of breakup. It is very difficult to obtain the true anomaly of the fragments at the time of breakup by propagating the element sets of debris pieces backward in time owing to uncertainties in orbital elements and drag parameters. Moreover, whenever the apogee of the fragment orbit becomes less than the perigee of the parent orbit due to various perturbations or errors, this method does not yield any solution for the radial component of the velocity increment. In this paper, first, a linear method is described to estimate the components of velocity addition, utilizing relations involving the differences in semi-major axis, eccentricity, inclination and argument of perigee in a particular combination and sequence, which avoids the singularity in the computation. Later, a fully nonlinear method to obtain velocity additions without any assumption is presented, which is found to be superior to any of the methods described in the literature so far.