The Efficient Analytic Computation of Fractional Reentering Debris from an Idealized Isotropic Explosion in General Elliptic Orbit

摘要

The efficient computation of the fraction of debris from an isotropic explosion in general elliptic orbit, that would fly in orbits whose perigees are below a certain given altitude is presented. Given an explosion velocity, a spacecraft idealized as a sphere breaks up into a myriad of small fragments, which either stay in earth orbit or reenter the atmosphere.rnTwo angles φ and θ_s uniquely define the orientation of the explosion velocity with respect to the pre-explosion velocity in a suitable rotating orbital reference frame. The locus of all points on the surface of the idealized sphere that separate the fragments that reenter the atmosphere from the fragments that stay in orbit has been obtained previously as an expression relating φ in terms of θ_s. The debris fragment counts on either side of the locus curve on the idealized sphere being proportional to the respective areas enclosed by the curve, it is shown that the area calculations are made possible if θ_s is instead cast in terms of φ because otherwise the angle φ can be multivalued in θ_s complicating the area evaluations. By contrast, an analytic expression in the form of a quartic in the cosine of θ_s is obtained and solved analytically for those values of φ called by the quadrature routine. Two real solution branches for cos θ_s are thus generated with one branch valid for a certain range in φ, while the other branch provides the required solution for the complement of that range.rnThe percentages of the debris fragments emanating from either side of the separating curve on the idealized sphere are thus evaluated rapidly through a few calls to the quartic solver routine. This analytic method provides identical accuracy percentages counts as a purely numerical method and it can readily be extended to analyze non-isotropic explosions as well.