Analytic theory of orbit contraction due to atmospheric drag

Abstract

The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by the same forces, namely, gravitational and aerodynamic. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. While previous authors, and in particular King-Hele, integrated this equation using various heuristic methods, the present authors present a rigorous analytic solution, with a high degree of accuracy, using Poincaré's method of small parameters. Next, using Lagrange's expansion, the major axis is expressed explicitly as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. This is a major achievement due to the discovery of a certain recurrence formula which facilitates the long and tedious analytic process. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. To obtain the same equation King-Hele must use an entirely different method. Again, while King-Hele only succeeded in obtaining an approximate solution to this case using a heuristic method of integration, the exact solution to the asymptotic equation has been obtained by the present authors. Numerical solutions have been generated to display the accuracy of the analytic theory. The explicit solution has been derived using a spherically symmetrical atmosphere with exponential variation of density with height but the basic equations developed and the technique for their integration apply to the case of an oblate atmosphere which is locally exponential.