Shannon L. Coffey

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ2 throughJ9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J2 only). In particular, we examine the manner in which the odd zonalJ3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vintis problem (J4+J22=0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vintis problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.

Elimination of the perigee in the satellite problem

Developed in this paper is a new approach to an analytic satellite theory which is based on Deprits elimination of the parallax. The first step in the theory is the elimination of the parallax canonical transformation which eliminates the short period terms in the perturbations to within a factor of (1/r)2. A new approach is then taken. The perigee terms are eliminated while retaining the short period terms in (1/r)2. A Delaunay normalization of the short period terms in the (1/x)2 factor is then constructed to complete the theory.

An analytical theory for tesseral gravitational harmonics

An analytical method has been developed for the treatment of tesseral harmonic perturbations. The procedure is an iterative Lie transformation technique which avoids the typical eccentricity expansions as well as the numerical singularities normally associated with resonance conditions. At each iteration, terms of the perturbing potential become multiplied by the ratio of the satellites orbital period to the earths rotational period. Following a suitable number of iterations, the potential is deemed to be sufficiently small that it may be ignored, with the tesseral effects captured in the transformation.

Applications of Parallel Processing to Astrodynamics

Parallel processing is being used to improve the catalog of earth orbiting satellites and for problems associated with the catalog. Initial efforts centered around using SIMD parallel processors to perform debris conjunction analysis and satellite dynamics studies. More recently, the availability of cheap supercomputing processors and parallel processing software such as PVM have enabled the reutilization of existing astrodynamics software in distributed parallel processing environments, Computations once taking many days with traditional mainframes are now being performed in only a few hours. Efforts underway for the US Naval Space Command include conjunction prediction, uncorrelated target processing and a new space object catalog based on orbit determination and prediction with special perturbations methods.

The Quadratic Zeeman Effect in Moderately Strong Magnetic Fieldsa

Over the last several years, celestial mechanics has made a fresh and candid assessment of the discipline’s resources and of the conditions facing its research. Out of this examination, there has come agreement on strategies that we believe hold the greatest promise for improving the value of our heritage in mathematical astronomy by moving the research to the front ranks of nonlinear dynamics. Implementing these strategies will require bold strokes of creativity and acts of difficult programming that will unfold in the years ahead. Our focus is on three strategic priorities:

Determination of TiPS libration using laser radar

This paper describes the use of satellite laser ranging data for determining the orbital and librational motion of the Tether Physics and Survivability (TiPS) experiment. We provide a description of new analytical tools and methodologies developed for determining the dynamics of the system. The results gleaned from a years worth of laser tracking are presented to provide a history of the librational and rotational motion of the TiPS system. Finally, the case for using SLR for libration determination on future tether experiments is presented.

The Nature of the Critical Inclinations in Artificial Satellite Theory

Mainfolds of orbits with stationary perigees are intrinsic features of the averaged main problem in aritifical satellite theory: they bifurcate off the manifold of circular orbits at the points where stability flips to instability and vice-versa.

On the Development of an Artificial Satellite Theory

An analytic theory for an artificial satellite is developed that can accomodate all of the perturbations due to the earth’s gravitational potential. The short period terms are eliminated by two canonical transformations of the Lie type while a third transformation produces the secular Hamiltonian. The literal expressions are explicitly developed by computer in two cases, for the main problem in the theory of an artificial satellite and for several of the tesseral harmonics.