David L. Richardson

Analytic construction of periodic orbits about the collinear points

A third-order analytical solution for halo-type periodic motion about the collinear points of the circular-restricted problem is presented. The three-dimensional equations of motion are obtained by a Lagrangian formulation. The solution is constructed using the method of successive approximations in conjunction with a technique similar to the Lindstedt-Poincaré method. The theory is applied to the Sun-Earth system.

A note on a Lagrangian formulation for motion about the collinear points

A lagrangian formulation for the three-dimensional motion of a satellite in the vicinity of the collinear points of the circular-restricted problem is reconsidered. It is shown that the influence of the primaries can be expressed in the form of two third-body disturbing functions. By use of this approach, the equations for the Lagrangian and for the motion itself are readily developed into highly compact expressions. All orders of the non-linear developments are shown to be easily obtainable using well-known recursive relationships. The resulting forms for these equations are well suited for use in the initial phase of canonical or non-canonical investigations.

Invariant Manifold Tracking for First-Order Nonlinear Hill's Equations

Abstract : An approach to provide nonlinear active control for the first-order nonlinear classical Hills equations is described. Both the linearized and nonlinear Hills equations are controlled to remain close to specific invariant manifolds defined through the various system Hamiltonians. It is then shown that trajectories similar to the periodic trajectories of the linearized system can be maintained by the nonlinear equations on invariant manifolds defined by the linearized system of equations. Forcing the nonlinear system trajectories onto an invariant manifold of the linearized system, with an appropriate choice of initial conditions, provides a significant reduction in the along-track drift of the first-order nonlinear Hills equations as compared to the linearized equations. There is also a small drift reduction in the radial coordinate direction. The cross-track position suffers only a slight increase in the maximum amplitude of its oscillation.

Chaos in the pitch equation of motion for the gravity-gradient satellite

The nonlinear dynamics of the pitch equation of motion for a gravity-gradient satellite in an elliptical orbit about a central body are investigated. This planar motion is shown to be either periodic, quasiperiodic, or chaotic, depending upon the values of eccentricity, satellite inertia ratio, and the initial conditions for pitch angle and its derivative with respect to the true anomaly. Bifurcation plots, PoincarC maps, and Lyapunov exponents are numerically calculated and presented. Chaos diagrams, which are computed from Lyapunov exponents and are dependent upon the satellites initial conditions, are also presented and may serve as a valuable satellite or orbit design tool. The sea of chaotic motion observed in the chaos diagrams has an interesting and complex structure. I t is found that the instability of the pitch angle for a gravity-gradient satellite generally increases for increasing values of orbit eccentricity.

A third-order intermediate orbit for planetary theory

By use of a new canonical transformation procedure, a third-order intermediary for planetary motion is developed. The intermediary contains all contributions that arise from the assumption of circular, coplanar orbits for the disturbing masses. The results are expressible in terms of elliptic integrals of the first, second, and third kinds.

Modification of the Richardson-Panovsky methods for precise integration of satellite orbits

Abstract A family of implicit methods based on intrastep Chebyshev interpolation has been developed to integrate initial value problems of the second-order harmonic oscillator form d 2 y dt 2 + ωy = f(y (t);t) . The procedure integrates the homogeneous part exactly (in the absence of roundoff errors). The Chebyshev approach uses stepsizes that are considerably larger than those typically used in Runge-Kutta or multistep methods. Computational overhead is comparable to that incurred by high-order conventional procedures. Chebyshev interpolation coupled with the iterative nature of the method substantially reduces local errors. Global error propagation rates are also reduced, making these procedures good candidates for use in long-term simulations of perturbed oscillator systems. The procedure is applied to integrations of the KS transformed equations for the oblateness-perturbed orbital motion of an artificial satellite.

A Simplified Variation of Parameters Approach to Euler’s Equations

A normalized form of Eulers equations is rewritten in a variation of parameters approach using amplitudes and angular displacement as parameters. This new form is compact and yields a more accurate numerically integrated solution over longer simulation times than does a conventional integration of the Euler equations.

The Hamiltonian transformation in quadratic lie transforms

The algorithm for Hamiltonian transformation in the quadratic perturbation technique of one of the authors admits of various equivalent forms. Using as a criterion the number of inter-term multiplications required for transformation, however, the amount of effort required to obtain the transformed Hamiltonian is not equivalent among these forms. Each is considered in some detail, and general guidelines for the choice of ‘most efficient’ algorithm to be used in a given problem are provided. Their utility is demonstrated by application to Duffings equation.

An analytical theory for the orbit of nereid

A complete analytical dynamic theory for the motion of Nereid has been constructed, accurate to approximately 0.01 arc second over several hundred years. The solution uses the Lie transform approach advanced by Deprit and is consistent with respect to the magnitudes of the disturbing functions, including all perturbations to an accuracy of 10−8 relative to the two-body potential (oblateness and third-body). Multiple short-period variables in the third-body perturbations are related via the ratio of their mean motions, reducing the number of independent variables. Extensive use is made of expansions giving trigonometric functions of the true anomaly as analytical Fourier series in the mean anomaly. Initial constants and mass parameters come from the data obtained during the Voyager II encounter with Neptune in 1989.