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Approximate Analytical Solutions for the Circular Restricted Three-Body Problem Including Non-Hamiltonian Solar Radiation Pressure

Hailee Hettrick, David W. Miller, Begum Cannataro

Abstract

The circular restricted three-body problem (CR3BP) with solar radiation pressure (SRP) has often been analyzed with assumptions made on a spacecraft's attitude, such that the problem remains Hamiltonian. These assumptions are unsatisfactorily limiting for a starshade mission since the starshade's attitude will inherently vary from the configuration that corresponds to Hamiltonian dynamics. This paper presents the derivation of the equations of motion for CR3BP with SRP that permit the application of the Lindstedt-Poincare method, such that approximate solutions are produced, which may serve as invaluable trajectory design tools. Examples of periodic orbits and manifolds corresponding to three sets of attitude angles are shown and the accuracy of their seventh-order approximations is considered.

Approximate Analytical Solutions for the Circular Restricted Three-Body Problem Including Non-Hamiltonian Solar Radiation Pressure

Abstract

The circular restricted three-body problem (CR3BP) with solar radiation pressure (SRP) has often been analyzed with assumptions made on a spacecraft's attitude, such that the problem remains Hamiltonian. These assumptions are unsatisfactorily limiting for a starshade mission since the starshade's attitude will inherently vary from the configuration that corresponds to Hamiltonian dynamics. This paper presents the derivation of the equations of motion for CR3BP with SRP that permit the application of the Lindstedt-Poincare method, such that approximate solutions are produced, which may serve as invaluable trajectory design tools. Examples of periodic orbits and manifolds corresponding to three sets of attitude angles are shown and the accuracy of their seventh-order approximations is considered.
Paper Structure (18 sections, 40 equations, 10 figures)

This paper contains 18 sections, 40 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Dynamical Model
  3. Artificial Equilibrium Points
  4. Finding Equivalent Formulation of the Equations of Motion
  5. Lagrangian
  6. Legendre Polynomials
  7. Applying Lagrange's Equations
  8. Recurrence Relations
  9. SRP Contributions
  10. Equations of Motion for Lindstedt-Poincaré Method
  11. Linearization
  12. Methodology for Finding Approximate Solutions
  13. Results
  14. Approximate Solutions for $\alpha = 80^\circ$, $\gamma = 0^\circ$
  15. Approximate Solutions for $\alpha = 0^\circ$, $\gamma = 40^\circ$
  16. ...and 3 more sections

Figures (10)

  • Figure 1: Cone and clock angles with respect to the Sun-Starshade line
  • Figure 2: Position of $H_2$ as it varies with respect to $\alpha$ and $\gamma$ for a constant $\beta$
  • Figure 3: Geometry of system with respect to $H_2$
  • Figure 4: Periodic Orbits for $\alpha = 80^\circ$, $\gamma = 0^\circ$
  • Figure 5: Manifolds for $\alpha = 80^\circ$, $\gamma = 0^\circ$
  • ...and 5 more figures