Accounting for Solar Radiation Pressure in the Hamiltonian Normal Form of the Elliptic Restricted Three-Body Problem

TL;DR

This work extends Hamiltonian normal-form techniques to the elliptic restricted three-body problem (ER3BP), deriving both Birkhoff and resonant normal forms about the collinear libration points to enable analytic trajectory and invariant-manifold characterization. It introduces a complete Floquet-based framework for the quadratic part, followed by real- and complex-coordinate transformations and Lie-series eliminations to obtain action-angle representations that reveal center-manifold structures and their dynamics, including Lyapunov, vertical, halo, Lissajous, and quasihalo trajectories. To increase realism, the authors incorporate a simple solar radiation pressure model by adjusting the SRP parameter $\Theta$, which shifts libration points and modifies the quadratic Hamiltonian, then validate the approach by projecting ephemeris data (e.g., JWST) into the normal-form spaces; results show ER3BP with SRP yields closer agreement to full ephemeris than CR3BP alone. The study demonstrates that SRP can be effectively integrated into the ER3BP normal form with only modest additional computational effort, improving fidelity for mission design and optimal-control applications near Sun–Earth libration points.

Abstract

Hamiltonian normal forms allow for the analytical approximation of center manifold trajectories and their invariant manifolds through the separation of the saddle and center subspaces that make up the dynamics at the collinear libration points within the elliptic restricted three-body problem. The circular restricted three-body problem is a special case of the elliptic problem-- one that does not take into account the eccentricity of the true orbits of the primaries and thus provides a dynamical model of varying accuracy depending on the true anomaly of the primaries. This paper first shows that the normal forms of the elliptic problem offer nearly identical trajectory characterization capabilities to those of the circular problem and then demonstrates the difference in fidelity by comparing the circular and elliptic normal form representations of ephemeris data for the James Webb Space Telescope. Furthermore, methodology for including solar radiation pressure within the normal form is introduced, and the same ephemeris data is used to demonstrate the resulting increase in fidelity of the dynamical model.

Figures (22)

• Figure 1: Radii of validity of the action-angle Hamiltonians for several systems.
• Figure 2: Earth-Moon $L_2$ ER3BP Birkhoff normal form horizontal Lyapunov trajectory $(I_2,I_3,\phi_2(0)) = (0.2,0,0)$.
• Figure 3: Earth-Moon $L_2$ ER3BP resonant normal form vertical Lyapunov trajectory $(\hat{I}_2,\hat{I}_3,\theta_3(0)) = (0,0.2,0)$.
• Figure 4: Northern Earth-Moon $L_2$ ER3BP resonant normal form halo trajectory $(\hat{I}_2,\hat{I}_3,\theta_2(0)) = (0.2398,0.3,\pi/2)$.
• Figure 5: Earth-Moon $L_2$ ER3BP resonant normal form Lissajous trajectory $(\hat{I}_2(0),\hat{I}_3,\theta_2(0)) = (0.5,0.7,0)$.