Elliptic Restricted Three-Body Problem (ER3BP)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The Elliptic Restricted Three-Body Problem (ER3BP, also ERTBP) is the natural generalization of the CR3BP to elliptical orbits. The two primaries no longer revolve in uniform circular motion but instead follow Keplerian ellipses about their common barycenter (eccentricity $e \in [0,1)$ ), with all other restricted assumptions held.

This seemingly minor change fundamentally alters the mathematical character of the problem (Szebehely 1967; Broucke 1969):

The inter-primary distance $r_{12}$ is no longer constant but a $2\pi$ -periodic function of true anomaly $f$ : $r_{12}(f) = a(1-e^2)/(1+e\cos f)$ , where $a$ is the semi-major axis.
The system is no longer autonomous — the equations of motion depend explicitly on time (or equivalently on $f$ ), with primary positions varying periodically.
The Jacobi integral ceases to exist — energy is not conserved in the time-varying system; a spacecraft can change its mechanical energy without expending propellant.

When eccentricity is non-negligible (e.g., lunar orbital eccentricity $\approx 0.0549$ ), the ER3BP is more faithful to real dynamics than the CR3BP, but the analysis is substantially harder due to the loss of autonomy and conservation.

Pulsating Rotating Frame

To keep the primaries stationary in the reference frame, the ER3BP typically employs a pulsating synodic frame (dimensionless pulsating coordinate system). Unlike the constant-distance CR3BP synodic frame, the ER3BP scales distance and time units by the instantaneous inter-primary distance $r_{12}(f)$ , with true anomaly $f$ (or time $t$ ) as the independent variable (Szebehely 1967; Gómez et al. 2001).

In this frame, the nondimensionalized equations take the form:

\begin{cases} \tilde{x}'' - 2\tilde{y}' = \dfrac{1}{1+e\cos f} \dfrac{\partial \tilde{\Omega}}{\partial \tilde{x}} \\[1em] \tilde{y}'' + 2\tilde{x}' = \dfrac{1}{1+e\cos f} \dfrac{\partial \tilde{\Omega}}{\partial \tilde{y}} \\[1em] \tilde{z}'' + \tilde{z} = \dfrac{1}{1+e\cos f} \dfrac{\partial \tilde{\Omega}}{\partial \tilde{z}} \end{cases}

where primes denote derivatives with respect to $f$ , and $\tilde{\Omega}$ is formally identical to the CR3BP effective potential $\Omega$ , but the left-hand side gains a time-varying factor depending on $e$ and $f$ , plus an additional $\tilde{z}$ term absent in the CR3BP.

Floquet Theory and Periodic-Orbit Stability

Periodic orbits in the ER3BP must be $T=2\pi$ (in the $f$ -domain) periodic solutions. Because the system is explicitly $f$ -dependent (non-autonomous), stability cannot be assessed via the CR3BP eigenvalue method and instead requires Floquet theory.

Given a periodic reference orbit $\mathbf{x}_0(f)$ , the variational dynamics are governed by a linear system with $2\pi$ -periodic coefficients:

\delta \mathbf{x}'(f) = \mathbf{A}(f)\,\delta \mathbf{x}(f), \quad \mathbf{A}(f+2\pi) = \mathbf{A}(f)

Integrating over one full period yields the monodromy matrix $\mathbf{M}$ : $\delta \mathbf{x}(2\pi) = \mathbf{M}\,\delta \mathbf{x}(0)$ . Its eigenvalues are the Floquet multipliers; stability requires $|\lambda| \le 1$ for all of them. Unlike the constant Jacobian of the CR3BP, the ER3BP's $\mathbf{A}(f)$ varies periodically and must be integrated numerically over the full period. See Floquet Multiplier and Monodromy Matrix.

Relationship to the CR3BP

Limit case: As $e \to 0$ , $r_{12}$ becomes constant, the factor $1/(1+e\cos f) \to 1$ , and the pulsating frame reduces to the ordinary CR3BP synodic frame; the Floquet multipliers approach the eigenvalues of the constant CR3BP Jacobian.
Planar vs. spatial ER3BP: The Planar ER3BP (PER3BP) confines motion to the orbital plane and is analyzed via $f$ -section Poincaré maps; the spatial ER3BP retains full 3D degrees of freedom, yielding more complex orbit families.
Resonance effects: At certain eccentricities, ER3BP periodic orbits can deviate significantly or bifurcate from their CR3BP counterparts — an effect that long-duration mission design must account for.

References

Szebehely, 1967, Theory of Orbits: The Restricted Problem of Three Bodies — Chapter 10 systematically treats the ER3BP pulsating frame and equations of motion.
Broucke, 1969, "Periodic Orbits in the Elliptic Restricted Three-Body Problem" — An early systematic study of ER3BP periodic orbits.
Gómez et al., 2001, Dynamics and Mission Design near Libration Points, Vol. III — Discusses continuation and Floquet stability of ER3BP libration-point periodic orbits.
Campagnola, 2010, New Techniques in Astrodynamics for Moon Systems Exploration, Ph.D. — Contains practical numerical methods for ER3BP trajectory design.