Quasi-Bicircular Problem

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

Definition

The Quasi-Bicircular Problem (QBCP) is a self-consistent dynamics model that accounts for solar gravitational perturbation in the Sun-Earth-Moon system. Proposed by Andreu and Simo in 1999, it is a refinement of the Bicircular Restricted Problem (BCP).

In the bicircular dynamics model, the two three-body models are independent of each other, which in fact does not satisfy Newton's second law, making the model non-self-consistent. To achieve self-consistency, the QBCP modifies the bicircular model so that the motions of the Sun, Earth, and Moon satisfy Newton's laws of motion, transforming the Hamiltonian function in the problem onto the central manifold, thereby obtaining a self-consistent dynamics description.

Hamiltonian Function

For the Sun-Earth-Moon system, the Hamiltonian of the Quasi-Bicircular model is:

H_{\mathrm{QBCP}} = \frac{1}{2}\alpha_1(p_x^2 + p_y^2) + \alpha_2(x p_x + y p_y) + \alpha_3(y p_x - x p_y) + \alpha_4 x + \alpha_5 y - \alpha_6\left( \frac{1 - \mu}{r_1} + \frac{\mu}{r_2} + \frac{m_{\mathrm{s}}}{r_{\mathrm{s}}} \right)

The parameters $\alpha_k$ are time-periodic:

\alpha_k(\theta) = \begin{cases} \alpha_{k0} + \sum\limits_{j \ge 1}\alpha_{kj}\cos(j\theta), & k = 1,3,4,6,7 \\ \sum\limits_{j \ge 1}\alpha_{kj}\sin(j\theta), & k = 2,5,8 \end{cases}

where $\theta = \omega_s t$ , and $\omega_s$ is the normalized solar motion frequency.

Relationship with Other Models

The QBCP occupies an intermediate position in the dynamics model precision hierarchy between CRTBP and the ephemeris model:

Model	Precision	Self-Consistency	Characteristics
CRTBP	Low	Self-consistent	Solar perturbation and lunar eccentricity ignored
BCP	Medium	Non-self-consistent	Solar influence introduced but two three-body models independent
QBCP	Medium-High	Self-consistent	Modified BCP to satisfy Newton's laws for three massive bodies
Ephemeris Model	High	Self-consistent	Uses real ephemeris data

Applications

Based on the QBCP model, Andreu discussed invariant manifolds and low-energy transfers near Earth-Moon libration points. Guzman used the multiple-shooting method to achieve periodic and quasi-periodic orbit design near Earth-Moon libration points in the QBCP model, with results close to real ephemeris cases, validating the feasibility of the quasi-bicircular model in orbit design.

The QBCP provides a good intermediate level between simplified models and high-precision ephemeris models, commonly used in model conversion strategies such as homotopy methods.

Circular Restricted Three-Body Problem (CR3BP)
Bicircular Restricted Problem (BCP)
Ephemeris Model
Homotopy Method

References

Andreu M A. The translunar halo orbit in the quasi-bicircular problem[D]. Universitat de Barcelona, 1999.
Andreu M A. Dynamics in the center manifold of the collinear points in the quasi-bicircular problem[D]. Universitat de Barcelona, 2002.
Guzman J. Spacecraft trajectory design in the context of a coherent restricted four-body problem[D]. Purdue University, 2001.