Barycentric Synodic Coordinate System

Author: 天疆说
Reference: 钱霙婧(2014)《地月空间拟周期轨道上航天器自主导航与轨道保持研究》
Website: https://cislunarspace.cn

Definition

The Barycentric Synodic Coordinate System has its origin at the Earth-Moon system barycenter and is a rotating coordinate system used as the standard coordinate system for studying the Circular Restricted Three-Body Problem (CR3BP). Under CR3BP conditions, this coordinate system is also called the Earth-Moon system barycentric synodic coordinate system.

In the barycentric synodic coordinate system, the $x_m$ axis points from the barycenter toward the Moon, the $z_m$ axis points in the direction of the system's angular velocity (perpendicular to the Earth-Moon orbital plane), and the $y_m$ axis forms a right-handed Cartesian coordinate system with $x_m$ and $z_m$ .

Coordinate Axis Definition

Axis	Direction	Description
$x_m$ axis	From Earth-Moon barycenter toward Moon	Rotates with lunar revolution
$z_m$ axis	In the direction of system angular velocity	Perpendicular to Earth-Moon orbital plane
$y_m$ axis	$x_m \times z_m$	Forms right-handed system with $x_m$ and $z_m$

Differences from GRC

Characteristic	Barycentric Synodic	Geocentric Rotating (GRC)
Origin	Earth-Moon system barycenter	Earth center
x-axis direction	Toward Moon	Toward Moon
Application	CR3BP analysis	Orbit design under actual ephemeris conditions

The barycentric synodic coordinate system uses the common barycenter of the Earth-Moon system as its origin, making it more suitable for analyzing the motion of two primary bodies under the gravitational influence of the third body.

Applications in CR3BP

Normalized Units

In the barycentric synodic coordinate system, common normalized units for CR3BP are:

Length unit [L]: Distance between Earth and Moon
Mass unit [M]: Sum of Earth and Moon masses
Time unit [T]: $\sqrt{[L]^3 / (G[M])}$

Equations of Motion

In the barycentric synodic coordinate system, the CR3BP equations of motion have standard form:

\ddot{\boldsymbol{\rho}} - 2\boldsymbol{\Omega} \times \dot{\boldsymbol{\rho}} = \nabla U^*(\boldsymbol{\rho})

Where $\boldsymbol{\Omega} = [0, 0, 1]^T$ is the angular velocity vector (normalized), and $U^*(\boldsymbol{\rho})$ is the pseudo-potential energy function.

Libration Point Positions

In the barycentric synodic coordinate system, L1, L2, and L3 points lie on the $x_m$ axis, while L4 and L5 points form an equilateral triangle with Earth and Moon.

Limitations

The barycentric synodic coordinate system is based on CR3BP assumptions:

Both primary bodies orbit the barycenter in circular motion
The barycenter is an inertial point
The orbital plane is fixed in inertial space

For the real Earth-Moon system, factors such as lunar orbital eccentricity and solar gravitational perturbations mean these assumptions are not fully satisfied, requiring analysis using GRC or ephemeris models.

References

钱霙婧. 地月空间拟周期轨道上航天器自主导航与轨道保持研究[D]. 哈尔滨工业大学, 2014.
Szebehely V. Theory of orbits: the restricted problem of three bodies[M]. Academic Press, 1968.