Geocentric Rotating Coordinate System (Geocentric Rotating Coordinate System, GRC)

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 Geocentric Rotating Coordinate System (GRC), also known as the Geocentric Synodic Coordinate System, is one of the commonly used non-inertial coordinate systems in cislunar orbital mechanics research. GRC takes the Earth center as the origin, with the x-axis pointing from Earth center toward the Moon, the z-axis pointing in the direction of instantaneous lunar orbital angular momentum, and the y-axis forming a right-handed Cartesian coordinate system with x and z axes.

In the Circular Restricted Three-Body Problem, GRC simplifies to the Geocentric Synodic Coordinate System with fixed angular velocity direction; under actual ephemeris conditions, GRC's angular velocity varies with time due to lunar orbital precession and nutation.

Coordinate Axis Definition

AxisDirectionDescription
x-axisFrom Earth center toward MoonVaries with lunar position in real-time
z-axisDirection of instantaneous lunar orbital angular momentumPerpendicular to lunar orbital plane
y-axisx × zForms right-handed system with x and z

Relationship with J2000 Inertial Coordinate System

The transformation between GRC and J2000 Geocentric Inertial Coordinate System involves:

  1. Lunar position: Obtain lunar position in J2000 system RM\mathbf{R}_M from ephemeris data
  2. Lunar velocity: Obtain lunar velocity in J2000 system VM\mathbf{V}_M from ephemeris data
  3. Angular velocity calculation: ω=RM×VM/∣RM∣2\boldsymbol{\omega} = \mathbf{R}_M \times \mathbf{V}_M / |\mathbf{R}_M|^2

The transformation matrix is determined by the Moon's position and velocity vectors.

Application in Orbit Design

Dynamical Advantages of Synodic Coordinate System

In the Geocentric Synodic Coordinate System, the libration point problem simplifies to gravitational attraction in a rotating coordinate system. CR3BP equations of motion have a concise form in GRC:

r¨−2ω×r˙=∇U\ddot{\mathbf{r}} - 2\boldsymbol{\omega} \times \dot{\mathbf{r}} = \nabla U

Where ω\boldsymbol{\omega} is the angular velocity vector, and UU is the gravitational potential function.

Instantaneous Libration Points

In GRC, libration point positions are functions of time (instantaneous libration points), because the lunar orbital motion causes the x-axis direction to continuously change. This differs from fixed libration point positions in CR3BP.

Coordinate Transformation Considerations

钱霙婧 (2014) pointed out that traditional methods make a two-body assumption for angular velocity during coordinate transformation between GRC and J2000, which may introduce errors in certain cases.

GRC and Other Coordinate Systems

Coordinate SystemOriginCharacteristics
GRCEarth centerx-axis toward Moon, z-axis toward lunar orbital angular momentum
LRCL2 pointx, z, y axes parallel to GRC
Barycentric SynodicEarth-Moon barycenterUsed for CR3BP analysis
J2000Earth centerInertial system, X-axis toward mean vernal equinox

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.