Jacobi Constant

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

Definition

The Jacobi constant (Jacobi integral) is the only conserved quantity (integral of motion) in the Circular Restricted Three-Body Problem (CR3BP). It is analogous to the combination of energy and angular momentum in the two-body problem and plays a central role in analyzing orbit topology.

Formula

C_J = 2U(x, y, z) - (\dot{x}^2 + \dot{y}^2 + \dot{z}^2)

where $U$ is the effective potential function:

U = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}

$r_1$ : distance to the larger primary body
$r_2$ : distance to the smaller primary body
$\mu$ : mass parameter

Physical Significance

The Jacobi constant determines the accessible region of an orbit through zero-velocity surfaces (ZVS):

When $C_J$ is large: the spacecraft is confined near one of the primary bodies or a libration point
When $C_J$ is small: the spacecraft can move freely throughout cislunar space
At critical values: the zero-velocity surfaces open at libration points, allowing transfers between regions

Role in Orbit Design

The Jacobi constant is a key parameter for orbit classification and design:

Different orbit families (DRO, Halo, Lissajous) occupy different ranges of $C_J$
Transfer orbits must satisfy energy constraints imposed by $C_J$
The value of $C_J$ determines which libration point regions are accessible

References

Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.