Two-Body Problem

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

Definition

The two-body problem is a fundamental problem in celestial mechanics concerning the motion of two bodies (point masses) under mutual gravitational attraction alone. For aerospace vehicles in near-Earth space, the on-orbit motion can be approximated as a two-body problem: the Earth is treated as a homogeneous sphere (point-mass model), the vehicle as a point mass, and only central gravity is considered. Under these assumptions, the vehicle's trajectory is a Keplerian orbit, and the characteristics of two-body motion serve as the foundation for studying real-world orbital mechanics.

Core Elements

Fundamental Assumptions

Assumption	Description
Earth as a homogeneous sphere	Gravitational effect is equivalent to a point mass concentrated at the center
Vehicle as a point mass	Dimensions are negligibly small compared to the radial distance
Central gravity only	Perturbing forces such as atmospheric drag and third-body gravity are neglected

Differential Equations of Motion

In an inertial reference frame, the equation of motion of the vehicle relative to the Earth is:

\ddot{\boldsymbol{r}} + \frac{\mu_E}{r^3}\boldsymbol{r} = 0

where $\mu_E = GM_E = 3.986 \times 10^{14} \, \text{m}^3/\text{s}^2$ is the Earth's gravitational parameter. This is a sixth-order nonlinear ordinary differential equation that requires six independent integration constants (i.e., orbital elements) for a complete solution.

First Integrals

The two-body equations of motion admit the following conserved quantities:

Conserved Quantity	Mathematical Expression	Physical Meaning
Specific angular momentum	$\boldsymbol{h} = \boldsymbol{r} \times \boldsymbol{v} = \text{const}$	Orbit plane orientation is fixed
Specific mechanical energy	$\varepsilon = \frac{1}{2}v^2 - \frac{\mu_E}{r} = \text{const}$	Orbit size is fixed
Eccentricity vector	$\boldsymbol{e} = \frac{1}{\mu_E}\left(\boldsymbol{v} \times \boldsymbol{h} - \frac{\mu_E}{r}\boldsymbol{r}\right) = \text{const}$	Orbit shape and orientation are fixed

Application Value

The two-body problem is the theoretical cornerstone of orbital mechanics. Although real orbits are perturbed by the Earth's oblateness, atmospheric drag, solar and lunar gravity, and other effects, the two-body solution provides a zeroth-order approximation and a reference orbit. Core problems such as orbit design, orbit prediction, and orbit determination all begin with the two-body model. The conclusions of the two-body problem can be generalized to motion around any central body by simply replacing the corresponding gravitational parameter.

References

郑伟, 安雪滢, 周祥, 何睿智. 空天飞行力学[M]. 国防科技大学, 2026.
贾沛然, 陈克俊, 等. 远程火箭弹道学[M]. 国防科技大学出版社.