Gaussian Perturbation Equations

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

Definition

The Gaussian perturbation equations are disturbed equations of motion expressed in terms of orbital elements. They represent the rate of change of each orbital element as a function of the six orbital elements and three orthogonal components of the perturbation acceleration. These equations were first established by Gauss during his study of the perturbation motion of the asteroid Pallas under Jupiter's gravitational influence. They are applicable to any conservative or non-conservative perturbation force, including thrust acceleration.

Core Elements

Gaussian Type I Equations

The perturbation acceleration is decomposed into a radial component $f_r$ , a transverse component $f_u$ , and an out-of-plane normal component $f_h$ :

\left\{\begin{array}{l} \dot{a} = \frac{2}{n\sqrt{1-e^2}}[e\sin f \cdot f_r + (1+e\cos f)f_u] \\ \dot{e} = \frac{\sqrt{1-e^2}}{na}[\sin f \cdot f_r + (\cos f + \cos E)f_u] \\ \dot{i} = \frac{r\cos u}{na^2\sqrt{1-e^2}}f_h \\ \dot{\Omega} = \frac{r\sin u}{na^2\sqrt{1-e^2}\sin i}f_h \\ \dot{\omega} = \frac{\sqrt{1-e^2}}{nae}[-\cos f \cdot f_r + (1+\frac{r}{p})\sin f \cdot f_u] - \cos i \cdot \dot{\Omega} \\ \dot{M} = n - \frac{1-e^2}{nae}[(2e\frac{r}{p}-\cos f)f_r + (1+\frac{r}{p})\sin f \cdot f_u] \end{array}\right.

Gaussian Type I Physical Characteristics

Characteristic	Description
Orbit size and shape	Determined jointly by $f_r$ and $f_u$ , independent of $f_h$
Orbital plane orientation change	Determined solely by $f_h$ , independent of $f_r$ and $f_u$
$f_h$ at the node	Most significant effect on $i$ , no effect on $\Omega$
$f_h$ at $u=\pm90°$	Most significant effect on $\Omega$ , no effect on $i$

Gaussian Type II Equations

The perturbation acceleration is decomposed into a tangential component $f_t$ , a principal normal component $f_n$ , and a binormal component $f_h$ . This form is suitable for atmospheric drag analysis:

Characteristic	Description
Orbit size change	Determined solely by the tangential component $f_t$
Orbit shape and mean position	Determined jointly by $f_t$ and $f_n$
Orbital plane orientation change	Determined solely by the binormal component $f_h$

Usage Constraints

For near-circular orbits, the rates of change of $\omega$ and $M$ are typically very large. For near-equatorial orbits, the rates of change of $\Omega$ and $\omega$ are typically very large. The cause is that poor choice of variation parameters leads to denominators containing eccentricity and the sine of orbital inclination.

Application Value

The Gaussian perturbation equations are the fundamental tool for analyzing the effects of arbitrary perturbation forces on orbits. Since they directly use perturbation acceleration components as input, they are suitable for analyzing non-conservative forces (such as atmospheric drag and thrust). During orbit capture and orbit maintenance, engine thrust acceleration can be approximately described using the Gaussian equations, and orbital element deviations can be eliminated through properly designed orbit control schemes.

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.
刘林. 航天器轨道理论[M]. 国防工业出版社.