Orbital Elements

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

Definition

Orbital elements are the six independent integration constants required to solve the two-body differential equations of motion. They describe the size, shape, orientation, direction, and position-time relationship of an orbit. The six classical orbital elements are $(a, e, i, \Omega, \omega, \tau)$ , also known as Keplerian elements.

Core Elements

Classical Orbital Elements

Element	Symbol	Definition	Property Described
Semi-major axis	$a$	Half the major axis of the conic section	Orbit size; determines energy and period
Eccentricity	$e$	Eccentricity of the conic section	Orbit shape
Inclination	$i$	Dihedral angle between the orbit plane and the equatorial plane	Tilt of the orbit plane
Right ascension of the ascending node (RAAN)	$\Omega$	Angle from the vernal equinox to the ascending node	Orientation of the orbit plane in the equatorial plane
Argument of perigee	$\omega$	Angle from the ascending node to perigee	Orientation of the orbit within its plane
Time of perigee passage	$\tau$	Instant at which the spacecraft passes perigee	Position of the spacecraft on the orbit

Singular Element Replacements for Special Orbits

Orbit Type	Singular Elements	Replacement Strategy
Non-equatorial circular orbit ( $e=0$ )	$\omega, \tau$ undefined	Use argument of latitude $u$ to describe position
Equatorial elliptical orbit ( $i=0°$ )	$\Omega, \omega$ undefined	Use longitude of perigee $\alpha_p$ to describe orientation
Equatorial circular orbit	$\Omega, \omega, \tau$ undefined	Use right ascension $\alpha$ to describe position

Equinoctial Elements

For near-circular or near-equatorial orbits, equinoctial elements can be used to avoid singularities:

\{a, i, \Omega, H=e\cos\omega, K=-e\sin\omega, L=M+\omega\}

where $H$ and $K$ together describe the orbit's shape and orientation.

Position-Velocity Conversion

Converting between orbital elements and position-velocity vectors is a core computation in orbital mechanics:

Forward conversion: Computes position $\boldsymbol{r}$ and velocity $\boldsymbol{v}$ from $(a,e,i,\Omega,\omega,f)$ , requiring coordinate rotations.
Inverse conversion: Computes orbital elements from $(\boldsymbol{r}, \boldsymbol{v})$ , requiring sequential calculation of the specific angular momentum, eccentricity vector, and other fundamental vectors.

Application Value

Orbital elements are the standard means of describing and communicating orbit information. Compared to Cartesian coordinates, orbital elements have clear physical meaning, making it easy to understand orbit characteristics. In orbit prediction, the first five elements are constants under the two-body assumption, and only the sixth (true anomaly) needs to be propagated. In orbit design, mission requirements directly map to constraints on the orbital elements. In orbit determination, observational data are compared with predicted values through the orbital elements.

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.