Orbital Equation

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

Definition

The orbital equation is a polar-coordinate equation that describes the relationship between the geocentric distance $r$ and the true anomaly $f$ of a spacecraft in two-body motion. Its standard form is:

r = \frac{p}{1 + e\cos f} = \frac{a(1-e^2)}{1 + e\cos f}

where $p = h^2/\mu_E$ is the semi-latus rectum and $e$ is the eccentricity. This equation is a conic section equation and constitutes the mathematical description of Kepler's first law, showing that the orbit of two-body motion is a conic section with the center of mass of the central body at one focus.

Core Elements

Eccentricity and Orbit Shape

Eccentricity	Orbit Type	Characteristics
$e = 0$	Circular orbit	$r = a$ , constant geocentric distance
$0 < e < 1$	Elliptical orbit	Periodic orbit with perigee and apogee
$e = 1$	Parabolic orbit	Escape orbit with apogee at infinity
$e > 1$	Hyperbolic orbit	Non-periodic escape orbit

Apsides and Apse Line

The two vertices of the major axis of a conic section are called apsides. For an Earth-orbiting orbit:

Periapsis (perigee): at $f = 0°$ , $r_{\min} = a(1-e)$
Apoapsis (apogee): at $f = 180°$ , $r_{\max} = a(1+e)$

The apse line (major axis) coincides with the eccentricity vector $\boldsymbol{e}$ and determines the orientation of the orbit within the orbital plane.

True Anomaly and Argument of Latitude

True anomaly $f$ : the geocentric angle between the spacecraft position and the perigee
Argument of latitude $u$ : the geocentric angle between the spacecraft position and the ascending node
Relationship between the two: $f = u - \omega$ , where $\omega$ is the argument of perigee

Application Value

The orbital equation is one of the core formulas in orbital mechanics, derived by integrating the eccentricity vector $\boldsymbol{e}$ . It establishes the mapping between orbit shape ( $e$ ), size ( $p$ or $a$ ), and the instantaneous position of the spacecraft ( $f$ ). The orbital equation can be used to calculate the geocentric distance corresponding to any true anomaly, making it a fundamental tool for orbit prediction, orbit design, and ballistic computation.

Specific Angular Momentum

References

Zheng W, An X Y, Zhou X, He R Z. Aerospace Flight Mechanics[M]. National University of Defense Technology, 2026.
Jia P R, Chen K J, et al. Long-Range Rocket Ballistics[M]. National University of Defense Technology Press.