Energy Parameter

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

Definition

The energy parameter $\gamma$ is a dimensionless quantity that describes the ratio of the spacecraft's velocity to the local circular orbit velocity at a given position:

\gamma = \frac{v^2 r}{\mu_E} = \frac{v^2}{v_{\text{circle}}^2}

where $v$ is the speed, $r$ is the geocentric distance, $\mu_E$ is Earth's gravitational parameter, and $v_{\text{circle}} = \sqrt{\mu_E/r}$ is the local circular orbit velocity. The energy parameter directly reflects the mechanical energy state of the spacecraft.

Core Elements

Relationship with Specific Mechanical Energy

The relationships between the energy parameter, specific mechanical energy $\varepsilon$ , and semi-major axis $a$ :

\gamma = 2 - \frac{r}{a}, \quad \varepsilon = -\frac{\mu_E}{2a} = \frac{v^2}{2} - \frac{\mu_E}{r}

Therefore $\gamma_k = 2r_k/r_{k,\text{circle}} = v_k^2 r_k / \mu_E$ .

Correspondence with Orbit Types

Orbit Type	Energy Parameter Range	Semi-major Axis	Specific Mechanical Energy
Circular orbit	$\gamma = 1$	$a = r$	$\varepsilon < 0$
Elliptical orbit	$0 < \gamma < 2$	$a > 0$	$\varepsilon < 0$
Parabolic orbit	$\gamma = 2$	$a \to \infty$	$\varepsilon = 0$
Hyperbolic orbit	$\gamma > 2$	$a < 0$	$\varepsilon > 0$

Role in Ballistic Parameter Computation

The energy parameter is a core intermediate variable in ballistic computation, appearing in several key formulas:

Formula	Expression
Semi-latus rectum	$p = r_k \gamma_k \cos^2\Theta_k$
Eccentricity	$e = \sqrt{1 - \gamma_k(2-\gamma_k)\cos^2\Theta_k}$
Semi-major axis	$a = r_k / (2 - \gamma_k)$
Free-flight angular range	$\sin(\beta_{ke}/2) = \gamma_k \sin 2\Theta_k / (2e)$

Relationship with Powered Phase Endpoint Parameters

At the powered phase endpoint K, the energy parameter is fully determined by the speed $v_k$ and geocentric distance $r_k$ :

\gamma_k = \frac{v_k^2 r_k}{\mu_E}

Its partial derivatives are:

\frac{\partial\gamma_k}{\partial v_k} = \frac{2\gamma_k}{v_k}, \quad \frac{\partial\gamma_k}{\partial r_k} = \frac{\gamma_k}{r_k}

Application Value

The energy parameter is the bridge connecting velocity, position, and orbit shape. Through the energy parameter, the orbit type of a spacecraft can be quickly determined and the expression of ballistic equations can be simplified. In computations for hit equations, range error coefficients, flight time, and more, the energy parameter serves as a core intermediate variable. In missile design, the magnitude of the energy parameter directly reflects the velocity-energy state at the cutoff point.

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.