Hit Equation

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

Definition

The hit equation is the core equation that relates the passive-phase angular range $\beta_{kc}$ of a ballistic missile to its burnout parameters:

\frac{r_k}{R_E} = \frac{1 - \cos\beta_{kc}}{\gamma_k \cos^2\Theta_k} + \frac{\cos(\beta_{kc} + \Theta_k)}{\cos\Theta_k}

where $r_k$ is the geocentric distance at burnout, $R_E$ is the Earth's radius, $\gamma_k$ is the energy parameter, and $\Theta_k$ is the local flight-path angle. This equation directly links the missile's range to its cutoff point parameters and serves as the foundation for trajectory design and impact accuracy analysis.

Core Elements

Solution Method

The hit equation can be transformed via the half-angle formula into a quadratic equation in $\tan(\beta_{kc}/2)$ :

A\tan^2\frac{\beta_{kc}}{2} - B\tan\frac{\beta_{kc}}{2} + C = 0

where the coefficients are:

Coefficient	Expression
$A$	$2R_E(1+\tan^2\Theta_k) - \gamma_k(R_E + r_k)$
$B$	$2\gamma_k R_E \tan\Theta_k$
$C$	$\gamma_k(R_E - r_k)$

Since $A \geq 0$ and $C \leq 0$ , the equation has a unique valid solution:

\tan\frac{\beta_{kc}}{2} = \frac{B + \sqrt{B^2 - 4AC}}{2A}

Passive-Phase Range and Free-Flight Range

Range Type	Definition	Geocentric Distance Substitution
Passive-phase range $L_{kc}$	Great-circle arc from the subsatellite point K' at burnout to the impact point C	Impact point $r_c = R_E$
Free-flight range $L_{ke}$	Great-circle arc from K' at burnout to the subsatellite point E' at reentry	Reentry point $r_e = r_k$

A compact formula for the free-flight angular range:

\sin\frac{\beta_{ke}}{2} = \frac{\gamma_k}{2e}\sin 2\Theta_k

Angular Range and Absolute Range

The angular range $\beta$ relates to the absolute range $L$ by $L = R_E \beta$ . When the energy parameter $\gamma_k$ and burnout altitude $h_k$ are given, the angular range always has a maximum value, and the corresponding flight-path angle is the optimal flight-path angle.

Application Value

The hit equation is the core tool for ballistic missile design. It enables direct computation of the missile's range from burnout parameters, as well as reverse computation of the required burnout parameters for a specified range. In guidance system design, the hit equation is used to establish the relationship between range deviations and burnout parameter deviations, serving as the starting point for error coefficient modeling.

Range Error Coefficient

References

Zheng Wei, An Xueying, Zhou Xiang, He Ruizhi. Aerospace Flight Mechanics (空天飞行力学)[M]. National University of Defense Technology, 2026.
Jia Peiran, Chen Kejun, et al. Long-Range Rocket Ballistics (远程火箭弹道学)[M]. National University of Defense Technology Press.