Optimal Velocity Inclination

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

Definition

The optimal velocity inclination $\Theta_{k,\mathrm{opt}}$ is the local velocity inclination angle that maximizes the trajectory range, given the geocentric distance $r_k$ and energy parameter $\gamma_k$ at the end of the powered phase. It satisfies the extremum condition $\partial\beta_{kc}/\partial\Theta_k = 0$ and serves as the common characteristic parameter of both maximum range trajectories and minimum energy trajectories.

Core Elements

Passive Phase Optimal Velocity Inclination

The relationship between the maximum angular range of the passive phase $\beta_{kc,\max}$ and its optimal velocity inclination $\Theta_{kc,\mathrm{opt}}$ :

\tan\Theta_{kc,\mathrm{opt}} = \sqrt{\frac{\gamma_k[2R_E - \gamma_k(R_E + r_k)]}{2R_E\gamma_k - 4(R_E - r_k)}}

\tan\frac{\beta_{kc,\max}}{2} = \sqrt{\frac{\gamma_k[R_E\gamma_k - 2(R_E - r_k)]}{2[2R_E - \gamma_k(R_E + r_k)]}}

Free-Flight Phase Optimal Velocity Inclination

For the free-flight phase ( $r_e = r_k$ ), the formulas simplify to:

\tan\Theta_{ke,\mathrm{opt}} = \sqrt{1 - \gamma_k}

\tan\frac{\beta_{ke,\max}}{2} = \frac{1}{2}\frac{\gamma_k}{\sqrt{1 - \gamma_k}}

Combining the two yields a concise relationship:

\Theta_{ke,\mathrm{opt}} = \frac{1}{4}(\pi - \beta_{ke,\max})

Physical Significance

Property	Description
Maximum range	At equal energy, $\Theta_{k,\mathrm{opt}}$ maximizes the range
Minimum energy	At equal range, $\Theta_{k,\mathrm{opt}}$ minimizes the required energy
Error sensitivity	At $\Theta_k = \Theta_{k,\mathrm{opt}}$ , $\partial L/\partial\Theta_k = 0$ , so velocity inclination deviations do not cause range deviations
Small range limit	When $\beta_{kc,\max} \to 0$ , $\Theta_{k,\mathrm{opt}} \to 45°$ , consistent with artillery weapons

Equivalence with Minimum Energy Trajectory

The maximum range trajectory is identical to the minimum energy trajectory for that range. When $\beta_{kc}$ and $h_k$ are given, the required $\gamma_k$ , $v_k$ , and $\varepsilon$ are minimized only when $\Theta_{k,\mathrm{opt}}$ is selected.

Application Value

The optimal velocity inclination is a key parameter in ballistic missile design. In practice, the velocity inclination at the end of the powered phase is typically set to a value near $\Theta_{k,\mathrm{opt}}$ . Selecting the optimal velocity inclination not only maximizes range but also reduces the impact of velocity inclination deviations on firing accuracy, thereby improving missile hit precision.

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.