Free-Flight Trajectory

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

Definition

The free-flight trajectory is the motion path of an aerospace vehicle during the unpowered, uncontrolled free-flight phase after engine cutoff. Since the free-flight phase occurs in near-vacuum with negligible aerodynamic forces, the vehicle can be treated as a point mass concentrated at its center of gravity, subject only to Earth's central gravitational attraction -- the two-body motion problem. The free-flight trajectory is a conic-section trajectory whose inertial spatial orientation, size, shape, and pointing remain constant.

Core Elements

Relationship with Powered-Phase Terminal Parameters

The inertial orientation, size, shape, and pointing of the free-flight trajectory are entirely determined by the position $\boldsymbol{r}_k$ and velocity $\boldsymbol{v}_k$ at the powered-phase terminal (engine cutoff point) K. Since the free-flight duration is less than one orbital period, the trajectory can be reduced from three dimensions to a two-dimensional planar trajectory, and the corresponding parameters simplify to:

Parameter	Symbol	Definition
Semi-latus rectum	$p$	$p = r_k \gamma_k \cos^2 \Theta_k$
Eccentricity	$e$	$e = \sqrt{1 - \gamma_k(2-\gamma_k)\cos^2\Theta_k}$
True anomaly	$f_k$	$f_k = \cos^{-1}\left[\frac{1}{e}\left(\frac{p}{r_k}-1\right)\right]$

Trajectory Shape Conditions

Trajectory Type	Energy Parameter Condition	Velocity Condition
Circular	$\gamma_k = 1$	$v_k = \sqrt{\mu_E/r_k}$
Elliptical	$0 < \gamma_k < 2$	$v_k < \sqrt{2\mu_E/r_k}$
Parabolic	$\gamma_k = 2$	$v_k = \sqrt{2\mu_E/r_k}$
Hyperbolic	$\gamma_k > 2$	$v_k > \sqrt{2\mu_E/r_k}$

Conditions for Satellite vs. Missile Trajectories

Satellite condition: $0 < \gamma_k < 2$ , with perigee radius $r_p > R_E + h_L$ ( $h_L$ is the survivability altitude)
Missile condition: $0 < \gamma_k < 2$ , with perigee radius $r_p < R_E$

These are distinguished by the critical velocity inclination $\Theta_{k,\lim}$ : when $\Theta_k \leq \Theta_{k,\lim}$ , the trajectory is a satellite orbit; otherwise, it is a missile trajectory.

Application Value

The free-flight trajectory constitutes the primary phase of ballistic missile flight, accounting for the vast majority of the total flight time. The free-flight trajectory characteristics are fully determined by the powered-phase terminal parameters, allowing the position and velocity at any point to be extrapolated. Free-flight trajectory analysis is the foundation of range calculation, flight time estimation, and error coefficient modeling.

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.