Velocity Inclination Angle

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

Definition

The velocity inclination angle ( $\theta$ ) is the angle between the velocity vector and the horizontal plane ( $xoz$ plane) of the launch coordinate system. It describes the direction of the velocity vector in the longitudinal plane. The velocity inclination angle is one of the core variables of the longitudinal equations of motion: at vertical launch, $\theta = 90°$ , and at the cutoff point, $\theta = \theta_k^*$ .

Core Elements

Description of Velocity Direction

The direction of the velocity vector in space is defined by two angles:

Angle	Symbol	Definition	Described Motion
Velocity inclination angle	$\theta$	Angle between the velocity vector and the horizontal plane	Ascent/descent in the longitudinal plane
Flight path yaw angle	$\sigma$	Angle of the velocity vector deviating from the firing plane	Lateral deviation

Differential Equation for Velocity Inclination Angle

The variation of the velocity inclination angle is governed by the normal acceleration equation:

\dot{\theta} = \frac{1}{mv}(P_e + C_y^\alpha q S_M)\alpha + \frac{g}{v}\cos\theta

where the first term is the angular acceleration produced by the normal components of thrust and lift, and the second term is the influence of the normal gravitational component.

Velocity Azimuth Angle

The velocity azimuth angle $A$ describes the angle between the projection of the velocity vector onto the horizontal plane and the meridian north direction:

\tan A = \frac{\dot{z}}{\dot{x}}

The relationship between the velocity azimuth angle, the flight path yaw angle $\sigma$ , and the firing azimuth angle $A_0$ is:

A = A_0 + \sigma

Cutoff-Point Velocity Inclination Angle

The cutoff-point velocity inclination angle $\theta_k^*$ is a key parameter in trajectory design:

For ballistic missiles, $\theta_k^*$ determines the passive-phase range
For launch vehicles, $\theta_k^*$ determines the insertion orbit inclination
The optimal velocity inclination angle varies with range: for short ranges it approaches 45°, and for long ranges it gradually decreases

Application Value

The velocity inclination angle is a core variable for powered-phase trajectory design and flight program optimization. The design of the pitch program angle is essentially the control of the velocity inclination angle from 90° (vertical launch) to the desired value $\theta_k^*$ at the cutoff point. For ballistic missiles, the accuracy of the velocity inclination angle directly affects the impact point accuracy. For launch vehicles, the velocity inclination angle, together with the speed magnitude, determines the insertion parameters.

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.