Ballistic Coefficient

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

Definition

The ballistic coefficient $B$ is a parameter that characterizes a vehicle's aerodynamic deceleration capability in the atmosphere, defined as:

B = \frac{C_x S_M}{2m}

where $C_x$ is the drag coefficient, $S_M$ is the reference area, and $m$ is the vehicle mass. The ballistic coefficient appears in the reentry equations of motion:

\frac{\mathrm{d}v}{\mathrm{d}t} = -B\rho_0 e^{-\beta h} v^2

Core Elements

Physical Significance

The ballistic coefficient comprehensively reflects the vehicle's aerodynamic shape and mass characteristics:

Parameter	Effect
Higher drag coefficient $C_x$	Larger ballistic coefficient, stronger aerodynamic deceleration
Larger reference area $S_M$	Larger ballistic coefficient, stronger aerodynamic deceleration
Greater mass $m$	Smaller ballistic coefficient, weaker aerodynamic deceleration

A larger ballistic coefficient means more significant deceleration of the vehicle in the atmosphere.

Relationship with Minimum Negative Acceleration

Under zero-angle-of-attack reentry conditions, the relationship between the minimum negative acceleration and the ballistic coefficient is:

\dot{v}_m = \frac{\beta v_e^2}{2e}\sin\Theta_e

This result shows that under given reentry conditions, the minimum negative acceleration is independent of the ballistic coefficient and depends only on the reentry velocity $v_e$ and reentry angle $\Theta_e$ .

Relationship with Altitude of Minimum Negative Acceleration

The altitude at which the minimum negative acceleration occurs:

h_m = \frac{1}{\beta}\ln\left(-\frac{C_x S_M \rho_0}{m\beta\sin\Theta_e}\right)

A greater mass $m$ or larger $|\Theta_e|$ results in a lower altitude at which the minimum negative acceleration occurs.

Velocity Decay Law

Under the assumption of negligible gravity, the velocity variation with altitude is:

v = v_e\exp\left(\frac{B\rho_0}{\beta\sin\Theta_e}e^{-\beta h}\right)

Application Value

The ballistic coefficient is a key parameter in reentry vehicle design, directly affecting the deceleration characteristics, peak load factor, and aerodynamic heating during reentry. In warhead design, increasing the ballistic coefficient (by increasing mass or reducing the drag area) lowers the altitude at which the minimum negative acceleration occurs, thereby improving penetration capability. In spacecraft reentry design, the ballistic coefficient is an important parameter for determining the reentry corridor.

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.