Range Error Coefficient

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

Range error coefficients are the partial derivatives of the passive-phase range angle βkc\beta_{kc} with respect to the burnout parameters vkv_k, Θk\Theta_k, and rkr_k. They describe the range deviation caused by a unit deviation in each burnout parameter. Neglecting higher-order terms, the passive-phase range deviation can be expressed as:

Δβkc=∂βkc∂vkΔvk+∂βkc∂ΘkΔΘk+∂βkc∂rkΔrk\Delta\beta_{kc} = \frac{\partial\beta_{kc}}{\partial v_k}\Delta v_k + \frac{\partial\beta_{kc}}{\partial\Theta_k}\Delta\Theta_k + \frac{\partial\beta_{kc}}{\partial r_k}\Delta r_k

Core Elements

Error Coefficient Types

TypeSymbolMeaning
Downrange error coefficient∂βkc/∂vk\partial\beta_{kc}/\partial v_kRange deviation due to velocity magnitude error
Downrange error coefficient∂βkc/∂Θk\partial\beta_{kc}/\partial\Theta_kRange deviation due to flight-path angle error
Downrange error coefficient∂βkc/∂rk\partial\beta_{kc}/\partial r_kRange deviation due to geocentric distance error
Cross-range error coefficient∂ζc/∂αk\partial\zeta_c/\partial\alpha_kLateral deviation due to azimuth error
Cross-range error coefficient∂ζc/∂ζk\partial\zeta_c/\partial\zeta_kLateral deviation due to lateral position error
Flight time error coefficient∂Tkc/∂vk\partial T_{kc}/\partial v_kTime deviation due to velocity error

First-Order Downrange Error Coefficients

Explicit expressions for the free-flight phase range error coefficients:

∂βke∂vk=4vkγk⋅1+tan⁡2Θktan⁡Θk⋅sin⁡2βke2\frac{\partial\beta_{ke}}{\partial v_k} = \frac{4}{v_k\gamma_k} \cdot \frac{1+\tan^2\Theta_k}{\tan\Theta_k} \cdot \sin^2\frac{\beta_{ke}}{2}

∂βke∂Θk=1γk⋅(1+tan⁡2Θk)(γk−2tan⁡βke2tan⁡Θk)tan⁡Θk⋅sin⁡βke\frac{\partial\beta_{ke}}{\partial\Theta_k} = \frac{1}{\gamma_k} \cdot \frac{(1+\tan^2\Theta_k)(\gamma_k - 2\tan\frac{\beta_{ke}}{2}\tan\Theta_k)}{\tan\Theta_k} \cdot \sin\beta_{ke}

Cross-Range Error Coefficients

Lateral deviation is caused jointly by velocity azimuth error and lateral position deviation:

Δζc=∂ζc∂αkΔαk+∂ζc∂ζkΔζk\Delta\zeta_c = \frac{\partial\zeta_c}{\partial\alpha_k}\Delta\alpha_k + \frac{\partial\zeta_c}{\partial\zeta_k}\Delta\zeta_k

where:

∂ζc∂αk=sin⁡βkc,∂ζc∂ζk=cos⁡βkc−tan⁡Θksin⁡βkc\frac{\partial\zeta_c}{\partial\alpha_k} = \sin\beta_{kc}, \quad \frac{\partial\zeta_c}{\partial\zeta_k} = \cos\beta_{kc} - \tan\Theta_k\sin\beta_{kc}

Properties of Error Coefficients

PropertyDescription
Optimal flight-path angle∂βke/∂Θk=0\partial\beta_{ke}/\partial\Theta_k = 0; flight-path angle error causes no range deviation
Velocity magnitude coefficientThe larger γk\gamma_k, the larger ∂Lke/∂vk\partial L_{ke}/\partial v_k
Second-order errorsFor long-range missiles, second-order error coefficients cannot be neglected

Cartesian Coordinate Representation

Error coefficients can be expressed in a Cartesian coordinate system (launch inertial frame). Through coordinate transformation matrices, polar error coefficients are converted to Cartesian error coefficients, facilitating integration with guidance system instrument error models.

Application Value

Range error coefficients are the core tool for ballistic missile accuracy analysis and guidance system design. Through error coefficients, one can quantitatively evaluate the impact of burnout parameter deviations on impact point accuracy, providing theoretical foundations for guidance method design, instrument error allocation, and firing accuracy assessment. Selecting the optimal flight-path angle eliminates the flight-path angle error coefficient, thereby improving firing accuracy.

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.