CHAMP型卫星定轨顾及非线性改正的轨道扰动方程

The orbital perturbation differential equations with the non-linear corrections for CHAMP-like satellite

本文针对CHAMP型卫星建立了顾及非线性改正的轨道扰动方程定轨理论与方法.首先从卫星运动的二阶微分方程出发,引入了正常引力位以及相应的参考轨道,然后分别推导了线性化轨道扰动方程与顾及非线性改正的轨道扰动方程,同时说明了建立的线性化轨道扰动方程与目前处理CHAMP卫星数据的动力学定轨方法是等价的.其次分别对线性化轨道扰动方程与顾及非线性改正的轨道扰动方程的精度进行了估计,在卫星定位精度为3cm与非惯性力测量精度为3×10^(-10)m·s^(-2)的前提下证明了下列结论:当参考轨道与实际轨道之间的距离ρ≤4.7m时线性化轨道扰动方程的精度能达到非惯性力的测量精度以及当ρ≤4.14×10~3m时顾及非线性改正的轨道扰动方程能达到非惯性力的测量精度.由此便可得出结论:相对于线性化轨道扰动方程,顾及非线性改正的轨道扰动方程具有更高的精度,且适合在更长的时间弧段上建立关于引力场位系数的法方程组,特别是针对CHAMP卫星计划进行的模拟计算也完全验证了该结论.最后利用叠加原理,给出了顾及非线性改正的轨道扰动方程的求解方法.此外,还针对GRACE卫星计划利用顾及非线性改正的轨道扰动方程进行了恢复引力场的模拟计算,结果表明:分段建立位系数的法方程组时子弧段分别取值2h、1d、6 d对恢复引力场的结果几乎不产生影响,这表明在处理GRACE数据时能够以6d的弧长来建立法方程组.

Different from the existed derivation methods, the linearized orbital perturbation differential equations for CHAMP-like satellite are derived directly from second order differential equations of satellite motion after introducing the normal gravitational field as well as the reference orbit, and then introducing the omitted terms into the linearized orbital perturbation differential equations, the orbital perturbation differential equations with the nonlinear corrections are proposed for CHAMP-like satellite.Since the omitted terms in derivations can be expressed clearly, the accuracies of the linearized orbital perturbation differential equations and the orbital perturbation differential equations with the nonlinear corrections are estimated respectively. By complicated computations, it is concluded from the above estimations that if the error for satellite position is less than 3cm and the error for non-gravitational accelerations is less than 3×10^-10m·s^-2, the linearized orbital perturbation differential equations can retain the measurement accuracy of non-gravitational accelerations when the distance ρ between the reference orbit and actual one is less than 4.7 m and the orbital perturbation differential equations with the nonlinear corrections can retain the same accuracy when ρ≤4.14×10^3 m. Hence, compared with the linearized orbital perturbation differential equations, the orbital perturbation differential equations with the nonlinear corrections have much higher accuracy if the same reference orbit is used. This means that the orbital perturbation differential equations with the nonlinear corrections are suitable to establish the normal system of linear equations about spherical harmonic coefficients of the gravitational field for longer time interval. In addition, with the help of superposition principle, solving methods for the orbital perturbation differential equations with the nonlinear corrections are given. Therefore, the complete theory and method about the orbital perturbation differential equations with the nonlinear corrections are established.Three arithmetic examples for CHAMP mission are imitated. By these examples, it is examined that the orbital perturbation differential equations with the nonlinear corrections have higher accuracies than the linearized orbital perturbation differential equations given the same reference orbit. In addition, it is examined by numerical imitations for GRACE mission that the span of time＇s segment making the orbital perturbation differential equations with the nonlinear corrections valid can be extended to 6 days at least if the measurement accuracy of the non-gravitational accelerations reaches to 3×10^-10m·s^-2. Obviously, the span of 6 days is much longer than the recommended span in dealing with GRACE data which is less than one day usually.GRACE-follow-on mission will be launched in the near future. The measurement accuracy of the non-gravitational accelerations in GRACE-follow-on mission may be raised to 10^-12 m·s^-2, and this means that the orbital perturbation differential equations with higher accuracy are required. Obviously compared with the linearized orbital perturbation differential equations, the orbital perturbation differential equations with the nonlinear corrections are more suitable to deal with GRACE-follow-on data.