Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

Abstract

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.

Appendix: Required partial derivatives for computing short-period variations due to tesseral harmonics

$$\begin{aligned} {\mathscr {W}}_ {2,gh}^T= & {} \frac{1}{\sin i}\left( \cos i\frac{\partial {\mathscr {W}}_2^T}{\partial g}-\frac{\partial {\mathscr {W}}_2^T}{\partial h}\right) \\= & {} -\,\frac{2!}{C_{20}^2}\frac{\mu R_e^n}{\tilde{n}\,a^{n+1}\eta ^{2n-1}}\sum _{p=0}^{p=n}F^\prime _{nmp}(i)\left[ CS_1\, \{-\cos (\beta )I_2^{nm} -\sin (\beta )I_1^{nmp}\} \right. \\&\left. +\, CS_2\, \{\cos (\beta )I_1^{nmp}-\sin (\beta )I_2^{nmp}\} \right] \\ \text {where } F^\prime _{nmp}(i)= & {} {\left\{ \begin{array}{ll} \frac{-m}{G}\,\tan \left( i/2\right) \,F_{nmp}(i) &{} \text {if } (n-2p)=m \\ \frac{1}{2}\frac{\cos (i)(n-2p)-m}{G\cos (i/2)}\,\left( \frac{1}{\sin (i/2)}F_{nmp}(i)\right) &{} \text {if } (n-2p)\ne m \end{array}\right. }\\ {\mathscr {W}}_{2,lg}^T= & {} \frac{1}{e}\left( \frac{\partial {\mathscr {W}}_2^T}{\partial l}-\frac{1}{\eta }\frac{\partial {\mathscr {W}}_2^T}{\partial g}\right) \\= & {} -\,\frac{2!}{C_{20}^2}\frac{\mu R_e^n}{\tilde{n}\,a^{n+1}\eta ^{2n-1}}\sum _{p=0}^{p=n}F_{nmp}(i)\\&\left[ CS_1\, \left\{ \cos (\beta )\left( -(n-2p)I_{2,lga}^{nmp} - \frac{n+1}{\eta ^3}I_{1,lgb}^{nmp}\right) \right. \right. \\&\left. \left. +\,\sin (\beta )\left( -(n-2p)I_{1,lga}^{nmp} + \frac{n+1}{\eta ^3}I_{2,lgb}^{nmp}\right) \right\} \right. \\&\left. +\, CS_2\, \left\{ \cos (\beta )\left( (n-2p)I_{1,lga}^{nmp} - \frac{n+1}{\eta ^3}I_{2,lgb}^{nmp}\right) \right. \right. \\&\left. \left. +\,\sin (\beta )\left( -(n-2p)I_{2,lga}^{nmp} - \frac{n+1}{\eta ^3}I_{1,lgb}^{nmp}\right) \right\} \right] \\ \frac{\partial I_1^{nmp}}{\partial a}= & {} m\frac{\hbox {d}\delta }{\hbox {d}a} \int l\,\sin ((n-2p)f-m\delta \,l)(1+e\cos f)^{n-1}\,\hbox {d}f\\ \frac{\partial I_2^{nmp}}{\partial a}= & {} -\,m\frac{\hbox {d}\delta }{\hbox {d}a} \int l\,\cos ((n-2p)f-m\delta \,l)(1+e\cos f)^{n-1}\,\hbox {d}f\\ \frac{\partial I_1^{nmp}}{\partial e}= & {} \frac{1}{\eta ^2}\int \left( \frac{-\,3e(n-1)}{2} \cos (\alpha (f)) +\frac{(-(n-1)+4p)}{2} \cos (\alpha (f)-f) \right. \\&\left. +\,\frac{((3n+1)-4p)}{2} \cos (\alpha (f)+f)+\frac{e(2p+1)}{4} \cos (\alpha (f)-2f) \right. \\&\left. +\, \frac{e(2n-2p+1)}{4} \cos (\alpha (f)+2f) \right) (1+e\cos f)^{n-1}\,\hbox {d}f\\ \frac{\partial I_2^{nmp}}{\partial e}= & {} \frac{1}{\eta ^2}\int \left( \frac{-3e(n-1)}{2} \sin (\alpha (f)) +\frac{(-(n-1)+4p)}{2} \sin (\alpha (f)-f) \right. \\&\left. +\,\frac{((3n+1)-4p)}{2} \sin (\alpha (f)+f)+\frac{e(2p+1)}{4} \sin (\alpha (f)-2f) \right. \\&\left. +\, \frac{e(2n-2p+1)}{4} \sin (\alpha (f)+2f) \right) (1+e\cos f)^{n-1}\,\hbox {d}f,\\ \text {where } \alpha (f)\equiv & {} (n-2p)f-m\delta \,l\\ \frac{\partial I_1^{nmp}}{\partial f}= & {} \cos ((n-2p)f-m\delta \,l)(1+e\cos f)^{n-1}\\ \frac{\partial I_2^{nmp}}{\partial f}= & {} \sin ((n-2p)f-m\delta \,l)(1+e\cos f)^{n-1}\\ I_{1,lga}^{nmp}= & {} \int \frac{1}{\eta ^3}\left( e + 2\cos (f) + e\cos ^2(f)\right) \cos ((n-2p)f-m\delta \,l)\\&(1+e\cos f)^{n-1}\,\hbox {d}f\\ I_{2,lga}^{nmp}= & {} \int \frac{1}{\eta ^3}\left( e + 2\cos (f) + e\cos ^2(f)\right) \sin ((n-2p)f-m\delta \,l)\\&\quad (1+e\cos f)^{n-1}\,\hbox {d}f\\ I_{1,lgb}^{nmp}= & {} \int \sin (f)\, \cos ((n-2p)f-m\delta \,l)(1+e\cos f)^{n}\,\hbox {d}f\\ I_{2,lgb}^{nmp}= & {} \int \sin (f)\, \sin ((n-2p)f-m\delta \,l)(1+e\cos f)^{n}\,\hbox {d}f. \end{aligned}$$