Impulsive orbit correction using second-order Gauss’s variational equations

Abstract

Based on Gauss’s variational equations (GVEs), the impulsive orbital-element corrections are investigated in orbit transfer problems. Both single impulse and multiple impulses are considered for the first-order and second-order GVEs. For the single impulse, a nonlinear least-squares iteration method for the minimum orbit error is provided to simultaneously solve for the impulse vector and the impulse position. For multiple impulses, a minimum-norm method for the energy cost is proposed to solve the three-impulse and two-impulse corrections for the in-plane and out-of-plane orbital elements, respectively. The impulse positions are analytically derived, and the impulse vectors are obtained by the minimum-norm method. Numerical examples are provided to verify the proposed single-impulse least-squares and multiple-impulse minimum-norm methods. The results show that the nonlinear solutions using the second-order GVEs are more accurate than those using the first-order GVEs.

Appendix: the partial derivatives \({\partial A}/{\partial f}\) and \({\partial \mathcal{G}}/{\partial f}\)

The nonzero components of the matrix ${\partial A}/{\partial f}$ are

$$\begin{aligned} \begin{array}{lll} \frac{\partial A_{11}}{\partial f} = \frac{2e\cos f}{n\sqrt{1-e^2}}, &{} \frac{\partial A_{12}}{\partial f} = \frac{-2e\sin f}{n\sqrt{1 - e^2}}, &{} ~ \\ \frac{\partial A_{21}}{\partial f} = \sqrt{\frac{p}{\mu }} \cos f, &{} \frac{\partial A_{22}}{\partial f} = - \sin f\sqrt{\frac{p}{\mu }} \left[ 1 + \frac{(1-e^2)}{(1 + e\cos f)^2} \right] , &{} ~ \\ \frac{\partial A_{33}}{\partial f} = \sqrt{\frac{p}{\mu }} \frac{-\sin u-e\sin \omega }{(1 + e\cos f)^2}, &{} \frac{\partial A_{43}}{\partial f} = \sqrt{\frac{p}{\mu }} \frac{\cos u+e\cos \omega }{(1 + e\cos f)^2 \sin i}, &{} ~ \\ \frac{\partial A_{51}}{\partial f} =\frac{1}{e} \sqrt{\frac{p}{\mu }} \sin f, &{} \frac{\partial A_{52}}{\partial f} = \frac{1}{e} \sqrt{\frac{p}{\mu }}\left[ \cos f\left( 1+\frac{1}{1 + e\cos f}\right) + \frac{e\sin ^2 f}{(1 + e\cos f)^2} \right] ,\\ &{} \frac{\partial A_{53}}{\partial f} =-\sqrt{\frac{p}{\mu }} \cot i\frac{\cos u + e\cos \omega }{(1 + e\cos f)^2}, \\ \frac{\partial A_{61}}{\partial f} =-\frac{1}{e} \sqrt{\frac{p}{\mu }} \sin f, &{} \frac{\partial A_{62}}{\partial f} = -\frac{1}{e} \sqrt{\frac{p}{\mu }}\left[ \cos f\left( 1+\frac{1}{1 + e\cos f}\right) + \frac{e\sin ^2 f}{(1 + e\cos f)^2} \right] . \end{array} \end{aligned}$$

The nonzero components of the tensor ${\partial \mathcal{G}}/{\partial f}$ (with ${\partial G_{ijk}}/{\partial f}={\partial G_{ikj}}/{\partial f}$) are

$$\begin{aligned} \frac{\partial G_{111}}{\partial f}= & {} {\frac{{4{a^3}}}{h^2}{e^2}\sin (2f)},\quad \frac{\partial G_{122}}{\partial f}=-\frac{{8{a^3}}}{h^2}e\sin f(1 + e\cos f),\\ \frac{\partial G_{112}}{\partial f}= & {} \frac{{4{a^3}}}{h^2}e\left[ {\cos f + e\cos (2f)} \right] ,\\ \frac{\partial G_{211}}{\partial f}= & {} -\frac{1}{2 e} \frac{p}{\mu } \sin (2f), \\ \frac{\partial G_{222}}{\partial f}= & {} \frac{p/\mu }{2e(1 + e\cos f)^2}\sin f\left[ {-6e - 2(3e^2-4)\cos f + 12e\cos ^2 f + 4e^2{\cos ^3}f} \right] \\&+ \frac{{p/\mu }}{(1 + e\cos f)^3}\sin f\left[ {4 + 6e\cos f + (3{e^2} - 4){{\cos }^2}f - 4e{{\cos }^3}f - {e^2}{{\cos }^4}f} \right] , \\ \frac{\partial G_{233}}{\partial f}= & {} - \frac{p/\mu }{2(1 + e\cos f)^2}[e\sin (2f) + 2\sin f] + \frac{p/\mu }{(1 + e\cos f)^3}e\sin f[e(1 +\cos ^2f)+2\cos f],\\ \frac{\partial G_{212}}{\partial f}= & {} - \frac{p/\mu }{2e(1+e\cos f)}[2\cos (2f)-3e{\sin ^2}f\cos f]-\frac{p/\mu }{2(1 + e\cos f)^2}\sin ^2 f(2\cos f-e\sin ^2 f),\\ \frac{\partial G_{333}}{\partial f}= & {} \frac{1}{2}\frac{p}{\mu }\cot i\left[ \frac{\sin (2u)}{(1 + e\cos f)^2} + 2\frac{e\sin f \sin ^2u}{(1 + e\cos f)^3}\right] ,\\ \frac{\partial G_{323}}{\partial f}= & {} -\frac{1}{2}\frac{p}{\mu }\left[ {\frac{-\sin u}{(1 + e\cos f)^2} + 2\frac{e\sin f\cos u}{(1 + e\cos f)^3}} \right] ,\\ \frac{\partial G_{433}}{\partial f}= & {} -\frac{p}{\mu }\frac{\cos i}{\sin ^2 i}\left[ \frac{\cos (2u)}{(1+e\cos f)^2}+\frac{e\sin f\sin (2u)}{(1+e\cos f)^3} \right] ,\\ \frac{\partial G_{423}}{\partial f}= & {} -\frac{1}{2}\frac{p}{\mu }\frac{1}{\sin i}\left[ \frac{\cos u}{(1+e\cos f)^2}+2\frac{e\sin f\sin u}{(1+e\cos f)^3}\right] ,\\ \frac{\partial G_{511}}{\partial f}= & {} \frac{1}{e^2}\frac{p}{\mu }\cos (2f), \\ \frac{\partial G_{522}}{\partial f}= & {} \frac{1}{e^2}\frac{p}{\mu }\left[ { -\cos (2f) - \frac{2\cos ^2 f}{1 + e\cos f} + \frac{3\sin ^2 f - \cos f(e+\cos f)}{(1 + e\cos f)^2} - \frac{2e\sin ^2 f(e + \cos f)}{(1 + e\cos f)^3}} \right] ,\\ \frac{\partial G_{533}}{\partial f}= & {} \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} =\frac{{p/\mu }}{{2{{(1 + e\cos f)}^2}}}\left[ {\frac{2}{e}\cos f + 2{{\cot }^2}i\cos (2u) + \sin (2u)\cot \omega - \sin (2f)\cot \omega } \right] \\ + \frac{{p/\mu }}{{{{(1 + e\cos f)}^3}}}e\sin f\left[ {\frac{2}{e}\sin f + {{\cot }^2}i\sin (2u) + ({{\sin }^2}u - {{\sin }^2}f)\cot \omega } \right] \quad \mathrm{{if}}\quad \sin \omega \ne 0 \\ \end{array} \\ ={\frac{p/\mu }{(1 + e\cos f)^2}{\left[ {\frac{1}{e}\cos f + \cot ^2 i\cos (2u) + \frac{e\sin f}{1 + e\cos f}\left( {\frac{2}{e}\sin f + \cot ^2 i\sin (2u)} \right) } \right] } \quad \mathrm{{if}}\quad \sin \omega = 0 } \\ \end{array}} \right. ,\\ \frac{\partial G_{512}}{\partial f}= & {} -\frac{1}{2e^2}\frac{p}{\mu }\left[ 2\sin (2f)\frac{2 + e\cos f}{1 + e\cos f}-\cos (2f)\frac{e\sin f}{(1+e\cos f)^2} \right] ,\\ \frac{\partial G_{523}}{\partial f}= & {} \frac{1}{2}\frac{p}{\mu }\cot i\left[ {\frac{\cos u}{(1 + e\cos f)^2} + \frac{2e\sin f\sin u}{(1 + e\cos f)^3}}\right] ,\\ \frac{\partial G_{611}}{\partial f}= & {} -\frac{\partial G_{511}}{\partial f}, \quad \frac{\partial G_{622}}{\partial f} = -\frac{\partial G_{522}}{\partial f}, \quad \frac{\partial G_{612}}{\partial f} = -\frac{\partial G_{512}}{\partial f},\\ \frac{\partial G_{633}}{\partial f}= & {} -\frac{1}{2e}\frac{p}{\mu }\left[ \frac{\cos f(2+e\cos f)}{(1+e\cos f)^2}+\frac{e\sin f^2(3+e\cos f)}{(1+e\cos f)^3}\right] . \end{aligned}$$