Relativistic satellite orbits: central body with higher zonal harmonics

Abstract

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order $(\mathrm{GM}/c^2R) \times J_k$ with $k \ge 2$ for the Earth are very small (of order $7 \times 10^{-10}\,\times \,J_k$) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit $J_k/c^2$-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian $J_k$-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node $<\!{\dot{\Omega }}\!>$ and argument of the pericenter $<\!{\dot{\omega }}\!>$. Finally orders of magnitude are given and discussed.

Appendices

Appendix

A. Orbital averages

In this paper, certain averages of functions F over one complete revolution in the unperturbed Keplerian orbit are employed:

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where M and f are the mean and true anomaly, respectively, $\eta \equiv \sqrt{1 - e^2}$ and

$$\begin{aligned} \frac{a}{r} = \frac{1 + e \cos f}{\eta ^2} \, . \end{aligned}$$

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The following averages are used:

(37a)

(37b)

From these relations, the following special cases can be derived:

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(38b)

(38c)

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(38f)

(38g)

(38h)

(38i)

(38j)

(38k)

with

$$\begin{aligned} \eta = \sqrt{1 - e^2} = \frac{G}{L}~, \end{aligned}$$

where G and L are Delaunay elements.

B. Solutions for the direct terms

Here, we list the results for all orbital elements. The subscript SP stands for short-periodic, LP for long-periodic and S for secular perturbations. To condense the short-periodic results, we use the functions

$$\begin{aligned} \Phi _{i,j,k,l}(f,\omega )&= \sum _{s=0}^{k+i} \sum _{q=0}^s \left( \begin{array}{c} k+i \\ s \end{array} \right) \left( \begin{array}{c} s \\ q \end{array} \right) \left( \frac{e}{2} \right) ^s \frac{1}{l+j+s-2q}\nonumber \\&\quad \times \,\left[ \begin{array}{c} \sin \\ \cos \end{array} \right] ^\text { k even}_\text { k odd} \left( (l+j+s-2q) f + l \omega \right) \end{aligned}$$

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$$\begin{aligned} \Phi ^*_{i,j,k,l}(f,\omega )&= \sum _{s=0}^{k+i} \sum _{q=0}^s \left( \begin{array}{c} k+i \\ s \end{array} \right) \left( \begin{array}{c} s \\ q \end{array} \right) \left( \frac{e}{2} \right) ^s\frac{1}{l+j+s-2q}\nonumber \\&\quad \times \,\left[ \begin{array}{c} \cos \\ \sin \end{array} \right] ^\text { k even}_\text { k odd} \left( (l+j+s-2q) f + l \omega \right) \end{aligned}$$

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$$\begin{aligned} \Psi _{i,j,k,l}(f,M)&= \sum _{s=0}^{k+i} \sum _{q=0}^s \left( \begin{array}{c} k+i \\ s \end{array} \right) \left( \begin{array}{c} s \\ q \end{array} \right) \left( \frac{e}{2} \right) ^s (f-M) \delta _{l,0}\delta _{s+j,2q} \end{aligned}$$

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Note, that there are poles in the denominator for certain combinations of indices. For the short-periodic solutions these correspond to long-periodic terms and for the long-periodic solutions they correspond to secular drifts, i.e. they have to be excluded from the sums.

1.1 Semi-major axis a

$$\begin{aligned} (\Delta a)_\text {SP}&= \frac{\mu }{c^2}\left( \frac{R}{a}\right) ^k\frac{J_k}{\eta ^{2k+4}}\cdot \Bigg [\sum _{m=0}^ke\cdot F_{k,0,m}(I)\nonumber \\&\quad \times \,\bigg (\left( 14+10k\right) \cdot \Big (\Phi ^*_{1,1,k,k'}-\Phi ^*_{1,-1,k,k'}\Big )\nonumber \\&\quad -3\left( 1+k\right) \eta ^2\cdot \Big (\Phi ^*_{0,1,k,k'}-\Phi ^*_{0,-1,k,k'}\Big )\bigg )\nonumber \\&\quad -2\sin I\sum _{j=0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}(-1)^{k}F_{k-2j-1,0,m}(I)\nonumber \\&\quad \times \,\bigg (4\Phi ^*_{3,0,k,k''}+2\Phi ^*_{2,0,k,k''}+\eta ^2\Phi ^*_{1,0,k,k''}\nonumber \\&\quad -e^2\cdot \Big (\Phi ^*_{1,2,k,k''}-2\Phi ^*_{1,0,k,k''}+\Phi ^*_{1,-2,k,k''}\Big )\bigg )\Bigg ]~, \end{aligned}$$

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$$\begin{aligned} (\Delta a)_{\text {LP}}&= J_k\frac{\mu }{c^2}\left( \frac{R}{a}\right) ^k\frac{2n}{\eta }\cdot \Bigg [\sum _{m=0}^ke\cdot \frac{F_{k,0,m}(I)}{k'\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k'\omega \right) \nonumber \\&\quad \times \,\bigg (\left( 14+10k\right) \cdot X_0^{-k-3,k'+1}-3\left( 1+k\right) \cdot X_0^{-k-2,k'+1}\bigg )\nonumber \\&\quad -(-1)^{k}\sin I\sum _{j=0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \nonumber \\&\quad \times \,\bigg (4\eta ^4X_0^{-k-5,k''}+2\eta ^2X_0^{-k-4,k''}+\eta ^2X_0^{-k-3,k''}\nonumber \\&\quad - e^2\bigg [X_0^{-k-3,k''+2}-2X_0^{-k-3,k''}+X_0^{-k-3,k''-2}\bigg ]\bigg )\Bigg ]~,\nonumber \\ \end{aligned}$$

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where $k' = k-2m\text { and }k'' = k-2j-2m~$.

1.2 Eccentricity e

$$\begin{aligned} \left( \Delta e\right) _\text {SP}&= \frac{\mu }{c^2a}\left( \frac{R}{a}\right) ^k \frac{J_k}{\eta ^{2k+2}}\cdot \Bigg [\sum _{m=0}^k F_{k,0,m}(I)\nonumber \\&\quad \times \,\bigg (\left( 7 + 5k\right) \cdot \Big (\Phi ^*_{1,1,k,k'}-\Phi ^*_{1,-1,k,k'}\Big )-\frac{3}{2}\left( 1+k\right) \eta ^2\cdot \Big (\Phi ^*_{0,1,k,k'}-\Phi ^*_{0,-1,k,k'}\Big )\nonumber \\&\quad -2\left( 1+k\right) \cdot \Big (\Phi ^*_{2,1,k,k'}-\Phi ^*_{2,-1,k,k'}\Big )+2e^2(1+k)\cdot \Big (\Phi ^*_{0,1,k,k'}-\Phi ^*_{0,-1,k,k'}\Big )\nonumber \\&\quad +e\left( 1+k\right) \cdot \Big (\Phi ^*_{1,2,k,k'}-\Phi ^*_{1,-2,k,k'}+\Phi ^*_{0,2,k,k'}-\Phi ^*_{0,-2,k,k'}\Big )\bigg )\nonumber \\&\quad -\sum _{j=0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}(-1)^{k}\alpha _{k,j}\cdot \sin I\cdot F_{k-2j-1,0,m}(I) \nonumber \\&\quad \times \,\bigg (\frac{\eta ^2}{2}\cdot \Big (\Phi ^*_{-1,1,k,k''}+\Phi ^*_{-1,-1,k,k''}\Big )+\left( 1+\frac{\eta ^2}{2}\right) \Big (\Phi ^*_{0,1,k,k''}+\Phi ^*_{0,-1,k,k''}\Big )\nonumber \\&\quad -e\cdot \Big (\Phi ^*_{1,2,k,k''} - 2\Phi ^*_{1,0,k,k''} + \Phi ^*_{1,-2,k,k''}\Big )\nonumber \\&\quad +3\cdot \Big (\Phi ^*_{1,1,k,k''} + \Phi ^*_{1,-1,k,k''}\Big )+2\cdot \Big (\Phi ^*_{2,1,k,k''} + \Phi ^*_{2,-1,k,k''}\Big )\nonumber \\&\quad +e\cdot \Big (\eta ^2\Phi ^*_{-1,0,k,k''} + 2\Phi ^*_{0,0,k,k''} + 4\Phi ^*_{1,0,k,k''}\Big )\bigg )\Bigg ]~, \end{aligned}$$

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$$\begin{aligned} (\Delta e)_\text {LP}&= \frac{\mu ^2}{c^2a}\left( \frac{R}{a}\right) ^k J_k \eta \cdot n\cdot \Bigg [\sum _{m=0}^k \frac{F_{k,0,m}(I)}{k'\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k'\omega \right) \nonumber \\&\quad \times \,\bigg (\left( 14 + 10k\right) X_0^{-k-3,k'+1}-3\left( 1+k\right) X_0^{-k-2,k'+1}-4\eta ^2\left( 1+k\right) X_0^{-k-4,k'+1}\nonumber \\&\quad +2e\left( 1+k\right) \Big [X_0^{-k-3,k'+2}+\frac{1}{\eta ^2}X_0^{-k-2,k'+2}\Big ]+\frac{4e^2}{\eta ^2}(1+k)X_0^{-k-2,k'+1}\bigg )\nonumber \\&\quad -(-1)^k\sum _{j=0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot \sin I\cdot \frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \nonumber \\&\quad \times \,\bigg (\frac{1}{2\eta ^2}\cdot \left[ X_0^{-k-1,k''+1}+X_0^{-k-1,k''-1}\right] \nonumber \\&\quad +\left( \frac{1}{2}+\frac{1}{\eta ^2}\right) \cdot \left[ X_0^{-k-2,k''+1}+X_0^{-k-2,k''-1}\right] \nonumber \\&\quad -e\cdot \left[ X_0^{-k-3,k''+2}-2X_0^{-k-3,k''}+X_0^{-k-3,k''-2}\right] \nonumber \\&\quad +3\cdot \left[ X_0^{-k-3,k''+1}+X_0^{-k-3,k''-1}\right] +2\eta ^2\cdot \left[ X_0^{-k-4,k''+1}+X_0^{-k-4,k''-1}\right] \nonumber \\&\quad +\frac{e}{\eta ^2}\cdot \left[ X_0^{-k-1,k''}+2X_0^{-k-2,k''}+4\eta ^2X_0^{-k-3,k''}\right] \bigg )\Bigg ]~,\nonumber \\ \text {where } k'&= k-2m \text { and } k'' = k-2j-2m~. \end{aligned}$$

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1.3 Inclination I

$$\begin{aligned} (\Delta I)_\text {SP}&= -\frac{\mu }{c^2a}J_k\left( \frac{R}{a}\right) ^k\frac{\cos I}{\eta ^{2k}}\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}(-1)^{k}\alpha _{k,j}F_{k-2j-1,0,m}(I)\nonumber \\&\quad \times \,\Bigg (\frac{2}{\eta ^2}\Phi ^*_{0,0,k,k''}+\Phi ^*_{-1,0,k,k''}\Bigg )~, \end{aligned}$$

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$$\begin{aligned} (\Delta I)_\text {LP}&= -\frac{\mu }{c^2a}J_k\left( \frac{R}{a}\right) ^k\frac{n\cos I}{\eta }\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}(-1)^{k}\alpha _{k,j}\frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\nonumber \\&\quad \times \,\left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \cdot \left( 2X_0^{-k-2,k''} + X_0^{-k-1,k''}\right) ~,\nonumber \\ \text {where } k''&= k-2j-2m~. \end{aligned}$$

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1.4 Argument of periapsis $\omega$

We give the results for $\omega ' = \omega + \Omega \cos I$. As mentioned above, one has to pay attention to $\omega$, if one considers only odd multipoles, in which case the lowest odd multipole will not give rise to long-periodic perturbations.

$$\begin{aligned} (\Delta \omega ')_\text {LP}&= \frac{\mu }{c^2a}J_k\left( \frac{R}{a}\right) ^k\frac{n\eta }{e}\cdot \Bigg [\sum _{m=0}^{k}\cdot (-1)^k\cdot \frac{F_{k,0,m}(I)}{k'\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k'\omega \right) \nonumber \\&\quad \times \,\bigg (\left( 14+10k\right) \cdot X_0^{-k-3,k'+1} -3\left( 1+k\right) \cdot X_0^{-k-2,k'+1} \nonumber \\&\quad -4\eta ^2\left( 1+k\right) \cdot X_0^{-k-4,k'+1}+2e\left( 1+k\right) \cdot \Big [\left( X_0^{-k-3,k'+2} - X_0^{-k-3,k'}\right) \nonumber \\&\quad +\frac{1}{\eta ^2}\left( X_0^{-k-2,k'+2} - X_0^{-k-2,k'}\right) \Big ]\bigg )\nonumber \\&\quad -\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\sin I \cdot \frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \nonumber \\&\quad \times \,\bigg (\frac{1}{2\eta ^2}\cdot \left[ X_0^{-k-1,k''+1}-X_0^{-k-1,k''-1}\right] \nonumber \\&\quad +\left( \frac{1}{2}+\frac{1}{\eta ^2}\right) \cdot \left[ X_0^{-k-2,k''+1}-X_0^{-k-2,k''-1}\right] \nonumber \\&\quad -e\cdot \left[ X_0^{-k-3,k''+2}-X_0^{-k-3,k''-2}\right] \nonumber \\&\quad +3\cdot \left[ X_0^{-k-3,k''+1}-X_0^{-k-3,k''-1}\right] \nonumber \\&\quad +2\eta ^2\cdot \left[ X_0^{-k-4,k''+1}-X_0^{-k-4,k''-1}\right] \bigg )~. \nonumber \\ \text {where } k'&= k-2m \text { and } k'' = k-2j-2m.\end{aligned}$$

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$$\begin{aligned} \left( \Delta \omega '\right) _\text {SP}&= \frac{\mu }{c^2a}J_k\left( \frac{R}{a}\right) ^k\frac{1}{e\eta ^{2k+2}}\cdot \Bigg [\sum _{m=0}^{k} (-1)^kF_{k,0,m}(I)\nonumber \\&\quad \times \,\bigg (\left( 14+10k\right) \cdot \left( \Phi _{1,1,k,k'} + \Psi _{1,1,k,k'}\right) \nonumber \\&\quad -3\left( 1+k\right) \eta ^2\cdot \left( \Phi _{0,1,k,k'} + \Psi _{0,1,k,k'}\right) \nonumber \\&\quad -4\left( 1+k\right) \cdot \left( \Phi _{2,1,k,k'} + \Psi _{2,1,k,k'}\right) \nonumber \\&\quad +2e\left( 1+k\right) \cdot \Big [\Phi _{1,2,k,k'} - \Phi _{1,0,k,k'} + \Psi _{1,2,k,k'} - \Psi _{1,0,k,k'}\nonumber \\&\quad +\Phi _{0,2,k,k'} - \Phi _{0,0,k,k'} + \Psi _{0,2,k,k'} - \Psi _{0,0,k,k'}\Big ]\bigg )\nonumber \\&\quad -\sum _{j=0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot \sin I\cdot F_{k-2j-1,0,m}(I) \nonumber \\&\quad \times \,\bigg (\frac{\eta ^2}{2}\cdot \Big (\Phi _{-1,1,k,k''}-\Phi _{-1,-1,k,k''})\nonumber \\&\quad +\left( 1+\frac{\eta ^2}{2}\right) \cdot \Big (\Phi _{0,1,k,k''}-\Phi _{0,-1,k,k''})\nonumber \\&\quad -e\cdot \Big (\Phi _{1,2,k,k''}-\Phi _{1,-2,k,k''})\nonumber \\&\quad +3\cdot \Big (\Phi _{1,1,k,k''}-\Phi _{1,-1,k,k''})\nonumber \\&\quad +2\cdot \Big (\Phi _{2,1,k,k''}-\Phi _{2,-1,k,k''})\bigg )\Bigg ]~,\nonumber \\ \text {where } k'&= k-2m\text { and } k'' = k-2j-2m~. \end{aligned}$$

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1.5 Longitude of the ascending node $\Omega$

$$\begin{aligned} (\Delta \Omega )_\text {SP}&= -\frac{\mu }{c^2 a}J_k \left( \frac{R}{a}\right) ^k\frac{\cot I}{\eta ^{2k}}\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot F_{k-2j-1,0,m}(I)\nonumber \\&\quad \times \,\Big (\Phi _{-1,0,k,k''} + \frac{2}{\eta ^2}\Phi _{0,0,k,k''}+\Psi _{-1,0,k,k''}+\frac{2}{\eta ^2}\Psi _{0,0,k,k''}\Big )~, \end{aligned}$$

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$$\begin{aligned} (\Delta \Omega )_\text {LP}&= -\frac{\mu }{c^2 a}J_k \left( \frac{R}{a}\right) ^k \frac{n\cot I}{\eta }\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot \frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\nonumber \\&\quad \times \,\left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \cdot \left( 2X_0^{-k-2,k''} + X_0^{-k-1,k''}\right) ~,\nonumber \\ \text {where } k''&= k-2j-2m~. \end{aligned}$$

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1.6 Mean anomaly M

We give results for $M' = M - n\cdot t + \omega \cdot \eta + \Omega \eta \cos I$. For k odd there are no secular perturbations in the mean anomaly.

$$\begin{aligned} \left( \Delta M'\right) _\text {S}&= \left( \frac{\mu }{ac^2}\right) J_k\left( \frac{R}{a}\right) ^k\cdot F_{k,0,k/2}(I)\nonumber \\&\quad \times \,\bigg [\left( 28+20k\right) X_0^{-k-2,0}-6(1+k)X_0^{-k-1,0}-8(1+k)\eta ^2X_0^{-k-3,0}\bigg ]\cdot nt~, \end{aligned}$$

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$$\begin{aligned} \left( \Delta M'\right) _\text {SP}&= \frac{J_k}{\eta ^{2k+1}}\frac{\mu }{ac^2}\left( \frac{R}{a}\right) ^k\cdot \Bigg [\sum _{m=0}^k (-1)^kF_{k,0,m}(I)\cdot \bigg (\left( 28+20k\right) \left( \Phi _{0,0,k,k'}+\Psi _{0,0,k,k'}\right) \nonumber \\&\quad -6\left( 1+k\right) \eta ^2\left( \Phi _{-1,0,k,k'}+\Psi _{-1,0,k,k'}\right) -8\left( 1+k\right) \left( \Phi _{1,0,k,k'}+\Psi _{1,0,k,k'}\right) \bigg )\nonumber \\&\quad +4e\sin I\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot F_{k-2j-1,0,m}(I)\cdot \bigg (\Phi _{0,1,k,k''} - \Phi _{0,-1,k,k''}\bigg )\Bigg ]~, \end{aligned}$$

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$$\begin{aligned} \left( \Delta M'\right) _\text {LP}&= \frac{\mu n}{c^2a}J_k\left( \frac{R}{a}\right) ^k\cdot \Bigg [(-1)^k\sum _{m=0}^k \frac{F_{k,0,m}(I)}{k'\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k'\omega \right) \nonumber \\&\quad \times \,\bigg (\left( 28+20k\right) X_0^{-k-2,k'}-6(1+k)X_0^{-k-1,k'}-8(1+k)\eta ^2X_0^{-k-3,k'}\bigg )\nonumber \\&\quad +4e\sin I\sum _{j = 0}^{\left[ {\frac{k-1}{2}} \right] }\sum _{m=0}^{k-2j-1}\alpha _{k,j}\cdot \frac{F_{k-2j-1,0,m}(I)}{k''\dot{\omega }}\cdot \left[ \begin{array}{c} \cos \\ \sin \end{array}\right] ^\text { k even}_\text { k odd}\left( k''\omega \right) \nonumber \\&\quad \times \,\Big (X_0^{-k-2,k''+1}-X_0^{-k-2,k''-1}\Big )\Bigg ]~,\nonumber \\ \text {where } k'&= k-2m \text { and } k'' = k-2j-2m~. \end{aligned}$$

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C. Expressions in the canonical approach

First, we list two remaining expressions from the calculation process in the canonical approach. From these the (quasi-)Newtonian results for arbitrary $J_k$ can be reproduced. Finally we list the expressions for the second-order drifts in $h = \Omega$ and $g = \omega$.

$$\begin{aligned} \mathcal {H}^*_1&= \frac{J_kR^k\mu ^{k+2}}{L'^3G'^{2k-1}}\sum _{m=0}^{\left[ {k/2} \right] }\left( 2-\delta _{k,2m}\right) \cdot b_{k-2m}\cdot \sum _{s=0}^{k-1}\sum _{q=0}^s \left( \frac{e'}{2}\right) ^s\left( {\begin{array}{c}k-1\\ s\end{array}}\right) \left( {\begin{array}{c}s\\ q\end{array}}\right) \nonumber \\&\quad \times \,\delta _{s-2q,k-2m}\cdot \cos \left( (k-2m\right) \cdot (g'-\frac{\pi }{2}))\nonumber \\&\quad -\frac{\mu ^4}{c^2L'^4}\left[ 3\frac{L'}{G'} - \frac{15}{8}\right] \end{aligned}$$

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$$\begin{aligned} W_1&= \frac{J_kR^k\mu ^k}{G'^{2k-1}}\sum _{m=0}^{\left[ {k/2} \right] }\left( 2-\delta _{k,2m}\right) \cdot b_{k-2m}\sum _{s=0}^{k-1}\sum _{q=0}^s \left( \frac{e'}{2}\right) ^s\left( {\begin{array}{c}k-1\\ s\end{array}}\right) \left( {\begin{array}{c}s\\ q\end{array}}\right) \nonumber \\&\quad \times \,\bigg [\delta _{s-2q,k-2m}(f'-l')\cdot \cos \left( (k-2m\right) \cdot (g'-\frac{\pi }{2}))\nonumber \\&\quad +\big (1-\delta _{s-2q,k-2m}\big )\frac{\sin \left( (s-2q+k-2m\right) f'+(k-2m)\cdot (g'-\frac{\pi }{2}))}{s-2q+k-2m}\bigg ]\nonumber \\&\quad -\frac{\mu ^2}{c^2L'}\left[ 3\frac{L'}{G'}(f'-l')-2(E'-l')\right] \end{aligned}$$

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Since the two Lie-transformations contain the short-periodic and long-periodic perturbations only, the secular drifts in $h = \Omega$ and $g = \omega$ are given by the drifts in $h''$ and $g''$, respectively. For these we find to second order

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and the derivatives, giving the second-order secular drifts, are

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The dots-term $\left( \dots \right)$ denotes the respective part in (Eq. 30).