Coefficients of some orbital variations
The coefficients \({\widehat{T}}_j,\,j=1,\,2,\,\dots 6\) entering Eqs. (39)–(43) and (55)–(60) in Sect. 2.2 are
$$\begin{aligned} {\widehat{T}}_1&:= 1, \end{aligned}$$
(A1)
$$\begin{aligned} {\widehat{T}}_2&:= \left[ \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{l}}}\right) ^2 + \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{m}}}\right) ^2\right] ,\end{aligned}$$
(A2)
$$\begin{aligned} {\widehat{T}}_3&:= \left[ \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{l}}}\right) ^2 - \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{m}}}\right) ^2\right] ,\end{aligned}$$
(A3)
$$\begin{aligned} {\widehat{T}}_4&:= \left[ \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{h}}}\right) \,\left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{l}}}\right) \right] ,\end{aligned}$$
(A4)
$$\begin{aligned} {\widehat{T}}_5&:= \left[ \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{h}}}\right) \,\left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{m}}}\right) \right] ,\end{aligned}$$
(A5)
$$\begin{aligned} {\widehat{T}}_6&:= \left[ \left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{l}}}\right) \,\left( \hat{\mathbf {{k}}}\mathbf \cdot \hat{\mathbf {{m}}}\right) \right] . \end{aligned}$$
(A6)
They depend on I and \(\Omega \), and on the polar angles in terms of which \({\hat{\textbf{k}}}\) is parameterized; see, e.g., Eqs. (83)–(86).
1.1 Coefficients of the instantaneous Newtonian shifts due to \(J_2\)
Here, we deal with the instantaneous Newtonian shifts induced by \(J_2\) calculated in Sect. 2.2. We display the explicit expressions of the coefficients \({\mathcal {A}}_1^{\left( J_2\right) },\ldots {\mathcal {P}}_6^{\left( J_2\right) }\) entering Eqs. (39)–(43), which are
$$\begin{aligned} {\mathcal {A}}_1^{\left( J_2\right) }&:= 4\,e\,\left[ -3\,\left( 4 + e^2\right) \,\cos f + e\,\left( -6\,\cos 2f - e\,\cos 3f\right) \right] -\left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.1)
$$\begin{aligned} {\mathcal {A}}_2^{\left( J_2\right) }&:= 6\,e\,\left[ 3\,\left( 4 + e^2\right) \,\cos f + e\,\left( 6\,\cos 2f + e\,\cos 3f\right) \right] - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.2)
$$\begin{aligned} {\mathcal {A}}_3^{\left( J_2\right) }&:= 3\,\left( e^3\,\cos \left( f - 2 \omega \right) + 6\,e\,\left\{ \left[ 2\,e\,+ \left( 4 + e^2\right) \,\cos f\right] \,\cos 2u + e\,\cos \left( 4f + 2\omega \right) \right\} \right. \nonumber \\&\quad + e^3\,\cos \left( 5 f + 2 \omega \right) - \nonumber \\&\left. - 16\,\sin f\,\sin \left( f + 2 \omega \right) \right) - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.3)
$$\begin{aligned} {\mathcal {A}}_4^{\left( J_2\right) }&:=0, \end{aligned}$$
(A.1.4)
$$\begin{aligned} {\mathcal {A}}_5^{\left( J_2\right) }&:=0, \end{aligned}$$
(A.1.5)
$$\begin{aligned} {\mathcal {A}}_6^{\left( J_2\right) }&:= 6\,\left( 16\,\cos \left( f + 2\omega \right) \,\sin f + e\,\left\{ -e^2\,\sin \left( f - 2\omega \right) + 6\,\left[ 2\,e\,+ \left( 4 + e^2\right) \,\cos f\right] \right. \right. \nonumber \\&\quad \,\sin 2u + \nonumber \\&\left. \left. + 6\,e\,\sin \left( 4f + 2\omega \right) + e^2\,\sin \left( 5 f + 2\omega \right) \right\} \right) - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.6)
$$\begin{aligned} {\mathcal {E}}_1^{\left( J_2\right) }&:= 4\,\left[ 3\,\left( 4 + e^2\right) \,\cos f + e\,\left( 6\,\cos 2f + e\,\cos 3f\right) \right] - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.7)
$$\begin{aligned} {\mathcal {E}}_2^{\left( J_2\right) }&:= -6\,\left[ 3\,\left( 4 + e^2\right) \,\cos f + e\,\left( 6\,\cos 2f + e\,\cos 3f\right) \right] - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.8)
$$\begin{aligned} {\mathcal {E}}_3^{\left( J_2\right) }&:= -4\,\left[ 3\,\cos \left( f + 2 \omega \right) + 7\,\cos \left( 3 f + 2 \omega \right) \right] + e \left\{ -e\,\left[ 3\,\cos \left( f - 2 \omega \right) \right. \right. \nonumber \\&\left. \left. + 33\,\cos \left( f + 2 \omega \right) + 17\,\cos \left( 3 f + 2 \omega \right) + 3\,\cos \left( 5 f + 2 \omega \right) \right] + 36\,\sin 2f\,\sin 2u \right. \nonumber \\&\left. + 120\,\sin f\,\sin \left( f + 2 \omega \right) \right\} - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.9)
$$\begin{aligned} {\mathcal {E}}_4^{\left( J_2\right) }&:= 0, \end{aligned}$$
(A.1.10)
$$\begin{aligned} {\mathcal {E}}_5^{\left( J_2\right) }&:= 0, \end{aligned}$$
(A.1.11)
$$\begin{aligned} {\mathcal {E}}_6^{\left( J_2\right) }&:= 6\,e^2\,\sin \left( f - 2 \omega \right) - 8\,\left[ 3\,\sin \left( f + 2 \omega \right) + 7\,\sin \left( 3 f + 2 \omega \right) \right] \nonumber \\&\quad - 2\,e\,\left\{ 24\,\left[ 3\,\cos f\,\cos 2u \right. \right. \nonumber \\&\left. + 5\,\cos \left( f + 2 \omega \right) \right] \,\sin f + e\,\left[ 33\,\sin \left( f + 2 \omega \right) \right. \nonumber \\&\left. \left. \quad + 17\,\sin \left( 3 f + 2 \omega \right) + 3\,\sin \left( 5 f + 2 \omega \right) \right] \right\} - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.12)
$$\begin{aligned} {\mathcal {I}}_1^{\left( J_2\right) }&:=0, \end{aligned}$$
(A.1.13)
$$\begin{aligned} {\mathcal {I}}_2^{\left( J_2\right) }&:=0, \end{aligned}$$
(A.1.14)
$$\begin{aligned} {\mathcal {I}}_3^{\left( J_2\right) }&:=0, \end{aligned}$$
(A.1.15)
$$\begin{aligned} {\mathcal {I}}_4^{\left( J_2\right) }&:= 6 f + 6\,e\,\sin f + 3\,\sin 2u + 3\,e\,\sin \left( f + 2 \omega \right) + e\,\sin \left( 3 f + 2 \omega \right) - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.16)
$$\begin{aligned} {\mathcal {I}}_5^{\left( J_2\right) }&:= -\left\{ 3\,\cos 2u + e\,\left[ 3\,\cos \left( f + 2 \omega \right) + \cos \left( 3 f + 2 \omega \right) \right] \right\} - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.17)
$$\begin{aligned} {\mathcal {I}}_6^{\left( J_2\right) }&:=0,\end{aligned}$$
(A.1.18)
$$\begin{aligned} {\mathcal {N}}_1^{\left( J_2\right) }&:=0,\end{aligned}$$
(A.1.19)
$$\begin{aligned} {\mathcal {N}}_2^{\left( J_2\right) }&:=0,\end{aligned}$$
(A.1.20)
$$\begin{aligned} {\mathcal {N}}_3^{\left( J_2\right) }&:=0,\end{aligned}$$
(A.1.21)
$$\begin{aligned} {\mathcal {N}}_4^{\left( J_2\right) }&:= - \left\{ 3\,\cos 2u + e\,\left[ 3\,\cos \left( f + 2 \omega \right) + \cos \left( 3 f + 2 \omega \right) \right] \right\} - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.22)
$$\begin{aligned} {\mathcal {N}}_5^{\left( J_2\right) }&:= 6 f + 6\,e\,\sin f - 3\,\sin 2u - e\,\left[ 3\,\sin \left( f + 2 \omega \right) + \sin \left( 3 f + 2 \omega \right) \right] - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.23)
$$\begin{aligned} {\mathcal {N}}_6^{\left( J_2\right) }&:=0,\end{aligned}$$
(A.1.24)
$$\begin{aligned} {\mathcal {P}}_1^{\left( J_2\right) }&:= 48\,e\,f + 8\,\left( 6 + 5\,e^2 + 6\,e\,\cos f + e^2\,\cos 2f\right) \,\sin f - \left[ f \rightarrow f_0\right] ,\end{aligned}$$
(A.1.25)
$$\begin{aligned} {\mathcal {P}}_2^{\left( J_2\right) }&:= 6\,\left[ -12\,e\,f - 2\,\left( 6 + 5\,e^2 + 6\,e\,\cos f + e^2\,\cos 2f\right) \,\sin f\right] - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.26)
$$\begin{aligned} {\mathcal {P}}_3^{\left( J_2\right) }&:= 4\,\left[ 3\,\sin \left( f + 2 \omega \right) - 7\,\sin \left( 3 f + 2 \omega \right) \right] \nonumber \\&\quad - e \left\{ 36\,\left[ 3\,\cos \left( f + 2 \omega \right) + \cos \left( 3 f + 2 \omega \right) \right] \,\sin f + \right. \nonumber \\&\left. + e\,\left[ 3\,\sin \left( f - 2 \omega \right) + 21\,\sin \left( f + 2 \omega \right) + 11\,\sin \left( 3 f + 2 \omega \right) + 3\,\sin \left( 5 f + 2 \omega \right) \right] \right\} \nonumber \\&\quad - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.27)
$$\begin{aligned} {\mathcal {P}}_4^{\left( J_2\right) }&:= -8\,e\,\left\{ 3\,\cos 2u + e\,\left[ 3\,\cos \left( f + 2 \omega \right) + \cos \left( 3 f + 2 \omega \right) \right] \right\} \cot I\nonumber \\&\quad - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.28)
$$\begin{aligned} {\mathcal {P}}_5^{\left( J_2\right) }&:= -8\,e\, \cot I \left\{ -6 f + 3\,\sin 2u + e\,\left[ -6\,\sin f + 3\,\sin \left( f + 2 \omega \right) + \sin \left( 3 f + 2 \omega \right) \right] \right\} \nonumber \\&\quad - \left[ f \rightarrow f_0\right] , \end{aligned}$$
(A.1.29)
$$\begin{aligned} {\mathcal {P}}_6^{\left( J_2\right) }&:= -6\,e^2\,\cos \left( f - 2 \omega \right) + 6 (-4 + 7\,e^2)\,\cos \left( f + 2 \omega \right) + 56\,\cos \left( 3 f + 2 \omega \right) \nonumber \\&\quad + 2\,e\,\left\{ 11\,e\,\cos \left( 3 f + 2 \omega \right) + \right. \nonumber \\&\left. + 3\,e\,\cos \left( 5 f + 2 \omega \right) - 36\,\sin f\,\left[ 3\,\sin \left( f + 2 \omega \right) + \sin \left( 3 f + 2 \omega \right) \right] \right\} - \left[ f \rightarrow f_0\right] . \end{aligned}$$
(A.1.30)
1.2 Coefficients of the total mixed shifts per orbit of order \(J_2/c^2\)
Here, the mixed averaged shifts per orbit of order \({\mathcal {O}}\left( J_2/c^2\right) \), calculated in Sect. 2.2, are treated. The explicit expressions of the coefficients \(\mathcal {{\overline{A}}}_1^{\left( J_2/c^2\right) },\ldots \mathcal {{\overline{H}}}_6^{\left( J_2/c^2\right) }\) entering Eqs. (55)–(60) are displayed below. They read
$$\begin{aligned} \mathcal {{\overline{A}}}_1^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.1)
$$\begin{aligned} \mathcal {{\overline{A}}}_2^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.2)
$$\begin{aligned} \mathcal {{\overline{A}}}_3^{\left( J_2/c^2\right) }&:= 8\,\left( 1 + e\,\cos f_0\right) ^3\,\cos 2\omega \,\sin 2f_0 \nonumber \\&\quad + \left\{ 4\,e\,\left( 3 + e^2\right) \,\cos f_0 + 4\,\left( 2 + 3\,e^2\right) \,\cos 2f_0 + e\,\left[ 3\,\left( 4 + e^2\right) \,\cos 3f_0 \right. \right. \nonumber \\&\left. \left. + e\,\left( 12 + e^2 + 6\,\cos 4f_0 + e\,\cos 5f_0\right) \right] \right\} \,\sin 2\omega , \end{aligned}$$
(A.2.3)
$$\begin{aligned} \mathcal {{\overline{A}}}_4^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.4)
$$\begin{aligned} \mathcal {{\overline{A}}}_5^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.5)
$$\begin{aligned} \mathcal {{\overline{A}}}_6^{\left( J_2/c^2\right) }&:= -2 \left\{ 4\,e\,\left( 3 + e^2\right) \,\cos f_0 + 4\,\left( 2 + 3\,e^2\right) \,\cos 2f_0\right. \nonumber \\&\quad \left. + e\,\left[ 3\,\left( 4 + e^2\right) \,\cos 3f_0 + e\,\left( 12 + e^2 + 6\,\cos 4f_0 + e\,\cos 5f_0\right) \right] \right\} \,\cos 2\omega + \nonumber \\&+ 16\,\left( 1 + e\,\cos f_0\right) ^3\,\sin 2f_0\,\sin 2\omega , \end{aligned}$$
(A.2.6)
$$\begin{aligned} \mathcal {{\overline{E}}}_1^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.7)
$$\begin{aligned} \mathcal {{\overline{E}}}_2^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.8)
$$\begin{aligned} \mathcal {{\overline{E}}}_3^{\left( J_2/c^2\right) }&:= - \left\{ 4\,\left[ 3\,\sin \left( f_0 + 2 \omega \right) + 7\,\sin \left( 3 f_0 + 2 \omega \right) \right] \right. \nonumber \\&\quad + e\,\left[ -3\,e\,\sin \left( f_0 - 2 \omega \right) + \left( 20 + 19\,e^2\right) \,\sin 2\omega + 60\,\sin u_0 + 18\,\sin \left( 4f_0 + 2\omega \right) \right. \nonumber \\&\left. \left. + 33\,e\,\sin \left( f_0 + 2 \omega \right) + 17\,e\,\sin \left( 3 f_0 + 2 \omega \right) + 3\,e\,\sin \left( 5 f_0 + 2 \omega \right) \right] \right\} , \end{aligned}$$
(A.2.9)
$$\begin{aligned} \mathcal {{\overline{E}}}_4^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.10)
$$\begin{aligned} \mathcal {{\overline{E}}}_5^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.11)
$$\begin{aligned} \mathcal {{\overline{E}}}_6^{\left( J_2/c^2\right) }&:= 8\,\left[ 3\,\cos \left( f_0 + 2 \omega \right) + 7\,\cos \left( 3 f_0 + 2 \omega \right) \right] \nonumber \\&\quad + 2\,e\,\left[ 3\,e\,\cos \left( f_0 - 2 \omega \right) + \left( 20 + 19\,e^2\right) \,\cos 2\omega + 60\,\cos u_0\right. \nonumber \\&\quad + 18\,\cos \left( 4f_0 + 2\omega \right) \nonumber \\&\left. + 33\,e\,\cos \left( f_0 + 2 \omega \right) + 17\,e\,\cos \left( 3 f_0 + 2 \omega \right) + 3\,e\,\cos \left( 5 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.12)
$$\begin{aligned} \mathcal {{\overline{I}}}_1^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.13)
$$\begin{aligned} \mathcal {{\overline{I}}}_2^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.14)
$$\begin{aligned} \mathcal {{\overline{I}}}_3^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.15)
$$\begin{aligned} \mathcal {{\overline{I}}}_4^{\left( J_2/c^2\right) }&:= 5\,e^2 + 3\,\cos u_0 + e\,\left[ -16\,\cos f_0 + 2\,e\,\cos 2\omega \right. \nonumber \\&\quad \left. + 3\,\cos \left( f_0 + 2 \omega \right) + \cos \left( 3 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.16)
$$\begin{aligned} \mathcal {{\overline{I}}}_5^{\left( J_2/c^2\right) }&:= 3\,\sin u_0 + e\,\left[ 2\,e\,\sin 2\omega + 3\,\sin \left( f_0 + 2 \omega \right) + \sin \left( 3 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.17)
$$\begin{aligned} \mathcal {{\overline{I}}}_6^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.18)
$$\begin{aligned} \mathcal {{\overline{N}}}_1^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.19)
$$\begin{aligned} \mathcal {{\overline{N}}}_2^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.20)
$$\begin{aligned} \mathcal {{\overline{N}}}_3^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.21)
$$\begin{aligned} \mathcal {{\overline{N}}}_4^{\left( J_2/c^2\right) }&:= 3\,\sin u_0 + e\,\left[ 2\,e\,\sin 2\omega + 3\,\sin \left( f_0 + 2 \omega \right) + \sin \left( 3 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.22)
$$\begin{aligned} \mathcal {{\overline{N}}}_5^{\left( J_2/c^2\right) }&:= 5\,e^2 - 3\,\cos u_0\nonumber \\&\quad - e\,\left[ 16\,\cos f_0 + 2\,e\,\cos 2\omega + 3\,\cos \left( f_0 + 2 \omega \right) + \cos \left( 3 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.23)
$$\begin{aligned} \mathcal {{\overline{N}}}_6^{\left( J_2/c^2\right) }&:=0,\end{aligned}$$
(A.2.24)
$$\begin{aligned} \mathcal {{\overline{P}}}_1^{\left( J_2/c^2\right) }&:= -4\,e\,\left( 44 + 17\,e^2 - 64\,e\,\cos f_0\right) \,\sin I, \end{aligned}$$
(A.2.25)
$$\begin{aligned} \mathcal {{\overline{P}}}_2^{\left( J_2/c^2\right) }&:= 6\,e\,\left( 44 + 17\,e^2 - 64\,e\,\cos f_0\right) \,\sin I, \end{aligned}$$
(A.2.26)
$$\begin{aligned} \mathcal {{\overline{P}}}_3^{\left( J_2/c^2\right) }&:= 2 \left\{ 4\,\left[ -3\,\cos \left( f_0 + 2 \omega \right) + 7\,\cos \left( 3 f_0 + 2 \omega \right) \right] \right. \nonumber \\&+ e\,\left[ -3\,e\,\cos \left( f_0 - 2 \omega \right) + 2\,\left( -10 + 9\,e^2\right) \,\cos 2\omega + 60\,\cos u_0 + \right. \nonumber \\&+ 18\,\cos \left( 4f_0 + 2\omega \right) + 45\,e\,\cos \left( f_0 + 2 \omega \right) + 19\,e\,\cos \left( 3 f_0 + 2 \omega \right) \nonumber \\&\left. \left. \quad + 3\,e\,\cos \left( 5 f_0 + 2 \omega \right) \right] \right\} \,\sin I, \end{aligned}$$
(A.2.27)
$$\begin{aligned} \mathcal {{\overline{P}}}_4^{\left( J_2/c^2\right) }&:= -16\,e\,\cos I \left\{ 3\,\sin u_0 + e\,\left[ 2\,e\,\sin 2\omega + 3\,\sin \left( f_0 + 2 \omega \right) + \sin \left( 3 f_0 + 2 \omega \right) \right] \right\} ,\end{aligned}$$
(A.2.28)
$$\begin{aligned} \mathcal {{\overline{P}}}_5^{\left( J_2/c^2\right) }&:= 16\,e\,\cos I \left\{ -5\,e^2 + 3\,\cos u_0 + e\,\left[ 16\,\cos f_0 + 2\,e\,\cos 2\omega \right. \right. \nonumber \\&\quad \left. \left. + 3\,\cos \left( f_0 + 2 \omega \right) + \cos \left( 3 f_0 + 2 \omega \right) \right] \right\} ,\end{aligned}$$
(A.2.29)
$$\begin{aligned} \mathcal {{\overline{P}}}_6^{\left( J_2/c^2\right) }&:= 4\,\sin I \left\{ 4\,\left[ -3\,\sin \left( f_0 + 2 \omega \right) + 7\,\sin \left( 3 f_0 + 2 \omega \right) \right] \right. \nonumber \\&+ e\,\left[ 3\,e\,\sin \left( f_0 - 2 \omega \right) + 2\,\left( -10 + 9\,e^2\right) \,\sin 2\omega + 60\,\sin u_0 \right. \nonumber \\&+ 18\,\sin \left( 4f_0 + 2\omega \right) + 45\,e\,\sin \left( f_0 + 2 \omega \right) + 19\,e\,\sin \left( 3 f_0 + 2 \omega \right) \nonumber \\&\left. \left. \quad + 3\,e\,\sin \left( 5 f_0 + 2 \omega \right) \right] \right\} ,\end{aligned}$$
(A.2.30)
$$\begin{aligned} \mathcal {{\overline{H}}}_1^{\left( J_2/c^2\right) }&:= 4\,e\,\left\{ 88 + 5\,e^4 - 16\,\sqrt{1 - e^2} - 3\,e^2\,\left( 21 + 8\,\sqrt{1 - e^2}\right) \right. \nonumber \\&\quad - e\,\left[ 3\,e^2\,\left( 7 + 4\,\sqrt{1 - e^2}\right) + 8\,\left( -17 + 6\,\sqrt{1 - e^2}\right) \right] \,\cos f_0 + \nonumber \\&\left. + e^2\,\left[ 8\,\left( 5 - 3\,\sqrt{1 - e^2}\right) \,\cos 2f_0 + e\,\left( 5 - 4\,\sqrt{1 - e^2}\right) \,\cos 3f_0\right] \right\} , \end{aligned}$$
(A.2.31)
$$\begin{aligned} \mathcal {{\overline{H}}}_2^{\left( J_2/c^2\right) }&:= 6\,e\,\left\{ -88 - 5\,e^4 + 16\,\sqrt{1 - e^2} + 3\,e^2\,\left( 21 + 8\,\sqrt{1 - e^2}\right) \right. \nonumber \\&\quad + e\,\left[ 3\,e^2\,\left( 7 + 4\,\sqrt{1 - e^2}\right) + 8\,\left( -17 + 6\,\sqrt{1 - e^2}\right) \right] \,\cos f_0 + \nonumber \\&\left. + 4\,e^2\,\sqrt{1 - e^2}\,\left( 6\,\cos 2f_0 + e\,\cos 3f_0\right) - 5\,e^2\,\left( 8\,\cos 2f_0 + e\,\cos 3f_0\right) \right\} , \end{aligned}$$
(A.2.32)
$$\begin{aligned} \mathcal {{\overline{H}}}_3^{\left( J_2/c^2\right) }&:= 3\,e^2\,\left( 2 - 7\,e^2\right) \,\cos \left( f_0 - 2 \omega \right) + 96\,e\,\sqrt{1 - e^2}\,\left( 1 + e\,\cos f_0\right) ^3\,\cos u_0 \nonumber \\&\quad + 8\,\left[ 3\,\cos \left( f_0 + 2 \omega \right) - 7\,\cos \left( 3 f_0 + 2 \omega \right) \right] \nonumber \\&+ e\,\left[ -2\,\left( -20 + 7\,e^2 + 13\,e^4\right) \,\cos 2\omega - 12\,\left( 14 + 11\,e^2\right) \,\cos u_0\right. \nonumber \\&- 18\,\left( 2 + 3\,e^2\right) \,\cos \left( 4f_0 + 2\omega \right) \nonumber \\&- 3\,e\,\left( 74 + 9\,e^2\right) \,\cos \left( f_0 + 2 \omega \right) - e\,\left( 138 + 31\,e^2\right) \,\cos \left( 3 f_0 + 2 \omega \right) \nonumber \\&\quad \left. - 3\,e\,\left( 2 + 3\,e^2\right) \,\cos \left( 5 f_0 + 2 \omega \right) \right] , \end{aligned}$$
(A.2.33)
$$\begin{aligned} \mathcal {{\overline{H}}}_4^{\left( J_2/c^2\right) }&:=0, \end{aligned}$$
(A.2.34)
$$\begin{aligned} \mathcal {{\overline{H}}}_5^{\left( J_2/c^2\right) }&:=0, \end{aligned}$$
(A.2.35)
$$\begin{aligned} \mathcal {{\overline{H}}}_6^{\left( J_2/c^2\right) }&:= -2\,\left[ 3\,e^2\,\left[ 2 + e^2\,\left( -7 + 4\,\sqrt{1 - e^2}\right) \right] \,\sin \left( f_0 - 2 \omega \right) \right. \nonumber \\&\quad + 2\,e\,\left[ -20 + 13\,e^4 + e^2\,\left( 7 - 36\,\sqrt{1 - e^2}\right) \right] \,\sin 2\omega + \nonumber \\&+ 8\,\left[ -3\,\sin \left( f_0 + 2 \omega \right) + 7\,\sin \left( 3 f_0 + 2 \omega \right) \right] \nonumber \\&\quad + e\,\left( 12\,\left[ 14 - 8\,\sqrt{1 - e^2} + e^2\,\left( 11 - 12\,\sqrt{1 - e^2}\right) \right] \,\sin u_0 \right. \nonumber \\&+ 18\,\left[ 2 + e^2\,\left( 3 - 4\,\sqrt{1 - e^2}\right) \right] \,\sin \left( 4f_0 + 2\omega \right) \nonumber \\&\quad + e \left\{ 3\,\left[ 74 - 48\,\sqrt{1 - e^2} + e^2\,\left( 9 - 12\,\sqrt{1 - e^2}\right) \right] \,\sin \left( f_0 + 2 \omega \right) \right. \nonumber \\&+ \left[ 138 - 144\,\sqrt{1 - e^2} + e^2\,\left( 31 - 36\,\sqrt{1 - e^2}\right) \right] \nonumber \\&\quad \left. \left. \left. \,\sin \left( 3 f_0 + 2 \omega \right) + 3\,\left[ 2 + e^2\,\left( 3 - 4\,\sqrt{1 - e^2}\right) \right] \,\sin \left( 5 f_0 + 2 \omega \right) \right\} \right) \right] . \end{aligned}$$
(A.2.36)
