## 3 Numerical Evaluation of the Dominant Terms of the Doubly Averaged Lunisolar Perturbation
### The Dominant Terms in the Long-Term Dynamics
\begin{table}
\begin{tabular}{l l c c c c c c} \hline  & **Argument** & \(\hat{g}\approx\hat{g}_{I_{2}}\) & \(\hat{g}\approx\hat{g}_{0}\) & \(\hat{g}\approx\hat{g}_{1}\) & \(\hat{g}\approx\hat{g}_{2}\) & \(\hat{g}\approx\hat{g}_{3}\) & \(\hat{g}\approx\hat{g}_{4}\) \\ \hline \(\odot\) & \(g-g_{\odot}\) & 23,669.36 & 1036.82 & 394.83 & 328.48 & 301.43 & 261.16 \\ \(\odot\) & \(g-g_{\odot}\) & 16,659.31 & 1018.06 & 392.08 & 326.57 & 299.83 & 259.95 \\ \hline \(\odot\) & \(3g-g_{\odot}\) & 6919.27 & 343.50 & 131.30 & 109.28 & 100.30 & 86.91 \\ \(\odot\) & \(3g+g_{\odot}\) & 6161.37 & 341.41 & 131.00 & 109.07 & 100.12 & 86.78 \\ \hline \(\ddagger\) & \(3g-g_{\mathcal{G}}-3h_{\mathcal{G}}\) & 184.42 & 367.55 & 488.04 & 279.03 & 227.11 & 168.40 \\ \(\ddagger\) & \(g-g_{\mathcal{G}}-3h_{\mathcal{G}}\) & 181.00 & 217.27 & 329.57 & 396.41 & 444.55 & 575.42 \\ \(\ddagger\) & \(g+g_{\mathcal{G}}+3h_{\mathcal{G}}\) & 177.71 & 152.69 & 123.19 & 115.89 & 112.33 & 106.23 \\ \(\ddagger\) & \(3g+g_{\mathcal{G}}+3h_{\mathcal{G}}\) & 174.54 & 117.70 & 75.75 & 67.86 & 64.29 & 58.51 \\ \hline \(\ddagger\) & \(3g-g_{\mathcal{G}}-2h+2h_{\mathcal{G}}\) & 108.11 & 154.24 & 562.28 & 4104.67 & 1736.91 & 473.80 \\ \(\ddagger\) & \(g-g_{\mathcal{G}}-2h+2h_{\mathcal{G}}\) & 106.93 & 118.62 & 145.73 & 157.48 & 164.55 & 179.68 \\ \(\ddagger\) & \(g+g_{\mathcal{G}}+2h-2h_{\mathcal{G}}\) & 105.77 & 96.37 & 83.72 & 80.28 & 78.55 & 75.52 \\ \(\ddagger\) & \(3g+g_{\mathcal{G}}+2h-2h_{\mathcal{G}}\) & 104.64 & 81.15 & 58.72 & 53.87 & 51.59 & 47.80 \\ \hline \(\ddagger\) & \(3g+g_{\mathcal{G}}+h+h_{\mathcal{G}}\) & 52.03 & 60.78 & 85.12 & 97.91 & 106.45 & 127.23 \\ \(\ddagger\) & \(g+g_{\mathcal{G}}+h+h_{\mathcal{G}}\) & 51.75 & 54.35 & 59.41 & 61.27 & 62.32 & 64.37 \\ \(\ddagger\) & \(g-g_{\mathcal{G}}-h-h_{\mathcal{G}}\) & 51.48 & 49.15 & 45.63 & 44.59 & 44.05 & 43.08 \\ \(\ddagger\) & \(3g-g_{\mathcal{G}}-h-h_{\mathcal{G}}\) & 51.21 & 44.86 & 34.07 & 37.03 & 35.04 & 32.37 \\ \hline \end{tabular}
\end{table} 
Table 6: Largest periods for the lunar and solar arguments appearing in the third-order doubly averaged disturbing potential. The first column helps to identify the third body. The periods (in years) are computed assuming the approximations of \(\hat{g}\) listed in Table 4 and \(h=h_{I_{2}}\). \begin{table}
\begin{tabular}{l l l l l l l l} \hline  & **Argument** & \(\hat{g}\approx\hat{g}_{I_{2}}\) & \(\hat{g}\approx\hat{g}_{0}\) & \(\hat{g}\approx\hat{g}_{1}\) & \(\hat{g}\approx\hat{g}_{2}\) & \(\hat{g}\approx\hat{g}_{3}\) & \(\hat{g}\approx\hat{g}_{4}\) \\ \hline \(\ddagger,\odot\) & \(2g\) & 9777.54 & 513.68 & 196.72 & 163.76 & 150.31 & 130.27 \\ \hline \(\ddagger\) & \(2h_{\mathcal{G}}-h-2g\) & 40.25 & 43.47 & 50.34 & 53.07 & 54.65 & 57.89 \\ \(\ddagger\) & \(2h_{\mathcal{G}}-h\) & 40.08 & 40.08 & 40.08 & 40.08 & 40.08 & 40.08 \\ \(\ddagger\) & \(2h_{\mathcal{G}}-h+2g\) & 39.92 & 37.18 & 33.30 & 32.20 & 31.64 & 30.65 \\ \hline \(\ddagger\) & \(h_{\mathcal{G}}+2g\) & 18.65 & 19.31 & 20.56 & 21.00 & 21.24 & 21.71 \\ \(\ddagger\) & \(h_{\mathcal{G}}\) & 18.61 & 18.61 & 18.61 & 18.61 & 18.61 & 18.61 \\ \(\ddagger\) & \(h_{\mathcal{G}}-2g\) & 18.58 & 17.96 & 17.00 & 16.71 & 16.56 & 16.28 \\ \hline \(\ddagger\) & \(h_{\mathcal{G}}-h-2g\) & 12.73 & 13.03 & 13.58 & 13.78 & 13.88 & 14.08 \\ \(\ddagger\) & \(h_{\mathcal{G}}-h\) & 12.71 & 12.71 & 12.71 & 12.71 & 12.71 & 12.71 \\ \(\ddagger\) & \(h_{\mathcal{G}}-h+2g\) & 12.69 & 12.40 & 11.94 & 11.79 & 11.72 & 11.58 \\ \hline \(\ddagger\) & \(2h_{\mathcal{G}}+2g\) & 9.31 & 9.47 & 9.77 & 9.87 & 9.92 & 10.02 \\ \(\ddagger\) & \(2h_{\mathcal{G}}\) & 9.31 & 9.31 & 9.31 & 9.31 & 9.31 & 9.31 \\ \(\ddagger\) & \(2h_{\mathcal{G}}-2g\) & 9.30 & 9.14 & 8.89 & 8.81 & 8.76 & 8.68 \\ \hline \(\ddagger\) & \(2g+h\) & 7.56 & 7.66 & 7.85 & 7.92 & 7.95 & 8.01 \\ \(\ddagger\) & \(h\) & 7.55 & 7.55 & 7.55 & 7.55 & 7.55 & 7.55 \\ \(\ddagger\) & \(2g-h\) & 7.55 & 7.55 & 7.27 & 7.21 & 7.19 & 7.13 \\ \hline \end{tabular}
\end{table}