## 3 Numerical Evaluation of the Dominant Terms of the Doubly Averaged Lunisolar Perturbation
### The Dominant Terms in the Long-Term Dynamics  
According to the theoretical considerations exposed in Section 2.2 and taking as a reference system (11), we evaluate the _amplitudes_ of the harmonic coefficients (see  
Table 1 shows all the amplitudes of the second-order solar harmonics. The number of the third-order solar harmonics computed is 28, but the corresponding coefficients are too small to be displayed: the largest values are of the order of \(10^{-11}\) km\({}^{2}\)/s\({}^{2}\). In the lunar case, the number of the second-order harmonics evaluated is 38, ranging from values of approximately \(10^{-5}\) km\({}^{2}\)/s\({}^{2}\) to \(10^{-11}\) km\({}^{2}\)/s\({}^{2}\). Instead, the third-order contribution consists of 196 harmonics, ranging from approximately \(10^{-8}\) km\({}^{2}\)/s\({}^{2}\) to \(10^{-17}\) km\({}^{2}\)/s\({}^{2}\). Of all these estimates, the largest amplitudes of both the second- and the third-order lunar potential are listed in Table 2.  
We notice that the largest lunar amplitudes in Table 2 are only one order of magnitude lower than the \(H_{I_{2}}\) contribution in Equation (7), which is of order \(10^{-4}\) km\({}^{2}\)/s\({}^{2}\) when evaluated at the Molinya parameter. This is due to the fact that a Molinya satellite attains high altitudes. Moreover, from the table, it is clear that, although considering the geocentric orbits of the Moon and of the Sun with their actual eccentricity (\(e_{\zeta}\,=\,0.0549\), \(e_{\odot}\,=\,0.0167\)), the third-order contribution in the Hamiltonian given by a third body is quite small, even if Molinya are highly eccentric and high-altitude orbits. In Table 3, we give the estimates of the largest values of the partial derivatives of the harmonic coefficients with respect to the actions, computed in the Molinya region. This analysis would indicate the dominant terms in the angular dynamics defined by the last two equations in Equation (11). to \(\dot{g}\), while, on the right, we report the terms determining the dynamics of \(h\). Only the second-order lunisolar contribution was taken into account. The precession caused by the lunar and solar mean terms (Table 3 on the left) is certainly larger than the secular drift of the argument of perigee due to the Earth's oblateness, which cancels out, by definition, at the critical inclination. Moreover, the amplitudes of the oscillations seem to be quite large. These facts imply that the third-body effects on the dynamics of the argument of perigee are small but not negligible if compared with the oblateness effect. If we change slightly the inclination--for example, \(i\approx 63.43\pm 2^{\circ}\)--the oblateness drift becomes only one order of magnitude larger than the ones produced by the third-body mean terms. For practical applications, because of the unavoidable errors in spacecraft maneuvering, the conclusion is that the gravitational attraction exerted by the Moon and the Sun is essential to study in depth the periods and the perigee resonance that characterize Molniya-like orbits (Sections 4.1 and 4). On the contrary, the partial derivatives of both the lunar and the solar mean terms in Table 3 on the right ensure that the oblateness effect is still the dominant perturbation for \(h\), as it usually happens in the case of a no frozen condition. Indeed,  
\[\dot{h}_{I_{2}}=\frac{\partial\mathcal{H}_{I_{2}}}{\partial H}|_{(L_{mol_{1}},G_{mol_{2}},H_{mol_{3}})}\approx-2.63\times 10^{-8}. \tag{16}\]  
Hence, to better capture how the lunisolar perturbation might affect the dynamics of the angular terms, the _periods_ of the arguments involved in the doubly averaged model are computed assuming \(\dot{h}\approx\dot{h}_{I_{2}}\) and different approximations of \(\dot{g}\) taking into account also the the lunisolar terms listed in Table 3.  
To handle the periodic effects, we need to set initial conditions also for the argument of pericenter and for the longitude of the ascending node of the satellite.