Fig. 5: Sensitivity on the ecliptic obliquity  (a= 18,000 km and = 10 m2/kg). A/m IV. QUASI-FROZEN ORBITS IN THE 3D DYNAMICS Fig. 5 suggests that frozen or solutions may exist as well in the three-dimensional and SRP case. J2 Despite that an exact solution (i.e., frozen orbit) does not exist if the SRP parameter is non-zero, the hypothesis of C orbits was verified numerically by minimising the cost function min    (7) w    de d w d d w di d e,  i , , e Sun  Sun i Sun the variation of eccentricity,  and inclination can be obtained from Eqs. (5) and (6) as  de h dh  k dk Sun Sun d d Sun  d k h 2 h2   k dh   k 2 (8) Sun d  d Sun d  h2 k Sun d 2  di p dp  q dq Sun Sun d d Sun 1  d p  q p  q 2 2 2 2 and are weight parameters introduced to treat this multi-objective minimisation problem (the three equations of we, wϕ, wi system Eq. (8) must be zero) as a single-objective minimisation. A global multi-start approach is adopted. A local algorithm is started from several points randomly distributed over the entire domain of eccentricity, arguments of displacement and inclination. Starting from each point on the mesh, a local minimisation is performed numerically, through a subspace trust-region method, based on the interior-reflective Newton method [19, 20]. Another variable of the problem would be the orbit semi-major axis; in fact, as seen in Fig. 2 this determines the eccentricity value of the stationary points. However, to keep the problem simple, we examine here the solution at a specific semi- major axis, leaving the extension to an extended range of orbit energies to a future work.
Fig. 5: Sensitivity on the ecliptic obliquity  (a= 18,000 km and A/m = 10 m2/kg). IV. QUASI-FROZEN ORBITS IN THE 3D DYNAMICS Fig. 5 suggests that frozen or solutions may exist as well in the three-dimensional J2 and SRP case. Despite that an exact solution (i.e., frozen orbit) does not exist if the SRP parameter C is non-zero, the hypothesis of orbits was verified numerically by minimising the cost function min w de d   w d d   w di d   (7) e, i , ,  e Sun  Sun i Sun the variation of eccentricity,  and inclination can be obtained from Eqs. (5) and (6) as de  h dh d Sun  k dk d Sun d Sun h  k 2  k dh  h2  k 2  d  d Sun d Sun d Sun h2  k 2 (8) di  p dp d Sun  q dq d Sun d Sun 1  p  q p  q 2 2 2 2 and we, wϕ, wi are weight parameters introduced to treat this multi-objective minimisation problem (the three equations of system Eq. (8) must be zero) as a single-objective minimisation. A global multi-start approach is adopted. A local algorithm is started from several points randomly distributed over the entire domain of eccentricity, arguments of displacement and inclination. Starting from each point on the mesh, a local minimisation is performed numerically, through a subspace trust-region method, based on the interior-reflective Newton method [19, 20]. Another variable of the problem would be the orbit semi-major axis; in fact, as seen in Fig. 2 this determines the eccentricity value of the stationary points. However, to keep the problem simple, we examine here the solution at a specific semi- major axis, leaving the extension to an extended range of orbit energies to a future work.
Colombo_C_Lucking_C_McInnes_CR_Pure_Orbital_dynamics_of_high_area_to_mass_ratio_spacecraft_with_J2_and_solar_radiation_pressure..._communications_services_Dec_2012.docx