d) a = 11,000 km, A/m = 10 m2/kg e) a = 14,864 km, A/m = 10 m2/kg f) a = 18,000 km, A/m = 10 m2/kg g) a = 11,000 km, A/m = 20 m2/kg h) a = 14,864 km, A/m = 20 m2/kg i) a = 18,000 km, A/m = 20 m2/kg Fig. 2: Eccentricity-ϕ phase space evolution for the planar simplified case of J2+SRP. The bold lines represent the separatrices in correspondence of the saddle point and zero eccentricity Hamiltonian, the shaded area marks orbit heights lower than one Earth radius. Semi-major axis of 11,000 km (first column), 14,864 km (second column) and 18,000 km (third column). Area-to-mass of 5 m2/kg (first row), 10 m2/kg (second row) and 20 m2/kg (third row).
dh d Sun  kW 5I 2  2I  2E 4 C 2 2  E 1  I   p  Hh  q cos Sun   E dk 5I 2  2I  1 1  I   p  p  Hh  cos  sin Sun   E 1  I  p  IKk sin  sin Sun  d Sun  hW C  2E 4 2 2 (6)  q  Hk  p cos  sin  E 1  I     I   q q  Hk  cos    E 1  I  q  IKh sin  sin   Sun Sun Sun dp  qW 1  C  Hp  (1  I )k   p cos   I sin   sin   q cos   d Sun E 4 E 1  I  Sun Sun dq   pW 1  C Hq  (1  I )k p cos   I sin  sin   q cos  d Sun E 4 E 1  I      Sun Sun  where E  1  h2  k 2 I  1  p2  q2 H  hp  kq K  hq  kp We note that the semi-major axis remains constant if we neglect eclipses. III.I. Three-dimensional phase space When we consider the actual value of the ecliptic obliquity, in the case of the Earth   23.4393 deg , this results in more complex coupled dynamics, because the solar radiation pressure causes a non-zero variation of all the four Lagrangian elements in Eq. (6) if acting on an orbit out of the ecliptic plane, whereas the effect of the J2 term presents a simplified form exclusively for orbits on the Earth’s equator, for which  and  can be simply summed up. However, if we a three-dimensional definition of the angle  as       Sun    the space dynamics observed in the simplified planar case can still be recognised for increasing but limited values of the angle  ( less than 30° [18]). This can be appreciated in Fig. 5 which shows the evolution in the eccentricity-  space for non-zero values of the ecliptic obliquity. Note that, the angles  ,  and Sun do not lie on the same plane (i.e., i and  are not zero), however the definition of the angle  is quite close to the correct three-dimensional trigonometric angle if i and  are not too high.
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.