5I 2I   2 dh  kW 2E Sun d 4 C 2 2 5I 2I 1 dk    2   E 1   cos Sun  E  I p  Hh q 1     cos  sin Sun  I  p p  Hh   E sin 1   sin Sun   I p IKk   hW Sun 2E d 4 C   sin (6) 2 2 1  q   sin        cos sin I q IKh Hk p  q    I   q  E 1  Hk cos   I  E Sun Sun Sun     (1 )k cos  sin     Hp I p I  qW 1  dp C  sin   cos  q 1 Sun  d E E I 4 Sun Sun  1 dq   pW  C (1 )k cos  sin    Hq I p I  sin  cos   q  1     Sun  d E E I  4 Sun Sun where 1    E h2 k 2 1 I   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 term presents a J2 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., and  are not zero), however the definition of the angle  is quite close to the correct three-dimensional i trigonometric angle if and  are not too high. i
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.
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