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Next: Precession of lunar perigee Up: Lunar motion Previous: Reduction to ecliptic

Summary of results

The results derived in the previous six sections are summarized in the following. The the overall modification to the lunar orbit is

    $\displaystyle \delta R$ $\displaystyle =-\left(e-\frac{11}{12}\,m^{\,2}\,e\right)\,\cos{\cal M} +\frac{1}{2}\,e^{\,2}-\frac{1}{2}\,e^{\,2}\,\cos(2\,{\cal M})$        
      $\displaystyle \phantom{=}-\left(\frac{15}{8}\,m\,e +\frac{155}{32}\,m^{\,2}\,e\right)\cos (2\,D-{\cal M}) -\frac{17}{16}\,m^{\,2}\,e\,\cos(2\,D+{\cal M})$    
      $\displaystyle \phantom{=}-\frac{1}{6}\,m^{\,2}+\frac{331}{288}\,m^{\,4}-\left(m...
...\,3}+\frac{125}{18}\,m^{\,4}\right)\cos(2\,D) -\frac{3}{8}\,m^{\,4}\,\cos(4\,D)$    
      $\displaystyle \phantom{=}+\frac{3}{2}\,m^{\,2}\,e'\,\cos{\cal M}' -\frac{7}{2}\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')+\frac{1}{2}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')$    
      $\displaystyle \phantom{=}+\left(\frac{E-M}{E+M}\right)\left(\frac{15}{16}\,m+\f...
...ght)\left(\frac{25}{64}\,m^{\,2}\right)\left(\frac{a}{a'}\right)\, \cos(3\, D),$ (11.334)
    $\displaystyle \delta \lambda$ $\displaystyle =2\,e\,\sin{\cal M} +\frac{5}{4}\,e^{\,2}\,\sin(2\,{\cal M})-\frac{1}{4}\,I^{\,2}\,\sin(2\,F)$        
      $\displaystyle \phantom{=}+\left(\frac{15}{4}\,m\,e +\frac{263}{16}\,m^{\,2}\,e\right)\sin (2\,D-{\cal M}) +\frac{17}{8}\,m^{\,2}\,e\,\cos(2\,D+{\cal M})$    
      $\displaystyle \phantom{=}+\left(\frac{11}{8}\,m^{\,2}+\frac{59}{12}\,m^{\,3}+\frac{893}{72}\,m^{\,4}\right)\sin(2\,D)+\frac{201}{256}\,m^{\,4}\,\sin(4\,D)$    
      $\displaystyle \phantom{=}-3\,m\,e'\,\sin{\cal M}'+\frac{77}{16}\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}') -\frac{11}{16}\,m^{\,2}\,e'\,\sin(2\,D + {\cal M}')$    
      $\displaystyle \phantom{=}-\left(\frac{E-M}{E+M}\right)\left(\frac{15}{8}\,m+\fr...
...ght)\left(\frac{15}{32}\,m^{\,2}\right)\left(\frac{a}{a'}\right)\, \sin(3\, D),$ (11.335)
    $\displaystyle \delta \beta$ $\displaystyle =I\,\sin F-e\,I\,\sin(F-{\cal M}) + e\,I\,\sin(F+{\cal M})$    
and     $\displaystyle \phantom{=}+\left(\frac{3}{8}\,m\,I +\frac{25}{32}\,m^{\,2}\,I\right)\sin(2\,D-F) + \frac{11}{16}\,m^{\,2}\,I\,\sin(2\,D+F).$ (11.336)

These results are accurate to second order in the small parameters $ m^{\,2}$ , $ e$ , $ I$ , $ e'$ , and $ a/a'$ .


next up previous
Next: Precession of lunar perigee Up: Lunar motion Previous: Reduction to ecliptic
Richard Fitzpatrick 2016-03-31