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Evaluation of disturbing function

It is convenient to evaluate the disturbing function,

$\displaystyle {\cal R}({\bf r},{\bf r}')= \mu'\left(\frac{1}{\vert{\bf r}-{\bf r}'\vert} - \frac{{\bf r}\cdot{\bf r}'}{r'^{\,3}}\right),$ (B.23)

in a frame of reference that is instantaneously aligned with the ecliptic plane, as described in Section 4.12. Let ($ x$ , $ y$ , $ z$ ) and ($ x'$ , $ y'$ , $ z'$ ) be the Cartesian components of $ {\bf r}$ and $ {\bf r}'$ , respectively, in this standard reference frame. It follows from Equations (4.38), and (4.72)-(4.74), that

    $\displaystyle x$ $\displaystyle = r\,\left[\cos{\mit\Omega}\,\cos(\omega+\theta) - \sin{\mit\Omega}\,\sin(\omega+\theta)\,\cos I\right],$ (B.24)
    $\displaystyle y$ $\displaystyle = r\,\left[ \sin{\mit\Omega}\,\cos(\omega+\theta) +\cos{\mit\Omega}\,\sin(\omega+\theta)\,\cos I\right],$ (B.25)
and   $\displaystyle z$ $\displaystyle = r\,\sin (\omega + \theta)\,\sin I,$     (B.26)

as well as

    $\displaystyle x'$ $\displaystyle = r'\,\left[\cos{\mit\Omega}'\,\cos(\omega'+\theta') - \sin{\mit\Omega}'\,\sin(\omega'+\theta')\,\cos I'\right],$ (B.27)
    $\displaystyle y'$ $\displaystyle = r'\,\left[ \sin{\mit\Omega}'\,\cos(\omega'+\theta') +\cos{\mit\Omega}'\,\sin(\omega'+\theta')\,\cos I'\right],$ (B.28)
and   $\displaystyle z'$ $\displaystyle = r'\,\sin (\omega' + \theta')\,\sin I',$ (B.29)

where

    $\displaystyle r (\theta)$ $\displaystyle = \frac{a\,(1-e^{\,2})}{1+e\,\cos\theta},$ (B.30)
and   $\displaystyle r'(\theta')$ $\displaystyle = \frac{a'\,(1-e'^{\,2})}{1+e'\,\cos\theta'}.$ (B.31)

Here, $ \theta $ , $ a$ , $ e$ , $ I$ , $ \omega$ , and $ {\mit \Omega }$ are the orbital true anomaly, major radius, eccentricity, inclination, argument of the perihelion, and longitude of the ascending node, respectively, of Mercury. Moreover, $ \theta'$ , $ a'$ , $ e'$ , $ I'$ , $ \omega'$ , and $ {\mit\Omega}'$ are the corresponding quantities for the perturbing planet.

It is helpful to define

    $\displaystyle \beta(\theta,\theta')$ $\displaystyle =\frac{{\bf r}\cdot{\bf r}'}{r\,r'},$ (B.32)
and   $\displaystyle \gamma(\theta,\theta')$ $\displaystyle = \frac{\partial\beta}{\partial\theta}.$     (B.33)

Making use of Equations (B.25)-(B.30), we deduce that

$\displaystyle \beta(\theta,\theta')$ $\displaystyle = +\cos({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\cos(\omega'+\theta')$    
  $\displaystyle \phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\cos(\omega'+\theta')\,\cos I$    
  $\displaystyle \phantom{=}+\sin({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I'$    
  $\displaystyle \phantom{=}+\cos({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I\,\cos I'$    
  $\displaystyle \phantom{=}+\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\sin I\,\sin I',$ (B.34)

and

$\displaystyle \gamma(\theta,\theta')$ $\displaystyle = -\cos({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\cos(\omega'+\theta')$    
  $\displaystyle \phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\cos(\omega'+\theta')\,\cos I$    
  $\displaystyle \phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I'$    
  $\displaystyle \phantom{=}+\cos({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I\,\cos I'$    
  $\displaystyle \phantom{=}+\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\sin I\,\sin I'.$ (B.35)

Hence,

$\displaystyle {\cal R}(r,\theta) = \mu'\left[\frac{1}{(r^{\,2} + r'^{\,2}-2\,r\,r'\,\beta)^{1/2}}-\frac{\beta\,r}{r'^{\,2}}\right],$ (B.36)

which implies that

$\displaystyle \frac{\partial{\cal R}}{\partial r} =- \mu'\,r'\left[\frac{r/r'-\beta}{(r^{\,2} + r'^{\,2}-2\,r\,r'\,\beta)^{3/2}} + \frac{\beta}{r'^{\,3}}\right],$ (B.37)

and

$\displaystyle \frac{1}{r}\,\frac{\partial{\cal R}}{\partial\theta} = \mu'\,r'\,...
...rac{1}{(r^{\,2} + r'^{\,2}-2\,r\,r'\,\beta)^{3/2}} - \frac{1}{r'^{\,3}}\right],$ (B.38)


next up previous
Next: Secular perihelion precession rate Up: Perihelion precession of Mercury Previous: Perihelion precession rate
Richard Fitzpatrick 2016-03-31