Evaluation of disturbing function

It is convenient to evaluate the disturbing function,

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

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

$$y = r\,\left[ \sin{\mit\Omega}\,\cos(\omega+\theta) +\cos{\mit\Omega}\,\sin(\omega+\theta)\,\cos I\right],$$ (B.25)

$$z = r\,\sin (\omega + \theta)\,\sin I,$$ (B.26)

as well as

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

$$y' = r'\,\left[ \sin{\mit\Omega}'\,\cos(\omega'+\theta') +\cos{\mit\Omega}'\,\sin(\omega'+\theta')\,\cos I'\right],$$ (B.28)

$$z' = r'\,\sin (\omega' + \theta')\,\sin I',$$ (B.29)

where

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

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

$$\beta(\theta,\theta') =\frac{{\bf r}\cdot{\bf r}'}{r\,r'},$$ (B.32)

$$\gamma(\theta,\theta') = \frac{\partial\beta}{\partial\theta}.$$ (B.33)

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

$$\beta(\theta,\theta') = +\cos({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\cos(\omega'+\theta')$$

$$\phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\cos(\omega'+\theta')\,\cos I$$

$$\phantom{=}+\sin({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I'$$

$$\phantom{=}+\cos({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I\,\cos I'$$

$$\phantom{=}+\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\sin I\,\sin I',$$ (B.34)

and

$$\gamma(\theta,\theta') = -\cos({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\cos(\omega'+\theta')$$

$$\phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\cos(\omega'+\theta')\,\cos I$$

$$\phantom{=}-\sin({\mit\Omega}-{\mit\Omega}')\,\sin(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I'$$

$$\phantom{=}+\cos({\mit\Omega}-{\mit\Omega}')\,\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\cos I\,\cos I'$$

$$\phantom{=}+\cos(\omega+\theta)\,\sin(\omega'+\theta')\,\sin I\,\sin I'.$$ (B.35)

Hence,

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

which implies that

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

and

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