Disturbing Function

For the moment, let us consider a simplified Solar System that consists of the Sun and two planets--see Figure 33. Let the Sun be of mass $M$, and position vector ${\bf R}_s$. Likewise, let the two planets have masses $m$ and $m'$, and position vectors ${\bf R}$ and ${\bf R}'$, respectively. Here, we are assuming that $m,\,m'\ll M$. Let ${\bf r} = {\bf R}-{\bf R}_s$ and ${\bf r}'={\bf R}'-{\bf R}_s$, be the position vector of each planet relative to the Sun. Without loss of generality, we can assume that $r'>r$.

Figure 33: A simplified model of the Solar System.

Now, the equations of motion of the various elements of our simplified Solar System are

$$M\,\ddot{\bf R}_s = G\,M\,m\,\frac{\bf r}{r^{\,3}} + G\,M\,m'\,\frac{{\bf r}'}{r'^{\,3}},$$ (730)

$$m\,\ddot{\bf R} = G\,m\,m'\,\frac{{\bf r}'-{\bf r}}{\vert{\bf r}'-{\bf r}\vert^{\,3}} - G\,m\,M\,\frac{{\bf r}}{r^{\,3}},$$ (731)

$$m'\,\ddot{\bf R}' = G\,m'\,m\,\frac{{\bf r}-{\bf r}'}{\vert{\bf r}-{\bf r}'\vert^{\,3}} - G\,m'\,M\,\frac{{\bf r}'}{r'^{\,3}},$$ (732)

It thus follows that

$$\ddot{\bf r} + \mu\,\frac{\bf r}{r^{\,3}} = G\,m'\left(\frac{{\bf r}'-{\bf r}}{\vert{\bf r}'-{\bf r}\vert^{\,3}} - \frac{{\bf r}'}{r'^{\,3}}\right),$$ (733)

$$\ddot{\bf r}' + \mu'\,\frac{{\bf r}'}{r'^{\,3}} = G\,m\left(\frac{{\bf r}-{\bf r}'}{\vert{\bf r}-{\bf r}'\vert^{\,3}} - \frac{{\bf r}}{r^{\,3}}\right),$$ (734)

where $\mu= G\,(M+m)$, and $\mu'=G\,(M+m')$. The right-hand sides of the above equations specify the interplanetary interaction forces that were neglected in our previous analysis. These right-hand sides can be conveniently expressed as the gradients of potentials: i.e.,

$$\ddot{\bf r} + \mu\,\frac{\bf r}{r^{\,3}} = \nabla {\cal R},$$ (735)

$$\ddot{\bf r}' + \mu'\,\frac{{\bf r}'}{r'^{\,3}} = \nabla' {\cal R}',$$ (736)

where

$${\cal R}({\bf r},{\bf r}') = \tilde{\mu}'\left(\frac{1}{\vert{\bf r}-{\bf r}'\vert} - \frac{{\bf r}\cdot{\bf r}'}{r'^{\,3}}\right),$$ (737)

$${\cal R}' ({\bf r},{\bf r}') = \tilde{\mu}\left(\frac{1}{\vert{\bf r}-{\bf r}'\vert} - \frac{{\bf r}\cdot{\bf r}'}{r^{\,3}}\right),$$ (738)

with $\tilde{\mu} = G\,m$, and $\tilde{\mu}'=G\,m'$. Here, ${\cal R}({\bf r},{\bf r}')$ and ${\cal R}'({\bf r}, {\bf r}')$ are termed disturbing functions. Moreover, $\nabla$ and $\nabla'$ are the gradient operators involving the unprimed and primed coordinates, respectively.