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Appendix 1: The time averaged potential due to a perturbing inner planet

The gravitational potential per unit mass due to a planet of mass $m_{\rm p}$located at $ {\vec r_{\rm p}} \equiv (r_{\rm p},\varphi_{\rm p})$at $ {\vec r} \equiv (r,\varphi)$ is

\begin{displaymath}\Phi = - {Gm_{\rm p}\over \sqrt{r^2+r_{\rm p}^2 -2rr_{\rm p}\cos(\theta)}},\end{displaymath} (64)

with $\theta =\varphi - \varphi_{\rm p}.$

In order to incorporate the effects of protoplanets orbiting interior to the disc we adopt a Jacobi coordinate system. In this system the coordinates of the innermost protoplanet are referred to the central star. The coordinates of the remainder are referred to the centre of gravity of the central mass and all interior protoplanets. The disc is referred to the centre of mass of central star and all inner protoplanets.

This has the following consequences:

For an object interior to the protoplanet with $r < r_{\rm p},$ the acceleration of the coordinate system due to the protoplanet must be allowed for. This gives rise, correct to first order in $m_{\rm p},$ to the indirect potential (e.g. Brouwer & Clemence 1961)

\begin{displaymath}\Phi_i = {Gm_{\rm p} r\cos(\theta)\over r_{\rm p}^2 },\end{displaymath} (65)

to be added to the potential $\Phi$.

For an object exterior to the protoplanet one must take account of the fact that the coordinate system is now based on the centre of mass of the inner protoplanets and central star. Assuming initially that $m_{\rm p}$ is the only such protoplanet, the central potential is modified to become

\begin{displaymath}\Phi = - {GM_* \over {\vert{\vec r} + m_{\rm p} {\vec r_{\rm p}}/M_*\vert }}\cdot\end{displaymath} (66)

To first order in $m_{\rm p}$ this gives

\begin{displaymath}\Phi = - {GM_* \over r} + {Gm_{\rm p} r_{\rm p} \cos\theta \over r^2} \cdot\end{displaymath} (67)

In this case too the additional potential on the right hand side of the above can be incorporated into the perturbing planet potential and viewed as giving rise to an indirect potential. Thus the form of the perturbing potential

\begin{displaymath}\Phi = - {Gm_{\rm p}\over \sqrt{r^2+r_{\rm p}^2 -2rr_{\rm p}\...
...{Gm_{\rm p} rr_{\rm p} \cos\theta \over \max(r^3,r_{\rm p}^3)},\end{displaymath} (68)

incorporates both cases $r < r_{\rm p}$ amd $r_{\rm p} <r.$

In addition, although we included just one inner perturbing protoplanet, because we work to first order in their masses, the principle of linear superposition is valid such that the contributions of many such objects may be linearly superposed.

The perturbing potential due to a single protoplanet may be decomposed as a Fourier expansion in $\theta.$

Thus

\begin{displaymath}{\Phi \over Gm_{\rm p}}
= -\sum_{m=0}^{\infty} { K_m(r,r_{\rm p}) \over \pi(1+ \delta_{m0})}
\cos m \theta ,\end{displaymath} (69)

with

\begin{displaymath}K_m(r,r_{\rm p}) = -{1\over G m_{\rm p}}
\int^{2\pi}_0 \Phi \cos m\theta {\rm d}\theta .\end{displaymath} (70)

For the problem on hand, namely the study of global m=1 modes of the disc planet system, we need only consider m=0 and m=1.Then from the above

\begin{displaymath}{\pi\Phi \over G m_{\rm p}} =
-{1\over 2}K_0(r,r_{\rm p}) - K_1(r,r_{\rm p})\cos \theta ,\end{displaymath} (71)

with

\begin{displaymath}K_0(r,r_{\rm p}) =
\int^{2\pi}_0 {1\over \sqrt{r^2+r_{\rm p}^2 -2rr_{\rm p}\cos(\theta)}}{\rm d}\theta \end{displaymath} (72)

and
$\displaystyle K_1(r,r_{\rm p}) = \bigg(
\int^{2\pi}_0
{1\over \sqrt{r^2+r_{\rm p}^2 -2rr_{\rm p}\cos(\theta)}}\cos\theta {\rm d}\theta$      
$\displaystyle - { \pi rr_{\rm p} \over \max(r^3,r_{\rm p}^3)}\bigg)\cdot$     (73)

Time averaged potential for a protoplanet with small eccentricity

We now suppose the protoplanet has a small eccentricity, $e_{\rm p}$and write for its motion $r_{\rm p} = a_{\rm p}(1+e_{\rm p}\cos\omega t),
\varphi_{\rm p} = \omega t -2e_{\rm p}\sin\omega t,$ where without loss of generality we assume the apsidal line to be along $\varphi =0$where the displacement $\xi_{\rm r}(r_{\rm p}) = e_{\rm p} a_{\rm p}$ at t=0.

Here the protoplanet semi-major axis is $a_{\rm p}$ and the orbital frequency is $\omega.$

Expanding to first order in $e_{\rm p},$ we find for the single protoplanet perturbing potential

\begin{eqnarray*}&&{\pi \Phi \over Gm_{\rm p}} = -{1\over 2}K_0(r,a_{\rm p}) -
...
...~~~~~~~~\times
\cos(\varphi -\omega t +2e_{\rm p}\sin\omega t) .
\end{eqnarray*}


Performing the time average then gives

\begin{displaymath}{\Phi \over Gm_{\rm p}} = -{1\over 2\pi}K_0(r,a_{\rm p})
-{e...
... p}^2 K_1(r,a_{\rm p})) \over \partial a_{\rm p}}
\cos\varphi .\end{displaymath} (74)

For small $e_{\rm p}$ we may replace $a_{\rm p}$ by $r_{\rm p}$ and expressing the result in terms of the radial displacement $\xi_{\rm r}(r_{\rm p}),$ we find

 \begin{displaymath}{\Phi \over Gm_{\rm p}} = -{1\over 2\pi}K_0(r,r_{\rm p})
-{\x...
... p}^2 K_1(r,r_{\rm p}))\over \partial r_{\rm p}}
\cos\varphi .
\end{displaymath} (75)

From Eq. (.75) the time averaged potential due to a protoplanet in circular orbit $(\xi_{\rm r}(r_{\rm p}) = 0 )$is

\begin{displaymath}\Phi_{\rm ext} = -G { m_{\rm p} \over 2\pi }
K_0(r,r_{\rm p}).
\end{displaymath} (76)

When $\xi_{\rm r}(r_{\rm p}) \ne 0,$ Eq.  (.75) gives the m=1 component of the perturbing potential as the real part of $\Phi_{\rm p}'(r)\exp(i\varphi),$ with

\begin{displaymath}\Phi_{\rm p}'(r) = -{G m_{\rm p} \xi_{\rm r}(r_{\rm p})\over ...
...l (r_{\rm p}^2 K_1(r,r_{\rm p}))\over \partial r_{\rm p}}\cdot \end{displaymath} (77)


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