Code-equations/Gravitational-Potential-including-indirect-terms.md

+The gravitational potential entering in the source terms of the Navier-Stokes equations is the sum of multiple terms:
+1. The potential from the central massive body (primary of mass $`M_*`$):
+
+   $`\Phi_*=-{{ G M_*} \over {r}}`$
+2. The potential form the planets of mass $m_p$. 
+   Below we consider the case of one single planet, easily extended to multiple planets.
+- In the 3 Dimensional case 
+we use  a cubic-potential of the form:
+
+$`\Phi _p = \left\lbrace \begin{array}{ll}
+-{m_pG\over d} &  d > r_{\rm sm} \\
+-{m_pG\over d}f({d\over r_{\rm sm}}) & d\leq r_{\rm sm}   
+\end{array} \right.`$
+
+with $`f({d\over r_{\rm sm}}) = \left [ \left( {d\over r_{\rm sm}}\right)^4-2\left( {d\over r_{\rm sm}}\right)^3+2{d\over r_{\rm sm}} \right]`$;
+  $`d`$ is the distance from the disc element to the planet, and $`r_{\rm sm}`$ the 
+smoothing length:
+$`r_{\rm sm} = \alpha _{sm} R_H`$, where $`\alpha _{sm}`$ is a smoothing parameter and
+ $`R_H`$ is the Hill radius of the planet.
+
+To select this choice for a simulation in the config file:
+
+```
+    Smoothing
+    {
+        Flat false   # select the cubic potential
+        Size 0.4     # smoothing parameter in units of the planet Hill radius
+    }
+```
+![SmoothingPotWiki](uploads/73ba8941570d358cbad84004cbd41548/SmoothingPotWiki.png)
+
+The plot shows the effect of smoothing the potential for various values of the smoothing length. In the 3 dimensional case we smooth the potential in order to avoid the singularity in the case
+$`d -> 0`$. The choice of the smoothing length depends mainly on the resolution: as a rule of the thumb the minimum smoothing length should correspond to the size of about 4 grid cells.
+
+- In the 2 Dimensional case 
+we use  a potential of the form:
+$`\Phi _p^{\epsilon} = -{m_pG\over {\sqrt {d^2+\epsilon^2}}}`$
+In 2 dimensional models we do not only have the problem of avoiding the singularity but we choose a potential that allows to mimic the average influence  that the planet would have  on the vertical gas column.
+A value often used for the smoothing is $`\epsilon = 0.7H`$ where $`H`$ is the disc  scale height.
+To select this choice for a simulation in the config file:
+```
+    Smoothing
+    {
+        Flat true   # select the  potential for 2D (flat) case
+        Size 0.4     # smoothing parameter in units of the planet Hill radius
+    }
+```
+
+We remark that the parameter $`\epsilon`$ is considered in the code in units of the Hill radius and not in disc scale height.
+ 
+
+