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

@@ -2,13 +2,12 @@
 
 The gravitational potential entering in the source terms of the Navier-Stokes equations is the sum of multiple terms:
 ## The potential from the central massive body
-Considering a primary of mass $`M_*`$:
+Considering a primary of mass $`M_*`$, the potential exerted on a disk element at distance $`r`$ from the primary is:
 
    $`\Phi_*=-{{ G M_*} \over {r}}`$
 ## The potential form the planets
-   Below we consider the case of one single planet  of mass $m_p$ (it is easily extended to multiple planets).
-- In the 3 Dimensional case 
-we use  a cubic-potential of the form:
+   We consider here the case of one single planet  of mass $`m_p`$ (it is easily extended to multiple planets).
+- **In the 3 Dimensional case** the potential exerted on a disk element at distance $`d`$ from the planet is expressed as:
 
 $`\Phi _p = \left\lbrace \begin{array}{ll}
 -{m_pG\over d} &  d > r_{\rm sm} \\
@@ -16,9 +15,9 @@ $`\Phi _p = \left\lbrace \begin{array}{ll}
 \end{array} \right.`$
 
 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 
+  $`r_{\rm sm}`$ is called the 
 smoothing length:
-$`r_{\rm sm} = \epsilon R_H`$, where $`\epsilon`$ is a smoothing parameter and
+$`r_{\rm sm} = \epsilon R_H`$, where $`\epsilon`$ is the smoothing parameter and
  $`R_H`$ is the Hill radius of the planet.
 
 To select this choice for a simulation in the config file:
@@ -38,10 +37,11 @@ To select this choice for a simulation in the config file:
 width="320" height="240">
 
 
-The plot shows the effect of smoothing the potential for various values of the smoothing parameter. 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.
+The plot shows the effect of smoothing the potential for various values of the smoothing parameter. In the 3 dimensional case we smooth the potential in order to avoid the singularity at
+$`d \rightarrow 0`$. 
+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 
+- **In the 2 Dimensional case** 
 we use  a potential of the form:
 $`\Phi _p^{\epsilon} = -{m_pG\over {\sqrt {d^2+(\epsilon R_H)^2}}}`$
 
@@ -59,26 +59,27 @@ To select this choice for a simulation in the config file:
 
 ## The potential due to indirect terms 
 We recall that we use a non inertial  frame : the origin of the
-system is centered on the primary body which is  
-accelerated by  the planet(s) and by the disk:
+system is centered on the primary body which is  accelerated by  the planet(s) and by the disk.
+
+ Therefore, we have to consider two indirect terms in the potential accounting for the acceleration of the primary induced respectively by the planet(s) and the disk:
 
 $`\tilde \Phi_p = {{ G m_p} \over {r_p^3}}(\vec r_p \cdot \vec r)`$
 
  
 with $`\vec r_p`$ and $`\vec r`$ indicating the distance between the
-planet and the star and a grid cell and the star
+planet and the star and a disk element and the star. The second term is:
 
 $`\tilde \Phi_d =   \vec a  \cdot \vec r `$
 
 where $`\vec a`$ is the acceleration of the primary due to the whole disk.
 
-When doing simulations in production mode such terms must be included in the computations. However, it may be useful to control the contribution from indirect terms. At this purpose the indirect terms may be de activated in the config file with the flag IndirectForces in the Referential group :
+When doing simulations in production mode such terms must be included in the computations. However, it may be useful to control the contribution from indirect terms. At this purpose the indirect terms may be de activated in the config file with the flag ``IndirectForces`` in the Referential group :
 
 ```
  Referential
     {
         Type Constant
-        IndirectForces true
+        IndirectForces false # true in production mode
         Omega 1
         NewOmega 0
     }