Editing Stellar dynamics (section)

=== A worked example of gravity vector field in a thick disk ===

First consider the vertical gravity at the boundary, 
<math display="block"> g_z(R,z) = - \partial_z \Phi(R,z) = -{G M_0 z \over 2z_0^2} \left[ {1 \over \sqrt{R_0^2+ R^2}} - { 1 \over \sqrt{(R_0+2z_0)^2 + R^2} } \right] , ~~ z= \pm z_0, </math>

Note that both the potential and the vertical gravity are continuous across the boundaries, hence no razor disk at the boundaries.
Thanks to the fact that at the boundary, <math>\partial_{|z|} (2 Q) - \partial_{|z|} Q_{-} = \partial_{|z|} \left(Q_{+} - \frac{2z_0}{R}\right) = {1 \over R} </math> is continuous.  Apply Gauss's theorem by integrating the vertical force over the entire disk upper and lower boundaries, we have 
<math display="block"> 2 \int_0^\infty (2 \pi R dR) |g_z(R,z_0)| = 4 \pi G M_0, </math>
confirming that <math>M_0</math> takes the meaning of the total disk mass.

The vertical gravity drops with <math display="block"> -g_z \rightarrow G M_0 z (1+R_0/z_0)/R^3 </math> at large radii, which is enhanced over the vertical gravity of a point mass <math> G M_0 z/R^3 </math> due to the self-gravity of the thick disk.