testing math functions and adjusted bulleted lists

docs/user-guide/gravitational-harmonics/index.rst

@@ -2,21 +2,37 @@
 Gravitational Harmonics
 ##########################
 
-Here, we show how to use Swiftest's Gravitational Harmonics capabilities. 
-Swiftest uses `SHTOOLS <https://shtools.github.io/SHTOOLS/>`__ to compute the gravitational harmonics coefficients for a given body and calculate it's associated acceleration kick. 
+Here, we show how to use Swiftest's Gravitational Harmonics capabilities. This is based on ``/spherical_harmonics_cb`` 
+in ``swiftest/examples``. Swiftest uses `SHTOOLS <https://shtools.github.io/SHTOOLS/>`__ to compute the gravitational 
+harmonics coefficients for a given body and calculate it's associated acceleration kick. The additional acceleration 
+kick is based on the gravitaional potential described `here <https://sseh.uchicago.edu/doc/Weiczorek_2015.pdf>`__.
+
+..math::
+
+    U(r) = \frac{GM}{r} \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \left( \frac{R_0}{r} \right)^l C_{lm} Y_{lm} (\theta, \phi) \label{grav_pot}
+
+* Gravitational potential:math:`U` at a point:math:`\Vec{r}`;:math:`\theta` is the polar angle;:math:`\phi` is the azimuthal angle;:math:`R_0` is the central body 
+radius;:math:`G` is the gravitational constant;:math:`Y_{lm}` is the spherical harmonic function at degree:math:`l` and order:math:`m`;:math:`C_{lm}` is the corresponding coefficient.
+
+
+Gravitational Harmonics coefficients
+=====================================
+
+Swiftest adopts the :math:`4\pi` or geodesy normalization for the gravitational harmonics coefficients.
 The coefficients can be computed in a number of ways: 
 
 - Using the axes measurements of the body ( :func:`clm_from_ellipsoid <swiftest.shgrav.clm_from_ellipsoid>`)
 
 - Using a surface relief grid ( :func:`clm_from_relief <swiftest.shgrav.clm_from_relief>`). 
 
-   * This function is still in development and may not work as expected.
+   - This function is still in development and may not work as expected.
 
 - Manually entering the coefficients when adding the central body ( :func:`add_body <swiftest.Simulation.add_body>`)
 
-This is based on ``/spherical_harmonics_cb`` in ``swiftest/examples``.
 
 
+Computing coefficients from axes measurements
+===============================================
 
 .. .. toctree::
 ..    :maxdepth: 2