Testing equations

docs/manual.rst

@@ -1,11 +1,43 @@
 .. _manual:
 
-orbitiz! Manual
+orbitize! Manual
 ==============
 
-Intro to orbitize!
+Intro to ``orbitize!``
 +++++++++++++++++
+Start with Section 4.2 of Sarah's thesis: https://thesis.library.caltech.edu/16076/
 
-Here is where the intro stuff will go! (written in markdown)
+At its core, ``orbitize!`` turns data into orbits. 
+This is done when relative kinematic measurements of a primary and secondary body are converted to posteriors over 
+orbital parameters through Bayesian analysis.
 
-Start with Section 4.2 of Sarah's thesis: https://thesis.library.caltech.edu/16076/
+``orbitize!`` hinges on the two-body problem, which describes the paths of two
+bodies gravitationally bound to each other. 
+The solution of the two-body problem describes the motion of each body as a 
+function of time, given parameters determining the position and velocity of both objects at a particular epoch. 
+
+
+
+There are many basis sets (orbital bases) that can be used to describe an orbit, 
+which can then be solved using Kepler’s equation. 
+
+It is important, then, to be explicit about coordinate systems. 
+
+For an interactive visualization to define and help users understand our coordinate system, 
+you can check out `this GitHub tutorial <https://github.com/sblunt/orbitize/blob/main/docs/tutorials/show-me-the-orbit.ipynb>`_. 
+
+There is also a `YouTube video <https://www.youtube.com/watch?v=0e24VUhQmbM>`_. 
+with use and explaination of the coordinate system.
+
+In its “standard” mode, ``orbitize!`` assumes that the user only has relative astrometric data to fit. 
+To obtain these measurements, an astronomer takes an image containing two point sources 
+and measures the position of the planet relative to the star in angular coordinates. 
+In the ``orbitize!`` coordinate system, relative R.A. and decl. can be expressed as the following functions 
+of orbital parameters 
+
+$$ \delta R.A. = \pi a(1-ecosE)[cos^2{i\over 2}sin(f+\omega_p+\Omega)-sin^2{i\over 2}sin(f+\omega_p-\Omega)] $$
+$$ \delta decl. = \pi a(1-ecosE)[cos^2{i\over 2}cos(f+\omega_p+\Omega)-sin^2{i\over 2}cos(f+\omega_p-\Omega)] $$
+
+where 𝑎, 𝑒, 𝜔p, Ω, and 𝑖 are orbital parameters, and 𝜋 is the system parallax. f is
+the true anomaly, and E is the eccentric anomaly, which are related to elapsed time
+through Kepler’s equation and Kepler’s third law: