From d267c17827cc101213677c711f6218acce9dc427 Mon Sep 17 00:00:00 2001
From: Sofia Covarrubias <44685425+sofiacovarrubias@users.noreply.github.com>
Date: Thu, 29 Feb 2024 11:40:00 -0800
Subject: [PATCH] formatting test

---
 docs/manual.rst | 49 ++++++++++++++++++++-----------------------------
 1 file changed, 20 insertions(+), 29 deletions(-)

diff --git a/docs/manual.rst b/docs/manual.rst
index 6d82ab48..3e53d60e 100644
--- a/docs/manual.rst
+++ b/docs/manual.rst
@@ -25,7 +25,7 @@ of orbital parameters
 
 .. math::
     \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, :math:`\omega_p`, Ω, and 𝑖 are orbital parameters, and 𝜋 is the system parallax. f is
@@ -34,14 +34,11 @@ through Kepler’s equation and Kepler’s third law
 
 .. math::
     M = 2\pi ({t\over P}-(\tau -\tau_{ref}))
-
-
+    \
     ({P\over yr})^2 =({a\over au})^3({M_\odot \over M_{tot}})
-
-
+    \
     M =E-esinE
-    
-
+    \
     f = 2tan^{-1}[\sqrt{{1+e\over 1-e}}tan{E\over 2}]
 
 ``orbitize!`` employs two Kepler solvers to convert between mean
@@ -52,29 +49,23 @@ orbit. See `Blunt et al. (2020) <https://iopscience.iop.org/article/10.3847/1538
 From scrutinizing the above sets of equations, one may observe
 a few important degeneracies:
 
-1. Individual component masses do not show up anywhere in this equation set. 
-
-2. The degeneracy between semimajor axis 𝑎, total mass :math:`𝑀_{tot}`, and
-parallax 𝜋. If we just had relative astrometric measurements and no external knowledge of the system parallax, 
-we would not be able to distinguish between a system
-that has larger distance and larger semimajor axis (and therefore larger total mass,
-assuming a fixed period) from a system that has smaller distance, smaller semimajor
-axis, and smaller total mass. 
-
-3. The argument of periastron :math:`\omega_p` and the position angle of nodes Ω. 
-The above defined R.A. and decl. functions are invariant to the transformation:
-
-.. math::
-    \omega_p' = \omega_p + \pi
-
-    \Omega' = \Omega - \pi
-
-which creates a 180◦ degeneracy between particular values of :math:`\omega_p` and Ω, and
-a characteristic “double-peaked” structure in marginalized 1D posteriors of these
-parameters. 
+    1. Individual component masses do not show up anywhere in this equation set. 
 
+    2. The degeneracy between semimajor axis 𝑎, total mass :math:`𝑀_{tot}`, and
+    parallax 𝜋. If we just had relative astrometric measurements and no external knowledge of the system parallax, 
+    we would not be able to distinguish between a system
+    that has larger distance and larger semimajor axis (and therefore larger total mass,
+    assuming a fixed period) from a system that has smaller distance, smaller semimajor
+    axis, and smaller total mass. 
 
+    3. The argument of periastron :math:`\omega_p` and the position angle of nodes Ω. 
+    The above defined R.A. and decl. functions are invariant to the transformation:
 
+    .. math::
+        \omega_p' = \omega_p + \pi
+        \
+        \Omega' = \Omega - \pi
 
-Including Radial Velocities 
-+++++++++++++++++
\ No newline at end of file
+    which creates a 180◦ degeneracy between particular values of :math:`\omega_p` and Ω, and
+    a characteristic “double-peaked” structure in marginalized 1D posteriors of these
+    parameters.