cut down for clarity

docs/manual.rst

@@ -26,52 +26,43 @@ 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)]
+    \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
 the true anomaly, and E is the eccentric anomaly, which are related to elapsed time
 through Kepler’s equation and Kepler’s third law
 
 .. math::
-    M = 2\pi ({t\over P}-(\tau -\tau_ref))
+    M = 2\pi ({t\over P}-(\tau -\tau_{ref}))
+
+
+    ({P\over yr})^2 =({a\over au})^3({M_\odot \over M_{tot}})
 
-    ({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}]
+    f = 2tan^{-1}[\sqrt{{1+e\over 1-e}}tan{E\over 2}]
 
 ``orbitize!`` employs two Kepler solvers to convert between mean
 and eccentric anomaly: one that is efficient for the highest eccentricities, and Newton’s method, which in other cases is more efficient for solving for the average
 orbit. See `Blunt et al. (2020) <https://iopscience.iop.org/article/10.3847/1538-3881/ab6663>`_ for more detail.
 
 
 From scrutinizing the above sets of equations, one may observe
-a few important inherent degeneracies. 
-
-First, notice that the individual component masses do not show up anywhere in this equation set. 
-While it is impossible to measure dynamical masses for either the primary or the secondary using just
-relative astrometry, there are methods to constrain the system. 
-If the mass of the planet can be safely assumed to be negligible compared to the mass of the star, 
-then the total mass derived from Keplerian analysis can be treated as a constraint on the dynamical mass 
-of the primary. 
-In practice, the reverse logic is often employed: an independent constraint
-on the mass of the primary (from e.g., spectroscopic analysis) is used as a prior on
-the total mass when the planet mass is small and can be ignored.
-
-A second important degeneracy is between semimajor axis 𝑎, total mass :math:`𝑀_{tot}`, and
+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. Luckily, we live in an era where parallax measurements
-are excellent overall thanks to the Gaia mission, with which strict priors can often be applied to, 
-breaking the degeneracy and enabling dynamical mass measurements of stars (when planet mass is negligible). 
-However, this degeneracy is important to take into account when considering the impact of potential
-biases in parallax or stellar mass measurements. 
+axis, and smaller total mass. 
 
-A final degeneracy I would like to point out concerns 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:
+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
@@ -80,11 +71,10 @@ and the position angle of nodes Ω. The above defined R.A. and decl. functions a
 
 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 (see Figure 4.2 for an example). 
-Physically, this degeneracy comes about
-because relative astrometry alone only constrains motion in the plane of the sky; an
-orbit tilted toward the observer, with the planet moving away from the observer has
-the same projection on the plane of the sky as an orbit tilted away from the observer,
-with the planet moving toward the observer. In practice, this degeneracy is handy,
-because if the :math:`\omega_p`/Ω posteriors do not appear identical before and after 180◦, 
-it is generally an indication that the MCMC chains are unconverged.
+parameters. 
+
+
+
+
+Including Radial Velocities 
++++++++++++++++++