Added RV section

docs/manual.rst

@@ -17,14 +17,14 @@ which can then be solved using Kepler’s equation, but first it is important to
     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 explanation of the coordinate system.
+    with use and explanation of the coordinate system by Sarah Blunt.
 
 In its “standard” mode, ``orbitize!`` assumes that the user only has relative astrometric data to fit. 
 In the ``orbitize!`` coordinate system, relative R.A. and declination can be expressed as the following functions 
 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 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)]
 
@@ -33,11 +33,11 @@ the true anomaly, and E is the eccentric anomaly, which are related to elapsed t
 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 \\
+    M =E-esinE 
     
     f = 2tan^{-1}[\sqrt{{1+e\over 1-e}}tan{E\over 2}]
 
@@ -49,25 +49,77 @@ 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:
 
-#. Individual component masses do not show up anywhere in this equation set. 
+1. Individual component masses do not show up anywhere in this equation set.
 
-#. The degeneracy between semimajor axis 𝑎, total mass :math:`𝑀_{tot}`, and
+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. 
 
-#. The argument of periastron :math:`\omega_p` and the position angle of nodes Ω. 
+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_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. 
 
+Solutions to breaking degeneracies 1 and 3 can be found in the next section. 
+
+
+Using Radial Velocities 
++++++++++++++++++
+In the ``orbitize!``coordinate system, and relative to the system barycenter, the
+radial velocity of the planet due to the gravitational influence of the star is:
+.. math::
+    rv_p(f) = [\sqrt{{G\over (1-e**2)}}]M_* sini(M_{tot})^{-1/2}a^{-1/2}(cos(\omega_p+f)+ecos\omega_p)
+
+    rv_*(f) = [\sqrt{{G\over (1-e**2)}}]M_p sini(M_{tot})^{-1/2}a^{-1/2}(cos(\omega_*+f)+ecos\omega_*)
+
+where 𝜔∗ is the argument of periastron of the star’s orbit, which is equal to 𝜔𝑝 +
+180◦.
+
+In these equations, the individual component masses m𝑝 and m∗ enter. This means
+radial velocity measurements break the total mass degeneracy and enable measurements of individual component masses 
+(“dynamical” masses). However it is crucial to keep in mind that radial velocities of a planet do not enable 
+dynamical mass measurements of the planet itself, but of the star. 
+Radial velocity measurements also break the Ω/𝜔 degeneracy discussed in the
+previous section, uniquely orienting the orbit in 3D space.
+
+``orbitize!``can perform joint fits of RV and astrometric data in two different
+ways, which have complementary applications. 
+
+The first method is automatically triggered when an
+``orbitize!``user inputs radial velocity data. ``orbitize!``automatically parses
+the data, sets up an appropriate model, then runs the user’s Bayesian computation
+algorithm of choice to jointly constrain all free parameters in the fit. ``orbitize!``
+can handle both primary and secondary RVs, and fits for the appropriate dynamical
+masses when RVs are present; when primary RVs are included, ``orbitize!``fits for
+the dynamical masses of secondary objects, and vice versa. 
+Instrumental nuisance parameters (RV zeropoint offset, 𝛾, and white noise jitter, 𝜎) for each RV instrument
+are also included as additional free parameters in the fit if the user specifies different
+instrument names in the data file.
+
+The second method of jointly fitting RV and astrometric data in ``orbitize!`` separates out the fitting of 
+radial velocities and astrometry, enabling a user to fit “one at a
+time,” and combine the results in a Bayesian framework. First, a user performs a
+fit to just the radial velocity data using, for example, radvel (but can be any radial
+velocity orbit-fitting code). The user then feeds the numerical posterior samples
+into ``orbitize!`` through the ``orbitize.priors.KDEPrior`` object. This prior
+creates a representation of the prior using kernel density estimation 
+(`kernel density estimation <https://mathisonian.github.io/kde/>`_),
+which can then be used to generate random prior samples or compute the prior
+probability of a sample orbit. Importantly, this prior preserves covariances between
+input parameters, allowing ``orbitize!``to use an accurate representation of the RV
+posterior to constrain the fit. This method can be referred to as the “posteriors as priors”
+method, since posteriors output from a RV fitting code are, through KDE sampling,
+being applied as priors in``orbitize!``.
+
 
 More coming soon!