boinor.core.propagation.vallado
===============================

.. py:module:: boinor.core.propagation.vallado


Functions
---------

.. autoapisummary::

   boinor.core.propagation.vallado.vallado


Module Contents
---------------

.. py:function:: vallado(k, r0, v0, tof, numiter)

   Solves Kepler's Equation by applying a Newton-Raphson method.

   If the position of a body along its orbit wants to be computed
   for a specific time, it can be solved by terms of the Kepler's Equation:

   .. math::
       E = M + e\sin{E}

   In this case, the equation is written in terms of the Universal Anomaly:

   .. math::

       \sqrt{\mu}\Delta t = \frac{r_{o}v_{o}}{\sqrt{\mu}}\chi^{2}C(\alpha \chi^{2}) + (1 - \alpha r_{o})\chi^{3}S(\alpha \chi^{2}) + r_{0}\chi

   This equation is solved for the universal anomaly by applying a Newton-Raphson numerical method.
   Once it is solved, the Lagrange coefficients are returned:

   .. math::

       \begin{aligned}
           f &= 1 \frac{\chi^{2}}{r_{o}}C(\alpha \chi^{2}) \\
           g &= \Delta t - \frac{1}{\sqrt{\mu}}\chi^{3}S(\alpha \chi^{2}) \\
           \dot{f} &= \frac{\sqrt{\mu}}{rr_{o}}(\alpha \chi^{3}S(\alpha \chi^{2}) - \chi) \\
           \dot{g} &= 1 - \frac{\chi^{2}}{r}C(\alpha \chi^{2}) \\
       \end{aligned}

   Lagrange coefficients can be related then with the position and velocity vectors:

   .. math::
       \begin{aligned}
           \vec{r} &= f\vec{r_{o}} + g\vec{v_{o}} \\
           \vec{v} &= \dot{f}\vec{r_{o}} + \dot{g}\vec{v_{o}} \\
       \end{aligned}

   :param k: Standard gravitational parameter.
   :type k: float
   :param r0: Initial position vector.
   :type r0: numpy.ndarray
   :param v0: Initial velocity vector.
   :type v0: numpy.ndarray
   :param tof: Time of flight.
   :type tof: float
   :param numiter: Number of iterations.
   :type numiter: int

   :returns: * **f** (*float*) -- First Lagrange coefficient
             * **g** (*float*) -- Second Lagrange coefficient
             * **fdot** (*float*) -- Derivative of the first coefficient
             * **gdot** (*float*) -- Derivative of the second coefficient

   .. rubric:: Notes

   The theoretical procedure is explained in section 3.7 of :cite:t:`Curtis2013` in really
   deep detail. For analytical example, check in the same book for example 3.6.