:mod:`lamberthub.p_solvers.gauss`
=================================

.. py:module:: lamberthub.p_solvers.gauss
   :noindex:

.. autoapi-nested-parse::

   This module holds all methods devised by Carl Fredrich Gauss.



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


Functions
~~~~~~~~~

.. autoapisummary::


   lamberthub.p_solvers.gauss.gauss1809


   lamberthub.p_solvers.gauss._get_s


   lamberthub.p_solvers.gauss._get_w


   lamberthub.p_solvers.gauss._gauss_first_equation


   lamberthub.p_solvers.gauss._gauss_second_equation


   lamberthub.p_solvers.gauss._X_at_x



.. py:function:: gauss1809(mu, r1, r2, tof, M=0, prograde=True, low_path=True, maxiter=250, atol=1e-05, rtol=1e-07, full_output=False)

   Lambert's problem solver devised by Carl Friedrich Gauss in 1809. The method
   has been implemented according to Bate's book (see [2]) and extended to the
   hyperbolic case. This method shows poor accuracy, being only suitable for
   low transfer angles.

   :param mu: Gravitational parameter, equivalent to :math:`GM` of attractor body.
   :type mu: float
   :param r1: Initial position vector.
   :type r1: numpy.array
   :param r2: Final position vector.
   :type r2: numpy.array
   :param M: Number of revolutions. Must be equal or greater than 0 value.
   :type M: int
   :param prograde: If `True`, specifies prograde motion. Otherwise, retrograde motion is imposed.
   :type prograde: bool
   :param low_path: If two solutions are available, it selects between high or low path.
   :type low_path: bool
   :param maxiter: Maximum number of iterations.
   :type maxiter: int
   :param atol: Absolute tolerance.
   :type atol: float
   :param rtol: Relative tolerance.
   :type rtol: float
   :param full_output: If True, the number of iterations and time per iteration are also returned.
   :type full_output: bool

   :returns: * **v1** (*numpy.array*) -- Initial velocity vector.
             * **v2** (*numpy.array*) -- Final velocity vector.
             * **numiter** (*int*) -- Number of iterations.
             * **tpi** (*float*) -- Time per iteration in seconds.

   .. rubric:: Notes

   The algorithm originally devised by Gauss exploits the so-called ratio of
   sector to triangle area, which is a numerical value related with the orbital
   parameter. This Algorithm was used to the discovery of the orbit of Ceres by
   the genius and adopted by many other authors of his time due to its
   simplicity. However, the Algorithm is found to be singular for transfer
   angles of 180 degrees and shows a low performance for really small angles.

   .. rubric:: References

   [1] Gauss, C. F. (1809). Theoria motus corporum coelestium in sectionibus
   conicis solem ambientium auctore Carolo Friderico Gauss. sumtibus Frid.
   Perthes et IH Besser.

   [2] Bate, R. R., Mueller, D. D., White, J. E., & Saylor, W. W. (2020).
   Fundamentals of astrodynamics. Courier Dover Publications.


.. py:function:: _get_s(r1_norm, r2_norm, dtheta)

   Returns the s auxiliary constant.

   :param r1_norm: The norm of the initial position vector.
   :type r1_norm: float
   :param r2_norm: The norm of the final position vector.
   :type r2_norm: float
   :param dtheta: The transfer angle.
   :type dtheta: float

   :returns: **s** -- An auxiliary constant.
   :rtype: float

   .. rubric:: Notes

   This is equation number (5.6-2) from Bate's book [2].


.. py:function:: _get_w(mu, tof, r1_norm, r2_norm, dtheta)

   Returns the w auxiliary constant.

   :param mu: The gravitational constant.
   :type mu: float
   :param tof: The time of flight.
   :type tof: float
   :param r1_norm: The norm of the initial position vector.
   :type r1_norm: float
   :param r2_norm: The norm of the final position vector.
   :type r2_norm: float
   :param dtheta: The transfer angle.
   :type dtheta: float

   :returns: **w** -- An auxiliary constant.
   :rtype: float

   .. rubric:: Notes

   This is equation number (5.6-3) from Bate's book [2].


.. py:function:: _gauss_first_equation(y, s, w)

   Evaluates Gauss' first equation.

   :param y: The dependent variable.
   :type y: float
   :param s: First auxiliary variable.
   :type s: float
   :param w: Second auxiliary variable.
   :type w: float

   :returns: **x** -- The independent variable or free-parameter.
   :rtype: float

   .. rubric:: Notes

   This is equation (5.6-13) from Bate's book [2].


.. py:function:: _gauss_second_equation(x, s)

   Evaluates Gauss' second equation.

   :param x: The independent variable.
   :type x: float
   :param s: First auxiliary variable
   :type s: float

   :returns: **y** -- Dependent variable.
   :rtype: float

   .. rubric:: Notes

   This is equation (5.6-14) from Bate's book, reference [2].


.. py:function:: _X_at_x(x, order=50)

   Computes capital X as function of lower x.

   :param x: The independent variable.
   :type x: float
   :param n_items: The number of terms to be considered in the series.
   :type n_items: int

   :returns: **X** -- The series summation.
   :rtype: float

   .. rubric:: Notes

   This is equation (5.6-15) from Bate's book, in reference [2].