Angle of rotation during gravitational maneuver
===============================================

A gravitational maneuver is a purposeful change in the trajectory and flight speed of a
spacecraft under the influence of the gravitational fields of celestial bodies. The
angle of the gravitational maneuver depends on the aiming range, the mass of the planet
and the velocity of the rocket relative to the planet.

**Notation:**

#. :math:`G` (:code:`G`) is :attr:`~symplyphysics.quantities.gravitational_constant`.

.. image:: https://upload.wikimedia.org/wikipedia/commons/a/ad/Gravity_assist\_-\_ru.svg
    :width: 240px

..
    TODO find link to law

.. py:currentmodule:: symplyphysics.laws.gravity.angle_of_rotation_during_gravitational_maneuver

.. py:data:: angle

    :attr:`~symplyphysics.symbols.basic.angle` of rotation during a gravitational maneuver (angle at which the velocity vector of the rocket rotates)

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: planet_mass

    The :attr:`~symplyphysics.symbols.basic.mass` of the planet.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: aiming_range

    The aiming range is the :attr:`~symplyphysics.symbols.classical_mechanics.euclidean_distance` between the asymptote of the hyperbolic trajectory
    of the circumnavigation of the planet and its focus coinciding with the center of the planet.

Symbol:
    :code:`d`

Latex:
    :math:`d`

Dimension:
    :code:`length`


.. py:data:: rocket_speed

    Rocket's :attr:`~symplyphysics.symbols.classical_mechanics.speed` relative to the planet.

Symbol:
    :code:`v`

Latex:
    :math:`v`

Dimension:
    :code:`velocity`


.. py:data:: law

    :code:`phi = 2 * atan(G * m / (d * v^2))`


    Latex:
        .. math::
            \varphi = 2 \operatorname{atan}{\left(\frac{G m}{d v^{2}} \right)}