Semimajor axis via Kepler's constant and total energy
=====================================================

The semi-major axis of an orbiting planet depends on the Kepler's constant of
the star—planet system and the total energy of the planet per unit of its mass.

**Notes:**

#. Works for both elliptical (:math:`\varepsilon < 0`) and hyperbolical
   (:math:`\varepsilon > 0`) orbits.

**Links:**

#. Sivukhin D.V. (1979), *Obshchiy kurs fiziki* [General course of Physics], vol. 1, p. 317, (58.2).

.. py:currentmodule:: symplyphysics.laws.gravity.radial_motion.semimajor_axis_via_kepler_constant_and_total_energy

.. py:data:: semimajor_axis

    The :attr:`~symplyphysics.symbols.classical_mechanics.semimajor_axis` of the planet's orbit.

Symbol:
    :code:`a`

Latex:
    :math:`a`

Dimension:
    :code:`length`


.. py:data:: kepler_constant

    The :attr:`~symplyphysics.symbols.classical_mechanics.kepler_constant`, whose value is determined by the :attr:`~symplyphysics.symbols.basic.mass`
    of the orbited star.

Symbol:
    :code:`K`

Latex:
    :math:`\mathfrak{K}`

Dimension:
    :code:`length**3/time**2`


.. py:data:: specific_energy

    :attr:`~symplyphysics.symbols.basic.specific_energy` of the planet. Can be negative or positive depending on the sign of
    the planet's energy.

Symbol:
    :code:`epsilon`

Latex:
    :math:`\varepsilon`

Dimension:
    :code:`energy/mass`


.. py:data:: law

    :code:`a = 2 * pi^2 * K / Abs(epsilon)`


    Latex:
        .. math::
            a = \frac{2 \pi^{2} \mathfrak{K}}{\left|{\varepsilon}\right|}