.. _sphx_glr_api_gallery_astro_orbital_mechanics:

Orbital mechanics
-------------------

Kepler's laws, Newton's inverse-square gravity, Gauss's orbit-determination
problem, and Hohmann's minimum-energy transfer -- the two-body machinery in
:mod:`physicskit.astro.orbital_mechanics`.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Kepler&#x27;s three empirical laws -- an elliptical orbit with the Sun at one focus, equal areas swept in equal times, and T^2\propto a^3 -- were purely descriptive when published (1609, 1619). Newton (1687) showed they follow as mathematical consequences of a single dynamical law, the inverse-square gravitational force">

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  .. image:: /api/gallery/astro/orbital_mechanics/images/thumb/sphx_glr_plot_01_kepler_newton_orbit_thumb.png
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  :doc:`/api/gallery/astro/orbital_mechanics/plot_01_kepler_newton_orbit`

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      <div class="sphx-glr-thumbnail-title">Kepler's laws and Newton's inverse-square gravity</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="When Piazzi lost Ceres in the Sun&#x27;s glare after only a few weeks of observation, Gauss showed how to recover a complete set of orbital elements from as few as three observed positions and times. The modern version of the last step of that problem -- given a single Cartesian state (position and velocity) at one instant, recover the six classical orbital elements (a, e, i, \Omega, \omega, \nu) -- is exactly what orbital_elements_from_state does; state_from_orbital_elements is its inverse. This example starts from a chosen (inclined, eccentric) orbit, converts to a state vector as if it were an observation, and recovers the original elements exactly -- verifying the round trip in both directions, then re-derives the whole 3D orbit from the recovered elements alone.">

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  .. image:: /api/gallery/astro/orbital_mechanics/images/thumb/sphx_glr_plot_02_orbit_determination_thumb.png
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  :doc:`/api/gallery/astro/orbital_mechanics/plot_02_orbit_determination`

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      <div class="sphx-glr-thumbnail-title">Gauss's problem: recovering orbital elements from a state vector</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hohmann (1925) showed that the cheapest way to move between two circular orbits is a transfer ellipse tangent to both, entered and left with a single impulsive burn at each tangent point:">

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  .. image:: /api/gallery/astro/orbital_mechanics/images/thumb/sphx_glr_plot_03_hohmann_transfer_thumb.png
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  :doc:`/api/gallery/astro/orbital_mechanics/plot_03_hohmann_transfer`

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      <div class="sphx-glr-thumbnail-title">Hohmann's minimum-energy transfer orbit</div>
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.. toctree::
   :hidden:

   /api/gallery/astro/orbital_mechanics/plot_01_kepler_newton_orbit
   /api/gallery/astro/orbital_mechanics/plot_02_orbit_determination
   /api/gallery/astro/orbital_mechanics/plot_03_hohmann_transfer