.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geophysics/plot_14_king_hele_satellite_flattening.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_geophysics_plot_14_king_hele_satellite_flattening.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_geophysics_plot_14_king_hele_satellite_flattening.py:


King-Hele's flattening from satellite orbits (1958)
===================================================

The equatorial bulge pulls on an inclined orbit and makes its node drift
westward. Desmond King-Hele and Robert Merson timed the drift of Sputnik 2's
orbit and found a J2 that gave a flattening of 1/298.3, not the 1/297 of the
International Ellipsoid that surveyors had used since 1924. A few months of
tracking one satellite had improved on two centuries of arc measurements.

.. GENERATED FROM PYTHON SOURCE LINES 11-42



.. image-sg:: /api/gallery/geophysics/images/sphx_glr_plot_14_king_hele_satellite_flattening_001.png
   :alt: The bulge turns the orbit
   :srcset: /api/gallery/geophysics/images/sphx_glr_plot_14_king_hele_satellite_flattening_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.constants import OMEGA_EARTH
    from earthsciencekit.geophysics import centrifugal_ratio, flattening_from_j2, nodal_precession_rate
    from earthsciencekit.units import convert

    gm, a, b, j2 = 3.986004418e14, 6378137.0, 6356752.3, 1.08263e-3
    perigee, apogee, inclination = 6378137.0 + 212e3, 6378137.0 + 1660e3, np.radians(65.33)  # Sputnik 2
    axis, eccentricity = (perigee + apogee) / 2, (apogee - perigee) / (apogee + perigee)
    observed = nodal_precession_rate(axis, eccentricity, inclination, j2, a, gm)
    recovered = observed / nodal_precession_rate(axis, eccentricity, inclination, 1, a, gm)
    flattening = flattening_from_j2(recovered, centrifugal_ratio(OMEGA_EARTH, a, gm / (a * b)))
    assert np.isclose(1 / flattening, 298.257, rtol=2e-3) and 1 / flattening - 297 > 1
    per_day = convert(observed, "rad", "degree") * convert(1, "day", "s")
    assert -3.5 < per_day < -2

    angle = np.linspace(0, np.pi, 181)
    fig, ax = plt.subplots()
    for height in [300e3, 1000e3, 3000e3]:
        rate = nodal_precession_rate(a + height, 0, angle, j2, a, gm)
        ax.plot(
            convert(angle, "rad", "degree"), convert(rate, "rad", "degree") * convert(1, "day", "s"), label=f"circular, {height / 1e3:.0f} km"
        )
    ax.plot(convert(inclination, "rad", "degree"), per_day, "ko", label=f"Sputnik 2: f = 1/{1 / flattening:.1f}")
    ax.axhline(0, color="k", lw=0.5)
    ax.set(xlabel="inclination [degrees]", ylabel="nodal drift [degrees/day]", title="The bulge turns the orbit")
    ax.legend()

    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.098 seconds)


.. _sphx_glr_download_api_gallery_geophysics_plot_14_king_hele_satellite_flattening.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../lite/lab/index.html?path=api/gallery/geophysics/plot_14_king_hele_satellite_flattening.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_14_king_hele_satellite_flattening.ipynb <plot_14_king_hele_satellite_flattening.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_14_king_hele_satellite_flattening.py <plot_14_king_hele_satellite_flattening.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_14_king_hele_satellite_flattening.zip <plot_14_king_hele_satellite_flattening.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_