.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geophysics/plot_03_clairaut_theorem.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_03_clairaut_theorem.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_03_clairaut_theorem.py:


Clairaut's theorem (1743)
=========================

Alexis Clairaut showed that a rotating Earth in hydrostatic equilibrium, of any
internal layering, has a gravity that grows from equator to pole by the
fraction ``5 m / 2 - f``. Pendulum gravity measured at a few latitudes therefore
gives the flattening without any arc survey. Applied to today's normal gravity
it returns the flattening to within about 1%, and the first-order formula
matches the exact Somigliana gravity of the reference ellipsoid to 50 parts per
million.

.. GENERATED FROM PYTHON SOURCE LINES 13-40



.. image-sg:: /api/gallery/geophysics/images/sphx_glr_plot_03_clairaut_theorem_001.png
   :alt: Flattening from gravity alone: 1/301
   :srcset: /api/gallery/geophysics/images/sphx_glr_plot_03_clairaut_theorem_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.geophysics import clairaut_gravity, normal_gravity
    from earthsciencekit.units import convert

    ratio = 0.00344978600308  # geodetic m = omega**2 a**2 b / GM of GRS80
    equator, pole = normal_gravity(0), normal_gravity(np.pi / 2)
    flattening = 5 * ratio / 2 - (pole - equator) / equator
    assert 295 < 1 / flattening < 305
    latitude = np.linspace(0, np.pi / 2, 91)
    clairaut = clairaut_gravity(latitude, equator, 1 / 298.257222101, ratio)
    somigliana = normal_gravity(latitude)
    np.testing.assert_allclose(clairaut, somigliana, rtol=5e-5)

    fig, (top, bottom) = plt.subplots(2, 1, sharex=True, figsize=(6, 6))
    degrees = convert(latitude, "rad", "degree")
    top.plot(degrees, somigliana, label="Somigliana (GRS80)")
    top.plot(degrees, clairaut, "--", label="Clairaut, first order")
    top.set(ylabel="gravity [m/s^2]", title=f"Flattening from gravity alone: 1/{1 / flattening:.0f}")
    top.legend()
    bottom.plot(degrees, convert(clairaut - somigliana, "m/s^2", "mGal"))
    bottom.set(xlabel="latitude [degrees]", ylabel="Clairaut - Somigliana [mGal]")
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.164 seconds)


.. _sphx_glr_download_api_gallery_geophysics_plot_03_clairaut_theorem.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_03_clairaut_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_clairaut_theorem.ipynb <plot_03_clairaut_theorem.ipynb>`

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

      :download:`Download Python source code: plot_03_clairaut_theorem.py <plot_03_clairaut_theorem.py>`

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

      :download:`Download zipped: plot_03_clairaut_theorem.zip <plot_03_clairaut_theorem.zip>`


.. only:: html

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

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