.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/seismology/plot_10_lehmann_inner_core.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_seismology_plot_10_lehmann_inner_core.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_seismology_plot_10_lehmann_inner_core.py:


Lehmann's inner core (1936)
===========================

Inge Lehmann noticed weak P arrivals inside the core shadow zone. She
explained them with an inner core faster than the outer core: rays meeting it
beyond the critical angle are totally reflected and emerge in the shadow. In
this straight-ray toy Earth (mantle 13 km/s, outer core 8 km/s, inner core
11 km/s, inner-core radius 1220 km) the post-critical reflections fill the
shadow zone of the 1914 example.

.. GENERATED FROM PYTHON SOURCE LINES 12-40



.. image-sg:: /api/gallery/seismology/images/sphx_glr_plot_10_lehmann_inner_core_001.png
   :alt: Inner-core reflections reach the shadow
   :srcset: /api/gallery/seismology/images/sphx_glr_plot_10_lehmann_inner_core_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    critical angle 46.7 deg; total reflections from 87 to 170 deg






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.seismology import critical_angle, ray_parameter

    R, rc, ri = 6371e3, 3480e3, 1220e3
    vm, vc, vi = 13e3, 8e3, 11e3
    shadow = (np.degrees(2 * np.arccos(rc / R)), 143.1)
    p = ray_parameter(R, vm, np.linspace(1e-4, np.arcsin(rc / R) - 1e-6, 4000))
    b_mantle, b_core = p * vm, p * vc
    hits = b_core < ri
    b_mantle, b_core = b_mantle[hits], b_core[hits]
    distance = 2 * (np.arccos(b_mantle / R) - np.arccos(b_mantle / rc)) + 2 * (np.arccos(b_core / rc) - np.arccos(b_core / ri))
    incidence = np.arcsin(b_core / ri)
    total = incidence >= critical_angle(vc, vi)
    lit = np.degrees(distance[total])
    assert lit.min() < shadow[0] and lit.max() > shadow[1]
    print(f"critical angle {np.degrees(critical_angle(vc, vi)):.1f} deg; total reflections from {lit.min():.0f} to {lit.max():.0f} deg")

    fig, ax = plt.subplots()
    ax.plot(np.degrees(incidence), np.degrees(distance))
    ax.axvline(np.degrees(critical_angle(vc, vi)), color="k", ls=":", label="critical angle")
    ax.axhspan(*shadow, color="0.85", label="shadow zone")
    ax.set(xlabel="incidence on the inner core [deg]", ylabel="epicentral distance [deg]", title="Inner-core reflections reach the shadow")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_seismology_plot_10_lehmann_inner_core.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/seismology/plot_10_lehmann_inner_core.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_10_lehmann_inner_core.ipynb <plot_10_lehmann_inner_core.ipynb>`

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

      :download:`Download Python source code: plot_10_lehmann_inner_core.py <plot_10_lehmann_inner_core.py>`

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

      :download:`Download zipped: plot_10_lehmann_inner_core.zip <plot_10_lehmann_inner_core.zip>`


.. only:: html

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

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