.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/seismology/plot_07_gutenberg_core_shadow.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_07_gutenberg_core_shadow.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_07_gutenberg_core_shadow.py:


Gutenberg's core shadow (1914)
==============================

Beno Gutenberg placed the core-mantle boundary at a depth of about 2900 km.
A slower core bends rays towards the centre, so in the real Earth direct P
waves fade near 103 degrees from the source and refracted core waves return
only beyond about 143 degrees. In between is a shadow. The ray parameter,
which is conserved along a ray, connects the two. This toy Earth has a uniform
13 km/s mantle over an 8 km/s core, so its rays are straight within each
shell and its shadow starts later, at about 114 degrees.

.. GENERATED FROM PYTHON SOURCE LINES 13-46



.. image-sg:: /api/gallery/seismology/images/sphx_glr_plot_07_gutenberg_core_shadow_001.png
   :alt: A slow core casts a shadow
   :srcset: /api/gallery/seismology/images/sphx_glr_plot_07_gutenberg_core_shadow_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    shadow zone from 113.7 to 143.1 degrees






|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.seismology import ray_parameter

    R, rc = 6371e3, 3480e3
    vm, vc = 13e3, 8e3
    takeoff = np.linspace(1e-4, np.pi / 2 - 1e-4, 4000)
    p = ray_parameter(R, vm, takeoff)
    direct = p * vm >= rc
    distance = np.empty_like(p)
    # Center angle of a straight chord between radii r1 > r2 with r sin(i) = b is arccos(b/r1) - arccos(b/r2).
    distance[direct] = 2 * np.arccos(p[direct] * vm / R)
    core = ~direct
    at_cmb = np.arcsin(p[core] * vm / rc)
    in_core = np.arcsin(p[core] * vc / rc)
    assert np.allclose(ray_parameter(rc, vm, at_cmb), ray_parameter(rc, vc, in_core))
    swept = 2 * (np.arccos(p[core] * vm / R) - np.arccos(p[core] * vm / rc)) + np.pi - 2 * in_core
    distance[core] = np.where(swept > np.pi, 2 * np.pi - swept, swept)

    edge, far = np.degrees(distance[direct].max()), np.degrees(distance[core].min())
    assert 110 < edge < 120 and 140 < far < 146
    print(f"shadow zone from {edge:.1f} to {far:.1f} degrees")

    fig, ax = plt.subplots()
    ax.plot(np.degrees(takeoff[direct]), np.degrees(distance[direct]), label="P through the mantle")
    ax.plot(np.degrees(takeoff[core]), np.degrees(distance[core]), label="PKP through the core")
    ax.axhspan(edge, far, color="0.85", label="shadow zone")
    ax.set(xlabel="takeoff angle from vertical [deg]", ylabel="epicentral distance [deg]", title="A slow core casts a shadow")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.087 seconds)


.. _sphx_glr_download_api_gallery_seismology_plot_07_gutenberg_core_shadow.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_07_gutenberg_core_shadow.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_07_gutenberg_core_shadow.ipynb <plot_07_gutenberg_core_shadow.ipynb>`

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

      :download:`Download Python source code: plot_07_gutenberg_core_shadow.py <plot_07_gutenberg_core_shadow.py>`

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

      :download:`Download zipped: plot_07_gutenberg_core_shadow.zip <plot_07_gutenberg_core_shadow.zip>`


.. only:: html

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

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