.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/relativity/kerr/plot_kerr_shadow_and_redshift.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_relativity_kerr_plot_kerr_shadow_and_redshift.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_relativity_kerr_plot_kerr_shadow_and_redshift.py:


The Kerr shadow: a rotating black hole's asymmetric silhouette
================================================================================

A rotating black hole's shadow is not a perfect circle like Schwarzschild's:
frame dragging shifts and flattens it along the direction of rotation, and
the accretion disk's approaching side is Doppler-boosted bright while the
receding side dims -- exactly the asymmetric brightness pattern seen in the
2019 Event Horizon Telescope image of M87*. This example ray-traces the full
(non-equatorial) Kerr null geodesic equations -- using Carter's separated
equations of motion in Mino time -- to render the shadow of a black hole of
mass :math:`M` and spin parameter :math:`a` at increasing spin, and shades
a thin, circularly orbiting equatorial accretion disk by its combined
gravitational and Doppler redshift

.. math::

    g = \frac{E_{\text{obs}}}{E_{\text{emit}}} = \frac{1}{u^t(r)\left[1 - b\,\Omega(r)\right]}

where :math:`b = L_{\text{ph}}/E_{\text{ph}}` is the received photon's
impact parameter, :math:`\Omega(r)` is the disk's Keplerian angular
velocity, and :math:`u^t(r)` is the emitting gas's time-dilation factor;
:math:`g>1` (blueshifted) marks the approaching side and :math:`g<1`
(redshifted) the receding side. Rendered brightness is further boosted by
:math:`g^3`, the relativistic Doppler-beaming factor for a locally
isotropic emitter.

.. GENERATED FROM PYTHON SOURCE LINES 28-33

.. code-block:: Python


    import matplotlib.pyplot as plt

    from physicskit.relativity.visualizers.shadow_render import plot_black_hole_shadow, render_black_hole_image








.. GENERATED FROM PYTHON SOURCE LINES 34-36

The shadow shrinks and shifts as spin increases
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-43

.. code-block:: Python

    fig, axes = plt.subplots(1, 3, figsize=(15, 5))
    for ax, a in zip(axes, [0.0, 0.7, 0.998]):
        result = render_black_hole_image(M=1.0, a=a, ny=180, nx=180, inclination=1.3, r_disk_inner=1.0, r_disk_outer=0.0)
        plot_black_hole_shadow(result, ax=ax)
        ax.set_title(f"a/M = {a}")
    plt.tight_layout()




.. image-sg:: /api/gallery/relativity/kerr/images/sphx_glr_plot_kerr_shadow_and_redshift_001.png
   :alt: a/M = 0.0, a/M = 0.7, a/M = 0.998
   :srcset: /api/gallery/relativity/kerr/images/sphx_glr_plot_kerr_shadow_and_redshift_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 44-46

A redshift-shaded accretion disk (near-extremal spin, oblique view)
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-54

.. code-block:: Python

    result = render_black_hole_image(M=1.0, a=0.9, ny=250, nx=250, inclination=1.35, screen_half_width=18.0)
    fig, axes = plt.subplots(1, 2, figsize=(13, 6))
    plot_black_hole_shadow(result, ax=axes[0], redshift=False)
    axes[0].set_title("Colored by radius only")
    plot_black_hole_shadow(result, ax=axes[1], redshift=True)
    axes[1].set_title("Doppler-shaded: approaching side brighter")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/relativity/kerr/images/sphx_glr_plot_kerr_shadow_and_redshift_002.png
   :alt: Colored by radius only, Doppler-shaded: approaching side brighter
   :srcset: /api/gallery/relativity/kerr/images/sphx_glr_plot_kerr_shadow_and_redshift_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 12.992 seconds)


.. _sphx_glr_download_api_gallery_relativity_kerr_plot_kerr_shadow_and_redshift.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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