.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "modeling/gallery/spatial/plot_shell2.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        Click :ref:`here <sphx_glr_download_modeling_gallery_spatial_plot_shell2.py>`
        to download the full example code

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

.. _sphx_glr_modeling_gallery_spatial_plot_shell2.py:


.. _shell2-spatial-model:

Shell2 spatial model
====================

This is a spatial model parametrizing a projected radiating shell.

The shell spatial model is defined by the following equations:

.. math::
    \phi(lon, lat) = \frac{3}{2 \pi (r_{out}^3 - r_{in}^3)} \cdot
            \begin{cases}
                \sqrt{r_{out}^2 - \theta^2} - \sqrt{r_{in}^2 - \theta^2} &
                             \text{for } \theta \lt r_{in} \\
                \sqrt{r_{out}^2 - \theta^2} &
                             \text{for } r_{in} \leq \theta \lt r_{out} \\
                0 & \text{for } \theta > r_{out}
            \end{cases}

where :math:`\theta` is the sky separation, :math:`r_{\text{out}}` is the outer radius
and  :math:`r_{\text{in}}` is the inner radius.

For Shell2SpatialModel, the radius parameter  r_0 correspond to :math:`r_{\text{out}}`
and the relative width parameter is given as eta = :math:`(r_{\text{out}} - r_{\text{in}})/r_{\text{out}}`
so we have :math:`r_{\text{in}} = (1-eta) r_{\text{out}}`.

Note that the normalization is a small angle approximation,
although that approximation is still very good even for 10 deg radius shells.

.. GENERATED FROM PYTHON SOURCE LINES 35-39

Example plot
------------
Here is an example plot of the shell model for the parametrization using outer radius and relative width.
In this case the relative width, eta, acts as a shape parameter.

.. GENERATED FROM PYTHON SOURCE LINES 39-69

.. code-block:: default


    import matplotlib.pyplot as plt
    from gammapy.modeling.models import (
        Models,
        PowerLawSpectralModel,
        Shell2SpatialModel,
        SkyModel,
    )

    tags = [
        r"Disk-like, $\eta \rightarrow 0$",
        r"Shell, $\eta=0.25$",
        r"Peaked, $\eta\rightarrow 1$",
    ]
    eta_range = [0.001, 0.25, 1]
    fig, axes = plt.subplots(1, 3, figsize=(9, 6))
    for ax, eta, tag in zip(axes, eta_range, tags):
        model = Shell2SpatialModel(
            lon_0="10 deg",
            lat_0="20 deg",
            r_0="2 deg",
            eta=eta,
            frame="galactic",
        )
        model.plot(ax=ax)
        ax.set_title(tag)
        ax.set_xticks([])
        ax.set_yticks([])
    plt.tight_layout()




.. image-sg:: /modeling/gallery/spatial/images/sphx_glr_plot_shell2_001.png
   :alt: Disk-like, $\eta \rightarrow 0$, Shell, $\eta=0.25$, Peaked, $\eta\rightarrow 1$
   :srcset: /modeling/gallery/spatial/images/sphx_glr_plot_shell2_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 70-73

YAML representation
-------------------
Here is an example YAML file using the model:

.. GENERATED FROM PYTHON SOURCE LINES 73-82

.. code-block:: default


    pwl = PowerLawSpectralModel()
    shell2 = Shell2SpatialModel()

    model = SkyModel(spectral_model=pwl, spatial_model=shell2, name="pwl-shell2-model")

    models = Models([model])

    print(models.to_yaml())




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

 Out:

 .. code-block:: none

    components:
    -   name: pwl-shell2-model
        type: SkyModel
        spectral:
            type: PowerLawSpectralModel
            parameters:
            -   name: index
                value: 2.0
            -   name: amplitude
                value: 1.0e-12
                unit: cm-2 s-1 TeV-1
            -   name: reference
                value: 1.0
                unit: TeV
                frozen: true
        spatial:
            type: Shell2SpatialModel
            frame: icrs
            parameters:
            -   name: lon_0
                value: 0.0
                unit: deg
            -   name: lat_0
                value: 0.0
                unit: deg
            -   name: r_0
                value: 1.0
                unit: deg
            -   name: eta
                value: 0.2






.. _sphx_glr_download_modeling_gallery_spatial_plot_shell2.py:


.. only :: html

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



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

     :download:`Download Python source code: plot_shell2.py <plot_shell2.py>`



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

     :download:`Download Jupyter notebook: plot_shell2.ipynb <plot_shell2.ipynb>`


.. only:: html

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

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