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

.. only:: html

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

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

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

.. _sphx_glr_examples_galaxies_plot_ellipticity.py:


Galaxy Ellipticity Distributions
================================

This example demonstrate how to sample ellipticity distributions
in SkyPy.

.. GENERATED FROM PYTHON SOURCE LINES 12-42

3D Ellipticity Distribution
---------------------------

In Ryden 2004 [1]_, the ellipticity is sampled by randomly projecting
a 3D ellipsoid with principal axes :math:`A > B > C`.

The distribution of the axis ratio :math:`\gamma = C/A` is a truncated
normal with mean :math:`\mu_\gamma` and standard deviation
:math:`\sigma_\gamma`.

The distribution of :math:`\epsilon = \log(1 - B/A)` is truncated normal
with mean :math:`\mu` and standard deviation :math:`\sigma`.


The func:`skypy.galaxies.morphology.ryden04_ellipticity()` model samples
projected 2D axis ratios. Specifically, it samples the axis ratios of the
3D ellipsoid according to the description above [1] and
then randomly projects them using
:func:`skypy.utils.random.triaxial_axis_ratio()`.


Validation plot with SDSS Data
------------------------------

Here we validate our function by comparing our ``SkyPy`` simulated galaxy
ellipticities against observational data from SDSS DR1 in Figure 1 [1].
You can download the data file
:download:`SDSS_ellipticity <../../../examples/galaxies/SDSS_ellipticity.npz>`,
stored as a 2D array: :math:`e_1`, :math:`e_2`.


.. GENERATED FROM PYTHON SOURCE LINES 42-54

.. code-block:: python3


    import numpy as np

    # Load SDSS data from Fig. 1 in Ryden 2004
    data = np.load('SDSS_ellipticity.npz')
    e1, e2 = data['sdss']['e1'], data['sdss']['e2']

    Ngal = len(e1)
    e = np.hypot(e1, e2)
    q_amSDSS = np.sqrt((1 - e)/(1 + e))









.. GENERATED FROM PYTHON SOURCE LINES 55-63

In this example, we generate 100 galaxy ellipticity samples using the
``SkyPy`` function
:func:`skypy.galaxies.morphology.ryden04_ellipticity()` and the
best fit parameters
given in Ryden 2004 [1]:
:math:`\mu_\gamma =0.222`, :math:`\sigma_\gamma=0.057`, :math:`\mu =-1.85`
and :math:`\sigma=0.89`.


.. GENERATED FROM PYTHON SOURCE LINES 64-78

.. code-block:: python3


    from skypy.galaxies.morphology import ryden04_ellipticity

    # Best fit parameters from Fig. 1 in Ryden 2004
    mu_gamma, sigma_gamma, mu, sigma = 0.222, 0.057, -1.85, 0.89

    # Binning scheme of Fig. 1
    bins = np.linspace(0, 1, 41)
    mid = 0.5 * (bins[:-1] + bins[1:])

    # Mean and variance of sampling
    mean = np.zeros(len(bins)-1)
    var = np.zeros(len(bins)-1)








.. GENERATED FROM PYTHON SOURCE LINES 80-101

.. code-block:: python3


    # Create 100 SkyPy realisations
    for i in range(100):
        # sample ellipticity
        e = ryden04_ellipticity(mu_gamma, sigma_gamma, mu, sigma, size=Ngal)
        # recover axis ratio from ellipticity
        q = (1 - e)/(1 + e)
        # bin
        n, _ = np.histogram(q, bins=bins)

        # update mean and variance
        x = n - mean
        mean += x/(i+1)
        y = n - mean
        var += x*y

    # finalise variance and standard deviation
    var = var/i
    std = np.sqrt(var)









.. GENERATED FROM PYTHON SOURCE LINES 102-107

We now plot the distribution of axis ratio :math:`𝑞_{am}`
using adaptive moments in the i band, for exponential galaxies in the SDSS DR1
(solid line). The data points with error bars represent
the `SkyPy` simulation:


.. GENERATED FROM PYTHON SOURCE LINES 108-121

.. code-block:: python3


    import matplotlib.pyplot as plt

    plt.hist(q_amSDSS, range=[0, 1], bins=40, histtype='step',
             ec='k', lw=0.5, label='SDSS data')
    plt.errorbar(mid, mean, yerr=std, fmt='.r', ms=4, capsize=3,
                 lw=0.5, mew=0.5, label='SkyPy model')

    plt.xlabel(r'Axis ratio, $q_{am}$')
    plt.ylabel(r'N')
    plt.legend(frameon=False)
    plt.show()




.. image:: /examples/galaxies/images/sphx_glr_plot_ellipticity_001.png
    :alt: plot ellipticity
    :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 122-129

References
----------


.. [1] Ryden, Barbara S., 2004, `The Astrophysical Journal, Volume 601, Issue 1, pp. 214-220`_

.. _The Astrophysical Journal, Volume 601, Issue 1, pp. 214-220: https://arxiv.org/abs/astro-ph/0310097


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

   **Total running time of the script:** ( 0 minutes  0.517 seconds)


.. _sphx_glr_download_examples_galaxies_plot_ellipticity.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_ellipticity.py <plot_ellipticity.py>`



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

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


.. only:: html

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

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