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

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_plot_ContributionFunction.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_plot_ContributionFunction.py:


===================================================
Contribution function and formation height of 854.2
===================================================

.. GENERATED FROM PYTHON SOURCE LINES 8-12

After converging a simulation, the depth-dependent opacity and emissivity
can be stored during a formal solution, and used to compute optical depths
and contribution functions. Here we look at where the Ca II 854.2 nm line
forms in the FAL C atmosphere.

.. GENERATED FROM PYTHON SOURCE LINES 12-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    import lightweaver as lw
    from lightweaver.fal import Falc82
    from lightweaver.rh_atoms import CaII_atom, H_6_atom
    from lightweaver.utils import compute_contribution_fn, compute_tau, tau_isosurface








.. GENERATED FROM PYTHON SOURCE LINES 21-22

Converge the NLTE problem with H and Ca II active.

.. GENERATED FROM PYTHON SOURCE LINES 22-31

.. code-block:: Python

    atmos = Falc82()
    atmos.quadrature(5)
    rad_set = lw.RadiativeSet([H_6_atom(), CaII_atom()])
    rad_set.set_active('H', 'Ca')
    spect = rad_set.compute_wavelength_grid()
    eq_pops = rad_set.compute_eq_pops(atmos)
    ctx = lw.Context(atmos, spect, eq_pops, Nthreads=2)
    lw.iterate_ctx_se(ctx, quiet=True)





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

 .. code-block:: none


    240



.. GENERATED FROM PYTHON SOURCE LINES 32-34

Ask the formal solver to store the full depth-dependent opacity and
emissivity (`ctx.depth_data`), then perform one more formal solution.

.. GENERATED FROM PYTHON SOURCE LINES 34-37

.. code-block:: Python

    ctx.depth_data.fill = True
    ctx.formal_sol_gamma_matrices()





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

 .. code-block:: none


    IterationUpdate(ctx=<lightweaver.Context object at 0x558d19ff3f30>, crsw=1.0, updated_J=True, dJ_max=0.004928092666125372, dJ_max_idx=308, updated_pops=False, dpops=[], dpops_max_idx=[], ng_accelerated=False, updated_ne=False, dne_max=0.0, dne_max_idx=0, updated_rho=False, Nprd_sub_iter=0, drho=[], drho_max_idx=[], updated_J_prd=False, dJ_prd_max=[], dJ_prd_max_idx=[])



.. GENERATED FROM PYTHON SOURCE LINES 38-42

Compute the contribution function and optical depth for the outgoing ray
with the largest mu (the last ray of the quadrature), and the height at
which tau = 1 along this ray. These are computed for every wavelength of
the simulation's grid, so we select the points around 854.2.

.. GENERATED FROM PYTHON SOURCE LINES 42-54

.. code-block:: Python

    mu = -1
    cfn = compute_contribution_fn(ctx, mu=mu)
    tau = compute_tau(ctx, mu=mu)
    z = np.asarray(atmos.height)
    tau1 = tau_isosurface(tau, z)

    wavelength = np.asarray(ctx.spect.wavelength)
    ca_8542 = 854.444
    mask = np.abs(wavelength - ca_8542) < 0.1
    wave = wavelength[mask]
    I = np.asarray(ctx.spect.I)[mask, mu]








.. GENERATED FROM PYTHON SOURCE LINES 55-57

Integrating the contribution function over height gives the emergent
intensity (up to the accuracy of this simple quadrature).

.. GENERATED FROM PYTHON SOURCE LINES 57-63

.. code-block:: Python

    integrated = -np.trapezoid(cfn[mask], z, axis=1)
    print(
        f'Max relative difference between integrated cfn and I: '
        f'{np.max(np.abs(integrated - I) / I):.2e}'
    )





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

 .. code-block:: none

    Max relative difference between integrated cfn and I: 5.48e-02




.. GENERATED FROM PYTHON SOURCE LINES 64-67

Plot the contribution function (normalised at each wavelength) with the
tau = 1 height overlaid, and the emergent profile below. The line core forms
in the chromosphere, around 1.2 Mm higher than the wings.

.. GENERATED FROM PYTHON SOURCE LINES 67-81

.. code-block:: Python

    z_mm = z / 1e6
    fig, ax = plt.subplots(
        2, 1, figsize=(7, 7), sharex=True, height_ratios=[2, 1], constrained_layout=True
    )
    cfn_norm = cfn[mask] / cfn[mask].max(axis=1, keepdims=True)
    ax[0].pcolormesh(wave, z_mm, cfn_norm.T, shading='auto', cmap='magma')
    ax[0].plot(wave, tau1[mask] / 1e6, c='C0', label=r'$\tau = 1$')
    ax[0].set_ylim(0, 2.2)
    ax[0].set_ylabel('Height [Mm]')
    ax[0].legend()
    ax[1].plot(wave, I)
    ax[1].set_xlabel('Wavelength [nm]')
    ax[1].set_ylabel(r'$I_\nu$ [J s$^{-1}$ m$^{-2}$ sr$^{-1}$ Hz$^{-1}$]')
    plt.show()



.. image-sg:: /auto_examples/images/sphx_glr_plot_ContributionFunction_001.png
   :alt: plot ContributionFunction
   :srcset: /auto_examples/images/sphx_glr_plot_ContributionFunction_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 3.482 seconds)


.. _sphx_glr_download_auto_examples_plot_ContributionFunction.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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