.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "tutorials/analysis-1d/ebl.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_tutorials_analysis-1d_ebl.py: Account for spectral absorption due to the EBL ============================================== Gamma rays emitted from extra-galactic objects, eg blazars, interact with the photons of the Extragalactic Background Light (EBL) through pair production and are attenuated, thus modifying the intrinsic spectrum. Various models of the EBL are supplied in `GAMMAPY_DATA`. This notebook shows how to use these models to correct for this interaction. .. GENERATED FROM PYTHON SOURCE LINES 16-21 Setup ----- As usual, we’ll start with the standard imports … .. GENERATED FROM PYTHON SOURCE LINES 21-36 .. code-block:: Python import astropy.units as u import matplotlib.pyplot as plt from gammapy.catalog import SourceCatalog4FGL from gammapy.datasets import SpectrumDatasetOnOff from gammapy.estimators import FluxPointsEstimator from gammapy.modeling import Fit from gammapy.modeling.models import ( EBL_DATA_BUILTIN, EBLAbsorptionNormSpectralModel, GaussianPrior, PowerLawSpectralModel, SkyModel, ) .. GENERATED FROM PYTHON SOURCE LINES 37-50 Load the data ------------- We will use 6 observations of the blazars PKS 2155-304 taken in 2008 by H.E.S.S. when it was in a steady state. The data have already been reduced to OGIP format `SpectrumDatasetOnOff` following the procedure :doc:`/tutorials/analysis-1d/spectral_analysis` tutorial using a `ReflectedRegions` background estimation. The spectra and IRFs from the 6 observations have been stacked together. We will load this dataset as a `~gammapy.datasets.SpectrumDatasetOnOff` and proceed with the modeling. You can do a 3D analysis as well. .. GENERATED FROM PYTHON SOURCE LINES 50-58 .. code-block:: Python dataset = SpectrumDatasetOnOff.read( "$GAMMAPY_DATA/PKS2155-steady/pks2155-304_steady.fits.gz" ) print(dataset) .. rst-class:: sphx-glr-script-out .. code-block:: none SpectrumDatasetOnOff -------------------- Name : stacked Total counts : 119 Total background counts : 37.75 Total excess counts : 81.25 Predicted counts : 44.00 Predicted background counts : 44.00 Predicted excess counts : nan Exposure min : 3.80e+05 m2 s Exposure max : 2.68e+09 m2 s Number of total bins : 10 Number of fit bins : 8 Fit statistic type : wstat Fit statistic value (-2 log(L)) : 109.21 Number of models : 0 Number of parameters : 0 Number of free parameters : 0 Total counts_off : 453 Acceptance : 8 Acceptance off : 96 .. GENERATED FROM PYTHON SOURCE LINES 59-67 Model the observed spectrum --------------------------- The observed spectrum is already attenuated due to the EBL. Assuming that the intrinsic spectrum is a power law, the observed spectrum is a `gammapy.modeling.models.CompoundSpectralModel` given by the product of an EBL model with the intrinsic model. .. GENERATED FROM PYTHON SOURCE LINES 70-73 For a list of available models, see :doc:`/api/gammapy.modeling.models.EBL_DATA_BUILTIN`. .. GENERATED FROM PYTHON SOURCE LINES 73-76 .. code-block:: Python print(EBL_DATA_BUILTIN.keys()) .. rst-class:: sphx-glr-script-out .. code-block:: none dict_keys(['franceschini', 'dominguez', 'finke', 'franceschini17', 'saldana-lopez21']) .. GENERATED FROM PYTHON SOURCE LINES 77-82 To use other EBL models, you need to save the optical depth as a function of energy and redshift as an XSPEC model. Alternatively, you can use packages like `ebltable `_ which shows how to interface other EBL models with Gammapy. .. GENERATED FROM PYTHON SOURCE LINES 84-86 Define the power law .. GENERATED FROM PYTHON SOURCE LINES 86-102 .. code-block:: Python index = 2.3 amplitude = 1.81 * 1e-12 * u.Unit("cm-2 s-1 TeV-1") reference = 1 * u.TeV pwl = PowerLawSpectralModel(index=index, amplitude=amplitude, reference=reference) pwl.index.frozen = False # Specify the redshift of the source redshift = 0.116 # Load the EBL model. Here we use the model from Dominguez, 2011 absorption = EBLAbsorptionNormSpectralModel.read_builtin("dominguez", redshift=redshift) # The power-law model is multiplied by the EBL to get the final model spectral_model = pwl * absorption print(spectral_model) .. rst-class:: sphx-glr-script-out .. code-block:: none CompoundSpectralModel Component 1 : PowerLawSpectralModel type name value unit error min max frozen link prior ---- --------- ---------- -------------- --------- --- --- ------ ---- ----- index 2.3000e+00 0.000e+00 nan nan False amplitude 1.8100e-12 cm-2 s-1 TeV-1 0.000e+00 nan nan False reference 1.0000e+00 TeV 0.000e+00 nan nan True Component 2 : EBLAbsorptionNormSpectralModel type name value unit error min max frozen link prior ---- ---------- ---------- ---- --------- --- --- ------ ---- ----- alpha_norm 1.0000e+00 0.000e+00 nan nan True redshift 1.1600e-01 0.000e+00 nan nan True Operator : mul .. GENERATED FROM PYTHON SOURCE LINES 103-105 Now, create a sky model and proceed with the fit .. GENERATED FROM PYTHON SOURCE LINES 105-109 .. code-block:: Python sky_model = SkyModel(spatial_model=None, spectral_model=spectral_model, name="pks2155") dataset.models = sky_model .. GENERATED FROM PYTHON SOURCE LINES 110-114 Note that since this dataset has been produced by a reflected region analysis, it uses ON-OFF statistic and does not require a background model. .. GENERATED FROM PYTHON SOURCE LINES 114-124 .. code-block:: Python fit = Fit() result = fit.run(datasets=[dataset]) # we make a copy here to compare it later model_best = sky_model.copy() print(result.models.to_parameters_table()) .. rst-class:: sphx-glr-script-out .. code-block:: none model type name value unit ... min max frozen link prior ------- ---- ---------- ---------- -------------- ... --- --- ------ ---- ----- pks2155 index 2.5531e+00 ... nan nan False pks2155 amplitude 1.2978e-11 cm-2 s-1 TeV-1 ... nan nan False pks2155 reference 1.0000e+00 TeV ... nan nan True pks2155 alpha_norm 1.0000e+00 ... nan nan True pks2155 redshift 1.1600e-01 ... nan nan True .. GENERATED FROM PYTHON SOURCE LINES 125-131 Get the flux points =================== To get the observed flux points, just run the `~gammapy.estimators.FluxPointsEstimator` normally .. GENERATED FROM PYTHON SOURCE LINES 131-139 .. code-block:: Python energy_edges = dataset.counts.geom.axes["energy"].edges fpe = FluxPointsEstimator( energy_edges=energy_edges, source="pks2155", selection_optional="all" ) flux_points_obs = fpe.run(datasets=[dataset]) .. GENERATED FROM PYTHON SOURCE LINES 140-144 To get the deabsorbed flux points (ie, intrinsic points), we simply need to set the reference model to the best fit power law instead of the compound model. .. GENERATED FROM PYTHON SOURCE LINES 144-156 .. code-block:: Python flux_points_intrinsic = flux_points_obs.copy( reference_model=SkyModel(spectral_model=pwl) ) # print(flux_points_obs.reference_model) # print(flux_points_intrinsic.reference_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SkyModel Name : UwRL5RB- Datasets names : None Spectral model type : CompoundSpectralModel Spatial model type : Temporal model type : Parameters: index : 2.553 +/- 0.30 amplitude : 1.30e-11 +/- 1.9e-12 1 / (cm2 s TeV) reference (frozen): 1.000 TeV alpha_norm (frozen): 1.000 redshift (frozen): 0.116 SkyModel Name : xBC7Q-dz Datasets names : None Spectral model type : PowerLawSpectralModel Spatial model type : Temporal model type : Parameters: index : 2.553 +/- 0.30 amplitude : 1.30e-11 +/- 1.9e-12 1 / (cm2 s TeV) reference (frozen): 1.000 TeV .. GENERATED FROM PYTHON SOURCE LINES 157-160 Plot the observed and intrinsic fluxes -------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 160-184 .. code-block:: Python plt.figure() sed_type = "e2dnde" energy_bounds = [0.2, 20] * u.TeV ax = flux_points_obs.plot(sed_type=sed_type, label="observed", color="navy") flux_points_intrinsic.plot(ax=ax, sed_type=sed_type, label="intrinsic", color="red") model_best.spectral_model.plot( ax=ax, energy_bounds=energy_bounds, sed_type=sed_type, color="blue" ) model_best.spectral_model.plot_error( ax=ax, energy_bounds=energy_bounds, sed_type="e2dnde", facecolor="blue" ) pwl.plot(ax=ax, energy_bounds=energy_bounds, sed_type=sed_type, color="tomato") pwl.plot_error( ax=ax, energy_bounds=energy_bounds, sed_type=sed_type, facecolor="tomato" ) plt.ylim(bottom=1e-13) plt.legend() plt.show() # sphinx_gallery_thumbnail_number = 2 .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_ebl_001.png :alt: ebl :srcset: /tutorials/analysis-1d/images/sphx_glr_ebl_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 185-201 Further extensions ------------------ In this notebook, we have kept the parameters of the EBL model, the `alpha_norm` and the `redshift` frozen. Under reasonable assumptions on the intrinsic spectrum, it can be possible to constrain these parameters. Example: We now assume that the FermiLAT 4FGL catalog spectrum of the source is a good assumption of the intrinsic spectrum. *NOTE*: This is a very simplified assumption and in reality, EBL absorption can affect the Fermi spectrum significantly. Also, blazar spectra vary with time and long term averaged states may not be representative of a specific steady state .. GENERATED FROM PYTHON SOURCE LINES 201-211 .. code-block:: Python catalog = SourceCatalog4FGL() src = catalog["PKS 2155-304"] # Get the intrinsic model intrinsic_model = src.spectral_model() print(intrinsic_model) .. rst-class:: sphx-glr-script-out .. code-block:: none LogParabolaSpectralModel type name value unit error min max frozen link prior ---- --------- ---------- -------------- --------- --- --- ------ ---- ----- amplitude 1.2591e-11 cm-2 MeV-1 s-1 1.317e-13 nan nan False reference 1.1610e+03 MeV 0.000e+00 nan nan True alpha 1.7733e+00 1.029e-02 nan nan False beta 4.1893e-02 3.743e-03 nan nan False .. GENERATED FROM PYTHON SOURCE LINES 212-216 We add Gaussian priors on the `alpha` and `beta` parameters based on the 4FGL measurements and the associated errors. For more details on using priors, see :doc:`/tutorials/api/priors` .. GENERATED FROM PYTHON SOURCE LINES 216-225 .. code-block:: Python intrinsic_model.alpha.prior = GaussianPrior( mu=intrinsic_model.alpha.value, sigma=intrinsic_model.alpha.error ) intrinsic_model.beta.prior = GaussianPrior( mu=intrinsic_model.beta.value, sigma=intrinsic_model.beta.error ) .. GENERATED FROM PYTHON SOURCE LINES 226-228 As before, multiply the intrinsic model with the EBL model .. GENERATED FROM PYTHON SOURCE LINES 228-232 .. code-block:: Python obs_model = intrinsic_model * absorption .. GENERATED FROM PYTHON SOURCE LINES 233-235 Now, free the redshift of the source .. GENERATED FROM PYTHON SOURCE LINES 235-248 .. code-block:: Python obs_model.parameters["redshift"].frozen = False print(obs_model.parameters.to_table()) sky_model = SkyModel(spectral_model=obs_model, name="observed") dataset.models = sky_model result1 = fit.run([dataset]) print(result1.parameters.to_table()) .. rst-class:: sphx-glr-script-out .. code-block:: none type name value unit ... max frozen link prior ---- ---------- ---------- -------------- ... --- ------ ---- ------------- amplitude 1.2591e-11 cm-2 MeV-1 s-1 ... nan False reference 1.1610e+03 MeV ... nan True alpha 1.7733e+00 ... nan False GaussianPrior beta 4.1893e-02 ... nan False GaussianPrior alpha_norm 1.0000e+00 ... nan True redshift 1.1600e-01 ... nan False type name value unit ... max frozen link prior ---- ---------- ---------- -------------- ... --- ------ ---- ------------- amplitude 1.9692e-11 cm-2 MeV-1 s-1 ... nan False reference 1.1610e+03 MeV ... nan True alpha 1.7733e+00 ... nan False GaussianPrior beta 4.1896e-02 ... nan False GaussianPrior alpha_norm 1.0000e+00 ... nan True redshift 1.4338e-01 ... nan False .. GENERATED FROM PYTHON SOURCE LINES 249-255 Get a fit stat profile for the redshift --------------------------------------- For more information about stat profiles, see :doc:`/tutorials/api/fitting` .. GENERATED FROM PYTHON SOURCE LINES 255-279 .. code-block:: Python total_stat = result1.total_stat par = sky_model.parameters["redshift"] par.scan_max = par.value + 5.0 * par.error par.scan_min = max(0, par.value - 5.0 * par.error) par.scan_n_values = 31 # %time profile = fit.stat_profile( datasets=[dataset], parameter=sky_model.parameters["redshift"], reoptimize=True ) plt.figure() ax = plt.gca() ax.plot( profile["observed.spectral.model2.redshift_scan"], profile["stat_scan"] - total_stat ) ax.set_title("TS profile") ax.set_xlabel("Redshift") ax.set_ylabel("$\Delta$ TS") plt.show() .. image-sg:: /tutorials/analysis-1d/images/sphx_glr_ebl_002.png :alt: TS profile :srcset: /tutorials/analysis-1d/images/sphx_glr_ebl_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 280-282 We see that the redshift is well constrained. .. _sphx_glr_download_tutorials_analysis-1d_ebl.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/main?urlpath=lab/tree/notebooks/dev/tutorials/analysis-1d/ebl.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: ebl.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: ebl.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: ebl.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_