This is a fixed-text formatted version of a Jupyter notebook

Flux point fitting in Gammapy

Introduction

In this tutorial we’re going to learn how to fit spectral models to combined Fermi-LAT and IACT flux points.

The central class we’re going to use for this example analysis is:

In addition we will work with the following data classes:

And the following spectral model classes:

Setup

Let us start with the usual IPython notebook and Python imports:

[1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
[2]:
from astropy import units as u
from gammapy.modeling.models import (
    PowerLaw,
    ExponentialCutoffPowerLaw,
    LogParabola,
)
from gammapy.spectrum import FluxPointsDataset, FluxPoints
from gammapy.catalog import (
    SourceCatalog3FGL,
    SourceCatalogGammaCat,
    SourceCatalog3FHL,
)
from gammapy.modeling import Fit

Load spectral points

For this analysis we choose to work with the source ‘HESS J1507-622’ and the associated Fermi-LAT sources ‘3FGL J1506.6-6219’ and ‘3FHL J1507.9-6228e’. We load the source catalogs, and then access source of interest by name:

[3]:
fermi_3fgl = SourceCatalog3FGL()
fermi_3fhl = SourceCatalog3FHL()
gammacat = SourceCatalogGammaCat("$GAMMAPY_DATA/gamma-cat/gammacat.fits.gz")
[4]:
source_gammacat = gammacat["HESS J1507-622"]
source_fermi_3fgl = fermi_3fgl["3FGL J1506.6-6219"]
source_fermi_3fhl = fermi_3fhl["3FHL J1507.9-6228e"]

The corresponding flux points data can be accessed with .flux_points attribute:

[5]:
flux_points_gammacat = source_gammacat.flux_points
flux_points_gammacat.table
[5]:
Table length=6
e_refdndednde_errndnde_errp
TeV1 / (cm2 s TeV)1 / (cm2 s TeV)1 / (cm2 s TeV)
float32float32float32float32
0.86090042.29119e-128.705427e-138.955021e-13
1.5615126.981717e-132.203541e-132.304066e-13
2.7637531.690615e-136.758698e-147.188384e-14
4.8915977.729249e-142.401318e-142.607487e-14
9.9885841.032534e-145.063147e-155.641954e-15
27.040357.449867e-165.72089e-167.259987e-16

In the Fermi-LAT catalogs, integral flux points are given. Currently the flux point fitter only works with differential flux points, so we apply the conversion here.

[6]:
flux_points_3fgl = source_fermi_3fgl.flux_points.to_sed_type(
    sed_type="dnde", model=source_fermi_3fgl.spectral_model
)
flux_points_3fhl = source_fermi_3fhl.flux_points.to_sed_type(
    sed_type="dnde", model=source_fermi_3fhl.spectral_model
)

Finally we stack the flux points into a single FluxPoints object and drop the upper limit values, because currently we can’t handle them in the fit:

[7]:
# stack flux point tables
flux_points = FluxPoints.stack(
    [flux_points_gammacat, flux_points_3fhl, flux_points_3fgl]
)

# drop the flux upper limit values
flux_points = flux_points.drop_ul()

Power Law Fit

First we start with fitting a simple power law.

[8]:
pwl = PowerLaw(index=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV")

After creating the model we run the fit by passing the 'flux_points' and 'pwl' objects:

[9]:
dataset_pwl = FluxPointsDataset(pwl, flux_points, likelihood="chi2assym")
fitter = Fit(dataset_pwl)
result_pwl = fitter.run()

And print the result:

[10]:
print(result_pwl)
OptimizeResult

        backend    : minuit
        method     : minuit
        success    : True
        message    : Optimization terminated successfully.
        nfev       : 40
        total stat : 33.68

[11]:
print(pwl)
PowerLaw

Parameters:

           name     value   error      unit      min max frozen
        --------- --------- ----- -------------- --- --- ------
            index 1.966e+00   nan                nan nan  False
        amplitude 1.345e-12   nan cm-2 s-1 TeV-1 nan nan  False
        reference 1.000e+00   nan            TeV nan nan   True

Finally we plot the data points and the best fit model:

[12]:
ax = flux_points.plot(energy_power=2)
pwl.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
pwl.parameters.covariance = result_pwl.parameters.covariance

pwl.plot_error(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)
ax.set_ylim(1e-13, 1e-11);
../_images/notebooks_sed_fitting_gammacat_fermi_22_0.png

Exponential Cut-Off Powerlaw Fit

Next we fit an exponential cut-off power law to the data.

[13]:
ecpl = ExponentialCutoffPowerLaw(
    index=1.8,
    amplitude="2e-12 cm-2 s-1 TeV-1",
    reference="1 TeV",
    lambda_="0.1 TeV-1",
)

We run the fitter again by passing the flux points and the ecpl model instance:

[14]:
dataset_ecpl = FluxPointsDataset(ecpl, flux_points, likelihood="chi2assym")
fitter = Fit(dataset_ecpl)
result_ecpl = fitter.run()
print(ecpl)
ExponentialCutoffPowerLaw

Parameters:

           name     value   error      unit      min max frozen
        --------- --------- ----- -------------- --- --- ------
            index 1.869e+00   nan                nan nan  False
        amplitude 2.126e-12   nan cm-2 s-1 TeV-1 nan nan  False
        reference 1.000e+00   nan            TeV nan nan   True
          lambda_ 1.000e-01   nan          TeV-1 nan nan  False

We plot the data and best fit model:

[15]:
ax = flux_points.plot(energy_power=2)
ecpl.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
ecpl.parameters.covariance = result_ecpl.parameters.covariance

ecpl.plot_error(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)
ax.set_ylim(1e-13, 1e-11)
[15]:
(1e-13, 1e-11)
../_images/notebooks_sed_fitting_gammacat_fermi_28_1.png

Log-Parabola Fit

Finally we try to fit a log-parabola model:

[16]:
log_parabola = LogParabola(
    alpha=2, amplitude="1e-12 cm-2 s-1 TeV-1", reference="1 TeV", beta=0.1
)
[17]:
dataset_log_parabola = FluxPointsDataset(
    log_parabola, flux_points, likelihood="chi2assym"
)
fitter = Fit(dataset_log_parabola)
result_log_parabola = fitter.run()
print(log_parabola)
LogParabola

Parameters:

           name     value   error      unit      min max frozen
        --------- --------- ----- -------------- --- --- ------
        amplitude 1.930e-12   nan cm-2 s-1 TeV-1 nan nan  False
        reference 1.000e+00   nan            TeV nan nan   True
            alpha 2.151e+00   nan                nan nan  False
             beta 5.259e-02   nan                nan nan  False
[18]:
ax = flux_points.plot(energy_power=2)
log_parabola.plot(energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2)

# assign covariance for plotting
log_parabola.parameters.covariance = result_log_parabola.parameters.covariance

log_parabola.plot_error(
    energy_range=[1e-4, 1e2] * u.TeV, ax=ax, energy_power=2
)
ax.set_ylim(1e-13, 1e-11);
../_images/notebooks_sed_fitting_gammacat_fermi_32_0.png

Exercises

  • Fit a PowerLaw2 and ExponentialCutoffPowerLaw3FGL to the same data.
  • Fit a ExponentialCutoffPowerLaw model to Vela X (‘HESS J0835-455’) only and check if the best fit values correspond to the values given in the Gammacat catalog

What next?

This was an introduction to SED fitting in Gammapy.

  • If you would like to learn how to perform a full Poisson maximum likelihood spectral fit, please check out the spectrum analysis tutorial.
  • To learn more about other parts of Gammapy (e.g. Fermi-LAT and TeV data analysis), check out the other tutorial notebooks.
  • To see what’s available in Gammapy, browse the Gammapy docs or use the full-text search.
  • If you have any questions, ask on the mailing list .
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