Super Exponential Cutoff Power Law Model used for 4FGL

This model parametrises super exponential cutoff power-law model spectrum used for 4FGL.

It is defined by the following equation:

\[\phi(E) = \phi_0 \cdot \left(\frac{E}{E_0}\right)^{-\Gamma_1} \exp \left( a \left( E_0 ^{\Gamma_2} - E^{\Gamma_2} \right) \right)\]

See Equation (3) in https://arxiv.org/pdf/1902.10045.pdf

Example plot

Here is an example plot of the model:

from astropy import units as u
import matplotlib.pyplot as plt
from gammapy.modeling.models import (
    Models,
    SkyModel,
    SuperExpCutoffPowerLaw4FGLSpectralModel,
)

energy_range = [0.1, 100] * u.TeV
model = SuperExpCutoffPowerLaw4FGLSpectralModel(
    index_1=1,
    index_2=2,
    amplitude="1e-12 TeV-1 cm-2 s-1",
    reference="1 TeV",
    expfactor=1e-2,
)
model.plot(energy_range)
plt.grid(which="both")
plt.ylim(1e-24, 1e-10)

YAML representation

Here is an example YAML file using the model:

model = SkyModel(spectral_model=model, name="super-exp-cutoff-power-law-4fgl-model")
models = Models([model])

print(models.to_yaml())

Out:

components:
-   name: super-exp-cutoff-power-law-4fgl-model
    type: SkyModel
    spectral:
        type: SuperExpCutoffPowerLaw4FGLSpectralModel
        parameters:
        - {name: amplitude, value: 1.0e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
            frozen: false, error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
        - {name: expfactor, value: 0.01, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: index_1, value: 1.0, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: index_2, value: 2.0, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}