Log Parabola Spectral Model

This model parametrises a log parabola spectrum.

It is defined by the following equation:

\[\phi(E) = \phi_0 \left( \frac{E}{E_0} \right) ^ { - \alpha - \beta \log{ \left( \frac{E}{E_0} \right) } }\]

Note that \(log\) refers to the natural logarithm. This is consistent with the Fermi Science Tools and ctools. The Sherpa package, however, uses \(log_{10}\). If you have parametrization based on \(log_{10}\) you can use the from_log10() method.

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 LogParabolaSpectralModel, Models, SkyModel

energy_range = [0.1, 100] * u.TeV
model = LogParabolaSpectralModel(
    alpha=2.3, amplitude="1e-12 cm-2 s-1 TeV-1", reference=1 * u.TeV, beta=0.5,
)
model.plot(energy_range)
plt.grid(which="both")

YAML representation

Here is an example YAML file using the model:

model = SkyModel(spectral_model=model, name="log-parabola-model")
models = Models([model])

print(models.to_yaml())

Out:

components:
-   name: log-parabola-model
    type: SkyModel
    spectral:
        type: LogParabolaSpectralModel
        parameters:
        -   name: amplitude
            value: 1.0e-12
            unit: cm-2 s-1 TeV-1
        -   name: reference
            value: 1.0
            unit: TeV
            frozen: true
        -   name: alpha
            value: 2.3
        -   name: beta
            value: 0.5