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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, min: .nan, max: .nan,
            frozen: false, error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
        - {name: alpha, value: 2.3, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: beta, value: 0.5, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}