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Generalized gaussian spatial model

This is a spatial model parametrising a generalized Gaussian function.

By default, the Generalized Gaussian is defined as :

$$\phi(\text{lon}, \text{lat}) = \phi(\text{r}) = N \times \exp \left[ - \left( \frac{r}{r_{\rm eff}} \right)^ \left( 1/\eta \right) \right] \,,$$

the normalization is expressed as:

$$N = \frac{1}{ 2 \pi \sqrt(1-e^2) r_{\rm eff}^2 \eta \Gamma(2\eta)}\,$$

where $\Gamma$ is the gamma function. This analytical norm is approximated so it may not integrate to unity in extreme cases if ellipticity tend to one and radius is large or $\eta$ much larger than one (outside the default range).

Example plot

Here is an example plot of the model for different shape parameter:

from astropy import units as u
import matplotlib.pyplot as plt
from gammapy.maps import Map, WcsGeom
from gammapy.modeling.models import (
    GeneralizedGaussianSpatialModel,
    Models,
    PowerLawSpectralModel,
    SkyModel,
)

lon_0 = 20
lat_0 = 0
reval = 3
dr = 0.02
geom = WcsGeom.create(
    skydir=(lon_0, lat_0),
    binsz=dr,
    width=(2 * reval, 2 * reval),
    frame="galactic",
)

tags = [r"Disk, $\eta=0.01$", r"Gaussian, $\eta=0.5$", r"Laplace, $\eta=1$"]
eta_range = [0.01, 0.5, 1]
r_0 = 1
e = 0.5
phi = 45 * u.deg
fig, axes = plt.subplots(1, 3, figsize=(9, 6))
for ax, eta, tag in zip(axes, eta_range, tags):
    model = GeneralizedGaussianSpatialModel(
        lon_0=lon_0 * u.deg,
        lat_0=lat_0 * u.deg,
        eta=eta,
        r_0=r_0 * u.deg,
        e=e,
        phi=phi,
        frame="galactic",
    )
    meval = model.evaluate_geom(geom)
    Map.from_geom(geom=geom, data=meval.value, unit=meval.unit).plot(ax=ax)
    pixreg = model.to_region().to_pixel(geom.wcs)
    pixreg.plot(ax=ax, edgecolor="g", facecolor="none", lw=2)
    ax.set_title(tag)
    ax.set_xticks([])
    ax.set_yticks([])
plt.tight_layout()

YAML representation

Here is an example YAML file using the model:

pwl = PowerLawSpectralModel()
gengauss = GeneralizedGaussianSpatialModel()

model = SkyModel(spectral_model=pwl, spatial_model=gengauss, name="pwl-gengauss-model")
models = Models([model])

print(models.to_yaml())

Out:

components:
-   name: pwl-gengauss-model
    type: SkyModel
    spectral:
        type: PowerLawSpectralModel
        parameters:
        -   name: index
            value: 2.0
        -   name: amplitude
            value: 1.0e-12
            unit: cm-2 s-1 TeV-1
        -   name: reference
            value: 1.0
            unit: TeV
            frozen: true
    spatial:
        type: GeneralizedGaussianSpatialModel
        frame: icrs
        parameters:
        -   name: lon_0
            value: 0.0
            unit: deg
        -   name: lat_0
            value: 0.0
            unit: deg
        -   name: r_0
            value: 1.0
            unit: deg
        -   name: eta
            value: 0.5
        -   name: e
            value: 0.0
            frozen: true
        -   name: phi
            value: 0.0
            unit: deg
            frozen: true