Note

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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 :

ϕ (lon, lat) = ϕ (r) = N \times \exp [- {(\frac{r}{r_{e f f}})}^{(1 / η)}],

the normalization is expressed as:

N = \frac{1}{2 π \sqrt{(} 1 - e^{2}) r_{0}^{2} η Γ (2 η)}

where $Γ$ 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 $η$ much larger than one (outside the default range).

The effective radius is given by:

r_{r m e f f} (lon, lat) = \sqrt{(r_{M} \sin (Δ ϕ))^{2} + (r_{m} \cos (Δ ϕ))^{2}} .

where $r_{M}$ ( $r_{m}$ ) is the major (minor) semiaxis, and $Δ ϕ$ is the difference between phi, the position angle of the model, and the position angle of the evaluation point. If the eccentricity ( $e$ ) is null it reduces to $r_{0}$ .

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()

$Disk, $\eta=0.01$, Gaussian, $\eta=0.5$, Laplace, $\eta=1$$

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())

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
    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
        -   name: phi
            value: 0.0
            unit: deg
metadata:
    creator: Gammapy 1.3
    date: '2024-11-26T10:08:39.908583'
    origin: null

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