Gaussian spatial model¶
This is a spatial model parametrising a Gaussian function.
By default, the Gaussian is symmetric:
where \(\theta\) is the sky separation to the model center. In this case, the Gaussian is normalized to 1 on the sphere:
In the limit of small \(\theta\) and \(\sigma\), this definition reduces to the usual form:
In case an eccentricity (\(e\)) and rotation angle (\(\phi\)) are passed, then the model is an elongated Gaussian, whose evaluation is performed as in the symmetric case but using the effective radius of the Gaussian:
Here, \(\sigma_M\) (\(\sigma_m\)) is the major (minor) semiaxis of the Gaussian, and
\(\Delta \phi\) is the difference between
phi, the position angle of the Gaussian, and the
position angle of the evaluation point.
Caveat: For the asymmetric Gaussian, the model is normalized to 1 on the plane, i.e. in small angle approximation: \(N = 1/(2 \pi \sigma_M \sigma_m)\). This means that for huge elongated Gaussians on the sky this model is not correctly normalized. However, this approximation is perfectly acceptable for the more common case of models with modest dimensions: indeed, the error introduced by normalizing on the plane rather than on the sphere is below 0.1% for Gaussians with radii smaller than ~ 5 deg.
Here is an example plot of the model:
import numpy as np from astropy.coordinates import Angle from gammapy.maps import WcsGeom from gammapy.modeling.models import ( GaussianSpatialModel, Models, PowerLawSpectralModel, SkyModel, ) phi = Angle("30 deg") model = GaussianSpatialModel( lon_0="2 deg", lat_0="2 deg", sigma="1 deg", e=0.7, phi=phi, frame="galactic", ) geom = WcsGeom.create( skydir=model.position, frame=model.frame, width=(4, 4), binsz=0.02 ) ax = model.plot(geom=geom, add_cbar=True) # illustrate size parameter region = model.to_region().to_pixel(ax.wcs) artist = region.as_artist(facecolor="none", edgecolor="red") ax.add_artist(artist) transform = ax.get_transform("galactic") ax.scatter(2, 2, transform=transform, s=20, edgecolor="red", facecolor="red") ax.text(1.5, 1.85, r"$(l_0, b_0)$", transform=transform, ha="center") ax.plot([2, 2 + np.sin(phi)], [2, 2 + np.cos(phi)], color="r", transform=transform) ax.vlines(x=2, color="r", linestyle="--", transform=transform, ymin=-5, ymax=5) ax.text(2.25, 2.45, r"$\phi$", transform=transform)
Here is an example YAML file using the model:
components: - name: pwl-gauss-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: GaussianSpatialModel frame: icrs parameters: - name: lon_0 value: 0.0 unit: deg - name: lat_0 value: 0.0 unit: deg - name: sigma value: 1.0 unit: deg - name: e value: 0.0 frozen: true - name: phi value: 0.0 unit: deg frozen: true