Disk Spatial Model

This is a spatial model parametrising a disk.

By default, the model is symmetric, i.e. a disk:

\[\phi(lon, lat) = \frac{1}{2 \pi (1 - \cos{r_0}) } \cdot \begin{cases} 1 & \text{for } \theta \leq r_0 \ 0 & \text{for } \theta > r_0 \end{cases}\]

where \(\theta\) is the sky separation. To improve fit convergence of the model, the sharp edges is smoothed using erf.

In case an eccentricity (e) and rotation angle (\(\phi\)) are passed, then the model is an elongated disk (i.e. an ellipse), with a major semiaxis of length \(r_0\) and position angle \(\phi\) (increaing counter-clockwise from the North direction).

The model is defined on the celestial sphere, with a normalization defined by:

\[\int_{4\pi}\phi(\text{lon}, \text{lat}) \,d\Omega = 1\,.\]

Example plot

Here is an example plot of the model:

import numpy as np
from astropy.coordinates import Angle
from gammapy.modeling.models import (
    DiskSpatialModel,
    Models,
    PowerLawSpectralModel,
    SkyModel,
)

phi = Angle("30 deg")
model = DiskSpatialModel(
    lon_0="2 deg", lat_0="2 deg", r_0="1 deg", e=0.8, phi="30 deg", frame="galactic",
)

ax = model.plot(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(1.7, 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=0, ymax=5)
ax.text(2.15, 2.3, r"$\phi$", transform=transform)

YAML representation

Here is an example YAML file using the model:

pwl = PowerLawSpectralModel()
gauss = DiskSpatialModel()

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

print(models.to_yaml())

Out:

components:
-   name: pwl-disk-model
    type: SkyModel
    spectral:
        type: PowerLawSpectralModel
        parameters:
        - {name: index, value: 2.0, unit: '', min: .nan, max: .nan, frozen: false}
        - {name: amplitude, value: 1.0e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
            frozen: false}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true}
    spatial:
        type: DiskSpatialModel
        frame: icrs
        parameters:
        - {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false}
        - {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false}
        - {name: r_0, value: 1.0, unit: deg, min: 0.0, max: .nan, frozen: false}
        - {name: e, value: 0.0, unit: '', min: 0.0, max: 1.0, frozen: true}
        - {name: phi, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: true}
        - {name: edge, value: 0.01, unit: deg, min: 0.01, max: .nan, frozen: true}