Shell2 spatial model

This is a spatial model parametrizing a projected radiating shell.

The shell spatial model is defined by the following equations:

\[\begin{split}\phi(lon, lat) = \frac{3}{2 \pi (r_{out}^3 - r_{in}^3)} \cdot \begin{cases} \sqrt{r_{out}^2 - \theta^2} - \sqrt{r_{in}^2 - \theta^2} & \text{for } \theta \lt r_{in} \\ \sqrt{r_{out}^2 - \theta^2} & \text{for } r_{in} \leq \theta \lt r_{out} \\ 0 & \text{for } \theta > r_{out} \end{cases}\end{split}\]

where \(\theta\) is the sky separation, \(r_{\text{out}}\) is the outer radius and \(r_{\text{in}}\) is the inner radius.

For Shell2SpatialModel, the radius parameter r_0 correspond to \(r_{\text{out}}\). The relative width parameter, eta, is given as eta = \((r_{\text{out}} - r_{\text{in}})/r_{\text{out}}\) so we have \(r_{\text{in}} = (1-\eta) r_{\text{out}}\).

Note that the normalization is a small angle approximation, although that approximation is still very good even for 10 deg radius shells.

Example plot

Here is an example plot of the shell model for the parametrization using outer radius and relative width. In this case the relative width, eta, acts as a shape parameter.

import matplotlib.pyplot as plt
from gammapy.modeling.models import (
    Models,
    PowerLawSpectralModel,
    Shell2SpatialModel,
    SkyModel,
)

tags = [
    r"Disk-like, $\eta \rightarrow 0$",
    r"Shell, $\eta=0.25$",
    r"Peaked, $\eta\rightarrow 1$",
]
eta_range = [0.001, 0.25, 1]
fig, axes = plt.subplots(1, 3, figsize=(9, 6))
for ax, eta, tag in zip(axes, eta_range, tags):
    model = Shell2SpatialModel(
        lon_0="10 deg",
        lat_0="20 deg",
        r_0="2 deg",
        eta=eta,
        frame="galactic",
    )
    model.plot(ax=ax)
    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()
shell2 = Shell2SpatialModel()

model = SkyModel(spectral_model=pwl, spatial_model=shell2, name="pwl-shell2-model")

models = Models([model])

print(models.to_yaml())

components:
-   name: pwl-shell2-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: Shell2SpatialModel
        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: 0.2