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Shell 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 and $r_{\text{out}} = r_{\text{in}}$ + width

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 model:

from gammapy.modeling.models import (
    Models,
    PowerLawSpectralModel,
    ShellSpatialModel,
    SkyModel,
)

model = ShellSpatialModel(
    lon_0="10 deg",
    lat_0="20 deg",
    radius="2 deg",
    width="0.5 deg",
    frame="galactic",
)

model.plot(add_cbar=True)

YAML representation

Here is an example YAML file using the model:

pwl = PowerLawSpectralModel()
shell = ShellSpatialModel()

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

print(models.to_yaml())

Out:

components:
-   name: pwl-shell-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: ShellSpatialModel
        frame: icrs
        parameters:
        -   name: lon_0
            value: 0.0
            unit: deg
        -   name: lat_0
            value: 0.0
            unit: deg
        -   name: radius
            value: 1.0
            unit: deg
        -   name: width
            value: 0.2
            unit: deg