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)

Out:

/Users/terrier/Code/anaconda3/envs/gammapy-dev/lib/python3.7/site-packages/astropy/units/quantity.py:1032: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  out = super().__getitem__(key)
/Users/terrier/Code/anaconda3/envs/gammapy-dev/lib/python3.7/site-packages/astropy/units/quantity.py:1054: FutureWarning: Using a non-tuple sequence for multidimensional indexing is deprecated; use `arr[tuple(seq)]` instead of `arr[seq]`. In the future this will be interpreted as an array index, `arr[np.array(seq)]`, which will result either in an error or a different result.
  self.view(np.ndarray).__setitem__(i, self._to_own_unit(value))

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, unit: '', min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: amplitude, value: 1.0e-12, unit: cm-2 s-1 TeV-1, min: .nan, max: .nan,
            frozen: false, error: 0}
        - {name: reference, value: 1.0, unit: TeV, min: .nan, max: .nan, frozen: true,
            error: 0}
    spatial:
        type: ShellSpatialModel
        frame: icrs
        parameters:
        - {name: lon_0, value: 0.0, unit: deg, min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: lat_0, value: 0.0, unit: deg, min: -90.0, max: 90.0, frozen: false,
            error: 0}
        - {name: radius, value: 1.0, unit: deg, min: .nan, max: .nan, frozen: false,
            error: 0}
        - {name: width, value: 0.2, unit: deg, min: .nan, max: .nan, frozen: false,
            error: 0}