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r"""
.. _shell-spatial-model:
Shell spatial model
===================
This is a spatial model parametrizing a projected radiating shell.
The shell spatial model is defined by the following equations:
.. math::
\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}
where :math:`\theta` is the sky separation and :math:`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())
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