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r"""
.. _shell2-spatial-model:
Shell2 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, :math:`r_{\text{out}}` is the outer radius
and :math:`r_{\text{in}}` is the inner radius.
For Shell2SpatialModel, the radius parameter r_0 correspond to :math:`r_{\text{out}}`.
The relative width parameter, eta, is given as \eta = :math:`(r_{\text{out}} - r_{\text{in}})/r_{\text{out}}`
so we have :math:`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())
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