# Disk spatial model#

This is a spatial model parametrising a disk.

By default, the model is symmetric, i.e. a disk:

$\begin{split}\phi(lon, lat) = \frac{1}{2 \pi (1 - \cos{r_0}) } \cdot \begin{cases} 1 & \text{for } \theta \leq r_0 \\ 0 & \text{for } \theta > r_0 \end{cases}\end{split}$

where $$\theta$$ is the sky separation. To improve fit convergence of the model, the sharp edges is smoothed using erf.

In case an eccentricity (e) and rotation angle ($$\phi$$) are passed, then the model is an elongated disk (i.e. an ellipse), with a major semiaxis of length $$r_0$$ and position angle $$\phi$$ (increasing counter-clockwise from the North direction).

The model is defined on the celestial sphere, with a normalization defined by:

$\int_{4\pi}\phi(\text{lon}, \text{lat}) \,d\Omega = 1\,.$

## Example plot#

Here is an example plot of the model:

import numpy as np
from astropy.coordinates import Angle
from gammapy.modeling.models import (
DiskSpatialModel,
Models,
PowerLawSpectralModel,
SkyModel,
)

phi = Angle("30 deg")
model = DiskSpatialModel(
lon_0="2 deg",
lat_0="2 deg",
r_0="1 deg",
e=0.8,
phi=phi,
edge_width=0.1,
frame="galactic",
)

# illustrate size parameter
region = model.to_region().to_pixel(ax.wcs)
artist = region.as_artist(facecolor="none", edgecolor="red")

transform = ax.get_transform("galactic")
ax.scatter(2, 2, transform=transform, s=20, edgecolor="red", facecolor="red")
ax.text(1.7, 1.85, r"$(l_0, b_0)$", transform=transform, ha="center")
ax.plot([2, 2 + np.sin(phi)], [2, 2 + np.cos(phi)], color="r", transform=transform)
ax.vlines(x=2, color="r", linestyle="--", transform=transform, ymin=0, ymax=5)
ax.text(2.15, 2.3, r"$\phi$", transform=transform)


This plot illustrates the definition of the edge parameter:

import numpy as np
from astropy import units as u
from astropy.visualization import quantity_support
import matplotlib.pyplot as plt
from gammapy.modeling.models import DiskSpatialModel

lons = np.linspace(0, 0.3, 500) * u.deg

r_0, edge_width = 0.2 * u.deg, 0.5

disk = DiskSpatialModel(lon_0="0 deg", lat_0="0 deg", r_0=r_0, edge_width=edge_width)
profile = disk(lons, 0 * u.deg)

plt.plot(lons, profile / profile.max(), alpha=0.5)
plt.ylabel("Profile (A.U.)")

edge_min, edge_max = r_0 * (1 - edge_width / 2.0), r_0 * (1 + edge_width / 2.0)
with quantity_support():
plt.vlines([edge_min, edge_max], 0, 1, linestyles=["--"], color="k")
plt.annotate(
"",
xy=(edge_min, 0.5),
xytext=(edge_min + r_0 * edge_width, 0.5),
arrowprops=dict(arrowstyle="<->", lw=2),
)
plt.text(0.2, 0.53, "Edge width", ha="center", size=12)
margin = 0.02 * u.deg
plt.hlines(
[0.95], edge_min - margin, edge_min + margin, linestyles=["-"], color="k"
)
plt.text(edge_min + margin, 0.95, "95%", size=12, va="center")
plt.hlines(
[0.05], edge_max - margin, edge_max + margin, linestyles=["-"], color="k"
)
plt.text(edge_max - margin, 0.05, "5%", size=12, va="center", ha="right")
plt.show()


## YAML representation#

Here is an example YAML file using the model:

pwl = PowerLawSpectralModel()
gauss = DiskSpatialModel()

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

print(models.to_yaml())


Out:

components:
-   name: pwl-disk-model
type: SkyModel
spectral:
type: PowerLawSpectralModel
parameters:
-   name: index
value: 2.0
is_norm: false
-   name: amplitude
value: 1.0e-12
unit: cm-2 s-1 TeV-1
is_norm: true
-   name: reference
value: 1.0
unit: TeV
frozen: true
is_norm: false
spatial:
type: DiskSpatialModel
frame: icrs
parameters:
-   name: lon_0
value: 0.0
unit: deg
is_norm: false
-   name: lat_0
value: 0.0
unit: deg
is_norm: false
-   name: r_0
value: 1.0
unit: deg
is_norm: false
-   name: e
value: 0.0
frozen: true
is_norm: false
-   name: phi
value: 0.0
unit: deg
frozen: true
is_norm: false
-   name: edge_width
value: 0.01
frozen: true
is_norm: false


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