GaussianSpatialModel

class gammapy.modeling.models.GaussianSpatialModel(lon_0, lat_0, sigma, e=0, phi='0 deg', frame='icrs')

Bases: gammapy.modeling.models.SpatialModel

Two-dimensional Gaussian model.

By default, the Gaussian is symmetric:

\[\phi(\text{lon}, \text{lat}) = N \times \exp\left\{-\frac{1}{2} \frac{1-\cos \theta}{1-\cos \sigma}\right\}\,,\]

where \(\theta\) is the sky separation to the model center. In this case, the Gaussian is normalized to 1 on the sphere:

\[N = \frac{1}{4\pi a\left[1-\exp(-1/a)\right]}\,,\,\,\,\, a = 1-\cos \sigma\,.\]

In the limit of small \(\theta\) and \(\sigma\), this definition reduces to the usual form:

\[\phi(\text{lon}, \text{lat}) = \frac{1}{2\pi\sigma^2} \exp{\left(-\frac{1}{2} \frac{\theta^2}{\sigma^2}\right)}\,.\]

In case an eccentricity (\(e\)) and rotation angle (\(\phi\)) are passed, then the model is an elongated Gaussian, whose evaluation is performed as in the symmetric case but using the effective radius of the Gaussian:

\[\sigma_{eff}(\text{lon}, \text{lat}) = \sqrt{ (\sigma_M \sin(\Delta \phi))^2 + (\sigma_m \cos(\Delta \phi))^2 }.\]

Here, \(\sigma_M\) (\(\sigma_m\)) is the major (minor) semiaxis of the Gaussian, and \(\Delta \phi\) is the difference between phi, the position angle of the Gaussian, and the position angle of the evaluation point.

Caveat: For the asymmetric Gaussian, the model is normalized to 1 on the plane, i.e. in small angle approximation: \(N = 1/(2 \pi \sigma_M \sigma_m)\). This means that for huge elongated Gaussians on the sky this model is not correctly normalized. However, this approximation is perfectly acceptable for the more common case of models with modest dimensions: indeed, the error introduced by normalizing on the plane rather than on the sphere is below 0.1% for Gaussians with radii smaller than ~ 5 deg.

Parameters:

lon_0, lat_0 : Angle

Center position

sigma : Angle

Length of the major semiaxis of the Gaussian, in angular units.

e : float

Eccentricity of the Gaussian (\(0< e< 1\)).

phi : Angle

Rotation angle \(\phi\): of the major semiaxis. Increases counter-clockwise from the North direction.

frame : {“icrs”, “galactic”}

Center position coordinate frame

Examples

import numpy as np
import matplotlib.pyplot as plt
from astropy.coordinates import Angle
from gammapy.modeling.models import GaussianSpatialModel
from gammapy.maps import Map, WcsGeom

m_geom = WcsGeom.create(
    binsz=0.01, width=(5, 5), skydir=(2, 2), coordsys="GAL", proj="AIT"
)
phi = Angle("30 deg")
model = GaussianSpatialModel("2 deg", "2 deg", "1 deg", 0.7, phi, frame="galactic")

coords = m_geom.get_coord()
vals = model(coords.lon, coords.lat)
skymap = Map.from_geom(m_geom, data=vals.value)

_, ax, _ = skymap.smooth("0.05 deg").plot()

transform = ax.get_transform("galactic")
ax.scatter(2, 2, transform=transform, s=20, edgecolor="red", facecolor="red")
ax.text(1.5, 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=-5, ymax=5)
ax.text(2.25, 2.45, r"$\phi$", transform=transform)
ax.contour(skymap.data, cmap="coolwarm", levels=10, alpha=0.6)

plt.show()

(png, hires.png, pdf)

Attributes Summary

e
evaluation_radius	Evaluation radius (Angle).
frame
lat_0
lon_0
parameters	Parameters (Parameters)
phi
position	Spatial model center position
sigma
tag

Methods Summary

__call__(self, lon, lat)	Call evaluate method
copy(self)	A deep copy.
create(tag, \*args, \*\*kwargs)	Create a model instance.
evaluate(lon, lat, lon_0, lat_0, sigma, e, phi)	Evaluate model.
evaluate_geom(self, geom)	Evaluate model on Geom.
from_dict(data)
to_dict(self)

Attributes Documentation

e

evaluation_radius

Evaluation radius (Angle).

Set as \(5\sigma\).

frame

lat_0

lon_0

parameters

Parameters (Parameters)

phi

position

Spatial model center position

sigma

tag = 'GaussianSpatialModel'

Methods Documentation

__call__(self, lon, lat)

Call evaluate method

copy(self)

A deep copy.

static create(tag, *args, **kwargs)

Create a model instance.

Examples

>>> from gammapy.modeling import Model
>>> spectral_model = Model.create("PowerLaw2SpectralModel", amplitude="1e-10 cm-2 s-1", index=3)
>>> type(spectral_model)
gammapy.modeling.models.spectral.PowerLaw2SpectralModel

static evaluate(lon, lat, lon_0, lat_0, sigma, e, phi)

Evaluate model.

evaluate_geom(self, geom)

Evaluate model on Geom.

classmethod from_dict(data)

to_dict(self)