Shell2D

class gammapy.image.models.Shell2D(amplitude, x_0, y_0, r_in, width=None, r_out=None, normed=True, **constraints)

Bases: astropy.modeling.Fittable2DModel

Projected homogeneous radiating shell model.

This model can be used for a shell type SNR source morphology.

Parameters:

amplitude : float

Value of the integral of the shell function.

x_0 : float

x position center of the shell

y_0 : float

y position center of the shell

r_in : float

Inner radius of the shell

width : float

Width of the shell

r_out : float (optional)

Outer radius of the shell

normed : bool (True)

If set the amplitude parameter corresponds to the integral of the function. If not set the ‘amplitude’ parameter corresponds to the peak value of the function (value at \(r = r_{in}\)).

See also

Sphere2D, Delta2D, astropy.modeling.models.Gaussian2D

Notes

Model formula with integral normalization:

\[\begin{split}f(r) = A \frac{3}{2 \pi (r_{out}^3 - r_{in}^3)} \cdot \left \{ \begin{array}{ll} \sqrt{r_{out}^2 - r^2} - \sqrt{r_{in}^2 - r^2} & : r < r_{in} \\ \sqrt{r_{out}^2 - r^2} & : r_{in} \leq r \leq r_{out} \\ 0 & : r > r_{out} \end{array} \right.\end{split}\]

Model formula with peak normalization:

\[\begin{split}f(r) = A \frac{1}{\sqrt{r_{out}^2 - r_{in}^2}} \cdot \left \{ \begin{array}{ll} \sqrt{r_{out}^2 - r^2} - \sqrt{r_{in}^2 - r^2} & : r < r_{in} \\ \sqrt{r_{out}^2 - r^2} & : r_{in} \leq r \leq r_{out} \\ 0 & : r > r_{out} \end{array} \right.\end{split}\]

With \(r_{out} = r_{in} + \mathrm{width}\).

Examples

import numpy as np
import matplotlib.pyplot as plt
from gammapy.image.models import Shell2D

shell = Shell2D(amplitude=100, x_0=25, y_0=25, r_in=10, width=5)
y, x = np.mgrid[0:50, 0:50]
plt.imshow(shell(x, y), origin='lower', interpolation='none')
plt.xlabel('x (pix)')
plt.ylabel('y (pix)')
plt.colorbar(label='Brightness (A.U.)')
plt.grid(False)
plt.show()

(Source code, png, hires.png, pdf)

Attributes Summary

amplitude
param_names
r_in
width
x_0
y_0

Methods Summary

evaluate(x, y, amplitude, x_0, y_0, r_in, width)	Two dimensional Shell model function normed to integral
evaluate_peak_norm(x, y, amplitude, x_0, ...)	Two dimensional Shell model function normed to peak value

Attributes Documentation

amplitude

param_names = ('amplitude', 'x_0', 'y_0', 'r_in', 'width')

r_in

width

x_0

y_0

Methods Documentation

static evaluate(x, y, amplitude, x_0, y_0, r_in, width)

Two dimensional Shell model function normed to integral

static evaluate_peak_norm(x, y, amplitude, x_0, y_0, r_in, width)

Two dimensional Shell model function normed to peak value