# Moffat1D¶

class astropy.modeling.functional_models.Moffat1D(amplitude=1, x_0=0, gamma=1, alpha=1, **kwargs)[source]

One dimensional Moffat model.

Parameters: Other Parameters: amplitude : float Amplitude of the model. x_0 : float x position of the maximum of the Moffat model. gamma : float Core width of the Moffat model. alpha : float Power index of the Moffat model. fixed : a dict, optional A dictionary {parameter_name: boolean} of parameters to not be varied during fitting. True means the parameter is held fixed. Alternatively the fixed property of a parameter may be used. tied : dict, optional A dictionary {parameter_name: callable} of parameters which are linked to some other parameter. The dictionary values are callables providing the linking relationship. Alternatively the tied property of a parameter may be used. bounds : dict, optional A dictionary {parameter_name: value} of lower and upper bounds of parameters. Keys are parameter names. Values are a list or a tuple of length 2 giving the desired range for the parameter. Alternatively, the min and max properties of a parameter may be used. eqcons : list, optional A list of functions of length n such that eqcons[j](x0,*args) == 0.0 in a successfully optimized problem. ineqcons : list, optional A list of functions of length n such that ieqcons[j](x0,*args) >= 0.0 is a successfully optimized problem.

Notes

Model formula:

$f(x) = A \left(1 + \frac{\left(x - x_{0}\right)^{2}}{\gamma^{2}}\right)^{- \alpha}$

Examples

import numpy as np
import matplotlib.pyplot as plt

from astropy.modeling.models import Moffat1D

plt.figure()
s1 = Moffat1D()
r = np.arange(-5, 5, .01)

for factor in range(1, 4):
s1.amplitude = factor
s1.width = factor
plt.plot(r, s1(r), color=str(0.25 * factor), lw=2)

plt.axis([-5, 5, -1, 4])
plt.show()


(png, svg, pdf) Attributes Summary

 alpha amplitude fwhm Moffat full width at half maximum. gamma input_units This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted). param_names x_0

Methods Summary

 evaluate(x, amplitude, x_0, gamma, alpha) One dimensional Moffat model function fit_deriv(x, amplitude, x_0, gamma, alpha) One dimensional Moffat model derivative with respect to parameters

Attributes Documentation

alpha = Parameter('alpha', value=1.0)
amplitude = Parameter('amplitude', value=1.0)
fwhm

Moffat full width at half maximum. Derivation of the formula is available in this notebook by Yoonsoo Bach.

gamma = Parameter('gamma', value=1.0)
input_units

This property is used to indicate what units or sets of units the evaluate method expects, and returns a dictionary mapping inputs to units (or None if any units are accepted).

Model sub-classes can also use function annotations in evaluate to indicate valid input units, in which case this property should not be overridden since it will return the input units based on the annotations.

param_names = ('amplitude', 'x_0', 'gamma', 'alpha')
x_0 = Parameter('x_0', value=0.0)

Methods Documentation

static evaluate(x, amplitude, x_0, gamma, alpha)[source]

One dimensional Moffat model function

static fit_deriv(x, amplitude, x_0, gamma, alpha)[source]

One dimensional Moffat model derivative with respect to parameters