The Moffat Function

Utilities for working with the Moffat function

The Moffat Function[14], a.k.a. the Moffat Profile, a.k.a. the Moffat Distribution, has the form

\[ I(x) = I_{pk}\left[ 1 + \frac{x^2}{\alpha^2}\right]^{-\beta} \]

With \(\beta=1\) it is the Lorentzian or Cauchy distribution. See also https://en.wikipedia.org/wiki/Moffat_distribution and https://en.wikipedia.org/wiki/Cauchy_distribution.

1-D and 2-D symmetric forms are provided. Utilities are provided for normalizing and calculating the full-width at half-maximum.

Functions
template<typename realT >
realT	mx::math::func::moffat (const realT x, const realT I0, const realT Ipk, const realT x0, const realT alpha, const realT beta)
	Find value at position (x) of the 1D arbitrarily-centered symmetric unnormalized Moffat function.
template<typename realT >
realT	mx::math::func::moffat2D (const realT x, const realT y, const realT I0, const realT Ipk, const realT x0, const realT y0, const realT alpha, const realT beta)
	Find value at position (x,y) of the 2D arbitrarily-centered unnormalized symmetric Moffat function.
template<typename realT >
realT	mx::math::func::moffatFWHM (realT alpha, realT beta)
	Compute the full-width at half-maximum of a Moffat profile.

Function Documentation

moffat()

template<typename realT >

realT mx::math::func::moffat	(	const realT	x,
		const realT	I0,
		const realT	Ipk,
		const realT	x0,
		const realT	alpha,
		const realT	beta
	)

Find value at position (x) of the 1D arbitrarily-centered symmetric unnormalized Moffat function.

The Moffat distribution is due to [14]. With \(\beta=1\) it is the Lorentzian or Cauchy distribution. See also https://en.wikipedia.org/wiki/Moffat_distribution and https://en.wikipedia.org/wiki/Cauchy_distribution.

Here we use the unnormalized general form, most useful for peak fitting.

This function computes:

\[ I(x) = I_0 + I_{pk}\left[ 1 + \frac{(x-x_0)^2}{\alpha^2}\right]^{-\beta} \]

Returns

the value of the 1D arbitrarily-centered unnormalized Moffat at (x)

Template Parameters

realT

is type to use for arithmetic

Test:

Scenario: compiling 1D Moffat function [test doc]

Parameters

[in]	x	is the x-position at which to evaluate the Moffat function
[in]	I0	is the constant to add to the Moffat function
[in]	Ipk	is the scaling factor (peak = A)
[in]	x0	is the x-coordinate of the center
[in]	alpha	is the width parameter of the Moffat function.
[in]	beta	is the shape parameter of the Moffat function

Definition at line 73 of file moffat.hpp.

References mx::math::six_fifths().

moffat2D()

template<typename realT >

realT mx::math::func::moffat2D	(	const realT	x,
		const realT	y,
		const realT	I0,
		const realT	Ipk,
		const realT	x0,
		const realT	y0,
		const realT	alpha,
		const realT	beta
	)

Find value at position (x,y) of the 2D arbitrarily-centered unnormalized symmetric Moffat function.

The Moffat distribution is due to [14]. With \(\beta=1\) it is the Lorentzian or Cauchy distribution. See also https://en.wikipedia.org/wiki/Moffat_distribution and https://en.wikipedia.org/wiki/Cauchy_distribution.

Here we use the unnormalized general form, most useful for peak fitting.

This function omputes:

\[ I(x) = I_0 + I_{pk}\left[ 1 + \frac{(x-x_0)^2 + (y-y_0)^2}{\alpha^2}\right]^{-\beta} \]

Returns

the value of the 2D arbitrarily-centered symmetric Moffat function at (x,y)

Template Parameters

realT

is type to use for arithmetic

Test:

Scenario: compiling 2D Moffat function [test doc]

Parameters

[in]	x	the x-position at which to evaluate the Moffat function
[in]	y	the y-positoin at which to evaluate the Moffat function
[in]	I0	the constant to add to the Moffat function
[in]	Ipk	the scaling factor (peak height is A-G0)
[in]	x0	the x-coordinate of the center
[in]	y0	the y-coordinate of the center
[in]	alpha	the width parameter of the Moffat function.
[in]	beta	the shape parameter of the Moffat function.

Definition at line 127 of file moffat.hpp.

References mx::math::six_fifths().

moffatFWHM()

template<typename realT >

realT mx::math::func::moffatFWHM	(	realT	alpha,
		realT	beta
	)

Compute the full-width at half-maximum of a Moffat profile.

This returns the value of

\[ FWHM = 2 \alpha \sqrt{2^{1/\beta} - 1} \]

Returns

the FWHM of the Moffat profile

Template Parameters

realT

is the type to use for arithmetic

Test:

Scenario: compiling Moffat FWHM [test doc]

Parameters

[in]	alpha	the width parameter of the Moffat function.
[in]	beta	the shape parameter of the Moffat function.

Definition at line 193 of file moffat.hpp.

References mx::math::six_fifths().

Referenced by mx::math::fit::fitMoffat1D< _realT >::fwhm(), and mx::math::fit::fitMoffat2D< fitterT >::fwhm().