ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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pm_distBand Module Reference

This module contains procedures and generic interfaces for computing the Band photon distribution widely used in modeling the spectra of a class of celestial objects known as Gamma-Ray Bursts.
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Data Types

type  distBand_type
 This is the derived type for signifying distributions that are of type Band as defined in the description of pm_distBand. More...
 
interface  getBandEbreak
 Generate and return the spectral break energy parameter of the Band spectral model/distribution from the corresponding spectral peak energy \(\epeak\) and the Band model spectral indices \((\alpha, \beta)\).
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interface  getBandEpeak
 Generate and return the spectral peak energy parameter of the Band spectral model/distribution from the corresponding spectral break energy \(\ebreak\) and the Band model spectral indices \((\alpha, \beta)\).
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interface  getBandUDF
 Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.
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interface  getBandZeta
 Generate and return the coefficient of continuity of the Band spectral model/distribution from the Band model parameters: the break energy \(\ebreak\) and the Band model spectral indices \((\alpha, \beta)\).
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interface  setBandEnergy
 Generate and return the energy integral (the energy fluence in units of the input break energy) of the Band model for the given distribution parameters from the corresponding photon integral of the distribution (the photon fluence in units of photon counts).
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interface  setBandMean
 Generate and return the mean of the Band distribution for an input set of parameters.
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interface  setBandPhoton
 Generate and return the photon integral (the photon fluence in units of photon counts) of the Band model for the given distribution parameters from the corresponding energy integral of the distribution (the energy fluence in units of the input break energy).
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interface  setBandUCDF
 Generate and return the unnormalized cumulative distribution function (UCDF) of the Band spectral model/distribution.
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Variables

character(*, SK), parameter MODULE_NAME = "@pm_distBand"
 
real(RKB), parameter MEAN_ALPHA = -1.1_RKB
 The scalar constant of type real of kind RKB, containing the average reported value for the low-energy exponent of the Band photon distribution model \(\alpha\).
This reported average value is primarily based on the analyses of data collected by the BATSE telescope onboard the Compton Gamma-Ray Observatory (CGRO).
More...
 
real(RKB), parameter MEAN_BETA = -2.3_RKB
 The scalar constant of type real of kind RKB, containing the average reported value for the high-energy exponent of the Band photon distribution model \(\beta\).
This reported average value is primarily based on the analyses of data collected by the BATSE telescope onboard the Compton Gamma-Ray Observatory (CGRO).
More...
 

Detailed Description

This module contains procedures and generic interfaces for computing the Band photon distribution widely used in modeling the spectra of a class of celestial objects known as Gamma-Ray Bursts.

The Band model is an empirical spectral model most widely used to fit GRB spectra, first proposed in Band et al. 1993, BATSE Observations of Gamma-Ray Burst Spectra. I. Spectral Diversity.

The model is continuously differentiable (i.e., its derivative is a continuous function).
It is characterized by four parameters:

  1. the amplitude \(A\) in \(\ms{photons} \ms{s}^{-1} \ms{cm}^{-2} \kev^{-1}\),
  2. a low-energy spectral index \(\alpha\),
  3. a high-energy spectral index \(\beta < \alpha\),
  4. a \(\nu F_\nu\) peak energy \(\epeak\) in units of \(\kev\) (or equivalently, the break energy, \(\ebreak\)).

The \(\nu F_\nu\) is the photon spectrum \(f(E)\) integrated twice over all energies ( \(E^2f(E)\)).
Therefore, \(\nu F_\nu\) represents the total energy flux per energy band (i.e., power density spectrum).
The \(\alpha\) index characterizes an asymptotic power-law (i.e., the tangential slope determined at \(E\rightarrow 0\) in a logarithmic scale).
This may not characterize the actual low-energy power-law, determined within the data energy range when the e-folding energy denoted by \(E_{0}\) approaches the lower energy bound.
Although the model was originally constructed based on the observed time-integrated BATSE catalog spectra, it has now become common practice to use the model to fit time-resolved GRB spectra as well.
There are, however, some time-resolved spectra that cannot be adequately fitted with this model.
The Band model has the following mathematical form,

\begin{equation} \large f_{\ms{BAND}}(E) = \begin{cases} A\left(\frac{E}{100\kev}\right)^\alpha \exp\left(-\frac{\alpha - \beta}{\ebreak}E\right) &,~ \ms{if} & E < \ebreak \\ A\left[\frac{\ebreak}{100\kev}\right]^{\alpha - \beta} \exp\left(\beta - \alpha\right) \left(\frac{E}{100\kev}\right)^\beta &,~ \ms{if} & E \geq \ebreak \\ \end{cases} \end{equation}

where \(\ebreak = \frac{\alpha - \beta}{2 + \alpha}\epeak = (\alpha - \beta)\efold\) is called the break energy of the model.

The above formulation takes a unit-full input value for the energy, \(E\), at which the spectrum must be computed.
Assuming the normalization energy is unity (i.e., \(1\kev\) instead of \(100\kev\) without loss of generality), the normalized bounded unit-less formulation, corresponding to the probability density function (PDF) for arbitrary \((\alpha, \beta)\)takes the form,

\begin{equation} \large \pi_{\ms{BAND}}(x | \alpha, \beta, \xbreak) = \eta(\alpha, \beta, \xbreak) \begin{cases} x^\alpha \exp\left(-\frac{x}{\sigma}\right) &,~ \ms{if} & 0 < \ms{lb} \leq x < \xbreak \\ \zeta x^\beta &,~ \ms{if} & \xbreak \leq x < \ms{ub} < +\infty \\ \end{cases} \end{equation}

where,

  1. \(x = \frac{E}{\kev}\),
  2. \(\xbreak = \frac{\ebreak}{\kev}\),
  3. \(\sigma = \frac{\efold}{\kev} = \frac{(\alpha - \beta)\kev}{\xbreak}\) is the normalized e-folding energy, a measure of the scale of the distribution below and above which the distribution approaches power-law behavior with exponents \(\alpha\) and \(\beta\) respectively,
  4. the factor \(\eta(\alpha, \beta, \xbreak)\) is a normalization constant that properly normalizes the PDF,
  5. the factor \(\zeta = \xbreak^{\alpha - \beta} \exp\left(\beta - \alpha\right)\) is the coefficient of continuity of the distribution that makes the distribution continuously differentiable.
  6. the constants \((\ms{lb}, \ms{ub})\) represent the lower and upper bounds of the PDF, respectively.

When the condition \(\alpha > -1\) holds, the lower component of the distribution follows the mathematical form of the PDF of the Gamma distribution,

\begin{eqnarray} \large x^\alpha \exp\left(-\frac{x}{\sigma}\right) &=& \sigma^\alpha \left(\frac{x}{\sigma}\right)^\alpha \exp\left(-\frac{x}{\sigma}\right) \\ &=& \sigma^{\alpha + 1}\Gamma(\alpha + 1)\pi_\mathcal{G}(x | \kappa = \alpha + 1, \sigma) ~, \end{eqnarray}

See also
Kaneko, 2005, Spectral studies of gamma-ray burst prompt emission.
Band et al. 1993, BATSE Observations of Gamma-Ray Burst Spectra. I. Spectral Diversity.
Eqn. A6 in Shahmoradi and Nemiroff, 2015, MNRAS, 451:4645-4662.
Test:
test_pm_distBand


Final Remarks


If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

  1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Fatemeh Bagheri, Tuesday, April 30, 2019, 12:58 PM, SEIR, UTA

Variable Documentation

◆ MEAN_ALPHA

real(RKB), parameter pm_distBand::MEAN_ALPHA = -1.1_RKB

The scalar constant of type real of kind RKB, containing the average reported value for the low-energy exponent of the Band photon distribution model \(\alpha\).
This reported average value is primarily based on the analyses of data collected by the BATSE telescope onboard the Compton Gamma-Ray Observatory (CGRO).

Warning
This average value makes the Band distribution unnormalized because the integral of the PDF does not converge for \(x\rightarrow0\).


Final Remarks


If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

  1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Fatemeh Bagheri, Tuesday, April 30, 2019, 12:58 PM, SEIR, UTA

Definition at line 118 of file pm_distBand.F90.

◆ MEAN_BETA

real(RKB), parameter pm_distBand::MEAN_BETA = -2.3_RKB

The scalar constant of type real of kind RKB, containing the average reported value for the high-energy exponent of the Band photon distribution model \(\beta\).
This reported average value is primarily based on the analyses of data collected by the BATSE telescope onboard the Compton Gamma-Ray Observatory (CGRO).


Final Remarks


If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

  1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Fatemeh Bagheri, Tuesday, April 30, 2019, 12:58 PM, SEIR, UTA

Definition at line 130 of file pm_distBand.F90.

◆ MODULE_NAME

character(*, SK), parameter pm_distBand::MODULE_NAME = "@pm_distBand"

Definition at line 103 of file pm_distBand.F90.