ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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 knowns 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 \((\alpha, \beta)\). More... | |
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 \((\alpha, \beta)\). More... | |
interface | getBandUDF |
Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution. More... | |
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 \((\alpha, \beta)\). More... | |
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). More... | |
interface | setBandMean |
Generate and return the mean of the Band distribution for an input set of parameters. More... | |
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). More... | |
interface | setBandUCDF |
Generate and return the unnormalized cumulative distribution function (UCDF) of the Band spectral model/distribution. More... | |
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... | |
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 knowns 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 4 parameters:
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,
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}
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.
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.
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).
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.
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.
Definition at line 118 of file pm_distBand.F90.
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.
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.
Definition at line 130 of file pm_distBand.F90.
character(*, SK), parameter pm_distBand::MODULE_NAME = "@pm_distBand" |
Definition at line 103 of file pm_distBand.F90.