Bpow - Broken Power Law

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(Fitting with Bpow: Edited description.)
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===Fitting with Bpow===
===Fitting with Bpow===
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There are four free parameters that are allowed to vary when using Bpow to fit the non-thermal portion of a spectrum (see parameters above). In practice, a[1], the negative power law index below the break energy is set to 1.5 and fixed. This is done because the broken power law is a phenomenological fit component. The non-thermal photon spectrum is due to the bremsstralung interaction of energetic electrons with plasma. The model of this interaction ([[Thick2 - Thick Target Bremsstrahlung Version 2|thick target]] or [[thin - Thin Target Bremsstrahlung|thin target]]) supposes that their is a low energy cutoff in the electron distribution function. This is necessary because the energy in the electrons becomes infinite otherwise. This cutoff means that there is a rollover in the resulting photon spectrum. In broken power law model this rollover is represented by the sharp change in slope at the break energy. The negative power law index of 1.5 below the break energy was arrived at by checking various values of the low energy cutoff in the electron spectrum and approximating the resulting slope of the photon spectrum. This value is not definitive, but it does prevent the energy in the electrons from becoming infinite.
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There are four free parameters that are allowed to vary when using Bpow to fit the non-thermal portion of a spectrum (see parameters above). In practice, a[1], the negative power law index below the break energy is set to 1.5 and fixed. This is done because the broken power law is a phenomenological fit component. The non-thermal photon spectrum is due to the bremsstralung interaction of energetic electrons with plasma. The model of this interaction ([[Thick2 - Thick Target Bremsstrahlung Version 2|thick target]] or [[thin - Thin Target Bremsstrahlung|thin target]]) supposes that their is a low energy cutoff in the electron distribution function. This is necessary because the energy in the electrons becomes infinite otherwise. This cutoff means that there is a rollover in the resulting photon spectrum. In the broken power-law model, this rollover is represented by the sharp change in slope at the break energy. The negative power-law index of 1.5 below the break energy was arrived at by checking the calculated slope of the photon spectrum below the electron cutoff energy. This value is not definitive. A better approximation of the photon spectrum from an electron spectrum with a low energy cutoff can be obtained using the thick or thin target function available in OSPEX.
The plot below shows an example of the broken power law in count flux (top panel) and photon flux (lower panel):
The plot below shows an example of the broken power law in count flux (top panel) and photon flux (lower panel):
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[[Image:bpow.jpg|center|thumb|600px|A broken power law fit to the count flux spectrum (top panel) and the photon flux spectrum (bottom panel)]]
[[Image:bpow.jpg|center|thumb|600px|A broken power law fit to the count flux spectrum (top panel) and the photon flux spectrum (bottom panel)]]
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In this case the break energy (a[2]) is ~15keV with a negative power law index above the break (a[3]) of 6.25. The normalization (a[0]) is 2.51, this value is determined by the value of the flux at 50keV supposing their is no break in the spectrum. Notice the different shapes of the two curves. This is due to detector effects. Altering the parameters of Bpow alters the shape of the non-thermal fit to the photon spectrum. The count rate model is the photon model convolved with the detector response matrix (DRM). The DRM accounts for detector effects on the spectrum thus changing the shape of the model.  
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In this case the break energy (a[2]) is ~15keV with a negative power law index above the break (a[3]) of 6.25. The normalization value (a[0]) of 2.51 photons cm^(-2) s^(-1) keV^(-1) is the flux at 50keV obtained from the low energy power-law component alone. The count-flux model is the photon model convolved with the detector response matrix (DRM). The DRM accounts for instrumental effects on the incident photon spectrum.  
*The broken power law is an approximation to the non-thermal photon spectrum
*The broken power law is an approximation to the non-thermal photon spectrum
*The negative power law index below the break, a[1], is usually fixed at 1.5, but this is not strictly necessary
*The negative power law index below the break, a[1], is usually fixed at 1.5, but this is not strictly necessary
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*The negative power law index above the break, a[3], contains information on the how hard the X-Ray spectrum is, the smaller the value the harder the spectrum
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*The negative power law index above the break, a[3], gives the hardness of the X-Ray spectrum, the smaller the value the harder the spectrum.
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**e.g. A value of 3 for a[3] means the spectrum is harder than a value of 5
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*Bpow does not contain information about the electron spectrum that gives rise to the non-thermal photon spectrum, for instance the  energy in the electrons cannot be calculated from Bpow.
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Revision as of 22:57, 9 September 2009

Contents

Broken Power Law

Introduction

The broken power law component models the non-thermal portion of the spectrum for a selected time interval during a flare. It fits the spectrum above approximately 25keV with a broken power law, normalized to the value of the count flux at 50keV.

Parameters

Fitting with Bpow

There are four free parameters that are allowed to vary when using Bpow to fit the non-thermal portion of a spectrum (see parameters above). In practice, a[1], the negative power law index below the break energy is set to 1.5 and fixed. This is done because the broken power law is a phenomenological fit component. The non-thermal photon spectrum is due to the bremsstralung interaction of energetic electrons with plasma. The model of this interaction (thick target or thin target) supposes that their is a low energy cutoff in the electron distribution function. This is necessary because the energy in the electrons becomes infinite otherwise. This cutoff means that there is a rollover in the resulting photon spectrum. In the broken power-law model, this rollover is represented by the sharp change in slope at the break energy. The negative power-law index of 1.5 below the break energy was arrived at by checking the calculated slope of the photon spectrum below the electron cutoff energy. This value is not definitive. A better approximation of the photon spectrum from an electron spectrum with a low energy cutoff can be obtained using the thick or thin target function available in OSPEX.

The plot below shows an example of the broken power law in count flux (top panel) and photon flux (lower panel):

A broken power law fit to the count flux spectrum (top panel) and the photon flux spectrum (bottom panel)

In this case the break energy (a[2]) is ~15keV with a negative power law index above the break (a[3]) of 6.25. The normalization value (a[0]) of 2.51 photons cm^(-2) s^(-1) keV^(-1) is the flux at 50keV obtained from the low energy power-law component alone. The count-flux model is the photon model convolved with the detector response matrix (DRM). The DRM accounts for instrumental effects on the incident photon spectrum.

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