Verication and evaluation of the DREAM model for subphotospheric dissipation in prompt GRB emission

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Verification and evaluation of the DREAM model for

subphotospheric dissipation in prompt GRB emission

Author:

Erik Ahlberg (900725-5079)

eahlb@kth.se

Department of Physics

Royal Institute of Technology (KTH)

Supervisor:

Josefin Larsson

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Typeset in LA_TEX

ISRN KTH/FYS/– – 16:45 – – SE

ISSN 0280-316X

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Abstract

Gamma-ray bursts (GRBs) are among the most energetic events in the universe, lasting only seconds and for that brief time outshining all other gamma-ray sources in the uni-verse. However, after decades of studies the emission processes of the prompt gamma-ray emission phase is still not well understood. A suggested model is the Dissipation with Radiative Emission as A table Model (DREAM). In this thesis a series of systematic tests were performed to test different implementations the DREAM model. The aim was to quantify different kinds of uncertainties in the model, and make suggestions for improvements.

The models were tested for their dependence on signal-to-noise ratio, how the interpo-lation of spectra affect the result, and any degeneracies. An optimum signal-to-noise ratio of 40_{− 100 was found. The systematic errors due to using interpolated spectra} were found to be 10 _{− 15 % in most of the parameter space and to never exceed the} estimated uncertainties arising from assumptions made in the physical scenario. In more than _{∼ 80 % of the parameter space, no strong degeneracies were found. The degenerate} region is characterized by a small fraction of the dissipated energy going to the electrons.

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Chapter 1 Introduction

Gamma-ray bursts (GRBs) are among the most energetic events in the universe, lasting only seconds and for that brief time outshining all other gamma-ray sources in the uni-verse. They have been extensively studied since their discovery in 1967. However, after decades of studies the emission processes of the prompt gamma-ray emission phase and the nature of the progenitor are still not well understood.

Several models have been suggested for the prompt emission phase. Among these is the Dissipation with Radiative Emission as A table Model (DREAM). The DREAM model is based on numerical simulations by Pe’er & Waxman (2005). The initial black-body spectra is broadened by some dissipation process occurring below the photosphere. The DREAM model is implemented as an xspec table model. These consist of a grid of model spectra that have been numerically calculated for each of the parameters. During the data fitting, xspec will interpolate over the grid to find the spectrum that provides the best fit. This method of calculating the model spectra in advance allows us to make relatively fast fits, despite the fact that the numerical calculations for finding the model spectra are computationally expensive. These models are very common in X-ray astronomy, but have not before been used in the gamma-ray band.

The purpose of this thesis is to verify and evaluate the DREAM model. In this thesis a series of systematic tests are performed to test the model on two criteria. First, do the fitted parameters of the table model predict the correct value? Second, is the uncertainty of the prediction within reasonable limits? The aim is to quantify different kinds of uncertainties in the model, and make suggestions for improvements.

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1.1 Outline of the thesis

In chapter 2 a theoretical background to gamma-ray bursts (GRBs), with emphasis on the prompt emission phase, is given. This chapter also describes the physical scenario and assumptions made in the model for subphotospheric dissipation considered in this thesis. In chapter 3, current instruments used for GRB observations are mentioned with a more detailed description of the Fermi Gamma-ray Space Telescope. The basics of time-resolved spectral analysis are also described.

In chapter 4 the basics of spectral fitting is discussed, concluding with a description of the Monte Carlo method developed in this thesis. Chapter 5 describes the methods used for evaluating the simulations and artificially modifying the signal-to-noise ratio.

Chapters 6–8 details the tests performed on the model. In chapter 6 the performance of the model on the grid-points of the model is tested. In chapter 7 the interpolation of the model spectra is tested. In chapter 8 the model is tested for degeneracies.

In chapter 9 the Monte Carlo method developed in this thesis is used to calculate the fitted parameters of an observed GRB.

In chapter 10 the results of chapters 6–8 are summarized and discussed.

1.2 Author’s contribution

In this thesis, a Monte Carlo based method for finding confidence intervals has been implemented, along with an extensive framework for verifying and evaluating an xspec table model. This framework includes all the data analysis in chapters 6-8, and the algorithm for automatically evaluating the simulations described in chapter 5. All testing and data analysis in chapters 6-8 were performed by the author. The program used to find the confidence intervals of GRB observations used in chapter 9 was developed by the author and is based on the Monte Carlo method.

The code for numerical calculations and creating table models, and GBM data were provided to the author.

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Chapter 2 Gamma-ray bursts

Gamma-ray bursts (GRBs) are among the most energetic events in the universe, last-ing only seconds and for that brief time outshinlast-ing all other gamma-ray sources in the universe. The first GRB was discovered in 1967 by the United State’s Vela spacecraft. However, the discovery wasn’t made public by the military until six years later (Klebe-sadel et al. 1973). In the past decades, several mission designed for the study of GRBs have been launched, these include the Dutch-Italian BeppoSAX, and NASA’s Compton Gamma Ray Observatory, Swift Gamma-Ray Burst Mission and most recently Fermi Gamma-ray Space Telescope. However, after decades of studies the emission processes of the prompt gamma-ray emission phase and the nature of the progenitor are still not well understood.

In this chapter I will describe some important fundamental properties of GRBs followed by a more detailed account of the prompt emission. Finally the physical scenario and implementation of the model investigated in thesis are described.

2.1 Fundamental properties

Early discoveries of GRBs only consisted of observations in the gamma-ray band. Later observations identifying emission at lower energies completed the picture of GRB emis-sion, with emission ranging from the highly energetic prompt phase in gamma-rays to the afterglow in X-ray, optical, infra-red and radio (van Paradijs et al. 1997). The first observation of X-rays associated with a GRB was made in 1997 by BeppoSAX a few hours after the initial gamma-ray emission. The longer duration X-ray emission made accurate measurement of the burst position possible, enabling follow-up observations in at lower energy bands (Costa et al. 1997). Identification of the host galaxy through spectroscopic observations in the optical spectrum suggested that the burst was indeed extra-galactic in origin (van Paradijs et al. 1997), something previously suspected due to the isotropic distribution of detected GRBs (Meegan et al. 1992). This claim has since then been confirmed by extensive observational evidence (Jakobsson et al. 2006).

The large distances to the observed GRBs, combined with the high observed fluences, suggest an isotropic energy release of the order 1054 _{erg (Kulkarni et al. 1999). This}

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Figure 2.1: A sample of GRB prompt emission light-curves observed by Swift Gamma-Ray Burst Mission with burst duration times ranging from milliseconds up to tens of seconds. Created by Daniel Perley with data from the public BATSE archive (http://gammaray.msfc.nasa.gov/batse/grb/catalog/).

energy is higher than the rest mass energy of a neutron star and is incompatible with current models for stellar deaths. The energy requirement can be reduced to the order of 1051_{erg if the emission is collimated in jets, instead of arising from a spherical surface} (Frail et al. 2001). This reduced energy is comparable to that released in a supernova. The idea that the outflow is jet-like is supported by the existence of late time breaks in the afterglow light curves (Fruchter et al. 1999, Kulkarni et al. 1999, Frail et al. 2001) and is now a part the preferred model for GRBs.

The nature of the progenitor objects are not well understood. There are two basic requirements imposed by the observational evidence. First, it must be able to produce a huge amount of energy _{∼ 10}51_{erg. Second, it must be able to explain the rapid} time variability observed. The variability in the light curves of some bursts lie in the millisecond range (figure 2.1). This implies, via a light-crossing argument, a radius of the fireball in the order of 107_{cm, compared with the radius of a neutron star in the order} of 106_cm.

Typically GRBs are divided in two classes, short and long, by the time T90, in which 90 % of the fluence is detected. Short-duration bursts (T90 < 2 s) are thought to be associated with the merger of compact objects (Lee & Ramirez-Ruiz 2007),while the long-duration bursts (T90 > 2 s) are thought to be associated with core-collapse of massive stars (Galama et al. 1998, Woosley et al. 2003, Cano et al. 2014). It is important to note that this somewhat arbitrary definition of T90 is highly dependent on the observation method used. It does not take into account the energy range observed, the shapes of the different light-curves, or the different redshifts of the bursts.

The most common model for describing GRBs is the so-called fireball model. In this model a huge amount of energy is released in a small region, likely gravitational potential

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energy released from a collapsing star. The released energy accelerates a relativistic expanding electron-photon fireball, eventually producing the prompt emission. The high energy release in electromagnetic-energy is accompanied by emission of neutrinos and gravitational waves. These two dominant modes of energy release are so far undetected.

2.2 Prompt emission

The brief, transient nature of the GRB prompt emission has made detailed studies diffi-cult. The low sensitivity of the detectors meant that for many years the time-integrated spectrum was the only means of examining the spectrum of the emission. This time-integrated spectrum will have a higher photon count and thus better statistics. However, since the time-integrated spectrum does not follow the time-evolution of the burst, im-portant time dependent features of the spectra may be lost or distorted, which may lead to the wrong theoretical conclusions. The improved sensitivity of recent satellites, most notably the Fermi Gama-ray Space Telescope has made the use of time-resolved spectral analysis more wide-spread. With time-resolved spectral analysis and the decom-position of the spectrum into different components, the time-evolution of each individual component in the spectrum can be followed.

GRB spectra cover a very broad energy spectrum, extending to very high energies in the MeV range. These high energies above the pair production peak at 511 keV suggest a relativistic expansion velocity of the outflow.

The most common model for describing the prompt GRB spectra is the Band model (Band et al. 1993). This is a purely analytical model with four free parameters – the low/high energy spectral slope, break energy and an overall normalization factor. Due to the empirical nature of the model, there is no preferred emission model. This means that simply fitting the Band model to data provides little insight in to the physical processes driving the emission. In order to deduce the physical properties of the emission, the fitted parameters must be interpreted within some physical model. Furthermore, since there are only four free parameters, it is not able to reproduce some of the complex spectral behavior observed. There are several similar models to the Band model, such as the broken power-law and smoothed broken power-law. These models produce similar results to the Band model.

The spectral slopes predicted by the Band model, especially the low energy slope, are not easily explained by any simple broad-band radiation process, such as synchrotron or synchrotron self-Compton (Preece et al. 1998). In particular, synchrotron radiation cannot by itself explain the observed spectrum in a large fraction of bursts. This result is consitent with the findings in Axelsson & Borgonovo (2015), who examined the full width at half maximum of GRB spectra fitted with Band functions. It was found that _{∼ 80 %} of the spectra were too narrow to be explained by the slow cooling sychrotron model, while for the fast cooling model the fraction was _{∼ 100 %. Physically motivated models} incorporating slow-cooled synchrotron radiation, with (Burgess et al. 2014) and without (Zhang et al. 2016) a black-body component, have been found to provide acceptable fits to some observed GRBs.

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Modifications to the Band model has been suggested. A hybrid model consisting a black-body component which provides a physical explanation for a part of the spectrum as thermal emission with a temperature∼ 100 kev (Ryde 2004, Iyyani et al. 2013) and a single power-law to account for the non-thermal component when considering a limited energy range. This power-law+black-body model still offer no physical explanation for the non-thermal component. Other suggestions include a Band function with an addi-tional black-body component. This Band+black-body model has provided good fits to observed data with a peak energy of the black-body lower then the Band function energy (Axelsson et al. 2012, Guiriec et al. 2013).

It is important to note that these modified Band functions does not predict the same peak energy as the simple Band model. This further illustrate the fact that the spectral analysis is model dependent and caution must be used when interpreting the result. The fitted Band function does not typically resemble a Planck spectrum, which predict much harder spectral slopes and narrow spectra. This is the reason to why the photosh-peric model was overlooked in the very early stages of GRB studies, even though some early observation were well described by photospheric emission (Ghirlanda et al. 2003, Ryde 2004). Interest in this model has, however, been renewed with the realization that broadening of the spectrum is possible (Rees & M´esz´aros 2005), along with the inability of synchrotron models in explaining the observed spectra.

Subphotospheric dissipation has been suggested as the explanation for some observed GRBs (Ryde et al. 2011, Iyyani et al. 2013). In the subphotospheric model the non-thermal spectrum found by the Band model is explained by broadening of the initial Planck spectrum, through dissipation of energy below the photosphere. Other theories, such as geometrical broadening, have also been shown to broaden the initial Planck spec-trum (Lundman et al. 2013). In this scenario the broadening is due to angle-dependency of the outflow Lorentz factor in collimated jets, combined with varying Doppler factors for different parts of the observed photosphere.

2.3 Dissipation with radiative emission as a table

model

A proposed model for the subphotospheric model is the Dissipation with Radiative Emis-sion as A table Model (DREAM). The DREAM model is based on the fireball model, with numerical simulations performed by the code described in Pe’er & Waxman (2005). A first proof-of-concept of fitting such a model showed that the DREAM model using four free parameters provides acceptable fits to some GRB data (Ahlgren et al. 2015). In this thesis a new version of this model is investigated. The physical scenario and assumptions of this new model is detailed in section 2.3.1 and an explanation to how the table model works are given in section 2.3.2.

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r₀ r_s r_d 2r_d rphot re

R

1

η Γ( R )

Jet

acceleration

Numerical

calculations

Spectrum

assumed

constant

Photons

released

Jet

deceleration

Figure 2.2: The evolution of the bulk Lorentz factor Γ as a function the fireball radius with key epochs noted. The evolution is followed from the nozzle radius r0, which lie very close to

the explosion, up to the point where all of the jet’s kinetic energy have been lost by shocks with the surrounding interstellar medium.

2.3.1 Physical scenario

The DREAM model is based on the fireball model, where a large amount of gravitational potential energy is released in a very short time. For comprehensive reviews on the fireball model, see e.g. M´esz´aros (2006) or Pe’er (2015).

In the fireball model, a large fraction of the gravitational energy is converted into ki-netic energy of the collimated relativistic jet of protons, electrons, positrons and pho-tons. Since the observed luminosity of the bursts far exceed the Eddington luminosity ∼ 1038_(M/M

) erg s−1 the fireball will initially expand and accelerate. The bulk Lorentz factor is limited by the energy available in the co-moving frame and has a maximum of Γmax = L0/ ˙M c2 = E/M c2 ≡ η where L0 is the luminosity in the co-moving frame, and η is the specific entropy per baryon. Note that L0 is not the observed luminosity. Since GRBs are observed with high Γ_{∼ 10}2 _{only a small fraction of the baryons in the initial} star are accelerated. The fireball will continue to expand until the baryon kinetic energy Γ(r)M c2 _{is comparable with the total energy released in the explosion at Γ(r)} _{≈ η at} the saturation radius rs ∼ ηr0 where r0 is the nozzle radius. The nozzle radius lie very close to the explosion itself. After the saturation radius most of the energy is in the form of kinetic energy and the flow cannot accelerate further. Instead it coast at a constant velocity Γ = η.

The numerical calculations assume the existence of a dissipation radius rd= L0σT 4πτ c3_Γ3_m p = rphot τ , (2.1)

where σT is the Thompson cross-section, τ the optical depth, mp the proton rest mass, and rphot is the photospheric radius. At the dissipation radius a fraction of the kinetic energy is dissipated by some unspecified dissipation process such as internal shocks or magnetic re-connection.

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Note that the version of the DREAM model in Ahlgren et al. (2015) assumes internal shocks as the dissipation process, thus giving r0 = rdΓ−2. In the new model, no assump-tions on dissipation mechanism is made, and the nozzle radius is an input parameter. The nozzle radius is set to r0 = 107cm, giving a saturation radius rs ∼ 1010cm. The assumed nozzle radius is the initial fireball radius implied by the light-crossing argument. The evolution of the plasma is calculated over one dynamical time tdyn = rd/c, or in terms of radii from r = rdto r = 2rd. A fraction e of the jet kinetic energy is transferred to electrons that take a Maxwellian distribution, and a fraction b is transferred to the magnetic fields. The method through which the energy is dissipated is not specified. This dissipated energy is used to accelerate particles that radiate energy in the form of photon that ultimately form the observed spectrum. The photons are released at the photosphere rphot where for this model of subphotosperic dissipation we assume rs < rd ≤ rphot. For simulations where 2rd< rphot, the interactions for radii 2rd< r < rphot are assumed have little effect on the final spectrum.

The kinetic energy in the jet not dissipated below the photosphere collides with the surrounding inter-stellar medium at re, resulting in external shocks that produce the afterglow. The afterglow is not described by the numerical simulations, since it does not follow the evolution of the plasma beyond the photosphere.

The evolution of the bulk Lorentz factor of the jet as a function of the fireball radius with relevant epochs noted is shown in figure 2.2.

The key component to the DREAM model is the numerical calculations during one dy-namical time at the dissipation radius. I will here state the assumptions made in the calulations, for the detialed calculations see Pe’er & Waxman (2005). The fireball is assumed to consist of protons, electrons, positrons and photons with isotropic and homo-geneous distribution permeated by a time-independent magnetic field. Some dissipation process taking place during one dynamical time that produces energetic electrons at a constant rate is assumed to exist. During the dynamical time no photons escape the fireball.

The photons, electrons and positrons in the plasma interact through pair production and annihilation, Compton scattering, synchrotron and synchrotron self-absorption in-teractions. All these interactions occur simultaneously during the dynamical time. In principle Coulomb interactions are also possible. These interactions are however negligi-ble compared to other interactions and is thus ignored (Pe’er & Waxman 2005).

The calculations are performed in the co-moving frame and transformed to the observer frame by the redshift z and bulk Lorentz factor Γ. For all calculations the nozzle radius is fixed to r0 = 107cm.

2.3.2 Table model implementation

A N -dimensional table model consist of a grid of model spectra that have been calculated for each of the N parameters. During the data fitting, xspec will interpolate over the grid to find the spectra for the set of parameters that is required at that point in the process. This method of calculating the model spectra in advance allows us to make relatively

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fast fits, despite the fact that the numerical calculations for finding the model spectra are computationally expensive. The table models are, however, significantly slower than the built-in models provided by xspec since the memory requirement increase as 2N_. Models with more than 3 or 4 parameters are not recommended.

Table models have been used for a long time and are very common in X-ray astronomy, e.g. the reflionx model for accretion disks in active galactic nuclei (Ross & Fabian 2005). Before the DREAM model, table models have not been used for observations in the gamma-ray spectrum.

The table model allows for two different interpolation methods, linear and logarithmic, for each parameter. It is probable that neither of these methods describe the true underlying distribution in the parameters. The interpolation may cause problems with the standard xspec Levenberg-Marquadt fitting algorithm that relies on the second derivatives. In the DREAM implementation five parameters are used, the optical depth τ , the fraction of the jet kinetic energy transferred to the electrons and magnetic field e and b, the luminosity in the co-moving frame L = L010−52, and the bulk Lorentz factor of the jet Γ. These parameters are selected to represent a physically motivated part of the parameter space. The parameters span the following space

τ : 1 5 10 35

e : 0.01 0.05 0.1 0.2 0.3 0.5 b : 10−6 0.01 0.05

L : 1 10 100

Γ : 100 150 200 250 300 400 .

The parameter value b = 10−6represent a scenario where the contribution of synchrotron radiation is negligible. Values higher than b = 0.05 are not part of the model due to computational constraints.

In addition to the five free parameters there are two fixed parameters; the redshift z and a overall normalization factor n. For each fit these parameters are constant, with the normalization set to correct for the lower observed flux for distant bursts. The redshift also shift the spectrum. The correct normalization as a function of the redshift is given by n(z) = dL(1) dL(z) 2 , (2.2) where dL(z) = (1 + z) c H0 Z z 0 ΩM(1 + z0)3+ ΩK(1 + z0)2 + Ωλ −1/2 dz0 (2.3)

is the luminosity distance at redshift z, c is the speed of light, H0 = 69.9 km s−1Mpc−1 is the Hubble constant, ΩM = 0.289, Ωλ = 0.714 and ΩK = 1− ΩM − Ωλ = 0.

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Chapter 3 Observational techniques

Since gamma-rays are absorbed in the Earth’s atmosphere we must use satellite mounted instruments to observe the prompt emission of GRBs. The most recent such satellite currently in use is the Fermi Gamma-ray Space Telescope1_{, which I will briefly describe} in section 3.1. Other current satellites include NASA’s Swift2_{, ESA’s INTEGRAL}3 _and the Russian KONUS4_.

High altitude balloon mission are also able to directly observe gamma-rays. These are, however, due to the relatively short flight-time not suitable for GRB observations. It is also possible to indirectly observe high energy gamma-rays, _{≥ 10 GeV through} ground based Cherenkov telescopes such as the HESS5_{. No GRB has been observed by} a Cherenkov telescope to date.

With time resolved spectral analysis the burst is divided in many time-bins, instead of using the full integrated spectrum. In section 3.2 the method and motivation behind time resolved analysis is explained.

3.1 Fermi Gamma-ray Space Telescope

In this section I will describe the Fermi Gamma-ray Space Telescope, which was launched in 2008. Fermi is equipped with two different instruments; the Large Area Telescope (LAT) operating in the 20 Mev–300 GeV energy range, and the Gamma-ray Burst Mon-itor (GBM) in the 8 keV–40 MeV energy range. The large field of view combined with the outstanding sensitivity allowed 954 GRBs to be identified during the first four years of operation (von Kienlin et al. 2014). Of these, only 43 were observed by the LAT (von Kienlin et al. 2014). The performance requirements of both instruments are given table 3.1. 1_{http://fermi.gsfc.nasa.gov} 2_{http://swift.gsfc.nasa.gov} 3_{http://sci.esa.int/integral/} 4_{http://asd.gsfc.nasa.gov/konus/} 5_{https://www.mpi-hd.mpg.de/hfm/HESS/}

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Figure 3.1: Illustration of the Fermi Gamma-ray Space Telescope. The large box at the top of the satellite is the LAT. The GBM detectors are highlighted in yellow/orange. Created by Aurore Simonnet at NASA Education and Public Outreach Group at Sonoma State University, CA

The LAT is a pair-production based tracking chamber where the direction of the incoming photon can be determined by tracking the resulting electron-positron pair. The detector has a relatively small field of view, observing only about 20 % of the sky. Due to the lower sky coverage and high energy range, the LAT will only observe _{∼ 5 % of the burst} seen by the GBM (von Kienlin et al. 2014).

Due to the isotropic distribution of GRBs over the sky, one of the most important feature of a telescope designed for GRB detection is a wide field of view. The brief transient nature of the prompt emission means that the detector must be pointed at the GRB when it appears in order to detect it. Thus if we want to maximize the number of detected GRBs we must maximize the part of the sky we observe a any one time. The GBM monitor the full sky not obscured by the Earth, compared to the LAT, which views only 20 % of the sky.

The GBM consist of 12 sodium iodide (NaI) scintillators operating in the 8 keV_{−1 MeV} range, and two bismuth germanate (BGO) scintillators operating in the 150 keV−30 MeV range. The scintillators produce optical light when the high energy photons interact and deposit energy. This optical light is detected by photo-multiplier tubes (PMTs) that produce a current proportional to the amount of energy deposited in the scintillator. GRBs detected by the GBM are seen as a significant rise in the count rate in a minimum of two NaI scintillators. In this thesis only data from the GBM is used in the analysis.

Quantity GBM LAT

Energy range 10 keV_{− 25 MeV 20 MeV − 300 GeV}

Energy resolution (∆E/E) < 10 % < 10 %

Sky coverage All sky _{∼ 20 % of the sky}

Time resolution < 10 µs < 100 µs

Location accuracy ∼ 3◦ _{< 0.5}0

Table 3.1: Summary of the Gamma-ray Burst Monitor and Large Area Telescope performance requirements. The energry resolution and location accuracy are energy dependent, the values given here are the minimum requirements for the full energy range.

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0 50 100 150 200 250 Time (s) 1000 1200 1400 1600 1800 2000 2200 2400 coun ts (s − 1 ) Lightcurve GRB 130925173 −20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 1000 1200 1400 1600 1800 2000 2200 2400 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 epsilon e Lightcurve GRB 130925173 Parameter evolution of epsilon e

Figure 3.2: Example of the GRB 130925 lightcurve binned using Bayesian Blocks (left), and the parameter evolution of e (right). The red lines in the left plot show the limits between

each bin. Note the varying width of each bin. Figures produced by Bj¨orn Ahlgren.

3.2 Time resolved spectral analysis

With time-resolved spectral analysis and the decomposition of the spectrum into different components, the time-evolution of each individual component in the spectrum can be followed. This method is necessary due to the rapid spectral evolution of the prompt GRB spectrum. Using the time-integrated spectrum will lead to important time dependent features of the spectra being lost or distorted, which may lead to the wrong theoretical conclusions.

The time-resolved analysis is sensitive to the binning method chosen. I will here describe three different methods used to bin the data; constant exposure time, constant signal-to-noise ratio, and Bayesian Blocks.

Common for each method is the figure of merit used to quantify the signal strength for each bin; the signal-to-noise ratio. It is defined as the number of observed counts from the source over the square root of the calculated background counts. The signal-to-noise ratio is discussed further in 5.2.

The simplest way to bin the data is by using a fixed time for each bin. With this method the signal-to-noise ratio will vary greatly among the different bins, possibly leading to bins with insufficient statistics. A more sophisticated way of binning the data is by requiring each bin to have a certain signal-to-noise ratio. With this method the width of each bin is different in order to have a sufficient number of observed counts in each bin. This guarantees that each bin have sufficient statistics. It is important to note that neither of these methods take the shape of the observed lightcurve into account.

The last method is Bayesian Blocks (Scargle et al. 2013). This is a non-parametric model, meaning that the binning is determined by the data instead of some predetermined condition. The goal of the method is to capture to local temporal variations in the count rate using Bayesian probability theory. This binning method will have different width and signal-to-noise ratio for each bin. While the method captures the flux evolution well, some bins may suffer from poor statistics due to a low number observed counts. In figure 3.2, the lightcurve of GRB 130925 using Bayesian Blocks, and the parameter evolution of e are shown.

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Chapter 4 Spectral fitting using XSPEC

xspec is an X-ray spectral fitting package developed by NASA’s High Energy Astro-physics Science Archive Research Center (Arnaud 1996). In this chapter I will introduce the basics of spectral fitting and explain the method used for finding the estimated values and confidence intervals of the model parameters. The method described illustrates the fundamental principle of spectral fitting. The actual implementation used by xspec is detailed in the xspec manual1_.

4.1 Fitting data

The spectrum measured by a spectrometer is not the true spectrum of the source. The observed count rate within one instrument channel C(I) is given by

C(I) = Z ∞

0

f (E)R(I, E) dE s−1_,

(4.1) where f (E) is the true spectrum of the source and R(I, E) is the instrumental response of the detector. Ideally we want to invert this relation to find the true model spectrum f (E). This is in general not possible and will lead to non-unique and unstable solutions. Instead a parametrized model f (E, p1, p2...) is introduced. For one set of parameters a predicted spectrum Cp(I) is calculated allowing the appropriate fit statistic to be calculated (section 4.1.1). The parameters are then varied until the most desirable fit statistic is found using the Levenberg-Marquardt algorithm (Marquardt 1963).

From eq. (4.1) we see that three components are needed to calculate the best-fit parame-ters, the observed count rate C(I), instrumental response R(I, E) and a model spectrum f (E). The total count rate is given by

C(I) = D(I) aDtD − bD bB B(I) aBtB s−1_, (4.2) where D(I) and B(I) is the total photon count and estimated background photon count, tD and tB the source and background exposure times, aD and aB are detector-dependent

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area scaling factors, and bB and bD background scaling factors used to ensure that the background flux is given for the same area as the signal flux.

The theoretical response function R(I, E) in eq. (4.1) is a continuous function propor-tional to the probability that an incoming photon of energy E is absorbed in instrument channel I. In practice however this will be a discrete function since any observation is made using a finite number of instrument channels. This continuous function is turned into a discrete matrix by dividing the energy spectrum in N discrete bins giving the response matrix for the I:th instrument channel and the J:th energy channel

RD(I, J) = REJ

EJ−1R(I, E) dE

EJ − EJ −1

cm2_. _(4.3)

The parametrized model is also an energy dependent continuous function that must be divided into discrete energy bins. The parametrized model for the J:th energy bin is given by

fD(J, p1, . . . , pn) = Z EJ

EJ−1

f (E, p1, . . . , pn) dE s−1cm2. (4.4) Using these quantities we can calculate the predicted photon count rate

CP(I) = N X

J =1

fD(J, p1, . . . , pn)RD(I, J) s−1. (4.5) Comparing CP(I) with C(I) from eq. (4.2) we can quantify how well our proposed parametrized model fit the data by calculating the appropriate fit statistic.

4.1.1 Fit statistic

The observed count rate is the sum of the source and background count rate. For GRB observations using Fermi’s Gamma-ray Burst Monitor these are drawn from two different distributions. The background can be observed before and after the burst allowing a much longer exposure time than the burst itself. These observations is used to create a model for the background in which the errors are Gaussian. The source photon count rate will however be relatively low due to the shorter exposure time, making the use of Poisson errors necessary. The suitable fit statistic for this type of distribution is the P G statistic given by P G = 2 N X i=1 ts(mi+ fi)− Siln(tsmi + tsfi) + 1 2σ2 i (Bi− tbfi)2− Si(1− ln Si) , (4.6) where ts and tb are the source and background exposure times, Si are the total observed counts, mi are the predicted total count rates based on the model and instrumental response, Bi are the background counts, σi are the errors in Bi and

fi =

tbBi− tsσ2i − t2bmi± di

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where

di = q

(tsσi2− tbBi + t2bmi)2− 4t2b(tsσi2mi− Siσ2i − tbBimi). (4.8)

4.2 Error estimation using Monte Carlo simulations

xspec provides two built in features for obtaining the confidence interval of the fit pa-rameters. First, using the covariance matrix. Second, varying the fit parameters until the fit statistic deviates from the best-fit value by some fixed amount depending on the certainty required. A more accurate method is to determine the confidence interval of the fit parameters using a Monte Carlo technique where a large number of simulated spectra with properties identical to the original spectra, with added statistical fluctuations, are fitted to the model.

The Monte Carlo method for finding the confidence interval used in this thesis have been developed by myself using xspec’s Python interface. The method consist of the following steps

1. Select one real observation as basis for the simulated spectra

2. Select a model Mn with n model parameters P ={p1, . . . , pn} that is assumed to describe the true spectrum

3. Simulate N number of spectra from the model Mn for the parameters P 4. Fit each of these simulated spectra to the model to be tested M∗

n giving a set of fitted parameters F =_{P∗

1, . . . , P ∗ n}

5. From the set F calculate the estimated value and confidence interval for each of the n free parameters of the model M∗

n

Each simulations is based on an actual observation made by the the Fermi space telescope. These observations are chosen to ensure realistic response matrices, energy resolution and background rates. The spectral shape of these observations do however not directly affect the simulated spectra. The simulated spectra are produced by using xspec’s fakeit command. This command simulates spectra by multiplying the response curve with the selected model and parameters, and adding background and statistical fluctuations. The statistical fluctuations introduced will depend on the magnitude and type of error used. After fitting each of the N simulated spectra the result is a set of fitted parameters F =_{P∗

1, . . . , P ∗

N}, where each set of fitted parameters contain n individual parameters, P∗

j ={p∗j,1, . . . , p∗j,n}. Thus for each model parameter pi there is N outcomes that can be represented as a ordered list for the i:th parameter _{p∗

1,i, . . . , p ∗

N,i}. The sample mean is used to estimate the parameter value.

The confidence intervals presented here are estimated at the 1σ level. Thus the lower and upper bound of the confidence interval is given by the 15.9th and 84.1th percentile, respectively. Three examples of estimated values and confidence intervals for different outcome distributions are shown in figure 5.1.

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Chapter 5 Method for evaluating the table

model

The main aim of this thesis is to evaluate the result produced by the DREAM model, and also suggest improvements to the table model. The first obstacle is to define what criteria are to be used to determine if the results are acceptable. In this thesis two main criteria are used to asses the table model. First, do the fitted parameters of the table model predict the correct value? Second, is the uncertainty of the prediction within reasonable limits? Neither of the criteria have a sharp boundary between what is acceptable and what is not. In section 5.1, the way in which these criteria are quantified is described. The main difficulty when examining the table model is differentiating between the errors inherent to the model and those caused by other factors. The purpose here is to find the errors inherent to the model so that these can be addressed. In this thesis mainly three sources of errors are addressed. First, the error caused by a low signal-to-noise ratio. Second, the difference in the interpolated model spectra and the true model spectra. Third, the effect of degeneracies in the table model.

The error caused by a low signal-to-noise ratio is separated by repeating the analysis for varying signal-to-noise ratios. The method for artificially modifying the signal-to-noise ratio is described in section 5.2. Of the three sources or errors considered, this is the only one not inherent to the table model. This is investigated in chapter 6

The error caused by the difference in the interpolated spectra and the true model spectra is investigated by simulating the true model spectra for non-grid-points and fitting these spectra to the table model. This is investigated in chapter 7.

The error caused degeneracies in the table model is tested by randomizing the initial guess when fitting the simulated spectra to the model. This is investigated in chapter 8.

5.1 Evaluating the outcome for each point

The previous chapter describes how to determine the confidence interval and estimated value of a model parameter. These values must, however, be interpreted when evaluating

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0 20 40 60 80 100

(a)

0.0

0.2

0.4

0.6

0.8

1.0 Norm

ali

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>5%

0 20 40 60 80 100

(b)

>40%

0 20 40 60 80 100

(c)

<1%

Figure 5.1: Example of three simulation outcomes that trigger one flag each. The solid blue lines show the determined values and the dashed blue lines the determined confidence intervals. The dash-dotted red line show the true values. (a) the first flag is triggered since the true value is not correctly predicted. (b) the second flag is triggered due to the wide distribution. (c) the third flag is triggered since it predicts no lower bound.

the table model. Each point is evaluated on how well it predicts the correct value and how certain that prediction is.

Due to the large number of points evaluated manual inspection of each result is imprac-tical. This inspection is automatized by defining three flags that are calculated for each point. These flags were developed to quantify the two main criteria used to asses the table model. The limits on these flags were chosen after extensive tests such that they in combination provide good criteria of how well the model is able to accurately deter-mine the true value. The three different flags are described below and an example of an outcome that triggers each of the flags is show in figure 5.1.

1. Do the tested model predict the correct value? This test is passed if the true value lies within the 1σ confidence interval determined by the Monte Carlo simulation. Alternatively, the test is also passed if the difference between the true and deter-mined value of the parameter is within 5 % of size of the allowed parameter space of that parameter.

This extra condition can also be interpreted as setting the minimum one-sided error at the 5 % limit, thus ensuring that very sharply peaked distribution of fitted values that predict the true value correctly do not trigger this flag.

If this test is failed, it may indicate that the tested model does not produce reliable results for this parameter in this part of the parameter space.

2. Is the prediction accurate enough? This test is passed if the width of the determined confidence interval is smaller than 40 % of the allowed parameter space of that parameter.

If this test is failed, it may indicate that the tested model is not sensitive to this parameter in this part of the parameter space.

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3. Is the parameter unbound? This test is passed if the determined confidence interval is no closer than 1 % of the allowed parameter space to any of the two boundaries of the allowed parameter space.

Typically, this test fails if the determined value is close to one of the parameter space boundaries and we can only predict a lower or upper bound of the parameter. It may also indicate that the tested model is insensitive to the parameter, since this may allow the one-sided error to be up to 40 % of the allowed parameters, softening the criteria of the second flag. An example of this false negative is shown in figure 5.1(c).

For points on the edge of the allowed parameter space this flag is equivalent of passing all the criteria, with the exception of the false negative stated above.

5.2 Modifying the signal-to-noise ratio

A widely used way to quantify the signal strength of an observation is the signal-to-noise ratio. It is defined as the signal counts divided by the square root of the background counts. Thus for a detector with N instrument channels the signal-to-noise ratio is given by SNR = PN i t(si− bi) q PN i tbi , (5.1)

where t is the exposure time, si and bi are the total and background photon rates respec-tively in the i:th bin.

By convention the signal-to-noise ratio of a Fermi GBM observation is given by the NaI detector with the best such ratio. Since only one detector is considered when calculating the to-noise ratio, the overall strength of the signal for a given calculated signal-to-noise ratio will vary with the spectral shape. This effect is stronger for those spectra with a strong high energy (> 1 MeV) contribution that will not increase the NaI signal-to-noise ratio since these energies are not detected by the NaI. The overall strength of the observed signal will however be greater due to the increased count rate in the BGO detectors.

In practice the signal-to-noise ratio of an observation is increased by increasing the ex-posure time since SNR∝√t, i.e. using a wider time bin. Thus brighter bursts will give a greater signal-to-noise ratio and/or allow for finer time binning.

When examining the effect of the background on the fitted parameters it is possible to artificially increase the signal-to-noise ratio by decreasing the background rate and the error in the background rate by a factor x. Since the signal count is independent of the background count rate, replacing the background count B with xB yield

SNR = √S B → S √ xB ∝ 1 √ x. (5.2)

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Chapter 6 Evaluating grid-points in the model

In this chapter the grid-points of three different implementations of the DREAM model are investigated. The first model has five free parameters, the optical depth at the dissipation radius τ , the luminosity in the co-moving frame L, the bulk Lorentz factor of the jet Γ, and the fraction of the jet kinetic energy transferred to the electrons and magnetic fields e and b, respectively. Two additional models are tested where b is fixed to 10−6 _{and 0.05, respectively. The models are tested for reliability and accuracy} in reproducing the parameter values of spectra simulated with known parameter values for varying signal-to-noise ratios.

For each grid-point and signal-to-noise ratio, 1000 spectra with properties identical to the grid-point spectra ,with the exception of the added statistical fluctuations, were simulated. Each of these spectra were then fitted to the model used to simulate the spectra. This test provides an elementary stability check of the the model. Since the spectra are simulated and fitted using the same model a robust model should be able to correctly predict the parameter values. The spread of the fitted parameters is also a measure of how sensitive the model is to that parameter, in that part of the parameter space.

The signal-to-noise ratio is varied by reducing the background rate to differentiate the errors inherent to the model from those caused by an overwhelming background rate. This variation of the signal-to-noise ratio allows for the separation of the effects from the model and the data in the analysis. The background rate is adjusted to a minimum of 1 % of the original count rate. This hard limit corresponds to an increase in the model predicted signal-to-noise ratio with the original background of a factor 10, see eq. (5.1). This increase in the signal-to-noise ratio is equivalent to an increase in the exposure time by a factor 100.

Each grid-point will be tested for the signal-to-noise ratios 20, 40, 100 and 150. These signal-to-noise ratios were not attainable for all grid-points with the maximum reduction limit of the background rate of 1 %. Reduction below this hard limit is not possible with the analysis method used here since some energy bins may receive a negative count rate. The 1 % limit is set to maximize the number of points used in the analysis since factors such as detector viewing angle and redshift will also affect the observed signal-to-noise ratio. In practice the background is not expected to vary more than_{∼ 20 %. Due to this,}

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Detector

Viewing angle SNR Background rate

Type No. (counts s−1₎

BGO 1 21.0◦ _47.5 ₁₈₈₈_{± 3}

NaI 7 45.1◦ 55.1 1274± 6

NaI 9 42.1◦ _56.4 ₁₂₀₃

± 9

NaI 11 22.0◦ _68.4 ₁₂₄₁_{± 5}

Table 6.1: Key observational properties of the time bin occurring∼ 25 s after the start of the burst of GRB 100414 with an exposure time t = 1.9 s. The background rates are calculated in the 5_{− 2000 keV and 0.1 − 50 MeV energy ranges for the NaI and BGO detectors respectively.}

the sets of parameters requiring a high correction factor are not expected to be visible at this level. Only the grid-points where the desired signal-to-noise ratio was obtained were used in the analysis.

All models evaluated in this chapter use logarithmic interpolation for all parameters. Comparison with linear interpolation showed no significant differences between the two methods. This result is expected for simulations from the grid-points since very little interpolation is required close to the grid-points. The effect of interpolation on the fitted parameters is examined in detail in chapter 7.

Due to performance considerations the actual parameter values were selected as initial guess for each fit. The effect of initial guess is analyzed separately in chapter 8.

The simulations are based on the time resolved spectra of GRB 100414 observed by the Fermi Gamma-ray Space Telescope. The time resolved analysis was done using Bayesian Blocks. The selected time bin corresponds to the peak luminosity with the highest signal-to-noise ratio occurring _{∼ 25 s after the start of the burst. The key properties of the} observation is summarized in table 6.1. It is important to note that the actual spectral properties of GRB 100414 has no effect on the simulated spectra. The only properties from the actual observation affecting the results are the response matrices, background rates, types of errors and exposure time. With the modifications to the background rate as described above. The spectra are fitted in the 8−1000 keV range for the NaI detectors, and in the 0.2_{− 40 MeV range for the BGO detector.}

6.1 Model with five free parameters

The model with five free parameters consist of 1296 different grid-points in total. These grid-points consist of each unique combination of the following parameter values

τ : 1 5 10 35

e : 0.01 0.05 0.1 0.2 0.3 0.5 b : 10−6 0.01 0.05

L : 1 10 100

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Grid-points Triggered flags Deviation from Confidence

SNR Used On edge No. 1 No. 2 No. 3 true value interval width

20 τ 917 457 0 69 463 < 0.1+0.3−0.0 3.5 +6.6 −3.0 e 299 4 55 311 0.002+0.013−0.002 0.051+0.090−0.047 B 603 1 477 712 < 0.0001+0.0032−0.0000 0.0215+0.0209−0.0215 L 511 13 64 511 < 0.1+0.6 −0.0 3.3+20.4−2.6 Γ 298 0 45 315 0.4+2.9 −0.3 39.6+52.3−31.4 40 τ 781 381 0 6 381 < 0.1+0.1 −0.0 2.1+4.3−1.7 e 247 3 4 248 < 0.001+0.007−0.001 0.030+0.051−0.027 B 507 2 239 538 < 0.0001+0.0018−0.0000 0.0114+0.0189−0.0114 L 430 9 38 430 < 0.1+0.4 −0.0 2.9+15.0−2.1 Γ 256 0 4 256 0.2+1.8 −0.2 22.5+37.8−17.4 100 τ 569 278 0 1 278 < 0.1+0.1 −0.0 1.3+2.9−1.1 e 156 0 0 156 < 0.001+0.004−0.001 0.020+0.041−0.017 B 356 2 92 359 < 0.0001+0.0010−0.0000 0.0071+0.0130−0.0071 L 395 7 26 395 < 0.1+0.5−0.0 3.2 +12.2 −2.7 Γ 174 0 0 174 0.2+1.3−0.2 14.9+29.1−11.2 150 τ 498 250 0 1 250 < 0.1+0.1 −0.0 1.1+2.5−0.9 e 147 0 0 147 < 0.001+0.004−0.001 0.017+0.040−0.015 B 316 3 69 318 < 0.0001+0.0010−0.0000 0.0048 +0.0137 −0.0048 L 383 6 15 383 < 0.1+0.5−0.0 3.8+11.7−3.3 Γ 147 0 0 147 0.2+1.2 −0.2 12.3+27.8−9.3

Table 6.2: Summary of the properties of the model with five free parameters and logarithmic interpolation. For each signal-to-noise ratio the total number of points used, and the number of points on the edge of the allowed parameter space for each parameter is given. Note that the points on the edge are expected to trigger flag no. 3. The different flags are described in section 5.1.

Out of the of 1296 grid-points the number of grid-points for which it was possible to attain the desired signal-to-noise ratio with the 1 % background reduction limit were

SNR 20 : 917 (71%) SNR 40 : 781 (60%) SNR 100 : 569 (44%) SNR 150 : 498 (38%) . The grid-points that are never checked are consistently those with lower L or the combi-nation of extreme values in e, Γ and τ . Plots showing the parts of the parameter space that is not checked are shown in appendix A. A clear decrease in the number of checked points for b = 10−6 is seen compared to b = 0.01, 0.05. This is motivated physically by the fact that more energy is transferred to the magnetic field for higher b, thus producing more photons through synchrotron emission.

The number of triggered flags, deviation of the fitted parameter values from the true values, and the width of the 1σ confidence interval are shown in table 6.2. Note that the stated values are absolute values.

From the number of triggered flags and width of the confidence intervals in table 6.2, it is clear that a narrower confidence interval is found when the we have a higher

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signal-to-0.00

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ǫb

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0.05

ǫb

0.00

0.05

ǫb

0.00

0.05

ǫb

Figure 6.1: A characteristic example of how the distribution for bis changed when the

signal-to-noise ratio is varied. The four plots correspond to SNR 20, 40, 100, 150 increasing from left to right. The solid blue line shows the median parameter value and the dashed blue lines show the confidence interval. Note that the two extra peaks clearly visible for the lower signal-to-noise ratios are grid-points in the table model. The other parameter values are τ = 1, e = 0.01,

L = 100, and Γ = 400. Note that the distribution is limited only by the size of the allowed parameter space.

noise ratio. This conclusion is perfectly in line with observational experience, and is very clear when examining the evolution of individual distributions as the signal-to-noise ratio is varied. A typical example of how the confidence interval narrows as the signal-to-noise ratio increases is show in figure 6.3.

From the data in table 6.2 it is clear that τ , e, L and Γ consistently predicts the true value correctly. The triggered flags at high signal-to-noise ratio for τ and L are found for the highest parameter values, where the distributions have a significant accumulation slightly below the true value. These tail-ended distributions shifts the lower bound of the confidence interval to lower values.

For b only a few points have true values lying outside the determined confidence interval, indicating that the table model correctly predicts the true value. This result is due to the wide confidence intervals found, resulting in a high number of triggered flags due to wide confidence intervals. From the median confidence interval width we also see that a appreciable number of points have confidence interval similar in size to the full allowed parameter space. An example of these wide distributions in b for different signal-to-noise ratios is shown in figure 6.1. From this result we draw the conclusion that the model spectra are not very sensitive to b in this part of the parameter space. This conclusion is further supported when examining the spectral dependency of b in figure 6.2. These indicate that the spectral features sensitive to b primarily lies outside the observed energy range 101_{− 10}4_keV.

When examining table 6.2 it is important to note that that the typical value of b ∼ 10−2 is orders of magnitude below the other parameters, with roughly a third at b ∼ 10−6. This will skew the values in table 6.2 to lower values than naively expected.

For SNR 20, the flag indicating unbound fitted parameters are triggered for 25 points, excluding b. Increasing the signal-to-noise ratio improves this number considerably. At SNR 40 only 1 is found, and none is found for higher signal-to-noise ratios. This result is consistent with figure 6.5, and indicate that a minimum signal-to-noise ratio of SNR 40 is needed.

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10−3 10−2 10−1 100 101 102 103 104 105 106 107 108

Energy(keV)

10−3 10−2 10−1 100 101 102

E

F

E

(k

eV

s

− 1

m

− 2

)

tau = 10.0 ee = 0.2 ed = 1.0 epl = 0.0 lum = 10.0 gamma = 200.0 eb 1e-06 0.01 0.05 0.1 0.2 0.3 0.5 10−3 10−2 10−1 100 101 102 103 104 105 106 107 108

Energy(keV)

10−3 10−2 10−1 100 101

E

F

E

(k

eV

s

− 1

m

− 2

)

tau = 5.0 ee = 0.01 ed = 1.0 epl = 0.0 lum = 100.0 gamma = 300.0

eb 1e-06 0.01 0.05

Figure 6.2: Comparison of the true spectra for varying values of b for the grid-point τ =

10 e = 0.2 L = 10 Γ = 200 (top) and b for the grid-point τ = 5 e = 0.01 L =

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6.2 Models with fixed

b

The two models with four free parameters consist of 432 different grid-points, each with a fixed value of bat 10−6and 0.05. These grid-points are the same as for the five parameter model where the b parameter is fixed. Out of the of 432 grid-points for b = 10−6 the number of grid-points for which it was possible to attain the desired signal-to-noise ratio with the 1 % background reduction limit were

SNR 20 : 277 (64%) SNR 40 : 235 (54%) SNR 100 : 130 (30%) SNR 150 : 123 (28%) , and for b = 0.05

SNR 20 : 326 (75%) SNR 40 : 272 (63%) SNR 100 : 226 (52%) SNR 150 : 193 (45%) . The grid-points that are never checked are the same as in the previous section, for the relevant values of b. Plots showing the parts of the parameter space that is not checked are shown in appendix A.

From the number of triggered flags and width of the confidence intervals in table 6.3 and 6.4, it is clear that a narrower confidence interval is found when the we have a higher signal-to-noise ratio. This conclusion, which is perfectly in line with observational experience, is very clear when examining the evolution of individual outcomes as the signal-to-noise ratio is varied. A typical example of how the confidence interval narrows as the signal-to-noise ratio increases is show in figure 6.3.

For SNR 20, the flag indicating unbound fitted parameters are triggered for 15 points, combined for both models. Increasing the signal-to-noise ratio improves this number considerably. At SNR 40 only 1 is found, and none is found for higher signal-to-noise ratios. This result is consistent with the findings in the previous section.

6.3 Comparison of the models

All three tested models show similar results in the parameters common to the three models. The deviation of the fitted values and the width of the confidence intervals as a function of the signal-to-noise ratio for each of the parameters is shown figure 6.4 and 6.5, respectively. In these figures only the grid-points for which a correction to SNR 150 was attainable are shown. This discrimination is made to eliminate the effect of varying sample size. In these figures no model can be seen to perform clearly better than any of the other, and that the deviation from the true values are minuscule and decrease with increased signal-to-noise ratio.

The one clear difference between the four- and five-parameter models is in the behavior of b. For the four-parameter models this parameter is fixed, while in the five-parameter model it is allowed to vary. In the five-parameter model b clearly performs worse than the other parameters, since very wide distributions (figure 6.1) are found even at highest signal-to-noise ratios. As noted in section 2.3.2, xspec table models with more than three or four parameters are not recommended. This is, however, a unlikely explanation since the other parameters produce results consistent with the four-parameter model to a great

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20 τ 277 137 0 20 137 < 0.1+0.6 −0.0 4.1+6.8−3.6 e 88 2 3 91 0.004+0.012−0.004 0.043+0.052−0.037 L 142 0 19 142 < 0.1+0.4 −0.0 4.3+16.2−3.0 Γ 90 0 3 91 0.9+5.4−0.9 38.0 +31.1 −28.0 40 τ 235 111 0 3 111 < 0.1+0.4 −0.0 2.8+5.0−2.4 e 74 2 1 75 0.002+0.010−0.002 0.028+0.049−0.021 L 139 0 9 139 < 0.1+0.3 −0.0 4.3+11.3−3.1 Γ 77 0 0 77 0.6+3.5−0.6 25.0+25.1−17.9 100 τ 130 63 0 0 63 < 0.1+0.3 −0.0 1.6+2.7−1.5 e 37 0 0 37 < 0.001+0.003−0.001 0.016+0.013−0.009 L 130 0 2 130 < 0.1+0.3−0.0 5.4 +11.9 −1.7 Γ 43 0 0 43 0.6+2.5−0.6 14.5+13.5−10.8 150 τ 123 61 0 0 61 < 0.1+0.2 −0.0 1.5+2.3−1.4 e 35 0 0 35 < 0.001+0.003−0.001 0.015+0.012−0.009 L 123 0 0 123 < 0.1+0.3−0.0 5.0 +9.7 −1.4 Γ 41 0 0 41 0.6+2.2 −0.6 13.4+11.8−9.7

Table 6.3: Summary of the properties of the model with four free parameters and b = 10−6

with logarithmic interpolation. For each signal-to-noise ratio the total number of points used, and the number of points on the edge of the allowed parameter space for each parameter is given. Note that the points on the edge are expected to trigger flag no. 3. The different flags are described in section 5.1.

20 τ 326 163 0 36 163 < 0.1+0.1 −0.0 2.9+7.0−2.4 e 106 0 46 116 < 0.001+0.021−0.001 0.063+0.123−0.058 L 192 3 22 192 < 0.1+0.2−0.0 3.6 +19.8 −3.1 Γ 104 0 12 105 0.3+2.0−0.3 37.9+59.5−31.7 40 τ 272 134 0 4 134 < 0.1+0.1 −0.0 1.6+4.0−1.4 e 87 0 2 87 < 0.001+0.005−0.001 0.031+0.065−0.029 L 146 3 6 146 < 0.1+0.2−0.0 3.2 +13.2 −2.2 Γ 88 0 0 88 0.1+1.5 −0.1 15.4+37.6−11.3 100 τ 226 111 0 0 111 < 0.1+0.1 −0.0 1.0+3.4−0.9 e 59 0 0 59 < 0.001+0.004−0.001 0.023 +0.054 −0.021 L 132 3 2 132 < 0.1+0.1−0.0 2.2+8.9−1.4 Γ 66 0 0 66 0.1+1.2 −0.1 10.5+27.0−7.5 150 τ 193 97 0 0 97 < 0.1+0.1 −0.0 0.8+2.6−0.7 e 57 0 0 57 < 0.001+0.003−0.000 0.018 +0.055 −0.016 L 128 1 1 128 < 0.1+0.1−0.0 2.5+7.2−1.7 Γ 53 0 0 53 0.1+1.1 −0.1 8.2+33.3−5.5

Table 6.4: Summary of the properties of the model with four free parameters and b = 0.05

with logarithmic interpolation. For each signal-to-noise ratio the total number of points used, and the number of points on the edge of the allowed parameter space for each parameter is

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200

300

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400

200

300

Γ

400

Figure 6.3: A characteristic example of how the distribution for Γ is changed when the signal-to-noise ratio is varied. The four plots correspond to SNR 20, 40, 100, 150 increasing from left to right. The solid blue line shows the median parameter value and the dashed blue lines show the confidence interval. Note that the two extra peaks clearly visible for the lower signal-to-noise ratios are grid-points in the table model. Note that the peaks in the distribution at Γ = 250, 300, 400 correspond to grid-points.

degree of accuracy. The likely explanation for this behavior is that the spectral properties are not very sensitive to b in the parameter space investigated. The explanation is consistent with the fact that both four-parameter models produce nearly identical results despite different b.

The model with the fewest grid-points unable to attain SNR 20 is the four parameter model with b = 0.05, while to one with b = 10−6 produce the most, relatively speak-ing, with the five parameter model in between since the average b lies between the two extremes. Physically this is explained by a greater contribution of synchrotron radia-tion in the final spectra, giving a better signal-to-noise ratio. It is important to note that predicting a higher signal-to-noise ratio does not imply that the model is better in explaining the prompt emission.

One can also note that that the grid-points giving the lower signal-to-noise ratios lie at the extremes of the allowed parameter space for the bulk Lorentz flow Γ. This indicate that the initial grid-points are indeed well selected.

From figure 6.5 it is clear that the rate at which the confidence interval width decrease with increased signal-to-noise ratio slows with increasing signal-to-noise ratio, and levels out at SNR 100. From this we draw the conclusion that increasing the signal-to-noise ratio beyond SNR 100 does little to increase the accuracy in the fitted parameters. In practice, the need for improved signal-to-noise ratio must be weighted against having a sufficient number of time bins. Since SNR∝ √t, reaching SNR > 100 will require a high exposure time. The increase in exposure time is made at expense of the number of time bins. In analysis where the time evolution of the prompt emission is the goal, a minimum of SNR 40 is needed. This is the lowest level at which the model produce bounded confidence intervals. The most appropriate selection would be the highest signal-to-noise ratio providing a sufficient number of bins, up to a maximum of SNR 100. Above

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20 40 100 150 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 τ 20 40 100 150 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 ǫe 20 40 100 150 SNR 0.0 0.2 0.4 0.6 0.8 L 20 40 100 150 SNR 0 1 2 3 4 5 6 7 Γ ǫb =10−6 ǫb =0.05 ǫb free

Figure 6.4: Median difference between simulated and true value on the grid-points. Only grid-points for which SNR 150 was attainable are shown in order to eliminate effects due to varying sample size.

this level the accuracy is not significantly improved.

For the very highest parameter values, both the L and e parameters show a tail-ended distribution. These tails rarely extend beyond the next grid-point. An example of this behavior is shown in figure 6.6. Since the grid is spaced relatively sparse in these regions, these distribution may span a significant portion of the allowed parameter space. This problem may be improved upon by adding extra grid-points in this region, thus creating a finer grid and hopefully a sharper distribution.

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20 40 100 150 0 2 4 6 8 10 12 τ 20 40 100 150 0.00 0.05 0.10 0.15 0.20 ǫe 20 40 100 150 SNR 0 5 10 15 20 25 30 35 L 20 40 100 150 SNR 0 20 40 60 80 100 120 Γ ǫ_b =10−6 ǫ_b =0.05 ǫ_b free

Figure 6.5: Median width of confidence interval on the grid-points. Only grid-points for which SNR 150 was attainable are shown in order to eliminate effects due to varying sample size.

10

20

30

τ

0.0

0.2

0.4

0.6

0.8

1.0 Norm

ali

ze

d f

re

que

ncy

0.2 0.3 0.4

ǫe

50

100

L

100 120 140 160

Γ

400 500 600

PG stat

Figure 6.6: Example of parameter distributions for the grid-point τ = 35 e = 0.3 L =

(34)

Chapter 7 Investigating the error introduced

by interpolation

In this chapter the effect the interpolation has on the fitted parameters is investigated. The model used has four free parameters with a fixed b = 10−6. Two different implemen-tations of this model are tested, one with linear and one with logarithmic interpolation, in all parameters.

For each tested point in the parameter space and signal-to-noise ratio, 1000 spectra are simulated and fitted to the model. These simulated spectra are assumed to represent the true underlying spectra with added statistical fluctuations. Unlike in chapter 6, these true spectra are not simulated from the same model as they are fitted to. The true spectra are instead found by using the code described in Pe’er & Waxman (2005). The difference between the true calculated spectra and the interpolated spectra should be small, provided that the grid-points in the model are tightly spaced. Thus investigating the errors caused by using the true calculated spectra away from the grid-points provide a good test of the spacing of the grid-points in the table model. An example of the difference between the true calculated spectra and the interpolated spectra is shown in figure 7.1.

In total 72 points are examined. These are all placed midway between two grid-points since closer to a grid-point the effect of interpolation becomes less noticeable. The points selected for analysis are

τ : 2.5 7.5 22.5 e: 0.025 0.15 0.4

L : 5 50

Γ : 125 175 275 350 .

The method used to correct the signal-to-noise ratio to the desired level, and selection of which points to use in the analysis is identical to that used in chapter 6. The points for which it was possible to attain the desired signal-to-noise ratio with the 1 % background reduction limit were the same for both interpolation methods. These were for each signal-to-noise ratio

(35)

10

100 1000

10

4

₁₀

5

100 1000

keV

2

(Photons cm

−2

s

−1

keV

−1

)

Energy (keV)

Figure 7.1: Comparison between the interpolated and true spectra for the grid-point τ = 2.5 e= 0.4 L = 50 Γ = 275. The true spectra is shown by the red line, and the interpolated

Verication and evaluation of the DREAM model for subphotospheric dissipation in prompt GRB emission

Verification and evaluation of the DREAM model for

subphotospheric dissipation in prompt GRB emission

Author:

Erik Ahlberg (900725-5079)

eahlb@kth.se

Department of Physics

Royal Institute of Technology (KTH)

Supervisor:

Josefin Larsson

Contents

Chapter 1

Introduction

1.1

Outline of the thesis

1.2

Author’s contribution

Chapter 2

Gamma-ray bursts

2.1

Fundamental properties

2.2

Prompt emission

2.3

Dissipation with radiative emission as a table

model

R

1

Jet

acceleration

Numerical

calculations

Spectrum

assumed

constant

Photons

released

Jet

deceleration

2.3.1

Physical scenario

2.3.2

Table model implementation

Chapter 3

Observational techniques

3.1

Fermi Gamma-ray Space Telescope

3.2

Time resolved spectral analysis

Chapter 4

Spectral fitting using XSPEC

4.1

Fitting data

4.1.1

Fit statistic

4.2

Error estimation using Monte Carlo simulations

Chapter 5

Method for evaluating the table

model

5.1

Evaluating the outcome for each point

0 20 40 60 80 100

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Norm

ali

ze

d f

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que

ncy

>5%

0 20 40 60 80 100

(b)