blowing in the wind the e physical properties of grb ... · populaire samenvatting gamma itsers...
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Universiteit van Amsterdam
Anton Pannekoek instituut
Blowing in the windThe e↵ects of an accretion disk wind on the spectrum of a
black hole binary system
Author:Niki Klop5956366
Supervisor:Dr. Phil Uttley
Report on a 12 EC Bachelor Project in Physics and Astrophysics,performed between June 4, 2012 and August 24, 2012.
Physical Properties of GRB Afterglows
Author: Supervisors:
Timo L.R. Halbesma, 6126561 Dr. Alexander J. van der Horst;
Version 3.14 (final) Prof. Dr. Ralph A.M.J. Wijers.
Report on a Bachelor Project of 12 ECTS in Physics and Astrophysics.
Research conducted between May 2013 and July 2013.
1
Abstract
Gamma Ray Burst (GRB) afterglows are caused by the broadband synchrotron ra-
diation emitted after the initial burst of gamma rays. The model adapted here is the
blast wave model, the standard model for GRB afterglows. From the power of a single
electron emitting synchrotron radiation in ultra relativistic blast waves, causing the GRB
afterglows, the dependence β of the spectral energy distribution of GRB afterglows on
the frequency is derived by integrating over the single electron power times the electron
distribution. From the time evolution of the peak frequency νm and cooling frequency
νc the time dependence α of the flux of GRB afterglow light curves is derived. The de-
rived equations are rewritten in terms of the particle distribution index p as a function
of α and/or β and the density parameter k as functions of α and β. Given α from fits
to the X-ray and optical light curves and β from fits to the X-ray and optical spectral
energy distribution obtained from the literature, p and k are calculated for 49 Swift GRB
afterglows. Assumptions made include a viewing angle equal to zero, adiabatic expansion
of the blast wave, collimated outflow without energy injection and a power law electron
energy distribution. The data looked at is in the slow cooling regime (νm < νc) for X-ray
and optical observations (ν > νm) before the jet break. This study shows shows that 36
GRBs are consistent with the derived equations and 13 are inconsistent. The 36 p values
found are in a broad range of 1.66 − 3.13 with a weighted mean of µ = 2.76(1). This is
consistent with the literature on calculations of p from observations and with theoretical
models showing a broad distribution but inconsistent with theoretical models showing
that p has one universal value. For 26 GRBs a value for k is determined. The majority is
consistent with a stellar wind (k = 2).
2
Populaire samenvatting
Gammaflitsers zijn een van de meest energetische processen in de sterrenkunde. Bij een
gammaflitser komt meer energie vrij per seconde dan de zon in zijn gehele levensduur
uit zal stralen. Een gammaflitser vind plaats in het eindstadium van de sterevolutie en
zijn op te delen in twee categorieen: lange en korte gammaflitsers, waarbij de categorie
bepaald wordt door een duur van meer of minder dan twee seconden. Bij een korte gam-
maflitser is het waarschijnlijk zo dat een neutronen ster en een andere neutronen ster
of zwart gat die om elkaar heen draaien uiteindelijk op elkaar botsen waarbij een ont-
ploffing ontstaat. Bij de lange gammaflitser ontploft een zware ster omdat de kernfusie
stopt waardoor er geen druk meer geleverd wordt om de zwaartekracht tegen te gaan.
Voor zowel lange als korte gammaflitsers wordt de materie die vrijkomt wordt in twee
jets uitgestuurd, waarbij wordt aangenomen dat de waarnemer vanaf aarde recht in de
jet kijkt. Omdat de ontploffing geen constante uitstroom van materie heeft ontstaan er
een soort schillen met verschillende snelheden die met elkaar in botsing zullen komen. De
botsingen van deze schillen in de jet zorgen ervoor dat er gammastraling wordt uitgezon-
den. De vele verschillende botsingen tussen schillen zorgen ervoor dat er een salvo van
gammaflitsen wordt uitgezonden; vandaar de naam Gamma Ray Burst, vrij vertaald als
gammaflitser salvo. De jet veegt materie (onder andere elektronen) uit de omgeving op
en versnelt de elektronen tot relativistische snelheden (bijna de lichtsnelheid). Een eigen-
schap van relativistische elektronen in de aanwezigheid van een hoog magnetisch veld is
dat er synchrotronstraling uitgezonden wordt. De elektronen geven hun energie weer af
door synchrotronstraling gedurende een langere periode van tijd. Dit verschijnsel is de
nagloeier van de gammaflitser. Een kenmerk van synchrotron straling is dat het breed-
bandig is. Daarom is de nagloeier van de gammaflitser waar te nemen in een groot deel
van het elektromagnetische spectrum: radio, optisch en rontgen.
In dit bachelorproject is de theorie onderzocht die de nagloeiers van de gammaflitsers
beschrijft. Met deze theorie zijn twee parameters afgeleid die uit optische en rontgen
waarnemingen te bepalen zijn en uitgedrukt in termen van twee natuurkundige parame-
ters. Dit zijn de index die de energieverdeling van de elektronen beschrijft en de index die
de structuur van de dichtheid van het medium om de gammaflitser heen beschrijft. De pa-
rameters die zijn bepaald in de waarnemingen van lange gammaflisers zijn uit de literatuur
gehaald om de twee natuurkundige paramaters te berekenen. Voor de energieverdeling is
een brede verdeling gevonden. Dat komt overeen met publicaties over de energieverdeling
berekend uit waarnemingen. De theoretische modellen voorspellen of dat er een universele
waarde is, of een brede verdeling. De resultaten komen dus niet met het eerste - en wel
met het tweede theoretische model overeen. De dichtheidsstructuur van het medium om
de lange gammaflitser heen is overwegend consistent met de dichtheidsstructuur om een
zware ster heen.
3
Contents
1 Introduction 5
2 Theory 8
2.1 The Blast Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The Compactness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Relativistic Beaming of Synchrotron Radiation . . . . . . . . . . . . . . . 11
2.5 The Spectral Energy Distribution . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Time Evolution of the Blast Wave . . . . . . . . . . . . . . . . . . . . . . 17
3 Method 19
4 Data 20
5 Results 21
6 Discussion 26
7 Conclusion 27
4
1 Introduction
Gamma Ray Bursts (GRBs) are brief flashes of γ-rays with extreme brightness that last
one tenth of a second up to hundreds of seconds with a typical duration of tens of seconds.
GRBs were discovered by defense satellites launched by the United States of America
in 1967 though data was not published straight away. The United States of America, still
afraid of nuclear bomb tests by the the former Soviet Union after said countries and the
United Kingdom agreed not to test nuclear warheads at the earth’s surface, launched the
Vela satellites to detect nuclear detonations both below the earth’s surface and in outer
space. The Vela satellites discovered the first GRB in July 1969. The origin of these
gamma rays was not some expected detonation of a nuclear warhead but an object of
cosmic origin. Manmade spacecraft, the earth and the sun were ruled out as the source
of the detected peak in gamma radiation based on information about the direction of the
gamma rays. It was not until four years after observing the first GRB scientists were
allowed to publish the discovery as it was no longer considered state secret (Klebesadel
et al. 1973).
Not earlier than 1997 a conclusive answer to the question where GRBs originate from
was published after a vast quarter century of speculation. Some studies showed that
GRBs might originate from relative nearby objects distributed in the halo surrounding
our Galaxy (Podsiadlowski et al. 1995) while other studies showed the origin of the GRBs
was cosmological (Paczynski 1995). Technological development provided better data and
made answering this question possible. The Compton Gamma Ray Observatory was
launched in 1991 with the Burst and Transient Source Experiment (BATSE) on board. If
GRBs originate from within the Milky Way an anisotropic distribution would be observed:
more GRBs would be found in the galactic plane or towards the Galactic Centre. How-
ever, BATSE observations showed that GRBs are distributed isotropically across the sky
(Meegan et al. 1992). This ruled out the Milky Way as the origin of the GRBs, although
the Halo of the Milky Way was still a possible origin.
On the 28th of February 1997, GRB 970228 (GRBs are named after the date of
discovery: GRB YYMMDD where YY are the last two digits of the year, MM the month
and DD the day) was discovered. X-ray satellite BeppoSAX determined the position of
the GRB with the onboard Wide-Field Cameras. With the position of the GRB two
ground-based telescopes on La Palma were able to observe a GRB afterglow at optical
and infrared wavelengths (van Paradijs et al. 1997). The BeppoSAX satellite observed a
GRB afterglow at X-ray wavelengths with the onboard Narrow-Field Instruments (Costa
et al. 1997). Because the light curve of GRB 970228 declined steeply the astronomers
knew that observations of GRBs had to be taken soon after the initial flash in order to
find the GRB afterglow in the other wave bands. This was done successfully in the optical
wave band for GRB 970508 (Metzger et al. 1997). The study showed that this GRB has a
redshift z = 0.835. This measurement shows that this GRBs is 1010 light years away from
the earth. Later many more GRBs with similar redshifts have been detected thus one
5
can conclude that GRBs are of cosmological origin. The discovery of GRB afterglows and
observing them is key to better understanding GRBs so GRB research shifted towards
afterglow studies. With the now known distance of GRBs one can also conclude that
GRBs are by far the most luminous sources of photons in the Universe (Paczynski 1986).
The next question to be answered was what could cause such an extremely energetic
explosion.
Analysis of BATSE data revealed that GRBs can be split up in two categories: long
GRBs (typical duration > 2 s) that are spectrally soft and short GRBs (typical duration
< 2 s) that are spectrally hard (Kouveliotou et al. 1993). Both short and long GRBs
do show an isotropic distribution across the sky but the spectral and temporal properties
differ quite a lot suggesting they have different progenitors. The 1968 prediction that long
GRBs originate from the collapse of massive stars that end in a supernova (SN) explosion
(Colgate 1968) became relevant in 1998. The GRB-SN connection was first indicated by
the discovery of SN 1998bw at the same location as GRB 980425 (Galama et al. 1998).
However GRB 980425 has a much lower redshift than the average GRB thus might not be
representative for all GRBs. The discovery of GRB 030329 and the associated supernova
SN 2003dh (Hjorth et al. 2003; Stanek et al. 2003) did confirm that a GRB-SN connection
exists and that a certain type of supernovae could be associated with long GRBs. Host
galaxies of long GRBs are typically blue indicating active star formation regions and
lots of young massive stars. Those young massive stars end their lives with a supernova
explosion. When a massive star has burnt through all its hydrogen, helium and higher
elements (up to iron) the nuclear fusion will no longer generate an outward pressure that
compensates for the massive internal pressure generated by the gravity of the massive
star. Because the star is massive enough even pressure generated by degenerate matter is
by far not enough to compensate for the gravity. As a result the star will collapse into a
neutron star or a black hole whilst creating a supernova explosion that powers the GRB.
This process is described by the collapsar model (Woosley 1993). However there are at
least two nearby GRBs that do not show signs of a supernova in the optical spectrum:
GRB 060505 and GRB 060614 (Fynbo et al. 2006). This challenges the collapsar model.
These GRBs have a redshift of 0.089 and 0.125 respectively. Furthermore both GRBs are
found in similar host galaxies as typical GRBs (Thone et al. 2008) so the lack of evidence
for a supernova in the GRB spectrum can not be explained based on redshift or host
galaxy arguments.
What the progenitor of short GRBs could be is still under debate, but the most favored
candidate is a neutron star merging with either another neutron star or with a black hole
(Paczynski 1986). The reason a definitive answer to the progenitor question is yet to be
given is the faintness of short GRBs making detection more difficult. Short GRBs have
lower energies than long GRBs causing a fainter afterglow (Berger 2007). On top of a
lower initial energy the short GRBs also take place in the outskirts of the host galaxy
(as is consistent with the binary model because the GRB progenitor moves away from
its host galaxy due to the force exerted by the supernova creating the compact objects
6
and it takes 105 − 1010 years for a binary to merge) where the density is much lower
causing a much fainter afterglow (Nakar 2007). Observational data shows differences in
the emission between short and long GRBs both in the initial gamma ray emissions and
in their afterglow properties. Combined with the different host galaxies where short and
long GRBs are found this confirms that both classes of GRBs have different progenitors.
There are several reasons to study GRBs. For one GRBs are the most luminous and
energetic phenomena known in the universe. A GRB will release more energy in a time
period of just a couple of seconds or (much) less than our sun emits in its entire lifetime.
Around GRBs high magnetic fields and particles accelerated to high Lorentz factors are
found. While at the LHC in CERN scientists are able to control the experiment that is not
the case for GRBs. Obtaining data about different GRBs is needed to test the models of
the various phenomena on a lot of datasets from different GRBs. A GRB could be seen as
a particle accelerator but yields much higher energies, magnetic fields and Lortenz factors
than we can ever achieve on earth. Not only does studying GRBs increase the knowledge
about GRBs itself but also late stage star evolution is studied. As GRBs are likely to play
a role in black hole or neutron star formation studying GRB afterglows might reveal lots
of interesting physics about black holes and neutron stars. To study faint stellar objects
technological breakthroughs are made that are not only applicable to astrophysics but
may have purposes in everyday life as well. For instance the technology developed to
detect GRBs could be used for medical and industrial imaging as well (Eisen et al. 1999).
Even string theory could be studied by looking at high redshift GRBs because they could
perhaps be explained by superconducting cosmic strings (Cheng et al. 2010). Even ideas
that the extreme amounts of radiation released by GRBs wipe out other intelligent life-
forms before they are able to start an interstellar journey to meet and greet us humans
have been published (Annis 1999).
With this background information about GRBs, GRB progenitors and GRB after-
glows in mind this study will focus on calculating physical parameters from optical and
spectral observations. First I will discuss the relevant theory to understand GRB after-
glows (Section 2). The derived equations (Section 2.5 and Section 2.6) are implemented
in a computer program (Section 3) to calculate the physical parameters from the obser-
vational data (Section 4). The results (Section 5) of the GRB afterglow analysis will be
studied and discussed (Section 6) to find answers (Section 7) to the following questions.
What is the explanation for the GRB afterglow and its time and frequency dependency?
What physical parameters are directly observable in the GRB afterglow spectrum? What
physical parameters can be derived from the observable parameters? What are the values
of the observable and derivable parameters for given datasets of known GRB afterglows?
7
2 Theory
2.1 The Blast Wave Model
The supernova explosion at the end of the life of a young massive star possibly powers
the long GRB. This supernova explosion will be referred to as ‘the inner engine’. The
inner engine for short GRBs is likely to be a binary merger. As for this model all that is
relevant that there is an internal engine but not what that inner engine is exactly.
The inner engine produces an ultra-relativistic jet of so-called internal shocks with
different Lorentz factors (speeds) inside the jet that will collide. This jet has to be ultra-
relativistic to solve the compactness problem (Section 2.2). The collisions produce a flash
of γ-rays. As there are multiple internal shock waves multiple γ-ray flashes are formed:
a burst of γ-rays. The GRB is caused by collisions of the internal shocks. The matter
in the jet interacts with the medium surrounding the inner engine (either stellar wind or
interstellar medium). The interaction with the surrounding medium causes a so-called
external shock: a forward shock that lasts relatively long. There is also a reverse shock
that lasts a shorter period of time. The forward shock excites the matter it encounters
and the excited matter will cause the broadband afterglow due to synchrotron radiation
(Section 2.3). The reverse shock will propagate back into the jet and will cause a flash of
optical and radio emission typically lasting minutes to hours after the burst. But recently
a GRB has been discovered for which this flash lasted for a couple of days (Laskar et al.
2013). A schematic view of the blast wave model is shown in Figure 1 (Piran 2003).
There is an undeclared dispute amongresearchers who study complex traits —physical or behavioural characteristics
of an organism that are dictated by combina-tions of more than one gene and the environ-ment. On one side are classical geneticists,who look at the differences in DNA sequencebetween individuals and attempt to correlatethem with physical and behavioural varia-tions. On the other are proponents of gene-expression analysis, who focus on variationsin the genes being switched on. On page 297of this issue, Schadt and colleagues1 describehow they brought these approaches togetherin a systematic, genome-wide analysis of the genetics of variation in gene expression.Their work provides support for a strategy by which complex traits could be studied at agreater scale and depth than is possible usingeither technique alone.
The idea of carrying out genome-widegenetic analyses of gene-expression data wasintroduced two years ago by Jansen andNap2, and Brem et al.3 were the first to applythis approach, in a study of budding yeast.The principles are outlined (for studyingmice) in Fig. 1, overleaf. Crossing two pro-genitor strains, and subsequently crossingtheir offspring among themselves, producesa genetically variable population. Variationsin DNA sequence across the genomes of this population are then analysed to identifytheir origin (that is, which one of the twoprogenitor strains). At the same time thepopulation is studied to find out which genesare being expressed in different individuals,and to what degree. The expression level ofeach gene is then treated as a quantitativetrait. Quantitative traits are determined bymore than one gene and show a graded
news and views
back into the jet, producing optical and radioemission that decays rather quickly.
How does this model account for theunusual light curve recorded by Fox et al.1? Inmost cases, the external shocks would beginaround one minute after the onset of the GRB while the internal shocks and the !-ray emission are still going on. Hence, the very early afterglow and the reverse-shock emission overlap the late part of theGRB. The early part of the light curve mayrepresent the emission from the reverseshock and the plateau a transition from thereverse to the forward shock5.
The late afterglow wiggles could be inter-preted as the result of the shock waveencountering an external medium of vari-able density6–8. But detailed calculations9
show that even an abrupt drop in densitycannot explain the very steep decay of theafterglow seen around 0.3 days after theburst. Furthermore, such density variationsare expected to have little influence on the X-ray band6,7 , and yet Fox et al.1 find thatfluctuations in the X-ray and optical lightcurves are correlated.
Alternatively, the shock wave’s energymay have varied7,8: an increase in shock-waveenergy could explain the early slow decay,and an energy decrease would naturally produce a steep decline. A shock wave thathas been slowed by the surrounding mediumcould be caught up by subsequent shocks,increasing the shock-wave energy10. If such‘refreshed’ shocks do account for the earlyslower decay, then the actual energy releaseof GRB 021004 was significantly larger thanimplied by the observed !-rays alone1 — so ifthis burst is typical, then GRBs are even moreenergetic than we thought.
There is another possible explanation forenergy modulation of the shock wave, whichstems from the ultra-relativistic motion ofthe GRB jet7,11. The emission from a radiatingobject moving at almost the speed of light isbeamed into a very narrow cone along theline of motion. As the object slows down, thecone opens up to wider and wider angles. Atfirst, during the GRB and the early afterglow,the shock wave is moving very fast and,because of this ‘relativistic beaming’, only atiny fraction of the expanding jet is observed(only the part that is moving practically head-on towards the observer). As time passes andthe blast wave slows down, a larger fraction ofthe jet is seen. The corresponding shock-waveenergy is the average over the observed regionand it may increase (or decrease) locally if thejet structure is inhomogeneous.
Both of these mechanisms are possibleexplanations for the energy increase seen by Fox et al.1 in the early afterglow period.But the light curve for the later afterglowshows a period of steep decline, correspond-ing to energy decreases. As refreshed shockscan only increase the blast wave’s energy,they cannot account for the later rapid decay.
It is more likely that a non-uniform jet structure is responsible for the observed pattern. Non-uniform structure in GRB jetshas been long expected11 — this may be thefirst evidence for its existence. ■
Tsvi Piran is at the Racah Institute for Physics,Hebrew University of Jerusalem, Jerusalem 91904, Israel.e-mail: [email protected]. Fox, D. W. et al. Nature 422, 284–286 (2003).
2. Akerlof, C. Nature 398, 400–402 (1999).3. http://space.mit.edu/HETE/Welcome.html4. Sari, R. & Piran, T. Astrophys. J. 520, 641–649 (1999).5. Kobayashi, S. & Zhang, B. Astrophys. J. 582, L75–L78
(2003).6. Lazzati, D. et al. Astron. Astrophys. 396, L5–L9 (2002).7. Nakar, E., Piran, T. & Granot, J. New Astron. (in the press). 8. Heyl, J. & Perna, R. Astrophys. J. (in the press);
Preprint astro-ph/0211256 at <http://arXiv.org> (2002).9. Nakar, E. & Piran, T. Preprint astro-ph/0303156 at
<http://arXiv.org> (2003). 10.Rees, M. J. & Mézáros, P. Astrophys. J. 496, L1–L4 (1998). 11.Kumar, P. & Piran, T. Astrophys. J. 535, 152–157 (2000).
Genomics
Gene expression meets geneticsAriel Darvasi
Genetic analyses look for differences in gene sequence that could explainvariation in physical traits. Gene-expression studies provide a snapshot ofactive genes. These approaches are now combined , to great effect.
NATURE | VOL 422 | 20 MARCH 2003 | www.nature.com/nature 269
Figure 1 The internal–external shocks model. A gamma-ray burst (GRB) is thought to be driven by an ‘inner engine’, a cataclysmic event such as the collapse of a massive star. Inside an ultra-relativisticjet of particles thrown out from the explosion, internal shocks release a vast amount of energy in aburst of !-rays. When the jet is slowed down by surrounding matter, external shocks are created: the forward shock that propagates further into space, and the reverse shock that is reflected backagainst the relativistic flow. Both types of shock waves heat the surrounding matter, producing an afterglow to the GRB.
Reverse
shock
Hot shocked
externalmatter
Forward
shock
Hot shocked
jetmatter
Innerengine
Ultra-relativistic
jet
Internal shocks External shocks
© 2003 Nature Publishing Group
Figure 1: Schematic view of the blast wave model. An inner engine powers the internal shocks triggering the
GRB. The relativistic external shock causes the afterglow (Piran 2003).
8
2.2 The Compactness Problem
Observations show fluctuations in the prompt emission of the gamma rays in the order of
milliseconds. Given the finite speed of light the radius of the compact source has to be
in the order of ∆R < c∆t is 106 m to be in causal contact. Given the observed energy
is typically 1052 erg (e.g. Metzger et al. 1997), such an energy-radius ratio is optically
thick to pair creations, meaning the spectrum should be thermal (blackbody) radiation.
However the observed spectra for GRBs are power-law spectra instead of the expected
blackbody spectra. Also very high-energy γ-ray photons should be absorbed yet they
have been detected from GRBs. This compactness problem can be solved by assuming
the source of the GRB is moving towards the observer with relativistic speeds (i.e. v ≈ c).Consider the situation where two photons are emitted at different times, as shown in the
schematic view below.
T1
T2
Moving shock
with speed v
Photon 1
X1(T
1) X
2(T
2)
Xshock (T1) Xshock (T
2)
Photon 1
Photon 2
Ƌtobserver
ƋR
Observer
Figure 2: Solution to the compactness problem. A shock moving towards the observer with speed v emits
photons at times T1 and T2. The observed millisecond time difference ∆t could still come from a region
optically thin for pair creation by assuming v ≈ c.
The displacement of the shock is xshock(T2) = xshock(T1)+v∆T and the displacement of
photon 1 is x1(T2) = x1(T1)+c∆T . Because x1(T1) = xshock(T1) and x2(T2) = xshock(T2)
also x2(T2) = x1(T1) + v∆T . The difference in location of the different photons at time
T2 leads to the equation for the radius:
x1(T2)− x2(T2) = c∆tobserver = (c− v)∆T =(1− v
c
)c∆T =
((1− v2
c2
)(1+ v
c )
)c∆T ≈ c∆T
2γ2
c∆T = ∆R ≈ 2γ2c∆tobserver (1)
If the shock waves are ultra relativistic with a Lorentz factor of the shock front γ ≈ 100
then the actual radius of the shock wave is in the order of 1010 m. This results in the
area being optically thin for pair creation and solves the compactness problem.
9
2.3 Synchrotron Radiation
The relativistic jet sweeps up matter (for instance electrons) and the magnetic field lines
along its path. The initial magnetic field (for instance caused by the progenitor) will
increase as the field lines are compressed by the shock. We start by looking at a single
electron that is accelerated by a magnetic field. Such particles are known to radiate. For
non-relativistic particles this process is called cyclotron radiation that has a frequency
equal to the frequency of gyration in the magnetic field. The relativistic equivalent of cy-
clotron radiation is synchrotron radiation that has a frequency distribution much broader
than the gyration frequency. Synchrotron radiation is the main emission mechanism used
to produce the GRB afterglow.
Rybicki & Lightman (1979) gives a detailed description of radiative processes in as-
trophysics including synchrotron radiation. I will summarize the main points. It is shown
from the equations of motion that the gyration frequency ωB for a single electron of mass
me and charge qe is:
ωB =~a⊥~v⊥
=qe| ~B|γemec
(2)
where ~a⊥ and ~v⊥ are the acceleration and velocity perpendicular to the , ~B the magnetic
field, c the speed of light and γe the Lorentz factor of the electron. Note that the gyration
frequency ωB is inversely proportional to the Lorentz factor. Compared to cyclotron
radiation this dependence on γe together with relativistic beaming (Section 2.4) cause
the synchrotron frequency distribution to be broad. The magnitude of the acceleration is
constant and the direction is perpendicular to the velocity. This circular motion in the
plane perpendicular to the direction of ~B and the constant motion in the direction of ~B
is called helical motion.
The property of this physical phenomena that is of interest to describe GRB afterglows
is the power that is emitted by this single electron, given by the Larmor formula:
P ′ =2q2e
3c3|~a ′|2 (3)
where |~a ′|2 = ~a · ~a = (a ′2⊥ + a ′
2‖). Both the power and the acceleration have to be
transformed to the frame of the observer (the primed quantities). The power is Lorentz
invariant (i.e. P = P ′). The acceleration of the electron is the second order time deriva-
tive of its place x, y or z. Both dx, dy, dz and dt are transformed using the Lorentz
transformations:
ax =d2x
dt2=
ax′
γ3eσ
3, (4)
ay =d2y
dt2=
ay′
γ2eσ
2− uy
′v
c2ax′
γ2eσ
3, (5)
az =d2z
dt2=
az′
γ2eσ
2− uz
′v
c2ax′
γ2eσ
3(6)
10
where σ = 1 + vux′
c2 . Note that the direction of x is parallel to the magnetic field and
the directions y and z are perpendicular to the magnetic field. In the instantaneous rest
frame σ = 1, ux′ = uy
′ = uz′ = 0 giving:
P =2q2e
3c3(a ′
2⊥ + a ′
2‖) =
2q2e
3c3γ4e (a2⊥ + γ2
ea2‖) (7)
Using equation (2) written in terms of a⊥ = v⊥ωB and a‖ = 0 the formula for the total
power for a single electron is derived:
P =4
3σT cβ
2γ2eUB (8)
where σT is the Thomson cross section σT = 8πr20/3 with r0 the classical electron
radius r0 =q2e
mec2, UB the magnetic energy density equal to UB = | ~B|2
8π and β = vc the ratio
of v to the speed of light∗. Integrating over the power for the single electron synchrotron
radiation times the electron distribution will give the total emitted synchrotron radiation
(Section 2.5).
2.4 Relativistic Beaming of Synchrotron Radiation
For radiation that has a velocity perpendicular to the acceleration, as is the case with
synchrotron radiation, the angular distribution of the emission is shown in Figure 3. The
aperture of the cone is derived from the ratio of the velocity perpendicular to the direction
of propagation of the electron to the parallel velocity:
tan θ =u⊥u‖
=u′⊥
γe(1 + vc2u′⊥)
(1 + vc2u′⊥)
u′‖ + v=
|u′| sin θ′
γe(|u′| cos θ′ + v)
Given that u′‖ = cos θ′|u′|:
cos θ =u‖
|u|=
u′‖ + v
(1 + vc2u′‖)
1
|u′|=
cos θ′ + v/|u′|1 + v|u′| cos θ′/c2
(9)
For a photon |u′| = c, and if it is emitted at right angles to v, then: θ′ = π/2, and:
tan θ =c
γev→ sin θ = tan θ cos θ =
c
γev
v
c=
1
γe(10)
Thus small angles of θ∼ γ−1e gives an aperture of 2γ−1
e . A small range of the total
emitted radiation will be visible to the observer because the radiation is emitted in these
cones by a particle in helical motion, see Figure 4. One can think of this as a lighthouse:
only if the beam is pointed at the observer, the radiation from the electron can be observed.
The observed time difference in arrival time of the pulses at the observer is derived
from the equations of motion and Figure 4, given that γe � 1:
∗In fact the average of the isotropic distribution of velocities over all angles is used < β2⊥ >= 2β2
3
11
“ I )
Figutv 4. I la Dipole mdiation pattern for patii& at mst.
(11)
Figwp 4.llb Angular dktribution of mdiatim emitted by a partic& with parollel accelerariosl and wlocity.
( r ) Figutv 4 .11~ Same as a
(11)
Figwp 4.11d Angular distribution of mdhtion emitted by a particle with perpendicular acceleration and wlocity.
3-Extreme Relativistic Limit. When y>> I , the quantity (1 - Pp) in the denominators becomes small in the forward direction, and the radiation becomes strongly peaked in this direction. Using the same arguments as before, we obtain
Figure 3: Schematic view of the angular distribution of radiation emitted by a particle that has a velocity
perpendicular to the acceleration (Rybicki & Lightman 1979).
170 Synchrotron Radhtion
Figurn 6.2 Emission cones at variouS points of an accelerated particle's trajectory.
the direction of observation. The distance As along the path can be computed from the radius of curvature of the path, a = As/AB.
From the geometry we have A0 = 2/y, so that As = 2 a / y . But the radius of curvature of the path follows from the equation of motion
AV 4 ym- = - v x B , A t c
Since (Av( = v A 0 and As = v At, we have
A 0 qBsina As ymcv '
wB sin a '
-=-
V a = -
(6.8a)
(6.8b)
Note that this differs by a factor sina from the radius of the circle of the projected motion in a plane normal to the field. Thus A s is given by
2u yw, sin a
As = (6.8~)
The times t, and t, at which the particle passes points 1 and 2 are such that A s = u(t, - t , ) so that
2 y o B sin a '
t , - t ,x
Let t f and tt be the arrival times of radiation at the point of observation
Figure 4: Two cones of the angular distribution of radiation at times t1 and t2 that are visible to the observer.
Radiation emitted at times not in the domain [t1, t2] are not visible to the observer (Rybicki & Lightman
1979).
∆tA ≈(γ3eωB sinα
)−1(11)
where the pitch angle α is the angle between the field and the velocity. Note that this
pulse is a factor γ3e smaller than the gyration frequency ωB . This means that the spectrum
will be broader than the gyration frequency by a factor γ3e . At a certain frequency which
is at the order of the inverse of the observed time difference (given by equation (11)) the
spectrum will be cut off. This so-called critical frequency ωe is defined as follows:
ωe ≡3
2γ3eωB sinα (12a)
νe =3
4πγ3eωB sinα =
3
4πγ2e
qe| ~B|mec
sinα (12b)
12
2.5 The Spectral Energy Distribution
The slope of a log-log plot of the total emitted flux as a function of the frequency is known
to change at certain frequencies. The spectral index s gives this slope and will be derived
in this section and the physical phenomena that drive said change is described.
The power times the electron distribution will be integrated to obtain the flux as a
function of the frequency. Rybicki & Lightman (1979) gives the particle distribution for
relativistic electrons N(γe) as a power law:
N(γe)dγe ∝ γ−pe dγe for γe ≥ γm (13)
where p is the particle distribution index and γm the minimum Lorentz factor for the
distribution to be valid. Ginzburg & Syrovatskii (1965) gives the dependence of the total
power on the frequency P (ν).
P (ν) ∝ F (x) ∝
x1/3
e−xx1/2
for x� 1
for x� 1(14)
with x = ν/νe. Since νe ∝ γ2e (see equation (12b)) the coordinates can be transformed to
x:
γe ∝ x−1/2ν1/2, (15)
dγe ∝ ν1/2x−3/2dx. (16)
N(γe) and dγe are rewritten in terms ofN(x) and dx by doing a coordinate transformation:
∫ ∞γm
P (ν)N(γe) ∝ ν−(p−1)/2
∫ ν/νm
0
x(p−3)/2+1/3dx
∝ ν−(p−1)/2[x(p−1)/2+1/3
]ν/νm0
∝ ν1/3 (17)
for ν < νm, and for ν > νm we get:
∫ ∞γm
P (ν)N(γe) ∝ ν−(p−1)/2
∫ ν/νm
0
e−xxp/2−1dx
∝ ν−(p−1)/2 [ Γ(p/2, ν/νm) ] ∝ ν−(p−1)/2. (18)
where Γ is the incomplete gamma function that yields a constant. The normal gamma
function would be the result of this integral from zero to infinity whereas the integral here
is from zero to ν/νm.
13
Equation (17) shows that the spectral index s = 1/3 for ν < νm and equation (18)
shows that s = −(p − 1)/2 for ν > νm. Apart from these spectral indices there are
spectral indices associated with synchrotron self-absorption. Synchrotron self-absorption
happens when a synchrotron photon is absorbed due to interaction with a charge in the
magnetic field. The slope for self-absorption is s = 2 and s = 5/2 depending on the order
of the self absorption frequency νa and νm (Sari et al. 1998; Rybicki & Lightman 1979).
There are different electron distributions as well. The electrons with the highest energy
lose their energy much faster than the electrons with lower energy because P ∝ γ2e , see
equation (8). Very energetic electrons with γe > γc cool down to γc in a certain cooling
time tc. For these electrons N(γe) ∝ γ−p−1e dγe. The same derivation as for equation (18)
can be followed to obtain s = −p/2. In the fast cooling regime (νm > νc) the electron
energy distribution is N(γe) ∝ γ−2e dγe. The same derivation as for equation (17) and
equation (18) yields s = 1/3 and s = −1/2 respectively.
To summarize: the spectrum of the afterglow yields four important observable pa-
rameters. The maximum flux and three characteristic or break frequencies: the self-
absorption frequency νa, the peak frequency νm and the cooling frequency νc
(Sari et al. 1998). The change in slope of the log-log plot of the flux versus the frequency
at the characteristic frequencies (i.e. the different spectral indices s) is given in Figure 5.
The self-absorption plays a part at the low end whereas the cooling frequency plays a part
at the high end of the frequency domain thus νa < νc. There are three possibilities to be
observed at different times since the burst: νa < νc < νm is known as fast cooling and is
observed after ∼1 hour; νa < νm < νc and νm < νa < νc is known as slow cooling and is
observed respectively after ∼1 day and ∼1 month to ∼1 year.
14
1.3 Broadband afterglow physics 13
Figure 1.6 — Schematic log-log spectral energy distributions (SEDs) which are typicallyobserved in GRB afterglows. The dominant emission mechanism is synchrotron radia-tion, caused by relativistic electrons which are accelerated by a relativistic blast wave.The SEDs are determined by the peak flux and three break frequencies: the peak fre-quency !m, the cooling frequency !c, and the synchrotron self-absorption frequency ! a.Some of the slopes are determined by the power-law index p of the electron energy dis-tribution. The shape of the SEDs depends on the ordering of the three break frequencies.Because of the temporal behaviour of the break frequencies, there are in practice threepossible orderings observed: !a < !c < !m (top panel; fast cooling regime; observed! 1 hour after the burst), !a < !m < !c (middle panel; slow cooling regime; ! 1 dayafter the burst), and !m < !a < !c (bottom panel; slow cooling regime; ! 1 month afterthe burst).
Figure 5: A schematic view of the log-log plot of the flux versus the frequency (the Spectral Energy Distribution,
SED) for the time development of GRB afterglows (from van der Horst (2007)). The plot shows the number
that the spectral index s equals above the line segments. The spectral index changes for three break frequencies:
the self-absorption frequency νa, the peak frequency νm and the cooling frequency νc. The top image shows
fast cooling while the middle and lower image shows slow cooling. Typical observation times for the SED are
respectively ∼1 hour, ∼1 day and ∼1 month after the burst.
Apart from obtaining νm and νc from observational data one could calculate their
theoretical values. The average energy < Ee > per electron is needed to calculate γm and
is as follows:
15
< Ee > =
∫∞γmEeN(γe)dγe∫∞
γmN(γe)dγe
=
∫∞γmγemec
2γ−pe dγe∫∞γmγ−pe dγe
=p− 1
p− 2γmmec
2 (19)
The electron number density and the nucleon energy density behind an ultra relativistic
shock is 4γne and 4γ2nmpc2 (Blandford & McKee 1976). The ratio εe of energy in
electrons to the energy in nucleons is given by:
εe ≡4γne < Ee >
4γ2nmpc2=
1 +X
2
me
mp
p− 1
p− 2
γmγ
(20)
where mp is the proton mass and γ is the Lorentz factor of the shock (note that the shock
may have a different Lorentz factor than the electrons in the shock!). The nucleon to
electron number density ratio equals its value before the shock nne
= 21+X where X is the
hydrogen mass fraction. Rewriting in terms of γm yields:
γm =2
1 +X
mp
me
p− 2
p− 1εeγ (21)
Similar to εe the ratio εB of the field energy density to a constant fraction of the postshock
nucleon energy density (Wijers & Galama 1999) is given by:
εB =UB
4γ2ρc2. (22)
Since UB = B2
8π the magnetic field (in the rest frame of the shock) is:
B = γc√
32πεBρ. (23)
The following equation gives the frequency νm as a function of γm (Sari et al. 1998):
νm = ν(γm) = γγ2m
qeB
2πmec= γ4qe
[2
1 +X
p− 2
p− 1εe
]2 [mp
me
]3√
8εBρ
πm2p
(24)
Note that this equation gives the frequency as measured in the frame of the observer,
explaining the factor γ.
The value for γc is defined as the Lorentz factor that electrons have when the dynamical
timescale and the cooling timescale equal t in the frame of the observer. The dynamical
timescale is the time it takes the shock to accelerate the electrons to this high energy. The
cooling timescale is the time it takes the electrons with γe > γc to cool down to γc. The
absorbed energy is equal to the synchrotron emission power per time where the power in
the rest frame of the shock is given by equation (8) with β = 1 and has to be multiplied
by γ2 to obtain the observed power (Rybicki & Lightman 1979):
γγcmec2 = P (γc)t =
4
3σT cγ
2γ2c
B2
8πt (25)
16
Rewriting in terms of γc and using the relation between γc and νc:
γc =6πmec
σT γB2t(26)
νc = ν(γc) = γγ2c
qeB
2πmec
= γ
[6πmec
σT γB2t
]2qeB
2πmec=
1
γ4
qeme
(σT tc)2
√81
8192π(εBρ)−3/2 (27)
The peak flux Fν,m is given by:
Fν,m =Ne · Pν,m
4πD2=
√32π3/2R3ρ3/2mec
3σT γ2ε
1/2B
9πD2qemp(28)
where Pν,m is the maximum power, Ne = 4πR3ρ/3mp and D is the distance to the object.
2.6 Time Evolution of the Blast Wave
In the previous sections we have determined the dependency of the various frequencies
and the peak flux on the Lorentz factor γ and the radius R. As the jet propagates through
the medium surrounding the GRB progenitor with density ρ = AR−k and it sweeps up
matter and the speed decreases. The parameter k has two interesting values: k = 0
for a homogeneous medium where A = nmp there is no R-dependence (e.g. interstellar
medium), and k = 2 for stellar wind. The radius R and the Lorentz factor γ both depend
on time. To derive the time-dependencies the evolution is assumed to be adiabatic. This
means that the energy is constant as compared to radiative evolution where the energy
decreases due to radiation losses. The energy of a spherical ultra-relativistic shock for the
adiabatic case is given by (Blandford & McKee 1976):
E =16
17πγ(t)2R(t)3ρc2 (29)
and is constant. Combined with equation (1) the time-dependence of γ and R can be
derived. Note however that the Lorentz factor behind the shock is a factor√
2 smaller
than the Lorentz factor of the shock front (Blandford & McKee 1976):
dR ≈ 4γ(t)2cdt (30)
Considering ρ = AR(t)−k and using equation (29) gives:
γ(t) =
(17E
(4c)(5−k)πAt(3−k)
)1/2(4−k)
∝ t3−k
2(4−k) (31)
R(t) =
(17Et
4πAc
)1/(4−k)
∝ t1
4−k (32)
17
The expressions for R(t) and γ(t) can be used to determine the time-dependence of
νm, νc and Fν,max.
For νm we get from equation (24):
νm ∝ γ4ρ1/2 ∝ γ4R−k/2 ∝ t−4(3−k)2(4−k) t
−k2(4−k) ∝ t
−3(4−k)2(4−k) ∝ t−3/2 (33)
From equation (27) we obtain νc:
νc ∝ γ−4ρ−3/2t−2 ∝ γ−4R3k/2t−2 ∝ t4(3−k)2(4−k) t
3k2(4−k) t
−4(4−k)2(4−k) ∝ t
−4+3k2(4−k) (34)
Fν,max is given by equation (28):
Fν,max ∝ γ2R3ρ3/2 ∝ γ2R3(2−k)
2 ∝ t−2(3−k)2(4−k) t
3(2−k)2(4−k) ∝ t
−k2(4−k) (35)
The above results are consistent with Starling et al. (2008). GRB afterglow data are
light curves where those parameters are measurable. The light curve in the slow cooling
regime (νm < νc) for X-ray and optical observations (νm < ν) as shown in the middle
plot of Figure 5 is described as follows (Sari et al. 1998):
Fν(t) = Fν,max
(ν
νm
)− p−12
for νm < ν < νc (36)
Fν(t) = Fν,max
(νcνm
)− p−12(ν
νc
)− p2
for ν > νc (37)
Plugging in the time-dependence:
Fν(t) ∝ t−k
2(4−k)
(1
t−3/2
)− p−12
∝ t−12(p−1)−k(3p−5)
4(4−k) for νm < ν < νc (38)
Fν(t) ∝ t−k
2(4−k)
(t−4+3k2(4−k)
t−3/2
)− p−12 (
1
t−4+3k2(4−k)
)− p2
∝ t−(3p−2)
4 for ν > νc (39)
The general dependence of the peak flux on the time and the frequency is parametrized
by the observable parameters α and β:
Fν(t) ∝ ν−βt−α. (40)
Then the parameters α and β are given by:
α =12(p− 1)− k(3p− 5)
4(4− k), β =
p− 1
2(41)
for νm < ν < νc. And for ν > νc:
α =(3p− 2)
4, β =
p
2(42)
18
The electron energy distribution index p is a function of the observable parameter α and
the parameter k and is given by:
p(α, k) =4(4− k)α+ 12− 5k
3(4− k)for νm < ν < νc (43)
p(α) =2(2α+ 1)
3for ν > νc (44)
How k depends on the observable parameters α and β can also be determined:
α =12(p− 1)− k(3p− 5)
4(4− k)=
24β − 6kβ + 2k
4(4− k)
⇒ k(α, β) =4(3β − 2α)
3β − 2α− 1(45)
Note that for ν > νc both α and β are independent of k so there is no equation for k in
that regime.
In summary: the dependence of the particle distribution index p and the density
parameter k on α and β given by Fν ∝ ν−βt−α in the slow cooling regime for X-ray and
optical data is derived. Equations (43), (44) and (45) give said relations between these
two physical properties of GRB afterglows and the observed light curves and spectra.
3 Method
I have written a computer program to calculate p and k from α and β. The computer
program takes input from text files in the form:
GRB βo βo-error βx βx-error αo αo-error αx αx-error
where the subscript indicates that the value is from optical or from X-ray observations.
There are three possible regimes that have a different order of the observed frequencies
and the cooling frequency νc. Both the optical and the X-ray frequencies could be lower
than the cooling break or both the optical and X-ray frequencies could be higher than the
cooling break. If this is the case then a single power law (PL) describes the observations.
The third options is that the optical frequency is lower and the X-ray frequency is higher
than the cooling break. In this case a broken power law (BR) describes the observations.
The density parameter k is calculated from measurements, but the p-values obtained
from the light curve below the cooling break are dependent of k. To calculate the p-values
from the light curve below the cooling break k is assumed to be either zero or two. Below
the cooling break the values and errors of p from the light curve are calculated using
equation (43). The notation of these values is as follows: p(αo, 0), p(αo, 2), p(αx, 0) and
p(αx, 2). The values of p from the spectrum are independent of k and are calculated using
equation (41). The notation of these values is as follows: pν<νc(βo) and pν<νc(βx). The
values of k are calculated using equation (45) for optical respectively X-ray observations
and their notation is as follows: k(αo, βo) and k(αx, βx).
19
Above the cooling break the values and errors of p(αo) and p(αx) are calculated using
equation (44) and pν>νc(β0) and pν>νc(βx) using equation (42).
The computer program calculates a comparison parameter nσ, given by:
nσ =|p1 − p2|
Error(p1) + Error(p2)(46)
that shows within how many standard deviations two different values p1 and p2 are con-
sistent. Table 1 shows the three regimes with two conditions that have to be satisfied for
the GRB to be in that regime. For both conditions the two values p1 and p2 that have to
be consistent are shown.
Table 1: Overview of p-values that have to be consistent for a regime to satisfy the equations.
Regime Type Condition 1 Condition 2
p1 p2 p1 p2
(1) νo < νx < νc Power Law (PL) p(αo, k) pν<νc(βo) p(αx, k) pν<νc(βx)
(2) νc < νo < νx Power Law (PL) p(αo) pν>νc(βo) p(αx) pν>νc(βx)
(3) νo < νc < νx Broken Power Law (BR) p(αo, k) pν<νc(βo) p(αx) pν>νc(βx)
The program then checks regime (1) and regime (3) for both k = 0 and k = 2.
Regime (2) does not depend on k and is checked as well. If both nσ-values are less
than one the p values are consistent within one sigma meaning that a regime satisfies the
equations. If none of the regimes satisfy the equation then the requirement is eased to
nσ values less than two. The GRB is inconsistent with the derived equations within two
sigma if still none of the regimes satisfy the equations.
The program outputs LATEX-tables showing all the calculated p and k values with
errors calculated using least squares error propagation and nσ values for each input line.
The program assists in finding the regime for each GRB and writes the findings to a text
file. That text file is analyzed to see if a GRB has one conclusive option. If there is more
than one option the input is examined to see which dataset is the most likely dataset.
For instance for GRBs that have several options the dataset for which the SED and the
LC are measured at the same time is preferred over a dataset with different times of
measurement.
4 Data
Zaninoni et al. (2013) collected data for 68 long GRBs observed by the Swift telescope
from the literature and determined the shapes of the optical light curves. Together with
the X-ray data from Margutti et al. (2013), Zaninoni et al. (2013) have modeled the
optical and X-ray spectra.
Zaninoni et al. (2013) published the values for αo obtained by fitting power laws to
the optical light curve and present them in Table C.1. of their publication. The values
20
for the slope of the X-ray light curves αx are published by Margutti et al. (2013). From
the slope of the SED it is not always clear if the slope does or does not show the cooling
break. Zaninoni et al. (2013) presents values for βo and βx for both a single power law
and a broken power law in Table C.2. respectively Table C.3. of their publication. In
Table C.8. of their publication it is shown if either a PL or a BR fit is better.
A subset of 49 of the GRBs is analyzed here. The GRBs used here have been selected
to only include the segment of the X-ray LC where normal decay is seen (X-ray flares,
steep decay and the plateau segment are not included).
The subset of data generally has two entries for each GRB: one for a PL fit and one
for a BR fit but may have more than two entries. The afterglow SED is influenced by
interactions of photons with dust and gas of the host galaxy; optical extinction influences
the value of βo. To correct for this phenomena assumptions have to be made about the
properties of the host galaxy. Said properties are assumed to be the same as found in
the Milky Way (MW), the Large Magellanic Cloud (LMC) or the Small Magellanic Cloud
(SMC). The corrections are done by Zaninoni et al. (2013).
5 Results
Table 2 shows the the GRBs in numerical order. If there is a solution then the p value
from the X-ray spectrum is shown. Compared to the optical spectrum the X-ray spectrum
has a lot more data points so the slope is better constrained. If the frequency lies below
the cooling break k is computable. For a broken power law this is the case for the optical
frequency. Therefore the k value calculated from the optical observations is shown unless
stated otherwise. The regime found is shown under comments in the table.
A total of 36 GRBs is consistent with the blast wave model for normal decay for which
the equations have been derived in Section 2, (referred to as ‘consistent’ from now on).
For those GRBs the value for p is plotted in Figure 6. Except for two values p with large
errors is distributed in the domain 1.66− 3.13.
Two methods of calculating the mean value of the p-distribution are used. The sum
over all p values divided by n = 36 yields µunweighted ≈ 2.46. The weighted mean is
calculated by:
µweighted =
n∑i=0
piσ(pi)2
n∑i=0
1σ(pi)2
≈ 2.76(1) (47)
where n = 36 is the total number of values in the sample. The error of the value is
included in calculating the mean value. A histogram is shown in Figure 7 to analyze
the distribution of p. A Gaussian fit has been tried to see if the p values are normal
distributed. This fit yields µ = 2.43 ± 0.079 and σ = 0.32 ± 0.079 with a reduced
χ2 = 7.42. For this fit the values of p larger than 3.13 have not been included because
of their large errors. The histogram does not show any information on the error of the
21
p value. Figure 8 shows the probability distribution of p where both the values and
the errors are taken into consideration. The calculated value of p has been used as the
mean value of a normalized Gaussian distribution with a standard deviation equal to the
calculated error of p. The sum of the p for all GRBs over the normalized Gauss function
gives the probability distribution and does include the error of the value. If a value has a
large error the Gaussian function will be very broad and low whereas a value with a small
error is narrow and has a high peak at the mean value. The probability distribution shows
a broad distribution with two high peaks (due to two measurements with small errors).
The outliers above p = 3.13 do not show a significant contribution because of their large
errors.
Of the 36 GRBs that are consistent 26 have an optical frequency below the cooling
break thus a value for k can be calculated (this means that 10 GRBs have frequencies
above the cooling break). Three have not been included in the plot seen in Figure 9
because the values are cut-off below or above 6 (and they have very large errors). The
histogram shown in Figure 10 shows the distribution of k where 3 has been used as cut-off
value (thus neglecting one more measurement). The histogram shows that the two most
likely values lie within the 0.25 to 0.75 and the 2.25 to 2.75 range, however the error
measurements are not included in this histogram so a probability distribution has been
calculated. The probability distribution in Figure 8 shows a peak around k = 0.5 and at
k = 2.
Here we discuss the individual GRBs that are uniquely consistent with k = 0 or
k = 2 and GRBs that are consistent or inconsistent with both k values. Nine GRBs
are consistent with k = 0 within 1σ and twelve within 2σ. Only one of those GRBs
is consistent uniquely with k = 0 within 1σ (thus also within 2σ): GRB090926A yields
k = −0.10± 0.37.
Eleven GRBs are consistent with k = 2 within 1σ and nineteen within 2σ. Five
of those GRBs are consistent uniquely with k = 2 within 1σ (thus also within 2σ):
GRB071010A yields k = 2.34± 0.66, GRB080319B yields k = 2.00± 0.07, GRB080413B
yields k = 2.38 ± 0.49, GRB090424 yields k = 1.60 ± 0.78, GRB090618 yields k =
2.39 ± 0.48. And three more are consistent uniquely with k = 2 within 2σ: GRB050319
yields k = 2.26±0.19, GRB080607 yields k = 2.67±0.56, GRB100901A yields 1.57±0.33.
Three GRBs are inconsistent with k = 0 and k = 2 within 2σ: GRB050820A yields
k = 0.67± 0.25, GRB050824 yields k = 5.24± 1.48, GRB060729 yields k = 0.37± 0.12.
Seven GRBs are consistent with both k = 0 and k = 2 within 1σ and ten within 2σ.
A total of 13 GRBs is inconsistent within 2 sigma. Five of the GRBs that are inconsis-
tent have relative high values for α meaning they could possibly be post jet break. That
would explain why this method does not yield consistent results as one of the assumptions
is that the input values are before the jet break. Two have relative low values for α and
two have relative low values for β. Those lower than average values influence the values
for p causing them to be inconsistent. These values might not be normal decay.
22
1
1.5
2
2.5
3
3.5
4
4.5
5
pµ = 2.76
Figure 6: Plot of the particle distribution index p for 36 GRBs consistent with the blast wave model. The
results as presented inTable 2 are plotted in numerical order from left to right.
0
2
4
6
8
10
12
14
1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8
N
p
Gaussian fit
Figure 7: Histogram (binwidth = 0.3) of the particle distribution index p for 36 GRBs consistent with the
blast wave model.
23
0 1 2 3 4 5
Pro
babi
lity
Dist
ribut
ion
p
Figure 8: Probability distribution of the particle distribution index p for 36 GRBs consistent with the blast
wave model.
-8
-6
-4
-2
0
2
4
6
8
k
Figure 9: Plot of parameter k for 23 GRBs consistent with the blast wave model. The results as presented in
Table 2 are plotted in numerical order from left to right. White space between points means for that GRB
ν > νc thus k is not computable. Three values are left out because the value for k is not well constrained.
24
0
1
2
3
4
5
6
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
N
k
Figure 10: Histogram (binwidth = 0.5) of parameter k for 22 GRBs consistent with the blast wave model.
-3 -2 -1 0 1 2 3 4 5
Pro
babi
lity
Dist
ribut
ion
k-3 -2 -1 0 1 2 3 4 5
Pro
babi
lity
Dist
ribut
ion
k-3 -2 -1 0 1 2 3 4 5
Pro
babi
lity
Dist
ribut
ion
k
Figure 11: Probability distribution of parameter k for 22 GRBs consistent with the blast wave model.
25
6 Discussion
This study shows a broad distribution of p from 1.66− 3.13 that is inconsistent with one
single value of p for a sample of 36 values. This study is consistent with observational
studies on the value of p for 10 BeppoSAX GRBs (e.g. Starling et al. 2008) and for 10 swift
GRBs (e.g. Curran et al. 2009) that show a broad distribution of p that is inconsistent
with one universal value. In comparison this study has a larger sample of results. A
detailed statistical analysis should be done to further investigate the results.
Theoretical studies of particles accelerated by ultra relativistic shocks show that the
value for p should be universal and equal to 2.2-2.3 (e.g. Kirk et al. 2000). Other theoretical
studies on the other hand indicate that the values for p should lie within a range of 1.5-4
(e.g. Baring 2004). This study is inconsistent with the former theoretical model and seems
to be consistent with the latter.
In the literature the density structure is usually assumed to be that of the interstellar
medium or that of a stellar wind. Here the density is parametrized by the parameter
k. The data used consists of long GRBs, where a density structure of a stellar wind
is expected based on the collapsar model. The majority of the calculated k values is
consistent with a stellar wind. However one GRB is uniquely consistent with k = 0 and
two GRBs show a k value around k = 0.5 that is well constrained. For those GRBs the
calculated value of p would significantly differ when k = 2 would have been assumed.
The results presented are calculated using the derived equations where some assump-
tions have been made. One of the assumptions is on-axis viewing of the jet. It is highly
unlikely this is true for one observation let alone all of the observations. The influence of
a viewing angle other than zero should be studied and taken into consideration in further
studies when calculating the results.
For the BR fit of β Zaninoni et al. (2013) have assumed that the difference ∆β between
β below and above the cooling break is equal to 1/2 for all but some GRBs. The ∆β is
given by the difference in the slope of the fit to the spectrum and assumes the model is
true. The presented values of p are calculated from βx. The influence of this assumption
should be examined more careful before drawing conclusions about the model. Future
studies could investigate the distribution of β and possible physical explanations for a
found distribution.
Zaninoni et al. (2013) provide information on which fit (PL/BR) describes the behavior
of the observations best. For ten GRBs the results of this study are inconsistent with their
findings. In that case Table 2 remarks this under comment. For twelve GRBs with both
PL and BR options the option that is consistent with the best fit according to Zaninoni
et al. (2013) is chosen as the best option. This is also remarked in Table 2 under comments.
The dataset used as input for this study includes data of the normal decay phase
only. Further studies could adapt the model such that other phases are included as well.
Other phases include the X-ray flares, the steep decay and the plateau phase found in the
canonical behavior of the GRB afterglow observations (Nousek et al. 2006).
26
7 Conclusion
The blast wave model describes the shock wave powered by an inner engine propagating
through some medium. The matter swept up by the ultra relativistic jet is accelerated
to ultra relativistic speeds. In the presence of high magnetic fields the ultra relativistic
electrons are known to emit synchrotron radiation. This synchrotron radiation causes the
GRB afterglows. The frequency dependence β is derived by integrating the single electron
power times the electron energy distribution and the time dependence α is the effect of
the time evolution of the blast wave. The electron energy distribution index p is derived
from fits to the X-ray and optical spectrum and light curves obtained from the literature.
The medium the shock wave propagates through is described by the density parameter
k that is calculated from the same observable parameters. A broad distribution of p in
the range 1.66− 3.13 is found. Most of the values of k are consistent with a stellar wind
density surrounding the GRB.
Acknowledgements
By doing this research I have had a lot of fun and I have learned a lot about the theory
describing gamma ray bursts. I have also learned quite a lot about processing large
amounts of data. This research has not been possible without the help, guidance and
feedback of my supervisor: dr. A.J. van der Horst. I am very grateful for your help and
support, Alexander. I would also like to thank my co-supervisor, prof. dr. R.A.M.J.
Wijers for his feedback on my work.
Without the support and love of my mom and dad I would not have been able to get
to the point where I am finishing my Bachelor of Science in physics. After long weeks
of studying I was always welcome to end the week with a pizza on Friday evening at my
mom and dad’s to recharge my energy levels for the week to come. Mom and dad, thanks!
Whilst looking for ways to improve referencing in BibTeX by using journal shortcuts
I stumbled upon a request by the ADS to thank them for providing a search engine for
publications by quoting the following line in publications: “This research has made use
of NASA’s Astrophysics Data System”.
The image at the title page of this report has been found online and is credited to
Nicolle Rager (National Science Foundation) (2005).
Lastly I would like to thank the Anton Pannenkoek Institute for providing an envi-
ronment where I was able to work quietly and fully focussed. Every now and then I was
able to relax for a couple of minutes in the nice chairs in the break room whilst enjoying
a lovely hot cup of tea before focussing on the next problem to solve.
27
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29
Table 2: Most likely values for p and k. Comments about regime and findings.
GRB Fit p k Comments
050319 BR 2.21 ± 0.09 2.26± 0.19 νo < νc < νx (between) for k = 2 within 1 sigma. Best solution of two.
050408 BR 2.42 ± 0.62 0.01± 3.74 νo < νc < νx (between) for k = 0 or k = 2. PL also possible within 1 sigma due to large p-errors. SED
best fit by BR∗.
050416A BR 2.05 ± 0.27 0.51± 1.54 νo < νc < νx (between) for k = 0 within 1 sigma, k=2 consistent within 2 sigma.
050730 – No consistent solution within 2 sigma. Very steep αx.
050820A PL 2.69 ± 0.04 0.67± 0.25 νo < νx < νc (below) for k = 0 within 2 sigma. However SED best fit by BR∗.
050824 BR 4.60 ± 2.54 5.24± 1.48 νo < νc < νx (between) for k = 2 within 2 sigma. But large error on p(bo). Large k-error, possibly because
βo and βx are about twice as big as average. However SED best fit by PL∗.
050908 PL 2.50 ± 0.71 νc < νo < νx (above) within 2 sigma. ν > νc thus k cannot be calculated.
050922C BR 2.12 ± 0.20 0.43± 0.97 νo < νc < νx (between) for k = 0 within 2 sigma. Best of 4 options.
051109A BR 2.12 ± 0.11 −22.67± 75.05 νo < νc < νx (between) for k = 0 within 2 sigma. The value for k is not well constrained.
051111 BR 2.42 ± 0.28 −2.36± 4.25 νo < νc < νx (between) for k = 0 within 1 sigma.
060124 BR 2.02 ± 0.25 −0.27± 1.88 νo < νc < νx (between) for k = 0 and k = 2 within 2 sigma. However SED best fit by PL∗.
060206 – No consistent solution within 2 sigma. Flat αx. p(a0, 2) consistent with p(βo) for BR but SED best fit by
PL∗.
060526 – No consistent solution within 2 sigma. Flat αo. p(αx) consistent with p(βx) for BR.
Continued on next page
30
Table 2: Continued from previous page.
GRB Fit p k Comments
060614 – No consistent solution within 2 sigma. Steep αo and αx. Possibly post jet break.
060729 PL 2.75 ± 0.01 0.37± 0.12 νo < νc < νc (below) for k = 0 within 2 sigma. k value from X-ray because k(αo, βo) is unconstrained.
However SED best fit by BR∗.
060904B PL 2.42 ± 0.35 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. Consistent below cooling break
for k = 0 within 2 sigma. For BR consistent for k = 0 and k = 2, but SED best fit by PL.
060908 – No consistent solution within 2 sigma. BR is consistent for k = 2 within 3 sigma. However SED best fit by
PL∗.
060912A BR 2.14 ± 0.36 1.53± 0.83 νo < νc < νx (between) for k = 2 within 1 sigma. Consistent with PL with WM properties too, but the
SED is best fit by BR∗ with SMC properties. So this is the best of the two.
061007 – No consistent solution within 2 sigma. Small errors on p due to small errors on α, β.
061126 – No consistent solution within 2 sigma. Flat βo and βx.
070125 – No consistent solution within 2 sigma. Steep αo and αx. Possibly post jet break
070208 BR 2.45 ± 0.28 −56.61± 419.84 νo < νc < νx (between) for k = 0 within 2 sigma. The value for k is not well constrained.
070318 PL 2.49 ± 0.74 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. SED best fit by PL. Here SED
is measured at the same time as LC, whereas for the other PL option the SED has been measured much
earlier. Best of four options.
Continued on next page
31
Table 2: Continued from previous page.
GRB Fit p k Comments
070411 PL 2.34 ± 0.54 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. SED best fit by PL. Here SED
is measured at the same time as LC, whereas for the other PL option the SED has been measured much
earlier.
070529 PL 4.26 ± 1.01 νc < νo < νx (above) within 2 sigma. ν > νc thus k cannot be calculated. SED best fit by PL but BR also
consistent within 2 sigma. Best of two options. High p value because both α and β are steep.
071003 PL 2.55 ± 0.10 νc < νo < νx (above) within 2 sigma. ν > νc thus k cannot be calculated.
071010A PL 3.05 ± 0.03 2.34± 0.66 νo < νx < νx (below) for k = 2 within 2 sigma. SED best fit by PL, but BR is consistent within 1 sigma.
k from X-ray is 0.83± 1.20. Best solution of three.
080310 PL 1.77 ± 0.07 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. SED best fit by PL. Only option
with normal decay.
080319B BR 1.97 ± 0.05 2.00± 0.07 νo < νc < νx (between) for k = 2 within 1 sigma.
080413B BR 2.48 ± 0.49 2.38± 0.49 νo < νc < νx (between) for k = 2 within 1 sigma. Only solution within 1 sigma, five options within 2 sigma.
080603A BR 3.05 ± 0.40 −0.20± 2.03 The only solution is at 2.12 sigma. The difference between the β values is 0.782 instead of the expected 0.5
but they are consistent within 1 sigma. p(βo) = 2.48± 0.31 but would normally be the same as p(βx).
080607 BR 2.60 ± 0.21 2.67± 0.56 νo < νc < νx (between) for k = 2 within 1 sigma.
080710 BR 2.12 ± 0.39 −1.17± 3.88 νo < νc < νx (between) for k = 0 within 2 sigma. However SED best fit by PL∗.
080721 BR 2.54 ± 0.46 0.65± 1.95 νo < νc < νx (between) for k = 0 within 1 sigma. However SED best fit by PL∗.
080810 BR 2.23 ± 0.24 1.12± 0.76 νo < νc < νx (between) for k = 0 or k = 2 both within 2 sigma. However SED best fit by PL∗.
Continued on next page
32
Table 2: Continued from previous page.
GRB Fit p k Comments
080913 PL 1.74 ± 0.34 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. Within 2 sigma also below
cooling break for k = 2. SED best fit by PL. Here SED is measured at the same time as LC, whereas for
the other PL option the SED has been measured much earlier.
081008 PL 2.87 ± 0.01 1.36± 0.70 νo < νx < νc (below) for k = 0 or k = 2 within 2 sigma. k value from X-ray because k(αo, βo) is
unconstrained. SED best fit by PL∗ but also consistent for BR within 2 sigma.
081029 – No consistent solution within 2 sigma. αo is steep and αx is either shallow or steep.
081203A – No consistent solution within 2 sigma. Steep αo and αx.Possibly post jet break.
090102 – No consistent solution within 2 sigma. Flat β, especially optical.
090313 BR 3.13 ± 0.47 14.15± 18.33 νo < νc < νx (between) for k = 0 within 2 sigma. Two inputs give these exact results. The value for k is
not well constrained.
090424 BR 1.66 ± 0.36 1.60± 0.78 νo < νc < νx (between) for k = 2 within 1 sigma. Best of three (though the other are 2.09 sigma) results.
However SED best fit by PL∗.
090426 PL 2.14 ± 0.35 νc < νo < νx (above) within 1 sigma. ν > νc thus k cannot be calculated. SED best fit by PL but BR
consistent within 1 sigma too. Best solution of two.
090618 BR 1.76 ± 0.50 2.39± 0.48 νo < νc < νx (between) for k = 2 within 1 sigma. However SED best fit by PL∗.
090926A PL 3.00 ± 0.06 −0.10± 0.37 νo < νx < νc (below) for k = 0 within 1 sigma. Best solution of three.
091018 – No consistent solution within 2 sigma.
091127 – No consistent solution within 2 sigma.
Continued on next page
33
Table 2: Continued from previous page.
GRB Fit p k Comments
100418A PL 2.38 ± 0.05 νc < νo < νx (above) within 2 sigma. ν > νc thus k cannot be calculated.
100901A BR 2.72 ± 0.14 1.57± 0.33 νo < νc < νx (between) for k = 2 within 2 sigma.
∗According to Table C.8. of (Zaninoni et al. 2013)
34