11affiliationtext: Department of Astronomy, University of Texas at Austin,
Austin, TX 78712, USA22affiliationtext: Racah Institute for Physics, The
Hebrew University, Jerusalem, 91904, Israel

# Magnetic Fields In Relativistic Collisionless Shocks

Rodolfo Santana1a1aaffiliationmark:  , Rodolfo Barniol
Duran2b2baffiliationmark:  , & Pawan Kumar1c1caffiliationmark:  (a)
<EMAIL_ADDRESS>(b<EMAIL_ADDRESS>(c)
<EMAIL_ADDRESS>
###### Abstract

We present a systematic study on magnetic fields in Gamma-Ray Burst (GRB)
external forward shocks (FSs). There are 60 (35) GRBs in our X-ray (optical)
sample, mostly from Swift. We use two methods to study $\epsilon_{B}$
(fraction of energy in magnetic field in the FS). 1. For the X-ray sample, we
use the constraint that the observed flux at the end of the steep decline is
$\geq$ X-ray FS flux. 2. For the optical sample, we use the condition that the
observed flux arises from the FS (optical sample light curves decline as $\sim
t^{-1}$, as expected for the FS). Making a reasonable assumption on $E$ (jet
isotropic equivalent kinetic energy), we converted these conditions into an
upper limit (measurement) on $\epsilon_{B}n^{2/(p+1)}$ for our X-ray (optical)
sample, where $n$ is the circumburst density and $p$ is the electron index.
Taking $n=1\mbox{ cm}^{-3}$, the distribution of $\epsilon_{B}$ measurements
(upper limits) for our optical (X-ray) sample has a range of $\sim
10^{-8}-10^{-3}$ ($\sim 10^{-6}-10^{-3}$) and median of $\sim\mbox{ few}\times
10^{-5}$ ($\sim\mbox{ few}\times 10^{-5}$). To characterize how much
amplification is needed, beyond shock compression of a seed magnetic field
$\sim 10\mu\mbox{G}$, we expressed our results in terms of an amplification
factor, $AF$, which is very weakly dependent on $n$ ($AF\propto n^{0.21}$).
The range of $AF$ measurements (upper limits) for our optical (X-ray) sample
is $\sim 1-1000$ ($\sim 10-300$) with a median of $\sim 50$ ($\sim 50$). These
results suggest that some amplification, in addition to shock compression, is
needed to explain the afterglow observations.

###### Subject headings:

gamma-ray burst: general -radiation mechanisms: non-thermal - methods:
analytical

##  1\. Introduction

Gamma-Ray Bursts (GRBs) are bright explosions occurring at cosmological
distances which release gamma-rays for a brief time, typically on a timescale
of $\sim\mbox{ few}\times$ 10 sec (e.g. Piran 2004, Gehrels et al. 2009, Zhang
2011). This short-lived emission of gamma-rays is known as the prompt
emission. After the prompt emission, long-lived emission in the X-ray,
optical, and radio bands (on timescales of days, months, or even years) is
also observed from what is called the “afterglow”. Although the mechanism for
the prompt emission is currently being debated, the afterglow emission has a
well-established model based on external shocks (Rees & Meszaros, 1992;
Meszaros & Rees, 1993; Paczynski & Rhoads, 1993). In this framework, a
relativistic jet emitted by the central engine interacts with the medium
surrounding the GRB progenitor. This interaction produces two shocks; the
external-reverse shock and the external-forward shock (Meszaros & Rees, 1997;
Sari & Piran, 1999b). The external-reverse shock heats up the jet while the
external-forward shock heats up the medium surrounding the explosion. The
external-reverse shock is believed to be short-lived in the optical band (Sari
& Piran, 1999a) and might have been observed, perhaps, in a few cases. The
long-lived afterglow emission is interpreted as synchrotron radiation from the
external-forward shock. This shock is taken to produce a power-law
distribution of high energy electrons and to amplify the pre-existing seed
magnetic field in the surrounding medium. These high energy electrons are then
accelerated by the amplified magnetic field and emit radiation by the
synchrotron process.

One of the open questions in the field of GRB afterglows is: what is the
dynamo mechanism amplifying magnetic fields in the collisionless relativistic
shocks involved for GRB external shocks? The magnetic field strength
downstream of the shock front is expressed in terms of $\epsilon_{B}$, which
is defined as the fraction of energy that is in the magnetic field downstream
of the shock front. With this definition, the explicit expression for
$\epsilon_{B}$ is

$\epsilon_{B}=\frac{B^{2}}{32\pi nm_{p}c^{2}\Gamma^{2}},$ (1)

where $B$ is the co-moving magnetic field downstream of the shock front, $n$
is the density surrounding the GRB progenitor, $m_{p}$ is the proton mass, $c$
is the speed of light, and $\Gamma$ is the Lorentz factor of the shocked fluid
downstream of the shock front (e.g. Sari et al., 1998; Wijers & Galama, 1999;
Panaitescu & Kumar, 2000). If shock compression is the only mechanism
amplifying the magnetic field downstream of the shock front, then $B$ is given
by $B=4\Gamma B_{0}$ (e.g. Achterberg et al., 2001), where $B_{0}$ is the seed
magnetic field in the medium surrounding the GRB progenitor. Using this
expression for $B$, $\epsilon_{B}$ simplifies to $\epsilon_{B}=B_{0}^{2}/2\pi
nm_{p}c^{2}$. Using the value for the ambient magnetic field of the Milky Way
galaxy $B_{0}\sim\mbox{ few }\mu\mbox{G}$ and a density for the surrounding
medium of $n=1\mbox{ cm}^{-3}$, $\epsilon_{B}$ is expected to be $\sim
10^{-9}$.

Several studies have modeled afterglow data to determine what values of the
afterglow parameters best describe the observations (e.g. Wijers & Galama,
1999; Panaitescu & Kumar, 2002; Yost et al., 2003; Panaitescu, 2005). The
results from previous studies show that $\epsilon_{B}$ ranges from
$\epsilon_{B}\sim 10^{-5}-10^{-1}$. These values for $\epsilon_{B}$ are much
larger than the $\epsilon_{B}\sim 10^{-9}$ expected from shock compression
alone and suggest that some additional amplification is needed to explain the
observations. There have been several theoretical and numerical studies that
have considered possible mechanisms, operating in the plasma in the medium
surrounding the GRB, that can generate extra amplification for the magnetic
field. The mechanisms that have been proposed are the two-stream Weibel
instability (Weibel, 1959; Medvedev & Loeb, 1999; Gruzinov & Waxman, 1999;
Silva et al., 2003; Medvedev et al., 2005) and dynamo generated by turbulence
(Milosavljević & Nakar, 2006; Milosavljevic et al., 2007; Sironi & Goodman,
2007; Goodman & MacFadyen, 2008; Couch et al., 2008; Zhang et al., 2009;
Mizuno et al., 2011; Inoue et al., 2011).

Recent results (Kumar & Barniol Duran, 2009, 2010) found surprisingly small
values of $\epsilon_{B}\sim 10^{-7}$ for 3 bright GRBs with Fermi$/$LAT
detections. These values of $\epsilon_{B}$ are $\sim 2$ orders of magnitude
smaller than the smallest previously reported $\epsilon_{B}$ value and they
can be explained with the only amplification coming from shock compression of
a seed magnetic field of a few $10\mu\mbox{G}$ 111The values given above of
$\epsilon_{B}\sim 10^{-7}$ are under the assumption of $n=1\mbox{ cm}^{-3}$.
It is important to note that when reaching the conclusion that shock
compression provides enough amplification to explain the afterglow data, Kumar
& Barniol Duran (2009, 2010) did not assume a value for $n$. Also, the results
of small $\epsilon_{B}\sim 10^{-7}$ values do not depend on whether or not the
LAT emission is produced by the external-forward shock. These small
$\epsilon_{B}$ values were inferred from the late time X-ray and optical data
and from the constraint that the external-forward shock does not produce flux
at 150 keV that exceeds the observed prompt emission flux at 50 seconds..
Although this seed magnetic field is stronger than the one of the Milky Way
galaxy by about a factor $\sim 10$, seed magnetic fields of a few
$10\mu\mbox{G}$ have been measured before. The seed magnetic fields in the
spiral arms of some gas-rich spiral galaxies with high star formation rates
have been measured to be 20-30 $\mu\mbox{G}$ (Beck, 2011). Seed magnetic
fields as high as $0.5-18$ mG were measured in starburst galaxies by measuring
the Zeeman splitting of the OH megamaser emission line at 1667 MHz (Robishaw
et al., 2008).

Given this disagreement between the recent and previous results, the question
regarding the amplification of magnetic fields in GRB external relativistic
collisionless shocks remains unanswered. The first goal of this study is to
provide a systematic determination of $\epsilon_{B}$ for a large sample of
GRBs by using the same method to determine $\epsilon_{B}$ for each burst in
our X-ray or optical sample. This is the first time such a large and
systematic study has been carried out for $\epsilon_{B}$. Knowing the value of
$\epsilon_{B}$ for large samples will help us determine how much amplification
of the magnetic field is needed to explain the afterglow observations. We
mostly limit our samples to GRBs detected by the Swift satellite, with
measured redshift. In this study, we determine an upper limit on
$\epsilon_{B}$ for our X-ray sample and a measurement of $\epsilon_{B}$ for
our optical sample. We use a new method to determine an upper limit on
$\epsilon_{B}$ with X-ray data, which relies on using the steep decline
observed by Swift in many X-ray light curves. We expect that the observed flux
at the end of the steep decline is larger than the predicted flux from the
external-forward shock. Making reasonable assumptions about the other
afterglow parameters, we are able to convert this constraint into an upper
limit on $\epsilon_{B}$. To determine a measurement of $\epsilon_{B}$ for our
optical sample, we restrict our sample to light curves that show a power law
decline with a temporal decay $\sim 1$ at early times, $\sim 10^{2}-10^{3}$
seconds, as expected for external-forward shock emission. We choose this
selection criteria so that the optical emission is most likely dominated by
external-forward shock emission. Making the same reasonable assumptions for
the other afterglow parameters and using the condition that the observed flux
from the optical light curve is equal to the external-forward shock flux, we
are able to convert this condition into a measurement of $\epsilon_{B}$. We
also applied a consistency check for the bursts that are in common to our
X-ray and optical samples to make sure our results for $\epsilon_{B}$ are
correct. The second goal of this study is to determine how much amplification,
in addition to shock compression, is needed to explain the results for the
$\epsilon_{B}$ upper limits$/$measurements. To quantify how much amplification
beyond shock compression is required by the observations, we also express the
results we found for the $\epsilon_{B}$ upper limits (measurements) for our
X-ray (optical) sample in terms of an amplification factor.

This paper is organized as follows. We begin in Section 2 by presenting a
review of the values previous studies have found for the microphysical
afterglow parameters $\epsilon_{e}$ (the fraction of energy in electrons in
the shocked plasma) and $\epsilon_{B}$. In Section 3 (Section 4), we present
the method we use to determine an upper limit (measurement) on $\epsilon_{B}$
and apply it to our X-ray (optical) sample of GRBs. In Section 5, we use the
GRBs that are in common to both samples to perform a consistency check. We
search for a possible correlation between the kinetic energy of the blastwave
and $\epsilon_{B}$ in Section 6. In Section 7, we write our results for
$\epsilon_{B}$ for our X-ray and optical samples in terms of an amplification
factor, which quantifies how much amplification – beyond shock compression –
is required by the observations. Lastly, in Section 8, we discuss our results
and present our conclusions. The convention we use for the specific flux
$f_{\nu}$, the flux per unit frequency $\nu$, is
$f_{\nu}\propto\nu^{-\beta}t^{-\alpha}$. In this convention, $\beta$ is the
spectral decay index and $\alpha$ is the temporal decay index. For a GRB at a
given redshift $z$, when calculating the luminosity distance to the GRB,
$d_{L}$, we used the Cosmological parameters $H_{0}=71\mbox{
km}/\mbox{sec}/\mbox{Mpc}$, $\Omega_{m}=0.27$, and $\Omega_{\Lambda}=0.73$.

##  2\. Literature Review Of Values Of $\epsilon_{e}$ and $\epsilon_{B}$

The flux observed from the external-forward shock depends on 6 parameters.
These parameters are $E$, $n$, $s$, $\epsilon_{e}$, $\epsilon_{B}$, and $p$.
$E$ is the isotropic equivalent kinetic energy of the jet and $n$ is the
number density of the surrounding medium. The density is taken to be
spherically symmetric and to decrease with $r$ as $n(r)\propto r^{-s}$, where
$s$ is a constant determining the density profile of the surrounding medium
and $r$ is the distance from the center of the explosion. Two cases are
usually considered for the density profile: $s=0$ and $s=2$, which
respectively correspond to a constant density medium and a wind medium. The
microphysical parameters are $\epsilon_{e}$ and $\epsilon_{B}$, where
$\epsilon_{e}$ ($\epsilon_{B}$) is the fraction of energy in the electrons
(magnetic field) in the shocked fluid. The microphysical parameters are taken
to be constant throughout the afterglow emission. A power law distribution of
electrons, $dN_{e}/d\gamma_{e}\propto\gamma_{e}^{-p}$ with
$\gamma_{e}\geq\gamma_{i}$, where $\gamma_{e}$ is the Lorentz factor of the
electrons, $\gamma_{i}$ is the minimum Lorentz factor of the electrons, and
$N_{e}$ is the number of electrons, is assumed to be produced when the
external-forward shock interacts with the surrounding medium. The power-law
index of the electron distribution, $p$, is a constant known as the electron
index.

In practice, it is very difficult to determine the values of the 6 afterglow
parameters. The value of $p$ and the density profile of the surrounding medium
(whether we have a $s=0$ or $s=2$ medium) can be determined from observations
of the afterglow spectral decay and temporal decay of the light curve with the
so-called “closure” relations (e.g. Zhang et al., 2006). The remaining 4
afterglow parameters are more difficult to determine. What is needed to
determine these 4 parameters is observations of the afterglow emission in the
4 different spectral regimes of the synchrotron afterglow spectrum (we will
discuss the afterglow synchrotron spectrum in more detail in Section 3.4). In
practice, most GRBs do not have this wealth of observations. In order to
determine these 4 parameters, previous works have either focused only on
determining the afterglow parameters for bursts with high quality data,
spanning all portions of the synchrotron spectrum, or have applied various
simplifying assumptions.

We performed a literature search for papers that determine values for
$\epsilon_{e}$ and $\epsilon_{B}$ to get an idea of what typical values
previous works have found. Different authors applied different techniques for
finding $\epsilon_{e}$ and $\epsilon_{B}$. When displaying the results from
the literature, we did not discriminate against any method and simply plotted
every value we found. However, we did not consider works that made simplifying
assumptions when determining $\epsilon_{e}$ and $\epsilon_{B}$, such as
equipartition of proton and electron energy ($\epsilon_{e}$). The GRBs for
which we found $\epsilon_{e}$ and $\epsilon_{B}$ values are shown in Table
LABEL:epsilon_e_and_epsilon_B_table in Appendix A. Except for GRB 080928, all
the GRBs in our sample have radio, optical, and X-ray observations, allowing
for a determination of all the afterglow parameters. We included GRB 080928 in
our sample because $\epsilon_{e}$ and $\epsilon_{B}$ were able to be uniquely
determined from optical and X-ray observations (Rossi et al., 2011).

The $\epsilon_{e}$ values we found in the literature for 29 GRBs are shown in
the histogram in the left panel of Figure 1. There is a narrow distribution
for $\epsilon_{e}$; it only varies over one order of magnitude, from $\sim
0.02-0.6$, with very few GRBs reported to have $\epsilon_{e}<0.1$. The mean of
this distribution is 0.24 and the median is 0.22. About 62% of the GRBs in
this sample have $\epsilon_{e}\sim 0.1-0.3$. These results for $\epsilon_{e}$
are also supported by recent simulations of relativistic magnetized
collisionless electron-ion shocks presented in Sironi & Spitkovsky 2011, where
they found $\epsilon_{e}\sim 0.2$. The narrow distribution of $\epsilon_{e}$
values from the literature and the results from recent simulations both show
that $\epsilon_{e}$ does not change by much from GRB to GRB.

The $\epsilon_{B}$ values we found in the literature for 30 GRBs are show in
the histogram in the right panel of Figure 1. Comparing the two histograms in
Figure 1, we can immediately see that there is a much wider range in the
distribution of $\epsilon_{B}$, with $\epsilon_{B}$ ranging from $\sim
3.5\times 10^{-5}-0.33$. A noticeable peak, containing about 24% of the
bursts, is seen in the bin with $-1<\mbox{log}_{10}(\epsilon_{B})<-0.5$. Two
other peaks, each containing about 17% of the GRBs, are seen in the bins with
$-2<\mbox{log}_{10}(\epsilon_{B})<-1.5$ and
$-4<\mbox{log}_{10}(\epsilon_{B})<-3.5$. The mean of this distribution is
$6.3\times 10^{-2}$ and the median is $1.0\times 10^{-2}$. The important point
of the $\epsilon_{B}$ histogram is that $\epsilon_{B}$ varies over 4 orders of
magnitude, showing that $\epsilon_{B}$ has a wide distribution and is an
uncertain parameter.

$\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{epsilon_e_histogram.eps}&\includegraphics[scale={0.6}]{epsilon_B_histogram.eps}\end{array}\end{array}$

Figure 1.— Left Panel: Histogram of the distribution of $\epsilon_{e}$ values
we found in the literature (Table LABEL:epsilon_e_and_epsilon_B_table in
Appendix A). Right Panel: Histogram of the distribution of $\epsilon_{B}$
values we found in the literature (Table LABEL:epsilon_e_and_epsilon_B_table
in Appendix A)

##  3\. Upper Limit On $\epsilon_{B}$ With Swift X-ray Light Curves

###  3.1. Constraining $\epsilon_{B}$ With The X-ray Light Curve Steep Decline

One interesting property found by Swift (Gehrels et al., 2004) is that at
early times, about 50% of the light curves detected by the XRT (X-ray
Telescope, Burrows et al., 2005) display a very rapid decline in flux, known
as the steep decline (Gehrels et al., 2009). The flux during the steep decline
typically decays as $t^{-3}$ and it usually lasts $\sim 10^{2}-10^{3}$ sec. By
extrapolating the BAT (Burst Alert Telescope, Barthelmy et al., 2005) emission
to the X-ray band, O’Brien et al. 2006 showed that there is a continuous
transition between the end of the prompt emission and the start of the steep
decline phase. This important conclusion lead to the interpretation that the
X-ray steep decline has an origin associated with the end of the prompt
emission.

The favored interpretation for the origin of the steep decline is high
latitude emission (Kumar & Panaitescu, 2000). Although high latitude emission
is able to explain most of the steep decline observations, some GRBs display
spectral evolution during the steep decline (Zhang et al., 2007b), which is
not expected. In any case, the steep decline cannot be produced by the
external-forward shock. Therefore, the observed flux during the steep decline
should be larger than or equal to the flux produced by the external-forward
shock. We do however assume that the time at which the steep decline typically
ends, at about $10^{2}-10^{3}$ seconds, is past the deceleration time222The
deceleration time marks the time when about half of the kinetic energy of the
blastwave has been transferred to the surrounding medium.. For our upper limit
on $\epsilon_{B}$ with X-ray data, we will use the expression for the flux
from the external-forward shock, which uses the kinetic energy of the
blastwave given by the Blandford and McKee solution (Blandford & McKee, 1976).
Since this solution is only valid for a decelerating blastwave, we need to be
past the deceleration time for it to be applicable.

Theoretically, it is expected that that the deceleration time occurs before
the end of of the steep decline. Depending on the density profile of the
surrounding medium, the deceleration time $t_{\mathrm{dec}}$ is given by

$t_{\mathrm{dec}}=\left\\{\begin{array}[]{ll}\hskip-7.0pt(220\mbox{
sec})E_{53}^{1/3}n_{0}^{-1/3}\Gamma_{2}^{-8/3}(1+z)&s=0\\\ &\\\
\hskip-7.0pt(67\mbox{
sec})E_{53}A_{*,-1}^{-1}\Gamma_{2}^{-4}(1+z)&s=2\end{array}\right.$ (2)

(e.g. Panaitescu & Kumar, 2000). In these expressions, $\Gamma$ is the Lorentz
factor of the shocked fluid, $z$ is the redshift, and we have adopted the
usual notation $Q_{n}\equiv Q/10^{n}$. For $s=2$, the proportionality constant
of the density, $A$, is normalized to the typical mass loss rate and stellar
wind velocity of a Wolf-Rayet star, which is denoted by $A_{*}$ and is defined
as $A_{*}\equiv A/(5\times 10^{11}\mbox{ g}\mbox{ cm}^{-1})$ (Chevalier & Li,
2000). For typical GRB afterglow parameters of $E_{53}=1$, $n_{0}=1$ (or
$A_{*}=0.1$ for $s=2$), $\Gamma_{2}=3$, and $z=2.5$, the deceleration time is
under $100$ seconds for both $s=0$ and $s=2$. Although there can be a large
uncertainty in the afterglow parameters $E$ and $n$, $\Gamma$ is the most
important parameter when calculating $t_{\mathrm{dec}}$ since
$t_{\mathrm{dec}}$ has a very strong dependence on $\Gamma$. For $s=0$
($s=2$), even if we take extreme parameters for $E$ and $n$, such as a high
kinetic energy of $E_{53}=100$ and a low density of $n=10^{-3}\mbox{ cm}^{-3}$
($A_{*}=10^{-2}$), with a typical $\Gamma_{2}=3-4$ (e.g. Molinari et al.,
2007; Xue et al., 2009; Liang et al., 2010), $t_{\mathrm{dec}}$ is still a few
hundred seconds. Thus, since the deceleration time is less than the typical
time at which the steep decline ends, the onset of the external-forward shock
emission occurs before the end of the steep decline.

Observationally, the deceleration time is also seen to occur before the end of
the steep decline for many GRBs. If the dominant contribution to the light
curves at early times is the external-forward shock, the light curve is
expected to rise as a power law, reach a peak, and then decline as a power
law, with the peak signifying the deceleration time. In Liang et al. 2010, a
sample of optical light curves that display this peak is studied. In their
Figure 1, for each GRB, they display both the optical light curve and the
X-ray light curve. For all their bursts that display a steep decline in the
X-ray light curve (except for GRB 080303 and GRB 081203A), it can be seen that
the peak of the optical light curve occurs before the end of the X-ray steep
decline. We did not include GRB 080330 and GRB 081203A in our samples.

Since an increase in $\epsilon_{B}$ increases $f_{\mathrm{ES}}$ (external-
forward shock flux), the condition that the X-ray flux during the steep
decline should be larger than or equal to $f_{\mathrm{ES}}$ gives an upper
limit on $\epsilon_{B}$. Since our goal is to attain the most stringent upper
limit on $\epsilon_{B}$, we take this constraint at the end of the steep
decline. Explicitly, the constraint we will use to find an upper limit on
$\epsilon_{B}$ with X-ray data is

$f_{\mathrm{EoSD}}\geq f_{\mathrm{ES}}(E,n,s,\epsilon_{e},\epsilon_{B},p).$
(3)

In this inequality, $f_{\mathrm{EoSD}}$ represents the observed flux at the
end of the steep decline (EoSD). We have also explicitly shown the dependence
of $f_{\mathrm{ES}}$ on the afterglow parameters. We will now discuss the
assumptions we make on the other afterglow parameters, which will allow us to
calculate an upper limit on $\epsilon_{B}$.

###  3.2. The Other Afterglow Parameters

####  3.2.1 $E$ and $\epsilon_{e}$

Although we do not know the isotropic equivalent kinetic energy of the
blastwave $E$, we can calculate the isotropic energy released in gamma-rays
during the prompt emission:

$E_{iso}^{\gamma}=\frac{\mbox{fluence}\times 4\pi d_{L}^{2}}{1+z}.$ (4)

In this equation, the fluence has units of ergs/$\mbox{cm}^{2}$ and it
represents the flux detected in gamma-rays, integrated over the duration of
the prompt emission. $d_{L}$ is the luminosity distance. Since we are
interested in the fluence radiated in gamma-rays, we use the fluence detected
in the BAT band, ranging from 15-150 keV (Barthelmy et al., 2005). The
fluences detected by BAT for each GRB can be found in NASA’s Swift GRB Table
and Lookup website333
http://heasarc.gsfc.nasa.gov/docs/swift/archive/grb_table/  .

To convert $E_{iso}^{\gamma}$ to $E$, we need to know the efficiency in the
conversion of kinetic energy of the jet to prompt gamma-ray emission. Recent
studies on the prompt emission efficiency, using X-ray light curves with
plateaus detected by Swift, were presented in Granot et al. (2006) and Zhang
et al. (2007a). For the 23 GRBs for which Zhang et al. (2007a) presented
results for the efficiency (see their Table 3), more than half of them were
found to have a high efficiency $\gtrsim$ 30% 444In Table 3 of Zhang et al.
(2007a), they present two different estimates for the efficiency. If the
shallow decay of plateaus seen in X-ray light curves is due to energy
injection, the more appropriate of the two estimates for the efficiency is
denoted as $\eta_{\gamma}(t_{\mathrm{dec}})$.
$\eta_{\gamma}(t_{\mathrm{dec}})$ represents the efficiency in gamma-ray
radiation ($\eta_{\gamma}$) calculated at the deceleration time
$t_{\mathrm{dec}}$., with a few being estimated to have an efficiency as high
as 90%. A high efficiency of $\sim$90% was also found in Granot et al. (2006).
However, Fan & Piran 2006 argue that bursts with X-ray plateaus should have
more moderate efficiencies $\sim 10\%$. In addition, Zhang et al. (2007a)
mention that the efficiencies they calculate for some bursts have large errors
due to the uncertainty in the microphysical parameters $\epsilon_{e}$ and
$\epsilon_{B}$. In this work, we will take a standard choice and calculate $E$
with the expression

$E=5E_{\mathrm{iso}}^{\gamma}.$ (5)

From the definition of the efficiency,
$\eta=E^{\gamma}_{\mathrm{iso}}/(E^{\gamma}_{\mathrm{iso}}+E)$, Equation 5
corresponds to an efficiency of $\sim 20$%. At the end of Section 3.5, we will
discuss in more detail how the uncertainty in the efficiency affects our
results. For our X-ray sample (see Section 3.3), from Equation 5, we found
values for $E$ in the range of $10^{51}-10^{54}$ ergs, with a typical value
$\sim 10^{53}$ ergs. The average value of $E_{53}$ for our X-ray sample is 2.8
and the median is 1.6. The fluence detected in the BAT band,
$E^{\gamma}_{\mathrm{iso,}52}$, $z$, and $d_{L28}$ for each GRB in our X-ray
sample are shown in Table LABEL:x-ray_properties_table 555Except for GRBs
060708, 060906, 061021, 061222A, 080906, and 081230, we obtained all the
redshifts from NASA’s Swift GRB Table and Lookup website. For the exceptions,
we obtained the redshifts from the website on GRB redshifts maintained by J.
Greiner, http://www.mpe.mpg.de/ jcg/grbgen.html
http://www.mpe.mpg.de/$\sim$jcg/grbgen.html.

For $\epsilon_{e}$, we assumed a value of 0.2 for all of the GRBs in our
sample. This choice for $\epsilon_{e}$ is justified from the results of
$\epsilon_{e}$ with previous afterglow studies (Figure 1) and with recent
results from simulations (Sironi & Spitkovsky, 2011), as discussed in Section
2.

####  3.2.2 Electron power-law index and density profile

In afterglow studies, $s$ and $p$ can be obtained by determining which closure
relation the observed temporal and spectral decay indices satisfy. We cannot
use this strategy to determine $p$ and $s$ for our X-ray sample since the
external-forward shock flux is below the observed steep decline emission.
Instead, we use a fixed $p$ for all GRBs in our X-ray sample. We consider a
small value of $p=2.2$, a typical value of $p=2.4$ for Swift GRBs (Curran et
al., 2010), and a larger value of $p=2.8$. Previous afterglow studies have
found that the majority of afterglow observations are better described by a
constant density medium (e.g. Panaitescu & Kumar, 2002; Schulze et al., 2011).
However, there are still a number of cases where the wind medium is a better
model for the afterglow observations. Therefore, we will consider both $s=0$
and $s=2$ when displaying the results for the upper limit on $\epsilon_{B}$.

####  3.2.3 Density

The density of the medium in the vicinity of GRBs is a highly uncertain
parameter. A histogram of values of $n$ determined by previous afterglow
modelling studies can be found in Figure 9 of Soderberg et al. (2006), which
shows $n$ to vary over 5 orders of magnitude, ranging from $\sim 10^{-3}\mbox{
cm}^{-3}$ to $\sim 10^{2}\mbox{ cm}^{-3}$. In Section 3.5, we will discuss in
more detail how the uncertainty in the density affects our results for the
$\epsilon_{B}$ upper limits.

###  3.3. The X-ray Sample

For our constraint on $\epsilon_{B}$ with X-ray data, we only consider X-ray
data detected by the XRT on board Swift. We used the X-ray light curves
presented in Butler & Kocevski (2007)666 http://butler.lab.asu.edu/swift/
http://butler.lab.asu.edu/swift/. With the exception of two cases, we only
consider bursts that display a steep decline in their X-ray light curve (see
below). After the end of the steep decline, GRBs display a variety of temporal
decays (Evans et al., 2009). Our sample can be divided into 4 different
subgroups, based on the temporal decay after the steep decline:

  1. 1.

Steep Decline To Plateau: In this subgroup, GRBs display a plateau after the
steep decline ($\sim$ 73% of our sample). In Table
LABEL:x-ray_properties_table, we display the time at the end of the steep
decline in units of $10^{2}$ sec, $t_{2,\mathrm{EoSD}}$, and the observed flux
at the end of the steep decline at 1 keV, $f_{1\mathrm{keV},\mathrm{EoSD}}$,
in $\mu$Jy.

  2. 2.

Steep Decline To Normal Decline: In this subgroup, GRBs display a temporal
decay of $\alpha\sim 1$ after the steep decline ($\sim$ 18% of our sample). In
Table LABEL:x-ray_properties_table, we show the time and the flux at the end
of the steep decline.

  3. 3.

Clear Steep Decline But Not A Clear End To The Steep Decline: In this
subgroup, it cannot be determined where the steep decline ends (the XRT
observations end before the steep decline ends). The following GRBs fall into
this subgroup: 050315, 060202, 070419A, 071122, and 090516. For these GRBs, in
Table LABEL:x-ray_properties_table, we show the flux and the time of the last
steep decline point observed. For these GRBs, to be sure that the last steep
decline point observed is past the deceleration time, we made sure that it is
at a few hundred seconds.

  4. 4.

No Clear Steep Decline Seen, Just Plateau: Two of the GRBs in our X-ray
sample, 050401 and 060927, do not display a steep decline. The first
observation of the X-ray light curve for these bursts is during the plateau.
We did not remove these GRBs from our X-ray sample because they are also part
of our optical sample (bursts with both X-ray and optical data are important
because they allow us to cross-check our results, see Section 5). For these 2
bursts, we considered the first observation in the X-ray light curve for our
constraint so that we have the least amount of energy injection. We made sure
that this point is at least at a few hundred seconds so that we can be
confident that the onset of the external-forward shock emission has occurred.
For these two bursts, we show the time and the flux of the first X-ray
observation of the plateau in Table LABEL:x-ray_properties_table.

In addition, 25% of the GRBs in our X-ray sample display X-ray flares during
the steep decline. We only consider bursts where the X-ray flare ends before
the end of the steep decline because it is difficult to determine the flux and
time at the end of the steep decline for bursts that show X-ray flares near
the end of the steep decline. It is fine to consider these bursts because
observationally, after the X-ray flare, the X-ray light curve is seen to
return to the same temporal decay prior to the X-ray flare (e.g. Chincarini et
al. 2007). Lastly, GRB 051221A is the only short GRB in our X-ray sample; all
the other bursts in our X-ray sample are long GRBs.

Table 1Properties of X-ray Sample  GRB | $z$ | $d_{L28}$ | Fluence | $E^{\gamma}_{\mathrm{iso,}52}$ | $t_{2,\mathrm{EoSD}}$ | $f_{1\mathrm{keV,EoSD}}$ | $\mbox{log}_{10}(\epsilon_{B})$ | $\mbox{log}_{10}(\epsilon_{B})$
---|---|---|---|---|---|---|---|---
|  |  |  $[\times 10^{-6}\mbox{ ergs}/\mbox{cm}^{2}$] |  |  |  [$\mu$Jy] |  ($s=0$) |  ($s=2$)
050315 | 1.949 | 4.71 | 3.22 | 3.04 | 4 | 3 | -5.2 | -6.0
050401 | 2.9 | 7.67 | 8.22 | 15.56 | 2 | 80 | -4.7 | -5.4
050721 | 2.5 | 6.40 | 3.62 | 5.32 | 4 | 30 | -4.2 | -5.0
050803 | 0.422 | 0.71 | 2.15 | 0.10 | 3 | 10 | -3.8 | -5.4
050814 | 5.3 | 15.76 | 2.01 | 9.95 | 9 | 1 | -5.4 | -6.0
051221A | 0.547 | 0.97 | 1.15 | 0.09 | 6 | 5 | -3.5 | -4.9
060108 | 2.03 | 4.95 | 0.37 | 0.38 | 6 | 0.8 | -4.2 | -5.5
060111B | 2 | 4.86 | 1.60 | 1.58 | 2 | 8 | -4.6 | -5.8
060115 | 3.53 | 9.72 | 1.71 | 4.48 | 8 | 1 | -5.2 | -5.9
060210 | 3.91 | 10.99 | 7.66 | 23.66 | 8 | 40 | -4.4 | -4.7
060418 | 1.49 | 3.37 | 8.33 | 4.79 | 5 | 40 | -4.3 | -4.9
060502A | 1.51 | 3.43 | 2.31 | 1.36 | 3 | 7 | -4.6 | -5.7
060607A | 3.082 | 8.25 | 2.55 | 5.35 | 5 | 60 | -3.6 | -4.3
060707 | 3.425 | 9.37 | 1.60 | 3.99 | 9 | 3 | -4.5 | -5.2
060708 | 1.92 | 4.62 | 0.49 | 0.45 | 2 | 10 | -3.7 | -5.2
060714 | 2.71 | 7.06 | 2.83 | 4.78 | 3 | 10 | -4.8 | -5.7
060729 | 0.54 | 0.96 | 2.61 | 0.20 | 6 | 7 | -3.8 | -5.1
060814 | 0.84 | 1.65 | 14.60 | 2.71 | 8 | 8 | -5.0 | -5.6
060906 | 3.686 | 10.24 | 2.21 | 6.21 | 4 | 0.8 | -5.9 | -6.7
060926 | 3.208 | 8.66 | 0.22 | 0.49 | 2 | 4 | -3.8 | -5.4
060927 | 5.6 | 16.81 | 1.13 | 6.08 | 0.8 | 8 | -5.3 | -6.6
061021 | 0.3463 | 0.56 | 2.96 | 0.09 | 3 | 10 | -3.9 | -5.5
061110A | 0.758 | 1.45 | 1.06 | 0.16 | 5 | 5 | -3.6 | -6.0
061121 | 1.314 | 2.88 | 13.70 | 6.19 | 2 | 40 | -5.1 | -5.8
061222A | 2.088 | 5.13 | 7.99 | 8.55 | 2 | 40 | -4.9 | -5.7
070110 | 2.352 | 5.94 | 1.62 | 2.14 | 4 | 8 | -4.3 | -5.3
070306 | 1.497 | 3.39 | 5.38 | 3.12 | 7 | 4 | -5.0 | -5.6
070714B | 0.92 | 1.85 | 0.72 | 0.16 | 4 | 6 | -3.5 | -5.0
070802 | 2.45 | 6.24 | 0.28 | 0.40 | 5 | 0.8 | -4.2 | -5.5
071122 | 1.14 | 2.41 | 0.58 | 0.20 | 8 | 0.8 | -4.1 | -5.4
080310 | 2.43 | 6.18 | 2.30 | 3.22 | 10 | 2 | -4.8 | -5.4
080413A | 2.433 | 6.19 | 3.50 | 4.91 | 2 | 10 | -5.1 | -6.0
080430 | 0.767 | 1.47 | 1.20 | 0.19 | 2 | 10 | -3.9 | -5.5
080607 | 3.036 | 8.10 | 24.00 | 49.07 | 3 | 90 | -5.2 | -5.5
080721 | 2.591 | 6.68 | 12.00 | 18.75 | 0.7 | 900 | -4.3 | -5.1
080905B | 2.374 | 6.00 | 1.80 | 2.42 | 2 | 20 | -4.3 | -5.4
080906 | 2.1 | 5.16 | 3.50 | 3.78 | 7 | 20 | -4.0 | -4.7
080916A | 0.689 | 1.29 | 4.00 | 0.50 | 3 | 10 | -4.4 | -5.7
081007 | 0.5295 | 0.93 | 0.71 | 0.05 | 2 | 8 | -3.5 | -5.3
081008 | 1.9685 | 4.77 | 4.30 | 4.14 | 6 | 20 | -4.2 | -4.9
081230 | 2 | 4.86 | 0.82 | 0.81 | 3 | 6 | -4.1 | -5.3
090418A | 1.608 | 3.71 | 4.60 | 3.05 | 2 | 20 | -4.8 | -5.8
090516A | 4.109 | 11.66 | 9.00 | 30.08 | 6 | 10 | -5.4 | -5.7
090519 | 3.85 | 10.79 | 1.20 | 3.62 | 5 | 0.8 | -5.4 | -6.2
090529 | 2.625 | 6.79 | 0.68 | 1.09 | 20 | 0.5 | -4.3 | -5.1
090618 | 0.54 | 0.96 | 105.00 | 7.86 | 4 | 200 | -4.9 | -5.3
090926B | 1.24 | 2.68 | 7.30 | 2.95 | 5 | 10 | -4.8 | -5.6
091029 | 2.752 | 7.19 | 2.40 | 4.16 | 6 | 1 | -5.5 | -6.2
091109A | 3.076 | 8.25 | 1.60 | 3.35 | 5 | 2 | -5.0 | -5.8
100302A | 4.813 | 14.06 | 0.31 | 1.33 | 8 | 1 | -4.2 | -5.2
100425A | 1.755 | 4.14 | 0.47 | 0.37 | 3 | 7 | -3.6 | -5.0
100513A | 4.772 | 13.92 | 1.40 | 5.91 | 9 | 7 | -4.1 | -4.8
100621A | 0.542 | 0.96 | 21.00 | 1.58 | 4 | 20 | -5.0 | -5.8
100704A | 3.6 | 9.95 | 6.00 | 16.23 | 6 | 9 | -5.1 | -5.5
100814A | 1.44 | 3.23 | 9.00 | 4.85 | 6 | 9 | -5.0 | -5.6
100906A | 1.727 | 4.05 | 12.00 | 9.09 | 3 | 20 | -5.2 | -5.9
110808A | 1.348 | 2.98 | 0.33 | 0.16 | 3 | 3 | -3.6 | -5.2
110818A | 3.36 | 9.16 | 4.00 | 9.67 | 20 | 1 | -5.3 | -5.5
111008A | 4.9898 | 14.68 | 5.30 | 23.95 | 3 | 9 | -5.5 | -6.1
111228A | 0.72 | 1.36 | 8.50 | 1.15 | 5 | 8 | -4.8 | -5.7
Table 1Properties of X-ray Sample (Continued)

###  3.4. Expected External-Forward Shock Emission At The End Of The Steep
Decline

The synchrotron afterglow spectrum consists of four power-law segments that
are smoothly joined together at three characteristic frequencies of
synchrotron emission (e.g. Sari, Piran, & Narayan, 1998; Granot & Sari, 2002).
These three characteristic frequencies are: $\nu_{a}$, the synchrotron self-
absorption frequency, $\nu_{i}$ (also commonly referred to as $\nu_{m}$), the
frequency of the photons emitted by the power-law distribution of injected
electrons with the smallest energy, and $\nu_{c}$, the cooling frequency
corresponding to electrons cooling on a dynamical time. For this work, we will
only consider the standard case for the ordering of the characteristic
frequencies, the slow cooling case, where $\nu_{a}<\nu_{i}<\nu_{c}$. One
argument against the fast cooling case ($\nu_{a}<\nu_{c}<\nu_{i}$) is that if
the observing frequency is between $\nu_{c}$ and $\nu_{i}$, the spectrum
should be $f_{\nu}\propto\nu^{-1/2}$; however, the spectral index $\beta=0.5$
disagrees with the typical observed afterglow spectral index $\beta\approx
0.9$ (e.g. Piro, 2001). The next question we need to consider is where the
X-ray band lies at the end of the steep decline (here, we consider 1 keV for
the X-ray band because the light curves we used are plotted at this energy).
The two possibilities for the spectral regime of the X-ray band are
$\nu_{i}<1\mbox{ keV}<\nu_{c}$ or $\nu_{c}<1\mbox{ keV}$. We rule out
$\nu_{c}<1\mbox{ keV}$ with the following two arguments.

First, we compare the observed flux at the end of the steep decline,
$f_{\mathrm{1keV,EoSD}}$, to the flux predicted by the external-forward shock
at the same time, if $\nu_{c}<1\mbox{ keV}$ (defined as $f_{\mathrm{pred}}$).
For $s=0$ , $f_{\mathrm{pred}}$ is given by (Granot & Sari, 2002)

$\displaystyle
f_{\mathrm{pred}}=0.855(p-0.98)10^{\frac{3p+2}{4}}8.64^{\frac{3p-2}{4}}e^{1.95p}$
$\displaystyle\times(1+z)^{\frac{p+2}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p-2}{4}}E_{53}^{\frac{p+2}{4}}(t_{\mathrm{2,EoSD}})^{-\frac{(3p-2)}{4}}$
$\displaystyle\times d_{L28}^{-2}\nu_{14}^{-\frac{p}{2}}\mbox{ mJy ,}$ (6)

where $\bar{\epsilon}_{e,-1}\equiv(p-2)/(p-1)\epsilon_{e,-1}$. For
$\nu_{c}<1\mbox{ keV}$, the external-forward shock flux is independent of the
density and the $s=2$ expression is almost identical. When calculating
$f_{\mathrm{pred}}$, for each of the bursts in our X-ray sample, we assumed a
standard $p=2.4$, $\epsilon_{e}=0.2$, and $\nu_{14}=2.4\times 10^{3}$, the
frequency corresponding to 1 keV. For the parameters $E$, $t$, $z$, and
$d_{L}$, we used the values given in Table LABEL:x-ray_properties_table for
each burst ($E=5E_{iso}^{\gamma}$). The remaining parameter we need to compute
$f_{\mathrm{pred}}$ is $\epsilon_{B}$. Since $\epsilon_{B}$ is raised to the
power of $(p-2)/4$, for a typical $p\sim 2-3$, the dependence on
$\epsilon_{B}$ is very weak. When calculating $f_{\mathrm{pred}}$, we assumed
a low value of $\epsilon_{B}=10^{-3}$.

We computed the ratio $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ for all the
GRBs in our X-ray sample and found that
$f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}>1$ for all the bursts and
$f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}>10$ for $54/60$ bursts. The mean
value of $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ is 50 and the median value
is 34. This means that the predicted flux from the external-forward shock,
when $\nu_{c}<\mbox{1 keV}$, over-predicts the observed flux at the end of the
steep decline by a factor that is larger than 10 for the majority of the
bursts. Therefore, the assumption that $\nu_{c}<1$ keV is incorrect. This is a
robust conclusion because the X-ray flux from the external-forward shock, when
$\nu_{c}<1$ keV, basically only depends on $\sim\epsilon_{e}E$ (see Equation
6), which cannot be decreased by a factor of $>10$ without introducing serious
efficiency problems in producing the prompt gamma-rays. Even if we allow for
an uncertainty of a factor of $\sim 2-3$ in both $\epsilon_{e}$ and $E$, this
is not enough to decrease $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ below 1
for the majority of bursts in our X-ray sample.

Before continuing, we want to add that $f_{\mathrm{pred}}$ (Equation 6) also
has a dependence on the Compton-$Y$ parameter:
$f_{\mathrm{pred}}\propto(1+Y)^{-1}$. With this dependence, if the Compton-$Y$
parameter is large, then it is possible for $f_{\mathrm{pred}}$ to decrease
below $f_{\mathrm{1keV,EoSD}}$. For a few bursts in our X-ray sample, we
performed a detailed numerical calculation of the external-forward shock flux
with the formalism presented in Barniol Duran & Kumar (2011), which includes a
detailed treatment of Compton-$Y$ with Klein-Nishina effects. From this
calculation, we also found that $f_{\mathrm{pred}}$ over-predicts
$f_{\mathrm{1keV,EoSD}}$ by a factor larger than 10, which means that the
Inverse Compton (IC) cooling of electrons producing 1 keV synchrotron photons
is a weak effect (even when $\epsilon_{B}$ is small). Numerically, we also
found that without making any assumption about the location of $\nu_{c}$,
solutions to the constraint $f_{\mathrm{ES,1keV}}\leq f_{1keV,EoSD}$ were only
found when 1 keV $<\nu_{c}$ (when $\epsilon_{B}$ is small such that 1 keV
$<\nu_{c}$, it turns out that IC cooling of electrons producing 1 keV photons
takes place in the Klein-Nishina regime at these early times of $\sim$ few
$\times 100$ sec).

Another argument in favor of the spectral regime of the X-ray band being
$\nu_{i}<\mbox{1 keV}<\nu_{c}$ at the end of the steep decline comes from the
extrapolation of $\nu_{c}$ at late times to the end of the steep decline. In
Liang et al. (2008), they made fits to late time XRT light curves during the
normal decline phase and they also provided the value of $\nu_{c}$ during the
geometrical midpoint of the normal decline phase777The geometrical midpoint of
the normal decline is defined by
$\mbox{log}_{10}t=(\mbox{log}_{10}t_{1}+\mbox{log}_{10}t_{2})/2$ , where
$t_{1}$ represents the time of the first observation of the normal decline and
$t_{2}$ represents the time of the last observation of the normal decline..
For this argument, we focus on GRBs that are in common to our X-ray sample and
to the sample of Liang et al. (2008). For these bursts, we extrapolate
$\nu_{c}$ at late times to the end of the steep decline. In Liang et al.
(2008), they only considered a constant density medium, where $\nu_{c}\propto
t^{-1/2}$. The results of the extrapolation of $\nu_{c}$ are shown in Table 2.

GRB | $t_{\mathrm{late},4}$ | $t_{\mathrm{EoSD},2}$ |  $\nu_{\mathrm{c,late}}$ (keV) |  $\nu_{\mathrm{c,EoSD}}$ (keV)
---|---|---|---|---
050315 | $3.5$ | 4 | 0.17 | 1.6
050401 | $6.9$ | 2 | 4.1 | 77
051221A | $1.7$ | 6 | 4.1 | 22
060210 | $3.2$ | 8 | 0.97 | 6.1
060502A | $0.42$ | 3 | 4.1 | 15
060714 | $3.5$ | 3 | 3.6 | 39
060729 | $40$ | 6 | 1.4 | 36
060814 | $2.0$ | 8 | 4.1 | 21
060926 | $0.26$ | 2 | 6.7 | 24
061121 | $1.1$ | 2 | 4.2 | 31
070110 | $14$ | 4 | 4 | 74
Table 2This table shows the extrapolation of $\nu_{c}$ from late times to the
end of the steep decline for GRBs in common to our sample and to the sample in
Liang et al. (2008) (first column). The second column shows
$t_{\mathrm{late},4}$, the late time in units of $10^{4}$ seconds at which
Liang et al. (2008) determined $\nu_{c}$. The third column shows
$t_{\mathrm{EoSD},2}$, the time at the end of the steep decline in units of
$10^{2}$ seconds. $\nu_{\mathrm{c,late}}$, given in keV, is the value found in
Liang et al. (2008) for $\nu_{c}$ at $t_{\mathrm{late}}$. The last column
shows $\nu_{\mathrm{c,EoSD}}$ in keV. $\nu_{\mathrm{c,EoSD}}$ is found by
extrapolating $\nu_{\mathrm{c,late}}$ to $t_{\mathrm{EoSD}}$. Since Liang et
al. (2008) assume a constant density medium, we take a constant density medium
for all the GRBs in this sample when making the extrapolation of $\nu_{c}$ to
$t_{\mathrm{EoSD}}$.

In Table 2, we find that at the end of the steep decline, 1 keV
$<\nu_{c,\mathrm{EoSD}}$ for all GRBs. This further confirms our choice that
$\nu_{i}<\mbox{ 1 keV}<\nu_{c}$ at the end of the steep decline888Including
energy injection will make the values of $\nu_{\mathrm{c,EoSD}}$ in Table 2
larger, making the conclusion that $\nu_{i}<\mbox{ 1 keV}<\nu_{c}$ at the end
of the steep decline more robust. For the bursts that have plateaus in their
X-ray light curve, energy injection needs to be considered. During the energy
injection episode, $E$ increases as $E\propto t^{1-q}$ (Zhang et al., 2006),
where $q$ is a positive constant that satisfies $0\leq q\leq 1$. Therefore,
between the end of the plateau and the end of the steep decline, since
$\nu_{c}\propto E^{-1/2}t^{-1/2}$, $\nu_{c}\propto t^{-1+(q/2)}$. This time
evolution of $\nu_{c}$ is steeper than $\nu_{c}\propto t^{-1/2}$ without
energy injection for $s=0$ (Liang et al. (2008) only considered $s=0)$..

The knowledge of the spectral regime regime at the end of the steep decline
allows us to write in an explicit expression for $f_{\mathrm{ES}}$ in Equation
3:

$\frac{f_{\mathrm{1keV,EoSD}}}{\mbox{mJy}}\geq\left\\{\begin{array}[]{ll}\hskip-7.0pt0.461(p-0.04)10^{\frac{3p+1}{4}}8.64^{\frac{3(p-1)}{4}}e^{2.53p}&\\\
\times(1+z)^{\frac{p+3}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p+1}{4}}n_{0}^{\frac{1}{2}}E_{53}^{\frac{p+3}{4}}&\\\
\times(t_{\rm
2,EoSD})^{-\frac{3(p-1)}{4}}d_{L28}^{-2}\nu_{14}^{-\frac{(p-1)}{2}}\mbox{
$s=0$}\\\ &\\\
\hskip-7.0pt3.82(p-0.18)10^{\frac{3p-1}{4}}8.64^{\frac{3p-1}{4}}e^{2.54p}&\\\
\times(1+z)^{\frac{p+5}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p+1}{4}}A_{*,-1}E_{53}^{(p+1)/4}&\\\
\times(t_{\rm
2,EoSD})^{-\frac{(3p-1)}{4}}d_{L28}^{-2}\nu_{14}^{-\frac{(p-1)}{2}}\mbox{
$s=2$}.\end{array}\right.$ (7)

On the left hand side of this inequality, we have the observed X-ray flux at
the end of the steep decline and on the right hand side we have the expression
for the external-forward shock flux when $\nu_{i}<\mbox{ 1 keV}<\nu_{c}$
(Granot & Sari, 2002). The notation used for $\bar{\epsilon}_{e}$ and $A_{*}$
is defined as $\bar{\epsilon}_{e,-1}\equiv\bar{\epsilon}_{e}/10^{-1}$ and
$A_{*,-1}\equiv A_{*}/10^{-1}$. The expressions in Equation 7 are only valid
for $p>2$, which we consider in this work (for $p<2$, see Bhattacharya 2001
and Resmi & Bhattacharya 2008).

Before displaying our results for the $\epsilon_{B}$ upper limit for our
entire X-ray sample, we show a simple calculation to get an idea of what
values to expect for the $\epsilon_{B}$ upper limits from the X-ray constraint
given in Equation 7. For a standard $p=2.4$, the X-ray constraint is

$\frac{f_{\mathrm{1keV,EoSD}}}{\mbox{mJy}}\geq\left\\{\begin{array}[]{ll}\hskip-7.0pt5.0\times
10^{1}t_{2,\mathrm{EoSD}}^{-1.05}\epsilon_{e,-1}^{1.4}E_{53}^{1.35}n_{0}^{0.5}\epsilon_{B}^{0.85}&s=0\\\
&\\\ \hskip-7.0pt7.0\times
10^{2}t_{2,\mathrm{EoSD}}^{-1.55}\epsilon_{e,-1}^{1.4}E_{53}^{0.85}A_{*,-1}\epsilon_{B}^{0.85}&s=2.\end{array}\right.$
(8)

For this calculation, we used the average $z=2.5$ for Swift GRBs (Gehrels et
al., 2009) (with a corresponding $d_{L28}=6.4$) and $\nu_{14}$ corresponding
to 1 keV. Solving for $\epsilon_{B}$, the upper limit depends on the afterglow
parameters as999For bursts with plateaus in their X-ray light curve, it is
possible that energy injection begins before the steep decline ends. However,
even with energy injection, there still exists a self-similar solution for the
energy (Equation 52 of Blandford & McKee 1976). For both $s=0$ and $s=2$, we
calculated the external-forward shock synchrotron flux with this new self-
similar solution and then calculated the upper limit on $\epsilon_{B}$ as
shown in Equation 9. We found that the $\epsilon_{B}$ upper limits are
affected by less than a factor of $\sim 2$. Thus, even if energy injection
begins before the end of the steep decline, it has very little to no effect on
our $\epsilon_{B}$ upper limit results.

$\epsilon_{B}\leq\left\\{\begin{array}[]{ll}\hskip-7.0pt1.0\times
10^{-2}\left(\frac{f_{1keV,\mathrm{EoSD}}}{\mbox{mJy}}\right)^{1.18}t_{2,\mathrm{EoSD}}^{1.24}&\\\
\times\epsilon_{e,-1}^{-1.65}E_{53}^{-1.59}n_{0}^{-0.59}&s=0\\\ &\\\
\hskip-7.0pt4.5\times
10^{-4}\left(\frac{f_{1keV,\mathrm{EoSD}}}{\mbox{mJy}}\right)^{1.18}t_{2,\mathrm{EoSD}}^{1.82}&\\\
\times\epsilon_{e,-1}^{-1.65}E_{53}^{-1}A_{*,-1}^{-1.18}&s=2.\end{array}\right.$
(9)

The median values for our X-ray sample for the parameters
$f_{\mathrm{1keV,EoSD}}$, $t_{2,\mathrm{EoSD}}$, and $E_{53}$ are $8\times
10^{-3}$ mJy, 4, and 1.6, respectively. Using these median values and
$\epsilon_{e,-1}=2$, the upper limit on $\epsilon_{B}$ becomes

$\epsilon_{B}\leq\left\\{\begin{array}[]{ll}\hskip-7.0pt2.8\times
10^{-5}\times n_{0}^{-0.59}&s=0\\\ &\\\ \hskip-7.0pt3.7\times 10^{-6}\times
A_{*-1}^{-1.18}&s=2.\end{array}\right.$ (10)

For a standard $n_{0}=1$ and $A_{*,-1}=1$, it can be seen that the
$\epsilon_{B}$ upper limit is lower for $s=2$. This is expected because for
$A_{*,-1}=1$, there is a larger density for the surrounding medium within a
typical deceleration radius of $10^{17}\mbox{ cm}$.

In Equation 10, the explicit dependence of the $\epsilon_{B}$ upper limit on
the density is shown for $p=2.4$. In the next subsection, we will display the
results of the $\epsilon_{B}$ upper limits for our entire X-ray sample. To
keep the density-dependence, we will display histograms of upper limits on the
quantity $\epsilon_{B}n_{0}^{0.59}$ ($\epsilon_{B}A_{*-1}^{1.18}$) for $s=0$
($s=2$) for $p=2.4$, or $\epsilon_{B}n_{0}^{2/(p+1)}$
($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$ ($s=2$) for a general $p$ (see
Equation 7).

###  3.5. $\epsilon_{B}$ Upper Limits For Our X-ray Sample

We display the results for the upper limits (from Equation 7) on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$
($s=2$) assuming all GRBs in our X-ray sample have $p=2.2$, 2.4, and 2.8 in
the Top-Left, Top-Right, and Bottom panels of Figure 2, respectively. Two
histograms are shown in each panel, one for $s=0$ and the other for $s=2$.
Table 3 shows the mean and median upper limits on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$
($s=2$) for each histogram.

X-ray |  |  |  |
---|---|---|---|---
$(s=0)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |
Mean | $1.1\times 10^{-4}$ | $7.2\times 10^{-5}$ | $6.1\times 10^{-5}$ |
Median | $4.2\times 10^{-5}$ | $2.8\times 10^{-5}$ | $2.0\times 10^{-5}$ |
X-ray |  |  |  |
$(s=2)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |
Mean | $7.9\times 10^{-6}$ | $5.5\times 10^{-6}$ | $5.6\times 10^{-6}$ |
Median | $4.0\times 10^{-6}$ | $3.2\times 10^{-6}$ | $3.4\times 10^{-6}$ |
Opt. |  |  |  |
$(s=0)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |  $p$ from $\alpha_{\mathrm{O}}$
Mean | $3.5\times 10^{-4}$ | $1.0\times 10^{-4}$ | $2.9\times 10^{-5}$ | $9.5\times 10^{-5}$
Median | $1.1\times 10^{-4}$ | $3.3\times 10^{-5}$ | $5.5\times 10^{-6}$ | $2.4\times 10^{-5}$
Opt. |  |  |  |
$(s=2)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |
Mean | $7.2\times 10^{-5}$ | $2.4\times 10^{-5}$ | $7.5\times 10^{-6}$ |
Median | $1.2\times 10^{-5}$ | $3.9\times 10^{-6}$ | $8.8\times 10^{-7}$ |
Table 3Mean and median $\epsilon_{B}$ values for the X-ray (upper limits on
$\epsilon_{B}$) and optical (measurements of $\epsilon_{B}$) histograms shown
in Figures 2 and 3. The section labeled “X-ray $(s=0)$” (“X-ray $(s=2)$”)
shows the mean and median $\epsilon_{B}$ upper limits assuming a constant
density (wind) medium with a standard $n=1\mbox{ cm}^{-3}$ ($A_{*}=0.1$). The
columns show the value of $p$ that was assumed. The section labeled “Opt.
$(s=0)$” (“Opt. $(s=2)$”) shows the mean and median $\epsilon_{B}$
measurements assuming a constant density (wind) medium with a standard
$n=1\mbox{ cm}^{-3}$ ($A_{*}=0.1$). The columns show the value of $p$ that was
assumed. The column labeled “$p$ from $\alpha_{O}$” shows the mean and median
$\epsilon_{B}$ measurements with $p$ determined from $\alpha_{O}$. There are
60 GRBs in our X-ray sample and 35 GRBs in our optical sample.

For the remainder of this section, we assume a standard $n_{0}=1$
($A_{*-1}=1$) for $s=0$ ($s=2$) when discussing our results for the
$\epsilon_{B}$ upper limits.

The $\epsilon_{B}$ upper limit histograms show a wide distribution. For a
constant density (wind) medium, all the histograms show a distribution ranging
from $\sim 10^{-6}-10^{-3}$ ($\sim 10^{-7}-10^{-4}$). For a constant density
(wind) medium, the mean and median $\epsilon_{B}$ upper limit values are
$\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-6}$). Assuming a
different value of $p$ does not have a significant effect on the distributions
of the $\epsilon_{B}$ upper limits for our X-ray sample. For both the $s=0$
and $s=2$ cases, when changing $p$, the mean and median $\epsilon_{B}$ upper
limit values change by less than a factor $\sim 2$. Although previous
afterglow studies also showed a wide distribution for $\epsilon_{B}$ (Figure
1), our distribution of $\epsilon_{B}$ upper limits is shifted towards lower
values. Unlike Figure 1, which shows that many GRBs have been reported to have
$\epsilon_{B}\sim 10^{-3}-10^{-1}$, none of our histograms of $\epsilon_{B}$
upper limits show an $\epsilon_{B}$ upper limit larger than $10^{-3}$.

We now discuss how our assumptions on the afterglow parameters can affect the
distribution of $\epsilon_{B}$ upper limits. For this discussion, we will take
a typical $p=2.4$; Equation 9 shows how the $\epsilon_{B}$ upper limit depends
on the other afterglow parameters. The strongest dependence is on
$\epsilon_{e}$, which is raised to the power of $-1.65$. However, as we
displayed in Figure 1, according to previous studies, the distribution of
$\epsilon_{e}$ values is narrow, with the $\epsilon_{e}$ values ranging only
over one order of magnitude. In addition, $\sim 62\%$ of the bursts have
$\epsilon_{e}\sim 0.1-0.3$. From Figure 1, a likely error in $\epsilon_{e}$
from our assumed $\epsilon_{e}=0.2$ is a factor $\sim 2$. From Equation 9, an
error in $\epsilon_{e}$ by a factor $\sim 2$ will only lead to an error in the
$\epsilon_{B}$ upper limit by a factor $\sim 3$. For a constant density (wind)
medium, the $\epsilon_{B}$ upper limit depends on $E$ as $E^{-1.59}$
($E^{-1}$). We assumed an efficiency of $\sim 20\%$ in the conversion of
kinetic energy to prompt gamma-ray radiation. Recent studies have found higher
values for the efficiency (Granot et al. 2006, Zhang et al. 2007a, see
however, Fan & Piran 2006). In Zhang et al. 2007a, the mean (median)
efficiency they reported is $\sim 37\%$ ($\sim 32\%$). Taking the efficiency
to be $\sim 30\%-40\%$ instead of $\sim 20\%$ would lead to an error in $E$ by
a factor $\sim 2-3$. From Equation 9, an error in $E$ by a factor $\sim 2-3$
would lead to an error in the $\epsilon_{B}$ upper limit by a factor $\sim
3-6$ ($\sim 2-3$) for a constant density (wind) medium. Lastly, the largest
source of uncertainty for the $\epsilon_{B}$ upper limits is the density,
since it has been reported to have a range $\sim 10^{-3}\mbox{
cm}^{-3}-10^{2}\mbox{ cm}^{-3}$. For a constant density (wind) medium, the
$\epsilon_{B}$ upper limit depends on the density as $n^{-0.59}$
($A_{*}^{-1.18}$). For $s=0$ ($s=2$), we assumed a standard $n=1\mbox{
cm}^{-3}$ ($A_{*}=0.1$). An error in the density by a factor $\sim 10^{3}$
($\sim 10^{2}$) will lead to an error in the $\epsilon_{B}$ upper limit by a
factor $\sim 60$ ($\sim 230$).

In summary, the expected errors in $\epsilon_{e}$ and $E$ of a factor $\sim
2-3$ will not change the $\epsilon_{B}$ upper limits by an order of magnitude.
On the other hand, the density is a very uncertain parameter and an error in
the density by $\sim 2-3$ orders of magnitude will lead to an error in the
$\epsilon_{B}$ upper limits by $\sim 1-2$ orders of magnitude.

$\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{x_ray_p_2_2_histogram.eps}&\includegraphics[scale={0.6}]{x_ray_p_2_4_histogram.eps}\end{array}\\\
\includegraphics[scale={0.6}]{x_ray_p_2_8_histogram.eps}\end{array}$

Figure 2.— The Top-Left, Top-Right, and Bottom panels show the histograms of
upper limits assuming $p=2.2$, $p=2.4$, and $p=2.8$ respectively, for all of
the GRBs in our X-ray sample (obtained with Equation 7). The filled-in (un-
filled) histograms show upper limits on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$
($s=2$) assuming all the GRBs in our X-ray sample are described by a constant
density (wind) medium.

Two additional parameters that can affect our $\epsilon_{B}$ upper limits are:
1. $\xi$, the fraction of electrons accelerated to a power-law distribution
and 2. $f$, which is a factor that takes into account the degeneracy for a set
of afterglow parameters. For a set of parameters $E$, $n$, $\epsilon_{e}$,
$\epsilon_{B}$, $\xi$ producing the observed external-forward shock flux,
another set of primed parameters $E^{\prime}=E/f$, $n^{\prime}=n/f$,
$\epsilon_{e}^{\prime}=f\epsilon_{e}$, $\epsilon_{B}^{\prime}=f\epsilon_{B}$,
$\xi^{\prime}=f\xi$ can also produce the observed external-forward shock flux
(Eichler & Waxman, 2005). Afterglow studies usually assume $\xi=f=1$ for
simplicity but $\xi\leq 1$ and $m_{e}/m_{p}\leq f\leq 1$ (Eichler & Waxman,
2005), where $m_{e}$ ($m_{p}$) is the electron (proton) mass. The external-
forward shock flux depends on $\xi$ and $\epsilon_{B}$ as
$f_{\nu}\propto\xi^{2-p}\epsilon_{B}^{(p+1)/4}$ (Leventis et al., 2012). From
this dependence, we find that the $\epsilon_{B}$ upper limit
$\propto\xi^{4(p-2)/(p+1)}$. Thus, including $\xi$ will decrease the values of
our $\epsilon_{B}$ upper limits. Values of $\xi$ have not been determined for
GRB external-forward shocks so we cannot quantify by how much the
$\epsilon_{B}$ upper limit values will decrease. Including $f$ will also
decrease the values of the $\epsilon_{B}$ upper limits since
$\epsilon_{B}^{\prime}=f\epsilon_{B}$ and $f<1$. Like $\xi$, values of $f$
have also not been determined from afterglow observations. The largest effect
$f$ can have on the $\epsilon_{B}$ upper limit values is decrease them by a
factor of $m_{p}/m_{e}\sim 2000$. For the remainer of this paper, we will be
conservative and continue to assume $\xi=f=1$, but we should keep in mind that
considering $\xi$ and $f$ will decrease the values of the $\epsilon_{B}$ upper
limits.

##  4\. Measurement Of $\epsilon_{B}$ With Optical Light Curves

###  4.1. $\epsilon_{B}$ Determination With Optical Data

The light curves we consider for our optical sample decline with a temporal
decay index $\alpha\sim 1$ from early times, $\sim 10^{2}-10^{3}\mbox{ sec}$,
as expected for the external-forward shock emission (see Section 4.2). Since
the light curves of these bursts are likely dominated by the external-forward
shock, this means that the observed optical flux is an actual measurement of
the external-forward shock flux, that is101010At late times, $\sim
10^{3}-10^{5}$ sec, many X-ray light curves decline with $\alpha\sim 1$: the
“normal” decline. If this segment arises from the external-forward shock, then
$\epsilon_{B}$ can be determined as described in this Section for our optical
sample. However, this is not straightforward since energy injection (as
evidenced by the plateau phase) should be considered. Also, one can attempt to
use the upper limit on $\epsilon_{B}$, found in Section 3, to calculate a
lower limit on $\nu_{c}$ during the normal decline and compare this to the
observed spectral regime ($\mbox{1 keV}<\nu_{c}$ or $\mbox{1 keV}>\nu_{c}$).
However, there are two difficulties with this $\nu_{c}$ consistency check: 1.
several “closure relations” are simultaneously satisfied within 2-$\sigma$ for
most of our sample (due to large errors in the spectral and temporal indices
at late times, which can be found in Evans et al. (2007, 2009)). 2\. for the
bursts that uniquely satisfy one closure relation, $\nu_{c}$ cannot be
determined precisely since Klein-Nishina suppression is weaker at late times
and $\nu_{c}$ depends strongly on $n$ when Compton-$Y$ is considered.

$f_{\mathrm{obs}}=f_{\mathrm{ES}}(E,n,\epsilon_{e},\epsilon_{B},p,s).$ (11)

Later in this section we will use this condition to determine $\epsilon_{B}$
for the bursts in our optical sample. We want to stress that we determine
$\epsilon_{B}$ for the optical sample. This is in contrast to the X-ray
sample, which only allowed us to determine an upper limit on $\epsilon_{B}$.

###  4.2. The Optical Sample

Our optical sample consists of 35 GRBs. $33/35$ of the bursts triggered Swift
and the remaining two bursts, 050502A and 080603A, were detected by INTEGRAL
(INTErnational Gamma-Ray Astrophysics Laboratory, Winkler et al. 2003). Table
LABEL:optical_properties_table shows properties of our optical sample. With a
few exceptions, most of the GRBs in our optical sample have a known
redshift111111The redshifts were taken from NASA’s Swift GRB Table and Lookup
website (exceptions are GRB 071003 (Perley et al., 2008) and GRBs 050502A and
071025 (J. Greiner’s website)). Three GRBs (050721, 070420, 060111B) in our
optical sample do not have a known redshift. For GRB 050721 and GRB 070420, we
assumed the average redshift of 2.5 for Swift GRBs (Gehrels et al., 2009). The
redshift for GRB 060111B was approximated as 2 in Stratta et al. (2009)..

Table 4Optical Sample Properties  GRB | $z$ | $d_{L28}$ | Fluence | $E^{\gamma}_{\mathrm{iso,}52}$ | $\alpha_{O}$ | Ref. | $t_{2}$ | $f_{2\mathrm{eV}}$ | $\mbox{log}_{10}(\epsilon_{B})$
---|---|---|---|---|---|---|---|---|---
|  |  | $[\times 10^{-6}\mbox{ ergs}/\mbox{cm}^{2}]$ |  |  |  |  | [mJy] |  ($s=0$)
050401 | 2.9 | 7.67 | 8.22 | 15.55 | $0.80\pm 0.03$ | [1] | 0.72 | 0.3 | -5.5
050502A | 3.793 | 10.59 | 1.4 | 4.12 | $1.16\pm 0.03$ | [2] | 1 | 5 | -4.5
050525A | 0.606 | 1.10 | 15.3 | 1.45 | $1.12\pm 0.05$ | [3] | 34.56 | 0.5 | -4.3
050721 | 2.5 | 6.40 | 3.62 | 5.32 | $1.29\pm 0.06$ | [4] | 20 | 0.2 | -5.0
050730 | 3.97 | 11.19 | 2.38 | 7.53 | $0.89\pm 0.05$ | [17] | 7.5 | 0.57 | -3.9
050802 | 1.71 | 4.00 | 2.00 | 1.49 | $0.82\pm 0.03$ | [1] | 3.6 | 0.5 | -3.5
051111 | 1.55 | 3.54 | 4.08 | 2.52 | $1.00\pm 0.02$ | [2] | 30 | 0.4 | -3.8
051221A | 0.5465 | 0.97 | 1.15 | 0.09 | $0.96\pm 0.03$ | [5] | 100 | 0.02 | -3.4
060111B | 2 | 4.86 | 1.60 | 1.58 | $1.18\pm 0.05$ | [6] | 2 | 0.4 | -5.2
060210 | 3.91 | 10.99 | 7.66 | 23.65 | $1.03\pm 0.06$ | [2] | 10 | 0.1 | -6.0
060418 | 1.49 | 3.37 | 8.33 | 4.78 | $1.13\pm 0.02$ | [7] | 2 | 8 | -4.6
060607A | 3.082 | 8.25 | 2.55 | 5.34 | $1.20\pm 0.03$ | [7] | 2 | 10 | -4.2
060904B | 0.703 | 1.32 | 1.62 | 0.21 | $1.00\pm 0.18$ | [17] | 5.5 | 0.58 | -3.5
060908 | 2.43 | 6.18 | 2.80 | 3.91 | $1.05^{+0.03}_{-0.03}$ | [8] | 2 | 2 | -4.5
060927 | 5.6 | 16.81 | 1.13 | 6.08 | $1.21\pm 0.06$ | [2] | 0.5 | 2 | -5.5
061007 | 1.26 | 2.74 | 44.4 | 18.48 | $1.70\pm 0.02$ | [7] | 2 | 50 | -6.0
061110B | 3.44 | 9.42 | 1.33 | 3.34 | $1.64\pm 0.08$ | [2] | 20 | 0.02 | -5.9
061121 | 1.314 | 2.88 | 13.7 | 6.19 | $0.82\pm 0.02$ | [7] | 4 | 0.5 | -4.7
061126 | 1.159 | 2.47 | 6.77 | 2.39 | $0.89\pm 0.02$ | [2] | 10 | 0.2 | -4.6
070318 | 0.84 | 1.65 | 2.48 | 0.46 | $0.96\pm 0.03$ | [7] | 20 | 0.2 | -3.7
070411 | 2.954 | 7.84 | 2.70 | 5.27 | $0.92\pm 0.04$ | [2] | 20 | 0.07 | -4.7
070420 | 2.5 | 6.40 | 14.0 | 20.56 | $0.81\pm 0.04$ | [2] | 3 | 0.8 | -4.7
070714B | 0.92 | 1.85 | 0.72 | 0.16 | $0.83\pm 0.04$ | [2] | 10 | 0.03 | -3.7
071003 | 1.6 | 3.69 | 8.3 | 5.45 | $1.466\pm 0.006$ | [9] | 0.6 | 20 | -5.7
071025 | 5.2 | 15.41 | 6.5 | 31.26 | $1.27\pm 0.04$ | [10] | 20 | 0.02 | -6.8
071031 | 2.692 | 7.00 | 0.9 | 1.5 | $0.97\pm 0.06$ | [11] | 10.5 | 0.4 | -3.4
071112C | 0.823 | 1.61 | 3.00 | 0.53 | $0.95\pm 0.02$ | [12] | 10.5 | 0.003 | -6.3
080603A | 1.688 | 3.94 | 1.1 | 0.80 | $0.99\pm 0.07$ | [13] | 30 | 0.1 | -3.6
080607 | 3.036 | 8.10 | 24.0 | 49.04 | $1.65$ | [14] | 3 | 0.2 | -8.0
080721 | 2.591 | 6.68 | 12.0 | 18.74 | $1.22\pm 0.01$ | [5] | 3 | 10 | -5.0
080810 | 3.35 | 9.13 | 4.60 | 11.06 | $1.23\pm 0.01$ | [7] | 3 | 30 | -3.9
080913 | 6.7 | 20.72 | 0.56 | 3.92 | $1.03\pm 0.02$ | [15] | 10 | 0.02 | -5.2
081008 | 1.967 | 4.76 | 4.30 | 4.13 | $0.96\pm 0.03$ | [16] | 2 | 3 | -4.1
090313 | 3.375 | 9.21 | 1.40 | 3.41 | $1.25\pm 0.08$ | [17] | 20 | 2 | -3.4
090418A | 1.608 | 3.71 | 4.60 | 3.05 | $1.21\pm 0.04$ | [7] | 2 | 0.8 | -5.5
Table 4Optical Sample Properties (Continued)

With the exception of the only short GRB in our optical sample, GRB 051221A,
all the optical light curves in our sample decline before 3500 seconds.
Considering early times has the advantage of minimizing possible energy
injection. Our optical sample can be separated into 4 different subgroups,
depending on the temporal behavior of the light curve before the $\alpha\sim
1$ decay as follows.

  1. 1.

Light Curves With A Peak At Early Times: The light curves of this subgroup are
characterized by a power law rise, reaching a peak, and then a power law
decline with $\alpha_{O}\sim 1$ ($\sim$43% of our sample). The peak of the
light curve is believed to be due to the deceleration time. For the bursts in
this subgroup, we show the temporal decay of the optical light curve after the
peak and the flux and the time of the second data point after the peak in
Table LABEL:optical_properties_table. We take the second data point to be
confident that the optical light curve is declining.

  2. 2.

Single Power Law Decay From Early Times: In this subgroup, the optical light
curve shows a decline as a single power law with $\alpha\sim 1$ from the
beginning of the observations ($\sim$ 40% of our sample). We display the
temporal decay of the optical light curve and the time and the flux of the
second data point observed in Table LABEL:optical_properties_table.

  3. 3.

Optical Light Curves With Plateaus At Early Times: The optical light curves of
3 bursts in our optical sample (GRBs 050525A, 060210, and 070411) display
plateaus at early times. The plateaus in our optical sample are short, with
the longest plateau lasting under 3500 seconds. After the plateau ends, the
light curves of these 3 bursts show a decay $\alpha_{O}\sim 1$, as expected
for the external forward-shock emission. In Table
LABEL:optical_properties_table, for these 3 bursts, we show the temporal decay
after the plateau and the time and the flux of the second data point after the
plateau.

  4. 4.

Light Curves With Possible Reverse Shock Emission At Early Times: 3 GRBs in
our optical sample (060111B, 060908, and 061126) show possible emission from
the reverse shock. The light curves in this subgroup show an initial steep
decline at early times, characteristic of the reverse shock, and then
transition to a more shallow decay of $\alpha\sim 1$ that is more typical for
the external-forward shock emission. For these GRBs, in Table
LABEL:optical_properties_table we show the temporal decay of the light curve
and the time and flux of the second data point after the possible reverse
shock emission.

###  4.3. Optical External-Forward Shock Spectral Regime and Afterglow
Parameter Assumptions

When referring to the optical band, we will use 2 eV since most of the light
curves in our optical sample are either plotted at 2eV or were observed in the
R filter. As we did with the X-ray sample, we will only consider the slow
cooling ordering of the synchrotron characteristic frequencies,
$\nu_{a}<\nu_{i}<\nu_{c}$. Because the optical light curve is declining at the
time we are considering, the optical band must be above $\nu_{i}$ at this
time. In Section 3.4, we argued that the X-ray band is between $\nu_{i}$ and
$\nu_{c}$ at the end of the steep decline at a few 100 sec; therefore, the
optical band must also be in this spectral regime at the early times $(\sim
10^{2}-10^{3})$ sec we are considering. The expression we will use to
determine the optical external-forward shock flux is also Equation 7; however,
we will have an equality (instead of an inequality), we replace
$f_{\mathrm{1keV,EoSD}}$ with $f_{\mathrm{2eV}}$ (which represents the
specific flux observed at 2 eV), and use $\nu_{14}$ corresponding to 2 eV.

The other afterglow parameters are determined as in Section 3.2:
$\epsilon_{e}=0.2$ and with $z$ and the fluence121212For the two bursts
detected by INTEGRAL, the fluence in Table LABEL:optical_properties_table is
in the 20-200 keV band of the instrument IBIS (Imager on-Board the INTEGRAL
Satellite, Ubertini et al. (2003)): GRB 050502A (Gotz & Mereghetti, 2005) and
GRB 080603A (Guidorzi et al., 2011)., we obtain $E^{\gamma}_{\mathrm{iso}}$
and use $E=5E^{\gamma}_{\mathrm{iso}}$. As with our X-ray sample, we will
display our $\epsilon_{B}$ results with $p=2.2$, 2.4, and 2.8. We can also
determine $p$ by using the temporal decay of the optical light curve,
$\alpha_{O}$, which is shown in Table LABEL:optical_properties_table for each
burst (optical spectrum is not always available, so we cannot use the closure
relations for the optical sample). In order to have $p>2$ for all of the
bursts in our optical sample, we only consider a constant density medium when
determining $p$ with $\alpha_{O}$ ($\alpha_{O}=3(p-1)/4$ for $s=0$ and
$\alpha_{O}=(3p-1)/4$ for $s=2$). Lastly, as we did for our X-ray sample, to
keep the density-dependence, we will plot the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$
($s=2$).

###  4.4. $\epsilon_{B}$ Results For Optical Sample

$\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{optical_p_2_2_histogram.eps}&\includegraphics[scale={0.6}]{optical_p_2_4_histogram.eps}\end{array}\\\
\begin{array}[]{cc}\includegraphics[scale={0.6}]{optical_p_2_8_histogram.eps}&\includegraphics[scale={0.6}]{s_0_optical_histogram_with_temporal_p.eps}\end{array}\end{array}$

Figure 3.— The Top-Left, Top-Right, and Bottom-Left panels show the histograms
of measurements assuming $p=2.2$, $p=2.4$, and $p=2.8$ respectively for all of
the GRBs in our optical sample. The filled-in (un-filled) histograms show
measurements of the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$
($\epsilon_{B}A_{*-1}^{4/(p+1)}$) assuming all the GRBs in our optical sample
are described by a constant density (wind) medium. Bottom-Right Panel: The
filled-in histogram shows the measurements on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ with $p$ determined from $\alpha_{O}$. The un-
filled histogram shows measurements on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$, assuming $p=2.4$ for all of the bursts in our
optical sample (this histogram was also shown in the Top-Right panel).

We display the results for the measurements (from Equation 7) on the quantity
$\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$
($s=2$) assuming all the GRBs in our optical sample have $p=2.2$, 2.4, and 2.8
in the Top-Left, Top-Right, and Bottom-Left panels of Figure 3, respectively.
Two histograms are shown in each panel, one for $s=0$ and the other for $s=2$.
We also use $\alpha_{O}$ to determine $p$ (assuming $s=0$) and compare the
results to the ones obtained with $p=2.4$ and $s=0$ (Bottom-Right panel of
Figure 3). In Table 3, we display a summary of the mean and median values of
the measurements of the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$
($\epsilon_{B}A_{*-1}^{4/(p+1)}$) for $s=0$ ($s=2$) for each histogram. For
the remainder of this section, we assume a standard $n_{0}=1$ ($A_{*-1}=1$)
for $s=0$ ($s=2$) when discussing our results for the $\epsilon_{B}$
measurements for our optical sample.

For a constant density (wind) medium, the mean and median $\epsilon_{B}$
measurements are $\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times
10^{-6}$) The mean and median $\epsilon_{B}$ measurements only change by a
factor of a few when assuming a different value of $p$. To determine if
assuming a standard $p=2.4$, as opposed to determining $p$ for each burst from
$\alpha_{O}$, significantly affects the distribution of $\epsilon_{B}$
measurements, we compared the two histograms in the Bottom-Right panel of
Figure 3 with a Kolmogorov-Smirnov (KS) test. The null hypothesis of the KS
test is that the two histograms are drawn from the same distribution. We test
this null hypothesis at the 5% significance level. The KS test confirmed the
null hypothesis that the two histograms are consistent with being drawn from
the same distribution.

As with the $\epsilon_{B}$ upper limits from X-ray data, the mean and median
$\epsilon_{B}$ measurements decrease by about an order of magnitude when
assuming a wind medium as opposed to a constant density medium. Compared to
the distribution of $\epsilon_{B}$ upper limits we attained from X-ray data,
the $\epsilon_{B}$ measurements from optical data show a much wider
distribution. For a constant density (wind) medium, the $\epsilon_{B}$
measurements range from $\epsilon_{B}\sim 10^{-8}-10^{-3}$ ($\epsilon_{B}\sim
10^{-9}-10^{-3}$). Also, since we used the same equation (Equation 7) to find
both the upper limits on $\epsilon_{B}$ with X-ray data and the $\epsilon_{B}$
measurements with optical data, the discussion at the end of Section 3.5 on
how the uncertainty in the afterglow parameters and the parameters $\xi$ and
$f$ can affect the distribution of $\epsilon_{B}$ upper limits also applies to
the distributions of $\epsilon_{B}$ measurements we presented in this section.
In addition, since $\xi$ and $f$ are less than unity, including these two
parameters will mean that our $\epsilon_{B}$ measurements are effectively
upper limits on $\epsilon_{B}$.

###  4.5. Comparison Of Our Results On $\epsilon_{B}$ To Previous Studies

We performed a KS test between our optical $\epsilon_{B}$ measurements and the
results from previous studies on $\epsilon_{B}$ (Figure 1). For our
$\epsilon_{B}$ results, we used the optical $\epsilon_{B}$ measurements with
$n=1\mbox{ cm}^{-3}$ and $p$ determined from $\alpha_{O}$ (filled-in histogram
in Bottom-Right panel of Figure 3; the $\epsilon_{B}$ values are shown in
Table LABEL:optical_properties_table). The result of the KS test is that the
null hypothesis is rejected. The P-value, which measures the probability that
the null hypothesis is still true, is $2.1\times 10^{-9}$. This result shows
that the rejection of the null hypothesis is statistically significant. It is
not surprising that the null hypothesis was rejected. The distribution from
the previous studies is very inhomogeneous, with the values for $\epsilon_{B}$
being drawn from many different studies with different methodologies. Also,
comparing the histogram in Figure 1 to the filled-in histogram in the Bottom-
Right panel of Figure 3, a couple of significant differences can be seen. The
range for the histogram of $\epsilon_{B}$ values found in the literature is
$\sim 10^{-5}-10^{-1}$, whereas the range for our $\epsilon_{B}$ results is
$\sim 10^{-8}-10^{-3}$. The mean and median values for these two histograms
are also significantly different. The mean (median) value for the
$\epsilon_{B}$ histogram from the literature, $6.3\times 10^{-2}$ ($1.4\times
10^{-2}$), is a factor $\sim 700$ ($\sim 600$) times larger than the mean
(median) $\epsilon_{B}$ value of the histogram with our results, which is
$9.5\times 10^{-5}$ ($2.4\times 10^{-5}$).

One assumption that is commonly made in afterglow modeling studies is
equipartition between $\epsilon_{e}$ and $\epsilon_{B}$. As we discussed in
Section 2, the results for $\epsilon_{e}$ from the literature and the results
from recent simulations of relativistic collisionless shocks support
$\epsilon_{e}\sim 0.2$. From this result, many works assume $\epsilon_{B}\sim
10^{-2}-10^{-1}$. However, there is no physical argument to expect
equipartition. Our distribution of $\epsilon_{B}$ upper limits and
measurements, although wide, supports that there is no equipartition between
electron and magnetic energies because none of the $\epsilon_{B}$ upper limits
or measurements in our samples has a value larger than $\epsilon_{B}\sim\mbox{
few }\times 10^{-3}$. Another source of error that can lead to differences in
$\epsilon_{B}$ values is differences in the determination of the spectral
regime for the optical band. We took it to be between $\nu_{i}$ and $\nu_{c}$,
but it is also possible for the optical band to be above $\nu_{c}$ at late
times (e.g. Panaitescu & Kumar, 2002; Cenko et al., 2010). Another source for
error is energy injection. We did not consider energy injection as a source of
error because only $3/35$ of the bursts in our optical sample show plateaus
(and these plateaus are short). The X-ray and optical light curves of many
bursts show plateaus and in these cases energy injection needs to be
considered. Also, errors in our determination of fluxes and times from X-ray
and optical light curves can also lead to small errors in $\epsilon_{B}$. In
summary, the main assumption we made when determining $\epsilon_{B}$ is an
efficiency of $\sim 20\%$ in the conversion of kinetic energy to gamma-ray
energy, and we did not assume equipartition between $\epsilon_{e}$ and
$\epsilon_{B}$. Different authors have made different assumptions that can
have a large effect on the results for $\epsilon_{B}$.

Lastly, for a few bursts, we checked if our method of determining
$\epsilon_{B}$ is consistent with the values determined for $\epsilon_{B}$
with other techniques. GRBs 980519 and 990123, discussed in the afterglow
modelling study of Panaitescu & Kumar 2002, have optical light curves that
decline as a power-law before the jet-break. The optical band for both of
these bursts was determined to be in the spectral regime $\nu_{i}<\mbox{ 2
eV}<\nu_{c}$. Applying our technique to find a $\epsilon_{B}$ measurement for
both of these bursts and using the value of $n$ reported in Panaitescu & Kumar
2002 for both of these bursts, we find that these bursts have
$\epsilon_{B}\sim 10^{-5}$, consistent with the results reported in Panaitescu
& Kumar 2002 for both of these bursts within a factor of a few. The small
differences in $\epsilon_{B}$ values can be accounted for by differences in
the coefficients used for the external-forward shock flux.

##  5\. GRBs In Common To Both Our X-ray And Optical Samples

14 bursts we considered are both in our X-ray and in our optical sample (GRB
number is in bold in the first columns of Tables LABEL:x-ray_properties_table
and LABEL:optical_properties_table). In this section, we verify for these
bursts that the X-ray $\epsilon_{B}$ upper limit is above the optical
$\epsilon_{B}$ measurement. For the optical data, we will use the
$\epsilon_{B}$ measurements with $p$ determined from $\alpha_{O}$ and
$n=1\mbox{ cm}^{-3}$ (filled-in histogram in the Bottom-Right panel of Figure
3). For the X-ray data, in this section, we will also consider $n=1\mbox{
cm}^{-3}$ ($s=0$) and we will use the value of $p$ determined from
$\alpha_{O}$ 13131310 out of 14 of these bursts have optical data before 1000
seconds. For these 10 bursts, we can use the optical data to check if they
satisfy the assumption we made in Section 3.1,
$t_{\mathrm{dec}}<t_{\mathrm{EoSD}}$. 9 of these 10 GRBs do satisfy this
assumption; for the remaining GRB (080721), we are not able to check this
assumption because the first optical observation (at 100 sec.) is after the
end of the steep decline ($t_{\mathrm{EoSD}}=70$ sec).. The comparison between
the X-ray $\epsilon_{B}$ upper limits and the optical $\epsilon_{B}$
measurements is shown in Figure 4. This plot shows that all the X-ray
$\epsilon_{B}$ upper limits are above the optical $\epsilon_{B}$ measurements.

Figure 4.— Comparison of the $\epsilon_{B}$ upper limits from X-ray data to
the $\epsilon_{B}$ measurements from optical data. The 14 dots correspond to
the 14 GRBs that are both in our X-ray and optical samples. The straight line
indicates where the $\epsilon_{B}$ measurements are equal to the
$\epsilon_{B}$ upper limits.

##  6\. $E$ vs. $\epsilon_{B}$ Correlation?

We now use the $\epsilon_{B}$ measurements from our optical sample to
determine if there is a correlation between $E$ and $\epsilon_{B}$. The 3
bursts studied in Kumar & Barniol Duran 2009, 2010 have small values of
$\epsilon_{B}$, consistent with shock compression of a seed magnetic field
$B_{0}\sim\mbox{ few}\times 10\mu\mbox{G}$. One property that distinguishes
these 3 GRBs is that they were particularly energetic, with
$E_{\mathrm{iso}}^{\gamma}\sim 10^{53}-10^{54}\mbox{ ergs}$. Could the large
energy intrinsic to these 3 bursts explain why these 3 bursts have low
$\epsilon_{B}$ values? We investigate this possibility in Figure 5 by plotting
the values of $E$ and $\epsilon_{B}$ for all the bursts in our optical sample.

Figure 5.— We plot the values of $E$ and the measurements of $\epsilon_{B}$ to
determine if they are correlated. The 35 points represent the GRBs in our
optical sample and the straight line is the best fit line:
$\mbox{log}_{10}(\epsilon_{B})=-1.02\mbox{log}_{10}(E_{53})-4.51$, with the
slope of the line being $-1.02\pm 0.23$ and the y-intercept of the line being
$-4.51\pm 0.16$. The correlation coefficient of the fit is 0.62 and the
P-value of the correlation is $1.2\times 10^{-4}$ (3.8$\sigma$ significance).
The $\epsilon_{B}$ measurements are for $n=1\mbox{ cm}^{-3}$ and $p$
determined from $\alpha_{O}$ (shown in the filled-in histogram in the Bottom-
Right panel of Figure 3 and Table LABEL:optical_properties_table) and the
values of $E$ were determined by assuming an efficiency $\sim 20\%$ for all
the GRBs in our optical samples.

For the energy of each burst, we assumed an efficiency of $\sim 20\%$
($E=5E^{\gamma}_{\mathrm{iso}}$) in the conversion of the kinetic energy of
the jet to gamma-ray radiation; the observables involved in calculating $E$
are the gamma-ray fluence and $z$. For the $\epsilon_{B}$ measurements, we
used the values with $p$ determined from $\alpha_{O}$ and $n=1\mbox{
cm}^{-3}$. The observables involved in determining $\epsilon_{B}$ are the
observed specific optical flux and the time. In Figure 5, we also show the
best fit line141414From Equation 7, it may be expected that $E$ and
$\epsilon_{B}$ are correlated. This is not necessarily true because each burst
has a different value for the observed quantities $f_{\nu}$ and $t$. In
addition, since $p$ was determined from $\alpha_{O}$, each burst has a
different $p$. We also checked if $E$ and $\epsilon_{B}$ were correlated when
assuming a fixed $p=2.4$. From Equation 9, we would expect the slope of the
line to be $-1.59$. The best fitting line for $p=2.4$ is
$\mbox{log}_{10}(\epsilon_{B})=-0.77\mbox{log}_{10}(E_{53})-4.53$, with the
slope of the line being $-0.77\pm 0.24$ and the y-intercept being $-4.53\pm
0.16$. The slope of this line is more than $3\sigma$ away from $-1.59$,
showing that there is no expected correlation between $E$ and $\epsilon_{B}$
for our methodology of determining $\epsilon_{B}$.. This fit shows that an
increase in $E$ leads to a decrease in $\epsilon_{B}$. The correlation
coefficient of the fit is $0.62$, indicating that there is a weak correlation
between $E$ and $\epsilon_{B}$. A value of the correlation coefficient close
to 1 would indicate a strong correlation. The P-value of the correlation is
$1.2\times 10^{-4}$ (3.8$\sigma$ significance), indicating that there is a
small probability that the correlation occurred by chance.

Although many points show large deviations from the best fit line, the scatter
of the points may be reduced or increased by the uncertainty in the afterglow
parameters. An error in the efficiency would affect the values of $E$ and an
error in $\epsilon_{e}$, the efficiency, or $n$ would affect the
$\epsilon_{B}$ measurements (see Section 3.5 for a discussion on how the
$\epsilon_{B}$ measurements would be affected when an error in a afterglow
parameter is made). It is possible that the uncertainty in $E$ and
$\epsilon_{B}$ can reduce or increase the scatter and make the correlation
between $E$ and $\epsilon_{B}$ stronger or weaker.

##  7\. Magnetic Field Amplification Factor For X-ray And Optical Results

In Sections 3-5, we presented our results for the strength of the magnetic
field downstream of the shock front in terms of the afterglow parameter
$\epsilon_{B}$. If shock compression was the only mechanism amplifying the
ambient magnetic field (assuming a standard $B_{0}\sim\mbox{few }\mu\mbox{G}$
and a standard $n=1\mbox{ cm}^{-3}$), then $\epsilon_{B}\sim 10^{-9}$ is
expected. Most of the bursts in our distributions of $\epsilon_{B}$ upper
limits and measurements have values larger than $\epsilon_{B}\sim 10^{-9}$.
These results suggest that amplification of the magnetic field, in addition to
shock compression, is needed to explain the afterglow observations. In this
section, we will present our results in terms of an amplification factor,
which quantifies the amplification that is needed, beyond shock compression,
to explain the observations.

If shock compression were the only mechanism amplifying the seed magnetic
field $B_{0}$, then $B=4\Gamma B_{0}$. To quantify how much additional
amplification of the ambient magnetic field is needed, beyond shock
compression, we define the amplification factor, $AF$, as

$AF\equiv\frac{B}{4B_{0}\Gamma}.$ (12)

$AF$ is a constant that satisfies $AF\geq 1$ since $B\geq 4B_{0}\Gamma$.
$AF=1$ means than the observed $B$ is consistent with the only amplification
arising from seed magnetic field shock compression. The expression for
$\epsilon_{B}$ is $\epsilon_{B}=B^{2}/32\pi m_{p}c^{2}n\Gamma^{2}$. With the
definition for $AF$, $\epsilon_{B}$ is

$\epsilon_{B}=(AF)^{2}\times\frac{B_{0}^{2}}{2\pi nm_{p}c^{2}}.$ (13)

We note that $\epsilon_{B}$ is given by $(AF)^{2}$ times the $\epsilon_{B}$ we
would get if shock compression were the only mechanism amplifying the magnetic
field.

We will now use Equation 13 and our previous results for the X-ray
$\epsilon_{B}$ upper limits to determine an upper limit on $AF$. In Section
3.5, if we assumed a standard $n=1\mbox{ cm}^{-3}$, we were able to attain an
upper limit on $\epsilon_{B}$ for a constant density medium 151515We will only
consider a constant density medium when displaying the results for the
amplification factor. We will show in Equation 15 that $AF$ has a weak
dependence on the density.. We will refer to these $\epsilon_{B}$ upper limits
as $\overline{\epsilon_{B}}(p,n_{0}=1)$. In the notation
$\overline{\epsilon_{B}}(p,n_{0}=1)$, the bar over $\epsilon_{B}$ signifies
that this is an upper limit on $\epsilon_{B}$, the $p$ in the parenthesis
shows that the $\epsilon_{B}$ upper limit depends on the value of $p$ we used,
and the $n_{0}=1$ shows that we assumed $n=1\mbox{ cm}^{-3}$. With this
notation, we can keep the dependence of the $\epsilon_{B}$ upper limit on $n$
(see Equation 7):

$\epsilon_{B}<\frac{\overline{\epsilon_{B}}(p,n_{0}=1)}{n_{0}^{2/(p+1)}}.$
(14)

Combining Equation 13 and Equation 14, the expression for the upper limit on
$AF$ is

$AF<\frac{1.0\times
10^{4}[\overline{\epsilon_{B}}(p,n_{0}=1)]^{1/2}}{B_{0,10\mu\mathrm{G}}n_{0}^{(1-p)/(2p+2)}},$
(15)

where $B_{0,10\mu\mathrm{G}}\equiv B_{0}/10\mu\mbox{G}$. One advantage to
expressing the results of the magnetic field downstream of the shock front in
terms of $AF$ is that $AF$ depends weakly on $n$. For a standard $p=2.4$,
$AF\propto n^{0.21}$. On the other hand, there is a strong dependence on
$B_{0}$, $AF\propto B_{0}^{-1}$.

###  7.1. Amplification Factor Upper Limit For Our X-ray Sample

We will now show the results for the $AF$ upper limits we obtained from
Equation 15. Since the amplification factor has a weak dependence on the
density, we will assume a standard $n_{0}=1$ when displaying the results for
the $AF$ upper limits. When plotting the results for $AF$, we will keep the
dependence on $B_{0}$ and plot the quantity $(AF)B_{0,10\mu\mathrm{G}}$. In
the left and right panels of Figure 6, we show the upper limits on the
quantity $(AF)B_{0,10\mu\mathrm{G}}$ for a fixed $p=2.4$ and $p=2.2$, 2.8,
where the values of $\overline{\epsilon_{B}}(p,n_{0}=1)$ used in Equation 15
were shown in Top-Right, Top-Left, and Bottom panels of Figure 2,
respectively. For the remainder of this section, we will assume
$B_{0}=10\mu\mbox{G}$ when discussing the results for the $AF$ upper limits.

The mean and median values of the $AF$ upper limits are summarized in Table 5.
The mean (median) $AF$ upper limits range from $AF\sim 60-80$ ($AF\sim
40-60$). The $AF$ upper limit histograms show a wide distribution, with a
range of $\sim 10$ to $\sim 300$. To determine if assuming a different value
of $p$ has a significant effect on the distribution of $AF$ upper limits, we
performed a KS test between the histograms in the right panel of Figure 6. The
KS test confirmed the null hypothesis, leading us to conclude that the $AF$
upper limit results are not sensitive to the value of $p$ we assume.

X-ray |  |  |  |
---|---|---|---|---
$(s=0)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |
Mean | $84$ | $67$ | $62$ |
Median | $63$ | $51$ | $44$ |
Opt. |  |  |  |
$(s=0)$ | $p=2.2$ | $p=2.4$ | $p=2.8$ |  $p$ from $\alpha_{\mathrm{O}}$
Mean | $130$ | $71$ | $36$ | $71$
Median | $100$ | $56$ | $23$ | $48$
Table 5Mean and median $AF$ values for the X-ray (upper limits on $AF$) and
optical (measurements of $AF$) histograms shown in Figures 6 and 7. All the
labels are the same as in Table 3. A constant density medium with $n=1\mbox{
cm}^{-3}$ (the amplification factor has a weak dependence on the density, see
Section 7 and Equation 15) and a seed magnetic field $B_{0}=10\mu\mbox{G}$
were assumed for all the bursts in our X-ray and optical samples.

$\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{amplification_factor_x_ray_p_2_4.eps}&\includegraphics[scale={0.6}]{amplification_factor_x_ray_p_2_2_p_2_8.eps}\end{array}\end{array}$

Figure 6.— Left: Upper limits on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for
our X-ray sample assuming $p=2.4$. Right: Upper limits on the quantity
$(AF)B_{0,10\mu\mathrm{G}}$ for our X-ray sample assuming $p=2.2$ and $p=2.8$.
A fixed $n=1\mbox{ cm}^{-3}$ was assumed for all of the histograms (the
precise value of $n$ is unimportant since $AF$ has a weak dependence on $n$,
see Section 7 and Equation 15).

We now discuss how an error in each of the afterglow parameters can affect our
results for the $AF$ upper limits. For this discussion, we will assume
$p=2.4$. From Equation 15,
$\mbox{AF}\propto\sqrt{\overline{\epsilon_{B}}(p=2.4,n_{0}=1)}B_{0}^{-1}n^{0.2}$
and from Equation 9, $\epsilon_{B}\propto\epsilon_{e}^{-1.6}E^{-1.6}n^{-0.6}$.
From these two expressions,
$\mbox{AF}\propto\epsilon_{e}^{-0.8}E^{-0.8}n^{0.2}B_{0}^{-1}$. We note that
compared to the $\epsilon_{B}$ upper limit (Equation 9), the $AF$ upper limit
has a weaker dependence on $\epsilon_{e}$, $E$, and $n$. As we discussed at
the end of Section 3.5, a likely error in $\epsilon_{e}$ is a factor of $\sim
2$; this error in $\epsilon_{e}$ will translate into an error in the $AF$
upper limits by a factor of only $\sim 2$. For the energy, we assumed an
efficiency $\sim 20\%$ and a likely error in the efficiency is a factor $\sim
2-3$ (see Section 3.5); this error in the efficiency would lead to an error in
the $AF$ upper limits by only a factor $\sim 2$. One advantage to expressing
the results of the magnetic field downstream of the shock front in terms of
$AF$ is that $AF$ has a very weak dependence on $n$. An error in $n$ by a
factor $\sim 10^{3}$ (see Section 3.5) from our assumed $n=1\mbox{ cm}^{-3}$
will only lead to an error in the $AF$ upper limits by a factor $\sim 4$. The
price to pay for a weak $n$ dependence is a linear dependence on $B_{0}$, with
$\mbox{AF}\propto B_{0}^{-1}$. $B_{0}$ is an uncertain parameter that likely
varies from GRB environment to GRB environment and it is the largest source of
uncertainty for $AF$.

###  7.2. Amplification Factor Measurement For Our Optical Sample

As we discussed in Section 4.2, for our optical sample, we found a measurement
for $\epsilon_{B}$ instead of an upper limit. This will allow us to determine
a measurement for $AF$. To do this, we will use Equation 15, but in this case
we have an equality instead and we have $\epsilon_{B}(p,n_{0}=1)$ instead of
$\overline{\epsilon_{B}}(p,n_{0}=1)$. The notation $\epsilon_{B}(p,n_{0}=1)$
denotes the $\epsilon_{B}$ measurements for our optical sample from Section
4.4 if we assume a standard $n=1\mbox{ cm}^{-3}$. Also, as with the X-ray
sample, we only consider $s=0$ when calculating the $AF$ measurements and
assume a fixed $n=1\mbox{ cm}^{-3}$. In the left panel of Figure 7, we show
the results for the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$
for $p$ determined from $\alpha_{O}$ and also assuming $p=2.4$. In the right
panel of Figure 7, we show the measurements on the quantity
$(AF)B_{0,10\mu\mathrm{G}}$ for $p=2.2$ and $p=2.8$. For the remainder of this
section, we will assume $B_{0}=10\mu\mbox{G}$ when discussing the results for
the $AF$ measurements.

A summary of the mean and median $AF$ measurements for our optical sample is
shown in Table 5. To determine if assuming a standard $p=2.4$, as opposed to
determining $p$ from $\alpha_{O}$ for each burst, has a statistically
significant effect on the distribution of $AF$ measurements, we performed a KS
test between the two histograms in the left panel of Figure 7. The KS test
confirmed the null hypothesis. The mean (median) $AF$ measurements for the
optical histograms range from $\sim 40$ to $\sim 130$ ($\sim 20$ to $\sim
100$). Compared to the $AF$ upper limit histograms, the $AF$ measurement
histograms show a wider distribution, ranging from $AF\sim 1$ to $AF\sim
1000$. Also, since we used the same expression to determine the $AF$ upper
limits and measurements (Equation 15), the discussion at the end of Section
7.1 on how an error in one of the afterglow parameters can affect the $AF$
upper limits also applies to the $AF$ measurements161616$\xi$ and $f$ also
affect our $AF$ results. To account for $\xi$, since the $\epsilon_{B}$ upper
limit$/$measurement is $\propto\xi^{4(p-2)/(p+1)}$ and the $AF$ upper
limit$/$measurement is $\propto(\overline{\epsilon_{B}}(p,n_{0}=1))^{1/2}$,
$AF\propto\xi^{2(p-2)/(p+1)}$. To account for $f$, since
$AF\propto(\overline{\epsilon_{B}}(p,n_{0}=1))^{1/2}$, $AF\propto f^{1/2}$.
Thus, including $\xi$ and $f$ will decrease the values of the $AF$ upper
limits$/$measurements. Taking the lowest possible value for $f$, the $AF$
upper limits$/$measurements can decrease by up to a factor $\sim 40$. This
would make $\sim 50\%$($\sim 60\%$) of the bursts in our X-ray (optical)
sample consistent with shock compression. In addition, as with the
$\epsilon_{B}$ measurements, since $\xi$ and $f$ are less than unity,
including these two parameters will mean that our $AF$ measurements will
become upper limits on $AF$..

$\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{amplification_factor_optical_p_temporal_p_2_4.eps}&\includegraphics[scale={0.6}]{amplification_factor_optical_p_2_2_p_2_8.eps}\end{array}\end{array}$

Figure 7.— Left Panel: The filled-in (un-filled) histogram shows the
measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p$ calculated
from $\alpha_{O}$ ($p=2.4$). Right Panel: The filled-in (un-filled) histogram
shows the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p=2.2$
($p=2.8$). A fixed $n=1\mbox{ cm}^{-3}$ was assumed for all of the histograms
(the precise value of $n$ is unimportant since $AF$ has a weak dependence on
$n$, see Section 7 and Equation 15).

##  8\. Discussion And Conclusions

In this work, we presented a systematic study on the magnetic field downstream
of the shock front for large samples of GRBs (60 in our X-ray sample and 35 in
our optical sample). We expressed the strength of the downstream magnetic
field in terms of both the afterglow parameter $\epsilon_{B}$ and a
amplification factor, denoted by $AF$, which quantifies how much
amplification, beyond shock compression of the seed magnetic field, is needed
to explain the downstream magnetic field. This is the first time a large and
systematic study has been carried out to study $\epsilon_{B}$ and to determine
how much amplification of the seed magnetic field is required by the
observations. For our X-ray (optical) sample, we determined an upper limit
(measurement) for both $\epsilon_{B}$ and $AF$. The upper limits on
$\epsilon_{B}$ and $AF$ for our X-ray sample were found from the constraint
that the observed flux at the end of the steep decline is greater than or
equal to the external-forward shock flux. This is a new method to constrain
$\epsilon_{B}$ that relies on the steep decline emission, which has been
observed by Swift for many GRBs. Our optical sample was restricted to light
curves that decline with $\alpha\sim 1$ from the early times $\sim
10^{2}-10^{3}$ sec, as expected for the external-forward shock emission. We
found the measurements for $\epsilon_{B}$ and $AF$ for our optical sample from
the condition that the observed flux is equal to the external-forward shock
flux.

The condition used for our X-ray (optical) sample was converted into an upper
limit (measurement) on the quantity $\epsilon_{B}n^{2/(p+1)}$ for $s=0$ or
$\epsilon_{B}A_{*-1}^{4/(p+1)}$ for $s=2$ by assuming a $\sim 20\%$ efficiency
in the conversion of kinetic energy to prompt gamma-ray radiation. To find an
upper limit (measurement) on $\epsilon_{B}$ for our X-ray (optical) sample, we
assumed a standard $n=1\mbox{ cm}^{-3}$ ($A_{*}=0.1$) for the density for a
constant density (wind) medium. A discussion on how the uncertainty in the
afterglow parameters affects our results for $\epsilon_{B}$ can be found at
the end of Section 3.5. The largest source of uncertainty for our results on
$\epsilon_{B}$ is the density, since the value of the density has been
observed to vary over many orders of magnitude and its precise value is not
known for each GRB (see Section 3.2). For the bursts that are both in our
X-ray and optical samples, we also applied a consistency check to make sure
our results for $\epsilon_{B}$ are correct (see Section 5).

From Table 3, for a constant density (wind) medium, most of the $\epsilon_{B}$
upper limit and measurement histograms have a median value $\sim\mbox{
few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-6}$). These results imply
that half of the bursts in both our X-ray and optical samples have a
$\epsilon_{B}$ value $\sim\mbox{ few}\times 10^{-5}$ or lower. Assuming
$n=1\mbox{ cm}^{-3}$ and $B_{0}\sim\mbox{few}\times\mu$G, shock compression is
only able to produce $\epsilon_{B}\sim 10^{-9}$. Although $\epsilon_{B}\sim
10^{-9}$ is 4 orders of magnitude lower than $\epsilon_{B}\sim 10^{-5}$,
$\epsilon_{B}\sim 10^{-5}$ is smaller by $\sim 2-4$ orders of magnitude
compared to the majority of previously reported $\epsilon_{B}$ values (Figure
1), which are $\epsilon_{B}\sim 10^{-3}-10^{-1}$. Assuming $B_{0}\sim
10\mu\mbox{G}$, $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$ corresponds to
$AF\sim 50$ (Equation 15). Our result of a median $\epsilon_{B}\sim\mbox{
few}\times 10^{-5}$ shows that the majority of the bursts in our X-ray and
optical samples only require a weak amplification beyond shock compression, by
a factor $\sim 50$ or lower.

The near equipartition $\epsilon_{B}\sim 0.01-0.1$ determined near the shock
front by theoretical studies and Particle-In-Cell (PIC) simulations (Medvedev
& Loeb, 1999; Chang et al., 2008; Martins et al., 2009; Keshet et al., 2009;
Lemoine, 2013; Sironi et al., 2013) stands in contrast with our median results
of $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$. PIC simulations of
relativistic collisionless shocks performed by Chang et al. 2008 and Keshet et
al. 2009 found that the magnetic field generated near the shock front decays
with distance downstream of the shock front. Lemoine 2013 and Lemoine et al.
2013 studied the effects that this decaying magnetic field has on the shock
accelerated electrons radiating afterglow emission downstream of the shock
front. The main effect is that electrons with different Lorentz factors cool
in regions with different magnetic fields, with the higher (lower) energy
photons being emitted by electrons that are closer (further) from the shock
front (Lemoine et al., 2013). Considering the decay of the downstream magnetic
field, Lemoine et al. (2013) modelled the afterglow data of 4 GRBs that have
extended emission at energies $>100\mbox{ MeV}$ (detected by Fermi-LAT) and
also X-ray, optical, and radio data. Their afterglow modelling results for the
X-ray, optical, and radio data found $\epsilon_{B}\sim 10^{-6}-10^{-4}$,
consistent with our results for the median $\epsilon_{B}$ upper limits and
measurements attained from X-ray and optical data.

Our next main result relates to the distribution of $\epsilon_{B}$ values. One
property the $\epsilon_{B}$ values from the literature shared with our optical
$\epsilon_{B}$ measurements is that they both show a wide distribution. The
literature compilation (Figure 1) showed $\epsilon_{B}\sim 10^{-5}-10^{-1}$
and our optical $\epsilon_{B}$ measurement histograms showed an even wider
distribution, ranging from $\epsilon_{B}\sim 10^{-8}-10^{-3}$
($\epsilon_{B}\sim 10^{-9}-10^{-3}$) for a constant density (wind) medium. One
possibility we investigated to explain the wide distribution of $\epsilon_{B}$
values is whether bursts with smaller $\epsilon_{B}$ values are more energetic
than bursts with larger $\epsilon_{B}$ values (Figure 5). Although the bursts
in our optical sample did show the trend that bursts with larger $E$ have a
smaller $\epsilon_{B}$, the correlation was weak, with a correlation
coefficient of $0.62$ and a P-value of $1.2\times 10^{-4}$ (3.8$\sigma$
significance).

Another possibility to explain the wide distribution of $\epsilon_{B}$ relates
to the uncertainty in the environmental parameters ($B_{0}$ and $n$) in the
medium surrounding GRBs. One possibility to explain bursts with values of
$\epsilon_{B}\sim 10^{-5}-10^{-3}$, under the interpretation of weak
amplification beyond shock compression, is that they occurred in environments
with particularly high seed magnetic fields. Since $\epsilon_{B}\propto
B_{0}^{2}$, an increase in $B_{0}$ by an order of magnitude will lead to an
increase in $\epsilon_{B}$ by two orders of magnitude. A weak amplification
beyond shock compression of $AF\sim 50$ (as inferred for the bursts with
$\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$) and a strong $B_{0}\sim
10\mu\mbox{G}-100\mu\mbox{G}$ yields $\epsilon_{B}\sim 10^{-5}-10^{-3}$ (see
Equation 15). We note that for this estimate we assumed $n=1\mbox{ cm}^{-3}$,
but as we discussed in Section 7, the amplification factor has a weak
dependence on the density. This possibility, that the larger values of
$\epsilon_{B}$ may be explained by bursts going off in environments with
particularly strong $B_{0}$, will be discussed further in Barniol Duran 2013
(in preparation).

In addition to many bursts in our optical sample having particularly large
$\epsilon_{B}\sim 10^{-5}-10^{-3}$, there were also some bursts in our optical
sample that have particularly low $\epsilon_{B}$ values. For $p$ determined
from $\alpha_{O}$ and $n=1\mbox{ cm}^{-3}$ (Bottom-right panel of Figure 3),
GRBs 071025 and 071112C have $\epsilon_{B}\sim 10^{-7}$ and GRB 080607 has
$\epsilon_{B}\sim 10^{-8}$. Assuming $n=1\mbox{ cm}^{-3}$ and
$B_{0}=10\mu\mbox{G}$, these bursts with $\epsilon_{B}\sim 10^{-8}-10^{-7}$
are consistent with shock compression of a seed magnetic field of a few 10
$\mu\mbox{G}$ being the only amplification necessary to explain the
observations.

Lastly, we mention that a similar conclusion for radio observations of
supernova remnants (SNRs) was reached in Thompson et al. (2009). Like GRB
afterglow emission, SNR emission arises from a blastwave interacting with the
surrounding medium, but at non-relativistic speeds. Thompson et al. (2009)
found that starburst galaxies have strong ambient magnetic fields $\sim 1$ mG
and they concluded that shock compression of this strong seed magnetic field
is enough to explain the radio emission from SNRs. On the other hand, for
normal spiral galaxies with ambient magnetic fields $\sim 5-10\mu$G, they
concluded that additional amplification beyond shock compression by a factor
$\sim 3-7$ was necessary to explain the radio emission from SNRs.

This work made use of data supplied by the UK Swift Science Data Centre at the
University of Leicester. RS dedicates this work to his friend, Jossue Colato,
thanks Patrick Crumley and Roberto Hernández for helpful discussions, and
Kevin Gately for his help with obtaining the values of $\epsilon_{e}$ and
$\epsilon_{B}$ from the literature. This work has been funded in part by NSF
grant ast-0909110. RBD was supported by an ERC advanced grant (GRB) and by the
I-CORE Program of the PBC and the ISF (grant 1829$/$12).

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##  Appendix A $\epsilon_{e}$ And $\epsilon_{B}$ Values From The Literature

In this Appendix, we show a table of the $\epsilon_{e}$ and $\epsilon_{B}$
values we used to make the histograms in Figure 1. For the first GRBs with
high quality afterglow data, different works (e.g. Wijers & Galama 1999;
Chevalier & Li 1999, 2000; Li & Chevalier 2001; Harrison et al. 2001; Yost et
al. 2003) have presented afterglow modeling studies on the same GRBs. In Table
LABEL:epsilon_e_and_epsilon_B_table, for these early afterglow studies, we
show the results from Panaitescu & Kumar 2001a, b; Panaitescu & Kumar 2002
since they have the largest afterglow modeling compilations. It is important
to note that in some cases, different groups determined significantly
different values for the afterglow parameters (e.g. GRB 970508 Wijers & Galama
1999; Chevalier & Li 2000; Frail et al. 2000; Panaitescu & Kumar 2002, GRB
000418 Berger et al. 2001; Panaitescu & Kumar 2002).

Table 6$\epsilon_{e}$ and $\epsilon_{B}$ Values From The Literature  GRB | $\epsilon_{e}$ | $\epsilon_{B}$ | Ref.