MAGNETIC FIELDS IN RELATIVISTIC COLLISIONLESS SHOCKS

One interesting property found by Swift (Gehrels et al. 2004) is that at early times, about 50% of the light curves detected by the X-ray Telescope (XRT; 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 ∼10²–10³ s. By extrapolating the Burst Alert Telescope (BAT; 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 led 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 10²–10³ s, is past the deceleration time.⁴ For our upper limit on _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 blast wave given by the Blandford and McKee solution (Blandford & McKee 1976). Since this solution is only valid for a decelerating blast wave, 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 the steep decline. Depending on the density profile of the surrounding medium, the deceleration time t_dec is given by

$$\begin{equation} t_{\mathrm{dec}} = \left\lbrace \begin{array}{@{}ll}(220 {\rm \,s}) E_{53} ^{1/3} n_{0} ^{-1/3} \Gamma _{2} ^{-8/3} (1+z) & s=0 \\ (67 {\rm \,s}) E_{53} A_{*,-1} ^{-1} \Gamma _{2} ^{-4} (1+z) & s=2 \end{array} \right. \end{equation} \tag{ 2 }$$

(e.g., Panaitescu & Kumar 2000). In these expressions, Γ is the Lorentz factor of the shocked fluid, z is the redshift, and we have adopted the usual notation Q_n ≡ Q/10ⁿ. 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_* ≡ A/(5 × 10¹¹ g cm⁻¹) (Chevalier & Li 2000). For typical GRB afterglow parameters of E₅₃ = 1, n₀ = 1 (or A_* = 0.1 for s = 2), Γ₂ = 3, and z = 2.5, the deceleration time is under 100 s for both s = 0 and s = 2. Although there can be a large uncertainty in the afterglow parameters E and n, Γ is the most important parameter when calculating t_dec since t_dec has a very strong dependence on Γ. For s = 0 (s = 2), even if we take extreme parameters for E and n, such as a high kinetic energy of E₅₃ = 100 and a low density of n = 10⁻³ cm⁻³ (A_* = 10⁻²), with a typical Γ₂ = 3–4 (e.g., Molinari et al. 2007; Xue et al. 2009; Liang et al. 2010), t_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 EoSD.

Observationally, the deceleration time is also seen to occur before the EoSD 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 _B increases f_ES (external-forward shock flux), the condition that the X-ray flux during the steep decline should be larger than or equal to f_ES gives an upper limit on _B. Since our goal is to attain the most stringent upper limit on _B, we take this constraint at the EoSD. Explicitly, the constraint we will use to find an upper limit on _B with X-ray data is

$$\begin{equation} f_{\mathrm{EoSD}} \ge f_{\mathrm{ES}} (E, n, s, \epsilon _{e}, \epsilon _{B}, p). \end{equation} \tag{ 3 }$$

In this inequality, f_EoSD represents the observed flux at the EoSD. We have also explicitly shown the dependence of f_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 _B.

3.2.1. E and _e

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

$$\begin{equation} E_{{\rm iso}} ^{\gamma } = \frac{{\rm fluence} \times 4 \pi d_{L}^{2}}{1+z}. \end{equation} \tag{ 4 }$$

In this equation, the fluence has units of erg cm⁻² and 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 to 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 Web site.⁵

To convert $E_{{\rm iso}} ^{\gamma }$ 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 ≳30%,⁶ with a few being estimated to have an efficiency as high as 90%. A high efficiency of ∼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 ∼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 _e and _B. In this work, we will take a standard choice and calculate E with the expression

$$\begin{equation} E = 5 E_{\mathrm{iso}} ^{\gamma }. \end{equation} \tag{ 5 }$$

From the definition of the efficiency, $\eta = E^{\gamma } _{\mathrm{iso}} / (E^{\gamma } _{\mathrm{iso}} + E)$, Equation (5) corresponds to an efficiency of ∼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 10⁵¹–10⁵⁴ erg, with a typical value ∼10⁵³ erg. The average value of E₅₃ 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 1.⁷

Table 1. Properties of X-Ray Sample

GRB	z	dL28	Fluence	$E ^{\gamma } _{\mathrm{iso, }52}$	t2, EoSD	f1 keV, EoSD	log10(B)	log10(B)
			(× 10−6 erg cm−2)			(μJy)	(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

Notes. This table displays the properties of our X-ray sample. The GRBs that are in bold are also part of our optical sample. The second and third columns show the redshift and the luminosity distance d_L (in units of 10²⁸ cm), respectively. The fourth column shows the fluence detected by BAT in units of 10⁻⁶ erg cm⁻². The next column shows $E ^{\gamma } _{\mathrm{iso, }52}$, the isotropic equivalent energy released in gamma rays during the prompt emission, in units of 10⁵² erg. t_{2, EoSD} represents the time at the end of the steep decline (EoSD) in units of 10² s. The column f_{1 keV, EoSD} shows the specific flux at 1 keV at the end of the steep decline, in units of μJy. The last two columns show the upper limits on _B, assuming p = 2.4. One column shows the results for a constant density medium (s = 0) assuming n = 1 cm⁻³ (filled-in histogram in the top right panel of Figure 2), and the other column shows the results for a wind medium (s = 2) assuming A_* = 0.1 (un-filled histogram in the top right panel of Figure 2).

Download table as:  ASCIITypeset image

For _e, we assumed a value of 0.2 for all of the GRBs in our sample. This choice for _e is justified from the results of _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

3.2.3. Density

For our constraint on _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).⁸ 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 EoSD, GRBs display a variety of temporal decays (Evans et al. 2009). Our sample can be divided into four different subgroups, based on the temporal decay after the steep decline.

• 1.  
  Steep decline to plateau. In this subgroup, GRBs display a plateau after the steep decline (∼73% of our sample). In Table 1, we display the time at the EoSD in units of 10² s, t_{2, EoSD}, and the observed flux at the EoSD at 1 keV, f_{1 keV, EoSD}, in μJy.
• 2.  
  Steep decline to normal decline. In this subgroup, GRBs display a temporal decay of α ∼ 1 after the steep decline (∼18% of our sample). In Table 1, we show the time and the flux at the EoSD.
• 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 1, 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.  
  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 two 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 1.

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 EoSD because it is difficult to determine the flux and time at the EoSD for bursts that show X-ray flares near the EoSD. 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.

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 et al. 1998; Granot & Sari 2002). These three characteristic frequencies are: ν_a, the synchrotron self-absorption frequency, ν_i (also commonly referred to as ν_m), the frequency of the photons emitted by the power-law distribution of injected electrons with the smallest energy, and ν_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 ν_a < ν_i < ν_c. One argument against the fast cooling case (ν_a < ν_c < ν_i) is that if the observing frequency is between ν_c and ν_i, the spectrum should be f_ν∝ν^−1/2; however, the spectral index β = 0.5 disagrees with the typical observed afterglow spectral index β ≈ 0.9 (e.g., Piro 2001). The next question we need to consider is where the X-ray band lies at the EoSD (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 ν_i < 1 keV < ν_c or ν_c < 1 keV. We rule out ν_c < 1 keV with the following two arguments.

First, we compare the observed flux at the EoSD, f_{1 keV, EoSD}, to the flux predicted by the external-forward shock at the same time, if ν_c < 1 keV (defined as f_pred). For s = 0, f_pred is given by (Granot & Sari 2002)

$$\begin{eqnarray} f_{\mathrm{pred}} &=& 0.855 (p-0.98) 10^{\frac{3p+2}{4}} 8.64^{\frac{3p-2}{4}} e^{1.95p} \nonumber \\ && \times (1+z)^{\frac{p+2}{4}} (\bar{\epsilon }_{e,-1}) ^{p-1} \epsilon _{B} ^{\frac{p-2}{4}} E_{53} ^{\frac{p+2}{4}} (t_{\mathrm{2, EoSD}}) ^{-\frac{(3p-2)}{4}} \nonumber \\ && \times d_{L28} ^{-2} \nu _{14} ^{-\frac{p}{2}} \,{\rm mJy,} \end{eqnarray} \tag{ 6 }$$

where $\bar{\epsilon }_{e,-1} \equiv (p-2)/(p-1) \epsilon _{e,-1}$. For ν_c < 1 keV, the external-forward shock flux is independent of the density and the s = 2 expression is almost identical. When calculating f_pred, for each of the bursts in our X-ray sample, we assumed a standard p = 2.4, _e = 0.2, and ν₁₄ = 2.4 × 10³, the frequency corresponding to 1 keV. For the parameters E, t, z, and d_L, we used the values given in Table 1 for each burst ($E = 5 E_{{\rm iso}} ^{\gamma }$). The remaining parameter we need to compute f_pred is _B. Since _B is raised to the power of (p − 2)/4, for a typical p ∼ 2–3, the dependence on _B is very weak. When calculating f_pred, we assumed a low value of _B = 10⁻³.

We computed the ratio f_pred/f_{1 keV, EoSD} for all the GRBs in our X-ray sample and found that f_pred/f_{1 keV, EoSD} > 1 for all the bursts and f_pred/f_{1 keV, EoSD} > 10 for 54/60 bursts. The mean value of f_pred/f_{1 keV, EoSD} is 50, and the median value is 34. This means that the predicted flux from the external-forward shock, when ν_c < 1 keV, overpredicts 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 ν_c < 1 keV is incorrect. This is a robust conclusion because the X-ray flux from the external-forward shock, when ν_c < 1 keV, basically only depends on ∼_eE (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 ∼2–3 in both _e and E, this is not enough to decrease f_pred/f_{1 keV, EoSD} below 1 for the majority of bursts in our X-ray sample.

Before continuing, we want to add that f_pred (Equation (6)) also has a dependence on the Compton-Y parameter: f_pred∝(1 + Y)⁻¹. With this dependence, if the Compton-Y parameter is large, then it is possible for f_pred to decrease below f_{1 keV, 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_pred overpredicts f_{1 keV, 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 _B is small). Numerically, we also found that without making any assumption about the location of ν_c, solutions to the constraint f_{ES, 1 keV} ⩽ f_{1 keV, EoSD} were only found when 1 keV <ν_c (when _B is small such that 1 keV <ν_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 ∼few × 100 s).

Another argument in favor of the spectral regime of the X-ray band being ν_i < 1 keV < ν_c at the EoSD comes from the extrapolation of ν_c at late times to the EoSD. In Liang et al. (2008), they made fits to late-time XRT light curves during the normal decline phase and also provided the value of ν_c during the geometrical midpoint of the normal decline phase.⁹ For this argument, we focus on GRBs that are in common with our X-ray sample and the sample of Liang et al. (2008). For these bursts, we extrapolate ν_c at late times to the EoSD. In Liang et al. (2008), they only considered a constant density medium, where ν_c∝t^−1/2. The results of the extrapolation of ν_c are shown in Table 2. In Table 2, we find that at the EoSD, 1 keV <ν_{c, EoSD} for all GRBs. This further confirms our choice that ν_i < 1 keV < ν_c at the EoSD.¹⁰

The knowledge of the spectral regime at the EoSD allows us to write in an explicit expression for f_ES in Equation (3):

$$\begin{equation} \frac{f_{\mathrm{1\,keV, EoSD}}}{{\rm mJy}} \ge \left\lbrace \begin{array}{@{}ll}0.461(p-0.04) 10^{\frac{3p+1}{4}} 8.64^{\frac{3(p-1)}{4}} e^{2.53p} & \\ \quad \times (1+z)^{\frac{p+3}{4}} (\bar{\epsilon }_{e,-1})^{p-1} \epsilon _B^{\frac{p+1}{4}} n_0^{\frac{1}{2}} E_{53}^{\frac{p+3}{4}} & \\ \quad \times (t_{\rm 2, EoSD}) ^{-\frac{3(p-1)}{4}} d_{L28}^{-2}\nu _{14}^{-\frac{(p-1)}{2}}{ s=0} \\ 3.82(p-0.18) 10^{\frac{3p-1}{4}} 8.64^{\frac{3p-1}{4}} e^{2.54p} & \\ \quad \times (1+z)^{\frac{p+5}{4}} (\bar{\epsilon }_{e,-1}) ^{p-1} \epsilon _B^{\frac{p+1}{4}} A_{*,-1} E_{53}^{(p+1)/4} & \\ \quad \times (t_{\rm 2, EoSD}) ^{-\frac{(3p-1)}{4}} d_{L28}^{-2}\nu _{14}^{-\frac{(p-1)}{2}}{s=2}. \end{array} \right. \end{equation} \tag{ 7 }$$

On the left-hand side of this inequality, we have the observed X-ray flux at the EoSD, and on the right-hand side we have the expression for the external-forward shock flux when ν_i < 1 keV < ν_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} ≡ 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; Resmi & Bhattacharya 2008).

Before displaying our results for the _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 _B upper limits from the X-ray constraint given in Equation (7). For a standard p = 2.4, the X-ray constraint is

$$\begin{equation} \frac{f_{\mathrm{1\,keV, EoSD}}}{{\rm mJy}} \ge \left\lbrace \begin{array}{@{}ll}5.0 \times 10^{1} t_{2, \mathrm{EoSD}} ^{-1.05} \epsilon _{e,-1} ^{1.4} E_{53} ^{1.35} n_{0} ^{0.5} \epsilon _{B} ^{0.85} & s=0 \\ 7.0 \times 10^{2} t_{2, \mathrm{EoSD}} ^{-1.55} \epsilon _{e,-1} ^{1.4} E_{53} ^{0.85} A_{*,-1} \epsilon _{B} ^{0.85} & s=2. \end{array} \right. \end{equation} \tag{ 8 }$$

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 ν₁₄ corresponding to 1 keV. Solving for _B, the upper limit depends on the afterglow parameters as¹¹

$$\begin{equation} \epsilon _{B} \le \left\lbrace \begin{array}{@{}ll}1.0 \times 10^{-2} \left(\frac{f_{1 \,{\rm keV}, \mathrm{EoSD}}}{{\rm mJy}} \right)^{1.18} t_{2, \mathrm{EoSD}} ^{1.24} & \\ \quad \times \epsilon _{e,-1} ^{-1.65} E_{53} ^{-1.59} n_{0} ^{-0.59} & s=0 \\ 4.5 \times 10^{-4} \left(\frac{f_{1 \,{\rm keV}, \mathrm{EoSD}}}{{\rm mJy}} \right)^{1.18} t_{2, \mathrm{EoSD}} ^{1.82} & \\ \quad \times \epsilon _{e,-1} ^{-1.65} E_{53} ^{-1} A_{*,-1} ^{-1.18} & s=2. \end{array} \right. \end{equation} \tag{ 9 }$$

The median values for our X-ray sample for the parameters f_{1 keV, EoSD}, t_{2, EoSD}, and E₅₃ are 8 × 10⁻³ mJy, 4, and 1.6, respectively. Using these median values and _{e, −1} = 2, the upper limit on _B becomes

$$\begin{equation} \epsilon _{B} \le \left\lbrace \begin{array}{@{}ll}2.8 \times 10^{-5} \times n_{0} ^{-0.59} & s=0 \\ 3.7 \times 10^{-6} \times A_{*-1} ^{-1.18} & s=2. \end{array} \right. \end{equation} \tag{ 10 }$$

For a standard n₀ = 1 and A_{*, −1} = 1, it can be seen that the _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 10¹⁷ cm.

In Equation (10), the explicit dependence of the _B upper limit on the density is shown for p = 2.4. In the next subsection, we will display the results of the _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)).

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 that 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. For the remainder of this section, we assume a standard n₀ = 1 (A_{* − 1} = 1) for s = 0 (s = 2) when discussing our results for the _B upper limits.

Zoom In Zoom Out Reset image size

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

We now discuss how our assumptions on the afterglow parameters can affect the distribution of _B upper limits. For this discussion, we will take a typical p = 2.4; Equation (9) shows how the _B upper limit depends on the other afterglow parameters. The strongest dependence is on _e, which is raised to the power of −1.65. However, as we displayed in Figure 1, according to previous studies, the distribution of _e values is narrow, with the _e values ranging only over one order of magnitude. In addition, ∼62% of the bursts have _e ∼ 0.1–0.3. From Figure 1, a likely error in _e from our assumed _e = 0.2 is a factor ∼2. From Equation (9), an error in _e by a factor ∼2 will only lead to an error in the _B upper limit by a factor ∼3. For a constant density (wind) medium, the _B upper limit depends on E as E^−1.59 (E⁻¹). We assumed an efficiency of ∼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 ∼37% (∼32%). Taking the efficiency to be ∼30%–40% instead of ∼20% would lead to an error in E by a factor ∼2–3. From Equation (9), an error in E by a factor ∼2–3 would lead to an error in the _B upper limit by a factor ∼3–6 (∼2–3) for a constant density (wind) medium. Lastly, the largest source of uncertainty for the _B upper limits is the density, since it has been reported to have a range ∼10⁻³ cm⁻³–10² cm⁻³. For a constant density (wind) medium, the _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 cm⁻³ (A_* = 0.1). An error in the density by a factor ∼10³ (∼10²) will lead to an error in the _B upper limit by a factor ∼60 (∼230).

In summary, the expected errors in _e and E of a factor ∼2–3 will not change the _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 ∼2–3 orders of magnitude will lead to an error in the _B upper limits by ∼1–2 orders of magnitude.

Two additional parameters that can affect our _B upper limits are (1) ξ, 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, _e, _B, ξ producing the observed external-forward shock flux, another set of primed parameters E' = E/f, n' = n/f, $\epsilon _{e}^{\prime } = f \epsilon _{e}$, $\epsilon _{B}^{\prime } = f \epsilon _{B}$, ξ' = fξ can also produce the observed external-forward shock flux (Eichler & Waxman 2005). Afterglow studies usually assume ξ = f = 1 for simplicity but ξ ⩽ 1 and m_e/m_p ⩽ f ⩽ 1 (Eichler & Waxman 2005), where m_e (m_p) is the electron (proton) mass. The external-forward shock flux depends on ξ and _B as $f_{\nu } \propto \xi ^{2-p} \epsilon _{B} ^{(p+1)/4}$ (Leventis et al. 2012). From this dependence, we find that the _B upper limit ∝ξ^{4(p − 2)/(p + 1)}. Thus, including ξ will decrease the values of our _B upper limits. Values of ξ have not been determined for GRB external-forward shocks so we cannot quantify by how much the _B upper limit values will decrease. Including f will also decrease the values of the _B upper limits since $\epsilon _{B}^{\prime } = f \epsilon _{B}$ and f < 1. Like ξ, values of f have also not been determined from afterglow observations. The largest effect f can have on the _B upper limit values is decrease them by a factor of m_p/m_e ∼ 2000. For the remainder of this paper, we will be conservative and continue to assume ξ = f = 1, but we should keep in mind that considering ξ and f will decrease the values of the _B upper limits.

The light curves we consider for our optical sample decline with a temporal decay index α ∼ 1 from early times, ∼10²–10³s, 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 is¹²

$$\begin{equation} f_{\mathrm{obs}} = f_{\mathrm{ES}} (E, n, \epsilon _{e}, \epsilon _{B}, p, s). \end{equation} \tag{ 11 }$$

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

Our optical sample consists of 35 GRBs. A total of 33/35 of the bursts triggered Swift, and the remaining two bursts, 050502A and 080603A, were detected by the International Gamma-Ray Astrophysics Laboratory (INTEGRAL; Winkler et al. 2003). Table 4 shows properties of our optical sample. With a few exceptions, most of the GRBs in our optical sample have a known redshift.¹³

Table 4. Optical Sample Properties

GRB	z	dL28	Fluence	$E ^{\gamma } _{\mathrm{iso, }52}$	αO	Ref.	t2	f2eV	log10(B)
			(× 10−6 erg cm−2)					(mJy)	(s = 0)
050401	2.9	7.67	8.22	15.55	0.80 ± 0.03	[1]	0.72	0.3	−5.5
050502A	3.793	10.59	1.4	4.12	1.16 ± 0.03	[2]	1	5	−4.5
050525A	0.606	1.10	15.3	1.45	1.12 ± 0.05	[3]	34.56	0.5	−4.3
050721	2.5	6.40	3.62	5.32	1.29 ± 0.06	[4]	20	0.2	−5.0
050730	3.97	11.19	2.38	7.53	0.89 ± 0.05	[17]	7.5	0.57	−3.9
050802	1.71	4.00	2.00	1.49	0.82 ± 0.03	[1]	3.6	0.5	−3.5
051111	1.55	3.54	4.08	2.52	1.00 ± 0.02	[2]	30	0.4	−3.8
051221A	0.5465	0.97	1.15	0.09	0.96 ± 0.03	[5]	100	0.02	−3.4
060111B	2	4.86	1.60	1.58	1.18 ± 0.05	[6]	2	0.4	−5.2
060210	3.91	10.99	7.66	23.65	1.03 ± 0.06	[2]	10	0.1	−6.0
060418	1.49	3.37	8.33	4.78	1.13 ± 0.02	[7]	2	8	−4.6
060607A	3.082	8.25	2.55	5.34	1.20 ± 0.03	[7]	2	10	−4.2
060904B	0.703	1.32	1.62	0.21	1.00 ± 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 ± 0.06	[2]	0.5	2	−5.5
061007	1.26	2.74	44.4	18.48	1.70 ± 0.02	[7]	2	50	−6.0
061110B	3.44	9.42	1.33	3.34	1.64 ± 0.08	[2]	20	0.02	−5.9
061121	1.314	2.88	13.7	6.19	0.82 ± 0.02	[7]	4	0.5	−4.7
061126	1.159	2.47	6.77	2.39	0.89 ± 0.02	[2]	10	0.2	−4.6
070318	0.84	1.65	2.48	0.46	0.96 ± 0.03	[7]	20	0.2	−3.7
070411	2.954	7.84	2.70	5.27	0.92 ± 0.04	[2]	20	0.07	−4.7
070420	2.5	6.40	14.0	20.56	0.81 ± 0.04	[2]	3	0.8	−4.7
070714B	0.92	1.85	0.72	0.16	0.83 ± 0.04	[2]	10	0.03	−3.7
071003	1.6	3.69	8.3	5.45	1.466 ± 0.006	[9]	0.6	20	−5.7
071025	5.2	15.41	6.5	31.26	1.27 ± 0.04	[10]	20	0.02	−6.8
071031	2.692	7.00	0.9	1.5	0.97 ± 0.06	[11]	10.5	0.4	−3.4
071112C	0.823	1.61	3.00	0.53	0.95 ± 0.02	[12]	10.5	0.003	−6.3
080603A	1.688	3.94	1.1	0.80	0.99 ± 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 ± 0.01	[5]	3	10	−5.0
080810	3.35	9.13	4.60	11.06	1.23 ± 0.01	[7]	3	30	−3.9
080913	6.7	20.72	0.56	3.92	1.03 ± 0.02	[15]	10	0.02	−5.2
081008	1.967	4.76	4.30	4.13	0.96 ± 0.03	[16]	2	3	−4.1
090313	3.375	9.21	1.40	3.41	1.25 ± 0.08	[17]	20	2	−3.4
090418A	1.608	3.71	4.60	3.05	1.21 ± 0.04	[7]	2	0.8	−5.5

Notes. The bursts in bold are also part of our X-ray sample, and the two bursts in italics were detected by INTEGRAL instead of Swift. The redshift and the corresponding luminosity distance in units of 10²⁸ cm, d_L28, are shown in the second and third columns, respectively. The fluence, in units of 10⁻⁶ erg cm⁻², is shown in the fourth column. In the next column we show the isotropic equivalent energy released in gamma rays during the prompt emission in the units of 10⁵² erg, $E ^{\gamma } _{\mathrm{iso, }52}$. The temporal decay of the optical light curve, α_O, is shown in the sixth column. The reference where we found each optical light curve and α_O is shown in the seventh column. The time in units of 10² s and the flux in mJy of the data point we used to determine _B are shown in the next two columns. For GRBs 050730 and 060904B, we display the time and flux at the peak of the optical light curve that are given in Melandri et al. (2010; we could not find an optical light curve in units of specific flux in the literature for these two bursts). In the last column we show the _B measurements for n = 1 cm⁻³ and p determined from α_O (see bottom right panel of Figure 3). References for light curves and α_O. (1) Panaitescu et al. 2006; (2) Melandri et al. 2008; (3) Panaitescu 2007; (4) Antonelli et al. 2006; (5) Schulze et al. 2011; (6) Stratta et al. 2009; (7) Panaitescu & Vestrand 2011; (8) Covino et al. 2010; (9) Perley et al. 2008; (10) Perley et al. 2009; (11) Krühler et al. 2009; (12) Uehara et al. 2010; (13) Guidorzi et al. 2011; (14) Perley et al. 2011; (15) Greiner et al. 2009; (16) Yuan et al. 2010; (17) Melandri et al. 2010.

Download table as:  ASCIITypeset image

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 s. Considering early times has the advantage of minimizing possible energy injection. Our optical sample can be separated into four different subgroups, depending on the temporal behavior of the light curve before the α ∼ 1 decay as follows.

• 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 α_O ∼ 1 (∼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 4. We take the second data point to be confident that the optical light curve is declining.
• 2.  
  Single power-law decay from early times. In this subgroup, the optical light curve shows a decline as a single power law with α ∼ 1 from the beginning of the observations (∼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 4.
• 3.  
  Optical light curves with plateaus at early times. The optical light curves of three 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 s. After the plateau ends, the light curves of these three bursts show a decay α_O ∼ 1, as expected for the external forward-shock emission. In Table 4, for these three bursts, we show the temporal decay after the plateau and the time and the flux of the second data point after the plateau.
• 4.  
  Light curves with possible reverse shock emission at early times. Three 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 α ∼ 1 that is more typical for the external-forward shock emission. For these GRBs, in Table 4 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.

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 2 eV 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, ν_a < ν_i < ν_c. Because the optical light curve is declining at the time we are considering, the optical band must be above ν_i at this time. In Section 3.4, we argued that the X-ray band is between ν_i and ν_c at the EoSD at a few × 100 s; therefore, the optical band must also be in this spectral regime at the early times (∼10²–10³ s) 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_{1 keV, EoSD} with f_2eV (which represents the specific flux observed at 2 eV), and use ν₁₄ corresponding to 2 eV.

The other afterglow parameters are determined as in Section 3.2: _e = 0.2 and with z and the fluence,¹⁴ we obtain $E^{\gamma } _{\mathrm{iso}}$ and use $E = 5 E^{\gamma } _{\mathrm{iso}}$. As with our X-ray sample, we will display our _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, α_O, which is shown in Table 4 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 α_O (α_O = 3(p − 1)/4 for s = 0 and α_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).

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 that 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 α_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 n₀ = 1 (A_{* − 1} = 1) for s = 0 (s = 2) when discussing our results for the _B measurements for our optical sample.

Zoom In Zoom Out Reset image size

For a constant density (wind) medium, the mean and median _B measurements are ∼few × 10⁻⁵ (∼few × 10⁻⁶). The mean and median _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 α_O, significantly affects the distribution of _B measurements, we compared the two histograms in the bottom right panel of Figure 3 with a Kolmogorov–Smirnov (K-S) test. The null hypothesis of the K-S test is that the two histograms are drawn from the same distribution. We test this null hypothesis at the 5% significance level. The K-S test confirmed the null hypothesis that the two histograms are consistent with being drawn from the same distribution.

As with the _B upper limits from X-ray data, the mean and median _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 _B upper limits we attained from X-ray data, the _B measurements from optical data show a much wider distribution. For a constant density (wind) medium, the _B measurements range from _B ∼ 10⁻⁸–10⁻³ (_B ∼ 10⁻⁹–10⁻³). Also, since we used the same equation (Equation (7)) to find both the upper limits on _B with X-ray data and the _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 ξ and f can affect the distribution of _B upper limits also applies to the distributions of _B measurements we presented in this section. In addition, since ξ and f are less than unity, including these two parameters will mean that our _B measurements are effectively upper limits on _B.

We performed a K-S test between our optical _B measurements and the results from previous studies on _B (Figure 1). For our _B results, we used the optical _B measurements with n = 1 cm⁻³ and p determined from α_O (filled-in histogram in the bottom right panel of Figure 3; the _B values are shown in Table 4). The result of the K-S 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 × 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 _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 _B values found in the literature is ∼10⁻⁵–10⁻¹, whereas the range for our _B results is ∼10⁻⁸–10⁻³. The mean and median values for these two histograms are also significantly different. The mean (median) value for the _B histogram from the literature, 6.3 × 10⁻² (1.4 × 10⁻²), is a factor ∼700 (∼600) times larger than the mean (median) _B value of the histogram with our results, which is 9.5 × 10⁻⁵ (2.4 × 10⁻⁵).

One assumption that is commonly made in afterglow modeling studies is equipartition between _e and _B. As we discussed in Section 2, the results for _e from the literature and the results from recent simulations of relativistic collisionless shocks support _e ∼ 0.2. From this result, many works assume _B ∼ 10⁻²–10⁻¹. However, there is no physical argument to expect equipartition. Our distribution of _B upper limits and measurements, although wide, supports that there is no equipartition between electron and magnetic energies because none of the _B upper limits or measurements in our samples have a value larger than _B ∼  few  × 10⁻³. Another source of error that can lead to differences in _B values is differences in the determination of the spectral regime for the optical band. We took it to be between ν_i and ν_c, but it is also possible for the optical band to be above ν_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 _B. In summary, the main assumption we made when determining _B is an efficiency of ∼20% in the conversion of kinetic energy to gamma-ray energy, and we did not assume equipartition between _e and _B. Different authors have made different assumptions that can have a large effect on the results for _B.

Lastly, for a few bursts, we checked whether our method of determining _B is consistent with the values determined for _B with other techniques. GRBs 980519 and 990123, discussed in the afterglow modeling 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 ν_i < 2 eV < ν_c. Applying our technique to find a _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 _B ∼ 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 _B values can be accounted for by differences in the coefficients used for the external-forward shock flux.

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 n₀ = 1 when displaying the results for the AF upper limits. When plotting the results for AF, we will keep the dependence on B₀ and plot the quantity (AF)B_{0, 10 μG}. In the left and right panels of Figure 6, we show the upper limits on the quantity (AF)B_{0, 10 μG} 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 B₀ = 10 μG when discussing the results for the AF upper limits.

Zoom In Zoom Out Reset image size

The mean and median values of the AF upper limits are summarized in Table 5. The mean (median) AF upper limits are in the range AF ∼ 60–80 (AF ∼ 40–60). The AF upper limit histograms show a wide distribution, with a range of ∼10 to ∼300. To determine if assuming a different value of p has a significant effect on the distribution of AF upper limits, we performed a K-S test between the histograms in the right panel of Figure 6. The K-S 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.

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), ${\rm AF} \propto \sqrt{\overline{\epsilon _{B}} (p=2.4, n_{0}=1)} B_{0} ^{-1} n^{0.2}$, and from Equation (9), $\epsilon _{B} \propto \epsilon _{e} ^{-1.6} E^{-1.6} n^{-0.6}$. From these two expressions, ${\rm AF} \propto \epsilon _{e} ^{-0.8} E^{-0.8} n^{0.2} B_{0} ^{-1}$. We note that compared to the _B upper limit (Equation (9)), the AF upper limit has a weaker dependence on _e, E, and n. As we discussed at the end of Section 3.5, a likely error in _e is a factor of ∼2; this error in _e will translate into an error in the AF upper limits by a factor of only ∼2. For the energy, we assumed an efficiency ∼20%, and a likely error in the efficiency is a factor ∼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 ∼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 ∼10³ (see Section 3.5) from our assumed n = 1 cm⁻³ will only lead to an error in the AF upper limits by a factor ∼4. The price to pay for a weak n dependence is a linear dependence on B₀, with ${\rm AF} \propto B_{0} ^{-1}$. B₀ is an uncertain parameter that likely varies from GRB environment to GRB environment, and it is the largest source of uncertainty for AF.

As we discussed in Section 4.2, for our optical sample, we found a measurement for _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 _B(p, n₀ = 1) instead of $\overline{\epsilon _{B}} (p, n_{0}=1)$. The notation _B(p, n₀ = 1) denotes the _B measurements for our optical sample from Section 4.4 if we assume a standard n = 1 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 cm⁻³. In the left panel of Figure 7, we show the results for the measurements on the quantity (AF)B_{0, 10 μG} for p determined from α_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 μG} for p = 2.2 and p = 2.8. For the remainder of this section, we will assume B₀ = 10 μG when discussing the results for the AF measurements.

Zoom In Zoom Out Reset image size

A summary of the mean and median AF measurements for our optical sample is shown in Table 5. To determine whether assuming a standard p = 2.4, as opposed to determining p from α_O for each burst, has a statistically significant effect on the distribution of AF measurements, we performed a K-S test between the two histograms in the left panel of Figure 7. The K-S test confirmed the null hypothesis. The mean (median) AF measurements for the optical histograms range from ∼40 to ∼130 (∼20 to ∼100). Compared to the AF upper limit histograms, the AF measurement histograms show a wider distribution, ranging from AF ∼ 1 to AF ∼ 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 measurements.¹⁸