VERITAS OBSERVATIONS OF THE UNUSUAL EXTRAGALACTIC TRANSIENT SWIFT J164449.3+573451

Swift J164449.3+573451 (hereafter Sw J1644+57) was first detected by the Swift Burst Alert Telescope (BAT) on 2011 March 28 at 12:57:45 UT. The Swift spacecraft slewed to the location of the source and began observations with the X-ray Telescope (XRT) and the UV/Optical Telescope (UVOT). These observations located a bright, uncataloged X-ray source but did not identify an optical afterglow typical of gamma-ray bursts (GRBs; Cummings et al. 2011). Less than one hour later, the BAT triggered a second time on Sw J1644+57, which ruled out a GRB origin and gave the first sign of the unusual nature of the source (Kennea et al. 2011a). This prompted multiwavelength follow-up observations at a number of observatories.

These follow-up observations identified an optical source consistent with the position of Sw J1644+57 (Cenko et al. 2011; Leloudas et al. 2011). Measurements obtained with the Gemini Observatory show an infrared (IR) source with a transient component at a location consistent with that of Sw J1644+57 and provide a redshift of z = 0.3534 from H_β and O iii emission lines (Levan et al. 2011a, 2011b). Radio observations with the Enhanced Very Large Array find an unresolved, variable radio source at a position consistent with the optical source detected by Cenko et al. (2011) and Leloudas et al. (2011), suggesting the optical source is most likely the host galaxy (Zauderer et al. 2011a, 2011b; Berger et al. 2011). Hubble Space Telescope observations show a nearly point-like IR source consistent with the location of Sw J1644+57 and, in an optical exposure, a resolved compact galaxy whose nucleus is consistent with the position of the IR point source (Fruchter et al. 2011). Temporal analysis of the Swift light curve, combined with the implied peak luminosity at a distance of z ∼ 0.35, provides evidence that the observed emission from Sw J1644+57 is likely beamed (Campana et al. 2011; Burrows et al. 2011; Bloom et al. 2011).

Because X-ray and very high energy (VHE; E > 100 GeV) gamma-ray emission are frequently correlated in other beamed sources, such as blazars (Böttcher 2010), it is reasonable to expect VHE emission from Sw J1644+57, depending on the parameters of the emission region and the surrounding environment. Here, we discuss deep VHE observations of Sw J1644+57 with the Very Energetic Radiation Imaging Telescope Array System (VERITAS) and the implications of our results for some possible emission scenarios for this unusual object.

VERITAS is an array of four imaging atmospheric-Cherenkov telescopes (IACTs) located at the Fred Lawrence Whipple Observatory in southern Arizona at an altitude of 1280 m above sea level (Holder et al. 2008). Imaging cameras, consisting of 499 photomutiplier tubes located in the focal plane of each telescope, detect Cherenkov light emitted by extensive air showers initiated in the upper atmosphere by gamma rays and cosmic rays. VERITAS has a field of view of 35 and is sensitive in the range of 100 GeV–30 TeV. The telescopes typically operate in "wobble" mode, where the location of the target is offset from the center of the field of view by 05, allowing for simultaneous background measurements (Fomin et al. 1994). The offset direction alternates between north, south, east, and west for each data segment (typically lasting 20 minutes) to reduce systematic errors in the background estimation.

On 2011 March 29 at 10:27 UT, approximately 22.5 hr after the first BAT trigger, VERITAS started observing Sw J1644+57. Subsequent daily observations with an average exposure of ∼2 hr/night were taken when weather conditions were favorable, continuing through 2011 April 15, after which observations were not possible because of the near-full Moon (exceeding ∼97% illumination). Zenith angles for our observations ranged from 25° to 40°. Due to temporary hardware issues, approximately 15% of the data were taken with an array of three telescopes. In total, VERITAS accumulated ∼28 hr of exposure on this source, of which ∼3.5 hr were taken within one day of the particularly intense flaring events observed in X-rays during 2011 March 28–31 (Kennea et al. 2011b).

For this analysis, about 90% of the data (∼25 hr) pass the quality selection criteria, with selection based primarily on weather conditions and trigger-rate stability. The selected data are processed through the standard VERITAS analysis package (Cogan 2007). Our cosmic-ray rejection procedure is based on applying selection criteria on standard image parameters (Hillas 1985): the size of the telescope images, the mean scaled width and mean scaled length parameters (Krawczynski et al. 2006), the height of maximum Cherenkov emission and the angular distance from the putative source position to the reconstructed arrival direction of the shower (θ). The standard selection criteria (see Table 1) were optimized using Monte Carlo simulations and real data from the Crab Nebula and the blazar PG 1553+113.

Table 1. Selection Criteria Used for VERITAS Analysis

Parameter	Selection Criteria
Image size	>400 digital counts (∼75 photoelectrons)
Mean scaled width	0.05 < MSW < 1.15
Mean scaled length	0.05 < MSL < 1.3
Height of shower maximum	>7 km
θ	<01

Note. For an explanation of these parameters see Section 2.

Download table as:  ASCIITypeset image

The remaining background is estimated using the "reflected-region" method described in Berge et al. (2007). The radii of the circular on- and off-source regions are 01. Statistical significances are computed using a modified version of Equation 17 from Li & Ma (1983) to allow for varying number of off-source regions due to the bright (V = 4.849) nearby star HR 6237 (Aharonian et al. 2004).

Significant VHE gamma-ray emission is not detected from the direction of Sw J1644+57 in the entire data set nor in subsets of the data (see Table 2). In order to look specifically for VHE emission contemporaneous with the intense X-ray flaring, the first subset consists of data that were taken within one day before or after periods where the XRT count rate exceeded 20 s⁻¹. This subset is denoted the "flaring" period and comprises the first three nights of observations. However, it is worth noting that VERITAS exposures during this "flaring" period fell between X-ray flares observed by Swift. Therefore, VERITAS observations were simultaneous with relatively low X-ray flux states during that period, characterized by an X-ray flux of νF_ν ∼ 10⁻¹⁰ erg cm⁻² s⁻¹, about two orders of magnitude lower than the major flares. The second subset, denoted the "low" period, comprises the remainder of the data.

Following the lack of signal in the data, we derive 99% confidence level upper limits over various time intervals (see Table 2).

• 1.  
  Total: 2011 March 29–April 15.
• 2.  
  Flaring: 2011 March 29–31.
• 3.  
  Low: 2011 April 1–15.
• 4.  
  Daily (by UT date, when observations available).

The procedure described by Rolke et al. (2005) is chosen for the upper-limit computation with the assumption of a Gaussian-distributed background. The total, flaring and low-flux upper limits on E*F(E) at 99% c.l. are 1.4 × 10⁻¹², 3.1 × 10⁻¹², and 1.5 × 10⁻¹² erg cm⁻² s⁻¹, respectively, where F(E) is the energy flux. The limits are calculated at 500 GeV assuming any emission follows a power-law spectrum with a photon index of −3.0. The decorrelation energy (500 GeV) is used to reduce the sensitivity of the limits to the choice of photon index; this energy is higher than the energy threshold of the observations (∼290 GeV). The flaring and low-state upper limits along with the daily upper limits are presented in Figure 1, superimposed on the Swift XRT light curve (Evans et al. 2007) for comparison.

Zoom In Zoom Out Reset image size

In this section, we provide some generic parameter constraints that can be derived from the observed X-ray properties of Sw J1644+57, along with the non-detection by Fermi/LAT (Omodei et al. 2011) and VERITAS. The X-ray flux varied on timescales of t_var = 100 s (Burrows et al. 2011), with a peak energy flux of F_X ∼ 10⁻⁸ erg cm⁻² s⁻¹, corresponding to a peak luminosity of L_pk ∼ 4.3 × 10⁴⁸ erg s⁻¹ if the emission were isotropic. In order to illustrate the dependence of the following estimates on the variability timescale, we parameterize t_var ≡ 100 t_var,2 s. The Eddington limit implies a central engine mass of M > 3.4 × 10¹⁰ M_☉, assuming unbeamed emission. Assuming that the emission-region size is not smaller than the Schwarzschild radius of the central engine, the observed variability implies M < 10⁷ M_☉. The two mass estimates can be reconciled by allowing for anisotropic and/or beamed emission, plausibly involving relativistic motion. Relativistic motion will result in Doppler boosting of the luminosity by a factor D⁴, along with variability time contraction by a factor D⁻¹, where D = (Γ[1 − β_Γcos θ])⁻¹ is the Doppler factor, Γ = (1 − β²_Γ)^−1/2 is the bulk Lorentz factor of the emission region, β_Γc is its velocity, and θ is the angle between the direction of motion and the line of sight. Reconciling the mass estimates above requires Doppler boosting by at least a factor D > 5.4.

Bloom et al. (2011) and Burrows et al. (2011) have argued that this event arises from the activation of a beamed jet and have hypothesized that this may be the result of tidal disruption of a star by a ∼10⁶–10⁷ M_☉ black hole. Both synchrotron-dominated (Burrows et al. 2011) and Compton-dominated (Bloom et al. 2011) origins have been proposed for the X-ray emission. Burrows et al. (2011) propose a Poynting-flux-dominated scenario, in which the X-ray emission is produced by synchrotron emission from relativistic electrons. Bloom et al. (2011) interpret the lack of variability of the radio–IR emission as evidence that the radio–IR emission is produced in a more extended region than the X-rays. They suggest inverse Compton scattering of external radiation as the mechanism producing the high-energy radiation.

In the following discussion, we present some general considerations to constrain the parameters of the X-ray emission region, including constraints placed by the VHE upper limits. We consider both synchrotron and inverse Compton as possible emission mechanisms.

4.1. Synchrotron Origin

We first consider a scenario in which the X-ray emission is synchrotron emission by relativistic electrons in a tangled magnetic field B. The electron Lorentz factor at which the non-thermal electron distribution has its peak radiative output is γ_p, and the electron density at that energy is n_p ≡ n_e(γ_p). The observed spectral variability suggests that the peak frequency might vary substantially during the various outbursts. For the following estimates, we scale the peak frequency as ν_pk ≡ 10¹⁹ ν_{p, 19} Hz and we base our estimates on the typical X-ray flux observed during VERITAS observations in the flaring state νF^sy_ν ≡ 10⁻¹⁰ f₋₁₀ erg cm⁻² s⁻¹, corresponding to νL_ν = 4.3 × 10⁴⁶ erg s⁻¹. We further assume that the variability timescale provides an estimate of the emission-region size, R_B = c t_var D/(1 + z). The observables can then be related to the emission-region parameters through (Rybicki & Lightman 1979)

$$\begin{equation} \nu _{\rm pk} = 4.2 \times 10^6 \, \gamma _p^2 \, {\left(\frac{B}{\rm 1\ G}\right)} \, {D \over 1 + z}\; {\rm Hz}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \nu L_{\nu } = {2 \over 9} \, c \, \sigma _T \, B^2 \, \gamma _p^2 \, n_p \, \left({c \, t_{\rm var} \over 1 + z} \right)^3\, D^7. \end{equation} \tag{ 2 }$$

A further constraint is derived from the condition that the synchrotron cooling timescale of electrons of energy γ_p should be of the order of the observed variability timescale. This corresponds to the assumption that the entire energy transferred to radiation throughout the duration of the flare is contained in the particle population at the onset of the flare. We define such a scenario as a particle-dominated scenario. Burrows et al. (2011) have shown that an alternative, Poynting-flux-dominated scenario with synchrotron cooling timescales of the order of t_sy ≲ 0.1 s can explain the spectral energy distribution (SED) of Sw J1644+57. Such a scenario requires continuous in situ re-acceleration of electrons to maintain a low-energy cutoff in the electron distribution, which is needed in order to reproduce the observed hard optical–X-ray spectral slope.

Assuming that the electron cooling timescale and the light-crossing timescale across the source are of the same order, we estimate

$$\begin{equation} t_{\rm var} \sim t_{\rm sy} \, {1 + z \over D} = {6 \, \pi \, m_e c^2 \over c \, \sigma _T \, B^2 \, \gamma _p} \, {1 + z \over D} \,. \end{equation} \tag{ 3 }$$

Parameterizing the Doppler factor in terms of D₁ ≡ D/10, we solve Equations (1)–(3) to find

$$\begin{eqnarray} B &=& 1.5 \, D_1^{-1/3} \, \nu _{p, 19}^{-1/3} \, t_{{\rm var,}2}^{-2/3}\;{\rm G}, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \gamma _p &=& 4.4 \times 10^5 \, D_1^{-1/3} \, t_{{\rm var,}2}^{1/3} \, \nu _{p, 19}^{2/3}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} n_p &=& 6.7 \times 10^4 \, D_1^{-17/3} \, \nu _{p, 19}^{-2/3} \, t_{{\rm var,}2}^{-7/3} \, f_{-10} \; {\rm cm}^{-3} \,. \end{eqnarray} \tag{ 6 }$$

These parameters correspond to a Thomson depth from electrons near the peak, τ_T, of

$$\begin{equation} \tau _T = n_e \, \sigma _T \, R_B = 10^{-6} \, D_1^{-14/3} \, \nu _{p, 19}^{-2/3} \, t_{{\rm var,}2}^{-4/3} \, f_{-10} \,. \end{equation} \tag{ 7 }$$

The expected synchrotron radiation energy density in the comoving frame, u'_sy, is

$$\begin{equation} u^{\prime }_{\rm sy} \sim {16 \over 9} \, \tau _T \, \gamma _p^2 \, u^{\prime }_B \,, \end{equation} \tag{ 8 }$$

where u'_B = B²/(8π) is the comoving magnetic-field energy density. Electrons with energy corresponding to γ_p will encounter synchrotron photons at frequency ν_pk (observer's frame) in the extreme Klein–Nishina limit, γ_p '_pk ∼ 3.5 × 10³ D^−4/3₁ t^1/3_var,2 ν^5/3_{p, 19} where '_pk = hν_pk/(D m_ec²). Therefore, the inverse Compton output from electrons at γ_p is strongly suppressed by a factor $f_{\rm KN} = \frac{3}{8} \frac{ln(2 \epsilon ^{\prime } \gamma) + 1/2}{\epsilon ^{\prime }\gamma } \sim 7.6 \times 10^{-4} \, D_1^{4/3} \, t_{{\rm var,}2}^{-1/3} \, \nu _{p, 19}^{-5/3}$. Hence, the ratio of synchrotron self-Compton (SSC) to synchrotron luminosities is expected to be

$$\begin{equation} \left({L_{\rm SSC} \over L_{\rm sy} } \right) = {u^{\prime }_{\rm sy} \over u^{\prime }_B} \times f_{\rm KN} \sim 2.6 \times 10^2 \, D_1^{-4} \, t_{{\rm var,}2}^{-1} \, \nu _{p, 19}^{-1} \, f_{-10} \,, \end{equation} \tag{ 9 }$$

which leads to an estimated intrinsic VHE flux of

$$\begin{eqnarray} \nu F_{\nu }^{\rm SSC, int} &=& \left({L_{\rm SSC} \over L_{\rm sy}} \right) \, \nu F_{\nu }^{\rm sy} \sim 2.6 \times 10^{-8} \nonumber\\ &&\times D_1^{-4} \, t_{{\rm var,}2}^{-1} \, \nu _{p, 19}^{-1} \, f_{-10}^2 \; {\rm erg} \; {\rm cm}^{-2} \; {\rm s}^{-1} \,, \end{eqnarray} \tag{ 10 }$$

with an emission peak around Dγ_pm_ec²/(1 + z) ∼ 1.7D^2/3₁ t^1/3_var,2 ν^2/3_{p, 19} TeV. We point out that if L_SSC/L_sy > 1, the effective electron cooling timescale will actually be shorter than the synchrotron cooling timescale by a factor ∼L_sy/L_SSC, in which case our initial assumption $t_{\rm var} \sim t_{\rm sy} \, {1 + z \over D}$ breaks down.

At VHE photon energies, the effect of γγ absorption internal to the emission region may become substantial (Burrows et al. 2011). Photons of comoving photon energy '_γ ≡ hν'_γ/(m_ec²) ≡ 10⁵ ₆ D⁻¹₁ are most efficiently absorbed by target photons of energy '_T ∼ 2/'_γ, corresponding to an observed target photon frequency of ν_T ∼ 10¹⁶ D²₁ ⁻¹₆ Hz, i.e., UV photons. We note that the UV flux appears to be very strongly absorbed by gas and dust local to the host galaxy, and the intrinsic UV flux may well be several orders of magnitude higher than the UVOT upper limits of ≲ 10⁻¹⁴ erg cm⁻² s⁻¹. Here we parameterize the intrinsic flux of photons at those frequencies as νF_ν(_T) ≡ 10⁻¹¹ f_{UV, −11} erg cm⁻² s⁻¹ since the X-ray to optical flux extrapolations from the SED, as well as other evidence about the extinction, show that f_UV is probably around 0.5 × 10⁻¹¹ erg cm⁻² s⁻¹ (Burrows et al. 2011). Based on a δ-function approximation to the γγ absorption cross section, the optical depth for γγ absorption is estimated as

$$\begin{eqnarray} \tau _{\gamma \gamma }^{\rm int} &\sim & {4 \over 3} \, {\sigma _T \, d_L^2 \, \nu F_{\nu } (\epsilon _T) \, (1 + z) \over \epsilon ^{\prime }_T \, D^5 \, m_e c^2 \, c^2 \, t_{\rm var} }\nonumber\\ &\sim & 5.5 \times 10^3 \, f_{{\rm UV}, -11} \, \epsilon _6 \, D_1^{-6} \, t_{{\rm var,}2}^{-1}. \end{eqnarray} \tag{ 11 }$$

For internal absorption, the suppression of the flux is given by $F_{\rm abs, internal} = F_{\rm internal} (1 - e^{-\tau _{\gamma \gamma }^{\rm int}}) / \tau _{\gamma \gamma }^{\rm int} \approx F_{\rm int} / \tau _{\gamma \gamma }^{\rm int}$ for τ^int_γγ ≫ 1. Consequently, after correction for extinction by the extragalactic background light (EBL), which amounts to a factor of $e^{-\tau _{\gamma \gamma }^{\rm EBL}} \sim 1/13$ at 500 GeV, using the Finke et al. (2010) EBL model, the particle-dominated synchrotron scenario predicts a VHE γ-ray flux of $\nu F_{\nu }^{\rm SSC, int} \, e^{-\tau _{\gamma \gamma }^{\rm EBL}} / \tau _{\gamma \gamma }^{\rm int} \sim 3.6 \times 10^{-13}$ erg cm⁻² s⁻¹, which is slightly below the VERITAS upper limits.

However, such an emission model would require the following equipartition ratio between the comoving energy densities in the magnetic field, u'_B, and the relativistic electrons, u'_e, based on the parameters of Equations (4)–(6):

$$\begin{equation} \epsilon _B \equiv u^{\prime }_B / u^{\prime }_e = 3.7 \times 10^{-6} D_1^{16/3} \nu _{p, 19}^{-2/3} t_{{\rm var,}2}^{2/3} f_{-10}^{-1}. \end{equation} \tag{ 12 }$$

Therefore, the particle-dominated synchrotron scenario, though possible, is disfavored as it requires an unusually large Doppler factor (≳ 100) to allow for equipartition to occur in the jet.

4.2. External Inverse Compton Origin

The X-rays may also be produced by inverse Compton scattering of low-energy radiation. We scale the peak frequency of the soft target photons for Compton scattering as ν_s ≡ 10¹³ ν_{s, 13} Hz. Assuming that the external radiation field is approximately isotropic in the rest frame of the host galaxy, the observed Compton peak frequency is ν_pk ∼ ν_s γ²_p D²/(1 + z) ∼ 10¹⁹ ν_{p, 19} Hz, yielding

$$\begin{equation} \gamma _p \sim 10^2 \, \nu _{s,13}^{-1/2} \, D_1^{-1} \, \nu _{p, 19}^{1/2}. \end{equation} \tag{ 13 }$$

Setting the observed variability timescale equal to the Compton cooling timescale (modulo D/(1 + z)) yields an estimate of the energy density of the external radiation field in the comoving frame,

$$\begin{equation} {u^{\prime }}_s \sim 415 \, \nu _{s,13}^{1/2} \, \nu _{p, 19}^{-1/2} \, t_{{\rm var,}2}^{-1} \; {\rm erg \; cm}^{-3}, \end{equation} \tag{ 14 }$$

which is related to the energy density in the rest frame of the host galaxy through u_s ≈ u'_s/Γ² ∼ u'_s/D². Assuming that this emission originates within a few hundred Schwarzschild radii of the central black hole (i.e., R_ext = 10¹⁵ R₁₅ cm), the above radiation energy density results in a luminosity of L_s ∼ 4π R²_ext c u_s ∼ 1.6 × 10⁴² R²₁₅ D⁻²₁ t_var,2⁻¹ ν^1/2_{s, 13} ν_{p, 19}^−1/2 erg s⁻¹, which corresponds to a flux of

$$\begin{equation} \nu F_{\nu }^{\rm s} \sim 3.8 \times 10^{-15} \, R_{15}^2 \, D_1^{-2} \, t_{{\rm var,}2}^{-1} \, \nu _{s,13}^{1/2} \, \nu _{p, 19}^{-1/2} \; {\rm erg \; cm}^{-2} \; {\rm s}^{-1}, \end{equation} \tag{ 15 }$$

which would be of the order of the observed flux of the IR peak for R₁₅ ∼ a few. The observed X-ray peak luminosity can be used analogous to Equation (2) to infer the density of electrons around γ_p:

$$\begin{equation} n_p \sim 8.5 \times 10^6 \, f_{-10} \, \nu _{s,13}^{1/2} \, D_1^{-7} \, \nu _{p, 19}^{-1/2} \, t_{{\rm var,}2}^{-2} \; {\rm cm}^{-3}, \end{equation} \tag{ 16 }$$

which yields a Thomson depth of

$$\begin{equation} \tau _T \sim 1.3 \times 10^{-4} \, f_{-4} \, \nu _{s,13}^{1/2} \, D_1^{-6} \, \nu _{p, 19}^{-1/2} \, t_{{\rm var,}2}^{-1}. \end{equation} \tag{ 17 }$$

Analogous to Equations (8) and (9), we can now compute the expected importance of higher-order Compton scatterings through the ratio of luminosities in second-order to first-order Compton luminosities:

$$\begin{equation} \left({L_{C2} \over L_{C1}} \right)_{\rm Thomson} \sim 2.2 f_{-10} \, D_1^{-8} \, \nu _{s,13}^{-1/2} \, \nu _{p, 19}^{1/2} \, t_{{\rm var,}2}^{-1}, \end{equation} \tag{ 18 }$$

and this emission would peak at

$$\begin{equation} \nu _{C2} \sim 10^{21} \, D_1^{-2} \, \nu _{s,13}^{-1} \, \nu _{p, 19}^2 \; {\rm Hz}, \end{equation} \tag{ 19 }$$

which corresponds to E_C2 ∼ 4 D⁻²₁ ν⁻¹_{s, 13} ν_{p, 19}² MeV and is substantially below the Fermi/LAT regime of E > 100 MeV. We therefore conclude that higher-order Compton scattering is not expected to lead to a detectable signal in the Fermi/LAT or VERITAS regimes.

Bloom et al. (2011) suggested that the target field for Compton scattering might be UV–soft X-ray emission from an accretion disk formed during the tidal disruption event. This would correspond to ν_{s, 13}  ∼  10⁴. Consequently, the X-rays could be produced through the bulk Compton process by cold (in the comoving frame) electrons. We would then infer an external radiation energy density of u_s  ∼  420 D⁻²₁ t⁻¹_var,2 ν_{p, 19}^−1/2 erg cm⁻³, corresponding to an observed flux of νF^s_ν ∼ 3.8 × 10⁻¹³ R₁₅² D⁻²₁ t_var,2⁻¹ ν^−1/2_{p, 19} erg cm⁻² s⁻¹. Considering the substantial UV extinction toward the emission region, this flux still appears consistent with the UVOT upper limits.

We conclude that among the scenarios discussed here, inverse Compton scattering of either the observed radio–IR radiation by relativistic electrons of γ_p ∼ 10² or of a putative accretion-related UV radiation field scattered by cold electrons in a relativistically moving emission region with D ∼ 10 are plausible mechanisms for the production of the observed rapidly-varying X-ray emission and are compatible with the Fermi-LAT and VERITAS upper limits. As discussed by Burrows et al. (2011), the observed SED and variability are also consistent with a synchrotron-dominated X-ray emission scenario, if the jet has a strong magnetic field (Poynting-flux-dominated) and has ongoing in situ acceleration of electrons. However, if the energy content of the emission region is dominated by relativistic particles, either far sub-equipartition magnetic fields or an uncomfortably large Doppler factor are required.

This research is supported by grants from the U.S. Department of Energy, the U.S. National Science Foundation and the Smithsonian Institution, by NSERC in Canada, by Science Foundation Ireland (SFI 10/RFP/AST2748), and by STFC in the UK. We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and the collaborating institutions in the construction and operation of the instrument. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.