GRB 110721A: PHOTOSPHERE "DEATH LINE" AND THE PHYSICAL ORIGIN OF THE GRB BAND FUNCTION

The prompt emission spectrum of a gamma-ray burst (GRB) is usually well described by a phenomenological function known as the Band function (Band et al. 1993). This model, which is essentially a broken power-law function with a smooth (exponential) transition, was traditionally invoked to model spectra of GRBs detected by BATSE on board the Compton Gamma-Ray Observatory. The function is found to be successful in describing most GRB spectra detected by later missions as long as the spectral band is wide enough (e.g., Abdo et al. 2009; Zhang et al. 2011).

The physical origin of this phenomenological Band function is not identified. The traditional model is synchrotron emission of non-thermal electrons in an optically thin region, e.g., internal shocks (ISs) or internal magnetic dissipation regions (Mészáros et al. 1994; Tavani 1996; Daigne & Mochkovitch 1998; Lloyd & Petrosian 2000; Bosnjak et al. 2009; Zhang & Yan 2011). Alternatively, a matter-dominated outflow (fireball) can have a bright photosphere (Paczynski 1986; Goodman 1986; Mészáros & Rees 2000; Mészáros et al. 2002), which may be enhanced by kinetic or magnetic dissipation processes near the photosphere (Thompson 1994; Rees & Mészáros 2005; Pe'er et al. 2006; Giannios 2008; Beloborodov 2010; Lazzati & Begelman 2010; Ioka 2010). It has been argued that due to geometrical and/or physical broadening, this quasi-thermal component may be modified to mimic a Band function (e.g., Beloborodov 2010; Lazzati & Begelman 2010; Pe'er & Ryde 2011; Lundman et al. 2012). The scalings of a non-dissipative photosphere model has been argued to be able to interpret various empirical correlations (Fan et al. 2012).

Within the framework of the standard fireball-shock model, a GRB prompt emission spectrum is expected to be the superposition of a quasi-thermal photosphere emission component and a non-thermal component in the optically thin IS region (Mészáros & Rees 2000; Zhang & Mészáros 2002; Toma et al. 2011; Pe'er et al. 2012). Such a superposition effect has been claimed in the BATSE data archive (e.g., Ryde 2005; Ryde & Pe'er 2009), and was confirmed more robustly recently with the Fermi data (e.g., Ryde et al. 2010; Zhang et al. 2011; Guiriec et al. 2011; Axelsson et al. 2012). On the other hand, most GRB spectra (e.g., GRB 080916C) are still well described by one single Band component (Abdo et al. 2009; Zhang et al. 2011). This sharpens the debate regarding the origin of the Band function. For GRB 080916C, the non-detection of a thermal component led to the suggestion of a Poynting-flux-dominated outflow (Zhang & Pe'er 2009; see also Daigne & Mochkovitch 2002; Zhang & Mészáros 2002). Alternatively, some authors attempted to interpret the entire Band function as emission from a dissipative photosphere (e.g., Beloborodov 2010; Vurm et al. 2011;Giannios 2008; Ioka 2010). These two interpretations invoke distinct assumptions regarding the composition of the GRB jets. Finding observational clues to differentiate between them is therefore essential in unveiling the physics of the GRB central engine, jet composition, and energy dissipation mechanisms, which are poorly constrained (e.g., Zhang 2011).

Here, we show that the time-resolved spectral information of GRB 110721A holds the key to address this open question.

For a hot fireball with total wind luminosity L_w launched from an initial fireball radius R₀, the initial temperature is

$$\begin{equation} T_0 \simeq \big(L_w/4\pi R_0^2 c a\big)^{1/4} \simeq 1.4 \times 10^{10} \ {\rm K} \, L_{w,52}^{1/4} R_{0,7}^{-1/2}, \end{equation} \tag{ 1 }$$

where c is the speed of light and a = 7.56 × 10⁻¹⁵ erg cm⁻³ K⁻⁴ is the Stefan–Boltzmann energy density constant. The observed photosphere temperature T_ph can be as high as T₀ if the fireball is clean enough so that the photosphere radius R_ph does not exceed the fireball coasting radius R_c, but is lower than T₀ for fireballs with a heavier baryon loading when R_ph > R_c. More specifically, one has (Mészáros & Rees 2000)

$$\begin{equation} \frac{T_{{\rm ph}}}{T_0} = \left\lbrace \begin{array}{@{}ll}\left(\frac{R_{{\rm ph}}}{R_c}\right)^{-2/3} = \left(\frac{\eta }{\eta _*}\right)^{8/3}, & \eta < \eta _*, R_{{\rm ph}} > R_c,\nonumber \\ 1, & \eta > \eta _*, R_{{\rm ph}} < R_c, \end{array} \right. \end{equation} \tag{ 2 }$$

where $\eta =L_w/ \dot{M}c^2$ is the dimensionless entropy of the fireball, and

$$\begin{equation} \eta _* = \left(\frac{L_w \sigma _{_T}}{4\pi m_p c^3 R_0}\right)^{1/4} \simeq 1.04\times 10^3 \left(\frac{L_{w,52}}{R_{0,7}}\right)^{1/4} \end{equation} \tag{ 3 }$$

is the critical value of η. The photosphere luminosity is L_ph π(1/Γ)²R²σT⁴∝(R/Γ)²T⁴∝R²Γ²T'⁴. For η > η_*, since Γ∝R and T'∝R⁻¹, one has L_ph∝R⁰ ~ const. For η < η_*, since Γ∝R⁰, T∝R^−2/3, L_ph∝R²R^−8/3∝R^−2/3. So the photosphere luminosity is

$$\begin{equation} \frac{L_{{\rm ph}}}{L_w} = \left\lbrace \begin{array}{@{}ll}\left(\frac{R_{{\rm ph}}}{R_c}\right)^{-2/3} = \left(\frac{\eta }{\eta _*}\right)^{8/3}, & \eta < \eta _*, R_{{\rm ph}} > R_c, \\ 1, & \eta > \eta _*, R_{{\rm ph}} < R_c. \end{array} \right. \end{equation} \tag{ 4 }$$

If a GRB spectrum is dominated by the photosphere emission, for a certain observed isotropic γ-ray luminosity L = L_ph, the spectral peak energy E_p should not exceed ζkT₀, where ζ is a factor to denote the νF_ν peak of the photosphere spectrum. Therefore, a baryonic photosphere emission has a "death line" defined by

$$\begin{equation} E_p \le \zeta k T_0 \simeq 1.2 {\rm MeV} \zeta L_{52}^{1/4} R_{0,7}^{-1/2}. \end{equation} \tag{ 5 }$$

The factor ζ is subject to the shape of the spectrum. For a strict blackbody, ζ ~ 3.92. For a relativistic outflow, the shape of the blackbody is modified to the form (see also Li & Sari 2008 and references therein)

$$\begin{equation} F_\nu \propto \frac{\nu ^2}{c^2} kT \int _{h\nu /kT}^\infty \frac{dx}{e^x-1}. \end{equation} \tag{ 6 }$$

The νF_ν spectrum has a maximum value at

$$\begin{equation} \zeta \sim 2.82. \end{equation} \tag{ 7 }$$

The death line (Equation (5)) is derived for a non-dissipative photosphere. For a thermally dominated jet, dissipation via ISs or neutron collisional heating near the photosphere would compensate for adiabatic cooling, so that the photosphere temperature would be maintained close to but will never exceed the maximum temperature (Beloborodov 2012), and so Equation (5) applies to a broad category of dissipative photosphere models as well. A possible exception would be the dissipative photosphere model that invokes continued magnetic dissipation at extended radii above the photosphere (Drenkhahn & Spruit 2002), which allows E_p to be above the death line (Equation (5)) if the bulk Lorentz factor is large enough and dissipation at high optical depth is suppressed (Giannios 2012). However, in order to thermalize the jet, the required Lorentz factor is very low (Vurm et al. 2012), making it essentially impossible to exceed the death line. Also this model may not be able to interpret the GRB 110721A phenomenology, including the existence of the shoulder thermal component and the rapid spectral softening during the rising phase of bolometric luminosity (see Section 3).

GRB 110721A was jointly detected by the Gamma-Ray Burst Monitor (GBM; Meegan et al. 2009) and the Large Area Telescope (LAT; Atwood et al. 2009) on board the Fermi Gamma-Ray Telescope (Axelsson et al. 2012 and references therein). A candidate optical counterpart was reported (Greiner et al. 2011). Assuming that the association is real, Berger (2011) suggested two possible redshifts, z = 0.382 or z = 3.512, with the former one preferred.

The time-dependent spectral evolution of GRB 110721A was reported by Axelsson et al. (2012). Two noticeable features of the burst are: (1) a thermal component identified to be superposed on the Band component in both the time-integrated spectrum and time-resolved spectra. The temperature of this "shoulder" thermal component evolves with time as a broken power law, which is consistent with the expectations of the photosphere model (Ryde & Pe'er 2009). (2) The Band component displays an extremely rapid spectral evolution. The E_p at the earliest epoch reaches a record-breaking value of ~15 MeV.

We have independently processed the Fermi data of GRB 110721A. We performed a joint spectral analysis using the data from the NaI 6, 7 and BGO 1 detectors on GBM, as well as the LAT data.⁴ For GBM, we used the time-tagged events data containing individual photons with time and energy tags. Background rates are estimated by fitting the light curve before and after the burst using a one-order background polynomial model. We pretreated the LAT data using the LAT ScienceTools-v9r27p1 package and the P7TRANSIENT_V6 response function (detailed information for the LAT GRB Analysis are available in the NASA Fermi Web site⁵). In the diffuse response calculation, a three-component model is used: GRB 100721A with a PowerLaw2 spectrum, the Galactic diffuse model of gal_2yearp7v6_v0.fits, and the extragalactic diffuse power-law model. We then extracted the background-subtracted light curves from the GBM and LAT and demonstrated them in Figure 1. It is obvious that higher energy photons arrive earlier than the lower energy photons.

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In order to better understand the cause of such a behavior, we carry out a detailed time-dependent spectral analysis. We adjust the size of the time bins so that each time bin contains enough photons to perform a statistically significant spectral analysis. We applied the software package RMFIT (version 3.3pr7) to carry out the analyses. We confirm the conclusion of Axelsson et al. (2012) that adding a shoulder thermal component can significantly improve the fits to the data. Since the evolution of the thermal component has been presented in Axelsson et al. (2012), in this Letter we focus on the Band component only in order to study its physical origin. We find that the Band model usually gives a reasonable fit to the data, with the reduced χ² in the range ~(0.9–1.1). Figure 2 gives an example of the Band model fitting the data in the time interval of [−0.512, 0.064] s. This earliest epoch indeed shows an extremely high E_p value ~19.6 ± 4.5 MeV, which is consistent with E_p = 15 ± 1.7 MeV reported by Axelsson et al. (2012). The larger error in our fit may be because the Fermi team has included the extra LLE (LAT Low Energy) data, which are currently unavailable to the public. We also present the best-fit νF_ν model curves in different time intervals in Figure 3.

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Figure 4 displays the rest-frame E_p − L plot for time-resolved spectra of Fermi GRBs with well-measured redshifts. The data are a sub-sample of Figure 9 of Lu et al. (2012), for which only the parameters during the rising phase of GRB pulses are adopted. This is because the decaying phase may be controlled by the high-latitude curvature effect, which does not directly reveal radiation physics. Two "death lines" (Equation (5)) are drawn that correspond to R₀ ~ 10⁷ cm (solid, typical value) and R₀ ~ 3 × 10⁶ cm (dashed, an extreme value to allow highest death line possible). GRB 110721A at the earliest epoch [−0.512. 0.064]s is plotted for two candidate redshifts. It is clearly seen from the figure that for both redshifts, the points are way above the death lines. This rules out a wide range of dissipative photosphere models at least for this earliest epoch.

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One may take one step further. Axelsson et al. (2012) showed that E_p evolution of the Band component can be fit as a power-law decay with time, suggesting that this component might have the same physical origin during the burst. Figure 3 displays the evolution of the spectral shape of the Band component, which shows a similar α value with a gradually shallowing β value. This suggests that the Band component should share the same physical origin in different epochs: it is not from the photosphere, but is likely from an optically thin region where non-thermal particles are accelerated.

We have shown that the Band function component of GRB 110721A is beyond the "death line" of thermally dominated dissipative photosphere models in the E_p–L plane. Together with the fact that an additional shoulder thermal component is consistent with the photosphere model, we reach the conclusion that the so-called Band component is not of photospheric origin, at least for this burst, and is formed via non-thermal dissipation processes in the optically thin regions. Veres et al. (2012) interpreted this emission to be synchrotron emission from a magnetically dominated jet.

Such a finding has profound implications in understanding the origin of other Band function spectra of GRBs. Zhang et al. (2011) identified three elemental spectral components through a detailed time-resolved spectral analysis of 17 GRBs co-detected with Fermi GBM and LAT. They found that there are two types of GRBs. GRB 080916C's spectra maintain the "Band" shape even though the time interval for the spectral analysis progressively reduces, reaching ~1 s in the rest frame. GRB 090902B, on the other hand, showed a clear "narrowing" feature as the time bin reduces, and the spectrum is much narrower at ~rest frame 1 s. The time-integrated spectrum of this burst is also narrower than other Band GRBs. This burst's "Band" component can indeed be de-composed as superposed photosphere emission (Ryde et al. 2010; Zhang et al. 2011; Pe'er et al. 2012; Mizuta et al. 2011). However, such bursts are not common. Most bursts are similar to GRB 080916C.

The spectral parameters of GRB 110721A are similar to those of GRB 080916C and many other GRBs. The conclusion that the Band component of GRB 110721A originates from the optically thin region also supports the suggestion that most Band components are non-thermal (synchrotron or SSC) emission in the optically thin region. Available data seem to suggest the following unified picture: the GRB central engine may have a range of magnetization parameter, σ₀. (1) For low σ₀ bursts, dissipation can occur at small radii, so that a bright photospheric emission component (such as the case of GRB 090902B) is detected. Such bursts are usually accompanied by a high energy component due to upscattering of the thermal photons (and probably also synchrotron self-Compton; Pe'er et al. 2012). (2) For intermediate σ₀ bursts, the photosphere component is weaker but still detectable. Examples of this category include GRB 110721A (Axelsson et al. 2012) and GRB 100724B (Guiriec et al. 2011). The Band component of these bursts are formed in the large radii via ISs or internal collision-induced magnetic reconnection and turbulence (ICMART). (3) Finally, if σ₀ is large enough, both the photosphere and IS components are suppressed. The Band component forms at even larger radii via the ICMART process (Zhang & Yan 2011).

Finally, very rapid hard-to-soft E_p evolution observed in GRB 110721A (Axelsson et al. 2012 and this work) and many other GRBs (Lu et al. 2010, 2012) challenge existing models. Such a rapid evolution is not expected in the IS model and the photosphere model. A possible interpretation may be made within the framework of the ICMART model. According to this model (Zhang & Yan 2011), σ in the emission region rapidly decreases during each ICMART event, since the magnetic energy is continuously dissipated. If a good fraction of local magnetic dissipation energy is deposited to electrons, the typical electron Lorentz factor γ_e would have a σ-dependence, so that E_p decreases with time as σ decreases. Also, relativistic turbulent reconnection would lead to locally Doppler-boosted mini-jets, whose Lorentz factors would also depend on the local σ value (Zhang & Zhang 2012). The time-dependent Doppler boosts for mini-jets would enhance the hard-to-soft E_p evolution.

We thank the referee for helpful comments; P. Mészáros, B.-B. Zhang, and P. Veres for discussing their work with us; and A. Beloborodov, F. Daigne, D. Giannios, P. Kumar, R. Mochkovitch, A. Pe'er, and F. Ryde for useful discussion. This work was partially supported by NSF AST-0908362, National Natural Science Foundation of China (Grants No. 11025313 and 11063001), the "973" Program of China (2009CB824800), the Guangxi Natural Science Foundation (2010GXNSFA013112 and 2010GXNSFC013011), and the special funding for national outstanding young scientist (Contract No. 2011-135). X.F.W. is supported by the One-Hundred-Talents Program of Chinese Academy of Sciences.