THE PHOTOSPHERIC RADIATION MODEL FOR THE PROMPT EMISSION OF GAMMA-RAY BURSTS: INTERPRETING FOUR OBSERVED CORRELATIONS

In the past 15 years, our understanding of gamma-ray bursts (GRBs) has been revolutionized. As usual, some aspects are understood better than others. For example, the detection of a bright supernova component in the afterglow of some nearby long GRBs establishes their collapsar origin and the late (∼10⁴ s after the trigger of the burst) afterglow data support the external forward shock model (Piran 2004; Zhang & Mészáros 2004). Yet the physical origin of the prompt emission of GRBs is still not clear. The "leading" internal shock model is found to have difficulty explaining some observational facts, motivating people to develop internal magnetic energy dissipation models and the photosphere models (see Piran 2004; Zhang & Mészáros 2004 for reviews). It is rather hard to distinguish among these models reliably. It is widely speculated that the polarimetry of the prompt emission, for example, by the POlarimeters for Energetic Transients (POET; Hill et al. 2008) and by POLAR (Orsi 2011), may play key roles in the future. In this Letter, we show that some empirical correlations of the prompt emission properties may shed valuable light on the underlying physics and that the photospheric model is favored.

The tight correlation E_p∝L^{0.5 ± 0.1} was discovered by Wei & Gao (2003; see Figure 6 therein) and has then been confirmed by many researchers (e.g., Liang et al. 2004; Yonetoku et al. 2004; Ghirlanda et al. 2009; Zhang et al. 2012). Recently, a tight correlation Γ∝L^{0.3 ± 0.002} was identified by Lü et al. (2012) and the correlation Γ∝E^{0.78 ± 0.18}_p was suggested by Ghirlanda et al. (2012). Very recently, Margutti et al. (2012) and Bernardini et al. (2012) discovered a tight correlation E_γ/E_x∝E^{0.66 ± 0.16}_p, where E_γ is the isotropic equivalent energy of the prompt emission and E_x is the total energy of the afterglow emission in the X-ray band. In the forward shock afterglow model, E_x is proportional to E_k, the kinetic energy of the outflow (Piran 2004; Zhang & Mészáros 2004). Therefore, E_γ/E_x(∝E_γ/E_k) is proportional to the GRB efficiency $\bar{\eta }_\gamma \equiv E_{\rm \gamma }/(E_{\rm \gamma }+E_{\rm k})$ as long as E_γ is considerably smaller than E_k. Hence, one has $\bar{\eta }_\gamma \propto E_{\rm p}^{0.7}$. Some possible interpretations of the E_p–L correlation can be found in the literature (e.g., Wei & Gao 2003; Rees & Mészáros 2005; Ghirlanda et al. 2012). In this Letter, we aim to interpret all the above four correlations together.⁵ The starting point is the extensively discussed speculation that the prompt emission of GRBs is mainly from the photosphere, which suffers significant modification, and its spectrum is normally no longer thermal-like (e.g., Rees & Mészáros 2005; Ioka et al. 2007; Beloborodov 2010; Lazzati et al. 2011; Giannios 2012).

First, we discuss the simplest scenario, in which the luminosity, spectral peak energy, and efficiency of the emission roughly resemble L_b, T_b, and Y_b, where L_b, T_b, and Y_b are the luminosity, temperature, and efficiency of the photospheric radiation, and Y_b and L_b are related to the total luminosity L₀ as Y_b = L_b/L₀. In such a scenario, if there are valid correlations among L_b, T_b, Γ, and Y_b, so will there be valid correlations among L, E_p, Γ, and $\bar{\eta }_\gamma$. For a relativistic baryonic fireball, the acceleration and the subsequent photospheric radiation have been initially investigated by Piran et al. (1993) and by Mészáros et al. (1993). Following these approaches, Fan & Wei (2011) have recently derived the expressions of the initial radius of the accelerated outflow (i.e., R₀) and the final Lorentz factor of the outflow (i.e., Γ)

$$\begin{equation} R_{0}\propto L_{\rm b}^{1/2}Y_{\rm b}^{3/2}T_b^{-2}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \Gamma \propto \big(Y_{\rm b}^{-1}-4/3\big)^{1/4}L_{\rm b}^{1/8}T_{\rm b}^{1/2}, \end{equation} \tag{ 2 }$$

respectively. For Y_b ≪ 1 (actually even for Y_b = 0.5, the difference between (Y⁻¹_b − 4/3)^1/4 and Y^−1/4_b is only by a factor of 1.3), Equation (2) reduces to the form obtained by Pe'er et al. (2007), i.e.,

$$\begin{equation} \Gamma \propto Y_{\rm b}^{-1/4}L_{\rm b}^{1/8}T_{\rm b}^{1/2}. \end{equation} \tag{ 3 }$$

As shown in Lü et al. (2012), for the outflow launched via the annihilation of neutrino pairs emitting from a hyper-accreting disk, the dimensionless entropy of the initial outflow is related to the total luminosity as η∝L^k₀ (k ∼ 7/27 is derived if the poorly understood collimation process is ignored (Lü et al. 2012). In the following derivation we regard k as a "free parameter"). The final Lorentz factor of the accelerated outflow is related to the initial dimensionless entropy as Γ ≈ 4(1 − 4Y_b/3)η/3. As long as the thermal radiation is not extremely efficient (say, Y_b ⩽ 0.25),⁶ we approximately have

$$\begin{equation} \Gamma \propto L_{\rm b}^{\rm k}Y_{\rm b}^{\rm -k}. \end{equation} \tag{ 4 }$$

Combining Equation (1) with Equation (3), we have

$$\begin{equation} \Gamma \propto L_{\rm b}^{1/4}R_0^{-1/4}Y_{\rm b}^{1/8}. \end{equation} \tag{ 5 }$$

Substituting this relation into Equation (4), we have

$$\begin{equation} Y_{\rm b}\propto L_{\rm b}^{8k-2\over 1+8k}R_0^{2\over 1+8k}. \end{equation} \tag{ 6 }$$

Hence Equations (4) and (1) give

$$\begin{equation} \Gamma \propto L_{\rm b}^{3k \over 1+8k}R_0^{-{2k\over 1+8k}}, \end{equation} \tag{ 7 }$$

and

$$\begin{equation} T_{\rm b} \propto L_{\rm b}^{{32k-5\over 4(1+8k)}}R_0^{1-4k \over 1+8k}, \end{equation} \tag{ 8 }$$

respectively. Finally we have

$$\begin{equation} \Gamma \propto T_{\rm b}^{12k \over 32k-5}R_0^{-{2k\over (32k-5)}}. \end{equation} \tag{ 9 }$$

So far we have shown that some correlations should be present.

In the current scenario, $E_{\rm p}, L, \;{\rm and }\; \bar{\eta }_\gamma$ largely resembles T_b, L_b, and Y_b, respectively. So if we take k  ∼  0.34, the expected relations are

$$\begin{eqnarray} \Gamma &\propto & E_{\rm p}^{0.7}R_0^{-0.11},\quad \Gamma \propto L^{0.27}R_0^{-0.18},\nonumber \\ E_{\rm p} &\propto & L^{0.4}R_0^{-0.1},\quad \bar{\eta }_{\gamma }\propto E_{\rm p}^{0.5}R_0^{0.5}, \end{eqnarray} \tag{ 10 }$$

respectively, which are nicely in agreement with the four correlations summarized in the first paragraph of this section and the only requirement is that R₀ depends on L insensitively. Interestingly, the required k ∼ 0.34 is close to that (k ∼ 7/27) found in a simple analytical approach (Lü et al. 2012). Actually, when adopting Equations (18) and (16) of Fan & Wei (2011), we have Γ  ≈  400(L/10⁵² erg s⁻¹)^1/4(Y_b/0.2)^1/8(R₀/10⁸ cm)^−1/4 and E_p ≈ 260 keV(L/10⁵² erg s⁻¹)^1/4(Y_b/0.2)^3/8(R₀/10⁸ cm)^−1/2, and the coefficients are consistent with those reported in the literature, as long as R₀ is in order of 10⁸ cm. These together with the plots in Figure 1 illustrate that the correlations found in the literature (including the normalization) are indeed interpretable within the photosphere model.

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Second, we adopt the so-called generic dissipative photospheric model developed by Giannios (2012), in which it is shown that at the radius R_eq (see Equation (5) therein), where radiation and electrons drop out of equilibrium, the spectral peak of the prompt emission forms⁷ and the Lorentz factor can be expressed as (see Equation (9) therein)

$$\begin{equation} \Gamma \propto E_{\rm p}^{3/5}\bar{\eta }_\gamma ^{-1/5}L^{1/10}f_{\pm }^{1/5}(\eta /\Gamma)^{-1/5}, \end{equation} \tag{ 11 }$$

where f_± is the number of electron+positron pairs per proton and is expected to be moderate. The acceleration calculation yields $R_{\rm eq} \propto \Gamma R_0 \bar{\eta }_\gamma ^{-3/2}$ (e.g., Piran et al. 1993; Fan & Wei 2011), with which we have⁸

$$\begin{equation} \Gamma \propto L^{1/4}\bar{\eta }_\gamma ^{1/4}R_0^{-3/10}f_\pm ^{1/5}(\eta /\Gamma)^{-1/5}. \end{equation} \tag{ 12 }$$

With the relation $\eta \propto L^{\rm k}\bar{\eta }_\gamma ^{\rm -k}$, Equations (11) and (12) give

$$\begin{eqnarray} E_{\rm p} &\propto & L^{(10k-1)\over 6}\bar{\eta }_\gamma ^{(1-5k)\over 3}f_\pm ^{-1/3}(\eta /\Gamma)^{-4/3} \end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray} \bar{\eta }_\gamma & \propto & L^{(4k-1)\over 4k+1}R_0^{6\over 5(4k+1)}f_\pm ^{-4\over 5(4k+1)}(\eta /\Gamma)^{-16\over 5(4k+1)}, \end{eqnarray} \tag{ 14 }$$

respectively. Substituting Equation (14) into Equations (12) and (13), we have

$$\begin{equation} \Gamma \propto L^{2k \over 4k+1}R_0^{-{6k\over 5(1+4k)}}f_{\pm }^{4k\over 5(4k+1)}(\eta /\Gamma)^{-{4k+5 \over 5(4k+1)}} \end{equation} \tag{ 15 }$$

and

$$\begin{equation} E_{\rm p}\propto L^{8k-1\over 2(4k+1)}R_0^{2(1-5k)\over 5(4k+1)}f_\pm ^{-{3\over 5(4k+1)}}(\eta /\Gamma)^{-{12\over 5(4k+1)}}, \end{equation} \tag{ 16 }$$

respectively. As long as the radiation efficiency is not very efficient (say $\bar{\eta }_\gamma <0.25$), one can take η/Γ ∼ 1 (Piran et al. 1993; Mészáros et al. 1993). For k ∼ 0.34 we have

$$\Gamma \propto L^{0.29}, E_{\rm p}\propto L^{0.37}, \Gamma \propto E_{\rm p}^{0.78}, \bar{\eta }_\gamma \propto E_{\rm p}^{0.4},$$

which are roughly consistent with the correlations summarized at the beginning of this section.

Both long and short GRBs follow the E_p–L correlation (Ghirlanda et al. 2009; Zhang et al. 2012) and the $\bar{\eta }_\gamma \hbox{--}E_{\rm p}$ correlation (Margutti et al. 2012; Bernardini et al. 2012). When taking the peak time of the GeV emission of the short GRB 090510 as the deceleration time of the forward shock, we found that the inferred bulk Lorentz factor also follows the Γ–L correlation. These suggest that the photospheric origin of the prompt emission may also apply to some short bursts.

Prominent thermal radiation components have been identified in GRB 090902B, a very bright burst at redshift z = 1.822 (Abdo et al. 2009; Pandey et al. 2010; Ryde et al. 2010; Zhang et al. 2011; Liu & Wang 2011; Barniol Duran & Kumar 2011; Pe'er et al. 2012). For example, Zhang et al. (2011) divided the whole data set of GRB 090902B into several time bins and showed that the spectrum in each bin can be nicely fitted by a thermal component plus a power-law spectral component. By applying the same technique, we redo the analysis using Fermi/GBM data and the newest Fermi/LAT PASS7 data. The thermal (blackbody) and non-thermal (power-law) spectral parameters and fluxes are derived in each time bin. Following Pe'er et al. (2007) and Fan & Wei (2011) and assuming a constant thermal radiation efficiency of ∼20%, the bulk Lorentz factors of the outflow shells can be evaluated in a straightforward manner. We plot the inferred Γ together with the simultaneous luminosity in the Γ–L diagram presented by Lü et al. (2012). As shown in Figure 1(a), these two sets of data are in agreement with each other. For most bursts discussed in Lü et al. (2012), the measurement of Γ was based on the modeling of the afterglow light curve(s). The physics involved in such a kind of estimation is completely different from that for GRB 090902B. The agreement between these two sets of data thus not only supports our hypothesis of a photospheric origin of the prompt emission but also validates the robustness of both methods of evaluating Γ. In Figure 1(b) we plot the time-resolved spectral peak energy versus the simultaneous luminosity of GRB 090902B in the E_p–L diagram presented by Zhang et al. (2012). Again, a nice agreement between these two sets of data is present, in support of the photospheric origin of the prompt emission of some GRBs.

Finally, we would like to point out that all these correlations have not been reasonably interpreted in either the internal shock models or the internal magnetic energy dissipation models (the outflow is magnetic). In the standard internal shock model, one has E_p∝L^1/2Γ⁻² (e.g., Zhang et al. 2002; Dai & Lu 2002; Fan & Wei 2005) and we expect no evident positive correlation between E_p and the luminosity after taking into account the correlation Γ∝L^0.3, which is at odds with the data. It is also straightforward to show that the correlation Γ∝L^0.3 predicts an extremely low internal shock efficiency unless the slow material shell has a width much wider than that of the fast shell (i.e., the duration to eject the slow shell needs to be a factor of ∼(Γ_f/Γ_s)^3.4 that of the duration needed to eject the fast shell, where Γ_f and Γ_s are the bulk Lorentz factor of the fast and slow shells, respectively). For a magnetic outflow, it was recognized by Lü et al. (2012) that an interpretation of the Γ–L correlation is not yet available, let alone an interpretation of the other correlations. All these facts strongly favor the suggestion that the dominant component of the prompt emission of some GRBs may be tightly relevant to the photospheric radiation process, though much work on getting a spectrum that nicely matches the data is still needed (P. Veres et al. 2012, in preparation).

We thank B. Zhang, D. Giannios, R. Margutti, and M. G. Bernardini for helpful communications and the referee for insightful comments. This work was supported in part by the National Natural Science of China under grants 10973041, 10921063, 11073057, and 11163003, and by the National Basic Research Program of China under grant 2009CB824800. Y.-Z.F. is also supported by the 100 Talents program of the Chinese Academy of Sciences.