SHORT GAMMA-RAY BURSTS: THE MASS OF THE ACCRETION DISK AND THE INITIAL RADIUS OF THE OUTFLOW

A nascent stellar black hole surrounded by a hyper-accreting disk, the widely adopted central engine of GRBs, is characterized by some important parameters, including the mass and the spin of a central black hole ($M_{_{\rm BH}}$ and a, where $a\equiv J/GM_{_{\rm BH}}^{2}c$, J is the angular momentum of the black hole and c is the speed of light), the mass of accretion disk M_disk, or, alternatively, the accretion rate $\dot{M}$.

The accretion disk is so hot that the energy loss may be mainly through neutrino and anti-neutrino radiation. The later neutrino and anti-neutrino annihilation may launch a baryonic fireball (Eichler et al. 1989). The annihilation luminosity has been extensively investigated in the literature (e.g., MacFadyen & Woosley 1999; Popham et al. 1999; Liu et al. 2007; Zalamea & Beloborodov 2011). In general the luminosity is a function of $M_{_{\rm BH}}$, a, $\dot{M}$, and possibly also the vertical structure of the disk.³ In this work we adopt an empirical relation proposed by Zalamea & Beloborodov (2011), which reads

$$\begin{eqnarray} L_{\nu \bar{\nu }} &\approx & 10^{52} \,{\rm erg\, s^{-1}}x_{\rm ms}^{-4.8}\left({M_{_{\rm BH}}\over 3\,M_\odot }\right)^{-3/2}\nonumber \\ &&\times\,\left\lbrace \begin{array}{@{}ll}0, & \hbox{for $\dot{M}<M_{\rm ign}$;} \\ \dot{m}^{9/4}, & \hbox{for $M_{\rm ign}<\dot{M}<\dot{M}_{\rm trap}$;} \\ \dot{m}_{\rm trap}^{9/4}, & \hbox{for $\dot{M}\ge \dot{M}_{\rm trap}$,} \\ \end{array}\right. \end{eqnarray} \tag{ 1 }$$

where the accretion rate $\dot{m}=\dot{M}/M_\odot \, {\rm s^{-1}}$, x_ms = r_ms(a)/r_g, M_ign = K_ign(α/0.1)^5/3, M_trap = K_trap(α/0.1)^1/3, and α is the viscosity. The coefficients K_ign and K_trap are functions of the black hole spin a. For a = 0, K_ign = 0.071 M_☉ s⁻¹ and K_trap = 9.3 M_☉ s⁻¹. For a = 0.95, K_ign = 0.021 M_☉ s⁻¹ and K_trap = 1.8 M_☉ s⁻¹ (Chen & Beloborodov 2007). The radius of last stable orbit r_ms is (Bardeen et al. 1972)

$$\begin{equation} r_{\rm ms}=r_{\rm g}\lbrace 3+Z_{2}\mp [(3-Z_{1})(3+Z_{1}+2Z_{2})]^{1/2}\rbrace /2, \end{equation} \tag{ 2 }$$

where

$$Z_{1}=1+(1-a^{2})^{1/3}[(1+a)^{1/3}+(1-a)^{1/3}]$$

and

$$Z_{2}=\big(3a^{2}+Z_{1}^{2}\big)^{1/2}.$$

For a = 0 we have r_ms = 3r_g, while for a = 1 we have r_ms = r_g/2 or r_ms = 9r_g/2 (retrograde). The retrograde case is irrelevant to the case of the accretion disk and will not be discussed further.

With the observational data the energy output of the central engine $\dot{E}_{\rm out}$ and hence $L_{\nu \bar{\nu }}$ can be reasonably inferred, which, however, is not enough to break the degeneracies among parameters a, $M_{_{\rm BH}}$, and $\dot{m}$. As shown in footnote 3, the same applies to the magnetic process of energy extraction. That is why it is rather hard to constrain the physical parameters of the central engine with current electromagnetic data and our knowledge of the central engine is mainly from numerical simulations. Fortunately, the specific scenario of binary–neutron-star merger could be an exception for which a preliminary probe is plausible.

In the specific binary–neutron-star merger model, also the leading one, for short GRBs (Eichler et al. 1989; Nakar 2007; Lee & Ramirez-Ruiz 2007), the mass of the formed black hole and its spin parameter can be relatively reasonably evaluated.

The mass of the central engine is expected to be close to the total mass of the progenitors since the mass ejection during the merger is expected to be tiny (Rosswog et al. 1999) and the mass range of neutron stars is relatively narrow. For the 10 neutron-star binaries well studied so far, the total mass of the binaries ranges from 2.57 M_☉ to 2.83 M_☉ (Kiziltan et al. 2011). Therefore, the mass of the nascent black hole formed in the merger is expected to be close to $M_{_{\rm BH}}\sim 2.7\,M_\odot$.

The spin parameter of the formed black hole can be semi-quantitatively estimated. Following Lee et al. (2000) and for simplicity, we assume that the double neutron stars have a similar mass $M_{_{\rm NS}}\sim 1.4\, M_\odot$, and that the orbital angular momentum of the binary system is $J_{\rm binary}=\sqrt{\vphantom{A^R}\smash{\hbox{${GM_{_{\rm NS}}^{3}r_{\rm t}/2}$}}}$, where r_t is the tidal radius. The spin parameter of the formed black hole can be expressed by $a\equiv xyJ_{\rm binary}/[(2yM_{\rm NS})^{2}G/c]=(x/4y)\sqrt{r_{\rm t}/r_{\rm g,_{\rm NS}}}$, where $r_{\rm g,_{\rm NS}}=2GM_{_{\rm NS}}/c^{2}$, assuming that a fraction (xy) of the orbital angular momentum of the neutron star binary goes into a nascent black hole keeping a fraction y of the total mass. For the binary–neutron-star merger scenario, the tidal radius is $r_{\rm t} \approx 6r_{\rm g,_{\rm NS}}$ (Haensel et al. 1991) and we have a = 0.61x/y, about 10% smaller than that suggested by Lee et al. (2000). As already mentioned, y is close to 1. Let us estimate x. The gravitational radiation changes the energy of the binary system at a rate $dE/dt\approx 1.9\times 10^{55}(r_{\rm t}/6r_{\rm g,_{\rm NS}})^{-5} \,{\rm erg \, s^{-1}}$ and the merger takes a time $\Delta t \sim 1.4\, {\rm ms}\ (r_{\rm t}/6r_{\rm g,_{\rm NS}})^{4}$ (Haensel et al. 1991). The corresponding change of the angular momentum $\Delta J_{\rm binary}\sim (dE/dt)\Delta t/\sqrt{\vphantom{A^R}\smash{\hbox{${2GM_{_{\rm NS}}/r_{\rm t}^{3}}$}}}$. We then have x = 1 − ΔJ_binary/J_binary ~ 0.9. Therefore, we have a ~ 0.55, i.e., the formed black hole rotates rapidly. As found in Lee et al. (2000), for the double neutron stars having different masses $M_{_{\rm NS-1}}$ and $M_{_{\rm NS-2}}$, roughly one has $a \propto {\cal R}\equiv M_{_{\rm NS-1}}M_{_{\rm NS-2}}^{5/6}/ (M_{_{\rm NS-1}}+M_{_{\rm NS-2}})^{11/6}$. For the 10 neutron-star binaries discussed in Kiziltan et al. (2011), one finds that ${\cal R}$ ranges from 0.275 to 0.282, i.e., a is insensitive to the mass ratio of the binaries. The above semi-quantitative analysis is in agreement with the numerical simulation, in which a typical a ~ 0.78 was found, depending weakly on the total mass and the mass ratio of the binary neutron stars (Kiuchi et al. 2009).

For a = 0.78, we have r_ms ≈ 1.45r_g, x_ms = 1.45, and (for $M_{\rm ign}<\dot{M}<\dot{M}_{\rm trap}$)

$$\begin{equation} L_{\nu \bar{\nu }} \approx 2 \times 10^{51} \,{\rm erg\, s^{-1}} \dot{m}^{9/4} \left({x_{\rm ms}\over 1.45}\right)^{-4.8} \left({M_{\rm BH} \over 2.7\,M_\odot }\right)^{-3/2}. \end{equation} \tag{ 3 }$$

A typical M_disk ~ 10⁻³–10⁻¹ M_☉ is found in the numerical simulation of the binary–neutron-star coalescence (see Kiuchi et al. 2009 and references therein). The binary–neutron-star merger hypothesis is supported if our M_disk estimated with the observational data is within such a mass range. In the foreseeable future the binary–neutron-star merger model for short GRBs could be directly tested by the gravitational wave data and the mass of the binaries as well as the formed disk could be inferred (Kiuchi et al. 2010). Therefore, the validity of our following approach will be directly tested.

A simple approach. Below we take the simplest approach to estimate $L_{\rm \nu \bar{\nu }}$. The isotropic-equivalent kinetic energy of the outflow powering long-lasting afterglow (E_{k, iso}) and the opening angle of the ejecta θ_j can be derived from the modeling of the multi-wavelength afterglow data (Panaitescu 2006). However, in a good fraction of short GRBs, such a goal is not achievable due to the lack of prompt observations. Fortunately, for the X-ray emission above both the typical synchrotron radiation frequency and the cooling frequency, the flux is independent of the poorly constrained number density of the medium n and the X-ray luminosity is a good probe of E_{k, iso} (Kumar 2000). Following Fan & Piran (2006) we take

$$\begin{eqnarray} E_{\rm k,iso} &\sim & 10^{53}\, {\rm erg}\, {\cal L}_{\rm X,46}^{4/(p+2)}\left({1+z \over 2}\right) \epsilon _{\rm B,-2}^{-(p-2)/(p+2)}\nonumber \\ &&\times\, \epsilon _{\rm e,-1}^{4(1-p)/(p+2)}(1+Y)^{4/(p+2)}, \end{eqnarray} \tag{ 4 }$$

where ${\cal L}_{\rm X}$ is the X-ray afterglow luminosity at t = 10 hr after the trigger of the burst, _e and _B are the fractions of shock energy given to the electrons and magnetic field, respectively, Y is the Compton parameter, and p ~ 2 is the energy distribution index of the shock-accelerated electrons and is constrained by the X-ray spectrum. The convention Q_n = Q/10ⁿ has been adopted here and throughout this work except for some specific notations.

The jet opening angle is estimated to be (Frail et al. 2001)

$$\begin{equation} \theta _{\rm j} \approx 0.076(t_{\rm j}/1 \,{\rm day})^{3/8}[(1+z)/2]^{-3/8}E_{\rm k,iso,51}^{-1/8}n_{-2}^{1/8}, \end{equation} \tag{ 5 }$$

where t_j is the jet break time. We then estimate the "intrinsic" power released by the GRB central engine as

$$\begin{equation} \dot{E}_{\rm out}\approx (1+z)(E_{\rm \gamma,iso}+E_{\rm k,iso})\theta _{\rm j}^{2}/(2T_{\rm act}), \end{equation} \tag{ 6 }$$

where T_act is the duration of the activity of the central engine. In reality, $\dot{E}_{\rm out}$ is just a fraction (${\cal F} \lesssim 0.3$) of the total neutrino–antineutrino annihilation luminosity outside the horizon of the rotating black hole, i.e., $\dot{E}_{\rm out}={\cal F}L_{\nu \bar{\nu }}$ (e.g., Aloy et al. 2005). The real duration of the main activity of the central engine might be shorter than the duration of the prompt emission T₉₀ by a factor ${\cal R} \gtrsim 1$, i.e., $T_{\rm act}=T_{90}/{\cal R}$, because different propagation velocities of the front and rear ends will lead to a radial stretching of the ultrarelativistic ejecta (see Section 4.1 of Aloy et al. 2005 for more details). Hence we have (for $M_{\rm ign}<\dot{M}<\dot{M}_{\rm trap}$)

$$\begin{eqnarray} \dot{m} &\approx & 0.53\, M_\odot \left[{{\cal R}(1+z)(E_{\rm \gamma,iso,51}+E_{\rm k,iso,51})\theta _{\rm j}^{2}\over {\cal F}T_{90}}\right]^{4/9}\nonumber \\ &&\times\, \left({x_{\rm ms}\over 1.45}\right)^{2.1} \left({M_{\rm BH}\over 2.7\,M_\odot }\right)^{2/3} \end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray} M_{\rm disk} &\approx & 0.53\, {M_{\odot }} \left[{(E_{\rm \gamma,iso,51}+E_{\rm k,iso,51})\theta _{\rm j}^{2}\over {\cal F}}\right]^{4/9}\nonumber \\ &&\times\, \left[{T_{90}\over (1+z){\cal R}}\right]^{5/9}\left({x_{\rm ms}\over 1.45}\right)^{2.1}\left({M_{\rm BH}\over 2.7\,M_\odot }\right)^{2/3}. \end{eqnarray} \tag{ 8 }$$

Since ${\cal R}\gtrsim 1$ and ${\cal F}<1$, their impacts on estimating M_disk are partly canceled. For a = 0.6, we have x_ms ≈ 1.9; M_disk given in Equation (8) will be enhanced by a factor of ~1.8.

Case studies. So far we have 10 short GRBs, as listed in Table 1, having relatively abundant afterglow data, with which we can estimate $\dot{E}_{\rm out}$ and then M_disk. For simplicity, instead of discussing all bursts one by one, below we focus on a few special events.

GRB 051221A, a burst with a duration z = 1.4 s, was at redshift z = 0.5465 and is distinguished by a long-lasting X-ray flat segment in the afterglow. Such a flat segment could be due to either the energy injection from the central engine (Soderberg et al. 2006; Burrows et al. 2006) or the forward shock emission of the emerging wide-component of the two component jet (Jin et al. 2007). The afterglow parameters reported in the literature are different. In the analysis we take the parameters obtained in the two-component jet modeling, in which the long activity of the central engine is not needed. If we take the somewhat more conservative estimate E_ej ≈ 3 × 10⁴⁹ erg (Soderberg et al. 2006; Burrows et al. 2006), the mass of the accretion disk will be reduced by a factor of 0.6.

GRB 090510, a burst at redshift z = 0.903, is the most energetic short event ever recorded and is also remarkable for its long-lasting GeV emission that is likely powered by an external forward shock (Gao et al. 2009; De Pasquale et al. 2010; Corsi et al. 2010). The self-consistent interpretation of the GeV/X-ray/optical data is not an easy task and the parameters are found to be somewhat unusual. At t ~ 1200 s after the trigger of the burst, the X-ray and the optical afterglow emission change the decline behaviors achromatically (De Pasquale et al. 2010). Such changes are most likely due to the jet effect, that is, the edge of the outflow enters our line of sight and the visible emitting region cannot be approximated as a spherical surface any longer. The jet opening angle is as small as ~0.006 (Gao et al. 2009; Corsi et al. 2010; He et al. 2011), which is about one order of magnitude smaller than that of other bursts (see Table 1) or that found in numerical simulation (Aloy et al. 2005). It is unclear how such a narrow collimation is reached. Nevertheless a similar narrow collimation was identified in the afterglow modeling of the naked-eye burst GRB 080319B (Racusin et al. 2008).

GRB 100816A, has a burst at z = 0.8035, a local duration of 2.8/(1 + z) = 1.55 s, consistent with being a short burst. The result of the Swift Burst Alert Telescope (BAT) spectral lag analysis is also consistent with being a short hard burst, but the error bars are too large to be definitive (Oates et al. 2010). With the X-ray light curve we take a jet break time t_j 2 × 10⁵ s. The most valuable information inferred from the afterglow data of this burst is likely the density profile of the circumburst medium. The preliminary white band flux decline is ~t^−1.15 for 100 s < t < 10⁴ s, steeper than the simultaneous t⁻¹-like X-ray decline (Oates et al. 2010). The spectral index of the X-ray afterglow photons is ≈1.03^+0.12_{− 0.16}, suggesting that the X-ray emission is above the cooling frequency of the forward shock electrons for p ~ 2. If the slow-cooling fireball was expanding into the interstellar medium and the typical synchrotron radiation frequency is below the observer's band, the optical afterglow emission should decline with the time as t^−0.75, shallower than the X-ray decline, which is at odds with the data. For a free-wind medium, the optical decline should be t^−1.25 steeper than the X-ray decline (Zhang & Mészáros 2004). Therefore the current data favor the free-wind medium model, in which the progenitor should be a massive star rather than a pair of compact objects. If our speculation could be confirmed by careful analysis of available afterglow data reported in GCNs, the collapsar origin of GRB 100816A with an intrinsic duration ~1.4 s would be established. In turn, such a result would support the hypothesis that collapsars can produce short events (Zhang et al. 2003) and the progenitors of short GRBs are diverse (Fan et al. 2005; Zhang et al. 2009). If so, the calculation made in Table 1 on these kind of bursts is likely invalid.

As shown in Table 1, for about half of short bursts in the sample, the accretion disk has a mass ~0.01–0.1 M_☉, clearly consistent with that found in the numerical simulations of a double neutron star merger (e.g., Rosswog et al. 2003; Kiuchi et al. 2009). For some other events, such as GRB 051221A and GRB 050724, the inferred M_disk ~ several 0.1 M_☉ may be a bit massive to form. This puzzle can be solved in either of the following scenarios. One is that the outflows of these short GRBs were launched via some more efficient magnetic processes rather than the neutrino mechanism, as speculated in the literature (e.g., Rosswog et al. 2003; Lee & Ramirez-Ruiz 2007). With footnote 3, for a ≈ 0.78, $M_{_{\rm BH}}=2.7\,M_\odot$, and $\dot{M}=(0.1, 1.0)M_\odot \, {\rm s^{-1}}$ we have $L_{_{\rm BZ}}\sim (100, 10)L_{\nu \bar{\nu }}$. Consequently, the disks with M_disk about 10 or more times smaller than those presented in Table 1 may be enough to power these short events. The outflow launched in this way is Poynting-flux dominated and the prompt emission due to the magnetic energy dissipation should have a high linear polarization degree. The other is that some short GRBs are from the neutron-star–black-hole merger for which a massive disk is possible. Another possibility that cannot be ruled out is that some short events might have a massive star origin. Since our estimated M_disk is close to that found in numerical simulations, the compact object merger scenario is likely viable, implying that some (possibly a considerable fraction of) short GRBs may really be driven by the coalescence of double neutron stars and are promising gravitational wave radiation sources.

The following approach is partly motivated by Pe'er et al. (2007). In their work, photospheric radiation taking place in the radiation-dominated phase and the matter-dominated phase has been investigated separately, whereas we solve the problems jointly. Furthermore, we focus on constraining R₀ in the absence of an ideal thermal signature, different from what has been done in the literature.

For a baryonic outflow, most of the initial thermal energy may have been converted into the kinetic energy of the baryons at the end of the acceleration (Shemi & Piran 1990), but a (quasi-)thermal emission component is likely inevitable (Paczyński 1990). The (quasi-)thermal emission is mainly from the photosphere at a radius R_ph, which satisfies the following relations.

Based on its definition, R_ph can be expressed as (e.g., Paczyński 1990; Daigne & Mochkovitch 2002; Jin et al. 2010)

$$\begin{eqnarray} R_{\rm ph} & \approx & 4.5\times 10^{11}\, {\rm cm}\, L_{54}\Gamma _{\rm ph,3}^{-2}\eta _{3}^{-1}\nonumber \\ &\approx & 3.7\times 10^{11}L_{54}f^{-2}\eta _{3}^{-3}, \end{eqnarray} \tag{ 9 }$$

where f ≡ 3Γ_ph/4η, η is the initial dimensionless entropy, and Γ_ph is the bulk Lorentz factor of the outflow at the photospheric radius.

The photospheric radius is also related to R₀. Following Piran et al. (1993) and Mészáros et al. (1993), we introduce an R_o, at which the bulk Lorentz factor of the outflow is Γ_o ~ a few. With the parameter

$$\begin{equation} {1\over D}={\Gamma _o \over \Gamma _{\rm ph}}+{3\Gamma _o \over 4 \eta _o \Gamma _{\rm ph}}-{3\over 4\eta _o}, \end{equation} \tag{ 10 }$$

the acceleration calculation yields (Piran et al. 1993)

$$\begin{equation} R_{\rm ph}=R_o{(\Gamma _o/\Gamma _{\rm ph})^{1/2}}D^{3/2}, \end{equation} \tag{ 11 }$$

where η_o = e'_o/n'_om_pc² ≈ η/Γ_o. We then have 1/D = Γ_o/Γ_ph + 3Γ²_o/4ηΓ_ph − 3Γ_o/4η = 3(1 − f)Γ_o/4fη + 9Γ²_o/16η²f, the first term will be dominant as long as 1 − f ≥ 3Γ_o/4η, which is usually satisfied. So we have

$$\begin{equation} 1/D \approx 3(1-f)\Gamma _o/4f\eta,\quad {D\Gamma _o/\eta }\approx 4f/[3(1-f)], \end{equation} \tag{ 12 }$$

with which we get

$$\begin{equation} R_{\rm ph}\approx {4\eta \over 3} R_0 {f \over (1-f)^{3/2}}\approx 1.3\times 10^{10} \,{\rm cm}\, \eta _{3}R_{0,7}{f \over (1-f)^{3/2}}, \end{equation} \tag{ 13 }$$

where the relation R₀ ≈ R_o/Γ_o has been used.

Finally the photospheric radius is constrained by the observational data. Suppose the (quasi-)thermal emission has a temperature T_obs and a flux F_bb, we have 4πΓ²_phR_ph²σT'_ph⁴ = L_bb = 4πD²_LF_bb, which can be simplified as

$$\begin{eqnarray} R_{\rm ph} &\approx & \left[F_{\rm bb}/\sigma T_{\rm obs}^{4}\right]^{1/2}(1+z)^{-2}\Gamma _{\rm ph}D_{\rm L}\nonumber \\ &\approx & 1.3\times 10^{11} \,{\rm cm}\, F_{\rm bb,-4}^{1/2}\left[{(1+z)T_{\rm obs}\over 1 \,{\rm MeV}}\right]^{-2} f\eta _{3}D_{\rm L,28},\nonumber\\ \end{eqnarray} \tag{ 14 }$$

where T_obs ≈ Γ_phT'_ph/(1 + z) has been taken into account and σ is the Stefan–Boltzmann constant.

Denoting the comoving thermal energy density and the number density of the outflow at R_ph as e'_ph and n'_ph, respectively, we have the (quasi-)thermal luminosity

$$\begin{eqnarray*} &&L_{\rm bb}\approx {e^{\prime }_{\rm ph} \over 4e^{\prime }_{\rm ph}/3+n^{\prime }_{\rm ph} m_{\rm p}c^{2}}L. \end{eqnarray*}$$

Since e'_ph = e'_o/D⁴ and n'_ph = n'_o/D³ (Piran et al. 1993), we have

$$\begin{equation} L_{\rm bb}\approx {e^{\prime }_o \over 4e^{\prime }_o/3+D n^{\prime }_o m_{\rm p} c^{2}}L \approx {L\over 4/3+D \Gamma _o/\eta }\approx {3(1-f)\over 4}L. \end{equation} \tag{ 15 }$$

With Y_bb ≡ L_bb/L < 3/4, the above relations yield f = 1–4Y_bb/3 and η = 3Γ_ph/[4(1–4Y_bb/3)]. Combing Equations (9), (13), and (14), we have

$$\begin{equation} R_{0}\approx 1.5\times 10^{8} \,{\rm cm}\, F_{\rm bb,-4}^{1/2}Y_{\rm bb}^{3/2}\left[{(1+z)T_{\rm obs}\over 1 \,{\rm MeV}}\right]^{-2}D_{\rm L,28}, \end{equation} \tag{ 16 }$$

$$\begin{equation} \Gamma _{\rm ph} \approx 10^{3} \left(Y_{\rm bb}^{-1}-4/3\right)^{1/4}F_{\rm bb,-4}^{1/8}\left[{(1+z)T_{\rm obs}\over 1 \,{\rm MeV}}\right]^{1/2}D_{\rm L,28}^{1/4}. \end{equation} \tag{ 17 }$$

If the baryon loading is so low that at R_ph the outflow is still radiation dominated, most of the initial energy of the outflow will be lost via the thermal radiation and Y_bb ~ 3/4, for which Γ_ph cannot be reliably inferred, reflecting the well-established fact that both the observed temperature and the thermal radiation luminosity are constant until most of the initial energy of the outflow has been transferred into the kinetic energy of the particles (Piran et al. 1993; Mészáros et al. 1993). For Y_bb 1 (i.e., R_ph is far above the coasting radius ~ηR₀), Equation (17) suggests Γ_ph∝Y^−1/4_bb, in agreement with Pe'er et al. (2007).

The two equations above finally give

$$\begin{equation} R_0 \approx 1.5\times 10^{8} \,{\rm cm}\, \Gamma _{\rm ph,3}^{-4}\big(Y_{\rm bb}^{-1}-4/3\big)Y_{\rm bb}^{3/2}F_{\rm bb,-4}D_{\rm L,28}^{2}. \end{equation} \tag{ 18 }$$

Therefore, if Γ_ph and Y_bb are obtainable with the observational data, Equation (18) provides us an independent estimate of R₀ without the need to identify a thermal component in the prompt spectrum. This is helpful since the physical processes taking place at R ≤ R_ph may be able to shape the thermal spectrum so significantly that the identification of an ideal thermal component would be very difficult (e.g., Beloborodov 2010). In fact, for short GRBs, no reliable thermal component has been identified so far. The disadvantage of our approach is that one can only get the time-averaged constraint.

GRB 090510 was detected by the Fermi γ-ray telescope and Swift satellite simultaneously. Though the physical origin of the prompt GeV emission is not clear yet, the long lasting GeV afterglow emission is most likely the synchrotron radiation of the external shock (e.g., Gao et al. 2009; Ghirlanda et al. 2010; Corsi et al. 2010; De Pasquale et al. 2010; Kumar & Barniol Duran 2010). Supposing the prompt GeV and MeV emission are from the same region, the observation of GeV photons sets an upper limit on the optical depth for pair production and then suggests a bulk Lorentz factor of the emitting region >1200 (De Pasquale et al. 2010; Abdo et al. 2009). To interpret the GeV emission at t > 2 s as the forward shock emission, a higher initial bulk Lorentz factor of the outflow Γ_int 1900 is required (He et al. 2011; Ghirlanda et al. 2010; Corsi et al. 2010). Obviously Γ_ph should be larger than Γ_int (it is straightforward to show that the outflow shells with higher bulk Lorentz factor contribute more to the thermal emission). In the following estimate we take Γ_ph ~ Γ_int ~ 2000. The time-averaged spectrum of GRB 090510 in the time interval 0.5–1.0 s can be nicely fitted by a Band function plus a power-law component. The Band function component has an isotropic-equivalent energy E_{Band, iso} ≈ 7 × 10⁵² erg while the very hard power-law component (with a spectrum F_ν∝ν^−0.62 in the energy range 10 keV–10 GeV) has an isotropic-equivalent energy E_{PL, iso} ≈ 5 × 10⁵² erg (Abdo et al. 2009). The afterglow modeling gives E_{k, iso} 5 × 10⁵³ erg (Gao et al. 2009; He et al. 2011). Clearly, only the Band function component may be relevant to the quasi-thermal radiation of the outflow. Hence we have a (quasi-)thermal radiation efficiency Y_bb ≤ E_{Band, iso}/(E_{k, iso} + E_{PL, iso} + E_{Band, iso}) ≈ 0.1 and the thermal radiation flux L_bb 6 × 10⁻⁵ erg s⁻¹ cm⁻². Substituting these values into Equation (18), we have

$$\begin{equation} R_0 \lesssim 6.5\times 10^{6} \,{\rm cm}\, \Gamma _{\rm ph,3.3}^{-4}Y_{\rm bb,-1}^{1/2}, \end{equation} \tag{ 19 }$$

which is about two or more orders of magnitude smaller than that reported in the literature (Pe'er et al. 2007; Ryde et al. 2010),⁴ suggesting that most energy has been deposited in a small cavity surrounding the nascent black hole. Since the collimation of the outflow in the double neutron star merger scenario should be mainly contributed by the interaction with accretion torus (Aloy et al. 2005), a small R₀ may be necessary to be consistent with the very small opening angle of the ejecta θ_j found in the afterglow modeling.

The GRB ejecta is mainly launched through the pole region of the rotating black hole. Clearly the outflow should be from a site above the horizon surface. Along the pole that means R₀ > r_g ≡ 2GM_BH/c² ≈ 3 km (M_BH/1 M_☉), regardless of the unknown spin of the black hole, where r_g is the Schwarzschild radius and G is the gravitational constant. In reality R₀ is larger than r_g by a factor of a few, as found in numerical simulations (e.g., Popham et al. 1999; Liu et al. 2007; Zalamea & Beloborodov 2011). With Equation (19), for GRB 090510 we then have M_BH < 22 M_☉Γ⁻⁴_{ph, 3.3}Y^1/2_{bb, -1}, i.e., it is likely a stellar black hole at the center, in agreement with the compact-object merger model.

For other short bursts, due to the lack of a robust estimate of Γ_int and then Γ_ph, a reliable constraint on R₀ is not possible.