SHORT HARD GAMMA-RAY BURSTS AND THEIR AFTERGLOWS

We have no novel suggestions regarding either the putative progenitors of cosmological SHBs or the ejection mechanism and the properties of their highly relativistic jets. As listed in Section 1, most conventionally, the progenitors may be the merger of a neutron star with another one or with a black hole, or large accretion episodes in microblazars (microquasars with their jets pointing accurately to us) or intermediate mass black holes. Accretion, cooling, or angular momentum loss may result in abrupt transitions of neutron stars to more compact objects (strange stars, quark stars, or stellar black holes). All these phenomena are natural candidates for SHB progenitors. The above putative progenitors meet the energy-budget requirements, even if the radiation of SHBs is not very highly collimated. The estimated merger rate of neutron stars is sufficient only if SHBs are not narrowly collimated. If they are, as in the CB model, some of the other mechanisms we cited must also be operative. Indeed, low-luminosity LGRBs detected by International Gamma-Ray Astrophysics Laboratory (INTEGRAL) led to the conclusion (Foley et al. 2008) that the rate of LGRBs with inferred low luminosity is ≳25% of that of Type Ib/c SNe.⁴ Since BATSE observed an SHB rate nearly 1/3 the rate of LGRBs, the rate of SHBs implied by the INTEGRAL observations must also be comparable with the rate of SNe of type Ib/c. Such a rate is a considerable fraction of the birth rate of neutron stars. It is much larger than the merger rate of neutron stars in close binaries. It favors other alternative sources of SHBs such as a phase transition in neutron stars or accretion episodes on stellar mass and intermediate mass, black holes.

The putative progenitors of SHBs are mostly located in the cores of GCs and the central region of SSCs. We shall assume that SHBs are produced in such an environment.

We shall assume that similar to LGRBs, the two dominant radiation mechanisms in SHBs are ICS and SR. ICS of glory light produced by a stellar companion or by an accretion disk, by electrons comoving with the CBS, dominates the prompt emission, while SR from the decelerating CBs in the ISM or the IGM dominates the afterglow emissions.

We shall also assume that ICS of ambient light in GCs of a low-density ISM or SR from the decelerating CBs in the high-density ISM of SSCs produces the ESEC.

We contend that the prompt γ- and X-rays of an SHB's pulse are made similar to those in LGRBs, that is, by ICS of glory light by the electrons contained within a CB. For LGRBs, the glory is formed by scattered and emitted radiations by the pre-SN wind/ejecta blown from the progenitor star, which was illuminated by light from the progenitor, the early flash of the SN explosion, and light from the jet itself. The burst environment of the putative progenitors of SHBs is probably complicated and can be estimated only very roughly. After ejection, the fast expanding CB probably propagates through a cavity produced by the ejecta/wind blown by the progenitor star, or by a close companion or from an accretion disk. The cavity contains a quasi-isotropic light from the companion star, the accretion disk and the jet itself, which illuminated the wind and was scattered/emitted by it.⁵ Outside a source of radius r_s, the intensity of light roughly decreases like 1/(r² + r²_s). The kinetic temperature of the ionized wind is roughly proportional to the intensity of the photoionizing light. The photoionized wind emits a quasi-isotropic light that presumably has a thin thermal bremsstrahlung spectrum with a temperature that decreases with distance beyond a radius r_g like 1/r², where the wind becomes transparent (optical thickness ∼1) to glory photons. Unable to theoretically evaluate n_γ(r), we will adopt a density of glory photons seen by a CB that is roughly given by n_γ(r) ∼ n_g(0) r²_g/(r² + r²_g), and treat r_g and n_g(0) as adjustable parameters to be directly determined from the data.

The launching of highly relativistic and narrowly collimated jets in mass accretion episodes, merger, or phase transitions is not sufficiently understood to predict the chaotic timing of CB emissions, nor the number, N_CB, of dominant CBs in a particular jet, nor the baryon number, N_B nor the initial Lorentz factor, γ₀, of observable CBs. But once the typical values of these parameters are inferred from the data, all properties of LGRBs and SHBs can be understood in simple terms. This also requires a number of items of information independent of the model itself. They are the angle between a CB jet and the observer, the properties of the ambient light a CB encounters in its flight and of the density of the ISM or intergalactic medium (IGM), which CBs pierce through. Naturally, the cosmological redshift, z, and the absorption or attenuation of light of various frequencies along the line of sight (LOS) play their usual roles, with the exception, in comparison with other models, that the LOS to a CB significantly moves (and hyperluminally) during the observation time. The standard cosmological model with Ω = 1, Ω_M = 0.27, Ω_Λ = 0.73, and H₀ = 71 km s^-1,  Mpc⁻¹, is assumed throughout this paper.

We hypothesize that a CB initially expands in its rest system at a speed comparable with the speed of sound in a relativistic plasma, $v\equiv \beta _s\, c/\sqrt{3}$ with β_s ≲ 1. A CB's initial radius must be of the order of the size of the progenitor compact object or its accretion disk, and rapidly becomes negligible as the CB expands. Thus, a CB traces a cone in the space of an initial opening angle $\theta _{\rm CB}=\beta _s/\sqrt{3}\, \gamma _0.$ The characteristic opening angle of the emitted radiation is 1/γ₀, much larger than θ_CB, so that the jet opening is not a concept that plays a very major role. In the AG phase, the effect of the deceleration of CBs in the ISM becomes important, and the radiation is spread over an angle 1/γ(t).

In the CB model, the Doppler factor, δ(t), relating times, energies, and fluxes in a CB's rest system to those in the observer's system plays a major role in the understanding of GRBs. Its form in terms of θ and the time-dependent Lorentz factor, γ(t), of a CB, is

$$\begin{equation} \delta (t)={1\over \gamma (t)\, (1-\beta (t)\, \cos \theta)}\approx {2\, \gamma (t)\over 1+[\gamma (t)\,\theta ]^2}\;, \end{equation} \tag{ 1 }$$

where the approximation is excellent for γ ≫ 1 and θ ≪ 1. Doppler boosting and relativistic beaming enhances the observed energy flux from a CB by a factor δ³, making the observations of GRBs increasingly improbable to observe GRBs at angles θ>1/γ. The angular phase space being dΩ ∝ θ dθ, the most probable angle of observation is ≈1/γ.

During the short phase of γ-ray emission in a SHB, the Lorentz factor γ of a CB stays put at its initial value γ₀, for the deceleration induced by the collisions with the ISM has not yet had a significant effect. The electrons comoving with the CB scatter the photons of the glory around the progenitor, which have a thin thermal-bremsstrahlung spectrum,

$$\begin{equation} \epsilon \, {dn_\gamma \over d\epsilon } \sim \left({\epsilon \over \epsilon _g }\right)^{-\beta _g}\, e^{-\epsilon /T_g}, \end{equation} \tag{ 2 }$$

with a temperature that decreases with distance beyond r_g like, T_g(r) ∼ T(0) r²_g/(r²_g + r²), with, T(0) ∼ 1 eV, for a glory produced by stellar light, and β_g ∼ 0 (photon spectral index Γ = β_g + 1 ∼ 1). The observed energy of a glory photon, which suffered an ICS by an electron comoving with a CB at redshift z, is then given by

$$\begin{equation} E={\gamma _0\, \delta _0\, \epsilon \, \over (1+z)}\, (1+\cos \theta _{\rm in}), \end{equation} \tag{ 3 }$$

where θ_in is the angle of incidence of the initial photon onto the CB in the SN rest system. For a quasi-spherical distribution of scattered light, cos θ_in in Equation (3) roughly averages to zero. The predicted time-dependent spectrum of the GRB pulse produced by ICS of the glory photons is given by (DD2004):

$$\begin{eqnarray} \hspace{-15pt}E\, {dN_\gamma \over dE} &\sim& \left({E\over E_p(t}\right)^{-\beta _g}\, e^{-E/E_p(t)}+ b\,(1-e^{-E/E_p(t)})\, \nonumber \\ &\times& \left({E \over E_p(t)}\right)^{-p/2}, \end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray} E_p(t)&\approx & E_p(0)\, {t_p^2 \over \big(t^2+t_p^2\big)}, \nonumber \\ E_p(0)&\approx & {\gamma _0\, \delta _0 \over 1+z}\, T(0), \end{eqnarray} \tag{ 5 }$$

with t_p ≈ (1 + z) r_g/c γ₀ δ₀ being the peak time of dN_γ/dt discussed in the following section.

The first term in Equation (4), with β_g ∼ 0, is the result of Compton scattering by the bulk of the CB's electrons, which are comoving with it. The second term in Equation (4) is induced by a very small fraction of "knocked-on" and Fermi-accelerated electrons, whose initial spectrum (before Compton and synchrotron cooling) is dN_e/dE ∝ E^−p, with p ≈ 2.2. For $b={\cal {O}}(1)$, the energy spectrum predicted by the CB model, Equation (4), bears a striking resemblance to the Band function (Band et al. 1993) traditionally used to model the energy spectra of GRBs, but GRBs where the spectral measurements extend over a much wider energy range than those of BATSE and Swift/BAT are better fitted by Equation (4) (e.g., Wigger et al. 2008). For many Swift GRBs, the spectral observations do not extend to energies bigger than E_p(0), or the value of b in Equation (4) is relatively small, so that the first term of the equation provides a very good approximation. This term coincides with the "cutoff power-law" spectrum, which has also been recently used to model many GRB/SHB spectra. For b ∼ 0 and β_g ∼ 0, it yields a peak value of E² dN/dE at E_p(t) whose pulse-averaged value is

$$\begin{equation} E_p\approx E_p(t_p)\approx 0.5\, E_p(0) \approx 330 \, {\gamma _0\,\delta _0\over 10^6}\; {T(0)\over 1\;{\rm eV}}\;{1.5\over 1+z}\;{\rm keV}. \end{equation} \tag{ 6 }$$

Let t be the time after the launch of a CB in the rest frame of the progenitor. After its launch, the fast expanding CB propagates in the progenitor's glory and wind on its way to the ISM. Its cross section increases, while both its density and opacity, as well as those of the wind, decrease. Approximating the CB geometry by a cylindrical slab of radius R and neglecting the spread in arrival times of scattered glory photons in the CB that entered it simultaneously, the rate of scattered glory photons is given by

$$\begin{equation} {dN_\gamma \over dt} = e^{-\tau _{_W}}\, n_g (t)\, \sigma _{_T}\, \pi \, R^2(t)\, {[1 - e^{-\tau _{{\rm CB}}}]\over \sigma _a}, \end{equation} \tag{ 7 }$$

where $\tau _{_W}$ is the opacity of the wind at the CB location, τ_CB is the effective opacity of the expanding CB encountered by a photon with energy E' = (1 + z) E/δ₀, which begins crossing it at a time t, σ_a(E') is the photoabsorption cross section at energy E', and σ_T is the Thomson cross section. At an early time, r ≈ c γ₀ δ₀ t/(1 + z). Consequently, n_g ∝ 1/(t² + Δt²), where Δt = (1 + z) r_g/c γ₀ δ₀. Thus, the shape of an ICS pulse produced by a CB is given approximately by

$$\begin{equation} E\,{d^2N_\gamma \over dt\, dE}\propto e^{-\tau _{_W}}\, n_g\, \pi \, R^2\, [1 - e^{-R_{\rm tr}^2/R^2}]\, E\,{dN_\gamma \over dE}, \end{equation} \tag{ 8 }$$

where R_tr is the radius of the CB at t = t_tr, when τ_CB ≈ 1, that is, when it becomes transparent to the scattered radiation, and E dN_γ/dE is given by Equation (4). The preburst wind from the progenitor system produces a density distribution, n(r) = n₀ r²₀/r² around it, which yields τ_W = a(E)/t with a(E) = σ_a n₀ r²₀(1 + z)/c γ₀ δ₀. At sufficiently high energies, the opacities of the wind and the CBs are mainly due to Compton scattering and then, $R_{\rm tr} \sim \sqrt{\sigma _{_T}\, N_b/\pi }$, where σ_T is the Thomson cross section and N_b is the baryon number of the expanding CB. At low energies, their opacity is dominated by free–free (ff) absorption because the CBs and the wind along their trajectory are completely ionized. In the CBs' rest frame, the glory photons have typical energies, E'≪ keV. At such low energies, the opacity of CBs with a uniform density behaves like $\tau _{{\rm CB}} \sim E^{\prime -3}\,(1 - e^{ - E^{\prime }/T^{\prime }})\,G(E^{\prime })\,R^{-5},$ where G(E') is the quantum mechanical Gaunt factor that logarithmically depends on E' (e.g., Lang 1999 and references therein). Thus, when the optical thickness of a CB is dominated by ff photoabsorption, its transparency radius increases with decreasing energy like R_tr ∝ E^−3/5 = E^−0.6 at E' ≫ T' and R_tr ∝ E^−2/5 = E^−0.4 at E' ≪ T', yielding R_tr ∼ E^{−0.5 ± 0.1}.

The initial rapid expansion of a CB slows down as it propagates through the wind and scatters its particles (DDD2002; DD2004). If this expansion is roughly described by R² ≈ R²_CB t²/(t² + t²_exp), where R_CB is the asymptotic radius of the CB and t_exp ≫ t_tr, then, Equation (8) can be approximated by

$$\begin{equation} E\,{d^2N_\gamma \over dt\, dE} \propto { e^{-a/t}\, \Delta t^2\, t^2\over (t^2 + \Delta t^2)\, \big(t^2 + t_{\rm tr}^2\big)}\, E\,{dN_\gamma \over dE}. \end{equation} \tag{ 9 }$$

For nearly transparent winds (a → 0) and for t_tr ∼ Δt, Equation (9) has an approximate shape:

$$\begin{equation} E\,{d^2N_\gamma \over dt\, dE} \propto {\Delta t^2\, t^2 \over (t^2+\Delta t^2)^2}\,E\,{dN_\gamma \over dE}, \end{equation} \tag{ 10 }$$

which peaks around t ≈ Δt. Except for very early times, this shape is almost undistinguishable from that of the "Master" formula of the CB model (DD2004):

$$\begin{equation} E\,{d^2N_\gamma \over dt\, dE} \propto e^{-\Delta t^2/t^2 } \, [1 - e^{-\Delta t^2/t^2} ]\,E\,{dN_\gamma \over dE}, \end{equation} \tag{ 11 }$$

which also took into account arrival time effects that depend on the geometry of the CB and the observer's viewing angle, and was shown to well describe the prompt emission pulses of LGRBs (DD2004) and the rapid decay with a fast spectral softening of their prompt emission (DDD2008a).

At the relatively low X-ray energies covered by Swift and, more so, at smaller ones, the first term on the rhs of Equation (4) usually dominates E dN_γ/dE. Consequently, the light curve generated by a sum of ICS pulses at a luminosity distance D_L is generally well approximated by

$$\begin{equation} E\,{d^2N_\gamma \over dt\, dE} \approx \Sigma _i\, A_i\Theta [t - t_i]\,{\Delta t_i^2\,(t - t_i)^2 \over \big((t - t_i)^2 + \Delta t_i^2\big)^2}\, e^{-E/E_{p,i}[t - t_i]}, \end{equation} \tag{ 12 }$$

where the index "i" denotes the ith pulse produced by a CB launched at an observer time t = t_i, or, alternatively,

$$\begin{eqnarray} \hspace{-15pt}E\,{d^2N_\gamma \over dt\, dE} &\approx& \Sigma _i\, A_i\Theta [t - t_i]\, e^{-\Delta t_i^2/(t - t_i)^2} \, [1 - e^{-\Delta t_i^2/(t - t_i)^2} ]\,\nonumber \\ &\times& e^{-E/E_{p,i}[t - t_i]}, \end{eqnarray} \tag{ 13 }$$

where E_p,i[t − t_i] is given by Equation (5) with t being replaced by t − t_i and

$$\begin{equation} A_i \approx {c\, n_g(0)\, \pi \, R_{\rm CB}^2\,\gamma _0 \delta _0^3\, (1+z) \over 4\, \pi \, D_L^2}. \end{equation} \tag{ 14 }$$

Thus, in the CB model, each ICS pulse in the SHB light curve is effectively described by four parameters, t_i,  A_i,  Δt_i, and E_p,i(0), which are best fitted to reproduce its observed light curve.

Setting t_i = 0, E_p(t) has the approximate form E_p(t) ≈ E_p t²_p/t². Such an evolution has been observed in the time-resolved spectrum of well isolated pulses (see, for instance, the insert in Figure 8 of Mangano et al. 2007), until the ICS emission is overtaken by the broadband synchrotron emission from the swept-in ISM electrons. Hence, the temporal behavior of the separate ICS peaks is given by

$$\begin{equation} E\, {d^2N_\gamma \over dt\,dE}(E,t) \propto {t^2/\Delta t^2 \over (1+t^2/\Delta t^2)^2} e^{-E\, t^2/\, E_p\,t_p^2} \approx F(E\,t^2), \end{equation} \tag{ 15 }$$

to which we shall refer as the "E t² law." A simple consequence of this law is that unabsorbed ICS peaks have approximately identical shapes at different energies when plotted as a function of E t².

A few other trivial but important consequences of Equation (15) for unabsorbed GRB peaks at E ≲ E_p are the following.

• 1.  
  The peak time of a pulse is at

  $$\begin{equation} t_p=t_i + \Delta t_i. \end{equation} \tag{ 16 }$$
• 2.  
  The FWHM of a pulse is

  $$\begin{equation} {\rm FWHM} \approx 2\, \Delta t_i, \end{equation} \tag{ 17 }$$

  and it extends from t ≈ t_i + 0.41 Δt_i to t ≈ t_i + 2.41 Δt_i.
• 3.  
  The rise time (RT) from the half peak value to the peak value satisfies

  $$\begin{equation} {\rm RT\approx 0.30\,FWHM}, \end{equation} \tag{ 18 }$$

  independent of energy. It agrees with the empirical relation that was inferred by Kocevski et al. (2003) from BATSE bright LGRBs, RT ≈ (0.32 ± 0.06) FWHM.
• 4.  
  The FWHM increases with decreasing energy approximately like a power law:

  $$\begin{equation} {\rm FWHM}(E)\sim E^{-0.5}. \end{equation} \tag{ 19 }$$

  This relation is consistent with the empirical relation, FWHM(E) ∝ E^{−0.42 ± 0.06}, satisfied by BATSE GRBs (Fenimore et al. 1995).
• 5.  
  The onset time, t_i, of a pulse is simultaneous at all energies. But the peak times t_p at different energies differ and the lower-energy ones "lag" behind the higher-energy ones:

  $$\begin{equation} t_p-t_i \propto E^{-0.5}. \end{equation} \tag{ 20 }$$
• 6.  
  The time-averaged value of E_p(t) for GRB peaks, which follows from Equation (5), satisfies

  $$\begin{equation} E_p= E_p(0)/2=E_p(t_p). \end{equation} \tag{ 21 }$$

The effective spectral index of unabsorbed ICS pulses in Equation (4) is given by (DDD2008a)

$$\begin{equation} \Gamma _i[E,t-t_i] =-E\; {d\,{\rm log} F_E\over dE}= 1+ \beta _g+{E\over E_{p,i}[t-t_i]}. \end{equation} \tag{ 22 }$$

The hardness, defined as the ratio between the number of events in two energy bands, is generally reported, uncorrected for absorption, for the Swift 25–150 keV and 15–25 keV bands, respectively. It can be approximated by (DDD2008a)

$$\begin{equation} {\rm HR}_i(t)=B_i\, e^{-\Delta E/ E_{p,i}[t-t_i]}, \end{equation} \tag{ 23 }$$

where ΔE is an effective interval between the bands.

For b ∼ 0 and β_g ∼ 0, Equation (4) yields a peak value of E² dN/dE at E = E_p(t). Observers usually report E_p(t) at the peak's maximum or its averaged value in a chosen time interval. For the approximate pulse shape given by Equation (10), E_p ≈ 0.5 E_p(0) over the FWHM of a pulse.

For LGRBs, Equations (5) give a good description of the observations, for _g ≈ 1 eV, γ₀ ∼ 10³, and δ₀ ∼ 10³, corresponding to the expected θ ∼ 1/γ₀. As their name reflects, SHBs typically have a larger E_p than LGRBs. One reason is model independent: SHBs are observed at smaller typical redshifts. In the CB model, according to Equations (5), larger values of (1 + z) E_p may be due to a larger value of the typical _g and/or γ₀. The glory of LGRBs being the very early light of a core-collapse SN, there are observational reasons to adopt _g ≈ 1 eV. For SHBs, the ambient light may be the light of a companion star or of an accretion disk. In the first case, there is also a reason to adopt _g ≈ 1 eV, which we will use for the sake of definiteness. The typical Lorentz and Doppler factors leading to the average value of (1 + z) E_p ∼ 1000 keV for the well-measured cases in the table of SHB properties of Kann et al. (2008) are

$$\begin{equation} \gamma _0[{\rm SHB}]\sim \delta _0[{\rm SHB}]\sim 1400, \end{equation} \tag{ 24 }$$

somewhat larger than the typical values, γ₀ ∼ δ₀ ∼ 10³, for LGRBs.

The peak time of dN/dt for the pulse shape given by Equation (10) is around t = Δt = t_tr, where

$$\begin{equation} t_{\rm tr}\approx {\sqrt{3}\, (1+z)\over \delta _0\, \beta _s}\, {R_{\rm tr}\over c} \approx {23\, {\rm ms}\over \beta _s}\, {10^{3}\over \delta _0}\, (1+z)\, \sqrt{N_B\over 10^{48}}. \end{equation} \tag{ 25 }$$

The full width of dN/dt at half-maximum is FWHM ∼2.38 t_tr and its rise time from the half-maximum to the peak value is t_rise ∼ 0.59 t_tr (FWHM ∼1.8 t_tr and t_rise ∼ 0.7 t_tr for the pulse shape used in Dado et al. 2004, hereafter DDD2004). The pulse shape is energy dependent because of the exponential factor in the pulse light curve:

$$\begin{equation} E {dN\over dE} \propto e^{-E/E_p(t)}\rightarrow e^{-2\,E\, t^2/E_p(0)\, \Delta t^2}, \end{equation} \tag{ 26 }$$

and the rise time, peak time, and FWHM of the ICS pulse are energy dependent and decrease with increasing energy. In the CB model, this explains the observed time-lag effect in LGRB pulses, which decreases with increasing energy. As can be seen from Equation (26), the time lag is proportional to the width parameter, Δt, and decreases with increasing E_p, yielding a vanishingly small time lag in SHBs with a very small width and a large E_p. For t² ≫ Δt², the pulse has a shape

$$\begin{equation} E\, {d^2N\over dt\, dE} \propto (E\,t^2)^{-1}\, e^{-2\,E\, t^2 /E_p(0)\, \Delta t^2}=F(E\,t^2). \end{equation} \tag{ 27 }$$

This "E t²" law is a test of ICS on a glory's light that becomes more radially directed at increasing radii and times: 〈1 + cos θ_in〉 → r²_g/2 r² ∼ Δt²/2 t² in Equation (10). For LGRBs and low energies, when the CB opacity is dominated by ff absorption, the law is also valid at early time. This law for LGRBs, in its form as a correlation between the FWHM of collections of pulses and the energy interval at which they are measured (FWHM ∼E^−1/2) has been known for a long time (e.g., Ramirez-Ruiz & Fenimore 2000). For well-measured single-pulse XRFs, Equation (27) can be tested with precision, from X-ray to optical frequencies, both as an (E, t) correlation and as a spectral form (DDD2007b). One example is XRF 060218, specifically discussed in De Rújula (2008). For SHBs, we do not have enough information to test Equation (27) and its underlying assumption that the source of the ambient light is localized close to, or around, the engine.

The measured redshifts of SHBs are relatively small, averaging 〈z〉 ∼ 0.5 compared with those of LGRBs; 〈z〉 ∼ 2. The widths of their pulses have been studied in more detail than their rise times, and range from 5 to 300 ms, with a broad peak at 50 ms (Nakar 2007; Kann et al. 2008). These numbers correspond to full widths at quarter maximum, FWQM ∼ 3.5 t_tr, for the pulse shapes. The typical values γ₀ = δ₀ = 1400, z = 0.5, and Δt = 10 ms in the observer's frame correspond to r_g = c γ(t) δ(t) Δt/(1 + z) ≈ 4 × 10¹⁴ cm in the progenitor's rest frame. Inverting Equation (25) and using γ₀ = δ₀ = 1400, we obtain N_B[SHBs] ∼ 10⁴⁸ for FWQM = 50 ms and β_S = 1/3, respectively. This is smaller than the typical N_B[LGRBs] ∼ 10⁵⁰ by two orders of magnitude, probably because core-collapse SNe have much more available mass for potential CB-generating accretion than any of the putative SHB progenitors.

Let L be the luminosity of the source (an accretion disk or a massive companion). For a transparent or semitransparent distribution of circumburst material, it is approximately the glory's luminosity. The peak luminosity of a CB is at t = t_tr, the time it becomes transparent, and it is given by (DD2004)

$$\begin{equation} (1 + z)^2\, L_p ={\delta _0^4\, \beta _s^2\, L\over 9}. \end{equation} \tag{ 28 }$$

In principle, Equation (28) could be used to estimate the typical L of the ambient light of SHBs. However, the results for L_p are often reported for binning times much larger than the peak rise times (e.g., Gehrels et al. 2006) and cannot be used to estimate L. For an SHB with N_CB prominent pulses of similar properties, the isotropic-equivalent energy, E_iso, is (DD2004)

$$\begin{eqnarray} E_{\rm iso}&=&{\delta _0^3\, L\, \beta _s\, N_{\rm CB}\over 6\, c}\, \sqrt{\sigma _{_T}\, N_B \over 4\, \pi }= (1.2\times 10^{50}\, {\rm erg})\, N_{\rm CB}\, \beta _s\,\nonumber \\ &&\times \left({\delta _0\over 10^3}\right)^3\, {L \over 10^{40}\;{\rm erg\;s^{-1}}}\, \sqrt{{N_b \over 10^{48}}}. \end{eqnarray} \tag{ 29 }$$

The typical E_iso of SHBs is a few 10⁵⁰ ergs, with a wide spread between 10⁴⁸ and 10⁵² erg (see, e.g., Kann et al. 2008). The typical values δ₀ = 1400, β_s = 1/3, N_CB = 3, and N_B = 10⁴⁸ yield the estimate, L ∼ 10⁴⁰ erg s^-1, which is quite normal for the putative progenitors of SHBs.

The pronounced dependence of LGRB observables on the Doppler factor, which ranges over a broad domain, led to simple correlations between them (Dar & De Rújula 2000, hereafter DD2000; DDD2004; DD2007a). Similar correlations between prompt emission observables are expected in SHBs if they are dominated by a single class of progenitors. But the scarcity of data and lack of statistics on SHBs with secured redshift do not yet allow conclusive tests of such correlations.

The simplest CB model correlations for single pulses are Δt ∝ E⁻¹_p and E_iso ∝ (1 + z)² L^4/3_p. They are well satisfied for LGRBs (DD2004). For other pairs of observables, the predicted correlations are slightly more elaborate (DDD2007a). For instance, given that (1 + z) E_p ∝ δ₀ γ₀ and E_iso ∝ δ³₀, one expects (1 + z) E_p ∝ E^1/3_iso. But this is precise only for low E_p and E_iso (large θ) cases at the XRF limit of the LGRB distribution. For θ ≲ 1/γ₀, (1 + z) E_p ∝ γ²₀ and E_iso ∝ γ³₀ implying that (1 + z) E_p ∝ E^2/3_iso. Since the observer's angle continuously varies, XRFs and GRBs lie, in the [(1 + z) E_p, E_iso] log–log plane, close to a line whose slope smoothly varies from 1/3 to 2/3 (DDD 2007a), as shown in Figure 5. In Figure 5, we also show the limited observational information on the [(1 + z) E_p,  E_iso] correlation for SHBs. The results, like those for LGRBs, are fitted to a power law varying from a 1/3 to a 2/3 slope (DDD2007a):

$$\begin{equation} (1+z)\, E_p\approx E_p^0\, \big(\big[E^\gamma _{\rm iso}\big/E_0\big]^{1/3}+\big[E^\gamma _{\rm iso}\big/E_0\big]^{2/3}\big), \end{equation} \tag{ 30 }$$

with two parameters E⁰_p and E₀, respectively. They are almost equally well fitted by just the higher power. This is not surprising, for selection effects may imply that, so far, only the most energetic SHBs have been observed. SHBs viewed far off-axis, are wider, and should look like GRBs of low luminosity without an associates SN.

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The correlations between E_p and other prompt observables (peak luminosity, peak rise-time, lag time, and variability), which are satisfied for LGRBs (DDD2007a), are also very straightforward tests of the hypothesis of ICS for SHB produced by a single class of progenitors, but the data on SHBs are not precise enough to reach a conclusion at the moment.

During its early-time emission, when both γ and δ stay put at their initial values γ₀ and δ₀, respectively, Equation (32) yields an early-time light curve $F_\nu \propto e^{-\tau _{_W}}\, R^2\, n^{(1+\beta)/2}\,\nu ^{-\beta }.$ If the progenitor is embedded inside a GC or an SSC, which blows a constant wind into the ISM or IGM with a density profile n ∝ 1/r² ∼ 1/t², the early-time SR light curve as given by Equation (32) has the form

$$\begin{equation} F_\nu \propto {e^{-a/t}\, t^{1-\beta }\, \nu ^{-\beta } \over t^2+t_{\rm exp}^2}, \end{equation} \tag{ 35 }$$

until the CB reaches the constant ISM or IGM density.

As it ploughs through the ionized ISM, the CB gathers and scatters its constituent ions, mainly protons. These encounters are "collisionless" since, at about the time it becomes transparent to radiation, a CB also becomes "transparent" to hadronic interactions (DD2004). The scattered and re-emitted protons exert an inward pressure on the CB, countering its expansion. In the approximation of isotropic re-emission in the CB's rest frame and a constant ISM density n ∼ n_e ∼ n_p, one finds that within a minute or so of the observer's time t, typical SHB generating CBs of the baryon number N_b ∼ 10⁴⁸ reach an approximately constant "coasting" radius R ∼ 10¹⁴ cm, before they finally stop and blow up, after a journey of years of the observer's time. During the coasting phase, and in a constant density ISM, γ(t) obeys (DDD2002a; DDD2006)

$$\begin{equation} ({\gamma _0/ \gamma })^4 + 2\,\theta ^2\,\gamma _0^2\,(\gamma _0/\gamma)^2 = 1+2\,\theta ^2\,\gamma _0^2+t/t_0, \end{equation} \tag{ 36 }$$

with

$$\begin{equation} t_0={(1+z)\, N_{{\rm B}}\over 8\,c\, n\,\pi \, R^2\, \gamma _0^3}. \end{equation} \tag{ 37 }$$

As can be seen from Equations (36) and (37), γ(t, t₀, θ, γ₀) and, consequently also, δ change little as long as t < t_b ≡ [1 + 2 γ²₀ θ²] t₀, where

$$\begin{eqnarray} t_b&=& (1300\;{\rm s})\, \big[1+2\,\gamma _0^2\, \theta ^2\big]\,(1+z) \left[{\gamma _0\over 10^3}\right]^{-3}\, \left[{n\over 10^{-3}\;{\rm cm}^{-3}}\right]^{-1}\nonumber \\ &&\times \left[{R\over 10^{14}\;{\rm cm}}\right]^{-2} \left[{N_{{\rm B}}\over 10^{48}}\right], \end{eqnarray} \tag{ 38 }$$

and they approach a power-law decay, δ ∝ γ ∼ t^−1/4 for t > t_b. The slow change in γ and δ for t ≲ t_b produces the "plateau" phase of canonical afterglows. Their asymptotic power-law decay for t ≫ t_b, when inserted in Equation (34), yields the asymptotic power-law decay (DDD2007b)

$$\begin{equation} F_\nu (t)\sim t^{-\beta -1/2}\, \nu ^{-\beta }=t^{-\Gamma +1/2}\, \nu ^{-\Gamma +1}, \end{equation} \tag{ 39 }$$

where β ≡ Γ − 1 = p/2 ∼ 1.1. Equation (39) is valid as long as the ISM has an approximately constant density. The gradual transition ("break") of the AG from the plateau phase to a power-law decay at t ≳ t_b takes place when the CB has swept in a mass comparable with its initial mass. This CB model bend/gradual break is different from the achromatic break predicted by the conical Fireball model when the ejecta decelerate to a point when the observer begins to see the entire opening angle of a conical ejecta (Rhoads 1997, 1999).

Density variations complicate the simple shape of AGs and will not be discussed here, except for noting that for an ISM with an approximate ∼1/r² density profile, such as in a GC, or in a galactic halo with an isothermal sphere density profile, which CBs may reach late in their motion, or in a windy environment created around SSCs by their blowing winds, the predicted X-ray and optical AG has a simple power-law form:

$$\begin{equation} F_\nu (t)\sim t^{-\beta -1}\, \nu ^{-\beta }=t^{-\Gamma }\, \nu ^{-\Gamma +1}. \end{equation} \tag{ 40 }$$

In ordinary LGRBs, t_b is usually larger than a few hundreds of seconds but, in intrinsically bright LGRBs, in particular in those that happen to take place in a dense ISM environment, the break in the X-ray AG may occur before the end of the prompt emission. Such breaks may be hidden under the much brighter ICS emission and only the power-law decaying tail of the AG is observed (DDD2008b). In SHBs, the low-density environment of GCs, in particular those with a large offset from the galactic center, yields relatively large t_b values as can be seen from Table 2. For SHBs in young SSCs with a strong wind, the AG has the simple power-law decay given by Equation (40) with no jet break.

For the measured average SHB redshift, z ∼ 0.5, and with the typical values, δ₀ γ₀ ∼ 1400, the distance traveled by a CB in the typical t_ESEC ∼ 100 s duration of an ESEC is x ∼ γ₀ δ₀ c t_ESEC/(1 + z) ≈ 1.2 pc, coincident with the typical core radius of a GC or SSC. This straightforward understanding of ESEC durations is independent of whether the mechanism generating the radiation is ICS or SR, provided the sources are CBs, moving relativistically, and not significantly decelerating in this short interval of the observer's time.

All the SHBs with ESECs (except perhaps 051210 with an unknown redshift) appear to reside in host galaxies. GCs and SSCs inside galaxies, probably when they are young, contain a considerable amount of gas and dust. Some may still be embedded in a molecular cloud. The light from the stars in such clusters is absorbed and re-emitted in the very far infrared. In a steady case, when the light of the stars is absorbed and re-emitted as a thermal radiation from the surface of the cluster core, L ≈ 4π σ R² T⁴. For R ∼ 1 pc, and L ∼ 10⁴⁰ erg s⁻¹, the temperature of the diffuse radiation in the cluster is typically T_c ∼ 0.003 eV.

As a CB moves through this microwave radiation of the host star cluster, its ICS produces a light curve, which traces the density of the radiation along the CB trajectory. This radiation is a sum of a smooth background radiation plus much harder light peaks from the passage near very luminous stars. For simplicity, consider the contribution to the ESEC light curve from ICS of background microwave radiation by a CB ejected near the center of an SSC. It is simply obtained by replacing the typical temperature T_g and density n_g(t) of the glory photons in the expression describing the prompt SHB with, respectively, the typical energy _c ∼ T_c and the density n_c(t) of the intracluster photons, where n_c(0) = 3 L_ssc/4 π c _c r²_c and L_ssc is the SCC luminosity. The resulting spectrum is an exponentially cutoff power law with a typical cutoff energy ≈ peak energy, E_p ∼ 2 keV (see Equations (4) and (5)).

The resulting ESEC light curve is given by Equations (10) and (4) with Δt replaced by Δt_c = r_c(1 + z)/c γ₀ δ₀. The isotropic-equivalent energy of the ESEC is then given by

$$\begin{eqnarray} \hspace{-15pt}E_{\rm iso}[ESEC]&\approx& \sigma _{_T}\, N_b\, N_{\rm CB}\,\gamma _0\, \delta _0^3\, {3\, L_{\rm ssc}\over 4\, c\, r_c} \sim (10^{49}\; {\rm erg})\,\nonumber \\ &\times& {N_b\over 10^{48}}\, {N_{\rm CB}\over 4} \, {pc\over r_c} {\gamma _0\, \delta _0^3\over 1400^4}\, {L_{\rm ssc}\over 10^8\; L_\odot }. \end{eqnarray} \tag{ 41 }$$

The result of Equation (41) is of the observed order of magnitude, but has more sources of accumulated uncertainty and variability than other CB-model results.

The ESEC after transparency time, for a spherical SSC, is a decreasing function of time from the beginning, if the progenitor is in the front hemisphere, and the CBs' distance from the cluster's center increases. Otherwise, the ESEC light curve first increases and then decreases with time. The data are consistent with these expectations, but so far insufficient to test them in detail.

Some young SSCs have a very large ISM density, 10³ ≲ n ≲ 10⁶ cm^-3. Consequently, the emission rate of SR radiation from CBs moving inside such SSCs, which is proportional to the ISM density (see Equation (32)), is very intense and relatively hard with a spectral index β ≃ 1/2, because ν_b ∝ n^1/2 (see Equation (31)) is well above the X-ray band. At early time, it is given by Equation (35) with β ≃ 1/2. As soon as the CB escapes out of a dense core of an SSC, the density decreases fast like 1/r², and the light curve decreases rapidly and becomes softer with its spectral index increasing to β ≈ 1.1,

$$\begin{equation} F_\nu \propto {e^{-a/t}\, t^{1/2}\, \nu ^{-1/2} \over t^2+t_{\rm exp}^2}\rightarrow t^{-2.1}\, \nu ^{-1.1}. \end{equation} \tag{ 42 }$$

Equation (42) is valid when the CBs encounter a very large density inside the SSC such that the CBs reach their coasting radius, R_CB ∝ N^1/3_b n^−1/3 γ^−2/3₀, well inside the SSC and t_exp ≪ Δ_ssc.

Observations. This SHB, which was localized by HETE (Villasenor et al. 2005), was relatively very faint (E_iso ∼ 2.4 × 10⁴⁸ erg). It had a multispike peak with T₉₀ = 70 ± 10 ms in the 30–400 keV band, T₉₀ = 220 ± 50 ms in the 2–25 keV band, and showed a fast hard to soft spectral evolution with E_p ∼ 86 ± 16 keV and a photon spectral index Γ = 0.82 ± 0.13. A soft extended emission component was detected 24 s after the burst with T₉₀ = 130 ± 7 s, Γ = 1.98 ± 0.18, and a fluence much larger than that of the short burst, unlike what is seen in the giant flares of SGRs. It was the first SHB for which an optical AG was discovered and was used to localize it in a star-forming dwarf galaxy at redshift z = 0.16 (Hjorth et al. 2005a; Fox et al. 2005; Covino et al 2006) at a projected distance of ∼3.8 kpc from its center. Its X-ray AG that was detected by Swift and Chandra (Fox et al. 2005) and its optical light curve as measured by Watson et al. (2006) are shown in Figure 1(a). The optical data do not shed light on a possible late-time flare suggested by the last two data points of Chandra (Fox et al. 2005),

Interpretation. The short burst shows all the properties expected from ICS of glory light by a jet of CBs with a relatively small baryon number emitted by the source. The spectral index of the ESEC, Γ = 1.98 ± 0.18, the lack of spectral evolution during the ESEC, and the duration of the ESEC are consistent with those expected from SR (Γ ∼ 2.1) of CB propagating in a r_c ∼ 1 pc core of a GC and t ∼ (1 + z) r_c/c γ₀, δ₀ ∼ 120 s. The optical afterglow is well explained by a CB propagating out of the dwarf galaxy with an isothermal sphere density distribution n ∝ 1/(r² + r²₀), as demonstrated in Figure 1(a). The 5 keV SR afterglow was calculated using the parameters of the optical light curve, assuming the theoretically expected spectral index, β = 1.1. No attempt was made to fit the the last two data points with a SR flare whose light curve, in the CB model, depends on four adjustable parameters.

Observations. This burst at redshift z = 0.257 was studied in detail in Campana et al. (2006b), Grupe et al. (2006), and Malesani et al. (2007). The BAT on board Swift triggered on the burst at 12:34:09 UT on 2005 July 24. The burst had T₉₀ = 3.0 ± 1.0 s, but most of the energy of the initial SHB was released in a hard spike with a duration of 0.25 s. The bulk of the burst energy was not emitted in the short initial spike but in an ESEC with Γ = 2.5 ± 0.2, which lasted ∼150 s. Swift's XRT began observing the afterglow 74 s after the BAT trigger. The Chandra X-ray observatory performed two observations: 2 days and about 3 weeks after the burst. The complete X-ray light curve is shown in Figure 1(b). It has the canonical shape observed in long GRBs, namely a rapid decay with a fast spectral softening ending with a sharp transition to a plateau phase with a much harder power-law spectrum, Γ = 1.79 ± 0.12, as shown in Figure 1(c). The AG gradually steepens into a late power-law decay. A large flare superimposed on the canonical light curve occurred around 50 ks after burst with a fluence of ∼7% of that of the prompt burst. The flare has also been detected in the optical and NIR bands (e.g., Malesani et al. 2007). Spectral analysis of the XRT data (Campana et al. 2006b) showed no evolution during the afterglow phase, including the large late flare. Spectral analysis of the Chandra observations from the fading tail of this flare confirmed this result (Grupe et al. 2006). The burst took place 2.5 kpc (in projection) from the center of an elliptical HG (Malesani et al. 2007).

Interpretation. The CB model X-ray light curve of SHB 050724 and its spectral index "light curve" are shown in Figures 1(b) and (c), respectively. The early-time emission is described by ICS of the progenitor's glory light and the ESEC by ICS of the light of the core of the assumed SSC or a GC environment. The fast decay of the X-ray light curve and the rapid spectral softening took place when the CB moved away from the quasi-isotropic light distribution in the dense stellar environment of the progenitor into the ISM of its elliptical HG. As can be seen in Figures 1(b) and (c), this fast decay and spectral softening stopped simultaneously when the AG was taken over by the SR from the decelerating CB in an ISM of a constant density. As expected, apart from normalization, the late-time SR afterglow is similar in shape to the SR afterglow of LGRBs. Also, the late flare superimposed on the canonical light curve is similar to those observed in many LGRBs. It was produced by enhancement of the emitted SR when the CB encountered a density bump in its voyage through the host's ISM. The Chandra data show that the canonical AG continued to decay after the flare with the same slope and the same spectral index, β_X = 0.79 ± 0.15. As shown in Figure 1(b), the complete XRT light curve is well fitted by the CB model. Moreover, the CB model relation α = β_X + 1/2 = p/2 is well satisfied. The temporal behavior of the canonical AG was best fitted with p = 1.56, implying an unabsorbed spectral index, β_X = p/2 = 0.78, and an asymptotic power-law decay with a power-law index, α = β_X + 1/2 = 1.28, in agreement with those observed.

The elliptical HG of SHB 050724 was argued to provide strong support for a neutron star merger origin of this SHB. But it was pointed out that neutron star mergers do not produce the late accretion episodes needed in the standard folklore to power a late central activity, which could produce the large flare around 50 ks after burst (Grupe et al. 2006). In the CB model, a late flare with a typical SR spectrum and little spectral evolution is produced by density bumps along the CB trajectory in the ISM. Such flares neither rule out nor support any specific origin of the SHB.

Observations. This was a faint two-spiked SHB localized by Swift. It had T₉₀ = 1.27 ± 0.05 s and no soft extended emission was detected (La Parola et al. 2006). The BAT spectrum was fitted with a power law with a photon spectral index Γ = 1.1 ± 0.3. The XRT started observation 79 s after the BAT trigger. It detected a rapidly decreasing light curve, shown in Figure 1(d), with a small flare superimposed around 134 s. A possible HG was discovered within the XRT error circle (Bloom et al. 2008), but no emission or absorption lines were detected. No optical afterglow was detected.

Interpretation. The spectrum of the burst is consistent with ICS of glory light (Γ ∼ 1 for E ≪ E_p). The fast declining XRT light curve was fitted as the tail of an ICS flare. The data are insufficient for a critical test of the CB model interpretation.

Observations. This burst was discussed in detail in Soderberg et al. (2006a) and Burrows et al. (2006). It was an intense, multispiked burst localized by Swift, The burst was also detected by Konus-Wind, Suzaku, INTEGRAL, and RHESSI and its late afterglow by Chandra. The burst had T₉₀ = 1.4 ± 0.2 s, Γ = 0.92 ± 0.13, and showed no signs of extended emission. The Swift/XRT observations of its X-ray light curve began 88 s after the Swift/BAT trigger. Its late-time X-ray emission was also detected by Chandra. The measured X-ray light curve (Figure 1(e)) of SHB 051221A shows the canonical behavior of X-ray light curves of LGRBs (Nousek et al. 2006) with an asymptotic decay index Γ ≈ 2. Its afterglow was also detected in the NIR and optical bands, and was localized near the center (∼1 kpc in projection) in a star-forming galaxy with z = 0.546.

Interpretation. The prompt SHB was produced by ICS of glory light as evident from its spectral index Γ ∼ 1. The XRT light curve is well fitted by the SR tail of prompt flares taken over by the canonical SR afterglow emitted by a CB moving in a constant density ISM of the HG, as shown in Figure 1(e). Its late afterglow with α = 1.6 ± 0.05 and that with Γ = 1.97 ± 0.13 are consistent within errors with the CB model prediction, α = Γ − 1/2. The wiggling of the light curve around the smooth theoretical line can be due to density variations along its trajectory, as was observed in afterglows of many LGRBs.

Observations. This was a multipeaked burst with a bright X-ray afterglow localized by Swift (Barbier et al. 2005). It was initially thought to be LGRB, since it has T₉₀ = 8.0 ± 0.2 s (Hullinger et al. 2005), but a spectral analysis that showed a negligible spectral lag and a broad softer bump between 30 and 50 s was claimed to indicate an SHB (e.g., Kann et al. 2008). An exceedingly faint optical afterglow was discovered within the XRT error circle (Malesani et al. 2005a, 2005b) with the Very Large Telescope (VLT) and also detected by Gemini.

Interpretation. The prompt burst is consistent with ICS of glory light (Γ ∼ 0.91 ± 0.28). The XRT light curve is consistent with an SR tail of a prompt emission taken over by a canonical SR afterglow, as shown in Figure 1(f). The XRT does not shed light on whether this GRB was or was not an SHB. Due to a gap in the data between 400 s and 4000 s, the theoretical shape and the values of the SR afterglow parameters are uncertain.

Observations: This intermediate-duration GRB was discussed in detail in Donaghy et al. (2006), de Ugarte Postigo et al. (2006), and Levan et al. (2006). It was localized by HETE-2. It consisted of two peaks with T₉₀ = 1.60 ± 0.07 s at high energies, which were followed by a faint soft emission for several hundreds of seconds. It was also detected by Konus-Wind, Suzaku, and RHESSI. Follow-up observations by Swift began 3 hr after the burst. Its XRT light curve is shown in Figure 1(f). Its faint optical afterglow was discovered by Malesani et al. (2006a), which led to the discovery of its extremely faint HG (Levan et al. 2006). The burst outshined its HG (by a factor of greater than 100). A photometric redshift for this event placed it at a most probable redshift of z = 4.6, with a less probable z = 1.7. In either case, GRB 060121 could be the farthermost short GRB detected to date with an isotropic-equivalent energy release in gamma rays comparable with that of LGRBs.

Interpretation. Its intermediate duration and its unusual large redshift Z = 4.7, relatively small E_p ≈ 120 keV, large equivalent isotropic gamma-ray energy, E_iso ≈ 2.2 × 10⁵³ erg (or E_iso ≈ 4.3 × 10⁵² erg for z = 1.7), vicinity (∼2 kpc) to the center of the HG, and its bright optical afterglow suggest that GRB 060121 may have been an ordinary LGRB, and was wrongly classified as SHB. In Figure 2(a), we show that the CB model fit to its late AG, which, although satisfactory, does not shed light on its true identity.

Observations. This unusual burst was reported and discussed in detail in Roming et al. (2006). It was detected by Swift, Konus-Wind, and INTEGRAL. It was a very intense burst, with many short spikes, each with FWHM smaller than 20 ms, with T₉₀ = 0.7 ± 0.1 s. It had a very unusual average initial photon index, Γ = −0.33(−0.29, + 0.25), below the spectral peak during the initial 0.192 s of the burst (Golenetskii et al. 2006). It had the highest fluence and the highest observed peak energy of all SHBs observed by Swift. No extended soft emission was detected. Swift XRT detected its X-ray afterglow 79 s after the BAT trigger and measured its light curve until 200 ks (Figure 2(b)) with a late afterglow photon index of Γ = 1.96 ± 0.09. The XRT light curve shows flaring activity superimposed on a canonical light curve during the first 1000 s after burst. The optical afterglow of SHB 060121 was detected and followed by the Swift UVOT and by the VLT following its localization by the Swift UVOT. A very faint HG was detected at the afterglow position by Berger et al. (2006a).

Interpretation. The unusual early light curve and spectrum of SHB 060313 suggest an unusual progenitor and environment of this unusual SHB. In the framework of the CB model, the large number of short pulses requires a progenitor, which fires many small CBs like a shot gun (R. Plaga 2008, private communication) or a machine gun, rather than a cannon. The unusual hard spectrum could be produced by ICS of self-absorbed radiation produced inside the CBs. In contrast to the unusual prompt emission, the afterglow is not unusual, and can be well explained as SR radiation from a CB propagating in a constant density ISM of the HG, as shown in Figure 2(b).

Observations. This SHB was detected and localized by Swift (Racusin et al. 2006a) and was also detected by Suzaku. It had T₉₀ = 0.5 ± 0.1 s. It consisted of two pulses, which peaked 60 ms and ∼ 100 ms after their beginning (Sato et al. 2006). No extended soft emission was detected. The XRT began taking data 63 s after the BAT trigger (Racusin et al. 2006b). The 0.3–10 keV X-ray light curve shows a plateau from 73 s until 115 s after the BAT trigger, which turns into an exponential decay. The X-ray emission was not detected after 1000 s. Follow-up optical observations did not detect an optical afterglow. A single galaxy lying within the XRT error box at redshift z = 1.1304 was suggested as its HG, making SHB 060801 the most distant SHB with a secured redshift.

Interpretation. The prompt emission pulses are well explained by ICS of a thin bremsstrahlung glory of two CBs. The extended soft emission at redshift 1.13 was too dim to be detected by BAT, but its ending and decay probably are the emission detected and followed with the XRT, respectively. In the CB model, it is well described by ICS of the GC's light when the CBs leave their dense core and move into the surrounding ISM, as shown in Figure 2(c).

Observations. The Swift observations of this burst are reported in Schady et al. 2006. This burst began with an intense double spike, which lasted 0.5 s. The SHB was also detected by RHESSI, Konus-Wind, and Suzaku (Hurley et al. 2006). It was followed by an extended soft emission with T₉₀ = 130 ± 10 s. The XRT began follow-up observations at 143 s after the burst. The XRT light curve (Figure 2(d)) initially decayed with a slope of α = 2.3 ± 0.3, followed by a shallow slope beginning at 300 s with a spectral index Γ = 1.7 ± 0.3. VLT observations (Malesani et al. 2006a) found a faint source that was subsequently found to fade, revealing a star-forming HG (Malesani et al. 2006b; Berger et al. 2006b) at redshift z = 0.4377. The burst location is about 8 kpc from the galactic center.

Interpretation. In the CB model, the prompt pulses can be explained by ICS of glory light of the SHB progenitor. The extended soft emission could be produced by SR from the CBs while they were crossing the SSC where the SHB took place. As shown in Figure 2(d), the initial decay of the XRT light curve can be explained by SR emission from the CBs while it was crossing the SSC wind, whereas the shallow decaying AG is the afterglow emitted when the CBs propagate in the ISM surrounding the SSC.

Observations. The Swift observations of this SHB are given in detail in Racusin et al. (2007). The SHB was a very bright multispiked event lasting about 3 s, which was followed by soft extended emission that lasted 64 ± 5 s. Swift/XRT began follow-up observations 61 s after trigger. A joint Swift–Suzaku spectral analysis yielded a hard spectrum and a high peak energy (Ohno et al. 2007). The photon spectral index integrated over the SHB was Γ = 0.99 ± 0.08. The 0.3–10 keV light curve (Figure 2) showed the canonical behavior observed in LGRBs, namely a steep decay with superimposed flares taken over around 400 s by an afterglow that gradually bends over into an asymptotic power-law decay with α ∼ 1.6. The fast decay of the soft extended emission was accompanied by a rapid spectral softening (Figure 2(f)) until the plateau phase took over. The photon spectral index during the AG phase was Γ = 1.95 ± 0.15 (Zhang et al. 2007). An optical afterglow was discovered 10 min after the GRB by the Liverpool Telescope (Melandri et al. 2007), which led to the discovery of an HG at a redshift z = 0.92 (Graham et al. 2008).

Interpretation. The initial fast decay and rapid spectral evolution of the early-time X-ray light curve of SHB 070714B are well reproduced by the CB model (Figures 2(e) and (f), respectively). The data beyond 400 s show a considerable flaring activity. This makes quite uncertain the CB model best-fitted parameters of the smooth SR afterglow component, reported in Table 2.

Observations. This short burst was a faint single-spiked GRB with T₉₀ = 0.40 ± 0.04 s localized by Swift with no detected ESEC (Ziaeepour et al. 2007). Swift XRT began its observations of the burst X-ray afterglow 72.1 s after trigger and followed its rapid decay and two superimposed X-ray flares peaking around 127 s and 200 s, respectively. The fast decay of the AG turned into a shallow decay around 400 s, which steepened around 40 ks. A faint source at redshift z = 0.457 within the XRT error circle was suggested as a possible star-forming HG. Deep imaging and image subtraction did not reveal an optical afterglow. An analysis of the BAT data yielded E_p ∼ 41 keV and a photon index of Γ = 2.2.

Interpretation. The CB model fit to the XRT light curve is shown in Figure 3(a). The light curve is well fitted by two ICS pulses taken over by an SR afterglow from a CB moving in a constant low-density ISM. The relatively large value, γ θ, indicates nearly a "far off-axis"' SHB consistent with its relatively small E_iso and E_p and a large pulse width compared with those of ordinary SHBs.

Observations. This short burst was detected and localized by Swift (Sakamoto et al. 2007; Sato et al. 2006). It was also observed by Suzaku and Konus-Wind. The SHB was a multipeaked structure with a duration of about 1.5 s and a time-averaged power-law spectrum with a photon spectral index Γ = 1.2 ± 0.2. The Swift XRT began observing GRB 071227 79.5 s after the BAT trigger. The XRT light curve shows the canonical behavior: soft flares superimposed on a decaying light curve that changes to a fast decay with a spectral softening, until it is taken over by a plateau/shallow decay around 1000 s.

Interpretation. The CB model fit to the XRT light curve is shown in Figure 3(c). The light curve and spectral evolution are well fitted by ICS of a GC light of a CB moving away from the GC into a constant low-density ISM where the AG is dominated by SR emission. However, the data after the fast decay phase are scarce and constrain only weakly the CB model parameters of the SR afterglow.

Observations. This short burst was detected and localized by Swift. The Swift observations of this burst are reported in detail in Ukwatta et al. (2008) and Copete et al. (2008). Its light curve shows two well-separated peaks. The first started at 0.3 s and is no wider than 64 msec. The second peak started at 0.6 s with a FRED-like shape and a duration of 256 ms. Its soft extended emission lasted 120 s after the prompt hard emission of the short GRB and included soft flares (Copete et al. 2008). The XRT observations started 108 s after the BAT trigger. The XRT light curve (Figure 3(d)) shows the canonical behavior: a fast decay with a rapid spectral softening, which was taken over by a plateau/shallow decay around 500 s with a typical photon index of Γ = 2.1 ± 0.2.

Interpretation. The fast decay of the XRT early-time light curve and its rapid spectral softening (not shown here) are well fitted by ICS of light from a GC/SSC. It is being taken over around 500 s by a plateau/shallow decay with a typical photon index of Γ = 2.1 ± 0.2, which is well fitted by SR emission from a CB deceleration in a medium of low constant density (Figure 3(d)). The scarce data on the late afterglow constrain only weakly the CB model parameters of the SR afterglow.

Observations. This short burst was detected and localized by Swift. The Swift observations of this burst are reported in detail in Mao et al. (2008). Its light curve (Figure 3(e)) shows an initial spike starting at 0.1 s after the BAT trigger with a fast rise to a peak at 0.2 s, and then a roughly exponential decay down to background at 0.7 s. It had an extended soft emission, which started at about 10 s, rose with two peaks at 26 s and 37 s, respectively, and then fell to background levels at 220 s. The Swift XRT began observing SHB 080503 81 s after the BAT trigger. The XRT light curve showed an exponential decay with a gradual spectral softening from Γ ∼ 1 at the beginning of the XRT observations to Γ ∼ 3 around 500 s. The X-ray AG (Figure 3(e)) decreased below the Swift detection sensitivity around 1000 s after the burst, but was detected later between 4.29 and 4.66 days after the burst by Chandra (Butler et al. 2008). A rising optical afterglow was detected on the second night after trigger by Perley et al. (2008a) with Gemini-North. Continuous monitoring of this optical counterpart on consecutive nights (Bloom et al. 2008; Perley et al. 2008b) found no further rebrightening and its fading (Figure 3(f)). The OT was not detected with HST 9.2 days after the BAT trigger (Perley et al. 2008c).

Interpretation. The exponential decay of the XRT light curve at the end of the ESEC is well described by ICS of a CB emerging from the dense stellar core of a SSC of a typical core radius r_c ∼ 1 pc, as shown in Figure 3(e). This is supported by the rapid spectral softening/decreasing hardness ratio during this decay. The decay is stretched over t > 900 s, probably because of a relatively large redshift and a very low surrounding density, which results in a very faint X-ray synchrotron afterglow and a delayed take-over time. The low extra cluster density yields a prolonged expansion time of the CB, t_exp ≈ 7553, in Equation (42) and an initially rising SR afterglow (Figure 3(f)). The asymptotic decay of the optical AG is well described by F_ν ∝ t^−(1+β) (Figure 3(f)) with β = β_X ≈ 1.1, typical of a SR emitted by a CB moving in the wind of a young SSC (a proto-GC). An SSC environment of the SHB is supported by the general properties of its ESEC.

Observations. This puzzling GRB was discussed in detail in Gehrels et al. (2006); Gal-Yam et al. (2006b); Cobb et al. (2006); Mangano et al. (2007); and Della Valle et al. (2006). Swift-BAT triggered on GRB 060614 on 2006 June 14 at 12:43:48 UT. The BAT light curve showed a 5 s series of short hard peaks followed by a fainter, softer, and highly-variable extended prompt emission with T₉₀ ∼ 102 s. Konus-Wind was also triggered by GRB 060614, about 4 s after the BAT trigger. For the initial group of short hard peaks, it measured E_p ∼ 300 keV. The XRT and UVO aboard Swift started observation 97 s after the BAT trigger, which continued up to 51 days after burst. The X-ray light curve showed the canonical behavior observed in many LGRBs and in practically all SHBs, that is, initial fast decay with a rapid spectral softening that is taken over by a plateau that later bends into an asymptotic power-law decay. The AG bend/break started around 30 ks and the asymptotic temporal decay had a power-law index α = 2.1 ± 0.07. The burst was located at the outskirts of a faint star-forming galaxy at redshift z = 0.125. Very deep searches (Gal-Yam et al. 2006a, 2006b; Mangano et al. 2008) did not discover an associated SN.

Interpretation. The initial short peaks of GRB 060614, the lack of an associate SN akin to those discovered in ordinary LGRBs, the strict limits on lag times in its early short pulses, its extended soft emission (well fitted by a power-law spectrum with a photon index Γ = 2.13 ± 0.05), and its low isotropic energy ((2.5 ± 0.7) × 10⁵¹ erg; Mangano et al. 2008) are typical of SHBs. However, the long T₉₀ ∼ 5 s of its initial complex of short pulses, its (1 + z) E_p ∼ 440(−281, + 923) keV, and its location inside a star-forming galaxy are typical of LGRBs. Various solutions of these apparent contradicting pieces of evidence were suggested. Gal-Yam et al. (2006a, 2006b) and Gehrels et al. (2006) suggested that GRB 060614 (and GRB 060505) belongs to a new class of long GRBs. Dado et al. (2008c) suggested that its location in the nearby HG could be a chance coincidence. DDD2008a, and DDD2008b suggested that GRB 060614 could be produced by an extremely faint core-collapse SN or by a "failed SN"—the original collapsar model (Woosley 1993).

Our CB-model analysis indicates that GRB 060614 probably was a very hard and energetic SHB in a very bright SSC, which was viewed far off-axis (γ θ 3.08). Its relatively large viewing angle yielded a relatively small Doppler factor δ, which stretched the observed durations of its prompt emission pulses and its ESEC relative to those of ordinary SHBs, and decreased their peak emission energy, equivalent isotropic energy, and peak luminosity.

In Figure 4(a), we present the CB model fit to the canonical X-ray light curve of GRB 060614. The decay of the ESEC is very well described by ICS of the light of a SSC as the CB leaves the cluster and enters the ISM. In Figure 4(b), we compare the measured R-band light curve (Della Valle et al. 2006 and references therein) and its CB model prediction obtained with the parameters of the X-ray afterglow. A contribution, F_ν = 3 μJy, from the HG was added to the SR afterglow light curve. In Figures 4(c) and (d), we compare the observed evolution of the XRT hardness ratio and of E_p(t) during the fast decay phase of the ESEC and the CB model predictions. The agreement between theory and observations is quite satisfactory. The XRT data on the decay of the prompt ICS emission extended only up to 480 s. The CB model estimates that it was taken over by SR only around 950 s. The second orbit XRT data began only at 4.5 ks after the BAT trigger but clearly show a stretched plateau until 30 ks when it begins to bend into an asymptotic power-law temporal decay. The plateau phase and the late decay of the X-ray afterglow are those expected for an SR afterglow from CB decelerating in an isothermal sphere density profile, n ∝ 1/(r² + r²_s), namely $F_\nu \sim t^{-(1+\beta _X)}\, \nu ^{-\beta _X}$ with β_X ≈ 1.1 Such a density profile is encountered by CBs, which move from inside the HG into its halo. The CB model best-fit value, γ₀ θ = 3.08, is typical of XRFs, which, in the CB model, are interpreted as far off-axis GRBs (e.g., DD2004, DDD2004). The corresponding value, δ = 2 γ/(1 + γ² θ²) ≈ γ/5, yields E_iso and L_p, which are, respectively, ∼125 and 625 times smaller than when they are measured at a typical viewing angle, θ ≈ 1/γ₀.