PRIOR EMISSION MODEL FOR X-RAY PLATEAU PHASE OF GAMMA-RAY BURST AFTERGLOWS

In this Letter, we have seen that for most events, phases II and III of the observed light curves of the X-ray afterglow can be well fitted with a very simple formula, Equation (2). In particular, the observational facts are that, for most events, the transition from phase II to III is slow and fairly smooth, and that the X-ray spectrum remains unchanged across the transition (Nousek et al. 2006); this can be naturally explained by our model because the transition from phase II to III is an artifact of the choice of the time zero. A typical exception is GRB 070110; at the end of the plateau phase II, the light curve shows an abrupt drop (Troja et al. 2007). Such events may need other explanations (e.g., Kumar et al. 2008).

After phase III, a subsequent fourth phase is occasionally observed: the so called post jet break phase (Phase IV; Zhang 2007). Its typical decay index is ∼−2, satisfying the predictions of the jetted afterglow model (Sari et al. 1999). This phase IV can also be clearly seen in our sample (e.g., GRB 060428A; upper right of Figure 2). If the prior emission is from an external shock of the prior outflow, the jetted afterglow model is applied to the present case. As will be discussed in Section 3.4, the origin of the optical afterglow is different from the X-ray one, which explains the fact that the epoch of the jet break in the X-ray band is not generally the same as the optical one (Sato et al. 2007; Liang et al. 2008).

Sato et al. (2008) investigated the characteristics of the transition from phase II to III for 11 events with known redshifts. They derived the transition time, T⁰_brk, and the isotropic luminosity at that time L_X,end, and found that L_X,end is well correlated with T⁰_brk as L_X,end ∝ (T⁰_brk)^−1.4. They adopt a broken power-law form to fit the light curve in phases II and III which is different from that considered in this paper. However, we expect that their T⁰_brk and L_X,end roughly correspond to T₀ and f₀(T₀), respectively, that are considered in Section 2. Indeed, one finds $f_0(T_0)\propto T_0{}^{-\alpha _0}$ from Equation (2). Hence, if α₀ ≈ 1.4, which is a typical number for the decay index of phase III, we can reproduce the observed result of Sato et al. (2008).

A similar analysis has been done but with Equation (5) as the fitting formula (Dainotti et al. 2008). For 32 events with measured redshifts, they found a correlation between T_a and the X-ray luminosity at the time T_a, L_X(T_a), as L_X(T_a) ∝ T^−β_a with β = 0.6–0.74. Since T_a ≈ T₀ and L_X(T_a) ≈ f₀(T₀), their correlation indicates f₀(T₀) ∝ T₀^−β. The index β is smaller than the typical value of α₀ (≈1.0–1.5). However, the claimed correlation has large scatter (see Figures 1 and 2 of Dainotti et al. (2008)), which may be explained by the scatter of α₀ in our model.

There is a difference between the results of Sato et al. (2008) and Dainotti et al. (2008). At present, the number of events with known redshifts may be small, so this discrepancy may be resolved if the number of events increases.

Nava et al. (2007) studied the properties of prompt emission and X-ray afterglows of 23 GRBs with known redshifts. They adopted Equation (5) in fitting phases II and III of the X-ray afterglow, and found that for events with measured spectral peak energy E_p, the time T_a weakly correlates with the isotropic equivalent energy E_γ,iso of the prompt GRB emission. One can find from Figure 6 of Nava et al. (2007) that T_a seems to be roughly proportional to E_γ,iso. At present, this correlation is not firmly established because, as noted by Nava et al. (2007), there is no correlation between T_a and the isotropic equivalent energy of prompt GRBs in the 15–150 keV band for a larger sample of GRBs with known redshift but unknown E_p (hence without k-correction).

The quantity E_γ,iso correlates with E_p (Amati et al. 2002). Hence, if the T_a–E_γ,iso correlation exists, the bright GRBs with large E_γ,iso and E_p have large T_a, which is responsible for the distinct plateau phase. On the other hand, the X-ray flashes or X-ray-rich GRBs (e.g., Heise et al. 2001; Barraud et al. 2003; Sakamoto et al. 2008) have small T_a (≈T₀), and have X-ray afterglow without phase II. This tendency could have been seen in Sakamoto et al. (2008).

The above arguments may lead a link between long GRBs and X-ray flashes/X-ray-rich GRBs. Suppose that the outflow ejection is not continuous but intermittent, i.e., the central engine ejects two distinct outflows with a time interval of ∼T₀. Just after the launch of the prior outflow, the central engine does not have enough energy for another outflow, so that it needs to store an additional one. During the time interval ∼T₀, matter surrounding the central engine is accreted, increasing the gravitational binding energy. This energy is released as the main outflow causing the prompt GRB. It is expected that the larger is T₀, the larger is the stored gravitational energy, resulting in a brighter burst with large E_γ,iso. On the other hand, if T₀ is small the energy of the main outflow becomes small; this is responsible for the X-ray flash or X-ray-rich GRBs. Further details will be discussed in the near future.

The main outflow that is responsible for the prompt GRB might cause external shock X-ray emission, f_X,ext(T). In the present two-component emission model, however, f_X,ext(T) must be dimmer than the prior X-ray emission f₀(T) throughout phases II and III. Let us consider the simplest model of external shock emission of the main outflow. The relativistically expanding shell with energy E_K interacts with the surrounding medium with uniform density n₀,² and emits synchrotron radiation with microphysics parameters at the shock, p, ε_e, and ε_B (Sari et al. 1998). Then, in the case of slow cooling and ν_c < ν_X, the X-ray light curves are analytically calculated as f_X,ext(T) ∝ E^(p+2)/4_Kε^p−1_eε^(p−2)/4_BT^(2−3p)/4ν^−p/2_X, which is independent of n₀ (e.g., Panaitescu & Kumar 2000). If p ≈ 2, then f_X,ext(T) hardly depends on ε_B. Before the Swift era, typical values had been E_K ∼ 10⁵²–10⁵³ erg, ε_e ∼ 10⁻¹, and ε_B ∼ 10⁻² so that the external shock emission reproduced the observed late-time X-ray afterglow. In the present case, f_X,ext(T) must be dim, which implies small E_K and/or ε_e. A similar discussion was made by Ghisellini et al. (2007, 2008). In some models, such as the energy injection model and the inhomogeneous jet model, prompt GRB emission needs high radiation efficiency, which is defined by ε_γ = E_γ,iso/(E_γ,iso + E_K), because E_K is small at the epoch of the prompt GRB emission (Fan & Piran 2006; Granot et al. 2006; Ioka et al. 2006; Zhang et al. 2007b). As discussed here, f_X,ext(T) can be dim if ε_e is small while E_K remains large, ∼10⁵²–10⁵³ erg. Hence, the present model could avoid a serious efficiency problem. On the other hand, if E_K of the outflow is small, the efficiency ε_γ should be high. Then the mechanism of prompt GRB emission is unlike a classical internal shock model (e.g., Thompson et al. 2007; Ioka et al. 2007).

The observed optical afterglow comes mainly from the prior outflow. For some events, the rising part of the early optical afterglow proceeds until T ∼ 10² s (Molinari et al. 2007), which is difficult to explain with prior emission. In the case of prior emission, the time zero would be shifted T₀ ∼ 10³–10⁴ s before the burst trigger, making the light curve extraordinarily spiky. Furthermore, in most cases, the transition from phase II to III is chromatic, i.e., the optical light curves do not show any break at that epoch (Panaitescu et al. 2006), although there exist a few exceptions (Liang et al. 2007; Grupe et al. 2007; Mangano et al. 2007). Hence, at least in the early epoch, the observed optical afterglow arises from the main outflow component or others, most likely an external shock emission. Indeed, there have been some observational facts that indicate different origins of X-ray and optical afterglows (e.g., Oates et al. 2007; Sato et al. 2007; Urata et al. 2007).³

A possible prediction of the present model is a bright X-ray precursor before the prompt GRB emission. Let us assume that the prior X-ray emission starts at t ∼ 10² s, although its onset time is fairly uncertain (see Section 2). Then, from Equation (1), the X-ray flux in the 2–10 keV band is estimated as f₀(t) ∼ 4 × 10⁻⁹(t/10² s)^−1.2 erg s⁻¹ cm⁻², where α₀ ≈ 1.2 is taken as a typical value and the flux normalization constant A₀ is determined so that f₀ ∼ 1 × 10⁻¹² erg s⁻¹cm⁻² at t ∼ 10⁵ s (Gendre et al. 2008). Such emission will be detected by Monitor of All-sky X-ray Image (MAXI).⁴ The expected event rate should be a few events per year (Suzuki et al. 2008).

However, if the emission starts with t ∼ 10² s, the predicted flux might be large enough to be detected by current instruments like BAT onboard Swift. In order to avoid this problem, the starting time of the emission should be comparable to or later than t ∼ 10³ s so that the peak flux is smaller than the detection limit of the prompt GRB emission monitors. One possibility is off-axis jet emission in the context of the external shock model (Granot & Kumar 2003; Granot 2005). Due to the relativistic beaming effect, the observed X-ray emission is dim as long as the bulk Lorentz factor of the emitting outflow is larger than the inverse of the angle between the emitting matter and the observer's line of sight. This effect shifts the peak time of the X-ray emission toward the later epoch. For an appropriate choice of parameters, we may adjust the starting time (the peak time) of the X-ray emission.

A signal of the onset of the prior emission might also be seen in the optical band. If the prior emission is extremal shock origin, the reverse shock emission might cause a bright optical flash (Sari & Piran 1999). So far, for some events, WIDGET⁵ has given an observational upper limit on the prior optical emission, V > 10 mag, about 750 s before the prompt emission (e.g., Abe et al. 2006). Further observations will constrain the model parameters. In summary, in order to test the model presented in this paper, the search for a signal in the data of sky monitors in the optical and X-ray bands is crucial.