Abstract
The central engine of gamma-ray bursts (GRBs) are believed to eject double-sided ultra-relativistic jets. For an observed GRB, one of the jets, the preceding one, should point toward us, responsible for the prompt gamma-ray and long-term afterglow emission. We discuss the other receding jet, which will give rise to late-time radio re-brightening (RRB) when it becomes non-relativistic (NR) and isotropically radiating. The RRB occurs at a time 5t N R = 2(E j,51 /n) 1/3 yr after the GRB, where t N R is the observed NR time for the preceding jet, E j is the jet energy and n is the ambient medium density, with a flux similar to the preceding jet emission at t N R . We expect the RRB of GRB 030329 1.7 yr after the burst with a flux ∼ 0.6 mJy at 15 GHz. The cases of GRBs 970508 and 980703 have also been discussed. The detection of RRBs, which needs dense monitoring campaign even a few years after a GRB, will be the direct evidence for the existence of double-sided jets in GRBs, and proves the black hole-disk system formations in the cores of progenitors.

1. introduction A standard shock model of gamma-ray burst (GRB) afterglows have now been well established, in which the GRB outflow (jets) drives a relativistic expanding blastwave that sweeps up and heats the GRB ambient medium to produce the long-term X-ray, optical and radio afterglow by synchrotron/inverse-Compton (IC) emission (see Zhang & Mészáros 2004 for recent reviews).

The recent spectroscopy observation of GRB 030329 afterglow (Stanek et al. 2003; Hjorth et al. 2003 ) has definitely proven its association with a type Ic supernova, and also confirmed, as suspected long before, that GRBs, at less the long class, are originated from explosions of massive stars in distant galaxies. The GRB progenitors are believed to form black hole-disk systems in their cores, and produce double-sided jets, which responsible for the GRB emission, along the spin axis of the black hole (Mészáros 2002) . For the observed hard GRB photons to escape freely, preventing from producing electron/positron pairs, the outflows need to be in ultra-relativistic speeds, with Lorentz factor Γ 100. Because of the relativistic beaming effect, the GRB emission is confined in a narrow cone, and can only be observed when the light of sigh is almost on the axis of the jets. Interestingly, it was this beaming effect that made many authors expect the present of "orphan afterglows" (Rhoads 1997; Dalal, Griest, & Pruet 2002; Granot et al. 2002 ) which correspond to off-axis GRBs with afterglows detected when the shock decelerates and without gamma-rays associated.

The previous works usually focus on the emission from the preceding jet (PJ), the one among the two pointing toward us, and neglect that from the other receding jet (RJ), since it always points away from us. However, in this Letter we discuss the emission from the RJ, which will emerge when the shock become non-relativistic (NR) and the radiation becomes isotropic. Because of the light travel time delay, the RJ contribution will overlay above the emission from the already decayed PJ. In the NR phase the afterglow emission has moved to radio frequencies, so the re-brightening from the RJ in NR phase is only expected in late-time radio afterglows. We predict the late-time radio re-brightening (RRB) of GRB 030329 and also discuss the long-term radio observations for two previous GRBs 970508 and 980703. The observation of RRBs will be particularly important due to its straightforward implication of the existence of double-sided jets in GRBs.

2. emission from the receding jet Consider two collimated outflows, jets, that are ejected in opposite directions from the central engine of a GRB. For an observed GRB, the observer should be almost on the jet symmetry axis. We assume the same characteristics for the two jets, such as the half opening angle θ j , the initial Lorentz factor Γ 0 , and the kinetic energy E j /2 (or with equivalent isotropic energy E iso = 2E j /θ 2 j ). Since most afterglow fits seem to favor uniform mediums ), we assume a constant particle density n for both jets.

In the above assumptions we have take the standard view of a homogeneous jet with sharp edge (Rhoads 1999) . Recently, a structured jet model (Mészáros et al. 1998; Dai & Gou 2001; Rossi et al. 2002; Zhang & Mészáros 2002) was proposed, but detailed fits to afterglow data are still needed to determine whether it is consistent with current observations (Granot & Kumar 2003) . Since the standard homogeneous jet model have been well successful in the data fits (e.g., , we only discuss the standard homogeneous jet model here.
Jet dynamics
The jet evolution can be divided into four phases. (1) At the beginning, Γ ≫ 1/θ j , the transverse size of the jet is larger than that of the causally connected region, therefore the jet evolves as if it were a conical section of a spherical relativistic blast wave. The PJ first undergoes a coasting phase with Γ = Γ 0 . (2) After the jet induced shock sweeps up enough medium, the jet kinetic energy is mostly transferred into the shocked medium matter, and the jet begins to decelerate significantly. This happens at the deceleration radius r d = 3E iso /(4πnΓ 2 0 m p c 2 ) at an observer time t d = r d /(2Γ 2 0 c), where a relation dr = 2Γ 2 cdt has been taken for surperluminal motions. At the deceleration phase, the dynamics may be well described by the Blandford & McKee (1976) self-similar solutions, where Γ ∝ r −3/2 ∝ t −3/8 . (3) As the jet continues to decelerate, the Lorentz factor drops to 1/θ j at a time
Jet dynamics
2/3 . After this time the transverse size of causally connected regions exceeds that of the jet, therefore the sphere approximation breaks down, and the jet starts expanding sideways. The dynamical evolution in this stage depends on the degree of sideways expansion. If the jet lateral velocity in the comoving frame equals the local sound speed, the jet will spread quickly with the opening angle growing and Γ dropping exponentially with r (Rhoads 1999) . At this stage, the radius hardly increases and may be regarded as a constant r ≈ r j , and Γ ∝ t −1/2 . (4) Finally the jet goes into NR phase with Γ ∼ 1 at t N R = t j /θ 2 j , where the evolution can be well described by Sedov-Taylor solution
Jet dynamics
. In summary, the evolution is: coasting:
Jet dynamics
(1) From the above simplified dynamical relation, the NR time and radius, defined as where Γ = 1, are easy to be calculated as
Jet dynamics
(2) They are independent of Γ 0 and θ j . With typical values E j = 10 51 E j,51 ergs and n = 1 cm −3 (e.g., Frail et al. 2001; Panaitescu & Kumar 2001) , we have
Jet dynamics
Note that this t N R refers to the PJ. As for the RJ, it also transit to NR phase at a radius r N R . However, due to the light travel time the observer time is delayed by a time 2r N R /c, so the observed NR time for the RJ is
Jet dynamics
5 times that of the PJ 2 . The lateral velocity of jets is not clear, and the jet may still be hardly expanding sideways. If we assume the extreme case that the lateral velocity is zero, then the jet remains conical geometry and continues to evolve as if it were a conical section of a spherical relativistic blast wave after Γ < 1/θ j . In this case, the NR time for the PJ is t N R = (1/2)(3E j /2πm p c 5 nθ
Jet dynamics
−1 , and the corresponding radius is r N R = 2ct N R , therefore for the RJ the NR time is t
Jet dynamics
1/3 yr. These may be regarded as upper limits, while the values in eqs. (2-4) for the fastest spreading jets as lower limits.
Non-relativistic phase emission
For the details of afterglow emission in NR phase, the readers might refer to Frail, Waxman & Kulkarni (2000) , Livio & Waxman (2000) and Waxman (2004) , we here only give brief introduction.
Non-relativistic phase emission
Consider also the PJ pbranch@example.com. After t N R , the evolution is described by the spherical NR self-similar evolution, with the PJ radius given by r = r N R (t/t N R ) 2/5 , and the swept-up particle number is N e ≃ (4/3)πnr 3 . Simply assume that the shocked material forms a thin and uniform shell, with the width ∆ = r/η ≪ r and the postshock thermal energy density U = E j /(4πr 2 ∆). The electron and magnetic field energy densities are assumed to be constant fraction ǫ e and ǫ B , respectively, of the postshock thermal energy density U . The electrons are believed to be accelerated to form a power law distribution, dn e /dγ e ∝ γ GHz,
Non-relativistic phase emission
where ǫ e,−1 = ǫ e /10 −1 , and ǫ B,−1 = ǫ B /10 −1 . The peak specific luminosity is L νm = N e P/ν m where P = (4/3)σ T cγ 2 m (B 2 /8π) is the synchrotron power of an electron with γ m . The observed radio frequency is usually beyond ν m at NR stage, the specific luminosity is therefore L ν = L νm (ν/ν m )
Non-relativistic phase emission
(1−p)/2 ∝ t 21/10−3p/2 , and given by
Non-relativistic phase emission
B,−1 n 3/4 E j,51
Non-relativistic phase emission
where p = 2 has been taken. The NR light curve flattens, compared with the decay, ∝ t −p , in the spreading phase (Rhoads 1999; Sari, Piran & Halpern 1999) .
Non-relativistic phase emission
As for the RJ, when it is still relativistic the emission is always beamed away from us and hence invisible. At a time t RJ N R after the GRB, the RJ becomes non-relativistic and the emission becomes isotropic. The observed RJ luminosity at this time should be comparable to that of the PJ at t N R , because of the same characteristics of the PJ and RJ. If neglect the NR expansion of jets with respect to the light speed, the time delay between the emissions from the PJ and RJ may be roughly fixed as 2r/c ≈ 2r N R /c = 4t N R (for r is not ≫ r N R ). The RJ specific luminosity, then, is written as, for
Non-relativistic phase emission
A schematic plot in figure 1 is given for both PJ and RJ radio light curves. After rapid increasing, the RJ emission reaches the peak at around t RJ N R , and then decreases as a delayed light curve of the NR-phase PJ, but showing a steeper decline in the logarithmic plot.
Non-relativistic phase emission
3. observation 3.1. GRB 970508 and GRB 980703
Non-relativistic phase emission
As discussed above, if a GRB can be monitored for a long time, e.g., a few years in radio bands, the RRB may be expected to be detected. Yet, there are till now two GRBs that have been reported in the literatures with radio monitoring longer than one year, i.e., GRB 970508 and GRB 980703 (Frail, Waxman & Kulkarni 2000; Berger, Kulkarni & Frail 2001; Frail et al. 2003) . Frail, Waxman & Kulkarni (2000) report an extensive monitoring of the radio afterglow of GRB 970508, lasting 450 days after the burst. In the data analysis they found that the spectral and temporal radio behavior indicate a transition to NR expansion at t N R ≈ 100 d after the burst, We, therefore, should expect the RRB at about 500 d. However, this is unfortunately behind the last observation of this burst.
Non-relativistic phase emission
GRB 980703 was even monitored ∼ 10 3 d long, as shown in figure 1 of Frail et al. (2003) . Two obvious flattening appear in the radio light curves: while the late-time transition to a constant flux is due to the host galaxy contribution, the earlier flattening at ≈ 40 d is due to the transition into NR expansion (Berger, Kulkarni & Frail 2001) . Modelling the Γ evolution has also inferred a similar value of t N R = 30 − 50 d . Therefore, the expected RRB should arise at ∼ 200 d, and then after another period of 200 d the radio flux should decline to a value similar to just before the RRB. At first glance, the radio light curves, somewhat more pbranch@example.com and 4.8 GHz, seem to show only decays into constant fluxes. However, we notice that there is no observation data collected between 210 d and ∼ 400 d (see figure 1 of Frail et al. 2003) , therefore the RRB may be missed in the observation.
Prediction for GRB 030329
GRB 030329 is a nearby burst (z = 0.1685). Besides its remarkable supernova signature, it has a bright afterglow at all wavelengths. Berger et al. (2003) have reported its bright radio afterglow up to ∼ 70 d after the burst. Since no radio flattening (due to transition to NR evolution) appears yet 3 , we should calculate t N R using E j /n derived from available data.
Prediction for GRB 030329
The early breaks (∼ 0.5 d) in the R band (Price et al. 2003 ) and X-ray (Tiengo et al. 2003 ) light curves infer a small jet opening angle, leading to a small jet-corrected energy release in GRB 030329, which is more than one order of magnitude below the average value around which most GRBs are narrowly clustered Panaitescu & Kumar 2001 ). Two models have been proposed to accommodate this problem: a refreshed-shock model with 10 times more energy injected later (Granot, Nakar & Piran 2003) ; and a two-component model with most energy in a wider jet component . Both models have the same result: the kinetic energy corresponding to the late-time radio afterglow is typical for all GRBs. Based mainly on the early multi-frequency data, a typical value for medium density, n = 1 cm −3 , was also derived by Willingale et al. (2004) . Therefore, the estimated value of t N R should be typical as in eq. (3), with t N R ∼ 130 d. In fact, the radio angular size measurement of this burst shows relativistic expanding 83 d after the burst (Taylor et al. 2004) , consistent with the estimation.
Prediction for GRB 030329
In details, Berger et al. (2003) have inferred E j,51 /n = 0.14ν −1/2 c,13 from a snapshot spectrum at t j ≈ 10 d, where
Prediction for GRB 030329
13 Hz is the cooling frequency. This parameter value leads to a too early NR phase, t N R ≈ 68 d (from eq. 3), conflicted with the superluminal expansion (Taylor et al. 2004) . However, in the parameter derivation, the assumption of jet break time at t j ≈ 10 d might be incorrect, because the millimeter observations show that the light curves have already steepened at ≈ 5 d at frequencies of 100 and 250 GHz (Sheth et al. 2003) , implying that t j 5 d. Moreover, the angular size evolution can also give constraints to the model parameters. For GRB 030329, Granot, Ramirez-Ruiz & Loeb (2004) derive E j,51 /n ∼ 0.8 (see also Oren, Nakar & Piran 2004 ) from the observation by Taylar et al. (2004) . This further yields a later transition at t N R ∼ 120 d.
Prediction for GRB 030329
We would like to take the more reasonable value t N R ≈ 120 d (It should be noticed that this might still be the lower limit since the most rapid sideways spreading has been assumed), and hence the expected RRB should arise at ∼ 1.7 yr after GRB 030329, with the fluxes comparable to that at the time t N R . Assuming the same temporal slope for late times, an extrapolation of the available radio light curves to ∼ 120 d gives a flux ∼ 0.6 mJy at 15 GHz, or ∼ 0.3 mJy at 44 GHz. Dense monitoring campaign around ∼ 1.7 yr after GRB 030329 is required to obtain a well-observed RRB profile.
summary and discussion
In this work we suggest that RRBs are common for GRBs with double-sided jets, which come from the RJ transition to NR evolution. If assuming the same properties for both PJ and RJ and also for the ambient mediums in both sides, the RRB would arise at a time t RJ N R = 5t N R = 1.8(E j,51 /n) 1/3 yr after the burst, with a flux comparable to that at the time t N R . After t RJ N R , the afterglow behaves as a delayed emission of that behind t N R , with the time lag of 4t N R (eq. 7). Unfortunately, no radio data could be collected when RRBs occur for GRB 970508 and GRB 980703, two longest observed GRBs so far. We suppose the RRB of GRB 030329 around 1.7 yr 3 Although the X-ray afterglow shows a flattening around day 37 (Tiengo et al. 2004) , we believe this is not relevant to the relativistic to NR transition of the radio jet, because of no simultaneous radio flattening , and also conflicted with the still superluminal expansion in the radio angular size measurement (Taylor et al. 2004 ). The X-ray flattening may need other explanations, e.g. an IC component; NR transition for the narrow jet in two-component model .
summary and discussion
after the burst with ∼ 0.6 (0.3) mJy at 15 (44) GHz, and urge dense monitoring campaign during that times.
summary and discussion
There has been growing evidence for collimated jets in GRBs over the past several years, which is coming mainly from observations of achromatic breaks in the afterglow light curves (e.g., Kulkarni et al. 1999; Stanek et al. 1999) . However, there are still other explanations for the light curve breaks, for example, the transition from the relativistic to NR phase of the blast wave at a few days due to highly dense medium (Dai & Lu 1999; Wang, Dai & Lu 2000) ; the effects of IC scattering flattening or steepening the light curves (Wei & Lu 1998 ; a sudden drop in the external density (Kumar & Panaitescu 2000) ; a break in the energy spectrum of radiating electrons (Li & Chevalier 2001) . However, since RRBs are only associated with collimated relativistic outflows, the detection of RRBs would provide straightforward evidences of doublesided jets in GRBs, and prove the black hole-disk system formations in the cores of progenitors. This makes RRB observations significantly important, which are encouraged in future Swift era.
summary and discussion
Z. Li thanks R. F. Shen for valuable discussions. This work was supported by the National 973 Project and the Special Funds for Major State Basic Research Projects. -The schematic log-log plot for the radio light curve at ν > νm. The solid and dashed lines correspond to the contribution from the PJ and RJ, respectively. The time scalings are marked, assuming p = 2. After a steep decline in the spreading phase, the PJ light curve flatten at t NR . The RJ emission rapidly increases to a peak around t RJ NR = 5t NR , and then declines following the way of the PJ unless a time delay of 4t NR , but exhibiting steep light curve in the log-log plot. The dashed-dot line indicates Lν (t NR ) = L RJ ν (t RJ NR ).

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