11institutetext: Department of Astronomy, Nanjing University, Nanjing 210093,
China

# On the afterglow from the receding jet of gamma-ray burst

Xin Wang  Y. F. Huang<EMAIL_ADDRESS>S. W. Kong

(Received month day 2009 / Accepted month day 2009)

According to popular progenitor models of gamma-ray bursts, twin jets should
be launched by the central engine, with a forward jet moving toward the
observer and a receding jet (or the counter jet) moving backwardly. However,
in calculating the afterglows, usually only the emission from the forward jet
is considered. Here we present a detailed numerical study on the afterglow
from the receding jet. Our calculation is based on a generic dynamical
description, and includes some delicate ingredients such as the effect of the
equal arrival time surface. It is found that the emission from the receding
jet is generally rather weak. In radio bands, it usually peaks at a time of
$t\geq 1000$ d, with the peak flux nearly 4 orders of magnitude lower than the
peak flux of the forward jet. Also, it usually manifests as a short plateau in
the total afterglow light curve, but not as an obvious rebrightening as once
expected. In optical bands, the contribution from the receding jet is even
weaker, with the peak flux being $\sim 23$ magnitudes lower than the peak flux
of the forward jet. We thus argue that the emission from the receding jet is
very difficult to detect. However, in some special cases, i.e., when the
circum-burst medium density is very high, or if the parameters of the receding
jet is quite different from those of the forward jet, the emission from the
receding jet can be significantly enhanced and may still emerge as a marked
rebrightening. We suggest that the search for receding jet emission should
mostly concentrate on nearby gamma-ray bursts, and the observation campaign
should last for at least several hundred days for each event.

###### Key Words.:

gamma rays: bursts — ISM: jets and outflows — stars: neutron

##  1 Introduction

Thanks to the discovery of X-ray, optical and radio afterglows of gamma-ray
bursts (GRBs), it is now clear that most GRBs are situated at cosmological
distances (Costa et al. 1997; van Paradijs et al. 1997; Frail et al. 1997). A
lot of progresses have been achieved during the past decade (Piran 2004;
Mészáros 2006). Especially, through the detection of GRB 030329, the
association of long GRBs with supernovae is firmly established (Hjorth et al.
2003), which strongly supports the collapsar model as the energy mechanism for
long GRBs (Woosley 1993; MacFadyen & Woosley 1999). Theoretically, the
collapse of a massive star will most likely give birth to a black hole,
surrounded by a temporal accretion disk. It is a common sense that the
accretion system will produce double-sided jets (MacFadyen & Woosley 1999;
Aloy et al. 2000; Rhoads 1999; Mészáros 2002). The GRB can be observed only
when our line of sight is right on one of the two jets. The collimation of GRB
ejecta can be tested observationally, through various beaming effects, such as
the achromatic break in GRB afterglow light curves (Sari et al. 1999; Liang et
al. 2008), the polarization in both the main burst phase and the afterglow
phase (Lazzati 2006), the predicted existence of orphan afterglows (Rhoads
1997; Huang et al. 2002; Granot & Loeb 2003), and the energy crisis already
noted in some GRBs (Frail et al. 2001). In fact, more and more observational
evidences have been accumulated today, supporting the idea that many GRB
ejecta might be highly collimated.

Current studies on the beaming effects are mostly concentrated on the emission
from the forward jet, i.e., the jet moving toward the observer. The emission
from the receding jet (or the counter jet) is generally omitted. It is
interesting to note that this ingredient recently has been studied by a few
authors (Granot & Loeb 2003; Li & Song 2004). By some simple analytical
derivations, Li & Song (2004) argued that the emission from the receding jet
can be detected in a few cases in the non-relativistic phase of GRB
afterglows. However, previous studies did not consider some important effects,
such as the action of the equal arrival time surface (EATS). Recently, Zhang &
MacFadyen (2009) presented a two-dimensional simulation of GRB outflow. The
emission from the receding jet has also been included in their calculations,
but they did not investigate the effects of various parameters on the receding
jet component.

In this paper, we will present our detailed numerical investigation on the
emission from the receding jet of GRB in the deep Newtonian stage. Although
the GRB jet may be complicatedly structured (Mészáros et al. 1998; Kumar &
Granot 2003; Huang et al. 2004), and the circum-burst environment may be wind
medium and even associated with some complex density variations (Mészáros et
al. 1998; Chevalier & Li 2000; Gou et al. 2001; Wu et al. 2004), here we will
only consider the simplest situation, i.e, the homogeneous double-sided jets
expanding into a homogeneous interstellar medium, which is favored by some
recent fits (Huang et al. 2000a; Yost et al. 2003).

The structure of our paper is organized as follows. §2 is mainly a review of
the dynamics and radiation model we used in our calculations. In §3 we present
the numerical results, together with our tentative explanations. §4 is our
conclusion and discussion.

##  2 Model Description

In the afterglow phase, the GRB ejecta expands into the interstellar medium
(ISM) and is decelerated continuously, giving rise to a strong external shock.
The swept-up electrons are accelerated by the blastwave, producing the
afterglow mainly through synchrotron radiation. In radio bands, the shell is
no longer optically thin, so that the synchrotron self-absorption should be
considered. In our study, we will use the simplified dynamical equations
suggested by Huang et al. (1999, 2000b), which is consistent with the self-
similar solution of Blandford & McKee (1976) in the ultra-relativistic phase,
and is consistent with the Sedov solution (Sedov 1969) in the non-relativistic
phase. The beaming effects (Rhoads 1997, 1999) can also be conveniently
simulated in this way. Here, for completeness, we first describe the dynamics
and the radiation process briefly.

###  2.1 Hydrodynamical Evolution

In our description, $t$ is the photon arrival time measured in the lab frame;
$R$ is the radial coordinate measured in the burst frame relative to the
initiation point; $m$ is the rest mass of the swept-up medium; $\theta$ is the
half-opening angle of the ejecta; $\gamma$ is the bulk Lorentz factor of the
moving material; $p$ is the electron distribution index which is typically
between 2 and 3; $n$ is the number density of ISM; $\xi_{\rm e}$ and
$\xi^{2}_{\rm B}$ are the energy equipartition factors for electrons and the
comoving magnetic field. We further denote the initial values of the rest
mass, the isotropic equivalent energy, the Lorentz factor and the half-opening
angle of the ejecta as $M_{\rm ej},E_{\rm 0,iso},\gamma_{0},\theta_{\rm j}$,
respectively.

The overall dynamical evolution of the GRB ejecta can be depicted by

$\displaystyle\frac{\mathrm{d}R}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle\beta c\gamma\left({\gamma+\sqrt{\gamma^{2}-1}}\right),$ (1)
$\displaystyle\frac{\mathrm{d}m}{\mathrm{d}R}$ $\displaystyle=$ $\displaystyle
2\pi\left({1-\cos\theta}\right)R^{2}nm_{\rm p},$ (2)
$\displaystyle\frac{\mathrm{d}\theta}{\mathrm{d}t}$ $\displaystyle=$
$\displaystyle\frac{c_{\rm s}}{R}\left({\gamma+\sqrt{\gamma^{2}-1}}\right),$
(3) $\displaystyle\frac{\mathrm{d}\gamma}{\mathrm{d}m}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}-1}{M_{\rm ej}+\varepsilon
m+2(1-\varepsilon)\gamma m},$ (4)

where $\beta=\sqrt{1-1/\gamma^{2}}$, $c$ is the speed of light, and $c_{\rm
s}$ is the comoving sound speed, which can be calculated by $c_{\rm
s}^{2}=\hat{\gamma}(\hat{\gamma}-1)(\gamma-1)c^{2}/\left[1+\hat{\gamma}(\gamma-1)\right]$
with $\hat{\gamma}\approx(4\gamma+1)/({3\gamma})$ being a reasonable
approximation for the adiabatic index. In Equation (4), $\varepsilon$ is the
radiative efficiency. In the extreme case, $\varepsilon=0$ means adiabatic
condition and $\varepsilon=1$ refers to highly radiative situation. Note that
in realistic case, $\varepsilon$ should evolve gradually from 1 to 0, in about
several hours.

Equations (1) — (4) is a convenient description of GRB afterglow dynamics that
is applicable in both the initial ultra-relativistic phase and the late
Newtonian phase.

###  2.2 Radiation Process

Basically, we assume that the shock-accelerated electrons follow a power-law
distribution according to their energies, $\mathrm{d}N^{\prime}_{\rm
e}/{\mathrm{d}\gamma_{\rm e}}\propto\gamma_{\rm e}^{-p}$, However, to ensure
that the calculation in the deep Newtonian phase is correct, we need to modify
the basic distribution function as $\mathrm{d}N^{\prime}_{\rm
e}/{\mathrm{d}\gamma_{\rm e}}\propto\left(\gamma_{\rm e}-1\right)^{-p}$ (Huang
& Cheng 2003). The minimum and maximum Lorentz factors of electrons can be
calculated as $\gamma_{\rm e,\min}=\xi_{e}(\gamma-1)m_{\rm p}(p-2)/[m_{\rm
e}(p-1)]+1$ and $\gamma_{\rm e,\max}=\sqrt{6\pi e/\left(\sigma_{\rm
T}B^{\prime}\right)}\approx 10^{8}(B^{\prime}/1{\rm G})^{-1/2}$, where
$B^{\prime}$ is the comoving magnetic field strength, $m_{\rm p}$ and $m_{\rm
e}$ are masses of proton and electron, respectively. As usual, we assume that
the energy ratio of magnetic field with respect to internal energy is
$\xi^{2}_{\rm B}$, so that the energy density of magnetic field is $B^{\prime
2}/(8\pi)=\xi^{2}_{\rm
B}\left(\hat{\gamma}-1\right)^{-1}(\hat{\gamma}\gamma+1)(\gamma-1)nm_{\rm
p}c^{2}$.

The cooling of electrons due to synchrotron radiation will lead to a steep
distribution function above a critical Lorentz factor, $\gamma_{\rm c}$. The
expression for $\gamma_{\rm c}$ can be derived as $\gamma_{\rm c}=6\pi m_{\rm
e}c/\left(\sigma_{\rm T}\gamma B^{\prime 2}t\right)$, where $\sigma_{\rm T}$
is the Thompson scattering cross section (Sari et al. 1998). Considering all
the above ingredients, we finally use the following electron distribution
function in our calculations (Huang & Cheng 2003):
1\. $\gamma_{\rm c}\leq\gamma_{\rm e,\min}$,

$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left\\{\begin{array}[]{l}\left(\gamma_{\rm e}-1\right)^{-2}\hskip
22.76228pt(\gamma_{\rm c}\leq\gamma_{\rm e}<\gamma_{\rm e,\min}),\\\
\left(\gamma_{\rm e}-1\right)^{-(p+1)}\hskip 9.53186pt(\gamma_{\rm
e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm e,\max});\\\ \end{array}\right.$ (5)

2\. $\gamma_{\rm e,\min}<\gamma_{\rm c}\leq\gamma_{\rm e,\max}$,

$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left\\{\begin{array}[]{l}\left(\gamma_{\rm e}-1\right)^{-p}\hskip
22.1931pt(\gamma_{\rm e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm c}),\\\
\left(\gamma_{\rm e}-1\right)^{-(p+1)}\hskip 9.67383pt(\gamma_{\rm
c}<\gamma_{\rm e}\leq\gamma_{\rm e,\max});\\\ \end{array}\right.$ (6)

3\. $\gamma_{\rm c}>\gamma_{\rm e,\max}$,

$\frac{\mathrm{d}N^{\prime}_{\rm e}}{\mathrm{d}\gamma_{\rm
e}}\propto\left(\gamma_{\rm e}-1\right)^{-p}\hskip 15.6491pt(\gamma_{\rm
e,\min}\leq\gamma_{\rm e}\leq\gamma_{\rm e,\max}).$ (7)

In the comoving frame, the synchrotron radiation power at $\nu^{\prime}$ is
(Rybicki & Lightman 1979)

$P^{\prime}(\nu^{\prime})=\frac{\sqrt{3}e^{3}B^{\prime}}{m_{\rm
e}c^{2}}\int_{\min\left(\gamma_{\rm e,\min}\ ,\ \gamma_{\rm
c}\right)}^{\gamma_{\rm e,\max}}{\left(\frac{\mathrm{d}N^{\prime}_{\rm
e}}{\mathrm{d}\gamma_{\rm e}}\right)}\
F\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\rm e}}\right)d\gamma_{\rm e},$ (8)

with $F(x)=x\int_{x}^{+\infty}{K_{5/3}(k)\,\mathrm{d}k}$ being the Bessel
function and $\nu^{\prime}_{\rm e}=3\gamma^{2}_{\rm e}eB^{\prime}/\left(4\pi
m_{\rm e}c\right)$ being the characteristic emission frequency (Shu 1991;
Longair 1992). To calculate the radio afterglows, we must consider the
synchrotron self-absorption. The optical depth of synchrotron self-absorption
can be obtained as

$\tau_{\nu^{\prime}}=\frac{\sqrt{3}e^{3}B^{\prime}}{8\pi m^{2}_{\rm
e}c^{2}\nu^{\prime 2}}\int_{\min\left(\gamma_{\rm e,\min}\ ,\ \gamma_{\rm
c}\right)}^{\gamma_{\rm e,\max}}{(q+2)\left(\frac{\mathrm{d}n^{\prime}_{\rm
e}}{\mathrm{d}\gamma_{\rm e}}\right)\frac{1}{\gamma_{\rm e}}}\
F\left(\frac{\nu^{\prime}}{\nu^{\prime}_{\rm e}}\right)d\gamma_{\rm e},$ (9)

where $\mathrm{d}n^{\prime}_{\rm e}/\mathrm{d}\gamma_{\rm e}$ denotes the
column density distribution of electrons measured in the comoving frame on the
line of sight; $q$ is the electron power-law distribution index which varies
from $2$ to $p+1$ for fast-cooling and from $p$ to $p+1$ for slow-cooling. The
synchrotron self-absorption will affect the radiation by a reduction-factor
(Waxman et al. 1998)

$f(\tau)=\frac{1-e^{-\tau_{\nu^{\prime}}}}{\tau_{\nu^{\prime}}}.$ (10)

Let us define the Doppler-factor as
$D=\left[\gamma\left(1-\beta\mu\right)\right]^{-1}$ (Mészáros 2006), where
$\mu=\cos\Theta$ and $\Theta$ is the angle between the velocity of the
emitting material and the line of sight. Also we denote the viewing angle as
$\theta_{\rm obs}$. Then the observed frequency is $\nu=D\nu^{\prime}/(1+z)$,
and the observed flux density from a point-like source is

$F_{\nu}=\frac{(1+z)D^{3}}{4\pi d_{\rm
L}^{2}}f(\tau)P^{\prime}\left[(1+z)D^{-1}\nu\right],$ (11)

where $d_{\rm L}$ is the luminosity distance. Finally, we can integrate the
flux density over the EATS (Waxman 1997; Sari 1998) determined by

$t_{\rm obs}=(1+z)\int\frac{\mathrm{d}R}{\beta\gamma cD}\equiv\rm{const}.$
(12)

##  3 Numerical Results

In this section, we present our numerical results concerning the emission from
the receding jet. First of all, for simplicity, we assume that the twin jets
have the same characteristics, i.e., the same initial energy, opening angle,
initial Lorentz factor, and the circum-burst ISM density. We also assume that
the microphysics shock parameters ($p$, $\xi_{\rm e}$, $\xi^{2}_{\rm B}$) are
the same for the receding and forward blastwaves. For convenience, we define a
set of parameter values as the “standard” condition: $n=1/{\rm cm}^{3}$,
$E_{\rm 0,iso}=10^{53}{\rm ergs}$, $\theta_{\rm j}=0.1$, $\varepsilon=0$,
$\xi_{\rm e}=0.1$, $\xi^{2}_{\rm B}=0.01$, $p=2.5$, $\theta_{\rm obs}=0$, and
$\gamma_{0}=300$. These values are typical in the study of GRB afterglows. For
redshift, we adopt the value of $z=0.1$ (which corresponds to $d_{\rm L}=454$
Mpc according to the popular cosmology model, Wright 2006).

Firstly we illustrate the evolution of the Lorentz factors of the twin jets in
Fig. 1. Note that the X-axis is observers’ time. For the observer, the
dynamical evolution of the receding jet is quite different from that of the
forward jet, especially in the relativistic phase. We see that in a rather
long time ($t\sim 50$ d), $\gamma$ of the receding jet remains almost
constant. This is due to the time delay induced by the long distance between
the twin jets. It also implies that the emission from the receding jet will be
very weak in this period, since it is highly beamed backwardly. At the
observers’ time of $t\sim 340$ d, the Lorentz factor of the receding jet is
still more than 10, while the forward jet’s Lorentz factor has already
decreased to less than 1.1 .

In Fig. 2, we show some examples of the equal arrival time surfaces (EATSes)
at three moments. As expected, at any particular moment, the typical radius of
the surface is much larger for the forward jet branch as compared with that
for the receding jet branch. Also, we notice that the curvature of the two
branches is quite different. Generally, the EATS is much flatter on the
receding jet. Another interesting feature is that the area of the EATS on the
forward branch is much larger than that of the corresponding receding branch.

Fig. 3 shows the radio and optical afterglow light curves under the “standard”
condition (thick lines). Here, the thick dotted line corresponds to emission
from the forward jet, the thick dashed line corresponds to emission from the
receding jet, and the thick sold line is the total light curve. Under the
“standard” condition, for the forward jet, the afterglow light curve (the
dotted line) becomes slightly flattened in the non-relativistic phase. It is
consistent with previous results in the deep Newtonian phase (Huang & Cheng
2003). Also it can be seen that the receding jet really can contribute a
significant portion in the total emission at very late stage. The role played
by the receding jet is reasonably more important in the radio band than in the
optical band. However, the dashed component is generally not very strong, so
that it can only lead to a plateau in the total light curve, but not an
obvious rebrightening or a marked peak as expected by Li & Song (2004).
Interestingly, our result is consistent with the simulation of Zhang &
Macfadyen (2009). We believe that the discrepancy between our numerical result
and Li & Song’s analytical result mainly comes from the effect of the EATS.
Below, we will give some detailed analyses on this point. Additionally, it
should be noted that in the radio band, the peak flux of the receding
component is about 4 orders of magnitude weaker than that of the forward
component. It essentially means that the receding component is very weak, and
is very difficult to detect. In the optical band, the condition is even more
awkward. The peak flux of the receding component is about 23 magnitudes dimmer
than that of the forward component in R band. Even comparing with the flux of
the forward jet at the jet break time, it is still 16 — 17 magnitudes weaker.
So, in optical band, it is even much more difficult to observe the receding
jet component.

According to Li & Song (2004), the time when the receding jet becomes notably
visible ($t_{\rm NR}^{\rm RJ}$) is relevant to the time when the forward jet
enters the non-relativistic phase ($t_{\rm NR}$), i.e.,

$t_{\rm NR}^{\rm RJ}=t_{\rm NR}+\frac{2r_{\rm NR}}{c},$ (13)

where $r_{\rm NR}$ is the radius of the forward jet at $t_{\rm NR}$. In the
standard frame work (Blandford & Mckee 1976; Rhoads 1999), the sphere-like
phase of a highly collimated GRB ejecta ends at the so called jet break time
determined by $\gamma_{\rm j}=1/\theta_{\rm j}$, with the shock radius being
$r_{\rm j}=\left(3E_{\rm 0,iso}\theta_{\rm j}^{2}/\left[{4\pi nm_{\rm
p}c^{2}}\right]\right)^{1/3}$. After the sphere-like phase, the jet spreads
laterally at the co-moving sound speed $c_{\rm s}$ so that we have
$\gamma\propto t^{-1/2}$ and $r_{\rm NR}\approx r_{\rm j}$ (Rhoads 1999). Then
finally we obtain

$t_{\rm NR}=\frac{1}{2c}\left(\frac{3E_{\rm 0,iso}\theta_{\rm j}^{2}}{4\pi
nm_{\rm p}c^{2}}\right)^{1/3},\\\ \qquad t_{\rm NR}^{\rm RJ}=5t_{\rm NR}.$
(14)

Adopting the standard values of our parameters, Equation (14) yields $t_{\rm
NR}\approx 104$ d and $t_{\rm NR}^{\rm RJ}\approx 520$ d. After correcting for
the cosmological time dilation ($z=0.1$), we get the corresponding observers’
time of $t_{1}=(1+z)\ t_{\rm NR}\approx 114$ d and $t_{3}=(1+z)\ t_{\rm
NR}^{\rm RJ}\approx 572$ d. In fact, in Fig. 2, the EATSes for these two
moments have been displayed. So, according to Li & Song’s suggestion, the
contribution from the receding jet should peak at $t_{3}\approx 572$ d. In our
Fig. 3, for the “standard” condition, the peak is postponed to $t_{\rm
peak}\sim 1140$ d for 8.46 GHz, and to $t_{\rm peak}\sim 1700$ d in R band.
So, the EATS effect and the deceleration of the external shock can lead to
some subtle difference between the analytical results and the numerical
results. Actually, Zhang & MacFadyen’s numerical results have clearly shown
that the observers’ time does not equal to the burst frame time at $t_{\rm
NR}$ (Zhang & MacFadyen 2009). Unfortunately, in previous analysises it is
usually assumed that these two times are equal.

Another reason that suppresses the rebrightening of the receding jet is as
follows. According to Li & Song’s analysis, at the observers’ time $t_{3}$,
the receding jet should be at the radius of $r_{\rm NR}$. However, from our
Fig. 2, we see that the typical radius of the EATS at $t_{3}$ on the receding
jet is much smaller than the radius of the forward jet at $t_{1}$. The reason
is again due to the EATS effect. It means that the receding jet still does not
decelerate enough at $t_{3}$ (actually, the bulk Lorentz factor is still
3.95), and its emission is still mainly directed forwardly (not backwardly
toward the observer). Additionally, Fig. 2 shows clearly that the area of the
receding jet at $t_{3}$ (corresponding to $t_{\rm NR}^{\rm RJ}$ ) is much
smaller than that of the forward jet at $t_{1}$ (corresponding to $t_{\rm
NR}$). So, the number of electrons involved in the radiation process is
typically much smaller on the receding jet at $t_{\rm NR}^{\rm RJ}$, as
compared to that on the forward jet at $t_{\rm NR}$. Due to the above reasons,
the contribution from the receding jet is naturally much weaker than that
deduced from $L_{\rm\nu}^{\rm RJ}(t)\approx L_{\rm\nu}(t-4t_{\rm NR}),\ (t\geq
t_{\rm NR}^{\rm RJ})$ (Equation (7) in Li & Song (2004) ).

However, although the receding jet emission is generally very weak in our
“standard” condition, we guess that in some special cases it still can be
enhanced. Obviously, a denser environment will help to decelerate the jet more
quickly, thus lead to a smaller $t_{\rm peak}$ and a higher intensity. In Fig.
3, we have also plotted in thin lines our numerical results for a double-sided
jet that locates in a dense circum-burst medium ($n=1000/{\rm cm}^{3}$). Note
that other parameters involved here are the same as the “standard” case.
Encouragingly, in Fig. 3(a) we see that the peak time of the receding jet can
be as early as $t_{\rm peak}\sim 150$ d, with the peak flux as large as a few
mJy in radio band (i.e., only several times less than the peak level of the
forward jet). In Fig. 3(b), the optical contribution from the receding jet is
still very weak, with the peak flux being about 28m.

In Fig. 4, we plot the afterglow light curves in more radio and
optical/infrared bands. Generally speaking, $t_{\rm peak}$ is about 1140 d in
radio bands and is about 1700 d in optical bands. Such a difference in the
peak time is insignificant, considering that the frequency difference between
radio and optical wavelengths is really huge. We notice that $t_{\rm peak}$
almost remains the same from radio to X-ray bands in Fig. 7 of Zhang &
MacFadyen (2009). Thus our results are roughly consistent with Zhang &
MacFadyen’s. Another interesting conclusion that can be drawn from our Figs. 3
and 4 is that at lower frequency, the relative intensity of the receding jet
component (its peak flux), as compared with the peak of the forward jet
component, becomes stronger. Such a tendency can also be roughly seen in Fig.
7 of Zhang & MacFadyen (2009).

Fig. 5 illustrates the effects of some parameters ($n$, $E_{\rm 0,iso}$,
$\theta_{\rm j}$, and $\varepsilon$) on the receding jet component in the
afterglow light curve. Fig. 5(a) shows that the circum-burst medium density
($n$) affects the peak time ($t_{\rm peak}$) of receding jet dramatically. A
larger number density usually leads to a smaller $t_{\rm peak}$. The strength
of the receding jet component is also obviously enhanced. It again hints that
the receding jet component is most likely detectable in a dense environment.
Similarly, the initial kinetic energy ($E_{\rm 0,iso}$) also affects $t_{\rm
peak}$ significantly, with larger $E_{\rm 0,iso}$ corresponding to a larger
$t_{\rm peak}$ (Fig. 5(b)). The effect of the initial jet opening angle
($\theta_{\rm j}$) on $t_{\rm peak}$ can also be clearly seen in Fig. 5(c). It
should be further noted that the receding jet component is more marked when
the opening angle is smaller. In Fig. 5(d), we can observe an obvious
rebrightening when the radiation efficiency ($\varepsilon$) is large. However,
in realistic case, $\varepsilon$ is unlikely to be so large. Actually, at such
late stages, the external shock should be adiabatic, so that $\varepsilon$
should be nearly zero.

In Fig. 5(d), we also plot the radio afterglow light curves for double-sided
jets under some special physical assumptions. The dash-dotted line is plotted
by assuming that both the forward jet and the receding jet do not experience
any lateral expansion. Since the deceleration of the jets is much slower in
this case, we see that the receding jet component emerges much later and is
also much less obvious as compared with our “standard” case. The dotted line
is plotted by assuming a much smaller initial Lorentz factor
($\gamma_{0}=30$), which may correspond to the so called failed GRBs (Huang et
al. 2002). The receding jet component emerges slightly earlier as compared
with the solid line, but its role becomes less significant correspondingly.

Fig. 6 illustrates the effects of other four parameters ($\xi_{\rm e}$,
$\xi_{\rm B}^{2}$, $p$, and $\theta_{\rm obs}$) on the receding jet component.
Generally speaking, a larger $\xi_{\rm e}$ and/or $\xi_{\rm B}^{2}$ can
enhance the receding jet component markedly. On the other hand, although $p$
has an important influence on the overall afterglow light curve, its impact on
the relative strength of the receding jet component is not significant. Again,
note that in all the cases, the contribution from the receding jet only
emerges as a plateau, but not as any obvious rebrightening. In Fig. 6(d), when
the observing angle ($\theta_{\rm obs}$) increases, the forward jet component
becomes weaker, while the receding jet component becomes stronger. It is in
good accord with our expectation (also see Granot & Loeb 2003). However, the
contribution from the receding jet still generally plays a minor role in the
total afterglow light curve. Additionally, for off-axis twin jets, the GRB
from the forward jet is un-observable, so that even the afterglow from the
forward jet itself (i.e., the orphan afterglow) is difficult to observe. Note
that in Fig. 6(d), when $\theta_{\rm obs}=\pi/2$ (i.e., the thick solid line),
the contribution from the receding jet and the forward jet are actually equal.

Equation (14) tells us that the peak time of the receding component should be
relevant to the 3 parameters of $n$, $E_{\rm 0,iso}$, $\theta_{\rm j}$; on the
other hand, other parameters such as $\xi_{\rm e}$, $\xi_{\rm B}^{2}$, $p$ do
not affect the peak time. These tendency can be clearly seen in Figs. 5 and 6.

In all the above calculations, we have assumed that the conditions and
parameters of the twin jets are the same. However, this may not be the case
for realistic GRBs. The circum-burst environment and the micro-physics
parameters may actually be different for the twin jets, as that may happen in
the two component jet structure (Huang et al. 2004; Jin et al. 2007; Racusin
et al. 2008). In Fig. 7, we have plotted the overall afterglow light curves by
assuming different parameters for the forward jet and the receding jet. In
each panel of Fig. 7, we first plot a common light curve (the solid line) by
adopting the standard parameter set, but change $\xi_{\rm e}$ to 0.01 and
change $\xi^{2}_{\rm B}$ to $10^{-4}$. We then increase the values of
$\xi_{\rm e}$, $\xi_{\rm B}^{2}$, and $n$ for the receding jet to see their
effects on the afterglow light curve. It is encouraging to see that the
emission from the receding jet really can be greatly enhanced, so that it can
manifest as an obvious rebrightening in the overall light curve. In Fig. 7(a),
7(b) and 7(d), the peak flux of the rebrightening can be nearly 100 times
larger than the “background” level in the best cases. It is imaginable that in
the most favorable cases, when all $\xi_{\rm e}$, $\xi_{\rm B}^{2}$ and $n$
are larger for the receding jet at the same time, the rebrightening will be
even more remarkable. However, note that the contrary condition may also exist
in realistic GRBs, i.e., these parameters may also be smaller for the receding
jet. Then the emission from the receding jet will be completely unnoticeable.

##  4 Conclusion and Discussion

We have studied the emission of the receding jet numerically. The effect of
the EATS is included in our calculations. Clearly, this effect plays an
important role in the process. It is found that the contribution from the
receding jet is generally quite weak. In most cases, it only manifests as a
short plateau in the overall afterglow light curve, but not a marked
rebrightening. The flux density of the plateau is usually much less than 100
$\mu$Jy in radio bands even at a small redshift of $z=0.1$ . If we place the
GRB at a more typical redshift of $z=1$, then the flux density of the plateau
will be less than 0.1 $\mu$Jy at 8.46 GHz. We noticed that the observed radio
afterglow emission is generally on the level of 0.1 — 1 mJy at about the peak
time. After several months, the radio afterglow usually decreases to a very
low level, and is submersed by the emission from the host galaxy, whose
strength can be 40 — 70 $\mu$Jy (Berger et al. 2001). Additionally, the error
bar of radio observations is usually $\sim$ 30 — 50 $\mu$Jy at very late
stages (Frail et al. 2003). Thus the contribution from the receding jet, i.e.
the plateau, is actually very difficult to detect currently, especially for
those GRBs at $z\sim 1$. Our results are consistent with a recent
observational report by van der Horst et al. (2008), who failed to detect any
clear clues of the receding jet emission.

However, as shown in our Fig. 7, if the micro-physics parameters of the
receding jet were different from the forward jet, or if the receding jet were
in a much denser environment, then it is still possible that the contribution
from the receding jet can be greatly enhanced. For example, if $\xi_{\rm e}$
and/or $\xi_{\rm B}^{2}$ of the receding jet is much larger than that of the
forward jet, then the receding jet can really manifest as an obvious
rebrightening.

Also, our Fig. 5(a) shows that a dense circum-burst environment can suppress
the emission of the forward jet, and enhance the contribution from the
receding jet. If the GRB occurs in a very dense molecular cloud with
$n>10^{3}/{\rm cm}^{3}$ (Dai & Lu 1999), the contribution from the receding
jet may be much easier to detect. Additionally, if the GRB is very near to us
at the same time, then the possibility of successfully detecting the receding
jet is very high (see the thin lines in Fig. 3(a)).

In short, we believe that the effort of trying to search for the afterglow
contribution from the receding jet is still meaningful. If observed, it would
provide useful clues to study the circum-burst environment and the micro-
physics of external shocks. We suggest that nearby GRBs (with redshift $z\leq
0.1$) should be good candidates for such studies.

###### Acknowledgements.

We would like to thank the anonymous referee for constructive suggestions that
lead to an overall improvement of this study. We also thank Z. Li for
stimulating discussion. This research was supported by the National Natural
Science Foundation of China (grant 10625313), and by the National Basic
Research Program of China (grant 2009CB824800). Xin Wang is also supported by
2008’ National Undergraduate Innovation Program of China (grant 081028441).

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Figure 1: The evolution of the Lorentz factors of the twin jets. The solid
line corresponds to the receding jet and the dashed line is plotted for the
forward jet. The twin jets are in “standard” condition as defined in Sect. 3.
The observers’ time has been corrected for the cosmological effect ($z=0.1$).
Figure 2: Schematic illustration of the EATSes at three moments, $t_{1}\approx
114$ d, $t_{2}\approx 286$ d and $t_{3}\approx 572$ d. In this calculation, we
have used the “standard” parameter set as defined in Sect. 3. “O” is the
position of the central engine, and the observer is on the far right side with
Y=0. The dotted lines indicate the jet boundary. For the receding jet, the
EATSes are plotted in thick solid lines, while for the forward jet the
surfaces are plotted in thin solid lines. Note that on the forward jet
branches, the bulk Lorentz factors of the material at the peak of the EATSes
are 1.17, 1.07, and 1.03 for $t_{1}$, $t_{2}$, and $t_{3}$, respectively. On
the receding jet branches, the bulk Lorentz factors of the material at the
peak of the EATSes are 56.07, 11.79, and 3.95 for $t_{1}$, $t_{2}$, and
$t_{3}$, respectively.

Figure 3: 8.46 GHz radio afterglow (a) and R-band optical afterglow (b) from
the forward jet and the receding jet. The thick lines are plotted for a
“standard” double-sided jet as defined in Sect. 3. The thin lines are plotted
for the double-sided jet with only one parameter altered as compared with the
“standard” condition, i.e. $n=1000/{\rm cm}^{3}$. In each group, the dotted
line reflects the emission from the forward jet, the dashed line reflects the
contribution from the receding jet, and the solid line is the total light
curve.

Figure 4: Multiwavelength afterglow light curves of a double-sided jet. Radio
afterglows are illustrated in panel (a), and optical/IR afterglows are plotted
in panel (b). In this calculation, we have used the “standard” parameter set
as defined in Sect. 3.

Figure 5: The effects of various parameters ($n$, $E_{\rm 0,iso}$,
$\theta_{\rm j}$, and $\varepsilon$) on the 8.46 GHz radio afterglow light
curves of double-sided jets. In each panel, “(s)” corresponds to the
“standard” condition as defined in Sect. 3, and other lines are drawn with
only one certain parameter altered or one condition changed. In panel (d), the
dash-dotted line is plotted for a double-sided jet without lateral expansion;
and the dotted line is plotted for a double-sided jet with a low initial
Lorentz factor ($\gamma_{\rm 0}=30$), which may correspond to the so called
failed GRBs.

Figure 6: The effects of various parameters ($\xi_{\rm e}$, $\xi_{\rm B}^{2}$,
$p$, and $\theta_{\rm obs}$) on the 8.46 GHz radio afterglow light curves of
double-sided jets. In each panel, “(s)” corresponds to the “standard”
condition as defined in Sect. 3, and other lines are drawn with only one
certain parameter altered.

Figure 7: 8.46 GHz radio afterglow light curves of double-sided jets. In this
figure, we assume that the parameters of the receding jet can be different
from those of the forward jet. In each panel, the solid line is plotted under
the “standard” condition, i.e., the parameters are completely the same for the
twin jets (but note that we have evaluated $\xi_{\rm e}$ as 0.01 and $\xi_{\rm
B}^{2}$ as $10^{-4}$ here). For other light curves, one or two parameters are
changed for the receding jet, to see its effect on the afterglows.