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arXiv:astro-ph/0508602v1 29 Aug 2005
Mon. Not. R. Astron. Soc. 000, 1–13 (2005) Printed 5 February 2008(MN LATEX style file v2.2)
Early Afterglows in Wind Environments Revisited
Y. C. Zou, X. F. Wu, and Z. G. Dai⋆
Department of Astronomy, Nanjing University, Nanjing 210093, China
5 February 2008
ABSTRACT
When a cold shell sweeps up the ambient medium, a forward shock and a reverse shock
will form. We analyze the reverse-forward shocks in a wind environment, including their
dynamics and emission. An early afterglow is emitted from the shocked shell, e.g., an optical
flash may emerge. The reverse shock behaves differently in two approximations: relativistic
and Newtonian cases, which depend on the parameters, e.g., the initial Lorentz factor of the
ejecta. If the initial Lorentz factor is much less than 114E1/4
shock is Newtonian. This may take place for the wider of a two-component jet, an orphan
afterglow caused by a low initial Lorentz factor, and so on. The synchrotron self absorption
effect is significant especially for the Newtonian reverse shock case, since the absorption
frequencyνais larger than the cooling frequencyνcand the minimum synchrotronfrequency
νmfor typical parameters. For the optical to X-ray band, the flux is nearly unchanged with
time during the early period, which may be a diagnostic for the low initial Lorentz factor of
theejecta in a wind environment.We also investigatethe earlylightcurveswith differentwind
densities, and compare them with these in the ISM model.
53∆−1/4
0,12A−1/4
∗,−1, the early reverse
Key words: shock waves - gamma rays: bursts - stars: wind
1 INTRODUCTION
Long-duration gamma-ray bursts (GRBs) may originate from the collapse of massive stars (Woosley 1993; Paczy´ nski 1998). The probable
associations between GRBs and supernovae have been detected in several cases, e.g., the most confirmed GRB 980425 / SN 1998bw (Galama
et al. 1998, Kulkarni et al. 1998) and GRB 030329 / SN 2003dh (Hjorth et al. 2003), which give a firm link to the collapsar model. The light
curves of several afterglows also show the supernova component, such as GRB 970228 (Reichart 1999), 980326 (Bloom et al. 1999), 011121
(Bloom et al. 2002, Greiner et al. 2003), 021004 (Schaefer et al. 2003), and so on. Thus, the surrounding environment is wind-type. Much
work about wind-type environment analyses has been done (Dai & Lu 1998; M´ esz´ aros, Rees & Wijers 1998; Panaitescu & Kumar 2000;
Chevalier & Li 2000), including the features of the afterglow light curves and the comparison with the interstellar medium (ISM) model.
At the beginning of the interaction between the ejected shell and environment, an early afterglow will emerge from the reverse shock, as
predicted by M´ esz´ aros & Rees (1997) and Sari & Piran (1999b). The prompt optical emission from GRB 990123 (Akerlof et al. 1999) and
GRB 021211 (Fox et al. 2003; Li et al. 2003) were observed, though observations of a very early afterglow were difficult before the Swift’s
launch (Gehrels et al. 2004). The optical flash of GRB 990123 was immediately analyzed (M´ esz´ aros & Rees 1999; Sari & Piran 1999a).
They pointed out that the optical flashes mainly come from the contribution of the reverse shock. Then, plenty of theoretical analyses were
advanced. The dynamics, numerical results of optical and radio emission, and analytical light curves, from the reverse and forward shocks in
uniform environments were discussed (Sari & Piran 1995; Kobayashi & Sari 2000; Kobayashi 2000). Early afterglows in wind environments
were also considered by several groups (Chevalier & Li 2000; Wu et al. 2003; Kobayashi & Zhang 2003). Panaitescu & Kumar (2004)
considered the reverse-forward shock scenario and the wind bubble scenario for the two observed optical flashes from GRB 990123 and
GRB 021211. However, little discussion comes to a Newtonian reverse shock in wind environment. We consider this case in the following
sections.
Kobayashi, M´ esz´ aros & Zhang (2004) noticed the synchrotron self-absorption (SSA) effect on the early afterglow. We find that other
⋆E-mail: zouyc@nju.edu.cn(YCZ); xfwu@nju.edu.cn(XFW); dzg@nju.edu.cn(ZGD)
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Y. C. Zou, X. F. Wu, and Z. G. Dai
parameters like the thickness of the shell and the initial isotropic kinetic energy can also influence the self absorption, even up to the optical
wavelength. In this paper, we derive the complete scaling-laws of the SSA frequency for all cases.
In the simulations of Zhang, Woosley & MacFadyen (2003), the initial Lorentz factor can be as low as about tens. For the structured
jet model (Kumar & Granot 2003), it is likely that the jet has low Lorentz factors at the wings of the jet. Huang et al. (2002) considered
that a jet with an initial Lorentz factor less than 50 may cause an orphan afterglow. Rhoads (2003) also pointed out that the fireball with a
low initial Lorentz factor will produce a detectable afterglow, though no gamma-ray emission is detectable. There are indications that some
GRBs ejecta have two components: a narrow ultra-relativistic inner core, and a wide mildly relativistic outer wing (Berger et al. 2003; Huang
et al. 2004; Wu et al. 2005; Peng et al. 2005). When the mildly relativistic shell collides with the wind environment, the reverse shock is
Newtonian. Kobayashi (2000) considered the Newtonian reverse shock in a uniform environment. However, no systematic analysis has come
into the Newtonian reverse shock in a wind environment. In this work, we discuss the Newtonian reverse shock, which is mainly caused by a
low initial Lorentz factor. In this case, the optical emission flux from the Newtonian shocked region exceeds that from the relativistic forward
shocked region.
Some authors have used the early afterglow as a diagnostic tool of gamma ray bursts’ parameters for the ISM case (Zhang, Kobayashi
& M´ esz´ aros 2003; Nakar et al. 2004) and for the wind case (Fan et al. 2004; Fan et al. 2005). Accordingly, the behavior of early reverse-
forward shocks should be completely described for the wind case. We derive the analytical scaling-laws of dynamics and radiations for both
relativistic and Newtonian reverse shock cases in the wind environment in §§2 and 3, and give the numerical results of radio to X-ray light
curves in §4. We present some discussions in §5.
2 HYDRODYNAMICS
Let’s consider a uniform and cold relativistic coasting shell with isotropic kinetic energy E0, Lorentz factor γ4 = η + 1 ≫ 1, and width in
observer’sframe∆0,ejectedfromtheprogenitor of theGRB.Thisshellsweeps upafreewindenvironment withnumber densityn1 = Ar−2,
where η is the initial ratio of E0 to the rest mass of the ejecta (Piran, Shemi & Narayan 1993). The interaction between the shell and the
wind develops a forward shock propagating into the wind and a reverse shock propagating into the shell. The two shocks separate the system
into four regions: (1) the unshocked approximately stationary wind (called region 1 hereafter), (2) the shocked wind (region 2), (3) the
shocked shell material (region 3), and (4) the unshocked shell material (region 4). By using the shock jump conditions (Blandford & McKee
1976, BM hereafter) and assuming the equality of pressures and velocities beside the surface of the contact discontinuity, the values of the
Lorentz factor γ, the pressure p, and the number density n in the shocked regions can be estimated as functions of n1, n4, and η, where
n4 = E0/(η4πr2γ4∆0mpc2) is the comoving number density of region 4.
Analytical results can be obtained in both relativistic and Newtonian reverse shock limit. These two cases are divided by comparison
between f and γ2
1995). As shown by Wu et al. (2003) for the wind environment case, f = l/(η2∆0), where l = E0/(4πAmpc2) is the Sedov length. If
f ≫ γ2
As discussed by Kobayashi & Sari (2000), even for NRS, the adiabatic index of the post-shocked fluid can be taken as a constant
ˆ γ = 4/3, because the electrons are still relativistic. Then the shock jump conditions can read (BM; Sari & Piran 1995)
4, where f ≡ n4/n1is the ratio of the number densities between the unshocked shell and the unshocked wind (Sari & Piran
4, the reverse shock is Newtonian (NRS), and if f ≪ γ2
4, the reverse shock is relativistic (RRS).
e2/n2mpc2= γ2− 1,n2/n1 = 4γ2+ 3,
e3/n3mpc2= ¯ γ3− 1,n3/n4 = 4 ¯ γ3+ 3,
where mp is the proton mass, e2 and e3 are the comoving energy densities of region 2 and region 3 respectively, and n2 and n3 are the
corresponding comoving number densities of particles, which are assumed to consist of protons and electrons. The relative Lorentz factor
between region 3 and region 4 is
(1)
(2)
¯ γ3 = γ3γ4(1 −
?
1 − 1/γ2
3
?
1 − 1/γ2
4).
(3)
Assuming γ2 = γ3, and γ2,γ4 ≫ 1, ¯ γ3 can be expressed as ¯ γ3 ≃ (γ4/γ2 + γ2/γ4)/2. The asymptotic solution is γ3 ≃
¯ γ3 ≃
The time it takes the reverse shock to cross the shell in the burster’s frame is given by (Sari & Piran 1995)
1
√2γ1/2
4
f1/4,
1
√2γ1/2
4
f−1/4for RRS, while γ3 ≃ γ4and ¯ γ3− 1 ≃4
7γ2
4f−1for NRS.
t∆ =
∆0
c(β4− β3)
?
1 −γ4n4
γ3n3
?
.
(4)
There are two simple limits involved in the problem: NRS and RRS, in which we can get analytical results. The relative Lorentz factor ¯ γ3is
constant in the whole reverse-shock period for RRS. t∆can be derived as t∆ = α∆0γ4f1/2/c, and the corresponding radius of the shell at
time t∆ is r∆ ≃ ct∆ = α∆0γ4f1/2≃ α√l∆0, where the coefficient α = 1/2 for RRS and α = 3/√14 for NRS. We will discuss both
cases separately in the following.
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Early Afterglows in Wind Environments Revisited
3
2.1 Relativistic Reverse Shock Case
In the RRS case, f ≪ γ2
z)r/2γ2
observer’s frame. We adopt the conventional denotation Q = Qk× 10kin this paper except for some special explanations. Using e2 = e3,
γ2 = γ3, together with the above equations, we get the scaling-laws of the hydrodynamic variables for time t⊕ < T,
4(i.e., η ≫ 114E1/4
53∆−1/4
0,12A−1/4
∗,−1), using the relation between the observer’s time and the radius t⊕ ≃ (1 +
3c, where z is the redshift of the GRB, we obtain T ≃ (1 + z)∆0/2c ≃ 16.7(1 + z)∆0,12s as the RRS-crossing time in the
¯ γ3 ≃ 1.9η2.5E−1/4
γ3 = γ2 ≃ 81.5E1/4
e3 = e2 ≃ 1.4 × 104E−1/2
53
∆1/4
0,12A1/4
∗,−1
(5)
53∆−1/4
0,12A−1/4
∗,−1
(6)
53
∆−3/2
0,12A3/2
∗,−1
?t⊕
T
?−2
erg cm−3
(7)
Ne,3 ≃ 2.1 × 1053E53η−1
Ne,2 ≃ 3.5 × 1051E1/2
2.5t⊕
T
(8)
53∆1/2
0,12A1/2
∗,−1
t⊕
T
(9)
where A∗ = 3 × 1035cm−1, and Ne,i is the number of electrons in the shocked region i. We note that γ3 and ¯ γ3 do not depend on time.
This is the property of wind environments, since the densities of the shell and the ambient environment have the same power-law relation
with radius r (n ∝ r−2).
After the reverse shock crosses the shell (t⊕ > T), the shocked shell can be roughly described by the BM solution (Wu et al. 2003;
Kobayashi & Zhang 2003; Kobayashi et al. 2004),
γ3 ∝ t−3/8
γ2 ∝ t−1/4
These variables can be scaled to the initial values (t⊕ = T), which are given by the expressions for the time t⊕ < T.
⊕
,n3 ∝ t−9/8
,n2 ∝ t−5/4
⊕
,e3 ∝ t−3/2
,e2 ∝ t−3/2
⊕
,r ∝ t1/4
,r ∝ t1/2
⊕,Ne,3 ∝ t0
⊕,Ne,2 ∝ t1/2
⊕,
⊕.
(10)
⊕⊕⊕
(11)
2.2Newtonian Reverse Shock Case
In the NRS case, f ≫ γ2
frame, if we consider the spreading of the cold shell (Piran et al. 1993). The evolution of the hydrodynamic variables before the time T′are
4, the time for the reverse shock crossing the shell is T′≃ tη ≃ 2.9 × 103(1 + z)E53η−4
1.5A−1
∗,−1s in the observer’s
¯ γ3− 1 ≃ 0.57t⊕
γ3 = γ2 ≃ γ4
e3 = e2 ≃ 5.8E−2
T′
(12)
(13)
53η6
1.5A3
∗,−1
?t⊕
?1/2
T′
?−2
erg cm−3
(14)
Ne,3 = 2.1 × 1054E53η−1
1.5
?t⊕
T′
(15)
Ne,2 ≃ 6.6 × 1052E53η−2
1.5t⊕
T′
(16)
What should be noted is that the values for NRS are not suitable for mildly relativistic reverse shock case. Nakar & Piran (2004) showed the
difference between the approximated analytical solution and the numerical results in the case of uniform environments. And for the spreading
of the shell, f decreases with radius. At the crossing time, ¯ γ3 ≃ 1.57 (see equation (12)), which deviates from the Newtonian reverse shock
approximation. More accurate values should be calculated numerically.
After the NRS crosses the shell, the Lorentz factor of the shocked shell can be assumed to be a general power-law relation γ3 ∝ r−g
(M´ eszar´ os & Rees 1999; Kabayashi & Sari 2000). However, the forward shock is still relativistic, and can be described by the BM solution.
The dynamic behavior is the same as the one in the RRS case. The scaling-law of the two regions are
γ3 ∝ t−g/(1+2g)
e3 ∝ t−8(3+g)/7(1+2g)
γ2 ∝ t−1/4
e2 ∝ t−3/2
⊕
,n3 ∝ t−6(3+g)/7(1+2g)
,r ∝ t1/(1+2g)
,n2 ∝ t−5/4
,r ∝ t1/2
⊕
,
⊕⊕
,Ne,3 ∝ t0
⊕,
(17)
⊕
⊕
,
⊕⊕,Ne,2 ∝ t1/2
⊕.
(18)
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Y. C. Zou, X. F. Wu, and Z. G. Dai
3 EMISSION
We now consider the synchrotron emission from the shocked material of region 2 and region 3. The shocks accelerate the electrons into
a power-law distribution: N(γe)dγe = Nγγ−p
Assuming that constant fractions ǫeand ǫBof the internal energy go into the electrons and the magnetic field, we have B =√8πǫBei, where
eiis the internal energy density of the shocked material. Regarding that the comoving internal energy of the electrons can also be written as
ǫee =?∞
The cooling Lorentz factor γcis defined when the electrons with γcapproximately radiate all their kinetic energy in the dynamical time,
i.e., (γc−1)mec2= P(γc)tco, where P(γe) = (4/3)σTc(γ2
of an electron with Lorentz factor γe in the magnetic field B, tco is the dynamical time in comoving frame (Sari et al. 1998; Panaitescu &
Kumar 2000). Then the cooling Lorentz factor γc = 6πmec/(σTB2tco) − 1.
The synchrotron radiation is taken to be monochromatic, and the corresponding frequency of an electron with Lorentz factor γe is
νe = (3/2)γ2
3(1 + z)−1γγ2
factor of the emitted region. Before the reverse shock crosses the shell, νc,2 = νc,3 is satisfied for the two regions having the same energy
density e, Lorentz factor γ, and the same comoving time tco.
The synchrotron self-absorption effect should not be ignored, especially at low frequencies, where the emission is modified enormously
for the large optical depth. Wu et al. (2003) have given the SSA coefficient and the corresponding spectral indices. We here quote the results
of Wu et al. (2003) and derive the SSA frequency for all six cases in the following.
The initial distribution of shock-accelerated electrons is
e dγe(γe > γm), where γm is the minimum Lorentz factor of the accelerated electrons.
γmNγγ−p
e γemec2dγe, and the comoving number density n =?∞
γmNγγ−p
e dγe, one can get γm = ǫe(¯ γ−1)(mp/me)(p−2)/(p−
m .
1), where ¯ γ = ¯ γ3or γ2corresponds to the reverse or forward shock, and Nγ = n(p − 1)γp−1
e−1)(B2/8π) (Rybicki & Lightman 1979) is the synchrotron radiation power
eνL, where νL = qeB/(2πmec) is the Larmor frequency, and qe is the electron charge. The critical frequencies are νm =
mqeB/(2πmec) and νc = 3(1 + z)−1γγ2
cqeB/(2πmec), in the observer’s frame respectively, where γ is the bulk Lorentz
N(γe) = Nγγ−p
e
γm < γe < γmax.
(19)
Taking into account the synchrotron radiation energy losses, the power-law distribution of electrons is divided into two segments (Sari et al.
1998), i.e.,
N(γe) = Nγ
?
γ−2
e
γ−(p+1)
e
γc < γe < γm
γe > γm,
(20)
for the fast-cooling case (γc < γm), and
N(γe) = Nγ
?
γ−p
e
γ−(p+1)
e
γm < γe < γc
γe > γc,
(21)
for the slow-cooling case (γc > γm).
The self-absorption coefficients in different frequency ranges are
kν =qe
BNγ
c1γ−(p+4)
1
?
?
?
ν
ν1
?−5/3
?−(p+4)/2
?−5/2
ν ≪ ν1
ν1 ≪ ν ≪ ν2,
ν ≫ ν2
c2γ−(p+4)
1
ν
ν1
c3γ−(p+4)
2
ν
ν2
e−ν/ν2
(22)
where c1 =
frequencies of electrons of Lorentz factor γ1and γ2, and Γ(x) is the Gamma function (Wu et al. 2003).
Electrons in both segments contribute to the SSA. For simplicity, the less important segment is neglected. In general, the absorption
coefficient is dominated by the electrons between γcand γm, for the frequency less than max(νc,νm); and dominated by the electrons with
Lorentz factor greater than max(γc,γm), for the frequency larger than max(νc,νm). So, the third expression in equation (22) is always
unimportant and can be neglected. We can obtain analytical expressions for the SSA frequency νaby taking kνL = τ0,
32π2
9×21/3Γ(1/3)
p+2
p+2/3, c2 =2√3π
9
2p/2(p +10
3)Γ(3p+2
12)Γ(3p+10
12
), c3 =2√6π3/2
9
(p + 2), ν1and ν2 are the typical synchrotron
νa =
?
?
?
c1qe
BNγγ−(p1+4)
1
ν5/3
1
L
τ0
? 3
5
νa ≪ ν1
ν1 ≪ νa ≪ ν2
νa ≫ ν2,
c2qe
BNγγ−(p1+4)
1
ν(p1+4)/2
1
L
τ0
?
?
2
p1+4
c2qe
BNγγ−(p1+4)
2
ν(p2+4)/2
2
L
τ0
2
p2+4
(23)
where L = Ne/(4πr2n) is thecomoving width of theemission region, τ0can be defined equal to 0.35 (Frailet al. 2000), γ1 = min(γc,γm),
γ2 = max(γc,γm), p1is the power-law index of the electron distribution between γ1 and γ2 (p1 = p for slowing cooling, p1 = 2 for fast
cooling), and p2 = p + 1 is the index of the electron distribution with Lorentz factor greater than γ2.
Because the peak spectral power Pν,max ≃ (1 + z)γmec2σTB/(3qe) in the observer’s frame is independent of γe, the peak observed
flux density can be given by Fν,max = NePν,max/(4πD2) at the frequency min(νc,νm), where D is the luminosity distance of the gamma-
ray burst.
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Early Afterglows in Wind Environments Revisited
5
3.1 Relativistic Reverse Shock Case
Using the above expressions, we obtain the typical frequencies and the peak flux density in the shocked shell and the shocked wind for the
RRS case,
νm,3 ≃ 5.9 × 1015(1 + z)−1¯ ǫ2
eǫ1/2
B,−1E−1/2
53
η2
2.5A∗,−1∆−1/2
0,12
?t⊕
T
?−1
Hz,
(24)
νm,2 ≃ 1.0 × 1019(1 + z)−1¯ ǫ2
eǫ1/2
B,−1E1/2
53∆−3/2
0,12
?t⊕
0,12A−2
T
?−1
Hz(25)
νc,2 = νc,3 ≃ 1.5 × 1012(1 + z)−1ǫ−3/2
Fν,max,3 ≃ 95.3(1 + z)ǫ1/2
Fν,max,2 ≃ 1.6(1 + z)ǫ1/2
B,−1E1/2
∗,−1∆−1
0,12D−2
53∆1/2
∗,−1
t⊕
T
Hz,
(26)
B,−1E53η−1
B,−1E1/2
2.5A1/2
0,12D−2
28Jy,
(27)
53A∗,−1∆−1/2
28Jy (28)
where ¯ ǫe ≡ ǫe,−0.5· 3(p − 2)/(p − 1). Note that Fν,max,3 > Fν,max,2, i.e. region 3 dominates the emission for the early afterglow, mainly
because the number of electrons in region 3 is much larger than that in region2.
We give the scaling-law of the SSA frequency in region 3,
νa,3 ≃
1.1 × 1016(1 + z)−1ǫ6/5
2.2 × 1014(1 + z)−1E1/6
4.7 × 1014(1 + z)−1¯ ǫ2/5
2.2 × 1014(1 + z)−1¯ ǫ−1
1.1 × 1015(1 + z)−1¯ ǫ6/13
5.2 × 1014(1 + z)−1¯ ǫ2/5
B,−1E−1/10
53η−1/3
2.5
53
η−3/5
2.5
A19/10
∗,−1∆−19/10
?t⊕
2.5A1/3
0,12
?−2/3
0,12
?t⊕
Hz
T
?−2
Hz
νa < νc < νm
A1/6
∗,−1∆−5/6
η2/15
0,12
T
νc < νa < νm
e
ǫ1/10
B,−1E1/30
e ǫ1/5
53
∗,−1∆−23/30
A2/5
?t⊕
T
T
?−11/15
?−1
?t⊕
?t⊕
Hz
νc < νm < νa,
B,−1E2/5
ǫ9/26
53η−8/5
2.5
∗,−1∆−7/5
η2/13
0,12
?t⊕
0,12
Hz
?−1
νa < νm < νc
e
B,−1E−1/26
ǫ1/10
532.5A9/13
∗,−1∆−25/26
∗,−1∆−23/30
T
?−11/15
Hz
νm < νa < νc
e
B,−1E1/30
53
η2/15
2.5A1/3
0,12
T
Hz
νm < νc < νa,
(29)
Here and in the following expressions for νa, we take p = 2.5. If more than one expressions above satisfy the followed restriction, the largest
νais the true value.
In region 2,
νa,2 ≃
9.6 × 1014(1 + z)−1ǫ6/5
5.7 × 1013(1 + z)−1A1/3
7.0 × 1014(1 + z)−1¯ ǫ2/5
4.5 × 1011(1 + z)−1¯ ǫ−1
1.9 × 1015(1 + z)−1¯ ǫ6/13
7.8 × 1014(1 + z)−1¯ ǫ2/5
B,−1E−2/5
∗,−1∆−2/3
ǫ1/10
53
A11/5
∗,−1∆−8/5
?t⊕
53
A4/15
0,12
?t⊕
Hz
T
?−2
Hz
νa < νc < νm
0,12
T
?−2/3
∗,−1∆−5/6
A6/5
νc < νa < νm
e
B,−1E1/10
e ǫ1/5
0,12
?t⊕
?t⊕
T
?−11/15
?−1
?t⊕
?t⊕
Hz
νc < νm < νa,
B,−1E−2/5
ǫ9/26
53
∗,−1∆−3/5
A8/13
0,12
T
Hz
νa < νm < νc
e
B,−1E1/26
ǫ1/10
53
∗,−1∆−27/26
A4/15
0,12
T
?−1
Hz
νm < νa < νc
e
B,−1E1/10
53
∗,−1∆−5/6
0,12
T
?−11/15
Hz
νm < νc < νa.
(30)
After the reverse shock crosses the shell (t⊕ > T), the behavior of both shocked regions can be described by the BM self-similar
solution. The power-law indices of emission variables with time are given in table 1. Another frequency νcut should be introduced here
(Kobayashi 2000) to substitute νcfor no fresh electrons. νcuthas the same time profile as νm. If νm > νcut, νmcomes down to νcutfor the
synchrotron cooling. And if νa > νcut, νa comes down to νcuttoo for no electrons distributed greater than corresponding γcut. These are
all represented in columns labeled 6 ? (for NRS case) and 8 ? in Table 1.
The scaling-law indices of flux densities with time are sophisticated, as they vary with time when any two of ν,νa,νc(or νcut),νm
cross each other, where ν is the observed frequency. These indices are given in Table 2. For the case ν > νcut, the flux density decreases
exponentially with observed frequency ν, then we take it to be zero, which is denoted by a short horizontal line in Table 2. The numerical
results will be given in §4.
For the typical parameters, the order of the frequencies at t⊕ = T is νm > ν > νa > νcfor both region 3 and region 2, if the considered
frequency is ν = 4.55 × 1014Hz. The flux density from the shocked shell and shocked environment are
Fν,3 ≃ 5.4(1 + z)1/2ǫ−1/4
B,−1E5/4
53η−1
2.5A−1/2
∗,−1∆−3/4
0,12D−2
28Jy.
(31)
Fν,2 ≃ 0.1(1 + z)1/2ǫ−1/4
B,−1E3/4
53∆−1/4
0,12D−2
28Jy.
(32)
Page 6
6
Y. C. Zou, X. F. Wu, and Z. G. Dai
3.2Newtonian Reverse Shock Case
For NRS case, before the reverse shock crosses the shell (t⊕ < T′),
νm,3 ≃ 4.1 × 1012(1 + z)−1¯ ǫ2
eǫ1/2
B,−1E−1
53η4
1.5A3/2
∗,−1
t⊕
T′Hz
?t⊕
(33)
νm,2 ≃ 1.3 × 1016(1 + z)−1¯ ǫ2
eǫ1/2
B,−1E−1
53η6
1.5A3/2
∗,−1
T′
?−1
t⊕
T′Hz
Hz(34)
νc,3 = νc,2 ≃ 2.7 × 1013(1 + z)−1ǫ−3/2
B,−1E53η−2
?t⊕
28Jy
1.5A−5/2
∗,−1
(35)
Fν,max,3 ≃ 7.6(1 + z)ǫ1/2
Fν,max,2 ≃ 0.2(1 + z)ǫ1/2
B,−1η3
1.5A3/2
∗,−1D−2
28
T′
?−1/2
Jy (36)
B,−1η2
1.5A3/2
∗,−1D−2
(37)
The SSA frequency in region 3,
νa,3 ≃
2.2 × 1012(1 + z)−1ǫ6/5
7.2 × 1012(1 + z)−1E−2/3
7.1 × 1012(1 + z)−1¯ ǫ2/5
7.3 × 1012(1 + z)−1¯ ǫ−1
5.8 × 1012(1 + z)−1¯ ǫ6/13
7.9 × 1012(1 + z)−1¯ ǫ2/5
B,−1E−2
53η7
1.5A19/5
∗,−1
?t⊕
η16/5
?t⊕
T′
T′
?−23/10
?−5/6
1.5A11/10
∗,−1
1.5A9/5
∗,−1
53η4
∗,−1
B,−1E−11/15
Hz
νa < νc < νm
53
η3
1.5A∗,−1
ǫ1/10
Hz
νc < νa < νm
e
B,−1E−11/15
e ǫ1/5
53
?t⊕
T′
?−7/15
Hz
Hz
νc < νm < νa,
B,−1E−1
ǫ9/26
53η4
?t⊕
T′
?t⊕
∗,−1
?−23/10
T′
?t⊕
νa < νm < νc
e
B,−1E−1
ǫ1/10
1.5A43/26
?−9/13
T′
Hz
νm < νa < νc
e
53
η16/5
1.5A11/10
?−7/15
Hz
νm < νc < νa,
(38)
and in region 2,
νa,2 ≃
2.8 × 1011(1 + z)−1ǫ6/5
2.3 × 1012(1 + z)−1E−2/3
1.4 × 1013(1 + z)−1¯ ǫ2/5
1.7 × 1010(1 + z)−1¯ ǫ−1
1.3 × 1013(1 + z)−1¯ ǫ6/13
1.6 × 1013(1 + z)−1¯ ǫ2/5
B,−1E−2
53η32/5
1.5A19/5
∗,−1
?t⊕
η10/3
?t⊕
?−2/3
1.5A11/10
T′
?−2
Hz
νa < νc < νm
53
η8/3
1.5A∗,−1
ǫ1/10
T′
Hz
νc < νa < νm
e
B,−1E−11/15
e ǫ1/5
53
∗,−1
?t⊕
∗,−1
?t⊕
?−1
?t⊕
?t⊕
T′
?−11/15
Hz
?−1
T′
Hz
νc < νm < νa,
B,−1E−1
ǫ9/26
53η12/5
1.5A9/5
∗,−1
A43/26
T′
νa < νm < νc
e
B,−1E−1
ǫ1/10
53η54/13
1.5
T′
Hz
νm < νa < νc
e
B,−1E−11/15
53
η10/3
1.5A11/10
∗,−1
?−11/15
Hz
νm < νc < νa,
(39)
After the reverse shock crosses the shell (t⊕ > T′), the temporal indices of the typical frequencies and the observed flux density are
also given in tables 1 and 2.
Thefrequency relationsat timet⊕ = T′areν > νc > νa > νmfor region 3andνm > ν > νc > νaforregion 2,if ν = 4.55×1014Hz.
The corresponding optical flux density from region 3 and 2 are
Fν,3 ≃ 65(1 + z)−1/4¯ ǫ3/2
e
ǫ1/8
B,−1E−1/4
53
η5
1.5A11/8
∗,−1D−2
28mJy,
(40)
Fν,2 ≃ 63(1 + z)1/2ǫ−1/4
B,−1E1/2
53η1.5A1/4
∗,−1D−2
28mJy.
(41)
4NUMERICAL RESULTS
The above analytical results can give approximate behaviors of variables as functions of time or frequency, but they are valid only in
relativistic or Newtonian limits. In the mildly relativistic case, the analytical values deviate from the actual ones very much (Nakar 2004).
For precise results, a numerical method should be engaged.
Combining equations (1), (2) and (3), and using the assumption of equalities between the Lorentz factors and pressures beside the
surface of the contact discontinuity, one can obtain solutions of γ2, γ3, ¯ γ3, e2, n2, n3numerically. Before the reverse shock crosses the shell,
Page 7
Early Afterglows in Wind Environments Revisited
7
Table 1. The temporal indices for the evolution of νm, νc, νaand Fν,max. The notations denote respectively: 1 ? early, NRS, region 2; 2 ? early, NRS, region
3; 3 ? early, RRS, region 2; 4 ? early, RRS, region 3; 5 ? lately, NRS, region 2; 6 ? lately, NRS, region 3; 7 ? lately, RRS, region 2; 8 ? lately, RRS, region 3. νc
is actually νcutat the columns 6 ? & 8 ?.
variable
t < t∆
1 ? 3 ? 4 ?
t > t∆
6 ?
notation
2 ?
5 ? 7 ?
8 ?
νm
νc
−11
−3
1
2
−8
−2
2
−15 g+24
14 g+7
−15 g+24
14 g+7
−33 g+36
70 g+35
−15 g+24
14 g+7
−15 g+24
14 g+7
−33 g+36
70 g+35
−15
−15
−3
−15
−15
−3
8
11
8
−2
−2
−p+3
p+5
−1
−23
−5
p−6
p+5
−23
p−7
p+4
p−6
p+5
−1
1055
νa≪ νc≪ νm
νc≪ νa≪ νm
νc≪ νm≪ νa
νa≪ νm≪ νc
νa
3638
(fast cooling)
−3 p+5
2 p+10
−3
−3 p+6
2 p+8
−3 p+5
2 p+10
−1
8
1055
νa
−1
−(15 g+24) p+32 g+40
(14 g+7) p+56 g+28
−15 g+24
14 g+7
−11 g+12
14 g+7
−15 p+26
8 p+32
−15
−9
νm≪ νa≪ νc
(slow cooling)
−p+3
p+5
0
8
νm≪ νc≪ νa
Fν,max
228
Table 2. The temporal indices for the evolution of flux density (Fν ∝ tα
radiation vanishes at those cases.
⊕). The notations are the same as in table 1. The short horizontal lines indicate the
t < t∆
1 ? 3 ? 4 ?
t > t∆
6 ?
case
2 ?
5 ? 7 ?
8 ?
ν < νa< νc< νm
νa< ν < νc< νm
νa< νc< ν < νm
νa< νc< νm< ν
ν < νc< νa < νm
νc< ν < νa< νm
νc< νa < ν < νm
νc< νa < νm< ν
ν < νc< νm< νa
νc< ν < νa
νc< νm< νa< ν
ν < νa< νm< νc
νa< ν < νm< νc
332
5 g+8
14 g+7
−6 g+4
14 g+7
–
–
19 g+36
14 g+7
–
–
–
–
–
–
5 g+8
14 g+7
−6 g+4
14 g+7
1
2
−1
1
2
−p−2
3
3
−5
6
−2
−1
3
−1
–
–
21
8
–
–
–
–
–
–
1
2
−1
2
0
4
2
p−1
2
3
−3 p−2
2
4
5
2
1
2
5
2
0
7
4
−1
4
−p−2
3
5
2
−p−2
2
2
p−1
2
3
5
2
p−1
2
3
−3 p−2
2
7
4
−3 p−2
1
4
24
1
3
−5
p−2
2
p−1
2
3
6
0
2
νa< νm< ν < νc
νa< νm< νc< ν
ν < νm< νa< νc
νm< ν < νa< νc
−p−1
−p−2
2
2
−3 p−1
−3 p−2
1
4
−(15 g+24) p+7 g
28 g+14
–
5 g+8
14 g+7
25 g+40
28 g+14
−(15 g+24) p+7 g
28 g+14
–
19 g+36
14 g+7
53 g+96
28 g+14
–
−15 p+3
–
1
2
23
16
−15 p+3
–
21
8
57
16
–
16
24
5
2
5
2
7
4
νm< νa< ν < νc
νm< νa< νc< ν
ν < νm< νc< νa
νm< ν < νa
νm< νc< νa< ν
−p−1
−p−2
2
2
p−2
2
p−1
2
3
−3 p−1
−3 p−2
1
416
24
5
2
5
2
7
4
−p−2
2
p−1
2
−3 p−2
4
the value of γ3should be solved from the following equation without approximation,
(γ3− 1)(4γ3+ 3) =
?
γ3γ4
?
1 −
?
1 −
1
γ2
3
?
1 −
1
γ2
4
?
− 1
?
×
?
4γ3γ4
?
1 −
?
1 −
1
γ2
3
?
1 −1
γ2
4
?
+ 3
?
f,
(42)
and then the other variables can be derived directly.
We take the parameters η = 300,E0 = 1.0 × 1052erg,A∗ = 0.1,∆0 = 5.0 × 1012cm,ǫe = 0.3,ǫB = 0.1, and D = 1.0 × 1028
cm, for the RRS case. For the NRS case, we set η = 30, while keeping the same other parameters as in the RRS case. Following the above
analysis, we can get the emission from the two shocked regions, of which the optical magnitude at frequency ν = 4.55 × 1014Hz is shown
in Figure 2 [later] for RRS case and NRS case respectively. The reverse shock dominates the emission at the beginning and fades after the
shock crosses the shell, which is identical for both RRS and NRS. This effect may be the cause of the so-called optical flash.
Page 8
8
Y. C. Zou, X. F. Wu, and Z. G. Dai
Figure 1. Flux density at ν = 8.46 GHz as function of time. Parameters are η = 300,E0 = 1.0 × 1052erg,A∗ = 0.1,∆0 = 5.0 × 1012cm,ǫe =
0.3,ǫB= 0.1,p = 2.5 for the upper panel (RRS case). Only η = 30 is different for the lower panel (NRS case). The long dashed and short dashed lines
represent the emission from region 3 and region 2, respectively, and the solid line is the total flux density from both regions.
After the reverse shock crosses the shell, we choose the parameter g = 1 for the dynamic evolution of NRS. Kobayashi and Sari (2000)
discussed that g should satisfy 3/2 < g < 7/2 in the ISM environment. A similar conclusion can be drawn for the dynamics of the ejecta in
the wind environment. As the NRS cannot decrease the velocity of the ejected shell effectively, the shocked ejecta should be quicker than the
one in RRS case, which satisfies γ3 ∝ r−3/2. On the other hand, the ejecta must lag behind the forward shock, which satisfies γ2 ∝ r−1/2.
So the range of g should then obey 1/2 < g < 3/2. What’s more, the evolutions of the hydrodynamics and the emission do not depend on
the value of g sensitively. The evolution of γ with the observer’s time has a narrow range from t−1/4
g.
Figures 1-3 show the light curves at radio (8.46 GHz), optical band (4.55×1014Hz) and X-ray (1.0×1018Hz) respectively. The upper
panel denotes the RRS case, and the lower panel denotes the NRS case. At low frequencies, νais always greater than the observed frequency,
so the emission at these frequencies is affected by the synchrotron self-absorption enormously, and can be estimated as thermal emission at
this band (Chevalier & Li 2000). The radio flux density increases with time before and shortly after the crossing time, as shown in Figure 1,
which comes from the increasing number of the accelerated electrons. The flux will be intense enough to be detected if the distance is not so
large, as the flux is inversely proportional to the square of the luminosity distance.
The numerical results are well consistent with the analytical ones. For the typical parameters and ν = 4.55 × 1014Hz as the observed
frequencies, at the crossing time, the orders of the typical frequencies are νc,3 < νa,3 < ν < νm,3 for RRS case, νc,3 < νm,3 < νa,3 < ν
for NRS case, and νc,2 < νa,2 < ν < νm,2for both cases. From Table 2, we find that the corresponding temporal indices are 1/2, −1/4 and
1/2 for the time before the reverse shock crosses the ejected shell, where p = 2.5. In Figure 2, the slopes can be seen from the four dashed
lines before the break point, which is the crossing time. The value of the flux density from region 3 at time t = T′is however not consistent
with the value (55 mJy) given by equation (40), which is about 3.5 mJy in the figure, since the reverse shock is mildly relativistic. The curves
are not accordant well with the approximated analytical slopes either.1
For the optical band, the reverse shock dominates the emission at the beginning, and decays quickly after the crossing time, since there
are no fresh shocked electrons to produce the emission. This is the same for both RRS and NRS cases, as seen in Figure 2. However, the
X-ray afterglow is always dominated by the forward shock, especially for the NRS case, since the reverse shock is not strong enough, and
⊕
to t−3/7
⊕
corresponding to the range of
1The reverse-forward shock is assumed to begin at the fireball’s coasting period, which is the initial time for the early-afterglow. However, the coasting radius
is not zero, though it can be neglected at late times, which has been adopted in the scaling-law analyses. Therefore, at early times, the curve in Figure 2 for
RRS case is not straight. As the curve for NRS in the figure begins at 10 s, the influence of the nonzero initial radius can be neglected now.
Page 9
Early Afterglows in Wind Environments Revisited
9
Figure 2. Optical magnitude as function of time. Parameters are the same as in Figure 1.
Figure 3. 2-10 kev flux as function of time. Parameters are the same as in Figure 1.
Page 10
10
Y. C. Zou, X. F. Wu, and Z. G. Dai
Figure 4. Magnitude as function of time at 4.55 × 1014Hz. Parameters are the same as in Figure 1 except for η. The η is 25,35, and 50 from top to down.
can’t accelerate the electrons to a high stochastic Lorentz factor to emit numerous X-ray band photons. Figure 3 shows the emission at X-ray
band for both RRS and NRS cases. From these three figures, we can see that the main emission is approximately at optical band.
As seen in equation (40), the flux density depends on η very sensitively. We plot the magnitude as function of time for different η values
in Figure 4. Taking into account the lower panel in Figure 2 and the first two in this figure, we can find that, with other parameters unchanged,
the larger η, the larger the flux density, as the flux density is proportional to about η8.5if p = 2.5. When η = 50, shown in the lowest panel
in Figure 4, the reverse shock becomes mildly relativistic. In the relativistic reverse shock case, it is approximately inversely proportional to
Page 11
Early Afterglows in Wind Environments Revisited
11
111213 14
1516
17
log10(ν)(Hz)
-28
-27
-26
-25
-24
-23
log (F )(Jy)
Fν,2
Fν,3
10 ν
νa,3
νm,3
νc,2
νa,2
νm,2
t-2t-1.6
t1
t0.5
t-0.5
t-1.8
t1
t-1.8
t-0.5
t-1.8
t-1t-1.5
νc,3
ν2
ν2
ν1/3
ν5/2
ν-1.25
ν-0.75
ν-1.25
Figure 5. Flux density in region 2 (solid line) and in region 3 (dashed line) as function of the observed frequency, at the crossing time. Parameters are the same
as in Figure 1 with η = 30. The arrows on the left of the vertical line denote the time behavior of the corresponding typical frequencies before the crossing
time, and the arrows on the right denote the time behavior after the crossing time.
Figure 6. Light curves in optical band (4.55 × 1014Hz) for different wind parameter A. Parameters are the same as in Figure 1 except A, and η = 30. The
crosses (×) indicate the crossing time of the reverse shock.
η. The flux density descends with the increase of η. Another phenomenon is that, with the increase of the η, the time for the emission from
region 2 to overtake the one from region 3 postpones, and then region 3 almost dominate the emission during the whole early period. Because
the number of the electrons in region 3 is much larger than that in region 2, the emission is dominated by region 3, when the reverse shock
is powerful enough to accelerate the electrons to emit enough optical band synchrotron photons. Thirdly, no distinct ascending of the optical
light curves appears before the crossing time for the NRS case. On the contrary, the light curves will descend at the beginning if the emission
is dominated by the region 3.
We plot the spectrum at the crossing time for reverse shock and forward shock respectively in Figure 5. The spectrum is a typical
synchrotron spectrum for the electron energy distribution with index p = 2.5. The breaks are smoothened by the time equal arrival effect
(Sari 1998). Both curves have three typical frequencies νa, νcand νm. The time behavior of the frequencies before and after the crossing time
isillustratedin thisfigure. Beforethecrossing time, νm,3 ∝ t−0.5, whichisdifferent from theanalytical result for NRScase νm,3 ∝ t1(listed
in Table 1). This comes from the fact that the term ¯ γ3− 1 can’t be taken to be much less than 1, especially for a shallow Newtonian reverse
shock. Consequently, νa,3becomes approximately ∝ t−0.5, not ∝ t−(p+3)/(p+5)in Table 1. For these parameters, νm,3is occasionally equal
to νc,3 at the crossing time. After the crossing time, as no fresh electrons supply, electrons with stochastic Lorentz factor greater than γc
disappears. The maximum electron Lorentz factor γcutvaries with time like γm(Kobayashi 2000). The corresponding νcutdoes so. As one
can see, the maximum typical frequency in region 3 is νa, so the synchrotron self-absorption effect is important for the reverse shock in NRS
case.
The wind parameter A is important for the reverse-forward shock. How do the light curves vary if the wind density varies? We give a set
Page 12
12
Y. C. Zou, X. F. Wu, and Z. G. Dai
Figure 7. Magnitude at 4.55×1014Hz as function of time in ISM environment. n1= 1cm−3is the number density of ISM material. η = 300 for the upper
panel is RRS case, and η = 30 for the lower panel is NRS case. Other parameters are the same as in Figure 1.
of light curves of the early afterglow with different A in Figure 6. The parameter A is taken from 3×1032cm−1to 3×1035cm−1(A∗ = 1).
With the increase of A, the reverse shock converts from Newtonian to relativistic. For the extreme NRS case, the emission is dominated by
the forward shock (the lower light curves in Figure 6). This makes the light curves (summation of region 2 and region 3) have no break at the
crossing time. But with the increase of A, the early emission are gradually dominated by the reverse shock, so the breaks (in the upper light
curves) appear at the crossing time. Another phenomenon is that, as A decreases, the crossing time becomes longer, which is mainly due to
the spreading of the ejected shell. With the approximation by ignoring the spreading effect, the analytical crossing time for RRS is a constant
(1 + z)∆0/2c. In Figure 6, we can see that the crossing time converges to (1 + z)∆0/2c = 167((1 + z)/2)∆0,5×1012s with the increase
of A.
The ISM environment case has been investigated enormously (Sari & Piran 1995; M´ esz´ aros & Rees 1997; Kobayashi & Sari 2000;
Kobayashi 2000; Zhang et al. 2003). We here calculate the light curves of the reverse-forward shock for the ISM density n1 = 1cm−3
and with the same other parameters as the typical values in the wind environment (see Figure 7). For these parameters, the flux densities
both for the RRS case (upper panel) and for the NRS case (lower panel) increase during the early period, which are different from the wind
environment case. We can see in Figure 2 that the light curves almost keep unchanged for early times at the optical band. This may be caused
by the decrease of the number density of the wind. Therefore, whether there exist a rapid increase of the early optical afterglow may be used
to distinguish between the ISM and wind environments.
5 CONCLUSIONS AND DISCUSSION
We have investigated the whole evolution of the dynamics and emission of the reverse-forward shock in a wind environment, by considering
both the RRS and NRS cases. The temporal indices of the physical quantities are given in Table 1 and 2, which cover all interrelations of the
typical frequencies νa, νm, νcand the observed frequency ν. The flux densities of the emission at radio, optical and X-ray bands as functions
of time are shown in Figures 1- 3 by numerical calculations.
For the ISM model, there exists a transition radius RN (satisfies f(RN) = γ2
shock becomes relativistic. An enormous difference between the wind model and the ISM model is that the Lorentz factor of the shocked
regions before the crossing time is constant with time for the wind model in RRS case. Since the transition should satisfy f/γ2
ratio f and γ4are both constant, for the wind environment, and thus no transition exists. Therefore, the relative Lorentz factor of the reverse
shock ¯ γ3 is also independent of time. Taking into account these properties, we find that the temporal indices are relatively reliable, even
if the estimates of the Lorentz factors ¯ γ3,γ2, and γ3 deviate from the actual values, which are caused by the Newtonian and relativistic
approximations.
An optical flash emitted from the shocked shell appears for the typical parameters, but perhaps no darkening can be observed at early
times. A rapid decay occurs after the reverse shock crosses the shell, and then the emission is dominated by the shocked environment
material. A radio flare lasts for a longer time. It increases continuously even shortly after the crossing time. The X-ray band emission is
always dominated by region 2. At the optical band and X-ray band, the flux is nearly unchanged at early times especially for the NRS case,
4) for thick shell (Sari & Piran 1995), where the reverse
4= 1, the
Page 13
Early Afterglows in Wind Environments Revisited
13
which may be used to diagnose the NRS in a wind environment. For the reverse shock, the synchrotron self-absorption can not be neglected,
since it may exceed the other two frequencies νcand νmas the number density increases.
There is also a possibility that no prompt optical emission is detected. It may be caused by a low initial energy, a low environmental
density, or strong absorption, and so on. These will decrease the flux density of the early afterglow to go beyond the detector’s limits. In the
Swift’s era, many early optical and X-ray afterglows will be detected by UVOT and XRT, like GRB050525A (Shao & Dai 2005, Klotz et al.
2005), and then the parameters may be determined by early afterglow data more precisely.
We would like to thank the anonymous referee for valuable suggestions. YCZ thanks T. Yan, H. L. Dai , Y. Z. Fan and Y. F. Huang for
helpful discussions. This work was supported by the National Natural Science Foundation of China (grants 10233010 and 10221001), and
the Ministry of Science and Technology of China (NKBRSF G19990754).
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