Ultra-relativistic double explosions

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by  Maxim Lyutikov

Ultra-relativistic double explosions
Maxim Lyutikov
Department of Physics, Purdue University,
525 Northwestern Avenue, West Lafayette, IN 47907-2036
ABSTRACT
7 We consider fluid dynamics of relativistic double explosion - when a point explosion
1
0 withenergyE1 isfollowedbyasecondexplosionwithenergyE2 aftertimetd (thesecond
2 explosion could be in a form of a long lasting wind). The primary explosion creates a

n self-similar relativistic blast wave propagating with Lorentz factor Γ (t). A sufficiently
1
a
J strong second explosion, with total energy E2 ≥ 10−2E1, creates a fast second shock in

3 the external fluid previously shocked by the primary shock. At times longer than the
2
interval between the explosions t , yet short compared with the time when the second
d

] shock catches up the primary shock at ∼ t Γ2, the structure of the second shock is
E d 1
approximately self-similar. Self-similar structure of the second shock exist for the case
H
. ofconstantexternaldensity(inthiscaseΓ2 ∝ t−7/3), butnotforthewindenvironment.
h
At early times the Lorentz factor of the second shock may exceed that of the primary
p
- shock and may boost the synchrotron emission of locally accelerated electrons into the
o
r Fermi LAT range.
t
s
a
[

1 1. Introduction
v
4
0 We are confident that Gamma Ray Bursts (GRBs) are produced in relativistic explosions
6
(Paczynski 1986; Piran 2004). The standard fireball model (Piran 2004) postulates that (i) the
6
0 prompt emission is produced within the ejecta; (ii) afterglows are generated in the relativistic blast
.
1 wave after the ejecta deposited much of its bulk energy into circumburst medium. This was a
0
well-established paradigm before the launch of Swift mission.
7
1
Swift satellites allowed us to probe early afterglows, on time scales shorter than ∼ a day
:
v
(Lien et al. 2016). Most surprising are the highly-variable early X-ray and optical afterglows, that
i
X
challenge the standard fireball model Kann et al. (2010). The launching mechanism, the nature
r
a of the central source, the composition of the primary wind, prompt radiation mechanisms remain
uncertain (Lyutikov 2009)
One of the key unexpected conclusions that emerged as a result of the observations of early
GRB afterglows is that the central source could remain active (keeps producing relativistic wind)
for long times after the explosion, ∼ 103−104 seconds, and perhaps even longer (Troja et al. 2007).
This motivates us to consider the following hydrodynamic problem: an initial powerful explosion
generates an ultra-relativistic shock wave. It is then followed by a second explosion or a wind,
– 2 –
which is could be subdominant energetically. What is the structure of outflows generated by such
double explosion?
2. A range of double self-similar outflows
Explosions and/or winds with power-law scaling of power with time produce self-similar struc-
tures of the flow behind a forward shock Sedov (1959); Blandford and McKee (1976); magnetized
forward shocks have been considered by Lyutikov (2002). Let’s assume that the primary explo-
sion was an instantaneous event. It create a self-similar post-primary shock flow described by the
Blandford-McKee, Blandford and McKee (1976), solution (B&Mc below). If there is a second ex-
plosions (which actually can be in a form of a long-lasting wind) after some time t , the structure
d
is not generally self-similar - since now there is a special point in time t and, in addition, there is
d
a special parameter - the ratio of the energies of the primary and second explosions, E /E . But -
1 2
this is the key point for the current approach - at times much smaller than the time it takes for the
second shock wave to catch with the primary one (or long afterward) the flows are approximately
self-similar. At very early times the second shock propagates through a velocity and pressure gra-
dients created by the primary shock; importantly, as long as the primary shock is in self-similar
stage, those velocity and pressure gradients are self-similar, power-law like. Thus, at early times
the second shock propagates in a time and coordinate scale-free environment, determined by the
primary shock (for the non-relativistic case, see Andriankin and Myagkov 1981).
In the case of GRBs the observer time t of early afterglow variability corresponds to physical
w
(coordinate) time after the explosion which is much longer, t ∼ t Γ2 ∼ months. On the other hand
w
the observed variability time t is of the same order as the intrinsic variability of the source. Thus,
w
in the case of GRBs with long-lasting central source we typically have a separation of temporal
scales: when the early afterglow emission is produced we have t (cid:28) t ≤ t Γ2. We expect that in
w w
the time range the double-explosion structure will be self-similar.
The long-lasting wind from the central source is most likely to be highly relativistic, with
Lorentz factor much larger than the Lorentz factor of the primary shock. (The models of the long
lasting engine include, for example, formation of a long-lasting neutron star, Metzger et al. (2011),
or a black hole that can retain magnetic field for times much longer than predicted by the no hair
theorem, Lyutikov and McKinney (2011); Lyutikov (2013a).) Also, the wind propagates through
a cavity cleared by the primary shock, so it does not decelerate until it starts interacting with the
material swept-up by the primary shock.
When the high-Lorentz factor secondary wind will catches up with the flow generated by the
primary shock wave it will launch a second shock in the swept-up material, Fig. (1). The flow
betweentheshocks(the“firstshockregion”inFig. (1))isself-similar, describedbyB&Mcsolution
with the self-similar parameter χ. As we demonstrate, the structure of the flow past the second
shock can be approximately self-similar.
– 3 –
Fig. 1.— Cartoon of the problem. The primary shock propagates with Lorentz factor Γ (t). The
1
post-primary shock flow (γ , p and n ) is self-similar, described by the self-similar coordinate χ
1 1 1
of B&Mc. The second shock propagates with Γ (t) and is located at χ = χ (t). Its instantaneous
2 2
Lorentz factor with respect to the shocked flow just in front of it is Γ . The flow parameters
1−2
behind the second shock (γ , p and n ) can be described by the self-similar parameter χ˜.
2 2 2
3. Double explosion in constant density environment
3.1. Governing equations
Assuming a spherically symmetric outflow, the relativistic fluid equations read Landau and
Lifshitz (1959)
∂ (cid:2)wγ2−p(cid:3)+ 1 ∂ (cid:2)r2wβγ2(cid:3) = 0 (1)
t r2 r
∂ (cid:2)wγ2β(cid:3)+ 1 ∂ (cid:2)r2(cid:0)wβ2γ2+p(cid:1)(cid:3)− 2p = 0 (2)
t r2 r r
– 4 –
∂ [ργ]+ 1 ∂ (cid:2)r2ρβγ(cid:3) = 0 (3)
t r2 r
where all notations are standard. (We choose to work consistently with proper quantities, i.e.
measured in the plasma rest frame. One should be careful in comparing our equation with B&M
below.)
Consider relativistic point explosion of energy E in a medium with constant density ρ =
1 ex
m n , followed by another explosion after time t . The second explosion can be a wind. The
p ex d
second shock propagates through relativistically hot and bulk-moving plasma with the self-similar
parameters created by the primary explosion (B&Mc)
(cid:114) (cid:115)
17 E
Γ = 1 t−3/2
1 8π ρ c5
ex
2
p = ρ c2Γ2f (χ)
1 3 ex 1 1
1
γ2 = Γ2g (χ)
1 2 1 1
n = 2n Γ n (χ)
1 ex 1 1
f (χ) = χ−17/12,
1
g (χ) = 1/χ
1
n (χ) = χ−5/4 (4)
1
(subscript 1 indicates that quantities are measured between the two shocks).
Suppose that the second explosion occurs at time t after the initial one and the second shock
d
is moving with the Lorentz factor Γ2 ∝ (t−t )−m (cid:29) Γ2. Then, the location of the second shock
2 d 1
at time t is
(cid:18) (cid:19)
1
R = (t−t ) 1− (5)
2 d 2Γ2(m+1)
2
The corresponding self-similar coordinate of the second shock in terms of the primary shock self-
similar parameter χ is
(cid:18) (cid:19)
R 8t 4
χ = (1+8Γ2)(1− ) ≈ d + Γ2 (6)
2 1 t t (m+1)Γ2 1
2
For t ≥ t/(2(m+1)Γ2)
d 2
8Γ2t
χ = 1 d ∝ t−4. (7)
2
t
Thus, the second shock with time approaches the primary (since χ decreases with time), but as
long as t ≤ 8t Γ2 the second shock is located far down stream of the primary shock, at χ (cid:29) 1.
d 1 2
Thus, the second shock does not affect the bulk of the flow created by the primary shock for a very
long time, much longer that the time between the two explosions. After the second shock catches
up with the primary shock, at time ∼ t Γ2 (cid:29) t , the system behaves as a single explosion with the
d d
– 5 –
sum of energies E +E . On the other hand, for t ≥ t , transient effects associated with the details
1 2 d
of the second explosion die out, so that for times t (cid:28) t (cid:28) 8t Γ2 the structure of the flow after the
d d
second explosion becomes self-similar. The purpose of this paper is to investigate this self-similar
structure.
3.2. Dynamics of the second shock
Before finding the behavior of the post-second shock variables let us do an order of magnitude
estimates of the temporal behavior of the second shock. The total amount of enthalpy created
by the first shock (enthalpy density for ultra-relativistic gas is four times pressure) that ends up
behind the second shock, at χ > χ , is
2
(cid:90) 4π (cid:90) ∞ dχ 16π17/12 (cid:18)c85m 17n17(cid:19)
w = 16π r2pdr = m c5n t3 = p ex t14/3 (8)
2 3 p ex χ17/12 5 175/12 E5t5
χ2 1 d
For a point second explosion, the energy E should equal this integrated enthalpy times Γ2,
2 2
E ≈ w Γ2 (9)
2 2 2
Assuming Γ2 ∝ t−m gives m = 14/3 and the scaling for the second shock
2
(cid:18) E 5E 12t 5 (cid:19)1/12
Γ2 ≈ 1 2 d t−14/3 (10)
2 c85(m n )17
p ex
Thus,
Γ2 E (cid:18)t Γ2(cid:19)5/12 E (cid:18)t (cid:19)5/12
2 ≈ 2 d 1 ≈ 2 d (11)
Γ2 E t E t
1 1 1 ob
where we defined the observer time associated with the first explosion, t ∼ t/(2Γ2). Γ ∝ t−7/12.
ob 1 2 ob
Condition t ≥ t/(2(m+1)Γ2) (for Eq. (7) to apply) requires
d 2
(cid:18)E (cid:19)12/17
t ≤ 0.47 2 t Γ2 . (12)
E d FS
1
This implies that as long as t (cid:28) t Γ2 (well before the second shock catches with the first)
d 1
even energetically subdominant second explosion produces a fast second shock, Γ ≥ Γ . (In order
2 1
to calculate the coefficient in front of this relation we need to know the post-second shock flow;
it is actually large ≈ 125, see (20) so that even energetically subdominant second explosion, with
E ≥ 10−2E can produce fast second shocks up to the catch-up time, ∼ t Γ2.)
2 1 d 1
We stress that the self-similar solution behind the second shock is only approximate. For
example,theenergywithinthesecondshockedregionincreasesevenforapointsecondaryexplosion
– 6 –
since the second shock advances in the self-similar coordinate tied to the first shock. The energy
associated with the first shock, that ends up behind the second shock is
2π (cid:90) ∞
E ≈ t3Γ2 χ−29/12dχ ∝ t17/3 (13)
1,2 3 1
χ2(t)
But for χ (cid:29) 1 only small fraction of the total energy resides behind the second shock; thus, we
2
assume that the energy (9) is much larger than (13)
3.3. Self-similar solutions for the second shock
The second shock is located in terms of the first self-similar coordinate at χ = χ (t); let us
2
consider the post-second shock quantities. Using strong relativistic shock conditions (Landau and
Lifshitz 1959), pressure and density immediately after the second shock (assuming Γ (cid:29) γ (χ ))
2 1 2
are
8 27/4 (cid:18) t (cid:19)5/12 Γ2
p(0) = p Γ2 = ρ c2 2
2 3 1 1−2 9 ex t 5/6
d Γ
1
ρ (cid:18) t (cid:19)3/4 Γ
(0) ex 2
n = 2n (χ )Γ =
2 1 2 1−2 23/4 td Γ3/2
1
Γ
2
Γ = (14)
1−2
2γ (χ )
1 2
Γ is the relative Lorentz factor of the second shock with respect to the local plasma flow in front
1−2
of it, downstream from the first shock.
Next we introduce new self-similar variable associated with the second shock
r r
χ˜ = (1+2(1+m)Γ2)(1− ) ≈ 2(1+m)Γ2(1− ) (15)
2 t−t 2 t
d
where
Γ2 ∝ (t−t )−m ≈ t−m (16)
2 d
(so that χ˜ = 1 at the location of the second shock), we parametrize
p = p(0)Γ2f (χ˜),
2 2 2 2
1
γ2 = Γ2g (χ˜)
2 2 2 2
(0)
n = n Γ n (χ˜) (17)
2 2 2 2
Using anzats (17), expanding in Γ (cid:29) 1 and taking a limit t (cid:29) t , we find
2 d
1 g (3m−17)χ˜−24m+44
2
∂ lnf = −
g χ˜ 2 3(m+1)(cid:0)g2χ˜2−8g χ˜+4(cid:1)
2 2 2
– 7 –
1 g (m+2)χ˜−7m+9
2
∂ lng = −
g χ˜ 2 (m+1)(cid:0)g2χ˜2−8g χ˜+4(cid:1)
2 2 2
1 g2(m−12)χ2+g (67−11m)χ+22m−58
∂ lnn = − 2 2 (18)
g χ˜ 2 2(m+1)(g χ−2)(cid:0)g2χ2−8g χ+4(cid:1)
2 2 2 2
with boundary conditions g (1) = f (1) = 1.
2 2
The point second explosion corresponds to m = 14/3. In this case the post-second shock
variables are
f (χ˜) = χ˜−71/51
2
g (χ˜) = χ˜−1
2
n (χ˜) = χ˜−53/34 (19)
2
Knowing the post-second shock solutions (19) we can calculate the coefficient in (10) for the case
of point secondary explosion, m = 14/3. We find
Γ2 E (cid:18)t Γ2(cid:19)5/12
2 = 0.35 2 d 1
Γ2 E t
1 1
Γ = (cid:114)71 (cid:18)17(cid:19)5/24(cid:18) E15td5 (cid:19)1/24(cid:112)E t−7/3 (20)
2 2 π c85(m n )17 2
p ex
Thus, for E ≥ 10−2E , there is a range for the self-similar solutions to be applicable all the way
2 1
to t ∼ Γ2t (the time when the second shock catches with the primary shock).
1 d
If expressed in terms of the observer time for the first shock t = t/(2Γ2),
ob 1
Γ (cid:114)E (cid:18)t (cid:19)5/24
2 2 d
= 9.6 , (21)
Γ E t
1 1 ob
see Fig. 3.
Thus, for observer’s time (defined with respect to the first explosion) shorter than the delay
between the two explosions, the Lorentz factor of the second may exceed the Lorentz factor of the
primary shock for E ≤ E . What is more, for observer times shorter than the delay time, and not
2 1
too weak second explosions the Lorentz factor of the second shock might exceed the first short by
∼ an order of magnitude.
3.4. Continuos energy injection
For a long-lasting central source producing luminosity L ∝ tq with t ∝ t/Γ2 the energy
w e e 2
deposited into the post-second shock flow scales as ∝ L t/Γ2 ∝ t(1+m)(1+q).Thus,
w 2
11−3q
m = (22)
3(2+q)
– 8 –
For constant luminosity source, q = 0, m = 11/6.
Qualitative estimates for the power absorbed by the shocked medium
t
L tq = w Γ2 (23)
0 emΓ2 2 2
2
with t = t/Γ2 and w given by (8) we confirm m = 11/6. The power balance gives
em 2 2
Γ2 (cid:114)L t (cid:18) t (cid:19)5/24 (cid:114)L t (cid:18) t (cid:19)5/24
2 ≈ w d = w ob d
Γ2 E tΓ2 E t 2
1 1 1 1 ob
√
5/24
L t
Γ = w d (24)
2
(E3n m c5)1/8t1/12
1 ex p ob
Thus, to have Γ ≥ Γ it is required for the applicability of the self-similar approach that the power
2 1
be sufficiently strong,
E
1
L ≥ (25)
w
5/12 7/12
t t
d ob
Since both t and t are ∼ 104 sec, then for the initial explosion of 1052 erg it is required that
d ob
L ∼ 1048 erg sec−1.)
w
For m (cid:54)= 14/3, introducing new variable x = χ˜g , Eqns (18) take the form
2
3m(x−8)−17x+44
∂ lnf =
x 2 3(m(x−4)+x2+17x−4)
m(x−7)+2x+9
∂ lng =
x 2 m(x−4)+x2+17x−4
m(cid:0)x2−11x+22(cid:1)−12x2+67x−58
∂ lnn = (26)
x 2 2(x−2)(m(x−4)+x2+17x−4)
with solutions
−mx+4m−x2−17x+4
Z =
1
3m−14
(cid:16) (cid:17)(cid:16) (cid:17)
1− √ m+19 1+ √ m+2x+17
m2+50m+305 m2+50m+305
Z =
2 (cid:16) (cid:17)(cid:16) (cid:17)
1+ √ m+19 1− √ m+2x+17
m2+50m+305 m2+50m+305
f = Z61(3m−17)Z−6−√3mm22−+8520mm++337075
2 1 2
m+2 √m2+33m+16
g = Z 2 Z2 m2+50m+305
2 1 2
m2−27m+218 m3+4m√2−737m+4636
n2 = (2−x)1m7−+m7 Z1 4(m−17) Z24(m−17) m2+50m+305 (27)
Thecontactdiscontinuity(CD)isatx = 2andthewindterminationshockisatx = 4. Sincefor
the point second explosion case x ≡ 1, these values are never reached in that case, naturally. Thus
– 9 –
solutions(27)shouldnotbeextendedbeyondx = 4. Form = 11/6,thepointx = 2isreachedwhen
χ = 1.82, g = 1.09, f = 0.73. At the termination shock x = 4, χ = 2.76, f = 0.47, g = 1.44.
2 2 2 2
Density is zero on the CD, Fig. 2.
Fig. 2.— Structure of the post-second shock flows for constant luminosity source following a point
explosion in a constant density environment (k = 0, m = 11/6). Second shock is located at χ˜ = 1.
CD denotes the location of the contact discontinuity, RS - of the reverse shock (for the case of fluid
wind, with zero magnetization.)
4. Applications to GRBs
4.1. Early GeV emission
Some GRBs show emission in the Fermi LAT detector, in the 100 MeV - 100 GeV range Abdo
et al. (2009); Ackermann et al. (2014). The LAT photons start arriving during the prompt phase,
and continue well after the prompt phase has ended. The earlier photons have typically lower
– 10 –
Fig. 3.— Evolution of the Lorentz factor of the second shock as a function of observer time t as-
ob
sociated with the primary shock, t = t/(2Γ2). Left Panel: point explosion, Right Panel: constant
ob 1
luminosity source. Lorentz factors are normalized to Γ (t ). Different curves are parametrized by
1 d
the ratio of second to first energies (for point explosions) and the total energy produced in time t
d
bythesource(forconstantluminositycase). Thus: (i)attimest ≤ t thesecondaryshockismuch
ob d
faster than the primary shock even for the case of energetically subdominant second explosion; (ii)
winds are ineffective in driving fast second shocks.
energy, few hundred MeV, while late photons can reach energies ∼ 100 GeV in the explosion rest
frame Ackermann et al. (2014). If the mechanism of GeV photon production is synchrotron, then
such high energy require both very fast rate of particle acceleration and large bulk Lorentz factors,
≥ 103 Lyutikov (2010, 2013b).
Late GeV photons are more naturally explained by the inverse Compton scattering (Be-
loborodov et al. 2014), but the early, lower energy ones can be synchrotron from the fast second
shock. For example, using a standard parametrization for the external shock, γ ∼ (m /m )(cid:15) Γ ,
e p e e 2
√
b2/(8π) = (cid:15) (8/3)m n c2, the typical frequency of the synchrotron photons in the second shock
2 B p ex
can be estimated as
5/6
eb √ √ t
(cid:15) = (cid:126)γ2Γ 2 = 6×10−3(cid:15)2 (cid:15) n Γ4 = 2.5×108 d eV (28)
s e 2m c e B ex 2 7/3
e t
ob
(for (cid:15) = (cid:15) = 0.1). This falls into the LAT range for t ∼ t .
e B d ob