8.7. Shocks with the ISM - External shocks
We turn now to the interaction of a relativistic shell with the
ISM. We have seen in section 7.4 that external
shocks cannot produce bursts with a complicated temporal structure.
Still it is worthwhile to explore this situation. First, there are
some smooth bursts that might be produced in this way. Second, one
needs to understand the evolution of external shocks in order to see
why they cannot satisfy the condition RE /
2
.
Third, it is possible that in some bursts emission is observed from both
internal and external shocks
[253].
Finally, as we see in the following section 9
the observed afterglow is most likely produced by external shocks.
8.7.1. Newtonian vs. Relativistic Reverse Shocks
The interaction between a relativistic flow and an external medium
depends, like in SNRs, on the Sedov length,
l
(E / nism mp
c2)1/3. The ISM rest mass energy within a
volume l3 equals the energy
of the GRB: E. For a canonical cosmological burst with
E
1052 ergs and a typical ISM density
nism = 1 particle/cm3 we have
l
1018 cm. A second length scale that appears in the problem is
, the width of the
relativistic shell in the observer's rest frame.
There are two possible types of external shocks
[233]. They are
characterized according to the nature of the reverse shock: Newtonian
Reverse Shock (NRS) vs. Relativistic Reverse Shock (RRS). If the
reverse shock is relativistic (RRS) then it reduces significantly the
kinetic energy of each layer that it crosses. Each layer within the
shell loses its energy independently from the rest of the shock. The
energy conversion process is over once the reverse shock crosses the
shell (see Fig. 13). A Newtonian
or even mildly
relativistic reverse shock (NRS) is comparatively weak. Such a shock
reduces the energy of the layer that it crosses by a relatively small
amount. Significant energy conversion takes place only after the
shock has crossed the shell several time after it has been reflected
as a rarefraction wave from the inner edge (see
Fig. 12). The shell behaves
practically like
a single object and it loses its energy only by the time that it
accumulates an external mass equal to M /
.
The question which scenario is taking place depends on the parameters of
the shell relative to the parameters of the ISM. As we see shortly it
depends on a single dimensionless parameter
constructed from
l,
and
:
[233]:
![]() |
(92) |
As the shell propagates outwards it is initially very dense and the
density ratio between the shell and the ISM,
f
n4 / n1, is
extremely large (more specifically f >
2).
The reverse shock is initially Newtonian (see Eq. 43). Such a shock
converts only
a small fraction of the kinetic energy to thermal energy. As the
shell propagates the density ratio, f, decreases (like
R-2 if the width of the shell is constant and like
R-3 if the shell is
spreading). Eventually the reverse shock becomes relativistic at
RN where f =
2.
The question where is the kinetic energy
converted depends on whether the reverse shock reaches the inner edge
of the shell before or after it becomes relativistic.
There are four different radii that should be considered. The following estimates assume a spherically symmetric shell, or that E and M are energy and rest mass divided by the fraction of a sphere into which they are launched. The reverse shock becomes relativistic at RN, where f = n4 / n1 = 1:
![]() |
(93) |
Using the expression for the velocity of the reverse shock into the
shell (Eq. 46) we find that the reverse
shock reaches the inner edge of the shell at
R
[233]:
![]() |
(94) |
A third radius is R, where the shell collects an ISM mass of
M /
[27,
18].
For NRS this is where an effective energy release occurs:
![]() |
(95) |
where we defined n1 = nism / 1
particle / cm3. Finally we have
R =
2,
(see Eq. 73). The
different radii are related by the dimensionless parameter
, and
this determines the character of the shock:
![]() |
(96) |
If > 1 then:
![]() |
(97) |
The reverse shock remains Newtonian or at best mildly relativistic
during the whole energy extraction process. The reverse shock reaches
the inner edge of the shock at R while
it is still Newtonian.
At this stage a reflected rarefraction wave begins to move
forwards. This wave is, in turn, reflected from the contact
discontinuity, between the shell's material and the ISM material, and
another reverse shock begins. The overall outcome of these waves is
that in this case the shell acts as a single fluid element of mass
M
E /
c2 that is interacting collectively with the ISM. It
follows from Eq. 39 that an external mass m = M /
is
required to reduce
to
/ 2 and to
convert half of the kinetic
energy to thermal energy. Energy conversion takes place at
R
. Comparison of
R
with Re (equation
27) shows that the optical depth is much smaller than unity.
If the shell propagates with a constant width then RN /
=
R
=
1/2
R
(see Fig. 25) and for
> 1 the
reverse shock remains Newtonian during the energy
extraction period. If there are significant variations in the
particles velocity within the shell it will spread during the
expansion. If the typical variation in
is of the
same order as
then the
shell width increases like R /
2.
Thus
changes with time in such a manner that at each moment the current
width,
(t),
satisfies
(t) ~
max[
(0), R /
2].
This delays the time that the reverse shock reaches the inner edge of
the shell and increases
R
. It also reduces the shell's density which, in
turn, reduces f and leads to a decrease in
RN. The overall result is a triple coincidence
RN
R
R
with a
mildly relativistic reverse shock and a significant energy conversion
in the reverse shock as well. This means that due to spreading a
shell which begins with a value of
> 1 adjusts
itself so as to
satisfy
= 1.
![]() |
Figure 25. (a) Schematic description of the
different radii for the case
|
For
1 we find that
Tradial ~ Tang ~
R
/
2
>
. Therefore,
NRS can produce only smooth bursts. The
bursts' duration is determined by the slowing down time of the shell.
In section 7 we have shown that only one time
scale is possible in this case. Given the typical radius of energy
conversion, R
this time scale is:
![]() |
(98) |
If or
are larger then
< 1. In this
case the order is reversed:
![]() |
(99) |
The reverse shock becomes relativistic very early (see
Fig. 25). Since
sh
=
2
<<
the
relativistic reverse shock converts very efficiently the kinetic
energy of the shell to thermal energy. Each layer of the shell that is
shocked loses effectively all its kinetic energy at once and the time
scale of converting the shell's kinetic energy to thermal energy is
the shell crossing time. The kinetic energy is consumed at
R
,
where the reverse shock reaches the inner edge of the shell. Using
Eq. 94 for R
and Eq. 45 we find that at
R
![]() |
(100) |
Note that
E
is independent of
. The
observed radial or angular time scales are:
![]() |
(101) |
Thus even for RRS we find that
T ~ T and
there is only one time scale. This time scale depends only on
and it is
independent of
! Spreading
does not affect this estimate since
for
< 1
spreading does not occur before the energy extraction.
In the following discussions we focus on the RRS case and we express
all results in terms of the parameter
. By setting
< 1 in
the expressions we obtain results corresponding to RRS, and by
choosing
= 1 in
the same expressions we obtain the spreading NRS
limit. We shall not discuss the case of non-spreading NRS
(
>> 1),
since spreading will always bring these shells to the mildly
relativistic limit
(
~
1). Therefore, in this way, the same formulae are valid for both the RRS
and NRS limits.
If > 1 it
follows from Eq. 97 that internal shocks will
take place before external shocks. If
< 1 then the
condition for internal shocks
R
< R
becomes Eq. 74:
3/2 >
.
As we have seen earlier (see
section 8.6.1) this sets an upper limit on
for
internal shocks.
8.7.2. Physical Conditions in External Shocks
The interaction between the outward moving shell and the ISM takes place in the form of two shocks: a forward shock that propagates into the ISM and a reverse shock that propagates into the relativistic shell. This results in four distinct regions: the ISM at rest (denoted by the subscript 1 when we consider properties in this region), the shocked ISM material which has passed through the forward shock (subscript 2 or f), the shocked shell material which has passed through the reverse shock (3 or r), and the unshocked material in the shell (4). See Fig. 21. The nature of the emitted radiation and the efficiency of the cooling processes depend on the conditions in the shocked regions 2 and 3. Both regions have the same energy density e. The particle densities n2 and n3 are, however, different and hence the effective "temperatures," i.e. the mean Lorentz factors of the random motions of the shocked protons and electrons, are different.
The bulk of the kinetic energy of the shell is converted to thermal
energy via the two shocks at around the time the shell has expanded to
the radius R. At this radius, the conditions at the
forward shock are as follows,
![]() |
(102) |
while at the reverse shock we have
![]() |
(103) |
Substitution of sh =
2 =
3/4 in
Eq. 49 yields:
![]() |
(104) |
If the magnetic field in region 2 behind the forward shock is obtained
purely by shock compression of the ISM field, the field would be very
weak, with
B
<< 1. Such low fields are incompatible with
observations of GRBs. We therefore consider the possibility that
there may be some kind of a turbulent instability which may bring the
magnetic field to approximate equipartition. In the case of the
reverse shock, magnetic fields of considerable strength might be
present in the pre-shock shell material if the original exploding
fireball was magnetic. The exact nature of magnetic field evolution
during fireball expansion depends on several assumptions. Thompson
[224]
found that the magnetic field will remain in equipartition
if it started off originally in equipartition. Mészáros, Laguna
& Rees [240]
on the other hand estimated that if the magnetic
field was initially in equipartition then it would be below
equipartition by a factor of 10-5 by the time the shell expands
to R
. It is uncertain which, if either, is right. As in
the forward shock, an instability could boost the field back to
equipartition. Thus, while both shocks may have
B << 1
with pure flux freezing, both could achieve
B
1
in the presence of instabilities. In principle,
B could be
different for the two shocks, but we limit ourselves to the same
B in both
shocks.
In both regions 2 and 3 the electrons have a power law distribution
with a minimal Lorentz factor
e,min given by Eq. 53 with the corresponding
Lorentz factors for the forward and the reverse shock.
8.7.3. Synchrotron Cooling in External Shocks
The typical energy of synchrotron photons as well as the synchrotron
cooling time depend on the Lorentz factor
e
of the relativistic electrons under consideration and on the strength of
the magnetic field. Using Eq. 53 for
e,min
we find the characteristic synchrotron energy for the forward shock:
![]() |
(105) |
and
![]() |
(106) |
The characteristic frequency and the corresponding cooling time for the "typical" electron are larger by a factor of [(p - 2) / (p - 1)]2 and shorter by a factor of [(p - 2) / (p - 1)]2, correspondingly.
These photons seems to be right in the observed soft
gamma-ray range. However, one should recall that the frequency
calculated in Eq. 105 depends on the forth power of
2.
An increase of the canonical
2
by a factor of 3(that is
2
= 300 instead of
2
= 100) will yield a
"typical" synchrotron emission at the 16 MeV instead of 160 keV.
The Lorentz factor of a "typical electron" in the reverse shock is
lower by a factor
3/2.
Therefore the observed energy is lower by a factor
3
while the cooling time scale is longer by a factor
-3/4.
Alternatively we can check the conditions in order that there are
electrons with a Lorentz factor
e
that be emitting soft gamma-rays with energies ~ 100 keV. Using Eq. 56 we
calculate
e:
![]() |
(107) |
Electrons with
e =
e are
available in the shocked material if
e,min <
e.
This corresponds to the condition
![]() |
(108) |
in the reverse shock, and the condition
![]() |
(109) |
in the forward shock. Since by definition
e
1, we see
that the reverse shock always has electrons with the right Lorentz
factors to produce soft gamma-ray synchrotron photons. However, the
situation is marginal in the case of the forward shock. If
> 100
and if the heating of the electrons is efficient, i.e. if
e |f ~ 1, then most of the electrons
may be too energetic. Of course, as an electron cools, it radiates at
progressively softer energies. Therefore, even if
min
is initially too large for the synchrotron radiation to be in soft
gamma-rays, the same electrons would at a later time have
e
~
e
and become visible. However, the energy
remaining in the electrons at the later time will also be lower (by a
factor
/
min), which means that the burst will
be inefficient. For simplicity, we ignore this radiation.
Substituting the value of
e
from equation 107 into the cooling rate Eq. 58
we obtain the cooling time scale as a function of the observed photon
energy to be
![]() |
(110) |
Eq. 110 is valid for both the forward and reverse shock, and is moreover independent of whether the reverse shock is relativistic or Newtonian.
The cooling time calculated above sets a lower limit to the variability time scale of a GRB since the burst cannot possibly contain spikes that are shorter than its cooling time. However, it is unlikely that this cooling time actually determines the observed time scales.