**B. Internal Shocks**

**1. Hydrodynamics of Internal Shocks**

Internal shocks take place when a faster shell catches a slower one, namely at:

(42) |

where _{100} is
the typical Lorentz factor in units of 10^{2} and
*t* is
the time difference between the emission of the two shells. I show later
that *t*
defined here is roughly equal to the observed fluctuations in the
light curve of the burst
*t*. Clearly
*R*_{int} < *R*_{ext}
must hold otherwise internal shocks won't take
place. *R*_{ext} is
defined as the location of efficient extraction of energy by
external shocks (see Section VIC). If
follows from the
discussion in Section VIC that the condition
*R*_{int} < *R*_{ext} implies:

(43) |

where *l* is defined by Eq. 58 and it is typically of
the order of 10^{18}cm, while
is the width of the
shell and it is of order 10^{12}cm. Both conditions set upper
limits on (of the
order of a few thousands) for internal
shocks. If the initial Lorentz factor is too large then internal
shocks will take place at large radii and external shocks will
take place before the internal shocks could take place. It is
possible that this fact plays an important role in limiting the
relevant Lorentz factors and hence the range of variability of
*E*_{p}, the peak energy observed in GRBs.

Internal shocks are characterized by a comparable Lorentz factor
of order of a few (1 <
< 10)
reflecting the relative motion
of the shells and by comparable densities *n* in both shells. In
this case, for an adiabatic index (4/3), the Loretz factor of the
shocked region
satisfies:

(44) |

The shocked density and energy are:

(45) |

Both shocks are mildly relativistic and their strength depends on the relative Lorentz factors of the two shells.

**2. The Efficiency of Internal Shocks**

Consider collision between two shells with masses *m*_{r} and
*m*_{s} that are moving at different relativistic velocities:
_{r}
_{s}
>> 1. The resulting bulk Lorentz factor,
_{m} in an
elastic collision is:

(46) |

The internal energy,
_{int}, in the
local frame and
*E*_{int}, in the frame of an external observer, of the merged
shell: *E*_{int} =
_{m}
_{int}, is the
difference of the kinetic energies before and after the collision:

(47) |

The conversion efficiency of kinetic energy into internal energy is [193]:

(48) |

As can be expected a conversion of a significant fraction of the
initial kinetic energy to internal energy requires that the
difference in velocities between the shells will be significant:
_{r} >>
_{s} and
that the two masses will be comparable
*m*_{r}
*m*_{s}
[67,
193].

Beloborodov
[19]
considered internal shocks
between shells with a lognormal distribution of
( - 1) /
(_{0} - 1),
where _{0}
is the average Lorentz
factor. The dimensionless parameter, *A*, measures the width of
the distribution. He shows that the efficiency increases and
reached unity when *A* is of order unity, that is typical
fluctuation in are
by a factor of 10 compared to the average. Similarly numerical
simulations of Guetta et al.
[154]
show that a significant fraction
of the wind kinetic energy, on the order of 20%, can be converted
to radiation, provided the distribution of Lorentz factors within
the wind has a large variance and the minimum Lorentz factor is
greater than
10^{2.5}
*L*^{2/9}_{52}, where *L*_{52} is the
(isotropic) wind luminosity in units of 10^{52} ergs/sec.

Another problem that involves the efficiency of GRBs is that not
all the internal energy generated is emitted. This depends further on
_{e}, the
fraction of energy given to the electron. If
this fraction is small and if there is no strong coupling between
the electrons and the protons the thermal energy of the shocked
particles (which is stored in this case mostly in the protons)
will not be radiated away. Instead it will be converted again to
kinetic energy by adiabatic cooling. Kobayashi and Sari
[195]
consider a more elaborated model in which colliding shells that do
not emit all their internal energy are reflected from each other,
causing subsequent collisions and thereby allowing more energy to
be emitted. In this case more energy is eventually emitted than
what would have been emitted if we considered only the first
collision. They obtain about 60% overall efficiency even if the
fraction of energy that goes to electrons is small
_{e} =
0.1. This is provided that the shells' Lorentz factor
varies between 10 and 10^{4}.

**3. Light Curves from Internal Shocks**

Both the similarity between the pulse width and
the pulse separation distribution and the correlation between
intervals and the subsequent pulses
[270,
327]
arise naturally within the internal shocks model
[268].
In this model both the
pulse duration and the separation between the pulses are
determined by the same parameter - the interval between the
emitted shells. I outline here the main argument (see
Nakar and Piran
[268]
for details). Consider two shells with a
separation *L*. The Lorentz factor of the slower outer shell is
_{S} =
and of the Lorentz
factor inner faster shell is
_{F} =
*a* (*a*
> 2 for an efficient collision).
Both are measured in the observer frame. The shells are ejected at
*t*_{1} and
*t*_{2}
*t*_{1} + *L* / *c*. The collision takes place
at a radius
*R*_{s}
2^{2}
*L* (Note that *R*_{s} does not depend on
^{2}). The
emitted photons from the
collision will reach the observer at time (omitting the photons
flight time, and assuming transparent shells):

(49) |

The photons from this pulse are observed almost simultaneously
with a (hypothetical) photon that was emitted from the "inner
engine" together with the second shell (at *t*_{2}). This
explains why various numerical simulations
[67,
193,
293]
find that for internal shocks the observed light curve replicates the
temporal activity of the source.

In order to determine the time between the bursts we should
consider multiple collisions. It turns out that there are just
three types of collisions, (i), (ii) and (iii), that characterize
the system and all combinations of multiple collisions can be
divided to these three types. Consider four shells emitted at
times *t*_{i} (*i* = 1, 2, 3, 4) with a separation of
the order of *L* between them. In type (i) there are two collisions -
between the first and the second shells and between the third and
the fourth shells. The first collision will be observed at
*t*_{2} while the second one will be observed at
*t*_{4}. Therefore,
*t*
*t*_{4}
- *t*_{2}
2*L* /
*c*. A different collision
scenario (ii) occurs if the second and the first shells collide,
and afterward the third shell takes over and collide with them
(the forth shell does not play any roll in this case). The first
collision will be observed at *t*_{2} while the second one will
be observed at *t*_{3}. Therefore,
*t*
*t*_{3}
- *t*_{2}
*L* /
*c*. Numerical simulations
[268]
show that more then 80% of the efficient
collisions follows one of these two scenarios ((i) or (ii)).
Therefore one can estimate:

(50) |

Note that this result is independent of the shells' masses.

A third type of a multiple collision (iii) arises if the third
shell collides first with the second shell. Then the merged shell
will collide with the first one (again the fourth shell does not
participate in this scenario). In this case the two pulses merge
and will arrive almost simultaneously, at the same time with a
(hypothetical) photon that would have been emitted from the inner
engine simultaneously with the third (fastest) shell.
*t* ~ *t*_{3}. Only a 20% fraction exhibits this type
of collision.

The pulse width is determined by the angular time (ignoring the
cooling time): *t* =
*R*_{s} / (2*c* _{s}^{2}) where
_{s} is the
Lorentz factor of the shocked emitting region. If the shells have an
equal mass (*m*_{1} = *m*_{2}) then
_{s} =
*a*
while if they have
equal energy (*m*_{1} = *a m*_{2}) then
_{s}
. Therefore:

(51) |

The ratio of the Lorentz factors *a*, determines the
collision's efficiency. For efficient collision the variations in
the shells Lorentz factor (and therefore *a*) must be large.

It follows from Eqs. 50 and 51 that for equal energy shells the
*t* -
*t* similarity and
correlation arises naturally from the reflection of the shells
initial separation in both variables. However, for equal mass
shells *t* is
shorter by a factor of *a* than
*t*. This
shortens the pulses relative to the intervals.
Additionally, the large variance of *a* would wipe off the
*t* -
*t*
correlation. This suggests that equal energy
shells are more likely to produce the observed light curves.