**C. Synchrotron**

Synchrotron radiation play, most likely, an important role in both the GRB and its afterglow. An important feature of synchrotron emission is its polarization (see Section VF). Observations of polarization in GRB afterglows and in one case in the prompt emission support the idea that synchrotron emission is indeed taking place there (note however that IC also produces polarized emission). I review here the basic features of synchrotron emission focusing on aspects relevant to GRBs. I refer the reader to Rybicki and Lightman [361] for a more detailed discussion.

The typical energy of synchrotron photons as well as the
synchrotron cooling time depend on the Lorentz factor
_{e}
of the relativistic electron under consideration and on the
strength of the magnetic field . If the emitting material moves
with a Lorentz factor
the photons are
blue shifted. The
characteristic photon energy in the observer frame is given by:

(16) |

where *q*_{e} is the electron's charge.

The power emitted, in the local frame, by a single electron due to synchrotron radiation is:

(17) |

where *U*_{B}
*B*^{2} / 8
_{B}
*e* is the magnetic energy density and
_{T} is the
Thompson cross section. The cooling time of the electron in the fluid
frame is then
_{e}
*m*_{e} *c*^{2} / *P*. The observed
cooling time *t*_{syn} is shorter by a
factor of :

(18) |

Substituting the value of
_{e}
from equation 16 into the cooling rate Eq. 18 one obtains the cooling
time scale as a function of the observed photon energy:

(19) |

Since
_{e}
does not appear explicitly in this equation
*t*_{syn} at a given observed frequency is independent of the
electrons' energy distribution within the shock. This is
provided, of course, that there are electrons with the required
_{e}
so that there will be emission in the frequency
considered. As long as there is such an electron the cooling
time is "universal". This equation shows a characteristic scaling of
*t*_{syn}()
^{-1/2}. This is not very
different from the observed relation
*T*
^{-0.4}
[93].
However, it is unlikely that
cooling and not a physical process determines the temporal profile.

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. Observations of GRBs typically show asymmetric spikes in the intensity variation, where a peak generally has a fast rise and a slower decay. A plausible explanation of this observation is that the shock heating of the electrons happens rapidly (though episodically), and that the rise time of a spike is related to the heating time. The decay time is then set by the cooling, so that the widths of spikes directly measure the cooling time. However, it seems that there are problems with this simple explanation. First when plugging reasonable parameters one finds that the decay time as implied by this equation is too short. Second, if the cooling time is long the shocked region would suffer adiabatic losses and this would reduce the efficiency of the process. Thus it is unlikely that the pulse shapes can be explained by Synchrotron physics alone.

**2. The Optically thin Synchrotron Spectrum**

The instantaneous synchrotron spectrum of a single relativistic
electron with an initial energy
_{e}
*m*_{e} *c*^{2} is approximately a power law with
*F*_{}
^{1/3} up to
_{syn}(_{e})
and an exponential decay above it. The peak power occurs at
_{syn}(_{e}),
where it has the approximate value

(20) |

Note that
*P*_{, max} does
not depend on
_{e},
whereas the position of the peak does.

If the electron is energetic it will cool rapidly until it will reach
_{e,c}, the Lorentz factor of an electron that
cools on a hydrodynamic time scale. For a rapidly cooling electron we
have to consider the time integrated spectrum. For an initial
Lorentz factor
_{e}:
*F*_{}
^{-1/2} for
_{syn}(_{e,c})
< <
_{syn}(_{e}).

To calculate the overall spectrum due to the electrons one needs
to integrate over the electron's Lorentz factor distribution. I
consider first, following
[375],
a power-law distribution a power index *p* and a minimal Lorentz factor
_{e,min}. This is, of course, the simplest
distribution and as discussed in Section VB
this is the expected distribution of shock accelerated particles:

(21) |

The condition *p* > 2 is required so that the energy does not
diverge at large
_{e}
(Bhattacharya
[28],
Dai and Cheng
[64]
consider also distributions with 2 > *p* > 1 with a maximal
energy cutoff, see below). The minimum Lorentz factor,
_{e,min},
of the distribution is related to the electron's energy density
*e*_{e} and the electron's number density
*n*_{e} as:

(22) |

The minimal Lorentz factor plays an important role as it
characterizes the `typical' electron's Lorentz factor and the
corresponding `typical' synchrotron frequency,
_{m}
_{syn}(_{e,min}). Interestingly the upper energy cutoff
(which essentially exists somewhere) does not play a critical
role in the spectrum for *p* > 2. Of course it will lead to a high
frequency cutoff of the spectrum around
_{syn} that
corresponds to this energy. However, quite generally, this happens
at the high energy tail far from where the peak flux or the peak
energy are emitted.

A simple modification of the above idea arises if only a fraction,
_{e}, of
the electrons is accelerated to high energies and the
rest of the electrons remain cold
[47,
154].
If a small fraction of electrons shares the energy *e*_{e}
then the typical Lorentz factor would be
_{e}^{-1}
_{e,min}, where
_{e,min} is
calculated from Eq. 22 above. All the corresponding
places where _{e,min} is used should be modified according to
this factor. At the same time fewer electrons will be radiating.
This will introduce a factor
_{e} that
should multiply the
total emitted flux. In the following discussion I will not add
this factors into the analysis. Similarly in situations when
multiple pair are formed
[124]
the electron's energy is shared by a larger number of electron. In
this case
_{e}
is larger than unity and similar modifications of the spectrum applies.

The lowest part of the spectrum (strictly speaking the lowest part
of the optically thin spectrum, as at very low frequencies self
absorption sets in, see Section VC3 below) is
always the sum of the contributions of the tails of all the electron's
emission: *F*_{}
^{1/3}. This is
typical to synchrotron
[55,
184,
258]
and is independent of the exact shape of the electron's distribution.
Tavani
[401,
402],
for example obtain such a low
energy spectrum both for a Gaussian or for a Gaussian and a high
energy power-law tail. The observation of bursts (about 1/5 of the
bursts) with steeper spectrum at the lower energy part, i.e. below
the "synchrotron line of death"
[321,
322]
is one of the problems that this model faces. The problem is even
more severe as in order that the GRB will be radiating
efficiently, otherwise the efficiency will be very low, it must be
in the fast cooling regime and the relevant low energy spectrum
will be
^{-1/2}
[55,
125].
However, as
stressed earlier (see Section IIA1) this
problem is not
seen in any of the HETE spectrum whose low energy tail is always
in the proper synchrotron range with a slope
[16]
and it might be an artifact of the low energy resolution of BATSE
in this energy range
[55].

On the other hand the most energetic electrons will always be
cooling rapidly (independently of the behavior of the "typical
electron"). These electrons emit practically all their energy
*m*_{e} *c*^{2}
, at their
synchrotron frequency. The number of electrons with Lorentz factors
~ is
^{1-p}
and their energy
^{2-p}.
As these electrons cool, they deposit most of their energy into a
frequency range
~ _{syn}()
^{2}
and therefore
*F*_{}
^{-p}
^{-p/2}. Thus the
uppermost part of the spectrum will satisfy:

(23) |

In the intermediate frequency region the spectrum differs between
a `slow cooling' if the `typical' electrons with
_{e,min} do
not cool on a hydrodynamic time scale and `fast cooling' if they
do. The critical parameter that determines if the electrons are
cooling fast or slow is _{e,c}, the Lorentz factor of an
electron that cools on a hydrodynamic time scale. To estimate
_{e,c} compare *t*_{syn} (Eq. 18) with
*t*_{hyd},
the hydrodynamic time scale (in the observer's rest frame):

(24) |

For fast cooling _{e,min} <
_{e,c}, while
_{e,min} >
_{e,c} for slow cooling. In the following
discussion two important frequencies play a dominant role:

(25) |

These are the synchrotron frequencies of electrons with
*e*,
*min* and with
_{e,c}.

**Fast cooling ( _{e,c} <
_{e,min}):** The typical electron is cooling
rapidly hence

(26) |

where _{m}
_{syn}(_{e,min}),
_{c}
_{syn}(_{e,c})
and *F*_{, max} is
the observed peak flux. The peak flux is at
_{c}
*F*_{, max}
*N*_{e}
*P*_{, max} /
4 *D*^{2} (where
D is the distance to the source and I
ignore cosmological corrections). The power emitted is simply the
power given to the electrons, that is
_{e} times the
power generated by the shock, *dE* / *dt*:

(27) |

The peak energy emitted (which corresponds to the peak of
*F*_{}) is at
_{m}. The resulting
spectrum is shown in Fig. 23.

**Slow cooling ( _{e,c} >
_{e,min}):** Now only the high
energy tail of the distribution (those electrons above

(28) |

where () is the Lorentz factor for which
the synchrotron frequency equals
,
*N*[]
is the number of electrons with a Lorentz factor
and
*P*[]
the power emitted by an electron with
. Overall
one finds:

(29) |

The peak flux is at
_{m} while the peak
energy emitted is at
_{c}. The emitted
power is determined by the ability of the electrons to radiate their energy:

(30) |

where, *N*_{e} is the number of electrons in the emitting
region and *P*_{syn}(_{e,min}), the synchrotron power of an
electron with _{e,min}, is given by Eq. 17.

Typical spectra corresponding to fast and slow cooling are shown
in Fig. 23. The light curve
depends on the
hydrodynamic evolution, which in turn determines the time
dependence of _{m},
_{c} and
*F*_{, max}. The
spectra presented here are composed of broken power laws.
Granot and Sari
[144]
present a more accurate spectra in which
the asymptotic power law segments are connected by smooth curves.
They fit the transitions by [( /
_{b})^{-n1}
+ ( /
_{b})^{-n2}]^{-1/n}. The parameter *n*
estimates the smoothness of the transition with
*n* 1 for all
transitions.

Fast cooling must take place during the GRB itself: the relativistic shocks must emit their energy effectively - otherwise there will be a serious inefficiency problem. Additionally the pulse won't be variable if the cooling time is too long. The electrons must cool rapidly and release all their energy. It is most likely that during the early stages of an external shock (that is within the afterglow phase - provided that it arises due to external shocks) there will be a transition from fast to slow cooling [187, 259, 261, 430, 431].

Tavani
[401,
402]
discusses the synchrotron spectrum
from a Gaussian electron distribution and from a Gaussian electron
distribution with a high energy tail. As mentioned earlier the
Gaussian (thermal) distribution has a typical low frequency
^{1/3}
spectrum. However, as expected, there is a sharp
exponential cutoff at high frequencies. Without a high energy
tail this spectrum does not fit the observed GRB spectra of most
GRBs (see Section IIA1).
Note, however, that it may fit a small subgroup with a NHE
[296].
With an electron distribution composed of a Gaussian and an added high
energy tail the resulting spectra has the typical
^{1/3}
component and an additional high energy tail which depends on the
electrons power law index. Such a spectra fits several observed GRB
spectra
[401,
402].

Another variant is the synchrotron spectrum from a power-law
electron distribution with 1 < *p* < 2
[28,
64].
In this case there must be a high energy cutoff
_{e,max} and the `typical' electron's
energy corresponds to this upper cutoff. A possible cutoff can
arise from Synchrotron losses at the energy where the
acceleration time equals to the energy loss time (see e.g.
de Jager et al.
[73]
and the discussion in Section VB):

(31) |

The resulting "typical" Lorentz factor
_{e,min} differs
now from the one given by Eq. 22. Bhattacharya
[28],
Dai and Cheng
[64]
find that it is replaced with:

(32) |

The resulting spectrum is now similar to the one obtained for
fast or slow cooling with the new critical frequencies
_{m}
given by plugging the result of Eq. 32 into Eq. 26.

**3. Synchrotron Self-Absorption**

At low frequencies synchrotron self-absorption may take place. It
leads to a steep cutoff of the low energy spectrum, either as the
commonly known ^{5/2}
or as
^{2}. To estimate the self
absorption frequency one needs the optical depth along the line of
sight. A simple approximation is:
'_{'} *R* /
where
'_{'} is the absorption
coefficient
[361]:

(33) |

The self absorption frequency
_{a} satisfies:
'_{'0}
*R* / = 1. It
can be estimates only once we
have a model for the hydrodynamics and how do *R* and
vary with time
[142,
439].

The spectrum below the the self-absorption frequency depends on
the electron distribution. One obtains the well known
[361],
^{5/2} when the
synchrotron
frequency of the electron emitting the self absorbed radiation is
inside the self absorption range. One obtains
^{2} if the
radiation within the self-absorption frequency range is due to
the low energy tail of electrons that are radiating effectively at
higher energies. For this latter case, which is more appropriate
for GRB afterglow (for slow cooling with
_{m} <
_{c})
[184,
187,
258,
288]:

(34) |

where *R* is the radius of the radiating shell and the factor
*k*_{B} *T*_{e} /
(
*m*_{p} *c*^{2}) describes the degree of electron
equipartition in the plasma shock-heated to an internal energy per
particle *m*_{p} *c*^{2} and moving with
Lorentz factor
.

The situation is slightly different for a shock heated fast
cooling i.e. if _{c} <
_{m}
[143].
In this case we
expect the electron's distribution to be inhomogeneous, as
electrons near the shock did not cool yet but electrons further
downstream are cool. This leads to a new spectral range
_{sa} <
<
_{sa'} with
*F*_{}
^{11/8} (see
Fig. 23).

Synchrotron self-absorption is probably irrelevant during the GRB itself. Note, however, that under extreme conditions the self absorption frequency might be in the low X-ray and this may explain the steep low energy spectra seen in some bursts. These extreme conditions are needed in order to make the system optically thick to synchrotron radiation but keeping it optically thin to Thompson scattering and pair creation [361]. Self absorption appears regularly during the afterglow and is observed typically in radio emission [142, 184, 187, 430, 439]. The expected fast cooling self-absorbed spectra may arise in the early radio afterglow. So far it was not observed.