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8.2. Synchrotron Emission from Relativistic Shocks

8.2.1. General Considerations

The most likely radiation process in GRBs is synchrotron emission [240, 18, 222, 103]. The observed low energy spectra provide an indication that this is indeed the case [95, 241].

The parameters that determine synchrotron emission are the magnetic field strength, B, and the electrons' energy distribution (characterized by the minimal Lorentz factor gammae,min and the index of the expected power-law electron energy distribution p). These parameters should be determined from the microscopic physical processes that take place in the shocks. However, it is difficult to estimate them from first principles. Instead we define two dimensionless parameters, epsilonB and epsilone, that incorporate our ignorance and uncertainties [222, 103].

The dimensionless parameter epsilonB measures the ratio of the magnetic field energy density to the total thermal energy e:

Equation 48 (48)

so that, after substituting the shock conditions we have:

Equation 49 (49)

There have been different attempts to estimate epsilonB [224, 240, 226]. We keep it as a free parameter. Additionally we assume that the magnetic field is randomly oriented in space.

The second parameter, epsilone, measures the fraction of the total thermal energy e which goes into random motions of the electrons:

Equation 50 (50)

8.2.2. The Electron Distribution

We call consider a "typical" electron as one that has the average gammae of the electrons distribution:

Equation 51 (51)

Collisionless acceleration of electrons can be efficient if they are tightly coupled to the protons and the magnetic field by mean of plasma waves [243]. Since the electrons receive their random motions through shock-heating, we assume (following the treatment of non-relativistic shocks) that they develop a power law distribution of Lorentz factors:

Equation 52 (52)

We require p > 2 so that the energy does not diverge at large gammae. Since the shocks are relativistic we assume that all the electrons participate in the power-law, not just a small fraction in the tail of the distribution as in the Newtonian case. An indication that this assumption is correct is given by the lower energy spectrum observed in some GRBs [95, 241]. The minimum Lorentz factor, gammae,min, of the distribution is related to epsilone and to the total energy e ~ gammash n mpc2:

Equation 53 (53)

where gammash is the relative Lorentz factor across the corresponding shock.

The energy index p can be fixed by requiring that the model should be able to explain the high energy spectra of GRBs. If we assume that most of the radiation observed in the soft gamma-rays is due to synchrotron cooling, then it is straightforward to relate p to the power-law index of the observed spectra of GRBs, beta. The mean spectral index of GRBs at high photon energies beta approx - 2.25, [84] corresponds to p approx 2.5. This agrees, as we see later (9.3.2) with the value inferred from afterglow observations (p ~ 2.25). We assume this value of p in what follows. The corresponding ratio that appears in Eq. 53 (p - 2) / (p - 1) equals 1/3 and we have gammae,min = 610 gammash.

The shock acceleration mechanisms cannot accelerate the electrons to arbitrary high energy. For the maximal electron's energy, with a corresponding gammae,max, the acceleration time equals to the cooling time. The acceleration time is determined by the Larmor radius RL and the Alfvén velocity vA [244]:

Equation 54 (54)

This time scale should be compared with the synchrotron cooling time gammae me c2 / Psyn (in the local frame). Using vA2 approx epsilonB2, Eq. 49 to estimate B and Eq. 57 below to estimate Psyn one finds:

Equation 55 (55)

cooling This value is quite large and generally it does not effect the observed spectrum in the soft gamma ray range.

8.2.3. Synchrotron Frequency and Synchrotron Power

The typical energy of synchrotron photons as well as the synchrotron cooling time depend on the Lorentz factor gammae of the relativistic electron under consideration and on the strength of the magnetic field (see e.g. [245]). Since the emitting material moves with a Lorentz factor gammaE the photons are blue shifted. The characteristic photon energy in the observer frame is given by:

Equation 56 (56)

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

Equation 57 (57)

where sigmaT is the Thomson cross section. The cooling time of the electron in the fluid frame is then gammae me c2 / P. The observed cooling time tsyn is shorter by a factor of gammaE:

Equation 58 (58)

Substituting the value of gammae from equation 56 into the cooling rate Eq. 58 we obtain the cooling time scale as a function of the observed photon energy:

Equation 59 (59)

Since gammae does not appear explicitly in this equation tsyn 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 gammae 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 tsyn(nu) propto nu-1/2. This is not very different from the observed relation deltaT propto nu-0.4 [102]. However, it is not clear if the cooling time and not another time scale determined 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 exponential decline (FRED). 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.

8.2.4. The Integrated Synchrotron Spectrum

The instantaneous synchrotron spectrum of a single electron with an initial energy gammae me c2 is a power law with Fnu propto nu1/3 up to nusyn(gammae) and an exponential decay above it. If the electron is energetic it will cool rapidly until it will reach gammae,c. This is 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 above hsyn(gammae,c): Fnu propto nu-1/2 from nusyn(gammae,c) up to nusyn(gammae).

To calculate the overall spectrum due to all the electrons we need to integrate over gammae. Our discussion here follows [246]. We consider a power-law electron distribution with a power index p and a minimal Lorentz factor gammae,min (see Eq. 52). Overall we expect a broken power law spectrum with a break frequency around the synchrotron frequency of the lowest energy electrons nusyn(gammae,min). These power law indices depend on the cooling rate. The most energetic electrons will always be cooling rapidly (independently of the behavior of the "typical electron"). Thus the highest spectrum is always satisfy:

Equation 60 (60)

similarly the low energy electrons will always be slow cooling and thus the lowest part of the spectrum will behave like Fnu propto nu1/3.

For slow cooling we have the instantaneous spectrum: Fnu propto nu1/3 for the lower part of the spectrum. For the upper part we have

Equation 61 (61)

where gamma(nu) is the Lorentz factor for which the synchrotron frequency equals nu. The most energetic electrons will cool rapidly even when the overall system is in slow cooling. These electrons emit practically all their energy me c2 gamma, at their synchrotron frequency. Thus the uppermost part of the spectrum will satisfy:

For fast cooling we have Fnu propto nu-1/2 for the lower part and Fnu propto nu-p/2 for the upper part. Here at the lower end the least energetic electrons will be cooling slowly even when the typical electron is cooling rapidly. Thus we will have fnu propto nu1/3 in the lowest part of the spectrum.

The critical parameter that determines if the electrons are cooling fast or slow is gammae,c, the Lorentz factor of an electron that cools on a hydrodynamic time scale. To estimate gammae,c we compare tsyn (Eq. 58) with thyd, the hydrodynamic time scale (in the observer's rest frame):

Equation 62 (62)

Fast cooling occurs if gammae,c < gammae,min. All the electrons cool rapidly and the electrons' distribution effectively extends down to gammae,c. If gammae,c > gammae,min only the high energy tail of the distribution (those electrons above gammae,c) cool and the system is in the slow cooling regime.

For the GRB itself we must impose the condition of fast cooling: the relativistic shocks must emit their energy effectively - otherwise there will be a serious inefficiency problem. Additionally we won't be able to explain the variability if the cooling time is too long. The electrons must cool rapidly and release all their energy. In this case gammae,min > gammae,c [103] and all the electrons cool down roughly to gammae,c. The observed flux, Fnu, is given by:

Equation 63 (63)

where num ident nusyn(gammae,min), nuc ident nusyn(gammae,c) and Fnu, max is the observed peak flux.

It is most likely that during the latter 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 [21, 23, 47, 247, 25]. When gammac > gammae,min, only those electrons with gammae > gammac can cool. We call this slow cooling, because the electrons with gammae ~ gammae,min, which form the bulk of the population, do not cool. Integration over the electron distribution gives in this case:

Equation 64 (64)

For fast cooling nuc < num. We find that the peak flux is at nuc while the peak energy emitted (which corresponds to the peak of nu Fnu) is at num. For slow cooling the situation reverses num < nuc. The peak flux is at num while the peak energy emitted is at nuc.

Typical spectra corresponding to fast and slow cooling are shown in Fig. 22. The light curve depends on the hydrodynamic evolution, which in turn determines the time dependence of num, nuc and Fnu, max.

Figure 22

Figure 22. Synchrotron spectrum of a relativistic shock with a power-law electron distribution. (a) Fast cooling, which is expected at early times (t < t0). The spectrum consists of four segments, identified as A, B, C, D. Self-absorption is important below nua. The frequencies, num, nuc, nua, decrease with time as indicated; the scalings above the arrows correspond to an adiabatic evolution, and the scalings below, in square brackets, to a fully radiative evolution. (b) Slow cooling, which is expected at late times (t > t0). The evolution is always adiabatic. The four segments are identified as E, F, G, H. From [246].

For fast cooling the power emitted is simply the power given to the electrons, that is epsilone times the power generated by the shock:

Equation 65 (65)

For slow cooling the emitted power is determined by the ability of the electrons to radiate their energy:

Equation 66 (66)

where, Ne is the number of electrons in the emitting region and Psyn(gammae,min), the synchrotron power of an electron with gammae,min, is given by Eq. 57.

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