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8.6. Internal Shocks

Internal shocks are the leading mechanism for energy conversion and production of the observed gamma-ray radiation. We discuss, in this section, the energy conversion process, the typical radiation frequency and its efficiency.

8.6.1. Parameters for Internal Shocks

Internal shocks take place when an inner shell overtakes a slower outer shell. Consider a fast inner shell with a Lorentz factor gammar that collides with a slower shell whose Lorentz factor is gammas. If gammar gtapprox gammas ~ gamma then the inner shell will overtake the outer one at:

Equation 73 (73)

where delta is the initial separation between the shells in the observer's rest frame and 10 = delta / 1010 cm and gamma100 = gamma / 100. Clearly internal shocks are relevant only if they appear before the external shock that is produced as the shell sweeps up the ISM. We show in section 8.7.1 that the necessary condition for internal shocks to occur before the external shock is:

Equation 74 (74)

where xi and zeta are two dimensionless parameters. The parameter, xi, characterizes the interaction of the flow with the external medium and it is defined in Eq. 92 (see section 8.7.1). The second parameter, zeta, characterizes the variability of the flow:

Equation 75 (75)

We have seen in section 7.4 that for internal shocks the duration of the burst T approx Delta / c and the duration of individual spikes deltaT approx delta / c. The observed ratio N defined in section 2.2 must equal 1 / zeta and this sets zeta approx 0.01.

The overall duration of a burst produced by internal shocks equals Delta / c. Thus, whereas external shocks require an extremely large value of gamma to produce a very short burst, internal shocks can produce a short burst with a modest value of the Lorentz factor gamma. This eases somewhat the baryon purity constraints on source models. The condition 74 can be turned into a condition that gamma is sufficiently small:

Equation 76 (76)

where we have used T = Delta / c and we have defined T10 = T / 10 s and zeta0.01 = zeta / 0.01. It follows that internal shocks take place in relatively "low" gamma regime. Fig. 23 depicts the regimes in the physical parameter space (Delta, gamma) in which various shocks are possible. It also depicts an example of a T = Delta / c = 1s line.

Figure 23

Figure 23. Different scenarios in the Delta (in cm) - gamma plane for zeta ident delta / Delta = 0.01. Relativistic ES occur for large Delta and large gamma - upper right -above the xi = 1 line (dark gray and light gray regions). Newtonian ES occur below xi = 1 - lower left - white region. IS occur, if there are sufficient variation in gamma below the xi = zeta2/3 line (light gray and white regions). The equal duration T = 1s curve is shown for Newtonian ES (solid line) a relativistic ES (dotted line) and IS (dashed line). Note that a relativistic ES and an internal shock with the same parameters have the same overall duration T but different temporal substructure depending on delta. From [232].

Too low a value of the Lorentz factor leads to a large optical depth in the internal shocks region. Using Eq. 27 for Re, at which the optical depth for Compton scattering of the photons on the shell's electrons equals one, Eq. 73 for Rdelta and the condition Re leq Rdelta we find:

Equation 77 (77)

In addition, the radius of emission should be large enough so that the optical depth for gamma gamma -> e+ e- will be less than unity (taugammagamma < 1). There are several ways to consider this constraint. The strongest constraint is obtained if one demands that the optical depth of an observed high energy, e.g. 100MeV photon will be less than unity [210, 211]. Following these calculations and using Eq. 73 to express Rdelta we find:

Equation 78 (78)

This constraint, which is due to the gamma gamma interaction, is generally more important than the constraint due to Compton scattering: that is taugammagamma > taue.

Eq. 76, and the more restrictive Eq. 78 constrains gamma to a relatively narrow range:

Equation 79 (79)

This can be translated to a rather narrow range of emission radii:

Equation 80 (80)

In Fig. 24, we plot the allowed regions in the gamma and delta parameter space. Using the less restrictive taue limit 77 we find:


Three main conclusions emerge from the discussion so far. First, if the spectrum of the observed photons extends beyond 100 MeV (as was the case in the bursts detected by EGRET [83]) and if those high energy photons are emitted in the same region as the low energy ones then the condition on the pair production, taugammagamma, Eq. 78 is stronger than the condition on Compton scattering Eq. 81. This increases the required Lorentz factors. Second, the Compton scattering limit (which is independent of the observed high energy tail of the spectrum) poses also a lower limit on gamma. However, this is usually less restrictive then the taugammagamma limit. Finally, one sees in Fig. 24 that optically thin internal shocks are produced only in a narrow region in the (delta, gamma) plane. The region is quite small if the stronger pair production limit holds. In this case there is no single value of gamma that can produce peaks over the whole range of observed durations. The allowed region is larger if we use the weaker limits on the opacity. But even with this limit there is no single value of gamma that produces peaks with all durations. The IS scenario suggests that bursts with very narrow peaks should not have very high energy tails and that very short bursts may have a softer spectrum.

Figure 24

Figure 24. Allowed regions for internal shocks in the delta (in cm), gamma plane. Note that the horizontal delta axis also corresponds to the typical peak duration, deltat multiplied by c. Internal shocks are impossible in the upper right (light gray) region. The lower boundary of this region depends on zeta ident delta / Delta and are marked by two solid curves, the lower one for zeta = 1 and the upper one for zeta = 0.01. Also shown are taugammagamma = 1 for an observed spectrum with no upper bound (dotted line), taugamma gamma = 1 for an observed spectrum with an upper bound of 100 MeV (dashed line) and taue = 1 (dashed-dotted). The optically thin internal shock region is above the tau = 1 curves and below the xi = zeta2/3 (solid) lines. From [232].

8.6.2. Physical Conditions and Emission from Internal Shocks

Provided that the different parts of the shell have comparable Lorentz factors differing by factor of ~ 2, the internal shocks are mildly relativistic. The protons' thermal Lorentz factor will be of order of unity, and the shocked regions will still move highly relativistically towards the observer with approximately the initial Lorentz factor gamma. In front of the shocks the particle density of the shell is given by the total number of baryons E / gamma mp c2 divided by the co-moving volume of the shell at the radius Rdelta which is 4pi Rdelta2 Delta gamma. The particle density behind the shock is higher by a factor of 7 which is the limiting compression possible by Newtonian shocks (assuming an adiabatic index of relativistic gas, i.e., 4/3). We estimate the pre-shock density of the particles in the shells as: [E / (gamma mp c2)] / (4pi(delta gamma2)2 gamma Delta). We introduce gammaint as the Lorentz factor of the internal shock. As this shock is relativistic (but not extremely relativistic) gammaint is of order of a few. Using Eq. 48 for the particle density n and the thermal energy density e behind the shocks we find:

Equation 81 (81)

Equation 82 (82)

We have defined here Delta12 = Delta / 1012 cm. Using Eq. 49 we find:

Equation 83 (83)

Using Eqs. 49, 51, 56 and 81 we can estimate the typical synchrotron frequency from an internal shock. This is the synchrotron frequency of an electron with a "typical" Lorentz factor:

Equation 84 (84)

The corresponding observed synchrotron cooling time is:

Equation 85 (85)

Using Eq. 53 we can express gammae,min in terms of gammaint to estimate the minimal synchrotron frequency:

Equation 86 (86)

The energy emitted by a "typical electron" is around 220keV. The energy emitted by a "minimal energy" electron is about one order of magnitude lower than the typical observed energy of ~ 100 keV. This should correspond to the break energy of the spectrum. This result seems in a good agreement with the observations. But this estimate might be misleading as both epsilonB and epsilone might be significantly lower than unity. Still these values of (h nusyn)obs are remarkably close to the observations. One might hope that this might explain the observed lower cutoff of the GRB spectrum. Note that a lower value of epsilonB or epsilone might be compensated by a higher value of gammaint. This is advantageous as shocks with higher gammaint are more efficient (see section 8.6.4).

The synchrotron cooling time at a given frequency (in the observer's frame) is given by:

Equation 87 (87)

We recover the general trend tsyn propto (h nu)-1/2 of synchrotron emission. However if (as we expect quite generally) this cooling time is much shorter than Tang it does not determine the width of the observed peaks. It will correspond to the observed time scales if, for example, epsilonB is small. But then the "typical" photon energy will be far below the observed range. Therefore, it is not clear this relation can explain the observed dependence of the width of the bursts on the observed energy.

8.6.3. Inverse Compton in Internal Shocks

The calculations of section 8.4 suggest that the typical Inverse Compton (IC) (actually synchrotron - self Compton) radiation from internal shocks will be at energy higher by a factor gammae2 then the typical synchrotron frequency. Since synchrotron emission is in the keV range and gammae,min approx mp / me, the expected IC emission should be in the GeV or even TeV range. This radiation might contribute to the prompt very high energy emission that accompanies some of the GRBs [83].

However, if the magnetic field is extremely low: epsilonB ~ 10-12 then we would expect the IC photons to be in the observed ~ 100 keV region:

Equation 88 (88)

Using Eqs. 71 and 83 we find that the cooling time for synchrotron-self Compton in this case is:

Equation 89 (89)

This is marginal. It is too large for some bursts and possibly adequate for others. It could possibly be adjusted by a proper choice of the parameters. It is more likely that if Inverse Compton is important then it contributes to the very high (GeV or even TeV) signal that accompanies the lower energy GRB (see also [250]).

8.6.4. Efficiency in Internal Shocks

The elementary unit in the internal shock model (see section 7.4) is a a binary (two shells) encounter between a rapid shell (denoted by the subscript r) that catches up a slower one (denoted s). The two shells merge to form a single shell (denoted m). The system behaves like an inelastic collision between two masses mr and ms.

The efficiency of a single collision between two shells was calculated earlier in section 8.1.1. For multiple collisions the efficiency depends on the nature of the random distribution. It is highest if the energy is distributed equally among the different shells. This can be explained analytically. Consider a situation in which the mass of the shell, mi is correlated with the (random) Lorentz factor, gammai as mi propto gammaieta. Let all the shells collide and merge and only then emit the thermal energy as radiation. Using conservation of energy and momentum we can calculate the overall efficiency:

Equation 90 (90)

Averaging over the random variables gammai, and assuming a large number of shells N -> infty we obtain:

Equation 91 (91)

This formula explains qualitatively the numerical results: the efficiency is maximal when the energy is distributed equally among the different shells (which corresponds to eta = - 1).

In a realistic situation we expect that the internal energy will be emitted after each collision, and not after all the shells have merged. In this case there is no simple analytical formula. However, numerical calculations show that the efficiency of this process is low (less than 2%) if the initial spread in gamma is only a factor of two [32]. However the efficiency could be much higher [33]. The most efficient case is when the shells have a comparable energy but very different Lorentz factors. In this case (eta = - 1, and spread of Lorentz factor gammamax / gammamin > 103) the efficiency is as high as 40%. For a moderate spread of Lorentz factor gammamax / gammamin = 10, with eta = - 1, the efficiency is 20%.

The efficiency discussed so far is the efficiency of conversion of kinetic energy to internal energy. One should multiply this by the radiative efficiency, discussed in 8.5 (Eq. 72) to obtain the overall efficiency of the process. The resulting values may be rather small and this indicates that some sort of beaming may be required in most GRB models in order not to come up with an unreasonable energy requirement.

8.6.5. Summary - Internal Shocks

Internal shocks provide the best way to explain the observed temporal structure in GRBs. These shocks, that take place at distances of ~ 1015 cm from the center, convert two to twenty percent of the kinetic energy of the flow to thermal energy. Under reasonable conditions the typical synchrotron frequency of the relativistic electrons in the internal shocks is around 100 keV, more or less in the observed region.

Internal shocks require a variable flow. The situation in which an inner shell is faster than an outer shell is unstable [251]. The instability develops before the shocks form and it may affect the energy conversion process. The full implications of this instability are not understood yet.

Internal shocks can extract at most half of the shell's energy [32, 33, 69]. Highly relativistic flow with a kinetic energy and a Lorentz factor comparable to the original one remains after the internal shocks. Sari & Piran [20] pointed out that if the shell is surrounded by ISM and collisionless shock occurs the relativistic shell will dissipate by "external shocks" as well. This predicts an additional smooth burst, with a comparable or possibly greater energy. This is most probably the source of the observed "afterglow" seen in some counterparts to GRBs which we discuss later. This leads to the Internal-External scenario [252, 20, 26] in which the GRB itself is produced by an Internal shock, while the "afterglow" that was observed to follows some GRBs is produced by an external shock.

The main concern with the internal shock model is its low efficiency of conversion of kinetic energy to gamma-rays. This could be of order twenty percent under favorable conditions and significantly lower otherwise. If we assume that the "inner engine" is powered by a gravitational binding energy of a compact object (see section 10.1) a low efficiency may require beaming to overcome an overall energy crisis.

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