2 Quasi-thermal comptonization

If the above conditions are not satisfied one can envisage an alternative scenario where the key role is instead played by the balance between the cooling and heating processes. This would be favored if the emitting region occupies the entire shell volume rather than the narrow region associated with a planar shock.

An immediate prediction is that the typical energy of the emitting electrons is mildly relativistic, and the main radiation process is quasi-thermal Comptonization. This also implies the existence of a characteristic observed frequency of a few MeV, controlled by the feedback introduced by the effect of e$^{\pm}$ pair production.

Let us assume that the heating process for a typical electron lasts for the duration of the shell-shell interaction, $\sim \Delta R^\prime/c$. The maximum amount of energy given to a single lepton, $\sim m_{\rm p} c^2 n^\prime_{\rm p}/n^\prime_{\rm e}$, corresponds to a total average heating rate $\dot E_{\rm heat} = n^\prime_{\rm p} m_{\rm p} c^3/\Delta R^\prime$. The characteristic electron energy is given by balancing $\dot E_{\rm heat}$and $\dot E_{\rm cool}$, i.e.:

$$\gamma^2\beta^2 \approx {n^\prime_{\rm p} m_{\rm p}\over n^\... ...{1\over 1+U_{\rm B}/U^\prime_{\rm r}} {3\pi \over \ell^\prime},$$ (1)

where $n^\prime_{\rm e}$ includes a possible contribution from ${\rm e}^\pm$pairs. A typical value for the compactness in this case is $\ell^\prime = 270$ $(L^\prime_{46}/ R_{13})(\Delta R^\prime_{11} /R_{13})$, and therefore e$^{\pm}$ are at most moderately relativistic. Even though the particle distribution may not have time to thermalize, it will be characterized by this mean energy, which can be expressed as an "effective temperature'' $\Theta\equiv kT/(m_{\rm e}c^2)$.

The small energy of the emitting particles implies:

1) the synchrotron emission is self-absorbed. It peaks at a comoving frequency $\nu^\prime_{\rm T}\sim 10^{14}$ Hz.

2) the main radiation mechanism is multiple Compton scattering and the self-absorbed synchrotron emission is the source of soft seed photons.

If one defines a generalized Comptonization parameter $y\, \equiv\, 4\tau_{\rm T}\Theta (1+\tau_{\rm T})(1+4\Theta)$, the ratio of the Compton to the synchrotron powers can then be approximated by ${\rm e}^y$, and thus in order to emit a Compton comoving luminosity $L^\prime_{\rm c} = 10^{46}L^\prime_{\rm c,46}$ erg s^-1, y must be $y\sim \ln(L_{\rm c}/L_{\rm s})\sim 11.5 \ln (L^\prime_{\rm c,46}/L^\prime_{\rm s,41})$. With this value of y, and $\tau_{\rm T}\sim 1$, the Comptonized high energy spectrum has a $F(\nu)\propto \nu^0$ shape, while the relatively modest optical depth prevents a strong Wien peak to form. Therefore, very schematically, the resulting observed spectrum would extend between $\nu_{\rm T}^\prime\Gamma$ and $\sim 2 \Gamma\ kT / h$, with $F(\nu)\sim$const.

The spectrum of a single shell will rapidly evolve: after the observed acceleration time, particles cool on a similar timescale, while the Comptonization spectrum steepens and the power decreases. In the time integrated emission any Wien hump and/or feature in the spectrum of individual shells will be smoothed out. The power-law continuum, if typical of all shells, would instead be preserved.

The e$^{\pm}$ pair production process can play a crucial role. This would both increase the optical depth and act as a thermostat, by maintaining the temperature in a narrow range. Additional pairs will be also produced outside the shell region, increasing the lepton content of the surrounding medium. The maximum equilibrium temperature in steady sources in pair equilibrium with $\ell^\prime$between 10 and 10³ is of the order 30-300 keV ([Svensson 1984]). Note that additional pairs will be also produced outside the shell region, increasing the lepton content of the surrounding medium.

We conclude that an "effective'' temperature $kT\sim 50$ keV and $\tau_{\rm T}\sim 4$ dominated by pairs, can be a consistent solution giving $y\sim 11$.