Stochastic Acceleration and the Evolution of Spectral Distributions in SSC Sources: A Self Consistent Modeling of Blazars' Flares
We investigated the role of the stochastic acceleration on the evolution of the particle energy distribution in relativistic jet, taking into account radiative losses in a SSC scenario. We find that in the acceleration dominated stage the distribution is well described by a logparabolic shape, and that the predicted trends among synchrotron curvature (b_{s}), peak energy (E_{s}), and peak luminosities (L_{s}), match those observed in a sample of 6 TeV detected BL Lacs.

ABSTRACT
The broadband spectral distributions of nonthermal sources, such as those of several known blazars, are well described by a logparabolic fit. The second degree term in these fits measures the curvature in the spectrum. In this paper we investigate whether the curvature parameter observed in the spectra of the synchrotron emission can be used as a fingerprint of stochastic acceleration. As a first approach we use the multiplicative Central Limit theorem to show how fluctuations in the energy gain result in the broadening of the spectral shape, introducing a curvature into the energy distribution. Then, by means of a MonteCarlo description, we investigate how the curvature produced in the electron distribution is linked to the diffusion in momentum space. To get a more generic description we turn to the diffusion equation in momentum space. We first study some "standard" scenarios, in order to understand the conditions that make the curvature in the spectra significant, and the relevance of the cooling over the acceleration process. We try to quantify the correlation between the curvature and the diffusive process in the preequilibrium stage, and investigate how the transition between the KleinNishina and the Thompson regime, in Inverse Compton cooling, can determine the curvature in the distribution at the equilibrium. We apply these results to some observed trends, such as the anticorrelation between the peak energy and the curvature term observed in the spectra of Mrk 421, and a sample of BL Lac objects whose synchrotron emission peaks at Xray energies.
Evolution of the particle energy distribution undergoing stochastic acceleration and SSC cooling
Left panels: evolution of the particle spectrum with impulsive injection and no escape for the case of R = 1 × 10^{15} cm and turbulence "hardsphere" turbulence. Upper panels represent the temporal evolution of n(γ), lower panels represent the temporal evolution of γ^{3}n(γ). Solid lines represent the case of SSC cooling. Red and blue solid lines, represent the final state for B = 1.0 G and B = 0.1 G, respectively. Green solid lines represent the temporal evolution, for B = 0.1 G, with step of 0.8×t_{D} (where t_{D} is the momentumdiffusion acceleration time scale). The dashed lines represent the final stage in the case of only synchrotron cooling. The vertical dotdashed lines represent the equilibrium energy in the case on only synchrotron cooling. Right panels: Evolution of the curvature as function of t/t_{D0} . Upper panel: curvature r evaluated at γ_{p}, for the case of SSC cooling (solid red and blue lines) and for the case of only synchrotron cooling (dashed red and blue lines). The solid green line represent the prediction from Eq. 19. Lower panel: the same as in the upper panel, for the curvature r_{3p} evaluated at γ_{3p} (empty and filled circles) compared to the case of r (solid lines).
Credits: ISDC/A. Tramacere

Comparison of the observed E_{s}b_{s} trend with the stochastic model prediction
Left panel: the E_{s}b_{s} trend observed for the six HBLs in our sample. The dashed green lines represent the trend reproduced by stochastic acceleration model, for the parameters reported in Tab. 3 of the paper, and for the trend driven by a change in the momentumdiffusion term (D). The different lines corresponding to three different values of B reported in Tab. 3 of the paper. The purple lines represent the trend obtained by fitting the numerically computed SED over a fixed spectral window in the range 0.5 − 100 keV.
Credits: ISDC/A. Tramacere

Comparison of the observed E_{s}L_{s} trend with the stochastic model prediction
Left panels: the E_{s}L_{s} trend observed for six HBLs in our sample, top panel corresponds to the case of an injected luminosity (L_{inj}) = 5×10^{39} erg/s, bottom panel corresponds to the case of L_{inj} = 5 × 10^{38} erg/s . The solid black lines represent the trend reproduced by stochastic acceleration model, for the parameters reported in Tab. 3 of the paper, and for the trend driven by a change in the momentumdiffusion term (D). The different lines corresponding to three different values of B reported in Tab. 3 of the paper. The dashed lines represent the trend obtained by fitting the numerically computed SED over a fixed spectral window in the range 0.5 − 100 keV.
Credits: ISDC/A. Tramacere

