On the statistical characterization of the synchrotron multi-zone polarization of blazars

TL;DR

The paper tackles the puzzle of energy-dependent blazar polarization by developing a Monte Carlo framework (JetSeT) for a turbulent, multi-zone jet with a spherical emission region populated by randomly distributed cells. It shows that polarization patterns from IXPE X-rays to RoboPol optical data can be reproduced without imposing correlations between cell size and EED parameters, provided large dispersions in the EED cutoff ($\gamma_{\rm cut}$, ≈90%) and the low-energy slope ($\Delta_p$, ≈0.5–1.5). The analysis demonstrates that the observed polarization degree is governed by the flux-weighted effective number of emitting cells, $N_{\nu}^{\rm eff}$, and that chromatic trends arise from dispersion in cell properties, especially $\gamma_{\rm cut}$ and $p$. By comparing with IXPE and RoboPol datasets, the work constrains the MC parameter space, favoring a log-uniform distribution of $\gamma_{\rm cut}$ and broad $p$-dispersions, with $\gamma_{\rm cut}^{\min ratio} \lesssim 0.1$, thereby providing quantitative insights into the turbulent, multi-zone structure of blazar jets and setting the stage for future investigations of shock–turbulence interplay in polarization signatures.

Abstract

Multiwavelength polarimetric observations of blazars reveal complex, energy-dependent polarization behavior, including a decrease in polarization fraction from X-rays to millimeter bands and significant variability in the electric vector position angle (EVPA). These trends challenge simple single-zone synchrotron models and suggest a more intricate, turbulent jet structure with multiple emission zones. We develop a statistical framework to model the observed energy-dependent polarization patterns in blazars, focusing on the behavior captured by IXPE in the X-ray band and RoboPol in the optical. The goal is to statistically characterize multi-zone models in terms of the distributions of cell size and the physical parameters of the electron energy distribution (EED). A Monte Carlo approach, implemented with the JetSeT code, is used to generate synthetic multi-zone synchrotron emission from a spherical region filled with turbulent cells with randomly distributed physical properties. Simulations explore scenarios ranging from identical cells to power-law distributions of cell sizes and EED parameters with variable cutoff and low-energy slopes. The results show that a purely turbulent, multi-zone model can reproduce the observed energy-dependent polarization without requiring correlations between cell size and EED parameters. The polarization degree is primarily determined by the effective, flux-weighted, number of emitting cells, modulated by the dispersion in cell properties, particularly the EED cutoff energy at high frequencies and the low-energy spectral index at low frequencies. With a fractional dispersion in cutoff energy of about 90% and a low-energy spectral index dispersion of ~0.5-1.5, the model reproduces the chromatic mm-to-X-ray polarization trends seen by IXPE and the optical polarization limiting envelope observed in the RoboPol dataset.

Figures (13)

• Figure 1: Left panels: The S SED (row a), and the corresponding fractional polarization (row b) as a function of the frequency, for a power-law cutoff EED, $\gamma_{\rm cut}=5\times 10^4$, and $p=[1.5,2.0,2.5]$. The solid lines in the row b panel mark the $\Pi_\nu^{\rm ord}$ evaluated using Eq.(\ref{['eq:pol_definition']}), the circles mark the $\delta$ approximation from Eq. (\ref{['eq:pol-delta-approx-nu']}) using $p_{\nu}^{(\delta)}$ from Eq. (\ref{['eq:p-delta-approx-n']}), and the crosses using $p_{\nu}^{(\delta)}$ from Eq. (\ref{['eq:p-delta-approx-p-log-deriv']}). The dashed lines mark the asymptotic power-law trend from Eq. (\ref{['eq:pol_asym']}). The relative error for the $\delta-$approximation methods, w.r.t. the exact method (row c). In the row d panel, we report the best-fit of $\Pi_\nu^{\rm ord}$ by means of a broken power-law function.
• Figure 2: Different colors identifying the corresponding $\nu^p$ bins, which are the same for all the panels. Top panels:Left column: case of identical cells (configuration ES1 in Table \ref{['tab:sim_conf_ss']}). Middle column: case of configuration ES2 in Table \ref{['tab:sim_conf_ss']}, i.e. using $f_{\gamma_{\rm cut}}$=log-uniform. Right column: panel, case of configuration ES3, same as ES2 but adding a flat PDF, for $p:f_{p}\sim\mathcal{U}[1.8,2.8]$. All the trends are reported versus $\nu/\nu_p$. Solid lines mark the MC 0.5 quantiles and shaded areas mark the $1$-$\sigma$ quantiles dispersion, for the MC trials. Row a: the trend of the ratio of $\langle \Pi_\nu \rangle$. Row b: same as for Row 1, but for trials-averaged trends for $\langle \Pi_\nu \rangle/\langle\Pi_\nu^{\rm ord}\rangle$. Row c: trials-averaged trends for $N_{\nu}$ (dashed lines) and $N_{\nu}^{\rm eff}$ (solid lines). Row d: trials-averaged trends for $\kappa_\nu$ (dashed line) and $\kappa_\nu^{\rm eff}$ (solid lines). Row e: SEDs for the $\nu_p$ bin =[$10^{16}-10^{17}$] Hz, the dashed vertical line shows the $\nu/\nu_p$ value, above which not all the cells contribute to the SED flux. The dot-dashed vertical lines mark $\nu=\nu_p^{\rm min}$, where $\nu_p^{\rm min}$ is the SED peak value for the lowest flux cell.
• Figure 3: Summary of the polarization slope trends for the PL-distributed parameter space reported in Table \ref{['tab:sim_conf_pl']}. Each subpanel refers to the three different values of $q=[0,-1.0,-1.5]$, as reported in the subpanel title. Top panels: Low-energy ($a_l$, left panels) and high-energy ($a_h$, right panels) polarization slopes as a function of the dispersion in the electron energy distribution index, $\Delta_p$. Middle panels:$a_l$ and $a_h$ as a function of the minimum cell size ratio, $r_{\rm min}$. Bottom panels:$a_l$ and $a_h$ as a function of the minimum cutoff ratio, $\gamma_{\rm cut}^{\rm min. ratio}$. Different colors correspond to the different $f_{\gamma_{\rm cut}}$ PDF reported in the top legend.
• Figure 4: Left panel: The limiting envelope, found by RoboPol Angelakis2016, for a large sample of $\gamma$-ray-loud blazars between $\nu_p^S$ and both the average fractional optical polarization and its dispersion. Right panel: The $\Pi_\nu$ trend observed in by IXPE for the HPS Mrk 501, Mrk 421, PKS2155-304, 1ES0229+200, and 1ES1959+650 collected from recent IXPE multiwavelength campaigns Liodakis2022DiGesu2023ixpedata1Middei2023.
• Figure 5: Summary of polarization slope trends for the PL-distributed parameter space reported in Table \ref{['tab:sim_conf_pl']}, for the HSP objects, selecting MC runs with $(10\times{15} {~\rm Hz}\leq \nu_p^S\leq 10\times{18} {~\rm Hz}$). Top panels: Low-energy ($a_{\rm mm-o}$, left panels) and high-energy ($a_{\rm o-X}$, right panels) polarization slopes as a function of the dispersion in the electron energy distribution index, $\Delta_p$. Middle panels:$a_l$ and $a_h$ as a function of the minimum cell size ratio, $r_{\rm min}$. Bottom panels:$a_l$ and $a_h$ as a function of the minimum cutoff ratio, $\gamma_{\rm cut}^{\rm min. ratio}$. Different colors and line styles correspond to the various PDFs parameter configurations as indicated in the legend and in the titles of the panels.