Recovering 3D Magnetic Turbulence from Single-Frequency Faraday Screens

Abstract

Statistics of polarized synchrotron radiation carry information about the properties of the underlying turbulence. Different statistical measures constructed from observables probe turbulence properties in different ways. We consider a setup in which synchrotron radiation is emitted in a distant volume and then passes through a turbulent screen that induces Faraday rotation. Using both MHD simulations and synthetic turbulence spectra, we explore the spectrum of observed polarization directions measured at a single frequency as a diagnostic for recovering the statistics of turbulence in both the emitting region and the Faraday-rotation screen. We compare these results with our analytical expectations. We also compare the spectrum of polarization direction (SPD) with the wavelength-derivative diagnostic introduced and analytically explored by Lazarian \& Pogosyan. We demonstrate that the SPD exhibits greater sensitivity to turbulence in the Faraday screen. We provide an observer-friendly criterion to determine whether the SPD samples turbulence in the synchrotron-emitting region or in the Faraday screen. These results open a practical pathway for extracting turbulence statistics from existing and forthcoming single-band radio polarimetry.

Figures (6)

• Figure 1: Schematic of the separated-screen geometry adopted in this work. Polarized synchrotron emission (gray cube), produced in a background region (e.g., a galactic halo with low thermal electron density), passes through a foreground Faraday-rotating screen (pink cylinder), where most of the rotation measure (RM) accumulates. Synchrotron emission and Faraday rotation are spatially separated. As shown, the polarization rotation due to the Faraday effect is represented by a change in the polarization angle, $\Delta \chi$, which depends on wavelength.
• Figure 2: Left: The analytical expectation of the structure function $D_u(R;\lambda)/2$, predicted by Eq. \ref{['eq:du']}, as a function of separation $R$ for the density-only synthetic Faraday screen at several values of $\eta$ (colored curves from $\eta=0.05$ in blue to $\eta=0.5$ in red). Dashed lines indicate the intrinsic ($\propto R^{m_i}$, blue) and Faraday ($\propto R^{m_\Phi}$, red) inertial-range asymptotics of the ansatz \ref{['eq:Du_inertial']}. The $\bigstar$ marks the crossover scales $R_\times$ and $k_\times$, predicted by Eqs. \ref{['eq:Rtimes']} and \ref{['eq:ktimes']}. As $\eta$ varies, $R_\times$ shifts accordingly and lies at the transition between the Faraday-dominated ($\propto m_\Phi$) and intrinsic ($\propto m_i$) regimes. Right: The smoothed power-spectrum proxy $\mathcal{M}(k) ~ \propto k^2 P_u(k)$ (Eq. \ref{['eq:Mk_def']}) for the same parameter set. Dashed lines show the corresponding $k^{-m_i}$ and $k^{-m_\Phi}$ scalings. The $\bigstar$ on the right panel denotes the crossover scale $k_\times$.
• Figure 3: Panel a: Results of the MHD AthenaK simulations for sub-Alfvénic turbulence ($M_A=0.8$) in the transonic ($M_s=1.0$, red) and highly supersonic ($M_s=10$, blue) regimes. The plotted $E_n(k)$ is the annularly integrated 2D spectrum of column-density fluctuations from the projected map, integrated over the emitting back three-quarter portion of the LOS; it is computed from the 2D Fourier power by summing over rings of constant projected wavenumber (so $E_n$ is a fluctuation power and variance spectrum, not kinetic energy). Panel b: Column-density spectrum $E_n(k)$, integrated over the emitting (back three-quarter) portion of the LOS, shown together with the projected magnetic-field spectrum $E_B(k)$ over the same integration depth. The plotted $E_B(k)$ is the annularly integrated (ring-summed) 2D spectrum of the projected magnetic-field map, computed by summing the 2D Fourier power over rings of constant projected wavenumber. Panel c: shows the energy spectrum of direct integration $\int n_e B_z\; dz$. We see a clear $k^{-8/3}$ where $k$ comes from the integration on the projection. Panel d: includes the directional spectrum generated by taking \ref{['eq:Pu_def']} for transonic and supersonic regimes, where the kolmogorov powerlaw $k^{-11/3}$ is followed and there are no slope break for $M_s=1$; with $M_s=10$, .
• Figure 4: Directional spectrum $P_u(k;\lambda)$ from $1024^3$ synthetic separated-screen experiments, illustrating the effect of mean-field and $B_\parallel$ fluctuations. Top row:$M_s=10$ ($\bar{n}/\sigma_n=0.54$, $\bar{B}_\perp/\sigma_{B_\perp}=6.33$). Bottom row:$M_s=1$ ($\bar{n}/\sigma_n=2.51$, $\bar{B}_\perp/\sigma_{B_\perp}=3.55$). Left: zero-mean case ($B_0=0$). Middle: density-only screen ($B_\parallel=B_0=\text{const}$). Right: full screen ($B_\parallel=B_0+\delta B$). The mean field does not qualitatively alter the two-slope structure, while reduced $B_\parallel$ fluctuations sharpen the spectral transition.
• Figure 5: Comparison of spectrum for directional spectrum with measures introduced in LazarianPogosyan2016: the spectrum of polarization $P$ and derivative polarization $dP/d\lambda^2$.