Modeling Differential Faraday Rotation in the Solar Corona

Abstract

For decades, radio remote-sensing techniques have been used to probe the plasma structure of the solar corona at distances of 2 – $20~\mathrm{R}_{\odot }$. Measurement of Faraday rotation, the change in the polarization position angle of linearly polarized radiation as it propagates through a magnetized plasma, has proven to be one of the best methods for determining the coronal magnetic-field strength and structure. Faraday-rotation observations of spatially extended radio sources provide the unique opportunity to measure differential Faraday rotation [$\Delta$RM] the difference in the Faraday-rotation measure between two closely spaced lines of sight (LOS) through the corona. $\Delta$RM is proportional to the electric current within an Ampèrian loop formed, in part, by the two closely spaced LOS. We report the expected $\Delta$RM for two sets of models for the corona: one set of models for the corona employs a spherically symmetric plasma density, while the other breaks this symmetry by assuming that the heliospheric current sheet (HCS) is a finite-width streamer-belt region containing a high-density plasma. For each plasma-density model, we evaluate the $\Delta \mathrm{RM}$ for three model coronal magnetic fields: a radial dipole and interplanetary magnetic field (DIMF), a dipole + current sheet (DCS), and a dipole + quadrupole + current sheet (DQCS). These models predict values of $0.01\lesssim \Delta \mathrm{RM}\lesssim 120~\mbox{rad}\,\mbox{m}^{-2}$ over the range of parameter space accessible by modern instruments such as the Karl G. Jansky Very Large Array. We conclude that the HCS contribution to $\Delta$RM is not negligible at moderate heliocentric distances ($<8~\mathrm{R}_{\odot }$) and may account for $\lesssim 20\,\%$ of previous observations of $\Delta$RM (e.g. made by Spangler, Astrophys. J. 670, 841, 2007).

Appendix

To better understand the $\Delta \mathrm{RM}$ as a function of $\beta _{c}$ for the DCS and DQCS models in Figures 6 and 9, it is constructive to examine the contributions to $\Delta \mathrm{RM}$ from the separate magnetic-field components $B_{\rho }$ and $B_{z}$. Figure 12 shows the $B_{\rho }$- and $B_{z}$-contributions to the DCS and DQCS model $\Delta \mathrm{RM}$ for the LDB and Streamer/Hole plasma densities (top and bottom panels, respectively). In this figure, we have selected the same impact-parameter offset as before ($\varepsilon =0.05$). We only demonstrate the behavior for a single impact parameter $R_{\mathrm{A}}=3~\mathrm{R}_{\odot }$, because the overall behavior of the $\Delta \mathrm{RM}$ for the LDB and Streamer/Hole models at larger impact parameters is similar (Section 3.1).

Figure 12

The contribution to differential Faraday rotation $[\Delta \mathrm{RM}]$ from the $\rho$- and $z$-components of the magnetic field as a function of the angle at which the LOS intersect the neutral line $[\beta _{c}]$. The impact-parameter offset between the two LOS is $\varepsilon =0.05$, approximately the offset between the two LOS discussed by Spangler (2007) and Kooi et al. (2014). The impact parameter of the LOS closest to the Sun is $R_{\mathrm{A}}=3~\mathrm{R}_{\odot }$. The plasma densities used to compute $\Delta \mathrm{RM}$ are the spherically symmetric LDB model (top panel, Leblanc, Dulk, and Bougeret, 1998) and the Streamer/Hole plasma-density model (bottom panel, Mancuso and Spangler, 2000).

Aside from the reduction in magnitude, the $B_{\rho }$-contribution to $\Delta \mathrm{RM}$ does not vary significantly in form between the two plasma-density models. It is clear that the difference between the $\Delta \mathrm{RM}$ resulting from the LDB and Streamer/Hole plasma-density models for $\beta _{c}\lesssim \pi /4$ results from the $B_{z}$-contribution (DCS$z$ and DQCS$z$ in Figure 12). In the case of the DCS model, when the impact parameter is small and $\beta _{c}\approx 0$, the LOS for small $\beta$ is largely parallel to the dipolar magnetic field (for a demonstration of this dipolar structure, see Figure 2, top panel in Banaszkiewicz, Axford, and McKenzie, 1998). This region of near-parallel magnetic-field lines is then enhanced by the Streamer density in the bottom panel. In the case of the DQCS model, the magnetic field has a quadrupolar structure (e.g. Figure 4); therefore, there is no such concentrated region of near-parallel field lines.