UNDERSTANDING THE GEOMETRY OF ASTROPHYSICAL MAGNETIC FIELDS

Magnetized plasmas are common in astrophysics, playing central roles in objects as diverse as galaxy clusters, the sites of star formation, the solar corona and accreting black holes, and their ultra-relativistic outflows. Despite their importance, however, there are few observational tools for assessing the physical conditions within them. This is especially true for strongly ionized or diffuse regions, in which molecular line emission is absent. In these the best evidence for significant magnetic fields, beyond theoretical arguments, is due to Faraday rotation observations of intrinsically polarized sources.

First discovered in optically active crystals (Faraday 1846), Faraday rotation has subsequently become one of the primary methods for measuring the strength and geometry of astrophysical magnetic fields. This has been primarily via the determination of the rotation measure (RM), defined in terms of the polarization angle, Ψ, by

$$\begin{equation} {\rm RM}= \frac{d\Psi }{d\lambda ^2} \simeq 8.12\times 10^5\int n_e \mbox{$\rm B$}\cdot d\mbox{$\rm \ell $}\,\,{\rm rad}\,{\rm m}^{-2}, \end{equation} \tag{ 1 }$$

where n_e, B, and ℓ are measured in cm⁻³, G, and pc, respectively. Thus, RMs provide a line-of-sight averaged measurement of the line-of-sight magnetic field, weighted by the local plasma density. Measured values for the RM range from nearly zero to nearly 10⁶ rad m⁻².

RM observations have played a critical role in determining the magnetic field strengths in the centers of galaxy clusters (e.g., Ge & Owen 1993, 1994), Galactic magnetic field strength and distribution (e.g., Men et al. 2008; Noutsos et al. 2008), magnetic field strength and structure within ultra-relativistic black hole jets (Zavala & Taylor 2004; Stirling et al. 2004; Miller-Jones et al. 2005; Brocksopp et al. 2007; Kharb et al. 2009), magnetic field strengths near, and accretion rate of, the supermassive black hole at the center of the Milky Way and other nearby low-luminosity active galactic nuclei (AGNs; e.g., Agol 2000; Quataert & Gruzinov 2000; Brunthaler et al. 2006; Macquart et al. 2006; Marrone et al. 2007), and even the solar corona magnetic field (e.g., Mancuso & Spangler 2000).

However, there are a number of significant ambiguities in Faraday rotation studies. The first is the degeneracy between density and magnetic field strength (i.e., high densities and weak fields versus low densities and strong fields). The second is the degeneracy between weak large-scale fields and strong tangled fields. In light of numerical simulations of magnetohydrodynamic turbulence, and the observation of supersonic (though sub-Alfvénic) turbulence in Galactic molecular clouds, these make interpreting measured RMs highly dependent upon models for the rotating medium. Since in many cases it is unclear if the Faraday rotation is occurring in situ or in a foreground region distant from the polarized source, there are large uncertainties in our understanding of the physical conditions in the source.

Here we show that at low frequencies Faraday rotation qualitatively changes, with the RM becoming proportional to ∫n_e|B · dℓ|. Comparisons between RMs measured within this "super-adiabatic" regime and at high frequencies can probe the line-of-sight geometry of the magnetic field.³ Furthermore, the frequency at which a given source transition between the standard and super-adiabatic regimes depends upon the local plasma properties at magnetic field reversals (when the magnetic field is orthogonal to the line of sight), and can thus distinguish between different candidates for the site of the rotating medium.

In Section 2, we review the plasma processes responsible for Faraday rotation and describe the super-adiabatic regime. Section 3 deals with the observational consequences unique to super-adiabatic Faraday rotation generally, while Sections 4 and 5 concern specific classes of sources in which ordinary Faraday rotation has been observed. Our conclusions and implications are summarized in Section 6. The mathematical details of an ab initio determination of the radiative transfer regimes is presented in Appendix A.

Faraday rotation is commonly presented in terms of the different phase velocities of the circularly polarized eigenmodes of magnetized plasmas. Specifically, a linearly polarized, monochromatic, incident wave is decomposed into the two plasma modes, which subsequently propagate independently, accruing a net phase difference, ΔΦ = ∫Δkdℓ (where Δk(ω) is the difference between the wavevectors of the two modes), as a consequence of the anisotropic nature of magnetized plasmas. This phase difference between circularly polarized modes results in a net rotation of the polarization angle by Ψ = ΔΦ/2. Frequently, this is extended to the case of elliptical plasma modes, which occurs when B is nearly orthogonal to the line of sight, or if the electrons are relativistic, producing Faraday conversion (also known as Faraday pulsations or generalized Faraday rotation), corresponding to oscillations between linear and circular polarizations as well as a rotation of the polarization angle.

Implicit in this description of the plasma processes underlying Faraday rotation is the assumption that the two plasma eigenmodes do indeed propagate independently, or "adiabatically," i.e., there is no mode crossing. In the case of uniform plasmas (constant n_e, B) this is true, and the foregoing explanation is complete. However, if the plasma is rapidly changing the situation is more complicated. For example, consider the propagation of the fast mode near the location of a magnetic field reversal, i.e., where k · B changes sign. Initially, the mode is nearly circularly polarized, with the electric vector rotating in the same sense as the gyrating electrons. As the component of the magnetic field decreases the mode becomes increasingly elliptical, changing into the linearly polarized extraordinary mode when k · B = 0. On the other side of the magnetic reversal the mode again becomes elliptical and eventually circular, though now with the sense of rotation reversed, following the sense of gyration of the electrons. Similarly, the slow mode reverses its polarization by evolving through the linearly polarized ordinary mode. The net Faraday rotation is then dependent upon ∫n_e|B · dℓ|. Clearly the conditions for the applicability of Equation (1) depend upon adiabaticity being broken at field reversals. That this must happen has been known for some time now; strong mode coupling at field reversals first having been discussed by Cohen (1960) and Zheleznyakov & Zlotnik (1964) in the context of type III solar radio bursts. Since that time a number of authors have investigated the moderately coupled regime, and the resulting residual mode conversion, in various environments and under various assumptions (Mätzler 1973; Jones & Odell 1977; Wilson 1980; Hodge 1982; Bjornsson 1988). Thus, we call this the "standard" regime, with the regions where the adiabatic condition fails referred to as "strongly coupled." However, the "super-adiabatic" regime, where the propagation is adiabatic throughout field reversals, can sometimes be observed with interesting consequences.

Since the failure of the adiabatic condition occurs near field reversals, we might expect that the standard and super-adiabatic regimes are distinguished by the properties of the eigenmodes when B · k = 0. Specifically, we might expect that when the eigenmodes change sufficiently slowly at field reversals, the propagation remains adiabatic. Near reversals, the evolution of the eigenmodes are dominated by the increase in ellipticity, becoming significantly elliptical only when |θ − π/2| ≲ Y ≡ ν_B/ν, where θ is the angle between the magnetic field and the line of sight and ν_B is the electron cyclotron frequency. Thus, the eigenmodes change substantially on length scales on the order of Y(dθ/dℓ)⁻¹ near reversals. To ignore mode conversion we require that this be large in comparison with Δk⁻¹, which at reversals is given approximately by c/πνXY², where X ≡ ν²_P/ν² in which ν_P is the electron plasma frequency. ⁴ Thus we arrive heuristically at the result that when cdθ/dℓ ≪ πνXY³ the propagation occurs in the super-adiabatic regime.

More explicitly, as shown rigorously in Appendix A, the necessary condition for the local plasma modes to propagate adiabatically is

$$\begin{equation} \left(\frac{\Delta k}{2}\right)^2 + \left|\frac{\Delta k^{\prime }}{2}\right| \gg \phi ^{\prime 2} + \chi ^{\prime 2}, \end{equation} \tag{ 2 }$$

where prime denotes differentiation with respect to ℓ, ϕ and χ are the orientation and ellipticity angle of the polarization ellipse, respectively. However, to second order in the plasma parameters Δk is given by

$$\begin{equation} \Delta k = 2\pi \frac{\nu }{c} {\it {X Y}} \sqrt{\cos ^2\theta +\left(\frac{Y}{2}\right)^2\sin ^2\theta }, \end{equation} \tag{ 3 }$$

where θ is the angle between the magnetic field and the line of sight. When B ⊥ k, Δk is reduced by a factor of Y/2 from its quasi-longitudinal value, and Δk' vanishes. On the other hand, near θ = π/2, the eigenmode ellipticity changes rapidly, giving

$$\begin{equation} \chi ^{\prime } \simeq \frac{Y}{4} \frac{\sin \theta \left(1+\cos ^2\theta \right)}{(Y/2)^2+\cos ^2\theta } \theta ^{\prime }, \end{equation} \tag{ 4 }$$

where we have assumed Y ≪ 1. If we further assume that the direction of rotation of the magnetic field is roughly isotropic, θ' ≃ ϕ', implying that at reversals χ' ≃ ϕ'/Y. Thus at the same time, the left side of Equation (2) falls substantially, and the right side increases by a similar factor. This is illustrated explicitly in the bottom panel of Figure 1, which shows the left side (blue) and right side (magenta) of the adiabatic condition through a reversal. As a consequence, at magnetic field reversals the mode propagation almost always becomes strongly coupled, and in the limit of small Y the polarization propagates as in vacuum.

Zoom In Zoom Out Reset image size

A direct result of the failure of the adiabatic condition at magnetic field reversals is that the left-handed (for concreteness) fast mode maps onto the left-handed slow mode on the other side of the reversal and vice versa. That is, there is an almost complete conversion of one mode to the other. Thus, we have the situation shown by the red line in Figure 2, as the polarized wave passes through regions of magnetic field with B · k>0, ΔΦ increases (solid lines), while when B · k < 0, ΔΦ decreases (dashed lines), yielding the well-known formula

$$\begin{equation} \Delta \Phi \propto \nu ^{-2} \int n_e\mbox{$\rm B$}\cdot d\mbox{$\rm \ell $}. \end{equation} \tag{ 5 }$$

Note that the failure of the adiabatic condition at field reversals was critical to obtaining the standard result.

Zoom In Zoom Out Reset image size

Nevertheless, at sufficiently low frequencies, the adiabatic condition, Equation (2), can be satisfied at reversals. That is, because |Δk'| vanishes and χ' ≫ ϕ' where B · k = 0, in order to remain in the adiabatic mode propagation regime throughout, what we call the "super-adiabatic" regime, we require

$$\begin{equation} \frac{\theta ^{\prime }}{Y} \ll \frac{\pi }{2} \frac{\nu }{c} X Y^2 \quad \Rightarrow \quad \nu \ll \nu _{\rm SA}\equiv \left[ \frac{\pi \nu _B^3\nu _P^2}{2 c \theta ^{\prime }} \right]^{1/4}, \end{equation} \tag{ 6 }$$

which is identical to the results of Cohen (1960) (see Equation (56); also the second equation in Zheleznyakov & Zlotnik 1964). Note that up to a factor of 2, this is precisely the condition we reached earlier by heuristic arguments. The critical frequency below which the propagation is super-adiabatic may be recast in terms of the local magnetic field reversal length scale, ℓ_B ≡ π/2θ' ≡ 10¹⁵ℓ_B,15 cm, and the local plasma parameters:

$$\begin{equation} \nu _{\rm SA}\simeq 87 n^{1/4} B^{3/4} \ell _{B,15}^{1/4}\,{\rm MHz}, \end{equation} \tag{ 7 }$$

where n and B are measured in cm⁻³ and G, respectively. Alternatively, this may be written in terms of the total number of reversals along the line of sight, N ≃ θ'L/2π, the RM determined at frequencies above ν_SA, ${\rm RM_S}\equiv \lambda ^{-2}\Delta \Phi \simeq \left.\lambda ^{-2}\Delta k\right|_{\theta =0}L /\sqrt{N}$ (where N enters due to the cancellations discussed previously) and the magnetic field strength:

$$\begin{equation} \begin{array}{rcl} \displaystyle \nu _{\rm SA}&\simeq & \displaystyle \left(\frac{\nu _Bc}{2\sqrt{2\pi }}\right)^{1/2} {\rm RM_S}^{1/4} \,N^{-1/8}\\ &&\\ &\simeq & \displaystyle 13 \,B^{1/2}\, {\rm RM_S}^{1/4}\, N^{-1/8} \,{\rm MHz}, \end{array} \end{equation} \tag{ 8 }$$

where in the final expression B and RM_S are in G and rad m⁻², respectively. Note that this is only a weak function of RM_S and N, and thus primarily indicative of the local magnetic field strength in the source. As we shall see in Section 4, for a number of sources this produces ν_SA near 100 MHz–1 GHz.

Since in the super-adiabatic regime there is no mode crossing at magnetic field reversals, the fast/slow mode maps onto the fast/slow mode, as illustrated in Figure 3. As a result, the phase difference between the modes only accrues, as seen by the green line in Figure 2. That is, for super-adiabatic Faraday rotation,

$$\begin{equation} \Delta \Phi _{\rm SA} \propto \nu ^{-2} \int n_e \left| \mbox{$\rm B$}\cdot d\mbox{$\rm \ell $} \right|. \end{equation} \tag{ 9 }$$

Zoom In Zoom Out Reset image size

At first it may appear surprising that coupling, or lack thereof, results in such a profound change in the polarization angle. Thus, while Equation (9) is completely sufficient to define the evolution of the position angle, we also present a qualitative description of the propagation of an initially linearly polarized wave through a magnetic field reversal in Appendix B.

There are three additional fundamental constraints upon observable values of ν_SA. The first is the requirement that within the Faraday screen, ν_SA must exceed the upper cutoff of the extraordinary mode:

$$\begin{equation} \nu _{\rm SA}>\frac{\nu _B}{2} \left(1+\sqrt{1+4\frac{\nu _P^2}{\nu _B^2}}\right). \end{equation} \tag{ 10 }$$

Inserting Equation (8) and assuming that there is considerable separation between the cyclotron and plasma scale gives two conditions upon the magnetic field,

$$\begin{equation} B > 2.1\times 10^{-5}\,{\rm RM_S}^{1/4} N^{-1/8} \ell _{B,15}^{1/2}\,{\rm G}, \end{equation} \tag{ 11 }$$

corresponding to ν_SA > ν_P, and

$$\begin{equation} B < 22\, {\rm RM_S}^{1/2}\, N^{-1/4}\,{\rm G}, \end{equation} \tag{ 12 }$$

corresponding to ν_SA > ν_B⁵. Note that this produces lower and upper and bound upon B as a function of RM_S. The former is shown for various values of N^−1/4ℓ_B by the dashed lines in Figure 4. The latter is denoted by the upper left grayed region in Figure 4.

Zoom In Zoom Out Reset image size

The second is that we must be able to ignore difference in the refraction of the two plasma eigenmodes. As the plasma eigenmodes propagate through the Faraday screen they necessarily follow different paths, dictated by their different indices of refraction. If the two paths pass through sufficiently distinct regions of the Faraday screen, Equation (1) is no longer valid, Φ depending upon the differences in the plasma conditions along the paths in addition to the difference in the phase velocities of the two modes, and the RM is no longer constant. As shown in Appendix C, the condition that the modes propagate through strongly correlated plasma regions is ν ≫ ν_R, where

$$\begin{equation} \begin{array}{rcl} {\nu _{\rm R}}&\simeq & \displaystyle 61 \frac{D^{1/3} L^{1/6}}{\ell _B^{1/2}} n^{1/3} B^{1/3} f^{1/3}\,{\rm kHz}\\ &&\\ &\simeq & \displaystyle 2.0 D_{22}^{1/3} \ell _{B,15}^{-2/3} {\rm RM}^{1/3} f^{1/3}\,{\rm MHz}, \end{array} \end{equation} \tag{ 13 }$$

in which f is the fractional variation in the plasma parameters, D = (D⁻¹_O + D⁻¹_S)⁻¹ = D₂₂10²² cm where D_O and D_S are the observer–screen and source–screen distances, respectively. The nature of this constraint depends upon the location of the Faraday screen. For example, when the rotation occurs in situ, D ≃ L implying that D/ℓ_B ≃ N, and giving ν_R that is a moderately weak function of N (∝N^1/2 or N^1/3 depending upon which of the above expressions one considers). For all of the sources we describe below, ν_SA > ν_R, justifying our neglect of refraction for these objects.

The third is the observational limits due to absorption in the ionosphere and heliopause (see, e.g., Bougeret 1996). The ionosphere limits ground observations to above roughly 15 MHz, though down to 10 MHz is possible at excellent sites. A fundamental limit for space missions is the plasma frequency at the heliopause, estimated at roughly 9 kHz. These are shown by the hatched and solid gray regions, respectively, in the lower left. While a number of interesting objects lie within reach of ground-based radio astronomy, including nearby AGNs and pulsars, pushing to space-based, ultra-low frequency radio observatories opens a new window upon a vast array of astrophysical environments.

If it can be observed, Faraday rotation in the super-adiabatic regime, what we term "super-adiabatic Faraday rotation," provides a wealth of new information about the magnitude and line-of-sight geometry of magnetic fields. First, unlike standard Faraday rotation the super-adiabatic RM:

$$\begin{equation} {\rm RM_{SA}}= C \int n_e \left|\mbox{$\rm B$}\cdot d\mbox{$\rm \ell $}\right|, \end{equation} \tag{ 14 }$$

does not suffer from a degeneracy between weak large-scale magnetic fields and strong tangled magnetic fields. Thus, RM_SA is far better suited for the determination of n_e and B via equipartition arguments than its high-frequency analog:

$$\begin{equation} {\rm RM_S}= C \int n_e \mbox{$\rm B$}\cdot d\mbox{$\rm \ell $}. \end{equation} \tag{ 15 }$$

Second, generally, RM_SA will be greater than RM_S by an amount depending upon the number of magnetic field reversals along the line of sight. Typically, we may expect that looking through N independent turbulent cells, the excess phase difference, and therefore RM_S, to scale as N^−1/2, i.e.,

$$\begin{equation} {\rm RM_{SA}}\simeq N^{1/2} {\rm RM_S}. \end{equation} \tag{ 16 }$$

Thus, measuring both RM_S (at ν>ν_SA) and RM_SA (at ν < ν_SA) provides an estimate for the number of magnetic field reversals along the line of sight:

$$\begin{equation} N \simeq \left(\frac{{\rm RM_{SA}}}{{\rm RM_S}}\right)^2. \end{equation} \tag{ 17 }$$

However, this estimate may not be appropriate in the presence of long-range order in the magnetic field, and more than a rough estimate requires a detailed analysis of the nature of the magnetic fields under consideration. For example, if there was some reason to not expect a randomly reversing field, but instead a band-structured field (e.g., as might be expected from pulsars or hydromagnetic turbulence within a given cell), then, RM_SA/RM_S ≃ N.

Third, as seen in Equation (8), the location of the super-adiabatic transition depends upon the local parameters of the plasma, though we have recast them in terms of the known RM_S and B by assuming that the intervening Faraday screen is not significantly clumpy. However, in many sources this assumption may not be appropriate. Measuring ν_SA provides a direct measurement of the local plasma properties at the magnetic field reversals, and thus the distribution of the electron density and magnetic field strength along the line of sight. On the other hand, if we can assume that these quantities are smoothly distributed, measuring RM_SA and ν_SA provides a method to estimate the magnetic field strength, independent of the intervening electron density:

$$\begin{equation} B \simeq 6 \,\nu _{\rm SA}^2 \left(\frac{{\rm RM_{SA}}}{{\rm RM_S}}\right)^{1/4} {\rm RM_S}^{-1/2} \,{\rm m}{\rm G}, \end{equation} \tag{ 18 }$$

where ν_SA and RM_S are measured in MHz and rad m⁻², respectively.

However, both bandwidth and beam depolarization are likely to be more severe problems for super-adiabatic Faraday rotation than its standard counterpart. Both are exacerbated simply because RM_SA > RM_S. Beam depolarization, however, is also more likely due to the nature of super-adiabatic propagation. At field reversals the polarization is effectively reflected about the magnetic field as the handedness of the polarization eigenmode switches. The observed polarization is therefore dependent in part upon the magnetic field direction in the last turbulent cell. This can produce large gradients in the polarization position angle on angular scales comparable to the turbulent scale. Furthermore, because probing the super-adiabatic regime requires pushing to longer wavelengths, it also generically results in larger beams. Nevertheless, for compact sources, those that subtend a small angular scale in comparison with the turbulence in the intervening Faraday screen, this effect is largely mitigated.

We now turn to the implications for specific classes of sources, and in particular estimate ν_SA. Figure 4 summarizes these. Here we discuss those sources for which the super-adiabatic regime is accessible from ground-based observations, i.e., ν_SA ≳ 15 MHz.

4.1. Ionospheric Faraday Rotation

4.2. Resolved Galactic Nuclei

4.2.1. Sagittarius A*

The supermassive black hole at the center of the Milky Way, Sagittarius A* (Sgr A*), is currently the best studied accreting black hole. Despite its comparatively small mass (4.7 × 10⁶ M_☉; Ghez et al. 2008; Gillessen et al. 2009) and extraordinarily low luminosity (L/L_Edd ≃ 10⁻¹⁰), due to its proximity it has the unique distinction of being the only black hole to have been probed on sub-horizon scales (Doeleman et al. 2008). It also has the largest RM observed, −5.6 ± 0.7 × 10⁵ rad m⁻² (Marrone et al. 2007; Macquart et al. 2006). Estimates of Sgr A*'s magnetic field arise from both fitting simple accretion models to both the RM results and the source spectrum (Agol 2000; Quataert & Gruzinov 2000; Falcke & Markoff 2000; Yuan et al. 2003; Marrone et al. 2007; Loeb & Waxman 2007), and range from 0.01–100 G. The considerable uncertainty reflects the uncertainty in the location region responsible for the Faraday rotation as well as the geometry of the magnetic field along the line of sight. Nevertheless, these estimates imply that Sgr A* should enter the super-adiabatic regime between 30 MHz and 3 GHz.

Typical length scales within the Faraday rotating medium surrounding Sgr A* are on the order of ℓ_B ≃ 10³GM/c² ≃ 10¹⁵ cm. Even with the large RMs observed in Sgr A*, this still results in a ν_R ≃ 1 MHz, and thus refractive decoherence is unimportant.

4.2.2. M81*

The supermassive black hole at the center of M81 is a second representative of a class of low-luminosity AGNs. At 7 × 10⁶ M_☉, it is moderately more massive than Sgr A*, and with L/L_Edd ≃ 10⁻⁵, considerably brighter, though still considerably sub-Eddington (Brunthaler et al. 2006). The lack of polarization in M81 at wavelengths above 3.6 mm implies lower limits upon the value of RM of 10⁴ rad m⁻² and 4 × 10⁵ rad m⁻², depending upon whether the source is beam or band depolarized (Brunthaler et al. 2006). Estimates for the magnetic field in the Faraday screen are again made by appealing to modeling the spectrum of M81* and in analogy with Sgr A* (see above), giving a range of roughly 2–2 × 10⁴ G, due to its larger luminosity and mass. This corresponds to somewhat larger ν_SA, near 3 GHz, though closer to the ν_SA ≃ ν_B cutoff. Again refraction is unimportant in this source.

4.2.3. Other Resolved AGNs

Determining the magnetic field strengths within the Faraday screen of nearby AGNs other than Sgr A* and M81* is the primary difficulty in assessing the likelihood of observing super-adiabatic Faraday rotation in these sources. For other resolved, sub-Eddington AGNs we may estimate the magnitude of the magnetic field in the accretion flow via scaling arguments, equipartition, and Sgr A*. Specifically, in terms of the luminosity in Eddington units, $\ell _{\rm Edd} = L/L_{\rm Edd} \propto \dot{M}/\dot{M}_{\rm Edd}$, the electron (and therefore ion) density near the black hole scales as

$$\begin{equation} n_e \propto \frac{\dot{M}}{M^2 c} \propto \ell _{\rm Edd} M^{-2} \dot{M}_{Edd} \propto \ell _{\rm Edd} M^{-1}, \end{equation} \tag{ 19 }$$

where we have assumed that the radiative efficiency is not a strong function of mass for substantially sub-Eddington sources. The ion temperature is largely independent of black hole mass, reaching 10¹¹ K in nearly all cases. Therefore, the equipartition magnetic field strength scales as

$$\begin{equation} B \propto \left(n_e k T \right)^{1/2} \propto \ell _{\rm Edd}^{1/2} M^{-1/2}. \end{equation} \tag{ 20 }$$

We can also use this to estimate the expected scaling of RMs:

$$\begin{equation} {\rm RM}\propto n_e B d\ell \propto \ell _{\rm Edd}^{3/2} M^{-1/2}. \end{equation} \tag{ 21 }$$

Note that these imply that brighter and lower-mass sources will have higher values of ν_SA. Since the rotation occurs locally in these sources, ν_SA/ν_R ∝ ℓ^1/2_EddM^1/4, and thus refractive decoherence is unlikely to ever preclude observing the super-adiabatic regime for resolved AGNs.

The black hole at the center of M31 is shown as an example. Despite being much more massive, M ≃ 1.4 × 10⁸ M_☉, it also is extraordinarily underluminous, with ℓ_Edd ≃ 10⁻¹⁰, similar to Sgr A* (Li et al. 2009). The correspondingly rescaled RM and B are shown in Figure 4. More generally, polarization observations of resolved nuclear regions will fill the upper right corner of the B–RM plane, corresponding to typical ν_SA ranging from the ground-based cutoff of 15 MHz to 10 GHz.

4.3. Unresolved AGNs

Faraday rotation has been one of the primary methods for studying the magnetic geometry of AGN radio jets and cores. RM gradients have even produced direct evidence for a helical field geometry in AGN jets (see, e.g., Kharb et al. 2009). Currently, RMs exist for more than 40 AGNs, ranging from roughly 10² rad m⁻² to nearly 10⁴ rad m⁻² (Zavala & Taylor 2004). These are usually determined via fitting VLBA observations at 8 GHz and 15 GHz, and in nearly all cases does not resolve the central source. In principle, these are associated with the accreting black hole or jet, and thus are in many respects similar to the systems discussed in Section 4.2, though generally with considerably higher ℓ_Edd.

Estimating the magnetic field is again the primary challenge. We obtain an approximate magnetic field strength based upon an equipartition argument with the emitting nonthermal electrons, assumed to be accelerated in relativistic shocks either in the accretion flow or in the jet. In this case we expect the electron power-law index to be roughly −2 between some minimum and maximum Lorentz factors, and L_ν is given locally by

$$\begin{equation} L_\nu \simeq 3\times 10^{32} \gamma _{\rm min} n_e B^{3/2} V \nu ^{-1/2} \,{\rm erg}\,{\rm s}^{-1}{\rm Hz}^{-1}, \end{equation} \tag{ 22 }$$

where n_e, B, V, and ν are given in cm⁻³, G, pc³, and GHz, respectively (the flat spectra of AGNs are then produced by spatial structure in these quantities, see, e.g., Blandford & Konigl 1979). The electron pressure is given by

$$\begin{equation} P_e \simeq \frac{1}{3} \epsilon _e = \frac{1}{3} \gamma _{\rm min} m_e c^2 n_e \ln \left(\frac{\gamma _{\rm max}}{\gamma _{\rm min}}\right). \end{equation} \tag{ 23 }$$

If the ion pressure is similar, equipartition gives P_e ≃ βB²/8π, for typical β of 10–100. Thus, upon combining Equations (22) and (23), given ν, L_ν, and V, we obtain a lower limit upon the field strength

$$\begin{equation} B \simeq 2\times 10^2\, \left(\Lambda /\beta \right)^{-2/7} L_{\nu,35}^{2/7}\, \nu ^{1/7} V^{-2/7}\,{\rm m}{\rm G}, \end{equation} \tag{ 24 }$$

where L_ν = 10³⁵L_ν,35 erg s⁻¹Hz⁻¹, Λ ≡ ln(γ_max/γ_min) and the other quantities are given in the units described previously. Note that this is weakly dependent upon β, and thus the system would need to be extraordinarily sub-equipartition for the magnetic field to be appreciably lower. For concreteness we choose Λ ≃ 10 and β ≃ 10².

For the AGN jets and cores for which we found RMs in the literature (Zavala & Taylor 2004; Kharb et al. 2009), at ν = 15 GHz, L_ν,35 is typically of order unity. Estimating V is more difficult, since the vast majority of AGN cores are unresolved, and we can therefore only place an upper limit upon V (lower limit upon the equipartition value of B). To estimate this, for each AGN we inspected high-frequency very long baseline interferometry (VLBI) images to determine approximate upper limits upon its angular size, typically comparable to beam sizes of ∼1 mas (Taylor 1998, 2000; Lister et al. 1998; Lister & Smith 2000; Lister 2001; Porcas & Rioja 2002; Zavala & Taylor 2003, 2004; Jorstad et al. 2005; Lister et al. 2009). Given the distances inferred from the source redshifts, this corresponds to limits upon the physical sizes of roughly 5 pc, or volumes of 10² pc³–10³ pc³. The resulting estimates for the magnetic field strength are 3 mG–30 mG, and shown by the dark-blue region labeled "AGN" in Figure 4.

In addition to the AGNs discussed above, there are a small number of high-redshift radio galaxies (HzRGs) exhibiting anomalously large rest-frame RMs. These include OQ 172 (z = 3.535, RM ≃ 4 × 10⁴ rad m⁻²; Udomprasert et al. 1997), 3C 295 (z = 3.377, RM ≃ 2 × 10⁴ rad m⁻²; Perley & Taylor 1991; Taylor & Perley 1992), SDSS 1624+3758 (z = 0.461, RM ≃ 1, 8 × 10⁴ rad m⁻²; Frey et al. 2008), and PKS B0529−549 (z = 2.575, RM ≃ 1.5 × 10⁴ rad m⁻²; Broderick et al. 2007). These RMs are an order of magnitude larger than those discussed previously, and thus may represent a distinct class. Their association with cooling-core clusters has led to speculation that the Faraday rotation is associated with the intracluster medium (ICM) and not intrinsic to the broad-line region. Nevertheless, if the rotation does occur within the core, this implies typical magnetic fields of 0.5 mG, somewhat lower than those for other AGNs due to their comparatively larger distances. This is shown in the dark-blue region labeled "HzRG-i."

Resolved AGNs exhibit strong RM gradients, frequently with RM dispersions that exceed the averaged RM. As a result, the RM for unresolved AGN cores is likely to be considerably smaller than the peak values. Furthermore, as we have stressed above, our estimates of the local magnetic field are highly uncertain. This is primarily a consequence of the unresolved natures of most AGNs, implying that our estimate of the equipartition magnetic field is only a lower limit. If the Faraday rotation occurs near the supermassive black hole, as appears to be the case in Sgr A*, then the AGN region should move up and to the right. Therefore, while for AGN typical ν_SA currently lie just at the frequency cutoff for ground-based observations, in reality it may be closer to those for their resolved counterparts. Furthermore, for in situ rotation, ν_R ≃ 0.3N^1/2ℓ^−1/3_B,18RM^1/3₃ MHz, still comfortably below ν_SA.

On the other hand, if the Faraday screen is far from the region responsible for the emission, the intrinsic magnetic field may be closer to 1 μG–100 μG, typical of the interstellar medium (ISM). In this case, the ν_SA would lie well below the ground-based cutoff, though above the heliospheric cutoff, implying that in these sources super-adiabatic Faraday rotation may be observable from ultra-low frequency, space-based radio observatories. Thus, even just the detection of super-adiabatic Faraday rotation would determine the region responsible for the observed Faraday rotation.

4.4. Pulsars: Intrinsic Rotation

Despite lying below the ionospheric cutoff, the super-adiabatic regime for many objects is potentially accessible from space (e.g., Jones et al. 2000). These necessarily have ν_SA ≳ 9 kHz, and are thus above the heliospheric cutoff. However, in many cases they are considerably closer to their respective refractive decoherence limits. Nevertheless, the observation of the super-adiabatic regime provides a prime motivation to push toward a space-based, low-frequency radio emission.

5.1. Solar Corona

5.2. Pulsars: Interstellar Medium

5.3. X-ray Binaries

5.4. Galactic Center

5.5. Galactic Molecular Clouds

5.6. Starburst Galaxies

5.7. Intracluster Medium

Despite commonly being described in terms of the independent propagation of plasma modes, non-adiabatic effects at magnetic field reversals force the mode crossings that are critical to understanding the standard expressions for Faraday rotation. For nearly all astrophysical systems, at frequencies above a few GHz, these effects dominate the propagation at magnetic field reversals. However, at sufficiently low frequencies it is possible to enter a "super-adiabatic" regime, in which the two plasma modes propagate independently throughout, resulting in RMs that are proportional to ∫n_e|B_∥|dℓ instead of ∫n_eB_∥dℓ.

For large-scale ordered magnetic fields, there is little difference between the standard and super-adiabatic regimes. However, for magnetic field geometries exhibiting many reversals along the line of sight, the RM in the super-adiabatic regime can be substantially larger than that determined at high frequencies. Thus, comparisons between the standard and super-adiabatic rotation provides a unique means to measure the magnetic field geometry along the line of sight.

The frequency at which Faraday rotation transitions between the standard and super-adiabatic regimes, ν_SA, is itself dependent upon the local plasma properties at the magnetic field reversals, though only weakly dependent upon the magnetic field geometry. Thus even the detection of super-adiabatic Faraday rotation, marked by a rapid increase in the RM at low frequencies, provides localized information about the plasma density and magnetic fields. This has the potential to remove the degeneracy between propagation path length and plasma density or magnetic field strength, allowing an unambiguous determination of the location of the Faraday screen in many systems.

For known systems, ν_SA ranges from the heliospheric cutoff, approximately 10 kHz, to nearly 10 GHz, with regions containing stronger magnetic fields and higher RMs transitioning at higher frequencies. For resolved AGNs, such as Sgr A*, M81*, and M31, this lies above the ionospheric cutoff at 15 GHz, and hence is observable from ground-based radio telescopes. Some unresolved AGNs, HzRGs, and pulsars may be observable from the ground as well. Detection of super-adiabatic Faraday rotation in any of these sources would provide striking confirmation of the presence of in situ Faraday rotation. For X-ray binaries, infrared galaxies, molecular clouds, the Galactic center region, ICM, ISM, and the solar corona, the super-adiabatic regime should be observable from space-based radio observatories.

The search for super-adiabatic Faraday rotation is especially timely given the proliferation of ultra-low frequency telescopes, such as the Murchison Widefield Array in Australia (80 MHz–300 MHz) and the Low Frequency Array in Europe (10 MHz–240 MHz), built ostensibly for the purpose of mapping 21 cm emission from the high-z universe, and the Frequency Agile Solar Radiotelescope in North America (50 MHz–21 GHz), currently under development (Bastian 2003). All of these devices will collect the full complement of Stokes parameters, and thus will be capable of searching for super-adiabatic Faraday rotation. In the long term, space-based radio observatories, such as the Astronomical Low Frequency Array, have been discussed for more than two decades now (Jones et al. 2000). They provide the only means to access the 10 kHz–15 MHz radio window, and beyond the confines of the Earth, can in principle do so with resolutions comparable to much shorter wavelengths at the Very Long Baseline Array. The ability to directly probe the geometry of astrophysical magnetic fields along the line of sight should add to the list of science motivators for such a capability.

The authors thank Donald B. Melrose for bringing the work by M.H. Cohen and V.V. Zheleznyakov to our attention and helping to sharpen our arguments considerably. This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al. 2009). This work was supported in part by the US Department of Energy under contract number DE-AC02-76SF00515.

In this Appendix, we derive the conditions identifying the "adiabatic" and "vacuum" propagation regimes. We do this by expanding Maxwell's equations in terms of the plasma eigenmodes, and identifying the specific mode coupling induced by variations of the underlying plasma parameters (the "adiabatic" and "vacuum" regimes being characterized by weak and strong couplings, respectively). Our analysis is similar to that of Ginzburg (1970) and equivalent to that of Cohen (1960), and within the appropriate limits we obtain an identical condition to that found in these. A discussion of the intermediate, partially coupled case can be found in Zheleznyakov & Zlotnik (1964) and Zheleznyakov (1996).

In general, Maxwell's equations give

$$\begin{equation} (\nabla ^2 - \bnabla \bnabla + \omega ^2 \bepsilon) \cdot {\bf E} = 0, \end{equation} \tag{ A1 }$$

for an electric field E, and a dielectric tensor . For plane waves propagating along the z-axis in a plane parallel medium (we address the possibility of refraction in Appendix C), this reduces to

$$\begin{equation} \frac{d^2 {\bf F}}{dz^2} + \omega ^2 \bepsilon \cdot {\bf F} = 0, \end{equation} \tag{ A2 }$$

where F is the Jones vector (i.e., a two-dimensional vector constructed from the transverse components of E). For an anisotropic dielectric tensor, there will exist two nondegenerate transverse modes defined such that

$$\begin{equation} \omega ^2 \bepsilon {\bf F}_i = k_i^2 {\bf F}_i. \end{equation} \tag{ A3 }$$

In the case of a plasma, these are in general elliptically polarized and aligned with the component of the background magnetic field normal to the direction of propagation, i.e.,

$$\begin{equation} {\bf F}_1 = {\bf Q} \left(\begin{array}{@{}c@{}} \sin \chi \\ i\cos \chi \end{array}\right) \quad {\rm and}\quad {\bf F}_2 = {\bf Q} \left(\begin{array}{@{}c@{}} \cos \chi \\ -i\sin \chi \end{array}\right), \end{equation} \tag{ A4 }$$

where the orientation of the polarization ellipses with respect to a set of axes fixed in space is determined by

$$\begin{equation} {\bf Q} = \left(\begin{array}{@{}cc@{}} \cos \phi &\sin \phi \\ -\sin \phi &\cos \phi \end{array}\right). \end{equation} \tag{ A5 }$$

Then, F = F₁F₁ + F₂F₂, may be inserted into Equation (A2) to give

$$\begin{eqnarray} F_1^{\prime \prime } +& 2 i s_2 \varphi F_1^{\prime } + (k_1^2-\varphi ^2-\psi ^2+is_2\varphi ^{\prime }+2ic_2\varphi \psi) F_1 \nonumber \\ &= 2\left(\psi -ic_2\varphi \right)F_2^{\prime } + (\psi ^{\prime }-ic_2\varphi ^{\prime }+2is_2\varphi \psi)F_2 \nonumber \\ F_2^{\prime \prime } -& 2 i s_2 \varphi F_2^{\prime } + (k_2^2-\varphi ^2-\psi ^2-is_2\varphi ^{\prime }-2ic_2\varphi \psi)F_2 \nonumber \\ &= -2\left(\psi +ic_2\varphi \right)F_1^{\prime } - (\psi ^{\prime }+ic_2\varphi ^{\prime }-2is_2\varphi \psi)F_1,\nonumber \\ \end{eqnarray} \tag{ A6 }$$

where a prime denotes differentiation with respect to z, c₂ ≡ cos 2χ, s₂ ≡ sin 2χ, φ ≡ ϕ', and ψ ≡ χ'. When φ = 0, these reproduce Försterling's coupled equations (see Budden 1961; Ginzburg 1970).

Thus far, no approximations have been made regarding the wave length or scale lengths of the plasma. From the form of Equations (A6), it is clear that if ϕ and ψ vanish, the two modes will propagate completely independently. In the limit that φ and ψ are small in comparison with k_1,2, we may look for WKB solutions of the form

$$\begin{equation} F_i = \frac{f_i}{\sqrt{k_i}}\,{e}^{i\int k_i dz}, \end{equation} \tag{ A7 }$$

and hence

$$\begin{eqnarray} f_1^{\prime } + i s_2 \varphi f_1 &= \left(\psi -ic_2\varphi \right)f_2\,{e}^{-i\int \Delta k dz}\nonumber \\ f_2^{\prime } - i s_2 \varphi f_2 &= -\left(\psi +ic_2\varphi \right)f_1\,{e}^{i\int \Delta k dz}, \end{eqnarray} \tag{ A8 }$$

where Δk ≡ k₁ − k₂ terms on the order of ψ², φ², ψ', φ', ψφ, and f''_i were ignored as they are small by assumption relative to those that remain. Further expand the f_i as

$$\begin{equation} f_1 = u_1 {e}^{-i\int (\Delta k/2) dz} \quad {\rm and}\quad f_2 = u_2 {e}^{i\int (\Delta k/2) dz}. \end{equation} \tag{ A9 }$$

Then,

$$\begin{eqnarray} u_1^{\prime } - i \left(\frac{\Delta k}{2} - s_2 \varphi \right) u_1 &= \left(\psi - i c_2 \varphi \right) u_2\nonumber \\ u_2^{\prime } + i \left(\frac{\Delta k}{2} - s_2 \varphi \right) u_2 &= -\left(\psi + i c_2 \varphi \right) u_1, \end{eqnarray} \tag{ A10 }$$

which may be combined to give

$$\begin{eqnarray} &u_1^{\prime \prime } + \left[ \left(\frac{\Delta k}{2}\right)^2 + \psi ^2 + \varphi ^2 - s_2 \varphi \Delta k - i\frac{\Delta k^{\prime }}{2} \right] u_1 = 0\nonumber \\ &u_2^{\prime \prime } + \left[ \left(\frac{\Delta k}{2}\right)^2 + \psi ^2 + \varphi ^2 - s_2 \varphi \Delta k + i\frac{\Delta k^{\prime }}{2} \right] u_2 = 0.\nonumber\\ \end{eqnarray} \tag{ A11 }$$

If the ψ and φ terms are dominated by the Δk terms, then

$$\begin{equation} u_{1,2} \simeq {\rm const} \times {e}^{\pm i \int \left(\Delta k/2\right) dz}, \end{equation} \tag{ A12 }$$

and thus the f_i are constant. Therefore, in this limit the modes propagate independently (the so-called adiabatic regime). In the opposing limit, when ψ and φ dominate Δk, then Equations (A11) are indistinguishable from the isotropic case (i.e., Δk = 0), and therefore the polarization propagates unaltered (the so-called vacuum regime). This can be directly proved by solving for u_1,2 in this limit and expressing the answer in terms of F. Because the φΔk term will only be relevant when

$$\begin{equation} \left(\frac{\Delta k}{2} \right)\varphi \sim \psi ^2 + \varphi ^2, \end{equation} \tag{ A13 }$$

this term may be neglected in determining the limiting regimes, yielding

$$\begin{equation} \begin{array}{lcl} \displaystyle \psi ^2+\varphi ^2 \ll \left(\frac{\Delta k}{2}\right)^2 + \left| \frac{\Delta k^{\prime }}{2}\right| &\quad &{\rm adiabatic}\\ &&\\ \displaystyle \psi ^2+\varphi ^2 \gg \left(\frac{\Delta k}{2}\right)^2 + \left| \frac{\Delta k^{\prime }}{2}\right| &\quad &{\rm vacuum.} \end{array} \end{equation} \tag{ A14 }$$

What remains is to identify these limits in terms of the physical characteristics of the plasma. That ν exceeds the upper cutoff of the extraordinary mode requires X + Y < 1, which in turn requires that the propagation occurs above both the electron cyclotron and electron plasma frequencies. Under these circumstances the ionic contribution to the dielectric tensor may be safely ignored. Thus, the k_1,2 are given by the standard Appleton–Hartree dispersion relations:

$$\begin{equation} k_{1,2} = \omega \sqrt{ 1 - \frac{2X(1-X)}{2(1-X)-Y^2\sin ^2\theta \mp \Gamma }}, \end{equation} \tag{ A15 }$$

where

$$\begin{equation} \Gamma = \sqrt{ Y^4 \sin ^4\theta + 4 Y^2(1-X)^2\cos ^2\theta }. \end{equation} \tag{ A16 }$$

To second order in Y, this gives

$$\begin{equation} \Delta k \simeq \frac{\omega }{c} X Y \sqrt{ \cos ^2\theta + \left(\frac{Y}{2}\right)^2\sin ^2\theta }. \end{equation} \tag{ A17 }$$

The ellipticity angle is given by (see, e.g., Ginzburg 1970; Budden 1964)

$$\begin{equation} \tan \chi = x + {\rm sgn}(x) \sqrt{1 + x^2}, \end{equation} \tag{ A18 }$$

where

$$\begin{equation} x \equiv \frac{Y \sin ^2\theta }{2 \left(1 - X \right)\cos \theta }. \end{equation} \tag{ A19 }$$

As a direct result,

$$\begin{equation} \chi ^{\prime } = \frac{x^{\prime }}{2(1+x^2)} \simeq \frac{Y}{4} \frac{(1+\cos ^2\theta)\sin \theta }{(Y/2)^2 + \cos ^2\theta } \theta ^{\prime }, \end{equation} \tag{ A20 }$$

where the final expression assumes that Y ≪ 1 and X', Y' ≪ θ'. Note that for θ ≃ π/2, χ' ≃ Y⁻¹θ' ≃ Y⁻¹ϕ' which at high frequencies (Y ≪ 1) is generally much larger than and ϕ', and thus ψ dominates φ near magnetic field reversals in the left-hand sides of Equations (A14), and in particular is maximized at θ = π/2.

On the other hand, as long as X', Y' ≪ ϕ near θ = π/2, |Δk'/2| vanishes identically, due to the symmetry of Δk. This does not mean that |Δk'/2| does not dominate (Δk/2)² elsewhere, which is clearly the case for the parameters shown in Figure 1. Nevertheless, sufficiently close to θ = π/2, the right-hand sides of Equations (A14) are dominated by the (Δk/2)² term, and reach their minimum.

Therefore, for the mode propagation to remain adiabatic for all magnetic field orientations (i.e., through an entire magnetic field reversal), it is both necessary and sufficient to have

$$\begin{equation} \psi \ll \left|\frac{\Delta k}{2}\right|. \end{equation} \tag{ A21 }$$

It is this condition that we use to define the "super-adiabatic" regime. Note that at magnetic reversals

$$\begin{equation} \left| \frac{\Delta k}{2} \right| \phi \ll \left| \frac{\Delta k}{2} \right| \psi < \max \left[\psi ^2,\left(\frac{\Delta k}{2}\right)^2\right], \end{equation} \tag{ A22 }$$

and therefore, much smaller than the dominant term in the above adiabatic condition, either ψ² or (Δk/2)², as was previously claimed.

Within Sections 2 and 3 we computed the RMs for ν ≪ ν_SA when many magnetic field reversals are present by considering the relative phase change between the two plasma eigenmodes. While sufficient for this purpose, it does not directly provide a clear picture of the evolution of the polarization angle, and more generally the polarization vector, of the propagating electromagnetic wave. Thus, here we provide an approximate description of this evolution for ν ≪ ν_SA, first by describing the propagation of an initially linearly polarized wave through a single reversal, how this qualitatively alters the measured RM due to more distant Faraday screens, and finally how many quasi-longitudinal regions can then be combined to produce the previously obtained result.

B.1. Adiabatic Propagation of Linearly Polarized Electromagnetic Waves at Field Reversals

Typically, the observed RM is dominated by contributions from quasi-longitudinal regimes, with contributions from propagation in the quasi-transverse regime being negligible in comparison. The reason for this is twofold: first, Δk is reduced by a factor of Y in the quasi-transverse regime, and second, the propagation through the quasi-transverse regime itself typically comprises only a fraction Y of the total path length. For this reason, we will ignore any Faraday rotation through the field reversal. Thus, assuming that the propagation is within the super-adiabatic regime, i.e., that it is adiabatic at the reversal, we may trivially write down the Jones vector (and hence the polarization) at all positions given the decomposition of the original polarization.

More concretely, let us imagine propagating an initially linearly polarized electromagnetic wave through a single reversal of the magnetic field, depicted in Figure 5. In this we assume that initially the field is parallel to the line of sight (χ = π/4), proceeds through the reversal such that it is vertical at orthogonality (χ = π/2), and then becomes antiparallel to the light of sight at the end (χ = 3π/4). Using Equations (A4), initially we have

$$\begin{equation} {\bf F}_i = \left(\begin{array}{@{}c@{}} \cos \Psi \\ \sin \Psi \end{array}\right) = \frac{{e}^{-i\Psi }}{\sqrt{2}} {\bf F}_1 + \frac{{e}^{i\Psi }}{\sqrt{2}} {\bf F}_2. \end{equation} \tag{ B1 }$$

Since the mode propagation is adiabatic, and thus the two eigenmode amplitudes are individually conserved (ignoring the minimal Faraday rotation in the quasi-transverse regime, as justified above, and assuming that k does not change substantially), we have for all subsequent χ,

$$\begin{equation} {\bf F} = \frac{{e}^{-i\Psi +i\Phi (z)}}{\sqrt{2}} {\bf F}_1[\chi (z)] + \frac{{e}^{i\Psi +i\Phi (z)}}{\sqrt{2}} {\bf F}_2[\chi (z)]. \end{equation} \tag{ B2 }$$

The intermediate polarizations are shown in Figure 5 throughout the reversal for a variety of initial polarizations, and are generally not linear. As a consequence, near θ = π/2 (χ = π/2) the tip of the electric field vector rotates rapidly and it is no longer meaningful to speak about the polarization angle. Nevertheless, it is possible to identify the asymptotic polarization on the other side of the reversal (χ = 3π/2), for which we have

$$\begin{equation} {\bf F}_f = \frac{{e}^{i\Phi }}{2} \left(\begin{array}{@{}c@{}} -{e}^{-i\Psi } + {e}^{i\Psi }\\ i{e}^{-i\Psi } + i{e}^{i\Psi } \end{array}\right) = i{e}^{i\Phi } \left(\begin{array}{@{}c@{}} \sin \Psi \\ \cos \Psi \end{array}\right), \end{equation} \tag{ B3 }$$

which is linear and, up to an overall phase, is simply F_i reflected about the x = y-axis (where ${\rm \hat{y}}\parallel {\rm k}\times {\rm B}$ and $\mbox{$\rm \hat{x}$}\parallel \mbox{$\rm k$}\times \mbox{$\rm \hat{y}$}$). That is, in the super-adiabatic regime we may think of the field reversal as an optical element that acts to reflect the polarization about some direction. While the preceding explicit calculation is necessary to obtain the axis of reflection, that some reflection occurs can be anticipated directly by the fact that the fast mode converts from left-handed to right-handed circular polarization (and vice versa for the slow mode), which are individually parity operations. Thus, propagation through the reversal induces a reflection upon the relevant basis vectors in the super-adiabatic regime, implying that it also does so on arbitrary polarizations as well.

Zoom In Zoom Out Reset image size

B.2. Intrinsic Versus Measured Rotation Measures

We can take the analogy with optical elements a step further by schematically replacing propagation through identical quasi-longitudinal regimes with an element that rotates the polarization by some amount, represented by the +RM₀ and −RM₀ boxes in Figure 6, the sign denoting the sense of the rotation (or equivalently the direction of the magnetic field). In the top line of Figure 6 we show the rotation induced by a +RM₀ region upon an initially vertically polarized wave at two wavelengths, from which it is obvious that the measured RM is positive as defined.

Zoom In Zoom Out Reset image size

We now introduce a reflecting element, represented by the R box, associated with a magnetic field reversal, and shown in the second line. With the results of the previous section, it is trivial to produce the resulting polarization, first rotated by the +RM₀ region and subsequently reflected about the appropriate axis by the R-element. In this case, the resulting observed polarization angles are indistinguishable from those produced by a single −RM₀ region, up to a wavelength-independent π/4 rotation. That is, we may again measure the RM by comparing the rotation of two different wavelength waves and find that in this case it is identical to that produced by a single −RM₀ region. The inverse is also true; a single −RM₀ region followed by an R-element produces an RM identical to a single +RM₀ region. It is important to note that locally within the +RM₀ region the rotation of the polarization vector proceeds in the normal direction; it is only the observed rotation after the reversing element that changes sign.

With this identification, it is now possible to schematically produce multiple quasi-longitudinal regions, separated by R-elements, and now reduce them to an analogous sequence of quasi-longitudinal regions without any R-elements. We do this simply by starting at the source and repeatedly applying our equivalence relation. This procedure is demonstrated in the third and following lines of Figure 6. From this it is clear that we recover the result in the main text, Equation (14): up to an overall sign that is set by the final rotating region, the observed RM is the sum of the absolute values of those in the intervening regions.

This can be seen explicitly as well. Figure 7 shows the polarization evolution of an initially linearly polarized electromagnetic wave as it propagates through three quasi-longitudinal regions, separated by two field reversals (i.e., the magnetic field returns to its initial direction), comparable to the example presented in Figure 6. As expected, reflections at the field reversals produce an appropriately enhanced rotation. That is, the net rotation of the plane of polarization in super-adiabatic case is three times that of the standard case.

Zoom In Zoom Out Reset image size

In Appendix A, we ignored the refraction of the radio wave. For propagation near the plasma and/or cyclotron frequencies this is not justified. Therefore, here we address the corrections and limits that refraction places upon the computation of the Faraday rotation.

Refraction potentially enters the problem in two ways. First, it provides a third scale over which the plasma eigenmodes change, though typically larger than 2π/χ' and 2π/ϕ', and thus not relevant. Second, and much more important, is the possibility that the two plasma eigenmodes refract differently, propagating through significantly different regions and sampling distinct plasma conditions. It is with this latter effect, illustrated in Figure 8, that we concern ourselves here.

Zoom In Zoom Out Reset image size

Let us define x_1,2(η) and k_1,2(η) be the trajectories and wavevectors of the two plasma eigenmodes, parameterized such that at η = 0 they begin at the source and at η = 1 they arrive at the observer. The phase difference between the two modes is then

$$\begin{equation} \int _0^1d\eta \left(\mbox{$\rm k$}_1\cdot \frac{d\mbox{$\rm x$}_1}{d\eta } - \mbox{$\rm k$}_2\cdot \frac{d\mbox{$\rm x$}_2}{d\eta } \right). \end{equation} \tag{ C1 }$$

That this may be reduced to an integral along a single curve, e.g., x = (x₁ + x₂)/2, requires that the two curves never separate further than the local plasma correlation length, roughly ℓ_B, leading to refractive decoherence. Note that in this case, it is not the total diffraction of the trajectories, but the differential trajectory that matters. Generally, this is a complicated global condition, requiring the complete construction of x_1,2 prior to addressing. However, in the "thin-screen" approximation, we can obtain a simple condition for when refractive decoherence may be safely ignored.

Within the thin-screen approximation we assume that the Faraday screen's width is much smaller than both the distance between the source and the screen, D_S, and the distance between the screen and the observer, D_O. Within this approximation, the trajectories are roughly straight lines, joined by a lensing angle at the screen, as shown in Figure 8. Geometrically, it is easy to see that D_Sα_i = D_Oγ_i and α_i + γ_i = β_i, from which it follows directly that,

$$\begin{equation} \delta = \frac{D_S D_O}{D_S+D_O} \left(\beta _1-\beta _2\right). \end{equation} \tag{ C2 }$$

That the maximum deviation between the paths taken by the two plasma eigenmodes is less than the characteristic plasma correlation length reduces to δ ≪ ℓ_B, or

$$\begin{equation} |\beta _1-\beta _2| \ll \frac{D_S+D_O}{D_S D_O} \ell _B = \frac{\ell _B}{D}, \end{equation} \tag{ C3 }$$

where D⁻¹ ≡ D⁻¹_S + D⁻¹_O. At this point, we must determine β_i.

It suffices to consider propagation in the quasi-longitudinal regime since this dominates the differential refraction (because the difference in the indices of refraction is largest and the propagation nearly always occurs in this regime). Therefore, the dispersion relation is given by

$$\begin{equation} D_{1,2} \simeq \frac{1}{2}\left[k_{1,2}^2 - \omega ^2 + \omega ^2 X(1\pm Y) \right], \end{equation} \tag{ C4 }$$

i.e., D_1,2 vanishes along the ray. The equations defining the ray are

$$\begin{equation} \begin{array}{c} \displaystyle \frac{d\mbox{$\rm x$}_{1,2}}{d\eta } = \frac{\partial D_{1,2}}{\partial \mbox{$\rm k$}_{1,2}} = \mbox{$\rm k$}_{1,2},\\ \\ \displaystyle \frac{d\mbox{$\rm k$}_{1,2}}{d\eta } = -\frac{\partial D_{1,2}}{\partial \mbox{$\rm x$}_{1,2}} = -\frac{\omega ^2}{2}\frac{\partial X(1\pm Y)}{\partial \mbox{$\rm x$}_{1,2}}. \end{array} \end{equation} \tag{ C5 }$$

Outside of the Faraday screen, X = Y = 0, and k²_i = ω². In the weak deflection limit, we may treat the component of k_i along the original direction as unchanged, the refraction simply introducing an orthogonal component. That is, we set

$$\begin{equation} \beta _{1,2} \simeq - \omega ^{-1} \int d\eta \frac{d\mbox{$\rm x$}_{1,2}}{d\eta } \times \frac{\partial X(1\pm Y)}{\partial \mbox{$\rm x$}_{1,2}} \simeq -\frac{1}{2} X(1\pm Y) f, \end{equation} \tag{ C6 }$$

where f is the characteristic fractional variation in the plasma parameters on the scale of ℓ_B. Therefore, for a Faraday screen of width ℓ_B we obtain |β₁ − β₂| ≃ XYf. Should the screen be composed of may turbulent cells, the trajectories will diffuse, with $|\beta _1-\beta _2|\simeq \sqrt{N}XYf\simeq \sqrt{L/\ell _B}XYf$. Therefore, for the maximum deviation between the paths of the two plasma eigenmodes to be sufficiently small we require

$$\begin{equation} {\it {XY}}\!f \ll \frac{\ell _B^{3/2}}{D L^{1/2}}. \end{equation} \tag{ C7 }$$

As with the super-adiabatic condition, we may view this physically, as a lower limit upon ℓ_B, or observationally, as an upper limit upon ν. In terms of the latter, we have

$$\begin{equation} \nu \gg \left(\nu _P^2 \nu _B f \frac{D L^{1/2}}{\ell _B^{3/2}}\right)^{1/3}, \end{equation} \tag{ C8 }$$

the expression underlying those in Equation (13).