THE CIRCULAR POLARIZATION OF SAGITTARIUS A* AT SUBMILLIMETER WAVELENGTHS

In an ideal interferometer with circularly polarized feeds, CP in a target source appears as a difference between the parallel-hand visibilities. Complications may arise due to imperfection in the interferometer response and calibration. The four polarized visibilities in a circularly polarized interferometer can be written (Thompson et al. 2001) as

$$\begin{eqnarray*} \mathcal {V}_{RR}&=&g_{Ra}g^*_{Rb} [(\mathcal {V}_I+\mathcal {V}_V)+d_{Ra}d^*_{Rb}(\mathcal {V}_I-\mathcal {V}_V) \\ &&+d_{Ra}(\mathcal {V}_Q-i\mathcal {V}_U)e^{2i\phi }+d^*_{Rb}(\mathcal {V}_Q+i\mathcal {V}_U)e^{-2i\phi }]\\ \\ \mathcal {V}_{RL}&=& g_{Ra}g^*_{Lb} [(\mathcal {V}_Q+i\mathcal {V}_U)e^{-2i\phi }+d_{Ra}(\mathcal {V}_I-\mathcal {V}_V)\\ &&-d^*_{Lb}(\mathcal {V}_I+\mathcal {V}_V) -d_{Ra}d^*_{Lb}(\mathcal {V}_Q-i\mathcal {V}_U)e^{2i\phi }]\\ \\ \mathcal {V}_{LR}&=& g_{La}g^*_{Rb} [(\mathcal {V}_Q-i\mathcal {V}_U)e^{2i\phi }-d_{La}(\mathcal {V}_I+\mathcal {V}_V)\\ &&+d^*_{Rb}(\mathcal {V}_I-\mathcal {V}_V) -d_{La}d^*_{Rb}(\mathcal {V}_Q+i\mathcal {V}_U)e^{-2i\phi }]\\ \\ \mathcal {V}_{LL}&=&g_{La}g^*_{Lb} [(\mathcal {V}_I-\mathcal {V}_V)+d_{La}d^*_{Lb}(\mathcal {V}_I+\mathcal {V}_V)\\ &&-d_{La}(\mathcal {V}_Q+i\mathcal {V}_U)e^{-2i\phi }-d^*_{Lb}(\mathcal {V}_Q-i\mathcal {V}_U)e^{2i\phi }], \end{eqnarray*}$$

where g_Ra and g_La (d_Ra and d_La) are the right and left circular feed gains (polarization leakage terms) of antenna a and ϕ is the parallactic angle of the feed. Solving the linear system of equations above for the Stokes I and the Stokes V visibilities, respectively, gives

$$\begin{eqnarray} \mathcal {V}_I&=&\frac{\mathcal {V}_{RR}}{2g_{Ra}g_{Rb}^*}\frac{(1+d_{La}d_{Lb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&+\frac{\mathcal {V}_{LL}}{2g_{La}g_{Lb}^*}\frac{(1+d_{Ra}d_{Rb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&+\frac{\mathcal {V}_{RL}}{2g_{Ra}g_{Lb}^*}\frac{(d_{La}-d_{Rb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&+\frac{\mathcal {V}_{LR}}{2g_{La}g_{Rb}^*}\frac{(d_{Lb}^*-d_{Ra})}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)} \end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray} \mathcal {V}_V&=&\frac{\mathcal {V}_{RR}}{2g_{Ra}g_{Rb}^*}\frac{(1-d_{La}d_{Lb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&-\frac{\mathcal {V}_{LL}}{2g_{La}g_{Lb}^*}\frac{(1-d_{Ra}d_{Rb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&-\frac{\mathcal {V}_{RL}}{2g_{Ra}g_{Lb}^*}\frac{(d_{La}+d_{Rb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}\nonumber\\ &&-\frac{\mathcal {V}_{LR}}{2g_{Rb}^*g_{La}}\frac{(d_{Ra}+d_{Lb}^*)}{(1+d_{Ra}d_{La})(1+d_{Rb}^*d_{Lb}^*)}. \end{eqnarray} \tag{ 2 }$$

To first order in the leakages d, and ignoring terms proportional to $\mathcal {V}_{LR}\,d$ and $\mathcal {V}_{RL}\,d$ (small LP as well as small leakages), we have

$$\begin{equation} \mathcal {V}_{I}\simeq \frac{1}{2}\lbrace\mathcal {V}_{RR}/(g_{Ra}g_{Rb}^*)+\mathcal {V}_{LL}/(g_{La}g_{Lb}^*)\rbrace \end{equation} \tag{ 3 }$$

and

$$\begin{equation} \mathcal {V}_{V}\simeq \frac{1}{2}\lbrace\mathcal {V}_{RR}/(g_{Ra}g_{Rb}^*)-\mathcal {V}_{LL}/(g_{La}g_{Lb}^*)\rbrace , \end{equation} \tag{ 4 }$$

and thus Stokes I and V are independent of the leakages to first order.

The MIRIAD reduction package (Sault et al. 1995) uses these first-order equations when solving for the polarized leakages, ignoring second-order terms in the leakages d and LP fraction. These terms contribute a systematic error in Stokes V of the form Id² and md, for LP fraction m. For the track of 2007 March 31, all leakage terms are of order ∼10⁻² (both real and imaginary parts), introducing a fractional contribution from Stokes I of ∼10⁻⁴. Likewise, the terms of order md are of order 10⁻³ and are unable to explain the observed CP fraction of 10⁻². Another uncertainty arises from the complex terms dependent on the parallactic angle in the form of a phase term in the full gain equations. A systematic phase effect could, in principle, explain offsets in the image plane; however, the low flux density contribution from these offset components (∼md) makes this possibility difficult to reconcile with the evident displacement of the entire point-source flux density in Figure 4.

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Errors in the relative calibration of the L and R gains of the antennas affect the precise cancellation of terms in Equation (4) and are the most natural way to generate false CP. The time variation of the antenna gains is monitored using 1733-130, which, at a flux density of ∼1 Jy, is much weaker than Sgr A*. In order to maximize the signal to noise of the gain measurements, we choose to average the amplitude of the LL and RR visibilities together and determine a single gain curve per antenna rather than separately measuring g_L and g_R. If there is an imbalance in the response of the two "feeds," which are really the same receiver looking through the same quarter-wave plate, this will introduce false CP.

The first test of our calibration method is to examine the calibrated 1733-130 data for CP. In Table 5, we report the polarization measured in 1733-130 after applying the same gains that are used with the Sgr A* data. Although we make no attempt to correct the difference in the L and R gains in our calibration procedure, we end up with little polarization in this source (<0.1%), with a fractional V sensitivity of ∼5 × 10⁻⁴. This suggests a systematic polarization limit of 10⁻³ for this calibration method. Performing the calibration with separate gains for L and R yields nearly identical results for the polarization fraction of Sgr A* but with slightly higher noise.

A second test of this calibration method is to examine the V/I measured for other quasars in the track. We expect no significant CP in quasars at these wavelengths. Figure 6 and Table 5 show that no CP is found in the quasars observed in the 2007 March 31 track. Fractional polarization limits are comparable to those obtained on 1733-130, except for 3C286, which is significantly fainter than the others and thus has a higher level of noise in V/I.

To guard against poorly understood instrumental effects that preferentially affect, for example, sources at low elevation or those with poor parallactic angle coverage, we undertook a supplementary test using the quasar 1924-292. The declination of this source is within half a degree of that of Sgr A*, ensuring that it follows the same path on the sky. The observation scheme was designed to precisely mimic that of the primary track on Sgr A*, with similar hour angle coverage and temporal sampling, identical polarization modulation, and a gain calibrator (1922-201) at a similar distance from the target (9° north compared to 16° for 1733-130 and Sgr A*). Because 1924-292 was brighter at the time of these observations, 7 Jy, than Sgr A* was during the other observations (3–4 Jy), and because of the long integration time in the track, this test also provides a more sensitive limit on instrumental CP than the that provided by observations of quasars on 2007 March 31.

The results of this test are shown in Figure 5 and in the last line of Table 5. Fitting the visibilities to a point source, as was done for other quasars and Sgr A*, we find V/I = (1–2) × 10⁻⁴, with an uncertainty of 10⁻³. This is a factor of more than 10 lower than the CP fraction observed in Sgr A*, including that measured in the same track in a short observation (last line, Table 3). That the polarization of 1924-292 was not found at the same time that the previously observed CP level in Sgr A* was reconfirmed enhances our confidence in our result.

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Figure 5 does show significant CP in an antisymmetric pattern around the pointing center, with a peak V/I values of +2.5 and −4.2 × 10⁻³. The negative peak is approximately five times the image rms. The antisymmetric pattern is clearly distinguishable from the offset CP detections in Sgr A*, which do not show peaks of opposite sign one resolution element away from the main peak. We conclude that this is a signature of an unrelated calibration problem. This calibration could mask a peak in the V map that is offset from the pointing center as seen in Sgr A* in Figure 4, with the allowed amplitude being approximately the difference in the positive and negative peaks, or −1.7 × 10⁻³. We take this as a limit on our uncorrected instrumental polarization, V/I < 2 × 10⁻³.

The LP of 1924-292 was measured to be 10.8% during this track, larger than was observed in the Sgr A* tracks in this work. The increased LP makes this source slightly more susceptible to errors from ignored second-order terms from feed imperfections (Section 4.1). Nevertheless, no significant CP is detected.

In the tests discussed thus far, we failed to find CP in any source other than Sgr A*, placing tight limits on the possible instrumental contribution to the observed signal. The previous section describes the most stringent test, which mimics everything about the Sgr A* observation and nevertheless finds insignificant CP in a bright test source.

A possible distinction between the test sources and Sgr A* is the presence of extended emission in our Galactic center. The test sources are nearly pointlike (with some low-level emission from jet knots in the case of 3C279), while Sgr A* is embedded in dust and free–free emission. To avoid contamination from larger-scale emission, we exclude baselines shorter than 20 kλ from our visibility fits and find that the same exclusion in other sources does not change our results. We are not aware of a mechanism by which the extended emission could introduce false CP, but we have also attempted to verify that no such effect is present. We examined a data set from 2006 July 17, which measures uv spacings of 27–390 kλ instead of the 5–50 kλ that is typical of the other tracks. This track should be more immune to the larger-scale emission, which is highly suppressed by the shorter fringe spacing. We see no significant difference in Table 3 between this track and the others.

Sgr A*, a stratified, self-absorbed synchrotron source (see de Bruyn 1976, for the basics of these models) with frequency- and time-variable polarization, clearly requires a sophisticated radiative transfer model. The Faraday rotation inferred from submillimeter observations (Marrone et al. 2007) has been assumed to be separate from the submillimeter emission regions, although it may instead occur deep inside the source (sometimes referred to as internal Faraday rotation, e.g., Wardle & Homan 2003). The production of CP may similarly be a signature of conversion processes within the emission region. The complete PRT equation is (e.g., Sazonov 1969)

$$\begin{eqnarray} &&\frac{d}{dl}\left(\begin{array}{@{}c@{}}I\\ Q\\ U\\ V\end{array}\right)= \left(\begin{array}{@{}c@{}}\eta _I\\ \eta _Q\\ 0 \\ \eta _V\end{array}\right) + \left(\begin{array}{@{}cccc@{}}-\kappa _I &-\kappa _Q&0&-\kappa _V\\ -\kappa _Q &-\kappa _I&-\kappa _V^*&0\\ 0&+\kappa _V^* &-\kappa _I&-\kappa _Q^*\\ -\kappa _V&0&+\kappa _Q^*&-\kappa _I\\ \end{array}\right)\left(\begin{array}{@{}c@{}}I\\ Q\\ U\\ V\end{array}\right),\nonumber\\ \end{eqnarray} \tag{ 5 }$$

where the Cartesian axes in the plane of the sky ($\hat{\mathbf {e}}_1$ and $\hat{\mathbf {e}}_2$)—which determine the precise definition of Stokes Q and U—are oriented such that $\hat{\mathbf {e}}_1$ is aligned with the local direction of the magnetic field. The absence of transport coefficients related to Stokes U in Equation (5) is a direct consequence of this convenient choice of coordinates, and thus it is a valid expression only locally.

Equation (5) is the vectorial generalization of the standard radiative transfer equation dI_ν/dl = j_ν − α_νI_ν. η_A (with A = I, Q, U, V) are the emission coefficients for each Stokes parameter. The absorption coefficients κ_A (one for each Stokes parameter) are related to η_A through detailed balance. The antisymmetric portion of the matrix comprises the Faraday transport coefficients κ*_A. They couple Q, U, and V, permitting exchanges of LP and CP along the path of radiation. This rotation in three-dimensional polarization space (Q, U, V) has been referred to as the generalized Faraday rotation effect (Melrose 1997). The general Faraday effect reduces to pure Faraday rotation (i.e., rotation in the Q–U plane) in the nonrelativistic limit and to pure Faraday conversion (i.e., rotation in a plane perpendicular to the Q–U plane) in the ultrarelativistic limit but, in general, is a combination of both.

The intrinsic emission term in Equation (5) (η_V) demonstrates the first potential source of CP. The intrinsic emission in Stokes V is weak: η_V is of order 1/γ smaller than the other components and is generally ignored in synchrotron-emitting plasma. On the other hand, predominantly CP emission can be expected in a cold cyclotron emitting plasma (e.g., Landau & Lifshitz 1975), in which the relativistic approximations intrinsic to synchrotron spectra (e.g., beaming, see Rybicki & Lightman 1979) are not valid (see also Mahadevan et al. 1996). The submillimeter emission from Sgr A* is expected to arise at very small radius where the plasma is hot, and therefore emission from cold electrons is not expected to be relevant.

The birefringence of the plasma may also transform LP to CP, as noted above, but this "Faraday conversion" is weak in a cold plasma. In this limit, the response of the medium depends exclusively on the plasma frequency, ν_p = (n_ee²)^1/2/(πm_e)^1/2, and the cyclotron frequency, ν_B = eB/(2πm_ec). In the high-frequency regime (i.e., ν_B/ν ≪ 1, applicable to the submillimeter emission considered here given plausible magnetic field strengths), the rotation and conversion coefficients are (Melrose & McPhedran 2005; Swanson 1989)

$$\begin{eqnarray} {\kappa _V^*}_\mathrm{cold}&=&(2\pi/c)\big(\nu _p^2\nu _B \big/\nu ^2\big)\cos \theta \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} {\kappa _Q^*}_\mathrm{cold}&=&{-}(\pi/c)\big(\nu _p^2\nu _B^2 \big/\nu ^3\big)\sin ^2\theta, \end{eqnarray} \tag{ 7 }$$

where θ is the angle between the magnetic field vector and the line of sight. The ratio of these equations shows Faraday conversion (κ*_V) to be a factor of ν_B/ν (∼10⁻³ for our data) smaller than rotation, inconsistent with our observations. Equations for higher temperature plasmas are presented in Section 5.6.

Homan et al. (2009) list a series of scenarios in which CP can be produced in quasars. By incorporating line-of-sight changes in the magnetic field geometry, they offer possibilities not present in the homogenous-field radiative transfer equation (Equation (5)). In a uniform field, Stokes Q needs to be converted into Stokes U before Stokes U can be converted into Stokes V. Changes in the field orientation along the line of sight break a fundamental assumption of Equation (5): the azimuthal angle (ϕ) between the field direction in the plane of the sky and the coordinate axis $\hat{\mathbf {e}}_1$ was defined to be zero. Changes in field direction will introduce a coupling of Q and V across the domains of magnetic orientation. The magnitude of the transfer coefficients also depends on the angle θ, introducing a small additional change in coupling. These field orientation changes, whether stochastic (Ruszkowski & Begelman 2002) or ordered (as in the helical fields of Beckert 2003), can also create significant CP. Both of these mechanisms are capable of producing the CP sign coherence observed in Sgr A*, although a small net bias to the field direction is important for the stochastic field.

The wavelength-dependent size of Sgr A* indicates the importance of synchrotron self-absorption in the observed spectral energy distribution. The source size and brightness are determined by the variation of density, magnetic field, and electron energy distribution with radius/distance, although there are unbreakable degeneracies between these quantities even assuming power laws in radius without additional information or assumptions. A specific choice of jet structure can determine the structure in density/field/energy and match both size and flux (e.g., Markoff et al. 2007) and even variability timing (Falcke et al. 2009). Similarly, the assumption of, e.g., an equipartition magnetic field can specify the structure for an accretion flow while matching observations.

Polarimetric observations add new information regarding the source structure, potentially reducing the need for unverifiable assumptions. The structure of Sgr A* is likely to be more complex than is captured by power-law models; this is particularly true at submillimeter wavelengths where the emission originates near the black hole. Complex magnetohydrodynamic simulations and general relativistic radiative transfer (e.g., Huang et al. 2008; Mościbrodzka et al. 2009; Shcherbakov et al. 2010; Shcherbakov & Huang 2011) will more faithfully represent the true source properties near 345 GHz than few-parameter analytic models. Nevertheless, the smoothness of the average CP spectrum, the exclusive left handedness of the polarization from 1.4 to 345 GHz, and the slow monotonic variation of the fractional polarization suggest that much insight can be gained from a simple conceptual model incorporating the ideas of self-absorbed synchrotron sources (e.g., de Bruyn 1976).

In the case of self-absorbed synchrotron sources, the specific intensity near frequency ν is dominated by the emission of electrons within a narrow energy range around γ_rad. An electron with Lorentz factor γ_rad emits most of its radiative power at a frequency ν_c ∼ γ²_radeBsin θ/(2πm_ec) = γ²_radν_B⊥. The brightness temperature associated with the emitting electrons is related to this effective Lorentz factor by (e.g., Pacholczyk 1970; Rybicki & Lightman 1979) k_BT_b = α(p)γ_radm_ec², where α(p) is a coefficient of order unity. The brightness temperature can be obtained directly from observations: T_b = 1.22 × 10¹²S_νν⁻²θ⁻²_s(ν)K; where S_ν is the observed flux density in Jy, θ_s is the source angular radius in milliarcseconds, and ν is in GHz. Assuming power-law dependencies of the form S_ν∝ν^−m and θ_s(ν)∝ν⁻ⁿ, we obtain γ_rad∝ν^{−m − 2 + 2n} (see, for example, Loeb & Waxman 2007).

In what follows, our main approximation is to assume that at each frequency, the synchrotron-emitting electrons, with energies γ_rad, are also responsible for Faraday conversion. We also assume that Faraday rotation and conversion can be considered to act in an alternating manner rather than attempting to treat the full, complicated PRT equation. The radiative transfer near the τ = 1 surface is particularly complicated because all matrix elements can be important. By assuming sequential effects, we can operator-split the differential equation. In this case, and for short propagation distances, the production of Stokes V is simply an angle of rotation in the Poincairé sphere corresponding to the cumulative phase shift between the polarization modes of the plasma near radius r₀ (Kennett & Melrose 1998). This can be calculated as an integral of κ*_Q,

$$\begin{equation} \Delta \psi (r_0,\nu)=\int _{r_0}^\infty \frac{\pi }{c}\frac{\nu _p^2\nu _B^2}{\nu ^3}\sin ^2\theta \Theta _\mathrm{FC}\, dr, \end{equation} \tag{ 8 }$$

where we have corrected the cold plasma Faraday conversion rate (Equation (7)) by a factor Θ_FC (see Section 5.6 below) that takes into account the effect of a power-law distribution of relativistic electrons on Faraday conversion.

Following the treatment of Faraday conversion presented by Kennett & Melrose (1998), we take the finite-temperature correction to be proportional to the local mean Lorentz factor of the electron distribution, and we further approximate this with our value of γ_rad (in the same approximation Θ_FR ∼ log 〈γ〉/〈γ〉; e.g., Sazonov 1969; Quataert & Gruzinov 2000),

$$\begin{equation} \Theta _\mathrm{FC}(\nu,T_e,\theta,B)\sim \langle \gamma (r)\rangle \sim \gamma _\mathrm{rad}. \end{equation} \tag{ 9 }$$

Assuming conversion in a narrow range in r, we write

$$\begin{eqnarray} \Delta \psi (r_0,\nu)&\propto& \int _{r_0}^\infty n_e(r)B(r)^2\sin ^2\theta \gamma _\mathrm{rad}(r) \nu ^{-3} dr\nonumber\\ &\sim& n_e(r_0)B(r_0)^2 \gamma _\mathrm{rad}(r_0)\nu ^{-3}r_0. \end{eqnarray} \tag{ 10 }$$

We can obtain Δψ as a function of frequency by using the fact that for each observed frequency ν₀, there is a corresponding radius r₀. If we assume, as well, that the electron density and magnetic field have radial profiles of power-law form, n_e ∼ r^−β and B ∼ r^−α, the Faraday conversion phase shift is

$$\begin{eqnarray} &&\Delta \psi (r_0,\nu _0)\sim r_0^{-\beta -2\alpha +1}\nu _0^{-m+2n-5}\sim \nu ^{\beta n+2\alpha n -m+n-5}. \end{eqnarray} \tag{ 11 }$$

Power-law relations of the form S_ν∝ν^−m—for the flux density—and θ_s(ν)∝ν⁻ⁿ—for the source angular size—are a natural consequence of stratified synchrotron sources (de Bruyn 1976). In this simple model, the size–frequency relation is obtained as follows. First, we take the synchrotron optical depth associated with a narrow shell of radius r (Rybicki & Lightman 1979): τ_ν(r) = α_νΔr ∼ n_e(r)B^{(p + 2)/2}(r)ν^{−(p + 4)/2}r ∼ r^{−β − α(p + 2)/2 + 1}ν^{−(p + 4)/2}, where p is the electron power-law index. Hence, for a given frequency ν₀, the τ = 1-surface occurs at r^{−β − α(p + 2)/2 + 1}₀ν₀^{−(p + 4)/2} ∼ 1 or r₀ ∼ ν^{−(p + 2)/(2β + α(p + 2) − 2)}₀. Similarly, the flux density can be shown to be (see de Bruyn 1976, for details) S_ν ∼ ν^{(13 − 5β − 3α − 2αp + 2p)/(2 − 2β − 2α − αp)}. The resulting spectral indices are

$$\begin{eqnarray} n&=&\frac{p+4}{2\beta +\alpha (p+2)-2}\quad {\rm and}\nonumber\\ m&=&{-}\frac{13-5\beta -3\alpha -2\alpha p+2p}{2-2\beta -2\alpha -\alpha p}. \end{eqnarray} \tag{ 12 }$$

Therefore, α and β can be solved for as functions of m and n:

$$\begin{eqnarray} \beta &=&\frac{-3-2m+5n-2p-mp+2np}{n}\quad {\rm and} \nonumber\\ \alpha &=&\frac{5+2m-4n}{n}, \end{eqnarray} \tag{ 13 }$$

which gives (dropping the subscript in ν₀ for clarity)

$$\begin{eqnarray} \Delta \psi \sim \nu ^l,\quad {\rm with}\quad l&=&(2n-m-2)(p-1)\nonumber\\ &=&\frac{(p-1)(-\alpha +\beta -1)}{\alpha (p+2)+2(\beta -1)}. \end{eqnarray} \tag{ 14 }$$

The indices α and β are very sensitive to n, which is close to 1. Falcke et al. (2009) report a value of n = 1.3  ±  0.1, obtained from observations between 22 GHz and 230 GHz, after correcting for interstellar scattering. These results are very sensitive to the scattering model. Another approach, used by Shen (2006), is to minimize the dependence on the scattering law by relying on the shortest wavelength data. For measurements at 43 and 86 GHz, their result is n = 1.09 ± 0.3. An even shallower slope can be obtained using the results from Doeleman et al. (2008). From the VLBI visibilities, the angular size of the source, assuming a circular Gaussian profile, is 37 μas, which is smaller than the last photon orbit for a Schwarzschild black hole (∼52 μas). If the source geometry is assumed to be an annulus (Doeleman et al. 2008), the mean diameter is 58 μas. Using this angular diameter at 230 GHz gives a value of n ∼ 0.8. In particular, for this value of n, with m = −0.43 (An et al. 2005), and assuming an electron power-law index of 2.4, we obtain α ∼ 1.2 and β ∼ 1.4. These values are close to those corresponding to spherical accretion and equipartition of energy (β = 3/2 and α = (β + 1)/2 = 5/4). Most importantly, these indices imply l ∼ 0.1, i.e., the CP production rate is almost independent of frequency, which is consistent with observations (ranging from 0.5% to 1.5%) across a range in frequency of almost two orders of magnitude. The main point here is that a mild increase of CP with frequency is possible. Although this is contrary to the decreasing efficiency of Faraday conversion with frequency that is expected for a uniform medium, it is a natural consequence of stratified emission.

The amount of CP generated from LP is proportional to this phase shift, i.e., m_c ∼ m_l Δψ for small Δψ. The weak dependence of Δψ on frequency implies that CP production in different layers of the source behaves similarly regardless of location respect to the SBMH. The value of Δψ should remain small to avoid conversion of CP back to LP at high frequencies. Starting with the ∼10% (e.g., Pacholczyk 1970) intrinsic polarization fraction of the optically thick synchrotron, achieving CP fractions of 0.1%–1% requires Δψ ∼ 0.01–0.1. Such small values are far from reversing the sign of CP, consistent with the fixed sign of CP at all frequencies, as long as the orientation of the orientation magnetic field does not fluctuate randomly. A variable orientation of the magnetic field with radius—i.e., field reversals—is inconsistent with the apparent self-similarity of the CP spectrum. If this self-similarity persists for more than two orders of magnitude in frequency (1–345 GHz), the orientation of the magnetic field must be highly coherent over spatial scales spanning nearly two orders of magnitude (since n ≈ 1), roughly between 1 R_S and 100 R_S.

In plane-of-the-sky coordinates aligned with $\mathbf {B}_\mathrm{sky}$ (total field equals plane-of-the-sky field $\mathbf {B}_\mathrm{sky}$ plus line-of-sight field $\mathbf {B}_\mathrm{los}$), Equation (5) shows that for intrinsic synchrotron emission only in I and Q, Faraday conversion proceeds as Q → U → V, i.e., conversion is driven by Faraday rotation within a narrow synchrotron shell (Beckert 2003). The sign of U generated by this mechanism depends on the sign of the Faraday rotation coefficient κ*_V (Equation (6)), which can change depending on the angle the magnetic field makes with respect to the line of sight θ. On the other hand, the sign of the Faraday conversion coefficient κ*_Q (Equation (7)) does not change with θ. This implies that the sign of Stokes V generated by conversion is exclusively determined by the sign of Stokes U generated from Q by local Faraday rotation, and is ultimately dependent on the local value of cos θ. Therefore, a constant sign in Stokes V needs a constant orientation of $\mathbf {B}_\mathrm{los}$. This requirement is already hinted at by the observed RM properties (Marrone et al. 2006a, 2007). On the other hand, the orientation of $\mathbf {B}_\mathrm{sky}$ is free to change in time without affecting the value of V. This is a consequence of the sequence Q → U → V given by Equation (5), where the first step is performed by Faraday rotation depending only on $\mathbf {B}_\mathrm{los}$ and the fact that Stokes V does not change with coordinate transformations in the plane of the sky. Changes of the orientation of $\mathbf {B}_\mathrm{sky}$ in time will only change the observer's definition of Stokes Q and U. Indeed, this is likely to be the explanation for the observed wandering values of Q and U (Marrone et al. 2006b), while Stokes V appears to be stable.

5.5.1. Relation to Observed Faraday Rotation Rates

A key assumption in the self-absorbed approximation above is that synchrotron emission, Faraday conversion, and Faraday rotation (which drives conversion) all take place within the same narrow shell of plasma. Previous work (e.g., Marrone et al. 2007) has assumed that Faraday rotation primarily occurs far from the emission region at a given frequency. This "Faraday screen" approximation may still operate in regions of the accretion envelope where the material is too optically thin to contribute any emission or absorption to the incoming radiation while still being dense and magnetized enough to produce Faraday rotation. For such cases, Marrone et al. (2006a) calculate the cumulative Faraday rotation acting upon polarized radiation emitted at radius r₀ (analogous to Equation (8)) as

$$\begin{equation} \Delta \chi (r_0,\nu)=\int _{r_0}^\infty \frac{2\pi }{c}\frac{\nu _p^2\nu _B}{\nu ^2}\cos \theta \Theta _\mathrm{FR}\, dr \end{equation} \tag{ 15 }$$

(except for the Θ_FR term, which is taken to be 1). The total emission (Stokes I, V, and total LP) is assumed to come from a region interior to r₀ and to be essentially unaffected as it emerges from the source, except for the Faraday rotation of the electric vector position angle (EVPA).

To be consistent with our self-absorbed Faraday conversion approach, Faraday rotation should act both internally (at τ ∼ 1, where it drives conversion) as well as externally (at τ ≪ 1, where the Faraday screen approximation is valid). To satisfy this requirement, the Faraday rotation scale lengths should extend far beyond the thickness of the emission/conversion shell. To confirm this, we define the differential Faraday rotation and conversion depths (differential forms of Equations (8) and (15)) as

$$\begin{equation} \frac{d\tau _\mathrm{FR}}{dr}\propto r^{-(\beta +\alpha)}\Theta _\mathrm{FR}(r)\nu ^{-2} \approx r^{-(\beta +\alpha)} \frac{\ln {\gamma _\mathrm{rad}(r)}}{\gamma _\mathrm{rad}(r)}r_0^{2n} \end{equation} \tag{ 16 }$$

and

$$\begin{equation} \frac{d\tau _\mathrm{FC}}{dr}\propto r^{-(\beta +2\alpha)}\Theta _\mathrm{FC}(r)\nu ^{-3}, \approx r^{-(\beta +2\alpha)} \gamma _\mathrm{rad}(r) r_0^{3n}, \end{equation} \tag{ 17 }$$

respectively, where γ_rad∝r^−δ (Section 5.3) is also a power law of r, with δ small. It can be readily seen that dτ_FC/dr is significantly steeper in radius than dτ_FR/dr. For comparison, the synchrotron optical depth can be written in differential form as dτ/dr ∼ r^{−β − α(p + 2)/2}ν^{−(p + 4)/2} (see Section 5.3). For values of p between 2 and 3, dτ/dr can be as steep as or steeper than dτ_FC/dr, and thus also steeper than dτ_FR/dr, confirming that significant Faraday rotation takes place outside of the τ = 1 surface (see Jones & Odell 1977). This means that Faraday rotation is still at work when Faraday conversion has stopped being effective. In other words, the source has only a limited spatial range to produce Stokes V—driven by the sequence Q → U → V—before the conversion rate decreases to negligible values. Therefore, while Faraday conversion is roughly local to the shell of radius r₀, Faraday rotation is not.

A consequence of having significant Faraday rotation within and immediately outside the photosphere is a frequency-dependent RM. The highest frequencies probe deeper layers in the stratified source and therefore integrate Faraday rotation contributions from the shells that define lower-frequency photospheres. There are, as of yet, few observational constraints on this possibility. The RM has been determined through the measurement of a change in χ across a narrow range in frequency (Marrone et al. 2007), and less securely through the comparison of position angles averaged over several observations at more widely separated frequencies (Macquart et al. 2006). A demonstration of the frequency dependence of the RM could be obtained through position angle measurements at multiple pairs of closely spaced frequencies, which should be possible very soon with improved millimeter interferometers and ALMA.

Although the Faraday screen approximation is reasonable for a stratified synchrotron source, it can be difficult to obtain an accurate RM observationally, due to the source's layered geometry. Assuming the cold, optically thin Faraday rotation relation χ(ν) = χ₀ + c²/ν²RM (e.g., Marrone et al. 2006a, Equation (1)), RM is observationally obtained by computing the quantity dχ/dλ² = dχ/d(c/ν²) ≈ Δχ/Δ(c²/ν²), using two neighboring frequencies and their respective observed EVPAs. The implied assumption, in addition to the validity of the screen, is that the emission at the two observed frequencies is originated in the same region and with the same initial EVPA χ₀. However, in the layered scenario, if these two frequencies correspond to two shells that do not overlap spatially, they will produce emission that will travel through two different Faraday screens. In that case, RM would be a function of the two frequencies observed and not a constant value. Thus far, observations suggest that this is not the case and that RM is indeed a constant, even when χ₀ varies in time (Marrone et al. 2007). Simultaneous observations at more than two frequencies will help resolve the question of whether Faraday rotation in Sgr A* is proportional to ν⁻² or not.

One unaddressed problem in the discussion above is that there apparently is not enough observed LP at low frequencies, (e.g., 4 and 8 GHz, where CP dominates) to generate the observed Stokes V via the Faraday conversion process. However, synchrotron flux density is highly polarized for ordered magnetic fields, even in the optically thick limit (Pacholczyk 1970). Therefore, it is highly likely that the observed LP suffers severe beam smoothing at these frequencies, at a point beyond the conversion region, if the Faraday rotation region shows fluctuations (e.g., stochastic or turbulent in nature) at spatial scales smaller than the source size. For the sake of argument, consider partially polarized radiation with constant polarization angle χ₀ entering a piecewise uniform Faraday screen. Radiation will exit the screen with rms fluctuations in EVPA equal to σ_χ ∼ σ_RM(c/ν)², where σ_RM contains the fluctuations in n_e, B, and θ. If the fluctuations in the outcoming Δχ are Gaussian, the observed LP will be $\langle m_l\rangle = m_{l,0} \,e^{-2\sigma _\chi ^2}$, where m_{l, 0} is the original LP fraction entering the screen. This occurs because, while Q and U are additive quantities, total LP is not, resulting in observed values of m_l that are severely sensitive to EVPA changes in the plane of the sky.

In an accretion flow, many of the physical quantities relevant for synchrotron emission are expected to increase inwardly. This is true for the electron density n_e and the temperature T. In particular, the temperature of the plasma near the event horizon can achieve mildly to highly relativistic temperatures. It is evident then that the cold plasma approximation should not apply in the interior regions where Faraday conversion could operate. A more general expression for the Faraday transport coefficients (Equations (6) and (7)) is

$$\begin{eqnarray} \kappa _V^*&=&\frac{2\pi }{c}\frac{\nu _p^2\nu _B}{\nu ^2}\cos \theta \, \Theta _\mathrm{FR}(\nu,T_e,\theta,B)\nonumber\\ &=&{\kappa _V^*}_\mathrm{cold} \, \Theta _\mathrm{FR}(\nu,T_e,\theta,B) \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} \kappa _Q^*&=&{-}\frac{\pi }{c}\frac{\nu _p^2\nu _B^2}{\nu ^3}\sin ^2\theta \, \Theta _\mathrm{FC}(\nu,T_e,\theta,B)\nonumber\\ &=&{\kappa _Q^*}_\mathrm{cold}\, \Theta _\mathrm{FC}(\nu,T_e,\theta,B), \end{eqnarray} \tag{ 19 }$$

where the functions Θ_FR(ν, T_e, θ, B) and Θ_FC(ν, T_e, θ, B) are finite-temperature corrections to the standard cold plasma transport coefficients. By definition, Θ_{FR, FC} → 1 when T → 0 and ν ≫ ν_B.

In a cold plasma, the electrons oscillate in response to electromagnetic waves but are effectively stationary otherwise. When the electron energy distribution is non-negligible, the interaction of electron motions and incident electromagnetic radiation is given by the Boltzmann equation in the presence of the Lorentz force: the Vlasov equation. The Vlasov equation describes the dielectric tensor of the plasma and thus its Faraday transport coefficients, which depend on the electron energy distribution (e.g., Melrose & McPhedran 2005).

One of the earliest approaches to this problem originated with Sazonov (1969). For a power-law distribution of electron energies, the ratio of the Faraday transport coefficients (see also Jones & Odell 1977) is

$$\begin{eqnarray} \left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{pl} &=&\left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{cold}\frac{\Theta _\mathrm{FC}^\mathrm{pl}}{\Theta _\mathrm{FR}^\mathrm{pl}}=\left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{cold}\nonumber\\ \ms &&\times\frac{2(p+1)}{(p+2){(p-2)}} \frac{\gamma _\mathrm{min}^{3}}{\ln \gamma _\mathrm{min}} \left[1-\left(\gamma _\mathrm{min}^2\frac{\nu _{B\perp }}{\nu }\right)^{\frac{(p-2)}{2}}\right],\nonumber\\ \end{eqnarray} \tag{ 20 }$$

which is an increasing function of γ_min (γ_min > 1 always) under the condition ν_B/ν ≪ 1. Equation (20) shows that even with (κ*_Q/κ_V*)_cold ≪ 1, for hot enough plasmas, the Faraday conversion coefficient κ*_Q can become a significant fraction of κ*_V or even exceed it in extreme cases.

For electrons in a relativistic thermal (Maxwell–Juttner) distribution, the plasma dielectric tensor is different. Melrose (1997) derived an ultrarelativistic approximation for the response tensor of a magnetized thermal plasma. Ballantyne et al. (2007) combined this result with that of a classical cold plasma to obtain an approximate continuous function of the polarization proper modes as a function of temperature. Shcherbakov (2008) extended these results, providing consistent analytic expressions valid for all temperatures. In this case, the ratio of the Faraday transport coefficients is

$$\begin{eqnarray} \left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{therm}&=&\left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{cold}\frac{\Theta _\mathrm{FC}^\mathrm{therm}}{\Theta _\mathrm{FR}^\mathrm{therm}}=\left|\frac{\kappa _Q^*}{\kappa _V^*}\right|_\mathrm{cold} \left[\frac{K_1\left(\frac{m_ec^2}{k_BT}\right)}{K_0\left(\frac{m_ec^2}{k_BT}\right)}\right.\nonumber\\ &&\left.+\, 6\frac{k_BT}{m_ec^2}\frac{K_2\left(\frac{m_ec^2}{k_BT}\right)}{K_0\left(\frac{m_ec^2}{k_BT}\right)}\right] \frac{g(T,\theta,\nu _B/\nu )}{f(T,\theta,\nu _B/\nu )}, \end{eqnarray} \tag{ 21 }$$

where K_n(x) is the modified Bessel function of the second kind. Equation (21) smoothly reduces to the cold plasma limit when T → 0, since the term in the square brackets becomes exactly 1 in this limit. At temperatures of 10⁹ K (kT/m_ec² = 1/6), the term in the square brackets takes a value of ∼2.5, and at 2 × 10¹¹ K (kT/m_ec² = 34), it reaches ∼1.3 × 10⁵.

The factors f and g, above, introduced by Shcherbakov (2008), take a value of 1 for ν_B/ν → 0 and provide the necessary corrections for higher-order terms in ν_B/ν. They can lead to significant modifications to the ratio of the Faraday transport coefficients in the moderately relativistic regime. For example, for intense magnetic fields, the cyclotron frequency can be written as ν_B = 0.35 GHz(B/20 G), and thus the ratio ν_B/ν takes values of ∼10⁻³ at submillimeter wavelengths. Following Shcherbakov (2008), at ν_B/ν = 10⁻³ and T = 2 × 10¹¹ K with θ = π/4, the correction factors become g ∼ 0.93 and f ∼ −0.003. The multipliers of the cold Faraday transport coefficients in Equation (21) can strongly change the expected behavior even at moderate temperatures. This clearly demonstrates the importance of including these factors in numerical simulations of Sgr A* in order to properly handle the PRT problem.