A CONSTRAINT ON THE ORGANIZATION OF THE GALACTIC CENTER MAGNETIC FIELD USING FARADAY ROTATION

To study the RM across this field, mosaics of the polarization angle were differenced between the two bands. The polarization angle difference image (hereafter Δθ image) was created by differencing the polarization angle images (θ_4.885 GHz − θ_4.835 GHz) and remapping each value of Δθ to the range −90° to 90°. Figure 3 shows the Δθ image and its error. Observationally, the rotation measure is defined as RM =Δθ/Δ(λ²). Assuming this λ² law, a P.A. difference of 1° corresponds to an RM of −220 rad m⁻².

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More generally, the observed polarization is the sum of polarized emission emitted with a range of RM (Burn 1966; Brentjens & de Bruyn 2005). Such complex sources can have non-quadratic changes in the polarization angle that can confuse a simple analysis. In these situations, the pertinent physical quantity is the "Faraday depth:"

$$\begin{equation} \phi = 0.81 \int _{{\rm there}}^{\rm here} \! n_e \, \vec{B} \, \cdot d\vec{l} \, \rm {rad} \, \rm {m}^{-2}, \end{equation} \tag{ 1 }$$

where n_e is in cm⁻³, $\vec{B}$ is in G, and $d\vec{l}$ is in pc. For simple physical distributions of n_e and $\vec{B}$, ϕ is equal to RM. However, robustly tying the RM to physical conditions in complex cases requires measurements at many wavelengths (Brentjens & de Bruyn 2005). Since we only have two wavelengths to study the polarization in this region, we instead use this formalism to define the limits of deriving physical conditions from the observed RM.

First, the formalism of Brentjens & de Bruyn (2005) shows that the spacing of the bands in wavelength determines the "RM resolution" and possible nπ ambiguities. For the two bands used here, the RM resolution is 4 × 10⁴ rad m⁻² and any aliasing occurs at RM =n*4 × 10⁴ rad m⁻², for an integer n. The RM expected in the GC region covered by this survey is typically <2000 rad m⁻² (Yusef-Zadeh et al. 1984; Tsuboi et al. 1986; Roy et al. 2005), so there is little chance of an nπ ambiguity. Second, the bandwidth determines the amount of Faraday rotation within a band, which limits the maximum Faraday depth detectable to ϕ < 2 × 10⁴ rad m⁻². Finally, a source that emits over a range of Faraday depths, known as "Faraday thick," can be internally depolarized. The maximum Faraday thickness detectable to the present observations is 830 rad m⁻². Some sources, such as the Radio Arc (Yusef-Zadeh & Morris 1987, 1988), have an RM that changes by more than 830 rad m⁻², so parts of the GC region may be Faraday thick to our observations.

Figure 4 shows histograms of Δθ (number of independent spatial beams per degree of Δθ) from the entire survey and a smaller region. Intrinsically, we expect the Δθ histogram to have contributions from many distinct regions of varying peak Δθ and width. We found that a Lorentzian profile fits these heterogeneous distributions better than a single Gaussian. For smaller regions, where Δθ has a single-valued, noise-like distribution, the Lorentzian can also approximate a single Gaussian. In the limit of a single-valued, noise-like Δθ distribution, the Gaussian noise is equivalent to a Poisson distribution in the large-N limit. We use this similarity to approximate the Δθ bin count errors as $\sigma _N = 1 + \sqrt{N + 0.75}$, where N is the number of independent beams in a bin (Gehrels 1986). In Section 3.2, we show that comparing RM measured histogram methods to previous work shows that the errors are conservative.

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Aside from theoretical expectations, the distribution of Δθ values shows that the apparent Δθ can generally be reliably converted to RM. Most values of Δθ measured have small offsets from 0°, with ∼50% within ±10°, ∼75% within ±20°, and ∼90% within ±45°. This is consistent with the ∼9° rotation expected for RM ≈±2000 rad m⁻². The typical angle change is ≪1 rad, so relative angles may be treated as roughly linearly distributed. The best constraint on the mean Δθ has a typical error of about 1° or 220 rad m⁻². For a change of 1° between our two bands, the Faraday rotation from λ = 0 cm is 49°. This typical uncertainty in Δθ makes the calculation of the intrinsic polarization angle highly uncertain, so no such results are presented here.

Visualizing the images of Δθ and RM is difficult, since the per-pixel sensitivity is poor and varies across the field of view. Convolution and other image processing techniques can be used to extract this information even in poorly calibrated VLA data (Rudnick & Brown 2009). We tested two statistical techniques to spatially smooth the RM: averaging and histogram fitting. These methods and a comparison of their results are described in the Appendix. In general, the two methods have similar results. The histogram-fitting method is less sensitive to outliers and has more conservative errors, so it is used in all results described below.

Since this work is applying a relatively new technique to a complex region, it is important to test the results against known sources. This section compares our results shown in Figures 1 and 3 to the RM for specific regions studied previously (LaRosa et al. 2001; Tsuboi et al. 1986; Haynes et al. 1992; Roy et al. 2005; Yusef-Zadeh et al. 1997). LaRosa et al. (2001) present images of 6 cm polarized intensity near the NRF G359.85+0.39 from VLA data with similar sensitivity and resolution as the present study. The two surveys have similar brightness distributions and structure in the polarized emission, particularly the depolarized regions on the southeast and northeast sides of G359.85+0.39 (see also Law et al. 2008a). The similarity shows that the calibration and imaging quality is similar to that of LaRosa et al. (2001).

Haynes et al. (1992) and Tsuboi et al. (1986) conducted independent, single-dish surveys near 3 cm, covering a few square degrees of the GC region. Although depolarization is weaker near 3 cm and their beam is larger, there is general agreement between our Figure 1 and their polarized intensity maps. Figure 5 shows a comparison of our RMs with the four-band measurements of Tsuboi et al. (1986). Near (017, 022) and (01, 035), Tsuboi et al. (1986) find that the RM has a maximum of +1000 rad m⁻², while the present survey finds a maximum of 770 ± 110 rad m⁻². The maps are similar moving north across (015, 04), where the RM switches from positive to negative values; Tsuboi et al. (1986) measure RM ≈−250 rad m⁻² while the present survey finds −220 ± 130 rad m⁻². There is some agreement at the northwestern edge of the polarized emission of the Radio Arc, shown in Figure 5, where the RM switches back to positive values. The exact location of this second RM sign change is slightly different and may reflect the different RM depths each survey is sensitive to.

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Yusef-Zadeh et al. (1997) presented a detailed study of the polarization properties of the nonthermal filament G359.54+0.18 (RF-C3) at 6 and 3.6 cm. Figure 3 of that work has a similar 6 cm brightness and RM distribution as the present work, both presented here and in Law et al. (2008a). The RM map of the filament shows three distinct, bright clumps each having relatively uniform values. The morphology seen in the present survey is similar to that of Yusef-Zadeh et al. (1997), although it had roughly three times better resolution (4'' compared to 12'' in the present work). The first clump, at R.A., decl. (B1950) = (17:40:41, −29:12:30) has RM ≈−2700 rad m⁻², compared to −3960 ± 1100 rad m⁻² in the present survey. The second clump, at (17:40:43, −29:12:40), has RM ≈−2000 rad m⁻², compared to −2200 ± 440 rad m⁻² in the present survey. The third clump, at (17:40:44, –29:12:45), has RM ≈−1500 rad m⁻², compared to −1540 ± 660 rad m⁻² in the present survey. We conclude that, in general, there is good agreement between the RM of the present survey and that of Yusef-Zadeh et al. (1997).

In summary, the polarized intensity and RM of the present 6 cm survey shows good agreement with those of other surveys. This is consistent with the fact that the polarimetric leakage is expected to have relatively little frequency structure for the VLA feed design (Cotton 1994, 1999); any systematic errors in the polarization angle are subtracted when forming the Δθ image. It also shows that histogram fitting of the Δθ values is a reasonable estimate of the RM and its uncertainty at 6 cm in this region.

The 6 cm polarized continuum intensity of the northern extension of the Radio Arc is several mJy beam⁻¹ and spans the entire eastern edge of the survey up to a latitude of b ∼ 08. To test for frequency structure in the polarized intensity, the polarized intensity maps in the two bands were differenced. The lack of diffuse emission in the difference map shows that the two maps have similar diffuse emission within roughly 1 mJy. The comparable 20 cm mosaic of polarized continuum shows no extended emission down to a level of about 0.1 mJy beam⁻¹ (more details in Law et al. 2008a).

For latitudes up to b = 03, the polarized continuum emission seen in the 6 cm interferometric maps (Figure 1) has a total intensity counterpart in the same data. However, north of b = 03, the total intensity counterpart is too extended to be detected by the VLA 6 cm observations. Since the polarized emission is broken into small spatial scales (as shown in Figure 2), it is detected throughout the region and the apparent polarization fraction often exceeds 100%.

As a final estimate of potential missing flux, we compare the VLA polarized intensity maps to continuum maps from the Green Bank Telescope (GBT; Law et al. 2008b). We convolve the VLA Stokes Q and U maps to the GBT resolution to estimate the polarization fraction. At 6 cm, the peak polarization fraction is 10% in the eastern half of the survey, which is within a factor of two of that seen in the 5 GHz survey of Sofue et al. (1987). The polarization fraction falls to a few percent in the western half of the survey, consistent with the large-scale, polarized structure seen at 8–10 GHz (Tsuboi et al. 1986; Haynes et al. 1992). This similarity shows that the VLA observations detect most of the polarized emission seen in single-dish surveys. This is consistent with the agreement between RM measured here and in other single-dish and interferometric observations (see Section 3.2). At 20 cm, the upper limit on the polarization fraction is roughly 1% of the total intensity measured by the GBT.

Figure 6 shows two maps of RM smoothed over 125 arcsec tiles with the histogram-fitting method. The images show that there is coherent structure on degree size scales. The east side of the survey tends to have RM greater than zero and the west side less than zero.

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The east–west structure is seen more clearly in averages calculated over all latitudes, as shown in Figure 7. In the east, for 02 < l < −03, RM ≈+330 rad m⁻², then the RM changes rapidly for −03 < l < −055, and in the west, for −095 < l < −055, RM ≈−880 ± 50 rad m⁻². Averaging RM over the top half of the survey (045 < b < 07) shows a similar structure as the average over all latitudes, but with a larger range. The maximum RM is +660 ± 75 rad m⁻² near l = −025 on the east side, while the minimum RM is −1320 ± 75 rad m⁻² near l = −065 on the west side.

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There are three arcminute-scale RM features that deviate from the simple structure described above. One of the regions with the largest positive RM is at the southern border of the survey, near (−01, 02). Figure 8 shows that the region with large RM covers a region about 8' across, just north of Sgr A. The average RM for this region is 1188 ± 198 rad m⁻². The average RM for all latitudes near l = −01 is ≈+330 ± 60 rad m⁻².

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The region with the most negative RM is at (−06, + 05), on the right side of Figure 6. Over an area about 8' across, the mean RM is −1320 ± 110 rad m⁻². Figure 7 shows that the mean RM at l ∼ −06 is ≈−880 ± 110 rad m⁻². This feature may partially explain why the amplitude of the east–west RM asymmetry is larger when averaging over the top half of the survey.

A third unusual RM structure is a ridge extending from (025, + 04) to (00, + 05), seen at the left of Figure 6. The feature has a negative RM, but the surrounding region has a positive RM. The average RM along this ridge is ≈−220 ± 110 rad m⁻², as compared to the mean value of ≈330 ± 60 rad m⁻² for all latitudes near l = 0°. This structure is seen in the RM map of Tsuboi et al. (1986), and a detailed comparison of that work to the present work is shown in Figure 5.

The complexity of the line of sight to—and through—the GC region argues for caution when interpreting RM patterns. The apparent 6 cm RM suggests that the line-of-sight magnetic field changes sign, as if the field was predominantly azimuthal. However, it is not immediately clear whether the observed polarized emission originates in the GC region or whether the observed RM can be used to measure properties of the GC magnetic field. This section addresses these issues with modeling of the Galactic electron density and magnetic field. We use the observed 6 cm RM longitude dependence to constrain parameters of the model and ultimately derive the expected polarimetric properties of the region over a range of wavelengths.

5.1.1. Galactic Model

The RM is calculated from a model of the electron density, n_e, and the Galactic horizontal magnetic field, $\vec{B}$. We use a Cartesian Galactic coordinate system with the origin at l = 0° at a distance of r_GC from the Sun, the x-axis pointing toward negative l, the y-axis pointing away from the Sun, and the z-axis pointing to positive b. This technique does not calculate the emissivity of the synchrotron radiation, so we effectively assume that the polarized emission originates behind the Faraday rotating medium on the xz-plane. The integral is done along a line to the GC distance, so the polarized emission is assumed to be at the peak of the electron distribution. For this model, the Faraday depth along a given line of sight is

$$\begin{equation} \phi = \phi _0 + 0.81 \int _{r_\mathrm{GC}-5w}^{r_\mathrm{GC}} n_e\vec{B}\cdot d\vec{r}, \end{equation} \tag{ 2 }$$

where ϕ₀ is the foreground Faraday depth, r is the distance from the Sun along the line of sight, and w is the horizontal FWHM of the GC electron density enhancement (Cordes & Lazio 2002). As described below, the Faraday rotation induced by the GC region dominates, so the limits of integration include only a path of length 5w on the front side of the GC. In Section 5.1.3, we relax this assumption and consider the emissivity of the plasma. The model electron density is

$$\begin{eqnarray} && n_e(x,y,z) = n_\mathrm{d} + n_{\mathrm{0}} \, \mathrm{exp}\nonumber\\ && \times \left\lbrace -4\log {2} \left[ \frac{(x-x_0)^2 +(y-y_0)^2}{w^2}+\frac{(z-z_0)^2}{h^2} \right] \right\rbrace\!, \end{eqnarray} \tag{ 3 }$$

where n_d represents the electron density of the disc, excluding the Gaussian enhancement in the GC. Here, n_d is assumed constant throughout the volume. The central enhancement is described by a three-dimensional Gaussian function, where x₀, y₀, and z₀ are its offset, h is its vertical FWHM, and n₀ is its maximum density.

Since the model is fit to RM measurements, the magnetic field model describes only the horizontal component. The field points counterclockwise as seen looking down on the plane, with the pitch angle p pointing slightly outward for positive p. The horizontal field strength is assumed to be a constant, b₀:

$$\begin{equation} B(x,y,z) = |b_0|\left(\begin{array}{@{}l@{}}- \sin (\theta -p)\\ \cos (\theta -p)\\ 0 \end{array} \right), \end{equation} \tag{ 4 }$$

where

$$\begin{equation} \theta = \tan ^{-1}\frac{y-y_0}{x-x_0}. \end{equation} \tag{ 5 }$$

5.1.2. Model Fit

The 11 model parameters must be estimated carefully: there are only 39 longitude bins over a narrow l, b range and only two frequency channels. To simplify the procedure, we only solve for four parameters: foreground RM, x-offset, magnetic field strength, and pitch angle. The parameters r_GC, w, h, n₀, y₀, and z₀ have default values taken from NE2001 (Cordes & Lazio 2002), while the disc electron density, n_d, is the sum of the NE2001 thin disc and the Gaensler et al. (2008) thick disc, evaluated at the GC. These default parameter values, shown in Table 1, are consistent with other observations (Spangler 1991; LaRosa et al. 2005; Lazio & Cordes 1998; Han et al. 2006).

Table 1. Model Parameters

Name	Value	±	Unit	Rangea	Description
b0	4.84	0.25	μG	[0.1, 20]	Magnetic field strength in plane
p	−11.6	3.9	°	[ − 25, 25]	Pitch angle
ϕ0	−210	44	rad m−2	[ − 1000, + 1000]	Foreground RM
x0	70	3	pc	[0, 100]	Offset in x
y0	0		pc		Offset in y
z0	−20		pc		Offset in z
n0	10		cm−3		Peak of GC electron density Gaussian
nd	0.019		cm−3		Electron density of disc
rGC	8000		pc		Distance to GC
w	240		pc		Horizontal FHWM
h	125		pc		Vertical FHWM

Note. ^aThe range of parameter values allowed during the constrained nonlinear model fit. If no range is shown, the parameter is fixed.

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Figure 9 shows a top-down view of the spatial distribution of the electron density and magnetic field. The shaded region demonstrates that the observed area is rather small compared to the model structure. This shows that the observations are limited to the center of the electron distribution.

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We determine the best-fit parameter values and uncertainties with a three-step Monte Carlo method. First, we randomly generate a list of 4000 mock observations. The RM value at each l is drawn from a Gaussian distribution with mean and standard deviation equal to the RM and its error measured over the entire latitude range of the survey (as shown in Figure 7). Second, we fit Equation (2) for b = +05 to each mock data set by minimizing the reduced χ² using the nonlinear, constrained, L-BFGS-B solver (Zhu et al. 1994). Third, the mean and standard deviation of each parameter is measured from the ensemble of fit results.

The parameters and fit results for the longitude dependence of the 6 cm RM are summarized in Table 1. Figure 10 shows the 68% and 95% confidence levels for all pairs of variables. The foreground RM is orthogonal to all other parameters, while the remaining three show mild degeneracies.

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The 6 cm RM predicted by the best-fit model is compared to the observed RM in Figure 11. Since the geometry of the magnetic field is not well known in the GC region, we compare the observations to the best-fit model with and without a pitch angle parameter. For the model with the pitch angle (and shown in Table 1), the reduced χ² is 1.6 with 34 degrees of freedom. The model with no pitch angle parameter is slightly worse, particularly at l < −06, with a reduced χ² of 2.2 and 35 degrees of freedom. A p of 0 is ruled out formally at the 3σ level, although as discussed in Section 5.1.3 considering emissivity of the model will change some details of the best-fit parameters.

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The quality of the fit shows that a realistic GC electron distribution and magnetic field geometry can explain the observed RM longitude pattern. In particular, the rapid change of RM with Galactic longitude is best explained by the rapidly changing magnetic field orientation within the central few hundred parsecs. This is consistent with our assumption that most of the polarized emission originates in the center and that the Faraday rotation happens on the near side of the GC.

Not only can the model fit the observed RM pattern, but the best-fit values are also consistent with other observations. The magnetic field strength and electron density are degenerate in the model, but we constrain the horizontal component of the magnetic field to be roughly $5 (10\, {\rm {cm}}^{-3}/n_e^{\rm {gc}})\,\mu$G at heights of about 05. Considering the predominantly vertical orientation of the magnetic field (Nishiyama et al. 2010), the implied total field strength is consistent with that measured previously (Haynes et al. 1992; Ferriére 2009; Crocker et al. 2010). The foreground RM is similar in magnitude to that observed toward the pulsars in the inner Galaxy, which have RM up to ∼200 rad m⁻² (Manchester et al. 2005; Han et al. 2006). We compare this model to other measurements of the foreground RM in Section 5.3.

The physical consistency of the model to the GC region argues for a GC origin of the polarized emission. The shape of the RM distribution requires a value of n_e · B_|| that exists only in the GC region (Lazio & Cordes 1998). Also, the typical n_e and B_|| in the Galactic plane are not changing enough with longitude to cause the observed RM pattern.

5.1.3. Faraday Depth Estimate

While the model is based on the observed 6 cm RM, reasonable assumptions about the synchrotron emission in our volume will allow us to predict RM and depolarization at all frequencies and positions. These assumptions are not parameterized in our initial model fit, so its predictions can comment on the reasonableness of our model. Below we show how the model predicts a Faraday depth distribution and how it affects the interpretation of the observed polarization at 6 cm.

The following treatment of synchrotron radiation closely follows that of Sun et al. (2008). The synchrotron intensity of a slice with thickness $d\vec{r}$ is given by

$$\begin{equation} I\ d\vec{r} \propto n_\mathrm{rel} \vec{B_\perp }^{2}\nu ^{-1}\ d\vec{r}, \end{equation} \tag{ 6 }$$

where n_rel is the relativistic electron density. We assume that n_rel is constant in the inner 3 kpc of the Milky Way, that the intrinsic fractional polarization of each volume element is constant in frequency and space, and that the shape of the synchrotron spectrum is the same in all volume elements. We therefore drop the frequency dependence and relativistic electron density from our simulation of the polarized intensity:

$$\begin{equation} fI\ d\vec{r} \propto \vec{B_\perp }^{2}\ d\vec{r}, \end{equation} \tag{ 7 }$$

where f is the fractional polarization. If the magnetic energy density is equal to the gas energy density, then, according to Murgia et al. (2004),

$$\begin{equation} \Vert \vec{B}\Vert \propto n_e^{1/2}. \end{equation} \tag{ 8 }$$

Because $\Vert \vec{B}\Vert$ is likely an order of magnitude stronger than the horizontal field (Crocker et al. 2010), we assume for the sake of simplicity that $B_\perp \approx \Vert \vec{B}\Vert$, and therefore

$$\begin{equation} fI\ d\vec{r} \propto n_e\ d\vec{r}. \end{equation} \tag{ 9 }$$

We can now combine the polarized emissivity from Equation (9) with the Faraday depth and Faraday thickness to create the Faraday dispersion function F(ϕ). The dispersion function is the polarized flux as a function of Faraday depth, which can be Fourier transformed into the complex fractional polarization as a function of λ² (Brentjens & de Bruyn 2005). We assume that the intrinsic polarization angle is the same throughout the GC region, which is reasonable given the large-scale organization reported elsewhere (Nishiyama et al. 2010). The integration was done from r_GC − 5w to r_GC + 5w to predict the emission across the entire GC region. All lines of sight are normalized to have a polarization fraction of 70% at λ² = 0.

Figure 12 shows several predictions of the model given the assumptions described above. Since the initial model was fit assuming emission at the GC distance and RM induced only in the foreground of the GC (i.e., 2), we expect this model to be more Faraday thick. Indeed, while the underlying model is smooth and simple, the RM and polarization fraction are highly structured as a function of latitude and frequency. It is also worth noting that the predictions of the model are idealized in the sense that they do not account for beam depolarization or finite bandwidth of the observations. As such, they likely overpredict the polarization fraction and frequency-dependent changes in RM.

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The relation between Faraday depth and physical distance is displayed in the top left panel of Figure 12. The Faraday depth changes little far from the GC because the medium is tenuous. However, it also changes little in dense regions where the field direction reverses, such as in front of and behind the GC. As a result, a significant amount of flux will end up at the Faraday depths of the vertical sections of the r versus ϕ plot. The flux per unit ϕ is large in these regions, as shown in the Faraday dispersion plot in the bottom left panel. Because these caustic-like features occur whenever the line-of-sight field reverses in a synchrotron emitting area (U.-L. Pen 2010, private communication), RM synthesis of Faraday thick areas is much more sensitive to these reversals than to the bulk emission at large ϕ scales.

If we think of the polarized emission as a complex Stokes vector that rotates according to its Faraday depth, we can imagine the effect on the observed RM. The peaks in F(ϕ) interfere to create complex Faraday effects as a function of λ². As shown in the top right panel, the RM as determined by observing only two nearby frequencies can vary widely and argues for caution when interpreting our physical model. This shows that considering the emission from the entire GC region (not just the Faraday rotation by the foreground) makes the region Faraday thick.

Despite the Faraday thickness, the model shows that the RM at 6 cm preserves the observed east–west gradient. Figure 13 compares our observed RM at 6 cm to the RM derived along many lines of sight through this model. This is similar to the RM derived from the model in Figure 11, which did not calculate the Faraday dispersion function, but instead assumed Faraday rotation occurred in the foreground. Figure 13 confirms that considering the emissivity produces a wide range of RM, but that there is a clear east–west gradient.⁶ The qualitative agreement between models with and without emissivity shows that the observed RM gradient is caused by the orientation of the magnetic field in the GC region.

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Figure 12 also shows the polarization fraction expected when considering the emissivity of the model. As we assume an intrinsic polarization fraction of 70%, the plot shows that nearly every line of sight will have significant depolarization. On average, the predicted polarization fraction is 20% at 4.8 GHz and 7% at 1.4 GHz. Since the simulation does not include beam depolarization and assumes a perfectly organized magnetic field, the predictions should overestimate the polarization fraction. We consider the predictions in agreement with the observed polarization fraction of 10% at 4.86 GHz and <1% at 1.4 GHz.

In summary, considering the emissivity of our best-fit model for the GC magnetized plasma shows that the observed RM structure has a complex connection to the actual physical properties. However, the RM trend with Galactic longitude is directly related to the magnetic field direction in the GC region. More detailed physical modeling will require polarimetry with hundreds of channels between 2 and 8 GHz, such as with the EVLA (Ulvestad et al. 2006).

Eight NRFs were detected in polarized emission at 6 cm and seven of these have reliable RM measurements (Law et al. 2008a). Interestingly, all but one of these have RM consistent with their surrounding diffuse emission. In other words, the RMs toward the NRFs largely follow the longitude dependence found toward the diffuse emission. If the RMs toward the filaments was unrelated to that of the diffuse emission, a binomial probability distribution predicts a 5% chance of 6/7 coincidences.

The similarity of the RMs in the diffuse and filamentary emission is consistent with observations of the brightest NRF, the Radio Arc. As shown in Section 3.2 and elsewhere (Yusef-Zadeh & Morris 1988), the morphology and RM measured toward the Radio Arc has a continuous connection into the polarized diffuse emission in the east of this survey. Since the Radio Arc is known to be within the central 100 pc of the GC (Yusef-Zadeh & Morris 1987; Lang et al. 1999b; Lasenby et al. 1989), some of the diffuse polarized emission must also be located in the GC.

The fact that the filaments and the diffuse polarized emission are physically near each other could explain their similar RMs. The simplest model to explain this coincidence is that the filaments and diffuse emission are behind the same Faraday screen. According to our modeling, such a screen would be located within the central kiloparsec. However, NRFs are known to have RM changes that coincide with physical changes, which argues that some RM is induced locally (Lang et al. 1999a). A locally induced RM is consistent with the strong magnetic fields inferred in NRFs (Yusef-Zadeh et al. 1997).

If this RM coincidence is not extrinsic, then it must be intrinsic: the diffuse emission and the NRFs have similar magnetic field orientations. Physically, this can be explained if the NRFs are local enhancements and perturbations of the global magnetic field. This concept is common to several models for generating NRFs (e.g., Benford 1988; Serabyn & Morris 1994; Boldyrev & Yusef-Zadeh 2006). However, it excludes the model of Shore & LaRosa (1999), which relies on an interaction of molecular clouds with a global wind to generate the NRFs. Other models (e.g., Rosner & Bodo 1996) do not clearly predict how filaments can enhance and perturb the global magnetic field.

The NRFs and the diffuse polarized emission also have a similar spatial distribution. As described earlier, the RM observed in the diffuse emission changes sign near l ≈ −035. Other observations, described in Section 5.3, show that most RM measurements in the GC region follow a pattern that is centered near l ≈ −035. High-resolution 20 cm survey of Yusef-Zadeh et al. (2004) showed that there are dozens of candidate NRFs in the GC region with the highest density between l = 02 and −07. This shows that the center of the distribution of NRFs is similar to the center of the RM pattern. A third coincidence is that the GC lobe, a shell of gas related to a mass outflow, is centered near the same longitude. These coincidences indicate a connection between the GC lobe outflow and the GC magnetic field.

Figure 14 shows a schematic of all RM measurements toward sources believed to be in or beyond the GC region. The RM values derived earlier in this work are shown along with observations of the extended polarized emission from the Radio Arc (Tsuboi et al. 1986), other NRFs (Lang et al. 1999a, 1999b; Gray et al. 1995; Reich 2003), and background sources (Roy et al. 2005). North of the plane, the east–west gradient in RM is seen toward diffuse, compact, and filamentary sources. The pattern is antisymmetric about l ≈ −035 and across the Galactic plane (esp. Tsuboi et al. 1986).

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The simplest pattern to describe the GC RM values is that of a checkerboard (four quadrants of alternating sign) shifted 035 (≈50 pc) west of the center. The possibility of a checkerboard pattern centered at l = 0° has been noted before (Uchida et al. 1985; Novak et al. 2003), but some RM values were not consistent with the pattern (Ferriére 2009). The RM reported in this work finds that the checkerboard pattern is robust if it is assumed to be shifted from the center.

The checkerboard pattern inspired the "flux-dragging" model for the GC magnetic field (Uchida et al. 1985; Novak et al. 2003). The model explains a large-scale pattern in the GC RM as the effect of Galactic rotation on a frozen-in, poloidal (vertical) magnetic field. As the disk rotates, the magnetic field in the disk is dragged away from us on the east side and toward us on the west side. This pull creates a line-of-sight component of the magnetic field. This perturbation to the magnetic field has a checkerboard pattern in the sign of the RM, such that the RM will have opposite signs toward any two adjacent quadrants formed about the center of rotation.

Interestingly, the parity of the checkerboard pattern constrains the orientation of the magnetic field, breaking the 180° ambiguity of observations of the polarization angle. If this scenario is valid, the parity of the observed pattern is consistent with the magnetic field pointing from south to north. This is the only known measurement that can break this ambiguity in the orientation of the magnetic field on the plane of the sky.

Roy et al. (2008) found that the flux-dragging scenario was inconsistent with the RM observed toward extragalactic sources seen through the central 12°. The RM toward their 60 background sources had an average RM =+413 rad m⁻² and no checkerboard pattern. While the RM values over the central 12° are predominantly positive, the three measurements in the central 2° studied here (G359.388+0.460, G359.604+0.306, G359.871+0.179) agree with the shifted checkerboard pattern.

However, there are clear differences between the RM structure seen here and that reported by Roy et al. (2008). For example, the average RM observed by Roy et al. (2008) in the central 12° is not consistent with the foreground RM in our model (ϕ₀). Assuming that half of the average RM is contributed by the foreground to the GC, we would expect ϕ₀ = 206.5 rad m⁻², but find roughly the opposite of that. One possibility is that the foreground RM fit by our model is poorly constrained by our data. Indeed, Section 5.1.3 shows that the absolute level of RM is poorly constrained by these narrowband observations, but that relative values are well constrained. Another possibility is that the central 2 degrees of the Galaxy is magnetically and dynamically different from the region beyond, making comparison with the RM measured in Roy et al. (2008) less meaningful. The presence of NRFs (Nord et al. 2004; Yusef-Zadeh et al. 2004) and the "central molecular zone" (Morris & Serabyn 1996) makes the central 2° notably different from the region beyond.

In summary, we argue that RM measured in the central 2° of the Galaxy is distinct from RM structure seen outside this region and that it reflects large-scale GC magnetic field structure. The RM structure is simplest to describe as a checkerboard pattern shifted ∼50 pc from the dynamical center of the Galaxy. The longitude shift requires some non-Keplerian motion of the ionized gas in the GC region. This is consistent with recent observations showing that the GC is host to a small starburst outflow centered ∼50 pc west of the GC (the GC lobe; Bland-Hawthorn & Cohen 2003; Law 2010). The distribution of NRFs, a more direct tracer of the GC magnetic field, is also centered tens of parsecs west of the GC (see Figure 29 of Yusef-Zadeh et al. 2004). It is clear that the electron and magnetic field distributions are not symmetric about the l = 0°; our new RM measurements confirm this.

To improve visualization of the Δθ images, the mosaics were smoothed using two independent methods. Both methods were implemented in IDL. The first method of smoothing Δθ was to measure the error-weighted mean value of Δθ over small regions. Hereafter, we refer to this as the mean method.

A major caveat to this method is that averaging angles is not proper, since they represent vector quantities. However, averaging angles is approximately correct for small angles, which is usually true for the Δθ images presented here (Figure 4). Furthermore, the large values in the Δθ map are generally in the noisiest parts of the image and are down weighted by large errors. There was no significant difference when using the median instead of the mean.

A second, more robust method of smoothing the Δθ images is to fit the distribution of Δθ values with a model. Since a model is fit to the histogram, this approach avoids the problems of averaging angles. The model can also parameterize the noise- and signal-like contributions to the Δθ distribution, making it robust to outliers.

As described in Section 3.1, histograms of Δθ were fit with a Lorentzian plus a constant background. The best-fit values and their errors were found by the Levenberg–Marquardt algorithm, implemented as MPFIT in IDL (Markwardt 2009). All errors reported are 1σ confidence intervals.

Figure 4 shows two examples of histograms extracted with best-fit models. A histogram bin size of 25 was used; varying the bin size by a factor of a few does not significantly change the fit. The smoothed maps have pixel sizes of 125'' × 125'', which are small enough to resolve structure in the mosaic, but large enough for accurate fit results.

An advantage of the histogram-fitting method is that it calculates the mean and error in Δθ from the distribution of Δθ; no noise image is required. A disadvantage of the histogram-fitting method is that it assumes that all pixels in the region sampled are a part of the same distribution. In fact, there are times when multiple sources, with different distributions of Δθ (i.e., different RMs) are sampled by the same histogram. In this case, the source that occupies the most pixels will dominate the histogram distribution and the best-fit value of Δθ. This is different from what happens when calculating an error-weighted average of Δθ, in which the average is dominated by the pixels with the lowest noise (i.e., the brightest sources).

Figure 15 compares images of Δθ using the mean and histogram-fitting methods for a grid of 125'' boxes. An image showing the significance of differences between the two methods in each pixel. The average difference between the two images is 0002 and the standard deviation is about 02, so there is no significant difference between the two methods.

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The image showing the difference in the two smoothing methods highlights two regions with greater than 1σ: (35955, 015) and (015, 02). These differences demonstrate the biases of the methods. In both locations, there is a small, polarized source with a different Δθ from the large, polarized background. Since the mean method favors bright sources and the histogram-fitting method favors large sources, these two regions are most likely to show a difference. While this is an important caveat, Figure 15 shows that, in general, the two methods are in good agreement.

The errors on Δθ differ according to the method used. When averaging over 125 arcsec tiles, both methods give errors within 50% of each other, if errors <3°. However, for errors greater than 3°, the errors found with the mean method are progressively less than those found by histogram fitting. The maximal error found by the mean is about 20°, while the histogram-fitting method has a maximal error of about 200°. This is expected, since the mean method assumes that angles are much smaller than 1 rad; this is not always true for the errors in Δθ. Thus, the histogram-fitting errors are a more accurate and more conservative estimate of the true errors in the mean value of Δθ.