A ROTATION MEASURE IMAGE OF THE SKY

To derive RM values from the NVSS observations, we downloaded the calibrated NVSS visibility data sets for each snap shot from the National Radio Astronomy Observatory data server and created mosaic images in Stokes I, Q, and U in each of the two bands. The images were created using AIPS. A mosaic weight image was also created to provide an estimate of the theoretical noise as a function of position. We also created images with combined data from both bands to measure the amount of depolarization. Since we use the peak intensity as an estimate of the source flux, a mild u–v taper was applied, degrading the resolution slightly compared to the original NVSS images to mitigate resolution effects. The peak flux density was measured in each image at the position of every NVSS catalog source with cataloged Stokes I intensity greater than 5 mJy. The noise level, σ, for the Q and U values was determined by calculating the rms variation about the mean in an annulus around each source in the mosaic images. The peak intensity was taken to be the map peak minus the local off-source mean. A source was retained for measurement of the RM if the signal-to-noise ratio for polarized intensity $p =\sqrt{Q^2 + U^2}$ was greater than 8σ. At this signal-to-noise level the noise bias correction to p is negligible (Simmons & Stewart 1985). We also required that the percent polarization is greater than 0.5% to ensure that the polarized intensity is not dominated by instrumental effects (Condon et al. 1998).

This process resulted in 37,543 sources for which RMs were derived from the change in polarization position angle between the two bands. From the change in position angle Δτ, the observed RM_o was calculated as

$$\begin{equation} {\rm RM_o} = C \frac{\Delta \tau }{\lambda ^2_2 - \lambda ^2_1}, \end{equation} \tag{ 1 }$$

where λ₁ and λ₂ are the center wavelengths of the bands at 1435 MHz and 1365 MHz, respectively, and C is a correction factor that accounts for the effect of the finite width of the bands on the effective center wavelength in λ² space, given by

$$\begin{equation} C = \left(1 + \frac{\Delta \lambda _2^2 - \Delta \lambda _1^2}{\lambda ^2_2 - \lambda ^2_1} \right) ^{-1} = 0.96, \end{equation} \tag{ 2 }$$

for the NVSS.

Given that we have only two data points in λ², there is a potential ambiguity between the true RM of the sources and the observed RM_o due to the possibility of wrapping of the polarization position angle between the two bands. The true RM can take on any value given by

$$\begin{equation} {\rm RM}= {\rm RM_o} + \frac{m\pi }{\lambda _2^2 - \lambda _1^2}, \end{equation} \tag{ 3 }$$

where m is an integer representing the number of wraps which occurred between λ₂ and λ₁. A single wrap (m = ±1), which can occur even for small RM, produces an increment in the observed RM of ±652.9 rad m⁻². A double wrap (m = ±2) would require |RM|>1306 rad m⁻². RMs of this magnitude are rare, and typically encountered only near the Galactic plane and toward the inner galaxy (Brown et al. 2007). In our analysis we assume |m| ⩽ 1.

We used the amount of depolarization to help resolve the RM ambiguity. Figure 1 shows the depolarization (ratio of observed to true polarized flux density) as a function of true RM for polarized flux densities measured in the individual bands and integrated over both bands. The polarized flux measured from images created by merging u–v data from both bands (p_i) suffers considerable depolarization for RM greater than about 100 rad m⁻². This will be the case for the NVSS catalog flux. Polarized flux densities measured in the individual bands (p₁ and p₂) are a good representation of the true flux density up to RM of 400 rad m⁻². Moreover, the ratio of polarized flux density in the individual bands to the band-integrated flux density, R_o = (p₁ + p₂)/2p_i, is a function of |RM|. For each polarized source, the predicted value of R(m) for each of the three possible values or RM (m = −1, 0, +1) was compared to the measured R. A likelihood was assigned to each value of RM based on the error on R_o and the difference between the measured and predicted value of R for each m. The RM value with the highest likelihood was selected as the true RM.

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There are two situations in which this analysis breaks down. First, the curves in Figure 1 show that for both |RM| < 50 and |RM|>520 the R_o ∼ 1.0. In the presence of errors on R_o our algorithm is thus unable to distinguish between these two cases. Second, the value R_o is insensitive to the sign of the RM, thus RM ± 326.5 which differ by m = ±1 and produce equal amounts (almost total) depolarization cannot be distinguished. To resolve these final ambiguities, we used the spatial distribution of RM values on the sky. For each source, we derived the median value of RM for all other sources within a radius of 3°. If the RM value of the source differed from the median value in the surrounding area by more than 520, the RM value was corrected by adding or subtracting 652.9. As a consequence, there are no RM values in the catalog that are more than 520 rad m⁻² from the median of the surrounding values. Any such objects have had their RM amplitude adjusted downward by 652.9. This should not in general be an issue for sources more than a few degrees from the Galactic plane.

Figure 2 shows our NVSS-derived RMs plotted against previously determined RMs for sources with |b|>10°. The comparison list was compiled by J. Brown using RM data published between 1988 and 2001 (Broten et al. 1988; Clegg et al. 1992; Oren and Wolf 1995; Minter and Spangler 1996; Gaensler et al. 2001). While a few outliers are seen, there is very good agreement for the majority of the sources. Differences between our results and previous observations could result from unwrapping errors in the older data which are typically taken at widely spaced frequencies, the effects of different angular resolution for slightly resolved sources, nonlinear behavior of RM with λ², or, possibly, changes in RM due to intrinsic variability in polarized properties of the radio source.

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Table 1 lists the data for the first 10 sources of the RM catalog. Columns 1–4 list the right ascension and declination (J2000) and galactic coordinates of the source. Column 5 is the integrated Stokes I flux density from the NVSS catalog. Column 6 is the average peak polarized intensity (p₁ + p₂)/2, and Column 7 the percent polarization (ratio of peak polarized intensity to peak Stokes I). Finally, Column 8 lists the derived RM for the source with error. All errors in the table are 1σ. The full catalog is available online at http://www.ucalgary.ca/ras/rmcatalogue.

The sky distribution of the catalog RMs in Galactic coordinates is shown in Figure 3. The value of RM for each of the 37,543 sources is represented by a circle plotted at its Galactic coordinates. The size of the circle scales linearly with the amplitude of RM. The black circle at l,  b = −60°, − 30° is a reference circle for |RM| = 300 rad m⁻². Positive RMs are shown as red circles and negative RMs as blue.

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The plot of individual RM values in Figure 3 shows large-scale coherent structures and high values of RM extending to several tens of degrees from the Galactic plane. Fainter structures can be seen all the way to the polar regions. To obtain a quantitative picture of the sky structure of RM on scales of several degrees, we created an image of the median RM as a function of l,  b. The sky was gridded on 05 pixels with the value at each pixel set to the median RM of all sources within a radius of 4° of the pixel location. The resulting median RM image is shown in Figure 4. Typically, 60–80 sources were used to derive each median value. The characteristic 1σ error on the median value is thus around 2 rad m⁻².

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The average value of the median RM over the region δ> − 40° is 16.7 rad m⁻². There is limited solid angle with very low RM on scales of several degrees. We have identified only five regions larger than 5° across where the amplitude of the median RM is consistently below 1 rad m⁻². These "holes" in the foreground RM structure of the Galaxy, listed in Table 2, may be useful for studies of the small-scale structure of cosmological magnetic fields (see, for example, Kolatt 1998).

Many of the RM structures in Figure 4 on scales of tens of degrees are due to the local Galactic interstellar medium. A more detailed analysis of these and other features in the all-sky RM data will be presented in a later paper. However, variations on the very largest scales can be attributed to the structure of the global Galactic magnetic field. Here we point out some initial implications for the Galactic magnetic field. The value of RM is the integral of the line-of-sight component of the magnetic field weighted by the line-of-sight distribution of electron density, given by

$$\begin{equation} {\rm RM} = 0.81 \int n_e\, {\bm {B}} \cdot {\bm {dl}} \quad \quad {\rm rad\; m^{-2}}, \end{equation} \tag{ 4 }$$

where B has units of μG, n_e has cm⁻³, and dl has pc. The all-sky RM images show large-scale RM structures centered around the Galactic plane extending to several tens of degrees from the plane. Using pulsar data and Hα emission, Gaensler et al. (2008) derive a scale height of h_e = 1.83^+0.12_−0.25 kpc for the free electron distribution. For this scale height, at b = 30° half of the total Galactic electron column along the line of sight is encountered within a distance whose projection onto the disk of the Galaxy is 2.2 kpc. For b = 40°, this decreases to 1.5 kpc. Large-scale RM structures at intermediate to high latitudes will thus be dominated by the geometry of the Galactic halo magnetic field within a cylinder that extends only a few kpc from the solar neighborhood.

We consider the effect of a simple magnetic field geometry within this cylinder given by

$$\begin{equation} \bm {B} = B_{xy} \cos \phi _o \hat{i} + B_{xy} \sin \phi _o \hat{j} + B_z \hat{k}, \end{equation} \tag{ 5 }$$

defined on a Cartesian coordinate frame with the $\hat{j}$-axis pointing away from the Galactic center and the $\hat{k}$-axis toward the north Galactic pole. B_xy is the amplitude of the field parallel to the plane of the Galaxy, assumed in this simple model to be independent of height, z. The angle ϕ_o is the inclination of B_xy to the solar circle, defined as zero in the direction of l = 90°. B_z is the poloidal component of the field perpendicular to the disk. With this model, and an electron distribution given by $n_e(z) = n_o e^{-z/h_e}$, the RM as a function of galactic coordinates will be

$$\begin{equation} {\rm RM}(l,b) = - 0.81 n_o h_e \left(\frac{B_{xy} \cos b\, \sin (l - \phi _o) + B_z \sin b}{\sin |b|} \right). \end{equation} \tag{ 6 }$$

RM is defined positive for the magnetic field directed toward the observer.

Figure 5 shows the mean and rms value of the distribution of RMs over all longitudes covered by the data as a function of Galactic latitude. Integrating Equation (6) over all values of l removes the contribution of the disk field from RM(l, b), yielding

$$\begin{equation} {\rm RM}(b) = -0.81 n_o h_e \cdot {\rm sign}(b) B_z. \end{equation} \tag{ 7 }$$

For b < −73° and b>19°, the data provide complete longitude coverage (see Figure 3). Over these ranges, Figure 5 shows an approximately constant value of average RM. It is striking that the average RM is positive in both the northern and southern hemispheres. The average RM of the 756 sources within 15° of the southern Galactic pole is 6.2 ± 0.68 rad m⁻². The corresponding value for the 1025 sources within 15° of the northern Galactic pole is 3.0 ± 0.52 rad m⁻². Adopting the Gaensler et al. (2008) values of n_o = 0.014 cm⁻³ and h_e = 1830 pc, the implied z-component magnetic fields in the solar neighborhood are B_z = +0.30 ± 0.03 μG for z < 0 and B_z = −0.14 ± 0.02 μG for z>0. The vertical component of the field changes sign across the plane. This is not consistent with a simple dipolar explanation for the vertical field (Han 2009), requiring instead a higher order multipole field.

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An estimate of the component of the halo field parallel to the plane was derived by fitting Equation (6) to the variation of the median RM(l) as a function of b. Figure 6 shows sample fits to data at b = 45° and b = −45°. The data show clear 360° period oscillations that are well represented by simple sinusoids, as predicted by Equation (6). The amplitude and phase of the family of sine fits as a function of latitude are shown in Figure 7. As can be seen in both Figures 6 and 7, the direction of B_xy reverses by 180° across the Galactic plane. The curves in the lower panel of Figure 7 are the coefficient of sin(l − ϕ_o) in Equation (6),

$$\begin{equation} {\rm RM} = -0.81\, n_o\, h_e\, B_{xy} \frac{\cos b}{\sin |b|}, \end{equation} \tag{ 8 }$$

with n_o and h_e from Gaensler et al. (2008) and B_xy = +0.83 μG below the plane (z < 0, dashed line) and B_xy = −0.39 μG above the plane (z>0, dot-dashed line). As for the B_z component, the fits imply that the field parallel to the plane changes sign across the plane and the magnitude below the plane is approximately twice as large as that above. The average value of ϕ_o (the upper panel of Figure 7) is 11° ± 30°. This result is generally consistent with an antisymmetric circular toroidal halo field component that reverses sign above and below the plane, as suggested by Han (2009). There is an indication in Figure 7 that ϕ_o increases at high positive latitudes, suggesting a more complex halo field configuration at higher z.

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