# Is There a Polarization Horizon?

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## Abstract

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## 1. Introduction

^{2}field, identifying multiple Faraday depth components and schematically tying them to two nearby primarily-neutral regions within ≈300 pc. Thomson showed 300–480-MHz data from the all-sky Galactic Magneto-Ionic Medium Survey (GMIMS) Low Band South (GMIMS-LBS) obtained with the CSIRO Parkes radio telescope. In his observations, there is clear depolarization associated with specific Hii regions, which I call a “depolarization wall”. Wolleben [6] used depolarization in a 1.4-GHz polarization survey to construct a model for Radio Loop I as the interaction of two shells. Sun et al. [7] used depolarization in GMIMS-High Band North (HBN) data to estimate the distance to the North Polar Spur. We also see clear evidence for depolarization in the 1280–1750-MHz GMIMS-HBN data, where the W4 superbubble reduces the intensity of Fan Region emission by ≈30% [8]. In this case, though depolarization clearly affects the polarized intensity and is associated with ionized gas at a discrete distance in a spiral arm, the depolarization does not appear to remove all information about the gas behind the depolarizing structure. Because depolarization is purely a polarization effect, there is no corresponding change in the morphology of total intensity images at the same wavelengths.

## 2. Propagation and Emission of Polarized Radiation

## 3. Methods

`pyrmsynth`package1 to calculate the Faraday depth spectrum. I performed RM synthesis in the two GMIMS frequency ranges, but not accounting for radio frequency interferences (RFI), other excluded frequency channels, noise, or any other instrumental realities. To distinguish the Faraday depth measured from RM synthesis from the true Faraday depth (which is known precisely as a function of s in these models but not in the real ISM), I use the notation $\tilde{\varphi}$ to denote the Faraday depth derived from RM synthesis. The input frequency ranges and output maximum value of $\tilde{\varphi}$, maximum scale of Faraday depth features, FWHM of the RM transfer function, and the range and chosen resolution $\delta \tilde{\varphi}$ of calculated Faraday depth spectra are listed in Table 2, with these values calculated following Schnitzeler et al. [24].

## 4. Results

`arm_5kpc`model, and in the

`HII_100pc`model, there are $\pi $ radian rotations down to ${\lambda}^{2}=0$, so this behavior is dependent on the depolarizing structures in the ISM.

`noarm`model, but is reached in the spiral arm at $s=5$ kpc in model

`arm_5kpc`and (marginally) in the Hii region at $s=100\text{\hspace{0.17em}}\mathrm{pc}$ in model

`HII_100pc`. At $\lambda =70\text{\hspace{0.17em}}\mathrm{cm}$, this threshold is reached in the Hii region in that model and by $s\approx 1\text{\hspace{0.17em}}\mathrm{kpc}$ in the other two models. Therefore, one expects the polarization horizon in the long-wavelength case to be quite nearby. At the other extreme, beam depolarization is not likely to cause a polarization horizon at the shorter wavelength at all in the

`noarm`model, although other depolarization effects such as variations in ${\psi}_{0}$ could.

`noarm`model compared to the predicted effects of depth (Equation (4)) and beam (Equations (6) and (7)) depolarization. At $\lambda =20\text{\hspace{0.17em}}\mathrm{cm}$ (bottom panel), the polarized intensity measured for the beam decreases from $100\%$ of the mean polarized intensity of the individual sightlines at $s=0$ to ≈75% of that of the individual sightlines at $s=8\text{\hspace{0.17em}}\mathrm{kpc}$. The depth depolarization model (brown dotted line) accurately describes the mean of the individual sightlines (red dashed line); beam depolarization without considering depth depolarization (purple dotted line) does not. At this wavelength, the first depth depolarization null is not reached in 8 $\mathrm{kpc}$. The combination of depth and beam depolarization (pink dotted line) produces a polarized intensity somewhat lower than seen in the simulation. This is unsurprising because Equation (6) assumes that the emission is behind the depolarizing slab and that $\varphi $ has a Gaussian distribution; neither assumption applies to the simulations.

`noarm`model, as one integrates further along the line of sight to emission regions at higher Faraday depths, the Faraday spectrum has power at Faraday depths ranging from zero to $\varphi \left(D\right)$, most clearly in the LBS band (Figure 6a). In the

`arm_5kpc`model, the effects of the arm are evident in Figure 6c,d in both frequency ranges: the arm shifts some of the emission to much higher $\tilde{\varphi}$, including Faraday depths much higher than the mean $\varphi \left(5\text{\hspace{0.17em}}\mathrm{kpc}\right)$. However, there is little impact on the total polarized intensity either at a single frequency (Figure 4), or in the Faraday depth spectra (Figure 6), or Moment 0 (Figure 7).

## 5. Caveats

`noarm`model as a background and add a region with a radius of several beams from the

`HII_100pc`model to see if the resulting image is qualitatively similar to observed depolarizing screens (see [4,28] and Thomson’s work at this meeting).

## 6. Discussion: Is There a Polarization Horizon?

`HII_100pc`observed at long wavelengths, that distance is very short, as predicted by the Burn [1] model: ${\sigma}_{\varphi}^{2}{\lambda}^{4}\gtrsim 1$ at $s\gtrsim 90\text{\hspace{0.17em}}\mathrm{pc}$. In that most extreme case, integrating further beyond the depolarizing structure leads to relatively little change in the polarized intensity at $70\text{\hspace{0.17em}}\mathrm{cm}$. However, although the polarized intensity at the single frequency shown no longer increases with distance, the Faraday spectrum (Figure 6e) and the moments (green dashed lines in Figure 7) do continue to evolve. This is the closest case in these models to a “depolarization wall”, but the wall is not opaque to polarized radiation. Similarly, in the other models and in both frequency ranges, there is no physical distance beyond which further integration does not affect at least some aspects of the observed spectrum. Therefore, though it is probably reasonable to say that there is a different (but difficult to precisely define) weighting function to the volume sampled by polarization observations at different frequencies (with longer-wavelength observations weighted more towards a more nearby volume, especially in the

`HII_100pc`model), these models suggest that longer wavelengths do not sample an entirely different volume than the shorter wavelengths.

`HII_100pc`model (Figure 6e,f), the first moment of $\tilde{\varphi}$ is approximately $\varphi $. This is expected because most of the Faraday rotation occurs in the Hii region at $s=100$ pc, so most of the emission occurs behind most of the Faraday rotation. However, evidently, the Hii region again does not fully block information about the ISM beyond it, even though ${\sigma}_{\varphi}^{2}{\lambda}^{4}\sim 300$ at $\lambda =70\text{\hspace{0.17em}}\mathrm{cm}$ (Figure 3): the measured first moment increases from $\tilde{\varphi}\left(200\text{\hspace{0.17em}}\mathrm{pc}\right)=+5.9\text{\hspace{0.17em}}{\mathrm{rad}\text{\hspace{0.17em}}\mathrm{m}}^{-2}$ to $\tilde{\varphi}\left(1\text{\hspace{0.17em}}\mathrm{kpc}\right)=+13.5\text{\hspace{0.17em}}{\mathrm{rad}\text{\hspace{0.17em}}\mathrm{m}}^{-2}$ to $\tilde{\varphi}\left(3\text{\hspace{0.17em}}\mathrm{kpc}\right)=+21.4\text{\hspace{0.17em}}{\mathrm{rad}\text{\hspace{0.17em}}\mathrm{m}}^{-2}$. With the moments, we lose the complexity in the Faraday spectrum. However, RM synthesis with moments still benefits from the wide ${\lambda}^{2}$ coverage, averaging over the $\mathrm{sinc}\left(\varphi {\lambda}^{2}\right)$ behavior of depth depolarization.

`noarm`model. There is no portion of any of the models presented here with a negative $\varphi $, yet $\tilde{\varphi}$ can have components with a negative centroid both at HBN and LBS frequencies (Figure 6). Therefore, depolarization effects lead not to a Faraday spectrum that samples a different volume, but measured values of $\tilde{\varphi}$ that do not map to $\varphi $ in an obvious way.

## Funding

## Acknowledgments

## Conflicts of Interest

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1 |

**Figure 1.**Input density along the line of sight for one sightline in each model, with parameters listed in Table 1. The “beam” in a given model consists of N sightlines, each chosen from the same distribution, as shown here. The fourth panel shows the line-of-sight component of the magnetic field for one model. Points show the individual cells; dashed lines show the median; vertical dotted lines show the position of the arm in each model. The bottom panel shows the mean Faraday depth across all the sightlines in each model.

**Figure 2.**Spectrum of the polarization angle as a function of wavelength integrated over a beam (the sum of $N=25$ sightlines) all the way through the

`noarm`model, calculated on 0.5-MHz intervals. Cyan lines show the slope $d\psi /d{\lambda}^{2}=\varphi /2$, as expected for a uniform slab, where $\varphi =37.1\text{\hspace{0.17em}}{\mathrm{rad}\text{\hspace{0.17em}}\mathrm{m}}^{2}$ is the mean Faraday depth integrated through all sightlines.

**Figure 3.**The factor in Equation (6), ${\sigma}_{\varphi}^{2}\text{\hspace{0.17em}}{\lambda}^{4}$, for each of the models. The left axis values are labeled for $\lambda =20\text{\hspace{0.17em}}\mathrm{cm}$; the right axis values are labeled for $\lambda =70\text{\hspace{0.17em}}\mathrm{cm}$. When ${\sigma}_{\varphi}^{2}{\lambda}^{4}\gg {10}^{0}$, total depolarization is expected in the Burn [1] model. Dotted vertical lines show the position of the arm in each model.

**Figure 4.**Polarized intensity as a function of D, the integration limit in Equation (1), at $\lambda =70\text{\hspace{0.17em}}\mathrm{cm}$ (

**a**) and $20\text{\hspace{0.17em}}\mathrm{cm}$ (

**b**). The solid lines show the polarized intensity over the beam, $|\langle {\mathcal{P}}_{i}\rangle |$. The dashed lines show the polarized intensity in one randomly-chosen individual sightline within the beam, $|{\mathcal{P}}_{i}|$. Vertical dotted lines show the position of the spiral arm in the model with the corresponding color. Dotted-dashed lines show the polarized intensity in the absence of depolarization effects (Equation (9)). Note that integrated polarized intensities (Moment 0) are shown in Figure 7 below. Dotted vertical lines show the position of the arm in each model.

**Figure 5.**Blue lines are as in Figure 4: polarized intensity for the full beam $|\langle {\mathcal{P}}_{i}\rangle |$ (blue solid line) and in a representative sightline within the beam $|{\mathcal{P}}_{0}|$ (blue dashed line) from the

`noarm`model as a function of D. I also show the mean of $|{\mathcal{P}}_{i}|$, $\langle \left|{\mathcal{P}}_{i}\right|\rangle $, of the $N=25$ individual sightlines (red dashed line). Dotted lines show analytic models of beam depolarization (Equations (6) and (7)), depth depolarization (Equation (4)), and both. Note that in the top ($70\text{\hspace{0.17em}}\mathrm{cm}$) panel, I show only the first $2\text{\hspace{0.17em}}\mathrm{kpc}$ because depolarization effects become dominant on shorter distance scales at the long wavelength.

**Figure 6.**Faraday depth spectra as a function of D as measured in the GMIMS-LBS frequency band (

**a**,

**c**,

**e**) and the GMIMS-HBN frequency band (

**b**,

**d**,

**f**). The simulated polarized intensity in K is indicated with the color bars. Solid lines show $\varphi $ integrated to $s=D$; dashed lines show the first moment of the Faraday depth spectra. Dotted vertical lines show the position of the arm in each model.

**Figure 7.**Moments of the Faraday depth spectrum as a function of distance along the sightline. Solid lines show moments of the GMIMS-HBN Faraday spectrum; dashed lines show moments of the GMIMS-LBS Faraday spectrum. The zeroth moment is the polarized intensity integrated across the Faraday spectrum, ${M}_{0}=\int L\text{\hspace{0.17em}}d\varphi $. The first moment is the intensity-weighted Faraday depth, ${M}_{1}=\int L\varphi \text{\hspace{0.17em}}d\varphi /{M}_{0}$; the dotted lines show the true $\varphi $ for the input model (from Figure 1), which is in principle the rotation measure one would measure towards a point source at distance D (or a background point source for the maximum value of D). The second moment is the intensity-weighted width of the Faraday depth spectrum, $\int L\xb7{(\varphi -{M}_{1})}^{2}d\varphi /{M}_{0}$; ${M}_{2}^{1/2}$ is shown. Dotted vertical lines show the position of the arm in each model.

Name | ${10}^{\mathsf{\mu}}$ | ${\mathit{\sigma}}_{\mathrm{log}\text{\hspace{0.17em}}\mathit{n}}$ | f | ${\mathit{n}}_{\mathrm{arm}}$ | ${\mathit{d}}_{\mathrm{arm}}$ | ${\mathit{w}}_{\mathrm{arm}}$ | $\mathrm{DM}$ | ${\mathit{B}}_{0}$ | ${\mathit{\sigma}}_{\mathit{B}}$ | $\mathit{\theta}$ | N |
---|---|---|---|---|---|---|---|---|---|---|---|

(cm^{−3}) | (cm^{−3}) | (pc) | (pc) | (pc cm^{−3}) | (μG) | (μG) | |||||

noarm | $0.016$ | $0.30$ | $0.5$ | $0.0$ | ... | ... | $78.6$ | $1.0$ | $3.0$ | ${45}^{\xb0}$ | 25 |

arm_5kpc | $0.016$ | $0.30$ | $0.5$ | $1.0$ | 5000 | 100 | $204.8$ | $1.0$ | $3.0$ | ${45}^{\xb0}$ | 25 |

HII_100pc | $0.016$ | $0.30$ | $0.5$ | $1.0$ | 100 | 10 | $42.2$ | $1.0$ | $3.0$ | ${45}^{\xb0}$ | 25 |

**Table 2.**Rotation measure (RM) synthesis parameters. GMIMS, Galactic Magneto-Ionic Medium Survey; LBS, Low Band South; HBN, High Band North.

Survey | Frequencies | $\mathit{\delta}\mathit{\nu}$ | ${\tilde{\mathit{\varphi}}}_{\mathrm{max}}$ | Max Scale | FWHM | Range | $\mathit{\delta}\tilde{\mathit{\varphi}}$ | |
---|---|---|---|---|---|---|---|---|

(MHz) | (rad m^{−2}) | |||||||

GMIMS-LBS | 300 | 480 | $0.5$ | 570 | $8.0$ | $6.2$ | $-300<\tilde{\varphi}<+300$ | 1 |

GMIMS-HBN | 1280 | 1750 | $0.5$ | 44,000 | 107 | 149 | $-600<\tilde{\varphi}<+600$ | 5 |

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Hill, A.S. Is There a Polarization Horizon? *Galaxies* **2018**, *6*, 129.
https://doi.org/10.3390/galaxies6040129

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Hill AS. Is There a Polarization Horizon? *Galaxies*. 2018; 6(4):129.
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Hill, Alex S. 2018. "Is There a Polarization Horizon?" *Galaxies* 6, no. 4: 129.
https://doi.org/10.3390/galaxies6040129