## Abstract

Narrow depolarized canals are common in maps of the polarized synchrotron emission of the Milky Way. Two physical effects that can produce these canals have been identified: the presence of Faraday rotation measure (RM) gradients in a foreground screen, and the cumulative cancellation of polarization known as differential Faraday rotation. We show that the behaviour of the Stokes parameters *Q* and *U* in the vicinity of a canal can be used to identify its origin. In the case of canals produced by a Faraday screen we demonstrate that, if the polarization angle changes by 90° across the canal, as is observed in all fields to-date, the gradients in RM must be discontinuous. Shocks are an obvious source of such discontinuities and we derive a relation of the expected mean separation of canals to the abundance and Mach number of supernova-driven shocks, and compare this with recent observations by Haverkorn, Katgert & de Bruyn. We also predict the existence of less common canals with polarization angle changes other than 90°. Differential Faraday rotation can produce canals in a uniform magneto-ionic medium, but as the emitting layer becomes less uniform the canals will disappear. We show that for moderate differences in emissivity in a two-layer medium, of up to 1/2, and for Faraday depth fluctuations of standard deviation ≲1 rad, canals produced by differential rotation will still be visible.

## Introduction

Polarized (synchrotron) radio emission is a rich source of information about the relativistic and thermal plasmas and magnetic fields in the interstellar medium (ISM). Recent observations have revealed an abundance of unexpected features that arise from the propagation of the emission through the turbulent ISM (Wieringa et al. 1993; Uyaniker et al. 1998; Duncan et al. 1999; Gray et al. 1999; Haverkorn, Katgert & de Bruyn 2000; Gaensler et al. 2001; Wolleben et al. 2006). Arguably the most eye-catching is the pattern of depolarized canals: a random network of dark narrow regions, clearly visible against a bright polarized background. These canals evidently carry information about the ISM, but it is still not quite clear how this information can be extracted. Two theories for the origin of the canals have been proposed; both attribute the canals to the effects of Faraday rotation, but one invokes steep gradients of the Faraday rotation measure (RM) across the telescope beam, in a Faraday screen (Haverkorn et al. 2000; Haverkorn, Katgert & de Bruyn 2004), whereas the other relies on the line-of-sight effects producing differential Faraday rotation (Beck 1999; Shukurov & Berkhuijsen 2003). In order to use the canal properties to derive parameters of the ISM, one must correctly identify their origin.

In this Letter we briefly discuss a few important aspects of the two theories describing the origin of canals. Detailed discussion of relevant depolarization mechanisms is presented in Fletcher & Shukurov (2006, and in preparation). Canals produced by Faraday rotation measure gradients are discussed in Section 2 where we show that these canals require a *discontinuous* distribution of free electrons and/or magnetic field; we further suggest an interpretation of the discontinuities in terms of interstellar shocks. The case of differential Faraday rotation is discussed in Section 3; in Section 4 we suggest an observational test that can be used to identify the specific mechanism that produces a given canal.

The defining features of the canals are as follows:

- (i)
the observed polarized emission

*P*approaches the polarization noise level σ_{P},*P*≲σ_{P}; - (ii)
the canal is about one beam wide;

- (iii)
the canal passes through a region of significant polarized intensity, say

*P*≳ 3σ_{P}; - (iv)
the canal is not related to any structure in total intensity, and so cannot be readily explained by e.g. an intervening gas or magnetic filament.

Polarized intensity vanishes when emission within the telescope beam consists of two equal parts with mutually orthogonal polarization planes. Thus feature (i) most often arises because the polarization angle Ψ changes by 90° across the canal. However, in Section 2 we predict a new type of canal across which the observed polarization angle does not change.

We only consider canals occurring in properly calibrated maps: Haverkorn et al. (2004) argue that the observations we discuss in Section 2.1 do not suffer from missing large-scale structure; Reich (2006) discusses the calibration of radio polarization maps in depth.

Polarized radiation is commonly described in terms of the complex polarization,

where*p*, the degree of polarization, is the fraction of the radiation flux that is polarized. When polarized emission passes through magnetized and ionized regions, the local polarization angle ψ (at position

*) changes by an amount depending on the wavelength λ due to the Faraday effect where*

**r***K*= 0.81 rad m

^{−2}cm

^{3}μG

^{−1}pc

^{−1}is a constant,

*n*

_{e}is the number density of free thermal electrons,

*B*

_{z}is the component of the magnetic field along the line of sight (here aligned with the

*z*-axis), and the observer is located at

*z*→∞. φ(

*z*) is known as the

*Faraday depth to a position z*and gives the change in polarization angle of a photon of wavelength λ as it propagates from

*z*to the observer. The maximum amount of Faraday rotation in a given direction is called the

*Faraday depth*

^{1}The observed amount of Faraday rotation, determined by the rotation measure RM = dΨ/dλ

^{2}, cannot exceed

*F*, i.e. |RM| ≤ |

*F*|λ

^{−2}.

The value of RM is related to *F*, but often in a complicated manner (see e.g. Burn 1966; Sokoloff et al. 1998). Simplest is the case of a Faraday screen, where the source of synchrotron emission is located behind a magneto-ionic region (e.g. because relativistic and thermal electrons occupy disjoint regions): then RM =*F*λ^{−2}. In a homogeneous region, where relativistic and thermal electrons are uniformly mixed, RM = 0.5*F*λ^{−2}.

Observations of linearly polarized emission provide the Stokes parameters *I*, *Q*, *U* which are related to *p* and Ψ via :

*P*= (

*Q*

^{2}+

*U*

^{2})

^{1/2}=

*pI*. The complex polarization can be written in terms of φ, where the integration extends over the volume of the telescope beam

*V*,

*W*(

*r*_{⊥}) defines the shape of the beam, a function of position in the sky plane

*r*_{⊥}= (

*x*,

*y*), and ε(

**) is the synchrotron emissivity. The total intensity is similarly given by ; the Faraday depth is a function of**

*r*

*r*_{⊥},

*F*=

*F*(

*r*_{⊥}).

## Canals Produced by a Faraday Screen

Consider polarized emission from a uniform background source passing through a Faraday screen – a layer where no further emission occurs, but which rotates the polarized plane. If *F* varies with *r*_{⊥}, adjacent lines of sight within the beam are subject to different amounts of Faraday rotation and the observed degree of polarization decreases. If the variation of *F* within the beam, Δ*F*_{D} (Fig. 1), produces a 90° difference in Ψ, one might expect that the depolarization will be complete as illustrated in Fig. 2(a), and a canal will be observed along contours defined by Δ*F*_{D}= (*n*+ 1/2)π with *n*= 0, 1, 2, … .

The situation is more subtle, though. Fig. 2(b) shows Ψ changing *smoothly* by 90° across a beam, as a result of a monotonic change in Faraday depth by the same amount, Δ*F*_{D}=π/2, as in panel (a). It is obvious that polarized emission does *not* cancel within the beam; the polarized intensity will be lower than for a uniform arrangement of angles but there will still be a polarized signal detected with a polarization angle of about −45°.

Thus gradients in *F* across the beam can produce complete depolarization in a Faraday screen if *F* changes by Δ*F*_{D}≃ (*n*+ 1/2)π within a small fraction of the beamwidth. As shown in Fig. 3 (solid line), obtained using equations (2) and (4), the depth of a canal will only be less than 10 per cent of the surrounding polarized emission if the gradient in *F* occurs over one-fifth or less of the beam: Δ*x*/*D*≲ 0.2, where Δ*x* is the extent of the region where the gradient occurs and *D* is the beamwidth. Haverkorn et al. (2004) reached a similar conclusion in an analysis of the depth and profile of canals observed at λ84 cm (see their section 4.4).

As illustrated in Figs 2(c) and 3 (dotted line), a variation in *F* of a magnitude Δ*F*_{D}=π across the FWHM of the Gaussian beam (i.e. Δ*F*=π and Δ*x*/*D*≃ 1) can also produce strong depolarization but now the change in polarization angle across the beam is ΔΨ≃ 0°. Similarly, other gradients in *F* can significantly reduce the degree of polarization: for example, Δ*F*= 0.7π (dashed line in Fig. 3) across a region that is about 0.7× the beamwidth will produce complete depolarization and a change in angle of 54° across the beam.

Larger gradients in *F* also result in strong depolarization and when Δ*F* > π the degree of depolarization is less sensitive to the resolution. This is illustrated in Fig. 3 for the case where Δ*F*= 1.5π (dash–dotted line); complete depolarization occurs when Δ*x*/*D*= 0, 1.5 but in the range 0 < Δ*x*/*D*≲ 2 the degree of polarization remains below the 10 per cent level. Thus an increment of Δ*F*= 1.5π, and more generally Δ*F*= (*n*+ 1/2)π, will produce one-beam-wide canals when the gradient occurs in a region roughly equal to or less than the beamwidth, i.e. where Δ*x*/*D*≲ 1. Other large gradients in *F*, such as Δ*F*=*n*π, depolarize to around 10 per cent of the background *p* and therefore can also produce canals; however, in this case one-beam-wide canals will only be formed when Δ*x*/*D*≃ 1.

So, if *F* is a continuous, random function of position (with sufficiently strong fluctuations on the scale of the beam) we predict the existence of canals with every change in polarization angle 0° < ΔΨ < 90° across the beam. On the other hand *a discontinuous change in F* by (*n*+ 1/2)π will produce canals across which the polarization angle changes by ΔΨ= 90°. This insight is important, as to-date all observed canals have ΔΨ≃ 90°. The only other way in which Faraday screens can consistently produce canals with this change in angle is if *all* the gradients in Faraday depth on the approximate scale of the beamwidth are of the magnitude Δ*F*≃ (*n*+ 1/2)π, *n*= 1, 2, … (*n*= 0 in this case only produces weak depolarization, as noted above); this is extremely unlikely in a random medium.

If a system of observed canals has ΔΨ= 90° and is caused by a foreground Faraday screen – see Section 4 for a diagnostic test for the origin of canals – then the distribution of rotation measure in the screen must be discontinuous on the scale of the beam. Haverkorn & Heitsch (2004) used magnetohydrodynamic simulations of Mach 10 turbulence to show that RM gradients can be steep enough to produce depolarized canals on smoothing with a sufficiently large simulated beam. Note that a finite beamsize is essential for the production of canals by this mechanism; with perfect resolution the width of the shock front will be resolved and no canal will appear. Furthermore, these canals will not fill up if the data are smoothed (solid and dash–dotted curves for Δ*x*/*D*→ 0 in Fig. 3) whereas canals produced by gradients other than Δ*F*= (*n*+ 1/2)π will disappear under smoothing at a rate shown with dashed and dotted curves in Fig. 3.

### The mean separation of canals produced by shocks

An obvious source of Faraday depth discontinuities is shocks in the ISM. Then the mean distance between the canals produced by a Faraday screen will be related to the distance between the shock fronts. The expected separation of suitable shock fronts can be derived using the model of interstellar shock-wave turbulence of Bykov & Toptygin (1987) and then compared with the separation of canals observed by Haverkorn et al. (2003), claimed to arise in a Faraday screen.

If the frequency distribution of interstellar shocks of Mach number is , the mean number of shocks of a strength exceeding that cross a given position in the ISM per unit time can be obtained as

and the mean separation between the shocks in three dimensions follows as where*c*

_{s}is the sound speed. The separation in the plane of the sky is reduced by because of projection effects and multiplied by a further factor of

*L*/

*d*if the shocks occur in a region of depth

*d*. At an average distance to the shocks of

*d*/2, the average angular separation in the sky plane is then

For supernova-driven shocks, Bykov & Toptygin (1987) derived

allowing for both primary and secondary shocks, where α is a numerical factor (α= 2 for a supernova remnant in the Sedov phase, and α= 4.5 for a three-phase ISM);*C*(α) ≈ 2.3 × 10

^{−2}and 4.1 × 10

^{−3}for α= 2 and 4.5 respectively;

*f*

_{cl}is the volume filling factor of diffuse clouds that reflect the primary shocks to produce the secondary ones; with

*S*the supernova rate per unit volume and

*r*

_{0}the maximum radius of a primary shock. We then obtain the following expression for the separation of shock fronts with in three dimensions: where ν

_{SN}is the supernova rate and

*R*and

*h*are the radius and scaleheight of the star-forming disc. The term in square brackets is approximately 1 for α= 4.5 and .

If the magnetic field is frozen into the gas and the gas density increases by a factor of ε_{ρ} at the shock then so will the field strength (we neglect a factor arising from the relative alignment of the shock and field). A canal forms where *F* increases discontinuously by Δ*F*_{D}= (*n*+ 1/2)π. So the required density compression ratio is

*F*〉 is the mean value of

*F*. The gas compression ratio depends on the shock Mach number as (e.g. Landau & Lifshitz 1960) where we have used γ≃ 5/3 for the ratio of specific heats, which yields the value of the shock Mach number required to produce a canal:

In the field of canals observed by Haverkorn et al. (2003) at λ84 cm, 〈RM〉≃−3.4 rad m^{−2} and a by-eye estimate of their mean separation gives arcmin. The most abundant canals are produced by the shocks which can generate Δ*F*_{D}=π/2 and from equation (12) this requires . Using *f*_{cl}= 0.25, taking α= 4.5 and the parameter values used to normalize equation (9), we obtain *L*≃ 10 pc for shocks with . Our estimate of *L* includes shocks with that will not produce canals, but the strong dependence of in equation (8) on Mach number means that *L* will underestimate the separation of canal-generating shocks insignificantly. Using equation (7), with *L*≃ 10 pc and arcmin, we find that the canals observed by Haverkorn et al. (2003) are compatible with a system of shocks occurring in a Faraday screen with a depth of *d*≃ 100 pc. The maximum distance in this field, beyond which emission is completely depolarized, is estimated to be ∼600 pc by Haverkorn et al. (2003). Thus this model for the origin of the canals requires that the nearest 100 pc in the direction (*l*, *b*) = (161°, 16°) is effectively devoid of cosmic ray electrons; if the cosmic rays normally spread over a distance *v*_{A}*t*∼ 1 kpc, where *v*_{A}≃ 20 km s^{−1} is the Alfvén speed and *t*≃ 3 × 10^{7} yr their lifetime, such a condition is difficult to explain.

## Canals Produced by Differential Faraday Rotation

In Section 2 we discussed the depolarization effects of Faraday rotation measure gradients transverse to the line of sight acting on smooth polarized background emission. Now we will consider a uniform layer in which both emission and Faraday rotation occur. This gives rise to the well-known effect of differential Faraday rotation (Burn 1966; Sokoloff et al. 1998), where polarized emission from two positions along the line of sight separated by a rotation of Δφ=π/2 exactly cancels, thus reducing the degree of polarization. When a line of sight has a total rotation of *F*=*n*π there is total cancellation of all polarized emission in the layer and the degree of polarization is *p*= 0. Canals produced by this mechanism are discussed by Shukurov & Berkhuijsen (2003).

### Differential Faraday rotation in a non-uniform medium

We now investigate how canals produced by differential Faraday rotation are affected by deviations from uniformity in the magneto-ionic medium. First we will discuss the case of a two-layer medium in which the synchrotron emissivity is different in each layer. Then we allow for random fluctuations of *F* (the latter case produces what is known as Faraday dispersion).

For a two-layer medium the fractional polarization can be written as

where we have assumed ψ_{0}= 0,

*I*

_{1}and

*I*

_{2}are the synchrotron fluxes from the furthest and nearest layers to the observer respectively,

*I*=

*I*

_{1}+

*I*

_{2}, and

*F*

_{1}and

*F*

_{2}are the Faraday depths through the two layers. [This is similar to equation (10) in Sokoloff et al. (1998), but with typos corrected.] We assume that

*F*=

*F*

_{1}+

*F*

_{2}=π, the condition for canals to form as a result of differential Faraday rotation, and investigate what will happen to the canals if

*I*

_{1}≠

*I*

_{2}. We parametrize the difference in the synchrotron emissivity in the two layers as ε= (

*I*

_{2}−

*I*

_{1})/

*I*

_{1}, choose equal Faraday rotation through each layer for simplicity so that

*F*

_{1}=

*F*

_{2}=π/2, and substitute these into equation (13). We then obtain the dependence of the minimum degree of polarization on ε: Thus, for a moderate difference in emissivity of ε= 1/2 between two layers, the canals will have a depth of

*p*/

*p*

_{0}≃ 1/6 compared with

*p*/

*p*

_{0}= 0 for a strictly uniform medium; these canals will still be clearly visible and can be interpreted as e.g. contours of

*F*or RM (Shukurov & Berkhuijsen 2003). Where the difference in emissivity is greater, say ε∼ 2 or more as one might expect viewing distant bright spiral arm emission near the Galactic plane, the canals will more readily fill up, and become much less distinct and more difficult to interpret confidently.

Now let us consider the effect of random fluctuations in Faraday rotation, sometimes called internal Faraday dispersion. We start with equation (34) of Sokoloff et al. (1998):

where*S*= 2σ

^{2}

_{F}− 2 i

*F*and σ

_{F}is the standard deviation of the Faraday depth

*F*. In order to produce canals we need

*F*=π and then The term in brackets shows the powerful depolarization resulting from internal Faraday dispersion (see Sokoloff et al. 1998): for large enough σ

_{F}depolarization is complete. However, as long as σ

_{F}< 1, canals produced by differential Faraday rotation will only fill up by one-third or less. For example, at λ20 cm the canals will not be destroyed as long as the dispersion in Faraday rotation measure is less than 30 rad m

^{−2}. This is why canals are still visible in the λ20-cm map studied by Shukurov & Berkhuijsen (2003) which has a dispersion in rotation measure of about σ

_{RM}≃ 10 rad m

^{−2}, i.e. σ

_{F}=σ

_{RM}λ

^{2}≃ 0.4.

## Observational Diagnostics for the Origin of a Canal

The complex polarization emitted by a layer producing differential Faraday rotation can be written as (Burn 1966)

We can see immediately that if ψ_{0}= 0 we have and . Since the orientation of the coordinate system in which

*Q*and

*U*are defined is arbitrary (i.e. the reference line from which we measure polarization angles can have any orientation), we can always choose a system in which ψ

_{0}= 0 near a canal. At the axis of a canal

*Q*=

*U*= 0, but in the case of a canal produced by differential Faraday rotation there exists a reference frame in the (

*Q*,

*U*) plane where one of the two Stokes parameters changes sign across a canal but the other does not.

If, otherwise, a canal is produced by discontinuities in Faraday rotation across the beam (Section 2), we have (Fletcher & Shukurov 2006)

in the vicinity of a canal and both*Q*and

*U*will change sign across a canal (given ψ

_{0}= 0) except in the special case Δ

*F*

_{D}=

*F*+ 2

*n*π. The change in sign occurs since

*Q*/

*I*=

*p*

_{0}cos [2(ψ

_{0}+

*F*)] on one side of the canal and

*Q*/

*I*=

*p*

_{0}cos [2(ψ

_{0}+

*F*) +π] on the other; a similar variation occurs in

*U*/

*I*. Therefore the behaviour of the observed Stokes parameters

*Q*and

*U*across a canal provides a way to distinguish between canals produced by foreground Faraday screens and those resulting from differential Faraday rotation.

## Summary

The main points of this paper can be summarized as follows.

- (i)
The behaviour of the Stokes parameters

*Q*and*U*in the vicinity of a canal allows one to identify whether a foreground Faraday screen or differential Faraday rotation is the cause of the canal (Section 4). - (ii)
A foreground Faraday screen can produce canals with any polarization angle change across the canal, 0 < ΔΨ < 90°. However, discontinuous jumps in the Faraday depth will only produce canals with ΔΨ≃ 90° (Section 2).

- (iii)
If shocks produce the discontinuities in a foreground Faraday screen that generate canals, the mean separation of the canals can provide information about the Mach number and separation of shocks in the screen (Section 2.1).

- (iv)
Canals produced by differential Faraday rotation are sensitive to non-uniformity in the medium along the line of sight, systematic or random. However, they will remain recognizable if the synchrotron emissivity varies by less than a factor of about 2 or if the standard deviation of the Faraday depth is σ

_{F}< 1 (Section 3.1).

## Acknowledgments

This work was supported by the Leverhulme Trust under research grant F/00 125/N.

## References

*F*/λ

^{2}, in our notation. However, it is more convenient, and physically better motivated, to define the Faraday depth, similarly to the optical depth, as a dimensionless quantity, as used by Spangler (1982) and Eilek (1989).