CHARACTERIZING MAGNETIZED TURBULENCE IN M51

We start by defining the difference $\Delta \Phi ({\boldsymbol \ell })$ in the orientation of the magnetic field (unless otherwise specified, in this paper we will only concern ourselves with the plane-of-the-sky component of the magnetic field) at two points separated by a distance ${\boldsymbol \ell }$

$$\begin{equation} \Delta \Phi ({\boldsymbol \ell })\equiv \Phi (\mathbf {r})-\Phi (\mathbf {r}+{\boldsymbol \ell }), \end{equation} \tag{ 1 }$$

with Φ(r) the magnetic field orientation at position r (both r and ${\boldsymbol \ell }$ are also understood to be located in the plane of the sky). Given a set of measurements Φ(r) on a polarization map, we can also define the following (second-order) angular structure function:

$$\begin{equation} \langle \Delta \Phi ^{2}(\ell )\rangle \equiv \frac{1}{N(\ell )}\sum _{i=1}^{N(\ell )}[\Phi (\mathbf {r})-\Phi (\mathbf {r}+{\boldsymbol \ell })]^{2}, \end{equation} \tag{ 2 }$$

where 〈⋅⋅⋅〉 denotes an average, N(ℓ) is the number of pairs of field orientation measurements separated by $\ell =|{\boldsymbol \ell }|$, and stationarity and isotropy were assumed (i.e., the structure function is only dependent on the magnitude of ${\boldsymbol \ell }$, and not on its orientation or r; see Falceta-Gonçalves et al. 2008; Hildebrand et al. 2009).

The main assumption in our analysis consists in modeling the magnetic field, and therefore its orientation through Φ(r), as being composed of a large-scale, ordered component Φ₀(r) and a smaller scale, zero-mean, turbulent component Φ_t(r). That is, we write

$$\begin{equation} \Phi (\mathbf {r})=\Phi _{\mathrm{t}}(\mathbf {r})+\Phi _{0}(\mathbf {r}), \end{equation} \tag{ 3 }$$

which, if we further assume these two components to be statistically independent, leads to

$$\begin{equation} \langle \Delta \Phi ^{2}(\ell )\rangle =\left\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\right\rangle +\left\langle \Delta \Phi _{0}^{2}(\ell )\right\rangle. \end{equation} \tag{ 4 }$$

We thus find that the structure function is composed of two angular components stemming from the contributions of the turbulent and ordered magnetic fields. Our assumption on the difference between the two scales therefore allows for their separation and analyses. This is exemplified in Figure 2, where we show (panel (a)) hypothetical turbulent and ordered contributions to the total angular structure function (panel (b), solid curve), as expressed in Equation (4). Panel (c) presents the same information as panel (b) for 〈ΔΦ²(ℓ)〉 but displayed as a function of ℓ² instead of ℓ. This is to show that the separation of the two length scales is sometimes easier to visualize, through the abrupt change in the slope of 〈ΔΦ²(ℓ)〉, by using the square of the distance for the abscissa; we will use both representations for the data analyzed later in this paper. The total angular structure function (of panels (b) or (c)) is the input to our problem as obtained from a polarization map, which we seek to analyze in order to characterize magnetized turbulence.

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The behavior of turbulent and ordered contributions to the total angular structure function of panel (a) can be qualitatively understood from the fact that, evidently, they must equal zero when ℓ = 0 and then initially increase with ℓ. The turbulent structure function will keep increasing until it reaches values for ℓ that sufficiently exceed the turbulence correlation length δ, at which point it will reach its maximum value. This can also be understood more quantitatively with the following relation:

$$\begin{eqnarray} \left\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\right\rangle & = & \langle \left[\Phi _{\mathrm{t}}(\mathbf {r})-\Phi _{\mathrm{t}}\left(\mathbf {r}+{\boldsymbol \ell }\right)\right]^{2}\rangle \nonumber \\ & = & 2\left[\left\langle \Phi _{\mathrm{t}}^{2}(0)\right\rangle -\left\langle \Phi _{\mathrm{t}}^{2}(\ell )\right\rangle \right], \end{eqnarray} \tag{ 5 }$$

where the (stationary and isotropic) turbulent autocorrelation function is defined with

$$\begin{equation} \left\langle \Phi _{\mathrm{t}}^{2}(\ell )\right\rangle \equiv \langle \Phi _{\mathrm{t}}(\mathbf {r})\Phi _{\mathrm{t}}\left(\mathbf {r}+{\boldsymbol \ell }\right)\rangle. \end{equation} \tag{ 6 }$$

It therefore follows that $\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\rangle =2\langle \Phi _{\mathrm{t}}^{2}(0)\rangle$ when ℓ ≫ δ, as the turbulence is not correlated on such scales and its autocorrelation vanishes. The ordered structure function is expected to rise, at first monotonically, with increasing values of ℓ in view of its larger-scale nature. For the example of Figure 2 we have characterized the turbulent component with a Gaussian autocorrelation function of width, or correlation length (i.e., its standard deviation) δ = 2'', while the ordered structure function was modeled with a Taylor expansion in powers of ℓ². This restriction to even powers in ℓ is dictated from the assumption of isotropy for the structure function.

With the previous assumption in the difference of the two length scales it becomes possible to model the ordered component $\langle \Delta \Phi _{0}^{2}(\ell )\rangle$ independently of $\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\rangle$ by using values of ℓ sufficiently large (i.e., sufficiently greater than δ) where any variation in $\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\rangle$ is negligible. For our example, we chose ℓ ⩾ 6'' to obtain the Taylor series fit given by the dot-broken curve in panels (b) and (c) of Figure 2. This curve is then representative of $\langle \Delta \Phi _{0}^{2}(\ell )\rangle$ but shifted up by the constant level of the turbulent component present in that range. More precisely, if we define a function χ(ℓ) for this curve, then we write

$$\begin{equation} \chi (\ell )=2\left\langle \Phi _{\mathrm{t}}^{2}(0)\right\rangle +\left\langle \Delta \Phi _{0}^{2}(\ell )\right\rangle. \end{equation} \tag{ 7 }$$

In this paper we will focus on characterizing magnetized turbulence, and we are thus interested in isolating the turbulent component of the structure function, or, alternatively, its autocorrelation. The latter (multiplied by a factor of two) is readily evaluated from Equations (4), (5), and (7) with

$$\begin{equation} 2\left\langle \Phi _{\mathrm{t}}^{2}(\ell )\right\rangle =\chi (\ell )-\langle \Delta \Phi ^{2}(\ell )\rangle \end{equation} \tag{ 8 }$$

and shown in panel (d) of Figure 2 (i.e., as the subtraction of the solid curve from the dot-broken curve in panel (b)).

Although we could very well use the angular structure function in the manner presented in this section for our analyses of the M51 data, we will nonetheless for the rest of our discussion focus instead on the angular dispersion function 1 − 〈cos [ΔΦ(ℓ)]〉. We note, however, that the properties, method, and technique discussed thus far for the structure function apply just as well to the dispersion function. In fact, we note that the two angular functions are simply related through

$$\begin{equation} 1-\left\langle \cos \left[\Delta \Phi (\ell )\right]\right\rangle \simeq \frac{1}{2}\langle \Delta \Phi ^{2}(\ell )\rangle \end{equation} \tag{ 9 }$$

when ΔΦ(ℓ) ≪ 1. The advantage of the dispersion function is its close connection to the autocorrelation of the magnetic field (see Equation (10) below), which then naturally leads to the study of the magnetized turbulent power spectrum through a simple Fourier transform (Houde et al. 2011).

Finally, it is important to note that in general the width of the turbulent autocorrelation function (and of the associated structure/dispersion function) is not solely due to the intrinsic correlation length of turbulence. A correlation scale brought about by the finite spatial resolution with which observations are realized will also combine with the intrinsic turbulent correlation length to set the overall width of the turbulent autocorrelation function. A further complication results from the related problem of signal integration through the line of sight. As we will soon see, a careful analysis of such effects will allow us not only to disentangle the intrinsic correlation length characterizing magnetized turbulence from the overall width of the turbulent autocorrelation function but also to determine the level of turbulent energy contained in the medium probed by the observations. For this to be feasible, however, some approximation must be made on the nature of the turbulent autocorrelation function. The case of isotropic Gaussian turbulence and beam profile functions was treated in the details in Houde et al. (2009; see their Section 2), and their main relations detailing the combination of the two length scales for the analysis of turbulence are given in the next section (see Equations (22)–(24) below). In this paper, we further provide an analysis of the more general case of anisotropic Gaussian turbulence in the Appendix, while the corresponding results are also summarized in Section 2.2 (see Equations (19)–(21)). This will in turn make possible the measurement of anisotropy in the turbulent autocorrelation function, which is another important parameter for the characterization of magnetized turbulence.

As previously mentioned, the analysis of the angular dispersion function found in the Appendix and summarized in this section follows that presented in Section 2 of Houde et al. (2009) with the difference that we now allow for the presence of anisotropy in the turbulence. As stated in Section 2.1, we are interested in the function $\cos [\Delta \Phi ({\boldsymbol \ell })]$ that is related to the magnetic field autocorrelation function through

$$\begin{equation} \left\langle \cos \left[\Delta \Phi ({\boldsymbol \ell })\right]\right\rangle =\frac{\langle \overline{\mathbf {B}}\mathbf {\cdot }\overline{\mathbf {B}}\mathbf {({\boldsymbol \ell })}\rangle }{\langle \overline{\mathbf {B}}\mathbf {\cdot }\overline{\mathbf {B}}(0)\rangle }, \end{equation} \tag{ 10 }$$

where 〈⋅⋅⋅〉 denotes an average and $\langle \overline{\mathbf {B}}\mathbf {\cdot }\overline{\mathbf {B}}\mathbf {({\boldsymbol \ell })}\rangle \equiv \langle \overline{\mathbf {B}}(\mathbf {r})\mathbf {\cdot }\overline{\mathbf {B}}\mathbf {(\mathbf {r}+{\boldsymbol \ell })}\rangle$. It is important to note that the magnetic field $\overline{\mathbf {B}}$ is a weighted average (with the polarized flux) through the thickness of the column of gas probed (i.e., the disk of M51) and across the telescope beam (see Equation (A1)). The local, non-averaged, magnetic field $\mathbf {B(\mathbf {x})}$ at a point x is composed of an ordered field $\mathbf {B}_{0}\mathbf {(\mathbf {x})}$ and a turbulent component $\mathbf {B_{\mathrm{t}}(\mathbf {x})}$ such that

$$\begin{equation} \mathbf {\mathbf {B\left(\mathbf {x}\right)}}=\mathbf {B}_{0}\mathbf {\mathbf {\mathbf {\left(\mathbf {x}\right)+\mathbf {B}_{\mathrm{t}}\left(\mathbf {x}\right)}}}. \end{equation} \tag{ 11 }$$

As was the case earlier, the displacement vector ${\boldsymbol \ell }$ in Equation (10), and others that will follow, is understood to be located in the plane of the sky. We will further break ${\boldsymbol \ell }$ down into two perpendicular components

$$\begin{equation} {\boldsymbol \ell }={\boldsymbol \ell }_{1}+{\boldsymbol \ell }_{2}, \end{equation} \tag{ 12 }$$

where, unless otherwise noted, ${\boldsymbol \ell }_{1}$ and ${\boldsymbol \ell }_{2}$ are taken to be, respectively, perpendicular and parallel to the projection of the ordered component of the magnetic field B₀ on the plane of the sky. For everything that follows, statistical independence between $\mathbf {B}_{0}\mathbf {(\mathbf {x})}$ and $\mathbf {B_{\mathrm{t}}(\mathbf {x})}$, homogeneity in the strength of the magnetic fields $\langle B_{0}^{2}\rangle \equiv \langle \mathbf {B}_{0}\mathbf {\cdot }\mathbf {B}_{0}(0)\rangle$ and $\langle B_{\mathrm{t}}^{2}\rangle \equiv \langle \mathbf {B}_{\mathrm{t}}\mathbf {\cdot }\mathbf {B}_{\mathrm{t}}(0)\rangle$, and, more generally, stationarity in the autocorrelation functions $\langle \mathbf {B}_{0}\mathbf {\cdot }\mathbf {B}_{0}\mathbf {({\boldsymbol \ell })}\rangle$ and $\langle \mathbf {B}_{\mathrm{t}}\mathbf {\cdot }\mathbf {B}_{\mathrm{t}}\mathbf {({\boldsymbol \ell })}\rangle$ are assumed. The assumption of homogeneity in the field strength in particular, while clearly an idealization, is needed for securing a quantitative measure of turbulence from our data (see the Appendix for more details).

Using these assumptions, it can be shown that, just as was the case for the structure function in Equation (4), the dispersion function $1-\langle \cos [\Delta \Phi ({\boldsymbol \ell })]\rangle$ can be decomposed into turbulent and ordered terms

$$\begin{eqnarray} 1-\left\langle \cos \left[\Delta \Phi ({\boldsymbol \ell })\right]\right\rangle & = & [b^{2}(0)-b^{2}({\boldsymbol \ell })]+[\alpha ^{2}(0)-\alpha ^{2}({\boldsymbol \ell })]\nonumber \\ & = & \lbrace b^{2}(0)+[\alpha ^{2}(0)-\alpha ^{2}({\boldsymbol \ell })]\rbrace -b^{2}({\boldsymbol \ell }),\nonumber\\ \end{eqnarray} \tag{ 13 }$$

with the normalized ordered and turbulent autocorrelation functions given by

$$\begin{eqnarray} \alpha ^{2}({\boldsymbol \ell }) & = & \frac{\langle \overline{\mathbf {B}}_{0}\mathbf {\cdot }\overline{\mathbf {B}}_{0}({\boldsymbol \ell })\rangle }{\langle \overline{\mathbf {B}}\mathbf {\cdot }\overline{\mathbf {B}}(0)\rangle } \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} b^{2}({\boldsymbol \ell }) & = & \frac{\langle \overline{\mathbf {B}}_{\mathrm{t}}\mathbf {\cdot }\overline{\mathbf {B}}_{\mathrm{t}}({\boldsymbol \ell })\rangle }{\langle \overline{\mathbf {B}}\mathbf {\cdot }\overline{\mathbf {B}}(0)\rangle }, \end{eqnarray} \tag{ 15 }$$

respectively. The quantity b²(0) in Equation (13) is simply, from Equation (15), the integrated turbulent-to-total magnetic energy ratio. It is also the equivalent of $\langle \Delta \Phi _{\mathrm{t}}^{2}(\ell )\rangle$ (see Equation (6)) when dealing with the angular structure function. The ordered function $\alpha ^{2}(0)-\alpha ^{2}({\boldsymbol \ell })$, which we assume to be of a larger spatial scale than $b^{2}({\boldsymbol \ell })$, can be advantageously modeled with a Taylor series. Since, as we will soon discuss, we adopt a model of turbulence where the autocorrelation function is even in directions parallel and perpendicular to the projection of B₀ on the plane of the sky, it follows that we can write

$$\begin{equation} \alpha ^{2}(0)-\alpha ^{2}({\boldsymbol \ell })=\sum _{\stackrel{{i+j=1}}{i,j\ge 0}}^{\infty }a_{2i,2j}\ell _{1}^{2i}\ell _{2}^{2j}. \end{equation} \tag{ 16 }$$

Accordingly, we will proceed by fitting the part within curly braces in Equation (13) using

$$\begin{equation} b^{2}(0)+[\alpha ^{2}(0)-\alpha ^{2}({\boldsymbol \ell })]=b^{2}(0)+\sum _{\stackrel{{i+j=1}}{i,j\ge 0}}^{\infty }a_{2i,2j}\ell _{1}^{2i}\ell _{2}^{2j} \end{equation} \tag{ 17 }$$

to the data for high enough values of $\ell \equiv \left|{\boldsymbol \ell }\right|$ where we expect $b^{2}({\boldsymbol \ell })$ to be negligible (i.e., $b^{2}({\boldsymbol \ell })$ dominates at lower values of ℓ). We will then obtain $b^{2}({\boldsymbol \ell })$ by subtracting the dispersion function data (i.e., the left-hand side of Equation (13)) from the aforementioned fit. This function is the equivalent to Equation (7) for χ(ℓ) defined in Section 2.1 for the angular structure function analysis.

We now adopt a picture for magnetized turbulence consistent with current models, either incompressible (Goldreich & Sridhar 1995) or compressible (Cho & Lazarian 2003; Kowal & Lazarian 2010). That is, we will assume that some anisotropy is present in the magnetized turbulent autocorrelation function, where it is expected that the correlation lengths in directions parallel and perpendicular to the ordered magnetic field are different. Although this is undoubtedly an idealization, we model the intrinsic autocorrelation function of the magnetized turbulence as an ellipsoid Gaussian function, with the symmetry axis of the ellipsoid aligned with the ordered magnetic field B₀ (see Figure 3). It is possible to analytically solve for $b^{2}({\boldsymbol \ell })$ for such cases, with the further assumption that the telescope beam profile is circular Gaussian in form

$$\begin{equation} H(\mathbf {r})=\frac{1}{2\pi W^{2}}e^{-r^{2}/2W^{2}}, \end{equation} \tag{ 18 }$$

with W = 0.425 FWHM the beam radius (i.e., it is not the beam's FWHM but its "standard deviation" equivalent). A general solution for $b^{2}({\boldsymbol \ell })$ when the magnetized autocorrelation function ellipsoid is at arbitrary inclination relative to the line of sight and arbitrary projected orientation on the plane of the sky can be derived and is given in the Appendix.

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For M51 we will limit ourselves to three cases. One has the ellipsoid symmetry axis (and the ordered magnetic field B₀) at an inclination angle α relative to the line of sight and its projection on the plane of the sky advantageously aligned with one of the observer's coordinate axes (still on the plane of the sky; see the Appendix). The integrated (or beam-broadened) autocorrelation function is then analytically expressed by

$$\begin{equation} b^{2}({\boldsymbol \ell })=\left[\frac{\left\langle B_{\mathrm{t}}^{2}\right\rangle }{N\left\langle B_{0}^{2}\right\rangle +\left\langle B_{\mathrm{t}}^{2}\right\rangle }\right]e^{-\frac{1}{2}\left[\ell _{1}^{2}/\left(\delta _{\bot }^{2}+2W^{2}\right)+\ell _{2}^{2}/\left(\mu _{2}^{2}+2W^{2}\right)\right]}, \end{equation} \tag{ 19 }$$

where the number of turbulence cells contained in the gas probed by the telescope beam is given by

$$\begin{equation} N=\frac{\sqrt{\left(\delta _{\bot }^{2}+2W^{2}\right)\left(\mu _{2}^{2}+2W^{2}\right)}\,\Delta ^{\prime }}{\sqrt{2\pi }\delta _{\Vert }\delta _{\bot }^{2}}, \end{equation} \tag{ 20 }$$

with

$$\begin{equation} \mu _{2}^{2}=\delta _{\Vert }^{2}\sin ^{2}\left(\alpha \right)+\delta _{\bot }^{2}\cos ^{2}\left(\alpha \right) \end{equation} \tag{ 21 }$$

and δ_|| and δ_⊥ the turbulent correlations lengths parallel and perpendicular to the (local) ordered magnetic field B₀, respectively. The inverses of these correlation lengths are, in effect, the corresponding widths of the associated turbulent power spectrum (see Equation (A7) and the associated discussion in the Appendix). We once again emphasize that, in Equation (19), ℓ₁ and ℓ₂ are the displacements, respectively, perpendicular and parallel to the projection of B₀ on the plane of the sky. In Equation (20) Δ' is the effective depth of the column of polarized gas probed by the telescope beam (Houde et al. 2009; see their Section 3.2 and Equation (45)). This elliptical Gaussian turbulence case will actually be the last one considered in Section 4.2.2.

The first case considered (in Section 4.1) will be for isotropic turbulence when δ_|| = δ_⊥ ≡ δ; Equations (19) and (20) then reduce to

$$\begin{equation} b^{2}(\ell )=\left[\frac{\left\langle B_{\mathrm{t}}^{2}\right\rangle }{N\left\langle B_{0}^{2}\right\rangle +\left\langle B_{\mathrm{t}}^{2}\right\rangle }\right]e^{-\ell ^{2}/2\left(\delta ^{2}+2W^{2}\right)} \end{equation} \tag{ 22 }$$

and

$$\begin{equation} N=\frac{(\delta ^{2}+2W^{2})\Delta ^{\prime }}{\sqrt{2\pi }\delta ^{3}}. \end{equation} \tag{ 23 }$$

Equation (16) is furthermore simplified to

$$\begin{equation} \alpha ^{2}(0)-\alpha ^{2}(\ell )=\sum _{j=1}^{\infty }a_{2j}\ell ^{2j}. \end{equation} \tag{ 24 }$$

This isotropic Gaussian turbulence is the model previously solved and used in Houde et al. (2009).

Also, in Section 4.2.1 we consider what could be termed as a "hybrid" model of anisotropic turbulence where Equations (22)–(24) for isotropic turbulence are applied to independent analyses on displacement vectors that are grouped in sets oriented approximately parallel or perpendicular to the local mean magnetic field. Differences in the width of the two turbulent autocorrelation functions can then reveal anisotropy in the magnetized turbulence.

For all three cases, the parameters measurable by fitting the integrated turbulent autocorrelation data with Equation (19) (or Equation (22)) are the correlation lengths δ_|| and δ_⊥ (or simply δ) and the intrinsic turbulent-to-ordered magnetic energy ratio $\left\langle B_{\mathrm{t}}^{2}\right\rangle /\left\langle B_{0}^{2}\right\rangle$ (we assume that Δ' and the inclination angle α are known a priori).

As stated in Sections 2.1 and 2.2, the polarization angle Φ is the basic observable needed for our analysis. Following the detailed discussion presented in Appendix B of Houde et al. (2009), given the angle difference between a pair of data points separated by a distance ℓ_ij ≡ |r_i − r_j|

$$\begin{equation} \Delta \Phi _{ij}=\Phi _{i}-\Phi _{j}, \end{equation} \tag{ 25 }$$

we calculate the (average) function 〈cos (ΔΦ_ij)〉_k from the data for all (ℓ_k − Δℓ/2) ⩽ ℓ_ij < (ℓ_k + Δℓ/2), with ℓ_k = kΔℓ corresponding to an integer multiple of the grid spacing Δℓ = 1''. This function is then corrected for measurement uncertainties according to

$$\begin{equation} \langle \cos (\Delta \Phi _{ij})\rangle _{k,0}\simeq \frac{\langle \cos (\Delta \Phi _{ij})\rangle _{k}}{1-\frac{1}{2}\langle \sigma ^{2}(\Delta \Phi _{ij})\rangle _{k}}, \end{equation} \tag{ 26 }$$

where the uncertainty on ΔΦ_ij is given by

$$\begin{equation} \sigma ^{2}(\Delta \Phi _{ij})\simeq \sigma ^{2}(\Phi _{i})+\sigma ^{2}(\Phi _{j})-2\sigma (\Phi _{i})\sigma (\Phi _{j})e^{-\ell _{ij}^{2}/4W^{2}} \end{equation} \tag{ 27 }$$

and σ²(Φ_i) is the uncertainty on Φ_i. Finally, the measurement uncertainties for the adopted dispersion function 1 − 〈cos (ΔΦ_ij)〉_{k, 0} are determined with

$$\begin{eqnarray} \sigma ^{2}[\langle \cos (\Delta \Phi _{ij})\rangle _{k,0}]&=&\langle \sin (\Delta \Phi _{ij})\rangle _{k}^{2}\langle \sigma ^{2}(\Delta \Phi _{ij})\rangle _{k}\nonumber\\ &&+\frac{3}{4}\langle \cos (\Delta \Phi _{ij})\rangle _{k}^{2}\langle \sigma ^{4}(\Delta \Phi _{ij})\rangle _{k},\quad \end{eqnarray} \tag{ 28 }$$

for all (ℓ_k − Δℓ/2) ⩽ ℓ_ij < (ℓ_k + Δℓ/2). The data and results presented in the figures and tables that follow are all based on these equations.

Our analysis assumes that the observed polarization angles at λ6 cm trace the orientation of the local magnetic field (we ignore the π/2 difference between the linear polarization plane of the observed electric field and the orientation of the magnetic field at the source of the emission). Faraday rotation can add an extra level of complexity to the distribution of angles, so here we estimate its contribution to the observed angles.

Faraday rotation can produce a systematic variation of Φ with position due to the presence of a mean magnetic field; if the mean field lies in the same plane as the galaxy disk, then the positional variation will occur due to the inclination of the galaxy to the line of sight. Fletcher et al. (2011) modeled the mean magnetic field in M51 and found that it does lie in the galaxy plane and is weak. The maximum observed RM due to the mean field of their model is RM ≈ 10 rad m², which rotates λ6 cm emission by ΔΦ ≈ 2°.

Random fluctuations of Faraday rotation, σ_RM, will also produce fluctuations in Φ. Fletcher et al. (2011) estimated that the intrinsic standard deviation of RM in M51 at 15'' resolution is σ_RM ≈ 10 rad m². At the 4'' resolution we are using σ_RM will be stronger, scaling as the ratio of the beam widths, so in our data σ_RM ≈ 40 rad m² corresponding to a rotation of ΔΦ ≈ 8° at λ6 cm.

Thus, Faraday rotation produces uncertainty in our dispersion functions of 1 − 〈cos (ΔΦ)〉 ≈ 0.01. This uncertainty is about an order of magnitude below the difference between the fits using Equation (16) (or Equation (24)) and the observations, until decorrelation of the angles occurs (e.g., see the middle panels of Figures 4–6). Note that decorrelation, i.e., where the autocorrelation function becomes zero, mostly occurs when the dispersion function is about 0.1, which corresponds to an angle difference of about 30°. Therefore, we will ignore the contribution of Faraday rotation to the polarization angles in our data, other than as a source of error. We also note that λ3 cm data also presented in Fletcher et al. (2011), which suffer less from Faraday rotation, were not used for this analysis because of lower spatial resolution.

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We first consider the case of isotropic turbulence and model our data for the three suitable regions in M51 with Equations (22)–(24). All the pertinent functions are therefore assumed to possess an azimuthal symmetry about the ℓ = 0 axis.

Figure 4 shows the result of the isotropic dispersion analysis for the northeast spiral arm as a function of ℓ² (top) and ℓ (middle). The broken curve ("ordered") is the least-squares fit for the sum of the integrated turbulent-to-total magnetic energy ratio (b²(0) in Equation (22)) and the ordered component of Equation (24) (i.e., Equation (17) while using Equation (24) on the right-hand side) to the data contained within 6 ⩽ ℓ ⩽ 10; data points are shown with symbols. The integrated magnetized turbulence autocorrelation function b²(ℓ), obtained by subtracting the data from the aforementioned fit of the middle graph, is shown at the bottom. The broken curve on the bottom graph shows the radial profile of the "autocorrelated synthesized beam." This represents the contribution of the synthesized beam to (the width of) b²(ℓ). That is, this is what b²(ℓ) would look like in the limit where the intrinsic turbulent correlation length δ was zero in the exponent of Equation (22) (i.e., disregarding its effect on the amplitude of b²(ℓ) through N). It follows from this and the fact that the data points for b²(ℓ) fall practically on top of the autocorrelated beam that the correlation length δ in this region of M51 is significantly smaller than the beam size W ≃ 170 ≃ 63 pc. We also find from these graphs that b²(0) ≃ 0.028; however, we cannot proceed any further in view of the impossibility of determining δ for this data set.

Figures 5 and 6 show the results of the dispersion analyses for the center and the southwest spiral arm of M51, respectively. In both cases we can clearly see a broadening of the integrated magnetized turbulence autocorrelation function beyond that due to the telescope beam (bottom graphs); this is an imprint of the turbulent correlation length intrinsic to the magnetized turbulence. Least-squares fitting a Gaussian function to these reveals that δ = 20 (74 pc) and 17 (62 pc), respectively. It therefore follows that we can provide estimates for the number of turbulent cells probed by the telescope beam and the intrinsic turbulent-to-ordered magnetic energy ratio for these two regions. The results are presented in Table 1, where we set Δ' = 800 pc from Fletcher et al. (2011). We thus find that our results are in good agreement with those of Fletcher et al. (2011), who estimated B_t/B₀ ≃ 1 in the neighborhood of the spiral arms using an RM dispersion analysis. Similar values have been reported in previous analyses for other sources (e.g., see Beck et al. 1999 for NGC 6964). As will be discussed in Section 5, their value of 2δ ≈ 50 pc is consistent with ours given the uncertainty in some of the parameters that enter the analysis.

Table 1. Results for Isotropic Turbulence

	Northeast Arm	Centre	Southwest Arm
δ (pc)a	...	67 ± 7	66 ± 8
Nb	...	13 ± 3	14 ± 4
$\langle \overline{B}_{\mathrm{t}}^{2}\rangle /\langle \overline{B}^{2}\rangle$c	0.028 ± 0.002	0.088 ± 0.026	0.072 ± 0.025
$\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$d	...	1.28 ± 0.29	1.08 ± 0.29
Bt/B0e	...	1.13 ± 0.13	1.04 ± 0.14

Notes. ^aTurbulent correlation length (1'' = 37 pc); from the fit of Equation (22) to the data. ^bNumber of turbulent cells probed by the telescope beam, using Δ' = 800 pc; from Equation (23). ^cMeasured value for the integrated turbulent-to-total magnetic energy ratio, corresponding to $b^2(\ell =0)=\langle B_{\mathrm{t}}^{2}\rangle /[N\langle B_{0}^{2}\rangle +\langle B_{\mathrm{t}}^{2}\rangle ]$ (see Equation (22)). ^dTurbulent-to-ordered magnetic energy ratio, corrected for signal integration; from the fit of Equation (22) to the data. ^eCalculated from the root of $\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$.

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4.2.1. "Hybrid" Model of Anisotropic Turbulence

We now abandon the isotropy assumption and make a first attempt at treating the more general case of anisotropic turbulence. To do so, we define two separate bins of data where the polarization angle differences used in Equation (10) are such that the displacement ${\boldsymbol \ell }$ is either oriented within ±45° of the (plane of the sky component of the) local mean magnetic field (and labeled ${\boldsymbol \ell }_{\Vert }$) or in a direction within ±45° from the axis normal to it (labeled ${\boldsymbol \ell }_{\bot }$). This is illustrated in Figure 7. The orientation of the mean field at a given position is simply approximated by averaging polarization angles contained within a radius of 2''. We then perform two separate dispersion analyses that will allow us to measure differences in the magnetized turbulence correlation lengths parallel and perpendicular to the local field, δ_|| and δ_⊥, respectively. Although this analysis is not based on the more rigorous model given in Equations (19) and (20), it will allow us to look for direct evidence of anisotropic turbulence in the same three regions of M51, as was done in the previous subsection. A more rigorous analysis applied to M51 as a whole will follow in Section 4.2.2.

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Figure 8 shows the results for the northeast spiral arm previously analyzed under the isotropy assumption in Figure 4. For such analysis the dispersion function parallel and perpendicular to the magnetic field must be treated simultaneously. That is, the least-squares fits for the sum of the integrated turbulent-to-total magnetic energy ratio and the ordered component (broken curves in the top two graphs) to the data contained within 6 ⩽ ℓ ⩽ 10 are not independent since they must meet at ℓ_|| = ℓ_⊥ = 0. These fits are thus performed simultaneously. The bottom graph shows the integrated magnetized turbulence autocorrelation functions parallel and perpendicular to the mean field as well as the mean autocorrelated telescope beam, as before. Although we see evidence for anisotropy from the separation of the two autocorrelation functions, we find that the perpendicular function has a width that is narrower than the contribution of the telescope beam, which is impossible. This is most likely due to the fact that our fits (on the top two graphs) are made with data points that are located at too low values for the displacements ℓ_|| and ℓ_⊥ and therefore to some extent fail to cleanly separate the ordered and turbulent dispersion functions (at the expense of the latter; see Section 5). At any rate, we can infer from this analysis that $\langle \overline{B}_{\mathrm{t}}^{2}\rangle /\langle \overline{B}_{0}^{2}\rangle \simeq 0.03$ (in agreement with the isotropic analysis) and that the intrinsic magnetized turbulence autocorrelation appears to be broader along the local magnetic field orientation than perpendicular to it.

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Figures 9 and 10 show the results of the same anisotropic dispersion analysis for the center and southwest spiral arm of M51, respectively. In these two cases, however, we clearly resolve the anisotropy in the turbulence. That is, we observe a separation in the integrated autocorrelation functions (bottom graphs) along the directions parallel and perpendicular to the local mean magnetic field, the former being the broader of the two, which is also consistent with what was observed for the northeast spiral arm in Figure 8. As we will discuss in Section 5 this result is consistent with theory and simulations of incompressible (Goldreich & Sridhar 1995; Cho et al. 2002) and compressible MHD turbulence (Cho & Lazarian 2003; Kowal & Lazarian 2010).

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The level of anisotropy in the turbulence can be gauged through the parallel-to-perpendicular correlation length ratio δ_||/δ_⊥, which is measured to be approximately 1.3 and 1.5 for the center and southwest spiral arm, respectively. To estimate the number of turbulent cells N we use Equations (20) and (21), with α = π/2. We once again find that the turbulent-to-ordered magnetic field strength ratio is significant and hovers around unity with 0.8 ≲ B_t/B₀ ≲ 1.7. A summary of the results is given in Table 2.

Table 2. Results for Anisotropic Turbulence

	Northeast Arm	Center	Southwest Arm
δ∥ (pc)a	25 ± 9	111 ± 7	61 ± 7
δ⊥ (pc)b	...	87 ± 8	41 ± 5
δ∥/δ⊥	...	1.27 ± 0.14	1.48 ± 0.25
Nc	...	7 ± 1	32 ± 8
$\langle \overline{B}_{\mathrm{t}}^{2}\rangle /\langle \overline{B}^{2}\rangle$d	0.028 ± 0.002	0.093 ± 0.003	0.082 ± 0.001
$\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$e	...	0.68 ± 0.10	2.86 ± 0.68
Bt/B0f	...	0.83 ± 0.06	1.69 ± 0.20

Notes. ^aTurbulent correlation length parallel to B₀ (1'' = 37 pc); from the fit of Equation (22) to the data. ^bSame as note a, but perpendicular to B₀. ^cNumber of turbulent cells probed by the telescope beam, using Δ' = 800 pc; from Equation (20). ^dMeasured value for the integrated turbulent-to-total magnetic energy ratio, corresponding to $b^2(\ell =0)=\langle B_{\mathrm{t}}^{2}\rangle /[N\langle B_{0}^{2}\rangle +\langle B_{\mathrm{t}}^{2}\rangle ]$ (see Equation (19)). ^eTurbulent-to-ordered magnetic energy ratio, corrected for signal integration; from the fit of Equation (22) to the data with $\delta =\sqrt{\delta _\parallel \delta _\perp }$. ^fCalculated from the root of $\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$.

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4.2.2. Two-dimensional, Anisotropic Gaussian Turbulence

We finally perform one last anisotropic analysis by taking advantage of the large number of reliable polarization measurements contained in the complete map of M51 shown in Figure 1. That is, we now consider the whole map at once without discriminating between the different regions (as long as p ⩾ 3σ_p). We hope in doing so that the large number of measurements will allow for the characterization of the intrinsic two-dimensional turbulence autocorrelation function, using the Gaussian anisotropic model given in Equations (19) and (20). One would expect that the previously measured anisotropy, quantified with δ_||/δ_⊥, would become more pronounced since we would do away with the cone-averages exemplified in Figure 7.

Figure 11 shows the result of the two-dimensional dispersion analysis. Only one quadrant of the contour plot of the dispersion function is displayed since it is assumed even in directions perpendicular and parallel to the mean magnetic field (i.e., even in powers of ${\boldsymbol \ell }_{1}$ and ${\boldsymbol \ell }_{2}$). Since we must now least-squares fit a two-dimensional surface corresponding to the sum of the turbulent-to-ordered magnetic energy ratio and the ordered component of the dispersion function (i.e., the right-hand side of Equation (13)), it is to be expected that this fitting process will be more challenging than before. This can be verified in Figure 12, where cuts through the two-dimensional dispersion and integrated turbulence autocorrelation functions along directions parallel and perpendicular to the local mean magnetic field are shown; data contained within 7 ⩽ ℓ ⩽ 10 were used to perform the aforementioned least-squares fit to the right-hand side of Equation (13). A comparison of Figure 12 with any such figures stemming from the previous isotropic or anisotropic analyses reveals that our fit to the two-dimensional dispersion function (top and middle graphs) is unable to perfectly match the data, the main consequence of this being the somewhat "ragged" appearance of the integrated two-dimensional turbulence autocorrelation function presented in the bottom graph of Figure 12 (symbols). Nonetheless, it is interesting to note that we once again find the same anisotropy as before, with the result that δ_|| > δ_⊥. We sought to quantify this by performing a two-dimensional (elliptical) Gaussian least-squares fit, using Equation (19), to the integrated turbulence autocorrelation function data. This is shown in the top panel of Figure 13, where a contour plot of the aforementioned Gaussian fit (red) is superposed on that of the integrated two-dimensional turbulence autocorrelation function (black). For this we used the known value of α = 70° for M51 (Tully 1974). The fit is forced to be even in directions perpendicular and parallel to the local mean magnetic field (i.e., even in powers of ${\boldsymbol \ell }_{1}$ and ${\boldsymbol \ell }_{2}$; see Equation (16)), as the dispersion function was also assumed to be. Although this Gaussian fit appears to be reasonable for ℓ ⩽ 4'', it is not expected to be a realistic representation since it is unlikely that the magnetized turbulence is Gaussian in nature in M51. Nonetheless, it allows us to extract a useful approximation to the intrinsic two-dimensional magnetized turbulence autocorrelation function; the resulting function is shown in the bottom panel of Figure 13. As stated earlier, such results are consistent with theory and simulations of incompressible (Goldreich & Sridhar 1995; Cho et al. 2002) and compressible MHD turbulence (Cho & Lazarian 2003; Kowal & Lazarian 2010). The parameters extracted from this anisotropic analysis are also consistent with our previous results for the three independent regions and are presented in Table 3.

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Table 3. Results for Two-dimensional Anisotropic Turbulence (α = 70°)

	All Regions
δ∥ (pc)a	98 ± 5
δ⊥ (pc)b	54 ± 3
δ∥/δ⊥	1.83 ± 0.13
Nc	15 ± 2
$\langle \overline{B}_{\mathrm{t}}^{2}\rangle /\langle \overline{B}^{2}\rangle$d	0.063 ± 0.008
$\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$e	1.02 ± 0.08
Bt/B0f	1.01 ± 0.04

Notes. ^aTurbulent correlation length parallel to B₀ (1'' = 37 pc); from the fit of Equation (19) to the data with α = 70°. ^bSame as note a, but perpendicular to B₀. ^cNumber of turbulent cells probed by the telescope beam, using Δ' = 800 pc; from Equation (20). ^dMeasured integrated turbulent-to-total magnetic energy ratio, corresponding to $b^2(\ell =0)=\langle B_{\mathrm{t}}^{2}\rangle /[N\langle B_{0}^{2}\rangle +\langle B_{\mathrm{t}}^{2}\rangle ]$ (see Equation (19)). ^eTurbulent-to-ordered magnetic energy ratio, corrected for signal integration; from the fit of Equation (19) to the data with α = 70°. ^fCalculated from the root of $\langle B_{\mathrm{t}}^{2}\rangle /\langle B_{0}^{2}\rangle$.

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Our results do not take into account any systematic uncertainties on some of the parameters used to characterize M51. For example, the effective depth Δ' = 800 pc, which comes in for all three cases treated in this section (isotropic, "hybrid" anisotropic, and anisotropic turbulence), enters linearly in the evaluation of the number of turbulent cells N contained in a telescope beam (see Equations (20) and (23)). In turn, the relative strength of the turbulent magnetic field component to the ordered magnetic field scales inversely with N^1/2. An overestimation by a factor of two in Δ', for example, would bring a corresponding underestimate of B_t/B₀ by $\sqrt{2}$. Furthermore, the fully anisotropic model is also dependent on the inclination of the galaxy, which, according to Tully (1974), spans α = 70° ± 5°. Unlike its dependency on Δ' discussed above, the relative level of turbulence is found to be largely insensitive to changes in α. On the other hand, the correlation lengths are somewhat affected by such uncertainties. For example, we find that 100 pc ⩾ ℓ_|| ⩾ 96 pc and 1.92 ⩾ ℓ_||/ℓ_⊥ ⩾ 1.78 when 65° ⩽ α ⩽ 75°.

Although the quality of the data and the high resolution with which they were obtained allowed us to determine some fundamental parameters that characterize magnetized turbulence in M51, we expect that a slightly higher resolution and sampling rate would result in an even more exhaustive analysis. As was shown by Houde et al. (2011) using submillimeter dust polarization data for Galactic molecular clouds, spatial resolutions resulting in smaller telescope beams such that $\delta \gtrsim \sqrt{2}W$ not only allow the determination of the same parameters uncovered by the present analysis but also can potentially reveal the underlying turbulent power spectrum. This is because the beam-broadened turbulent autocorrelation function $b^{2}({\boldsymbol \ell })$ is related to the turbulent power spectrum b²(k_v) through a simple Fourier transform. It is then found that

$$\begin{eqnarray} b^{2}\left(\mathbf {k}_{v}\right)&=&\frac{1}{\langle \overline{B}^{2}\rangle }\left\Vert H\left(\mathbf {k}_{v}\right)\right\Vert ^{2}\nonumber\\ &&\times \left[\frac{1}{2\pi }\int \mathcal {R}_{\mathrm{t}}\left(\mathbf {k}_{v},k_{u}\right)\mathrm{sinc}^{2}\left(\frac{k_{u}\Delta }{2}\right)dk_{u}\right],\qquad \end{eqnarray} \tag{ 29 }$$

with $\mathcal {R}_{\mathrm{t}}\left(\mathbf {k}_{v},k_{u}\right)$ and H(k_v) the Fourier transforms of the intrinsic turbulent autocorrelation function (i.e., not beam-broadened) and the telescope beam, respectively (see the Appendix and Houde et al. 2011 for a detailed discussion). It follows that beams of smaller spatial extent than the intrinsic turbulent autocorrelation function will have a broader spectral coverage that will reduce their filtering effect on the power spectrum. It then becomes possible to effectively invert Equation (29) to reveal the underlying power spectrum (through some "deconvolution" techniques, for example). In such cases, it is not necessary to assume any model for the turbulence, such as the Gaussian form used in our analysis. The measured turbulent power spectrum could thus be modeled directly from the data and compared to candidate theories for magnetized turbulence.

As can be seen from our results for $b^{2}({\boldsymbol \ell })$ in Figures 4–6, 8–10, and 12, however, the contribution of the correlation length to the width of the beam-broadened turbulent autocorrelation function (approximately gauged through the ratio δ²/(δ² + 2W²); see Equations (20) and (23)) is typically modest, implying that the spectral filtering of the telescope beam is too severe to recover the intrinsic turbulent power spectrum. But even a relatively modest increase in spatial resolution, e.g., by a factor of a few, could allow us to recover the power spectrum in future observations.

Another negative impact of a larger telescope beam and its broadening of the autocorrelation function $b^{2}({\boldsymbol \ell })$ is that it renders more difficult the separation of the small- and large-scale components present in the dispersion function. For M51 this means that the scale of the turbulence, quantified with δ, can get "mixed up" with the larger scale of the spiral structure through its artificial broadening to δ² + 2W² caused by the beam. As alluded to in Section 4.1, this may be a reason why we were unable to see any contribution from δ to the width of $b^{2}({\boldsymbol \ell })$ in our analyses of the northeast arm (see Figures 4 and 8). More precisely, we were unable to cleanly separate the large from the small scale using our Taylor expansions, i.e., Equations (16) and (24). This is probably also true, but to a lesser extent, for the other two regions studied, as can be seen from the absence of significant "tails" for ℓ ≳ 5'' in $b^{2}({\boldsymbol \ell })$. An increase in spatial resolution would resolve this issue, which is likely to bring some errors in our determination of the correlation length scales and turbulent-to-total energy ratios. We expect this error to be small, but it is not possible to quantify it at this point.

Finally, we wish to once again emphasize that the choice of a Gaussian turbulence model is unlikely to be realistic for M51. But in view of the aforementioned impossibility to uncover the underlying turbulent power spectrum because of the significant spectral beam filtering, this model, which can be solved analytically, allows us to quantify key parameters that characterize magnetized turbulence. Furthermore, our more comprehensive model for anisotropic Gaussian turbulence defined with Equations (19) and (20) implicitly assumes that the N ellipsoid turbulent cells contained in the column of gas probed by the telescope beam have the same spatial orientation in relation to the local magnetic field. This is clearly unlikely to be true across the beam (FWHM ≃ 148 pc), or through the thickness (∼800 pc) and the extent of the studied regions on the galactic disk. It is therefore more realistic to view the correlation lengths δ_|| and δ_⊥ as some averages representative for magnetized turbulence in M51.