Spectrum and Anisotropy of Turbulence from Multi-Frequency Measurement of Synchrotron Polarization

Spectrum and Anisotropy of Turbulence from Multi-Frequency Measurement of Synchrotron Polarization

Abstract

We consider turbulent synchrotron emitting media that also exhibits Faraday rotation and provide a statistical description of synchrotron polarization fluctuations. In particular, we consider these fluctuations as a function of the spatial separation of the direction of measurements and as a function of wavelength for the same line-of-sight. On the basis of our general analytical approach, we introduce several measures that can be used to obtain the spectral slopes and correlation scales of both the underlying magnetic turbulence responsible for emission and the spectrum of the Faraday rotation fluctuations. We show the synergetic nature of these measures and discuss how the study can be performed using sparsely sampled interferometric data. We also discuss how additional characteristics of turbulence can be obtained, including the turbulence anisotropy, the three dimensional direction of the mean magnetic field. We consider both cases when the synchrotron emission and Faraday rotation regions coincide and when they are spatially separated. Appealing to our earlier study in Lazarian & Pogosyan (2012) we explain that our new results are applicable to a wide range of spectral indexes of relativistic electrons responsible for synchrotron emission. We expect wide application of our techniques both with existing synchrotron data sets as well as with big forthcoming data sets from LOFAR and SKA.

Subject headings:
turbulence – ISM: general, structure – MHD – radio lines: ISM.

1. Introduction

Radio observations of synchrotron emission is an important source of information about astrophysical magnetic fields (see Ginzburg 1981). Diffuse synchrotron emission is observed throughout the ISM and the ICM, as well as in the lobes of radio galaxies (e.g. Westerhout et al. 1962, Carilli et al. 1994, Reich et al. 2001, Wolleben et al. 2006, Haverkorn et al. 2006, Clarke & Enßlin 2006, Schnitzeler et al. 2007, Laing et al. 2008). Observations testify that turbulence is ubiquitous in astrophysics (see Armstrong et al. 1994, Lazarian 2009, Chepurnov & Lazarian 2010). As most astrophysical environments are magnetized and relativistic electrons are in most cases are present, the turbulence results in synchrotron fluctuations, which carry important information, but at the same time, interfere with attempts to measure Cosmic Microwave Background (CMB) with high precision. In addition, synchrotron fluctuations present an impediment for studying fluctuations of atomic hydrogen distribution in the early Universe. The latter has become a direction of intensive discussion recently (see Loeb & Zaldarriaga 2004, Pen et al. 2008, Loeb & Wyithe 2008, Liu, et al. 2009, Ferdinandez et al. 2014). If we know the spectrum of underlying turbulence, these fluctuations can be separated from the CMB signal (see Cho & Lazarian 2010). Better cleaning of the CMB maps is particularly important while analyzing polarized radiation in the search of enigmatic B-modes produced by gravitational waves in the Early Universe. The polarized synchrotron present a important foreground that such studies have to deal with.

A number of earlier studies tried to utilize synchrotron intensity fluctuation to obtain the spectrum and anisotropies of underlying magnetic turbulence (see Getmantsev 1959, Chibisov & Ptuskin 1981, Lazarian & Shutenkov 1990, Lazarian & Chibisov 1991, Chepurnov 1998). In addition, polarization fluctuations were proposed to address the complex issues of measuring magnetic field helicity (Waelkens et al. 2009, Junklewitz & Enßlin 2011). The serious limitation of all the above studies was that it was done for a single spectral index of relativistic electrons that allowed to write the synchrotron emissivity not as generally applicable , where is a perpendicular component of magnetic field and depends on the spectrum of emitting electrons, but only for . In a way, the studies were limited to a single point of a parameter space.

The above deficiency was addressed in our recent study (Lazarian & Pogosyan 2012, henceforth LP12) where we provided the statistical description synchrotron fluctuations for an arbitrary index corresponding to the actual energy distribution of relativistic electrons. Very importantly, rather than taking a usual ad hoc and incorrect assumption that magnetic field in turbulent media can be presented as , i.e. as a superposition of a regular magnetic field and isotropic stochastic magnetic field we used the model of the turbulence for realistic anisotropic magnetic turbulence which corresponds to theoretical expectations (Goldreich & Sridhar 1995, see Brandenburg & Lazarian 2013 for a review) and supported by numerical simulations in Cho & Lazarian (2003) and Kowal & Lazarian (2010). These two advances brought the studies of magnetic turbulence using synchrotron to a new stage. Testing of the expressions obtained in LP12 has been performed with synthetic data in Heron et al. (2015).

The study in LP12 was mostly dealing with synchrotron intensities. Present day telescopes present opportunities to get detailed maps of polarization. In fact, the Position-Position Frequency (PPF) data cubes are getting available with high spatial and spectral resolution. Such data cubes present a good opportunity for studying magnetic turbulence, provided that the description of the relation of the synchrotron polarization statistics and the statistics of the underlying magnetic fields is available.

We also derived correlations of synchrotron polarization but did not deal with the important effect of Faraday rotation of the polarized radiation that arises as radiation propagates in the magnetized plasmas. The angle of polarized radiation rotation is proportional to the , where is the wavelength of the radiation, the integration is done along the line-of-sight, while is the component of magnetic field along the line-of-sight and is the density of electrons in thermal plasmas. In terms of synchrotron polarization the effects of Faraday rotation decrease the polarization and introduce additional fluctuations arising from both fluctuations of parallel component of magnetic field as well as electron density. Therefore ignoring the Faraday rotation while dealing with polarized intensity can only be justified for sufficiently short wavelengths.

Faraday rotation measurements have been extensively used for studying regular and fluctuating components of magnetic fields using radio emission of external sources, e.g. point radio sources. In addition, the effect of Faraday depolarization was used to probe magnetic field at different distances from the observer. Indeed, by changing the wavelength of the radiation one can vary the contribution of polarized synchrotron emission from the regions at different distances along the line of sight. Indeed, using longer the wavelengths one can sample emission from closer emitting volumes. In fact, our present study shows that the criterium for sampling the turbulence with synchrotron polarization is different from the one for intensity studies.

More recently there have been renewed interest to getting detailed maps of diffuse synchrotron emission that experiences Faraday rotation within the emitting volume (Beck et al. 2013). These new studies provide Position-Position-Frequency (PPF) data cubes which exhibit an intricate structure of fluctuations that arise from both the fluctuations of magnetic field and the fluctuations in the Faraday measure. Our paper opens new ways of using these PPF data cubes for studying turbulence by providing the analytical description of fluctuations in these data cubes. In particular, we below we describe techniques for studying polarization fluctuations at a given wavelength as a function of spatial separation. We also explore the potential of the dispersion of the polarized signal when it is studies as a function of frequency. The first technique with separated lines-of-sight some has similarities to the Velocity Channel Analysis (VCA) technique that employs spectral Doppler-shifted lines to study velocity turbulence introduced by us some time ago (Lazarian & Pogosyan 2000, 2004), while the studies of the frequency dependence of the dispersion has some similarities to the Velocity Correlation Spectrum (VCS) technique that was suggested by us later, i.e. in Lazarian & Pogosyan (2006, 2008). Both VCA and VCS make use of Position Position Velocity (PPV) spectral data which is an analog of PPF in the present analysis. Both techniques have been successfully employed to study velocity turbulence data (see Lazarian 2009 for a review). In analogy with these techniques we term the technique based on the analysis of spatial fluctuations of polarization Polarization Spatial Analysis (PSA), which is an analog of VCA for velocity data cubes, and on the analysis of frequency dependence of the polarization variance, Polarization Variance Analysis (PVA), which is an analog of VCS. In view of the revival of interest to the Faraday rotation synthesis technique (Brentjens & Bruyn 2005)1 we discuss how to use this technique within the PVA approach.

We would like to stress that there are two major advantages of using different techniques for studying turbulence. First of all, they measure different components of turbulent cascade. For instance, it is advantageous to measure independently both the spectrum of velocity and the spectrum of magnetic field. This, for instance, is possible combining VCA and PSA measurements for the same media. Second, combining different techniques it is possible to study whether properties of magnetic turbulence in different media, e.g. to explore the continuity the turbulent cascade in different phases of the ISM and test whether the cascade is these phases corresponds to the Big Power Law in the Sky (Armstrong et al. 1994, Chepurnov & Lazarian 2010).

The present paper follows the pattern of our earlier publications on studying spectrum of turbulence from observations (see Lazarian & Pogosyan 2000, 2004, 2006). We obtain general expressions, but are focused on obtaining the asymptotic regimes for turbulence statistics. While, as we discuss in the paper, these asymptotic expressions are informative, the full expressions may have advantages for the analysis of observational data as was shown in Chepurnov et al. (2010). Indeed, in the latter paper, apart from the spectral slope, the injection scale of turbulence and the turbulence Mach number were obtained. We expect that additional measures, e.g. injection scale and variations of turbulence intensities along different directions, can be available.

In what follows, we discuss the basic statistics of MHD turbulence that we seek to obtain using synchrotron polarization fluctuations in §2, introduce the measures and explore the properties of synchrotron statistics in §3, introduce the correlations of polarization at different spatial points, i.e. PSA technique in §4 and discuss the statistics of measures along the same line of sight in §5. In §6 we discuss additional measures, including spatial correlations of the derivatives of polarization wrt to wavelength, and application to interferometers. On the basis of our formalism we formulate new techniques of turbulence study with synchrotron polarization in §7 and provide the discussion of our results and comparison of the different ways to study magnetic turbulence in §8. The latter section may be the most useful for researchers interested in practical application of the techniques. Our findings are summarized in §9.

2. Spectrum of MHD turbulence

2.1. Importance

This paper deals with developing the technique for obtaining properties of magnetic turbulence from observations. Turbulence in magnetized plasmas plays a crucial role for the processes of cosmic ray propagation (see Schlickeiser 2003, Longair 2011), star formation (see Elmegreen & Scalo 2005, McKee & Ostriker 2007), heat transfer in magnetized plasmas (see Narayan & Medvedev 2001, Lazarian 2006), magnetic reconnection (see Lazarian & Vishniac 1999, Kowal et al. 2009, Eyink et al. 2011, see review Lazarian et al. 2015 and ref. therein). The advantage of statistical description of turbulence is that it allows to reveal regular features within chaotic picture of turbulent fluctuations. Recent reviews on MHD turbulence include Brandenburg & Lazarian (2013) and Beresnyak & Lazarian (2015).

In this paper we use the statistical description of turbulence and claim that it is an adequate and concise way to characterize many essential properties of interstellar turbulence. Indeed, while turbulence is an extremely complex chaotic non-linear phenomenon, it allows for a remarkably simple statistical description (see Biskamp 2003). If the injections and sinks of the energy are correctly identified, we can describe turbulence for arbitrary and . The simplest description of the complex spatial variations of any physical variable, , is related to the amount of change of between points separated by a chosen displacement , averaged over the entire volume of interest. Usually the result is given in terms of the Fourier transform of this average, with the displacement being replaced by the wavenumber parallel to and . For example, for isotropic turbulence the kinetic energy spectrum, , characterizes how much energy resides at the interval . At some large scale (i.e., small ), one expects to observe features reflecting energy injection. At small scales, energy dissipation should be seen. Between these two scales we expect to see a self-similar power-law scaling reflecting the process of non-linear energy transfer, which for Kolmogorov turbulence results in the famous relation. However, from the point of view of astrophysics both the injection scale or multiple injection scales (see Yoo & Cho 2014), which are also expected in interstellar medium, as well as dissipation scale are of great interest.

For our statistical description, we need to know what are expected properties of MHD turbulence, e.g. to know which range of spectral indexes we should consider deriving our asymptotic solutions and what other effects, e.g. related to anisotropy we should consider. Apparently, MHD turbulence is more complex than the hydrodynamical one. Magnetic field defines the chosen direction of anisotropy (Montgomery & Turner 1981, Shebalin, Matthaeus & Montgomery 1983, Higdon 1984). For small scale motions this is true even in the absence of the mean magnetic field in the system. In this situation the magnetic field of large eddies defines the direction of anisotropy for smaller eddies. This observation brings us to the notion of local system of reference, which is one of the major pillows of the modern theory of MHD turbulence 2. Therefore a correct formulation of the theory requires wavelet description (see Kowal & Lazarian 2010). Indeed, a customary description of anisotropic turbulence using parallel and perpendicular wavenumbers assumes that the direction is fixed in space. In this situation, however, the turbulence loses its universality in the sense that, for instance, the critical balance condition of the widely accepted model incompressible MHD turbulence (Goldreich & Sridhar 1995, henceforth GS95) which expresses the equality of the time of wave transfer along the magnetic field lines and the eddy turnover time is not satisfied.

The existence of the local scale dependent anisotropy does not mean that describing synchrotron fluctuations as they are seen by the observer one should use the GS95 description. In fact, the local system of reference in most cases is not available to the observer who measures turbulence in the system of mean magnetic field. In such a system, the anisotropy is also present, but it is independent of scale (see the discussion in Cho et al. 2002) and the perpendicular Alfvénic perturbations absolutely dominate the spectrum of fluctuations from Alfvénic turbulence3. Therefore the expected spectrum from GS95 is Kolmogorov-type with (see Cho, Lazarian & Vishniac 2002). The tensor for turbulence with the scale-independent anisotropy in the global system of reference is given in LP12 and discussed for realizations of compressible MHD turbulence.

It is important to understand that, contrary to the entrenched in the community notion, MHD compressible turbulence can be described in simple terms. Numerical studies support the notion that the turbulence of fast modes develops mostly on its own and corresponds to the spectrum of acoustic turbulence, i.e. (Cho & Lazarian 2002, 2003, Kowal & Lazarian 2010). The slow mode fluctuations are expected to follow the spectrum of the Alfvén mode (GS95, Lithwick & Goldreich 2001, Cho & Lazarian 2002). Shocks, which are inevitable for highly supersonic turbulence, are expected to induce steeper turbulence with spectrum . Steepening of observed velocity fluctuations was reported in Padoan et al. (2009) and Chepurnov et al. (2010). These groups used for their studies the VCS technique ( Lazarian & Pogosyan 2000, 2006)4. It is interesting to know whether magnetic field fluctuations will also demonstrate the corresponding steepening. Thus the development of the corresponding techniques is important from the point of view of establishing the correct spectral slope of magnetic fluctuations in MHD turbulence.

We should mention that the theory of MHD turbulence is a developing field with its ongoing debates5. While we feel that among all the existing models the GS95 provides the best correspondence to the existing numerical and observational data (see Beresnyak & Lazarian 2010, Chepurnov & Lazarian 2010, Beresnyak 2011, 2013), the issue of the actual nature of MHD turbulence requires further research for which observational techniques will play important role. Indeed, the inertial range provided by astrophysical turbulence is much larger than that of numerical simulations. Thus the synchrotron studies may be very useful for getting insight into the nature of MHD turbulence.

The issues of the spectrum of magnetic fluctuations are important for turbulence theory and its implications, e.g. cosmic ray and heat propagation (see Brandenburg & Lazarian 2013). However, for describing many astrophysical processes, the issues of intensity of turbulent fluctuations, degree of turbulence compressibility, degree of magnetization of turbulence, injection and dissipation scales of turbulence are essential. In particular, interstellar medium is a very complex system and therefore one cannot be a priori sure that simple models of isothermal turbulence can be directly applicable to its parts not to speak about the interstellar medium in general. Additional physics, as well as multiple sources of energy injection can affect the shape of the turbulence spectrum and this makes obtaining the turbulence spectrum from observations essential. Within our treatment we do not assume that the spectral index of magnetic fluctuations is , but treat it as a parameter that should be established from observations.

All in all, the techniques that we propose in this paper are important for establishing (a) sources and sinks of turbulent energy, (b) the distribution of turbulence in Galaxy and other astrophysical objects, (c) clarification of the properties of compressible MHD turbulence.

2.2. Shallow and Steep spectra

Magnetic fields that are sampled by synchrotron polarization are turbulent. The prediction for the magnetic turbulence within the GS95 theory is the Kolmogorov spectrum, which is in terms of 3D spectrum corresponds to . Note, when direction averaged, the spectrum gets another , which provides the usual value , which is a more common reference to the Kolmogorov spectrum. We, however, will use in this paper, similar to our other publications (see LP00) the 3D spectra.

At the same time, in some cases the spectrum of turbulence may be more shallow. This, for instance, corresponds to the magnetic field in the viscosity-damped turbulence, which is the high regime of turbulence in the fluid where the ratio of viscosity to resistivity is much larger than one (Cho, Lazarian & Vishniac 2002, 2003, Lazarian, Vishniac & Cho 2004). There it was shown that the one-dimensional spectrum can be which corresponds to the 3D spectrum of magnetic field of . The spectrum of corresponded to the border-line spectrum of turbulence between the steep and shallow regimes (see LP00) with shallow regime corresponding to most of the turbulent energy being at small scales, while the steep regime corresponds to most of the energy being at large scales. We are not aware of any expectations of the turbulent magnetic spectrum with the index more shallow than this borderline value. Therefore we shall consider only steep magnetic field spectra.

The fluctuations of synchrotron polarization are affected not only by magnetic perturbations, but also by Faraday rotation fluctuations that are proportional to the product of the parallel to the line of sight component of magnetic field and density. Shallow and steep spectra are, however, a confirmed reality of the spectrum of density in MHD turbulence (see Beresnyak, Lazarian & Cho 2005, Kowal, Lazarian & Beresnyak 2007). The spectrum gets shallow as turbulence gets supersonic and more density fluctuations are localized in corrugated structures of shock-compressed gas. This regime in case of interstellar medium is relevant to cold phases of the medium, e.g. to molecular clouds (see Draine & Lazarian 1998 for the list of the idealized phases), but localized ionization sources may result also in a shallow spectrum of random density. The warm interstellar medium responsible for the synchrotron radiation corresponds to transonic turbulence with Mach number of the order of unity (see Burkhart, Lazarian & Gaensler 2010). Nevertheless, Faraday rotation may take place in any media between the observer and the region of region of synchrotron emission, which may include cold high Mach number turbulence. Therefore, in our treatment we consider both shallow and steep spectra of turbulence.

The anisotropy of the density correlations depends on the sonic Mach number. In MHD turbulence at low Mach numbers density follows the velocity scaling and exhibit GS95 type anisotropy, while at large Mach numbers the density gets isotropic (Cho & Lazarian 2003, Beresnyak et al. 2005, Kowal et al. 2007). In this paper we will use only very general spectral properties of the RM correlations, while we continue elsewhere our studies of anisotropies (see Lazarian, Esquivel & Pogosyan 2001, Esquivel & Lazarian 2005, 2009, Lazarian & Pogosyan 2012).

There can be processes that create correlations between the magnetic field and the density of thermal electrons. Such correlations are possible for shocked regions. We consider such correlations. However, as we discuss further, the correlations of the vector, i.e. the line of sight component of magnetic field, and a scalar, i.e. thermal electron density are always zero. Our assumption within the paper is that the squared perpendicular component of magnetic field and the density of cosmic rays are not correlated, as observations are indicative of more isotropic distribution of cosmic electrons. However, this is not a crucial assumption for our work, as we discuss below.

2.3. Statistical description of magnetic fields

MHD turbulence is more complex than the hydrodynamical one. Magnetic field defines the preferred direction (Montgomery & Turner 1981, Shebalin et al. 1983, Higdon 1984) and the statistical properties of magnetized turbulence are anisotropic. For small scale motions this is true even in the absence of the mean magnetic field in the system. The local system of reference that, as we discussed earlier, is fundamental for modern theory of Alfvénic turbulence (see Lazarian & Vishniac 1999, Cho & Vishniac 2000, Maron & Godreich 2001) is not accessible to an observer who deals with projection of magnetic fields from the volume to the pictorial plane. The projection effects inevitably mask the actual direction of magnetic field within individual eddies along the line-of-sight. As the observer maps the projected magnetic field in the global reference frame, e.g. system of reference of the mean field, the anisotropy of eddies becomes scale-independent and the degree of anisotropy gets determined by the anisotropy of the largest eddies which projections are mapped (Cho et al. 2002, Esquivel & Lazarian 2005).

In the presence of the mean magnetic field in the volume under study, an observer will see anisotropic turbulence, where statistical properties of magnetic field differ in the directions orthogonal and parallel to the mean magnetic field that defines the symmetry axis. The description of axisymmetric turbulence was given by Batchelor (1946), Chandrasekhar (1950) and later Matthaeus & Smith (1981) and Oughton (1997). This is the description that was employed in our earlier paper (LP12) for the description of anisotropic fluctuations of the synchrotron emission. The index-symmetric part of the correlation tensor can then be presented in the following form:

(1)

where the separation vector has the magnitude and the direction specified by the unit vector . The direction of the symmetry axis set by the mean magnetic field is given by the unit vector and . The magnetic field correlation tensor may also have antisymmetric, helical part (see Appendix A.3). This part was not considered in LP12 as its contribution to synchrotron intensity fluctuations may be shown to be small. However, as we will discuss in the present paper, this part provides a very distinct response within the polarization studies that we discuss. Therefore, while we do not dwell upon this part within this paper, we would like to stress, that, as we discuss below, with polarization correlations it is feasible to detect the helical part of the tensor. Such a detection would be very important understanding of many problems of magnetic dynamo.

The structure function of the field has the same representation

(2)

with coefficients etc. In case of power-law spectra, one can use either the correlation function or the structure function, depending on the spectral slope. For the shallow spectrum it is natural to use the correlation function, while for steep one the structure function (see e.g. Lazarian & Pogosyan 2004). In this case the structure function coefficients can be thought of as renormalized correlation coefficients.

Magnetic field spectrum can be obtained by a Fourier transform of correlation and structure functions of magnetic fields (see Monin & Yaglom 1975). First we consider the case when statistics of the magnetic field is isotropic, which may, for instance, correspond to the super-Alfvénic turbulence, i.e. for the turbulence with the injection velocity much in excess of the Alfvénic one. The structure tensor of a Gaussian isotropic vector field, a special case of Eq. (2), is usually written in the form

(3)

where and are structure functions that describe, respectively, the correlation of the vector components normal and orthogonal to point separation . In case of solenoidal vector field, in particular the magnetic field, two structure functions are related by

(4)

which in the regime of the power-law behaviour leads to both functions being proportional to each other .

From the point of view of observations, our considerations in §2.1 suggest that the slope of the spectrum is not expected to change when the measurements are done in the system of reference of the mean magnetic field, which is the only system of reference available to the observer. Therefore, if we are interested only in the crude description of turbulence, which includes the spectral slope and approximate measures of the injection/dissipation scales, to use the isotropic description of turbulence. This was the description that we adopted in our earlier papers dealing with velocity spectra (Lazarian & Pogosyan 2000, 2004, 2006, 2008), but it is different from the description adopted for describing anisotropy of synchrotron fluctuations in LP12.

Potentially, our treatment of synchrotron fluctuations may be done for the case of relativistic electrons correlating with the strength of squared component of the magnetic field perpendicular to the line of sight. Then the correlated quantities should be not , but .

3. Statistical description of the polarization signal from emitting distributed medium

3.1. Basic definitions

To characterize fluctuations of the synchrotron polarization one can use different combinations of Stokes parameters (see our discussion in LV12). In this paper we shall focus on the complex measure of the linear polarization

(5)

Other combinations may have their advantages and should be discussed elsewhere.

In case of an extended synchrotron sources, the polarization of the synchrotron emission at the source is characterized by the polarized intensity density , where marks the two-dimensional position of the source on a sky and is a line-of-sight distance. The polarized intensity detected by an observer in the direction at wavelength

(6)

is a line-of-sight integral over emission at the sources modified by the Faraday rotation of the polarization plane (see Brentjens & Bruyn 2005). Here is the extent of the source along the line-of-sight and the Faraday rotation measure (RM) is given by

(7)

where is the density of thermal electrons in , is the strength of the parallel to the line-of-sight component of magnetic field in Gauss, and the radial distance is in .

In general, synchrotron emission intensity depends on the wavelength , as discussed in Appendix A.1. In this study we consider the polarization measure in which this dependence has been scaled out. This can be accomplished, for instance, by determining the mean wavelength scaling from the total intensity measurements. With such rescaling, polarization at the source is treated as wavelength independent, while the observed contains the residual wavelength dependence due to Faraday rotation only.

While Faraday rotation reflects the line-of-sight component of the magnetic field, the intrinsic synchrotron emission at the source is determined locally by its transverse, , part (see § A.1). Clearly both polarization at the source and Faraday rotation influence the observed signal. This makes the analysis of polarization fluctuations much more complicated compared to pure intensity that we studied in LP12.

3.2. Correlations of the Faraday rotation measure

Let us write the Faraday rotation measure as

(8)

where denotes the RM per unit length along the line-of-sight.

Electron density , the line-of-sight component of the magnetic field and, correspondingly, are spatially local quantities which we assume to be statistically homogeneous. Both density and magnetic field can be presented as a sum of the mean value and a fluctuation. Therefore, the average value of the RM linear density is

(9)

where and are zero mean fluctuations of the electron density and the magnetic field. Note that these fluctuations taken at the same point are generally uncorrelated due to the vector nature of the magnetic field and the symmetry under the local reversal of its direction. Thus, it is just the product of the mean that defines the mean RM density.

The variance of fluctuations in Faraday RM density is

(10)

Correlation properties of the RM density at two points in space are described by the correlation or structure functions

(11)
(12)

Assumption of statistical homogeneity of the medium is reflected in the fact that and depend only on the coordinate difference between the two positions. In what follows we illustrate our treatment of the problem using power-law correlation model

(13)
(14)

where is the scaling slope and is the correlation length of RM density, and . These expressions are not the most general expressions for the correlations in the magnetic field (compare with Eqs. (1),(2)), but they are adequate if we discuss measuring the scaling properties. Similar models were employed e.g. in Lazarian & Pogosyan (2006). In several regimes obtained in this paper, models with produce similar asymptotical behaviour as model, thus we will frequently use the notation

(15)

The total RM , in contrast to RM density, is not a local but an integral quantity. Its behaviour in coordinate is not statistically homogeneous, rather it depends on the length of the integration path, which becomes a critical feature in our studies. At the same time the behaviour transverse to the line-of-sight remains statistically homogeneous. In particular, the mean RM is proportional to the distance along the line-of-sight from the emitter at to the observer

(16)

but does not depend on X.

Due to inhomogeneity in , one has to separate the mean Faraday RM and its fluctuations , even when studying the structure functions (normally, insensitive to the mean). The variance of the fluctuation is

(17)

while the structure function for fluctuations in the RM we define as

Note the non-standard factor in the definition, which we introduced to simplify the subsequent equations.

Let us study geometrical properties of the above structure function in plane. In Figure 1 its behaviour is demonstrated for the power law model given by Eq. (14) and a fixed .

Figure 1.— Left: profile of the structure function of RM fluctuations in and space for , . Right: cross-sections of the left panel surface at fixed (from lower to upper curve, respectively).

It demonstrates a valley shape with the bottom at line that slowly rises with and steep walls in direction. Main conclusion is that the dependence in close to the local minima of is primarily simply quadratic due to geometrical reason of different integration lengths.

Analytic considerations in Appendix B suggest the following quadratic approximation

(19)

where along the bottom of the valley

(20)

and the curvature in is

(21)

The residual dependence of coefficients on is one more manifestation of inhomogeneity of statistical measures in direction. Figure 2 shows that as a function of is initially quadratic but then becomes linear at larger . However, for both low and high asymptotics have dependence

(22)
(23)

while for at low there is no dependence on and at high scaling is inverted

(24)
(25)
(26)

In the subsequent sections we shall use these results extensively to analyze the asymptotics for synchrotron polarization structure functions.

Figure 2.— valley bottom profile Left: as the function of for several in the range from to (from bottom to the top, dashed , solid with the thickest curve corresponding to ), Right: as the function of for several in the range from to (from bottom to the top, dashed , solid with the thickest curve corresponding to ),

Important special case is that of a single line-of-sight, . The approximation Eq. (19) is reduced to , demonstrating that within its range of validity Faraday effect is dominated by a purely geometrical factor, insensitive to correlations of quantity. We can study this case in more detail using the exact formula

(27)

which explicitly shows that the correlated terms are further and further subdominant to the first geometrical one as decreases. The exact criterium is that the quadratic geometrical term dominates at , where is the correlation length of the product of the electron density fluctuations and the parallel component of the magnetic field. At the Faraday structure function tends to another, linear, universal behaviour that represents a random walk in the value of the Faraday RM accumulated over different intervals of the line-of-sight. This tendency to random walk at large is also seen in Figure 2 in the general case of separated lines-of-sight of greatly non-equal lengths. Transition for to behaviour depends on the details of correlation of the RM density. Note that statistics of RM fluctuations are homogeneous along a single line-of-sight.

3.3. Correlation of the synchrotron polarization at the source

Magnetic field at the source can be decomposed into regular and random components. The regular component provides mean polarization, while the random component provides fluctuations of polarization. Our study is mostly devoted to the statistical description of the random component of polarization as it is measured by the observer being averaged along the line-of-sight and rotated through Faraday rotation, although the effect of the regular magnetic field is also discussed where appropriate.

The polarization at the source provides an initial polarization in our study, which is described by polarized intensity density denoted as in this paper. As we discuss in Appendix A, polarized emissivity depends on the transverse to the line of observation magnetic field and the wavelength , . In this paper we shall consider observational measures in which the underlying dependence on the wavelength is scaled out. This can be accomplished, for instance, by measuring the wavelength dependence of the mean intensity of the synchrotron radiation. Thus we consider that is wavelength independent.

Fractional power dependence on the magnetic field is the same for the intensity and polarized intensity. This allows us to apply the results of LP12 to the polarized intensity and express the fluctuation of polarization at the source for an arbitrary index using the fluctuations of magnetic field obtained for .

(28)

Here is a factor given by the ratio of the variances which dependence on is similar to that for the intensity correlations discussed in LP12. In isotropic turbulence, average polarization is zero, unless there is a uniform average component to the magnetic field. If the turbulence is anisotropic, difference between the variances of different components of the magnetic field may contribute to the mean polarization as well.

In terms of and Stokes parameters in the observers frame, the correlation between polarizations at two sources is, in general,

(29)

Two parts of the correlation, real and imaginary, describe correlation invariants with respect to rotation of the observers frame. Explicit expressions via the magnetic field components for are given in Appendix A.3. The real part is the trace of the polarization correlation matrix (see LP12) and imaginary part is the antisymmetric contribution to the correlation. For synchrotron signal, the latter one can be present only if the magnetic field correlation tensor has index antisymmetric part, which, in general, is related to the helical correlations (Oughton et al. 1997, also see Appendix A.3). Although we shall not consider these antisymmetric correlations in this paper, we stress that the very detection of helical correlations will be a major discovery.

The main parameters of the correlation function of the polarization at the source is the correlation length and the characteristic scaling slope of its fluctuations, and the relative contribution from the mean and fluctuating polarization. While our subsequent analysis does not rely on a specific shape of , for numerical illustrations we adopt a saturated isotropic power law similar to Eq. (14)

(30)

The mean polarization dominates on all scales if , in which case the functional form for intrinsic correlation effectively corresponds to the infinite correlation length . Otherwise, the mean contribution can be neglected for separations which covers all the separations within the correlation length of intrinsic fluctuations if .

3.4. Correlation of the observed polarization

The observed polarization is subject to both integration along the line-of-sight and to the Faraday rotation. As a result, the invariant over frame rotation measure of the observed correlation is

(31)

We shall consider all quantities to be statistically homogeneous in real space, however we do not have homogeneity property in the square-of-wavelength “direction” . With the mean effect separated, Eq. (31) becomes

(32)

The formula represents the general expression for correlation function in PPF (position-position-frequency) data cube and is the starting point for our further study.

Observable correlation function in terms of the Stokes parameters is split again into real and imaginary parts that are separately invariant with respect to frame rotation

(33)

In this paper we focus on the symmetric real part which is easier to determine and which carries the most straightforward information about the magnetized turbulent medium. Antisymmetric imaginary part potentially reflects helical correlations of the magnetic field, but, as will be shown, can be also generated by Faraday rotation in the anisotropic MHD turbulence. Its measurement in data provides valuable observational constraints on such contributions 6.

Let us summarize the parameters and scales of the problem that determine the observed synchrotron polarization correlations, subject to Faraday rotation. Long list of parameters and notations is summarized in Table 1, however not all of them determine the results independently. Our problem contains the correlation length of the rotation measure , the correlation length of the transverse magnetic field , the line-of-sight size of the emitting region and the separation between two line-of-sight over which we correlate two polarization polarization measurements. As well we have scaling slopes for RM measure and intrinsic correlations , amplitude of fluctuations in RM and intrinsic correlations , possible mean rotation and mean intrinsic polarization , and the wavelength of observations . Among them, is trivial to account for separately, is a simple coefficient the signal is proportional to, while the magnitude of RM, either random or mean together with observation wavelength determine the characteristic distance (see next section for exact definition) over which Faraday effect rotates the polarization by one radian. As the final tally, we have five scales, , , , , and two scaling slopes and .

Parameter Meaning First appearance
Scales:
wavelength of observations Eq.   6
line-of-sight extent of the emitting region Eq.   6
separation between lines-of-sight Eq. 14
correlation length for Faraday Rotation Measure density Eq. 14
correlation length for polarization at the source Eq. 30
distance of one rad revolution by random Faraday rotation Eq. 37
distance of one rad revolution by mean Faraday rotation Eq. 39
the smallest of and Eq. 38
Spectral indexes:
Correlation index for Faraday RM density Eq. 14
Correlation index for polarization at the source Eq. 30
Basic statistical:
Mean Faraday RM density Eq. 9
rms Faraday RM density fluctuation Eq. 10
Mean polarization at the source Eq. 30
rms polarization fluctuation at the source Eq. 30
Table 1Parameters for correlation studies of the synchrotron polarization from an extended emitting region with Faraday rotation

4. Statistics of the turbulence from single wavelength PPF slice

In this section we study how spatial correlation properties of the observed polarization of synchrotron emission reflect the underlying statistical properties of magnetic and electron density turbulence. Observed polarization correlation properties depend on the separation between the lines-of-sight and the wavelengths of the observation.

Let us consider spatial correlations in polarization maps for measurements at the fixed wavelength. Such approach we shall call Polarization Spatial Analysis (PSA). The signal is accumulated along pairs of lines-of-sight, separated by . The main effect of the Faraday rotation in the sufficiently turbulent (criterium to follow) medium is to suppress the observed correlations by establishing an effective narrow line-of-sight depth over which correlated part of the signal is accumulated. As we shall show, at small separations , this depth depends on , resulting in modified scaling of the polarization correlations that reflects the correlation of the Faraday RM density. At large separations, the suppression is uniform, synchrotron correlations are accumulated over an effectively thin slice and reflect the underlaying correlations of the magnetic field.

We make two approximations in our quantitative treatment. First we take to be a Gaussian quantity, definitely good approximation when its fluctuations are dominated by the fluctuations in the magnetic field. Second, we neglect the correlations between the fluctuations in intrinsic polarization at the source and the Faraday RM. Here we note that when both are dominated by fluctuations of magnetic field, which may give the most of cross-correlation, these are different (perpendicular and parallel to line-of-sight) components of the magnetic field that define intrinsic polarization and Faraday RM. At small separations between the lines-of-sight the correlation between them is suppressed (and is formally zero along coincident lines-of-sight or between sources at the same distance when turbulence is isotropic or is a strong turbulent mix of Alfvén and slow modes as this is the case of nearly incompressible turbulence (see Goldreich & Sridhar 1995)). 7 Whereas at large separations effect of Faraday rotation is, as we’ll see below, mostly amounts to providing a window over which synchrotron polarization fluctuations are sampled.

Under stated assumptions

(35)

For observations done at sufficiently long wavelength (criterium to follow), we can use quadratic approximation of Eq. (19)

(36)

According to this formulae, both the mean field and the fluctuating, turbulent Faraday rotation establish an effective width in separation over which the polarization correlations are accumulated over. For the mean field, the effective width is

(37)

while the one for the fluctuative rotation is , both windows decreasing with the increase in the wavelength of the observations. These scales have the meaning of a line-of-sight distance over which polarization direction rotates by approximately a radian. In what follows we consider the spatial extent of the emitting region to be much larger that the smallest of these two scales, , where

(38)

In the opposite case the effect of Faraday decorrelation can be neglected.

Note that the effect of turbulent rotation can be more dramatic, leading to Gaussian window in comparison to slower oscillatory cutoff from the mean field Faraday rotation. The quadratic approximation Eq. (36) is sufficient when this effective window produced by turbulent component of Faraday rotation is narrower than the intrinsic correlation length of synchrotron fluctuations arising from the magnetic field component , i.e. when . Following Eq. (21), is bounded from below by , i.e for and at . Thus the required criterium is with line-of-sight separation . This criterium can also be written in terms of scales as and where we define

(39)

which is the quantity that will be used through the rest of our paper.

4.1. Dominance of turbulent rotation,

We first consider the case when . This is the case of either weak regular magnetic field, with respect to its fluctuations, or of strongly inhomogeneous distribution of electron density, or both. The problem is complex, having five scales involved, namely the scale for Faraday rotation , the correlation length of the rotation measure , the correlation length of the transverse magnetic field , the line-of-sight size of the emitting region and the separation between two line-of-sight over which we correlate to polarization signal. We shall always consider the extent to exceed the correlation length of the polarization fluctuations at the source, and we limit our studies to . This leaves us with two parameters and to study polarization correlation as a function of . We expect which in case of inequality will give rise to the intermediate regime which can be potentially used to investigate two correlation lengths separately. In the limiting case when polarization at the source is dominated by the mean contribution, we should replace by in all criteria and results that follow.

Now two basic regimes can be distinguished:

(a) the regime of strong Faraday rotation, . In this regime, Faraday rotation does not decorrelate the polarization only from sources with . In the approximation of Eq. (36), the integral over such narrow window of gives

(40)

The remaining line-of-sight integral provides the effective depth along the line-of-sight over which the signal is accumulated. It depends on and warrants a detailed examination. For it evaluates simply to as and . At finite , however, it is shortened, since Faraday rotation decorrelates the signal as we integrate along two non-coincident lines-of-sight. Mathematically, increases with with coefficients that increase with as described by Eqs. (22,23, 25). To compute the Faraday effective depth, is exponentiated and then integrated over . Since is growing in both and , small behaviour of the will be defined by the functional form of at large , while large dependence will be determined by small . As the result the effective depth decreases with

(41)

from it’s maximum value of as dictated by Eq. (23) until it becomes effectively constant (with weak dependence on and ) at as follows from Eq. (25). This behaviour of the Faraday window is summarized in Figure 3.

Figure 3.— Effective Faraday depth in units of as the function of separation between lines-of-sight given in units of the Faraday correlation length . Dashed lines correspond to and . Solid lines are asymptotical scalings, at and at . Here . Note that for , the small asymptotics behaves as if .

If one may detect the intermediate asymptotics , over the range of scales , as governed by Eq. (22).

Thus, at we have asymptotic behaviour of polarization correlation