Polarization of synchrotron turbulent bubbles

Polarization properties of turbulent synchrotron bubbles: an approach based on Chandrasekhar-Kendall functions

N. Bucciantini
INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy
Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via G. Sansone 1, I-50019 Sesto F. no (Firenze), Italy
INFN - Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto F. no (Firenze), Italy
E-mail: niccolo@arcetri.astro.it
Accepted / Received
Abstract

Synchrotron emitting bubbles arise when the outflow from a compact relativistic engine, either a Black Hole or a Neutron Star, impacts on the environment. The emission properties of synchrotron radiation are widely used to infer the dynamical properties of these bubbles, and from them the injection conditions of the engine. Radio polarization offers an important tool to investigate the level and spectrum of turbulence, the magnetic field configuration, and possibly the degree of mixing. Here we introduce a formalism based on Chandrasekhar-Kendall functions that allows us to properly take into account the geometry of the bubble, going beyond standard analysis based on periodic cartesian domains. We investigate how different turbulent spectra, magnetic helicity and particle distribution function, impact on global properties that are easily accessible to observations, even at low resolution, and we provide fitting formulae to relate observed quantities to the underlying magnetic field structure.

keywords:
radiation mechanisms: non-thermal - polarization - turbulence - radio continuum: ISM - ISM: supernova remnants - ISM: bubbles

1 Introduction

Synchrotron emission originates from relativistic pairs, spiraling in a magnetic field. In astrophysics synchrotron emission is a powerful tool to investigate relativistic plasmas, and non-thermal particles distributions. These are the signatures of acceleration processes related often to relativistic engines, like Neutron Stars (as in the case of Pulsars) and Black Holes (as in the case of AGNs).

One of the key properties of synchrotron emission, is the high level of linear polarization (Westfold, 1959; Legg & Westfold, 1968), that theoretically can be as high as 70%, for the typical particles distribution functions that are observed (quite often power-laws). It was indeed thanks to its polarization, that synchrotron emission was recognized for the first time in an astrophysical source, the Crab nebula (Baade, 1956; Oort & Walraven, 1956; Woltjer, 1958a). When relativistic outflows interact with the surrounding environment, they tend to form synchrotron emitting bubbles. This happens for Pulsars inside Supernova Remnants (Pacini & Salvati, 1973; Rees & Gunn, 1974; Kennel & Coroniti, 1984; Gaensler & Slane, 2006; Bucciantini, 2008), leading to pulsar wind nebulae (often referred as plerions), that are characterized by a broad-band emission ranging from Radio to X-ray (and TeV due to Inverse Compton). It also happens for Radio Lobes fed by the jet produced in AGNs (Scheuer, 1982; Begelman, Blandford & Rees, 1984; Begelman & Cioffi, 1989; Carilli, Perley & Harris, 1994; Reich, Testori & Reich, 2001). The study of radio emission and polarization enable us to infer information on the strength and structure of the magnetic field, that are relevant to understand the dynamics of these systems. For example, in pulsar wind nebulae, radio emission traces old particles, that fill the entire volume of the nebula, and it is detectable also in old systems, where the injection from the pulsar has faded away (Frail et al., 1996; Giacani et al., 1997; Roberts et al., 1999; Ma et al., 2016). Recent observations (e.g. Ma et al. (2016)), and a set of theoretical indications based either on numerical simulations (Jun, 1998; Blondin, Chevalier & Frierson, 2001; Porth, Komissarov & Keppens, 2014) or on spectral modeling (Tang & Chevalier, 2012; Olmi et al., 2015, 2016; Tanaka & Asano, 2016) suggest that a non negligible level of turbulence should be present in these systems. Some simplified model for the turbulent field has already been presented attempting to explain some of the observed results (Ma et al., 2016; Bucciantini et al., 2017). The level of turbulence in wind bubbles could be important to understand how particles diffuse (Tang & Chevalier, 2012), the level of mixing and penetration of the ambient medium (Blondin, Chevalier & Frierson, 2001), the strength of the magnetic field (its ability to quench the cascade) and its role in the nebular dynamics (Bucciantini et al., 2004).

There is a vast literature that in the past years has focused on modeling the polarized properties of synchrotron emission, in a turbulent magnetic field, but most of it has focused on the galactic background emission, associated to cosmic rays [e.g. Lazaryan & Shutenkov (1990); Lazaryan & Chibisov (1991); Waelkens, Schekochihin & Enßlin (2009); Junklewitz & Enßlin (2011); Lazarian & Pogosyan (2012a, 2016)]. Far less has been done in the case of confined systems, like wind bubbles. Part of the problem is due to the fact that the techniques developed for the former are of little use in the latter. In general the approaches to modeling the polarized properties of synchrotron emission, in a turbulent magnetic field, are based on the idea of an infinitely extending volume, where the turbulence is homogenous (Thiébaut et al., 2010; Lazarian & Pogosyan, 2012b; Zhang et al., 2016; Herron et al., 2016), and make use of internal Faraday rotation, which is not present for a pair plasma (as expected in pulsar wind nebulae). Assuming homogeneous, infinitely extending turbulence, is equivalent to neglecting any physical scale of the system under investigation (they are all much larger that any of the scale of the turbulent cascade). Unfortunately this is not the case for wind bubbles, where the scales of the turbulent cascade are comparable with the bubble dimension itself. On top of this we need to recall that several mathematical tools have been developed through the years to deal with turbulence in an homogenous infinite space, while the case of a confined volume (where surface effects are important), being more dependent on the specific boundary conditions, has been hardly touched. Finally, numerical tools exist that are efficient, and easy to implement to work in cartesian domains with periodic boundaries (think the Fast Fourier Transform), that are not available on other domains or for other geometries.

In this work we present a new approach to the study of turbulent magnetic field configurations inside confined domains, and in particular we select a spherical domain, which can be though of as a good first-order approximation for an emitting bubble, based on an alternative set of harmonic functions with respect to Fourier plane waves. Our approach is based on Chandrasekhar Kendall functions (CK) (Chandrasekhar & Kendall, 1957; Chandra, 1987). CK functions have been used for modeling spheromak configurations (Vandas et al., 1997; Dasgupta et al., 2002), plasmoid in the solar corona (Burlaga, 1988; Chandra & Prasad, 1991; Farrugia, Osherovich & Burlaga, 1995; Vandas, Fischer & Geranios, 1991), and for dynamos in confined domains (Mininni & Montgomery, 2006; Mininni, Montgomery & Turner, 2007; Brandenburg, 2011). They are the natural extension of Fourier modes to spherical geometry, but they can be extended also to other geometry (Rasband & Turner, 1981) as in the case of shells (Morse, 2007), cylinders (Marsh, 1992; Morse, 2005) or ellipsoids (Ivanov & Kharshiladze, 1985). They allow us to set the proper boundary conditions for a confined field, and to take into account the geometry of the emitting region in a formally correct way (instead of truncating the emission at the edge of a given volume without care of the magnetic field structure). This allows us to take into account volumetric effects. Given that the scope of this paper is to characterize the signature of a turbulent field, providing estimates that could easily help guide the interpretation of observations, we focus our attention on global quantities that are easily measurable even at the typical low resolution, at which synchrotron bubbles are observed.

In Sect.2, we introduce the formalism that we adopt to model a turbulent magnetic field confined within an emitting bubble. In Sect. 3 we derive some simple scaling of quantities that will be analyzed in Sect. 4 where we present the result of a statistical study of the global polarized properties for synchrotron emitting bubbles. In Sect. 5 we present our conclusions.

2 Magnetic field realization

We are going to discuss here how to implement a magnetic field realization on a spherical domain, such that we can enforce the specific boundary condition of full confinement at the spherical surface bounding the domain itself, in a rigorous manner. The idea is to use spherically adapted harmonic function, instead of simple cartesian Fourier modes. Note that the use of adapted harmonic functions can be extended to a variety of different geometries, for which they are defined.

2.1 Chandrasekar-Kendall functions

In spherical coordinates , given the scalar function:

(1)

where is the spherical Bessel function of first kind of degree , is the real spherical harmonic of degree , and is the -th zero of , the vector field defined by:

(2)

is a force free field , with vanishing radial component of the unitary 2-sphere : at . Being force free (Woltjer, 1958b; Molodensky, 1974), this vector field is also solenoidal, and it is known as Chandrasekhar Kendall functions. It can be shown that CK functions form a complete basis for solenoidal vector fields in the unitary ball , with vanishing component on the boundary [Yoshida (1992); Mininni & Montgomery (2006); Mininni, Montgomery & Turner (2007), see for example Deredtsov, Kasantsev & Schuster (2007) for other basis]. CK functions can be written in terms of real vector spherical harmonics:

(3)

where , and . CK functions are, moreover, orthogonal in the sense that:

(4)

where and only if and are the same sign.

2.2 Magnetic field, power spectrum and CK functions

Given the properties of CK functions any magnetic field bound to be confined into a spherical region of unitary radius can be written as the sum of CK modes:

(5)

where the first sum is done on CK functions of different sign. Being the modes orthogonal, for the magnetic field defined in Eq. 5, it is possible to write the total energy , the total helicity , the total current helicity in as:

(6)
(7)
(8)

Note that modes with the same but different are degenerate in energy. It is immediately obvious that the decomposition in CK functions, allows one to control the helicity of the field and to build magnetic field configurations having the same total energy but with various possible prescription for the helicity:

  • maximal helicity realization if (or equivalently ) for all . In this case all CK modes contribute an helicity of the same sign, and the total helicity is maximal;

  • zero helicity realization if for all . In this case the same power goes into CK modes of opposite helicity, and the total helicity vanishes;

  • definite helicity realization if for all . In this case all the power for each CK mode goes into a definite helicity state, even if the sign differs for different modes;

  • random helicity realization if no contraints are imposed on for all . In this case the power for each CK mode is randomly distributed in the two possible helicity states.

In general, when using Fourier decomposition in finite cartesian domains, turbulent magnetic field realizations are buit assigning at each mode of wave number an amplitude such that when integrating the energy contribution in the volume off all modes up to any arbitrary value of the wave number norm one finds:

(9)

where is the energy associated with the Fourier mode of wave number and is the on-shell power spectrum (representing all the energy at a wave number ), and is the first allowed mode. Usually Fourier modes are quantized, given that in a finite domain only modes that are periodic are allowed. Fourier decomposition is however just a special case of decomposition into harmonic functions, corresponding to plane waves, adapted to a cartesian domain. In different geometries other harmonic decompositions might prove more suitable (one can for example define properly the boundary conditions on a spherical shell). In spherical geometry, one can use for example spherical waves defined in term of Bessel functions and spherical harmonics as Eq. 1. It can be shown that it is always possible to transform from one to the other (Mininni & Montgomery, 2006; Mininni, Montgomery & Turner, 2007; Liao & Su, 2015). CK function can be seen as generalization to spherical geometry of Fourier modes. In particular given that each CK mode has maximal helicity, they are the spherical equivalent of Beltrami waves. Boundary conditions set the quantization of the modes [].

If , with then:

(10)

This condition translates into a condition on the coefficients of the realization:

(11)

How this power is distributed among modes of different sign, depends on the helicity prescription. On the other hand the distribution among the various depends on the isotropy of turbulence. It is well known that MHD turbulence is anisotropic (Goldreich & Sridhar, 1995; Brandenburg & Lazarian, 2013), with the anisotropy dependent on scale. For simplicity, in our realizations we asume isotropy. For a solenoidal magnetic field, isotropy implies that all directions are equally probable for the orientation of each multipole of degree . Given the known rotational properties of spherical harmonics, in each realization we achive this by randomly distributing the power of the multipole among the varios subject to the normalization given by Eq. 11). In practice for each value of and we generate random number in the interval , (where the factor 2 comes from the presence of modes with two sign: the part), we set each of the equal to one of these random numbers, and then we renormalize them such that the sum of their squared over and satisfies the condition of Eq. 11. This ensures that: 1- on a single realization, the spatial orientation of the various multipoles , are uncorrelated; 2- that averaged over many realizations, the orientation of each of the various multipoles, over which the field is decomposed, is uniform on the 2-sphere. So, fur us “isotropy” means that in each realization the various multipoles are randomly oriented, and there is not preferential alignement among them. Please not that the uniform assumption hold for the ensemble of realizations.

2.3 Numerical setup and simulated synchrotron maps

Turbulent magnetic field realizations are built on a 3D cartesian grid containing the unitary sphere. The resolution has been chosen such that the results on the quantities of interest are converged. Assuming a power law distribution of emitting pairs:

(12)

where is the energy in units , and in principle could be a function of position. The emissivity toward the observer (assumed to be located along the -axis) at a frequency is:

(13)

where , and is given by synchrotron theory

(14)

Assuming the and coordinates to lay on the plane of the sky, it is possible to compute the maps of the various Stokes parameters integrating the contribution of each fluid element along the line of sight through the domain of the realization, according to

(15)
(16)
(17)

where the local polarization position angle is the angle of the emitted electric field vector in the plane of the sky, such that

(18)

The total intensity and total polarized intensity , can be obtained integrating the various Stokes parameters over the plane of the sky.

3 Cascade energy and total brightness

If the turbulent cascade extends up to a maximum value one can model the total emission from the unitary sphere of volume as the sum of the emission coming from regions inside which the magnetic field can be taken as uniform. For a random gaussian realization, in each of these regions, the field orientation and strength will be randomly distributed. If the emitting particles are uniformly distributed, then the total intensity , integrated over the emitting area, will just be function of the magnetic field:

(19)

where is the angle between the magnetic field and the direction of the observer , we have used the definition of mean and the fact that direction and strength of the magnetic field are uncorrelated variables, for isotropic turbulence. For any quantity having a Maxwellian distribution (like the strength of the magnetic field), it can be shown that , hence we find:

(20)

Then on can estimate how the total brightness depends on :

(21)
(22)

where and are the magnetic energy and total intensity of a realization extending to respectively, and we have introduced the quantity which represents the difference of the magnetic energy between a turbulent cascade extending to and one up to . For the polarized intensity instead, one must recall that unlike , and are not positively defined. For a realization extending up to , there will be , domains that contribute to the polarized intensity incoherently. So we expect that the contribution to the polarized intensity from these domains should scale as the variance of the total intensity. The difference in polarized intensity between a turbulent cascade extending to and one up to .

(23)

suggesting that for typical values of and , the polarized intensity should typically saturate within few percents already at .

4 Results

We have computed models for various values of the turbulent spectral index, , and particle distribution function . The turbulent index has been chosen in order to include both Kolmogorov and Kraichnan scaling. The particle distribution function, has been chosen to target the typical radio spectrum of PWNe (for the Crab nebula ). For each value of and we computed synchrotron maps for one hundred different random realizations, for which the uniform assumptions holds as an ensemble, which allowed us to compute the ensemble mean values of observable quantities and to estimate their typical variances . In order to get converged results on the quantities of interest, we found that each realization must include at least CK modes (this means that we have to compute at least coefficnets and the related Bessel functions and spherical harmonics). Taking the turbulent spectrum to be a pure power-law, corresponds to the assumption that the coherence scale of the turbulence is the size of the bubble itself. We focus our attention to global quantities that are easily accessible to simple measures. Quantities like the two-points correlation function, or the Fourier spectrum of the emission map, while far more informative, in general require the object to be observed at very high resolution, which is often not the case for our typical targets (PWNe).

4.1 Maximal helicity

We present here the results in the simplest case where the emitting particles (pairs) are distributed uniformly in space ( is constant in space), and the magnetic field follows a realization with maximal helicity: . In Fig. 1 we compare brightness maps in polarized intensity obtained from realizations with different turbulent power spectra but where the power at each is distributed in the same proportion among the degenerate modes (the relative orientation of the modes is the same), such that the final maps have the same structure. As expected, realizations with smaller values of tend to show more fine structures, however the difference is quite small, and it would require very high resolution and high signal-to-noise observations to be detected. On the contrary we found that the polarized intensity, once normalized to the average unpolarized brightness is quite sensitive to the value of . Another feature that is commonly found in our realizations is the presence of one or two highly polarized spots, where the typical polarized intensity is about twice the average nebular value, and which in general are not coincident with the brightest regions in total intensity. As shown in the figure, these highly polarized spots, are not necessarily located close to the center, first because synchrotron emissivity strongly depends on the orientation of the magnetic field, and in general there is no reason for the magnetic field to be preferentially orthogonal to the line of sight close to the center of the bubble. Second being maps in polarized intensity, one should remember that there are depolarization effects once one integrates along the line-of-sight over regions with different magnetic orientations. So polarized emission will peak where, by chance of the realization, these depolarization effects are minimal, and these regions might or might not be close to the center. On the other hand the polarized fraction tends to rise at the very edges of the bubble where it can reach values close to the theoretical maximum , in the body of the emitting bubble one finds either unpolarized regions and regions with polarization as high as 50% of the theoretical maximum. Please note that the maps shown correspond just to one realization, in order to provide an idea of the results to the reader, but are not meant to represent any typical average. They are are mean to illustrate how a different choice of the turbulent index , quenching or instrumental resolution affects the appearance of map in polarized intensity.

In Fig.2 we show the mean (ensemble averaged over many realizations) of the polarized fraction defined as the ratio of total polarized intensity over total intensity: with error-bars indicating the range of the results computed over a hundred different realizations. The point is set at , according to the discussion of Sect. 3, because for the total intensity is dominated by the small scale components which carry most of the magnetic energy, hence the polarized fraction, which is instead due to the large scale components, vanishes.


Figure 1: Polarized surface brightness maps, normalized to the mean total surface brightens, for the same magnetic realization (same orientations of the modes) but with various turbulent indexes, in the case of maximal helicity. The spectral index is set to

We see immediately that the integrated polarized fraction shows a clear power-law dependence on the turbulent index, while the dependence on the distribution functions of the emitting pairs is, within the statistical uncertainty of our realizations, simply given by the standard synchrotron factor . Indeed one can model our result with the simple relation:

(24)

with the limits given by:

(25)
(26)

Obviously these fits apply only to the range of we have investigated. In particular, extrapolation to would lead to which is unphysical. Indeed we find that, if only the mode is present, then the mean normalized to the spectral index factor is .


Figure 2: Integrated polarized fraction normalized to the synchrotron factor , as a function of the power spectral index of the turbulence . Black diamonds are the mean values for , blue triangles for , and red squares for . The error-bars represent the range.

4.2 Zero helicity

At the other extreme with respect to the maximal helicity case, there are the zero helicity realizations with . Attempts to define observables able to constrain the level of the helicity of the magnetic field, have been developed in the past, making use of higher-order statistics and rotation measure (Waelkens, Schekochihin & Enßlin, 2009; Junklewitz & Enßlin, 2011). Here we focus on a simple estimate of the possible role of helicity over integrated quantities. It can easily be shown that for a magnetic field described as a simple linearly polarized plane wave, the ratio of the polarized intensity over the total intensity is a constant for every direction of the observer, and in every position of the emitting domain, and the direction of polarization is also constant. For a magnetic field described as a circularly polarized plane wave, there are depolarization effects along the line of sight (the direction of changes throughout the emitting domain). So it is reasonable to expect that a magnetic realization, with zero helicity, should show larger polarized fractions. How large would depend on the power spectrum of the turbulent realization. If the polarization properties of the simulated bubble, are dominated by magnetic fluctuations of the largest scale, to which only few modes of small contribute, then the dependence on the helicity will be pronounced. If on the other hand polarization is dominated by magnetic fluctuations on smaller scales, where many high modes contribute incoherently, than one would expect to see only small differences.

Given the power spectrum we have adopted for our magnetic field realizations, we found that the polarized fraction is almost completely set by the first few () modes at small , for all the values of we have simulated. As a result the effects of a different helicity prescription are substantial. On Fig. 3 we show the same result of Fig. 2, but for . Note that again we find that the integrated polarized fraction can be described as a power-law function of the turbulent index, but the typical values are 50% higher that the previous case and well beyond the range of variance. Even the variance of the distribution is now larger. Despite the fact that the case has a flatter spectrum of magnetic turbulence than the case, we do not see any appreciable, difference associated to the choice of helicity. This suggests that, even at , the large scales are still dominant.


Figure 3: Integrated polarized fraction normalized to the synchrotron factor , as a function of the power spectral index of the turbulence , for the zero helicity case: . The spectral index is . The solid curve is a power-law fit to the averages, and is just given by Eq. 24 multiplied by a factor 1.52. The dotted curves represent power-law fit to the range.

4.3 Equipartition

The magnetic field given by Eq. 5 as the sum of force-free modes, is itself not force free, and as such it is out of dynamical equilibrium. In many situations of interest, like wind bubbles, to which our approach is targeted, the energy of the magnetic field is comparable to the one in the emitting plasma. It is common in these systems to assume equipartition between the particles and the field, in order to infer physical informations from the emission properties (Pacholczyk, 1970). The validity of such assumption is often confirmed by more sophisticate models of the spectral energy distribution. This requires the distribution of emitting particles to be highly correlated to the magnetic field. In general however, when dealing with possible variations of the density of emitting particles, it is customary to assume a complete uncorrelation. In order to model a distribution of emitting particles that follows an equipartition recipe, we modified Eq. 14 using a variable such that:

(27)

where is the magnetic field strength at position and is the maximum strength of the magnetic field in the unitary ball (which differs in each realization). This ensures that the sum of the magnetic and particle energy is constant throughout the emitting volume.

Interestingly we find that the mean polarized fraction and its variance are unchanged with respect to the case of a uniform electron distribution, even if the surface brightness in the emission maps, appears shallower. Given that emission scales linearly with the particle number density, one can safely conclude that for an electron distribution given as:

(28)

the polarized fraction is independent of the value of .

4.4 Cascade quenching and instrumental resolution

The results discussed previously, assume that the turbulent cascade extends all the way to , and that maps have infinite spatial resolution. We can however take into account both the effect of turbulent quenching (if the cascade is truncated at a given ) and of instrumental resolution. In Fig. 4 we show how the map of polarized intensity changes depending where the cascade is truncated. In Fig. 5 we plot the total intensity and polarized intensity, for some of our realizations, showing how they change depending on the truncation of the power-law distribution of the magnetic field. In particular the cutoff is parametrized in terms of the relative cascade energy: .

We find that in general the polarized intensity saturates already for values of , coherently with our estimate Eq. 23, on the other hand the total intensity keeps increasing quasi-linearly according to Eq. 22. This, as discussed, reflects tha fact that while the total intensity is always a positively defined quantity, and so it increases with the addition of small scale modes, the polarized intensity is derived from Stokes parameters and which are not positively defined, and so undergo cancellation effects. Interestingly we find that, for the following relation for the polarized fraction holds with a few percent accuracy:

(29)

where corresponds to the polarized fraction of a cascade extending to and whose mean and variance are given in Eq.s 24-26.


Figure 4: Polarized surface brightness maps, normalized to the mean total surface brightens, for the same magnetic realization (with and ) in the case of maximal helicity, but for turbulent magnetic cascades truncated at varios valued of .

Figure 5: Trend of the total intensity and polarized intensity, as a function of cascade quenching, for the maximal helicity case and . The upper points correspond to the total intensity of several realizations as a function of (each point-style/color selects a realization set with a given configuration of the modes, progressively truncated at higher ). The lower points refer to the corresponding polarized intensity . and are normalized to the total intensity in the limit . The solid line represents the function , that provides a good fit to the trend of the intensity. The dashed line represents the mean polarized fraction as given by Eq. 24, and the dotted lines the range as given by Eq. 26-25, for the limit .

Figure 6: Polarized surface brightness maps, normalized to the mean total surface brightens, for the same magnetic realization (with and ) in the case of maximal helicity, but for different full-width half maximum (FWHM) of the point spread function. From left to right: , and times the diameter of the bubble.

Figure 7: Best fits of the mean of the integrated polarized fraction PF normalized to the synchrotron factor , as a function of the power spectral index of the turbulence . The spectral index is . The curves refer to different instrumental resolution. From top to bottom the full-width half-maximum of the point spread function is times the diameter of the emitting bubble.

On the other hand, instrumental resolution acts to decrease the level of polarization, because, while not altering the total intensity, it introduces cancellation effects of the Stoke’s parmantes and . In Fig. 6 we show how the polarized emission maps change due to instrumental resolution assuming a gaussian point spread function of varying width. Interestingly by comparing the first panel in Fig. 6 with the last one in Fig. 4, we see that on the morphology of the polarized emission, the effect of a truncated magnetic cascade is analogous to the one due to instrumental resolution. The two maps are hardly distinguishable, especially with reference to the brightest parts. What really distinguishes the two cases, is the net polarized fraction, because quenching and instrumental resolution, act differently on the total intensity. In Fig. 7, we show how the mean polarized fraction of our realizations changes, depending on instrumental resolution and for different values of . Interestingly we find that even in this case, we can fit the dependence of the mean value on with a power law. The variance instead shows only minor changes (at most %) with respect to the case of infinite resolution.

By comparing Fig. 1 with Fig. 6, or Fig. 4, we see that the structure of the polarized emission changes. While for an infinite cascade, at very high resolution, the polarized intensity, show many small scale features, in the other cases, these do not smear out into a more uniform map, but ends up merging into larger structures that have the general shape of loops. Interestingly a similar pattern was seen in the body of G327.1-1.1 (Ma et al., 2016). It was also observed that the magnetic field direction seems to follow the loops. In that paper a model was put forward to explain the level of polarization and the typical scales observed, however due to the simplicity of the approach it was not possible to reproduce the loopy structure of the magnetic field, and the correlation between direction and bright features. The authors suggested that a possible reason for this discrepancy was to be looked for in the fact that the model did not enforce the solenoidal condition on the magnetic field. In our model where the solenoidal condition is enforced by construction, we indeed recover exactly this king of behavior, with the direction of the magnetic field inferred from Stoke’s parameter aligned with the bright features.

4.5 Two points correlation

In the previous sections we have investigated the trends of global integrated quantities, like the total polarized flux and fraction, which are easily accessible even to simple observations. Here, in order to provide a quantitative estimate of the small scale properties of our simulated maps, we investigate the two-points correlation function of the Stoke’s parameters defined (in the case of and analogously for ), as:

(30)

where and label the two points located respectively at and , and . In the top panel of Fig. 8 we show, for a Komogorov distribution , the probability distribution of the value , normalized for convenience to the mean total surface brightness squared as a function of the separation . It is evident the skewness of the distribution for , where the two point correlation function is dominated by positive values suggesting that the polarized features are correlated, while for , the distribution is symmetric pointing to uncorrelated features. This is made more evident in the bottom panel of Fig. 8, where the probability distribution is shown for various values of , and where the mean value of the two point correlation function is also shown.

Figure 8: Upper panel: probability distribution function of the two-point correlation normalized to the mean squared total surface brightness , as a function of the separation between the two points . Lower panel: probability distribution function of the two-point correlation for selected values of : 0.02 solid-black line, 0.1 dotted-blue line, 0.2 dashed-green line, 0.5 dash-dotted-red line. The insert shows the average value of (multiplied times 100), as a function of .

5 Conclusions

In this work we have introduced a novel approach to the study of synchrotron emission from a turbulent magnetic field realization, based on a set of harmonic functions (the Chandrasekhar-Kendall functions) that allows us to take into account the geometrical properties of the source, and to enforce correct surface-bounday conditions, going beyond the standard small scale approximation, typical of many previous studies. This is particularly relevant for synchrotron emitting bubbles, where current instrumental resolution does not allow us to sample structures that are much smaller than the bubble size itself. The approach offers moreover the possibility to take into account anisotropy in the turbulent cascade, and to control the level and distribution of helicity among the modes.

We try to investigare how global properties, like the integrated polarized fraction, scale depending on the spectrum of turbulence, helicity distribution, spectral index of the emitted radiation, resolution and cascade quenching. Interestingly we found that the spatial distribution of the emitting particles (either uniform or anticorrelated with the magnetic field strength) does not affect such global properties. While the dependence on the spectral index (tied to the particle distribution function) follows the standard synchrotron law. We found that the mean polarized fraction of our realizations can be fitted with a power-law with respect to the turbulent index and that this trend holds also for different helicity and instrumental resolution.

To illustrate an application of our results, let us consider for example G327.1-1.1 (Ma et al., 2016). This is middle age Pulsar Wind Nebula, with the pulsar outside the main radio body of the nebula. In radio a loopy structure is observed, with the magnetic field inferred from linear polarization aligned with the bright features. The spectral index in radio is . The body observed at 3cm and 6cm, with a resolution corresponding to a FWHM times the nebular diameter, shows an average polarized fraction , with a single highly polarized spot at the center, and an increasing polarization toward the edges, mostly an artifact due to the low local surface brightness. With reference to Fig.7 we find that the lower value coincides with the expected mean of PF in the Kolmogorov case , with maximal helicity, while the upper limit, higher than the range, would suggest some cascade quenching with . Obviuolsy, taking into account the typical variance of our realizations, the detected polarized fraction, is compatible with turbulent spectra in the range , and the upper limit cound be due to a different helicity. It is howerver interesting that our model allows us, to set some constraint on the tubulence inside such system: the preferred model suggests a Kolmogorov turbulence, either with low helicity or some quenching.

Acknowledgements

The author acknowledge support from the PRIN-MIUR project prot. 2015L5EE2Y Multi-scale simulations of high-energy astrophysical plasmas.

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