Obtaining magnetic field strength using differential measure approach and velocity channel maps
DD RAFT VERSION F EBRUARY
20, 2020Typeset using L A TEX twocolumn style in AASTeX63
Obtaining magnetic field strength using differential measure approach and velocity channel maps A LEX L AZARIAN ,
1, 2 K A H O Y UEN , AND D MITRI P OGOSYAN
3, 41
Department of Astronomy, University of Wisconsin-Madison, USA Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea Department of Physics, University of Alberta, Edmonton, Canada Korea Institute for Advanced Studies, Seoul, Republic of Korea
Submitted to ApJABSTRACTWe introduce two new ways of obtaining the strength of plane-of-sky (POS) magnetic field by simultaneoususe of spectroscopic Doppler-shifted lines and the information on magnetic field direction. The latter canbe obtained either through polarization measurements or using the velocity gradient technique. We show theadvantages that our techniques have compared to the traditional Davis-Chandrasekhar-Fermi (DCF) techniqueof estimating magnetic field strength from observations. The first technique that we describe in detail employsstructure functions of velocity centroids and structure functions of Stokes parameters. We provide analyticalexpressions for obtaining magnetic field strength from observational data. We successfully test our results usingsynthetic observations obtained with results of MHD turbulence simulations. We measure velocity and magneticfield fluctuations at small scales using two, three and four point structure functions and compare the performanceof these tools. We show that, unlike the DCF, our technique is capable of providing the detailed distribution ofPOS magnetic field and it can measure magnetic field strength in the presence of both velocity and magneticfield distortions arising from external shear and self-gravity. The second technique applies the velocity gradienttechnique to velocity channel maps in order to obtain the Alfven Mach number and uses the amplitudes of thegradients to obtain the sonic Mach number. The ratio of these two Mach numbers provides the intensity ofmagnetic field in the region contributing to the emission in the channel map. We test the technique and discussobtaining the 3D distribution of POS galactic Magnetic field with it. We discuss the application of the secondtechnique to synchrotron data.
Keywords:
Interstellar magnetic fields (845); Interstellar medium (847); Interstellar dynamics (839); INTRODUCTIONThe role of magnetic fields in astrophysics is difficult tooverestimate. Magnetic force is the second most importantforce in the present day Universe after gravity. The mag-netic field plays an important role at different stages of starformation (e.g. Mestel & Spitzer 1956; Galli et al. 2006;Mouschovias et al. 2006; Johns-Krull 2007). In view of as-trophysical flows with large Reynolds numbers the magneticfields are turbulent (see Elmegreen & Scalo 2004; McKee &Ostriker 2007; Xu & Zhang 2016a,b). The evidence of tur-bulent magnetic field is coming from observations of density [email protected]@[email protected] structure of the interstellar medium (e.g. Armstrong et al.1995; Chepurnov & Lazarian 2009) and velocity fluctuationstudies (Larson 1981; Heyer & Brunt 2004; Chepurnov &Lazarian 2010).Davis (1951) and Chandrasekhar & Fermi (1953) pro-posed the technique (Davis-Chandrasekhar-Fermi technique,henceforth DCF technique) that allows to estimate the mag-nitude of the mean magnetic field in interstellar medium andmolecular clouds by measuring both the variations of the ob-served magnetic field directions and the dispersion of veloc-ities. The technique has been revised and improved by thecommunity (e.g. Heitsch et al. 2001; Crutcher 2004; Houde2004; Girart et al. 2006; Falceta-Gonc¸alves et al. 2008), butthe foundations of the technique stayed the same. In par-ticular, the DCF assumes that the perturbations of magneticfield direction arise from the collection of Alfven waves atall scales with waves at the largest scale dominating the ob-served variations. This, however, is not true for most of the a r X i v : . [ a s t r o - ph . GA ] F e b L AZARIAN , Y
UEN & P
OGOSYAN astrophysical settings with large scale magnetic field fluctu-ations being affected by the factors not related to turbulence,e.g. by gravity. As a result, the accuracy of the DCF tech-nique is low.Improvements of the DCF technique was taken care first byreplacing the dispersion of polarization angles to the struc-ture functions of it in Hildebrand et al. (2009). The authorsused the structure functions of magnetic field angle intro-duced in Falceta-Gonc¸alves et al. (2008) and discussed howthe dispersion of magnetic field can be obtained from obser-vations. Hildebrand et al. (2009) models the structure func-tion of magnetic field angles (not polarization angles) by thefirst two terms of its Taylor expansion SF { φ } ( R ) ∼ b + m R to obtain the ratio between the turbulent-to-regularmagnetic field strength. The calculations, however, assumedthat the correlation length scale of turbulence is smaller thanthe separation between the line of sights at which the struc-ture function is calculated. This assumption of very smallscale turbulence was applied further in the subsequent study(Houde et al. 2009). This assumption, as we discuss further,is not applicable for interstellar studies as there is good evi-dence that we resolve turbulence both in diffuse media (seeCrutcher 2010) and in molecular clouds (Padoan et al. 2009;Houde et al. 2011). Therefore it is advantageous to explorewhat we can get using the small scale differential measuresthat are influenced mostly by turbulent motions.In a separate development, improvements on the dispersionof velocity has not been considered until the work by Cho &Yoo (2016) and the subsequent works (Cho 2017; Yoon &Cho 2019; Cho 2019). Cho & Yoo (2016) discuss the ori-gin of magnetic field strength overestimation when using theDCF technique due to the multiple sampling of largest turbu-lent eddies along the line of sight. A suggestion of replacingthe velocity dispersive measure from the velocity line width δv los to the dispersion of velocity centroids δC was tested inCho & Yoo (2016) and show that the replacement of velocitycentroid correctly estimate the mean magnetic field strengthon the plane of sky. The problem of dealing with the largescale variations of magnetic field while statistically determin-ing the magnetic field strength was addressed in Cho (2019)by replacing the dispersion of velocity centroid to its multi-point structure function variants.A natural improvement of the DCF technique would be re-placing both δφ and δv by their structure functions. Thiscame from the energy balance of the Alfven waves, namelythe Alfven relation. The reason of why structure functionof the observables are crucial in estimating magnetic fieldstrength can be understood as follows assuming we are hav-ing an idealized fully driven incompressible magnetized tur-bulence with a constant density (cid:104) ρ (cid:105) :We can conjecture thatthe three-dimensional structure functions for both magneticfield and velocities be: SF , D { B } ( r ) = 4 π (cid:104) ρ (cid:105) SF , D { v } ( r ) (1)where SF {∗} denotes the 2nd order structure function forthe variable ∗ and r = ( x, y, z ) the three-dimensional vec-tor. In incompressible turbulence this conjecture makes sense since we expect magnetic fields and velocities are driven co-herently. The structure function formalism would be veryuseful in the aspect of theoretical point of view since Lazar-ian & Pogosyan (2012) and Kandel et al. (2017a) tackle whatphysical properties are stored in the polarization angle andvelocity structure functions respectively (See also Lazarian& Pogosyan 2008, 2016).In what follows we propose a new technique that is basedon the both the structure function measurements of both ve-locities and polarization angles from dust grain alignments(see Andersson et al. 2015 for a review) at small scales. In§2 we review the foundation of the CF technique and its lim-itations. In §3 we review the required input for the DCFtechnique. In §4 we shall discuss the theoretical formula-tion of the differential measure method that allows the esti-mation of magnetic field strength in small scales. In particu-lar, we developed a detailed formulation based on the respec-tive structure function analysis (Lazarian & Pogosyan 2008,2012, 2016; Kandel et al. 2017a) in estimating the magneticfield strength when we have different ratios of MHD modes.In §5 we discuss our numerical methods in testing §4. In§6 we perform our numerical tests and provide the recipe inapplying §5 in observations. In §7 we discuss the potentialuse of the multipoint statistics in applying our new differ-ential measure method. In §8 we compare our technique toother viable magnetic field strength estimation techniques.We discuss the possible application of the Velocity Gradientobservables in acquiring magnetic field strength in §9. In §10we discuss the achievements and the existing limitations. Wediscuss the potential applications of our technique in §11 and§12 we summarize our work. DAVIS-CHANDRASEKHAR-FERMI TECHNIQUEAND ITS LIMITATIONS2.1.
Basics of the DCF approach
In Alfvenic motions the magnetic field fluctuations andthose of the velocity are directly related through the averagedAlfven relation (Alfv´en 1942): δB = δv (4 π (cid:104) ρ (cid:105) ) / , (2)where (cid:104) ρ (cid:105) is the mean density, which can be obtained throughindependent observations. Therefore, by measuring δv onecan obtain the strength of magnetic field perturbation. Theseperturbations induce the deviations of the underlying field B mean by an angle θ , which is δθ ≈ δB/B mean . As a result,if one can measure θ it is possible to evaluate the strength ofthe underlying magnetic field in the system B mean .This simple physical mechanism is behind the DCF tech-nique (Davis 1951; Chandrasekhar & Fermi 1953) that usesglobal dispersion of magnetic field directions δθ and the ve-locity dispersion δv . The deficiency of the DCF technique isthat it is sensitive to large scale magnetic field distortions thatdo not arise from turbulent velocities as well as to large scaleshear.The Alfvenic Mach number is connected to the DCF tech-nique since M A ∼ δθ . As we shall discuss in §3 & §11 EASURING
B-F
IELD S TRENGTH M A . If the AlfvenMach number is known and less than unity , it is possibleto obtain the mean magnetic field strength from the relation M A = δBB mean . Assuming that the relation of δB and themeasured linewidth δB obey the Alfvenic relation, i.e. seeEq.(2) one can express the mean magnetic field strength as B mean = C (cid:112) π (cid:104) ρ (cid:105) δvM A (3)where δV is the 3D velocity dispersion, which is to be de-termined from observations, while C is an adjustable factor,reflecting the ambiguities associated with this simplified ap-proach.We took into account that it is the plane of sky (POS) meancomponent of magnetic field that is being explored with thetechnique. In the time being we shall assume the total meanfield are completely resides into the POS, then we can sim-ply replace B mean to B P OS . As only line of sight (LOS)velocity δv los is available in observation, for the practicaluse of the Eq. (3) the velocity dispersion δv there shouldbe associated with δv los , i.e. δv = C δv los , where C is acoefficient that relates the dispersions of the turbulence POSvelocities with the available LOS ones. For an uniformlydistributed Alfven wave that moves along the mean magneticfield line, C = √ due to the 2 degrees of freedom theAlfven wave enjoy. The coefficient C grows up when theangle between the mean magnetic field direction and the lineof sight is smaller. In the limiting case of magnetic field par-allel to the line of sight, C is not defined, as no line of sightvelocities can be associated with the Alfven motions.2.2. DCF and MHD turbulence
It is well known that the actual interstellar turbulence isdifferent from the superposition of Alfvenic waves that is dis-cussed in the pioneering studies. The simplest approximationis the incompressible turbulence. The extensive studies ofthis regime of turbulence during last two decades revealed afew distinct regimes (see Beresnyak & Lazarian 2019) whichwe describe below. We have a short summary of propertiesof MHD turbulence in Appendix A.2.2.1.
Super-Alfvenic Turbulence
If the velocity at the injection scale L inj is larger thanAlfven velocity, the turbulence is super-Alfvenic and M A > The hydrodynamic motions easily bend magnetic field atthe injection scale and the observed distribution of magneticfield is random. However, at as turbulence cascades themotions at the scale l A = L inj M − A become Alfvenic andthe are dominated by magnetic forces. Starting from thisthe amplitudes the velocity perturbations and the amplitudesmagnetic field perturbations are related to the magnetic fieldstrength.It is clear that the DCF approach is not applicable to super-Alfvenic turbulence. Indeed, the dispersion of magnetic fielddirections on the large scale is determined by hydrodynamicmotions that are marginally affected magnetic field strength. 2.2.2. Sub-Alfvenic Turbulence
In the opposite case, i.e. when the injection velocity is lessthan the Alfven velocity, the turbulence is sub-Alfvenic andmagnetic fields are strong enough to affect the magnetizedfluid motions from the injection scale. Therefore again theamplitudes of velocities and magnetic field perturbations arerelated to the magnetic field strength.It is also clear that even for sub-Alfvenic turbulence theDCF approach that ignores the actual properties of magne-tized turbulence cannot provide accurate magnetic strengthmeasurements.2.2.3.
Applying DCF in compressible turbulence
One should remember, however, that even in the case ofincompressible MHD turbulence Alfvenic turbulence is notacting alone. The fluctuations of magnetic field compres-sion which are the degenerate limiting case of slow waves,pseudo-Alfven waves, exist in this case. The scaling prop-erties of pseudo-Alfven and Alfven modes are similar, as theAlfven modes shear pseudo-Alfven perturbations and imposetheir scaling on them (Goldreich & Sridhar 1995).The situation gets more complex in the compressible MHDturbulence. In compressible media MHD turbulence can bedecomposed into Alfven, slow and fast modes. These modescascade, but the properties of them are very different. For in-stance, Alfven modes are are mostly responsible for the devi-ations of magnetic field directions (see Lazarian & Vishniac1999).2.3.
Linewidth and the dispersion of velocity centroids
Some issues with the DCF are so self-evident that it is a bitsurprising that in its simplest incarnation the approach safelysurvived till now. The most obvious problem is related tothe use of the linewidth in Eq. (3). The measured linewidthdoes not change if the emitting region extends for more thanone injection scale L inj along the line of sight. At the sametime, as it was correctly pointed out by Cho & Yoo (2016),the magnetic fields on the scale larger than L inj are beingsummed up in a random walk manner. The authors providedtheir solution that we briefly discuss in Appendix C.Cho & Yoo (2016) uses the dispersion for the velocitycentroids δC (see Eq.4) for the expression of velocity cen-troid) instead of the linewidth that also can be obtained fromobservations. The elementary transformation to the line ofsight integration involve the Jacobian change according to ρ v dv = ρ ( z ) dz , where ρ ( z ) is the density of emitters inalong the line of sight. Thus the integral in the numeratoris equal to (cid:82) L v ( z ) ρ ( z ) dz , where L is the integration lengthalong the line of sight/ In the limit of constant density, i.e. ρ ( z ) = ρ , provides the mean velocity along the line of sightmultiplied by ρ L . The integral in the denominator providesthe column density ρ L which is proportional to intensity ofmeasured radiation. Therefore, for incompressible turbu-lence C provides the turbulent velocity averaged along theline of sight. The dispersion of C at the injection scale arecompared to the dispersion of polarization directions to ob- L AZARIAN , Y
UEN & P
OGOSYAN tain the strength of the mean magnetic field as suggested inCho & Yoo (2016).In the Cho & Yoo (2016) approach both the velocities andmagnetic fields are summed up in the same way along theline of sight (see Appendix §C) and therefore the modifiedtechnique is applicable to L > L inj cases. Nevertheless, themodification of the technique shares with the DCF approachthe limitations related to the nature of magnetic fluctuations.2.4. Other limitations of the DCF and attempt to improvethe technique
Apart from using the over-simplified model of magneticand velocity fluctuations in the interstellar medium, the DCFapproach has additional deficiencies. For instance, determin-ing the velocity dispersion arising in realistic astrophysicalsettings can be problematic for the DCF. In many cases theline broadening is affected by the shear not related to turbu-lence. A typical example is diffuse HI, for which the non-thermal broadening mostly arise from galactic rotation andtherefore the measured line widths are not meaningful withinthe DCF approach.Determining the magnetic field dispersion can also beproblematic for the DCF. Self-gravity presents a seriousproblem for the DCF technique as it induces the dispersionof magnetic field directions that does not related to the effectof turbulence. Separating of the two withing the DCF ap-proach that uses the global dispersions of the magnetic fielddirections may not be possible.There have been numerous attempts to improve the accu-racy of the DCF technique, most of them based on numericaltesting with attempts to adjust the factor C in Eq. (3) (see e.g.Cho & Yoo 2016) or extend the technique to larger range of M A (Falceta-Gonc¸alves et al. 2008). We also mentioned ap-proach in (Cho & Yoo 2016) (see Appendix C). Nevertheless,all these studies have not attempted to change the nature ofthe DCF technique, namely, its use of global dispersions ofthe measures employed.In this paper we advocate the technique that uses the sameobservables, but in a different way. The new technique thatemploys the differential measures of increments of δθ and δv at small scale where the contribution of global inhomo-geneities is mitigated. To distinguish the two approaches wewill use the term Differential Measure Approach (henceforth,DMA). Note, that as we explain later in §8 our approach isvery different from that proposed in Hildebrand et al. (2009)and developed in subsequent publications. THE INPUT INFORMATION FOR OBTAININGMAGNETIC FIELD STRENGTHAs we discuss in §2 the DCF technique (Davis 1951; Chan-drasekhar & Fermi 1953) is based on the assumption that theobserved fluctuations are Alfven waves. In this simplifiedmodel the amplitude of magnetic fluctuations for a given ve-locity perturbation depends on the strength of the magneticfield. Thus, it was suggested that the amplitude of velocitycan be measured due to the Doppler shift and the magnetic field perturbation can be measured with dust polarization andthis would provide the magnetic field strength.The techniques that we discuss in the paper similarly usethe information about the magnetic field and the non-thermalvelocities, but in a different way. Below we list the measuresthat provide the required information.3.1.
Velocity centroids and channel maps as an observableof velocity information
The velocity information on astrophysical turbulent vol-ume is available from observations in the form of Position-Position-Velocity (PPV) cubes where intensity I ( X , v ) thatis the measure of the plane of sky coordinate X and theDoppler-shifted velocity v . There can be different ways ofstudy of this quantity. One can study the intensities in chan-nel maps, integrating I ( X , v ) over the thickness of the veloc-ity channel. This is the basis of the Velocity Channel Anal-ysis (VCA) technique introduced in (Lazarian & Pogosyan2000). However, dealing with the DMA we shall focus onthe velocity moments of ρ ( X , v ) , e.g. the normalised firstmoment, i.e. the velocity centroid, C ( X ) ∝ (cid:90) ba dvvρ ( X , v ) / (cid:90) ba dvρ ( X , v ) , (4)where depending on the choice of the integration limits onecan get different measures. For instance, integrating over theentire spectral line width one gets a measure known as a ve-locity centroid. If the integration limits are chosen over a partof the line, we are dealing with the reduced centroids (Lazar-ian & Yuen 2018a). For the incompressible fluid the velocitycentroids provide the value of velocity averaged along theline of sight. The reduced centroids are valuable for prob-ing turbulence in the presence of galactic rotational curve.Then, in the incompressible limit the reduced centroids pro-vide the estimate of the mean velocity over a selected part ofthe galactic media.In addition to velocity centroids, fluctuations of intensityin velocity channels carry the information about the turbu-lent velocity field. The use of velocity channels is based ontheory of turbulent fluctuations in Position-Position-Velocity(PPV) space introduced in Lazarian & Pogosyan (2000) andelaborated in subsequent publications (Lazarian & Pogosyan2006; or by the deconvolution method described from et al.2016). 3.2. Tracing magnetic fields with polarization
Stokes parameters:
In polarization observations the mag-netic field is measured using the Stokes parameters for syn-thetic observations are given by Q ∝ (cid:90) dzn cos(2 θ ) sin γ inc U ∝ (cid:90) dzn sin(2 θ ) sin γ inc θ pol = 12 tan − ( U/Q ) (5) EASURING
B-F
IELD S TRENGTH n is the number density, θ, γ inc are the POS posi-tional angle of the magnetic field and the inclination angle ofmagnetic field with respect to the line of sight, respectively. Polarization from aligned dust:
Dust polarization arisesfrom emission of non-spherical grains aligned with long axesperpendicular to the ambient magnetic field (see Anderssonet al. 2015). Similarly, polarization of starlight arises fromthe differential extinction by aligned grains. The processesof dust alignment is generally believed to happen due to ra-diative torques (RATs) (see Dolginov & Mytrophanov 1976;Draine & Weingartner 1996 ). The theory of the RAT align-ment have is based on the analytical model in Lazarian &Hoang (2007) and further studies e.g. in Hoang & Lazarian(2008, 2016).The RAT alignment theory at its present form (see Lazar-ian & Hoang 2019) can account for the major observationalfeatures of grain alignment. In particular, in typical con-ditions of diffuse ISM the silicate grains are nearly per-fectly aligned, while in dense molecular clouds the degreeof alignment depends on the grain illumination mostly byembedded stars. In other words, the existing grain align-ment theory can evaluate in what conditions one shouldexpect the polarization arising due to the aligned dust totrace magnetic fields. With more polarization measurementsobtained using starlight and with more distances to starsmeasured there is a possibility to trace magnetic field in 3D.
Goldreich-Kylafis Effect:
Goldreich, & Kylafis (1981;1982, henceforth GK) effect provides a viable way of tracingmagnetic fields in molecular clouds. The polarization arisesdue to the differences of the radiation transfer in the mediawith anisotropies or shear. The resulting polarization is ei-ther parallel or perpendicular to the magnetic field. In spiteof this ambiguity, the effect has been successfully employedto trace magnetic field structure of molecular clouds (Li et.al2011). Combining GK with velocity gradients one can re-move the 90 degree ambiguity in the magnetic field direction.
Ground State Alignment:
A promising developmentin terms of magnetic field tracing is presented by theatomic/ionic ground state alignment (GSA) effect sug-gested and quantified for use in astrophysical conditionsby (Yan & Lazarian 2006, 2007, 2008, 2012). The GSAemploys atoms/ions with fine and hyperfine split levels. Theatoms/ions get aligned in the ground or metastable stateby external anisotropic radiation. The Larmor precessionin the ambient magnetic field re-aligns the atoms/ions im-printing its direction on polarization. The atoms/ions stayin ground or metastable state long and thus they can tracevery weak magnetic fields. The effect has been recentlyconfirmed with observations (Zhang et al. 2019), openinga wide avenue of applying it for tracing magnetic fieldsin various environments. The difference in distribution ofatoms and conditions for atomic alignment in space pro-vides a way to get the 3D distribution of magnetic fieldin diffuse medium. The technique is especially interest- ing for probing magnetic field direction near bright sources.3.3.
Tracing magnetic field from velocity gradients
In a recent series of papers we introduced velocity gra-dients as a way of tracing magnetic field (see Gonz´alez-Casanova & Lazarian 2017; Yuen & Lazarian 2017a,b;Lazarian et al. 2017; Lazarian & Yuen 2018a; Hu et al.2019a). The physical explanation why velocity gradients indiffuse media are perpendicular to the LOS projected mag-netic field is routed in the theory of MHD turbulence (Gol-dreich & Sridhar 1995) and turbulent reconnection (Lazarian& Vishniac 1999).In particular, the theory of magnetic turbulent reconnec-tion (Lazarian & Vishniac 1999) predicts that the turbu-lent motions perpendicular to the magnetic field are notconstrained by the back-reaction of magnetic field. Thispresents the favorable way of turbulent cascading with mostenergy concentrated in the form of eddies perpendicular tothe local direction of magnetic field. The notion of ”lo-cal magnetic field of eddies” is the key for understandinghow the gradient technique works. Indeed, if the rotationof turbulent eddies is aligned with magnetic field, then thegradients of velocity amplitudes are perpendicular to themagnetic field and therefore they can trace the magneticfield direction. As the magnetic field reconnects in oneeddy turnover time (Lazarian & Vishniac 1999) the eddymotions are Kolmogorov-like with the scaling of turbulentvelocities v l ∼ l / ⊥ , where l ⊥ is eddy diameter perpendic-ular to local direction of magnetic field. As a result, thegradients of velocity amplitude scale as v l /l ⊥ ∼ l − / ,meaning that the maximal gradients are produced by thesmallest resolved eddies. Due to this scaling, regular sharingmotions do not affect the velocity gradient measurements. Tracing B-fields with gradients:
A formal discussion of thevelocity gradient technique is provided in Lazarian & Yuen(2018a). There it is shown that the gradients arising fromAlfven and slow modes are perpendicular to the magneticfield, while the gradients arising from fast modes are parallelto magnetic fields. Studies of compressible MHD turbulenceshow that the dominant contribution in most cases arises fromAlfven and slow modes.Velocity gradients present a possibility of 3D studiesif different molecular lines are used. Indeed, differentmolecules are produce and survive at different depth inmolecular clouds. This opens a possibility of studying mag-netic fields in molecular clouds at different depths (Yuen& Lazarian 2017b; Hu et al. 2019b) In addition, galac-tic rotation provides a way to probe magnetic field at dif- A common misconception about the MHD theory is related to the fact thatthe concept of local direction of magnetic field is not a part of the originalGoldreich & Sridhar (1995) idea. As we discuss this concept naturallyfollows from turbulent reconnection, which is proved in the subsequentnumerical studies (Cho & Vishniac 2000; Maron & Goldreich 2001; Cho& Lazarian 2002). L AZARIAN , Y
UEN & P
OGOSYAN ferent distances from the observer (Gonz´alez-Casanova &Lazarian 2018). Note, that due to the galactic rotation, ve-locity gradients can sample magnetic fields in many moreclouds in the galactic disc compared to far infrared po-larimetry. For the latter the confusion of emission fromdifferent clouds along the line is sight is detrimental.
Dispersion of Velocity Gradient Orientation:
Lazarian etal. (2018) discuss the possibility of obtaining M A throughthe Velocity Gradient Technique (Gonz´alez-Casanova &Lazarian 2017; Yuen & Lazarian 2017a,b; Lazarian et al.2017; Lazarian & Yuen 2018a). The gradients of spectro-scopic observables in diffuse interstellar media are test bothnumerically and observationally that they are perpendicularto the local magnetic field directions. Moreover, Lazarian etal. (2018) showed that the dispersion of the velocity gradientorientation is correlated to the local Alfvenic Mach number.Employing this technique Hu et al. (2019a) is possible to es-timate the magnetization of a number of molecular clouds onthe sky and obtain previously unachievable magnetic infor-mation on a high velocity cloud hid behind the galactic arm. NEW TECHNIQUE: DIFFERENTIAL MEASUREANALYSIS4.1.
Simplified approach
Consider the variations of the observed magnetic field di-rection within a volume with size L measured along the lineof sight and the turbulence injection scale L inj . The varia-tions of the magnetic field angle can be characterized by tan δθ ≈ (cid:82) δBdz (cid:82) B P OS dz (6)where the integration is done along the line of sight andwhere, without losing generality, we assumed that δB ismeasured along the line of sight and perpendicular to themean magnetic field B P OS in the plane of the sky. Naturally,for sufficiently small (cid:82) δBdz/ (cid:82) B P OS dz an approximation tan δθ ≈ δθ is valid (See Falceta-Gonc¸alves et al. 2008).However, in this study we do not need to use this approxima-tion.If we are interested in the spatial variations of the observedmagnetic field directions at the scale l , those can be obtainedusing the second-order structure functions ( SF ) of the po-larization angle φ : SF D { φ } ( R ) = (cid:104) [ φ ( X + R ) − φ ( X )] (cid:105) X (7)where X is a two dimensional vector Plane of Sky (POS), (cid:104) ... (cid:105) X denotes an ensemble averaging on the variable X . Forpractical applications this means averaging for different X We are here to use θ to denote the magnetic field angle and φ as the po-larization angle because there is a subtle difference between them in somespecial geometry of magnetic field lines. See Lazarian & Yuen (2018a) fora discussion. over the area (cid:29) l . If within this area the (cid:82) B P OS ds doesnot significantly change, the averaging in Eq. (7) amounts toaveraging of the integrals of structure functions SF D { B } ( R ) = (cid:90) (cid:90) SF D { b } ( r ) dz dz (8)where SF D { b } ( r ) is the structure function of the POS mag-netic field SF D { b } ( r ) = (cid:104) [ b turb ( x + r ) − b turb ( x )] (cid:105) x . (9)with x be the 3D position vector; z and z denote the line ofsights along which the integration of the structure function of3D fluctuating magnetic field b turb is performed.In the system of the mean magnetic field, which is the onlysystem that is available in the absence of 3D data, there isno scale-dependent anisotropy that is predicted in GS95 rela-tions. The anisotropy at all scales is determined by the vari-ations of the magnetic field direction at the injection scale,as it was demonstrated in Cho et al. (2002). Therefore, thesame spectral slope of the fluctuations can be measured par-allel and perpendicular to the mean magnetic field. In this sit-uation, for the sake of simplicity, we will use structure func-tions averaged over the positional angle, which will makethese functions only dependent on the line of sight distance l separating the points.Observing that fluctuations of turbulent field are accumu-lated along the line of sight L in a random walk fashion onegets the structure function of the polarization angle φSF D { φ } ( l ) ≈ SF D { b } ( l ) l L (10)where l is the separation between the lines of sight. In statis-tical sense, the turbulence has the axial symmetry around thedirection of mean magnetic field (see discussion in Lazarian& Pogosyan 2012). As a result, when we observe perpendic-ular to the magnetic field the eddies that have cross-section l in the POS plane, have also the extension l along the lineof sight. These eddies are independent entities at the scale l and therefore their summation happens in the random walkfashion. This provides the physical justification of Eq. (10).In fact, the problem at hand has 3 scales - separation onthe sky l , integration/cloud depth L and the injection scale L inj , which is also the line-of-sight correlation length. Sothe answer is expressible via those three. If L (cid:29) l , we shouldget SF D ( l ) ∝ SF D ( l ) l L , so extra length factor is variablewith l , which makes the slope steeper by unity as long as l is sufficiently small compared to L inj and L . This is theprincipal case that we are interested to explore in this paper.Note, that in the limiting case of l (cid:29) L i.e we just takea narrow slice, we get SF D ( l ) = SF D ( l ) L . This is aspecial case of studies when only a narrow surface area ofthe turbulent volume being proved by observations. This casecan be realized in the presence of strong dust absorption asdiscussed in Kandel et al. (2018). We do not discuss this casein the present work. EASURING
B-F
IELD S TRENGTH B P OS L . As a result, the mea-sured structure function is (cid:112) SF D { θ } ( l ) ≈ SF / D { b } B P OS (cid:114) l L . (11)For Alfvenic turbulence the fluctuations of velocityand magnetic field are symmetric. Therefore v turb = b turb / √ πρ , where ρ is the plasma density. In terms ofstructure functions this means that the structure function ofvelocity: SF D { v } ( l ) = (cid:104) [ v turb ( x + l ) − v turb ( x )] (cid:105) x . (12)is related to the structure function of magnetic field in Eq. (9)as SF D { b } = 4 π (cid:104) ρ (cid:105) SF D { v } . (13)where the averaging variable x is suppressed.With observational spectral line data, one can measure thestructure function of velocity centroids: SF D { C } ( R ) = (cid:104) [ C ( X + R ) − C ( X )] (cid:105) X , (14)which presents the proxy of the structure function of the ve-locities, averaged along the line of sight. Due to this sum-ming up of velocities procedure, the addition of velocity fluc-tuations happens similar similar to summing up of magneticperturbations δb turb that we deal with earlier. As a result, thesummation process of the velocity fluctuations is a randomwalk process, i.e. SF D { C } ( l ) ≈ (cid:90) L SF D { v } ≈ SF D { v } l L (15)Combining Eqs. (15), (13) and (11), one gets the expressionfor the mean magnetic field: B ⊥ ≈ f (cid:112) π (cid:104) ρ (cid:105) SF / D { C } ( l ) SF / D { φ } ( l ) (16)where both SF centroid ( l ) and SF θ ( l ) are available fromobservations and f is constant of order unity that depends onthe percentage of fundamental modes (see Cho & Lazarian2002) that compose the MHD turbulence. One can argue thatfor Alfvenic motions f should be ≈ as this case the fluc-tuations of velocity and magnetic field are identical in ampli-tude. In our simplified approach f is a factor that should bedetermined from numerical simulations. In general, can alsohave the dependence on the angle between the line of sightand the mean magnetic field direction.We would like to stress that Eq. (16) is applicable to sit-uations that the turbulence injection scale L inj is larger orsmaller L , as long as L (cid:29) l , the correlation length. The onlyrequirement is that L should be the same for the calculationsof SF / D { C } ( l ) and SF / D { φ } ( l ) . This requirement is au-tomatically fulfilled if we use velocity gradients or ground state alignment are used to find SF / D { φ } ( l ) . The case ofdust polarization requires more care to be sure that the po-larization is collected from the same column of gas that con-tributes to the line emission. For instance, if the used line is13CO, it is necessary to make sure that the column densityof gas associated with CO emission is much larger than thecolumn density of the of HI along the same line of sight.If turbulence is uniform and homogeneous Eq. (16) isequivalent to Eq. (7) as for l → ∞ the structure functions getproportional to the total dispersion. However, in astrophys-ical situations, we have to deal with inhomogeneous sam-ples for which differential measurements that reveal smallscale inhomonogeneities are advantageous. We shall call themethod of differential measures the Differential MeasureAnalysis (DMA) .The advantages of using the new DMA compared to DCFcan be briefly summarized as follows:• While dispersion of velocities and magnetic field di-rections that are employed by DCF are distorted by thelinear large-scale shear, the structure functions used inthe DMA are not sensitive to it.• On large scales the structure of observed magnetic andvelocity field is determined by gravity, outflows andother galactic processes determining the contours ofindividual molecular cloud, this is not a problem forthe DMA that focuses only on the small scale differ-ences in magnetic and velocity properties.• Self-gravity induces additional distortions of magneticfield making the classical DCF approach not applica-ble. The DMA is expected to work in the case of thedistorted magnetic field.A clear illustration of the first point in the list above is thatEq. (16) is applicable to studies of magnetic field using the 21cm line of atomic hydrogen. This line is broadened by boththermal motions and also galactic rotation, but one can stilluse structure functions of velocities using Reduced VelocityCentroids (RVCs) introduced in Lazarian & Yuen (2018a).Using the gradients of the RVCs one can trace the distributionof magnetic fields as a function of distance from the observer(see Gonz´alez-Casanova & Lazarian 2018). As a result, Eq.(16) allows one to get the 3D distribution of B P OS in theGalactic disk.We note that if we apply Eq. (16) to the turbulence at largescale for l comparable with the turbulence injection scale L inj , we are getting not the DCF classical expression, but itsgeneralization obtained in Cho & Yoo (2016). We show inAppendix C that in this limit we can obtain can obtain mag-netic for cases that L > L inj , which is beyond the domain ofthe traditional DCF formula.4.2. Detailed calculations
DCF approach does not take into account the properties ofMHD turbulence. Based on the idea of linear Alfven waves,it was assumed to be applicable to more realistic turbulent L
AZARIAN , Y
UEN & P
OGOSYAN settings. Our estimates in the previous section went one stepfurther by taking into account that turbulent eddies producerandom walk when their contributions are summed up alongthe line of sight. However, the actual MHD turbulence ismore than that. In Lazarian & Pogosyan (2012, henceforthLP12) we described the statistics of magnetic fluctuationsarising from MHD turbulence. In the subsequent study byKandel et al. (2017a, henceforth KLP17) the fluctuationsof velocities have been described following the approach inLP12. These papers provide the basis for our detailed calcu-lations.In what follows, we use the results of LP12 and KLP17to have a derivation valid in the case of correlated magneticfield and velocity fluctuations, without reliance on the sim-plified considerations of random walk integrating over theline of sight that we used in the previous section. To do thiswe use ”synchrotron polarization” formalism from Appendixof LP12 to describe the direction of magnetic field. Note,that the approach LP12 does not depend on the way we tracemagnetic field. It can be synchrotron polarization or dustpolarization, or velocity gradients etc since the mathemati-cal structure of structure functions computed by the magneticfield directions traced by these methods exhibit the same be-havior.The strategy below can be literally summarized as follows:(1) We would first discuss what are a legitimate structurefunctions for velocity and magnetic field angles that couldbe measured observationally. (2) We then derive the expres-sion of the structure function in 2D in relation to its 3D vari-ant (3) We perform multipole expansion for each of the 2Dstructure functions according to Lazarian & Pogosyan (2012)for magnetic field angles and Kandel et al. (2017a) for veloc-ity centroids. (4) We shall see how the ratio of the multipoleterms of the structure functions would resemble the magneticfield strength of a given localized volume.4.2.1.
Multipole expansion of Stokes Parameter structurefunctions
For angle of polarization signal that traces the magneticfield direction (synchrotron, dust polarization, synthetic po-larization from gradient maps) we can quite generally write cos(2 φ ) = (cid:82) dz ( H x − H y ) (cid:82) dz ( H x + H y ) ∝ Q/I (17) sin(2 φ ) = (cid:82) dz H x H y (cid:82) dz ( H x + H y ) ∝ U/I (18)where
I, Q, U are the Stokes parameters (See §6) from whichwe can construct the structure function. (cid:42)(cid:18) Q I − Q I (cid:19) (cid:43) + (cid:42)(cid:18) U I − U I (cid:19) (cid:43) = 2 (cid:104) − cos(2( φ − φ ) (cid:105) (19) We drop here the degree of polarization p , since we are just interested inthe structure functions of polarization angles. In the limit of small angle differences this structure functionis proportional to one given by Eq. (7). In fact, in the caseof small angle differences Eq.19 reduces to (cid:104) ( φ − φ ) (cid:105) .However, it is a more general expression that is better definedfrom observations, and also applicable to the case when anglefluctuations are large, e.g., when the Alfvenic Mach numberis large.If we assume that the denominator is dominated by themean field, we get (cid:104) − cos(2( φ − φ ) (cid:105) ≈ D QQ + D UU ( ¯ I ) (20)where, using the notation listed in Table 3 (See also Lazarian& Pogosyan 2012): ¯ I = (cid:90) dz ( H x + H y ) = L σ H ⊥ (21)Following Appendix C & D in Lazarian & Pogosyan (2012), D QQ + D UU ≈ L σ H ⊥ (cid:90) dz ( D + ( R, z ) − D + (0 , z )) (22)thus D φ ≡ (cid:104) − cos(2( θ − θ ) (cid:105)≈ (cid:82) dz ( D + ( R , z ) − D + (0 , z )) L σ H ⊥ (23)We note, that this derivation assumed that the mean magneticfield dominates the perturbations, so in the same spirit onecan replace the second moment by the square of the magneticfield, σ H ⊥ ≈ ¯ H ⊥ .The statistics of centroids was recently discussed in Kandelet al. (2017a). There, for the sake of theoretical conveniencethe definition of centroids was modified compared with thestandard one given by Eq. (1). In particular, the numeratorwas divided not by the intensity at the given point, but by themean intensity. A numerical study in Esquivel & Lazarian(2005) shows that this change does not significantly alter thestatistics of the centroids. At the same time, this significantlysimplifies the analytical treatment of the centroids.In multipole representation (cid:82) dzD + ( R , z ) has coefficients D + n = A ( A,F,S ) B C n ( m ) R m ∞ (cid:88) s = −∞ (cid:98) E s G ( A,F,S ) n − s ( γ ) (24)where the amplitude A B that has dimensions of (cid:2) H L − m (cid:3) appears in the definition of the power spectrum of a giventurbulent mode E ( k, µ = ˆ k · ˆ B ) = A ( A,F,S ) B k − − m (cid:98) E ( µ ) (25) We denoted with hat all non-obviously dimensionless quantities
EASURING
B-F
IELD S TRENGTH A A,F,SB L minj = (cid:10) δB (cid:11) × (2 π ) (cid:82) ∞ d ln kk − m (cid:82) − dµ (cid:98) E ( µ ) (26)where L inj is the energy injection scale. Accurate model in-volve smooth truncation of the power spectrum at this scale,rather than a sharp cutoff at dimensionless wavenumber sat-isfying kL inj = 1 , but the dimensionless integral factors willanyway drop out from the subsequent consideration.For the structure function of angle fluctuations we there-fore obtain D φn = A A,F,SB L ¯ H ⊥ C n ( m ) R m ∞ (cid:88) s = −∞ (cid:98) E s G ( A,F,S ) n − s ( γ ) (27)where effective L inj / L factor expresses the suppression ofstructure function amplitude due to random walk in the iner-tial range that starts with L inj < R < L range. Note that theLazarian & Pogosyan (2004) and or by the deconvolutionmethod described from et al. (2016) formalism uses k z = 0 approximation for evaluating z integral on z-coordinate dif-ferences along two line of sights, assumes that the integrationrange exceeds k − z of any scale of interest. Thus it is not ap-plicable near L inj if L inj > L since L − > k z ∼ L − inj will not average out, so we assume that the integration depthexceeds the injection scale.4.2.2. Multipole expansion of velocity centroid structure functions
The subsequent step is to evaluate the structure function inthe nominator via the structure function of velocity centroidswhich has a very similar behaviour, given that the magneticfield and velocity scales in the same way. Indeed the multiplemoments of the structure function of centroids (Kandel et al.2017a), normalized by the mean column intensity of the gasalong the line of sight ¯ I = (cid:15) L ¯ ρ , have the form (cid:101) D n = D n / ( (cid:15) ¯ ρ L ) == ( L inj / L ) A v C n ( m ) ∞ (cid:88) p = −∞ ˆ E s W n − s ( R/L inj ) m , (28)where we have used the fact that ˆ A p of KLP16 is equal to (cid:98) E p of LP12 to change the notation to that of Equation (24).4.2.3. The ratio of the multipole expansions of magnetic fieldangle and velocity centroid structure functions
The amplitude of velocity perturbations A v has the samerelation to the variance of (cid:104) δV (cid:105) as A B has to (cid:104) δB (cid:105) . Thusratios of the centroids and angle structure function multipolecoefficients can be expressed with the help of variances as D n D θn = ¯ B ⊥ (cid:10) δV (cid:11) (cid:104) δB (cid:105) × (cid:80) p ˆ E p W ( A,F,S ) n − p ( γ ) (cid:80) p ˆ E p G ( A,F,S ) n − p ( γ ) (29) We note that the residual dependence on the orientation ofthe magnetic field with respect to the line of sight is arisingprimarily due to different geometrical structure of the veloc-ity and perpendicular magnetic fields as expressed in distinctgeometrical weights W ( γ ) and G ( γ ) .4.2.4. The effect of the composition of MHD modes
Now we need to choose the composition of MHD tur-bulence in terms of energies in different modes.As we dis-cussed in Appendix A the velocity gradients of slow andAlfven modes trace magnetic field the same way, while thefast modes produce a perpendicular orientation of gradients.To have a discussion relevant to both to polarization and togradients we defer considering the fast mode to the subse-quent publication. This partly justified by the fact that fastmodes do not dominate in MHD turbulence and they aresubject to damping that is usually stronger than for Alfvenmodes. Therefore, dealing with turbulence at small scaleswe may frequently disregard the contribution of fast modes.For incompressible driving that we employ in our numeri-cal simulation in order to test our expressions, fast modesare subdominant at all scales (see Cho & Lazarian 2002).We shall discuss two of the simple cases here and workon the low β case with numerical analysis instead (See §6). Pure Alfven case:
Note that purely Alfvenic case in our ap-proximation has zero fluctuations if the mean field is per-pendicular to the line of sight due to our formal integra-tion over the line of sight that set k z = 0 . In LP12 weexplained that in the mean magnetic field system of ref-erence for finite turbulent Alfven Mach number M A oneshould account for magnetic field wandering which increaseswith M A . This allows avoiding degeneracies that arise dueto the excessively idealized setting. However, some sup-pression of Alfvenic perturbation in line-of-sight projectionfor perpendicular field should be real effect. Accountingfor the mean magnetic field changes along the line of sightleads to partial isotropization of the geometrical effects inEquation (29) which can be modeled by weighted additionof an isotropic term to geometrical functions as G ( A ) n − p → W I ( M A ) δ np + W L ( M A ) G ( A ) n − p and W ( A ) n − p → W I ( M A ) δ np + W L ( M A ) W ( A ) n − p . Following suggestion of LP12, one canadopt a simple model W I ≈ M A / M A , W L ≈
11 + / M A . (30) This model corresponds to assuming that at low Alfv´enic Mach num-bers, the tangent of the typical deviation ∆ φ of the local direction of themagnetic field from the global mean one is given by M a and therefore cos (∆ φ ) ≈ / (1 + M A ) , while at large M a the field wandering an-gle covers all the values from 0 to π/ , thus cos (∆ φ ) ≈ / . We usethis opportunity to note an inconsistency in LP12 where W I as used inEquation (71) is twice the one introduced in Equation (45). AZARIAN , Y
UEN & P
OGOSYAN
Retaining only the monopole ˆ E and quadrupole ˆ E ± in thepower spectrum expansion, we obtain D D θ ≈ ℵ − ¯ B ⊥ W I + W L (1 − cos γ (1 + K ( γ ))) W I + W L cos γ (1 + K ( γ )) (31)where K ( γ ) = 2 ˆ E ˆ E − cos γ γ ℵ = (cid:10) δB (cid:11) (cid:104) δv (cid:105) ≈ π ¯ ρ (32)Then, using Eq. (30), one can obtain the strength of perpen-dicular magnetic field as ¯ B ⊥ = (cid:112) πρ (cid:115) D D θ M − A cos γ (1 + K ( γ ))1 + M − A [1 − cos γ (1 + K ( γ ))] (33) High β ( M A < ) case: The situation is much simplerfor the case of strong turbulence in high β ∝ M A /M s plasma, where both Alfvenic and slow modes are excitedwith the same power. Physically this corresponds to the caseof incompressible MHD turbulence. In this case W A + Sp − n = G A + Sn − p = δ np so that we obtain D n D θn = ¯ B ⊥ ℵ − (34)Angular dependencies disappear despite the anisotropic dis-tribution of power in each mode, since the motion structure isisotropic for such a mix. Same relation between variances ofvelocity and magnetic field fluctuations is also true for themix of Alfvenic and slow modes (high β ) since in this casethese modes are two just polarization of the same motionsand have equal power. We then conclude that ¯ B ⊥ = (cid:112) πρ (cid:115) D n D θn , (35)which holds for all possible n = 0 , , ... . Eq.35is a very simple expression for magnetic fieldstrength and very similar to that given by Eq. (16).4.3. Uncertainties and applicability
Due to its simplicity Eq.(35) can be considered as our ma-jor result that we can recommend for the practical obser-vational studies. The equal admixture of Alfven and slowmodes is a good approximation to the weakly compressibleMHD turbulence. To move further one requires to know amore detailed composition of MHD turbulence in terms of We expect that with contributions of the fast modes, we shall have a ”f”factor as we have in Eq.16. fundamental modes. This faces both theoretical and practicaldifficulties. On the practical side, the procedures of decom-position of contributions from different modes (see Kandelet al. 2017a) have not been applied to observations. On thetheoretical side, the shocks formed by turbulent motions donot fit well into the picture of fast modes. All these issuesdeserve a rigorous study to be done elsewhere. We shouldjust add here that qualitatively one expects to overestimatethe strength of magnetic field if fast modes are present.As we mentioned earlier, Eq.(35) is formally very similarto the one obtained via our simplified approach in Eq. (16).The difference, however, that with our detailed approach weunderstand nature of the approximation that is used to obtainthis expression. We also can see the nature of the uncertain-ties that are related to the practical use of Eq. (35).Incidentally, Eq. (35) provides the estimate of magneticfield strength without the requirement of turbulence to havepower law for all scales. By measuring the structure func-tions one localizes the contribution of the scales correspond-ing to the separation of the line of sight. Therefore it isenough to have the turbulence around this scale.Note, that Eq.(35) uses only the monopole part of the mul-tipole decomposition in LP12. This monopole part can beeasily obtained via isotropic averaging of observational data.In this paper we did not use the higher moments of the LP12multipole decomposition, in particular, we did not use thequadropole term. This term carries the information about theanisotropy imposed by the on turbulence by the presence ofthe mean field. The amplitude of this term is another sourceof the information on the Alfven Mach number of turbulence.Naturally, this provides synergy and additional testing of theway of evaluating the magnetic field strength that we discussin this paper. Making use of this quadropole term is the goalof our further studies. METHODMost of the numerical data cubes are obtained by 3DMHD simulations that is from a single fluid, operator-split, staggered grid MHD Eulerian code ZEUS-MP/3D toset up a three dimensional, uniform, isothermal turbulentmedium. To simulate the part of the interstellar cloud, pe-riodic boundary conditions are applied. These simulationsuse the Fourier-space forced driving solenoidal driving. Forisothermal MHD simulation without gravity, the simulationsare scale-free. If V inj is the injection velocity, while V A and V s are the Alfven and sonic velocities respectively, then thetwo parameters, namely, the Alfven Mach numbers M A = V inj /V A and sonic Mach numbers M s = V inj /V s , deter-mine all properties of the numerical cubes and the resultantsimulation is universal in the inertial range. That means Our choice of force stirring over the other popular choice, i.e. of the decay-ing turbulence, is preferable because only the former exhibits the full char-acteristics of turbulence statistics, e.g power law, turbulence anisotropy,extended from k = 2 to a dissipation scale of pixels in a simulation ,and matches with what we see in observations (e.g. Armstrong et al. 1995;Chepurnov & Lazarian 2010). EASURING
B-F
IELD S TRENGTH Table 1.
Description of MHD simulation cubes which some ofthem have been used in the series of papers about VGT (Yuen &Lazarian 2017a,b; Lazarian & Yuen 2018a,b). M s and M A are theR.M.S values at each the snapshots are taken.Model M S M A β = 2 M A /M S Resolutionhuge-0 6.17 0.22 0.0025 huge-1 5.65 0.42 0.011 huge-2 5.81 0.61 0.022 huge-3 5.66 0.82 0.042 huge-4 5.62 1.01 0.065 huge-5 5.63 1.19 0.089 huge-6 5.70 1.38 0.12 huge-7 5.56 1.55 0.16 huge-8 5.50 1.67 0.18 huge-9 5.39 1.71 0.20 e6r3 (time-series) 5.45 0.24 0.0019 Ms0.2Ma0.2 0.2 0.2 2 Ms0.4Ma0.2 0.4 0.2 0.5 Ms4.0Ma0.2 4.0 0.2 0.005 Ms20.0Ma0.2 20.0 0.2 0.0002 incompressible 0 0.7 ∞ one can easily transform to any arbitrary units as long asthe dimensionless parameters M A , M s are not changed. Thechosen M A and M s are listed in Table 1. For the case of M A < M s , it corresponds to the simulations of turbulentplasma with thermal pressure smaller than the magnetic pres-sure, i.e. plasma with β/ V s /V A < . In contrast, thecase that is M A > M s corresponds to the magnetic pressuredominated plasma with β/ > . To investigate the behav-ior of the incompressible case, we adopt the incompressiblecube our previous work Lazarian et al. (2017).Further we refer to the simulations in Table 1 by theirmodel name. For example, the figures with model name indi-cate which data cube was used to plot the corresponding fig-ure. Each simulation name follows the rule that is the nameis with respect to the varied M s & M A in ascending order ofconfinement coefficient β . The selected ranges of M s , M A , β are determined by possible scenarios of astrophysical turbu-lence from subsonic to supersonic cases. NUMERICAL TESTS6.1.
Building up the numerical recipe for incompressibleMHD turbulence
To use Eq.35 practically, the two structure functions D n and D θn need to have the same power law with respect to dis-tance, i.e. D V ∝ D φ ∝ r m for some m with distance r smaller than the injection scale . This requirement is easilyfulfilled in the inertial range of incompressible sub-Alfvenic magnetized turbulence. For instance, Fig 1 shows the be-havior of structure functions for both velocity and magneticvariables in 3D and projected 2D space in an incompress-ible magnetized turbulence with M A = 0 . . For easier vi-sual comparisons we normalize the structure functions bythe variance of the respective variables since SF { v } ( R →∞ ) → (cid:104) δv (cid:105) .In Fig 1 we plot the angular average structurefunctions, i.e the monopole term of their angular dependence,for instance for the velocity one SF D { V } ( R ) = 12 π (cid:90) dθSF D { V } ( R ) (36)The angular averaged structure functions are plotted in thedistance range of , L/ where L is the size of the simulationregion, in our case L = 512 pixels. Fig 1 shows the 2Dstructure functions for projected velocities V = (cid:82) dzv andpolarization angles φ = 0 . − ( U/Q ) , in which they havethe same power-law slope as a function of r when r ∼ − pixels. We shall utilize the range of scales that the twostructure functions have the same power-law slope for theestimation of magnetic field strength.Traditional DCF technique uses the ratio of δv to δφ asan estimation of magnetic field strength (weighted by √ π ¯ ρ .The use of the dispersions of v and φ correspond to the partof their respective structure functions in Fig.1 that has a flat slope and has r ∼ L inj . Hence the ratio of the dispersionfunctions, aka the structure functions with r ≥ L inj wouldnot be a function of distance. However the Alfven relation(Eq2) develops only in scales smaller than the characteristicscales of the magnetized turbulence (See §A) and these scalesare smaller than L inj . The fundamental physical issue for theDCF technique that utilizes the dispersion of observables inan unphysical length scale could be addressed properly by thestructure function treatment which we are delivering below.We shall seek for the part of the two structure functions SF { V } ( R ) and SF { φ } ( R ) that have the same power-law slope. The length scale needs to be smaller than L inj and larger than the numerical dissipation scale, which isusually − pixels depending on the properties of thenumerical solvers. We shall use the upper bound of thenumerical dissipation scales for our current analysis. Aswe show in Fig.2, the part of the two structure functionsthat carry the same power-law slope would be r = 20 − pixels. That means we could examine whether thequantity (cid:112) π ¯ ρSF { C } /SF { φ } would be approximatelya constant of r to obtain the magnetic field strength atthe length scales of r = 20 − pixels. From Eq.2,the (cid:112) π ¯ ρSF { C } /SF { φ } is flat at in the length scale of r ∼ and giving B estimated ∼ . , which is close to theglobal mean value of the magnetic field strength. We shallcall the condition of obtaining magnetic field strength bycomparing the ratio the structure functions of SF { C } ( R ) and SF { φ } ( R ) that have the same power-law dependencieswith respect to the distance r be the flat criterion for DMA.The use of the structure functions for both velocities andmagnetic field observables is advantageous compared to the2 L AZARIAN , Y
UEN & P
OGOSYAN dispersion method and also the Hildebrand-Houde methodsince it provides a unique treatment of obtaining local mag-netic field strength with less sampling points. For instance,one needs to compute only the structure functions with thedistance lag r ≥ pixels in our sample synthetic obser-vations (Fig.1). This allows observers to acquire the mag-netic field strength using smaller number of spectral informa-tion compared to the traditional DCF technique (Davis 1951;Chandrasekhar & Fermi 1953). Figure 1.
The angular averaged structure functions for the density-constant velocity centroid V and the polarization angles φ normal-ized by times their variance respectively in an incompressibleMHD simulation with M A = 0 . . Figure 2.
The estimated magnetic field strength using DMAmethod as a function of distance by Eq.33 (blue). The exact valueof ¯ B is drawn as a red dash line while the estimated value from theDMA method is marked with a blue dash line. Proceeding to compressible magnetized turbulence
Using Eq.35 in the case in compressible turbulence be-comes more complicated because of the existence of thetwo compressible modes. §4.2 discussed already how thecombination of Alfven and slow modes would contributeto the structure functions and also the differential treatmentin Eq.35. Indeed, in the presence of the compressiblemodes, the structure functions of velocities and polarizationangles are expected to behave differently from what we seefrom the incompressible counterpart since the slope of struc-ture functions are closely related to the slope of the powerspectrum, and the fast modes have different power spectralslopes ( P F ( k ) ∝ k − / ) than that of Alfven and slow modes( P A,S ( k ⊥ , k (cid:107) ∼ k / ⊥ ) ∝ k − / ⊥ ) even in small M s case (SeeCho & Lazarian 2003). Therefore the recipe that we devel-oped in §6.1 would not work unless we have an adjustmenton the case when the two structure functions SF { C } ( R ) and SF { φ } ( R ) have different power-law slope.We use the method of compensated structure functions asa workaround for using Eq.33 or Eq.35 and extends the lat-ter equations to local structure functions and also local dis-persions. The idea is illustrated in Fig. 3 with the introduc-tion of the local statistical quantities such as the local cir-cular dispersions. The local circular dispersion is simply anextreme case of the structure function since one should re-call SF ( R → L ) = 2 σ if L is the size of the region. For Figure 3.
A figure showing how the structure function of polar-ization angle is related to the its circular dispersion when sampledlocally. We first randomly select a square block of size r (pixels)and compute the circular dispersion within it. The averaged valueof 100 such selections are plotted as the ”local circular dispersion”value as the blue points in this figure as a function of block size r .For reader’s comparison, we also plot the square root of the angu-lar averaged structure function (black curve) and the global circulardispersion (red dash line). reader’s comparison, we also plot the square root of the an-gular averaged structure function (black curve of Fig. 3) and EASURING
B-F
IELD S TRENGTH r while thecharacteristic scale for the cloud to be L cloud , then we canestimate locally the dispersion of angles by δφ local = σ φ,local (cid:18) L cloud r (cid:19) ν φ / (37)The respective CF method is, formally: B ∼ (cid:112) π ¯ ρ δvσ φ,local (cid:18) L cloud r (cid:19) − ν φ / (38)The formula should subject to the theoretical correction in§4.2. We also expect if we take the total differential measureapproach , then both σ v and σ θ would have a distance com-pensation factor of (cid:0) L cloud r (cid:1) ν φ / for some structure functionpower indices ν V,φ , which accounts for the insufficient sta-tistical sampling on the sky. To utilize Eq.33, the respectiveCF method should be B ∼ (cid:115) π ¯ ρ SF V SF φ (cid:18) L cloud r (cid:19) ( ν V − ν φ ) / (39)If accidentally ν V − ν φ = 0 (most likely in sub-sonic,sub-Alfvenic or incompressible case), then there is no compensa-tion term needed. Fig. 4 shows an example on how to utilizeEq.39 when the structure functions of the two observableshave different power-law dependencies with respect to r in acompressible magnetized turbulence. The compressible sim-ulation “e6r3” used here is a super-sonic ( M S = 5 . )sub-Alfvenic ( M A = 0 . ) saturated turbulence simulation,which as a plasma β (cid:28) .In this scenario the wave-vectorof the slow mode is expect to be parallel to the local meanmagnetic field direction (Cho & Lazarian 2003).We shall apply the flat criterion for the compensated struc-ture functions. Notice that there is a particular length scale l A = L inj M A ∼ . pixels here for the Alfven relation(Eq.2) to develop (See Appendix). We are therefore seek-ing for the flat criterion to hold with length scales r ≤ l A .Here we assume that we have the knowledge of L inj = L sat = 400 pixels (green dash line of Fig.4) where L sat isthe length scale for the structure functions SF { C } ( R ) and SF { φ } ( R ) to be saturated. Using the flat criterion as de-livered in §6.1, we see that the DMA method with the lengthscale correction (Eq.39) has a very nice estimation of mag-netic field strength (40.7) compared to the actual value (38.5),despite that we do not have the information of ratio of MHDmodes on hand.6.3. Dependencies on M s , M A To apply Eq. (16) or Eq. (35) in observations, we need toknow how the constant f is related to the global properties of MHD turbulence (i.e. M s , M A ). Knowing how the conver-sion factor is related to the sonic and Alfvenic Mach numberis crucial for the DMA technique. Here we use the densityweighted centroid (c.f. Eq.4) as the density is not constantanymore in compressible turbulence for our testing of Eq.16.Fig. 5 shows how the conversion factor is related to the sonicMach number M s (left) and Alfvenic mach number M A . Onecould see that while there is a tiny fluctuation on the value of f as a function of M s and M A , the fraction of fluctuation isrelatively small ( ∼ − ) compared to the mean value.Therefore we conclude that we can take a range of value of f ∼ . − . in observation. USE OF MULTI-POINT STATISTICS ANDSUPPRESSION OF THE EFFECTS OF TO SHEARAND SELF-GRAVITYWe have used for our study two point second order struc-ture functions. Compared to correlation functions, those al-low removing the constant shifts of the foreground. In thepresence of shear we provided the procedure for removingthe shear contribution. However, there is a more robust wayof dealing with the problem that was explored for the statisti-cal studies of emission lines in Lazarian & Pogosyan (2008),namely, the use of multi-point structure functions. A detaileddescription of three and four point second order structurefunctions is given in Chepurnov & Lazarian (2009, see alsoFalcon et al. 2007; Lazarian & Pogosyan 2008; Cho 2019).For our approach the number of points does not matter,as the magnetic field strength enters the expression via theAlfvenic relation between the perturbations of magnetic fieldand velocity. Therefore with the multi-point structure func-tions we can use Eq. 33 or 35 to determine the magnetic fieldstrength.7.1.
Theoretical description of DMA in the presence ofgalactic shear and regular velocity components
One of the advantage of the DMA is the ability of tacklingregular shear flows through treatments of the structure func-tions on velocity observables. In this subsection we illustratehow to tackle this self-consistently. Let us suppose that a reg-ular velocity field v ( rg ) is added on top of the 3D turbulentvelocity v (0) , so that the total velocity is v ( t ) i ( r ) = v (0) i ( r ) + v ( rg ) i ( r ) (40)If we approximate the regular velocities to linear order inexpansion around the center r of the emitting volume, theeffect of the regular motions v ( t ) i ( r ) = v (0) i ( r ) + v ( rg ) ( r ) + v ( rg ) ij ( r j − r j ) (41)is determined by the linear shear tensor v ( rg ) i,j = ∂v ( rg ) i ( r ) ∂r j (42)Here Einstein summation over repeated indices is used and i, j = x, y, z .4 L AZARIAN , Y
UEN & P
OGOSYAN
Figure 4. (Left) The structure functions of the normalized velocity centroid C (blue) and the polarization angle φ (red). The respective trendlines ( ν φ = 2 / , ν v = 1 ) are added at the distance range of r = 20 − pixels. (Right) A plot showing the value computed by Eq.39 as afunction of distance r (blue scatter points). We search for the part of the curve that has a flat slope, which is indicated by the blue dash line.The inferred magnetic field strength by Eq.39 is close to the exact value which is indicated with the red dash line. Figure 5.
The response of the constant C as defined in Eq.16 (c.f. Eq.4) as a function of sonic and Alfvenic Mach number. For the group withvarying sonic Mach number, their M A ∼ . . While for that of varying Alfvenic Mach number, their M s are ∼ . . Assuming incompressible turbulence at constant density,the velocity centroid C ( R ) = (cid:82) dzv ( t ) z ( R , z ) is the in-tegral along the line of sight of total velocity z -component.Taking the difference of the values of two centroids at skyseparation R C ( R + R ) − C ( R ) = (cid:90) dzv (0) z ( R , z ) + L v ( rg ) zl R l (43)where L is the depth of the emitting volume and l = x, y ,eliminates all terms in the regular velocity contribution thatare constant across the sky.The standard 2-point structure function of the centroids isobtained by averaging of the square of this difference and is easily shown to be given by SF { C } ( R ) ≡ (cid:104) ( C ( R + R ) − C ( R )] (cid:105) = L (cid:90) dz (cid:16) SF { v (0) } ( R , z ) − SF { v (0) } ( , z ) (cid:17) + L S ( rg ) lm R l R m (44)where first term is the standard structure function of turbu-lent centroids, which behaves as R m if SF { v } ∼ r m .and the second term is due to regular shearing velocities with S ( rg ) lm = v ( rg ) zl v ( rg ) zm . We see that the turbulent term accruedone extra power of R in projection, while the regular termremains quadratic in R . Thus, a general model for the angleaverage two-point structure function of centroids that is con-sistent with MHD theory in the presence of regular shear has EASURING
B-F
IELD S TRENGTH SF { C } ( R ) = qR m + pR (45)where q and p are constants. For a realistic turbulence the firstterm saturates at the energy injection scale L inj . In belowwe discuss how the multipoint structure functions would aidremoving the shear contribution from the centroid statistics.7.2. Application of 3 and 4 point statistics to regions withshear
The three & four point second order structure functions aredefined as : SF pt { A } ( r ) ∝ (cid:104) [ A ( x + r ) − A ( x ) + A ( x − r )] (cid:105) SF pt { A } ( r ) ∝ (cid:104) [ A ( x + 2 r ) − A ( x + r ) + 3 A ( x ) − A ( x − r )] (cid:105) . (46)Their ability to remove shear velocity field and recover theoriginal structure functions was illustrated in Cho (2019).In a similar manner to 2-point structure function that can-celled constant velocity contribution,, 3-point and higher or-der structure functions cancel any linear and respectivelyhigher power regular contributions to velocity field. The costof using them is an increased noise contribution when appliedto noisy data.Here we test whether this is the case. Fig. 6 shows howthe 3-point and 4-point angular averaged structure functionsbehave as a function of the correlation lag r on the projectedvelocities of incompressible cube. Notice that due to the non-periodicity of the map, using more points within a region de-creases the possible number of statistical samples. We cansee from Fig 6 that the multi-point structure functions are in-deed not altered by the presence of constant velocity shearfield. This could potentially replace the method used in §7.1with a cost of reducing ranges of the correlation lag r .7.3. Applications of DMA to self-gravitating media
One of the issue of the DCF technique is its questionableapplicability in self-gravitating regions, through it is exten-sively applied to observations. Therefore we would like toexamine whether the DMA equations (Eq.33 or Eq.35) needto be modified in the presence of strongly gravitating regions.Assuming we are given a vector field of polarization angle ˆ φ defined at a space A ∈ R , then the ”polarization anglestreamlines”, which resembles the geometry of the magneticfield lines (See Yuen & Lazarian 2020b for a detailed discus-sion). Following Yuen & Lazarian (2020b),the definition ofunsigned polarization angle curvature κ would be κ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˆ φdl (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (47)where dl is the line element of the ”polarization angle stream-lines”. In the case of structure functions, the effect of curva-ture would not accumulate until κr ∼ . Therefore the struc-ture functions would have different power-law dependencewhen r < /κ and r > /κ . To study this effect, we select a Figure 6.
A figure showing the behavior of the 3 point and 4 pointprojected velocity structure function with and without the presenceof shear in the incompressible cube. To display the differences weintroduce a small horizontal offset of the structure functions of un-modified velocity field, and also a vertical offset of 3 for both the 4point structure function. pixels × pixel region from a (1200 pixels ) syntheticobservation map from ”e6r3” (See Table 1) to investigate theeffect the magnetic field line curvature to the structure func-tions.For instance, the left of Fig. 7 we show a self-gravitatingsimulation with its magnetic field streamlines (pink) andgravitational potential contours (blue). The simulation’ssnapshot is taken at t = 0 . t ff where t ff is the free falltime. This particular snapshot is taken right before the Tru-elove criterion is violated (See Truelove et al. 1997). Thecurvature distribution is displayed in the right of Fig. 7 whichshows a significant area of the synthetically observed regionshas the radius of curvature /κ < pixels.We are interested to see how the change of number ofpoints in computing structure functions would change the be-havior of the structure functions as a function of distance.Fig. 8 shows the 2-point (red), 3-point (blue) and 4-pointstructure functions (green) computed in the area of interest(the (200 pixel ) ) region and also the 2-point structure func-tion computed globally (black). We can see that all variantsof the structure functions (2-point, 3-point, 4-point) are gen-erally linear in log-log space until r = 40 pixels, whichcan be visually seen by comparing the structure functionsto that of the global structure functions computed in the (1200 pixels ) area (black). The special scale r = 40 pixelsrepresent the bending that we can visually see on the rightof Fig.7. In fact, for the 3-point and 4-point structure func-tions there is a significant change of the slope of the struc-ture functions that can hardly be seen in in the 2-point struc-ture functions. This suggests that one potentially separate thelarge scale curvature contribution to that of the small scalefluctuation of magnetic field if one compares the local mul-6 L AZARIAN , Y
UEN & P
OGOSYAN tipoint structure functions to the global structure functions,especially when applying the DMA to the self-gravitating re-gions. COMPARISON DMA WITH THE EARLIER USE OFSTRUCTURE FUNCTIONSIn Hildebrand et al. (2009) the structure functions of fluc-tuations of the polarization angle directions introduced inFalceta-Gonc¸alves et al. (2008) were used in order to findmagnetic field strength of molecular clouds. The modelwithin Hildebrand et al. (2009) which the results were ob-tained assumes that the correlation scale of magnetic turbu-lence is smaller than the lag l between the points for whichthe structure function is calculated. Such calculations are ap-plicable to very low resolution studies, e.g. the studies appli-cable to magnetic field in other galaxies.In Milky Way molecular clouds the turbulence spectrumcan be resolved in molecular clouds. For instance, Houde etal. (2011) found spectra of turbulence k − α with α = 1 . ± . . Note, that the Kolmogorov value of / is getting withinthis range. As for evaluating the strength of Houde et al.(2011)) adopted the traditional DCF technique and did notmake use of the advantages of local measured provided bystructure functions that we employ here.The idea of using differential measures in estimating mag-netic field properties is not new and has been explored by thecommunity in different ways. Esquivel & Lazarian (2005)uses the anisotropy of spectroscopic observables, e.g. veloc-ity centroids, to estimate the orientation In the work of ofmagnetic fields, while Hildebrand et al. (2009) and the sub-sequent publications (Houde et al. 2009, 2011; Chitsazzadehet al. 2012; Houde et al. 2016) investigate a model of struc-ture functions of observed polarization angles to estimate thethe turbulent-to-regular magnetic field strength ratio and thusthe strength of the regular part of magnetic field. In a sep-arate development, Lazarian & Pogosyan (2012, 2016) dis-cusses the properties of the structure function of synchrotronobservables (intensity and polarization) bases on the theoryof MHD turbulence. The analysis framework of Lazarian &Pogosyan (2012, 2016) has also been used to the studies ofvelocity centroid structure functions (Kandel et al. 2017a).In the work of Hildebrand et al. (2009), they only replace δθ → SF / D { φ } and investigate its properties under severalimportant assumptions that lead to the estimation of (cid:104) B (cid:105) :(1) The turbulence is isotropic; (2) There exist two lengthscales : the turbulent correlation scale δ and the large scalemagnetic field scale L ; (3) The magnetic field can be writtenas the sum of the regular part and turbulent part B = (cid:104) B (cid:105) + B t with a special properties that (cid:104) B t ( r (cid:48) ) B t ( r + r (cid:48) ) (cid:105) r (cid:48) = 0 for all r ≥ δ ; (4) The two dimensional polarization angle (theobservable of magnetic field angle in 2D) can be modelled bythe Taylor expansions of structure functions of polarizationangles: SF , D { φ } ( R ) ∼ b + m R (48)where b, m are some fitting factors. It is shown in Hilde-brand et al. (2009) that the turbulent to regular magnetic field strength ratio to be: (cid:104) B t (cid:105)(cid:104) B (cid:105) ∼ b − b (49)and thus combining the DCF method, 48,49 and writing SF , D { V } ( R ) → δv , (cid:104) B t (cid:105) / (cid:104) B (cid:105) → δφ , , Hildebrandet al. (2009) arrives (cid:104) B (cid:105) ∼ (2 − b )4 π (cid:104) ρ (cid:105) δv b (50)Houde et al. (2009) further expands the method developedin Hildebrand et al. (2009) by considering the telescope beameffect and also introduces a Gaussian model for both the auto-correlation function and the beam profile function. Undersuch formalism, not only could they obtain the turbulent toregular magnetic field strength ratio but also the number ofturbulent eddies N (which they call ”independent turbulentcells”) along the line of sight which could be obtained by fit-ting the structure function similar to that in Hildebrand et al.(2009). It’s worth to note that Cho & Yoo (2016) argues sepa-rately √ N ∼ δC/δv los . Similar idea is behind our derivationin Eq.(14).The approach described in (Hildebrand et al. 2009; Houdeet al. 2009) was applied to observational data to both molec-ular clouds (Houde et al. 2011; Chitsazzadeh et al. 2012;Houde et al. 2016) and also galactic disks (Houde et al.2013). However, we claim that the assumptions made withinthis approach are inconsistent to the theory of MHD turbu-lence (See Goldreich & Sridhar 1995; Lazarian & Vishniac1999; Cho & Lazarian 2002; Cho 2019) and the subse-quent theoretical studies of properties of structure functionsthat arise from MHD turbulence (see Lazarian & Pogosyan2012, 2016; Kandel et al. 2017a, §4). For instance, Eq. (45)presents what sort of structure functions we expect to see inturbulence in the presence of shear. In view of that, belowwe propose an alternative explanation of some of the obser-vational data.As we discuss in Appendix A2, if turbulence is super-Alfvenic, i.e. M A > , the magnetic fields are correlated upto the scale l A given by Eq. (A4), i.e. l A = L inj M − A . Thisscale can be found by correlating the the structure functionsof polarization angle directions, for instance (see Falceta-Gonc¸alves et al. 2008; Hildebrand et al. 2009). The turbulentinjection scale L inj can be found by correlating velocity fluc-tuations. A more sophisticated ways of measuring L inj arealso possible (see Chepurnov & Lazarian (2010)). With someadditional assumptions one can identify L inj by the analysisof density fluctuations.Some observational results were interpreted withinHildebrand-Houde approach. Below we provide an alterna-tive interpretation of the data. For instance, in Chitsazzadehet al. (2012) it was claimed that magnetic field fluctuationsin OMC-1 have the correlation scales ∼ ” and ∼ ” forthe Stokes Q and U parameters and ∼ ” for density fluc-tuations. From the theory of super-Alfvenic turbulence (seeAppendix A2), the first two numbers can be associated with EASURING
B-F
IELD S TRENGTH Figure 7. (Left) An image showing the intensity map (the Grey back-image) self-gravitating cloud from synthetic observations of a late stage ofthe time series of e6r3 with its magnetic field streamlines mapped by polarization angles plot as pink while the projected gravitational potentialdrawn with contours as blue. (Right) A map showing the radius of curvature /κ in the same area with color bar adjusted to − pixels,showing the region that has strong curvature as dark green. the angular size associated with l A , and the third number withthe injection size L inj . Using Eq. (A4) one can estimate M A for OMC-1 as (13 / / ≈ . . This means that OMC-1 is amildly super-Alfvenic object. MAGNETIC FIELD STRENGTH FROM COMBINING M S AND M A Application to channel maps
The method that we developed in §4 can also be resembledwith the turbulent Mach numbers that we could obtain fromdifferent techniques. From §4.2 we see that it might be possi-ble that there are extra weighting factors as a function of M A depending on the mode composition. In the current sectionwe shall stick with the simplest form of the DMA, i.e. theDCF technique form Eq.3 for our discussion.As we discussed in §2, in observations the measure that isdirectly available with polarization measurement is M A, ⊥ = δv/V A, ⊥ , where δv is the injection velocity and V A, ⊥ = B ⊥ / √ π ¯ ρ is the plane of sky Alfven velocity. The samevalue can be presented as M A, ⊥ = δB ⊥ /B ⊥ , where ⊥ de-notes the plane of sky magnetic field component.Notice that the value of the perpendicular component themagnetic field can be obtained from the ratio M s M A, ⊥ = B ⊥ c s (4 πρ ) − / . (51)Writing Ω = δv (cid:107) /δv and noticing that M A ∼ δv/v A, ⊥ , wewould have B ⊥ = Ω c s (cid:112) πρM s M − A, ⊥ (52) The emergence of the geometric term Ω δv (cid:107) /δv suggests thatthere is a geometrical factor that affects the measurable mag-netic field strength on the plane of sky. For Alfvenic wavesperpendicular and magnetic field perpendicular to the line ofsight Ω ≈ . In general, it depends on the nature of turbu-lence and the angle γ given by Ω ∼ sin γ . However, the geo-metrical factor is not trivial since it is a complex function ofthe Alfvenic Mach number. We shall discuss how these geo-metrical factor would affect the measurement in a later workby Yuen & Lazarian (2020c). In the time being, we shallstudy the case when Ω = 1 , i.e. B ⊥ LOS . The techniquethat we introduce here uses two different Mach numbers, theAlfven and sonic one. Therefore we will term this techniqueMM2.One special property about the use of Eq.52 is that, both M s and M A can be obtained purely from spectroscopic chan-nel map and c s can be obtained from the accepted measure-ments of temperature of emitting gas (Draine 2006) or by thethermal deconvolution method described in(Yuen & Lazarian2018). As for the value of M A , it can be obtained both polar-ization and non-polarization method, e.g. using the width ofthe probability distribution function of the gradient directionswithin a sub-block (see Lazarian et al. 2018) or the curvatureof either polarization or gradient orientations (see Yuen &Lazarian 2020b). As a result the value of M A, ⊥ can be ob-tained in a more localized fashion compared to the observa-tions of polarization. The most striking advantage of MM2 isthat applying velocity gradients to channel maps it is possibleto find the distribution of M A, ⊥ for channel maps. The corre-sponding distributions of M A, ⊥ were obtained with channel8 L AZARIAN , Y
UEN & P
OGOSYAN
Figure 8.
A figure showing how the 2-point (red,2pt), 3-point(blue,3pt) and 4-point (green,4pt) structure functions behave as afunction of r compared to the case when we compute the globalstructure function (black,global) which contains zero curvature inaverage. The turning point suggests that there is a large scale mag-netic field curvature with the radius of curvature of r ∼ pixelscontributing to the dispersion of structure functions. Two trend linesare added to show the slope differences between the 3/4 point struc-ture functions (green dash line, 1/3) and the global/2-point structurefunction (black-dash line). A vertical dash line marking r = 40 isdrawn to signify the effect of radius of curvature of pixels wehave seen in Fig.7. Vertical offsets are introduced for reader’s easyvisual comparisons between different structure functions. maps for galactic HI in Lazarian et al. (2018) and for molec-ular CO lines in Hu et al. (2019b).The sonic Mach number M s = δv/c s , where c s is thesound velocity, can also be obtained using the statistical prop-erties of the velocity channel maps. Various techniques ofobtaining M s are suggested. For instance, Burkhart et al.(2010) & Burkhart & Lazarian (2012) successfully used theskewness and kurtosis of the intensity PDFs and establishedthe relation between these quantities and M s . This techniquewas successfully applied it to HI in Small Magellanic Cloudto find the POS distribution of M s .Other ways of obtaining M s include the analysis of Tsallisstatistics (Tofflemire et al. 2011) and a more recent techniquebased on using the distribution of amplitudes of velocity gra-dients (Yuen & Lazarian 2018). Similar to PDFs, the cal-culation of M A with velocity gradients can be done locally,as it is demonstrated in Lazarian et al. (2018). The results of M s measurements are very robust and they are marginally af-fected either shear or by large-scale magnetic field curvature.Similarly, the calculation of M s is not much influenced bythe galactic shear or any other large-scale shear induced bynon-turbulent motions. Therefore, like the DMA, the MM2technique is local and, compared to the DCF, it can be usedfor a wider variety of astrophysical settings. In the regime of the Velocity Gradient Technique, the ratioof gradient amplitude to gradient dispersion provides the ex-pression M s /M A, ⊥ . In particular, the gradient observablescould be related to the Mach numbers : σ ( ∇ I ¯ I ) ∝ (cid:40) M s ( M s < M s ( M s > − R ∝ (cid:40) M − . ± . A, ⊥ ( M A, ⊥ < M − . ± . A, ⊥ ( M A, ⊥ > (53)where ∇ I/ ¯ I is the gradient amplitude of normalized inten-sity (Yuen & Lazarian 2018) and − R is the inverted vari-ance for twice of the gradient angle orientation (Lazarian etal. 2018) . This approach requires to use the unique proper-ties of velocity gradients, namely, that the same volume ofemitting gas is used both to find M s and M A, ⊥ . As a result,using the Galactic rotation curve one can obtain the distri-bution of the value of the plane of sky component of galac-tic magnetic field at different distances from the observer. Infact, we can approximate Eq.52 into the combinations of gra-dient amplitude and gradient dispersion assuming M s > : B ⊥ ∝ C Ω c s (cid:112) π ¯ ρσ ( ∇ I ¯ I )(1 − R ) β ( M s > (54)where the constant C contains the proportionality constantsrelated to the techniques in Yuen & Lazarian (2018); Lazar-ian et al. (2018). Moreover, β = 1 / . for M A < and β = 1 / . for M A > (See Eq.53). The sub-sonic for-mula can be obtained from similar manner. To test the rela-tion Eq.54, we use the set of simulations ”huge-0” to ”huge-3” in Table 1 that has constant c s , ¯ ρ , M s > , M A < andwe also put Ω = 1 . We plot the quantity Ω c s M s /M A ∼ c s σ ( ∇ I ¯ I )(1 − R ) β as a function of the Alfven speed v A = (cid:104) B (cid:105) / √ π ¯ ρ in Fig.9. From our expectation in Eq.54, we ex-pect the slope of this plot to be exactly , and the fitting slopefrom the data points computed is 1.07. Moreover, we seethat the relation holds true both for sub-Alfvenic and super-Alfvenic cases. This indicates that Eq.54, i.e. a combinationof Eq.52 with Eq.3, would predict the magnetic field strengtheven only with the spectroscopic data available.9.2. Application to synchrotron gradient measures
The traditional DCF technique and the new DMA one re-quire the spectroscopic data. However, one may notice thatthe MM2 approach requires just the ratio of two Mach num-bers, namely, M A and M s . These Mach numbers were stud-ied earlier in number of papers, including those employingsynchrotron emission from turbulent volumes. For instance,fluctuations of turbulent magnetic field can be studied both The circular standard deviation of VGT is defined as σ V GT = (cid:112) − R ) if R is known, i.e. − R = 1 − e − σ V GT / . When σ V GT is small, − R ∼ σ V GT / EASURING
B-F
IELD S TRENGTH Figure 9.
A figure showing how the quantity Ω c s M s /M A ∼ c s σ ( ∇ I ¯ I )(1 − R ) β should be related to the Alfven speed v A = (cid:104) B (cid:105) / √ π ¯ ρ based in Eq.54 in the set of simulations huge-0 to huge-9. We draw the points that are sub-Alfvenic as blue while those whoare super-Alfvenic as green. with the Synchrotron Intensity Gradients (SIGs, Lazarianet al. 2017) or
Synchrotron Polarization Gradients (SPGs,Lazarian & Yuen 2018b). These techniques were success-fully used to trace magnetic field. As discussed in Lazarianet al. (2018), similar to velocity gradients, the distributionof gradients of synchrotron or synchrotron polarization canbe used to obtain the distribution of ”perpendicular” AlfvenMach numbers M A, ⊥ .The distribution of sonic Mach numbers M s have been ob-tained with the PDFs of synchrotron polarization gradients(Gaensler et al. 2011; Burkhart & Lazarian 2012). A moreelaborate approach was proposed in Yuen & Lazarian (2018)and it is applicable to both synchrotron and synchrotron po-larization gradients. As a result, one can directly use the Eq.(52) and assume that Ω ≈ there. SIGs can be applied tofind sampling the distribution of magnetic field intensitiesthrough the entire volume. At the same time SPGs can beapplied to obtain the 3D distribution of the POS componentsof magnetic field by measuring SPGs at different frequen-cies. The corresponding procedure of magnetic field tomog-raphy using polarization gradients was described in Lazarian& Yuen (2018b). There it was applied to tracing the POS di-rection of magnetic field in 3D. However, it is obvious thatthe same approach can deliver the distribution of M A, ⊥ and M s in 3D volume. Therefore, applying Eq. (52) one shouldbe able to map not only magnetic field directions, but alsomagnetic field intensities.One may wonder why it may be interesting to measuremagnetic field intensities this way while synchrotron emis-sion is itself provide the information about the magnetic fieldstrength. The caveat here that the synchrotron intensities pro-vide the product of magnetic intensity and cosmic relativisticelectron densities. To evaluate the magnetic field strength one frequently has to make an assumption about the equi-partition of cosmic ray energy density and magnetic field en-ergy density as well as the assumption of the fraction of CRenergy in cosmic ray relativistic electrons. These assump-tions are far from trivial and in many cases they are not ex-pected to be true. In comparison, using Eq. (52) one canobtain the magnetic field strength directly. Having this esti-mate, by comparing the results with the synchrotron intensi-ties, one can get insight into the energy density of relativisticelectron distribution.9.3. Application to density gradients
In some situations the only available information is inten-sity. Turbulent density relation to the MHD turbulence prop-erties is somewhat more complicated (see Kowal & Lazar-ian 2007). Thus the intensity gradients (IGs ,see Yuen &Lazarian 2017b; Lazarian & Yuen 2018a, and also a com-parison to the Histogram of Relative Orientation in Hu et al.2019c) reflect not only the magnetic field directions, but alsoshocks. For low sonic Mach numbers M s the IGs can alsocan be used for magnetic field tracing. Obtaining M A withintensity gradients was explored in Hu et al. 2019c and theanalysing either the PDFs of intensities (see Burkhart et al.2010) or amplitudes of the intensity gradients similar to Yuen& Lazarian (2018)it is possible to find M s . As a result, Eq.(52) can again be used to estimate the magnetic field inten-sity.Naturally, due to IGs being an inferior tool for describ-ing magnetic field properties compared to velocity of syn-chrotron gradients, we expect a lower level of accuracy fordetermining the magnetic field strength. However, in thecases where no other sources of information are available,this can be a valuable way of magnetic field study. ACHIEVEMENTS AND EXISTING LIMITATIONSThe DCF technique is widely used technique with wellknown limitations. It is an empirical technique with seri-ous problems related to its accuracy of obtaining the value ofmagnetic field strength.In this paper two new techniques were considered. TheDMA technique uses the ratio of the structure functions ofthe Stokes parameters and velocity centroids in order to cal-culate the magnetic field. The technique is based on the the-ory of MHD turbulence. We demonstrated that for weaklycompressible turbulence the DMA can return accurate valuesof magnetic field strength. The DMA technique does not suf-fer from many limitations of the DCF technique (see §11.1)and its analytical formulation based on the modern theory ofMHD turbulence allows further improving its accuracy if thecomposition of MHD cascade in terms of fundamental MHDmodes is known.The ability to deal with weakly compressible media (seeFigure 1) is already is an achievement that opens a way toobtain maps of the POS magnetic field strengths in many as-trophysical media, e.g. in warm phase of the ISM.Using structure functions with 3 and 4 points one can ob-tain magnetic field strength in the systems that are subject to0 L
AZARIAN , Y
UEN & P
OGOSYAN velocity shear or magnetic field distortion of non-turbulentnature (see Figure 8).To deal with the compressible media we proposed correct-ing procedures that account for velocity centroids represent-ing the actual media velocities in a way that is affected bydensity fluctuations. This procedure requires more knowl-edge of the system, e.g. of the turbulence injection scale.This calls for the approach that includes the simultaneousstudies of turbulence and magnetic field strengths. With therequired input data, the DMA delivers accurate results alsofor compressible media (see Figure 5).At the same time, our second approach to finding magneticfield strength by dividing the Alfven and sonic Mach num-bers obtained in channel maps opens a way of using galac-tic shear to map the 3D distribution of POS magnetic fieldstrength. Our Figure 9 illustrates good ability of the MM2technique to measure the strength of magnetic field in spc-troscopic channel maps using velocity gradient data. It isadvantageous that the MM2 technique can be also appliedto synchrotron data in order to find the value of magneticfield which does not hinge on the assumption about relativis-tic electron energy density.
DISCUSSION11.1.
Relation to the Davis-Chandrasekhar-Fermitechnique
If we study Eq. (35), we can notice that for the line ofsight separations L larger that the turbulent injection scale L inj , this equation reverts to the DCF traditional expressionas the structure function asymptotically approach the value ofdispersions at scales larger than the turbulent injection scale.This is the limiting case proving that the transition from ournew expression to the old DCF formulae takes place for largeseparations.Provided that the structure functions of fluctuations of an-gle and velocity follow the same power-law one can see thatthe same ratio of the structure functions that existed at thelarge scale, i.e. the scale for which the traditional DCF tech-nique works, should be present also at the small scales. Thisprovision is not guaranteed, however. For example, for thecase of super-Alfvenic turbulence, i.e. M A > the mag-netic field structure function is expected to saturate at thescale L inj M − A , while the velocity structure function sat-urates only at the scale L inj . While measuring these twosaturation scales provides a new way of evaluating M A , thepractical determination of these scales may not be observa-tionally easy. Indeed, the injection scales in most cases arecomparable with the scales of the systems, e.g. galactic scaleheight for the galactic turbulence, molecular cloud size, for amolecular cloud. At the scales of the system it is difficult toget reliable statistics and, moreover, large-scale perturbationsof non-turbulent nature are important.An additional advantage of the DMA compared to the DCFis that it can be successfully used for studying cases whenline broadening is sub-thermal. The separation of the ve-locity components into the thermal and non-thermal part israther complicated for the lines which are dominated by ther- mal broadening. This limits the accuracy at which this pro-cess can be performed withing the DCF. At the same time theDMA does not require such separation, as the structure func-tion of centroids is not sensitive to the thermal part of the line(see Lazarian & Esquivel 2003; Esquivel & Lazarian 2005;Kandel et al. 2017a).At the same time, the limitations of the DMA techniqueare related to the accuracy at which the fluctuations of veloc-ity centroids and the fluctuations of polarization angles cor-rectly reproduce the statistics of the velocities and magneticfield, respectively. To answer this question for a variety ofinterstellar conditions a detailed study is necessary. This isbeyond the scope of this paper which aims at introducing thenew technique. Within this paper, however, using numeri-cal simulations we have demonstrated that the new techniqueprovides a reliable recovery of the value of magnetic fieldfor for both incompressible media and compressible mediaas well as to the cases where the traditional DCF techniquefails, namely, to the media subject velocity and magnetic fieldshear, as well as to the clouds where magnetic field directionsare perturbed by self-gravity. We also suggested a recipe forcorrecting our estimates if more information about the turbu-lence at hand is available. Detailed studies of the effects ofcompressibility on the new technique will be provided else-where.To avoid any misunderstanding, we would like to stressthat the expression given by Eq. (35) provides a more gen-eral relation that is valid provided that centroids representthe turbulent velocity averaged along the line of sight. Thesituations the two statistics differ, additional studies shouldprovide the correcting factors/functions f .11.2. Measuring magnetic field within channel maps
In our quest for the ways of measuring magnetic fieldstrength in a way different from the DCF approach, we alsoexplored studying magnetic fields using channel maps in §9.Our DMA technique is not directly applicable to channelmaps as it requires centroids that suppose integrating overvelocities.At the same time our earlier studies employing velocitygradients provide a way to use channel maps in order obtainthe plane of the sky Alfven Mach number and the sonic Machnumber of turbulence. Using this new approach that we termChannel Magnetic Fields (CMF), by combining these twoMach numbers one can easily get the distribution of planeof sky magnetic field intensity.We feel that this is very promising alternative way of ob-taining magnetic 3D magnetic field distribution in interstellarmedium with the 3D information coming from the galacticrotation curve.11.3.
Prospects of the DMA for inhomogeneous clouds
The traditional technique based on the DCF approach isdeveloped to be applied to molecular clouds which have welldefined boundaries. In this case the cloud has a well definedDoppler-broadened line with which the dispersion of veloc-ities can be easily calculated and used together with the dis-
EASURING
B-F
IELD S TRENGTH
Studying super-Alfvenic turbulence
The DCF technique was suggested assuming that the fluc-tuations δB are smaller than the mean field B . In terms ofmodern MHD turbulence theory this corresponds to the caseof sub-Alfvenic turbulence (see Appendix A1).If turbulent perturbations δB > B , the turbulence is super-Alfvenic (i.e. M A > , see Appendix A2). In this case themeasured directions of projected magnetic field are expectedto be uniformly distributed. Naturally the dispersion of an-gles that is employed in DCF, in this case, is not meaningful. MM2 technique keeps can measure M A > . However,the sensitivity of the technique is going to drop with the in-crease of M A . At the same time, the DMA is not expectedto have limitations in obtaining magnetic field strength for M A > studies. Indeed, as it is discussed in Appendix A2,at the scale l A = L inj M − A the turbulence transfers to theMHD regime and therefore the velocity and magnetic fieldfluctuations get related by the Alfvenic relation. Therefore,but measuring the velocity and magnetic field fluctuationsat the scales l < l A one can successfully find the magneticfield strength. Note, that the structure functions of centroidsand angles measured at the separation of lines of sight l aremostly influenced by the correlations at the scale l . To in-crease the accuracy of measuring magnetic field strength ad-ditional procedures can be employed. Those include filter-ing of large scale contributions similar to what was appliedto synthetic maps of super-Alfvenic turbulence in Lazarianet al. (2017). Use of the multi-point structure functions thatwe demonstrated in §7 can also improve the accuracy of themagnetic strength measurements.This theoretical paper does not present detailed calcula-tions relevant to magnetic field studies for super-Alfvenic tur- The modification of the DCF proposed in Cho & Yoo (2016) is not infor-mative for super-Alfvenic turbulence either. Unlike the assumptions in thetechnique, the correlation functions for the magnetic field and velocity havedifferent characteristic scales, i.e. L inj M − A and L inj , respectively. bulence. This is an important avenue for our new techniquesto be explored numerically in future.11.5. Measuring magnetic field in galactic HI
Apart of non-homogeneity of the sample, additional prob-lems plug the measurements for the astrophysical objects.For instance, any regular motion of media with the velocitycomponent that changes along the line of sight contributesto the velocity dispersion. Those motions can be caused bythe rotation of the cloud or due to the media participating inthe galactic rotation. The latter is the case of galactic atomichydrogen, i.e. HI gas.The velocity dispersion of LOS velocities in the case ofgalactic HI is determined by the Galactic rotation curverather than the turbulent velocities of HI gas. In this situation,the DCF approach is not meaningful. At the same time, thestatistics of the DMA measures is affected only at very largescales. Indeed, the latter measures the difference of velocitiesarising from the shear. The estimates in LP00 show that theshear of the turbulence and that of galactic rotation could getcomparable at a scale of several kilo-parsecs, which is muchlarger than the expected scale of galactic turbulent motions.As a result, the galactic rotation can be disregarded and theDMA can be applied to get magnetic field strength in galacticHI. 11.6.
Use of dust polarization and lines
Dust polarized emission is currently the major way ofstudying the directions of magnetic field in cold, warm ISMas well as in molecular clouds. We mentioned in this pa-per that this monopoly is coming to the end with the newway of magnetic field studies that use spectral lines. Someof the processes, e.g. Goldreich-Kylafis effect and GroundState Alignment, require measuring line polarization, someof them, e.g. velocity gradients, require just data on Doppler-shifted spectroscopic lines, e.g. velocity gradients. At themoment, the latter is the better studied way of measuringmagnetic fields and is the main competitor to the dust po-larization studies.The advantages of obtaining the information about themagnetic field directions using lines is self-evident withinthe techniques of extracting the magnetic field strengths thatare discussed in this paper. Indeed, both the DMA and thebriefly discussed technique that uses velocity channels, i.e.CMF, use the same spectroscopic information both to obtainthe variations of magnetic field directions and variations ofturbulent velocity. This is in contrast to the use of dust po-larization which distribution may not spatially coincide withthe distribution of emitters used to study magnetic field.When combined with spectroscopic data, measurementsof magnetic field using polarization and gradients providean enormous field for measuring the strength of magneticfield. In hot diffuse media the measurements of ground statealignment and velocity gradients may be most advantageous.In cold media, dust polarization and velocity gradients arepromising.2 L
AZARIAN , Y
UEN & P
OGOSYAN
Synergy with Velocity Gradients Technique
Velocity Gradient Technique (VGT) is a new very promis-ing development in the way magnetic fields can be studied.As we discuss in Appendix A, for Alfven and slow modesthat usually dominate MHD turbulence the VGT providesthe directions that are coincident with the directions shownby the far-infrared polarization measurements. VGT has sig-nificant advantages compared to the traditional polarimetry.First of all, the regions corresponding to different spectrallines are spatially separate. For instance, some spectral linesare excited around luminous stars. In molecular clouds, dif-ferent molecules are produced at different optical depths andthis allows a way of obtaining the 3D distribution of themagnetic field structure within molecular clouds (Hu et al.2019b). In addition, the galactic rotation provides a way tostudy 3D structure of magnetic field in galactic HI Gonz´alez-Casanova & Lazarian (2018) as well as to study separatelymagnetic fields of molecular clouds along the same line ofsight. In fact, this is the problem for polarimetric observa-tions of most of molecular clouds within the galactic disk.The VGT employs the sub-block averaging approach(Yuen & Lazarian 2017a) which degrades the spatial reso-lution of the original maps. However, this resolution canbe high using ground-based observations and, especially, ifinterferometers are employed. The missing low frequencyharmonics were considered as an impediment for the statis-tical obtaining the statistical estimates of the magnetic fieldstrength (Houde et al. 2016). For our analysis this is not animpediment, as the structure functions are dominated by theturbulent signal at the scales of the study. Therefore the con-tributions from the large scales that are sampled by the inter-ferometric data measured at small baselines is not important.11.8.
Domain of Applicability
The approach that we discuss in the paper is applicableonly for the regions where both velocity dispersion and mag-netic field bending arises from magnetic turbulence. For theparts of the molecular cloud that are dominated by the grav-itational collapse one should not apply either our techniqueor the traditional DCF analysis. It is advantageous that us-ing velocity gradients one can identify such regions. Indeed,the velocity gradients turn 90 degrees in the presence of thegravitational collapse (Yuen & Lazarian 2017b). This effectcan be identified either by the 90 degree shift of the direc-tions measured by polarization and the velocity gradients orby the changes of the properties of the distribution of gradi-ents calculated within data block (Lazarian & Yuen 2018a).11.9.
DMA in high resolution data
The new technique is really timely these days where bothpolarimetry and velocity gradient field measurements canhave high spacial resolution. This allows to measure moredetailed statistics compared to the earlier days. In the paperabove we show that using structure functions of both the fluc-tuations of projected magnetic field and the structure func-tions of velocity centroids one can get much more precise and detailed information about the magnetic field and its dis-tribution over the turbulent astrophysical volume.11.10.
Importance of mode separation
Our study shows that the outcome of the magnetic fieldmeasurements by the technique depends on the compositionof turbulence in terms of Alfven, slow and fast modes. Thisis natural, as Alfven modes dominate the bending of mag-netic field lines, while all modes contribute to velocity fluc-tuations. Therefore to improve the accuracy of the tech-nique it is advantageous to find the relative contribution ofthe modes. This is an important direction of further work,the foundations of which laid by the theoretical studies ofthe anisotropies induced by different MHD turbulence modes(LP12, KLP16).
CONCLUSIONThe paper seeks the ways to measure the strength of mag-netic field. Most of the study is devoted to a new way ofmeasuring magnetic field strength that is based on using dif-ferential measures of both velocity and magnetic field fluctu-ations. For these differential measures we use the structurefunctions of velocity centroids and the structure functions ofvariations of the direction of magnetic field. These variationsof projected magnetic field can be obtained through polar-ization measurement or by velocity gradients. We derivedanalytical expressions for the strength of magnetic field forAlfvenic modes of MHD turbulence as well as the admixtureof Alfvenic and slow modes.We demonstrate that the differential measures provide sig-nificant advantages compared to the global values of dis-persion that is used in the traditional Davis-Chandrasekhar-Fermi (DCF) approach to measuring magnetic field strength.The technique shows further promise when use of multi-pointstructure functions. These advantages of the new technique,that we termed
Differential Measure Analysis (DMA) , canbe briefly summarized in the following way:• DMA can be applied to data for which the dispersionof dispersion at the injection scale is not available ordata inhomgeneity and interfering processes not re-lated to the turbulent cascade are present. As the DMAis applied to smaller patches of the sky, unlike DCF, itcan provide a detailed distribution of the plane of thesky component of magnetic field.• This type of measurements is much less affected ei-ther by the large scale variations of magnetic field di-rections. This opens a way to getting magnetic fieldstrength in the settings for which the DCF approach isnot applicable, i.e. to highly inhomogeneous clouds,to clouds where magnetic field geometry is affected byself-gravity, in clouds with super-Alfvenic turbulence.• The new technique is capable of measuring magneticfield strength in the situations when the Doppler broad-ening is dominated by the the shear arising from ve-locities of non-turbulent nature, as it is the case of HI
EASURING
B-F
IELD S TRENGTH Table 2.
Regimes and ranges of MHD turbulence.Type Injection Range Motion Waysof MHD turbulence velocity of scales type of studyWeak V L < V A [ L inj , l trans ] wave-like analyticalStrongsub-Alfv´enic V L < V A [ l trans , l diss ] eddy-like numericalStrongsuper-Alfv´enic V L > V A [ l A , l min ] eddy-like numerical L inj and l diss are injection and dissipation scales, respectively M A ≡ u L /V A , l trans = L inj M A for M A < and l a = L inj M − A for M A > . in galactic disk. If velocity gradients are used to mapmagnetic field, this provides a unique way for studyingthe 3D distribution of magnetic field strengths.In addition, in the paper we explored another way of prob-ing the strength of POS magnetic field by using the ratio ofsonic and Alfven Mach numbers, i.e. M s and M A, ⊥ . Thistechnique that we termed MM2, is very promising for find-ing the distribution of magnetic field strength using spec-troscopic velocity channel maps. The VGT approach wasdemonstrated to be capable of obtaining the distribution of M A, ⊥ related to the POS component of magnetic field. Tofind B ⊥ we proposed to combine this with the distributionof sonic Mach number that we obtain either by using veloc-ity gradients or other PDF-based techniques. The ratio ofthe two Mach numbers provides us with the magnetic fieldstrength B ⊥ . Compared to the DCF technique this way ofmagnetic field study provides• a detailed distribution of the plane of sky magneticfield strength; • 3D distribution of plane of sky magnetic field galacticdisk magnetic fields, if galactic rotation curve is em-ployed;• the 3D distribution B ⊥ strength in molecular clouds ifa combination of emission lines arising from molecularspecies formed at different optical depths is used.We argued that the extension of the MM2 technique forstudies of magnetic field strength using synchrotron in-tensity and synchortron polarization gradients, as wellas density gradients can bring new ways of probingthe distribution of magnetic field in turbulent media. Acknowledgment
We thank Jungyeon Cho for providing theset of incompressible MHD simulation data and the inspir-ing discussions. A.L. and K.H.Y. acknowledge the supportthe NSF AST 1816234 and NASA TCAN 144AAG1967.The numerical part of the research used resources of bothCenter for High Throughput Computing (CHTC) at the Uni-versity of Wisconsin and National Energy Research Scien-tific Computing Center (NERSC), a U.S. Department of En-ergy Office of Science User Facility operated under Con-tract No. DE-AC02-05CH11231, as allocated by TCAN144AAG1967. D.P. thanks Theoretical Group at Korea As-tronomy and Space Science Institute (KASI) for hospitality.APPENDIX A. DESCRIPTION OF COMPRESSIBLE MHD TURBULENCEIn this section we briefly summarize the scaling laws for compressible MHD turbulence as we did in Lazarian et al. (2018). Ifthe energy is injected with the injection velocity V L that is less than the Alfven speed V A , the turbulence is sub-Alfvenic . Inthe opposite case it is super-Alfvenic . The illustration of turbulence scalings for different regimes can be found in Table 2. Webriefly describe the regimes below. A more extensive discussion can be found in the review by Brandenburg & Lazarian (2013).A.1. Sub-Alfvenic Turbulence
In the case the Alfvenic Mach number M A = V L /V A < . The turbulence in the range from the injection scale L inj to thetransition scale l trans = L inj M A (A1)4 L AZARIAN , Y
UEN & P
OGOSYAN is termed the weak Alfvenic turbulence. This type of turbulence keeps the l (cid:107) scale stays the same while the velocities changeas v ⊥ ≈ V L ( l ⊥ /L inj ) / (Lazarian & Vishniac 1999) The cascading results in the change of the perpendicular scale of eddies l ⊥ only. With the decrease of l ⊥ the turbulent velocities v ⊥ decreases. Nevertheless, the strength of non-linear interactions ofAlfvenic wave packets increases (see Lazarian 2016). Eventually, at the scale l trans , the turbulence turns into the strong regimewhich obeys the GS95 critical balance.The situations when the l trans is less than the turbulence dissipation scale l diss require M A that is unrealistically small forthe typical ISM conditions. Therefore, typically the ISM turbulence transits to the strong regime. If the telescope resolution isenough to resolve scales less than l trans then we should observe the signature of strong turbulence in observation.The anisotropy of the eddies for sub-Alfvenic turbulence is larger than in the case of trans-Alfvenic turbulence described byGS95. The following expression was derived in LV99: l (cid:107) ≈ L inj (cid:18) l ⊥ L inj (cid:19) / M − / A (A2)where l (cid:107) and l ⊥ are given in the local system of reference. For M A = 1 one returns to the GS95 scaling. The turbulent motionsat scales less than l trans obey: v ⊥ = V L (cid:18) l ⊥ L inj (cid:19) / M / A , (A3)i.e. they demonstrate Kolmogorov-type cascade perpendicular to local magnetic field.In the range of [ L inj , l trans ] the direction of magnetic field is weakly perturbed and the local and global system of referenceare identical. Therefore the velocity gradients calculated at scales larger than l trans are perpendicular to the large scale magneticfield. While at scales smaller than l trans the velocity gradients follow the direction of the local magnetic fields, similar to thecase of trans-Alfvenic turbulence that we discuss in the main text.A.2. Super-Alfvenic Turbulence If V L > V A , at large scales magnetic back-reaction is not important and up to the scale l A = L inj M − A , (A4)the turbulent cascade is essentially hydrodynamic Kolmogorov cascade. At the scale l A , the turbulence transfers to the sub-Alfvenic turbulence described by GS95 scalings , i.e. anisotropy of turbulent eddies start to occur at scales smaller than l A .The velocity gradients at the range from the injection scale L inj to l A are determined by hydrodynamic motions and thereforeare not sensitive to magnetic field. The contribution from these scales is better to remove using spacial filtering. For scales lessthan l A the gradients reveal the local direction of magnetic field , as we described e.g. in Yuen & Lazarian (2017b); Lazarianet al. (2017). For our numerical testing we are limited in the range of M A > that we can employ. two In the case when M A is sufficiently small, the scale l A will be comparable to the dissipation scale l dis and therefore the inertial range will beentirely eliminated. From the theoretical point of view, there are no limitations for tracing magnetic field within super-Alfvenicturbulence provided that the telescope or interferometer employed resolves scales less than l A and l A > l diss .A.3. Cascades of fast and slow MHD modes
In compressible turbulence, apart from Alfvenic motions, slow and fast fundamental motion modes are present (see Biskamp2003). These are compressible modes and their basic properties are described e.g. in Brandenburg & Lazarian (2013).In short, the three modes, Alfven, slow and fast modes have their own cascades (see Cho & Lazarian 2002, 2003). Alfvenic eddymotions shear density perturbations corresponding to the slow modes and imprint their structure on the slow modes. Thereforethe anisotropy of the slow modes mimic the anisotropy of Alfven modes, the fact that is confirmed by numerical simulations forboth gas pressure and magnetic pressure dominated media (Cho & Lazarian 2003; Kowal & Lazarian 2010). Therefore the bothvelocity and magnetic field gradients are perpendicular to the local direction of magnetic field. This is confirmed in numericaltesting in (Lazarian & Yuen 2018a).Fast modes for gas pressure dominated media are similar to the sound waves, while for the media dominated by magneticpressure are waves corresponding to magnetic field compressions. In the latter case, the properties of the fast mode cascadewere identified in Cho & Lazarian (2002). The gradients arising from fast modes are different from those by Alfven and slowmodes as shown in (Lazarian & Yuen 2018a). However, both theoretical considerations and numerical modeling (see Branden-burg & Lazarian 2013) indicate the subdominance of the fast mode cascade compared to that of Alfven and slow modes. Inaddition, in realistic ISM at small scales fast modes are subject to higher damping (see Yan & Lazarian 2004; Brunetti & Lazar-ian 2007). In numerical simulations (Lazarian & Yuen 2018a) the velocity gradients calculated with Alfvenic modes only wereindistinguishable from those obtained with all 3 modes present.
EASURING
B-F
IELD S TRENGTH B. GRADIENT TECHNIQUE AND RELATION TO POLARIZATIONThe gradient technique has different branches. To study magnetic field structure one can use gradients of velocities (Yuen& Lazarian 2017a,b; Lazarian & Yuen 2018a), gradients of synchrotron intensities (Lazarian et al. 2017) and gradients of syn-chrotron polarization (Lazarian & Yuen 2018b). In addition, to get additional information about interstellar processes densitygradients can also be used (Yuen & Lazarian 2017b; Hu et al. 2019c).In the original development of Yuen & Lazarian (2017a), they modelled the gradient orientation distribution as a Gaussian-likefunction. In Lu et al. (2019) they point out that the Gaussian modelling is not accurate in the theoretical point of view. Wefeel that the gradient approach from Lu et al. (2019) would be better describing the behavior of velocity gradients and wouldbe complementary in the discussion of tracing magnetic field strength by the products on VGT. Therefore we shall discuss howthe approach in Lu et al. (2019) would be beneficial in improving the gradient technique.We follow the approach that we firstintroduced in Lu et al. (2019).For a gradient of a random field f ( X ) one can consider the gradient covariance tensor σ ∇ i ∇ j ≡ (cid:104)∇ i f ( X ) ∇ j f ( X ) (cid:105) = ∇ i ∇ j D ( R ) | R → , (B5)which is the zero separation limit of the second derivatives of the field structure function D ( R ) ≡ (cid:68) ( f ( X + R ) − f ( X )) (cid:69) .MHD turbulence is anisotropic and this makes the structure function of the corresponding observables dependent on the anglebetween R and the projected direction of the magnetic field. This was studied in Lazarian & Pogosyan (2012) for synchrotron,or by the deconvolution method described from et al. (2016) for velocity channel intensities and Kandel et al. (2017a) for velocitycentroids. This anisotropy is present in the limit R → and results in non-vanishing traceless part of the gradient co-variancetensor σ ∇ i ∇ j − (cid:88) i =1 , σ ∇ i ∇ i =12 (cid:32) (cid:0) ∇ x − ∇ y (cid:1) D (R) 2 ∇ x ∇ y D (R)2 ∇ x ∇ y D (R) (cid:0) ∇ y − ∇ x (cid:1) D (R) (cid:33) R → (cid:54) = 0 (B6)The eigen-direction of the tensor corresponding to the largest eigenvalue provides the direction of the gradient that makes anangle θ with the coordinate x-axis tan θ = 2 ∇ x ∇ y D (cid:113)(cid:0) ∇ x D − ∇ y D (cid:1) + 4 ( ∇ x ∇ y D ) + (cid:0) ∇ x − ∇ y (cid:1) D . (B7)The structure function can be presented as a Fourier integral D ( R ) = − (cid:90) d K P ( K ) e i K · R , (B8)where P ( K ) is a power spectrum and K is a 2D wave vector. If the direction of K is defined by angle θ K and that of the projectedmagnetic field by angle θ H , one can write for the spectrum P ( K ) = (cid:88) n P n ( K ) e in ( θ H − θ K ) (B9)and for the derivatives of the structure function ∇ i ∇ j D ( R ) == (cid:88) n (cid:90) K P n ( K ) (cid:90) dθ K e in ( θ H − θ K ) e iKR cos( θ R − θ K ) ˆ K i ˆ K j , (B10)6 L AZARIAN , Y
UEN & P
OGOSYAN where hat denotes unit vectors, namely ˆ K x = cos θ K and ˆ K y = sin θ K . Integrating over θ K , one obtains the anisotropic part ( ∇ x − ∇ y ) D ( R ) = 2 π (cid:88) n i n e in ( θ − θ H ) × (B11) × (cid:90) dKK J n ( kR ) (cid:0) P n − ( K ) e i θ H + P n +2 ( K ) e − i θ H (cid:1) ∇ x ∇ y D ( R ) = π (cid:88) n i n +1 e in ( θ − θ H ) × (B12) × (cid:90) dKK J n ( kR ) (cid:0) − P n − ( K ) e i θ H + P n +2 ( K ) e − i θ H (cid:1) In the limit R → , only n = 0 term for which J (0) = 1 survives and ( ∇ x − ∇ y ) D ( R ) = (cid:20) π (cid:90) dKK P ( K ) (cid:21) cos 2 θ H (B13) ∇ x ∇ y D ( R ) = (cid:20) π (cid:90) dKK P ( K ) (cid:21) sin 2 θ H (B14)Notice that anisotropy of the gradient variance is determined by the quadrupole of the power spectrum (and structure function).Substituting this result into Eq.B7, we find that the eigen-direction of the gradient variance has the form tan θ = A sin 2 θ H | A | + A cos 2 θ H = (cid:40) tan θ H A > − cot θ H A < (B15)and is either parallel or perpendicular to the direction of the magnetic field, depending on the sign of A ∝ (cid:82) dKK P ( K ) , i.ethe sign of the spectral quadrupole P . Results of Kandel et al. (2017a) show that A is negative for Alfv´en and slow modes,which thus give gradients orthogonal to the magnetic field. In contrast, fast modes in low- β plasma produce positive A plasmaand gradients parallel to the magnetic field. Fast modes in high- β plasma are purely potential and isotropic and have no preferreddirection for the gradients.Since the direction of the magnetic field that we aim to track is unsigned, it is appropriate to describe it as an eigen-directionof the rank-2 tensor, rather than a vector. This naturally leads to the mathematical formalism of Stokes parameters. As the localestimator of the angle θ via the gradients, we can introduce pseudo-Stokes parameters (cid:101) Q ∝ ( ∇ x f ) − ( ∇ y f ) ∝ cos 2 θ (B16) (cid:101) U ∝ ∇ x f ∇ y f ∝ sin 2 θ (B17)so that (cid:101) U (cid:101) Q = tan 2 θ ∼ tan 2 θ H (B18)In the next section, we describe the exact procedure for the estimator that we use in this paper.The pseudo Stokes parameters naturally connect the gradient techniques with polarization studies. More exactly, both forsynchrotron (Lazarian & Pogosyan 2012; Kandel et al. 2018) and thermal dust emission (Clark et al. 2015; Caldwell et al. 2017;Kandel et al. 2018, see Crutcher 2010 and ref. therein), we expect the true polarization Stokes parameters to be Q ∝ (cid:90) dz ( H x − H y ) ∝ cos 2 θ H (B19) U ∝ (cid:90) dz H x H y ∝ sin 2 θ H (B20)Thus, the pseudo Stokes parameters constructed from the gradients can be directly compared with Stokes parameters that probepolarized emission in magnetized medium. C. Cho & Yoo (2016) MODIFICATION TO THE DCF TECHNIQUEIf magnetic field variations are measured from polarization measurements and the δv los is determined through spectroscopicDoppler shift measurements, the corresponding expression is given by DCF expression. Their expression trivially follows fromEq.(3) substituting M A ∼ δB/B ∼ δθ pol : B P OS = f (cid:112) πρ δv los δθ pol (C21) EASURING
B-F
IELD S TRENGTH f is ∼ . − (See §6.3).The study by Cho & Yoo (2016) was intended to improve the accuracy of the DCF approach without changing the nature ofthe measurements to be performed. Similar to DCF, the authors were considering the magnetic and velocity fluctuations at theinjection scale. However, it was noted by Cho & Yoo (2016) that Eq C21 must be corrected to deal with the case when theinjection scale of turbulence L inj is less than the extend of the line of sight L within the emitting turbulent volume. To explainthe problem, consider a setting with mean magnetic field being along x -direction in the plane of the sky and the magnetic fieldfluctuation δB is along y -direction. If the 3D magnetic field is b reg , it is adds up linearly along the line of sight and therefore theobserved B x is (cid:82) L b reg dx ≈ b reg L . On the contrary, the fluctuating magnetic field b turb with correlation scale L inj is added upin the random walk fashion with δB y providing (cid:82) L b turb dx ≈ b turb (cid:112) L inj L . As a result an additional factor enters the δB y /B x ratio, namely, the observed fluctuation gets reduced by a factor ≈ (cid:112) L inj / L .To account for this factor, Cho & Yoo (2016) considered the ratio of the line of sight velocity and the centroid velocity. Thelatter is given by Eq. (4), while the former is the usual δv los arising from the velocity dispersion at the scale L inj . The velocitymeasured by centroids is, on the contrary δC = (cid:82) L δv los dx/ L ≈ δv los (cid:112) L inj / L . As a result, if δv los is substituted by thedispersion of Velocity Centroid δC the Eq. (C21) can be used both for the case of L ∼ L inj and L (cid:29) L inj . In other words, theexpression B P OS ≈ f (cid:48) (cid:112) πρ δCδθ pol (C22)with some other constant f (cid:48) related to the angle of projections. Eq.C22 has a wider range of applications than the original DCFexpression as they show in the series of numerical works (Cho & Yoo 2016; Yoon & Cho 2019; Cho 2019). In particular, themagnetic field strength computed based on Eq.C22 would not depend on L inj / L ratio. In comparison, the DCA technique usesthe differential measures and it does not require measurements at the turbulence injection scale. D. DEPENDENCE ON γ IN DCF FORMULA FOR ALFVENIC TURBULENCE
Figure 10.
A figure showing how does the traditional CF method (blue) and the Eq.D23 (red) behave as a function of the inclination angle γ inc by rotating the numerical cube ”Ms20.0Ma0.2”. We mark the cut-off angle γ inc = 4 tan − ( M A / √ as the green dash line while theexpected total magnetic field strength as the red dash line. In this section we shall discuss how the line of sight angle come into play in estimating the total magnetic field strengthprovided that the mean magnetic field inclination angle is given. We shall discuss the possibility of obtaining this inclinationangle in a full manner in Yuen & Lazarian (2020c) but in fact in Yuen & Lazarian (2020c) our study shows a rather non-trivialdependence on the angle γ = cos − ( ˆB · ˆ z ) between the mean magnetic field and the line of sight ˆ z . In fact, the condition thatthe following argument could hold is to have the inclination angle γ inc > − ( M A / √ . Below we shall discuss how DCFapproach should be modified if we assume that the turbulence has only Alfvenic component.A natural modification on estimating the total magnetic field strength is to introduce a sin γ inc factor to compensate the pro-jection effect: B = (cid:112) πρ δv z δφ γ inc (D23)8 L AZARIAN , Y
UEN & P
OGOSYAN
However as we shall discuss in Yuen & Lazarian (2020c) in detail, the aforementioned formula is correct only when γ inc > − ( M A / √ . Figure .10 shows an example on using Eq.D23 when we rotate the numerical cube ”Ms20.0Ma0.2” by 5degrees each. We can see that the total magnetic field strength could be estimated only when γ inc > − ( M A / √ . Thereason on why there is a lower bound for γ inc is because the turbulent component shall dominate over the mean field componentwhen γ inc < − ( M A / √ , resulting an underestimation of mean magnetic field strength in this regime.REFERENCES Alfv´en, H. 1942, Nature, 150, 405Andersson, B.-G., Lazarian, A., & Vaillancourt, J. E. 2015,ARA&A, 53, 501Armstrong, J. W., Rickett, B. J., & Spangler, S. R. 1995, ApJ, 443,209Brandenburg, A., & Lazarian, A. 2013, SSRv, 178, 163Beresnyak, A., & Lazarian, A. 2019, Turbulence inMagnetohydrodynamicsBiskamp, D. 2003, Magnetohydrodynamic TurbulenceBrunetti, G., & Lazarian, A. 2007, MNRAS, 378, 245Burkhart, B., Stanimirovi´c, S., Lazarian, A., et al. 2010, ApJ, 708,1204Burkhart, B., & Lazarian, A. 2012, ApJL, 755, L19Burkhart, B., Lazarian, A., Le˜ao, I. C., de Medeiros, J. R., &Esquivel, A. 2014, ApJ, 790, 130Caldwell, R. R., Hirata, C., & Kamionkowski, M. 2017, ApJ, 839,91Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113Chepurnov, A., & Lazarian, A. 2009, ApJ, 693, 1074Chepurnov, A., & Lazarian, A. 2010, ApJ, 710, 853Chitsazzadeh, S., Houde, M., Hildebrand, R. H., & Vaillancourt, J.2012, ApJ, 749, 45Cho, J., & Lazarian, A. 2002, Physical Review Letters, 88, 245001MNRAS, 2003, 345, 325Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273Cho, J., & Yoo, H. 2016, ApJ, 821, 21Cho, J. 2017, Journal of Physics Conference Series, 012002Cho, J. 2019, ApJ, 874, 75Cho, J., Lazarian, A., & Vishniac, E. T. 2002, ApJL, 566, L49Clark, S. E., Hill, J. C., Peek, J. E. G., Putman, M. E., & Babler,B. L. 2015, Physical Review Letters, 115, 241302Crutcher, R. M. 2004, Ap&SS, 292, 225Crutcher, R. 2010, From Stars to Galaxies: Connecting ourUnderstanding of Star and Galaxy Formation, 3Davis, L. 1951, Physical Review, 81, 890Dolginov, A. Z., & Mytrophanov, I. G. 1976, Ap&SS, 43, 257Draine, B. T., & Weingartner, J. C. 1996, American AstronomicalSociety Meeting Abstracts 189, 16.02Draine, B. T. 2006, ApJ, 636, 1114Elmegreen, B. G., & Scalo, J. 2004, ARA&A, 42, 211Esquivel, A., & Lazarian, A. 2005, ApJ, 631, 320Esquivel, A., & Lazarian, A. 2010, ApJ, 710, 125Esquivel, A., & Lazarian, A. 2011, ApJ, 740, 117 Esquivel, A., Lazarian, A., & Pogosyan, D. 2015, ApJ, 814, 77Falceta-Gonc¸alves, D., Lazarian, A., & Kowal, G. 2008, ApJ, 679,537-551Falcon, E., Fauve, S., & Laroche, C. 2007, PhRvL, 98, 154501Gaensler, B. M., Haverkorn, M., Burkhart, B., et al. 2011, Nature,478, 214Galli, D., Lizano, S., Shu, F. H., et al. 2006, ApJ, 647, 374Girart, J. M., Rao, R., & Marrone, D. P. 2006, Science, 313, 812Goldreich, P., & Kylafis, N. D. 1981, ApJL, 243, L75Goldreich, P., & Kylafis, N. D. 1982, ApJ, 253, 606Goldreich, P., & Sridhar, S. 1995, ApJ, 438, 763Gonz´alez-Casanova, D. F., & Lazarian, A. 2017, ApJ, 835, 41Gonz´alez-Casanova, D. F., & Lazarian, A. 2018, ApJHeitsch, F., Zweibel, E. G., Mac Low, M.-M., Li, P., & Norman,M. L. 2001, ApJ, 561, 800Heyer, M. H., & Brunt, C. M. 2004, ApJL, 615, L45Hildebrand, R. H., Kirby, L., Dotson, J. L., Houde, M., &Vaillancourt, J. E. 2009, ApJ, 696, 567Hoang, T., & Lazarian, A. 2008, MNRAS, 388, 117Hoang, T., & Lazarian, A. 2016, ApJ, 831, 159Houde, M. 2004, ApJL, 616, L111Houde, M., Vaillancourt, J. E., Hildebrand, R. H., Chitsazzadeh, S.,& Kirby, L. 2009, ApJ, 706, 1504de, M., Rao, R., Vaillancourt, J. E., & Hildebrand, R. H. 2011, ApJ,733, 109Houde, M., Fletcher, A., Beck, R., et al. 2013, ApJ, 766, 49Houde, M., Hull, C. L. H., Plambeck, R. L., Vaillancourt, J. E., &Hildebrand, R. H. 2016, ApJ, 820, 38Hu, Y., Yuen, K. H., Lazarian V., et al. 2019, Nature AstronomyHu, Y., Yuen, K. H., Lazarian, A., et al. 2019, ApJ,arXiv:1904.04391.Hu, Y., Yuen, K. H., & Lazarian, A. 2019, ApJ, 886, 17Johns-Krull, C. M. 2007, Star-disk Interaction in Young Stars, 31Kandel, D., Lazarian, A., & Pogosyan, D. 2016, MNRAS, 461,1227Kandel, D., Lazarian, A., & Pogosyan, D. 2017, MNRAS, 464,3617Kandel, D., Lazarian, A., & Pogosyan, D. 2017, MNRAS, 470,3103Kandel, D., Lazarian, A., & Pogosyan, D. 2018, MNRAS, 478, 530Kowal, G., & Lazarian, A. 2007, ApJL, 666, L69Kowal, G., & Lazarian, A. 2010, ApJ, 720, 742Larson, R. B. 1981, MNRAS, 194, 809
EASURING
B-F
IELD S TRENGTH Table 3.
List of notations used in this work
Parameter Meaning First appearance r x − x Eq. (1) R X − X Eq. (1) z Line of sight variable Eq. (6) x X l Distance of the 3d separation | r | Eq. (9) L Size of a turbulent cloud Eq. (10) L inj Turbulence injection scale Eq. (10) ρ ( r ) ρ ( X , v ) Density of emitters in the PPV space Eq. (4) B b turb Turbulent part of the magnetic field Eq.(9) B pos = H ⊥ / √ π Projected magnetic field Eq.(11) H x,y The x & y component of magnetic field Eq.(18)
Q, U
Stokes Q & U Eq.(18) v C Velocity Centroid Eq.(15) θ Magnetic field angle Eq. (6) φ Polarization angle Eq. (5) M s Sonic Mach number Eq.(2) M A Alfvenic Mach number Eq.(3) f Weighting factor of the DCF Equation Eq.(16) κ unsigned polarization angle curvature Eq.(47) (cid:104) A (cid:105) x average of the quantity A over variable x Eq.(7) SF D/ D { A } A Eq (1) γ Angle between line of sight and symmetry axis Eq. (24) µ = k · ˆ B Eq. (25) D QQ , D UU D + D + = D xx + D yy Eq.(22) D φ ( R ) = SF { φ } ( R ) , of polarization angle structure function Eq. (23) D n ( R ) Multipole moment of centroid structure function ( SF { C } ( R ) ) Eq. (28) D φn ( R ) Multipole moment of polarization angle structure function Eq. (27) A ( A,F,S ) B Amplitude of the power spectrum for Alfven, Fast, Slow modes Eq.(24) C n ( m ) − i n Γ [ ( | n |− m − ] m Γ [ ( | n | + m +3) ] Eq.(24) G ( A,F,Sn ( γ ) Multipole decomposition of the geometric functions of polarization angles, defined in Lazarian & Pogosyan (2012) Eq.(24) W ( A,F,Sn ( γ ) Multipole decomposition of the geometric functions of velocity centroids, defined in Kandel et al. (2017a) Eq.(24) W I ( M A ) weight of the isotropized spectral part Eq.(30) W L ( M A ) weight of the local anisotropic spectral part Eq.(30) AZARIAN , Y
UEN & P