Draft version September 10, 2018Preprint typeset using LATEX style emulateapj v. 5/2/11

THE GALACTIC MAGNETIC FIELD

Ronnie Jansson and Glennys R. FarrarCenter for Cosmology and Particle Physics, Department of Physics

New York University, New York, NY 10003, USA

Draft version September 10, 2018

ABSTRACT

With this Letter, we complete our model of the Galactic magnetic field (GMF), by using the WMAP722 GHz total synchrotron intensity map and our earlier results to obtain a 13-parameter model of theGalactic random field, and to determine the strength of the striated random field. In combinationwith our 22-parameter description of the regular GMF, we obtain a very good fit to more than fortythousand extragalactic Faraday Rotation Measures (RMs) and the WMAP7 22 GHz polarized andtotal intensity synchrotron emission maps. The data calls for a striated component to the randomfield whose orientation is aligned with the regular field, having zero mean and rms strength ≈20%larger than the regular field. A noteworthy feature of the new model is that the regular field hasa significant out-of-plane component, which had not been considered earlier. The new GMF modelgives a much better description of the totality of data than previous models in the literature.

1. INTRODUCTION

The magnetic field of the Galaxy (GMF) eludes ourunderstanding in many respects, in spite of a great dealof effort measuring and modeling its various compo-nents. The most familiar components of the GMF arethe large-scale regular fields and the small-scale randomfields, reviewed in Beuermann et al. (1985) and Sun et al.(2008). The random fields are due to a number of phe-nomena including supernovae and other outflows, pos-sibly compounded by hydrodynamic turbulence, whichare expected to result in randomly-oriented fields with acoherence length λ of order 100 pc or less (Gaensler &Johnston 1995; Haverkorn et al. 2008). In addition tothese, there may be “striated” random fields whose ori-entation is aligned over a large scale, but whose strengthand sign varies on a small scale. Such striated fields canbe produced by differential rotation of a medium contain-ing small scale random fields, the levitation of bubblesof hot plasma carrying trapped randomly oriented fieldsaway from the disk, or both. Striated fields are a specialcase of the more generic possibility of anisotropic ran-dom fields introduced in Sokoloff et al. (1998), which canbe considered a superposition of multiple striated andpurely random fields.

In Jansson & Farrar (2012) (JF12 below) we intro-duced a 22-parameter model of the large-scale, coherentor regular Galactic Magnetic Field, constrained by a si-multaneous fit to the WMAP7 Galactic Polarized Syn-chrotron Emission (Stokes Q and U , or, collectively, PI)maps (Gold et al. 2011) and more than forty thousandextragalactic Faraday rotation measures (RMs). Themodel includes an out-of-plane component and takes intoaccount the possible presence of a striated random fieldand/or need for rescaling the assumed density of rela-tivistic electrons. The new model gives a much better ac-counting of the observational data than achieved earlier.Key to the feasibility of the undertaking is the fact thatthe different data types probe different aspects of the fieldand are sensitive to different sources of uncertainty. TheFaraday rotation measure is the integrated line-of-sight

component of ~B, weighted by the thermal electron den-

sity, for which we adopt the standard NE2001 Cordes &Lazio (2002) model. The polarized and total synchrotronintensity are also line-of-sight integrals, but sensitive tothe transverse rather than longitudinal component of theGMF and weighted by the relativistic (cosmic ray) elec-tron density, ncre.

We present here a 13-parameter model of the spatialvariation of the strength of the Galactic random field(GRF), obtained by applying the methodology and in-frastructure of JF12 to the total, unpolarized Galacticsynchrotron emission from WMAP. The new random-field model provides a good fit to the large-scale structureof the total synchrotron emission I of the galaxy. Thisapproach determines the rms value of Brand as a func-tion of position within the Galaxy but is not sensitive tothe “internal structure” of the random field, i.e., maxi-mum coherence length and power-spectrum. Combiningthese results with data on the variance can be used toconstrain the distribution of coherence lengths, as will bepresented elsewhere.

A global parameterization of the GMF including bothcoherent and random fields is needed for many practicalpurposes. One important application is predicting thepattern of ultra-high energy cosmic ray (UHECR) de-flections, to identify sources and gain insight into com-position; an early effort to characterize the Galactic ran-dom field for UHECR studies was made by Tinyakov& Tkachev (2005). Another important application is tounderstand the spatial variation and anisotropy of thediffusion coefficients for Galactic cosmic ray propagation,needed to obtain more realistic predictions for the diffusegamma ray background and Galactic cosmic ray spectraand thus to have more accurate models for the back-grounds to possible Dark Matter signals.

2. MODEL FIELD

We allow for three distinct types of magnetic structuresin our analysis: large-scale regular fields, striated fields,and small-scale random fields. These different structurescan be disentangled because they contribute in generalto different observables: the large-scale regular field can

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contribute to all the observables, I, PI and RM; the stri-ated field can contribute to I and PI but not to RMs, inleading order, due to its changing sign; while the purelyrandom field contributes only to the total synchrotronemission, I.

In JF12 we considered two forms of ncre and adoptedthe GALPROP model provided by A. Strong (privatecommunication, 2009), allowing for a possible overallrescaling by a factor α. Increasing the scale heightand rescaling the GALPROP ncre, as could arise ifanisotropic diffusion due to the out-of-plane field wereincorporated in GALPROP, further improves the fit;Strong et al. (2011) and Jaffe et al. (2011) have alsoshown evidence of a need for a modification in the in-tersteller electron spectrum. In future work we intendto self-consistently solve for the GMF and ncre simul-taneously, but for the present we adopt JF12’s partial-improvement of optimizing the normalization but not theshape of the GALPROP ncre.

The analysis of JF12 describes the coherent field as asuperposition of disk, toroidal halo, and “X-field” com-ponents, with a total of 21 parameters. Each of thesefield components was initially allowed to have a separateamount of striation, with B2d,h,X striβd,h,X B

2d,h,X reg and

the orientation of a given component of the striated fieldtaken to be aligned with corresponding component of thelocal coherent field. This introduces 3 more parametersthan used to describe the coherent field alone. Howeverthere was not a significant improvement in the fit usingthe freedom of separate βd,h,X , so a single β value wasused for all components in the final parameter optimiza-tion. When there is a single β for all components, itmeans that the possible striated field is aligned with thetotal local regular field resulting in a degeneracy betweenthe strength of the striated component of the magneticfield and the relativistic electron density: if we write themultiplicative factor as γ = α(1 + β), we can interpretα as being a rescaling factor of the nominal relativisticelectron density, with B2stri = βB

2reg. The combined fit to

Q, U , and RMs in JF12 established that γ = 2.92±0.14;our present fit to I allows us to break the degeneracy anddetermine α and β separately. A future improvement willbe to allow α and β to be spatially varying.

After some experimentation, to get a good fit to theobservations with a minimum of model complexity, wefound that a good fit to the WMAP7 I map can beobtained by taking the random field to be a superpo-sition of a disk component, with a central region andeight spiral arms having the same geometry as for thecoherent field, plus an extended random halo compo-nent. The total random field has an rms strength ofBrand =

√B2disk +B

2halo.

The disk component of the GRF is modeled as theproduct of a radial factor times a vertical profile, takento be a Gaussian of width zdisk0 . In the inner 5 kpc,the radial factor is a constant: bint. Starting at 5 kpc,the rms strength is different from one arm to another.Its value at 5 kpc in the ith spiral arm is denoted bi,and it decreases as ∼ 1/r at larger radii. The extendedrandom halo field is simply the product of an exponentialin the radius and a Gaussian in the vertical direction,respectively: Bhalo = B0 exp[−r/r0] exp[−z2/2z20 ].

Figure 1. Top left: WMAP 22 GHz unpolarized synchrotronradiation. Top right: Simulated 22 GHz synchrotron radiationfor the optimized model. Second row: As top row, in logarithmicscale. Third row, left panel: The estimated σ of the unpolarizedsynchrotron data. Third row, right panel: The difference betweenobserved and simulated data. Bottom left: The χ2 of the fit, with101 pixels removed (see section 3). Bottom right: The χ2 of themasked-out pixels. Skymaps are in the Mollweide projection withgalactic longitude zero in the center, increasing to the left.

3. METHOD AND DATA

Our approach to constraining the parameters of theGMF is described in detail in JF12 but we review itbriefly here, since the present analysis uses the samemethodology. The observables used in the present anal-ysis are the average values of I in 13.4 square degreeHEALpix cells(Górski et al. 2005) and we minimize χ2I =∑

i(Idata,i − Imodel,i)2/σ2I,i as a function of the GalacticRandom Field (GRF) parameters, taking the JF12 largescale GMF parameters and value of γ as given. The con-ceptual foundation on which this and the JF12 projectis based, is the recognition that the σi’s are themselvesmeasurable observables. The variances in the primaryobservables are not merely due to observational uncer-tainties, but include and are dominated by the astro-physical variance caused by turbulent magnetic fields,inhomogeneities in the interstellar medium, and sourcecontributions to the RMs. We measure σi for a given ob-servable, from the variance in values of that observable inthe 16 sub-pixels of the ith 13.4 square degree HEALPixpixel.

It is possible to estimate the unpolarized synchrotronemission without implementing a fully stochastic randomfield, which is computationally expensive, by generaliz-ing a trick introduced in Jansson & Farrar (2012) fora striated random field. Assuming a spectral index of3 for the relativistic electrons, the synchrotron emissiv-

ity of a volume element in a region with local field ~B is

3

0 2 4 6 8 10

χ2

0

200

400

600

800

1000

1200

Figure 2. A histogram of χ2 for the 3072 pixels covering thesky. To reduce the impact of mainly nearby structures the 101pixels with χ2 greater than 10 are removed from the data, and asecond optimization is performed to obtain the best-fit parameterspresented in Table 1.

∝ ncreB2⊥ = ncreB2sin2θ where θ is the angle between

the line of sight and the direction of ~B. If we add a ran-

dom field of rms strength Brand to ~Breg and average overthe direction the random field may take, represented by< >, then < B2⊥ >= B

2reg sin

2θ + 23 B2rand. To simulate

the total synchrotron emission from a superposition ofrandom, striated and regular fields, without ever creat-ing realizations of the random field, we use Hammurabi(Waelkens et al. 2009) with the model ncre, but replacing

B2reg → α (1+β)B2reg,model

(1 +

2

3

B2rand(1 + β)B2reg,model sin

2 θ

),

(1)with α (1 + β) = γ fixed from JF12. Since α and 1 + βenter the expression for the total intensity differently, fit-ting the total intensity data allows them to be separatelydetermined. As in JF12, we use a Metropolis MarkovChain Monte Carlo (MCMC) algorithm to find best-fitparameters and confidence levels.

The WMAP total Galactic synchrotron intensity mapused for the present analysis is shown in linear and log-arithmic scales in the top two rows of the left-hand col-umn in Fig. 1, and the map of measured σ values foreach 13.4 square degree HEALPix pixel is shown in theleft panel of the third row. In order that fluctuations dueto nearby strong individual contributions do not distortthe determination of the global properties of the Galac-tic random field, after performing an initial fit for themodel parameters we mask-out the pixels with χ2i > 10,and then perform a second (and final) parameter opti-mization. The distribution of χ2i values from the initialmodel fit is shown in Fig. 2. The lower panels of Fig. 1show the χ2i per pixel maps (left) with the masked pixelsremoved and (right) showing the 101 masked-out pixels;the χ2i ’s shown in these maps are from the second modeloptimization.

4. RESULTS OF THE FIT

The best-fit parameters for this model are given in Ta-ble 1. After finding the minimum chi-squared, an ad-ditional 100k MCMC iterations were performed to ob-tain a probability distribution in parameter space, re-flecting the extent to which the observations do a betterjob constraining some parameters than others; the errorsgiven in Table 1 are obtained from that distribution, bymarginalizing over the other parameters.

Table 1Best-fit parameters of the random field, with 1 − σ intervals.

Field Best fit Parameters DescriptionDisk b1 = 10.81 ± 2.33µG field strengths at r = 5 kpccomponent b2 = 6.96 ± 1.58µG

b3 = 9.59 ± 1.10µGb4 = 6.96 ± 0.87µGb5 = 1.96 ± 1.32µGb6 = 16.34 ± 2.53µGb7 = 37.29 ± 2.39µGb8 = 10.35 ± 4.43µGbint = 7.63 ± 1.39µG field strength at r < 5 kpczdisk0 = 0.61 ± 0.04 kpc Gaussian scale height of disk

Halo B0 = 4.68 ± 1.39µG field strengthcomponent r0 = 10.97 ± 3.80 kpc exponential scale length

z0 = 2.84 ± 1.30 kpc Gaussian scale heightStriation β = 1.36 ± 0.36 striated field B2stri ≡ βB

2reg

Figure 3. Left panel: The random field in the disk. Right panel:The disk component of the JF12 coherent field model for compar-ison; it is clockwise in rings 3-6 and counterclockwise in 1,2,7, and8.

The top two rows of Fig. 1 show the comparison be-tween data and predicted synchrotron intensity for thisbest-fit random field. The right-hand panel in the thirdrow shows the residual between the WMAP 22-GHz to-tal synchrotron emission map and that produced by therandom field model. It is concentrated near the Galac-tic plane and in isolated other regions, and is compatiblewith being due to individual sources such as Cen A, theNorth Polar Spur and other nearby structures not mod-eled here. The magnitude of the residual between modeland data is small compared to the total signal, as mosteasily seen from the left panel of the second row, wherethe total intensity is shown in a logarithmic scale.

The 13-parameter form of the Galactic random fieldadopted here proves to be sufficiently general to give avery good accounting of the data. In combination withthe JF12 coherent field and the striated random field, itprovides an excellent fit to the total intensity, I, with areduced χ2 (χ2 per degree of freedom, χ2dof) after the sec-ond optimization of 1.064 with 2957 degrees of freedom.(One should not attach too much significance to the ex-act value of χ2dof for the GRF model, since it depends onthe arbitrary cut used to remove pixels with large “in-dividualistic” contributions; as evident from the map ofresiduals in Fig. 1, additional pixels could be placed inthis category which would decrease χ2dof .) As noted, thefit to I breaks the degeneracy between rescaling ncre andthe presence of a striated random field.

The left panel of Fig. 3 shows the disk componentof the random field, with the magnitude of the coherentdisk field from JF12 in the right panel for comparison.

4

The average rms strength of the disk component of therandom field at the solar circle is 6.6 µG, but it variesstrongly from arm to arm. The spiral-arm model itselfshould not be taken too literally – it is presumably nomore than a simplified encoding of some important fea-tures of the structure.

Due to the large value of the random and striated fieldscompared to the coherent field, we cannot predict thevalue of the field at a particular position since the fluctu-ations dominate the mean value. Nonetheless, we shouldcheck whether the predicted range of values for the totalfield in the solar neighborhood is consistent with observa-tions – keeping in mind that the rms component has O(1)variance locally, so the actual local field at any particu-lar position can be expected to differ significantly fromthe result of combining the local coherent, striated andrandom components in quadrature. The estimate is addi-tionally uncertain due to our position near the boundarybetween the 4th and 5th arms, since the simple arm ge-ometry assumed in the present models is only expectedto be valid in some average sense and may be locallymodified, and the parameters specifying the geometry ofthe arms in JF12 were taken from the NE2001 model ofne rather than being free parameters of the GMF model.With those caveats, the vertical component of the localcoherent field is 0.2 µG and the horizontal component is0.5-1.2 µG in the 4th and 5th arms. Combining the co-herent, striated and random components in quadraturegives an estimate of the magnitude of the local field of3-5 µG, with the range reflecting the values obtained forthe 5th and 4th arms. Given the O(1) variance from therandom field, this estimated local GMF value is consis-tent with the 6 µG value commonly cited (Beck 2008).The more recent studies of Taylor et al. (2009) and Maoet al. (2010) are complementary, with the former hav-ing larger sky coverage and the latter a higher densityof well-measured extragalactic sources but restricted tothe polar caps. Our results are consistent with both,within the observational uncertainties and predicted fluc-tuations. For instance, Mao et al. (2010) finds that thelocal random field is larger than the local coherent field,as we do, and estimates the halo random field above thesolar system based on the variance in RMs in the po-lar caps to be ≈ 1µG in some average sense, consistentwith our fit which decreases slowly from ≈ 2µG in theGalactic plane, and is 1µG about 3 kpc above the plane.

5. SUMMARY

We have presented here a 13-parameter model of therandom component of the Galactic magnetic field, tocomplete our characterization of the Galactic magneticfield. Taken together, our comprehensive GMF model fitutilizes 36 parameters, including the 21-parameter JF12model for the coherent field, the strength of a striatedrandom field proportional to and aligned with the localcoherent field, and an overall rescaling of the GALPROPcosmic ray electron density. This model provides an ex-cellent fit to more than 40,000 extragalactic RMs and theWMAP polarized and unpolarized Galactic synchrotronemission maps – a total of nearly 10,000 independent ob-servables, each with a measured variance. The striatedrandom field has vanishing mean and is locally alignedwith the regular field; its energy density is found to be≈ 35% larger than that of the regular field. The fit in-

dicates that the GALPROP cosmic ray electron densityprovided by A. Strong in 2009 is ≈ 20% too low in therelevant energy regime, and/or its scale height may needto be revised; indeed, direct comparison to the currentobservations shows that including the correction factorinferred by our method gives a better fit in the relevantenergy region.

The GMF model provided here and in JF12 is a sub-stantial improvement over the previous state-of-the-artGMF models in several ways. First, the coherent com-ponent gives a much better fit to the RM, Q and U datathan any other model. The JF12 coherent field modeland γ rescaling has a χ2 per degree of freedom of 1.096for the 6605 observables (pixels of RM, Q and U) whilethe fit is much worse for other recent models in the lit-erature, with χ2dof of 2.663 & 4.971 for Pshirkov et al.(2011) BSS & ASS respectively and 1.672 for Sun et al.(2008), with their 2010 parameter update. Second, thefunctional form used in our model is better and moregeneral: we enforce that the field is divergenceless andinclude out-of-plane and striated components. The im-provement due to our functional form is evident by re-optimizing the parameters of the other models with ourMCMC procedure and Hammurabi; then their χ2dof val-ues become 1.452, 1.591 and 1.325 respectively – stillvastly worse than the 1.096 of JF12, considering thereare nearly 6600 degrees of freedom. Finally, we pro-vide for the first time a detailed model of the randomfield strength. The final fit to 2971 pixels of I includingthe contributions of all three components of the GMF(coherent, striated and random) has χ2dof = 1.064. Weemphasize that there are two additional sources of uncer-tainty which we cannot quantify: the uncertainty in therelativistic (and thermal) electron distributions, neededto predict the RM and synchrotron emission observablesfor a given GMF model, and a possible inadequacy inthe field model which could arise due to not having cho-sen a sufficiently general functional form. Future workwill aim to address both of these issues, through self-consistent determination of the electron thermal and rel-ativistic electron distributions, and exploration of addi-tional field models.

The total energy in the random and striated fields inthis model is roughly six times that of the coherent field.Taking all three components together – random, stri-ated and regular – we find that the magnetic energy ofthe Galaxy is ≈ 5 × 1055 erg. The local magnetic fieldstrength is predicted to be 3− 5µG, with an uncertaintyof order of the rms local random field, 3µG; this is con-sistent with the 6 µG value of (Beck 2008). The averagerms strength of the disk component of the random fieldat the solar circle is 6.6 µG, but has significant arm-to-arm variations. The vertical component of the JF12coherent local field is 0.2 µG, which is compatible withobservational estimates (Han & Qiao 1994; Taylor et al.2009; Mao et al. 2010).

The improved determination of the Galactic randomand coherent fields provided by this work represents asignificant advance in our understanding of the Galaxy.Explaining the Galactic magnetic field will be an impor-tant challenge to theory, and using this more exact modelwill allow greater fidelity in making prediction for a hostof important applications, from the deflections of ultra-high energy cosmic rays to the spectra and distribution

5

of Galactic cosmic rays.This research has been supported in part by NSF-

PHY-0701451 and NASA grant NNX10AC96G.

REFERENCES

Beck, R. 2008, in AIP Conference Series, Vol. 1085, ed. F. A.Aharonian, W. Hofmann, & F. Rieger, 83–96

Beuermann, K., Kanbach, G., & Berkhuijsen, E. M. 1985, A&A,153, 17

Cordes, J. M., & Lazio, T. J. W. 2002Gaensler, B. M., & Johnston, S. 1995, MNRAS, 277, 1243Gold, B. et al. 2011, ApJS, 192, 15Górski, K. M., et al. 2005, ApJ, 622, 759

Han, J. L., & Qiao, G. J. 1994, A&A, 288, 759Haverkorn, M., et al.2008, ApJ, 680, 362Jaffe, T. R., et al.2011, MNRAS, 416, 1152Jansson, R., & Farrar, G. R. 2012, ApJ, 757Mao, S. A., et al. 2010, ApJ, 714, 1170Pshirkov, M. S., , et al. 2011, ApJ, 738, 192Sokoloff, D. D., , et al. 1998, MNRAS, 299, 189Strong, A. W., Orlando, E., & Jaffe, T. R. 2011, A&A, 534, A54Sun, X. H., , et al. 2008, A&A, 477, 573Taylor, A. R., Stil, J. M., & Sunstrum, C. 2009, ApJ, 702, 1230Tinyakov, P. G., & Tkachev, I. I. 2005, Astroparticle Physics, 24,

32Waelkens, A., , et al. 2009, A&A, 495, 697

ABSTRACT1 Introduction2 Model Field3 Method and Data4 Results of the Fit5 Summary