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arXiv:0807.1532v1 [astro-ph] 9 Jul 2008

ACCEPTED TO The Astrophysical Journal

Preprint typeset using LATEX style emulateapj v. 08/13/06

A RADIO AND OPTICAL POLARIZATION STUDY OF THE MAGNETIC FIELD IN THE SMALL MAGELLANIC

CLOUD

S. A. MAO,1B. M. GAENSLER,2,8S. STANIMIROVI´C,3M. HAVERKORN,4,9N. M. MCCLURE-GRIFFITHS,5L. STAVELEY-SMITH6,10AND

J. M. DICKEY7

Accepted to The Astrophysical Journal

ABSTRACT

We present a study of the magnetic field of the Small Magellanic Cloud (SMC), carried out using radio

Faraday rotation and optical starlight polarization data. Consistent negative rotation measures (RMs) across

the SMC indicate that the line-of-sight magnetic field is directed uniformly away from us with a strength

0.19±0.06 µG. Applying the Chandrasekhar-Fermi method to starlight polarization data yields an ordered

magnetic field in the plane of the sky of strength 1.6±0.4 µG oriented at a position angle 4◦±12◦, measured

counter-clockwise from the great circle on the sky joining the SMC to the Large Magellanic Cloud (LMC).

We construct a three-dimensional magnetic field model of the SMC, under the assumption that the RMs and

starlight polarization probe the same underlyinglarge-scale field. The vector defining the overall orientation of

the SMC magnetic field shows a potential alignment with the vector joining the center of the SMC to the center

of the LMC, suggesting the possibility of a “pan-Magellanic” magnetic field. A cosmic-ray driven dynamo

is the most viable explanation of the observed field geometry, but has difficulties accounting for the observed

uni-directional field lines. A study of Faraday rotation through the Magellanic Bridge is needed to further test

the pan-Magellanic field hypothesis.

Subject headings: magnetic fields —Faraday rotation—polarization—galaxies: Small Magellanic Cloud

1. INTRODUCTION

Magnetic fields play key roles in many astrophysical pro-

cesses in theinterstellar medium(ISM) — theyaccelerateand

confine cosmic rays, trigger star formation and exert pressure

to balance against gravity (Beck 2007). Therefore, to better

understand galaxy evolution, investigating the structure, ori-

gin and evolution of galactic magnetic fields is necessary.

It is useful to picture the total magnetic field at any location

in a galaxy as a superposition of an ordered large-scale com-

ponent and a random small-scale component. An ordered (or

uniform)field caneitherbecoherentorincoherent: a coherent

field has unidirectionalfield lines, whereas an incoherentfield

has fieldlines of thesame orientationbut hasfrequentfield re-

versals. A large-scale dynamo is the only known mechanism

that can generate large-scale coherent fields (Beck 2004).

Coherentmagneticfields havebeenobservedin normalspi-

ral galaxies such as the Milky Way and M31. The fields are

typically in spiral-like configurations with field strengths of

a few µG (Beck 2007). Because these galaxies have signifi-

cant differential rotation, such observations can be explained

by the standard α-ω dynamo which amplifies and orders the

field by small scale turbulent motion (the α-effect) as well

as differential rotation (the ω-effect) in the galactic disk on a

1Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138;

samao@cfa.harvard.edu

2School of Physics, The University of Sydney, NSW 2006, Australia

3Department of Astronomy, University of Wisconsin, Madison, WI 53706

4Astronomy Department, University of California, Berkeley, CA 94720

5Australia Telescope National Facility, CSIRO, Epping, NSW 1710, Aus-

tralia

6School of Physics, University of Western Australia, Crawley, WA 6009,

Australia

7Physics Department, University of Tasmania, Hobart, TAS 7001, Aus-

tralia

8Alfred P. Sloan Research Fellow, Australian Research Council Federa-

tion Fellow

9Jansky Fellow, National Radio Astronomy Observatory

10Premier’s Fellow

global e-folding time of ∼ 109yrs (Shukurov 2007). Despite

its success in accountingforthelargescale coherentfieldseen

in spiral galaxies, the standard dynamo theory fails to explain

the presence of coherent magnetic fields discovered in several

irregular galaxies such as NGC 4449 and the Large Magel-

lanic Cloud (LMC), due to its long amplification time scale

(Klein et al. 1996; Chy˙ zy et al. 2000; Gaensler et al. 2005).

The polarized radio continuum emission of NGC 4449,

a dwarf irregular galaxy, at 4.9 and 8.6 GHz reveals large

scale spiral like structure in the magnetic field. Moreover,

this slowly rotating galaxy shows regions of coherent mag-

netic field from Faraday rotation studies (Klein et al. 1996).

The long field amplification time scale of the classical mean

field dynamo argues against this being the underlying mecha-

nism that produces the magnetic field observed in NGC 4449.

A Faraday rotation measure study of extragalactic polarized

sources behind the LMC carried out by Gaensler et al. (2005)

suggests that the LMC hosts a coherent axisymmetric mag-

netic field of strength ∼ 1µG. The random component domi-

nates over the ordered component with a strength of ∼ 3 µG.

It is believed that close encounters between the Magellanic

Clouds and the Milky Way have triggered episodes of star

formation in the LMC over the past 4 billion years (Bekki

& Chiba 2005). Any coherent field built up by the standard

dynamo would have been disrupted by the outflow from ac-

tive star forming regions. Hence, the existence of a coherent

field in the LMC suggests that a field generation mechanism

with a much faster amplification time scale is at work.

A much more efficient process, the Parker (1992) dynamo,

could account for the large-scale magnetic fields detected in

irregular galaxies such as NGC 4449 and the LMC (Hanasz

et al. 2004). Vertical pressure from cosmic rays can force

magnetic field lines into the galactic halo and form loops

which reconnect and then are amplified by the ω effect. This

process can significantly increase the α effect and can operate

over a much shorter amplification time scale than the α−ω

dynamo (Hanasz et al. 2004).

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2 Mao et al.

As a close neighbor of the Milky Way, the large angular ex-

tent of the Small Magellanic Cloud (SMC) on the sky allows

us to determine RMs of polarized background radio sources

whose projections lie behindit. Since RM is an integral of the

line of sight magnetic field strength weighted by the thermal

electron density, only a coherent field can produce a consis-

tent sign in RM as a function of position on the sky. An RM

study can distinguish between coherent and incoherent fields

andcanindicatethegeometryofanycoherentfield, andthere-

fore can potentially reveal the field generation mechanism.

The alignment of non-spherical dust grains with the mag-

netic field in the ISM linearly polarizes optical radiation that

travels through it. Therefore, measuring the optical polariza-

tion of stars in the SMC enables us to estimate the orienta-

tion of the ordered magnetic field in the plane of the sky.

The spread in polarization position angle of an ensemble of

starlightpolarizationmeasurementsallowsonetoestimate the

mean strength of the ordered componentof the magnetic field

using the Chandrasekhar-Fermi (1953), or C-F method. As-

suming that the field is unidirectional and knowing both the

line-of-sight and the plane-of-the-sky magnetic field strength

and orientation, one can construct a three dimensional mag-

netic field vectorfor the SMC, whichcan furtherconstrainthe

field generation mechanism.

In this paper, we present the results of a radio and optical

polarization study of the magnetic field in the SMC. We use

RMs of polarized extragalactic background radio sources to

determine the magnetic field strength and direction along the

line of sight. The orientation and strength of the plane-of-the-

skymagneticfieldis studiedusingopticalpolarizationofstars

in the SMC. We start in § 1.1 by reviewing the properties of

the SMC and we summarize previous studies on SMC’s mag-

netism in § 1.2. In § 1.3 and § 1.4, we summarize the physics

behind the RM method and the C-F method respectively. We

then describe the observation, data reduction procedures and

present results in § 2. We derive the line-of-sight magnetic

field of the SMC in § 3. In § 4, we estimate the plane-of-the-

sky magnetic field and the random field strength in the SMC.

A 3D magnetic field vector of the SMC is constructed in § 5.

A discussion of possible field generation mechanisms is pro-

vided in § 6.

Throughout this paper, we represent physical quantities in

the plane of the sky by the subscript ⊥ and those along the

line of sight by the subscript ?. We denote the average of a

quantity x over the plane of the sky and that averaged along

the line of sight by ?x? and ¯ x respectively. Table 1 contains a

glossary of the variables used in this paper.

1.1. The Small Magellanic Cloud

The Small Magellanic Cloud is a nearby gas rich dwarf

irregular galaxy. Recent precise measurements of ap-

parent magnitudes of stars at the tip of the red giant

branch in the SMC yield a distance modulus of 18.99±

0.03(formal)±0.08(systematic)(Cionietal.2000),whichcor-

respondstoa distanceofroughly63±1kpc. Inthispaper,we

adopt a distance to the SMC of 60 kpc. Basic parameters of

theSMC arelisted inTable 2. Both theSMC andthe LMC are

thought to be satellite galaxies of the Milky Way. However,

a recent study by Besla et al. (2007) suggests that the Clouds

are not bound to the Milky Way but are on their first passage

about the Galaxy. It is still of great debate as to whether the

Magellanic Clouds formed as a binary, or whether they be-

came dynamicallycoupledto each other ∼ 4 Gyrs ago (Bekki

&Chiba2005). Themostrecentpropermotionmeasurements

of the Clouds suggest that both scenarios are equally proba-

ble (Kallivayalil et al. 2006; Piatek et al. 2008). It is believed

that the last close encounter of the Magellanic Clouds ∼ 0.2

Gyrs ago triggered star formation in the SMC and created the

morphological and kinematic features seen in the present day

SMC (Yoshizawa & Noguchi 2003).

Stanimirovi´ c et al. (2004) found that the depth of the SMC

is within its tidal radius (∼ 4 − 9 kpc). Lah et al. (2005) have

measured distances to pulsating red giants in the SMC and

found a distance scatter of 3.2 ± 1.6 kpc, which agrees with

the results of Stanimirovi´ c et al. (2004). N-body simulations

of the gravitational interaction between the LMC, SMC and

the Milky Way have been able to reproduce the large line-of-

sight extent of the SMC and its two tidal arms (Gardiner et al.

1994; Yoshizawa & Noguchi 2003).

The gas component in the SMC shows signs of rotation

whereas the old stellar component does not (Hatzidimitriou

et al. 1997). The gas kinematics of the SMC were investi-

gated by Stanimirovi´ c et al. (2004), who found a strong ve-

locity gradient in HI across the SMC from the southwest to

the northeast. Stanimirovi´ c et al. (2004) constructed a rota-

tion curve of the gas disk, and derived a maximum rotation

velocity of 50 km s−1.

1.2. Previous Studies of Magnetism in the SMC

The most common way to study magnetic fields in external

galaxies is by observing synchrotron emission at radio wave-

lengths. Haynes et al. (1986) examined linear polarization

maps of the SMC at 1.4 GHz. They found, without any Fara-

day rotation correction, an ordered magnetic field directed

along the SMC’s bar in the plane of the sky. Loiseau et al.

(1987) analyzed radio continuum maps of the SMC at 408

MHz, 1.4 GHz and 2.3 GHz and obtained a total equiparti-

tion field strength of ∼ 5 µG by using an average non-thermal

spectral index α of 0.87 (specific intensity of synchrotron

emission Iν∝ ν−α) and a depth of the synchrotron emit-

ting region of 6 kpc. Haynes et al. (1990) observed the SMC

at 2.5, 4.8 and 8.6 GHz and concluded that the SMC has a

large-scale magnetic field, since weak polarized emission is

detected across the whole SMC body.

Chi & Wolfendale (1993) measured the γ-ray flux from the

SMC to determine the field strength from radio synchrotron

emission without needing to invoke the equipartition assump-

tion. Theyobtainedanestimatewhichexceededtheequiparti-

tion value and concludedthat energy equipartitionis not valid

in the SMC. Pohl (1993), however, took the energy density in

cosmic ray electrons into account, and demonstrated that en-

ergy equipartition between magnetic field and cosmic rays is

not necessarily violated.

One should note that calculating the total equipartitionfield

requires knowledge of the depth of the synchrotron emitting

layer of the SMC and the inclination of magnetic field with

respect to the plane of the sky, which are both poorly con-

strained in the SMC. Also, as explained by Beck & Krause

(2005), the classical equipartition energy formula underesti-

mates the true equipartition field strength, since the former

involvesintegratingthe radiospectrum with a fixed frequency

interval instead of a fixed energy interval, and with insuffi-

cient knowledgeof the ratio of the total energydensity of cos-

mic ray nuclei to that of the electrons and positrons. We will

further explore this issue in § 4.2.

Optical polarization from stars in the SMC can be used

to map the geometry of the plane-of-the-sky component of

the magnetic field, assuming that the observed polarization

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3

is due to scattering by non-spherical foreground dust grains

aligned by the local magnetic field. Polarization measure-

ments for 147 SMC stars have been made by Mathewson &

Ford (1970a,b), Schmidt (1970, 1976), and Magalhães et al.

(1990). Since the polarization “vectors"1, after removal of

Galactic foregroundpolarization, appear to run parallel to the

direction connecting the Magellanic Clouds (Figure 1), the

“Pan-Magellanic" magnetic field hypothesis emerged, which

suggests the existence of a large scale magnetic field associ-

ated with the entire Magellanic system. Wayte ’s (1990) re-

analysis of previouslyobtained starlight polarization data sets

appeared to support the idea of this Pan-Magellanic magnetic

field. However, as the Galactic foreground polarization also

runs along the projection of the line joining the Magellanic

Clouds (Schmidt 1970), any contribution from Galactic fore-

ground that has not been correctly subtracted could be misin-

terpreted as an intrinsic magnetic field connecting the Mag-

ellanic Clouds. In addition, anisotropic scattering in the ISM

may also polarize starlight (Widrow 2002). Therefore, one

has to be cautious interpreting these results. In § 4.1, we will

furtheranalyze the opticalstarlight polarizationdata using the

C-F method to derive the ordered magnetic field strength of

the SMC in the plane of the sky.

1.3. Faraday Rotation

When linearly polarized light travels through a magnetized

plasma, the plane of polarization rotates due to birefringence.

The change in the polarization position angle ∆φ in radians

is given by

∆φ = RMλ2

(1)

where λ is the wavelengthof the radiationmeasuredin meters

and RM is the rotation measure, defined by

RM = 0.812

?observer

source

ne(l)B?(l)dl rad m−2

(2)

In the above equation, ne(l) (in cm−3) is the thermal elec-

tron density, B?(l) (in µG) is the line of sight magnetic field

strength and dl (in pc) is a line element along the line of sight.

The sign of the RM gives the direction of the line of sight

component of the average field. For example, a negative RM

represents a field whose line of sight component is directed

away from us.

RMs for extragalactic radio sources behind the SMC can

be decomposed into various contributions along the line of

sight: the intrinsic RM of the source, the RM through the

intergalactic medium (IGM), the RM through the SMC, and

the foregroundMilky Way RM.

RMobserved= RMIntrinsic+RMIGM+RMSMC+RMMilkyWay (3)

RMMilkyWaycan be estimated by observing RMs of extra-

galactic sources whose projections on the sky lie outside, but

close to the SMC. RMs of extragalactic sources at Galactic

latitudes |b| > 30◦have a standard deviation ∼ 10 rad m−2

(Johnston-Hollittet al. 2004). Inaddition,Brotenet al. (1988)

showed that the extragalactic RMs in the neighborhoodof the

SMC have |RMIntrinsic+RMIGM+RMMilkyWay| ≤ 25 rad m−2.

This implies that the intrinsic RM and the RM through the

IGM are both small compared to the statistical errors of our

RM measurements (See Table 3). After the removal of the

1Position angles provide information on the orientation of the polarization

plane, but not the direction. Hence, there is a 180◦direction ambiguity.

Galactic foreground,the observedRM shouldadequatelyrep-

resenttheRM throughthe SMC. IonosphericFaradayrotation

may also contaminate our data. However, since the magni-

tude of RM induced by the ionosphere is typically only ∼ 1

rad m−2(Tinbergen 1996), this is not of great concern in our

experiment as the statistical errors of our RM measurements

greatly exceed this value (See Table 3).

Faraday rotation is complementary to other measurement

techniques such as equipartition, synchrotron intensity and

starlight polarization since RMs provide the direction of the

magnetic field (and hence the field coherency), while other

techniques only provide the field orientation and estimates

of the field strength, but not its direction. With independent

knowledgeofthethermalelectrondensityandtheline ofsight

depth of the SMC, one can estimate the average line-of-sight

magneticfield strengthusingEquation(2)assumingthat there

is no correlation between electron density and magnetic field

on small scales. If such correlation or anti-correlation exists,

it will result in either underestimation or overestimation of

the field strength by a factor of up to two to three (Beck et al.

2003).

1.4. Optical Starlight Polarization and the

Chandrasekhar-Fermi Method

Starlight polarization alone does not directly give the mag-

netic field strength. However, measuring the spread in po-

larization position angles for an ensemble of stars allows one

to estimate the mean strength of the ordered component of

the magnetic field (Chandrasekhar & Fermi 1953). This tech-

nique assumes that the magnetic field is frozen into the gas

and that turbulence leads to isotropic fluctuation of the mag-

neticfieldaroundthemeanfielddirection(Heitschetal.2001;

Sandstrom 2001). Assuming equipartition between the turbu-

lent kinetic and the magnetic energy, the ordered magnetic

field strength averaged over the plane of the sky, ?Bo,⊥?, is

given by (Heitsch et al. 2001):

?Bo,⊥?2= 4πρ

σ2

vlos

σ(tan δp)2

(4)

where ρ is the density of the medium, θpis the measured po-

larization position angle, ?θp? is the weighted mean of the

measured position angles, δp≡ θp−?θp?, and σvlosis the dis-

persion of the line-of-sight velocity in the medium.

2. OBSERVATIONS, DATA REDUCTION AND RESULTS

2.1. Radio Observations

RM data were acquiredat the Australia Telescope Compact

Array (ATCA) over the period 2004 July 10th –18th, using

the 6A array configuration spanning baselines from 336.7 m

to 5938.8 m, with a total of 32 adjacent frequency channels

each of bandwidth 4 MHz centered on 1384 MHz. The stan-

dard primary flux calibrator PKS B1934−638, whose flux at

1384 MHz was assumed to be 14.94 Jy, was observed at the

beginning and the end of each observation. The secondary

calibrator PKS B0252-712 was observed every hour and was

used to correct for polarization leakages and to calibrate the

time-dependent antenna gains. To cover the whole SMC as

well as the region around it, we scanned a 40-square-degree

region divided into 440 pointings. For each pointing, we ob-

tained 30 cuts of 30 seconds, resulting in a total observing

time of 110 hours. These observations have poor sensitivity

on scales larger than ∼ 30 arcseconds. Therefore, extended

sources in the SMC and diffuse emission from the SMC itself

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4Mao et al.

are not detected; what we mainly see are background point

sources.

The MIRIAD packagewas used for data reduction(Sault &

Killeen 2003). Data were first flagged and calibrated. Flag-

ging and rebinningthe 32 4MHz-widechannels resulted in 13

8-MHzwide channels. Foreachpointingand frequencychan-

nel, maps of Stokes parameters Q and U were made. These

maps were then deconvolved using the CLEAN algorithm. A

final map was generated by convolving the sky model with a

Gaussian beam of dimensions 13”×8" . We produced a re-

stored image for each pointing, for Stokes Q and U at each

of the 13 frequency channels. This results in a total of 11440

images, each with a sensitivity of ∼ 1 mJy per beam.

For each pointing and each channel, a linearly polarized in-

tensity (PI) map, corrected for positive bias was made. To

ensure that no source was lost through bandwidth depolariza-

tion, PI maps over all channels were then averaged together

to make a single polarization map for each pointing. A lin-

early polarized intensity map with sensitivity of 0.4 mJy per

beam, covering all 440 pointings, was created using the task

LINMOS. Polarized point sources were identified from the

mosaicked polarized intensity image using the task SFIND,

which implements the False Discovery Rate (FDR) algorithm

(Hopkinset al.2002). Thesepolarizedpointsourcesarelikely

to be extragalactic as their positions do not coincide with

known supernova remnants (SNRs) (Filipovi´ c et al. 2005).

Figure 2 shows two examples of linear polarization detected

from extragalactic background sources in the field. For each

of the 13 channel maps for each source, values of Stokes Q

and U were extracted for the peak pixel and the eight bright-

est pixels (in polarized intensity) surrounding it.

The RM of each source was computed following the al-

gorithm developed by Brown et al. (2003). As long as the

RMs have magnitudes less than ∼ 2700 rad m−2, our data

do not suffer from an nπ ambiguity because of the closely

spaced frequencychannels. For each pixel of each source, the

RM per pixel was calculated by least-squares fitting the un-

wrappedpolarizationpositionangle(seeBrownetal.2003)as

a functionofthe wavelengthsquared. Figure3shows theleast

squares fit for one of our background sources. These RMs

were then passed through tests to ensure sufficient signal-to-

noise and a reasonable quality of fit. A source was accepted if

more than half of the pixels yield reliable RMs (quality of fit

2Q > 0.1) . The source RM (weighted by the error in the RM

for each pixel) and its uncertainty were computed from the

good pixels. If the scatter of RM from pixel to pixel within

the same source was larger than twice the average statistical

error of the source pixels, the source was rejected.

The data reduction procedures described above produce 70

reliable and accurate RMs as listed in Table 3. After compar-

ing catalogued positions of HII regions (Henize 1956) with

those of the extragalactic background sources, we find that

source 134 has a projectionthat coincides with N90, an active

star forming region in the wing of the SMC. The RM through

this particularsight line traces magneticfield andelectrondis-

tribution through the HII region as well as through the diffuse

ISM.

As mentioned in § 1.3, RMMilkyWay can be estimated us-

ing the RM values of extragalactic sources whose projected

positions lie close to, but outside the SMC. We define the

boundary of the SMC to be where the neutral hydrogen col-

2The probability of a random distribution generating a value of χ2greater

than the observed value, for ν degrees of freedom

umndensitydropsbelow2×1021atomscm−2ortheextinction

corrected intrinsic Hα intensity of the SMC drops below 25

deci-Rayleigh (dR), where 1 R = 106photons per 4π stera-

dian = 2.42 × 10−7ergs cm−2s−1sr−1(see § 2.3). A source’s

projection is considered to be inside the SMC if it lies inside

eithertheHIcolumndensityortheHα threshold. We findthat

10 extragalactic sources satisfy this criteria and are indicated

with * in Table 3.

The data are insufficient to constrain a foreground RM de-

pendence on declination as there are very few background

sourcesat moresoutherlydeclinations. However,it is obvious

that background sources to the west of the SMC have values

of RM which are more positive than those to the east, hence,

we perform a least square fit to the value of the foreground

rotation measure as a function of right ascension in degrees

(Figure 4). The best fit has the form

RMMilkyWay= (46.1±4.1)−(4.9±0.9)×a rad m−2

where a is the offset in degrees eastward from zero right as-

cension.

After subtracting the fit to the foreground RM as given in

Equation(5) and propagatingthe associated uncertainties into

RMSMC, the distribution of RM through the SMC is shown

in Figure 5 and listed in Table 5. The RMs of the 10 ex-

tragalactic sources which lie directly behind the SMC range

from −400±60rad m−2(source 135) to 0±50 rad m−2(source

136), with a weighted mean of −30 rad m−2, a weighted stan-

dard deviation(calculatedusing Equation(4.22)in Bevington

& Robinson (2003)) of 40 rad m−2and a median of −75 rad

m−2. After the foregroundsubtraction, RMs of sources whose

projections lie outside the SMC should be zero by construc-

tion. We find a residual RM of 0 rad m−2with a weighted

standard deviation of 20 rad m−2.

From the fact that 9 out of 10 extragalactic sources behind

the SMC have negative RMs and the other has a RM consis-

tent with zero, we argue that the underlying field is unlikely

to be random in direction as this would produce equal num-

bers of positive and negative RMs across the galaxy with a

mean close to zero. If we are observing a random field, the

probability of getting at least nine out of ten RMs of the same

sign is 0.4%. In other words, the magnetic field across the

entire SMC is coherently directed away from us at a 99.6%

confidence level. The measured large RMs through the SMC

alsocast doubtontheorientationoftheplane-of-the-skymag-

netic field obtained by Loiseau et al. (1987) and Haynes et al.

(1991) from linearly polarized radio synchrotron emission,

because our observed mean RM of −30 rad m−2rotates the

polarization position angle by ∼ 70◦at 20 cm. Since Loiseau

et al. (1987)and Haynes et al. (1991) did not correct for Fara-

day rotation, their angles do not correspondto intrinsic angles

in the SMC.

(5)

2.2. Optical Starlight Polarization Data

Mathewson & Ford (1970a) observed 76 stars in the SMC,

along with 60 Galactic stars towards the SMC at distances

from 50 pc to 2 kpc to correct for the foregroundpolarization.

They found that the foreground signal has a fractional polar-

ization of 0.2%. The distribution of SMC stars and their raw

optical polarization position angles are plotted in Figure 1.

As pointed out in § 1.4, the Galactic foreground polariza-

tion is directed along the SMC-LMC connection, so a careful

foregroundcorrectionis required. Schmidt (1976) subdivided

the SMC’s projection onto the celestial sphere into five re-

gions and calculated the foreground correction for each re-

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5

gion by studying a large number of Galactic foreground stars

at differentdistances. We haveappliedthis improvedGalactic

foregroundcorrection to the 76 stars observed by Mathewson

& Ford (1970a). Because of the large angular extent of the

SMC, measuring the deviation of polarization position angles

with respect to the north is not useful. Instead, we choose

our reference direction to be along the great circle joining the

SMC and the LMC on the celestial sphere. The positions, po-

larization position angles and associated errors of the polar-

ization vectors of 76 SMC stars, after foregroundsubtraction,

are listed in Table 6.

2.3. Hα and HI data

In order to estimate the thermal electron density in the

SMC, we have used the continuum subtracted SHASSA Hα

map of the SMC smoothed to 4 arcminutes (Gaustad et al.

2001). The image has a sensitivity of 5 dR.

Knowing the Hα intensity of the SMC and the foreground

extinction allows one to evaluate the emission measure, as

will be shown in § 3.2. To correct the observed Hα inten-

sity for interstellar extinction, we use the integrated neutral

hydrogen (HI) column density map of the SMC presented by

Stanimirovi´ cetal.(1999)fromATCAandParkesspectralline

observations. The column density was derived by integrating

the 21cm HI signal over the heliocentric velocity range +90

to +215 km s−1, and the resulting column density map has an

angular resolution of 1.6 arcminutes.

3. THE LINE-OF-SIGHT MAGNETIC FIELD STRENGTH IN THE

WARM IONIZED MEDIUM OF THE SMC

In this section, we construct three ionized gas models from

the extinction corrected Hα intensity of the SMC and from

pulsar dispersion measures. These models allow us to es-

timate the average magnetic field strength along the line of

sight, B?, from the RMs presented in § 2.1.

3.1. Pulsar Dispersion Measure and Rotation Measure

The dispersion measure (DM) of a pulsar is an integral of

the electron density content along the line of sight, defined as:

?L

where neis the average electron density along the total path

length L. There are 5 known radio pulsars in the SMC. Their

positions, measured DMs and an RM for the one source with

Faraday rotation information, are listed in Table 4.

We subtract the Galactic contribution to DMs of SMC pul-

sars using the NE2001 Galactic free electron model devel-

oped by Cordes & Lazio (2002). The average DM of pulsars

in the SMC after the removal of the Galactic contribution is

?DMSMC,pulsar? = 80.9 pc cm−3. If we assume that the pul-

sars are evenly distributed through the SMC, the total DM

through the SMC is approximately twice the mean value, that

is, ?DMSMC? ≈ 162 pc cm−3.

Following the treatment of Manchester et al. (2006), the

mean electron density ?ne? in the SMC can be estimated by

computing the dispersion of DMs and the dispersion of pul-

sars’ spatial coordinates. The underlying assumption is that

the SMC is spherically symmetric. We assume that the mean

distance to the SMC pulsars is 60 kpc, instead of 50 kpc as in

Manchester et al. (2006), and that the offsets of pulsar loca-

tions in RA and DEC directions are independent. The mean

electron density in the SMC is given by

?ne? =

σspatial,1D

DM =

0

ne(l)dl = neL pc cm−3

(6)

σDM

(7)

where σDM≈ 48 pc cm−3is the dispersion of pulsar DMs,

after foreground subtraction; and σspatial,1D≈ 1230 pc is the

one-dimensionalspatial dispersion of their positions. This es-

timation gives a mean electron density of ?ne? ≈ 0.039 cm−3

in the SMC.

Outofthe5knownradiopulsarsintheSMC,onlyone(PSR

J0045-7319)has a measured rotationmeasure, with a value of

−14 ± 27 rad m−2(Crawfordet al. 2001). Comparing the DM

of this pulsar from Table 4 with ?DMSMC?, one can conclude

that this pulsar is located approximately half way through the

galaxy. Following the foreground subtraction procedure de-

scribed in § 2.1, the component of the RM from this pulsar

thatresults fromthe magnetizedmediuminthe SMC is −40±

30 rad m−2. This negative value is consistent with negative

signs of RMs of extragalactic sources through the SMC as

given in Table 5.

3.2. Emission Measure

The emission measure (EM) of ionized gas along the line of

sight is defined as

?L

where n2

along the total path length L.

We derive an emission measure map of the SMC from

the smoothed and star-subtracted Hα emission in this region

(Gaustad et al. 2001) by correcting for both foreground ex-

tinction caused by dust in the Milky Way and internal ex-

tinction in the SMC. The foreground Milky Way contribution

to the observed Hα emission is estimated by the off source

Hα intensity in regions surrounding the SMC. We assume a

constant Galactic foreground HI column density of (4.3 ±

1.3)×1020atoms cm−2(Schwering & Israel 1991) and a di-

mensionless Galactic dust-to-gas ratio k of 0.78 (Pei 1992),

where k is defined as:

k ≡ 1021(τB/NHI)

where τBdenotes the optical depth in the optical B band and

NHIdenotes the neutral hydrogen column density. For the in-

ternal extinction of the SMC, the correction is derived from

the HI column density map (Stanimirovi´ c et al. 2004) and a

dust-to-gas ratio k of 0.08 (Pei 1992). We have used the em-

pirical extinctioncurves of the Milky Way and the SMC at the

wavelength of Hα ( λHα= 6563A)

ξ(λHα) = τHα/τB= 0.6

(Pei 1992), where τHαis the optical depth at 6563A. The op-

tical depth at the wavelength of Hα can thus be expressed as

τHα= k(NHI/(1021cm−2))ξ(λHα)

The intrinsic Hα intensity of the SMC is calculated assum-

ing3that the Hα emitting gas is uniformly mixed with dust

in a region of optical depth τHα. The EM for Hα intensity

IHα,intrinsic,SMCproduced by gas at electron temperature Teis

(see for example Lequeux 2005):

EM =

0

ne(l)2dl = n2

eL pc cm−6

(8)

eis the average of the square of the electron density

cm−2

(9)

(10)

(11)

EM =

IHα,intrinsic,SMC(Te/10000K)0.5

0.39(0.92−0.34ln(Te/10000K))

(12)

where Teis the electron temperature of the diffused ionized

medium in the SMC and IHα,intrinsic,SMC in Rayleighs is the

intrinsic Hα intensity of the SMC .

3See Appendix A for details

Page 6

6Mao et al.

As no measurement of the temperature of SMC’s diffuse

ionized medium exists in the literature, we estimate Te by

adding 2,000K to the average temperature in HII regions (∼

12,000K (Dufour & Harlow 1977)) in the SMC, by analogy

with the diffused ionized medium in the Milky Way, which

are ∼ 2,000K hotter than Galactic HII regions (Madsen et al.

2006). We thus adopt Te∼ 14,000 K. The resulting emission

measure map is shown in Figure 6.

3.3. Diffuse Ionized Gas Models

Ourmodelsarebasedontheassumptionthatthereis nocor-

relationbetweenthe fluctuationsin the electrondensity andin

the magnetic field. From Equation (2) the average magnetic

field strength along the line of sight B?is then:

B?=

RMSMC

0.812 neL

(13)

For gas densities lower than 103cm−3, there is no obser-

vational evidence of correlation between the magnetic field

strength and gas density (Crutcher et al. 2003). As discussed

by Beck et al. (2003), if pressure equilibrium is maintained,

oneexpectsananti-correlationbetweenthemagneticfieldand

the thermal electron density on small scales. This would lead

to an underestimation of B?. On the other hand, neand B?

might be correlated by compression in SNR shocks in se-

lected regions. This would lead to an overestimation of B?.

Depending on the property of the turbulent ISM in the SMC,

our estimates of B?presented in the following are potentially

subject to systematic bias by a factor of two to three.

The following models ignore the presence of individualHII

regions and only focus on diffused ionized regions. As men-

tioned in § 2.1, source 134 appears to coincide with N90, an

active star formation region in the SMC. Estimations of B?

in the following models along this particular sight line might

not reflect the true value and it is excluded when calculating

the average line-of-sight magnetic field strength of the SMC,

?Bc,??.

3.3.1. Model 1: Constant Dispersion Measure

We can estimate the line of sight magnetic field strength

(B?) by assuming that the dispersion measure through the

SMC is constant across the galaxy, with ?DMSMC?=neL =

162 pc cm−3. Combining Equations (6) and (13), we find

B?=

RMSMC

0.812?DMSMC?

(14)

EstimationsofB?obtainedusingthis modelarelisted in the

second columnof Table 7. The weighted magnetic field along

the line of sight averaged across the SMC is −0.20 µG. This

model gives crude estimates of B?, since in reality, both the

depth of the SMC (L) and the line of sight average of electron

density nevary from one sight line to the other, while their

product need not stay the same.

3.3.2. Model 2: Constant ne, varying L

The motivation behind this model is the observational evi-

dence for the variation of the line-of-sight depth as a function

of position in the SMC, as seen in both the HI velocity dis-

persion (Stanimirovi´ cet al. 2004) and the variation of the dis-

tance modulus to Cepheid variables (Lah et al. 2005) across

the SMC.

It is useful to define filling factor f, the fraction of the total

path length occupied by thermal electrons, as the following

(see for example Reynolds 1991; Berkhuijsen et al. 2006)

f =

ne

ncloud

(15)

where ncloud is the average electron density in ionized gas

clouds along the line of sight and neis the mean electron den-

sity along the line of sight. For the special case where the

electron densities in the individual ionized gas clouds along

the line of sight are the same, the filling factor can be ex-

pressed as

f =ne2

ne2

(16)

In this model, we assume that the filling factor f, and the

mean electron density along the line of sight neremain the

same across the galaxy, while the depth L of the SMC varies.

Additionally, we assume that all ionized gas clouds along the

line of sight have the same electron density ncloud. Using the

definition of the filling factor f given in Equations 15 and 16,

one can express the average DM and EM through the SMC as

?DMSMC? = ne?LSMC? = ncloudf?LSMC? = ncloudf?LSMC? (17)

?EMSMC? = n2

where ?LSMC? denotes the average depth of the SMC. Com-

bining the two equations above, we obtain

ncloud=?EMSMC?

?DMSMC?

We assumed in the previous sections that ?DMSMC? = 162

pc cm−3. ?EMSMC? is estimated by the average emission mea-

sure through the SMC defined by the HI column density con-

tour in Figure 5. Individual HII regions with intrinsic Hα

intensity higher than 500 dR are masked out before taking the

average, resulting in ?EMSMC? ≈ 16 pc cm−6. This yields a

mean density in an ionized cloud ncloud≈ 0.10 cm−3. The fill-

ing factor f is estimated using Equation 15 assuming a mean

electron density ne= ?ne? ≈ 0.039 cm−3obtained from the

pulsar DM analysis in § 3.1, to yield f = 0.39. Using the def-

inition of filling factor given in Equation 16, we can express

the EM towards an extragalactic background source as

EMsource= n2

Solving for Lsource, the path length through the SMC to an

extragalactic source, yields

Lsource=EMsource

e?LSMC? = n2

cloudf?LSMC? = n2

cloudf?LSMC? (18)

(19)

cloudfLsource

(20)

fncloud2

(21)

Usingthe aboveequation,we havecomputedtheaveragepath

length through the SMC to the extragalactic sources to be ?L?

≈ 10 kpc with a standard deviation ∆L ≈ 6 kpc, which is

roughly consistent with Stanimirovi´ c et al. (2004).

Using additional information from Hα, the mean magnetic

field strength parallel to the line of sight can be found by

RMSMC

0.812?DMSMC?

This equation has a similar form to Equation (14) with an ad-

ditional correction factor for the variation of EM across the

galaxy. Values for B?using this estimate are listed in the third

columnofTable7. Theweightedmagneticfieldalongtheline

of sight averaged across the SMC is −0.16 µG.

B?=

??EMSMC?

EMsource

?

(22)

Page 7

7

3.3.3. Model 3: Constant Occupation Length fL

In this model, we assume that the occupationlength fL (the

effective length occupied by thermal electrons along a sight

line) is constant while the line-of-sight mean electron density

neis allowed to vary between sight lines. In addition, we

assume that ncloud, the density of all ionized clouds along the

line of sight, is the same. One can manipulate Equations (17)

& (18) and solve for the occupation length fL:

fL =?DMSMC?2

?EMSMC?

(23)

We found in § 3.1 that ?DMSMC? = 162 pc cm−3. ?EMSMC? ≈

16 pc cm−6was estimated in § 3.3.2. This yields an occupa-

tion length fL ≈ 1.6 kpc.

Using the definition of filling factor given in Equation 16,

we can express EM towards an extragalactic background

source as

EMsource= n2

eL = ne2L/f

(24)

Solving for neyields

ne=

?

EMsourcef

L

(25)

Substituting the above expressions into Equation (13) and

solving for the mean magnetic field strength through differ-

ent sight lines gives

B?=

RMSMC

0.812?DMSMC?

?

?EMSMC?

EMsource

(26)

This equation has a similar form to Equation (22), but with a

square root rather than linear dependence on EMs. Values of

B?derived using this method are listed in the fourth column

ofTable 7. The weightedmagneticfieldalongthe lineof sight

averaged across the SMC is −0.19 µG.

3.3.4. Summary of the Models

All models yield field strengths that are mostly consistent

with each other within their uncertainties. Model 1 is a sim-

plified picture of the physical situation which does not make

use of all the known information, and thus it provides rough

estimates of B?. Model 2 and 3 are more sophisticated, and

thus they provide estimates that probably better describe the

true B?. Out of the three models, model 2 makes use of the

most information one can get from pulsar dispersion measure

analysis and the Hα intensity of the SMC. Model 3 has the

most degrees of freedom: both f and L are allowed to vary

from one line of sight to the other as long as their product

stays the same, and the averageelectron density along the line

of sight neis allowed to change depending on which sight

line one is looking through. However, one should note that

both model 2 and 3 assume that the electron density in ion-

ized clouds along the line of sight is either ncloud(a constant)

or 0, that is, a smooth fluctuationof electron density along the

line of sight is forbidden. The following discussions refer to

estimations from model 3 unless specified otherwise.

The 10 extragalactic sources that lie behind the SMC yield

line-of-sightmagneticfield strengths rangingfrom−1.5± 0.6

(source135)to 0.0± 0.2µG (source136),where the negative

sign denotes a magnetic field directed away from us. Only

three out of the 10 sources have a B?consistent with zero,

while all the others are negative at at least the 2σ and four

are negative at the 3 σ confidence level. The distribution of

the magnetic field through the SMC is plotted in Figure 6 on

the emission measure map of the galaxy. The weighted mean

of the line-of-sightstrength of the magnetic field is −0.19 µG,

whichinsubsequentdiscussionweadoptas thecoherentmag-

netic field strength of the SMC parallel to the line of sight,

?Bc,??.

Using the standard error in the weighted mean prescription

in Cochran (1977) and assuming that there is one overall un-

derlying field in the SMC; we find that ?Bc,?? = −0 .19 µG

with a standard error in the weighted mean of 0.06 µG. We

can also quantify the scatter of the field by quoting at 68 %

confidence level that the coherent magnetic field strength of

the SMC, ?Bc,?? is −0.2+0.1

magnetic field that we derive in this section is independent of

the temperature of the WIM one picks to convert the Hα in-

tensity into an EM (see Equation 12) because the expressions

of the field strength (Equation 14, 22 & 26) only involve the

ratio of emission measures.

−0.3µG. Note that the strength of the

4. THE PLANE-OF-THE-SKY MAGNETIC FIELD OF THE SMC

4.1. Estimation of ?Bo,⊥?using the C-F method

In § 3, we have derived the line-of-sight magnetic field

strength of ionized gas in the SMC using the RMs of extra-

galactic radio sources. Now, we would like to estimate the

strengthofthemagneticfield perpendicularto thelineofsight

by applying the C-F method to the starlight polarization data

presented in § 2.2. Since the infrared dust emission of the

SMC has a similar morphology to its Hα intensity, we as-

sume that dust is in the warm ionized medium (WIM) of the

SMC. In addition, Rodrigues et al. (1997) found, from analy-

sis of extinction and polarization data of SMC stars, that the

SMC has smaller grain sizes than those in the Milky Way

(where dust lie mostly in the warm neutral medium (WNM),

see Lockman & Condon (2005)). One expects smaller dust

grains in the WIM than in the WNM due to grain shattering

and grain-grain collisions (Jones et al. 1996). The fact that

the SMC has smaller grain sizes than those in the Milky Way

furthersupports our assumptionthat dust is in the WIM of the

SMC. To estimatethe strengthofthe plane-of-the-skycompo-

nentoffield, we use Equation(4). The densityofthe medium,

ρ, is

ρ = γHnHmH

where γH ≈ 1.22 is the equivalent molecular weight of the

ISM for SMC abundances (Russell & Dopita 1992; Peimbert

& Peimbert 2000), nHis the number density of hydrogen and

mH is the mass of a hydrogen atom. We assume that at a

temperature of 14,000K, hydrogen in the WIM is completely

ionized, with a negligible ionization fraction for heavier el-

ements. Hence, the mean ISM hydrogen number density is

nH= ncloud∼ 0.1 cm−3(see § 3.3.2). The average HI line-of-

sight velocity dispersion is 22 ± 2 km s−1(Stanimirovi´ c et al.

2004),which we adopt as the line-of-sightvelocity dispersion

in the WIM also.

To estimate the ordered field strength in the plane of the

sky, we eliminate stars with large uncertainty in the polar-

ized fraction as well as stars that lie outside the main body of

the SMC as defined by the column density contour value of

2×1021atoms cm−2(see Figure 5). The starlight polarization

measurementsthat are used in this calculationare listed in Ta-

ble 6. The average polarization position angle ?θp? deviates

+4◦±12◦from the great circle joining the SMC to the LMC,

measured counterclockwise. The standard deviation of tanδp

(27)

Page 8

8 Mao et al.

is thencalculated. UsingEquation(4),theorderedcomponent

of the magnetic field in the plane of the sky averaged over the

whole SMC is ?Bo,⊥? = 1.6 ± 0.4 µG. We choose not to break

the SMC up into sub-regions and estimate the plane-of-the-

sky field in each because there are not a sufficient number

of polarization measurements in each sub-region to sensibly

estimate ?θp? and σ(tanθp). This analysis is complementary

to the rotation measure study since it provides information on

the orderedcomponentofthe plane-of-the-skymagneticfield.

We assume that the fields obtained using the C-F method and

the RM method are orthogonal components of the same large

scale field, so for a magnetic field whose line-of-sight com-

ponent is coherent and whose plane-of-the-sky component is

ordered, the 3D magnetic field vector is likely to be coherent

as well. Hence, we write ?Bc,⊥? = ?Bo,⊥?.

However, this calculation is subjected to various uncertain-

ties. In the Milky Way, it is found that dust exists mainly in

theWNM(Lockman&Condon2005)butWIMdustemission

has also been detected(Lagacheet al. 1999). In the abovecal-

culation, we have assume that all dust lie in SMC’s WIM. In

reality, some dust must be present in the WNM of the SMC

(due to the correlation of IR dust emission and HI column

density, see Stanimirovi´ c et al. (2000)) and hence when esti-

mating ρ and σvlosin the SMC, one should take into account

of the contribution from the neutral medium as well as the

ionized medium. Also, if the polarization measurements are

fromstars onthe nearside of the SMC, we are merelyprobing

the “surface” magnetic field of the galaxy (Magalhães et al.

1990). Furthermore, since dust regions entrenched in oppo-

sitely directed magnetic fields would polarize starlight in the

same fashion, the plane-of-the-skymagnetic field strengthde-

rived is correct only if the ordered magnetic field direction

does not change appreciablyalong the entire line of sight. We

will overestimate the field strength when the plane-of-the-sky

field reverses direction along the line of sight. A correction

factor was introduced by Myers & Goodman (1991) to ac-

count for this effect. We do not need to correct for this here

if the C-F method and the RM method probe the same large

scale field, since RM data demonstrate that the field does not

reverse on large scales along the line-of-sight.

4.2. Estimation of ?Btotal,⊥?

We now compute the total plane-of-the-sky (i.e.

dom and ordered fields combined) magnetic field strength

?Btotal,⊥? using equipartition energy arguments. If we assume

that the cosmic ray energy density is the same as the mag-

netic field energy density, one can estimate ?Btotal,⊥? using

the relations given in Pacholczyk (1970) and Melrose (1980)

between the specific intensity of synchrotron emission, the

total plane-of-the-sky field and the synchrotron emitting path

length through the galaxy. We assume that the synchrotron

emitting layer of the SMC has the same thickness as the Fara-

day rotating layer, i.e. Lsynchrotron= ?L? ≈ 10 kpc (§ 3.3.2).

Using a spectral index α = 0.87, a cosmic ray energy density

K= 5× 10−17erg−1cm−3(Beck 1982), and a non-thermal in-

tensity Iν= 6.4 × 10−20erg s−1cm−2Hz−1sr−1at ν = 2.3 GHz

(Loiseau et al. 1987), we obtain ? Btotal,⊥? ≈ 2.2 µG.

As pointed out by Beck & Krause (2005), the above cal-

culation is likely to underestimate ?Btotal,⊥? due to the uncer-

tainty in K, and the integration of the radio spectrum over a

fixed frequency range (instead of a fixed energy range to ap-

proximatethecosmicrayspectrum). We haveusedtherevised

equipartition estimate of the magnetic field given in Beck &

Krause (2005) to compute ?Btotal,⊥?. Using the ratio of num-

ran-

ber densities of protons to electrons for cosmic rays acceler-

ated in SNRs K0≈ 100 and Lsynchrotron≈ 10 kpc, we obtain an

equipartition field strength ?Btotal,⊥? = 3.2 µG.

4.3. The Random Magnetic Field in the SMC

Therandomcomponentsofthe magneticfields inthe Milky

Way and in the LMC are found to dominate over the ordered

components (Beck 2000; Gaensler et al. 2005). From the dis-

persion of RMs in the SMC, one can estimate the strength of

the random component of the magnetic field5.

To allow comparison of the random field derived by com-

bining the synchrotron intensity and starlight polarization

measurements (see the next paragraph), which has the same

assumptions as ionized gas model 2 in § 3.3.2, we construct

the random magnetic field model of the SMC based on the

same ionized gas model. We assume that the average electron

densityalong the line of sight, ne, is the same throughall lines

of sight but that the depth of the SMC, L, changes from one

sight line to another. We decompose the magnetic field along

each sight line into coherent and random components, such

that the coherent component does not vary across the SMC;

thedifferencesbetweenthemagneticfieldstrengthsalongdif-

ferent sight lines are only due to the random component. In

AppendixB weshowthatthecorrespondingdispersioninRM

is:

?

σRM= 0.812lone

?Bc,??2(∆L

lo

)2+B2

r(?L?

3lo)(28)

whereσRM∼ 40 radm−2is the weightedstandarddeviationin

RMfortheextragalacticsourcesthatlie behindtheSMC; lo∼

90 pc is the typical cell size along the line of sight, which we

take to be similar to that in the LMC (Gaensler et al. 2005); ne

= 0.039 cm−3, is the mean electron density in the SMC as de-

rivedin§ 3.1,?Bc,?? ≈ −0.16µG is the averageSMC coherent

field strength along the line of sight as obtained using ionized

gas model 2; ?L? ≈ 10 kpc is the average depth of the SMC

alongdifferent sight lines; and ∆L ≈ 6 kpc is the standard de-

viation of the depth of the SMC between different sight lines

(see § 3.3.2). Using the above method, we find Br= 19/√l0

∼ 2 µG. Therefore, in the SMC, the random component of

the magnetic field dominates over the coherent magnetic field

along the line of sight.

A key prediction of our assumption that the RMs, optical

starlight polarization and synchrotron intensity probe differ-

ent projections of the same large-scale magnetic field is that

the independently derived measurements of the random mag-

netic field must agree. Since the total synchrotron intensity

probes the total magnetic field in the plane of the sky while

the C-F method probes the ordered component in the plane of

the sky, one can write:

?B2

total,⊥? = ?B2

o,⊥?+?B2

r,⊥?

(29)

where ?B2

plane of the sky.

isotropic, then its strength is given by

r,⊥? is the random magnetic field strength in the

If we assume that the random field is

B2

r=3

2?B2

r,⊥?

(30)

5If there is no random field and the uniform component is coherent

throughout the galaxy, there will still be an RM gradient across the galaxy

due to projection onto the curved celestial sphere. We ignore this small effect

and assume that the patch of celestial sphere towards the SMC is flat.

Page 9

9

Using ?Btotal,⊥? ≈ 3.2 µG (see § 4.2) and ?Bo,⊥? ≈ 1.6 µG

(see § 4.1) leads to a random magnetic field strength of ∼

3.4 µG. Since the estimate of the random field strength using

the scatter of rotation measure agrees well with that obtained

by combiningthe synchrotronintensity and starlight polariza-

tion measurements, our data demonstrate that our underlying

assumptions are self-consistent.

5. THE 3D MAGNETIC FIELD STRUCTURE OF THE SMC

We cancombinethe results of theRM study(§ 3)andof the

C-F method(§4.1) to constructa 3D magneticfield vectorfor

the SMC, assuming that the two methods probethe same field

(in terms of strength, overall geometry and fluctuations).

The strength of the coherent magnetic field in the SMC is

Btotal,c=

?

?Bc,??2+?B2

c,⊥? = 1.7±0.4µG

(31)

where ?Bc,?? = −0.19 ± 0.06 µG and ?Bc,⊥? ≈ 1.6 ± 0.4 µG

denote the coherent fields found from Faraday rotation and

optical starlight polarization, respectively. The three dimen-

sional field is almost entirely in the plane of the sky.

In order to more precisely determine the direction of the

coherent magnetic field in the SMC, we need to transform

into a cartesian coordinate system with the center of the SMC

at the origin. We define our coordinate system such that the

x-y plane is the sky plane, the negative x axis points towards

the LMC’s projection onto the sky plane, and the positive z

axis points along the vector joining the center of the SMC to

the observer. In this coordinate system, the earth is located at

(0, 0, 60) kpc and the LMC is located at (−17, 0, 13) kpc. At

the center of the SMC, the line-of-sight magnetic field is in

the negative z direction and has a strength of 0.19± 0.06 µG,

while the plane-of-the-sky magnetic field, with a magnitude

of 1.6 ± 0.4 µG, makes an angle 4◦(counterclockwise) with

the positive x axis as shown in § 4.1. Taking into account the

ambiguity of the magnetic field direction in the plane of the

sky, the coherent magnetic field vector in the SMC could be

either

?

Bc,1= 1.6ˆ x+0.1ˆ y−0.19ˆ z

or

?

Bc,2= −1.6ˆ x−0.1ˆ y−0.19ˆ z µG

Equations (32) & (33) allow us to compute the possible an-

gles that the magnetic field vector makes with the character-

istic axes of the Magellanic System. We consider two such

axes: that defined by the path from the LMC along the Mag-

ellanic Bridge to the SMC, and that defined by the normal to

the SMC disk.

Since the 3D structure of the Magellanic Bridge is not well

known, we here assume that the Bridge is parallel to?C, the

vector which runs from the center of the SMC to that of the

LMC.?C lies in the x-z plane and has the form:

µG (32)

(33)

?C = 17ˆ x−13ˆ z kpc (34)

This is a crude approximation, since the interaction between

the Magellanic Clouds most likely does not follow a straight

line.

Separately, the normal to the plane of the SMC’s disk is

given by the unit vector ˆ n:

ˆ n = −0.62ˆ x−0.16ˆ y−0.77ˆ z

(35)

Note that the angle between?C and ˆ n is 92◦(i.e., the SMC disk

is inclined by 2◦degrees from the SMC-LMC axis).

We now consider the extent to which each of?Bc,1and?Bc,2

are aligned with?C or are normal to ˆ n. In the following dis-

cussion, we quote 90% confidence intervals in the statistical

uncertainties in angles. We consider any angle between vec-

tors of less than 20◦to represent broad alignment, and an-

gles in the range70◦−110◦to indicateroughperpendicularity

(reflecting the additional systematic uncertainties in our esti-

mates of?C and ˆ n).

We find that the angle between?Bc,1and?C is 31◦+8◦

have used Monte Carlo simulations with 50,000random sam-

plings to delineate the full probability distribution and find

that the angle between?Bc,1and?C is consistent with alignment

within 2.6σ. On the other hand,?Bc,2makes an angle 136◦+4◦

with?C. Monte Carlo simulations as described above show

that any alignment between?Bc,2and?C is ruled out at >3.1σ.

Comparing the magnetic field vectors with ˆ n, we find an

angle between?Bc,1and ˆ n of 123◦+4◦

Carlo simulations to find that the angle between?Bc,1and ˆ n is

consistent with 90◦at ∼ 2.4σ. The angle between?Bc,2and ˆ n

is 44◦+7◦

larity between?Bc,2and ˆ n at 4.2σ.

The above calculations show that while at 90% confidence

level the SMC magnetic field vector does not orient itself ei-

ther with the Magellanic Bridge or with the SMC disk, at a

slightly higher confidence, the vector?Bc,1does indeed align

with both the Bridge and the disk. We thus favor?Bc,1as the

more likely true magnetic field vector of the SMC over?Bc,2.

In this case, the possible alignment between?Bc,1and?C leaves

open the Pan-Magellanic hypothesis proposed by Schmidt

(1970) and Magalhães et al. (1990), i.e., that the SMC field

orientation is an imprint of the geometry of the overall Mag-

ellanic system.

To further test this Pan-Magellanic idea, additional RM

studies of extragalactic polarized sources behind the Magel-

lanic Bridge will be needed, to see whether the magnetic field

in the Bridge potentially also aligns with the vector?C. Mean-

while, the separate possibility that?Bc,1lies in the SMC disk

(which as noted above, lies in a plane only 2◦from the axis

defined by the Bridge) provides an important constraint on

the origin of the magnetic field in the SMC, as we will dis-

cuss fully in § 6 below. We stress that the above analysis is

based on the assumption that the RMs and optical starlight

polarization probe the same large-scale field in the SMC.

−5◦ . We

−8◦

−8◦ . We have used Monte

−5◦ . Monte Carlo simulations rule out any perpendicu-

6. DISCUSSION

Our observations of the SMC demonstrate the existence of

a large-scale coherent magnetic field. A coherent field cannot

be explained by compression or stretching of a preexisting

random field. The large scale dynamo is the usual mecha-

nism invoked to producea coherentmagnetic field on galactic

scales (Beck 2000). In this section, we explorewhichdynamo

(orother)mechanismsmightberesponsibleforproducingthis

coherent field.

6.1. Ram Pressure Effects

When galaxies with large scale magnetic fields move

rapidlythroughtheintra-clustermedium(ICM),thefieldlines

can be compressed, increasing the total magnetic energy of

the system without dynamo action(Otmianowska-Mazur &

Vollmer 2003). Therefore, it is reasonable to consider ram

Page 10

10Mao et al.

pressure as a mechanism that amplifies galactic magnetic

fields.The maximum ram pressure considered in the 3D

MHD model of Otmianowska-Mazur & Vollmer (2003) cor-

responds to a galaxy moving at a velocity of 1500 km s−1

through an ICM of density 2 × 10−3cm−3. The total mag-

netic energy is increased by a factor of ∼ 5 in their optimal

model during the ram pressure event. Simulated polarized in-

tensity maps show characteristic features during different in-

teraction phases with the ICM. Bright ridges are seen in the

compressed region during the compression/stripping phase,

while a large scale “ring" field, resembling the field created

by a dynamo mechanism, is seen during the gas re-accretion

phasein the polarizedintensitymaps. No suchfeaturescan be

seen in single dish continuumdata of the SMC (Loiseau et al.

1987; Haynes et al. 1991). Furthermore, the space velocity

of the galaxy used in the model of Otmianowska-Mazur &

Vollmer (2003) is approximately three times larger than that

of the velocity of the SMC with respect to the Galactic center

(Kallivayalil et al. 2006), while the density of the Milky Way

halo is ∼ 10−5to 10−4cm−3at the distance of the SMC (Stan-

imirovi´ cet al. 2002;Sembach2006). Therefore,the rampres-

sure effect on the SMC would be roughly 2 orders of magni-

tude weaker than for the simulations of Otmianowska-Mazur

&Vollmer(2003). Also, it is unclearhowrampressureeffects

could generate a coherent large scale field from an initial field

which might be incoherent. Therefore, we rule out the possi-

bility of ram pressure effects generating the field in the SMC.

6.2. The Mean-Field Dynamo

The α-ω or mean-field dynamo requires turbulence to rise

above or below the galactic disk to transform an azimuthal

field into a poloidal one (Beck et al. 1996). The radial com-

ponent of the poloidal field is then transformed back into an

azimuthal component by differential rotation of the disk. Al-

though conservation of magnetic helicity can strongly sup-

press the α effect, it has been shown that this constraint on

the mean field dynamo can be alleviated by flows between

the disk and the halo, or by galactic outflows, which in turn

allow the mean magnetic field to grow to a strength compara-

ble to the equipartition value (see for example Vishniac 2004;

Shukurov et al. 2006).

Dynamoactioncanbe characterizedbytwo parameters: Rα

and Rω, given by Ruzmaikin et al. (1988)

Rα=3loΩ

u0

(36)

Rω=3s∂Ω

∂sh2

lou0

0

(37)

where lois the outer scale of the turbulence,s is the radial dis-

tance from the center of the galaxy, h0is the scale height of

the gas disk and Ω is the angular velocity of the rotating disk.

The typical speed, u0, of turbulent motion of gas in the SMC

can then be approximated by the velocity dispersion in HI, u0

= 22 ± 2 km s−1(Stanimirovi´ c et al. 2004). It is generally be-

lieved that supernovae and superbubbles are the main drivers

of turbulence in the Galactic disk (McCray & Snow 1979), so

lois approximately the size of a supernova remnant or a su-

perbubble. We assume that the ISM in the Milky Way, SMC

and the LMC have comparable outer scales of turbulence, lo

∼ 90 pc (Gaensler et al. 2005)and gas disk scale heights h0∼

500 pc (see for example Shukurov 2007). We use the SMC’s

HI rotation curve obtained by Stanimirovi´ c et al. (2004) to

characterize its degree of differential rotation. Under the con-

dition6that Rω≫ Rα, we can compute the dynamo number,

D, a dimensionless parameter which determines the growth

rate of the magnetic field (Ruzmaikin et al. 1988):

D = RαRω=9Ωsh2

0

u2

0

∂Ω

∂s.

(38)

Note that the above equation is independent of the turbulent

outer scale. Dynamo numbers at radii ranging from s = 0.5

to 3.2 kpc are computed. In this range, the amount of shear

in the disk of the SMC is given by s∂Ω

is comparable to the shear in the Galactic disk near the sun7

∼ 5×10−16s−1. We obtain values of |D| ranging from 0 to 4

in the SMC, while for the Milky Way, |D| ∼ 20 in the solar

vincinity (Shukurov2007). The critical value for an exponen-

tial growth of the field is given by |Dcritical| ∼ 8 − 10, while

a sub-critical dynamo number implies no growth (Shukurov

2007). We thus concludethat for the SMC, the classical mean

field dynamo is not at work.

Using statistical studies of the SMC’s neutral hydrogen,

Stanimirovi´ c & Lazarian (2001) found no characteristic scale

of turbulence up to the size of the galaxy. This implies that

the turbulent outer scale lo could be up to a few kpcs and

the value Rωwould then be much smaller than Rα. In this

case, the dynamo number obtained using Equation (38) is no

longer a good description of the field growth rate, since both

the α-ω and α2dynamos (the latter is a dynamo driven by

helical turbulence action alone) will operate. In this case the

dynamo number |D| would increase by ∼ 30% (Ruzmaikin

et al. 1988), which is not enough to rise the dynamo number

abovethecriticallevel. Moreover,sincetheSMC experienced

bursts of star formation triggered by tidal interactions ∼ 0.4

and 2.5 Gyrs ago (Zaritsky & Harris 2004), the additional en-

ergy injected into the ISM could have created outflows that

wouldconstantly disrupt the buildupof a largescale magnetic

field produced by the α-ω dynamo. We draw the conclusion

that the mean-field dynamo is likely not responsible for the

observed coherent field in the SMC.

∂s∼ 10−16s−1, which

6.3. The Fluctuating Dynamo

It is thoughtthat whenthelargescale dynamois ineffective,

as may occur in weakly rotating galaxies such as the SMC,

the fluctuating dynamo (or the small-scale dynamo) can be-

come important. The fluctuating dynamo, unlike the large-

scale dynamo, can work without differential rotation in the

galactic disk and can generate magnetic field with a correla-

tion length similar to the energy carrying scale of the turbu-

lence(Shukurov2007). The fluctuatingdynamois believedto

operate in small and slowly rotating galaxies with enhanced

star formation, such as IC 10 (Chy˙ zy et al. 2003). The typical

fieldamplificationtime scale is 106to 107years, muchshorter

than the standard dynamo growth rate. Signatures of ran-

dom magnetic fields created by a fluctuating dynamo are iso-

lated polarized non-themal regions coinciding with locations

of star formation (Chy˙ zy et al. 2003). Since magnetic fields

produced by a fluctuating dynamo are incoherent on galactic

scales, they cannot be responsible for producing the observed

coherentfield in theSMC. However,therandomfieldstrength

6For an energy injection scale of value lo∼ 90pc, the condition that Rω

≫ Rαis satisfied.

7Adopting a value for Oort’s constant A ∼ 15 km s−1kpc−1

Page 11

11

(∼ 3µG) estimated in the SMC in § 4.3 suggests that the fluc-

tuating dynamo could be responsible for producing the ran-

dom field component. Single dish radio continuum data of

the SMC at multiple wavelengths show global diffuse syn-

chrotron emission (Loiseau et al. 1987; Haynes et al. 1991),

which also suggests that the randomfield strengthin the SMC

might be relatively high.

6.4. The Cosmic-Ray Driven Dynamo

Parker (1992) proposed a cosmic-ray driven dynamo that

has a much shorter amplification time scale than the stan-

dard mean-field dynamo. In this model, the driving force

comes from cosmic rays injected into the galactic disk from

the acceleration of charged particles in SNR shocks. Un-

like the standard dynamo, this model incorporates a set of

interacting forces including the buoyancy of cosmic rays, the

Coriolisforce,differentialrotationandmagneticreconnection

(Hanasz & Lesch 1998). Differential rotation of the galactic

diskisstill requiredbuttheconsiderablylargerα effectallows

weakly rotating galaxies to achieve a supercritical dynamo

number. The first numerical magneto-hydrodynamic (MHD)

model of the CR-driven dynamo was developed by Hanasz &

Lesch (2004). They modeled a differentially rotating galaxy

withaconstantsupplyofcosmicraysandfoundthatthelarge-

scale magnetic field amplification time scale was about 250

Myrs. OB associations and frequent supernova explosions

duringtheburstsofstarformationintheSMCcouldresultina

large cosmic ray flux, allowing the amplification of magnetic

field in the SMC via the Parker-type dynamo. If the fast dy-

namois responsibleforthe observedfield due to the tidal trig-

gered star formation episode ∼ 0.2 Gyrs ago, it would have

just enough time to build up a galactic scale field before the

tidal velocity field damps the dynamo effect (Kronberg 1994;

Chy˙ zy & Beck 2004). This can potentially also explain the

coherent spiral field seen in the LMC (Gaensler et al. 2005).

Otmianowska-Mazur et al. (2000) modeled the magnetic

field in the LMC-type irregular galaxy NGC 4449 using a

value of Rαcomparable to that of a fast dynamo. NGC 4449

is found to display “fan" like structures that mimic magnetic

spiral arms in polarized intensity. The Faraday rotation map

of NGC 4449 suggests that the galaxy hosts a coherent field

(Klein et al. 1996; Chy˙ zy et al. 2000). Otmianowska-Mazur

et al. (2000) consider a model galaxy with a radius of ∼ 2.5

kpc and a maximum rotational velocity of about 30 km s−1,

which is similar to the SMC. No outflow from a bar and no

random field were included. The value for l2

s−1andtheturbulentdiffusivity(η ∼l0u0/3) was chosento be

1.5×1026cm2s−1. These parameters are typical of a cosmic

ray dynamo as shown by Hanasz et al. (2004). Evolving the

modeled galaxy using the above parameters over ∼ 0.1 Gyr

leads to an increase in the total magnetic energy. Also, this

model is able to reproduce spiral-like field structure resem-

bling the observation of NGC 4449. However, it does not in-

clude several possibly importantphysical processes. First, the

SMC is likely to be subjected to an injection of random field

into the ISM due to a fluctuating dynamo (see § 6.3), which

this model does not account for. Second, the SMC has a rota-

tion curve which peaks at ∼ 50 km s−1rather than the 30 km

s−1used by Otmianowska-Mazuret al. (2000). This results in

a more effective ω effect, which increases the growth rate of

the magnetic field, while random field injection increases the

total magneticenergyof the galaxyfaster. No simulated Fara-

day rotation map was producedby Otmianowska-Mazuret al.

0Ω/h0was 5 km

(2000), therefore, no direct comparison can be made between

their model and our data. MHD models devoted to simulate

the growth of the magnetic field in the SMC are needed in

order to provide a definitive conclusion.

We haveestablishedabovethat it is possible forthe cosmic-

ray driven dynamo to produce the observed magnetic field in

the SMC in terms of time scale arguments. Let us now con-

sider whether this dynamo can explain the observed field ge-

ometry. In§ 5, weshowedthat the3Dmagneticfield vectorof

theSMC maylie in thedisk ofthegalaxyandthat it mayalign

with the vector joining the Magellanic Clouds. A dynamo

produces an azimuthal magnetic field that predominantly lies

in the disk of a galaxy (see Ruzmaikin et al. 1988), and this

could account for the potential alignment of the SMC’s mag-

netic field with the SMC disk as calculated in § 5. If the field

is alignedwith the Bridgerather thanthe disk (as also allowed

by the range of angles calculated in § 5), this could be under-

stood as resulting from ongoingtidal interactions between the

Magellanic Clouds, which could provide a slight realignment

of the overall field orientation.

If the cosmic-ray driven dynamo is the underlying mech-

anism that produces the magnetic field in the SMC, it also

needs to explain the unidirectional field lines seen across the

galaxy. The magnetic field configuration in a galaxy can be

decomposed into different dynamo modes (Beck et al. 1996).

The strongest dynamo mode in an axisymmetric disk is the

m=0 mode, followedby a weaker bisymmetric(m=1) mode.

It has been suggested by Moss (1995) that tidal interactions

can generate bisymmetric magnetic fields in galaxies, pro-

vided that the axisymmetric mode is already at work. Ob-

servations show that in interacting galaxies, such as M51 and

M81, the bisymmetric mode can be important (Krause et al.

1989). According to the 3D mean field dynamo model stud-

ied by Vögler & Schmitt (2001), non-axisymmetric gas mo-

tion is induced in galactic disks during tidal interaction, and

can damp the usual dominant m = 0 mode and excite the

m = 1 mode when the induced tidal velocity is small. An

axisymmetric magnetic field would exhibit a change in the

sign of RM across the disk of the galaxy when viewed edge

on whereas a bisymmetric magnetic field would vary double-

periodically with the azimuthal angle (Krause et al. 1989). It

is unclear how the superposition of a m = 0 mode and a m = 1

mode could produce unidirectional field lines with negative

RM across the SMC, because a superposition of higher order

modes will result in more RM sign changes across the galaxy

disk when viewed edge on.

The observed unidirectional magnetic field lines and the

possible alignment of the field with the Magellanic Bridge

could be explained as follows. Cosmic ray driven dynamo

produces a predominately azimuthal magnetic field in the

SMC disk; this field is then stretched tidally along the SMC-

LMC axis, maintaining its orientation when projected onto

the plane of the sky (to produce starlight polarization vectors

of similar orientation). Note that RM is non-zero only when

the average line of sight electron density is non-zero. It is

possible that the field lines in the SMC do close, that is, there

are sight lines along which field lines do point towards us, but

only at locations with low EM off the main body of the SMC.

Only half of the displaced magnetic loop, whose line of sight

componentis directed away from us, would then be observed.

Theotherhalfofthe loopwhose lineofsight componentis di-

rected towards us would not show positive RMs, as it should

coincide with regions of low EM .

To summarize, the cosmic-ray driven dynamo is a possible

Page 12

12Mao et al.

field generation mechanism for the SMC but has difficulties

explaining the observed magnetic field geometry. One has to

explain the fact that the observed field is unidirectional and

that it potentially lies in the disk of the SMC and aligns with

theMagellanicBridge. Currentobservationaldataarenotsuf-

ficient to rule out/prove the cosmic-ray driven dynamo; fur-

ther observational tests are needed.

7. CONCLUSIONS

We have measured the Faraday rotation of extragalacticpo-

larized sources behind the Small Magellanic Cloud to deter-

mine the SMC’s magnetic field strength and geometry. Our

study reveals that the SMC has a galactic-scale field of 0.19

± 0.06 µG directed coherently away from us along the line

of sight. Optical polarization data on stars in the SMC are re-

analyzed using the Chandradsekhar-Fermi method to give an

ordered component of the magnetic field in the plane of the

sky, of strength 1.6 ± 0.4 µG. Under the assumption that the

Faraday rotation measures and optical starlight polarization

probe the same underlying large scale field in the SMC, we

have constructed a 3D magnetic field vector of the SMC. It

is found that this magnetic field vector possibly aligns with

the Magellanic Bridge. This potential alignment needs to

be verified by future studies of RMs towards extragalactic

sources behind the Magellanic Bridge. The random magnetic

field strength in the SMC derived from RM data alone and

that derived by combining the results of the C-F method with

equipartition were found to be in agreement (∼ 3 µG). This

implies that our underlying assumption, that these 3 indepen-

dent methods probe different components of the same large

scale field, is self-consistent.

The SMC is a slowly rotating galaxy, for which the stan-

dard mean-field dynamo is not expected to be at work be-

cause of the subcritical dynamo number. The cosmic-ray

driven dynamo has a short enough amplification time scale

to explain the observed coherent field. With modifications

by tidal interactions, the field generated by the cosmic-ray

driven dynamo could potentially be aligned with the Magel-

lanic Bridge. However, this model faces difficulties in ex-

plaining the observed uni-directional field lines. Therefore,

the origin of the magnetic field in the SMC is still an open

question which needs to be followed up with more observa-

tions. The relatively small number of background rotation

measures makes it difficult to interpret the observed RMs in

detail. Future observations of the SMC with the Square Kilo-

metre Array will provide ∼ 105RMs in a field of 40 square

degrees surrounding the SMC (Beck & Gaensler 2004); with

which different possible origins of the magnetic field in the

SMC can be fully evaluated (Stepanov et al. 2008).

Acknowledgements We thank Joseph Gelfand for carrying

out the ATCA observations, Anvar Shukurov, Erik Muller,

Alyssa Goodman and Douglas Finkbeiner for useful discus-

sions, and Rainer Beck, Marita Krause and Ellie Berkhuijsen

for their help and hospitality during S. A. M. ’s visit to the

Max Planck Institute for Radio Astronomy. M. H. acknowl-

edges support from the National Radio Astronomy Observa-

tory (NRAO), which is operated by Associated Universities

Inc., under cooperative agreement with the National Science

Foundation. This research was supportedby the National Sci-

ence Foundationthroughgrant AST-0307358to HarvardCol-

lege Observatory. The Australia Telescope Compact Array is

partof the Australian Telescope,which is fundedby the Com-

monwealth of Australia for operation as a National Facility

managed by CSIRO. The Southern H-Alpha Sky Survey At-

las (SHASSA) is supported by the NSF.

Facilities: ATCA

APPENDIX

A. EXTINCTION CORRECTION OF Hα EMISSION

We here describe the procedure to derive the intrinsic Hα intensity of the SMC. We assume that dust is well mixed with Hα

emitting gas for both the SMC and the Milky Way. The observed Hα emission is given by

Iobserved=Iinstrinic,SMC

τHα,SMC

where τHα,SMCis the optical depth of Hα in the SMC, τHα,MWis the optical depth of Hα in the Milky Way, Iintrinsic,MWis the

intrinsic Hα emission of the Milky Way, and Iintrinsic,SMCis the intrinsic Hα emission of the SMC. The second term in the above

equation can be estimated by the observed off source Hα intensity in the regions surrounding the SMC.

The uncertainty in estimating the intrinsic Hα intensity mainly results from the location of the dust with respect to the Hα

emitting regions. The upper estimate of Iinstrinic,SMCcan be found by placing all the dust behind the Hα emitting region in the

SMC, so that what we observe is the intrinsic Hα intensity extincted only by the foregroundMilky Way dust. The lower estimate

of Iinstrinic,SMCcan be found by placing all the dust in front of the Hα emitting region in the SMC.

(1−eτHα,SMC)e−τHα,MW+Iintrinsic,MW

τHα,MW

(1−e−τHα,MW)(A1)

B. A MODEL TO ESTIMATE THE RANDOM MAGNETIC FIELD STRENGTH

We construct this model based on Gaensler et al. (2001,2005)and ionized gas model 2 (see § 3.3.2), for which case we assume

that the average electron density (ne) along different lines of sight is the same but the depth of the SMC varies from one sight line

to the other. From model 3, the mean depth through the SMC is ?L? ≈ 10 kpc with a standard deviation ∆L ≈ 6 kpc. Suppose

that the depth of the SMC through a particular sight line is L, divided up into cells of linear size lo. The total number of cells one

looks through along the line of sight is given by

N =L

lo

(B1)

Within each cell, we suppose that the magnetic field is composed of a coherent component of strength Bc(same direction and

strength from cell to cell), whose strength along the line of sight is ?Bc,?? ≈ 0.16µG, and a random component of strength Br

oriented at an angle θcell,iwith respect to the line of sight. The component of the random field along the line of sight is

Br,?= Brcosθcell,i

(B2)

Page 13

13

The line of sight magnetic field strength in a cell is given by

B?= ?Bc,??+Br,?= ?Bc,??+Brcosθcell,i

(B3)

In addition, we assume that the random component is coherent within each cell but that θ varies randomly from cell to cell.

Different levels of Faraday rotation will be experienced by the incident light rays because they pass through different series of

cells and differentnumbersof cells. Linearly polarizedlight which passes througha single cell in the SMC experiencesa Faraday

rotation given by

RM1−cell= 0.812necell,iloB?= 0.812necell,ilo(?Bc,??+Brcosθcell,i)

After passing through N cells, the incident radiation experiences a Faraday rotation of

(B4)

RMN−cells= 0.812loBr

N

?

i=1

necell,icosθcell,i+0.812lo?Bc,??

N

?

i=1

necell,i

(B5)

where

N

?

i=1

necell,i= neN

(B6)

Since the electron density does not correlate with the orientation of the random field in individual cells,

N

?

i=1

necell,icosθcell,i= ne

N

?

i=1

cosθcell,i.

(B7)

One can rewrite the expression for the rotation measure of the radiation after passing through N cells as

RMN−cells= 0.812loBrne

N

?

i=1

cosθcell,i+0.812lo?Bc,??Nne

(B8)

Averaging across different sight lines, the mean RM through the SMC is given by

?RM? = 0.812lo?Bc,???N?

(B9)

where ?N? = ?L?/lois the average number of cells along different sight lines.

Using the central limit theorem for large N, the standard deviation of RM through the SMC can be expressed as

σRM= 0.812lone

?

?Bc,??2(∆L

lo

)2+Br2(?L?

3lo) .

(B10)

Page 14

14Mao et al.

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Page 15

15

TABLE 1

A LIST OF SYMBOLS USED IN THIS PAPER

SymbolPhysical QuantitySection in which first used

α

a

B?(l)

B?

?Bo,⊥?

B?,i

?Bc,⊥?

?Btotal,⊥?

Br

σB,i

Br,?

?Bc,??

?Br,⊥?

Btotal,c

?Bc,1,?Bc,2

spectral index of synchrotron emission defined by Iν∝ ν−α

angular offset in degrees eastward from RA = 0

magnetic field as a function of path length l along the line of sight in units of µG

average magnetic field strength along the line of sight in µG

average magnetic field strength in the plane of the sky in units of µG

individual measurement of line-of-sight magnetic field B?through the SMC in units of µG

strength of the coherent component of the magnetic field in the plane of the sky in units of µG

strength of the total equipartition magnetic field in the plane of the sky in units of µG

strength of random magnetic field in units of µG

uncertainty associated with an individual magnetic field measurement B?,iin units of µG

strength of random magnetic field parallel to the line of sight in units of µG

coherent magnetic field parallel to the line of sight averaged across the SMC in units of µG

random magnetic field strength perpendicular to the line of sight in units of µG

total coherent magnetic field strength in units of µG

the 3D magnetic field vector of the SMC, subscripts 1 and 2 indicates the two vectors whose

line-of-sight component is the same but have oppositely directed plane-of-the-sky component

equivalent molecular weight of the ISM for SMC abundance

power law index of electron energy distribution in cosmic rays

the vector joining the center of the LMC to the center of the SMC

angle that the polarization plane rotates through in radians

angular deviation of θpfrom ?θp?

dynamo number

critical dynamo number

dispersion measure in units of pc cm−3

average dispersion measure towards SMC pulsars, after foreground subtraction

average dispersion measure through the SMC

standard deviation of pulsar DMs in the SMC

emission measure in units of pc cm−6

average emission measure through the SMC in pc cm−6

emission measure towards an individual extragalactic source behind the SMC

filling factor of thermal electrons along the line of sight

occupation length in units of pc of thermal electrons along the line of sight

scale height of galactic disk in pc

turbulent diffusivity in units of cm2s−1

orientation of the random magnetic field with respect to the line of sight in cell i

specific intensity of synchrotron emission in erg s−1cm−2Hz−1sr−1

intrinsic Hα intensity of the SMC in units of Rayleighs

observed Hα intensity towards the SMC

Hα intensity of the SMC after extinction correction for both the Milky Way and the SMC

Hα intensity of the Milky Way after extinction correction

dimensionless galactic dust-to-gas ratio defined in Equation 9

energy density of cosmic rays in the galaxy in units of erg−1cm−3

ratio of number densities of protons to electrons in cosmic rays accelerated in SNRs

differential path length along the line of sight in units of pc

path length along the line of sight in units of pc

total path length along the line of sight

average depth of the SMC in pc

path length through the SMC to an extragalactic source in pc

depth of the synchrotron emitting layer in the SMC in cm

average path length through the SMC to extragalactic sources in units of pc

standard deviation of Lsourcethrough different sight lines in the SMC

Turbulence outer scale/ typical cell size in the SMC in units of pc

wavelength in meters

§ 1.2

§ 2.1

§ 1.3

§ 3.3

§ 1.4

§ 3.3.4

§ 4.1

§ 4.2

§ 4.3

§ 3.3.4

§ B

§ 3.3.4

§ 4.3

§ 5

§ 5

§ 4.1

§ 4.2

§ 5

§ 1.3

§ 1.4

§ 6.2

§ 6.2

§ 3.1

§ 3.1

§ 3.1

§ 3.1

§ 3.2

§ 3.3.2

§ 3.3.2

§ 3.3.2

§ 3.3.3

§ 6.2

§ 6.4

§ B

§ 1.2

§ 3.2

§ A

§ A

§ A

§ 3.2

§ 4.2

§ 4.2

§ 1.3

§ 1.3

§ 3.1

§ 3.3.2

§ 3.3.2

§ 4.2

§ 3.3.2

§ 3.3.2

§ 4.3, § 6.2

§ 1.3

γH

γe

?C

∆φ

δp

D

Dcritical

DM

?DMSMC,pulsar?

?DMSMC?

σDM

EM

?EMSMC?

EMsource

f

fL

h0

η

θcell,i

Iν

IHα,intrinsic,SMC

Iobserved

Iinstrinic,SMC

Iintrinsic,MW

k

K

K0

dl

l

L

?LSMC?

Lsource

Lsynchrotron

?L?

∆L

lo

λ