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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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2Mao 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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4 Mao 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
6 Mao 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
12 Mao 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
Symbol Physical Quantity Section 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
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