Table Of ContentMon.Not.R.Astron.Soc.000,000–000 (0000) Printed4July2014 (MNLATEXstylefilev2.2)
Magnetic field tomography, helical magnetic fields and
Faraday depolarization
C. Horellou1⋆ and A. Fletcher2
4
1 1 Department of Earth and Space Sciences, Chalmers Universityof Technology, Onsala Space Observatory,SE-439 92 Onsala, Sweden
0 2 School of Mathematics and Statistics, Newcastle University,Newcastle-upon-Tyne NE1 7RU, U.K.
2
l
u
Accepted 2014April6.Received2014March24;inoriginalform2014January16
J
3
ABSTRACT
]
O Wide-band radio polarization observations offer the possibility to recover informa-
tion about the magnetic fields in synchrotron sources, such as details of their three-
C
dimensional configuration, that has previously been inaccessible. The key physical
.
h process involved is the Faraday rotation of the polarized emission in the source (and
p elsewherealongthe wave’spropagationpathtothe observer).Inorderto proceed,re-
- liablemethodsarerequiredforinvertingthesignalsobservedinwavelengthspaceinto
o
usefuldatainFaradayspace,withrobustestimatesoftheiruncertainty.Inthispaper,
r
t we examine how variations of the intrinsic angle of polarized emission ψ0 with the
s
a Faradaydepthφwithinasourceaffecttheobservablequantities.Usingsimplemodels
[ for the FaradaydispersionF(φ) and ψ0(φ), alongwith the currentand plannedprop-
erties of the main radio interferometers,we demonstrate how degeneraciesamong the
2
parametersdescribing the magneto-ionicmedium canbe minimised by combining ob-
v
servationsindifferentwavebands.We alsodiscuss howdepolarizationby Faradaydis-
2
persiondue to a randomcomponentofthe magnetic field attenuates the variationsin
5
1 thespectralenergydistributionofthe polarizationandshiftsitspeaktowardsshorter
4 wavelengths.This additionaleffect reducesthe prospectofrecoveringthe characteris-
. tics of the magnetic field helicity in magneto-ionic media dominated by the turbulent
1
component of the magnetic field.
0
4 Key words: polarization–methods:dataanalysis–techniques:polarimetric–ISM:
1
magnetic fields – galaxies: magnetic fields – radio continuum: galaxies
:
v
i
X
r
a 1 INTRODUCTION z oftheproductofthedensityofthermalelectrons,ne,and
the component of the magnetic field parallel to the line of
A new generation of radio telescopes will map the polar-
sight:
ization of cosmic radio sources over a large range of wave-
lengths,from afewcentimetres toseveral metres.Sincethe
plane of polarization of a linearly polarized wave is rotated +∞
by an amount that depends on the magnetic field and free- φ(z)∝ ne(z′)Bk(z′)dz′, (1)
electron distributions and thewavelength (λ),the resulting Zz
data will probe both the synchrotron-emitting sources and
anyinterveningmagneto-ionicmediuminunprecedentedde- hence, in principle, F(φ) can be used to obtain both the
tail. A useful way to characterize the intrinsic properties of perpendicular and the parallel components of the three-
magneto-ionic media is the Faraday dispersion function1, dimensional magnetic field. (Our system of coordinates is
F(φ),whichcontainsinformation onthetransverseorienta- such that the origin is at the far end of the source and the
tionofthemagneticfield(B )andontheintrinsicpolarized observerislocatedat+ .Amagneticfieldpointingtowards
⊥ ∞
emission as a function of Faraday depth, φ. The Faraday theobserver yields a positive Faraday depth.)
depth is proportional to the integral along the line of sight Reconstruction of F(φ) is usually done by taking ad-
vantage of the Fourier-transform type relationship between
theobservedpolarized emission andtheFaradaydispersion
⋆ E-mail:[email protected] function. The observed complex polarization P(λ2) can be
1 Theterm‘Faradayspectrum’issometimesusedfortheFaraday expressed as the integral over all Faraday depths of the in-
dispersionfunction, which can be misleadingbecause it is not a trinsiccomplexpolarizationF(φ)modulatedbytheFaraday
truespectrum(functionoffrequency). rotation (Burn 1966):
2
P(λ2)= +∞F(φ)e2iφλ2dφ, (2) phisticated statistical methods have been devised to try to
inferitspresence,althoughwithout inclusionofFaradayef-
Z−∞
fects (Oppermannet al. 2011, Junklewitz & Enßlin 2011).
so that F(φ)can be expressed in a similar way:
Anomalousdepolarization(anincreaseratherthantheusual
F(φ)= 1 +∞P(λ2)e−2iφλ2dλ2. (3) ddeuccreedasbeyofatnhehdeleigcarelefioefldpowlaarsizadtisiocnuswseidthbwyavSeoleknoglotffh)eptraol-.
π
Z−∞ (1998). Helical fields have been invoked to explain the
F(φ) is a complex-valued function: anomalous depolarization properties of the nearby galaxy
NGC 6946 (Urbanik,Elstner, & Beck 1997) and polariza-
F(φ)= F(φ)e2iψ0(φ), (4)
| | tion characteristics of the central part of the starburst
where F(φ)dφ is the fraction of polarized flux that comes galaxyNGC253(Heesen et al.2011).Helicalmagneticfields
from re|gions|of Faraday depthbetween φ and φ+dφ, ψ is are also important in galactic and protostellar jets (e.g.
0
theintrinsic polarization angle (perpendicularto thetrans- Keppens& Meliani2009,Fendt2011).Bi-helicalfields(with
verse component of the magnetic field, B~ ) and may itself opposite signs of helicity on small and large scales) are
⊥
depend on φ. produced in simulations of galactic dynamos and the sig-
Equation (3) lies at the heart of methods to re- natures of such fields are discussed in a recent paper by
cover F(φ) from multi-frequency observations of the com- Brandenburg& Stepanov(2014).Inthispaper,wefocuson
plex polarized intensity (called Rotation Measure, RM, thedetectability of single-helical magnetic fields.
synthesis; Brentjens & deBruyn 2005). The RM synthe-
sis has been used to recover Faraday components of com-
pact sources (e.g. Mao et al. 2010) and diffuse structures
in the Milky Way (e.g. Schnitzeler, Katgert, & deBruyn 2 ANALYSIS
2009), in nearby galaxies (e.g. Heald, Braun, & Edmonds
2.1 Observables
2009) and in galaxy clusters (e.g. Pizzo et al. 2011). Sev-
eraltechniqueshavebeenproposedtodealwiththelimited WeconsiderobservationsoftheStokesparametersQandU
λ2coverageprovidedbyrealtelescopes(RM-CLEAN;Heald withtheinstrumentslistedinTable1.Weusedanominalin-
2009;sparseanalysisandcompressivesensing;Li et al.2011, tegrationtimeof1hforthelow-frequencyobservations(Gi-
Andrecut,Stil, & Taylor2012;multiplesignalclassification; ant Meterwave Radio Telescope, GMRT, Westerbork Syn-
Andrecut 2013) and with the missing negative λ2 (e.g. us- thesis Radio Telescope, WSRT, Low Frequency Array, LO-
ing wavelet transforms; Frick et al. 2010, 2011). Beck et al. FAR) and 10 min for observations with the more sensitive
(2012) also used wavelets to analyze the scales of struc- instruments (Jansky Very Large Array, JVLA, Australian
tures in Faraday space and emphasized the need to com- Square Kilometre Array Pathfinder, ASKAP and Square
bine data at high and low frequencies. Because of the dif- KilometreArray 1,SKA1).Thisallows an easy comparison
ficulty of the RM synthesis technique to recover multi- ofthesensitivitiesandmakesitpossible todisplay thecon-
ple Faraday components, it has been suggested to use di- fidenceintervals of theparameters of interest on a common
rect q(λ2) and u(λ2) fitting, where q and u are the Q graph (Figs 2 and 3). We used a channel width of 1 MHz
and U Stokes parameters normalised to the total intensity for all instruments except the JVLA for which a channel
I (Farnsworth, Rudnick,& Brown 2011; O’Sullivan et al. width of 2 MHz is more than sufficient in the wide S-band
2012). (2000–4000MHz)toresolvethemainfeaturesofthespectral
In this paper we show how observations, performed energydistributionofthepolarization. Notethatallinstru-
in the various wavelength ranges available at existing and mentsallow theuse of narrower channels; however, there is
plannedradiotelescopes,canbeusedtoconstrainthevaria- anobvioustrade-offbetweensensitivityperchannelandto-
tionofψ0(andthereforetheorientationofthemagneticfield talintegration time.Wehavevariedthechannelwidthover
component perpendicular to the line of sight) with φ. We two orders of magnitude between 0.1 and 10 MHz and ob-
use a Fisher matrix analysis to quantify the precision that servedthattheresultingprecisiononthemainparameterof
can be achieved for fitted parameters and investigate the interest, β (equation (16)), changes by less than 10−2 for a
degeneracies that exist between the different constituents sametotalintegrationtimeoftheSKA1-Survey.Thequoted
of our model. Recently, Ideguchiet al. (2014) performed a sensitivitiesareindicativesinceseveralinstrumentslistedin
similaranalysistoevaluatethecapabilityofnewradiotele- Table 1 are still in their design phase. Also, some bands,
scopes to constrain the properties of intergalactic magnetic especially the low-frequency ones, will be affected by radio
fieldsthroughobservationsofbackgroundpolarizedsources. frequencyinterferencesandafractionofthechannelswillbe
Their work assumed two Faraday components, each with a missing. With real data at hand it will be straightforward
constant ψ0, a narrow one (the compact radio source) and toincludetheactualfrequencycoverageandsensitivitiesin
abroadone(possiblyassociated withtheMilkyWay).Here themodeling of a particular data set.
we consider a linear variation of ψ0 with φ and show how We scaled the sensitivities σlit quoted in the literature
thedegeneracies between pairs of model parameters can be for a given effective bandwidth BW and integration time
lit
brokenusingcomplementary datasets from differentinstru- t to new values of the channel width ∆ν and integration
lit
mentsinordertorecoverψ0(φ),usingtwosimplemodelsof timetint,asgiveninthetable,foragivennumberoftunings
F(φ),a constant and a Gaussian. N to cover thewhole bandwidth:
tunings
In the simple cases we consider, the variation of ψ (φ)
0
can be produced by a helical magnetic field. Magnetic he- BW t
σ=σ lit litN . (5)
licity is a natural consequence of dynamo action and so- lit ∆ν t tunings
r int
3
Table 1.Frequencyrangeandsensitivityofthelistedinstruments.MoredetailsaregiveninSect. 2.1.
Instrument Note Frequency band Sensitivity Channelwidth Integration time Reference
[MHz] [mJy] [MHz]
JVLA S-band 2000-4000 0.3 2 10min (1)
JVLA L-band 1000-2000 0.6 1 10min (1)
ASKAP 700–1800 2.5 1 10min (2)
SKA1 Survey 650–1670 0.3 1 10min (3)
SKA1 Mid 350–3050 0.1 1 10min (3)
SKA1 Low 50–350 0.08 1 10min (3)
GMRT 580–640 0.5 1 1hr (4)
GMRT 305–345 3.8 1 1hr (5)
WSRT 92cm 310–390 3.9 1 1hr (6)
LOFAR HBA2 210–250 2.6–6.0 1 1hr (7)
LOFAR HBA1 110–190 1.6 1 1hr (7)
(1)science.nrao.edu/science/surveys/vlass/VLASkySurveyProspectus WP.pdf
(2)www.atnf.csiro.au/projects/askap/spec.html
(3)www.skatelescope.org/wp-content/uploads/2013/03/SKA-TEL-SKO-DD-001-1 BaselineDesign1.pdf, page18.
(4)www.ncra.tifr.res.in/ncra/gmrt/gmrt-users
(5)Farnes (privatecommunication).
(6)www.astron.nl/ smits/exposure/expCalc.html
(7)www.astron.nl/radio-observatory/astronomers/lofar-imaging-capabilities-sensitivity/sensitivity-lofar-array/sensiti,
table4,fora40-stationDutcharray.Thosenumbersarepreliminary,especiallyinthehighestband.
The JVLA will be used to carry out sensitive surveys 500MHz,sotwotuningswillbeneeded.Thecorresponding
of large parts of thesky.Weusefigures providedby Steven noise levels per 1 MHz bandwidth and after 10 minutes of
T.Myers[NationalRadioAstronomyObservatory(NRAO)] observations are given in Table 1.
in the Karl Jansky VLA Sky Survey Prospectus (see Ta-
For the GMRT 610 MHz band, our sensitiv-
ble1)fortheJVLAinitsB-configuration.IntheS-band(2–
ity estimate is based on the figures quoted by
4 GHz), the effective bandwidth (free from radio frequency
Farnes, Green, & Kantharia (2013) who reached a noise
interferences) is 1500 MHz, and a noise level of 0.1 mJy level in Q and U of 36 µJy per beam of 24′′ in 180 minutes
can be achieved in 7.7 seconds. The size of the synthesized
ina16MHzbandcenteredat610MHz.Fourtuningswould
beam is 2.7′′ at the centre of the band. In the L-band (1–
berequired to cover thewhole band.
2 GHz), the effective bandwidth is 600 MHz and a noise
Our estimate of the sensitivity of the GMRT in the
level of 0.1 mJy can be achieved in 37 seconds of integra-
tion.Thesizeofthesynthesizedbeamis5.6′′.Inbothcases 325 MHz band relies on a noise level of 2.7 mJy per beam
per1MHzchannelin1hour,basedonpolarizationobserva-
weassumedasingletuning.WenotethatMao et al.(2014)
tions of a pulsar done in 2011 (Farnes, private communica-
recently submitted a science white paper for a JVLA sky
tion)andassumingthatall30antennaswouldbeavailable.
surveyintheS-band(intheC-configuration),inwhichthey
Assuming that 2 tunings would be necessary to cover the
consider several alternatives ranging from a shallow all-sky
whole band, this gives a noise level of 3.8 mJy in a total of
survey to ultra-deep fields of a few tens of square degrees.
1 h.
They estimate that a shallow all-sky survey of a total of
about3000hourswouldleadtothedetectionofover2 105 The WSRT also operates in the 320 MHz band (called
×
polarized sources. the92cmband).After1hourofobservation,thetheoretical
According to the ASKAP website, a continuum sensi- noise level in Stokes I is 1.2254 mJy beam−1 in a 10 MHz
tivity of 29 to 37 µJy beam−1 for beams between 10′′ and band.Thiscorrespondstoabout3.9mJybeam−1ina1MHz
30′′ can be reached in 1 hour for a bandwidth of 300 MHz. channel.Notethatconfusion noiseisexpectedtobesignifi-
Fourtuningswouldberequiredtocoverthewholefrequency cantinobservationsoftheStokesparameterI butitcanbe
bandfrom700to1800MHz,soinatotalof1hanoiselevel neglectedQandU.Gießu¨bel et al.(2013)recentlydetected
of 1– 1.3 mJy per 1 MHz channel would be reached. A ma- polarizationwiththeWSRTtowardstheAndromedagalaxy
jor polarization survey with ASKAP (POSSUM) is in the at 350 MHz.
design study phase. Thehigh-bandarray(HBA)oftheLOFARoperatesat
In its first phase, theSKA will observe at low frequen- frequenciesbetween110MHzand250MHzwithafilterbe-
cies (50 – 350 MHz, SKA1-Low), mid-frequencies (0.35 – tween 190 and 210 MHz. LOFAR has detected polarization
3.05 GHz, SKA1-Mid) and in a survey mode in the 0.65 in the HBA and rotation measures could be inferred (e.g.
– 1.67 GHz range (SKA1-Survey). In 1 hour of observa- in pulsars, Sotomayor-Beltran et al. 2013, and in polarized
tion and per 0.1 MHz channel, the sensitivity is expected sourcesin thefield ofM51, Mulcahyet al., in prep.).How-
to be 63 µJy for SKA1-Mid, 103 µJy for SKA1-Low, and ever,at theLOFARfrequenciesdepolarization isextremely
263 µJy for SKA1-Survey. We have assumed a single tun- strong and for the fiducial models presented in this paper
ingforSKA1-LowandthatforSKA1-Midfourtunings(ina themeasurementsatthequotedsensitivitiesdonotprovide
770MHzbandwidtheach)willbeneededtocoverthewhole improvedconstraintsontheparametersrelatedtothemag-
band; for SKA1-Survey, the maximum bandwidth will be netic field. The LOFAR frequency coverage is displayed in
4
JVLA−S JVLA−L GMRT LOFAR JVLA−S JVLA−L GMRT LOFAR
SKA1−MidASKAPSKA1−Survey WSRT 92 cmSKA1−Low SKA1−MidASKAPSKA1−Survey WSRT 92 cmSKA1−Low
1 1
β = 0 m2 β = 0 m2
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
y] y]
mJ0.5 β = −0.2 m2 mJ0.5
P [ P [
0.4 β = +0.2 m2 0.4
0.3 0.3
0.2 0.2 β = −0.2 m2
0.1 0.1 β = +0.2 m2
0 0
10−3 10−2 10−1 100 101 10−3 10−2 10−1 100 101
λ2 [m2] λ2 [m2]
ν [MHz]
ν [MHz]
1400 610 325
1 1400 610 325
1
0.9
P [mJy]0000....5678 β= 0 m2 β= −0.2 m2 Q, U [mJy] 0.05 U β = −0.2 m2
0.4 P, Q
β
0.3 =
+
0.2 0.2 m −0.5
2
0.1
0 −1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1
λ2 [m2] λ2 [m2]
Figure 1.Polarizedintensityversusλ2 forthetop-hatmodeloftheFaradaydispersionfunction(left)andtheGaussianmodel(right)
fordifferentvaluesoftheβ parameter:β=0(solidline),β=−0.2m2 (dashedline),andβ=+0.2m2 (dottedline).Thegraphsonthe
toprowareinsemilogarithmicscaleandextendouttothefrequencybandsofLOFAR,whilethegraphsonthebottomrowareinlinear
scaleand extend to awavelength of about 1metre, aband covered now by theGMRT and theWSRT andto becovered bythe SKA.
Thewavebandsofseveralinstrumentsareshownincolour.FortheGaussianmodel,thevariationsoftheStokesparametersQandU are
also shown (bottom right panel, brown dotted-dashed and blue dotted lines) for the case with β =−0.2 m2. The pattern of polarized
intensityisshiftedhorizontallyasβ varies,peakingatλ2=−β,asillustratedbytheredarrowsinthebottom leftpanel.Anegativeβ
(B|| andhandednessofB⊥ ofoppositesigns)resultsinapeakatλ2>0,whereasapositiveβ resultsinincreaseddepolarisationdueto
thecombineddepolarizationeffects ofintrinsichelicityandFaradayrotation.
Fig.1buttheLOFARconfidenceintervalsarethereforenot ∂2ln 1 ∂2χ2
shown in the otherfigures. Fjk =−∂pj∂pLk = 2∂pj∂pk , (6)
whereχ2 isdefinedinEq.(7).DenotingQ andU the
mod mod
values of the Stokes parameters Q and U for the assumed
2.2 The Fisher matrix model,estimatedatwavelengthsλi andwithnoiselevelsσi,
we have
The Fisher analysis is often used in cosmology (e.g.
Albrecht et al. 2006, page 94). Consider a set of N data
pointsandamodelwithP parameters,p1,...pP.TheFisher χ2 = N Qi−Qmod(λi;p1,...,pP) 2
matrix elements Fjk are proportional to the second partial i=1 σi (7)
derivativeswithrespecttotwogivenparametersofthelike- P (cid:0)+ Ui−Umod(λi;p1,...,pP(cid:1)) 2 .
lihood function that the data set derives from the given σi
L
model. If the measurement errors follow a Gaussian proba- (cid:0) (cid:1)
bility distribution, then The Fisher matrix elements can be written as
5
= N 1 ∂Qmod(λ2i;p1,...,pP)∂Qmod(λ2i;p1,...,pP) rotationofB⊥ havethesamedirection.Anegativeβ means
Fjk i=1σi2 ∂pj ∂pk (8) thattheFaradayrotation effectivelycounteracts theintrin-
P (cid:16)+ ∂Umod(λ2i;p1,...,pP)∂Umod(λ2i;p1,...,pP) . sic rotation of the plane-of-sky magnetic field. This effect
∂pj ∂pk will bediscussed further in Sect. 3.
(cid:17)
The covariance matrix is theinverse of the Fisher matrix:
σ2 =( −1) . (9) 2.3.2 Gaussian Faraday dispersion function
jk F jk
As a simple alternative to the top-hat parametrisation of
2.3 Models of F(φ) with linearly varying intrinsic F(φ)we also consider a Gaussian form,
polarization angle
2
φ φ
vWaerycionngsiψderatswaofsuimncptlieonmoofdFelasrfaodraFy(dφe)p,teha:chwithalinearly F(φ;p)=F0 exp"−0.5(cid:18) −σφ 0(cid:19) #e2iψ0(φ;p), (18)
0
where p and ψ (φ,p) are defined as before. This gives a
ψ (φ)=α+βφ, (10) 0
0 complex polarization of
with some constants α and β. This is a parametrisation of
ψ0 as a first-order polynomial and, as discussed below, it P(λ2;p)=√2πσ F exp 0.5 λ2+β 2 e2iψ(λ2;p),(19)
can also beinterpreted as a helical magnetic field. φ 0 "− (cid:18)(2σφ)−1(cid:19) #
where ψ(λ2;p) is given by Eq. (13). The modulus of P(λ2)
2.3.1 Constant Faraday dispersion in the source is a Gaussian centered at λ2 = β with variance σ2 =
(2σ )−2. −
OneofthesimplestpossiblemodelsforF(φ)isthetop-hat, φ
F(φ;p)=F T(φ;p)e2iψ0(φ;p), (11)
0
2.3.3 General case
where the set of parameters is p = (F ,φ ,φ ,α,β),
0 0 m InthetwoprevioussectionswecalculatedP(λ2)byintegrat-
ψ (φ;p)isgivenbyEq.(10)andT(φ;p)isthetop-hatfunc-
0
ing Eq. (2) analytically. Using the properties of the Fourier
tion, with T =1 in the range φ φ <φ <φ +φ and
0 m 0 m
− transforms,wenowshowwhyanylinearvariationofψ with
T =0 elsewhere. The complex polarization is 0
φproducesatranslation oftheobservedpolarized intensity
P(λ2;p)=2φ F sinc[2φ (λ2+β)]e2iψ(λ2;p), (12) in the λ2 space.
m 0 m
UsingthestandardexpressionforFouriertransform(in-
where tegralovertfrom to+ ofafunctionf(t)timese−2πjνt
ψ(λ2;p)=α+(λ2+β)φ (13) for the direct tra−ns∞form, a∞nd times e2πjνt for its inverse),
0
Eq.(2) can be written as
and sinc(x)=sin(x)/x.
Inauniformslab,theFaradaydepthvarieslinearlywith P(πλ2)=FT−1 F(φ) , (20)
{ }
thez-coordinate for φ between φ0±φm: where FT−1 is theinverse Fourier transform.
φ=0.81neBkz, (14) Using|F(φ)|=Fc(φ)∗δ(φ−φ0) whereFc(φ) isa real-
valued function centered at φ=0,
where ne is in cm−3, Bk in µG, z in pcand φ in rad m−2.
Consider a magnetic field with a constant line-of-sight P(πλ2)=FT−1 Fc(φ) δ(φ φ0) e2i(α+βφ) . (21)
{ ∗ − · }
component, but with a rotating component in the plane of The factor e2iα is independent of φ and can be taken out
thesky:
oftheintegral. Multiplication becomes aconvolution in the
B⊥cos(α+kHz) Fourier (λ2) space and convolution becomes a multiplica-
B= B⊥sin(α+kHz) . (15) tion, so
Bk ! P(πλ2)=e2iαFT−1 Fc(φ) FT−1 δ(φ φ0) FT−1 e2iβφ .(22)
{ }· { − }∗ { }
TheintrinsicpolarizationangleclearlyvarieswiththeFara-
Translation in theφ-spacegivesa rotation in λ2-space, and
day depth:
the inverse transform of the term involving βφ becomes a
ψ0 =α+kHz=α+βφ, (16) delta-function, giving
where P(πλ2) =e2iα FT−1{Fc(φ)}·e2iπφ0λ2 ∗δ(λ2+ πβ). (23)
Fβor a==n h00..e80l1ik8cnH6aemlBkfi2e(cid:16)ld2πwraitdkhHkpkcH−1≃(cid:17)(cid:16)2π0.0r3ancdem−kp3(cid:17)c−(cid:16)1,3Bµw|G|e(cid:17)ha.ve (β17≃) WPe(πthλe2n)ob==tain(cid:0)PFβ(cid:16)=T0−(1π{(Fλc2(+φ)πβe2)i)αe}2i∗φ0δ((πλλ22++(cid:17)β)βπ,)(cid:1)e2iπφ0(λ2+πβ) (24)
0.1 m2.
where P is the complex polarization corresponding to
Notethatthesignofβ dependsontherelativeorienta- β=0
Fc(φ) when β = 0. Changing the variable from πλ2 to λ2,
tion of themagnetic field component along theline of sight
we obtain
and the handedness of the helix. A positive β means that
B , which produces the Faraday rotation, and the intrinsic P(λ2)=P (λ2+β)e2iφ0(λ2+β), (25)
k β=0
6
F0 000...123 Φm105 α 024 β 0.02 s(λ)= λλ0 −αnth , (29)
−0.01 0 −−42 −0.2 with th(cid:16)e no(cid:17)rmalisation wavelength λ = 0.2 m and a fixed
141516 141516 141516 141516 0
Φ [rad m−2] Φ [rad m−2] Φ [rad m−2] Φ [rad m−2] non-thermal spectral index, α = 1. The values of the
0 0 0 0 nth −
Φm105 α 024 β 0.02 bpyolasr(iλze).dTinhtiesnseifftyecctiavlecluylaitnecdreuassiensgtEhqe.fl(2u)xwdeernesmitiuelstimpleiead-
0 −−42 −0.2 suredatλ>λ comparedtothevariationsshowninFig.1,
0
−0.100.10.20.3 −0.100.10.20.3 −0.100.10.20.3
F [mJy/(rad m−2)] F [mJy/(rad m−2)] F [mJy/(rad m−2)] where nospectral dependencewas included.
0 0 0
α 24 β 0.2
0 0
−−42 −0.2 3 RESULTS
0 5 10 0 5 10
Φ [rad m−2] Φ [rad m−2]
m m We use a fiducial top-hat model of the Faraday dispersion
β 0.2 function with the following parameters: φ0 = 15 rad m−2,
0 φ =5 rad m−2, F = 0.1 mJy/(rad m−2), α=0 rad, and
−0.2 m 0
−4−2 0 2 4 three values of β, 0 and 0.2 m2. The total intrinsic polar-
α [rad] izedfluxdensity(integrat±edoverallFaradaydepths)isthus
2φ F = 1 mJy. Because of depolarization, the signal ex-
Figure2. 68.3percentconfidenceregionsforatop-hatmodelof m 0
pectedatagivenfrequencywillbeweaker(seeFigure1)but
theFaradaydispersionfunctionwhenβ=0.TheASKAPregions
shouldbedetectablewithin areasonable amountof observ-
areinblueandtheGMRTonesingreen.
ing time by current and future instruments (see Table 1).
For comparison, we also use a Gaussian model of F(φ)
| |
so that with the same total flux. The dispersion of the Gaussian is
σ =φ =5rad m−2 andthepeakfluxdensityperunitof
φ m
|P(λ2)|=|Pβ=0(λ2+β)|. (26) Faraday depth 2/√2πF0 ≃0.08 mJy/(rad m−2). Note that
σ characterises theGaussian profileoftheFaradaydisper-
This shows that the observed modulus of the polarized in- φ
sionfunctionforamodelwitharegularfield,nottobecon-
tensity of a medium with a given β is simply a translation
byβ in λ2-spaceof what would beobserved if β wereequal fused with the dispersion in rotation measure (RM) caused
bypossibleRMfluctuationsacrossanobservingbeam(usu-
to zero.
ally denoted σ ). In Sect. 3.2 we discuss the additional
Fc(φ) is real-valued, by definition; if it is even in φ, RM
| | depolarization effect by Faraday dispersion produced by a
itsinverseFouriertransform isalsoreal-valued,andtheob-
random field. The width of the Faraday structure (that is,
served polarization angle will be
thetotalFaradaydepth,sometimesdenoted ,seeSect3.2)
ψ(λ2)=α+φ0(λ2+β) (27) inourfiducialtop-hatmodelis2φm =10radRm−2.Inmany
astrophysical cases this quantity can be larger (up to 80
as found in Sect. 2.3.1 and 2.3.2 for the top-hat and the rad m−2 in spiral arms, e.g. Arshakian & Beck 2011∼). A
Gaussian cases.
largertotalFaradaydepthtranslatesintoanarrower“main
peak”oftheP(λ2)distributionandweakeremissionatlong
wavelengths.
2.4 Spectral dependence
TheexpressionsofP(λ2)usedintheprevioussectionsdonot
3.1 Differential Faraday rotation versus magnetic
include any spectral dependence of the synchrotron emis-
field helicity
sion.Ingeneral,theobservedpolarization canbewrittenas
theintegralalongthelineofsight(los)andonacertainsolid Fig.1showsthevariationofthepolarizedintensitywithλ2
angle Ω on thesky of the intrinsic polarization modulated for a top-hat (left column) and a Gaussian (right column)
b
by the Faraday rotation, where the intrinsic polarization is Faraday dispersion function. In the former case, the solid
a fraction of the total emissivity of thesource: line (β = 0) is the well-known sinc function produced by a
uniform slab, which is more usually shown using the linear
P(λ2)= dldΩǫ(r,λ)p0(r)e2i(ψ0(r)+φ(r)λ2) (28) horizontal axis used in the bottom left panel. Note that,
for clarity, the graphs in Fig. 1 do not include any spectral
Z Z
losΩb dependenceoftheintrinsicpolarization.Figures2and3(the
where theemissivity in thedirection r, ǫ(r,λ),may depend confidenceregionsoftheparameters)do,ontheotherhand,
on λ,and p (r) is theintrinsic degree of polarization. includeaspectraldependenceoftheformgivenbyEq.(29).
0
Iftheemissivitycanbeexpressedastheproductoftwo In the Gaussian case, the polarized intensity decreases
functions where one of them contains the spectral depen- monotonically and no emission is produced in the longer
dence (e.g., ǫ(r,λ)=ˆǫ(r)s(λ)), then s(λ) can be taken out wavebands at which the GMRT and LOFAR operate
oftheintegralandEq.(28)isinvertible,following thesame (Fig. 1). In the bottom right panel we show the variation
formalism atinEqs(2)and(3)(foradiscussion,seeSect.3 of the Q and U Stokes parameters (in brown and blue) for
of Brentjens & deBruyn 2005). the Gaussian F(φ), which are the direct observables. They
Inthecalculationoftheconfidenceintervalsthatcanbe oscillate with a 90◦ phase shift with respect to each other.
obtained from observations in various wavebands, we mod- In therest of thepaper we focus on the top-hat model
elled thespectral dependenceas a power law, because it gives stronger emission in longer wavebands for
7
β = +0.2 m2 β = −0.2 m2
10 0.8 5
0.6 0
5
α [rad] 0 2β [m] 00..024 α [rad] 0 2β [m]−−−000...321
−5
−0.2 −0.4
−10 −0.2 F [m0Jy/(rad0 .m2−2)] 0.4 −0.2 F [m0Jy/(rad0 .m2−2)] 0.4 −5 −0.2 F [m0Jy/(rad0 .m2−2)] 0.4 −0.5 −0.2 F [m0Jy/(rad0 .m2−2)] 0.4
0 0 0 0
10 0.8 5
0.6 0
5
0.4 −0.1
α [rad] 0 2β [m] 0.2 α [rad] 0 2β [m]−0.2
0 −0.3
−5
−0.2 −0.4
−10 −5 −0.5
2 Φ4 [rad m6−2] 8 2 Φ4 [rad m6−2] 8 2 Φ4 [rad m6−2] 8 2 Φ4 [rad m6−2] 8
m m m m
Figure 3. 68.3percentconfidenceregionsforatop-hatmodeloftheFaradaydispersionfunctionforsomeoftheinstrumentslistedin
Table1.Leftpanel(fourfigures):β=+0.2m2;rightpanel(fourfigures):β=−0.2m2.Theyellowellipsesshowtheconfidenceregions
obtainedbycombininganASKAP(blueellipses)andaGMRT(greenellipses)dataset.TheSKA1-Surveyconfidenceellipsesareshown
inblack,andtheJVLAones inpurple(S-band)andpink(L-band).
the parameters selected here and it includes the standard selected waveband. This is what makes it possible to break
case for Faraday depolarization calculations of a uniform parameterdegeneraciesbycombiningshort-wavelength(like
slab as a special case. It is most interesting to compare the ASKAP)andlongwavelength(likeGMRT)datasetsinor-
variationsofthepolarizedintensityforapositiveandaneg- der to better constrain thederived parameter values, as we
ative β. When β > 0, the intrinsic helicity of the magnetic discuss next.
field and the Faraday rotation act in the same direction. Fig. 3 shows the one-sigma confidence ellipses for α,
This results in an increased depolarization at short wave- β, F and φ for β = +0.2 m2 (four left panels) and
0 m
lengths. Even for λ = 0, where Faraday depolarization is β = 0.2 m2 (four right panels). As expected, the orien-
−
absent, the emission is significantly depolarized compared tation of the ellipses is similar for the short-wavelength in-
tothecase ofaconstant magneticfieldorientation (β =0). struments JVLA (S-band in purple and L-band in pink),
When β <0, Faraday rotation counteracts the intrinsic ro- ASKAP (blue) and SKA1-Survey (black). On the other
tation of thefield,which meansthat thepolarized emission hand,thelong-wavelengthGMRTdataset(green)produces
peaks at a wavelength different from zero (dashed lines), a different correlation between parameters; in some cases
whereλ2 = β (eqs.(12)and(19)).Thiseffectissimilarto the confidence ellipses at short and long wavelengths are
−
the “anomalous depolarization” discussed by Sokoloff et al. almost orthogonal, making it possible to reduce the confi-
(1998, Section 9). denceintervalsonthederivedparametervaluesconsiderably
Fig. 2 shows the 68.3% confidence regions of the five byusing bothwavelength ranges together (such asASKAP
different parameters obtained from the Fisher analysis for and GMRT, shown in yellow). This is further illustrated in
β = 0. The colour code is the same as in Fig. 1, with the Fig. 4 where the precision that can be achieved in β for
ASKAP confidence regions in blue and the GMRT ones in different instruments is shown. A combination of ASKAP
green.TheplotsinthefirstrowshowthatthecentralFara- and GMRT observations makes it possible to reach an un-
day depth φ is mostly uncorrelated with the other param- certainty ∆β < 0.25 m2 if β < 0, for the integration times
0
eters. On the other hand, φ , which describes the extent shown in Table 1 and the set of model parameters used. If
m
oftheFaradaycomponent inφ-space,isstrongly correlated β>0,allsignalsareweakerbecauseofincreaseddepolariza-
withtheparametersrelatedtotheintrinsicpolarization an- tionandtheprecisiononβ(andallothermodelparameters)
gle (α and β) and at short wavelengths (in the ASKAP is lower.
band)with thenormalization of F(φ) (F ). This is because
0
the crucial effect is Faraday differential rotation across the
Faraday component and not the magnitude of the central
Faraday depth. An increase in φ means stronger depolar-
m 3.2 Faraday dispersion
ization due to differential rotation which must be counter-
actedbyamorenegativeβ inordertoproduceasimilar fit In this section we discuss the effect of depolarization by
tothedata.Viceversa,alowerφ meansweakerdepolariza- Faraday dispersion, due to possible random fluctuations of
m
tionandβneedstobecomepositivetoincreasethedepolar- the magnetic field inside the synchrotron-emitting source.
ization.Inotherwords,φ andβareanti-correlatedaround This effect was first discussed by Sokoloff et al. (1998)
m
β=0.Mostimportantly,thestrengthofthecorrelation be- (Sect. 9) but they considered only thecase of twisted fields
tween pairs of parameters including α or β varies with the wheretherotationduetothehelicityiscounteractedbythe
8
104 1
JVLA−S band
JVLA−L band 0.9 β = 0 m2
103 AGSMKRATP 0.8 (∆ψ0 = 0o)
ASKAP+GMRT
SKA1−Survey 0.7 β = −0.2 m2
102
(∆ψ = −115o)
0.6 0
2∆β [m]101 P [mJy]0.5
0.4
100
0.3
10−1 0.2 β = +0.2 m2
0.1 (∆ψ = +115o)
10−−02.5 0 0.5 0 0
0 0.2 0.4 0.6 0.8 1
β [m2] λ [m]
Figure 4. Precision that can be achieved on the β parameter Figure 5. Variation of the polarized intensity with wavelength
as afunction ofβ forobservations withsomeof theinstruments inthe caseof a uniformslabof Faraday depth R=10 radm−2
listedinTable1.ThecombinationofASKAPandGMRT(shown
andFaradaydispersioncharacterised byσRM =0,2.5,5,7.5and
here in dotted-dashed red) provides the smallest dependence of 10 rad m−2 (curves from top to bottom) for β = 0 (no he-
the uncertainty on β, comparable to what can be achieved with licity; black solid lines), β = +0.2 m2 (red dotted lines) and
SKA1-Survey. Note, however, the difference in integration time β=−0.2m2(bluedashedlines).Thetwolattercasescorrespond
(a total of 3 h for the joint ASKAP/GMRT observations versus
toagradualchangeoftheperpendicularcomponentofthemag-
10minforSKA1-Survey). netic field by about 115◦ along the line of sight across the slab.
Thepresenceofarandomfieldattenuatesthevariationsandshifts
thepeakofthepolarizedemissiontowardsshorterwavelengths.
Faraday rotation (β < 0 in our notation). As detailed in
AppendixA,the observed complex polarized emission is
the magneto-ionic medium can be minimised by combining
1 e−SH
P(λ2) =e2i∆ψ0 − (30) observations in different wavebands. In particular we have
h i SH shown that it may be possible to recover the sign and the
where magnitudeof β,aparameter thatwehavedefinedandthat
SH =−2i Rλ2−∆ψ0 +2σR2Mλ4. (31) ivserrseelamteadgntoettichefireeldlataivnedetffheectFaorfatdhaeyhreolitcaittiyonofdtuheettoratnhse-
∆ψ0 = k(cid:0)HL is the to(cid:1)tal rotation due to the helicity of parallel component of the magnetic field. Since the direc-
the intrinsic polarization angle ψ across the thickness L tion of B can be easily inferred from RM measurements,
0 ||
of the slab, is the total Faraday depth of the source it should be possible to recover the sign (and, under some
R
and σ is the dispersion of the total Faraday depth. In assumptions themagnitude) of the helicity of themagnetic
RM
the previous sections we had = 2φ = 10 rad m−2 and field. However, the additional effect of Faraday dispersion
m
∆ψ =2φ β =0 and 2 radR 115◦. Figure 5 shows the byarandomcomponentofthemagneticfieldattenuatesthe
0 m
± ≃±
variation of the polarized intensity with wavelength for the variations of the polarized emission as a function of wave-
three cases considered before (a non-helical magnetic field lengthandmayshiftthepeakofpolarizedemissiontowards
(β = 0) and helical magnetic fields with β = 0.2 m2) to shorterwavelengthsifβ isnegative.Faradaydepolarization
±
which theeffect of arandom magnetic fieldcomponent was effects(both bydifferential rotation and bydispersion) will
added. The random fluctuations are described by values of have to be included in the modelling of real data in order
σ increasing from 0 (top, thicker curves) to 10 rad m−2 to recover information on thehelicity of themagnetic field.
RM
by step of 2.5 rad m−2. The Faraday dispersion caused by Thisapproachiscomplementarytostatisticalstudiesofthe
the random fluctuations attenuates the variations and, for correlation betweenthedegreeofpolarization andtherota-
negative values of β where the intrinsic helicity acts in the tion measures of cosmic sources which may also provide in-
opposite direction of the Faraday rotation, shifts the peak formation onmagnetichelicity (Volegova& Stepanov2010,
of the polarized intensity towards shorter wavelengths. Brandenburg& Stepanov 2014).
Planned surveys of fixed sensitivity will be biased to-
wards radio sources with negative β, because of the depo-
larization produced when β > 0. Detection of β > 0 will
4 SUMMARY
be more difficult both through Q(λ2) and U(λ2) model-
We examined how variations of the intrinsic polarization fittingandRMsynthesisbecausemostofthesignalisshifted
angle ψ with the Faraday depth φ within a source af- toward negative λ2. Brandenburg& Stepanov (2014) show
0
fect the observable quantities. Using simple models for the that restricting the integral in Eq. (3) to the positive λ2
Faraday dispersion F(φ) and ψ (φ), along with the current yields an erroneous reconstruction of F(φ) when β>0.
0
and planned properties of the main radio interferometers, In this work we used a first-order parametrisation of
weshowhowdegeneraciesamongtheparametersdescribing the variation of intrinsic polarization angle with Faraday
9
depth. Higher-order representations could be used (or e.g. SchnitzelerD.H.F.M.,KatgertP.,deBruynA.G.,2009,
Chebyshevpolynomials) ifthedataareofsufficientquality. A&A,494, 611
Including a second-order term implies a convolution with Sokoloff D. D., Bykov A. A., Shukurov A., Berkhuijsen
animaginaryGaussianandasignificantlymorecomplicated E. M., Beck R., Poezd A. D.,1998, MNRAS,299, 189
expression for P(λ2). Sotomayor-Beltran, C., Sobey,C., Hessels, J. W.T., et al.
2013, A&A,552, A58
Urbanik M., Elstner D., Beck R., 1997, A&A,326, 465
Volegova A. A.,Stepanov R. A., 2010, JETPL, 90, 637
ACKNOWLEDGMENTS
We thank Oliver Gressel for organizing the stimulating
Nordita workshop ‘Galactic Magnetism in the Era of LO-
FAR and SKA’ held in Stockholm in 2013 September . We APPENDIX A: FARADAY DISPERSION
also thank Axel Brandenburg and Rodion Stepanov for in- AROUND A MEAN HELICAL FIELD
teresting discussions on the topics discussed in this paper
Ingeneral,Eq.(2)canberewrittenasanintegralalongthe
and for sharing their related results with us. We are grate-
lineofsightoftheintrinsiccomplexpolarization modulated
ful to Rainer Beck, John H. Black, Jamie Farnes, George
bythe Faraday rotation:
Heald, Anvar Shukurov and to the referee for constructive
comments.AFisgratefultotheLeverhulmeTrustforfinan- P(λ2)= observerdzF(z)e2iφ(z)λ2. (A1)
cial support undergrant RPG-097.
Zsource
For a slab-like source of intrinsic polarized emission
F(z) = 1/L between 0 6 z 6 L and intrinsic polariza-
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therandomfieldvariesisdenotedd.Thecomponentsofthe
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2011, A&A,530, A89 = m(LR−z) + 0.81ne dzbk (A6)
Pizzo, R. F., de Bruyn, A. G., Bernardi, G., Brentjens, Zz
M. A. 2011, A&A,525, A104 regular random
| {z }
| {z }
10
where m = 0.81neB|| = R/L, where R is the total Fara-
day depth of the source. If b is a Gaussian variate, then
||
φ(z) is also a Gaussian variate2, pG(φ), of mean m(L z)
−
and variance (0.81nebkd)2 L−dz = v2(L−z), where v2 =
(0.81nebk)2d. Equation (A4(cid:0)) bec(cid:1)omes
e2iφ(z)λ2 = +∞ dφe2iφλ2
h i −∞
R pG φ; R(L z); v2(L z) . (A7)
L − −
(cid:0) mean variance(cid:1)
| {z }
Thiscanbeexpressedastheinver|seF{ozurie}rtransformofthe
GaussianPDF,whichcanberewritteninamoreconvenient
mannerusing the properties of the Fourier transform:
he2πiφ(z)λ2i = FT−1 pG φ;0;v2(L−z) ∗δ φ− RL(L−z)
(A8)
= exp[ (cid:8)(2πλ(cid:0)2)2v2(L−z)] e2π(cid:1)iRL(L(cid:0)−z)λ2 (cid:1)(cid:9)
− 2 ·
Replacing πλ2 byλ2, and simplifying:
e2iφ(z)λ2 =exp[ S(L z)], (A9)
h i −L −
where
S = 2i λ2 + 2v2Lλ4 .
− R (A10)
regular random
Finally Eq|. ({Az3)}become|s {z }
P(λ2) = 1 Ldze2ikHz exp[ S(L z)]
h i L 0 −L −
(A11)
R
= e2i∆ψ0 1−e−SH ,
SH
(cid:16) (cid:17)
where ∆ψ0 = kHL is the total rotation of the polarization
angle across theslab and
SH = 2i( λ2 ∆ψ0)+2v2Lλ4. (A12)
− R −
Equation (A11) is a generalization of the expressions pro-
vided by Burn (1966) and Sokoloff et al. (1998) to the case
ofFaradaydispersionaroundanhelicalfield.v2Listhevari-
anceofthetotalFaradaydepthofthesourceandisusually
denoted σ2 .
RM
2 NotethatBurn(1966)hadafactord/2insteadofdinhisex-
pressionofthevariance,whichresultedinafactorof2inthe“ran-
dom” term of S (see Eq. (A12)), as also noted by Sokoloffetal.
(1998).
