Tracing interstellar magnetic field using velocity gradient technique: Application to Atomic Hydrogen data

0.91 MB·

by  Ka Ho Yuen

DRAFTVERSIONJANUARY30,2017
PreprinttypesetusingLATEXstyleAASTeX6v.1.0
TRACINGINTERSTELLARMAGNETICFIELDUSINGVELOCITYGRADIENTTECHNIQUE:
APPLICATIONTOATOMICHYDROGENDATA
KAHOYUEN1,2,A.LAZARIAN1
1DepartmentofAstronomy,UniversityofWisconsin-Madison
2DepartmentofPhysics,TheChineseUniversityofHongKong
7
ABSTRACT
1
0 The advancement of our understanding of MHD turbulence opens ways to develop new techniques to probe
2 magnetic fields. In MHD turbulence, the velocity gradients are expected to be perpendicular to magnetic
fieldsandthisfactwasusedbyGonza´lez-Casanova&Lazarian(2016)tointroduceanewtechniquetotrace
n
a magneticfieldsusingvelocitycentroidgradients. Thelattercanbeobtainedfromspectroscopicobservations.
J WeapplythetechniquetoGALFAHIsurveydataandcomparethedirectionsofmagneticfieldsobtainedwith

7 our technique with the direction of magnetic fields obtained using PLANCK polarization. We find excellent
2 correspondencebetweenthetwowaysofmagneticfieldtracing, whichisobviousviavisualcomparisonand
through measuring of the statistics of magnetic field fluctuations obtained with the polarization data and our
]
A technique. This suggests that the velocity centroid gradients has a potential for measuring of the foreground
magneticfieldfluctuationsandthusprovideanewwayofseparatingforegroundandCMBpolarizationsignals.
G
Keywords:ISM:general—ISM:structure—magnetohydrodynamics(MHD)—radiolines: ISM—turbu-
.
h lence
p
-
o 1. INTRODUCTION neticfield. Thispropertyofvelocitygradientswasemployed
r inGonza´lez-Casanova&Lazarian(2016,hereafterGL16)to
t Turbulence is ubiquitous in astrophysics. The Big Power
s introduce a radically new way of tracing magnetic fields us-
a LawintheSky(Armstrongetal.1995;Chepurnov&Lazar-
ingspectroscopicdata.Insteadofusingalignedgrainsorsyn-
[ ian2010)showsclearevidencethatinterstellarturbulenceex-
chrotronpolarization(seeDraine2011),GL16appliedveloc-
tendsover10ordersofmagnitudeofscalesintheinterstellar
1 ity centroid gradients (henceforth VCGs) to synthetic maps
media(ISM).TheISMismagnetizedandthereforetheturbu-
v
obtained via MHD simulations and obtained a good agree-
4 lenceismagnetohydrodynamic(MHD)innature,e.g. see(Li
mentbetweentheprojectedmagneticfieldsandthedirections
4 etal.2014;Zhangetal.2014;Pillaietal.2015).
traced by the VCGs. As the velocity centroids can be read-
9 The modern theory of turbulence has been developed on
7 the basis of the prophetic work by Goldreich, (1995, hence- ilyavailablefromspectroscopicobservations(seeEsquivel&
0 Lazarian2005),thisprovidedawaynotonlyforobservational
forthGS95).Theoriginalideasweremodifiedandaugmented
. tracing of magnetic fields but also for finding its strength
1 in subsequent theoretical and numerical studies (Lazarian &
0 Vishniac 1999; Cho & Vishniac 2000; Maron & Goldreich using the GL16 technique that is similar to the well-known
7 Chardrasechar-Fermimethod.
2000; Lithwick & Goldreich 2001; Cho et al. 2001; Cho &
1 Motivated by the GL16 study, in this paper we calculate
Lazarian2002,2003;Kowal&Lazarian2010, seeBranden-
v: burg&Lazarian2013foraareview).1 TheAlfvenicincom- theVCGsusingHIdatafromtheGALFAsurvey(Peeketal.
i pressiblemotionsdominatethecascade. Thiscascadecanbe 2011) and compare the directions of the magnetic fields that
X
we trace using the gradients with the directions of magnetic
visualized as a cascade of elongated eddies rotating perpen-
ar dicular tothe local direction ofthe field.2 Naturally, thisin- fields that are available from the PLANCK polarization sur-
vey (Adam et al. 2016).3 To do this, we first significantly
ducesthestrongestgradientsofvelocityperpendiculartothe
improvetheprocedureofcalculatingoftheVCGsandtestit
magnetic field. Thus one can expect that measuring the gra-
with numerical data. Our recipe for calculating the VCGs is
dientinturbulentmediacanrevealthelocaldirectionofmag-
presentedin§2, whilein§3, weapplythetechniquetotrace
[email protected],[email protected] magneticfields. Wediscussourresultsin§4,andourconclu-
1 WedonotconsiderthemodificationsoftheGS95modelthatwerein- sionsarepresentedin§5.
tendedtoexplainthespectrumk−3/2thatwasreportedinsomenumerical
studies(e.g.Boldyrev(2006)).Webelievethatthereasonforthedeviations
fromtheGS95predictionsisthenumericalbottleneckeffect,whichismore
extendedintheMHDcomparedtohydroturbulence(Beresnyak&Lazarian 2. IMPROVEDPROCEDUREFORCALCULATING
2010). Thisexplanationissupportedbyhighresolutionnumericalsimula- VELOCITYGRADIENTS
tionsthatcorrespondtoGS95predictions(seeBeresnyak&Andrey(2014)).
ThesimulationsalsostronglysupporttheanisotropypredictedinGS95and
ruleouttheanisotropypredictionintheaforementionedalternativemodel.
2 ThenotionofthelocaldirectionwasnotapartoftheoriginalGS95 3BasedonobservationsobtainedwithPlanck(http://www.esa.int/Planck),
model.Itwasintroducedandjustifiedinmorerecentpublications(seeLazar- anESAsciencemissionwithinstrumentsandcontributionsdirectlyfunded
ian&Vishniac1999;Cho&Vishniac2000;Maron&Goldreich2000). byESAMemberStates,NASA,andCanada.
2
GL16 established that the VCGs can trace magnetic field
inMHDturbulence. However,thisexploratorystudylacksa
criteriononjudgingonhowwellgradientscantracemagnetic
fields. Thereforeitisdifficulttojudgewhatistheresolution
requirementtotracemagneticfieldvectorsandandwhatare
the uncertainties. Therefore our first goal is to introduce a
more robust procedure of the VCGs calculation which is to
returnthetracingthatisindependentontheresolutionofthe
simulationsandonlydependsontheparametersofMHDtur-
bulence.
Weusedasinglefluid,operator-split,staggeredgridMHD
Eulerian code ZEUS-MP/HK,4 a variant of the well-tested
code ZEUS-MP (Norman 2000; Hayes et al. 2006), to set
upathree-dimensional,uniform,isothermal,supersonic,sub-
Alfvenic turbulent medium. We adopted periodic boundary
conditions. The initial cube was set with a uniform den-
sity, and an initial uniform field. Turbulence was injected
solenoidallycontinuously,e.g. see(Ostrikeretal.2000),see
alsoAppendixofOttoetal.(2017). Oursimulationshadthe
resolution of 7923. We selected two cubes with sonic Mach
number M = 5 andAlfvenic Machnumber M = 0.6 but
s A
differentinitialmagneticfieldorientation(onewasparallelto
the z-axis, another is at the angle π/7 to the z-axis). Com-
paredtotheGL16,weusedhigherresolutionsimulationsand
studiedtheeffectofvaryingmagnetic-fielddirectionrelative
tothelineofsight.
To trace magnetic field we generated polarization maps
by projecting our data cubes along the x-axis and assum-
ing that the dust producing the polarization followed the gas
and was perfectly aligned by the magnetic field. Let φ =
tan−1(B /B ),whereB aretheyandzdirectionofmag- Figure1. (Upperfour)Thedistributionofabsoluteangle(red)
y z y,z
netic field. The intensity I, velocity centroid C and stokes and relative angle (blue) in a synthetic map of size 792x792
parametersQ,U werecomputedby: for sub-regions of size 33x33, 50x50, 99x99, 198x198, re-
spectively. The Gaussian profile emerges when the patch is
(cid:90)
1/8ofthetotallengthofthemap. Theprofileiswell-defined
I(r)= ρ(r,x)dx
when it is 1/4 of the map. (Lower four) The distributions of
(cid:90) absolute angle (red) and relative angle (blue) from observa-
C(r)=I−1 ρ(r,x)v (r,x)dx
x tion data for sub-region of size 50x50, 100x100, 200x200,
(cid:90) 300x300(relativetoGALFA-HIdataresolution)respectively.
(1)
Q(r)∝ ρ(r,x)cos2φdx
(cid:90)
totheSoberoperatorusedinSoleretal.(2013).Wesmoothed
U(r)∝ ρ(r,x)sin2φdx
ourdatawithaσ =2pixelsGaussiankernel.
The statistical properties of gradient fields can determine
themeandirectionofmagneticfieldsinasub-regionofinter-
where r is the vector on the y − z plane. The polarization est. We divided our synthetic maps into sub-regions and ex-
angleisgivenbyφ2d =0.5tan−1(U/Q). Polarizationtraces amined the statistical behavior of gradient vector orientation
themagneticfieldprojectedalongthelineofsight. (hereafterabsoluteangle(AA))andrelativeangleφbetween
WecalculatedvelocitycentroidsfollowingGL16butmod- gradientsandfields(hereafterrelativeangle(RA))withinthe
ified the VCGs calculations to increase the accuracy of the region. The upper four panels of 1 shows what distributions
procedure. In particular, we performed cubic spline interpo- oftheAAandRAlooklikewhensizeoftheblockdecreases.
lation,whichusesathree-pointestimatetoprovidethemaps As the block size increases, the mean gradient direction be-
forgradientstudy. Theresultingmapis10timeslargerthan comesmorewell-defined. Thealignmentbetweenthegradi-
the original one. To search for maximum gradient direction entandmagneticfieldalsobecomesmoreclearasblocksize
in each data point, we selected a neighborhood of the radius increases. Wefindthatastheblocksizearrivesat 100×100,
vectorr ∈(0.9,1.1)pixelsintheinterpolatedmap.Theinter- asharpdistributionemergeswithwell-definedmeananddis-
polationprocessisaccuratewitha3oerror,andiscomparable persion. BymeasuringthemeanoftheAAdistributions,we
determine the mean magnetic field direction within the re-
4MaintainedbyOtto&Yuen,(https://bitbucket.org/cuhksfg/zeusmp-hk/) spectiveblock.TheRAdistributionstellsushowaccuratethis
3
Figure2. (Upper16panels)ThedistributionofAA(red)andRA(blue)inasyntheticmapfromrun-2withblocksize198×198.
By detecting the peak of the AA distribution, we determined the mean magnetic field direction within the block. (Lower) The
predictedmeanmagneticfieldvector(red)comparedwiththerealmagneticfieldvector(blue). Thebackgroundistheintensity
ofthesyntheticmap.
4
prediction of magnetic fields is. We shall call this treatment High Frequency Instrument (HFI).5 We performed the same
sub-block averaging in the following sections. Notice that, procedureasindicatedinSection2. WecheckedtheAAand
sub-block averaging is not a smoothing method. It is used RA,asshowninthelower4panelsof1,topickanappropriate
to increase the emphasis of important statistics and suppress blocksizeforagradientvector. Forthegivencase, a100×
noise in a region, and provide an estimate on how accurate 100blocksatisfiestherequirementintherecipe. Thevelocity
thisaveragingisbytheAA-RAdiagram. Ontheotherhand, gradientvectorsareplottedwithpolarizationvectorsinFigure
smoothingdoesnotprovidesuchanestimate. Adetaileddis- 3. Inthisregion,mostofthegradientvectorsalignverywell
cussion of how white noise affects the sub-block averaging withpolarizationvectors. Thedetailedstudyoftheobserved
andsmoothingisprovidedinanextendedpaperbyLazarian deviationsfromtheperfectalignmentwillbeprovidedinour
et al. (2017), where the a companion new measure, namely, subsequentpublication.
synchrotronintensitygradientsarestudied. Following GL16 we provide a comparison with the align-
ThebenefitsofourapproachcanbeseeninFigure2.Wedi- mentmagneticfieldastracedbypolarizationandtheintensity
videdthewholesimulationdomaininto16blockswithequal gradients. Theemissionintensityofatomicisproportionalto
size,andpredictedthemagneticfielddirectionineachblock. itscolumndensity.Thecolumndensitygradientswereshown
Asonecanseefromthesefigures,theVCGstracewellmag- toactastracersofmagneticfieldsis(Soleretal.2013).Figure
neticfields. Wealsoconfirmedthisforsyntheticobservations 5 shows the histograms of relative orientations between ve-
whenthelineofsightwasatdifferentanglestothemeandi- locityandintensitygradientvectorstopolarization. Inagree-
rectionofmagneticfield. ment with the theoretical expectations as well as the results
Chandrasekhar & Fermi (1953, C-F) provides an expres- in GL16, our improved procedure of calculating the VCGs
sionrelatingthestrengthofofplane-of-skymagneticfieldby shows that the latter are much better aligned with polariza-
dispersionofturbulentvelocitiesδv andpolarizationvectors tioncomparedtotheintensitygradients. Indeed,nearly80%
δθ in magnetized turbulence (For an improved C-F method, of the VCGs are within 45o deviation from the polarization
seeFalceta-Goncalvesetal.2008): directioncomparedto61%oftheintensitygradients.
(cid:112) δv 4. DISCUSSION
δB ∼ 4πρ (2)
δφ
4.1. Structurefunctionsofvelocitygradients
Themeanmagneticfieldstrengthcanalsobecalculatedus- The structure functions of polarization and gradient fields
ing the same concept in sub-block averaging.The dispersion can also allow us to study how well-aligned they are. As
ofVCGsandthatofmagnetic-fielddirectionsarenotexactly the statistics of polarization are dependent on the Alfvenic
thesame,butthedifferenceissmall. GL16introducedafac- MachnumberM (Falceta-Goncalvesetal.2008),theclose
A
torγ of∼1.29toaccountforthisdifference. Inourcase,us- relationship between rotated the VCGs and magnetic fields
ingourimprovedprocedureofgradientcalculationwegetthe suggests that gradient statistics should have similar behavior
dispersionoftheVCGsinblocksthatisjust1.07-timesthatof to the polarization statistics. To compare the VCGs to po-
polarization. Thestandarddeviationoftheratioofthedisper- larization in synthetic maps, we extended the sub-block av-
sionsis0.05. AsillustratedinGL16,thefactorγ varieswith eraging algorithm to every point of our map, and computed
parameters of MHD turbulence. Elsewhere we shall provide the structure function in terms of the orientation θ of gradi-
afittingexpressionforγasthefunctionofMsandMA. This ent/polarizationvectors:
shouldfurtherincreasetheaccuracyofobtainingthevalueof
SF (r)=(cid:104)(θ(r(cid:48))−θ(r(cid:48)+r))2(cid:105) (3)
magneticfieldstrength. Moredetailsonthetechniqueofob- 2
taining magnetic field intensity using only spectroscopic in- Thestatisticsofdustpolarizationareimportantforstudying
formation and no polarimetry will be provided in our forth- magneticfieldturbulence(Falceta-Goncalvesetal.2008)and
comingpaper(Yuen&Lazarian,inpreparation). forcleaningtheCMBpolarizationmaps. Ifwewanttodothe
sameusingVCGs,itisimportanttesttowhatextentthestatis-
3. APPLICATIONTOOBSERVATIONDATA ticsoftheVCGsaresimilartothoserevealedbypolarization.
The left and the middle panels of 4 show the power spectra
With the tested procedure in hand, we selected diffuse re-
P (k) and second order structure functions SF (r), respec-
gions from observation surveys. We acquired data from the φ 2
tively,oftheVCGsorientationandthepolarizationangle. In
Galactic Arecibo L-Band Feed Array HI Survey (GALFA-
termsofthespectra,bothVCGsorientationsandpolarization
HI).WecomparetheVCGsdirectionstothePLANCKpolar-
angles exhbitit a −2 slope. We also examined the structure
ization data. In diffuse media, polarization of emitted radia-
functionsforpolarizationandtheVCGdistributionsfromthe
tionisperpendiculartolocalmagneticfielddirection(Lazar-
observationdatausingthesameprocedure.Therightpanelof
ian 2007; Andersson et al. 2015), i.e. the same way as the
Figure4showsthestructurefunctioncomputedusingobser-
VCGs. To adapt the difference of resolutions, we adjust the
vationdata,the+1slopealsoemerged.
block size used in Planck to reflect the same physical block
GALFAisreferringto.
4.2. Comparisonwithothertechniquesandearlierpapers
TheregionweselectedfromGALFA-HIsurveydataspans
right ascension 15o to 35o and declination 4o to 16o. The
5 We use the planckpy module to extract polarization
bin size along the velocity axis is 0.18 km/s. We analyzed
data in a particular region with J2000 equatorial coordinate:
353GHz polarization data obtained by the Planck satellite’s (https://bitbucket.org/ezbc/planckpy/src)
5
Figure3. Rotated VCGs (Yellow) map obtained using GALFA-HI data. Red vectors are polarization directions obtained from
thePLANCKdata. Thedirectionspresentedinthisfigureshowthedirectionbutnotthemagnitude. Thebackgroundshowsthe
columndensityofatomichydrogen.
ThispaperpresentsthefirstapplicationoftheVCGstoob- ploredbySoleretal.(2013).Thealignmentofthesegradients
servational data arising from diffuse media. By comparing withmagneticfieldisalsoduetothepropertiesofturbulence.
theresultsobtainedwiththeVCGsandPLANCKpolarimetry For instance, Beresnyak et al. (2005) showed that GS95 tur-
data, we have demonstrated the practical utility of the VCG bulence can in some situations imprint its structure on den-
for tracing of magnetic fields and obtaining statistical infor- sity. However,densitydoesnottraceturbulenceasdirectlyas
mationaboutmagneticfieldinthisdiffuseregion. velocitydoes. Therefore, weexpectmoredeviationsofden-
Thegradienttechniqueshavebigadvantageoverothertech- sity gradients from the magnetic field direction compared to
niques for estimating magnetic field direction and strengths: thevelocitygradients. Ourstudyconfirmstheconclusionsin
These techniques only require an easily available centroid. GL16 that the VCGs provide a better tracer. We expect that
Unlike the PLANCK map, the VCG maps do not require thedensitygradientsarerelatedtothefilamentswhichalign
uniquemulti-billiondollarsatellitesbutcanberoutinelyob- withmagneticfieldsasreportedinClarketal.(2015). There-
tainedwiththeexistingspectroscopicsurveys. Byusingdif- foreweexpectthattheVCGstracemagneticfieldsbetterthan
ferent species, one can distinguish and study separately dif- thefilaments.
ferent regions along the line of sight. Combining the VCGs We,however,havetostressthatthisregionisonlyapartic-
thattracemagneticfieldsindiffusegaswithpolarimetry,e.g. ularexampleonhowVCGworks,whichdoesnotrepresentit
ALMA polarimetry, that traces magnetic fields in molecular isapplicableeverywherewithoutcautionsonthelimitations.
clouds,onecanstudywhatishappeningwithmagneticfields One should understand that both density and velocity prop-
asstarformationtakesplace.Thismaybeawaytotestdiffer- ertiesareimportantcomponentsofMHDturbulentcascades.
ent predictions, e.g. the prediction of magnetic flux removal Therefore, the deviations of the gradients from the magnetic
through the reconnection diffusion process (Lazarian & A. field direction are informative. For instance, we observe an
2005,2014;Lazarianetal.2012). adifferentbehaviorofVCGsanddensitygradientsinthere-
The alignment of density gradients were previously ex- gions of strong shocks as well as in self-gravitating regions
6
Figure4. The power spectrum (Left) and second order structure function (Middle) of the sub-block averaged velocity gradient
(blue) and polarization (red) from the synthetic map.. Both power spectra and structure functions show very similar behavior.
Structurefunctions(Right)ofthesub-blockaveragedvelocitygradientsandpolarizationanglefromobservationdata.
OurworkprovideapromisingexampleonhowtheVeloc-
1.0 ity Centroid Gradient (VCG) technique introduced in GL16
VCG tracesmagneticfieldsininterstellarmedia. Inthepaper:
IG
0.8 1. We provide a new robust prescription for calculating the
nt VCGsandtestthisnewapproachusingthesyntheticdataob-
u
o
e c tainedwithMHDsimulations.
ativ0.6
mul 2. We show that with the new prescription the estimates of
u
d c magneticfieldstrengthbasedontheC-Fapproachcanbeim-
alize0.4 proved.
m
or
n 3. We apply the VCGs to the available high latitude HI
0.2
GALFA data and demonstrate an excellent alignment of the
direction of the VCGs and those measured by PLANCK po-
0.0 larization.
0 10 20 30 40 50 60 70 80 90
Relative pseudo-angle between VCG/IG to polarization (degrees)
4. We show that the statistics of the fluctuations measured
Figure5.Acumulativehistogramshowingtherelativepseudo- by the VCGs and polarization have the same slope for both
angle between sub-block averaged VCG and intensity gradi- synthetic and observational data, which suggests that VCGs
ent(IG)topolarization. Totracethemagneticfielddirection, couldpotentiallybepromisingtoolforaccountingforpolar-
werotatedgradientvectorsby90degrees. izedforegroundswithinCMBstudies.
5. Thedifferencesbetweenthedirectionsdefinedbythepo-
larization, the VCGs and the intensity gradients carry infor-
(Yuen&Lazarian,inprep.).Thereforethereisimportantsyn- mation about the turbulent interstellar medium and this calls
ergyofthesimultaneoususeofVCGs,density/intensitygra- forthesynergeticuseofthethreemeasures.
dientsandpolarimetry.Addingtothelistthenewlysuggested
techniqueofsynchrotronintensitygradientsthatisdiscussed WethankSusanClarkforherhelpwithGALFAdata. We
inanewpaperbyLazarianetal. (2017)increasesthewealth thankAviLoebandDiegoF.Gonzalez-Casanovausefuldis-
oftheavailabletools.Thisopensnewwaysofexploringmag- cussions. WealsothankPaulLawforhisgeneroushelpwith
neticfieldsinthemulti-phaseISM. PLANCK data. The stay of KHY at UW-Madison is sup-
Wewouldalsoliketopointoutthatwhilethepolarimetry ported by the Fulbright-Lee Hysan research fellowship and
directions in Figure 3 seem to be well aligned over signifi- DepartmentofPhysics,CUHK.ALacknowledgesthesupport
cant patches of the sky, this does not mean that there is no theNSFgrantAST1212096,NASAgrantNNX14AJ53Gas
turbulence there. The correspondence of the VCGs and po- well as a distinguished visitor PVE/CAPES appointment at
larization directions can be understood only if the media is the Physics Graduate Program of the Federal University of
turbulent. Thepowerlawbehaviorofthestatisticsrelatedto RioGrandedoNorte,theINCTINEspaoandPhysicsGradu-
boththeVCGsandpolarizationdirectionsconfirmsthis. The ateProgram/UFRN.
factthatthepowerlawdoesnotcorrespondtotheGS95slope
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