Spectro-polarimetric Imaging Reveals Helical Magnetic Fields in Solar Prominence Feet

0.92 MB·English

by  M. J. Martinez Gonzalez

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SPECTRO-POLARIMETRICIMAGINGREVEALSHELICALMAGNETICFIELDSINSOLARPROMINENCEFEET
M.J.Mart´ınezGonza´lez1,2,R.MansoSainz1,2,A.AsensioRamos1,2,C.Beck1,2,4,J.delaCruzRodr´ıguez3,A.J.D´ıaz1,2,5
SubmittedtotheAstrophysicalJournal
ABSTRACT
Solarprominencesarecloudsofcoolplasmalevitatingabovethesolarsurfaceandinsulatedfromthemillion-
degreecoronabymagneticfields. Theyforminregionsofcomplexmagnetictopology,characterizedbynon-
potential fields, which can evolve abruptly, disintegrating the prominence and ejecting magnetized material
intotheheliosphere. However,theirphysicsisnotyetfullyunderstoodbecausemappingsuchcomplexmag-
5
netic configurations and their evolution is extremely challenging, and must often be guessed by proxy from
1
photometricobservations. Usingstate-of-the-artspectro-polarimetricdata,wereconstructthestructureofthe
0
magnetic field in a prominence. We find that prominence feet harbor helical magnetic fields connecting the
2
prominencetothesolarsurfacebelow.

n
Subjectheadings:Sun: magnetictopology—Sun: chromosphere—Sun: corona—Polarization
a
J

4 1. THEEMPIRICALSTUDYOFMAGNETICFIELDSINSOLAR longitudinal or transversal Zeeman effects, respectively (e.g.
1 PROMINENCES LandiDegl’Innocenti&Landolfi2004). Iflightisscattered,
Solar prominences are seen as bright translucent clouds at additionalsymmetriesarebrokenandthedependenciesofpo-
R] ∼10Mmoverthesolarlimbbecausetheymainlyscatterlight larization are more involved. In solar prominences and fila-
fromtheunderlyingdisk. Whenseenonthesolardisk, they ments,spectrallinesarepolarizedbyscatteringandtheZee-
S appear as dark, long filamentary structures, hence called fil- maneffect,andfuthermodifiedbytheHanleeffect(Tandberg-
.
h aments. The main body of the filament (spine) often has Hanssen1995;LandiDegl’Innocenti&Landolfi2004), pro-
p shorter side-wards extensions (filament barbs) which, when vidingdirectinformationonthemagneticfieldvector.
- seen at the limb, give the impression of extending from the Many studies have observed prominences with the aim to
o
spine to the photosphere below (prominence feet) (Mackay determinemagneticfields. Somemapsofthemagneticfield
r
t etal.2010). vector in quiescent prominences show horizontal magnetic
s It has long been clear that a magnetic field supports the fields of ∼ 10 − 20 G (Casini et al. 2005; Orozco Sua´rez
a
[ dense material of prominences against gravity and prevents etal.2014),inagreementwithresultsobtainedinthe1970’s
them from dissipating into the faint, extremely hot corona. and 1980’s from observations with limited spatial resolution

1 Local dips on magnetic field lines can support the plasma, (Sahal-Brechot et al. 1977; Leroy 1989). In contrast, verti-
v andcouldbeinducedbythedense,heavyprominenceplasma calfieldshavebeenalsodiagnosedinprominences(Merenda
5 itself (Kippenhahn & Schlu¨ter 1957), or they can exist in etal.2006). Consideringprominencefeet,theobservationof
9 force-free(Aulanier&Demoulin1998;Antiochosetal.1994) verticalvelocitiesinthesestructuressuggestverticalfieldsdi-
2 or stochastic magnetic fields (van Ballegooijen & Cranmer rectlyconnectingthespinewiththephotosphere(Zirkeretal.
3 2010). Yet, all these theoretical claims must be constrained 1998). Also, from observations of photospheric magnetic
0
by the empirical determination of magnetic fields in promi- fields(nottheprominenceitself),thebarbsareinterpretedas
1. nences. aseriesoflocalhorizontaldipssustainingplasmaatdifferent
0 In the Sun, and in any astrophysical plasma in general, heights(Lo´pezAristeetal.2006).
5 we are not able to directly measure these fields, but we are Inthelightoftheseresultsitisclearthat,fromtheobserva-
1 obliged to infer them from the light they emit. Spectro- tionalpointofview,theprecisetopologyofmagneticfieldsin
v: polarimetry, the measurement of the polarized spectrum of prominencesisstillamatterofdebate. Themainreasonsare
i light allows us to recover quantitative information on the that1)measuringthepolarizedspectrumofsolarprominences
X magneticfieldvector. Thepolarizationstateofobservedlight isanobservationalchallenge,and2)theinferenceofthemag-
r is compatible with the intrinsic (broken) symmetries of the netic field vector in the Hanle regime is subject to potential
a emitting plasma, in particular, with the presence of a mag- ambiguities. Inthispaper,wereconstructthetopologyofthe
netic field. Thus, for example, the emission by an isotropic feetofaquiescentprominencefromspectro-polarimetricdata
(andthereforeunmagnetized)mediumisunpolarized. Polar- attheHei1083.0nmline. Westudythepossiblesolutionsto
ized emission along a magnetic field is circularly polarized the inverse problem and, in contrast to previous results, we
while, normallytothefield, itislinearlypolarized, asinthe discard some of them using a physical constraint. We also
propose an analytical method to be used to find the multiple
1InstitutodeAstrof´ısicadeCanarias,V´ıaLa´cteas/n,E-38205LaLa- solutionsinthecaseofHanlediagnosticsinprominencesand
guna,Tenerife,Spain writetheexplicitequationsintheAppendix.
2Dept. Astrof´ısica,UniversidaddeLaLaguna,E-38206,LaLaguna,
Tenerife,Spain 2. NEAR-INFRAREDSPECTRO-POLARIMETRYAND
3InstituteforSolarPhysics,Dept. ofAstronomy,StockholmUniver- MULTIWAVELENGTHIMAGING
sity,AlbanovaUniversityCenter,SE-10691Stockholm,Sweden
4Presentaddress: NationalSolarObservatory, SacramentoPeakP.O. On April 24 2011 (10:00-13:00 UT), we performed four
Box62,Sunspot,NM88349,USA consecutivespectro-polarimetricscansofaquiescentpromi-
5Present address: Departament de F´ısica, Universitat de les Illes nence(locatedat90E42S)focusingatthe1083.0nmmulti-
Balears,E-07071PalmadeMallorca,Spain plet using the Tenerife Infrared Polarimeter (Collados 1999)
2 Mart´ınezGonza´lezetal.
a b
5 Mm 10 Mm
c d
0.005
0.000
-0.005
5 Mm
0.004
e f
f2 0.002
f2a
f1 0.000
f2b
-0.002
f1 f2a f2b
Figure1. a,ProminenceseenatthecoreofHα. Thissnapshotwasrecordedhalfwayofthespectropolarimetricscan. b,Prominenceasseenat17.1nm. This
snapshotwasrecorded10hafterthespectro-polarimetricscan,whenthepillarsofthetornadoesweremoreprominent(noneedforsharp-marskingfilter)thanat
thetimeoftheobservations. c,ReconstructedintensitymapatthecoreoftheHeI1083.0nmline. d,e,MapsoftheamplitudeoflinearpolarizationStokesQ
andU.ThereferencedirectionforpositiveStokesQisparalleltothelimb.f,MapoftheamplitudeofcircularpolarizationStokesV.Insetsshowdetailedviews
ofthecircularpolarizationinbothfeet.Insetsf1-f2b:StokesVintheregionaftersubtractingtheaveragevalue(f1andf2b)
attheGermanVacuumTowerTelescopeintheObservatorio 1b)observedwiththeAtmosphericImagingAssembly(AIA;
delTeide. Weintegrated30sperscansteptoreachapolari- Lemen et al. 2012) instrument onboard the Solar Dynamics
metric sensitivity of 7× 10−4 times the maximum intensity. Observatory(Pesnelletal.2012).
Eachscanoftheprominenecetookaround30min. Thisdata In this paper, we focus on the study of the third scan be-
setconstituteauniquetimeseriesofhighpolarimetricsensi- causeofthedouble-helixappearenceoftheprominencefeet.
tivityandunprecedentedspatialresolution(∼ 470kmonthe Figure 1 displays the multiwavelength intensity, and polari-
Sun) of a prominence. We applied standard reduction pro- metric imaging of the observed prominence. Both in the He
cedurestotherawspectra(biasandflat-fieldcorrection, and I 1083.0 nm scan and H intensity images, one of the two
α
polarizationdemodulation)toobtainmapsofthefourStokes prominencefeet(f2fromFig. 1f)showsacleardouble-helix
parameters I, Q, U, and V (Fig. 1c-f). The slit was always structureformedbytwofibrils. Amorecompact,twistedhe-
keptacrossthesolarlimb(horizontaldirectionintheimages), licalstructurecanalsobeguessedinfootf1. Thetwopromi-
whichallowedustocorrectforseeing-inducedcross-talkand nencefeetcorrespondtovertical,dark(absorption)structures
straylight(Mart´ınezGonza´lezetal.2012).Theseeingcondi- observed at 17.1 nm that resemble those recently named so-
tionswereexcellent,theadaptiveopticssystemoftenreaching lartornadoes(Suetal.2012;Wedemeyeretal.2013). They
anapparentmirrordiameterof20cm. Thismadetheapplied wereobservableintheAIAdataforalmostthreemoredays,
seeing-inducedcorrectionstobeverysmall,i.e.,closetothe andonApril26(23:38UT),theysuddenlyerupted,showing
limb,wheretheeffectsofseeingareexpectedtobethelargest, aclearhelicalshape. Interestingly,thefootf2presentsoppo-
theaveragecorrectionsappliedwere5×10−5timesthemaxi- sitepolaritiesoftheStokesV parameteratbothsidesofone
mumintensity. fibril (f2a). This means that the magnetic field has reversed
Simultaneousimagesweretakenwithanarrow-bandLyot polarities along the line of sight at both sides of this fibril.
filter centered at the core of the H line with a cadence of 1 Moreover,whensubtractingtheaveragecircularpolarization
α
s. Theseimagesweretreatedwithblinddeconvolutiontech- (i.e., themeanlongitudinalmagneticfield)to theotherfibril
niques(vanNoortetal.2005)toresolveveryfinespatialde- offeetf2(panelf2b)andtotheotherfeet(f1),wefindsimilar
tailsofthetemporalevolutionoftheprominence(Fig. 1aand patterns.
theonlineversionoftheH movie). Furthercontextwaspro-
α 3. INFERENCEOFTHEMAGNETICANDDYNAMICSTRUCTURE
videdbyimaginginthecoronallineofFeIXat17.1nm(Fig.
OFTHEPROMINENCEFEET
3
a
a
5 Mm
b
50
40
B [G]2300
10
LV
5 Mm
b c
LOS
Figure2. a,Inferredmagneticfieldtopologyoftheprominencefeet.Direc-
tionofthefieldprojectedontotheplaneofthesky(shortlines)overplotted
totheintensityoftheHeI1083.0nmlineandmagneticfieldstrength(b).
Short-dashedblacklinesinpanelatracetheaxesofthefibrilsinthetwofeet.
Inthesketchesbelow,bluearrowsrepresenttheactualinferredfieldvectorin
Figure3. Artisticrepresentationofthethree-dimensionalmagneticfieldin
f2afibril. Notetheoppositepolaritiesofthemagneticfieldinbothsidesof
theobserved.Ina,blueandredlinesrepresentarbitrary,recoveredfieldlines
thelocalvertical.This,andtherelativeinclinationoftheprojectedfieldwith
intheprominencefeet; graylinesfollowfieldlinesinthespineaccording
respecttothefibrilsimplyahelicalfield(redlines).
to overall models for prominence structure and are only drawn as context
(i.e. thetwistoffieldlinesdoesnotrepresentreality). Backgroundimage:
Weanalyzethespectro-polarimetricdatausingthenumeri-
intensityintheHeI1083.0nmline. b,c,Sideviews(seetheonlinemovie
calcodeHaZeL(HanleZeemanLight;AsensioRamosetal. ofthethree-dimensionalreconstruction).
2008)torecoverthefullmagneticfieldvectorandthethermo-
dynamicalproperties oftheplasma. Facinganinverse prob-
lengthofthehorizontallinefromthefoottothelimb,which
lemwithobservationaldataandalargenumberofdimensions
isagoodapproximationclosetothelimb. Theinferredscat-
isalwaysill-posed. AclassicalinversioncodesuchasHaZeL
teringangle(theanglebetweenthelineofsightandthelocal
retrieves one atmospheric model that fits the observed pro-
vertical)isapproximatelygivenby
files, though others may exist. In our case, the number of
theseambiguoussolutionsdependontheregimeofthemag- θ∼−cosl(2d/R) 1/2. (2)
neticfieldand,moreimportantly,onthescatteringgeometry. obs
Our approach is to find compatible solutions of the same in- The distance (d/R) obtained by substituting t = t , i.e.,
obs obs
verseproblemineachpixelandthenselecttheglobalscenario the time of the observations, in Eq. 1. After this procedure,
physically compatible with the context. This procedure step we obtain that the spectro-polarimetric scan was taken at a
isessentialtoreconstructtheglobaltopologyofthemagnetic scattering geometry of θ = 98◦, i.e. while the prominence
fieldintheprominence. wasslightlybehindthelimb.
3.1. Determinationofthescatteringgeometry 3.2. Determinationofmultiplesolutions
Theangleoftheemittingatominthelocalvertical(orthe Wefollowa2-stepinversionschemetoobtainarobustcon-
observedstrcutureifweassumeitinthesameplane)withre- vergence of the code. Since the magnetic field is of second
specttotheline-of-sight(thescatteringangleθ)isaveryim- order to the intensity profile, we first use the intensity pro-
portantparametertocorrectlyinferthemagneticfieldvector filealonetoinferthethermodynamicalquantities. Onasec-
from spectro-polarimetric signals generated from scattering ondstep,wefixthethermodynamicalparametersandfindthe
processes. Inordertodetermineit,weusedtheimagesofthe magneticfieldvectorusingtheinformationofthepolarization
Atmospheric Imaging Assembly (AIA) at 17.1 nm in which profiles.
weidentifiedthefeetofourobservedprominenceastwodark Asstatedbefore,thismaynotbetheuniquesolutiontothe
(absorption) vertical filaments (Fig. 1b). We followed these problem. In order to capture all possible solutions we fol-
filamentsastheyenteredontothediskanddetectedtheirpo- low,again,a2-stepprocedure. First,wesamplethespaceof
sitions(asaprojectionontotheplaneofthesky).Wemeasure parameterswithapproximate(thoughappropriate)analytical
theprojecteddistanceofthefilamentstothelimbdovertime expressions. Finally,weusetheseanalyticalsolutionsasini-
tandfititwiththeapproximateexpression tial guesses for a second HaZeL inversion. This will allow
ustorefinethesolutionsinthegeneralunsaturatedregimeto
d 1
∼ ω2(t−t )2, (1) overcometheapproximationswehavemade.
R 2 limb
In the case of an optically thin plasma, a normal Zeeman
inferringthevalueoft ,thetimeatwhichtheprominence triplet, and a magnetic field in the saturated Hanle effect
limb
feetwereatthelimb. ThesymbolRistheradiusoftheparal- regime–whentheLarmorfrequencyismuchlargerthanthe
lelatlatitudel = 42◦,andω = 0.55◦ h−1 isthesolarangular inverseofthecharacteristictimeforscattering–,thegeomet-
velocity at that latitude. The distance d was obtained as the ricdependenciesofpolarizationaresimplyexpressedthrough
4 Mart´ınezGonza´lezetal.
theStokesparametersas(Casinietal.2005):
Q∝(3cos2θ −1)sin2Θ cos2Φ
B B B
U∝(3cos2θ −1)sin2Θ sin2Φ
B B B
V∝cosΘ , (3)
B
showingadependenceontheinclinationΘ andazimuthΦ
B B
ofthemagneticfieldwithrespecttotheline-of-sight(LOS),
likeintheZeemaneffect,andanadditionaldependenceonthe
geometryofthescatteringeventthroughtheinclinationθ of
B
themagneticfieldwithrespecttothelocalvertical(LV).The
linearly polarized components (Stokes Q and U) are domi-
nated by scattering polarization and the Hanle effect, while
the circularly polarized component (Stokes V) is generated
bythelongitudinalZeemaneffect,anddefinesthepolarityof
the magnetic field along the LOS (Landi Degl’Innocenti &
Landolfi2004).
Equations3showadependenceofpolarizationonΘ and Figure4. HanleeffectdiagramfortheHeI1083.0nmlinecomputedwith
Φ thatyieldthewell-known180◦-ambiguityforΦ ofBclas- the HaZeL code assuming a 98◦ scattering geometry and the saturation
sicBalZeemandiagnostics. TheadditionaldependenBceonθB, rLeVg.imTehe(Bχ=B3=0G0).coTrhreesipnocnlidnsattioonth(eθBli)naenodfasziigmhut,than(φdBχ)Bar=er0efeforrredatroadtihael
characteristic of scattering processes, may yield two addi- vectorawayfromtheSun(AsensioRamosetal.2008,see). Thesolidline
tionalambiguoussolutionsforthemagneticfield(Casinietal. representthecurvesatconstantinclination.TheHanlediagramfortherange
ofinclinationsbetween90◦ and180◦ isthesamebutwithareversedcolor
2005). TheHeI1083.0nmlineoftenformsclosetothesat-
palette. Theorangedotsrepresenttheobservedpolarizationamplitudes.All
uration regime described by Eqs. 3 and all (up to four) am- theobservedpointshaveambiguousverticalandhorizontalsolutions.How-
biguous field configurations can be determined analytically ever,notethatthesignofthecircularpolarizationisonlycompatiblewith
(seetheAppendix)(seealsoJudge 2007). oneofthesolutionsrepresentedbyfilledoremptydots.
Inordertohaveanideaofthepotentialmultiplesolutions
weface,wehaverepresentedinFigure4theso-calledHanle a 6
diagram: the amplitude of Stokes Q and U in terms of the 4
inclination(θ )andazimuth(φ )ofthemagneticfieldinthe
B B
LsoVl.arTlhiembdia(mgriadmdleis-ucpopmerpuptaerdtsfoofr tahepooibnsteartve1d5(cid:48)p(cid:48)raobmoivneenthcee 02 -1[]km s
faefiete)l.dWstreenhgatvheoafss3u0mGe)d,athnedsthateusractaitotenrirneggiamnegl(eindeptaerrtmiciunleadr -2 vLOS
byourobservations(98◦). Thezeroinclinationisdefinedas
-4
afieldintheLV,andthezeroazimuthisdefinedintheLOS, 5 Mm
beingφ =180◦afielddirectedawayfromtheobserver. -6
B
The more vertical fields (θ ≤ 35◦ and θ ≥ 145◦) have 0
B B b
fourpossiblesolutionsirrespectiveoftheazimuthvalue: two
5
of them (the ones represented by filled and empty circles)
are not ambiguous if circular polarization is observed, since 10
itprovidesthesignofthelongitudinalcomponentofthemag-
m]
netic field. The other two solutions have opposite polarities M 15
[
of the field in the LV. Fields with inclinations between 35◦
and145◦ haveeightpossiblesolutions: amoreverticalincli- 20 0 20 40 60 80
nation with the four solutions stated above and another four [min]
25
solutions with more inclined fields. In the case of our ob-
servations,thesensitivityofcircularpolarizationallowsusto
30
constraintheazimuthrange,andhenceonlyfourambiguous
0 20 40 60 80
solutionsneedtobedetermined. Thepolarityofthefield(in [min]
theLV)cannotbedetermined. However,aswewillsee,this Figure5. a,LOSvelocityfieldfromtheinversionoftheHeI1083.0nmob-
ambiguityisunimportantforthepurposesofthispaper. servationswithintensitycontours(whitelines)overplottedforreference. b,
Space-timevariationsofHαintensityforanartificialslitat5.8Mm,parallel
3.3. Thehelicalmagneticfield tothelimbacrossthehelicalstructure(grayline). Bracketsmarkthetime
intervalduringthescanoff2. Dottedyellowlinestracetheperiodicmove-
The ambiguities apply at each pixel of our observations. mentofthetwofibrilsinthedouble-helix.Thecompositionofbothmotions
However, we assume that there are no tangential discontinu- indicaterotationofthestructure.
itiesorshocksandhencethemagneticfieldintheprominence
drawcontinuouslines. Wereconstructedfourglobaltopolo- ity reversal at opposite sides of its axis. Similar LOS polar-
gies of the magnetic field which could be grouped into two ityinversionsappearacrosstheaxesoftheotherfibrilswhen
broadcategories: onewithfieldsinclinedbyθ ∼ 30◦,150◦, theaveragevalueofStokesVintheregion,correspondingto
B
andanotheronewithmoreinclinedfields(θ ∼90◦). Inboth themeanLOScomponentofthemagneticfield,issubtracted
B
cases,theprojectionofthefieldontotheplaneoftheskyisat (Figs. 1f1 and 1f2b). Put together, this points to an helical
ananglewiththeaxisofeachfibrilthatformtheprominence global topology in both families of solutions, with the more
feet. Thefibrilf2a(displayedinFig. 1)showsaLOSpolar- horizontalconfigurationsshowingamoretwistedfield.
5
Inordertodisambiguatetheproblem,weintroduceaphys- consistentwiththeperiodsreportedinSuetal.(2012)forso-
ical constraint through a stability analysis. According to the lar tornadoes. Assuming that the change in H brightness is
α
Kruskal-Shafranovcriterion(Hood&Priest1979)akinkin- onlyduetoplasmamovements,theobservedperiodicmotion
stabilitydevelopswhentheamountofmagnetictwistexceeds could be also interpreted in terms of plasma flowing along
acriticalvalue,sothatastructureisstableif helicalfieldlines. Rotationofthemagneticstructureisvery
unlikely since the observed ∼ 50 min period would imply a
2πr B
z ≥1, (4) fullturnofthemagneticfieldinlessthanonehour. Thiswill
L B
θ considerablyincreasethetwistofthestructureandwillsoon
where r and L are the radius and the length of the fibril, make it unstable under magnetohydrodynamical instabilities
respectively, and B and B are the vertical and azimuthal (easilywithinaday). Theplane-of-the-skyperiodicmotions
z θ
components of the magnetic field, respectively. Estimating show maximum tangential velocities of ∼ 9 km s−1, which
r=1.7MmandL=11.6Mmforourprominence,thestabil- arefarlargerthantheonesinferredfromtheDopplereffectin
itycriterionyields0.09forthesolutionwiththefieldsaround theHeiline. Wemustbecarefultodirectlyrelatethosetwo
90◦and1.28forthefieldswithinclinationsof30◦,150◦. velocitiessinceDopplermeasurementsaregeneratedonlyby
Figure2showsthemagneticfieldstrengthandtopologyof plasmamotionswhilechangesinH brightnesscannotonly
α
themagneticfield. Themagneticfieldstrengthhasverysim- beassignedtoplasmamovementsbuttochangesinthether-
ilar values throughout all families of solutions. In most of malconditions.
theprominencethemagneticfieldstrengthisbelow20G,but
showsfilamentarystructures,paralleltothehelixstructure,of 4. DISCUSSION
higherfieldstrength(∼ 60G;Fig. 2b). Theprojectionofthe The feet of the observed prominence harbor helical mag-
field onto the plane of the sky is always roughly perpendic- netic fields. After assuming a simple stability criterion, we
ular to the solar limb, and at an angle (∼ 30◦-50◦) with the haveapreferenceforthemoreverticalsolutionandconclude
axisofthefibrilsthatformtheprominencefeet(Fig.2a). All thatthemagneticfieldsinprominencefeetconnectthespine
fibrils present magnetic field polarity reversals at both sides withtheunderlyingatmosphere. Theseresultsareincontrast
oftheiraxis. Thisimpliesahelicalfieldalongthefibrilswith tothescenarioinwhichthesestructuresareformedbyase-
theaxisintheplaneoftheskyinf2a(sincetheaveragelon- ries of local horizontal dips that sustain the plasma at differ-
gitudinal magnetic field is close to zero), and slightly tilted ent heights (Lo´pez Ariste et al. 2006; Aulanier et al. 1999).
relativetotheLVintheotherfibrils(sincepolarityreversals Assumingauniformlytwistedstraightcylinder,themagnetic
are only seen when the mean longitudinal magnetic field is field displays a twist (number of turns over its length L) of
subtracted). Thehelicalmagneticfieldsofthetwofibrilsare 1.32 at each fibril. We speculate that the connectivity of the
interlacedtoformadouble-helixthatconstitutesonefootof prominence spine, with a well-defined helicity, and the pho-
theprominence(seeFig.3andtheonlinemovieoftheartistic tospheric magnetic field below, with a fluctuating topology,
representation of the field topology). The feet magnetically maynaturallyyieldthekindofhelicalstructureswefind.
connect the spine with lower layers, in contrast to previous The Hei Doppler velocities display opposite velocities
workssuggestingthatfeetarejustacollectionofdipsatdif- alongtheLOSatbothsidesofprominencefeet.TheH inten-
α
ferentheightsLo´pezAristeetal.(2006). sitydisplaysperiodicmotionsintheplane-of-the-sky. Using
only the Doppler velocities or the plane-of-the-sky motions
3.4. Motionsoftheprominencematerial
alone do not allow to reach any conclusion on the plasma
TheintensityprofilesoftheHeIlinecarryinformationon motions and can lead to controversies in the literature (e.g.
the LOS velocity, opacity, temperature, and density of the Panasencoetal.2014;OrozcoSua´rezetal.2012). Ifweput
structure. We recover all these parameters along with the the Hei and the H information together, we could interpret
α
magnetic topology using the HaZeL code. They have the theseobservationsasthematerialoftheprominenceflowing
samevaluesforallfamiliesofsolutionsofthemagneticfield. along helical field lines. However, the LOS Doppler veloci-
Figure5ashowstheinferredLOSvelocities. Interestingly, tiesinferredfromtheHeilineandthetangentialvelocitiesex-
they have opposite signs at both sides of both feet of the pectedfortheobservedperiodoftheH imagesdonotmatch.
prominence (f1 and f2), with values up to 2kms−1 (similar It could be that neutral H and He clouαds have different ther-
valuesasOrozcoSua´rezetal.2012). Assumingthattheneu- modynamicalpropertiesoritcouldbethatthechangesinthe
tralatomsofHestilltracethefieldlines,thispatternofveloc- H brightnesshaveanimportantcontributionfromchanging
α
itiescouldbeanaturalconsequenceiftheprominencemate- thermal conditions, but we need more data to really under-
rialisflowingalongtheinferreddouble-helixstructureofthe standthisissue.
prominencefeet. Notethatf1hasapositive-negativeDoppler The prominence remains stable for several days, and we
pattern at lower heights, while it has a negative-positive one think that the magnetic topology of the prominence feet re-
higher in the feet, as we would expect for a more compact portedinthispaperplaysafundamentalroleonthestability
double-helix,whichimpliesalargertwist. oftheprominenceasawhole.
Ifthematerialisflowingalongthehelicalfieldlinesofthe
fibrils of each feet, a Doppler pattern similar to that of large
scalesshouldbeobservedatsmallerscales. Theupperparts The authors are especially grateful to Eric Priest, Manuel
of f2a show a positive-negative pattern. The rest of the fib- Collados, and In˜igo Arregui for very interesting discussions
rilshaveacontinuousincreaseoftheDopplervelocityaswe onsolarprominences,andfortheirenthusiasmonthiswork.
movefromlefttorightacrossthestructure. Thiscouldbein- We are also grateful to Arturo Lo´pez Ariste, Manuel Luna,
terpretedasanegative-positivepatternifthefibrilsarenotat DavidOrozcoSua´rez,andJavierTrujilloBuenoforveryhelp-
a90◦scatteringgeometry,whichisverylikely. ful discussions. This work is based on observations made
The two fibrils of f2 exhibit a periodic motion in the H with the Vacuum Tower Telescope operated on Tenerife by
α
space-time diagram, with a period of ∼ 60 min (Fig. 5b), theKiepenheuer-Institutfu¨rSonnenphysikintheSpanishOb-
6 Mart´ınezGonza´lezetal.
servatorio del Teide of the Instituto de Astrof´ısica de Ca- Leroy,J.L.1989,inAstrophysicsandSpaceScienceLibrary,Vol.150,
narias.FinancialsupportbytheSpanishMinistryofEconomy DynamicsandStructureofQuiescentSolarProminences,ed.E.R.Priest,
andCompetitivenessandtheEuropeanFEDERFundthrough 77–113
Lo´pezAriste,A.,Aulanier,G.,Schmieder,B.,&SainzDalda,A.2006,
project AYA2010-18029 (Solar Magnetism and Astrophysi-
A&A,456,725
calSpectropolarimetry)isgratefullyacknowledged.AARac-
Mackay,D.H.,Karpen,J.T.,Ballester,J.L.,Schmieder,B.,&Aulanier,G.
knowledgesfinancialsupportthroughtheRamo´nyCajalfel- 2010,SpaceSci.Rev.,151,333
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APPENDIX
A. AMBIGUITIESINTHEHANLEEFFECTINTHESATURATIONREGIME
InthesaturationregimeoftheHanleeffect,StokesQandUareinsensitivetothefieldstrength,butaresensitivetothegeometry
ofthefield. Foratwo-levelatomwitha J = 0 → J = 1transition,theopticallythinlimit,andthesaturationregimeforthe
up low
Hanleeffect,thelinearpolarizationcanbewrittenas:
q(cid:16) (cid:17)
Q= 3cos2θ −1 sin2Θ cos2Φ
B B B
2
q(cid:16) (cid:17)
U= 3cos2θ −1 sin2Θ sin2Φ . (A1)
B B B
2
TheinclinationandazimuthofthemagneticfieldintheLVarerepresentedbythesymbolsΘ andΦ ,respectively. IntheLOS,
B B
theinclinationandazimutharedisplayedasθ andφ . Theseequationsareformallythesameirrespectiveofthescatteringangle
B B
θ. Thedependenceonthescatteringgeometryisimplicitintheamplitudeintheabscenceofamagneticfieldq.
ThecoordinatesofthemagneticfieldvectorBinthereferencesystemoftheverticalandthereferencesystemoftheLOSare:
B=B(sinθ cosφ i+sinθ sinφ j+cosθ k)
B B B B B
B=B(cid:0)sinΘ cosΦ i(cid:48)+sinΘ sinΦ j(cid:48)+cosΘ k(cid:48)(cid:1), (A2)
B B B B B
wheretheunitvectorsarerelatedbyasimplerotation:
i(cid:48)=cosθi−sinθk
k(cid:48)=sinθi+cosθk. (A3)
Giventhatthemagneticfieldvectormustbethesameinbothreferencesystems,wefindthatthefollowingrelationsapply:
sinθ cosφ =sinΘ cosΦ cosθ+cosΘ +sinθ
B B B B B
sinθ sinφ =sinΘ sinΦ
B B B B
cosθ =cosΘ cosθ−sinΘ cosΦ sinθ. (A4)
B B B B
7
Solvingthepreviousthreeequationsinthetwodirections,wefindthefollowingtransformationsbetweentheanglesinthevertical
referencesystemandtheLOSreferencesystem:
cosΘ =cosθcosθ +sinθsinθ cosφ
B B B B
(cid:112)
sinΘ =+ 1−cos2Θ
B B
cosθsinθ cosφ −cosθ sinθ
cosΦ = B B B
B sinΘ
B
sinθ sinφ
sinΦ = B B (A5)
B sinΘ
B
and
cosθ =cosθcosΘ −sinθsinΘ cosΦ
B B B B
(cid:112)
sinθ =+ 1−cos2θ
B B
cosθsinΘ cosΦ +cosΘ sinθ
cosφ = B B B
B sinθ
B
sinΘ sinΦ
sinφ = B B. (A6)
B sinθ
B
Note that, since Θ ∈ [0,π], we can safely use the square root and take the positive value. In order to transform from one
B
referencesystemtotheother,wecancomputetheinclinationeasilybyinvertingthesinusorthecosinus. However,thesituation
is different for the azimuth, because the range of variation is [−π,π]. Therefore, one has to compute the cosinus and the sinus
separatelyandthedecidewhichisthecorrectquadrantfotheangleintermsofthesignsofbothquantities.
FourmultiplesolutionscanexistfortheStokes QandU parameters. TheideaisthatΦ canbemodifiedandstillobtainthe
B
sameQandU byproperlyadjustingthevalueofΘ . Itisclearthat,giventhatthetermthatcanbeusedtocompensateforthe
B
changeintheazimuthontheLOSreferencesystemisthesameforStokesQandU,wecanonlycompensateforchangesinthe
sign. Therefore,wehavethefollowingpotentialambiguities:
Φ(cid:48) =Φ
B B
Φ(cid:48) =Φ −π/2
B B
Φ(cid:48) =Φ +π/2
B B
Φ(cid:48) =Φ +π. (A7)
B B
We have to compute the value of Θ(cid:48) that keeps the value of Q and U unchanged. Therefore, once we find a solution to
B
the inversion problem in the form of the pair (θ ,φ ), we can find the remaining solutions in the saturation regime following
B B
the recipes that we present now. Remember that, unless one knows the sign of cosΘ (given by the observation of circular
B
polarization), the number of potential ambiguous solutions is 8. If the polarity of the field is known, the number is typically
reducedto4(or2ifno90◦ambiguityispresent).
A.1. Φ(cid:48)B =ΦB
Underthischange,wehavethat
cos2Φ(cid:48) =cos2Φ , (A8)
B B
sin2Φ(cid:48) =sin2Φ ,
B B
cosΦ(cid:48) =cosΦ ,
B B
sinΦ(cid:48) =sinΦ .
B B
MakinguseofthepreviousrelationsbetweentheangleswrttotheverticalandtheLOS,wehavetosolvethefollowingequation:
(cid:16) (cid:17) (cid:16) (cid:17)
3cos2θ(cid:48) −1 sin2Θ(cid:48) = 3cos2θ −1 sin2Θ , (A9)
B B B B
whichcanbewrittenas:
(cid:104)3(cid:0)cosΘ(cid:48) cosθ−sinθsinΘ(cid:48) cosΦ (cid:1)2−1(cid:105)sin2Θ(cid:48) (A10)
B B B B
(cid:104) (cid:105)
= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ .
B B B B
Aftersomealgebraanddoingthesubstitutiont=sinΘ(cid:48),weendupwiththefollowingequationtobesolved:
B
√
At4+Bt2+Ct3 1−t2 = K, (A11)
8 Mart´ınezGonza´lezetal.
where
A=−3cos2θ+3sin2θcos2Φ
B
B=3cos2θ−1
C=−6cosθsinθcosΦ
B
(cid:104) (cid:105)
K= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ . (A12)
B B B B
√
Thepreviousequationcanbesolvedifwemakethechangeofvariablest=± Z,resultingin:
(C2+A2)Z4+(−C2+2AB)Z3+(−2AK+B2)Z2 (A13)
−2BKZ+K2 =0.
This polynomial of 4-th order can have four different solutions. From these solutions, we have to take only the real solutions
whicharelargerthan0,giventherangeofvariationofΘ :
B
t∈R, 0≤t≤1. (A14)
Oncethesolutionsfort arefound,wemakeΘ(cid:48) = arcsint. Notethat,forafixedvalueoft,twovaluesofΘ(cid:48) arepossible. We
B B
choosethecorrectonebyevaluatingtheexpressionsfor QandU andtestingwhichofthetwopossiblechoicesgivethevalues
equal(orverysimilar)totheoriginalones.
Theangles(θ ,φ )areobtainedbydoingthetransformationfrom(Θ(cid:48),Φ )totheverticalreferencesystem.
B B B B
A.2. Φ(cid:48)B =ΦB+π
Underthischange,wehave:
cos2Φ(cid:48) =cos2Φ , (A15)
B B
sin2Φ(cid:48) =sin2Φ ,
B B
cosΦ(cid:48) =−cosΦ ,
B B
sinΦ(cid:48) =−sinΦ .
B B
Followingthesameapproach,wehavetosolveforΘ(cid:48) in
B
(cid:104)3(cid:0)cosΘ(cid:48) cosθ+sinθsinΘ(cid:48) cosΦ (cid:1)2−1(cid:105)sin2Θ(cid:48) (A16)
B B B B
(cid:104) (cid:105)
= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ .
B B B B
Thesolutionareobtainedastherootsofthesameequationsasbeforebutnow
A=−3cos2θ+3sin2θcos2Φ
B
B=3cos2θ−1
C=6cosθsinθcosΦ
B
(cid:104) (cid:105)
K= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ . (A17)
B B B B
Theangles(θ ,φ )areobtainedbydoingthetransformationfrom(Θ(cid:48),Φ +π)totheverticalreferencesystem.
B B B B
A.3. Φ(cid:48)B =ΦB+π/2
Underthischange,wehave:
cos2Φ(cid:48) =−cos2Φ , (A18)
B B
sin2Φ(cid:48) =−sin2Φ ,
B B
cosΦ(cid:48) =−sinΦ ,
B B
sinΦ(cid:48) =cosΦ .
B B
Followingthesameapproach,wehavetosolveforΘ(cid:48) in
B
(cid:104)3(cid:0)cosΘ(cid:48) cosθ+sinθsinΘ(cid:48) sinΦ (cid:1)2−1(cid:105)sin2Θ(cid:48) (A19)
B B B B
(cid:104) (cid:105)
= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ .
B B B B
Thesolutionareobtainedastherootsofthesameequationsasbeforebutnow
A=−3cos2θ+3sin2θsin2Φ
B
B=3cos2θ−1
C=6cosθsinθsinΦ
B
(cid:104) (cid:105)
K=− 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ . (A20)
B B B B
9
Theangles(θ ,φ )areobtainedbydoingthetransformationfrom(Θ(cid:48),Φ +π/2)totheverticalreferencesystem.
B B B B
A.4. Φ(cid:48)B =ΦB−π/2
Underthischange,wehave:
cos2Φ(cid:48) =−cos2Φ , (A21)
B B
sin2Φ(cid:48) =−sin2Φ ,
B B
cosΦ(cid:48) =sinΦ ,
B B
sinΦ(cid:48) =−cosΦ .
B B
Followingthesameapproach,wehavetosolveforΘ(cid:48) in
B
(cid:104)3(cid:0)cosΘ(cid:48) cosθ+sinθsinΘ(cid:48) sinΦ (cid:1)2−1(cid:105)sin2Θ(cid:48) (A22)
B B B B
(cid:104) (cid:105)
= 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ .
B B B B
Thesolutionareobtainedastherootsofthesameequationsasbeforebutnow
A=−3cos2θ+3sin2θsin2Φ
B
B=3cos2θ−1
C=−6cosθsinθsinΦ
B
(cid:104) (cid:105)
K=− 3(cosΘ cosθ−sinθsinΘ cosΦ )2−1 sin2Θ . (A23)
B B B B
Theangles(θ ,φ )areobtainedbydoingthetransformationfrom(Θ(cid:48),Φ −π/2)totheverticalreferencesystem.
B B B B