Radiative transfer in cylindrical threads with incident radiation VII. Multi-thread models

0.77 MB·

by  N. Labrosse

Astronomy&Astrophysicsmanuscriptno.26897 (cid:13)cESO2016
January12,2016
Radiative transfer in cylindrical threads with incident radiation (cid:63)
VII. Multi-thread models
N.LabrosseandA.S.Rodger
SUPA,SchoolofPhysicsandAstronomy,UniversityofGlasgow,GlasgowG128QQ,Scotland
e-mail:[email protected]
ABSTRACT
6
1 Aims. Our aim is to improve on previous radiative transfer calculations in illuminated cylindrical threads to better understand the
0 physicalconditionsincoolsolarchromosphericandcoronalstructurescommonlyobservedinhydrogenandheliumlines.
2 Methods. Wesolvedtheradiativetransferandstatisticalequilibriumequationsinatwo-dimensionalcross-sectionofacylindrical
structureorientedhorizontallyandlyingabovethesolarsurface.Thecylinderisfilledwithamixtureofhydrogenandheliumandis
n illuminatedatagivenaltitudefromthesolardisc.Weconstructedsimplemodelsmadefromasinglethreadorfromanensembleof
a severalthreadsalongthelineofsight.Thisfirstuseoftwo-dimensional,multi-threadfinestructuremodellingcombininghydrogen
J and helium radiative transfer allowed us to compute synthetic emergent spectra from cylindrical structures and to study the effect

1 ofline-of-sightintegrationofanensembleofthreadsunderarangeofphysicalconditions.Weanalysedtheeffectsofvariationsin
1 temperaturedistributionandingaspressure.Weconsideredtheeffectofmulti-threadstructureswithinagivenfieldofviewandthe
effectofpeculiarvelocitiesbetweenthestructuresinamulti-threadmodel.Wecomparedthesenewmodelstothesinglethreadmodel

] andtestedthemwithvaryingparameters.
R
Results.Thepresenceofatemperaturegradient,withtemperatureincreasingtowardstheedgeofthecylindricalthread,reducesthe
S relativeimportanceoftheincidentradiation comingfromthesolardiscontheemergentintensitiesofmost hydrogenandhelium
. lines. We also find that when assuming randomly displaced threads in a given field of view, the integrated intensities of optically
h thickandthintransitionsbehaveconsiderablydifferently.Inopticallythinlines,theemergentintensityincreasesproportionallywith
p
thenumberofthreads,andthespatialvariationoftheintensitybecomesincreasinglyhomogeneous.Opticallythicklines,however,
-
o saturateafteronlyafewthreads.Asaconsequence,thespatialvariationoftheintensityretainsmuchsimilaritywiththatofthefirst
r fewthreads.Themulti-threadmodelproducescomplexlineprofileswithsignificantasymmetriesifrandomlygeneratedline-of-sight
t velocitiesareaddedforeachthread.
as Conclusions.Thesenewcomputationsshow,forthefirsttime,theeffectofintegratingtheradiationemittedinHandHelinesby
[ severalcylindricalthreadsthatarestaticormovingalongthelineofsight.Theycanbeusedtointerprethigh-spatialandspectral
resolutionsofcylindricalstructuresfoundinthesolaratmosphere,suchascoolcoronalloopsorprominencethreads.

1
v Keywords. radiativetransfer–line:formation–line:profiles–Sun:chromosphere–Sun:corona
2
9
31. Introduction model. The code was then capable of simulating variations in
2
the radiation field from both radial and azimuthal coordinates.
0Tomodelcoolhorizontalstructuresinthesolaratmosphere(e.g.
The incident radiation on the cylinder comes from a uniformly
.cool coronal loops, fine filament, or prominence threads), we
1 radiatingsphereatagivendistanceandinclination.
0use a two-dimensional cylindrical radiative transfer model de- Gouttebroze(2006,paperIII)combinedtheresultsfromthe
6partingfromlocalthermodynamicequilibrium(NLTE).Atwo-
first two papers to create a working model for a 10-level plus
1dimensionalcylinderherereferstoacylindricalmodelwithonly
continuum hydrogen atom with all lines calculated using com-
:two active variables out of the three cylindrical coordinates.
v pleteredistributioninfrequency(CRD).Anarbitraryinclination
The development of this model is described in a series of pa-
i of the cylinder with respect to the solar surface can be chosen.
Xpers(Gouttebroze2004,2005,2006,2007,2008;Gouttebroze&
TheauthorinvestigatedtheformationofseveralLymanlinesand
rLabrosse2009,hereafterpapersItoVI).Below,wesummarise the Lyman continuum under different temperature and pressure
athe results of the most relevant studies in this series (see also
conditions,includingtwo-dimensional(2D)distributionsoftem-
Labrosse2015).
perature and pressure to model intensities emitted by complex
Gouttebroze (2004, paper I) started with a basic one-
structures. The behaviours of line profiles and integrated inten-
dimensional model for a pure hydrogen structure (a vertical
sitieswerediscussed.
cylinderinthesolaratmosphere).Gouttebrozeinvestigatedtwo
Next,Gouttebroze(2007,paperIV)andGouttebroze(2008,
acceleratedlambda-iteration(ALI)methodswiththeaimtotest
their strengths and to create an accurate, efficient numerical paper V) added various improvements such as time-dependent
solutionstostudythermalequilibriumandtheeffectsof3Dve-
model.ThemodelwasthenexpandedinGouttebroze(2005,pa-
locityfieldsontheemergenthydrogenspectrum,respectively.
per II) to a more general two-dimensional, azimuth-dependent
Finally,Gouttebroze&Labrosse(2009,paperVI)expanded
(cid:63) AppendixfiguresA1-A5areonlyavailableinelectronicformvia ontheworkdescribedinpaperIIIandaddedaHei–Heii–Heiii
http://www.edpsciences.org systemtomodelchromosphericandcoronalloop-likestructures
Articlenumber,page1of18
A&Aproofs:manuscriptno.26897
visible in hydrogen and helium lines. In Paper VI, the electron offinestructureofsolarprominences,andsolarcoolloopobser-
densityisdeterminedbytheionisationequilibriaofbothhydro- vations are discussed in Parenti (2014). Labrosse et al. (2010)
gen and helium. In that paper the authors successfully studied provides an overview of the concepts and techniques used in
theeffectsoftemperature,pressure,andheliumabundanceusing NLTEradiativetransfermodellingappliedtoprominences(e.g.,
isothermalandnon-isothermalisobaricmodels.Themodelsde- asolarstructureilluminatedfromthesolardisc).
scribedtherepresentagoodmethodformodellingsinglethread Ouraimistopresentnewcalculationsthatcanimprovethe
structures.However,inpaperVItherewasnoattempttoinvesti- understanding of the physical conditions in cool solar chromo-
gatetheeffectsofseveralthreadspresentalongthelineofsight sphericandcoronalstructures.Toaccomplishthis,wedetailthe
(LOS)ontheemergentspectra. models we use and the results we obtain in the following sec-
Highspatialresolutionobservationsclearlyshowthatpromi- tions.Section2startsbyprovidingthemainfeaturesofthemod-
nences are made of very fine threads (see, for example, Berger elling.Section3presentstheset-upforconsideringthesuperpo-
et al. 2008; Chae et al. 2006; Gunár et al. 2007; Lin et al. sitionofseveralstaticindividualthreadsandtheresultsfromthat
2005, 2012; Parenti 2014; Schmieder et al. 2014; Vial & En- investigation.Section4detailshowvelocitiesareintroducedin
gvold2015).Thenumberofsmall,finestructurethreadsinclose themodelling.Finally,Section5offerssomeconclusionsabout
proximitytoeachotherislikelygreaterbeyondtheresolutionof thiswork.
currentinstrumentation.Similarly,coolloopsinthesolaratmo-
sphereareusuallyobservedtobeingroups.Hence,itisimpor-
tanttobeabletomodeltheradiationemergingfromabundleof 2. Modelling
threads.
In this section we briefly summarise the main features of the
Gunáretal.(2007)useda2DthreadmodelinCartesiange-
models that we have used. The modelling details are given in
ometrytocomputetheemergentradiationinhydrogenlines,and
Paper VI. The model atom for hydrogen contains five discrete
extended the modelling to a multi-thread configuration to rep-
energylevelsandacontinuum.Thisallowsfortendiscretetran-
resent a series of threads along the LOS. The authors investi-
sitionsandfivebound-freetransitions.Theheliummodelatom,
gated the emission in the hydrogen Lyman lines from an arbi-
ontheotherhand,has34discretestates;with29forHei,fourfor
trarynumberofthreads,eachwithagivenprominence-to-corona
Heii,andoneforHeiii.Thenumberofpermittedtransitionsfor
transition region. By comparing their computed line profiles to
the entire helium system is 76. Under usual conditions, a large
observationsfromtheSUMERinstrumentonSOHO,theycon-
proportion of these are optically thin. The models for both the
cludedthatthespectraproducedfromprominencefinestructures
hydrogenandheliumatomsassumeCRDforalldiscretetransi-
arebettersimulatedwithamulti-threadmodelascomparedtoa
tions.
single thread model. Their model assumes magnetohydrostatic
(MHS) equilibrium (Heinzel & Anzer 2001). This allows them
toinferrealisticpressureprofilesacrossthe2Dthreads.Thisin
2.1. Parametersdefinedforeachsinglethread
turndeterminesanydensityfluctuationsalongtheLOS.
Line profiles of optically thick transitions often display a Theinclinationofacylindricalthreadisdefinedbytheanglebe-
substantial central reversal, where the intensity at the central tweentheaxisofthecylinderandthevertical,α.Owingtosym-
wavelengthissignificantlyabsorbedbythematerialintheLOS. metry, α is considered between 0 and 90◦. We consider cylin-
Theresultingprofilebecomesdoublepeaked.Variouseffectscan dricalthreadsorientedhorizontallyinthesolaratmosphere,i.e.
causethesetwopeakstobecomeasymmetric,forexampleeither α = 90◦.TheLOSintersectsthecylinderperpendicularlytothe
theredorbluepeakbecomeshigherinintensitywithrespectto cylinderaxes.Forafullschematicviewofthemodelgeometry,
theother.Theseasymmetriclineprofileshavebeenobservedfre- seeFigs.1and2inPaperII.
quentlyinsolarprominencesandtheirfinestructureforthehy- Thesizeofthethread(i.e.thediameterofthecross-section
drogenLymanlineswiththeSUMERinstrumentaboardSOHO ofthecylindricalthread)canbechosentorepresentalarge-scale
byHeinzeletal.(2001);Schmiederetal.(1999),andSchmieder prominence modelled as a whole (thickness 103–104 km), or a
etal.(2007),andearlierwiththeOSO-8(OrbitingsolarObser- significantly smaller fine structure of thickness ∼ 102 km. We
vatory)missionbyVial(1982). focusonthreadsofdiameter1000−2000km.
InGunáretal.(2008),theauthorsexpandontheirprevious Theimportantphysicalparameterswithinthethreadaregas
modeltoincluderandomLOSvelocitiesforeachofthethreads. pressureP ,temperatureT,heliumabundance,A ,andmicro-
g He
TheyfoundthattheLOSvelocitiesproduceasymmetriesinthe turbulentvelocity.Theeffectsofabundanceandofmicroturbu-
HLymanlineprofilesthatagreewithSUMERprominenceob- lent velocity are not investigated here. In all models the micro-
servations.InPaperV,asinglethreadwasconsideredwithvary- turbulentvelocityis5kms−1 withaheliumabundanceratioof
ing plasma velocity distributions (radial, tangential, and longi- A =0.1.
He
tudinalflows).Thedifferenttypesofvelocitydistributionswere To understand the effects of gas pressure and temperature
found to have different effects on the synthetic hydrogen line ontheoverallsystem,twogroupsofmodelsareused:t models
profiles. Plasma oscillations insingle thread with Cartesian ge- (isothermal and isobaric) and p models (isobaric models with
ometryhavebeensuccessfullymodelledinHeinzeletal.(2014). a radial temperature distribution). The same models were first
Thisstudyfindsthatdifferentoscillatorymodesproducespectral used in Paper III, as well as in Paper VI, and we therefore use
indicator variations with differing magnitudes and that a seis- thesamenotationhere.
mology analysis of the hydrogen Hα and Hβ line parameters All of the t models in this paper use a constant gas pres-
could be used to find and diagnose oscillatory modes in solar sure of 0.1 dyn cm−2. By varying the isothermal temperature
prominences.ItshouldbenotedthattheLOSvelocitiesusedin inthesemodels,theeffectoflarge-scaletemperaturevariations
thisstudyareglobalquantitiesandonlyvarybetweenthreadsin can be investigated. It is, however, often necessary to take the
multiplethreadmodels. presence of a distribution of temperatures into account, which
Several review articles have been written on solar promi- isexpectedtoresultinasignificanttemperaturegradient,espe-
nenceobservationsandmodelling.Heinzel(2007)givesareview cially near the cylinder edges. The importance of the transition
Articlenumber,page2of18
Labrosse&Rodger:Radiativetransferincylindricalthreadswithincidentradiation
Radial Temperature Distribution Isothermal Distribution
105 105
104
104
103
y y103
t102 t
si si
n n
e e
Int101 He I 10830 Int102
He I 584
100 He I 5876
He I 6878
101
He I 537
10-1
He II 304
10-2 100
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
r/R r/R
Fig.1.Effectofincidentradiationonheliumlineintensityacrossasectionofacylindricalthread.Thefrequency-integratedintensityismeasured
inergcm−2s−1sr−1.Thepositionisgiveninfractionalunitsofthecylinderradius.Thehorizontalaxisrepresentsthepositionverticallyawayfrom
thesolarsurfacewiththezeropointatthethreadcentre.Theleft-handgraphusedthep4model,whilsttheright-handgraphusedtheisothermal
t1model.
region in the case of prominences was discussed by Anzer & determinestheboundaryconditionsforthesolutionoftheradia-
Heinzel (1999). The profile of the temperature variation is not tivetransferequation.
well known, though it is possible to make some assumptions. For p models, the excitation and ionisation state of the
Anzer & Heinzel (1999) proposed an analytical expression of plasma may be primarily driven by the gas temperature, which
thetemperatureandpressuredependingonthecolumnmassin- can be large at the outer edge of the thread illuminated by the
side a 1D quiescent prominence model. This was extended to incident radiation. In a model with a typical temperature dis-
2DbyHeinzel&Anzer(2001).Here,thetemperaturedistribu- tribution, the effect of incident radiation from the Sun or other
tion chosen to model this change in p models is the same as in sources can easily be masked by the local temperature gradi-
PaperIIIandisgivenby ents. This effect can be seen in Fig. 1, which compares the
frequency-integrated emergent intensity in several helium lines
r−r
logT(r)=logT +(logT −logT ) 0 . (1) foranisothermalmodelandamodelwithatemperaturegradi-
0 1 0 r1−r0 ent. Fig. 1 shows that in an isothermal model (t1, right panel),
thevariationoftheemergentintegratedintensityalongavertical
The parameter T defines the temperature at the centre of the
0 cutthroughthecross-sectionofthecylinderisprimarilydriven
thread and T represents the temperature of the surrounding
1 bythechangeintheradiationincidentonthestructureaswego
corona. The inner and outer edges of the transition region are
fromthebottomtothetopofthehorizontalthread.Ontheother
definedbyr andr ,respectively.Thetemperatureatthethread
0 1 hand,thevariationoftheemergentintegratedintensitiesinmost
centreis6×103K,whilstthetemperatureatthecoronalbound-
lines for a model with a temperature gradient (p4, left panel)
ary is 105 K. This distribution is not created from a theoretical
shows a lesser difference between the bottom and top edges of
basis and is designed only to depict the effect of a radial tem-
thecylindricalthreadwithlocalvariationsduetoplasmacondi-
perature gradient. A plot showing such a temperature variation
tions.However,theHeii304Ålinestillshowstheinfluenceof
acrossthethreadcanbefoundinFig.1ofPaperVI.
theanisotropicincidentradiationonthecylindricalthread,since
Thecylinderdiameteris1000kmintmodelsand2000kmin
pmodels(e.g.forourpmodels,r =500kmandr =1000km). thislineismostlyformedbythescatteringoftheincidentradia-
Thischoicemakestheeffectofad0dingatemperatu1regradienton tionunderthephysicalconditionsconsideredhere.
The model parameters we used are summarised in Table 1.
topoftheisothermalthreadcoreclearerwhencomparingthere-
We frequently use the p4 model since it corresponds to a typi-
sultsbetweentandpmodels.Thealtitudeofthecylinderinflu-
cal gas pressure (0.1 dyn cm−2), which is also adopted in all t
enceshowtheradiationcomingfromthesolardiscisilluminat-
models,makingcomparisonsbetweenthesemodelseasier.
ing the structure (through, most notably, the height-dependent
dilution factor). We use a fixed height of 10000 km above the
solar surface. The effect that incident radiation may have on
2.2. Multi-threadmodels
the emergent intensities depends on the plasma parameters and
on the lines under consideration. For instance, cool isothermal Thisarticleaimstoapplythemethodsofthemulti-threadmodel
threads are comparatively more sensitive to the incident radia- of Gunár et al. (2007) in Cartesian geometry to the cylindrical
tion than non-isothermal models. Indeed, the incident radiation modelsofPaperVI.Indoingthis,theeffectofmulti-threadmod-
Articlenumber,page3of18
A&Aproofs:manuscriptno.26897
Table1.Parametersforp(withtemperaturegradient)andt(isothermal)
models:gaspressure,temperature,andcylinderdiameter. LOS
Model P (dyncm−2) T (103K) D(km)
g
{0.02,0.03,0.05, T(centre)=6 1st 2nd 3rd Nth
p1–p7 2000
0.1,0.2,0.3,0.5} T(surface)=100
SolarDisc
{6,8,10,15,20,30
t1–t11 0.1 1000
40,50,65,80,100}
Fig.2.Schematicdiagramshowinglayoutofsimplefullyalignedmulti-
threadmodel.Thedashedlinesrepresenttheupperandloweredgesof
thefieldofview,withtheloweredgefacingthesolardisc.
ellingisalsoappliedtoheliumlines,whichhas beenrelatively
unexploredsofar(forapreliminarystudyseeLéger&Paletou
2009).Ourfocusisoninvestigatingtheverticalcross-sectional
intensity, to try and discern any important geometrical effects.
The method used to do this is described below, and the results
aregiveninSection3. LOS
The multi-thread model considers an arbitrary number of
cool threads, N, oriented horizontally in the solar atmosphere
with an inclination angle α equal to 90◦. In the single thread
model, the field of view was automatically set by the physical
boundariesofthecylinderinconsideration,i.e.theedgeclosest 1st 2nd 3rd Nth
andfurthestfromtheSunintheverticaldirection.Inthemulti- SolarDisc
threadcasethisdefinitioniskept,usingtheclosestthreadtothe
observer(hereafterreferredtoastheforemostthread).
In the multi-thread model, the final emergent intensity af- Fig. 3. Schematic diagram showing layout of a randomly positioned
ter N threads is calculated as the sum of the constituent inten- (unaligned) multi-thread model. The dashed lines represent the upper
andloweredgesofthefieldofview,definedbythefirstthreadalong
sitiesfromeachindividualthread.However,photonsemittedby
thefurthestthread(Nth)havetotravelthroughallotherthreads theLOS,withtheloweredgefacingthesolardisc.
alongtheLOS(N−1threads).Itsintensityishencereducedby
thecumulativeeffectofalltheotherthreadsopticalthicknesses.
models. To ensure that at least some of each thread was within
The equation for the total emergent intensity is similar to the
thefieldofview,therandomlygenerateddisplacementvaluewas
followingequationusedbyGunáretal.(2007): cappedat+3/4and-3/4ofthediameterofthethread.Thevalues
Itotal = I1+I2e−τ1 +···+INe−(cid:80)N1−1τi , (2) frourntahreed{i0s,p-l1ac1e9m,-e4n0t,s1i2n0p,o9s,it-i4o5n,a5r5r,a-y8p6o,3in7t,s3u}s,ewdhinertehtehmetoodtaell
where I isthefrequencyandposition-dependenttotalemer- width of the field of view is 201 points across, centred at 0. A
total
gent intensity from N threads; and I and τ are the frequency schematicdiagramforthissetupcanbeseeninFig.3.
i i
andposition-dependentemergentintensityandopticalthickness
oftheith thread,respectively.Thenumber1hereisusedtorep-
2.3. Assumptions
resenttheforemostthread.
After a brief inspection of Eq. (2), it is is easy to see that In creating the simple fully aligned case and the randomly
there is a special case of this scenario, in which all threads are unaligned, multi-thread models there are several important as-
identicalandperfectlyalignedwithrespecttoeachotherandthe sumptions for the system that should be noted. As mentioned
LOS(Fig.2).Inthiscase,foreachfrequencyandpositionacross previously,ourassumptionsforthesimplecaserequirethateach
thecross-sectionofthecylinder,allI andτ areidentical.Inthis thread in the model has identical physical properties (tempera-
i i
case,thetotalemergentintensityisgreatlysimplifiedas turedistribution,gaspressure,etc).Thereisnoneedforthisre-
quirementintheunalignedmodel,althoughthisstudyislimited
i=(cid:88)N−1 toidenticalthreads.
I = I e−iτ, (3)
total Anotherassumptionisthat,inbothcases,theincidentradia-
i=0 tiononeachthreadmustbeidentical.Thatistosaythatradiation
whereIandτaretheidenticalintensitiesandopticalthicknesses from the Sun must appear uniform across the distance between
fromeachsinglethreadinthesimplifiedmulti-threadmodel. thefirstthreadandthelastintheLOS.Thisplaceslimitsonboth
Thesimplefullyalignedmodelisusefulasitgivestheabil- themaximumlengthoftheLOSand/ortheminimumaltitudeof
itytoverifythevalidityofthemulti-threadexpansionthroughan theindividualthreads.Thisassumptionisparticularlyimportant
easycomparisonwiththesinglethreadmodel.Itishoweverun- intheunalignedcaseasitrequiresthealtitudedisplacementbe-
realistic,sothelesstrivialcasewherethethreadsarerandomly tweenthethreadstobesignificantlysmallwhencomparedtothe
unalignedwithrespecttoeachotherhastobeconsidered.Inthis altitudeofthefieldofviewabovethesolarsurface.
case, the field of view was arbitrarily chosen to again encom- The final assumption we make is that the threads within
pass the upper and lower bounds of the foremost thread. Each the multi-thread models are not radiatively interacting. This
threadaftertheforemostthreadwasgivenarandomlygenerated means that no incident radiation from one thread upon another
verticaldisplacementwithrespecttotheforemostthread,which is considered. This assumption makes the model much simpler
effectively has a displacement of 0. The total width of the field as it keeps the equation for the emergent intensity in the form
ofviewwastakentocorrespondtoadistanceofonediameterof of Eq. (2) as well as negating any relevance of the horizontal
thecylindricalthread,1000kmfortmodels,and2000kmforp distance between the threads. This assumption is acceptable
Articlenumber,page4of18
Labrosse&Rodger:Radiativetransferincylindricalthreadswithincidentradiation
Table2.Position-averagedtotalopticalthickness(τ)andopticalthick-
106 He I 584 - Single Thread 106 He I 584- Multi-thread ness at line centre (τ ) for given transition lines in single thread p4
0
105 105 model.
nsity104 nsity104 Line τ0 τ
Inte103 Inte103 LLyyαβ 23..1455EE++0054 12..1907EE++0043
102 102 Lyγ 1.20E+04 9.79E+02
101 101 Hα 4.31E-01 1.01E-01
0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
∆λ ∆λ Hβ 5.96E-02 1.29E-02
106 He I 5876 - Single Thread 106 He I 5876 - Multi-thread Pα 5.93E-03 1.99E-03
105 105 Hei10830 8.30E-02 1.94E-02
104 104 Hei584 1.60E+04 5.08E+02
Intensity111000123 Intensity111000123 0000....0001235 HHHeeeiii56588377768 386...916021EEE+--000343 211...252843EEE+--000342
100 100 00..23 Heii304 1.33E+03 6.23E+01
10-1 10-1 0.5
10-2 10-2
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
∆λ ∆λ
Thecombinedeffectsofhighopticalthicknessatlinecentreand
Fig. 4. Changes in line profiles for helium lines Hei 584 Å and loweropticalthicknessinthewingscreatestheprofilesseenfor
Hei 5876 Å for a single thread model and for a multi-thread model Hei584Å.Thisissimilartowhatisobservedinotheroptically
with ten perfectly aligned threads. Intensities are measured in erg thick lines. Table 2 shows the optical thicknesses for all transi-
s−1cm−2sr−1Å−1. The legend gives the isobaric gas pressure in units tions we considered. The values of the optical thickness at line
ofdyncm−2.DeltalambdaismeasuredinÅ. centre and total optical thickness are averaged across all points
inthefieldofviewobtainedfromasinglethreadp4model.
as the intensity of the incident radiation of the Sun should be
considerably larger than any possible incident radiation from 3.2. Randomlypositionedthreads
onethreadtoanother.
The multi-thread model was then studied for ten randomly
positioned threads with the displacement values given in Sec-
3. Staticmulti-threadconfigurations
tion 2.2. When compared with the simple, fully aligned case
3.1. Fullyalignedthreads (Fig. 4), the position-averaged line profile shapes and intensity
values from the unaligned, multi-thread model are not greatly
Themulti-threadconfigurationwasfirstinvestigatedbycompar- affected. This is not unexpected since the displacements were
ing the synthetic spectra for ten perfectly aligned threads using
randomlygeneratedandthelineprofileshapeonlychangessig-
the p4 model with the previously studied single thread spectra.
nificantlytowardstheveryedgeofthethread.
Thiswasdonebytakingtheposition-averagedlinespectrafrom The clearer effects of the multi-thread configuration with
the single thread model and comparing it with the profiles pro-
randomlypositionedthreadswerefoundbyplottingtheposition-
duced by the fully aligned, multi-thread model. The change in
dependentintensityacrosstheverticalfieldofview.Foreachpo-
Hei 584 Å and Hei 5876 Å line profiles from the single and
sition in this vertical cross-section, the intensity was integrated
multi-threadmodelscanbeseeninFig.4.Thesegraphsshowthe
overfrequency.Figure5showsthechangeinintensityacrossthe
changesinlineprofileswithgaspressure,wherethegaspressure
field of view, with increasing numbers of randomly unaligned
wasisobaricacrossthecylinderasdictatedbymodelsp1through (p4 model) threads, for several different hydrogen and helium
p7(seeTable1).
lines.
FromFig.4itisclearthattherelationshipbetweenlinepro-
It is clear from Fig. 5 that the variation of the frequency-
fileshapeandgaspressuredoesnotchangesignificantlybetween
integrated intensity along the vertical direction can have a no-
asinglethreadandanensembleoftenthreadsforeitherhelium ticeably different behaviour when the number of threads in the
line.However,theshapeofHei5876Åisalsounaffectedbythe fieldofviewisincreased,dependingontheparticulartransition
changefromasingletoamulti-threadmodel.Thisisbecausethe in consideration. Optically thin lines increase in intensity with
plasma is optically thin for Hei 5876 Å photons (see Table 2): increasing numbers of threads, and any variation of the inten-
the emergent intensity from N, fully aligned threads, is simply sity along the vertical direction resulting from the presence of
∼ N timestheintensityofthesinglethread. severalrandomly-positionedthreadsalongtheLOSissmoothed
The shape of the line profiles for Hei 584 Å changes sig- towardshomogeneity.ThiseffectisseeninHα,Hβ(notshown
nificantly between the single and multi-thread models. This is infigure),andHei5876Å.
because Hei 584 Å is optically thick at line centre, whilst re- In contrast, in optically thick lines only the photons from
maining optically thin in the wings. Its high optical thickness the closest threads reach the observer, while photons coming
at line centre prevents photons at that wavelength from travel- from threads further away along the LOS are lost through
ling far within the medium; they are scattered very efficiently. multiple scattering. This saturation effect allows the structure
This causes the line centre to saturate as only the light from of the nearest threads in the field of view to be maintained,
theforemostthreademerges.Thelinehoweverbecomesincreas- independentofthenumberofthreadsthatmayliefurtheraway
inglylessopticallythickfurtherawayfromthelinecentre.This intheLOS.Thismeansthatcomparisonsbetweenobservations
causestheintensityoftheopticallythinlinewingstoscalepro- inopticallythicklinesandopticallythinlinescouldbeusedto
portionallywiththenumberofthreads,asseeninHei5876Å. inferthenumberofthreadstructuresintheLOS.Figure5shows
Articlenumber,page5of18
A&Aproofs:manuscriptno.26897
Lyman-Alpha Lyman-Beta H-alpha
120000 900 200000
800
100000
700
150000
80000 600
y
sit 500
n 60000 100000
e
nt 400
I
40000 300
50000
200
20000
100
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
He I 584 He I 5876 He I 537
6000 80000 200
70000
5000
60000 150
4000
50000
y
sit
n3000 40000 100
e
nt
I 1 thread
30000
2000 2 threads
3 threads
20000 50
5 threads
1000 10 threads
10000
20 threads
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
Position Position Position
Fig.5.Changesinfrequency-integratedintensityofvarioushydrogenandheliumlinesasafunctionofdistancealongtheverticaldirectionthrough
thefieldofviewdefinedbythefirstthread(frombottomtotop),fordifferentnumbersofrandomlyunalignedthreads(usingthep4model).The
intensityisgiveninergs−1cm−2sr−1 andtheverticalpositioninthefieldofviewisgiveninMm.Thehorizontalaxisrepresentstheposition
verticallyawayfromthesolarsurfacewiththezeropointatthecentreofthefirstthread.
theopticallythickLymanα,Lymanβ,andHei537Ålinesall pressures,electroncollisionalprocessestakealargerroleinthe
show this saturation effect after only a few threads. However, formation of theline, and the linecentre intensity begins to in-
Hei584Å,whichisanopticallythickline,doesnotdisplaythis creasewithpressure.
behaviour.ThisisdiscussedfurtherinSection3.2.1.
To verify this, the change in frequency-integrated intensity
along the vertical direction with increasing number of threads
3.2.1. Variationofintegratedintensityalongthevertical
was compared for all the p1–p7 models. Figure 6 shows the
directionasafunctionofpressure
changeinintensitywithpressureforthehydrogenLymanαline
(toptworows),whilethebottomtworowsshowtheresultsfor
Herewelookinmoredetailsatwhysomeopticallythicklines,
theHei584Åline.
mostprominentlyHei584Å,appeartoshowlessintegratedin-
tensity saturation with increased number of threads than other At low pressures, the integrated intensity does not saturate
optically thick lines. We suggest that this is an effect of the significantly, and hence much of the foremost thread structure
line’s relationship with gas pressure. In Fig. 5 we used the p4 islost,andtheintensityvariationalongtheverticaldirectionis
model.Thismodel’sradialtemperaturedistributionisdescribed smoothedbytheadditionofmorethreads.Aspressureincreases,
byEq.(1)andthegaspressureisfixedat0.1dyncm−2.Several thevariationoftheintensityalongtheverticaldirectionreaches
opticallythicklinesbegintoshowachangeinbehaviouraround saturation with fewer and fewer threads. At high pressures the
thatvalueofthepressure;thiscanbeseen,forexampleforHei integrated intensity almost immediately saturates, masking the
584ÅinFig.4.AsimilarchangecanbeseenclearlyinLymanα contributionofalloftheinnerthreads.Thiseffectisseeninall
andtoalesserextentinLymanγandHei537Å.Atlowerpres- optically thick lines, including Hei 584 Å, however the effect
sures,radiationscatteringprocessesaredominantandincreasing ismostpronouncedinthehydrogenLymanαline,especiallyat
thepressurehaslittleeffect:thelinecentreissaturated.Atlarger lowpressures.
Articlenumber,page6of18
Labrosse&Rodger:Radiativetransferincylindricalthreadswithincidentradiation
Lyman-alpha at 0.03 dyn cm−2 Lyman-alpha at 0.05 dyn cm−2 Lyman-alpha at 0.10 dyn cm−2
50000 70000 120000
60000
100000
40000
50000
80000
y30000
sit 40000
en 60000
nt 30000
I20000
40000
20000
10000
20000
10000
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
Lyman-alpha at 0.20 dyn cm−2 Lyman-alpha at 0.30 dyn cm−2 Lyman-alpha at 0.50 dyn cm−2
350000 600000 1200000
300000
500000 1000000
250000
400000 800000
y
sit200000
en 300000 600000
nt150000
I
200000 400000 1 thread
100000 2 threads
3 threads
50000 100000 200000 5 threads
10 threads
20 threads
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
Position Position Position
He I 584 at 0.03 dyn cm−2 He I 584 at 0.05 dyn cm−2 He I 584 at 0.10 dyn cm−2
2500 3500 6000
3000
5000
2000
2500
4000
y1500
sit 2000
en 3000
nt 1500
I1000
2000
1000
500
1000
500
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
He I 584 at 0.20 dyn cm−2 He I 584 at 0.30 dyn cm−2 He I 584 at 0.50 dyn cm−2
12000 20000 50000
10000
40000
15000
8000
y 30000
sit
en 6000 10000
nt
I 20000
1 thread
4000
2 threads
5000 3 threads
2000 10000 5 threads
10 threads
20 threads
0 0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
Position Position Position
Fig.6.Changesincross-sectionalfrequency-integratedintensityofthehydrogenLymanαline(toptworows)andtheHei584Åline(bottomtwo
rows)withanincreasingnumberofunalignedthreadsforisobaricmodelsp1–p7.Theintensityisgiveninergs−1cm−2sr−1andthepositioninthe
fieldofviewisgiveninMm.Thehorizontalaxisrepresentsthepositionverticallyawayfromthesolarsurfacewiththezeropointatthecentreof
thefirstthread.
Articlenumber,page7of18
A&Aproofs:manuscriptno.26897
Hydrogen line FWHM at varying pressures Helium line FWHM at varying pressures
1.8 1.2
Lyman-alpha He I 10830
Lyman-beta He I 584
1.6 H-alpha He I 5876
Lyman-gamma 1.0 He I 6878
H-beta He I 537
1.4
P-alpha He II 304
1.2 0.8
M 1.0
H
W 0.6
F0.8
0.6 0.4
0.4
0.2
0.2
0.0 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
Pressure Pressure
Fig.7.Changesinfullwidthathalfmaximum(FWHM)withpressure(modelsp1–p7)forvarioushydrogenandheliumlines.TheFWHMis
giveninÅandthepressureisgivenindyncm−2.
Thechangeinbehaviourwithpressureisrelatedtothecom- thread configuration quickly saturates, and the features in the
parative widths of the emission line profiles at these pressures. spatialintensityvariationduetothespecificphysicalconditions
Thechangeinlinewidthwithpressurewasinvestigatedbycal- inthethreadsalongtheLOSaresmoothedout.Thisallowsthe
culatingthefullwidthathalfmaximum(FWHM).TheFWHM structureoftheclosestthreadsintheLOStobereflectedinthe
was calculated simply as the width at the point where the line spatial variation of the intensity, regardless of the total number
presents half its maximum intensity, which may or may not be ofthreads.Thisstructuredependsonthelocationsofthenearest
the central peak (in case the line has a central reversal). The threadsintheLOSandmostprominentlyontheforemostthread.
change in FWHM with pressure for several hydrogen and he- Thecreationofthisstructureisbestseenbycomparingthecu-
liumlinescanbeseeninFig.7. mulative effect of each thread on the total field of view after N
Ingeneral,theintensitiesofopticallythicklinesthathavea threads,anexampleofthiseffectcanbeseeninFig.8.InFig.8
lowerFWHMsaturatemorethanbroaderlines.Broadoptically the solid blue lines show the combined effect of N unaligned
thick lines have extended wings where the optical thickness is threads,whilstdashedgreenlinesshowtheindividualemission
muchlowerthanatlinecentre,andwhichthenallowstheradi- fromtheNththread.EachsuccessivethreadisaddedtotheLOS
ationfromadditionalthreadsalongtheLOStobedetected.For behind the foremost thread. Many of the panels of Fig. 8 illus-
narroweropticallythicklines,onlytheforemostthreadscanbe tratethatsignificantpartsofthestructuralfeaturesshowninthe
detected. As shown in Fig. 7, the increased FWHM for Lyman bluecombinedviewfollowthesamepatternsasthoseshownby
αandHei584Åatlowpressuresexplainstheirapparentlower theforemostthreadgiveninpanel1.Mostrecognizablefeatures
saturation,however,athigherpressurestheirFWHMdecreases in the variations of the integrated intensity computed using the
andtheirintensityacrossthefieldofviewsaturates. unaligned multiple thread models come from the foremost and
The effect discussed here may form a spectral diagnostic closestthreadsintheLOS.
tool,asthedegreeofvariationoftheintegratedintensityinthe
field of view for optically thick lines gives an indication of the
4. Multi-threadconfigurationswithsmall-scale
numberofthreadsinthesystemifthegaspressureispreviously
velocitiesalongtheLOS
known. This is the case as long as the pressure values for all
threadsarenottoodifferent.Thepressurevariationacrosseach As discussed in Section 1, line profiles of optically thick tran-
threaddoesnothaveahugeimpactontheeffectdiscussedhere. sitions often display a substantial central reversal together with
Fromanobservationalpointofview,however,theusefulnessof asymmetries in the wings, where one intensity peak is stronger
this diagnostic is limited by the spatial resolution of the instru- thantheother.AsymmetricprofilesofthehydrogenLymanlines
ment. have been modelled by Gunár et al. (2008), who assumed the
asymmetries occur because of the presence of random peculiar
velocities between individual threads. That is to say that each
3.2.2. Effectofthenumberofthreadsonintensitiesin
threadinamulti-threadsystemismovingwithagivenvelocity
opticallythicklines
with respect to all the other threads. If a proportion of a thread
Asdiscussedintheprevioussection,thevariationofemittedin- velocityisdirectedintheLOS,theemergentspectrumfromthat
tensitiesofopticallythicklineswithverticaldirectioninamulti- threadisDopplershiftedbyanamountproportionaltothemag-
Articlenumber,page8of18
Labrosse&Rodger:Radiativetransferincylindricalthreadswithincidentradiation
50000 50000
1 2
40000 40000
sity30000 30000 LOS
n
e
nt20000 20000
I
10000 10000
0 0 1st 2nd 3rd Nth
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
60000 60000 SolarDisc
3 4
50000 50000
y40000 40000 Fig.9.LayoutofunalignedmultiplethreadswithrandomLOSveloci-
sit ties.Thedashedlinesrepresenttheupperandloweredgesofthefield
nten30000 30000 ofviewwiththeloweredgefacingthesolardisc.Thearrowswithinthe
I20000 20000 threadscorrespondtotheLOSvelocities,whilstthecolourofthethread
representsitsDopplershift,i.e.redshiftorblueshift.
10000 10000
0 0
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
ForLOSvelocitiestohaveanyeffectonthesymmetryofthe
70000 70000
5 6 emergent line profiles, a multi-thread system is required. The
60000 60000
reasonforthisisthatwhenconsideringathreadinisolation,any
50000 50000
y Doppler shift along the LOS simply moves the profile in either
nsit40000 40000 theblueorreddirection.Inamulti-threadsystem,photonsemit-
e
nt30000 30000 tedbyamovingthreadare“seen”byotherthreadsatadifferent
I
20000 20000 frequency, and therefore at a different optical thickness than if
10000 10000 they had been emitted by a thread at rest. This effect results in
0 0 the formation of significantly asymmetrical profiles from only
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
fairly small velocities, as demonstrated by Gunár et al. (2008).
70000 70000
7 8 The work from these authors shows that observed asymmetries
60000 60000
in the line profiles of the hydrogen Lyman lines can be repro-
50000 50000
ducedwithatwo-dimensional,multi-threadmodelinCartesian
y
nsit40000 40000 geometry.
e
nt30000 30000 Thissectionofthestudyaimstodescribehowtheconceptof
I
20000 20000 randomLOSvelocitiescanbeextendedtoworkinacylindrical
10000 10000 multi-thread model for both hydrogen and helium line transi-
0 0 tions.
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
80000 80000
70000 9 70000 10 4.1. AddingLOSvelocities
60000 60000
ensity4500000000 4500000000 Texhpeacnydleinddtroicainlcmluudltei-rthanredaodmmisoeddelp,edceuslciarirbevdelionciStieecstiaolnon3g,wthaes
Int30000 30000 LOS between each thread. The velocities described in this sec-
20000 20000 tion are specifically global velocities, which are experienced
10000 10000 identically across the entirety of each thread and are not local
0 0 flows within the structure. The velocity given for each thread
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
is constant across the whole cylinder. The shift in wavelength
Position Position
causedbytheresultingDopplershiftfromtheLOSvelocitiesis
Fig.8.Cumulativeeffectofunalignedthreadsonthetotalemergentin- ∆λ = λ u/c,where∆λ istheDopplershiftinwavelength,λ
i 0 i i 0
tensity for the Lyman α line. The solid blue line shows the emergent
isthecentralwavelengthforagiventransition,u isthevelocity
intensity after integration through N threads, whilst the dashed green component in the LOS direction for the ith threaid, and c is the
lineshowstheemergentintensityoftheN-ththreadintheLOSonits
speedoflight.
own. N is given by the number in the top right-hand corner of each
panel.Theintensityisgiveninergs−1cm−2sr−1 andthepositionisin Each thread in the multi-thread model was assigned a ran-
Mm.Thepositiondirectionisorientedverticallyawayfromthesolar domly produced LOS velocity in a ±10 km s−1 interval. These
surfacewiththezeropointatthecentreofthefieldofview.Theradia- velocitiesare,forthepurposesofthisstudy,adhocandnophys-
tionemergentfromeachthreadiscomputedwiththep4model. ical reason for them is given as of yet. Assuming that the only
change caused by the LOS velocities is the Doppler shift at-
tributedtoit,thestationaryvaluesfortheemergentintensityand
opticalthicknesschangeaccordingtothethread’sDopplershift,
nitude of the velocity in the LOS direction. LOS velocities in
prominence fine structures can occur as a result of several dif-
ferentmechanisms,includingflowsofplasmaalongthepromi- Ii(λ)→ Ii(λ−∆λi), (4)
nence’s magnetic field lines or by oscillations of the magnetic
and
structure itself (Gunár et al. 2008; Gouttebroze 2008; Heinzel
etal.2014). τ(λ)→τ(λ−∆λ). (5)
i i i
Articlenumber,page9of18
A&Aproofs:manuscriptno.26897
1e5 Lyman-Alpha 1e3 Lyman-Beta 5 5
5 3.0 1 2
4 4
2.5
4
y y
y y2.0 sit3 sit3
sit3 sit en en
en en1.5 nt2 nt2
nt2 nt I I
I I1.0 1 1
1
0.5
0 0
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3
0 0.0 ∆λ ∆λ
1.0 0.5 0.0 0.5 1.0 0.4 0.2 0.0 0.2 0.4
∆λ ∆λ 5 5
3 4
1e2 Lyman-Gamma 1e4 He I 584
9 1.6 4 4
8 1.4 y y
7 1.2 nsit3 nsit3
y6 y e e
nsit5 nsit01..80 Int2 Int2
te4 te 1 1
n n0.6
I3 I
2 0.4 00.3 0.2 0.1 0.0 0.1 0.2 0.3 00.3 0.2 0.1 0.0 0.1 0.2 0.3
1 0.2 ∆λ ∆λ
0 0.0 5 5
0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4
5 6
∆λ ∆λ
4 4
81e2 He I 537 3.01e4 He II 304 y y
7 2.5 ensit3 ensit3
6 nt2 nt2
y y2.0 I I
t5 t
si si 1 1
n4 n1.5
e e
t t
n3 n 0 0
I I1.0 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3
2 ∆λ ∆λ
0.5
1 5 5
7 8
0 0.0
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3 4 4
∆λ ∆λ
y y
sit3 sit3
Fig. 10. Asymmetrical line profiles of various optically thick hydro- n n
e e
gen and helium spectral lines from four unaligned threads with ran- nt2 nt2
domly generated velocities. The blue line shows the profile after in- I I
tegration through all four moving threads. The dashed red line shows 1 1
theprofileofasingular,stationarythread.Theintensityisgiveninerg
0 0
s−1cm−2sr−1Å−1.Thenumberatthetopleft-handsideofeachpanel 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3
∆λ ∆λ
is a multiplication factor for the intensity values. The horizontal axis
givesthedistancefromlinecentre∆λinÅ.Theintensitiesarespatially 5 5
9 10
averagedoverthefullfieldofviewdefinedinFig.9.
4 4
y y
sit3 sit3
n n
e e
nt2 nt2
I I
1 1
0 0
0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.3 0.2 0.1 0.0 0.1 0.2 0.3
TherandomisedLOSvelocitiesusedinSection4.2werespecif- ∆λ ∆λ
ically{u}={7,-7,9,-2,-7,-9,4,-3,4,-7}kms−1.Eachthread
i Fig. 11. Cumulative effect of randomly moving, randomly positioned
wasdisplacedbyarandomnumberofpositionarraypointsand
threads on the position-averaged emergent intensity for the Lyman α
thedisplacementsusedwere{0,-119,-40,120,9,-45,55,-86,
line in a multi-thread model. The solid blue line shows the emergent
37, 3}, where the total number of array points was 201 cover-
intensity after N threads. The number of threads is indicated at the
ing a thread of diameter specified by the model used, e.g. a p
top right of each panel. The dashed green line shows the line profile
modelwouldhavedisplacementvaluesof{0,-1190,-400,1200, of the (N +1)th thread in the LOS. The intensity is given in 105 erg
900, -450, 550, -860, 370, 30} km. For models with less than s−1cm−2sr−1Å−1.Thedistancefromlinecentre∆λisgiveninÅ.The
tenthreadsthesamevelocityanddisplacementarrayswereused radiationemergentfromeachthreadiscomputedwiththep4model.
withthevaluespastthenumberofthreadsignored.Aschematic
diagram of the multi-thread system with velocities in the LOS
directioncanbeseeninFig.9.Themodelusedthroughoutthis
sectionisthep4model(seeTable1).
Articlenumber,page10of18