Millimeter radiation from a 3D model of the solar atmosphere I. Diagnosing chromospheric thermal structure

2.3 MB·English

by  Maria Loukitcheva

Astronomy&Astrophysicsmanuscriptno.loukitcheva_arxiv (cid:13)cESO2015
January14,2015
Millimeter radiation from a 3D model of the solar atmosphere
I. Diagnosing chromospheric thermal structure
M.Loukitcheva1,2,S.K.Solanki1,3,M.Carlsson4,andS.M.White5
1 Max-Planck-InstitutforSonnensystemforschung,Justus-von-Liebig-Weg3,37077Göttingen,Germany
e-mail:[email protected]
2 AstronomicalInstitute,St.PetersburgUniversity,Universitetskiipr.28,
198504St.Petersburg,Russia
3 SchoolofSpaceResearch,KyungHeeUniversity,Yongin,Gyeonggi446-701,Korea
4 InstituteofTheoreticalAstrophysics,UniversityofOslo,P.O.Box1029,Blindern,N-0315Oslo,Norway
5 5 SpaceVehiclesDirectorate,AirForceResearchLaboratory,KirtlandAFB,NM,UnitedStates
1
0 Received;accepted
2

n
ABSTRACT
a
J
Aims.Weuseadvanced3DNLTEradiativemagnetohydrodynamicsimulationsofthesolaratmospheretocarryoutdetailedtestsof

3 chromosphericdiagnosticsatmillimeterandsubmillimeterwavelengths.
1 Methods.Wefocusedonthediagnosticsofthethermalstructureofthechromosphereinthewavelengthbandsfrom0.4mmupto
9.6mmthatcanbeaccessedwiththeAtacamaLargeMillimeter/SubmillimeterArray(ALMA)andinvestigatedhowthesediagnostics

] areaffectedbytheinstrumentalresolution.
R
Results.Wefindthattheformationheightrangeofthemillimeterradiationdependsonthelocationinthesimulationdomainandis
S relatedtotheunderlyingmagneticstructure.Nonetheless,thebrightnesstemperatureisareasonablemeasureofthegastemperature
. attheeffectiveformationheightatagivenlocationonthesolarsurface.Thereisconsiderablescatterinthisrelationship,butthisis
h
significantlyreducedwhenveryweakmagneticfieldsareavoided.Ourresultsindicatethatalthoughinstrumentalsmearingreduces
p
thecorrelationbetweenbrightnessandtemperature,millimeterbrightnesscanstillbeusedtoreliablydiagnoseelectrontemperature
o- uptoaresolutionof1(cid:48)(cid:48).Iftheresolutionismoredegraded,thenthevalueofthediagnosticdiminishesrapidly.
r Conclusions. We conclude that millimeter brightness can image the chromospheric thermal structure at the height at which the
t radiation is formed. Thus multiwavelength observations with ALMA with a narrow step in wavelength should provide sufficient
s
a informationforatomographicimagingofthechromosphere.
[
Keywords. Sun:atmosphere–Sun:chromosphere–Sun:radioradiation–Sun:magneticfields

1
v
81. Introduction ter and millimeter wavelengths (Loukitcheva et al. 2004). The
9 intensities of submm/mm continua, whose source function can
8
As a result of its particularly complicated physics, the chro- betreatedinLTE(Rutten 2003),dependlinearlyontemperature
2
mosphere remains the least understood layer of the solar atmo- andthereforemaybeabletoprovideasensitivetestofnumeri-
0sphere.Thepastdecadeofmodelingeffortshasenormouslyad- calmodels.Theadditionaladvantageofsubmm/mmcontinuais
.
1vanced our understanding of this enigmatic layer, however. In thattheirintensitiescanbeeasilysynthesizedfromthemodels.
0recentyears,3Dmodelingofthechromosphere,includingmuch Loukitchevaetal. (2004)comparedalargecollectionofsub-
5ofthenecessaryphysics,hasfinallystartedtobecomefeasible. millimeterandmillimeterobservationswithsyntheticbrightness
1
Advanced3Dmagneto-hydrodynamicmodelsrunfromtheup- temperaturescalculatedinclassicalstandardmodelsbyFontenla
:
vper convection zone through the chromosphere into the lower et al. (1993) and the 1D dynamic model of Carlsson & Stein
icorona(Gudiksenetal. 2011)andtakeintoaccountphysicalef- (1995)anddemonstratedthatboththeclassicalandthedynamic
X
fectssuchasmagneticfielddynamics,thermalconduction,am- modelprovideareasonablefittoobservedtemporallyandspa-
arbipolardiffusion,andtheHalleffect,non-equilibriumionization, tiallyaveragedmillimeterdata.Theanalysisofthe1Ddynamic
andNLTEeffects(see,e.g.,Carlsson 2009,forareview).These simulations of Carlsson and Stein further revealed that radio
simulations provide the opportunity to model the formation of emissionatmillimeterwavelengthsisextremelysensitivetody-
variouschromosphericlinesandcontinuaandtostudythediag- namic processes in the chromosphere if these are spatially and
nosticpotentialofchromosphericdata(Leenaartsetal. 2013a,b; temporallyresolved(Loukitchevaetal. 2004,2006).
Pereiraetal. 2013;Stepanetal. 2012;Leenaartsetal. 2012).
Wedemeyer-Böhm et al. (2007) were the first to study
Chromospheric diagnostics, such as Ca ii H&K, Hα, Ca ii submm/mmbrightnesssynthesizedfrom3Dradiation(magneto-
854.2 nm, and Mg ii h&k lines, suffer from the fact that they )hydrodynamic simulations, using those of Wedemeyer et al.
are formed out of local thermodynamic equilibrium (LTE) and (2004) and Leenaarts & Wedemeyer-Böhm (2006). These 3D
thus decouple from the local conditions. In addition, their for- (M)HDmodelswereconstructedundertheassumptionsofLTE
mation is complex. An alternative chromospheric diagnostic is for the radiative transfer and the equation of state to make
providedbyobservationsoftheradiocontinuumatsubmillime- thecomputationsfeasible.Inaddition,Wedemeyer-Böhmetal.
Articlenumber,page1of12page.12
A&Aproofs:manuscriptno.loukitcheva_arxiv
Fig.1.Horizontalcutsthrougha3DMHDsimulationsnapshot.Plottedaretheelectrontemperature(toppanel),electronnumberdensity(middle
panel),andmagneticfieldstrength(bottom)atgeometricalheightsof650,900,1500,1700,and2000km.Toenhancethetemperaturecontrast,
thedisplayedtemperaturerangeissettothatofthesyntheticbrightnessmapsshowninFig.2.Therangeoftemperaturesisgiveninthelower
rightcorneroftheframes.Thefieldsizeis24Mmx24Mm.
(2007)studiedsubmm/mmbrightnessfromrepresentativemodel with the actual temperature distributions to assess the accuracy
snapshots with non-equilibrium electron densities and a weak ofthemm-wavelengthdiagnostics.
magnetic field. The authors reported mm brightness maps with Thestructureofthepaperisasfollows:InSects.2and3we
filamentarybrightenings,resultingfromshock-inducedthermal describethe3DmodelatmospherebasedontheBifrostcodeand
structure,andfainterregionsinbetween.Theaveragebrightness thesubmm/mmradiativetransfercomputations.Theanalysisof
temperature and relative contrast were found to increase with thesyntheticbrightnessispresentedanddiscussedinSect.4.In
wavelength. thissectionwealsoinvestigatetheeffectofspatialsmearingof
themodelbrightnessinthelightoffutureinterferometricobser-
Thispaperisthefirstinaseriesinwhichweuseadvanced3D vations with ALMA. We summarize our findings and conclude
radiative magnetohydrodynamic simulations of the solar atmo- inSect.5.
sphereperformedwiththeBifrostcode(Gudiksenetal. 2011)
to carry out detailed tests of chromospheric diagnostics at mil-
limeter and submillimeter wavelengths. In this paper we focus
2. 3DModelatmosphere
onthediagnosticsofthechromosphericthermalstructureinthe
wavelength bands accessible to the advanced radio interferom- We studied the formation of millimeter and submillimeter con-
eter Atacama Large Millimeter/Submillimeter Array (ALMA). tinuainasnapshotofa3Dradiation-MHDsimulationperformed
Located on the Chilean Chajnantor plateau above 5000 m alti- withtheBifrostcode(Gudiksenetal. 2011).Weusedsnapshot
tude,ALMAconsistsofagiantarrayof12mantennas(the12 385ofthesimulation“en024048_hion”madeavailablethrough
m array) with baselines of up to 16 km, which aims at achiev- the Interface Region Imaging Spectrograph (IRIS, De Pontieu
ing high spatial resolution, and an additional compact array of et al. 2014) project at the Hinode Science Data Centre Europe
7mand12mantennasthatgreatlyenhanceALMA’sabilityto (http://sdc.uio.no).ThesamesnapshotwasusedbyLeenaartset
imageextendedtargets.Atthetimeofwriting,ALMAobserves al. (2012)toinvestigatetheHαlineformationandbydelaCruz
atwavelengthsintherange3.6mmto0.4mm(84to720GHz) Rodriguezetal. (2013)tostudyCaii8542spectra.Itwasalso
indualpolarizationmodewithprospectsofgoingtolonger(up used by Leenaarts et al. (2013a) and other papers in that se-
to 10 mm) and shorter (down to 0.3 mm) wavelengths in the ries to investigate the formation of the Mg ii h and k lines in
future.Wesimulatethemm/submmbrightnessatALMAwave- the solar atmosphere. Bifrost solves the equations of resistive
lengthsfromthe3DMHDsimulationsandcomparetheresults MHD on a staggered Cartesian grid. The simulation includes
Articlenumber,page2of12page.12
M.Loukitchevaetal.:MillimeterRadiationfroma3DModel
optically thick radiative transfer in the photosphere and low (Zlotnik 1968)
chromosphere,parameterizedradiativelossesintheupperchro-
χ0
mosphere, transition region, and corona (Carlsson & Leenaarts χe,o (cid:39) (cid:113) e,o , (1)
2012),thermalconductionalongmagneticfieldlines(Gudiksen 1∓ ωB|cosα|
etal. 2011),andanequationofstatethatincludestheeffectsof ω
non-equilibriumionizationofhydrogen(Leenaartsetal. 2007).
where χ0 characterizes absorption (χ or χ ) in the absence
The simulation covers a physical extent of 24 x 24 x 16.8 e,o ep eH
of the magnetic field, ω is the observing frequency, ω is the
Mm, with a grid of 504 x 504 x 496 cells, extending from B
gyrofrequency, α is the angle between the magnetic field and
2.4 Mm below the average height of τ = 1, which corre-
500 the line of sight, the minus sign corresponds to the e-mode,
sponds to Z = 0, to 14.4 Mm above, so that it covers the up-
and the plus sign to the o-mode. For a weak magnetic field
perconvectionzone,photosphere,chromosphere,andthelower (cid:113)
corona.Thehorizontalaxeshaveanequidistantgridspacingof (| ωB cosα|(cid:28)1)thequasilongitudinalapproximationisvalid
ω
48 km (0.06(cid:48)(cid:48)), the vertical grid spacing is nonuniform, with a for all angles between the magnetic field and the line of sight,
spacing of 19 km between Z = −1 Mm and Z = 5 Mm. The except for narrow intervals of angles close to transverse propa-
spacing increases toward the bottom and top of the computa- gation(Zheleznyakov 1996).
tionaldomaintoamaximumof98km.Thesimulationcontains We considered the radiative transfer equation in terms of
amagneticfieldwithanaverageunsignedstrengthof50Ginthe brightnesstemperatureTe,oseparatelyforthee-ando-mode,
b
photosphere,concentratedinthephotosphereintwoclustersof
oppositepolaritylying8Mmapart,representingtwopatchesof dTe,o
b =T −Te,o, (2)
quiet-Sunnetwork.Werefertothesnapshotfromthissimulation dτ e b
weanalyzedsimplyasthe3DMHDsnapshot.Formoredetails
whereT = T (l)istheprofileofthekinetictemperaturealong
ofthInisFsinga.p1shwoetwsheorwefeexratomCplaerslsosfonhoertizaol.nt(a2l0s1l3ic,e2s0t1h5ro).ughthe the lightepath,eτ(l) = (cid:82)l0lχe,o(l)dl is the optical depth, l is geo-
3D MHD snapshot of electron temperature (upper panel), elec- metricaldistancealongthelightpath,andχ isthecorrespond-
e,o
tronnumberdensity(middlepanel),andmagneticfieldstrength ingabsorptioncoefficient.Thesolutionoftheradiativetransfer
(bottom) taken at the geometrical heights Z = 650, 900, 1500, equation on a discrete grid can be expressed in terms of T as
e
1700, and 2000 km. These heights very roughly correspond to (Hagen 1951)
the effective formation heights of the radiation at 0.4, 1.0, 3.0,
4n.e5ti,canpdat1ch0emsmth,eretesmpepcetirvaetulyre(sseheowSescfit.la4m.2e).nBtaerytwseterunctthueremtahga-t Tbe,o =(cid:88)n (1−exp−χre,o∆h)Terexp−(cid:80)rs−=11χs∆h, (3)
r=1
followsthegeneralorientationofthemagneticfield.Ahotchro-
mosphericcanopyliesaboveandbetweenthephotosphericmag- where χre,o stands for χe or χo at position r, r = 1 corresponds
neticfieldconcentrations,andcoronaltemperaturesarereached tothelayeratthetopoftheatmosphere,∆histhegriddistance.
already at heights of around 1700 km. The electron density The items within the sum represent the contribution of various
displays a similarly complex pattern, with a clear anticorrela- layerstotheemergingradiationintensity.Werefertothemasthe
tion with the temperature above magnetic patches at heights valuesofthe(unnormalized)contributionfunction(CF)toTe,o.
b
above1700km.Themagneticfieldconcentrationsexpandwith ThetotalbrightnessTb andthedegreeofcircularpolarizationP
heightfromtheirphotosphericfootpoints(withamaximumfield atwavelengthλaredefinedas
strength of 2200 G) to a volume-filling field higher up with a Te+To Te−To
maximumstrengthof60Gatageometricalheightof2000km Tλ = b b, P= b b. (4)
b 2 Te+To
(seealsoFig.5). b b
Hereafter we discuss the total brightness at millimeter wave-
lengths as a primary diagnostics of the chromospheric thermal
3. Synthesisofmmbrightness
structure, leaving the discussion of the circular polarization for
Wecalculatedthesubmillimeterandmillimeterradiationemerg- thenextpaperoftheseries,whichwillbeonthesubjectofthe
ingfromthe3DMHDsnapshotat32wavelengthsfrom0.4mm chromosphericmagneticfield.
to10mm.Inthecalculationsweassumedeachverticalcolumn
inthe3Dsnapshottobeanindependent1Dplane-parallelatmo-
4. Results
sphere.Theradiativetransferwascomputedundertheassump-
tionthatbremsstrahlungopacityisresponsibleforthemmcon- 4.1. Distributionofbrightnesstemperature
tinuumradiationinthequietSun.Wedistinguishedtwosources
In Fig. 2 we show the results of the brightness calculations for
ofopacity:opacityduetoencountersbetweenfreeelectronsand
0.4mm(ALMAband9),1mm(band7),3mm(band3),4.5mm
protons, and opacity due to encounters between free electrons
and neutral hydrogen. They are commonly referred to as H0 (band 2) and 10 mm (band 1). The brightness temperature at
opacity and H− opacity. The corresponding absorption coeffi- all considered wavelengths exhibits a complex pattern of inter-
cients depend on the effective number of collisions undergone mittent bright and dark regions (top row of Fig. 2) similar to
thoseseenintheelectrontemperatureanddensityhorizontalcuts
by an electron per unit time, including collisions with protons
throughthemodelatmosphere(seeFig.1).Inthecentralpartof
(χep ∝ TN3e/N2νp2)andwithneutralhydrogenatoms(χeH ∝ NeNνH2Te1/2). the field of view the most prominent features are bright elon-
e
Expressions for calculating the radiative transfer were taken gated fibrils that extend outward from the network patches and
fromZheleznyakov (1996)andcanalsobefoundinLoukitcheva arealignedalongthechromosphericmagneticfieldlines.Atthe
etal. (2004).Inthepresenceofamagneticfieldinquasilongi- shortest wavelengths the imprint of the central loop-like mag-
tudinalapproximation,theabsorptioncoefficientsforextraordi- netically formed structures is weak, but it becomes more pro-
nary (e) and ordinary (o) modes are determined by the relation nouncedtowardlongerwavelengths.
Articlenumber,page3of12page.12
A&Aproofs:manuscriptno.loukitcheva_arxiv
Fig. 2. Maps of simulated mm brightness at the resolution of the model for the wavelengths 0.4, 1.0, 3.0, 4.5, and 10 mm (labeled in the top
row).ThehorizontalwhitelineinthetoprowmarksthepositionoftheverticalcutatY=0.9MmdisplayedinFig.5.Wealsoplottheeffective
formationheightsofthemmradiation(bottomrow)andthecorrespondingelectrontemperatures,electronnumberdensities,andmagneticfield
strengthtakenattheeffectiveformationheights(threemiddlerows).Thevaluesofthemostextreme1%ofpointsareclippedinthebrightness
maps.Thedisplayedtemperaturerangeforelectrontemperaturesissettothatofthemmbrightnessmaps.Therangeofcoveredtemperaturesis
givennearthebottomoftheframes.Thefieldsizeis24Mmx24Mm.
Toquantifythebrightnessdistributionsatmmwavelengths, and11,whicharediscussedindetaillater).TheRMSvariation
welistinTable1somestatisticsincludingthelowestandhighest Trms also steadily increases with λ. A more complex behavior
b
brightnesstemperature,averagebrightnesstemperature< T >, is displayed by the relative brightness contrast, which initially
b
RMSvariationTrms,andtherelativebrightnesstemperaturecon- increases with wavelength, reaches a value of 0.22 at around
b
Trms λ=3 mm, and stays almost constant at longer wavelengths. In
trast b for a number of analyzed wavelengths. The average
bright<nTebs>stemperatureaswellasthehighestbrightnessincrease Fig. 3 we plot average brightness temperatures < Tb >, with
Trms as error bars, overlaid on the observed millimeter bright-
withwavelengthasaresultoftheshiftingcontributingheightsto b
ness spectra from the compilation made by Loukitcheva et al.
higher(hotter)layersintheatmosphere(seeFigs.6,7,8,9,10,
Articlenumber,page4of12page.12
M.Loukitchevaetal.:MillimeterRadiationfroma3DModel
Table1.Lowestandhighestbrightnesstemperature,averagebrightness
temperature< T >,RMSvariationTrms,andrelativebrightnesstem-
b b
Trms
peraturecontrast b foranumberofanalyzedwavelengths.
<Tb> contributing heights
a
λ,mm Tbmin,K Tbmax,K <Tb >,K Tbrms,K <TTbrmb>s 1•106
0.4 3336 8446 4507 503 0.11 8•105
1.0 2706 11094 4786 823 0.17 ce
n
33..06 23914051 1133120247 66039904 11343032 00..2222 curre 6•105
c
4.5 3054 13637 6786 1485 0.22 o 4•105
10.0 2924 20082 8615 1788 0.21
2•105
0
(2004). Gratifyingly, < T > calculated from the snapshot of 0 700 1400 2100 2800 3500
b
height (km)
the3DMHDmodel(redsquaresinFig.3)areconsistentwithin
theerrorbarswiththeavailableobservationsofmmbrightness.
effective heights
The only small discrepancy occurs at 1 mm, where, in spite of
some overlap, the simulated T appears to lie somewhat below 1•105 b
b
theobservationaldata.Longwardof3mmthesimulatedvalues
aresignificantlyhigherthanthosereportedinWedemeyer-Böhm 8•104
etal. (2007). e
nc 6•104
e
urr
c
oc 4•104
12
2•104
K)
310 0
e ( 10 0 700 1400 2100 2800 3500
ur height (km)
at
er
p
m 8
Te Fig. 4. (a) Histograms of the heights contributing more than 1% of
ss thetotalcontributiontotheemergingintensityat0.4mm(dot-dashed),
e
htn 6 1mm(dashed),3.6mm(solid),and10mm(dotted).(b)Thesamefor
Brig theeffectiveformationheights.
4
enhancedtemperature,wenormallyseedowntolowerchromo-
sphericheights(e.g.,1200-1500kmat3mm),whileinloop-like
0.1 1.0 10.0
wavelength (mm) structures that connect the magnetic concentrations, we sample
higher layers (1500-2000 km at 3 mm). Details on the forma-
Fig. 3. Simulated millimeter brightness (red squares) overlaid on the tion heights of individual brightness features can be found in
observedbrightnesstemperaturesfromLoukitchevaetal. (2004).Data Sects.4.3and 4.4.
obtained around solar cycle minimum, solar maximum, and at inter- Tocheckhowrepresentativeofthetrueformationlayersthe
mediate phases are depicted by filled circles, open circles, and stars, effective height is, we plot in the second top row of Fig. 2 the
respectively. electrontemperatureT takenattheeffectiveformationheights.
e
Forcompleteness,weshowinthethirdandfourthrowsofFig.2
the same for the electron number density n and the magnetic
e
field strength B. Qualitatively, the properties of the atmosphere
4.2. Effectiveformationheights
takenattheeffectiveformationheights(Fig.2)lookquitesimilar
The bottom row of Fig. 2 shows maps of the effective forma- totheatmosphericpropertiesofthelayerscorrespondingtothe
tionheightforfivewavelengths.Wedefinetheeffectiveforma- averageeffectiveheights(Fig.1),butthereisasignificantdiffer-
tion height as the height corresponding to the centroid of the ence in the range of values, specifically for the magnetic field.
contributionfunction.At0.4mmtheradiationoriginatesmainly FromthecomparisonofthetoprowofFig.1,whichshowsT at
e
in layers close to the traditional temperature minimum region theaverageformationheightsofmmradiationinthe3DMHD
and lower chromosphere with a few, very localized excursions snapshot, with the two upper rows of Fig. 2, which depict syn-
toheightsabove1000km.Theaverageformationheightofthis theticT andT attheeffectiveformationheights,weconclude
b e
wavelengthisaround650km.Withincreasingwavelengths,we thatingeneral,mmbrightnesscanserveasameasureoftemper-
clearlysamplehigherchromosphericlayers.Effectiveformation ature in the solar atmosphere, albeit not a perfect measure. We
heights,averagedoverallspatiallocations,reachvaluesofabout report the results of a correlation analysis that provides a more
900 km for 1 mm, 1500 km for 3 mm, 1700 km for 4.5 mm, quantitativecomparisoninSect.4.5.
and around 2000 km for 10 mm. In the expanding magnetic Statistical distributions of the heights that contribute to the
patchesthatoverliephotosphericmagneticfootpointsandshow emerging intensity are presented in Fig. 4 in the form of his-
Articlenumber,page5of12page.12
A&Aproofs:manuscriptno.loukitcheva_arxiv
largerextent, however,they aredefinedby thetemperature and
X-profile at Y=270 numberdensitystructurealongthelineofsight(inparticular,the
3000 a stronglocalgradientsoftemperatureanddensitythatinfluence
2500 3.4 3.7 3.9 4.2 4.5 4.7 5.0
opacity peaks and therefore the form of the contribution func-
m) 2000
ht (k 1500 tion)andthusmixcontributionsfrombothhighandlowlayers.
g
ei 1000
h
500
0
3000
b
2500 8.7 9.9 11.112.413.614.816.0
m) 2000
ht (k 1500 2.0 0.4 mm a
g
hei 1000 m) 1.5 1 2 3 4 5
500
M
30000 c ght ( 1.0
2500 -0.4 0.2 0.8 1.5 2.1 2.7 3.3 ei
h
m) 2000
ht (k 1500 0.5
g
hei 1000 2.5 1.0 mm b
500
0 2.0
m)
3ness (10 K) 1105 d height (M 11..05
ht
brig 5 0.5
-10 -5 0 5 10
x-location (megameters) 3.0 3.6 mm c
Fig.5.XZ-profileofthe3DMHDsnapshotalongthecutatY=0.9Mm, 2.5
markedbyahorizontallineinFig.2.(a)Electrontemperatureonalog- m)
M 2.0
(abri)thEmleicctrsocnalenucmlipbpeerddaetn1si0ty5Kontoaelnohgaanricthemthieccsocnaltera.s(tca)tMloawgenrehtiecigfihetlsd. ght ( 1.5
strengthonalogarithmicscale.Overplotteddot-dashed,dashed,solid, hei
anddottedlinesinpanels(a)-(c)depicttheeffectiveformationheights 1.0
fortheradiationat0.4mm,1.0mm,3.6mm,and10mm,respectively. 0.5
(d) Brightness temperature at the wavelengths 0.4 mm (dot-dashed), 3.5 10 mm d
1.0 mm (dashed), 3.6 mm (solid), and 10 mm (dotted). In panels (a)-
3.0
(c)they-axisscaleisexpanded.Inallpanelsverticaldashedlinesmark m)
thelocationsofthedifferentstructurespresentedinFigs.7,8,9,10,and M 2.5
11. ght ( 2.0
ei
h
1.5
tograms forfour ofthe analyzedwavelengths togetherwith the
distributions of the effective formation heights for comparison. 1.0
-10 -5 0 5 10
The corresponding histogram peaks in the two panels coincide X-location (Mm)
withanaccuracyof50kmforallwavelengthsexceptforthatat
1mm.At1mm,thediscrepancybetweenthepeaksinthecon-
Fig. 6. Gray-scale figures of the contribution function along the cut
tributing and effective height histograms reaches 160 km, and atY= 0.9displayedinFig.5.Darkcolorcorrespondstohigheram-
the histograms differ in form, which suggests that statistically plitudes.Contributionfunctionsineachcolumnarenormalizedtothe
weneedtobemorecarefulat1mmininterpretingtheeffective highestvalueofthecolumn.Heightswithcontributionfunctionampli-
formationheights. tudesequalto10%ofthehighestcontributiontotheemergingintensity
aremarkedwithbluecontours.Redsolidlinesrepresenteffectivefor-
mationheights.Thevariouspanelsshowresultsfor0.4mm(a),1mm
4.3. Analysisofaverticalslicethroughthesnapshot (b), 3.6 mm (c), and 10 mm (d). In all panels the y-axis scale is ex-
panded.ThedashedlinesarethesameasinFig.5.
InFig.5weprovideanexampleofanXZ-slicethroughtheat-
mospherealongthecutatY=0.9Mmmarkedinthetoprowof
Fig. 2 by the horizontal white line. The panels (a-c) of Fig. 5 Awayfromtheregionswithastrongmagneticfield,theef-
displaytheelectrontemperatureTe,theelectronnumberdensity fective formation height tends to be lower than in the vicinity
ne ,andmagneticfieldstrength Balongthecut,withtheeffec- ofmagneticfieldconcentrations(seeFig.5c),althoughthereare
tiveformationheightsatfourselectedwavelengthsoverplotted. exceptions.Directlyabovethefieldconcentrationsweseearela-
Thebottompanel(d)showsthesynthesizedbrightnessalongthe tiveshiftoftheiso-Kelvintemperaturecurves(andconsequently
cutatλ=0.4mm(dot-dashed),1.0mm(dashed),3.6mm(solid), oftheeffectiveformationheights)tohigherlayersthanthesur-
and10mm(dotted). rounding locations. Synthesized brightness at mm wavelengths
Tosomeextent,theeffectiveformationheightsofmmradia- does not necessarily follow the effective formation height pro-
tionreplicatethedistributionofthelocalspatialinhomogeneities file,butthereisatendencyforlocalbrighteningsabovethefield
in electron temperature and number density along the cut. To a concentrations(seecurvesofthesametypeinFig.5a-cand5d).
Articlenumber,page6of12page.12
M.Loukitchevaetal.:MillimeterRadiationfroma3DModel
snapshot 385 x=95 y=270 contribution function mm brightness
10000 107 1013 1.6•104
1.0
0.4 mm
1.4•104
1 mm
1000 106 1012 0.8 3.6 mm
10 mm 1.2•104
K)
B (G) 100 T (K)e105 1011-3N (cm)e CF 00..46 Brightness ( 81..00••110034
10 104 1010 6.0•103
0.2
4.0•103
1 103 109 0.0
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.1 1.0 10.0
h (km) h (km) Wavelength (mm)
Fig.7.Spatiallocation(x,y)=(-8.1,0.9)Mm,marked”1”inFig.6.(a)Electrontemperature(solidcurve),electronnumberdensity(dashedcurve),
andmagneticfieldstrength(dotted)asafunctionofheight.(b)Contributionfunctionstoemergingintensity(normalizedtotheirmaximum)as
labeledinthefigure.Thehorizontallinemarks10%ofthehighestvalueforcomparisonwithFig.6.(c)Resultingbrightnessspectrum.Starsmark
thefourwavelengthsanalyzedindetail.
snapshot 385 x=142 y=270 contribution function mm brightness
10000 107 1013 1.6•104
1.0
1.4•104
1000 106 1012 0.8
1.2•104
K)
B (G) 100 T (K)e105 1011-3N (cm)e CF 00..46 Brightness ( 81..00••110034
10 104 1010 6.0•103
0.2
4.0•103
1 103 109 0.0
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.1 1.0 10.0
h (km) h (km) Wavelength (mm)
Fig.8.SameasinFig.7forthespatiallocation(-5.2,0.9)Mm,marked”2”inFig.6.
snapshot 385 x=266 y=270 contribution function mm brightness
10000 107 1013 1.6•104
1.0
1.4•104
1000 106 1012 0.8
1.2•104
K)
B (G) 100 T (K)e105 1011-3N (cm)e CF 00..46 Brightness ( 81..00••110034
10 104 1010 6.0•103
0.2
4.0•103
1 103 109 0.0
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.1 1.0 10.0
h (km) h (km) Wavelength (mm)
Fig.9.SameasinFig.7forthespatiallocation(0.7,0.9)Mm,marked”3”inFig.6.
InFig.6weshowtherangeofgeometricalheightsthatcon- locations the contribution achieves significant values over two
tributetotheemergingintensityasafunctionoflocationalong distinctheightrangeswithdouble-peakedcontributionfunctions
the vertical slice from Fig. 5. The heights marked in Fig. 6 (see,e.g.,λ=1mminFig.8andλ=3.6mminFigs.9and10).
with blue contours at a level of 10% of the CF maximum can
serve as a measure of the height range sampled by each of the
four analyzed wavelengths. For comparison, we plot the corre- 4.4. Analysisofindividualprofiles
spondingeffectiveformationheight(redsolidline)ateachwave-
In this section, we analyze the formation of the mm brightness
length.Owingtothecomplexthermalstructure,abroadheight
spectraatfivelocationsalongtheXZ-slicethatrepresentdiffer-
range is involved in emitting radiation at all considered wave-
ent features in the solar atmosphere and display a broad range
lengths, resulting in quite complicated contribution functions
ofbehavior.InFig.5theselocationsaremarkedbydashedver-
(see Sect. 4.4 for details). The figure demonstrates the limita-
tionsofusingeffectiveheightstoassignthesynthesizedbright- tical lines, and in the top panel of Fig. 6 they are additionally
labeled with numbers from 1 to 5. The properties of the spa-
nesstothethermalstructureoftheatmosphere.Atanumberof
tiallyresolvedatmospheres(electrontemperature,electronnum-
Articlenumber,page7of12page.12
A&Aproofs:manuscriptno.loukitcheva_arxiv
snapshot 385 x=450 y=270 contribution function mm brightness
10000 107 1013 1.6•104
1.0
1.4•104
1000 106 1012 0.8
1.2•104
K)
B (G) 100 T (K)e105 1011-3N (cm)e CF 00..46 Brightness ( 81..00••110034
10 104 1010 6.0•103
0.2
4.0•103
1 103 109 0.0
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.1 1.0 10.0
h (km) h (km) Wavelength (mm)
Fig.10.SameasinFig.7forthespatiallocation(9.5,0.9)Mm,marked”4”inFig.6.
snapshot 385 x=465 y=270 contribution function mm brightness
10000 107 1013 1.6•104
1.0
1.4•104
1000 106 1012 0.8
1.2•104
K)
B (G) 100 T (K)e105 1011-3N (cm)e CF 00..46 Brightness ( 81..00••110034
10 104 1010 6.0•103
0.2
4.0•103
1 103 109 0.0
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0.1 1.0 10.0
h (km) h (km) Wavelength (mm)
Fig.11.SameasinFig.7forthespatiallocation(10.2,0.9)Mm,marked”5”inFig.6.
ber density, magnetic field strength) as a function of geometri- height of the temperature minimum with a secondary peak
calheight,contributionfunctionsatfourwavelengthsandcorre- in the low chromosphere, whereas at 1 mm the main peak
sponding brightness spectra are shown in Figs. 7, 8, 9, 10, and moves to chromospheric heights, but a small contribution
11,forfeatures1to5.Notethatforbettercomparisonthescales from the temperature minimum region remains. At longer
remainunchangedfromonefiguretothenext. wavelengths the contribution heights move higher into the
chromosphere,thecontributionfunctionsaresharplydefined
– Thespatiallocation(x,y) = (−8.1,0.9)Mm(marked"1"in with a single peak close to the boundary between the chro-
Fig.6)isdetailedinFig.7:Atthesolarsurfacethelocation mosphereandtheTR.Theresultingmmbrightnessesaresig-
corresponds to an intergranular feature with a rather weak nificantlyhigherthantheaveragevaluesgiveninTable1and
magneticfield(<100G).Itstemperaturedistributionischar- increasealmostlinearlywithwavelength.
acterizedbyalow-lyingtransitionregion(at1500km).The – Spatial location (x,y) = (0.7,0.9) Mm (Fig. 9, marked ”3”
electron density manifests a localized increase (by an order in Fig. 6): This location corresponds to the central part of
ofmagnitude)ataround1400km.Thecontributionfunctions theloopconnectingthetwomainfieldconcentrationsandis
at0.4mmand1mmareverysharplydefined,andthecon- characterized by complicated temperature and density pro-
tributing formation heights are in the range 600−700 km, filesshowinganumberoflocalenhancementsalongtheline
where temperature decreases with height, leading to a tem- of sight. This leads to a compound contribution function at
peratureminimumataround800km.Atλ=3.6mmthecon- 3.6mm.Thebrightnessspectraachieveaveragevalues.
tribution function is double peaked, with the main peak at – Spatiallocation(x,y) = (9.5,0.9)Mm(Fig.10,marked”4”
lower height covering the extended temperature minimum in Fig. 6): This is a peculiar location at the boundary be-
region, and the narrow, weak secondary peak in the chro- tweenanintergranularlaneandagranule.Itischaracterized
mosphereandlowertransitionregion.Asaconsequence,the byabubbleofveryhot(basicallycoronaltemperature),low-
synthesizedbrightnessesareextremelylow(around4000K) density gas at a height of about 2000 km, somewhat like a
andbrightnesstemperaturesat1mmand3.6mmarelower piece of corona, separated from the main corona. This fea-
than that at 0.4 mm. Radiation at 10 mm is formed over a ture, due to low electron density, produces low opacity and
rangeofheightswithasharpandclearmaximumcontribu- doesnotcontributetotheemergingintensity.Asaresult,the
tion from around 1450 km. The resulting brightnesses are contribution function at 3.6 mm has peaks on either side of
alsosignificantlylowerthantheaveragevaluesforthiswave- this abrupt temperature enhancement. Radiation at 0.4 mm
length. and 1 mm forms below these layers and is not affected by
– Spatiallocation(x,y) = (−5.2,0.9)Mm(Fig.8,marked”2” thisinhomogeneity,whileradiationat10mmisformedonly
in Fig. 6): Here B reaches 1900 G in the photospheric in- above it. The brightness at 0.4 mm and 1 mm is typical for
tergranularlane.Atshortwavelengthsthecontributionfunc- thesewavelengths.TheT at3.6mmand10mmexceedsthe
b
tions are double peaked, at 0.4 mm the main peak is at the averagevalues.
Articlenumber,page8of12page.12
M.Loukitchevaetal.:MillimeterRadiationfroma3DModel
– Spatial location (x,y) = (10.2,0.9) Mm (Fig. 11, marked InFig.12wecorrelatethesyntheticmmbrightnesswiththe
”5”inFig.6):Thisisanexampleoftheatmosphereabovea modeltemperaturetakenattheheightscorrespondingtotheef-
granule.Asteeptransitionregionat2000kmheightresults fectiveformationheightsatwavelengths0.4mm,1mm,3mm,
in a narrow contribution function for 10 mm and in an ex- and 4.5 mm. Because of the many pixels in the snapshot, we
tremelylowsynthesizedbrightness.Theradiationatshorter showthecorrelationsbybinningthedatainto2Dhistograms.In
wavelengthsreceivesitsmaincontributionfromtheheights thedensityscatterplotswedistinguishbetweenaweakmagnetic
wherethetemperatureisbelow5000K,andasexpected,it field with |B| < 30 G at the effective formation height (darker
also shows very low brightness. Both this location and the graycontoursintheprintedversionandbluecontoursintheon-
previousonedisplaysignsofamagneticcanopywithabase line version), and an enhanced magnetic field with |B| ≥ 30 G
ataround500-700km. (light gray contours in the printed version, red contours in the
onlineversion). Thescatter seeninFig. 12illustratesthe influ-
enceof theextendedformationheight ranges,whichinstead of
sampling the temperature at a fixed geometrical height, effec-
0.4 mm 1 mm
10 12 tivelymixescontributionsfromdifferentlayers.
ForregionswithlargeB,thelargestdifferencesbetweenT
10 gas
8 and T occur at the high and low end of the temperature range
b
K) K) 8 sampled at each λ. In the cool part of the range mm brightness
Tb (k 6 Tb (k 6 tendstooverestimateTgas,whilewhenTgasishighthesynthetic
4 brightnessunderestimatesit.Forweakmagneticfeaturesawider
4
”scatterrectangle”aroundtheT =T curveinthescatterplots
2 2 gas b
2 4 6 8 10 2 4 6 8 10 12 is seen. When confining the analysis to fields with |B| ≥ 30 G,
Tgas (kK) Tgas (kK) thescatterbecomesmuchsmallerandmmbrightnessreproduces
3 mm 4.5 mm temperature at different chromospheric heights relatively well.
14 14 Theagreement for |B| ≥ 30G isbetter forlongerwavelengths,
12 12 whichhastworeasons:applyinga30Gmagneticfieldstrength
K) 10 K) 10 thresholdattheformationheightmeansthatweconsiderfewer
b (k 8 b (k 8 regionsatlongerλ,sinceonaverageBdecreaseswithheight.In
T 6 T 6 themorestronglymagnetizedpartsoftheatmospherecontribu-
4 4 tionfunctionsatlongerwavelengthshowasimpleformwithone
2 2 peak,seeFig.8,sothattheeffectiveformationheightprovidesa
2 4 6 8 10 12 14 2 4 6 8 10 12 14
goodmeasureoftherealformationheight.
Tgas (kK) Tgas (kK)
The relation between brightness temperature and atmo-
spheric temperature can also be seen in Fig. 13, where we
Fig. 12. Density scatter plots (2D histograms) of the brightness tem- show brightness at 3.6 mm (solid), the longest of the cur-
peratureTbversusgastemperatureTgastakenattheeffectiveformation rently available ALMA wavelengths, as a function of position
heightsfor0.4mm,1mm,3mm,and4.5mm.Darkercolorindicates along the analyzed XZ-cut, together with the model tempera-
morepixelsinthebin.SolidlinesdenoteTb=Tgas.Bluecontour(dark ture(dashed)takenattheheightscorrespondingtotheeffective
grayintheprintedversion)encloses95%ofthespatiallocationswith
formationheights.IntheT profilethereareexcursionstoval-
weakmagneticfield(|B| < 30Gattheeffectiveformationheight),red gas
ues lower and higher than the synthesized T . Furthermore, at
contour(lightgrayintheprintedversion)includes95%ofthelocations b
ofstrongmagneticfield(|B|≥30Gattheeffectiveformationheight). X = +9.5 Mm, there is a huge discrepancy between the two
quantities. It corresponds to an intergranule-granule boundary.
ThislocationwasanalyzedinFig.10.Atthislocationtheeffec-
tiveheightevaluatedfromthecentroidofthecontributionfunc-
tion does not reflect the real formation heights because of the
complex form of the CF that has two peaks of similar strength
with negligible contribution from in between. For strong mag-
netic elements (e.g., spatial location X = −5.2 Mm) the agree-
mentbetweensyntheticbrightnessandatmospherictemperature
isnearlyperfect.
4.6. Effectofspatialsmearing
Depending on array configuration (maximum baseline) and
Fig. 13. Brightness temperature at 3.6 mm (solid), electron tempera- wavelength,ALMAcanachieveaspatialresolutionintherange
ture at the effective formation height (dashed) along the X-profile at 0.005(cid:48)(cid:48)-5(cid:48)(cid:48).InTable2welisttheestimatesoftheFWHMofthe
Y=0.9Mm.TheverticaldashedlinesarethesameasinFigs.5and 6.
synthesized beam (point spread function), which is the inverse
Fouriertransformofa(weighted)u−vsamplingdistribution.To
account for this finite resolution, we spatially smeared the syn-
theticbrightnessmapsbyconvolvingwithaGaussiankernelof
4.5. Thermaldiagnostics
corresponding FWHM to mimic the instrumental profile. Note
Inthissectionweinvestigatethepotentialofsubmm/mmbright- that we did not change the pixel size, so that in the following
nessspectratodiagnosechromosphericplasma.Wepresentcor- plotsweoversamplethespatiallysmeareddatasets.
relations between brightness temperature and the properties of The effect of spatial smearing on the mm brightness is il-
theatmosphere,inparticularitsthermalstructure. lustrated in Fig. 14 for three wavelengths: 0.1 mm, 1 mm, and
Articlenumber,page9of12page.12
A&Aproofs:manuscriptno.loukitcheva_arxiv
model res=0.2" res=0.4" res=1" res=4"

m
m
4
0.

m

m
1

m

m
3
5Mm

Fig.14.Spatiallysmearedsyntheticbrightnessat0.4mm(toprow),1mm(middlerow),and3mm(bottomrow)ataspatialresolutionof0.2(cid:48)(cid:48),
0.4(cid:48)(cid:48),1(cid:48)(cid:48)and4(cid:48)(cid:48).Thesamecolorscaleisusedforallpanelsinarow.Thefieldsizeis24Mmx24Mm.
0.4 mm
5•104
3.6mm 0.2" 0.4"
4•104
0.2" 14 14
ence 3•104 01."4" 12 12
occurr 2•104 4" kK) 10 kK) 10
1•104 b ( 8 b ( 8
T T
6 6
0
2.0•103 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 4 4
brightness (K)
2 2
2 4 6 8 10 12 14 2 4 6 8 10 12 14
1 mm
3•104 Tgas (kK) Tgas (kK)
1" 4"
e 2•104 14 14
enc 12 12
occurr 1•104 kK) 10 kK) 10
b ( 8 b ( 8
T T
6 6
0
2.0•103 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 4 4
brightness (K) 2 2
2 4 6 8 10 12 14 2 4 6 8 10 12 14
3 mm
Tgas (kK) Tgas (kK)
1.5•104
nce 1.0•104 Fig. 16. Density scatter plots of brightness temperature at 3.6 mm at
occurre 5.0•103 raetstohleuetiffoencstoivfe0f.o2r(cid:48)(cid:48)m,0at.i4o(cid:48)(cid:48)n,h1e(cid:48)(cid:48)ig,hatnsd.D4(cid:48)a(cid:48)rvkeerrscuoslgoarsintedmicpateersatmuroereTgpaisxtealkseinn
thebin.Thedataarerestrictedtothelocationsofstrongmagneticfield
(|B| ≥ 30 G) at the effective formation height. Red contours enclose
0
2.0•103 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104 95%oftheselectedpixels.
brightness (K)
Fig. 15. Spatial resolution effect on the brightness histogram. Top:
brightnesshistogramsat0.4mmforaspatialresolutionof0.2(cid:48)(cid:48)(solid), 3 mm, and for four values of FWHM: 0.2(cid:48)(cid:48), 0.4(cid:48)(cid:48), 1(cid:48)(cid:48) , and
0.4(cid:48)(cid:48) (dotted),1(cid:48)(cid:48) (dashed),and4(cid:48)(cid:48) (dot-dashed).Middle:thesamefor 4(cid:48)(cid:48). While at 0.2(cid:48)(cid:48) much of the fine structure of the original-
1mm.Bottom:thesamefor3mm. resolutionimagesispreserved,itisincreasinglylostastheres-
olution is reduced further. At the same time, the intensity con-
trast is rapidly reduced, as can be deduced from the intensity
histograms in Fig. 15 plotted for the same three wavelengths
Articlenumber,page10of12page.12