The dynamic atmospheres of Mira stars: comparing the CODEX models to PTI time series of TU And

1.4 MB·English

by  M. Hillen

Astronomy&Astrophysicsmanuscriptno.aa˙TUAnd5˙adapted (cid:13)c ESO2012
January30,2012
The dynamic atmospheres of Mira stars: comparing the CODEX
models to PTI time series of TUAnd
M.Hillen1,T.Verhoelst1,2,P.Degroote1⋆,B.Acke1⋆,andH.vanWinckel1
1 InstituutvoorSterrenkunde(IvS),K.U.Leuven,Celestijnenlaan200D,B-3001Leuven,Belgium
e-mail:[email protected]
2 BelgianInstituteforSpaceAeronomy,Brussels,Belgium
Received??,2011;accepted??,2011
2
1 ABSTRACT
0
2 Context.OurcomprehensionofstellarevolutionontheAGBstillfacesmanydifficulties.Toimproveonthis,aquantifiedunderstand-
ingoflarge-amplitudepulsatoratmospheresandinterpretationintermsoftheirfundamentalstellarparametersareessential.
n Aims.WewishtoevaluatetheeffectivenessoftherecentlyreleasedCODEXdynamicalmodelatmospheresinrepresentingM-type
a
Miravariablesthroughaconfrontationwiththetime-resolvedspectro-photometricandinterferometricPTIdatasetofTUAnd.
J
Methods.WecalibratedtheinterferometricK-bandtimeseriestohighprecision.Thisresultsin50nightsofobservations,covering

7 8subsequentpulsationcycles.Ateachphase,thefluxat2.2µmisobtained,alongwiththespectralshapeandvisibilitypointsin5
2 channelsacrosstheK-band.Wecomparedthedatasettotherelevantdynamical,self-excitedCODEXmodels.
Results.Bothspectrumandvisibilitiesareconsistentlyreproducedatvisualminimumphases.Nearmaximum,ourobservationsshow

] thatthecurrentmodelspredictaphotospherethatistoocompactandhot,andwefindthattheextendedatmospherelacksH Oopacity.
R 2
Sincecoverageinmodelparameter spaceiscurrentlypoor,moremodelsareneededtomakefirmconclusions onthecauseofthe
S discrepancies.WearguethatforTUAnd,thediscrepancycouldbeliftedbyadoptingalowervalueofthemixinglengthparameter
. combinedwithanincreaseinthestellarmassand/oradecreaseinmetallicity,butthisrequiresthereleaseofanextendedmodelgrid.
h
p Key words. Stars: AGB and post-AGB – Stars: oscillations – Stars: atmospheres – Stars: fundamental parameters – Techniques:
- interferometric–Infrared:stars
o
r
t
s
a 1. Introduction tionofthesourceofthisdiscrepancyishamperedbythelimited
[ coverageinparameterspaceofthemodels,butwediscusspos-
In spite of their great astrophysical importance (Muzzinetal.
sible avenuesof furtherresearch,and stress the need forlarger
1 2009),manyaspectsofasymptoticgiantbranch(AGB)starsre-
publishedmodelgridsandforobservationalcampaignscovering
v main poorly understood (e.g. Kerschbaumetal. 2011, and ref-
thevisualmaximumphaseindetail.
5 erences therein). Among the AGB stars, the luminous Mira
1
variables might be the most enigmatic with their long-period,
8
large-amplitudepulsationsandhighmass-lossrates.Inparticu- 2. Observations
5
larourunderstandingoftheiratmosphericstructure,whichisthe
1. linkbetweenenrichedstellar interiorandevolution-dominating TUAndwasextensivelyobservedaspartofthelargePTIMira
0 wind,isataturningpoint,thankstoadvancesonboththeobser- programme (Thompson 2002) during 60 nights from 1999 to
2 2005, over eight pulsation cycles. The publicly available PTI
vationalside(IRinterferometry),andthemodellingsidewitha
1 data archive provides both squared visibility amplitudes (here-
newgenerationofself-excitedatmospheremodels.
: aftervisibilities)andintegratedphotoncounts.Wedescribethe
v ThepublicreleaseoffourCODEXmodelseriesforM-type
datareductionandcalibrationoftheseobservablesbelow.
i MiravariablesbyIrelandetal.(2011, hereafterISW11),greatly
X
increasestheopportunitiesforthegeneralcommunitytoassess
ar theperformanceofsuchdynamicmodels.Here,wepresentthe 2.1.Interferometry
firstconfrontationoftheCODEXmodels(seeSect.3)toafully
phase-resolved 2 µm spectro-photometric and interferometric The now dismounted Palomar Testbed Interferometer (PTI)
dataset covering eight pulsation cycles, as obtained with the was a Y-shaped, long-baseline near-IR interferometer located
Palomar Testbed Interferometeron the Mira TUAnd. Thisstar at Palomar Observatory and operated by JPL/Caltech. It con-
hasaspectraltypeM5IIIe,aprogenitormassM =1.15−1.4M , sisted of three 40 cm apertures with separations of 85-110m
⊙
metallicity Z = 0.004 − 0.02 (Mennessieretal. 2001), and a and used in pairwise combination(see Colavitaetal. 1999, for
GCVSperiodP∼317d(Samusetal.2009). a detailed description). PTI thus measured only one interfero-
metricobservableperspectralchannel:thesquareofthe“fringe
The most intriguing result of this comparison is that there
contrast” or visibility amplitude. Visibilities were recorded in
seems to be a systematic discrepancy between the models and
fivechannelswithintheK-band(2.0-2.4µm).Processingofthe
our observations around visual maximum: the star appears too
rawdata(correctingfordetectorbiasesandread-outnoise,and
hot,anditsouteratmospheretoodevoidofwater.Anidentifica-
averagingthe125sscansintofringeamplitudes)wasautomati-
⋆ Postdoctoral Fellow, Fund for Scientific Research of Flanders cally done on site, following procedures described in Colavita
(FWO) (1999). Calibrating for the system visibility, i.e. the response
1
Hillenetal.:ComparingtheCODEXmodelstoPTItimeseriesofTUAnd
of the measurement system (interferometer + atmosphere) to phases(hereaftersimplyphases)werecomputedbyfittingasine
a point source, was done using the nbCalib tool (Bodenetal. totencyclesaroundthetimesofthePTIobservations.
1998) provided by the NExScI which uses a weighted-mean
formalism to estimate the system visibility at the time and po-
3. TheCODEXmodels
sition of the science observation from all calibrator measure-
ments that fit the user-defined temporal and spatial constraints The CODEX atmosphere models (Irelandetal. 2008) are the
(see vanBelleetal. 1999, for more details). To ensure calibra- mostadvancedpubliclyavailabledynamicalmodelsforM-type
tion stability and consistency, essential for making full use of Mira variables. The pulsationsin these models are self-excited
thisdatasetasaninterferometrictimeseries,severalcalibrators and limited in amplitude by the non-lineareffects of shocks in
wereobservedeachnightandwereintercomparedbynbCalib. theouteratmosphereandbyturbulentviscosity.Foreachmodel
Thestellarparametersofourcalibratorsweredeterminedby series, a few typical cycles were selected for detailed radiative
fitting a reddened Kurucz model atmosphere (Kurucz 1993) to transfer calculations with an opacity-sampling method in LTE
high-qualityarchivalphotometry.Theerrorswereestimatedus- with a phase-resolution of ∼ 0.1. This results in the centre-to-
ingMonteCarlosimulations.Table 1(onlyavailableelectroni- limb variation (CLV) and spectrum, with ∆λ = 0.0002µmbe-
cally)liststhecalibratorsandtheiradoptedparameters.Allcal- tween λ = 0.45−2.50µm.The currentmodelseries are made
ibrators have angular diameters well below the PTI resolution to match o Cet, R Leo, and R Cas, although even these well-
limitof1mas(vanBelle&vanBelle2005)andare,onaverage, observedMiras have large uncertaintieson their basic parame-
10◦awayfromthesciencetarget. ters(e.g.the2verydifferentmodelsforRCas).Theywerepre-
The calibrated visibilities of TUAnd from 50 nights were sented to the communityand comparedto photometricand in-
retained,havingamedianrelativeprecisionrangingfrom5%at terferometricobservationsinISW11.Themodelparametersare
2.2µm,7.5%at2.4µmand10%at2.0µm. mass M, luminosity L, metallicity Z, mixing length parameter
α , andturbulentviscosity parameterα . For a detailed justifi-
m ν
cationofthecurrentsetofparametervalues,werefertoISW11.
2.2.Spectro-photometry
With a slightly meandering (± 5d) pulsation period of
Togetherwiththeuncalibratedvisibilities,PTIdeliveredtheraw ∼317 d, TU And is compared to the R52 (M = 1.1M , L =
⊙
photoncountsofeachobservation,typicallyinblocksoffiveob- 5200L ,Z = 0.02,α = 3.5,α = 0.25)ando54(M = 1.1M ,
⊙ m ν ⊙
servationstakenwithinminutesandinterleavingsciencetargets L = 5400L , Z = 0.02, α = 3.5, α = 0.25) model series,
⊙ m ν
withcalibrators.WederivedthespectralshapeovertheK-band whichhavederivedperiodsof307and330d,respectively.
andflux-calibratedthe2.2µmchannelwithabootstrapmethod. When comparing a model series to observations, the only
The spectral shape calibration procedure consists of divid- free parameter is the distance, or equivalently, the phase-
ing the science-target normalised photon counts by the ones averagedRosselandangulardiameterθ .Consideringtheangu-
R
of the calibrator and multiplying the result with the intrinsic larresolutionofPTI,weconvertedthemodelintensitiestosyn-
spectral shape of the calibrator. To obtain an absolute flux in theticobservablesforθ ∈ [1.5,4.5]mas,instepsof0.05mas.
R
the 2.2 µm channel, one calibrator measurementon either side Convertingthemodelintensitiestothespectro-photometricob-
of the science observation with respect to airmass is required servableswasdonewithEq.1.Thesquaredvisibilitiesarecom-
to solve for the atmospheric extinction per unit airmass A = putedfollowingWittkowskietal.(2004):
v
(log[I /I ]+log[S /S ])/(m −m )andtheinstrumentspectral
respon2se1R =S /(1I ×2exp(−2A m1)),withI theintrinsicandS λuk(ν(B/λ)·F)⊗Gk2λdλ
tchaelibmreaatesdurflveudxflius1xt,hea1nndcamlciuthlaeteadviram1saIss,o=fSthie/iRthc×aleixbpra(Atorm.Th)ei, V2(B/λeff)= Rλl λuk(ν(0)·F)⊗Gk2λdλ (2)
whereS denotesthemeasuredphostcoincouscnitsavndm thvesaciir- Rλl
sci sci withF,G,λ,andλ asinEq.1,and
massofthesciencetarget.Foreachobservingblock,theabove l u
procedure is applied 5000 times to a randomly picked science
ν(B/λ)= IµJ (πθ qB/λ)qdq (3)
observation,correspondingrandomcalibrator observationsand Z λ 0 R
calibratorintrinsicfluxes.Finally,thedistributionsofallobserv-
ing blocks within a single night are co-added, and the average themonochromaticHankeltransformoftheintensityprofile Iµ
λ
andstandarddeviationaretakenfromtheresultingnormaldis- atthebaselineBandwavelengthλ.
tribution,asthefinalcalibratedfluxanditserror.
Toobtainthecalibratorintrinsicfluxes,thePTIopticalcon-
4. Results
figuration was taken into account:after beam combinationand
dispersionbyaprism,thewavefrontwassentthrougha broad- Figure 1 shows the resulting photometric time series of TU
bandKBarrfilterontothedetectorpixels(Colavitaetal.1999). And asa functionof phase,with two consecutivecyclesof the
EachoftheKuruczmodelsintheMonteCarlodistributionsob- R52 model series repeatedly overplotted. The two other avail-
tained with the spectralenergydistributionfitting, is converted able cycles of this series give qualitatively identical results, so
intotheeffectivefluxIineachofthePTIchannelsas: are not shown. The Rosseland angular diameter is chosen to
λu[(M·F)⊗G]λdλ match the absolute flux at 2.2 µm, with a resulting value of
I = Rλl (1) θR = 2.20± 0.15 mas or d = 1.13± 0.07 kpc. Although the
λu[F⊗G]λdλ error on the absolute calibration in this channelis rather large,
Rλl the contamination from upper atmospheric molecular layers at
with M theKuruczmodel, F the filter profile,G a Gaussianof these wavelengths is smallest. This channel thus provides the
FWHM =0.7µm(M.Colavita,privatecommunication),andλ, best view at the continuum photosphere in the spectral region
l
λ theedgesofthespectralchannelunderconsideration. wherethebulkoftheluminosityisemitted.Moreover,fixingthe
u
Finally,weconstructedavisuallightcurvebyretrievingob- angulardiameterlike thisgivesa consistentpicturein boththe
servedV-bandphotometryfromtheAAVSOandAFOEV.Visual spectro-photometryandtheinterferometry.
2
Hillenetal.:ComparingtheCODEXmodelstoPTItimeseriesofTUAnd
1.1
Flux/Flux2.20001....8970
0.6 2.0µm 2.1µm
1.1
Flux/Flux2.200001.....68970 2.3µm 2.4µm Fig.1.Spectro-photometry.Thetwoupperpan-
Flux (100 Jy)2.201121.....64028 ersloehlswsopsweehrscotpwtahtontehetealhbvetshao2reil.au2vttiieµosumnflaiulncxhflthauinnexnsteiphnlee,cmttlhraaaetgltetnshrih,itaruadpdneepdswa.tnihItenhel
6
8 each panel two cycles of the R52 model se-
mag10 ries are consecutively overplotted as the solid
V-12
line.Ontheleft,thecyclesareshownasafunc-
14
0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 tionofvisualphasewhileontherighttheyare
φV φV
phase-folded.
TheR52modelseriesshowsreasonableagreementwiththe
observedvisuallightcurvein the lowerpanel,consideringthat
2.4µm
0.6
(1) the amplitude was not tuned to match this light curve and 2.3µm
2.2µm
(2)non-LTEeffectsplayanimportantroleatthesewavelengths 0.4 2.1µm
2.0µm
(Irelandetal.2008),causinganincreaseinvisualbrightnessof
0.5to1.0magnitude.Incasethehighervalueofthiscorrection 0.2
is valid, the perceived difference between model and observa- φV=0.0 φV=0.1
0.0
tionscould pointto the presenceof an excessin visuallightat 0.6
maximumphasestogetherwithatoolargemodelamplitude.
0.4
Thetwoupperpanelsshowthevariationinthespectralshape
with time. A clear discrepancy occurs around times of visual 0.2
maximum, then the model predicts too much flux. Moreover, φV=0.2 φV=0.3
the amplitude of this excess shows an interesting wavelength- 0.0
0.6
dependence:thestrongesteffectisobservedat2.0µm,asmaller
discrepancyat2.1and2.4µmandagoodagreementat2.3µm. 0.4
2V
The visibility time series is presentedin Fig. 2, overplotted
0.2
with the same cycles of the R52 model series. Each panel dis-
plays the visibilities at a given phase, using different symbols 0.0 φV=0.4 φV=0.5
for observationscoming from differentcycles (at least two per 0.6
panel). The model reproduces the observations well at phases
0.4
0.3-0.6,butnotaroundvisualmaximum.Comparisonwithother
cycles does not give better results. In fact, this reveals that 0.2
the model has larger cycle-to-cyclevariationsat pre-maximum φV=0.6 φV=0.7
phases than are seen in the observations. A closer look at the 0.0
0.6
wavelengthdependenceshowsthatthediscrepancyisleastpro-
nouncedat2.3and2.4µm.Between2.0and2.2µm,themodel 0.4
shows very little wavelength-dependence, unlike the observa-
tionswhichshowacleardecreaseinvisibilitytowards2.0µm. 0.2
This observation is supported by the more quantitative as- 0.0 φV=0.8 φV=0.9
200 300 200 300
sessment presented in Figs. 3 and 4 (available electronically Spatial frequency (cycles/arcsec)
only). Epochswith observationson at least two differentbase-
Fig.2.Visibilitytimeseries.Eachpanelshowsallsquaredvisi-
lineswereselected,toresolvethedegeneracybetweenCLVba-
bilitiesasafunctionofspatialfrequency,obtainedatthevisual
sicsizeandshape.Eachofthesedatasetswithgooduv-coverage
phasegiveninthelowerleftcornerofthepanel.Differentsym-
wascomparedtoeachmodel,independentlyofmodelphaseand
forarangeofθ .Thefiguresshowtheresultingχ2-maps.Inall bols are used for observationsfrom different cycles. The same
R
two model cycles as in Fig. 1 are overplotted as the full and
panelsacurvedvalleyshowsup:everydatasetcanbefittedwith
dashedlinesineachpanel.
amodelofanyphase,albeitwithasignificantlylargerdiameter
for visual maximum models. However, within these valleys a
better χ2 is typically foundwith a modelat the correct(photo-
5. Discussion
metric)diameterbutwithaphasenearertovisualminimumthan
theactualvalueoftheobservations.Inconclusion,wefindthat Irelandetal.(2008)demonstratethattheCODEX modelshave
themodelsaroundvisualmaximumshouldresemblethoseofvi- reachedalevelofmaturity,concerningcomputationalissuesand
sualminimummoreinCLVshapeandwavelength-dependence. inputphysics,that allowsquantitativecomparisonswith obser-
Finally,wealsocomparedtheobservationstothesimilaro54 vations.Pulsationsareself-excitedandmolecularlayersarepro-
model series, except for its luminosity. The results for o54 are ducednaturallyandinaphysicallyself-consistentway.
alike and thus not shown here. The visibilities seem to match Nevertheless,ourobservationsofTUAndshowacleardis-
slightlybetter,butthelightcurvediscrepanciesbecomelarger. crepancy with the models around phases of visual maximum,
3
Hillenetal.:ComparingtheCODEXmodelstoPTItimeseriesofTUAnd
ter values that fits all observables. Accepting the LMC period-
1.8 3.5 luminosityrelationandassumingthattheeffectofmetallicityis
φV=6.6 φV=7.8 3.0 insufficient(seeISW11),theonlyfreeparametersareM,αm,and
1.4 2.5 α .Sincethelastmainlyinfluencesthepulsationamplitude,but
ν
odel φV 1.0 12..50 2logχ ncloutdtehethpaetrtihoedporrotbhleem‘paartenhtanstdari’sroandeiuosf,tchaeliabbraotvinegauththeomrsixcoinng-
M 0.6 1.0
length parameter as a function of mass. Decreasing the rather
0.5
0.2 0.0 high αm = 3.5 to a more ‘realistic’ value requires an increase
−0.5 in mass or a decrease in metallicity (albeit with a smaller de-
−0.21.6 2.0 2.4 2.8 3.2 3.6 4.0θR4. 4(m1a.6s)2.0 2.4 2.8 3.2 3.6 4.0 4.4 pendency)topreservetheperiod,whichwouldincaseofoCet
be inconsistentwith its kinematicallydeterminedage. TU And
Fig.3. The χ2 maps of two data sets with a good uv-coverage though, was classified as a member of a population with ini-
(see text). The log χ2 is shown as a function of Rosseland an- tial masses in the 1.15 − 1.4M range and Z ∈ [0.004,0.02]
⊙
gular diameter and phase, of the same two model cycles as in (Mennessieretal. 2001), which makes it a better candidate to
Figs. 1 and 2. The phase of the data set is given in the upper testtheabovehypothesis.
rightcornerof the paneland is plottedas the horizontaldotted We can at this point not gauge the impact of using an α
ν
line. The vertical dotted line marks the photometric diameter. whichisnottunedtotheamplitudeofTUAnd.Asaresult,we
The intersection of the dotted lines shows where the minimum feelthatthemodelgridshouldspansomescopeinthisparameter
χ2shouldbeifthemodeliscorrect. toallowmorecomparisonswith‘small-amplitude’pulsators.
Could either change in the model parametersalso induce a
stabilizingeffectonthe H O layer,asrequiredbyourobserva-
2
bothinthespectrumandinthevisibilities.Themostnaturalin- tions,oristhereaneedtoaddmorephysicstothemodels?
terpretation of the wavelength-dependence of this discrepancy
seems to be in terms of the shape of the H O opacity curve,
2
whichshowsanincreasefrom2.2µm towards2.0and2.4µm. 6. Conclusions
TheformationofH Olayersdependssensitivelyonthedensity,
2 The combination of a spectro-photometric and interferometric
temperature,andpressurestratificationintheupperatmosphere.
time series is a powerfultool for testing the validity of pulsat-
In the models, the conditions in the upper atmosphere are set
ing AGB model atmospheres, even at low spectral resolution.
bythedynamics.Shockfrontsthatemergefromthecontinuum
PTIobservedalargesampleofM-typeMirasandsemi-regulars,
photosphere between phases -0.3 and -0.1 and then propagate
severalwith a similar samplingto the one presentedhere,cov-
outwards lead to the formation of an H O layer around phase
2 ering periods between 150 and 470d. To fully exploit this rich
0.2-0.3,whichthenstronglydecreasesinstrengthatphase0.7-
database, better coverage in the model parameter space is ur-
0.8.Atφ = 0.5bothvisibilitiesandspectrashowgoodagree-
V gentlyneeded,eventhoughitrequiresalargecomputationalef-
ment.However,as demonstratedby Figs. 1, 2, and3, thenear-
fort. Only in this way can the model parameters be calibrated
maximumobservationsare representedbetter by modelphases
and(simply)linkedtoMiraobservableproperties.Fromtheob-
nearertominimum,andthuswiththepresenceofanH Olayer
2 servationalpointofview,interferometricstudieswithbetteruv-
intheatmosphere.Thisinterpretationissupportedbythebetter
coverageand spectral resolution should focus on phases 0.0 to
agreementat2.3and 2.4µm. ThereCO is animportantsource
0.15, since none of the current model series predicts any ex-
of opacity,in the photosphereand in the extendedatmosphere,
tendedmolecularatmosphereinthatphaserange.
evenatvisualmaximum.Asamuchmorestablemolecule,CO
dependslessontheexactconditionsandpartlyhidesthelackof
Acknowledgements. ThePalomarTestbed Interferometer was operated bythe
anH2Omolecularlayerinthespectrumandvisibilities. NASA Exoplanet Science Institute and the PTI collaboration. It was devel-
In retrospect, it is interesting to note that the image recon- opedbytheJetPropoulsionLaboratory,CaliforniaInstituteofTechnologywith
structionofTLep(P∼380d, LeBouquinetal.2009),madeata funding provided from the National Aeronautics and Space Administration.
Weacknowledge with thanks the variable starobservations fromthe AAVSO
phaseof0.8-0.9,showstheclearsignatureofanextendedmolec-
InternationalDatabasecontributedbyobserversworldwideandusedinthisre-
ular layer with a brightness ratio of ∼10%, unlike its closest search.
model series o54, which supports our findings. Woodruffetal.
(2009) found fair agreement between the wavelength depen-
dence of their measured 1.1-3.8 µm UDs and the predictions References
of the CODEX models, for several Miras at phases 0.5 to 0.8,
Boden, A. F., Colavita, M. M., van Belle, G. T., & Shao, M. 1998, in
butneededslightlylargermodelcontinuumdiametersthanpre- PresentedattheSocietyofPhoto-OpticalInstrumentationEngineers(SPIE)
dicted.Ontheotherhand,Wittkowskietal.(2011)haverecently Conference,Vol.3350,SocietyofPhoto-OpticalInstrumentationEngineers
comparedAMBERmediumspectralresolutionobservationsof (SPIE)ConferenceSeries,ed.R.D.Reasenberg,872–880
Colavita,M.M.1999,PASP,111,111
four Miras to the CODEX models and found good agreement,
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althoughtheycomparedthemodelstostarsofadifferentperiod, Ireland,M.J.,Scholz,M.,&Wood,P.R.2008,MNRAS,391,1994
henceluminosity. Ireland,M.J.,Scholz,M.,&Wood,P.R.2011,MNRAS,1633
In summary,atvisualmaximumthe currentmodelspredict Kerschbaum,F.,Lebzelter,T.,&Wing,R.F.,eds.2011,AstronomicalSociety
ofthePacificConferenceSeries,Vol.445,WhyGalaxies CareaboutAGB
acontinuumphotospherethatistoocompactandhotasalready
StarsII:ShiningExamplesandCommonInhabitants
suggestedbyISW11,incombinationwithtoosmallanH Oop-
2 Kurucz,R.L.1993,VizieROnlineDataCatalog,6039,0
ticaldepthintheouteratmosphere,asfoundhere. LeBouquin,J.-B.,Lacour,S.,Renard,S.,etal.2009,A&A,496,L1
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Muzzin,A.,Marchesini,D.,vanDokkum,P.G.,etal.2009,ApJ,701,1839
currentmodellingframework?Isitjustaquestionoftuningthe
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parameterstomorerealisticvalues?Followingthediscussionof
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5
Hillenetal.:ComparingtheCODEXmodelstoPTItimeseriesofTUAnd,OnlineMaterialp1
Table1.Thederivedcalibratorparameters.
Calibrator SpT T (K) UDDiameter(mas)
eff
HD166 K0V 5540±90 0.61±0.05
HD1404 A2V 8840±175 0.47±0.02
HD2628 A7III 7200±50 0.48±0.05
HD3651 K0V 5250±50 0.74±0.06
HD6920 F8V 6350±200 0.61±0.04
HD7034 F0V 7600±450 0.49±0.04
HD7229 G9III 5550±150 0.80±0.12
HD9712 K1III 4750±240 0.78±0.20
1.8
3.5
φV=1.8 φV=3.1
1.4
3.0
1.0
2.5
0.6
2.0
0.2
φV
Model −10..82 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 1.5 2logχ
φV=3.2 φV=3.3 1.0
1.4
1.0 0.5
0.6 0.0
0.2
−0.5
−0.21.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4
θR (mas)
Fig.4.SimilartoFig.3,butforthefourotherdatasetswithgood
uv-coverage. The log χ2 is shown as a function of Rosseland
angulardiameterandphase,ofthesametwomodelcyclesasin
Figs. 1 and 2. The phase of the data set is given in the upper
rightcornerof the paneland is plottedas the horizontaldotted
line. The vertical dotted line marks the photometric diameter.
The intersection of the dotted lines shows where the minimum
χ2shouldbeifthemodeliscorrect.