Convective line shifts for the Gaia RVS from the CIFIST 3D model atmosphere grid

2.2 MB·English

by  C. Allende Prieto

Astronomy&Astrophysicsmanuscriptno.allende˙final c ESO2013
(cid:13)
January17,2013
Convective line shifts for the Gaia RVS from the CIFIST 3D model
⋆
atmosphere grid
C.AllendePrieto1,2,L.Koesterke3,H.-G.Ludwig4,B.Freytag5,andE.Caffau4
1 InstitutodeAstrof´ısicadeCanarias,E-38205LaLaguna,Tenerife,Spain
2 DepartamentodeAstrof´ısica,UniversidaddeLaLaguna,E-38206LaLaguna,Tenerife,Spain
e-mail:[email protected]
3 TexasAdvancedComputingCenter,UniversityofTexas,Austin,TX78759,USA
e-mail:[email protected]
3 4 Zentrumfu¨rAstronomiederUniversita¨tHeidelberg,Landessternwarte,Ko¨nigstuhl12,D-69117Heidelberg,Germany
1
e-mail:[email protected], [email protected]
0 5 CentredeRechercheAstrophysiquedeLyon,UMR5574:CNRS,Universite´deLyon,E´coleNormaleSupe´rieuredeLyon,46alle´e
2
d’Italie,F-69364LyonCedex07,France

n e-mail:[email protected]
a
J SubmittedJuly20,2012

6
ABSTRACT
1

Context.Toderivespacevelocitiesofstarsalongthelineofsightfromwavelengthshiftsinstellarspectrarequiresaccountingfor
]
R anumberofsecond-ordereffects.Formoststars,gravitationalredshifts,convective blueshifts,andtransversestellarmotionarethe
dominantcontributors.
S
Aims.WeprovidetheoreticalcorrectionsforthenetvelocityshiftsduetoconvectionexpectedforthemeasurementsfromtheGaia
.
h RadialVelocitySpectrometer(RVS).
p Methods.Weusedasetofthree-dimensionaltime-dependentsimulationsofstellarsurfaceconvectioncomputedwithCO5BOLDto
- calculatespectraoflate-typestarsintheGaiaRVSrangeandtoinferthenetvelocityoffsetthatconvectivemotionswillinducein
o radialvelocitiesderivedbycross-correlation.
r Results.The net velocity shifts derived by cross-correlation depend both on the wavelength range and spectral resolution of the
t
s observations.ConvectiveshiftsforGaiaRVSobservationsarelessthan0.1kms−1forlate-K-typestars,andtheyincreasewithstellar
a mass,reachingabout0.3kms 1ormoreforearlyF-typedwarfs.Thistendencyistheresultofanincreasewitheffectivetemperaturein
−
[ bothtemperatureandvelocityfluctuationsintheline-formingregion.OursimulationsalsoindicatethatthenetRVSconvectiveshifts

canbepositive(i.e.redshifts)insomecases.Overall,theblueshiftsweakenslightlywithincreasingsurfacegravity,andareenhanced
1
atlowmetallicity.Gravitationalredshiftsamountupto0.7kms 1 anddominateconvectiveblueshiftsfordwarfs,butbecomemuch
v −
weakerforgiants.
3
0 Keywords.Stars:atmospheres–Stars:solar-type–Stars:late-type–Line:formation–convection–Techniques:radialvelocities
7
3
.
1 1. Introduction zone and the velocity field weakens with height. This causes
0 thenetconvectiveshifttobestrongestforlinesformedindeep
3 Stellar atmospheres are not in hydrostatic equilibrium.
layers,progressivelyweakeningandthendisappearingfor lines
1 Convection, magnetic fields, rotation, oscillations, and other
formed higherup in the photosphere(Allende Prieto & Garc´ıa
: phenomenainducetimeandspatiallychangingstructureonthe
v Lo´pez1998;Pierce & Lopresto2000;Ram´ırez, Allende Prieto
surfaces of stars. In the case of late-type stars, surface convec-
i &Lambert2008;Gray2009).
X tionis responsibleforthe granulationpatterndirectlyobserved
r inhigh-resolutionpicturesoftheSun. Stellar spectra are often used to measure the velocities of
a Granulation broadens absorption line profiles in observed stars projected along the line of sight. The interpretation of
solar spectra, making them asymmetric and introducing a net the measured line wavelength shifts in terms of Dopplershifts
blueshift in their central wavelengths. Even though the typical purely associated with bulk stellar motion results in biased ra-
granularvelocitiesreach severalkilometerspersecond, the net dial velocities owing to the presence of photospheric velocity
blueshiftobservedin spatially-averagedspectra of solar photo- fields. Furthermore, motions in stellar atmospheres are not the
spheric lines rarely exceeds 1 km s−1. In solar-type stars, the onlymechanismthatcanbiasradialvelocitymeasurements,and
photosphereis usually located above the convectivelyunstable a number of other effects, most notably gravitational redshifts
(e.g., Pasquini et al. 2011), frustrate the direct derivation of
⋆ Theauthorsacknowledge theTexasAdvanced Computing Center
the line-of-sightcomponentofthe velocityof a star’scenterof
(TACC) at The University of Texas at Austin for providing HPC
massfromthespectrum.Correctingforalltheseeffectsinvolves
resources that have contributed to the research results reported
knowledge that is usually uncertain and, in most cases, model
within this paper (www.tacc.utexas.edu). The model spectra dis-
cussed in this paper are available in electronic form at the CDS dependent, and therefore the IAU recommends using the term
via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via ’radial-velocitymeasure’torefertothevelocitynaivelyinferred
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ fromthespectralshiftsunderclassicalNewtonianmechanics,ig-
1
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
noringthe correctionsmentionedearlier(Lindegren& Dravins a Cartesian grid with periodic horizontal boundary conditions
2003). andaccountingfortheeffectoftheradiationfieldontheenergy
TheESAmissionGaiaisscheduledforlaunchinlate2013 balance.
with the purpose of measuring propermotions, parallaxes, and Themodelstypicallyhavearesolutionof140 140 150(X-
spectrophotometryforsome109starsdowntoV 20mag,and × ×
Y-Z) grid points, corresponding to physical sizes significantly
spectraforabout108 starsdownto V 17 mag∼(deBruijneet larger than the typical size of granules, and range from a few
∼
al.2009;Lindegren2010).Thetypicaluncertaintyoftheradial- MmfordwarfstothousandsofMmforgiantstars.Theradiation
velocity measures provided by Radial Velocity Spectrometer
fieldapplyingamulti-grouptechnique(Nordlund1982)using5
(RVS) onboard Gaia (Katz et al. 2004; Wilkinson et al. 2005) groupsformodelsofsolarand6groupsformodelsofsubsolar
willtypicallybein therange1–20kms−1, buttakingtheaver- metallicity.Thenumberofgroupswaschosenbytrial-and-error
ageforlargegroupsofstarswilllargelyreducetheuncertainties,
toensureanacceptablerepresentationoftheradiativecoolingor
andthesystematicerrorsdiscussedabovecanbecomedominant heatingeffects.Thiswasdoneprimarilybylookingatsolar-like
if neglected. More important, the RVS does not have any on- stars. It turned out, however, that also giants, as well as hotter
board calibration source to rely on for wavelength calibration,
and cooler dwarfs could be treated in the same way obtaining
and systematic effects need to be considered,since wavelength similaraccuracies.Itwasfoundthatatsubsolarmetallicitiesthe
calibrationisbasedonasubsetofthe verysame stellarspectra treatmentoftheradiativetransferinthecontinuumismorecrit-
obtainedbyRVS(Katzetal.2011). ical,sothatanextragroupwasintroduced.
Asnaturefavorstheformationofstarswithlowmasses,we
Webelievethatthetypicallyemployedfiveorsixgroupsare
focusourattentiononlate-typestars,forwhichsystematicerrors
sufficientforobtainingaccuratelineshifts,butthisisdifficultto
affectingthetranslationfromspectralshiftstobulkvelocitiesare
demonstrateingeneralsinceonlyforfewcasesaremodelswith
mainlygravitationalredshifts,convectiveshifts,andtheeffectof
moregroupsavailable.However,thelineshiftshingeverymuch
thetransversalmotionontherelativisticDopplereffect.Withthe
onthevelocityfieldwhichisdrivenindeeperlayers,wherethe
parallaxesmeasuredbyGaia,throughthecomparisonofcolors
wavelengthdependenceoftheradiativetransferbecomesofsec-
and absolute brightnesswith stellar evolution theory,it will be
ondaryimportance.Inthefewmodelsemploying12groupsthat
possible to estimate masses and radii to calculate gravitational
areavailable,changesinthelevelofthetemperaturefluctuations
redshiftstobetterthan10%(AllendePrieto&Lambert1999),
arenotlarge.Wethinkthatratherspatialresolutionandsampling
orsome50ms 1 forasolar-likestar.Accuratepropermotions
− ofthetimeseriesdominatetheuncertaintiesinthelineshifts.
and parallaxes will also provide accurate transversal motions.
WiththeexceptionoftheSun,itisverydifficulttodisentangle The equation of state takes into account H and He ioniza-
convective shifts from the bulk stellar motion, and therefore it tion and H2 formation, while the opacities are calculated with
becomesnecessarytorelyonmodels. the Uppsala package (Gustafsson et al. 2008). The physical
Wehaveshownthatmodernthree-dimensionalhydrodynam- size of the computational box is the same for models with the
ical modelsof the solarsurface can accuratelypredictthe con- same surface temperature and gravity. The upper boundary of
vective shifts of individualweak and moderate-strengthphoto- the models is open, and extended beyond a Rosseland optical
sphericironlinestowithin70ms−1(AllendePrietoetal.2009). depth of 10−6, while the lower boundary is well into optically
Thesamemodels,however,predictnetredshifts,whicharenot thick,convectivelyunstablelayers.Eachsimulationwasrunun-
observed,forverystrongironlines(AllendePrietoetal.2009; til fully relaxed. Chemical abundances are from Grevesse &
seealsoAsplundetal.2000).Convectiveshiftsneedtobeeval- Sauval (1998), with the exception of CNO, which are updated
uatedforspecificspectralwindowsusedinthederivationofra- followingAsplund(2005).MoredetailsareprovidedbyLudwig
dial velocities, ensuring that models are appropriate for those etal.(2009).
windows. Foreachsimulation,asetofuncorrelatedsnapshotswasse-
In this paper,we focusourattention on the observationsto lected for spectral synthesis. The numberof snapshots and the
beobtainedbytheGaiaRVSandpredictconvectivewavelength total time span of the simulations varied from star to star, ac-
shifts for stars of spectral types M through A (3500 < Teff < cordingtothegranularturn-overtimescaleforeachmodel.The
7000 K) from a collection of hydrodynamicalsimulations pre- snapshotswerehandchoseninordertoberepresentativeofthe
sentedbyLudwigetal.(2009).Assuchcorrectionsarespecifi- wholesimulatedtime series,avoidingstatistical anomalies.We
callyderivedfortheGaiaRVS,andinparticularforradialveloc- foundfromteststhatabout20uncorrelatedsnapshotsareenough
ity measures derived by cross-correlation of RVS spectra with to represent a model, but in some cases a smaller numberwas
template spectra calculated from classical (hydrostatic) model adequate.
atmospheres, we will refer to them, indistinctively, as net con-
In order to determine the net convective shift predicted for
vectiveshiftsoreffectiveconvectiveshiftsforRVSspectra.
each3Dmodel,wecalculateda1Dmodelatmospherewiththe
Section2describesthehydrodynamicalsimulations.Section
sameeffectivetemperature,surfacegravityandmetallicity.The
3 is devoted to the spectral synthesis calculations, and 4 de-
effective temperatureforeach simulation was determinedfrom
§
scribestheresults.In 5wecommentbrieflyonthesize ofthe
thetemporallyandspatiallyaveragedemergentbolometricflux.
§
gravitationalshiftsexpectedforthesamestarspreviouslymod-
The1DmodelswerecalculatedwiththeLHDpackage(Caffau
eled,andSection6presentsasummaryofourfindings.
& Ludwig2007),and share the basic ingredients(mostimpor-
tantly the opacities and equation of state) with the 3D simula-
2. Modelatmospheres tions. A second comparison set of 1D models was derived by
linear interpolationof the ODFNEW model grid by Castelli &
Simulationsofstellarsurfaceconvectionfor83combinationsof Kurucz1 (2004), based on the solar abundancesof Grevesse &
surfacegravity,entropyfluxatthebottomoftheatmosphere,and Sauval(1998).
chemicalcompositionwerecomputedusingthecodeCO5BOLD
(Freytag,Steffen&Dorch2002;Wedemeyeretal.2004;Freytag
etal.2012).Thecodesolvestheequationsofhydrodynamicsin 1 kurucz.harvard.edu
2
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
Fig.1. Observed (black line) and synthetic (1D in red, 3D in blue) spectra for Arcturus and the Sun – all data smoothed with a
Gaussiankerneltohavearesolvingpowerofabout11,500asappropriatefortheGaiaRVS. Boththenormalizedspectraand the
residualsareshownforeachstar.Amicroturbulenceof1.5kms 1 wasadoptedforthe1Dmodels,whichwerecalculatedwiththe
−
LHDpackage.
3. Spectralsynthesis typical strong Fe I line. For the continuum and weak lines the
differencestendtobemuchsmallerthanthat.
3.1.3Dspectralsynthesis
Theadoptedsolarabundancesareverysimilartothoseused
We computed synthetic spectra covering the Gaia RVS wave- inthe3Dsimulations.Thesmallexistinginconsistenciesareex-
lengthrange(847–874nm)withwavelengthstepsequivalentto pected to have a negligible impact in the predicted convective
0.1kms−1 (oraresolvingpower R λ/δλ 106).Thecal- shifts. ContinuousabsorptionfromH andH− is included.Line
≤culations were carried out with the 3D≡synthe∼sis code ASSǫT absorption is included in detail from the atomic and molecu-
(Koesterke 2009; Koesterke, Allende Prieto & Lambert 2008). lar files compiled by Kurucz. Opacities are precomputed in a
The referencesolarabundancesare from Asplund,Grevesse & temperature-densitygridbracketingallthesnapshotswithsteps
Sauval(2005),andwhiletemperature,densityandvelocityfields of 250 K and 0.25 dex, and subsequently derived for the grid
were taken from the model atmospheres, the electron density pointsusingcubicBe´zierinterpolation.
wasrecalculatedwithanequationofstateincludingthefirst99 Foreachindividualsnapshotaspectrumisderivedin2steps.
elements in the periodic table and 338 molecules (Tsuji 1973, Firstabackgroundradiationfieldiscalculated.Thisisusedfor
withpartitionfunctionsfromIrwin1981). the scattering term in the subsequentspectrum synthesis. Note
Thenumberofsnapshotsusedvariedbetween8and21,but thatactually2spectraarecalculatedforeachsnapshot.Thesec-
in the vast majority of cases was about 20, and the total time ond one, which is derived from opacity data without spectral
coveredrangedbetween0.6and19500hours,spanningatleast lines,isusedforcontinuumnormalization.
10timesthetypicallifetimesofconvectivecells.Thenumberof Theradiativetransfercalculationsaccountforatomichydro-
pixelsconsideredforeachsnapshotwasreducedbytakingone genRayleighandelectronscattering.Themeanbackgroundra-
out of three points in the X- and Y-directions, and at least the diationfieldiscalculatedfrom24angles(8azimuthalwithlin-
uppermost90verticalpoints.Thesechoicesarebasedonprevi- earweights,and3zenithperhalf-spherewithGaussianweights)
ous experience and limited testing with the set of models used usingashortcharacteristicsscheme,whiletheemergentfluxis
in this paper. These tests indicate that dropping spatial points integratedfor21angles(verticalrayplus3zenithalraysand8
would lead only to differencesat the level of about 0.2% fora azimuthal angles again with linear weights, except that only 4
3
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
zenithalanglesareconsideredfortheshallowestrays).Theab-
scissaeandweightsforeachcasecanbefoundinAbramowitz&
Stegun(1972).Thissetupevenlycoversaspherewith8angles
for the azimuth (0-360 deg) and 3-4 angles for the inclination
(0-180deg).Theintegrationalongaraywasperformedfromthe
top layerdownto opticaldepthsof atleast 20.The finalemer-
gentfluxisanaverageofthefluxforallthesnapshotsconsidered
foreachsimulation.
Rotationwasnotaccountedforinthisstep.Itiswell-known
thatrotationaffectstheobservedlineasymmetries(Gray1986),
but low rotational velocities should only have a modest effect
onthenetshiftsderivedbycross-correlationfortheentireGaia
RVSspectralwindow.Inpart,thisisrelatedtothespectral res-
olutionofthe RVS instrument,correspondingto 26 kms 1 (de
−
Bruijne2012).
ThecalculationswereperformedontheLonghornclusterat
theTexasAdvancedComputingCenter(TACC).Theintensities Fig.2. The spectra for a single snapshot of model
forabout150millionfrequencieswerederivedacrossallsnap- d3t59g45mm00n01 (black) and its 1D counterpart (black),
shots. The total computationaleffortis roughlyequivalentto 3 with near solar atmospheric parameters, are shown in the
monthsonamodernworkstationwith4cores. left-handpanelsfortwodifferentvaluesoftheresolvingpower
Fig. 1 illustrates the spectra in our 1D and 3D grids that (R). The right-hand panels illustrate how the corresponding
are closest to the atmospheric parameters for Arcturus and the shifts(1Drelativeto3D)areobtainedbyfittingthepeakofthe
Sun. This comparison is done after smoothing the data to a cross-correlationfunctionwithaparabola.
FWHM resolving power of R = 11,500 as appropriate for
the RVS; we adopted a microturbulent velocity of 1.5 km s 1
−
4. EffectiveconvectiveshiftsfortheGaiaRVS
for the 1D calculations. The agreement between models and
observations is significantly better in the solar case (observed Wedividedthespectrabythepseudocontinua,estimatedbyit-
spectrum from Kurucz et al. 1984) than for Arcturus (Hinkle erativelyfittingaloworderpolynomialtotheupperenvelopeof
et al. 2000). Overall, predicted absorption features are slightly eachspectrum.Wethencalculatedthenetoffsetbetweenthe3D
strongerin1Dthanin3D.Theagreementisreasonable,consid- and1Dspectratoderivetheoverallconvectiveshiftpredictedby
eringthatthemodelsarenottailoredtothesestars.Arcturushas thehydrodynamicalsimulationsbycross-correlation.
an Teff 4300K, logg 1.7 (with g in cm s−2) and a metal- The cross-correlation of two arrays (o spectra) T and S of
∼ ≃
licityabout[Fe/H] 0.5(see,e.g.,Ram´ırez&AllendePrieto equalandevennumberofelements N isdefinedasanewarray
≃ −
2011),andtheSunhasanTeff = 5777K,logg 4.437and,by Cwith
≃
definition,[Fe/H]=0(see,e.g.Stix2004).Themodelcompared
herewithArcturus(d3t40g15mm10n01)hasan T of4040K, N
logg = 1.5 and a metallicity of [Fe/H]= 1.0, aenffd that com- Ci =XTkSk+i N, (1)
paredwiththeSun(d3t59g45mm00n01)ha−sanT of5865K, k=1 −2
eff
logg=4.5andsolarmetallicity. where i runs from 1 to N. If the spectrum T is identical to S,
but shifted by an an integer numberof pixels p, the maximum
value in the array C will correspondto its element i = p+ N.
3.2.1Dspectralsynthesis 2
Cross-correlation can be similarly used to measure shifts that
The 1D counterpart radiative transfer calculations were per- correspondtonon-integernumbers.Inthiscase, findingthe lo-
formedwiththe1DversionofASSǫT,andthereforetheresults cation of the maximum value of the cross-correlation function
can be directly compared between the 1D and 3D models. In canbeperformedwithavastchoiceofalgorithms.
particular,chemicalcompositions,opacities,andradiativetrans- We calculated the cross-correlation of 1D and 3D model
fercalculationfortheemergentintensitiesaresharedbythe1D spectra using the software by Allende Prieto (2007), which al-
and3Dbranches.Theonlydifferencesbetweenthe1Dand3D lows fitting the peak of the cross-correlation function with a
calculationsare thatthe opacitiesarecalculated exactly forthe parabola, a cubic polynomial, or a Gaussian. Based on tests
gridpointsin1D(everydepthinthemodelatmospheres),while addingnoisetosimulatedspectra,wechosetouseparabolicfit-
they are interpolated in 3D (from a precomputed grid in tem- tings to the central3 pointsaroundthe maximumof the cross-
perature and density, as described in 3.1), and the radiative correlationfunction.
§
transfersolverforthecalculationofthe meanintensities in the Thenetconvectiveshiftderivedbycross-correlationnotonly
grid, which is based on Feautrier’s method (Feautrier 1964) in depends on the chosen spectral range, but also on the resolv-
1Dandanintegralmethodin3D.Ourtestsnonethelessindicate ing power of the spectrograph. A degraded resolution will in-
thatthesedifferenceshaveanegligibleeffectonthecontinuum- troduce, in addition to random noise, a systematic error in the
normalizedfluxes.ThedifferencebetweentheFeautrierandthe derived velocity shifts. This is illustrated in Fig. 2 for a single
directmethodin1Disabout10 5.Theeffectoftheinterpolation 3D snapshot and its 1D counterpart at two different values of
−
oftheopacitiesin3Disestimatedtobeontheorderof10 4.The the resolving power: resolution does change the apparent con-
−
different choices for the opacity calculation and the radiation vectiveshift.Fig.3showsthederivedshifts(1Drelativeto3D)
transfer method are made to accommodate the very expensive formodeld3t59g45mm00n01(seeTable1forreference),which
calculationsin3D,whicharenotfeasiblewithoutinterpolating was compared with the solar spectrum in Fig. 1, as a function
theopacitiesandbenefitfromthefaster1stordermethodforthe oftheresolvingpower.AttheresolvingpoweroftheGaiaRVS
integrationoftheintensities. (R =11,500),the inferred convective blueshift is about 0.1 km
4
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
metallicities. Fig. 4 shows the results obtained when the LHD
1D models,which are fully consistentwith the 3D simulations
in terms of opacitiesand the equationof state, are used as ref-
erence. However, very similar results are found when Kurucz
modelsareusedasareference(seebelow).
Weadoptedamicro-turbulenceof2.0kms 1,andamacro-
−
turbulence of 1.5 km s 1 for the reference 1D calculations,
−
but our experiments indicate that these parameters would only
lead to a modest impact on the spectral shifts determined by
cross-correlation. For instance, completely neglecting macro-
turbulenceleadstoameancorrectioninthepredictedconvective
blueshifts of just 0.01 (σ = 0.032) km s 1. Adopting a micro-
−
turbulenceof1.5kms 1 insteadof2.0kms 1 will,onaverage,
− −
reduce the convective blueshifts by about 0.007 (σ = 0.011)
kms 1.Finally,adoptingKuruczmodelatmospheresinsteadof
−
LHDmodelsasour1Dreferencewill,onaverage,enhancethe
blueshiftsbyabout0.01(σ=0.03)kms 1.
−
Fig.3. Derived convective shift for model d3t59g45mm00n01,
Thesmallest(inabsoluteterms)convectiveshiftsinthegrid
withnearsolaratmosphericparameters,asafunctionofthe re-
are for the coolest dwarfs of low metal content, and amount
solvingpower.Theseshiftsareobtainedbycross-correlationus- barely to < 0.1 km s 1. The most vigorous net velocities
−
ing the Gaia RVS spectral window, and fitting the peak of the
are found for low metallicity F-type subgiants, practically the
cross-correlationfunction with a parabola. The dashed vertical
warmeststarsconsideredinthegrid,wheretheyexceed0.3km
linemarkstheRVSresolvingpower. s 1.
−
There are a few simulations (mostly with [Fe/H]= 1) for
−
which redshifts are predicted. While perhaps not very evident
s 1, while that measured when the spectral lines are fully re-
− in Fig. 5 the drop is clearly apparent in a global analytical fit
solved (R > 200,000)approaches0.24km s 1. Our results for
− oftheconvectiveshiftsofthe3Dsimulations(seeAppendixA).
thesimulationsareincludedinTable1.Allthevaluespresented
Theseredshiftsareaclearpredictionofthesimulations.Theyare
inthetable,anddiscussedbelowcorrespondtotheRVSresolu-
causedbystrongredshiftsofthecoresoftheCaIItripletlines,
tion.
buttheydominatetheoverallcross-correlationsignal.Thecores
Inadditiontocalculatingthenetconvectiveshiftsfortheav-
getredshiftedwhenthe linesarestrongandthe flow exhibits a
erage spectra from the selected snapshots for each simulation,
pronounced”reversegranulation”pattern.Still,theoverallshift
we also calculated the shifts for each snapshot and averaged
oftheCaIItripletlinecoresisasuperpositionofredshiftedand
themout.Thesecorrespondtothe’mean’shiftsinTable1.The
blueshiftedcontributionsandthereforeitisdifficulttopredicta
variance(σ2)amongsnapshotswasusedtoestimatetheintrinsic
prioriwhichonesaredominating.
uncertaintiesas σ/√n, withnthenumberofsnapshotsconsid-
ered,andthesearealsoincludedinthetable.Webelievethatthe Wetriedtointerpretourfindingsintermsofthetemperature
uncertainties calculated in this way are conservative since the andvelocityfluctuationspresentinthe3Dsimulations.Weaver-
subsampleofsnapshotsselectedforspectralsynthesisischosen aged the temperatureand verticalvelocity componenthorizon-
to represent the properties of the complete run (e.g., in effec- tally and temporally on surfaces of constant Rosseland optical
tivetemperatureandmeanvelocities).Moreover,theseparation depth.We tookvaluesatRosselanddepthτ = 2/3asrepresen-
in time among the snapshots is large enough that they can be tativefortheconditionsprevailinginthelineformingregions–
consideredasindependentdespitebeingselectedfromonetime atleastforweaklines.Theseareshowninthetwobottompan-
series. The agreement between the shifts of the average spec- elsofFig.5,whereamildcorrelationbetweentemperatureand
traandtheaverageshiftsfromindividualsnapshotsisexcellent, velocityfluctuationsisapparent.
with an average of 0.0004 km s 1 and a standard deviation of Figure 6 shows how the temperature and velocity fluctua-
−
0.001kms 1. tionsdependoneffectivetemperature,surfacegravityandmetal-
−
Fig. 4 showsthe convectiveshifts forall models.Note that licity.Interestingly,wefindthatthevelocityfluctuationsdepend
therearemodelsforfourdifferentmetallicities,color-coded,and on metallicity for T < 5000K but they do not for warmer
eff
that the resulting effective temperatures of the simulations are stars.Onlyclosetothemain-sequenceasystematicdependence
notalwaysexactlythesameforanycombinationofsurfacegrav- appearsbelow 5000K with highermetallicity implyinggreater
ityandmetallicity,asthisisnotaninputparameter–theentropy fluctuations. For main-sequence stars, T = 5000K also sep-
eff
of the material entering the simulations box through the open arates regions of different correlation between metallicity and
bottomboundaryisachoseninputinstead. temperaturefluctuations.Above5000K,highermetallicitycor-
Thechangesofthenetconvectiveshiftswiththeatmospheric responds to lower temperature fluctuations but this is reversed
parameters are perhaps most obvious in Fig. 5. The net shifts below 5000 K. While Fig. 6 certainly motivates the increase
show a mild correlation with the effective temperature of the in convective shifts towards higher effective temperatures and
models, in the sense that warmer stars tend to exhibit larger lowergravity,itcannotexplainalltrendsoftheconvectiveshifts.
blueshifts. As found in previous studies (Dravins & Nordlund What is indeed missing is information on the mutual correla-
1990a,b;Nordlund&Dravins1990),thenetenergyfluxtravers- tion between temperature and velocity fluctuations, as well as
ing the atmosphere is the most important parameter. Surface onchangesofthelineformationconditionswithatmosphericpa-
gravity has a small effect on the convectiveshifts predicted by rameters–asdiscussedfurtherbelow.Inparticular,Fig.6 does
our simulations – overall the shifts intensify at lower gravities notexplainthefastdropofthepredictedconvectiveshifts near
–, and so has metallicity, with lower net blueshifts at higher thehighesteffectivetemperaturesinourgrid.
5
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
Fig.4.Effectiveconvectiveshiftspredictedbythe3DhydrodynamicalmodelsforGaiaRVSobservations.
Nevertheless, our results resemble earlier high-resolution locityand/orthetemperaturefluctuations(Dravins&Nordlund
studiesofspectrallineshapesandconvectiveshifts,butthenet 1990b; Dravins et al. 1993; Collet, Asplund & Trampedach
velocity shifts predicted for the RVS observations are smaller 2007; Gray 2009). As a result, convective blueshifts should
than for individual, resolved, spectral lines (see, e.g., Bigot & strenghten–aslongasthesurfaceareafractionscoveredby up-
Thevenin2006, 2008;Chiavassa et al. 2011). As expected,the flowsanddown-flowsdonotchange.AsillustratedinFig.6,our
netshiftmeasuredbycross-correlationisaweightedaverageof detailedsimulationsroughlysupportthissimplepicture.
thelineshiftsfromallspectralfeatures.Athighmetallicity,many
Atlowermetallicityweexpectthereducedcontinuumopac-
lines contribute,with strong lines giving verysmall convective
ity to make visible deeper atmospheric layers with more vig-
shifts,andweakerfeaturescontributinglargershifts.Forexam-
orous convection (Allende Prieto et al. 1999), and the simula-
ple, for a solar-like atmosphere, if we apply cross-correlation
tions indicate that the convective shifts of individual lines be-
byisolatingwindowsaroundtheCaIIlines,wefindverysmall
come stronger, but the span of the line bisectors weakens dra-
convectiveshiftsontheorderof0.030kms 1,whilelimitingthe
− maticallywith metallicity.ThisisillustratedforthestrongFe I
spectralwindowtothestretchbetweentheCaIIlines(857-863
lineat868.86nminsolar-like(T 5900K andlogg = 4.5)
nm), the net shift is about 0.18 km s 1, double than what we eff ≃
− stars in Fig. 7. In this figure, we show line profiles (top panel)
−
obtainforthewholeRVSspectralrange.
and the corresponding bisectors (lower panel), with the solid
Forasolar3Dmodelnotincludedinthegriddiscussedhere, lines for the calculations based on 3D models and the dashed
wefoundinapreviouspaper(AllendePrietoetal.2009)acon- lines for 1D models. This analysis is performed at full resolu-
vectiveshift of 0.26km s−1 , in excellentagreementwith the tion,sincelinebisectorsareerasedattheRVSresolution,using
−
value directly derivedfrom the solarflux atlas of Kurucz et al. amicro-turbulenceof2kms 1andnomacro-turbulence.Asthe
−
(1984). These two values correspond to observations at much featureweakens,therangeofatmosphericheightssampledfrom
higherspectral resolution,and can be comparedwith those we the wings to the core is reduced, which is a likely explanation
find(seeFig.2)forthemodelinthegridwithparametersclosest forthesmallerbisectorspans.Forthisline,wealsofindthatthe
tosolar,d3t59g45mm00n01,namely 0.24kms−1 whenthe predictedconvectiveshiftis 0.00, 0.10, 0.12,and 0.13km
resolvingpowerisR > 200,000butn≃ot−ethatchangesto 0.09 s 1 at [Fe/H]= 0, 1, 2, an−d 3, r−especti−vely. Note −that these
kms−1atR=11,500. − b−isectors are calcu−late−d using−the standard definition, i.e. sub-
From a theoreticalstandpoint, the main effect of a reduced tractingthevelocityinthebluewingfromthatatthesamenor-
gravityisalowersurfacepressure,aswellasanincreaseinboth malized flux level in the red wing, and thus the net line shifts
the vertical and horizontal scales. At a given effective temper- arenotincludedandtheyapproachzerovelocityinthelinecore
ature the flow is constrained to transport the same energy flux (Gray1992).Inadditiontothecharacteristicsofsurfaceconvec-
sothatareducedpressureanddensity(sincethetemperatureis tion changing as a function of metallicity, the strengtheningof
almostfixed)hastobecompensatedforbyanincreaseintheve- the line blueshifts with decreasing metallicity is related to the
6
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
Fig.5.Threeuppermostpanels:Predictedconvectiveshiftsandatmosphericparametersofthe3DhydrodynamicalmodelsforGaia
RVSobservations.Twolowermostpanels:Convectiveshiftsversusrelativehorizontaltemperaturefluctuationsandthermsofthe
verticalvelocityatRosselandopticaldepthτ=2/3.InallpanelsdatapointsarecolorcodedbymetallicityasinFig.4.
line formation shifting deeper into the photosphere as the line dictionsagainsthighqualityobservationsofmetal-poorstarsis
weakens.Thisisconsistentwithourfindingsfortheoverallnet verymuchneeded.
blueshiftforGaiaRVSobservations.
Ourcalculationsarelikelyleastreliableforcoolgiants, and
ThefilledcirclesintheupperpanelofFig.7showtherenor- infactthepredictedconvectiveshiftsforstarsapproaching4000
malizedobservationsfortheSunofKuruczetal.(1984).Asig- K are unlikelytobe realistic. Gray,Carney& Yong(2008)ex-
nificantimprovementin the agreementbetweentheoryand ob- aminedbisectorsforanumberofredgiantstarswithlowmetal-
servationisobviousforthelinecorewhengoingfrom1Dto3D licity, finding that those with temperatures of about 4100 K or
models (dashed and solid black lines, respectively). A similar lesstendtoexhibitreversedbisectors(withan’inverseC’shape,
effectisfoundusingaKuruczmodel,insteadofitsLHDcoun- ratherthanwitha’C’-likeshapeseeninsolar-typestars). More
terpart, for the 1D spectrum. It is also noticeable the extreme recent results for the supergiant γCyg confirm the same trend
discrepancies between the 1D and 3D cases at [Fe/H]= 2, as (Gray2010).Thesestarsalsotendtoexhibitradialvelocity’jit-
−
itisthelargereductioninstrengthforthelinecalculated in3D ter’(see,e.g.,Carneyetal.2008,andthetheoreticalpredictions
between[Fe/H]= 2and 3.Anexhaustivecheckofthesepre- byChiavassaetal.2011).
− −
7
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
0.06
0.05
0.04
T
s(T)/ 0.03
m
r
0.02
0.01
0.00

3
−1s)
m
v) (k 2 log(g)=
ms( 4.5
r 4.25
4.0
1 3.5 [M/H]=
2.5 0
2.0 −1
1.5 −2
−3
0
7000 6500 6000 5500 5000 4500 4000
Teff (K)
Fig.6. Relative horizontal temperature fluctuation (top panel)
and absolute vertical velocity fluctuations (bottom panel) at
Rosseland optical depth τ = 2/3 in the 3D models. The sur- Fig.7.LineprofilesandvelocitybisectorsfortheFeItransition
face gravity is color coded, metallicity is depicted by different at868.6nm.Thefilledcirclesintheupperpanelcorrespondto
plotsymbols. solar observations. Solid lines are for 3D simulations and bro-
ken lines for1D models. Note that the 1D calculation at solar-
metallicityforthisline(brokensolidline)showsanon-zerove-
We haveidentifiedanexpressionthatcanreproducethenet locitybisector,duetooverlappingCNtransitionsincludedinall
convectiveblueshiftstheoreticallypredictedfortheRVS obser- our calculations that distort the otherwise perfectly symmetric
vationsinTable1withanrmsof0.021kms−1;andamaximum 1Dprofiles.
deviation of 0.074km s 1. We have coded this expressioninto
−
anIDLfunctionprovidedintheappendix.
6. DiscussionandSummary
5. Gravitationalredshifts The radial velocities inferred from the wavelength shifts mea-
sured in stellar spectra are systematically offset from the true
Photonsescapingfroma stellarphotospheretoinfinityarered-
spacevelocitiesofstarsprojectedalongthelineofsightdueto
shifted as a result of climbing up the gravitational field by
a number of effects. Quantifying the impact of such effects is
GcM/R, where G is the Gravitational constant, c the speed of
modeldependent,andthereforetheIAUhasintroducedtheterm
light, M the stellar mass, and R the stellar radius. Using solar
’radial-velocitymeasure’(Lindegren&Dravins2003)toreferto
units for the stellar mass and radius, the gravitational redshift
theradialvelocitynaivelyinferredfromtherelativewavelength
amounts to 0.6365 M/R km s 1, and for light intercepted at 1
− shiftsoflinescδλ/λ,andencouragesobserverstokeepmeasure-
AU0.6335M/Rkms−1(Lindegren&Dravins2003).Thesame mentsdisentangledfromsubsequentcorrections2.Nevertheless,
effectactswhenphotonsenterintheEarth’sgravitationalfield,
for most stars, two relatively clean corrections are dominant:
which reduces the redshift by about 0.2 m s 1, but this small
− photosphericgravitationalredshiftsandconvectiveblueshifts.In
correctioncanbeignoredformostpurposes.Withtheparallaxes
the case of stars movingat high speed in the direction perpen-
and spectrophotometryprovidedby Gaia, it should be possible
diculartotheobserver,therelativisticcontributiontotheDopper
toobtainfairlyaccuratecustomizedpredictionsforthegravita-
shift needs to be considered at a precision level of hundred of
tionalredshiftsforindividualGaiatargets.
Gravitationalshiftsforlate-typedwarfsamounttoabout0.7- 2 Wenotethatthe’radial-velocitymeasure’doesnothaveaunique
0.8 km s−1 at logg 4.5, but they dramatically decrease with definition,sincethevaluederiveddependsonthespectralfeaturesand
surface gravity down∼to 0.02–0.03km s−1 for the giants in our theactualmethodused.Therefore,suchmeasuresmustbeaccompanied
grid(logg 1.5). byadetaileddescriptionoftheprocedureusedtoderiveit.
∼
8
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
meterspersecond;the spectrum of a star with a transverse ve- References
locityof300kms 1 andnoradialvelocitywillberedshiftedas
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muchasthatofastarwitharecedingradialvelocityof0.15km
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K-typestars,whichincreasewitheffectivetemperaturetoreach
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theGaiaRVSisaself-calibratedinstrument,inthesensethatthe
Caffau,E.,&Ludwig,H.-G.2007,A&A,467,L11
wavelengthsolutionisderivedfromstellarobservations,system-
Carney,B.W.,Latham,D.W.,Stefanik,R.P.,&Laird,J.B.2008,AJ,135,196
atic offsetsfromstar to star such as those relatedto convective Castelli,F.,&Kurucz,R.L.2004,arXiv:astro-ph/0405087AvailableasPoster
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timateconvectiveblueshiftsforstarswithspectraltypesFtoM,
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Grevesse,N.,&Sauval,A.J.1998,SpaceScienceReviews,85,161
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Dravins,D.,Lindegren,L.,Nordlund,A.,&Vandenberg,D.A.1993,ApJ,403,
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particularforlaterspectraltypes(KandM),wheretheconvec- Dravins,D.,&Nordlund,A.1990b,A&A,228,203
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Freytag,B.,Steffen,M.,&Dorch,B.2002,AstronomischeNachrichten, 323,
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213
nounced,isexpectedfortheconvectiveblueshifts,insuchaway Freytag,B.,Steffen,M.,Ludwig,H.-G.,Wedemeyer-Bo¨hm,S.,Schaffenberger,
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Gray, D. F. 1992, The Observation and Analysis of Stellar Photospheres,
When adding together convective and gravitational shifts,
Camb.Astrophys.Ser.,Vol.20
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ourstudy,witheffectivetemperaturesaround4000K,wherethe Gray,D.F.2010,AJ,140,1329
net effect, dominated by gravitational redshifts, amounts up to Gray,D.F.,Carney,B.W.,&Yong,D.2008,AJ,135,2033
nearly 0.6 km s 1. By using the corrections provided here and Gustafsson,B.,Edvardsson,B.,Eriksson,K.,Jørgensen,U.G.,Nordlund,Å.,&
− Plez,B.2008,A&A,486,951
custom-madeestimatesofthegravitationalshifts,thesesources
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rangingfromabout0.1kms 1 forF-typedwarfs,toa fewtens Wallace,JeffValenti,andDianneHarmer.(SanFrancisco:ASP)ISBN:1-
−
ofmeterspersecondforlate-K-typestars. 58381-037-4,2000
Irwin,A.W.1981,ApJS,45,621
Inadditiontocalculatedconvectiveshifts,weprovideFITS
Katz,D.,etal.2004,MNRAS,354,1223
filesincludingthe1Dand3Dmodelfluxeswehavecomputed. Katz,D.,Cropper,M.,Meynadier,F.,etal.2011,EASPublicationsSeries,45,
These may turn useful to build radial velocity templates for 189
derivationofradialvelocitiesfromobservations. Koesterke,L.2009,AmericanInstituteofPhysicsConferenceSeries,1171,73
Koesterke,L.,AllendePrieto,C.,&Lambert,D.L.2008,ApJ,680,764
Weendwithawordofcaution.Thetheoreticalcalculations
Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, National Solar
presented here only have been checkedagainsta limited set of Observatory Atlas, Sunspot, New Mexico: National Solar Observatory,
observations, mainly for the Sun. An exhaustive check against 1984,
high-quality observed spectra for warmer and cooler tempera- Lindegren,L.2010,IAUSymposium,261,296
Lindegren,L.,&Dravins,D.2003,A&A,401,1185
tures, especially at very low metallicity, is still needed before
Ludwig,H.-G.,Caffau,E.,Steffen,M.,Freytag,B.,Bonifacio,P.,&Kucˇinskas,
onecanensurethatthepredictedshiftsarecompletelytrustwor-
A.2009,MemoriedellaSocietaAstronomicaItaliana,80,711
thy. Nordlund,A.1982,A&A,107,1
Nordlund,A.,&Dravins,D.1990,A&A,228,155
Acknowledgements. WeareindebtedtoIvanHubenyforassistancecomputing PasquiniL.,MeloC.,ChaveroC.,DravinsD.,LudwigH.-G.,BonifacioP.,de
radiative opacities. Carlos Allende Prieto is thankful toMarkCropper, David LaRezaR.,2011,A&A,526,A127
Katz,andFre´de´ricThe´veninforfruitfuldiscussions.B.F.acknowledgesfinancial Pierce,A.K.,&Lopresto,J.C.2000,Sol.Phys.,196,41
supportfromtheAgenceNationaledelaRecherche(ANR),andthe“Programme Ram´ırez,I.,AllendePrieto,C.,&Lambert,D.L.2008,A&A,492,841
NationaldePhysiqueStellaire”(PNPS)ofCNRS/INSU,France.HGLacknowl- Ram´ırez,I.,&AllendePrieto,C.,2011,AJ,submitted
edges financial support bythe Sonderforschungsbereich SFB881“The Milky Sbordone,L.,Bonifacio,P.,Caffau,E.,etal.2010,A&A,522,A26
WaySystem”(subprojectA4)oftheGermanResearchFoundation(DFG).
9
C.AllendePrietoetal.:ConvectivelineshiftsfortheGaiaRVSfromtheCIFIST3Dmodelatmospheregrid
Stix,M.2004,TheSun:anIntroduction,2nded.,byMichaelStix.Astronomy ; (<0 for blueshifts)
andastrophysicslibrary,Berlin:Springer,2004. ISBN:3540207414 ;
Tsuji,T.1973,A&A,23,411 ; --- Thu Oct 20 20:09:50 2011 --- B.F. ---
Wedemeyer,S.,Freytag,B.,Steffen,M.,Ludwig,H.-G.,&Holweger,H.2004,
;check that 3 parameters are input
A&A,414,1121
if n params() lt 3 then begin
Wilkinson,M.I.,etal.2005,MNRAS,359,1306
print,’
return,-10000000000.d0
AppendixA: Anempiricalfittingfunctionto
endif
describetheblueshiftsasafunctionof ;check that we’re within limits
atmosphericparmeters if (min(Teff) lt 3790.00 or $
max(Teff) gt 6730.00 or $
To derive from the tabulated data an easily applicable function
min(FeH) lt -3.0 or $
wefollowtherecursivefittingprocedureoutlinedbySbordoneet
max(FeH) gt 0. or $
al.(2010):wesearchedforananalyticalfunctionfortheconvec-
max(logg) gt 4.5 or $
tive blueshiftwith three independentparameters(logT , logg
eff min(logg-(teff*9.184e-4-2.482)) lt 0.0) $
and[Fe/H]),writtenasy=y(x ,x ,x ;A).Thefunctionandthe
0 1 2 then begin
vectoroffreeparameters A = (A ,A ,...) thatgivethe bestfit
0 1 print,’% ConvShift: Params. out of range:’
tothetabulateddataistobefound.
print,’% 3790 <= Teff <= 6730. K’
Thefunction-findroutine(“fufi”,writteninIDL)startswith
print,’% 9.184e-4*Teff-2.482<=logg<= 4.5’
themostsimple“function”,aconstantA –theweightedaverage
0 print,’% -3.0 <= [Fe/H] <= 0.0 dex’
ofthetabulatedblueshiftdata.Ateachrecursiveiterationwelet
return,-10000000000.d0
theroutinereplaceeachparameterA ofacandidatefittingfunc-
i endif
tionwithapolynomialA A +A x +A x +A x or,alterna-
i 0 1 0 2 1 3 2 logTeff=alog10(Teff*1d-3)
→
tively,anexponentialA A +A exp(A +A x +A x +A x )
i 0 1 2 3 0 4 1 5 2 M=-FeH
→
to reach the next level of complexity. For each of these func-
A=dindgen(24)
tions,theoptimumparametersetisthensearched,usingthepa-
A(00)= 3.9921954E-01 & A(01)=-7.1438992E-01
rameters from the previous level as starting point. We use the
A(02)= 2.0028012E-02 & A(03)=-5.2146580E-02
inverseof thermsfluctuationsofthe convectiveshiftsbetween
A(04)=-4.1100047E-06 & A(05)= 0.0000000E+00
thesnapshotsforagivenstellarparametercombination(or0.01
A(06)= 2.2334658E+01 & A(07)=-1.7070215E+00
km s 1 when the fluctuations are below this value) as fitting
A(08)=-1.7445524E-01 & A(09)= 1.4332379E-02
−
weights.Thefunctionwiththefitting-parametersetthatprovides
A(10)= 0.0000000E+00 & A(11)=-1.3239030E+01
thesmallestoverallerror(amongthousandsofcandidates)isfi-
A(12)= 8.2077384E-01 & A(13)=-9.6311294E-02
nallywrittentoafileasaFortranorIDLfunction.Thefunction
A(14)=-1.7150490E+03 & A(15)= 0.0000000E+00
we supply has been slightly edited to improve readability. We
A(16)=-9.3405848E+00 & A(17)= 7.7603954E-01
emphasize that the functionalform and fitting parametershave
A(18)= 1.3519228E+00 & A(19)=-5.0111694E+00
nophysicalmeaningwhatsoever.
A(20)= 0.0000000E+00 & A(21)=-8.2077414E-01
We varied some control parameters (the list of candidate
A(22)= 1.0457402E-01 & A(23)= 4.2118204E-01
terms and the recursion depth), and compared errors and the
Shift=A(00)+A(01)*logTeff+(A(02) + $
overall functional dependence of the results. Fitting functions
(A(04)+A(09)*exp(A(10)+(A(11)+ $
composedexclusivelyof simplepolynomialtermsalreadygive
A(14)*exp(A(15)+ $
decentfits(andwouldrequireamuchsimplerproceduretode-
(A(16)+A(19)*exp(A(20)+ $
rive than the one outlined above). Still, fits including the ex-
A(21)*logTeff+ $
ponentialterm are superiorand behave much better and closer
A(22)*logg+ A(23)*M))*logTeff+ $
to our expectations at – and even slightly beyond – the bor-
A(17)*logg+A(18)*M))*logTeff+ $
dersofthesetofstellarparametersforwhichmodelresults are
A(12)*logg+A(13)*M))*exp(A(05)+ $
available. The overall rms scatter between the direct measure-
A(06)*logTeff+A(07)*logg+A(08)*M))*logg+A(03)*M
mentsfromthesimulationsandthefittingformulaisjust0.021
Shift=-Shift
km s 1, with a maximum deviation of 0.074 km s 1. We give
− − return, Shift
an implementation of the fitting function in IDL below. It in-
end
cludes a check that the input parameters are within the range
coveredbythe3Dmodels.A fewofthefreefittingparameters
arezero,butareleftintheroutinetoallowforamoresystematic
writing of the final expression. Figs. A.1, A.2, A.3, A.4 illus-
trate the convective shifts predicted by the fitting equation for
[Fe/H]=0, 1, 2,and 3,respectively.
− − −
function ConvShift, Teff, logg, FeH
;
; IN: Teff -- effective temperature (K)
; logg -- surface gravity (g in cm/s2)
; FeH -- [Fe/H] metallicity relative to solar
;
; OUT: Convective shift (m/s)
10