Microturbulent velocity from stellar spectra: a comparison between different approaches (Research Note)

0.43 MB·English

by  Alessio Mucciarelli

Astronomy&Astrophysicsmanuscriptno.mucciarelli˙paper c ESO2011
(cid:13)
January19,2011
Microturbulent velocity from stellar spectra:
a comparison between different approaches
(ResearchNote)
A.Mucciarelli1
DipartimentodiAstronomia,Universita´degliStudidiBologna,viaRanzani1,40126,Bologna,Italy
e-mail:[email protected]
1 Received....;accepted.....
1
0 ABSTRACT
2
Context.Theclassicalmethodtoinfermicroturbulentvelocityinstellarspectrarequiresthattheabundancesoftheironlinesarenot
n
correlatedwiththeobservedequivalentwidths.Analternativemethod,requiringtheuseoftheexpectedlinestrength,isoftenusedto
a
J by-passtheriskofspuriousslopesduetothecorrelationbetweentheerrorsinabundanceandequivalentwidth.
Aims.Tocomparethetwomethodsandidentifyprosandconsandapplicabilitytothetypicalpracticalcases.
8
Methods.Iperformedatestwithagridofsyntheticspectra,includinginstrumentalbroadeningandPoissoniannoise.Forallthese
1
spectra,microturbulentvelocityhasbeenderivedbyusingthetwoapproachesandcomparedwiththeoriginalvaluewithwhichthe

syntheticspectrahavebeengenerated.
]
R Results.Thetwo methods provide similarresults for spectra withSNR 70, whilefor lower SNR both approaches underestimate
≥
thetruemicroturbulentvelocity,dependingoftheSNRandthepossibleselectionofthelinesbasedontheequivalentwidtherrors.
S
Basically, thevalues inferredby usingtheobserved equivalent widthsbetter agree withthoseof thesynthetic spectra. Infact, the
h. methodbasedontheexpectedlinestrengthisnottotallyfreefromabiasthatcanheavilyaffectthedeterminationofmicroturbulent
p velocity. Finally, I recommend to use the classical approach (based on the observed equivalent widths) to infer this parameter. In
- casesoffullspectroscopicaldeterminationofalltheatmosphericalparameters,thedifferencebetweenthetwoapproachesisreduced,
o leadingtoanabsolutedifferenceinthederivedironabundancesoflessthan0.1dex.
r
t Keywords. Stars:fundamentalparameters–Techniques:spectroscopic–
s
a
[

1. Introduction (2) the 1-D modelatmospheres (usually adopted in the chemi-
1
v calanalysis)includedepth-independentmicroturbulentvelocity.
6 One of the major limitations in the model atmospheres based Observationalhintspointtowardadifferentscenario,seee.g.the
2 on1-dimensionalgeometryisanunrealistictreatmentofsurface worksaboutArcturusbyGray(1981)andTakeda(1992);
4 convection.Usually,theconvectiveenergytransferiscomputed (3) the turbulent velocity distribution is generally modeled as
3 by adoptingthe classical mixing-lengththeory (Bo¨hm-Vitense, isotropic and Gaussian; this is contradicted by the observa-
1. 1958), a parameterizationaimed to by-passthe lack of a rigor- tion of the granulation on the solar surface; as pointed out
0 ousmathematicalformulationfortheconvectiveenergytransfer. by Nordlundetal. (1997) and Asplundetal. (2000), the phe-
1 Ontheotherhand,thisparameterizationcannotdescriberealis- nomenon of photospheric granulation is characterized by lam-
1 ticallytheconvectivephenomena,ignoringtheirtime-dependent inar(andnotturbulent)velocities,duetostrongdensitystratifi-
: andnon-localnature. cation;
v
i Basically,photosphericturbulentmotionsaredividedintwoty- (4) mixing-length theory describes the general temperature
X pologies,accordingtothesizeoftheturbulentelement:ifthisis structure of the photosphere, but it is unable to model some
r smallerthanthemeanfreepathofthephotonswecallitmicro- relevant convective effects, as the photospheric velocity field
a turbulence,otherwisemacroturbulence. and the horizontal fluctuations in the temperature structure
Such motions broaden the spectral lines, and the necessity to (Steffen&Holweger,2002).
parametrize these effects arises from the discrepancy between Theclassicalandempiricalwaytoestimatev istoeraseany
t
empiricalandtheoreticalCurvesofGrowth(COG)ontheirflat slopebetweentheabundanceA(Fe)1andthereducedequivalent
part. Microturbulent velocity (hereafter v) is a free-parameter widthlog (EW/λ)(hereafterEWR,whereEWistheequivalent
t 10
introducedtoremovethisdiscrepancyandtocompensateforthe width),becausev workspreferentiallyonthemoderated/strong
t
missing broadening.Usually, v is included during the spectral lines,whiletheweakonesaremoresensitivetoabundance.Even
t
synthesisin theDopplershiftcalculation,addingin quadrature iftheleastsquarefittingofthesetwoquantitiesisusuallyused
v andthermalDopplerlinebroadening. toinferv,thismethodrequiresthatthetwoquantitieshaveun-
t t
Somegeneralobjectionscanbeadvancedonthissimplification: correlated errors. This condition is not respected in this case,
(1)thedistinctionbetweenmicroandmacroturbulenceisonlyan becausetheerrorsinEWRandA(Fe)areobviouslycorrelated,
arbitrary boundary, because based on the uncorrected assump-
tionthatthevelocityfieldsworkonadiscretescale,whilethese 1 Iadopted thecanonical formalismA(Fe)=log (N(Fe)/N(H)) +
10
velocityfieldsratherrunonacontinuousscale; 12.
2 A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN)
with the risk to introduce a spurious slope in the A(Fe)–EWR
plane.
Tobypassthisobstacleandprovideamorerobustdiagnostic
to infer v, Magain (1984) proposed to derive v by using the
t t
theoreticalEW(hereafterEWT)insteadoftheobservedone.In
thispaper,I investigatedtheimpactofthese twoapproachesto
derivev,inordertoevaluatetheirreliabilityandtoestimatethe
t
bestonetouseinthetypicalchemicalabundanceanalysis.
2. Artificialspectra:generalapproach
Thebasicideaistogeneratesetsofsyntheticspectrawithagiven
andaprioriknownv (with inclusion of instrumentalbroaden-
t
ingandnoise)andtoanalyzethemwiththetwomethods.Such
aprocedureallowstoestimatetheabsolutegoodnessofthetwo
methods (comparing the derived v with the original one) but
t
alsotherelativeaccuracybetweenthetwoapproaches,inorder
tohighlightsystematiceffects,ifany.
Differentsets of artificial spectra have been generatedwith the
followingmethod:
(1)syntheticspectraforagiantstarwithT =4500K,logg=2.0,
eff
[M/H]= –1.0 dex and v=2 km/s have been computed with the
t
spectro-synthesis code SYNTHE by R. L. Kurucz in its Linux
version(Sbordoneetal.,2004,2005),startingfromLTE,plane-
Fig.1. Behavior of the trueequivalent widths measured in the
parallelATLAS9 modelatmospheresandby adoptingthe line
synthetic spectrum (without Poissonian noise) computed with
compilation by R. L. Kurucz 2; the synthetic spectra cover a
the atmosphericparametersof a giantstar, as a functionof the
wavelengthrangebetween5500and8000A˚;
quantitylog (gf) θχ .
(2) the spectra have been convolvedwith a Gaussian profile in 10 − ex
ordertomimictheinstrumentalresolutionofthemodernechelle
spectrographs. A spectral resolution of λ/δλ= 45000 has been
Themostadoptedproxyforthetheoreticallinestrengthcan
adopted (analogous to that of several spectrographs as UVES,
bewrittenaslog (gf) θχ (seealsoEq.14.4inGray,1992),
CRIRES,SARG); 10 − ex
wherelog (gf)istheoscillatorstrength,θtheBoltzmannfactor
(3) the broadened spectra have been re-sampled to a constant 10
(definedas θ= 5040/T ) and χ the excitationpotential.Note
wavelength step of δx= 0.02 A˚/pixel (similar to that of UVES thatEWTdoesnotincleuffdetermsexstrictlyinferredbytheobserva-
echellespectra); tionsbutonlyrelatedtotheadoptedmodelatmospheres(θ)and
(4)poissoniannoisehasbeenaddedtotherebinnedspectra,by toatomicdataoftheinvolvedtransitions(log (gf)andχ ).
usingtheIRAFtaskMKNOISE,inordertoreproducedifferent 10 ex
Fig. 1 shows the comparison between the true equivalent
noiseconditions.SNRof25,40,50,70,100and200havebeen widthcomputedforthesyntheticspectrumofthegiantstar(be-
simulated. Due to the randomnature of the noise, I resorted to foreaddingnoise)andtheproxyusedforEWT.Asappreciable
MonteCarlosimulations,andforeachvalueofSNR,200artifi- infigure,thequantitylog (gf) θχ isproportionaltothetrue
cialspectrahavebeengeneratedandanalyzedindependently. equivalent width of the s1y0ntheti−c speexctrum (at least in the EW
I selected from the Kurucz linelist 250 Fe I transitions in the rangeshown in Fig. 1), beinga monotonicfunctionof the true
∼
spectrum,takingintoaccountonlyunblendedlinesorthosewith EW. In this work, I adopt the quantity log (gf) θχ as the
negligibleblending,similarly to what is done on real observed proxy for EWT because this is the formali1s0m usu−allyexadopted
spectra.Forallspectra,EWshavebeenmeasuredwiththeauto- in the chemical analysis that follow the method described by
maticcodeDAOSPEC(Stetson&Pancino,2008).
Magain(1984).
A threshold for the strongest lines has been applied: only
linesweakerthanlog(EW/λ)=–4.85havebeenincludedin the
3. Determinationofv andcaveats
t analysis, correspondingto 78 mA˚ at 5500 A˚ and 113 mA˚ at
For all the synthetic spectra described above, least square fits 8000 A˚. Strongerlines(bas∼icallylocatedalongthe∼flat/damped
have been computedboth in the A(Fe)–EWRand A(Fe)–EWT partof the COG) have been excludedin order to notintroduce
planes, choosing the value of vt that minimizes the slope. All additionaleffects(bothobservationalandtheoretical)thatcould
the other parameters (Teff, log g and [M/H]) were kept fixed, affect the determination of vt. In fact: (i) strong features devi-
accordingtotheoriginalonesofthesyntheticspectra3. ate from the Gaussian approximation (due to the development
of Lorenztian wings), with the risk to under-estimate the EW
2 http://kurucz.harvard.edu/linelists.html
(asdiscussedinStetson&Pancino,2008,see theirFig.7),and
3 Notethatthisisasimplifyingassumption.Infact,themostadopted (ii)theselinesareaffectedbyothereffects,inadditiontothemi-
methodtoperformachemicalanalysis(theso-calledfullspectroscopic
croturbulent velocity, i.e. hyperfine/isotopic splitting, damping
method) is based on the following requirements: (i) the iron abun-
treatmentandmodelingofoutermostatmosphericlayers:thein-
dance is independent of χ (constraint for T ); (ii) the same abun-
ex eff clusionofsuchlinescouldleadtoadifferentv,verysensitiveto
danceshouldbederivedfromsingleandionizedironlines(constraint t
for logg);(iii)theironabundance isindependent of thelinestrength thetreatmentoftheseeffectsfortheironlines,butitisnotsure
(constraint for vt);(iv) themodel metallicitywellreproduces theiron that the derived vt will be appropriate also for other elements
abundance(constraintfor[M/H]). (as discussed by Ryan, 1998). Note that, in principle, stronger
A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN) 3
lines can be taken into accountin the abundanceanalysis (and
forsomeelementslikeCaorBaonlystrong/dampedtransitions
areoftenavailable),buttheyshouldbeexcludedfromthedeter-
minationofv.
t
It is noteworthy that the approach used in this work repre-
sentsonlyasimplecasewithrespecttotheanalysisofrealspec-
tra,becauseitisbasedonsomesimplifications:
(i) large number of unblended lines. In real spectra, the num-
ber of lines is limited by the quality of the available oscillator
strengthsandbythewavelengthcoverage;
(ii)onlyv isadoptedasafreeparametertodetermine,because
t
theotheratmosphericalparametersareknownapriorifromthe
computationofthesyntheticspectra;
(iii)theusedspectraareflatandtheircontinuum(virtually)set
totheunity.Hence,distortionsofthecontinuumduetoresidual
fringing, uncorrected merging of the echelle orders or residual
ofblazefunctionarenotincluded;
(iv) this analysis is limited to features that can be well repro-
ducedbyaGaussianfunction,whileinrealcasessomefeatures
with prominentdampingwingscan be includedin the analysis
by adopting a Voigt profile fitting (but taking into account the
caveatsdiscussedaboveforstronglines).
4. ImpactofSNR Fig.2.DifferencebetweenthemeasuredandtrueEWinasyn-
thetic spectrum with SNR=25 as a function of the true EW. In
For each SNR, 200 artificial spectra were generated and ana-
theleftinsetisshownthebehavioroftheerrorinpercentageof
lyzedindependently,andthemeanvalueofv ofeachgroupof
t themeasuredEWasafunctionofthetrueEW,whileintheright
SNRwasderived.Animportantstepinthiskindofanalysisisa
insetisshownthesamebehaviorasafunctionofthemeasured
carefulselectionofthelinesusedtoinferv,excludingfeatures
t EW.
notwellmeasuredthatcouldintroduceabiasinthev determi-
t
nation.InordertoquantifytheimpactofinaccurateEWsinthe
v estimate,Ideterminedv byconsideringdifferenterrorthresh-
t t
oldsandbytakingintoaccountonlylineswitherrorsσ less
EW
than 10, 20, 30 and 50%; these errors have been computed by
theDAOSPECcodefromthestandarddeviationofthelocalflux
residualsandrepresenta68%confidenceintervalonthederived
valueoftheEW(seeStetson&Pancino,2008,fordetails).The
discardoflineswithinaccurateEWshavebeenperformedindi-
viduallyfor each artificialspectrum.As sanity check, the EWs
measuredinasyntheticspectrumwithSNR=25havebeencom-
paredwiththetrueEWsoftheoriginalspectrum(beforeadding
noise).TheresultsareshowninFig.2;theinnerpanelsshowthe
behaviorof the relativeerrorin percentageas a functionof the
trueandmeasuredEW(leftandrightinset,respectively).These
plotsconfirmthereliabilityofthemeasuredEWsandshowthat
theerrorsintheEWmeasurementaffectmainlytheweaklines,
while the strong (>40 mA˚) lines are well measured with small
uncertainties.
Theresultsconcerningthedeterminationofv with thetwo
t
methodsforthesampleofsyntheticspectradescribedaboveare
plottedinFig.3,whereemptyandfilledcirclesindicatetheval-
uesof v derivedby usingthe EWR methodandEWT one,re-
t
spectively.Theerrorbarsindicatethedispersionaroundthemean
ofeachsetofsimulationsofagivenSNR.Forsakeofclarity,the
originalvalueofv ofthesyntheticspectraisplottedasadashed Fig.3. Behavior of the measured microturbulent velocity as a
t
line. Typically, the dispersion around the mean in each group functionofSNR, derivedwiththeclassicalmethod(emptycir-
rangesfromafewhundredthsofkm/s(forSNR=100and200) cles)andtheprescriptionbyMagain(1984)(filledcircles)fora
up to 0.2 km/s for SNR= 25 and considering lines with resid- giantstar.Eachpointistheaveragevalueobtainedbyusing200
ualerrorslessthan50%;notethatsuchdispersionsarebasically syntheticspectra ofa givenSNR and the errorbarsindicatethe
compatiblewiththecorrespondingerrorsofthecomputedslopes dispersion around the mean. Dashed lines indicate the original
4. valueofv ofthesyntheticspectra.Eachpanelshowstheresults
t
obtainedbyconsideringdifferentthresholdsfortheresidualer-
4 For a sub-sample of synthetic spectra, the analysis has been re-
rorsintheEWmeasurement.
peated by using the true EWsinstead of EWT as defined previously.
4 A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN)
Incaseofrelativelyhigh( 70)SNR,thetwomethodspro-
≥
videverysimilarresultsandagreewiththev oftheoriginalsyn-
t
theticspectrum.Infact,increasingtheSNR,theerrorsintheEW
measurementsarereducedandthediscrepancybetweenthetwo
approachesis erased.Onthe otherhand,spectrawith SNR<70
showv alwayslowerthantheoriginalvalueandthediscrepancy
t
increaseswhenalsolineswithhighuncertaintiesaretakeninto
account.Generallyv inferredbyusingEWTturnsouttobesys-
t
tematically lower than that obtained with the classical method
by0.1–0.2km/s,approximately.Thereasonsofthissystematic
under-estimatewillbeexplainedinSect.5.
ThesefindingshighlightalsotheimpactofmeasuredEWswith
highuncertainties:the inclusionof lines with EWs veryuncer-
tain(generally,theweakestlinesinspectrawithlowSNR)wors-
ensthediscrepancywiththetruev oftheoriginalsyntheticspec-
t
tra. Thus, a careful selection of the used lines, considering the
accuracyoftheirEWs,iscrucialinthederivationofv,inorder
t
toexcludeadditionalbias.
A simple analytical model can be derived by fitting the data
shown in Fig. 3. The absolute value δ of the difference be-
vt
tween the derived v and the true one can be expressed as
t
δ = f /SNR,where f isalinearfunctionoftheadopteder-
vt σEW σEW
rorthresholdσ (expressedin%).Thefitofthepresentdataset
EW
providesδ =(0.14 σ +2.64)/SNRforv derivedbyusing
EWRandδvt =(0.19· σEW +5.16)/SNRwhent EWTisadopted. Fig.4. Behavior of the iron abundance as a function of EWR
I repeated tvhtese che·cksEWalso with synthetic spectra computed (upperpanel) and EWT (lower panel) for a synthetic spectrum
with the typical parameters of a dwarf star (Teff=6000 K, computedwithvt=2km/s.Filledpointsshowtheshiftofagiven
log g=4.0, [M/H]= –1.0 dex and v=1 km/s), finding basically lineinthetwocasesduetotheEWerror.
t
thesameresult.
5. IstheEWTapproachfreefrombiases?
An useful example to clarify the systematic offset between the
twomethodsistomeasureEWsoftheredgiantsyntheticspec-
trumbeforeaddingnoise:alllines,regardlessoftheirstrengths,
providethesameabundance.TheseEWshavebeenvariedran-
domly by 1σ, where σ was computed with the classical for-
±
mula by Cayrel (1988) (σ = 1.5 (SNR 1) √FWHM δx,
−
adopting SNR= 25, the wavelength·step δx= 0·.02 A˚/pixel·and
a FWHM of λ/R, correspondingto 0.12 A˚ at 5500 A˚ and 0.18
at 8000 A˚) and the analysis repeated as described above. The
error is assumed the same regardless of the size of the EW.
Fig. 4 shows an example of this test, with the distribution of
theFeIlinesintheA(Fe)–EWRandA(Fe)–EWTplanes(upper
andlowerpanel,respectively).Filledpointsshowhowageneric
weak line (with initial A(Fe)= 6.5 dex) moves in the two dia-
gramsbecauseoftheeffectofEWerrors.Intheupperpanelof
Fig.4theeffectduetothecorrelationbetweentheerrorinEW
measures and derived abundances is clearly visible, while the
correlationisavoidedbydefinitioninA(Fe)–EWTplane:thisis
themainreasonbehindtheuseofEWT.
The mean slope, computed by taking into account 200
Fig.5. Same of Fig. 4 but highlighting the lines detectable at
MonteCarlo simulations of random error distribution, is of
+0.13 0.02 in the A(Fe)–EWR plane, and –0.01 0.02 in the SNR=25(greypoints).Greyregionsindicatethelocationofthe
± ± undetectablelines.
A(Fe)–EWT plane (ever keeping fixed the v value at the orig-
t
inal value of the synthetic spectrum, v= 2 km/s). Such slopes
t
point out the need to increase v (of at least 0.2 - 0.3 km/s)
t
to erase the residual slope when v is optimized by using the cussedbyMagain(1984)(andshowninhisFig.2)butitseems
t
EWR.Ontheotherside,adoptingtheEWT,norelevantchange to be contrary to the findings shown in Fig. 3, where the two
inv arenecessary.Thisfindingreproducesexactlythebiasdis- methodsprovidelowerv withrespecttothetruevalue.Infact,
t t
Magain (1984) demonstrated that v is systematically overesti-
t
Theresultsareveryconsistent withthoseobtained by usingEWT,in matedwhenderivedwiththeclassicalmethod.
agreementwiththediscussionofSect.3andthefindingofFig.1. How can we explain such a discrepancy? In the real case (and
A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN) 5
also in the test performed with synthetic spectra, see Sect. 2), (b) errors in A(Fe) and EWT are still not totally independent
not all the lines can be measured and used to infer v. The ef- from each other, because both quantities suffer in uncertainties
t
fectoftheslantingmovementofthelinesshowninFig.4(upper from the adopted log (gf) values. In fact, an overestimate of
10
panel)iscriticalforveryweaklines(wheretheerrorsaremore the oscillator strengthfor a givenline will providean overesti-
relevant) but at low SNR not all these very weak features are mateofEWTbutalsoanunderestimateofA(Fe),introducinga
detectable,becausetheyaretoonoisyorindiscerniblefromthe negative,spuriousslopewhenEWTisusedtoinferv.Notethat
t
noise envelope. Hence, many weak lines that contribute to the thiseffectissecondary,ifaccuratelaboratoryoscillatorstrengths
positive slope in the A(Fe)–EWR plane will go out of the de- aretakenintoaccount.Forinstance,thetypicalerrorsoftheFeI
tectabilitythreshold,reducingthepositiveslope. log (gf)inthecriticalcompilationbyFuhr&Wiese(2006)are
10
ForagivenSNR, areasonableandgenerallyadoptedthreshold less than25%.Sucherrorstranslate inan ironabundanceerror
to measure spectral lines is to assume 3 times the uncertainty of the same order of magnitude obtained with an error of EW
providedbytheCayrelformula.Thus,forSNR=25wecanmea- of 25%orless.Thecorrelationbetweentheerrorsinlog (gf)
sure absorption lines of at least 9 mA˚ and 11 mA˚ at 5500 and and∼abundancesintroducea spuriousslope in the plane A10(Fe)-
8000A˚,respectively.Becausethedifferenceisnegligiblealong EWT,inasimilarwaytothespuriousslopeintheA(Fe)-EWR
the entire coverageof the spectrum,I adoptedconservativelya plane due to the correlationbetween abundanceand EWR dis-
thresholdof11mA˚.Fig.5reportsthesameresultsofFig.4but cussed above. Otherwise, one needs to consider also this sec-
ondary bias only if inaccurate/inhomogeneousatomic data for
highlightingthelinesreallydetectableassumingsucha thresh-
ironlinesareincludedinthelinelist;
old(greypoints),while thelineslocatedinthegreyshadedre-
(c)inthecasewherethetemperaturehasbeeninferredbyimpos-
gionsareexcludedfromtheanalysis.
ingtheexcitationalequilibrium(erasinganyslopeinA(Fe)–χ
The mean slope of the 200 MonteCarlo simulations, com- ex
plane), also θ will be affected (in a stronger way for low SNR
putedonthesurvivinglines,turnsouttobeof-0.10 0.02inthe
± spectra)bytheerrorsinEWandthereforealsoEWTwillbeaf-
A(Fe)–EWR plane, and -0.14 0.02 in the A(Fe)–EWT plane.
± fectedbythesamebias,bydefinition.
Whenwetakeintoaccounttheeffectivepossibilitytodetectvery
Note that these last two aspects have not been deeply investi-
weaklineswith largeuncertainties,the situationturnsinto the
gatedinthepresentwork
oppositeofthecasedescribedinFig.4:
– thethresholdintheA(Fe)–EWTplaneresultstobediagonal,
6. Impactofthenumberoflines
introducinganegativeslopeandtheneedtolowerv toerase
t
anytrend5: PreviousresultsarebasedonalargenumberofFeIlines(250),
allowing to well sample both weak and moderatelystrong fea-
– in both cases the derived vt will be lower than the original tures. Such an assumption allows to avoid spurious slopes (re-
valueofthesyntheticspectrum(thesamefindingdiscussed gardlessto the adoptedmethod)dueto inadequatesamplingof
aboveandshowninFig.3). linestrengths.
Some mid/high-resolution multi-object spectrographs offer
In principle,all these effectsare validalso for higherSNR,
wavelength coverages that limit the number of available Fe I
but become negligible because the quoted uncertainties in the
lines.
EWmeasurementareofonlyafewpercent.
A simple test has been performed in order to clarify how
Thisexperimentclarifiestheoriginofthesystematicunder-
thenumberoflinesiscritical.50FeIlineshavebeenextracted
estimateofv withthetwoapproaches(showninFig.3).When
t randomlyfromthemeasuredlinesinaredgiantsyntheticspec-
v isderivedfromEWR, the correlationbetweenthe EWRand
t trumwithnoiseSNR=25andv wasredeterminedwiththetwo
abundanceerrorsmovesthelinesin an asymmetricway onthe t
methods. Such a procedure has been repeated by using 200
A(Fe)–EWRplane(upperpanelofFig.4),butthethresholdof
MonteCarlo extractions. The average values of the v distribu-
detectability of the lines will populate mainly the region with t
tions is very similar to those obtained above but with a larger
A(Fe)>6.5dex with respect to other one (as appreciable in the
dispersion (σ 0.3–0.4 km/s) with respect to that obtained in
upperpanelofFig.5),leadingtoanegativeslope(andanunder- ∼
thecasedescribedinSection.4.
estimateofv).
t If the same check is performedwith a synthetic spectrum with
Whenv isderivedfromEWT,thethresholdofthelineselec-
t SNR=100, the final distributions share dispersions around the
tionshowninthelowerpanelofFig.5increasestheasymmetric
mean of about σ 0.15–0.2 km/s, in both cases, while in the
distributionofthelinesintheplane,everleadinganegativeslope ∼
case discussed in Section 4 the dispersion was very small (see
(asdemonstratedby the averageslopes in the two cases) anda
Fig.3).Alsointhiscase,suchdispersionsarelargerthanthose
slightlylowerv value.
t inferredfromEWRandEWTwhenalargenumberoftransitions
Obviously,theinclusionofweaklineswithinaccuratemea-
isused.
sureswillworsenthediscrepancy(lowerpanelsinFig.3).
Themostdiscrepantv valuesinthederiveddistributionsare
Summarizing,v inferredbyusingtheA(Fe)–EWTplaneare t
t obtained when an inadequatesampling of the weak and strong
nottotallyfreefromobservationalbiases.Infact:
linesisadopted.Infact,iftheexperimentisrepeatedimposing
(a)theuseofA(Fe)–EWTplanepartiallyerasestheeffectdueto
theconditionthatthe50randomlinesareextractedbysampling
the correlationbetween errorsin EW and abundances,but also
equallytheentireEWrange(thusprovidingastatisticallysignif-
introduces another bias due to the threshold of detectability of
icantsampleofweakandstronglines),thedispersionaroundthe
the lines (more relevant and crucial at low SNR), as described
meanofthederiveddistributionofv isreducedbyaboutafactor
above; t
of1.5and2forSNR=25and100respectively.Inthiscase,the
5 Note that the slope of the boundary between detectable and un- distributions show dispersions around the mean that are barely
detectable linesisroughly orthogonal to theshift of a linedue tothe consistentwiththoseobtainedwiththeentirelinelist.Icancon-
correlationbetweentheerrorsinEWandabundance.Basically,thetwo clude that the number of used lines and the relative sampling
slopeshaveoppositesign. betweenweakandstronglinesarecrucialelementsthatcanaf-
6 A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN)
fectthedeterminationofv moreseverelythanthechoiceofthe 9. Determinationofv:thebeststrategy
t t
EWRorEWTmethodintheanalysis.
Iproposeaneasystrategytominimizetheeffectofthebiasdis-
cussedabove,derivingthev fromEWR,beingtheapproachthat
t
givesthebetterresultswithrespecttothetruev ofthesynthetic
t
7. Impactofthecontinuumplacement spectrausedinSect.4:
(i)beingthedeterminationofv stronglyaffectedbythecor-
t
Asalreadypointedoutabove,thetestperformedwithsynthetic
related errors in abundances and EWs, the lines with high un-
spectra neglectsuncertaintiesarising fromthe continuumloca-
certaintiesintheEWmeasurementneedtobediscarded.Infact,
tion, because the continuum in artificial spectra is set at unity
asdemonstratedinFig.3,theinclusionoflineswithinaccurate
andtotallyflat,whileinrealspectraseveraladditionaleffectsaf-
EWsleadstohighlyuncorrectedv withrespecttothetruevalue.
t
fectits correctidentification.In order to clarify how muchthis
Basically,theuseoflineswithEWerrorslessthan10%provides
sourceofuncertaintyintheEWmeasurementsimpactsv,Iper-
t agoodagreementwiththev valueofthesyntheticspectra.The
t
formed two simple tests by considering synthetic spectra with
reader can use the relations providedin Sect. 4 to estimate the
SNR=25:
amplitude of the bias in v as a function of the adopted σ
t EW
(1) the EWs have been remeasured by adopting a higher and
cutoff. Note that the computationof the EW uncertainty is not
lowercontinuumlevel,withrespecttothatderivedintheprevi-
providedbyallthecodesdevelopedtomeasureEWs,butastan-
ousexperiment.Thevariationsinthecontinuumlevelareof 5
dard erroraboutthe goodnessof the fit for each measuredline
±
and 10%, translating to a systematic (positiveor negativede-
isrecommended,throughthestandarddeviationofthelocalflux
±
pendingbythesignofcontinuumvariation)offsetofthederived
residualsaftertheGaussianfitting(asperformedbyDAOSPEC)
EWs(thesamefindinghasbeendiscussedinStetson&Pancino
or by adopting other formulae, i.e. the Cayrel formula or that
(2008),seetheirFig.2).Thedifferencesbetweenv derivedwith
t proposedbyRamirezetal.(2001);
theEWR andEWT methodsare similar tothose obtainedwith
(ii)asshowninFig.3,evenifanadequatelinesselectionis
theoriginalcontinuumlevel;
applied,a residualdiscrepancybetween the derivedandtrue v
(2)thesyntheticspectrahavebeenmultipliedwithasinusoidal t
of 0.15km/sremainsinthemostcriticalcaseofSNR=25.A
curve, in order to mimic a fringing-likepattern, similar to that ∼
possible method to reduce this residual discrepancy is to per-
observed in real spectrum. Also in this case, no relevant dif-
form a linear fit obtained taking into account the uncertain-
ference (within the uncertainties) with the original analysis is
ties in both EWs and abundances (following the approach by
found.
Pressetal. (1992) as implemented in the FORTRAN routine
The continuum tracement (at least in case where the con- FITEXY). Errors in EWs have been estimated as described in
tinuum is defined globally along the entire spectrum) does not Sect.4,whiletheerrorintheabundanceofeachindividualline
affectdramaticallytheresultsdescribedinSect.4abouttherela- hasbeenderivedbycomputingtheabundancewiththeEWvar-
tivedifferencebetweenvtderivedwithEWRandEWTmethods. iedof 1σEW.Thismethodtoperformthelinearfitallowstoob-
±
tainvaluesofv consistentwithintheerrorswiththetruevalueof
t
thesyntheticspectra(seethefirstpanelofFig.3),reducingthe
8. Arealcase:UVESspectraofLMCgiants residualdiscrepancy.Usuallyinthechemicalanalysis,theslope
inA(Fe)—EWRplaneiscomputedneglectingtheuncertainties
In order to show a case based on real spectra, I derived v inthetwovariables(becausethisprocedureneedsthecomputa-
t
with the two methodsfor some LargeMagellanicCloud giants tion ofthe errorin bothEW andabundancefor eachmeasured
starsobservedwith UVES(Mucciarellietal., 2009), with typi- line)butthisapproach(orotherapproachestofindthelinearre-
calSNR=50andoriginallyanalyzedfollowingtheprescriptions lationship in data with uncertaintiesin both directions,see e.g.
byMagain(1984). Hogg,Bovy&Lang,2010)isrecommended.
Microturbulent velocities computed with the classical
method (and fixing the other parameters) are higher by 0.2 –
∼ 10. Conclusions
0.25km/sthanthose obtainedwith the prescriptionby Magain
(1984);thisdifferencetranslatestoanironabundance 0.15dex
Idrawthefollowingresultsfromtheabovetest:
∼
lowerwhentheclassicalapproachisadopted.
(1) the method to infer the microturbulent velocity from the
ApointtorecallhereisthattheTeffandvtcanbecorrelatedif EWTprovidesthesameresults(withinthequoteduncertainties)
thesequantitiesarederivedwiththeclassicalfullspectroscopic oftheclassicalmethodincaseofmoderate/highSNR( 70);
method.Infact,themajorityofweaklineshavehighχex,while (2) in case of SNR<70 both methodsunderestimateth≥e micro-
severalstrongtransitionshavelowexcitationpotentials,leading turbulentvelocityandwhenlow(25)SNRspectraareanalyzeda
toacorrelationbetweenχex andthestrengthofthelines.Also, betteragreementwiththetruevtofthesyntheticspectraisfound
variationsinTeffandvtchangedifferentlysingleandionizediron withtheclassicalmethodbasedonEWR;
lines abundances, needing to re-adjust the surface gravity. The (3) the discrepancy with the true v value (regardless of the
t
highervtcomputedwiththeclassicalmethodimpliestheneedto adopted method) worsens when EWs with high uncertainties
adjustspectroscopicallybothTeff andlogg,byincreasingthem (generally,theweakestlinesinlowSNRspectra)aretakeninto
and hence reducingthe iron abundancedifference(to less than account.Thisfindingpointsouttheneedtoestimatethequality
0.1dex). ofthemeasuredEWsforeachindividualline(aprocedureusu-
Summarizing, the absolute difference in iron abundance allynotperformedinthechemicalanalysis),inordertoreduce
computedwiththetwoapproachesisreduced(lessthan0.1dex) thissystematicunderestimate;
ifafullspectroscopicdeterminationofallatmosphericalparam- (4) these results are valid for both giantand dwarf stars, under
etersisperformed. Therefore, if one or more parameters are the assumptions that the strong lines that deviate significantly
fixed to a certain value, thiscompensationdoes not take place fromtheGaussian approximationare excludedfromthe analy-
andtheabsolutedifferencecanbehigher. sis;
A.Mucciarelli:Microturbulentvelocityinstellarspectra(RN) 7
(5)thenumberofusedlinesandtheadequatesamplingofweak
andstronglinesinthecaseofa smallnumberofavailableiron
lines affect the determination of v more than the adopted ap-
t
proachtoinferthisparameter;
(6)thespectroscopicoptimizationofalltheatmosphericparam-
eters simultaneously is recommended, in order to compensate
errorsinthedeterminationsoftheparameters.
Acknowledgements. Ithankthereferee(MatthiasSteffen)whosecommentsand
suggestionshaveimprovedsignificantlythepaper.IwarmlythankElenaPancino
forherusefulsuggestionsandacarefulreadingofthemanuscript.
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