Calibrating convective-core overshooting with eclipsing binary systems. The case of low-mass main-sequence stars

0.23 MB·

by  G. Valle

Astronomy&Astrophysicsmanuscriptno.ov-language cESO2016
(cid:13)
January8,2016
Calibrating convective-core overshooting with eclipsing binary
systems
The case of low-mass main-sequence stars.
G.Valle1,2,3,M.Dell’Omodarme3,P.G.PradaMoroni2,3,S.Degl’Innocenti2,3
1 INAF-OsservatorioAstronomicodiCollurania,ViaMaggini,I-64100,Teramo,Italy
2 INFN,SezionediPisa,LargoPontecorvo3,I-56127,Pisa,Italy
6 3 DipartimentodiFisica"EnricoFermi”,UniversitàdiPisa,LargoPontecorvo3,I-56127,Pisa,Italy
1
0 Received17/09/2015;accepted21/12/2015
2

n ABSTRACT
a
J Context.Double-linedeclipsingbinarieshaveoftenbeenadoptedinliteraturetocalibratetheextensionoftheconvective-coreover-
7 shootingbeyondtheborderdefinedbytheSchwarzschildcriterion.
Aims.Inarobuststatisticalway,wequantifythemagnitudeoftheuncertaintythataffectsthecalibrationoftheovershootingefficiency
] parameter β that is owing tothe uncertainty on the observational data. Wealso quantify the biases on the β determination that is
R causedbythelackofconstraintsontheinitialheliumcontentandontheefficienciesofthesuperadiabaticconvectionandmicroscopic
S diffusion.
Methods.Weadoptedamodifiedgrid-basedSCEPtERpipelinetorecovertheβparameterfromsyntheticstellardata.Ourgridspans
.
h themassrange[1.1;1.6] M andevolutionarystagesfromthezero-agemainsequence(MS)tothecentralhydrogendepletion.The
p β estimates were obtained b⊙y generalising the maximum likelihood technique described in our previous works. As observational
- constraint,weadoptedtheeffectivetemperatures,[Fe/H],masses,andradiiofthetwostars.
o
Results.BymeansofMonteCarlosimulations,adoptingareferencescenarioofmildovershootingβ = 0.2forthesyntheticdata,
r
t andtakingtypicalobservationalerrorsintoaccount,wefoundbothlargestatisticaluncertaintiesandbiasesontheestimatedvalues
s of β. For thefirst 80% of the MS evolution, β isbiased byabout 0.04, withthe1σ error practically unconstrained in thewhole
a −
exploredrange[0.0;0.4].Inthelast5%oftheevolutionthebiasvanishesandthe1σerrorisabout0.05.The1σerrorsaresimilar
[
whenadoptingdifferentreferencevaluesofβ.Interestingly,forsyntheticdatacomputedwithoutconvective-coreovershooting,the

1 estimatedβisbiasedbyabout0.12inthefirst80%oftheMSevolution,andby0.05afterwards.Assuminganuncertaintyof 1in
v thehelium-to-metalenrichment ratio∆Y/∆Z,wefoundalargesystematicuncertaintyintherecoveredβvalue,reaching0.2a±tthe
5 60%oftheMSevolution.Takingintoaccountboththeheliumabundanceindeterminationand1σstatisticaluncertainty,wefound
3 that intheterminalpart of theMS evolutiontheerror on theestimatedβvalues ranges from 0.05to+0.10, whileβisbasically
−
5 unconstrainedthroughouttheexploredrangeatearlierevolutionarystages.Wequantifiedtheimpactofauniformvariationof 0.24
±
1 inthemixing-lengthparameterαml aroundthesolar-calibratedvalue.Thelargestbiasoccursinthelast5%oftheevolutionwithan
0 errorontheestimatedmedianβfrom 0.03to+0.07.Inthislastpart,the1σuncertaintythataddresses statisticalandsystematic
−
. errorsourcesrangesfrom 0.09to+0.15.Finally,wequantifiedtheimpactofacompleteneglectofdiffusioninthestellarevolution
1 computations.Inthiscase−,the1σuncertaintythataddressesstatisticalandsystematicerrorsourcesrangesfrom 0.08to+0.08in
0 −
theterminal5%oftheMS,whileitispracticallyunconstrainedinthefirst80%oftheMS.
6
Conclusions.Thecalibrationoftheconvectivecoreovershootingwithdouble-linedeclipsingbinaries-intheexploredmassrange
1
and withboth components stillintheir MS phase -appears tobe poorly reliable,at least until further stellarobservables, suchas
:
v asteroseismicones,andmoreaccuratemodelsareavailable.
i
X Keywords. Binaries:eclipsing–methods:statistical–stars:evolution–stars:low-mass–stars:interiors
r
a
1. Introduction in the envelope.We focuson the convective-coreextentduring
thecentralhydrogenburningphaseoflow-massstars.
Astheresultofthehugeeffortmadeinthepastdecadestorefine
In classical modelsthe border of the convectivecore is de-
the accuracy and reliability of the stellar evolutionary predic-
terminedbymeansoftheSchwarzschildcriterion.However,this
tions,stellarevolutiontheoryhasbecomeoneofthemostrobust
borderidentifiesthelocuswheretheacceleration,nottheveloc-
areas of astrophysical research. However, several mechanisms
ity,oftheconvectiveelementvanishes,thusraisingthequestion
involvedintheevolutionofstarsarestillpoorlyunderstood.
of the extension of the overshootbeyondthe borderitself. The
Thelackofaself-consistenttreatmentofconvectioninstel- extensionoftheextra-mixingregionbeyondtheSchwarzschild
larevolutionarycodesisoneofthemajorweaknessesaffecting border is usually parametrized in terms of the pressure scale
stellarmodels.Thislackpreventsa firmandreliableprediction heightH :l =βH ,whereβisafreeparameter.
p ov p
ofthe extensionofthe convectiveregions,bothin thecoreand
Many efforts to constrain the values of β have been per-
formed since the first investigations (among them we recall
Sendoffprintrequeststo:G.Valle,[email protected] Saslaw&Schwarzschild1965;Shaviv&Salpeter1973).Anob-
Articlenumber,page1of13
A&Aproofs:manuscriptno.ov-language
vious way to obtain clues on its value and on the possible 2. Methods
dependence on the stellar mass is to compare the theoreti-
The value of the β parameter was determined by means of the
cal models computedwith differentconvective-coreovershoot-
SCEPtERpipeline,amaximum-likelihoodtechniquerelyingon
ing efficiencies with proper observations. Several methods for
a pre-computedgrid of stellar modelsand on a set of observa-
calibration have been explored in the literature. They include
tional constraints (see e.g. Valleetal. 2015b). A first applica-
the isochrone fitting of stellar cluster colour-magnitude dia-
tiontoeclipsingbinarysystemshasbeenextensivelydescribed
grams (see e.g. Bertellietal. 1992; PradaMoronietal. 2001;
inValleetal.(2015a).Webrieflysummarizethetechniquehere
Barminaetal. 2002; Brocatoetal. 2003; VandenBergetal.
andfocusonthemodificationsadoptedtoestimateβ.
2006; Bressanetal. 2012) and the comparison with recently
We assume and to be two stars in a de-
available asteroseismic constraints (e.g. Montalbánetal. 2013; S1 S2
tached binary system. The observed quantities adopted
Guentheretal.2014;Tkachenkoetal.2014;Aerts2015).
to estimate the core overshooting efficiency are
AnasriOeasnfaterhxeeaommftpaelnney,aptdhooespstriebedlleeavscaanancndeiiddoaeftaetlsh,teedsoctoubrbeeldeo-lfvionerersβhecocloaipltiisbnirgnagteioffibni--. {iqnσS1(t,T2heeff,oSb≡1,2s)e,rσve({[dTFeeqff/u,SHa1n,2]S,ti[1t,F2ie)e,s/σ.HF(]MoSr1S,21e,,2aM)c,hSσ1(,p2R,oRSin1S,2t1),2}j}.boenLthteheteuenσsct1ie,m2rtaatiinot=ny
ciency was explored in Andersenetal. (1990). By considering grid of stellar models, we define qj ≡ {Teff,j,[Fe/H]j,Mj,Rj}.
eightbinarysystems,theyfoundthatnon-standardmodelswere Let 1,2 bethesingle-starlikelihoodfunctionsdefinedas
j
required for masses higher than 1.5 M . Other investigations, L
whichtriedtoexplorethedependenceof⊙βonstellarmass,were 4 1 χ2
pieesrfroearmcheeddbsyomReibwahsaettdaiffl.e(r2e0n0t0c)onanclduCsiloanrse.tT(2h0e0in7v).eTsthigeasteiosntubdy- L1,2j =Yi=1 √2πσ1i,2×exp− 21,2, (1)
Claret(2007)indicatesthatthedependenceofβonmassismore
where
uncertainandlesssteepthanstatedbyRibasetal.(2000),reach-
itranyengpcdeiacnSapttlalilannytvcecleaoisffuntiesogifeadttβeiaro≃eln.ds(0e2w.n20eo1fruo5eg)r.phaOetsrovtfaoemrrrammaltleoc,dhraebtlmhoyewaMossβbeisvnveegarlvt&uhaeatiZn(oih2n.aeaMn.l.gd⊙a(.0t2Ma.02.1o)4ries) χ21,2 =Xi=41 qSi1,2σ−i qij2. (2)
In the computation of the likelihood functions we consider
However, we recall that the β value provided by the fitting
only the grid points within 3σ of all the variables from .
1,2
procedure depends on the input physics actually implemented S
Moreover, in the estimation process we explicitly assume that
in the evolutionary code that is used to compute stellar mod- the stars are coeval1. We also imposethatthe two stars share a
els,evenwhentheparametrizationadoptedtodescribetheover-
commonvalueofβsincethesamplingisperformedatconstant
shootingisthesame.Thus,itwouldbemoremeaningfultocom- valueofovershootingefficiency;inSect.3.2weexploretheef-
pare the convective-core extension instead of the β values ob- fect of relaxing this constraint. Then, the joint likelihood ˜ of
tainedbydifferentsetsofstellarmodels. L
the system is computed as the productof the single star likeli-
hoodfunctions.Let ˜ bethemaximumvalueobtainedinthis
Moreover, the aforementioned calibrations suffer from at max
L
step.Thejoint-starsestimatedβisobtainedbyaveragingthecor-
least two shortcomings. The first, which makes a direct com-
respondingquantitiesofallthemodelswithalikelihoodgreater
parisonofthe resultsfromdifferentstudiesdifficult,isthe lack
than0.95 ˜ .
ofhomogeneityinthetreatmentofthestatisticalerrorsowingto max
×L
In the computation we assumed as standard deviations 100
the stellar observational uncertainties. A chief difficulty is that
K in T , 0.1 dex in [Fe/H], 1% in mass, and 0.5% in radius.
the uncertainty in the derived masses is often neglected in the eff
Thecovariancematrixadoptedin the perturbationaccountsfor
finalerrorbudget.Thesecondproblemisthelackoftheoretical
a correlation of 0.95 between effective temperatures, 0.95 be-
investigationsonthepossiblebiasinthevalueofβobtainedby
tween metallicities, 0.8 between masses, and 0.0 between radii
afitofthesystemswithstellarisochrones.
(seeValleetal.2015a,fordetails).
Asafirstefforttopartiallyfillthisgap,wehereaddressmany
aspects of these questions. By means of Monte Carlo simula-
2.1.Stellarmodelgrid
tions,weprovidesomecluesonthereliabilityofconvective-core
overshooting calibration for low-mass binary systems, restrict- The grid of models covers the evolution from the zero-age
ingthe analysisto themass range[1.1;1.6] M and tothe MS main-sequence (ZAMS) up to the central hydrogen depletion
phase. This mass range contains several well-⊙studied systems of stars with mass in the range[1.1;1.6] M and initial metal-
(e.g.AQSer,GXGem,BKPeg,BWAqr,V442Cyg,ADBoo, licity [Fe/H] from 0.55 dex to 0.55 dex. T⊙he grid was com-
VZHya,V570Per,HD71636)thathavebeenusedinliterature puted by means o−f the FRANEC stellar evolutionary code
tocalibratetheβparameter(e.g.Lacyetal.2008;Clausenetal. (Degl’Innocentietal. 2008; Tognellietal. 2011) in the same
2010;Torresetal.2014). configurationaswasadoptedtocomputethePisaStellarEvolu-
tionDataBase2forlow-massstars(Dell’Omodarmeetal.2012;
WeadopttheSCEPtERpipeline(Valleetal.2014,2015b,a),
Dell’Omodarme&Valle 2013). Seventeensets of modelswere
modifiedforestimatingthecoreovershootingparameter.Wead-
computedwithavaryingcoreovershootingparameterβfrom0.0
dress in a statistical robust way the propagation of the uncer-
to0.4(thelastrepresentingapossiblemaximumvalue,seee.g.
tainty on the observationalvalues to the fitted β parameterand
the results in Claret 2007) with a step of 0.025. The details on
explore the possible biases that are due to some unconstrained
mechanismsorinputthatinfluencethe evolutioninthe consid- 1 Thisassumptionisjustifiedbecauseinbuildingthesyntheticdatasets
eredmassrange:the initialheliumcontentof thestars, andthe weimposedthattheagedifferencebetweenthebinarycomponentsmust
efficiency of microscopic diffusion and of the external convec- be 10Myr.
tion,parametrizedthroughthemixing-lengthvalueα . 2 h≤ttp://astro.df.unipi.it/stellar-models/
ml
Articlenumber,page2of13
Valle,G.etal.:Calibratingconvective-coreovershootingwitheclipsingbinarysystems
5 5 5
0.2 b = 0 0.2 0.2
b = 0.2
0 b = 0.4 0 0
2 2 2
0. 0. 0.
)M⊙0.15 )M⊙0.15 )M⊙0.15
( ( (
cc0 cc0 cc0
M 1 M 1 M 1
0. 0. 0.
5 5 5
0 0 0
0. 0. 0.
0 0 0
0 0 0
0. 0. 0.
0 1 2 3t (Gy4r) 5 6 7 0 1 t (G2yr) 3 0.0 0.5 1t. 0(Gyr)1.5 2.0
Fig.1.Left:extensionoftheconvectivecore(inM ),fromZAMStocentralhydrogenexhaustion,forthreedifferentvaluesoftheovershooting
parameterβ=0.0,0.2,and0.4foramodelofM=⊙1.12M atinitial[Fe/H]=0.0.Middle:sameasintheleftpanel,butforM=1.36M .Right:
sameasintheleftpanel,butforM=1.60M . ⊙ ⊙
⊙
thestandardgridofstellarmodels,thesamplingprocedure,and theeffectofadoptingdifferentβreferencevaluesinSect.3.1.We
the age estimation technique are fully described in Valleetal. thereforegeneratedasyntheticsampleofN =50000binarysys-
(2015b) and Valleetal. (2015a); the adopted input and the re- temsfromthegridofstellarmodelswithβ = 0.2andsubjected
lateduncertaintiesarediscussedinValleetal.(2009,2013a,b). all these systems to random perturbationon the observablesto
simulatetheeffectofmeasurementerrors.Theovershootingpa-
rameterwasthenestimatedfromadoptingthefullgridofstellar
2.1.1. Influenceofβontheextensionoftheconvectivecore
modelswithseventeendifferentβvaluesintherange[0.0;0.4].
Before we proceed with analysing the theoretical capability of It is important that that the procedure accounts simultaneously
recovering an unknown value of β from a synthetic dataset, it foralltheerrorsin theobservables,so thatitis possibleto ob-
is worth showing the effect of the differentchoices of β on the tainarobustestimateofβandofitsuncertainty.
extension of the convectivecore for the models adopted in the The results of the simulations are presented in Table 1 and
presentwork.Figure1showsforaninitial[Fe/H]=0.0theevo- in Fig. 2. The table reports the median (q ) of the estimated
50
lutionoftheconvectivecoremassversusthe evolutionarytime coreovershootingparameterandthe1σrandomerrorenvelope
forstellarmodelsofmasses1.12,1.36,and1.60M ,whichcov- (q and q ) as a function of the mass of the primary star, of
16 84
ers the range we explore here. The extensionof th⊙e convective theprimarystarrelativeager (definedasthe ratiobetweenthe
core for the lightest model computed without overshooting is ageofthestar andtheageofthesame star atcentralhydrogen
clearly negligible.Moreover,the impact of increasing β on the exhaustion), and of the mass ratio q = M /M . The envelope
2 1
maximum core extension is lower at higher masses. These re- was obtained as in Valleetal. (2015b) by computing the 16th
sults, coupled with the fact that more than 95% of the Kepler and84thquantilesoftherelativeerrorsovera movingwindow
targets in the KOI catalogue have a mass lower than about 1.6 inmassandrelativeage.
M ,arethemainreasonsforthechoiceoftheupperandlower
Themainresultisthattheestimatedβvaluesareaffectedby
lim⊙itsofthemassrangeexploredhere.
largerandomerrors,with a mean envelopehalf-widthof about
0.10. We note a very mild increase of the envelope width as
a function of the mass of the primary star; with respect to the
3. Intrinsicaccuracyandprecisionofthe
massratioq,wenoteamildshrinkingoftheenvelopeathighq.
overshootingcalibration
Themostsignificanttrendisduetotheevolutionarystageofthe
The first key point to discuss is the intrinsic precision and ac- primary star. While the overshooting parameter is consistently
curacyofthecoreovershootingcalibrationprocedure,thatis,in underestimated by about 0.04 when the primary star is in the
the ideal case in which the stellar models used in the recovery early part of its MS phase (r ≦ 0.5), in a more advanced MS
procedureperfectlymatchthedata.Inotherwords,itisimpor- stage (r & 0.7) we observe a trend similar to that described in
tanttoestablishthemagnitudeofboththerandomerrorandthe Valleetal.(2015b)formassestimates.Inthiszonethemorphol-
possiblebiasintherecoveredovershootingparameterthatisdue ogyofthegridscomputedwithvariousovershootingefficiencies
alone to the observational uncertainties that affect the observ- aremostdifferent,sincetheoverallcontractionstartsindifferent
ablesusedto constrainthe overshootingitself. Tothispurpose, regionsof the grids. As a consequence,it is more likely that a
we followed a procedure similar to that outlined in Valleetal. model computed with overshooting efficiency β is nearer to a
(2015a),asdetailedbelow. modelwithβ˜ < βthantoamodelwithβ˜ > β.Thiseffectleads
We determinedthe expectederrorsin the β estimation pro- totheobservedunderestimationofβ.Intheverylastevolution-
arystages(r 0.9)thebiasvanishessinceallthestellartracks
cessbyMonteCarlosimulations.Inthefollowingweassumea
≥
true value of β = 0.2, regardlessof the mass of the stars. This returntoevolveparallelaftertheoverallcontractionend.Inthis
assumption would not affect the validity of the results if there zonetheerrorenvelopehalf-widthshrinksfromameanvalueof
0.15toabout0.06.
wereatrendoftheovershootingparameterwiththestellarmass
(e.g.Ribasetal.2000;Claret2007)becauseweareinterestedin The apparentlack of bias in the left panel of Fig. 2 should
adifferentialeffect,thatis,inthetheoreticaldifferenceexpected be considered with caution. It is due to the adopted sampling
betweenthetrueandtherecoveredvaluesofβ.Wealsodiscuss method, which favours systems with primary stars in the later
Articlenumber,page3of13
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4 4
0. 0.
3 3
0. 0.
2 2
b 0. b 0.
1 1
0. 0.
0 0
0. 0.
1.1 1.2 1.3 1.4 1.5 1.6 0.0 0.2 0.4 0.6 0.8 1.0
M (M ) primary relative age
1 ⊙
Fig.2.Left:1σenvelopeofβestimatesthatareduetotheobservationalerrors(solidline)asafunctionofthemassoftheprimarystar.Thedashed
linemarksthepositionofthemedian.Right:sameasintheleftpanel,butasafunctionoftherelativeageoftheprimarystar.
Table1.Median(q )and1σrandomenvelope(q ,q )oftheestimatedcoreovershootingparameter,asafunctionofthemassoftheprimary
50 16 84
star,ofitsrelativeager,andofthemassratioofthesystemq.Inallthreecases,theenvelopeisobtainedbymarginalizationoveralltheremaining
parameters.
primarystarmass(M )
1.1 1.2 1.3 1.4 1.5 1.6 ⊙
q 0.13 0.10 0.10 0.10 0.07 0.07
16
q 0.21 0.20 0.20 0.20 0.20 0.20
50
q 0.30 0.28 0.30 0.30 0.30 0.32
84
primarystarrelativeager
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
q 0.03 0.04 0.05 0.04 0.04 0.04 0.04 0.03 0.03 0.10 0.17
16
q 0.15 0.15 0.16 0.16 0.16 0.17 0.20 0.17 0.12 0.19 0.20
50
q 0.34 0.33 0.31 0.33 0.33 0.35 0.35 0.35 0.33 0.28 0.28
84
massratioq
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0
q 0.07 0.07 0.06 0.06 0.06 0.06 0.10 0.11
16
q 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
50
q 0.35 0.33 0.32 0.31 0.31 0.30 0.30 0.30
84
evolutionarystages,wherethebiasinβisabsent(seeValleetal. 3.1.Effectofchangingtheβreferencevalue
2015a,foradetaileddiscussionontheadoptedsampling).Adif-
ferentsamplingwouldshowthebiasespresentatlowr. Inthissectionwebrieflypresentsomeresultsobtainedbyadopt-
ingdifferentβreferencevaluesanddiscussthedifferenceswith
In summary, even in this ideal case of a perfect agreement respecttothestandardscenarioofβ=0.2.Figure3–analogous
between the stellar tracks adopted for the recovery procedure to the right panels of Fig. 2 – shows the resulting error enve-
and the artificial stars, which are sampled from the same grid lopeobtainedbyassumingβ=0.0,0.1,0.3,and0.4asreference
ofmodels,the estimatedvalueofβis generallybiasedtowards values.Thedifferencesbetweenthepanelareapparent.
lowervaluesandshowsalargeuncertainty. Inthecaseofsamplingfromthegridwithβ=0.0,nounder-
estimationcan occur;thereforethe resultsare influencedbyan
Suchaconsiderablerandomerrorunderminesthepossibility edge effect similar to those discussed in other works for mass,
ofobtainingareliableestimateofthecoreovershootingparam- radius,andageestimates(Valleetal.2014,2015b,a).Amedian
eter from a single binary system in the considered mass range biasofabout0.12wasfounduptor 0.8.Inthelastpartofthe
≈
withstarsintheMSbecausetheprobabilityofcalibratingona evolutionthebiasisofabout0.05.Forcomparisonwealsoplot
statisticalfluctuationishigh.Moreover,thepresenceofthebias inthe figuretheenvelopeobtainedby halvingtheuncertainties
sheds doubtseven on the statistical calibrationof β that is per- on the observational parameters. In this case the bias is lower
formedbycombiningtheresultsobtainedinseveralsystems. andthe envelopeshrinks, butonly for r 0.4. Concerningthe
≥
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Valle,G.etal.:Calibratingconvective-coreovershootingwitheclipsingbinarysystems
4 reference β = 0.0 4 reference β = 0.1
0. 0.
3 3
0. 0.
β 0.2 β 0.2
1 1
0. 0.
0 0
0. 0.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
primary relative age primary relative age
4 4
0. 0.
3 3
0. 0.
β 0.2 β 0.2
1 1
0. 0.
0 reference β = 0.3 0 reference β = 0.4
0. 0.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
primary relative age primary relative age
Fig.3.SameasintherightpanelofFig.2,butsamplingfromthegridwithdifferentβvaluesreportedinthelabels.Thegreendot-dashedlines
refertothescenarioinwhichthenominalerrorsontheobservablesarehalved.
effectofthesingleerrorsources,weverifiedthattheirindividual the 1σ error envelope is wide (from overestimation of +0.1 to
halvingcontributesinnearlyequalmannertothefinalresult. underestimationof 0.025).
−
Thisscenarioisparticularlyinterestingbecauseitshowsthe
tendencytoprefermodelswithmoderateovershootingefficiency The cases of reference β = 0.3,0.4are more similar to the
even in the fit of binary systems sampled from a grid without standard scenario, showing a strong underestimation (greater
overshooting.Theresultingmedianbiasisrelevantsincein the than 0.2 for β = 0.4) in the first part of the evolution, and the
recentliterature,asdiscussedinSect.1,thereisageneralfeeling characteristicunderestimationaroundr = 0.8.Inbothcasesthe
thatβ 0.2isadequatetodescribetheobservations. latterunderestimationispartiallysuppressedwhentheerrorson
≤
The presence of a distortion in the estimates of the theobservablesarehalved.
convective-coreovershootingparametersshedssomedoubtson
theprecisionoftheestimatesatlowβ,inparticularforβ 0.1. Inconclusion,whiletheexpectedbiasdependsontheactual
≤
Thecaseofreferenceβ=0.1providesalowerbiasesthanall valueoftheovershootingefficiencyinthereferenceset,thereis
otherscenariosintheestimatedovershootingefficiency;thetrue clear evidence that all the scenarios show a large variability in
value is overestimated by about 0.05 up to r 0.8. However, theβestimatesat1σlevel.Furthermore,thebiastowardhigher
≈
theestimateisaffectedbynon-negligiblerandomerrors;evenin valuesofβthatwefoundwhensamplingfromβ=0.0suggests
the last partof the evolution– where the precisionis highest– greatcareintheevaluationoffittedβ.0.1.
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3.2.Effectofrelaxingtheassumptionofacommonβforboth systems because the convectiveenvelopevanishesand because
stars thetimescaleevolvesveryfast,thefirstisunavoidableinevery
massrange.
In the previous analyses we assumed that the two stellar com-
Toquantifythebiasesduetotheconsideredfactors,wefol-
ponentsshareacommonvalueofβ.Sinceithasbeenproposed
lowed the same procedure as described in Valleetal. (2015a).
in the literature that a possible trend of β with the stellar mass
We built several synthetic datasets of artificial binary systems
exists(Ribasetal.2000;Claret2007),weverifyinthissection
samplingfromgridscomputedwithβ=0.2,butvaryingtheini-
thatnobiasesmarourresultsthatarecausedby theestimation
tial helium content, the mixing-length value, or neglecting mi-
procedureassumptions.
croscopic diffusion. These systems were subjected to random
Tothispurposewerepeatedtheestimationoftheovershoot-
perturbationto simulate the observationalerrors, following the
ingparameterforN =10000binarysystems,sampledfromthe
sameprocedureasoutlinedinSect.3.Finally,weestimatedthe
grid of β = 0.2, without imposing the constraint that the solu-
overshootingparameterbyadoptingthestandardmulti-grid.
tionsfortheprimaryandthesecondarystarsshareacommonβ
value.Inotherwords,onlythecoevalitywasimposedduringthe
recovery.Thisprocedurewouldshowpossiblesystematicdistor- 4.1.Initialheliumabundance
tionsintheestimatedovershootingparameterforthetwostarsin
The evolution of stars strongly depends on the initial helium
thesystems.Alackofbiaswouldsupporttheconclusionthatthe
abundance.Asaconsequence,foragivenmassandage,theob-
individualβisrecoveredwellbyourtechnique.
servablestellarquantitiesdependonitsvalue.Ontheotherhand,
The results of the analysis are displayed in Fig. 4. The left
thestarsweexamineherearetoocoldtoallowaspectroscopic
panelofthefigureshowstheprobabilitydensityfunction(eval-
measurement of the surface helium content. Moreover, such a
uatedbymeansofakerneldensityestimator,seee.g.Scott1992;
value would not be representative of the initial value owing to
Venables&Ripley2002andAppendixAinValleetal.2014)of
theeffectofmicroscopicdiffusion.
the recovered β values for the whole systems – that is, impos-
This means that stellar modellers have to assume an ini-
ingthatthe twostarssharea commonvalue,asintheprevious
tial value Y for the helium abundance to adopt in stellar evo-
sections–andindividuallyforprimaryandsecondarystars.Itis
lution computations. A linear relationship between the primor-
apparentthattherecoveredvaluesarealwaysconsistentinme-
dialhelium abundanceY and metallicity Z is usually assumed.
dianwiththetruevalueβ = 0.2andthatthedistributionsofthe
Unfortunately,the value of the helium-to-metalenrichmentra-
individualestimatesarewiderthanthecommonone,especially
tio ∆Y/∆Z is still quite uncertain (e.g. Pagel&Portinari 1998;
forthesecondarystars,whicharesampledinearlierstagesofthe
Jimenezetal. 2003; Gennaroetal. 2010). This in turn leads to
their evolution.The rightpanel of Fig. 4 shows the probability
anuncertaintyintheinitialheliumabundanceadoptedinstellar
density functionof the β differencesbetween primary and sec-
computationsforagiveninitialmetallicity,whichaffectsstellar
ondarystarsfor the N binarysystems. Thefunctionis strongly
modelpredictionsand,consequently,thevalueoftheestimated
peakedatzerodifference,withthe16thand84thquantile 0.12
∓ βparameter.
respectively.
To assess the biasthatis dueto the uncertaininitialhelium
These results are reassuring for the capability of the tech-
content,wecomputedtwoadditionalgridsofstellarmodelswith
niqueofrecoveringthecorrectvalueoftheovershootingparam-
β = 0.2, the same metallicity values Z as in the standard grid,
eterforthetwostars,withoutsufferingfromdistortionsthatare
butbychangingthehelium-to-metalenrichmentratio∆Y/∆Z to
duetothedifferentmassesandevolutionarystagesfromthepri-
values 1 and 3. Then, we built two synthetic datasets, each of
maryandsecondarystar.
N = 50000 artificial binary systems by sampling the objects
In the light of these results, it is safe to conclude that the
from these two non-standard grids, and subjected the observ-
individualβforthetwostarscanbeestimatedwithoutsignificant
ablesoftheobtainedsystemstorandomperturbations.Asalast
biaswithrespecttothescenarioofconstrainedcommonvalue,at
step, the core overshooting efficiency was reconstructed using
theexpenseofanextralossofprecisionontheindividualvalues.
thestandardmulti-gridwith∆Y/∆Z =2.
However,sincenodefinitivetrendofβwiththestellarmass
The resulting error envelopes are presented in Fig. 5 as a
isfirmlyestablished,weprefertoadoptinthesamplingscheme
functionoftherelativeageoftheprimarystar.Themedianesti-
that we followed to produce the synthetic dataset a common β
matedβinthelow-heliumscenariotendstodecreasewithr up
value for both components and impose the same constraint in
to r 0.7,while the oppositetrendappearsfor the highinitial
the recoveryphaseas well.Inturn,thischoicetranslatesintoa ≃
heliumcase.Wealsonoteashiftintheestimatedβforstarsnear
conservative evaluation of the uncertainties in the estimated β
totheZAMS,positiveforthe∆Y/∆Z =1case,andnegativefor
values.
∆Y/∆Z =3.Thetrendsvisibleinthefigureareobviouslycaused
by the differential effects on the stellar evolution timescale of
4. Stellarmodelsinducedbiasesinthecalibration changingthe initialhelium contentandof assuming a different
overshootingefficiencyβ.Thegreatdifferencebetweenthetwo
procedure
panelsofFig.5posesaseriousproblemwhenevertheβvalueis
In addition to the randomfluctuation discussed in the previous attemptedtobeconstrainedbecausetheoriginalheliumcontent
section,theestimatedβvalueispronetosystematicbiasesthat ofasystemisgenerallypoorlyconstrained.
are due to stellar models adopted in the calibration procedure. The final uncertainty on β for a change of 1 in ∆Y/∆Z is
±
Several inputs or assumptions are routinely adopted in stellar presentedinFig.6andTables2and3.Thefigureandtablesshow
evolutionarycodes,whicharestillpoorlyconstrainedbyobser- thepositionofthemedianandofthe1σand2σerrorboundaries
vations. In this section we focus on three bias sources that are ontheβestimates,accountingforthechoseninitialheliumvari-
relevantinthemassrangeconsideredinthispaper:theinitialhe- ability. They were constructed by computing for each relative
liumcontent,themixing-lengthvalue,andtheefficiencyof the ager themaximumandminimumvaluesassumedbytheenve-
elementdiffusionusedtocomputestellarmodels.Whilethein- lope boundariesand by the median in all the considered initial
fluenceofthelasttwoisexpectedtodisappearformoremassive helium scenarios. As an example, the ql (qu ) value was ob-
50 50
Articlenumber,page6of13
Valle,G.etal.:Calibratingconvective-coreovershootingwitheclipsingbinarysystems
7 common 7
primaries
6 secondaries 6
5
5
y y
t t4
i i
s s
n4 n
e e
d d3
3
2
2
1
1 0
0.0 0.1 0.2 0.3 0.4 −0.4 −0.2 0.0 0.2 0.4
b difference of b
Fig.4.Left:probabilitydensityfunctionsfortheestimatedβfromN=10000systemssampledattrueβ=0.2.Theblacksolidlinecorresponds
totheestimationperformedassumingacommonvalueofβforprimaryandsecondarystar;thereddashedlinecorrespondstotheβvaluesforthe
primarystaralone;thegreeddot-dashedlinetoβvaluesforthesecondarystaralone.Right:probabilitydensityfunctionforthedifferencesofthe
estimatedβforprimaryandsecondarystars.
4 4
0. 0.
3 3
0. 0.
2 2
b 0. b 0.
1 1
0. 0.
0 0
0. 0.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
primary relative age primary relative age
Fig.5.Left:asintherightpanelofFig.2,butsamplingfromagridwith∆Y/∆Z = 1.Right:sameasintheleftpanel,butsamplingfromagrid
with∆Y/∆Z=3.Inbothcasesthereconstructionwasperformedwiththestandardgridwith∆Y/∆Z=2.
tainedbyconsideringtheminimum(maximum)valueatagiven 4.2.Mixing-lengthvalue
mass(Table2)orr(Table3)ofthemedianβobtainedsampling
from∆Y/∆Z =1,2,3grids. One of the weakest point in stellar model computations is the
treatmentoftheconvectivetransportinsuperadiabaticregimes,
whichactuallypreventsafirmpredictionoftheeffectivetemper-
atureandradiusofstarswithanouterconvectiveenvelope.
Wenoteamildincreaseoftheuncertaintywiththemassof Stellar evolutionary codes usually implement the mixing-
the primary star M ; the trend with r is clearly more problem- lengthformalism(Böhm-Vitense1958),wheretheefficiencyof
1
atic and suggeststhatforrelativeagesr 0.8the effectof the theconvectivetransportdependsonafreeparameterα thatis
ml
≤
unknowninitialheliumcontentevenpreventsthe possibilityof routinelycalibratedontheSun,aprocedurethatinourstandard
determiningthe sign of the bias. For the very last evolutionary caseprovidesα =1.74.Nevertheless,thereareseveraldoubts
ml
phases the 1σ random error on β ranges from 0.05 to +0.10, that the solar-calibrated α value is also suitable for stars of
ml
−
while at the 2σ level the value of β is almost unconstrainedin differentmassesand/orindifferentevolutionarystages(seee.g.
thewholeexploredrangeofovershootingefficiency. Ludwigetal.1999;Trampedachetal.2014).
Articlenumber,page7of13
A&Aproofs:manuscriptno.ov-language
median 1 s 2 s
4 4
0. 0.
3 3
0. 0.
2 2
b 0. b 0.
1 1
0. 0.
0 0
0. 0.
1.1 1.2 1.3 1.4 1.5 1.6 0.0 0.2 0.4 0.6 0.8 1.0
M (M ) primary relative age
1 ⊙
Fig.6.Left:overalluncertainty(statisticalandsystematic)thatisduetotheconsideredvariationsin∆Y/∆Z,independenceonthemassofthe
primarystar.Thelighterregionshowstheuncertaintyonthemedianoftheestimatedβcausedbyignoringthecorrectinitialheliumvalue.The
intermediatecolourregionsshowtheoverallerrorupto1σthatisduetothecumulatedcontributionofobservationalerrorsandsystematicbiasas
aresultoftheunknowninitialheliumvalue.Thedarkerregionscorrespondtocumulatederrorsupto2σ.Right:sameasintheleftpanel,butin
dependenceontherelativeageoftheprimarystar.
Table3.SameasTable2,butasafunctionoftherelativeageoftheprimarystar.
primarystarrelativeage
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
unknowninitialheliumcontent
q 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05
0.025
q 0.03 0.03 0.05 0.03 0.03 0.03 0.01 0.00 0.00 0.07 0.15
0.16
ql 0.08 0.11 0.16 0.16 0.14 0.12 0.09 0.07 0.07 0.16 0.20
50
qu 0.28 0.26 0.23 0.20 0.25 0.28 0.30 0.27 0.12 0.20 0.23
50
q 0.38 0.38 0.38 0.35 0.38 0.38 0.38 0.38 0.35 0.30 0.30
0.84
q 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.38
0.975
unknownmixing-length
q 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03
0.025
q 0.01 0.03 0.04 0.03 0.03 0.04 0.03 0.03 0.03 0.07 0.11
0.16
ql 0.12 0.12 0.14 0.13 0.12 0.17 0.19 0.15 0.12 0.15 0.17
50
qu 0.15 0.17 0.19 0.19 0.20 0.23 0.20 0.17 0.14 0.23 0.27
50
q 0.35 0.35 0.35 0.34 0.35 0.36 0.35 0.35 0.34 0.30 0.35
0.84
q 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
0.975
unknownelementdiffusionefficiency
q 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01
0.025
q 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.03 0.03 0.05 0.12
0.16
ql 0.13 0.13 0.15 0.16 0.16 0.17 0.20 0.17 0.12 0.17 0.17
50
qu 0.15 0.15 0.16 0.17 0.19 0.23 0.28 0.26 0.20 0.19 0.20
50
q 0.34 0.33 0.31 0.33 0.35 0.36 0.38 0.38 0.36 0.31 0.28
0.84
q 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.38
0.975
To establish the impact of assuming different values of su- jectedtheobservablestorandomperturbations.Finally,theover-
peradiabaticconvectiveefficiency,we computedtwo additional shootingefficiencywas reconstructedusingthe standardmulti-
grids of stellar models with β = 0.2, but with α = 1.50 and gridwithα =1.74.
ml ml
α = 1.98. As in the previous section, we built two synthetic
ml Theassumedscenarioisappropriateforsystemswithq 1;
datasets, each of N = 50000artificialbinarysystems, by sam- ≈
thesestarsareexpectedtoshareacommonvalue–whateveritis
plingtheobjectsfromthesetwonon-standardgrids,andwesub-
–ofmixing-length.Thehypothesisismorequestionableforsys-
Articlenumber,page8of13
Valle,G.etal.:Calibratingconvective-coreovershootingwitheclipsingbinarysystems
Table2.Overalluncertainty,statistical(1σand2σ)andsystematic(ql , 4.3.Heliumandheavyelementdiffusion
50
qu )asaresultoftheconsideredvariationsin∆Y/∆Z,α ,andmicro-
50 ml
scopicdiffusionefficiencyasafunctionofthemassoftheprimarystar.
Anothertrickyprocesstoimplementinstellarevolutioncodesis
the diffusion of helium and heavy elements. Depending on the
primarystarmass(M )
1.1 1.2 1.3 1.4⊙ 1.5 1.6 efficiency of microscopic diffusion, the surface abundances of
the chemical elements change with time. As an example, dur-
unknowninitialheliumcontent
ingthecentralhydrogenburning,thesurface[Fe/H]dropsfrom
q 0.01 0.00 0.00 0.00 0.00 0.00
0.025 the ZAMS value and reaches a minimum at about 90% of its
q 0.12 0.07 0.07 0.07 0.05 0.03
0.16 evolutionbeforecentralhydrogendepletion(seee.g.Fig.12in
ql 0.21 0.20 0.20 0.20 0.20 0.17
50 Valleetal.2014).Afterthispointtheconvectiveenvelopesinks
qu 0.23 0.22 0.21 0.23 0.23 0.23
50 inwardsinmoreinternalregions,wheremetalswerepreviously
q 0.32 0.30 0.30 0.33 0.34 0.35
0.84 accumulatedby gravitationalsettling, leading to an increase of
q 0.39 0.38 0.39 0.40 0.40 0.40
0.975 thesurfacemetallicity.Thesizeoftheeffectdependsonboththe
unknownmixing-length
mass of the star and its initial metallicity. The lower the initial
q 0.01 0.00 0.00 0.00 0.00 0.00
0.025 metallicity and the thinner the convective envelope, the higher
q 0.09 0.07 0.07 0.07 0.07 0.07
0.16 thesurfacemetallicitydrop.
ql 0.17 0.16 0.17 0.17 0.18 0.20
50
qu 0.30 0.28 0.25 0.23 0.23 0.20 IntheconsideredmassrangeofMSstarstheobserved[Fe/H]
50
q0.84 0.38 0.36 0.35 0.33 0.33 0.33 mightthereforebequitedifferentfromtheinitialone–ifinhibit-
q0.975 0.40 0.40 0.40 0.40 0.40 0.40 ing processesare inefficient– dependingon the stellar age and
unknownelementdiffusionefficiency mass. Hence the microscopic diffusion effect should be taken
q 0.01 0.00 0.00 0.00 0.00 0.00 intoaccountwheneverthestructuralcharacteristicsoflow-mass
0.025
q 0.10 0.05 0.05 0.07 0.07 0.07 MS stars have to be determined.Neglectingmicroscopicdiffu-
0.16
ql 0.17 0.17 0.17 0.17 0.18 0.20 sion in the stellar models used in the recovery procedure and
50
qu 0.21 0.20 0.20 0.20 0.20 0.20 assumingthattheobserved[Fe/H]is coincidentwith theinitial
50
q 0.30 0.28 0.30 0.30 0.32 0.33 onewillintroduceasystematicbias.
0.84
q 0.38 0.38 0.38 0.40 0.40 0.40
0.975 However, the efficiency of diffusion has been questioned
bysome authors(see e.g.Kornetal. 2007; Grattonetal. 2011;
Nordlanderetal. 2012; Gruytersetal. 2014). Moreover, some
temswithsignificantlydifferentmasses,sinceapossibletrendof widelyusedstellar modelgridsin theliterature,namelyBaSTI
(Pietrinfernietal. 2004, 2006) and STEV (Bertellietal. 2008,
α withthestellarmassisstilluncertain(Trampedach&Stein
ml
2009), do not implement diffusion. The same occurs in stel-
2011;Bonacaetal.2012;Mathuretal.2012;Tanneretal.2014;
lar models used in some grid-based techniques, such as RA-
Magicetal. 2015). Therefore the results presented here can be
DIUS (Stelloetal. 2009) and SEEK, which both adopt a grid
consideredastheextremevariationontheβcalibrationfordif-
of models computed with the Aarhus STellar Evolution Code
ferencesinthemixing-lengthparameterof 0.24.
± (Christensen-Dalsgaard2008).
The error envelopesas a function of the relative age of the
primary star are shown in Fig. 7. The trends caused by assum- Thus,itisworthwhiletoanalysethedistortionintheinferred
ing differentmixing-lengthvalues are slightly different; lower- value of the convective-coreovershootingefficiency that arises
ingα hastheeffectofunderestimatingtheβvalueforr 0.7, whenevertheeffectsofthediffusionareneglectedinstellarmod-
ml
whileforr 0.9theovershootingefficiencyisoverestimat≤ed.In els. The maximumbias caused by the elementdiffusionuncer-
contrast,an≥highervalueofα leadstoaβunderestimateinthe taintyontheestimatedβvaluewasassessedbycomputingagrid
ml
lastpartoftheMSevolution. ofmodelswithβ = 0.2andnoelementdiffusion.Asinthepre-
vioussection,asyntheticdatasetof N = 50000binarysystems
Thefinaluncertaintyonβthatisduetotheignoranceofα
ml was sampled from this grid and subjected to random perturba-
is presentedinFig. 8 andTables2 and3.As expected(seeleft
tions.Theβvalueswerethenestimatedbyadoptingthestandard
panelin Fig. 8), the bias caused by the exploreduncertaintyin
multi-overshootinggrid.
the mixing-lengthvalue vanishesformassive systems. In these
casestheprimarystarsdonotpresentaconvectiveenvelope. TheresultsarepresentedinFig.9;thefigureshowsthe1σβ
envelope as a function of the relative age of the primary star.
Finally(rightpanelinFig.8),amoderatebiasofabout 0.05
± Overall, the diffusion-induced bias is the lowest of those ex-
in the last stages of the evolution is apparent, an unfortunate
ploredinthepaper.Thelargestdifferencefromthestandardsce-
event,since this zonehas been foundin the precedingsections
nariooccursforsystemswhoseprimarystarhasarelativeagein
toofferthebestopportunityfortheβestimate.
therange0.5<r<0.8.Intheterminalphaseoftheevolution,a
Toexploreingreaterdetailthebiasthatisduetothelackof smallunderestimationof0.025occurs.Thefinaluncertaintyon
informationaboutthe correctmixing-lengthvalueto adopt,we βthat is dueto the lack of knowledgeofdiffusionefficiencyis
present in Table 4 the cumulated envelopeobtained by consid- presentedinFig.10andTables2and3.Asforthemixing-length
eringonlysystemswithasecondarystarmoremassivethan1.4 variability, we present in Table 4 the detail about the expected
M and than 1.5 M . These stars have a progressively thinner bias and random uncertainty in the subsets of masses obtained
co⊙nvective envelope⊙, and therefore they are increasingly more byrestrictingtosystemswithsecondarystarmoremassivethan
independentof the efficiency of the superadiabatic convection. 1.4 M and than 1.5 M . As a result of two competing effects
Asexpected,inthesesubsetsofmassesthewidthofthebiasin- (i.e.th⊙e thinningofthe⊙convectiveenvelopeandthe faster evo-
tervalinβ,forr=1.0shrinksfrom0.10inthestandardscenario lutionarytimescale),weobtainedthatthedifferencesamongall
to0.05and0.03,respectively. thesescenariosarealmostnegligible.
Articlenumber,page9of13
A&Aproofs:manuscriptno.ov-language
4 4
0. 0.
3 3
0. 0.
2 2
b 0. b 0.
1 1
0. 0.
0 0
0. 0.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
primary relative age primary relative age
Fig.7.Left:asintherightpanelofFig.2,butsamplingfromagridwithα =1.50.Right:sameasintheleftpanel,butsamplingfromagridwith
ml
α =1.98.Inbothcasesthereconstructionwasperformedwiththestandardgridwithα =1.74.
ml ml
median 1 s 2 s
4 4
0. 0.
3 3
0. 0.
2 2
b 0. b 0.
1 1
0. 0.
0 0
0. 0.
1.1 1.2 1.3 1.4 1.5 1.6 0.0 0.2 0.4 0.6 0.8 1.0
M (M ) primary relative age
1 ⊙
Fig.8.Left:overalluncertainty(statisticalandsystematic)thatisduetotheconsideredvariationsinα ,independenceonthemassoftheprimary
ml
star.Thelighterregionshowstheuncertaintyonthemedianoftheestimatedβthatisaresultofignoringthecorrectmixing-lengthvalue.The
intermediatecolourregionsshowtheoverallerrorupto1σcausedbythecumulatedcontributionofobservationalerrorsandsystematicbiasthat
isduetotheunknownmixing-lengthvalue.Thedarkerregionscorrespondtoerrorsupto2σ.Right:sameasintheleftpanel,butindependence
ontherelativeageoftheprimarystar.
5. Conclusions themass,andtheradiusofthetwostars.Thegridofstellarmod-
elswascomputedfortheevolutionaryphasesfromtheZAMSto
We theoretically investigated the statistical and systematic er-
thecentralhydrogendepletioninthemassrange[1.1;1.6] M ,
rors in the calibration of the convective-coreovershootingeffi- with 17 different values of the core overshooting parameter⊙β
ciencyonlow-massbinarysystemswhosetwo componentsare
from0.0to0.4.
in the MS phase. We used the grid-based pipeline SCEPtER
(Valleetal. 2014, 2015b,a) and adopted as observational con- Thestatisticaluncertaintythataffectstheβvalueprovidedby
straints the stellar effective temperature,the metallicity [Fe/H], thecalibrationprocedurearisingfromtheobservationalerrorsis
Articlenumber,page10of13