Confronting uncertainties in stellar physics: calibrating convective overshooting with eclipsing binaries

0.54 MB·English

by  Richard J. Stancliffe

Astronomy&Astrophysicsmanuscriptno.OSCalibration (cid:13)c ESO2015
January23,2015
Confronting uncertainties in stellar physics:
calibrating convective overshooting with eclipsing binaries
R.J.Stancliffe1,L.Fossati1,J.-C.Passy1,andF.R.N.Schneider1,2
1 Argelander-Institutfu¨rAstronomie,UniversityofBonn,AufdemHu¨gel71,D-53121Bonn,Germany
2 DepartmentofPhysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,U.K.
5 ABSTRACT
1
0 Aspartofalargerprogramaimedatbetterquantifyingtheuncertaintiesinstellarcomputations,weattempttocalibratetheextent
2 ofconvectiveovershootinginlowtointermediatemassstarsbymeansofeclipsingbinarysystems.Wemodel12suchsystems,with
componentmassesbetween1.3and6.2M⊙,usingthedetailedbinarystellarevolutioncodestars,producinggridsofmodelsinboth
n metallicityandovershootingparameter.Fromthese,wedeterminethebestfitparametersforeachofoursystems.Forthreesystems,
a noneofourmodelsproduceasatisfactoryfit.Fortheremainingsystems,nosinglevaluefortheconvectiveovershootingparameterfits
J
allthesystems,butmostofoursystemscanbewelldescribedwithanovershootingparameterbetween0.09and0.15,corresponding

1 toanextensionofthemixedregionabovethecoreofabout0.1-0.3pressurescaleheights.Oftheninesystemswhereweareableto
2 obtainagoodfit,sevencanbereasonablywellfitwithasingleparameterof0.15.Wefindnoevidenceforatrendoftheextentof
overshootingwitheithermassormetallicity,thoughthedatasetisoflimitedsize.Werepeatourcalculationswithasecondevolution

] code,mesa,andwefindgeneralagreementbetweenthetwocodes.Fortheextensionofthemixedregionabovetheconvectivecore
R requiredbythemesamodelsisabout0.15-0.4pressurescaleheights.ForthesystemEICep,wefindthatmesagivesanovershooting
S regionthatislargerthanthestarsonebyabout0.1pressurescaleheightsfortheprimary,whileforthesecondarythedifferenceis
. only0.05pressurescaleheights.
h
p Keywords.stars:evolution,binaries:eclipsing,stars:interiors,stars:low-mass
-
o
r
t 1. Introduction lutionarytracks,suchasthederivationofthemassesandradiiof
s
a transitingexoplanets.
[ Using observations to constrain stellar evolution models is In its current status, bonnsai contains sets of evolutionary
one of the primary drivers of stellar astrophysics research.
1 tracks only for intermediate to massive stars (M>5M⊙) calcu-
Nevertheless, it is not straightforward to directly link observa-
v lated for three differentmetallicities: Galactic, LMC and SMC
tions and theory in a fully consistent manner, properly taking
2 (Brottetal. 2011; Ko¨hleretal. 2015). We aim at extendingthe
2 into accountthe uncertainties,particularlythe theoreticalones. modeldatabasetolowermassstarsfrom0.8to10M ,andfora
⊙
3 Asaresult,observersoftenconsidertheirpreferredsetofstellar varietyofdifferentmetallicities. Thisextensionof bonnsaiwill
5 evolutiontracksasintrinsicallycorrect,andonlyrecentlycom-
bemadebycalculatingnewlargegridsofmodelsinitiallywith
0 parisonsbetweenthestellar parametersgatheredfromdifferent twodifferentcodesstars(Eggleton1971;Stancliffe&Eldridge
1. setsofmodelshavebeguntobeperformed,mostlyforlow-mass 2009) and mesa (Paxtonetal. 2011), with the intention to add
stars (e.g. Casagrandeetal. 2011). This “exercise” should be
0 furthergridscalculatedwithothercodes.However,thefirststep
5 regularly performed, but in order to do that, particularly for a inthecreationofthesegridsisthecalibrationofcertainparame-
1 large number of stars, one needs to develop a dedicated auto-
ters.Inthiswork,wefocusontheissueofconvectiveovershoot-
: matictool.
v ing.
i To this end, we have set up the “ConfrontingUncertainties One of the key uncertainties in the evolution of main se-
X
in Stellar Physics’ (CUSP) project, the aim of which is to bet-
quence stars is the size of the convectivecore. It is widely ac-
r ter quantify the impact of theoretical uncertainties in the use
a ceptedthatmodelsbasedoneithertheSchwarzchildorLedoux
ofstellarevolutionmodelstodeterminefundamentalstellarpa-
criterionproducecoresthataretoosmalltomatchobservations
rametersfrom observables. We intend to make use of bonnsai1
andhencesomeformofovershootingmustbeappliedtostellar
(Schneideretal. 2014), a publicly available tool, which allows
models(e.g.Maeder&Meynet1991).Byovershootingwesim-
onetoderivestellar parameters(e.g.,mass,radius,age)froma
plymeanthatthechemicallymixedregioninthestar’scorehas
setofobservationalparameters(e.g.,effectivetemperature,sur-
been extendedbeyondthe convectiveboundarypredicted from
facegravity,rotationalvelocity),properlyaccountingfortheob-
standardstellar theory.Thisadditionalmixing couldbe caused
servationaluncertainties.Wewillthenexploretheimpactofboth
byany numberofphenomena,and is notnecessarilyrelated to
observationalandtheoreticaluncertaintiesintheanalysisofpar-
the motionof materialdrivenbyconvection‘overshooting’the
ticular astrophysical problems which are directly linked to the
formalconvectiveboundary.
estimationofstellarfundamentalparametersonthebasisofevo-
Convective overshooting is a key component of canonical
stellarmodelsbutitrequirescalibration.Manypossiblemethods
1 The bonnsai web-service is available at for calibration exist, including the fitting of isochrones to stel-
www.astro.uni-bonn.de/stars/bonnsai. lar cluster colour-magnitude diagrams (e.g. VandenBergetal.
1
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
2006). More recently, asteroseismology has opened up a new
avenue for calibration of mixing properties in stellar interiors
(e.g.Montalba´netal.2013;Guentheretal.2014).Aerts(2015)
presents 16 OB dwarfs for which asteroseismic determinations
of the extent of overshooting have been made. Similar anal-
yses have also been carried out by Neineretal. (2012) and
Tkachenkoetal.(2014).
Here we focus only on calibrations using binary systems.
Schro¨deretal. (1997) used ζ Aurigae-type systems to attempt
tocalibratetheextentofovershooting.Theycouldfindadequate
fits to their systems (with masses between 2.5 and 6.5M ) us-
⊙
ingovershootingequivalentto0.24-0.32pressurescaleheights.
Subsequently,Polsetal.(1997)usedthesameevolutionarycode
and overshooting prescription to look at 49 eclipsing binary
systems taken from the compilation of Andersen (1991). They
found that models with and without overshooting could ad-
equately fit the observations of the majority of the systems.
However,for three systems (namely AI Hya, WX Cep and TZ
For)modelswithenhancedmixingprovidedabetterfit.
Further attempts to calibrate convective overshooting have Fig.1.Hertzsprung-Russelldiagramshowingeclipsingbinaries
been made by Claret (2007). He used 13 double-lined eclips- fromthe Torresetal. (2010) sample,with the systemswe have
ingbinariescoveringa rangeofevolutionarystatesandmasses selected shown in red. Stellar evolutionary tracks computed
(from1.3to nearly30M⊙).Moderateamountsofovershooting withoutovershootingareshowntogiveanapproximateindica-
ofaround0.2timesthepressurescaleheightwerefoundtobest tionofmassandevolutionarystate.
fit the data, with little evidencefor a mass dependency.This is
in contrast to the earlier work of Ribasetal. (2000), who sug-
gestedamassdependenceforovershootingmayexist,basedon
eter). Thisformalismensuresa smooth transitionbetween sys-
asampleof8starsbetween2and12M .Morerecentworkby
⊙
temswithandwithoutconvectivecores.
Meng&Zhang(2014),usingfoureclipsingbinariesinthemass
Eachmodelisevolvedfromthepremainsequenceusing999
range1.3-3.6M ,alsofindsnoevidenceforamassdependency
⊙ mesh points2. The mixinglength α is set to 2.0, based on cali-
toovershooting,thoughthemassrangeismuchsmallerthanthe
brationtoaSolarmodel.Nomasslossisincluded.Inaddition,
twostudiesmentionedabove.
eachbinaryisplacedinawideorbitsothatthereisnointeraction
In this work, we revisit the issue of calibrating convective
between the components– we do not attempt to reproducethe
overshootingusing eclipsing binaries. Ultimately, the aim is to
observedorbitalperiod.Similarly,thesemodelsarenon-rotating
arriveatareliabledeterminationthatcanbeusedforthecompu-
andwe donottrytoreproducetheobservedrotationalvelocity
tationoflargegridsoflow-massstellarmodelsforusewiththe
bonnsaitool. forthose systems whereit hasbeen measured.Rotationalmix-
ingwouldalso acttoincreasethesize ofthechemicallymixed
region.Providedourtargetsystemsdonotrotaterapidly,theuse
2. Stellarmodels ofnon-rotatingmodelsshouldsuffice.Eachsystemisevolvedto
Computations in this work were made using the stars the point where the primary has comfortably exceeded the ob-
servedprimaryradius.
stellar evolution code originally developed by Eggleton
(1971) and updated by many authors (e.g. Polsetal.
1995). The code is freely available for download from
http://www.ast.cam.ac.uk/∼stars. This code solves the 3. Results
equations of stellar structure and chemical evolution in a fully
We attempted to model11 binarysystems: V364Lac, AI Hya,
simultaneous manner, iterating on all variables at the same
EI Cep, TZ For, WX Cep, V1031 Ori, SZ Cen, AY Cam, AQ
time in order to converge a model (see Stancliffe 2006, for
Ser,V539AraandCVVel.Thepropertiesofeachofthesesys-
a detailed discussion). The version employed here is that of
temsarelistedinTable3.Theseparticularsystemswereselected
Stancliffe&Eldridge (2009) which was developed for doing
fromthesampleofTorres,Andersen,&Gime´nez(2010),which
binarystellar evolution.Thecodetreatsallformsofmixingby
lists the known eclipsing binaries whose parametershave been
meansofadiffusiveformalism(Eggleton1972).
determinedtobetterthan3%.Wechosethosesystemswhichap-
Schro¨der,Pols,&Eggleton (1997) describe the implemen-
pear the mostevolvedin the Hertzsprung-Russelldiagram(see
tationof overshootingin the code.Rather thanapplyingan ex-
Fig.1),withbothcomponentsofthesystembeingclearlysepa-
tensiontotheconvectiveregionthatissomefractionofapres-
ratedfromthezeroagemainsequence.Evolvedsystemsshould
sure scale height,thisimplementationmakesanadjustmentdi-
bemoresensitivetotheeffectsofovershootingastheyhavebeen
rectlyto the convectivecriterion(inthiscase theSchwarzchild
influenced by the process for longer. Where possible, we have
criterion).Aregionisdeterminedtobeconvectivelyunstableif
avoided short periodsystems as tidal forcescould have altered
▽ >▽ −δ,where
rad ad thestellarstructureandevolution.
δ
δ= ov (1)
2.5+20ζ+16ζ2 2 Select model sequences have been constructed using 1999 mesh
points,andseparately,twicethenumberoftimesteps.Theevolutionary
with ζ being the ratio of radiation to gas pressure and δov is a tracksareindistinguishablefromtheonespresented.Wearetherefore
constantthat must be determined(i.e. the overshootingparam- satisfiedthatthecomputationsarenumericallyconverged.
2
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
System Period Spectral Mass σ Radius σ T σ logg σ logL/L σ
eff ⊙
(d) type (M ) (R ) (K)
⊙ ⊙
V539Ara 3.17 B3V 6.240 0.066 4.516 0.084 18100 500 3.924 0.016 3.293 0.051
B4V 5.314 0.060 3.428 0.083 17100 500 4.093 0.021 2.955 0.055
CVVel 6.89 B2.5V 6.086 0.044 4.089 0.036 18100 500 3.999 0.008 3.207 0.049
B2.5V 5.982 0.035 3.950 0.036 17900 500 4.022 0.008 3.158 0.049
V364Lac 7.35 A4m 2.333 0.014 3.309 0.021 8250 150 3.766 0.005 1.658 0.032
A3m 2.295 0.024 2.986 0.020 8500 150 3.849 0.006 1.621 0.031
AIHya 8.29 F2m 2.140 0.038 3.916 0.031 6700 60 3.583 0.006 1.443 0.017
F0V 1.973 0.036 2.767 0.019 7100 65 3.849 0.005 1.242 0.017
EICep 8.44 F3V 1.772 0.007 2.897 0.048 6750 100 3.763 0.014 1.194 0.030
F1m 1.680 0.006 2.330 0.044 6950 100 3.929 0.016 1.056 0.030
TZFor 75.67 G8III 2.045 0.055 8.320 0.120 5000 100 2.908 0.013 1.589 0.037
F7IV 1.945 0.027 3.965 0.088 6350 100 3.531 0.018 1.361 0.033
WXCep 3.38 A5V 2.533 0.050 3.996 0.030 8150 250 3.638 0.005 1.801 0.054
A2V 2.324 0.045 2.712 0.023 8900 250 3.938 0.006 1.617 0.050
V1031Ori 3.41 A6V 2.468 0.018 4.323 0.034 7850 500 3.559 0.007 1.804 0.112
A3V 2.281 0.016 2.978 0.064 8400 500 3.848 0.019 1.598 0.105
SZCen 4.11 A7V 2.311 0.026 4.556 0.032 8100 300 3.485 0.006 1.904 0.065
A7V 2.272 0.021 3.626 0.026 8380 300 3.676 0.006 1.765 0.062
AYCam 2.73 A0V 1.905 0.040 2.772 0.020 7250 100 3.832 0.004 1.280 0.025
F0V 1.707 0.036 2.026 0.017 7395 100 4.058 0.005 1.042 0.025
AQSer 1.69 F5 1.346 0.024 2.281 0.014 6430 100 3.850 0.009 0.901 0.027
F6 1.417 0.022 2.451 0.027 6340 100 3.810 0.012 0.939 0.042
Table 1. Parameters of the eclipsing binary systems used in this study. All the data comes from the compilation of Torresetal.
(2010).
Foreachsystem,werunagridofmodelsacross4metallic-
4
ities (namely Z=0.01, 0.02, 0.03, 0.04) and with overshooting
parametersδ from0to0.30instepsof0.03.Theheliumcon-
ov
tentispresumedtovaryasY=0.25+15Z.Toassessthequality
3.5
of fit, for each timestep in the model sequence we calculate a
goodness-of-fitviatheformula:
(cid:12)
P= exp −[xi−µi]2 (2) R/R 3
 2σ2 
wherYeix istherelevaintparameterfromthestellarmodel(R,or
i 2.5
T )foreachstarinthesystem,andµ andσ aretheobserved
eff i i
quantity and its error bar respectively. Note this means we are
fitting both the primary and secondary simultaneously at each
2
timestep.If ourmodelfits perfectlywe obtainP=1. We specif- 7400 7200 7000 6800 6600 6400 6200 6000
ically use only R and Teff in this fit (and not surface gravity or Teff/K
luminosity)asthesearethedirectlymeasuredquantities.Forthe
purposes of this study, we have not investigated the effects of
Fig.3. Evolutionary tracks in the radius-T plane for the pri-
eff
uncertaintiesinthemassdeterminations.
mary star of EI Cep computed with different metallicities:
Z=0.015(lightgrey),0.02(mediumgrey)and0.025(darkgrey).
3.1.EICep Crosses representthe best fit modelsfor each track and the er-
rorbarsdenotetheobservedpropertiesofthesystem.
A best fit for EI Cep is obtainedwith our Z= 0.02 modelwith
δ = 0.15 model, where we obtain P = 0.9066. In Fig. 2, we
ov
showtheevolutionarytracksforbothcomponentsofthissystem
intheHRdiagram,togetherwiththeobservedcharacteristicsof valuesofδ becausethesemodelsarealreadyhotterandmore
ov
the system. Our best fit timestep is displayed by the crosses in luminous. The hook at the end of the main sequence occurs
thefigure. at higher temperatures at lower metallicity and for sufficiently
Forthissystem,wehavealsocomputedmodelsformetallic- small δ , the observed temperatures cannot be reached while
ov
itiesofZ=0.015andZ=0.025inordertogetsomeideaofhow thestarisonthemainsequence.
sensitive δ is to changes in metallicity. The best fit probabil- Asanadditionaltestforthissystem,wealsocomputemod-
ov
ityfallsoffrapidlyasmetallicityofthemodelsisdecreased.At els where the initial helium content is varied by ±0.05 (with a
Z=0.015,P=3.292×10−2,whereasthedeclineismuchslower corresponding variation in the hydrogen abundance – i.e. the
at higher metallicites, falling from P = 0.6880 at Z=0.025 to metalabundance,Z,isheldconstant).Byreducing(increasing)
P = 0.3929 at Z=0.03. The reason for this is shown in Fig. 3. the helium abundance, a model with the same Z and δ be-
ov
Overshootinghastwoeffects:primarilyitextendsthelengthof comes larger (smaller) at a given temperature but the effect is
the main sequence, but it also makes the star slightly more lu- quite small. This means that models with lower helium abun-
minousatagiventemperature.Lowermetallicitiesfavourlower dancetendtorequiresmallervaluesofδ .ForZ=0.02,amodel
ov
3
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
4 1
0.9
0.8
3.5
0.7
0.6
(cid:12)
R/R 3 P 0.5
0.4
2.5 0.3
0.2
0.1
2
7400 7200 7000 6800 6600 6400 6200 6000 0
T /K 1.2e+09 1.3e+09 1.4e+09 1.5e+09 1.6e+09
eff
Age (years)
1
4.5
0.9
0.8
4
0.7
0.6
(cid:12)
R
R/ 3.5 P 0.5
0.4
0.3
3
0.2
0.1
2.5
7400 7200 7000 6800 6600 6400 6200 0
T /K 9e+08 9.2e+08 9.4e+08 9.6e+08 9.8e+08 1e+09
eff
Age (years)
5 1
0.9
4.5
0.8
4 0.7
0.6
(cid:12)
R/R 3.5 P 0.5
0.4
3
0.3
2.5 0.2
0.1
2
9200 9000 8800 8600 8400 8200 8000 7800 0
T /K 5e+08 5.1e+08 5.2e+08 5.3e+08 5.4e+08 5.5e+08
eff
Age (years)
Fig.2. Left column: Evolutionary tracks in the radius-T plane for the systems EI Cep (top), AI Hya (middle) and WX Cep
eff
(bottom).Theprimaryisdenotedbythedarkgreylineandthesecondarybythelightgreyline.Crossesrepresentthebestfitmodels
foreachtrackandtheerrorbarsdenotetheobservedsystem.Rightcolumn:P-valuesforthesystemsasafunctionofageforthe
systemsEICep(top),AIHya(middle)andWX Cep(bottom).Theprimaryisdenotedbythedarkgreylineandthesecondaryis
denotedbythelightgreyline.TheblacklineistheP-valueforthesystemasawhole.
with Y = 0.275 gives a best fit at δ = 0.12 with P = 0.8595, 3.1.1. Comparisonofevolutioncodes
ov
whereas one with Y = 0.285 gives δ = 0.18 and P = 0.9092.
ov
Notethatthisisamarginallybetterfitthanourstandardcase.
In addition to our stars models, we have also computed a set
of models using the stellar evolution code mesa (Paxtonetal.
2011).FollowingtheprescriptionofHerwig(2000),convective
overshooting in this code is implemented by means of a dif-
4
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
fusive exponential formalism whereby the extent of mixing is
4
computedviatheequation
2z
D = D exp − , (3) 3.5
OV 0 fH !
p
wtivheerreegDio0n,isztihsethMeLdTistdainffcuesfioronmcotheefficcoiennvtecintisviedebothuendcaornyv,eHc- R/R(cid:12) 3
p
is the pressure scale height at the convective boundary and f
is a dimensionlessfreeparameter.Motivatedbythe workfrom
2.5
VandenBergetal.(2006),theovershootingparameterfollowsa
rampequation:
2
f = f0 1−cos π M∗−Mmin (4) 7400 7200 7000 6800 6600 6400 6200 6000
2 " Mmax−Mmin!# Teff/K
where f0 is a constant, M∗ is the current stellar mass of the Fig.4. Evolutionary tracks in the radius-Teff plane for EI Cep.
model, M is the stellar mass below which overshoot mix- Solid lines denote models computed with stars, while dashed
min
ing does not occur, and M the stellar mass above which linesdenotemodelscomputedwithmesa.Darkgreydenotesthe
max
f = f0. VandenBergetal. (2006) gives Mmin = 1.1M⊙ and primary and light grey the secondary.The best fit stars model
Mmax =1.8M⊙.Previously,aparameterof f0 ≃0.014wasfound isdenotedwithcrosses,whilethebestfitmesamodelisdenoted
to reproducethe width of the main sequence(for more details, withstars.Theblackerrorbarsdenotetheobservedpropertiesof
seeHerwig2000). thesystem.
Themesa modelgridisrunoverthesamemetallicityrange
as the stars grid (namely Z=0.01, 0.02 and 0.03) and the free
parameter f isvariedfrom0to0.05instepsof0.005.
0
ThebestfitmodelforEICepascomputedwithmesa isfor
Z = 0.02 and f = 0.04, with P = 0.399. As with the stars 3.3.SZCen
0
models, the quality of fit for the modelsfalls off rapidlyas the
ForSZCen,weobtainourbestfitforZ=0.01andδ =0.12,with
metallicity is changed such that no good fit is obtained at ei- ov
P= 0.7932.Thisplacestheprimarybeyondtheendofthemain
ther Z = 0.01 or 0.03. In Fig. 4, we show the evolutionary
tracksforthebestfitmodelsofbothmesaandstars.Theinter- sequence. Polsetal. (1997) give a best fit metallicity between
0.018and0.024,whileClaret(2007)obtainsabestfitatZ=0.02
nalstructureofbothstarsforthebestfittingmodelsisdisplayed
inFig.5.Themesamodelhasalargerovershootingregionthan and with 0.1 pressure scale heights of overshooting. However,
the stars model which is entirely consistent with the fact that it should be noted that Torresetal. (2010) revised the param-
the mesa evolutionary track has an end to the main sequence eters for this system, compared to the values used in both of
theaboveworks.Theyadoptedatemperaturebasedonthebeta
thatis redwardandbrighterthan the end of the main sequence
in the stars track. If we expressthe extentof the overshooting index, rather than on an assumed spectral type coupled with a
colour-T calibration(G. Torres,privatecommunication).The
region as a fraction of the pressure scale height at the convec- eff
tive boundary, the primary and secondary for the stars model resultofthisisthatthetemperaturesgivenbyTorresetal.(2010)
have f = 0.251and0.244,respectively.Forthemesamodels, aresomewhathotterthanthoseusedinearlierworks.Ifweadopt
we obtHapin f = 0.350 and 0.306 respectively. The stars hy- the parameters for SZ Cen used by Claret (2007), we obtain a
Hp bestfitatZ=0.03forδ =0.03withP=0.0121.Thisdoesnot
drogenprofileshowsmuchsmootherfeaturesattheedgeofthe ov
representagoodfit.Ontheotherhand,Ribasetal.(2000)obtain
zonefromwhichthefullymixedregionhasretreated.Thisisnot
a bestfittothissystemwith Z=0.007,Y=0.2and0.1-0.2pres-
due to any physicalcause, but is a computationalartefact. The
stars code employs a non-Lagrangianmesh which is prone to sure scale heights of overshooting.Aside from the low helium
value,ourresultisconsistentwiththeseparameters.
numerical diffusion which acts to smooth out any composition
discontinuities (for further details, see Stancliffe 2006). Above
thisregion,thetwomodelsshowverysimilarstructure.
3.4.AYCam,CVVelandV539Ara
These three systems all give excellent fits. For AY Cam, we
3.2.V1031Ori
obtain a best fit for Z=0.02 and δ = 0.09, with a P-value of
ov
ForV1031Ori,ourbestfitsystemhasametallicityofZ=0.03. 0.9267. We have been unable to find any other attempts to fit
Bothδ =0.21and0.24givethesameP-value,namely0.9297 this system in the literature.CV Vel and V539Ara are consid-
ov
anditispresumedthatthetruebestfitliesbetweenthesetwoval- erably more massive than the other binaries in our sample and
ues.Claret(2007)givesabestfitforZ=0.015and0.15pressure both have primary masses around 6M . For both systems, we
⊙
scale heights of overshooting. Ribasetal. (2000) find similar find a bestfitat Z=0.02,thoughtheδ valuesare differentfor
ov
parameters,with Z=0.016,Y=0.25and over0.2 pressure scale both.ForCVVel,wepreferδ =0.09,whileforV539Araδ
ov ov
heights of overshooting. Polsetal. (1997), on the other hand, = 0.15givesthe bestfit. Polsetal. (1997) also attempted to fit
prefer a model with overshooting and a metallicity of 0.023, both these systems. For CV Vel, they give a slight preference
though their model without overshooting and with a metallic- to models without overshootingand suggest the best fit metal-
ityofZ=0.029isalsoagoodfit.Itisdifficulttoreconcilethese licityisZ=0.016.ForV539Ara,theirovershootingmodelsare
differentmodelsforthesamesystem. preferredandZ=0.016isagainthebestfitmetallicity.
5
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
Primary Secondary
0.7
STARS Mass(M⊙) 3.86±0.15 2.95±0.09
0.65 MESA Radius(R ) 32.9±1.5 3.0±0.2
⊙
0.6 Teff (K) 4850±100 11000±500
0.55
Table 2. Parameters of the system V2291 Oph. These data are
0.5
takenfromSchro¨deretal.(1997).
0.45
X
0.4
0.35
3.5.AIHya,TZForandV2291Oph
0.3
0.25 Though not included in the Torres et al. list, we have also at-
temptedtomodelthesystemV2291Oph.Thisζ Aurigasystem
0.2
was used by Schro¨deretal. (1997) for their overshooting cal-
0.15
ibration and it was a key system in their determination of the
0 0.1 0.2 0.3 0.4 0.5 0.6
overshootingparameter.Theparametersofthissystemarelisted
M/M
(cid:12) inTable2.
0.7
STARS Weareabletoobtainreasonable,ifnotexcellent,fitstothe
0.65 MESA systemsAIHya,TZForandV2291Oph.Theselattertwosys-
tems are significantly evolved. Schro¨deretal. (1997) obtained
0.6
δ =0.12,foramodelwithX=0.70,Y=0.28andZ=0.02.Claret
ov
0.55 (2009)choseahighermetallicitybasedonchemicalanalysisby
Marshall (1996), preferring Z=0.03, and finds a best fit when
0.5
0.2pressurescaleheightsofovershootingareused.Ourbestfit
X
0.45 isforZ = 0.03andδ = 0.24,withP=0.4777.Thisplacesthe
ov
primaryonthefirstascentofthegiantbranch.However,wealso
0.4
obtain a fit of P = 0.2240 for Z = 0.02 and δ = 0.15 which
ov
0.35 would place the primary in the core helium burning phase, in
agreementwiththePolsetal.solution.
0.3
TZ For is another system in which the primary is highly
0.25 evolved.For this system, we obtain a bestfit for Z = 0.03 and
0 0.1 0.2 0.3 0.4 0.5 0.6
δ =0.24,withP=0.7036.Thisplacestheprimaryonthefirst
ov
M/M
(cid:12) ascent of the red giant branch. The P-values for neighbouring
Fig.5. Hydrogen profiles as a function of mass for the best fit δov and Z-values in our grid are all less than 0.1, so this solu-
modelsforEICep.Theupperpanelshowstheprimarystarand tionisstronglyfavoured.Previousattemptstoobtainasolution
thelowerpanelthesecondarystar.Darkgreylinesrepresentthe forthissystemhavealsohadissues. Claret(2007)suggestsus-
starsmodel,whilethemesamodelisinlightgrey.Thestarrep- ing0.6pressurescaleheightsofovershooting,butnotesthatthis
resents the edge of the (Schwarzchild)convectivecore and the seemstobetoolargeforstarsofmassescomparabletoTZFor.
squaretheedgeoftheovershootingregion. Polsetal. (1997) find slight evidence in favour of modelswith
overshooting, but their solution is also less than ideal (see the
right-handpaneloftheirfigure11).
5
Unlikethe previoustwo systems,AI Hyahasbothitscom-
ponentsonthemainsequence.Forthissystem,wefindabestfit
forZ=0.04andδ =0.15.Fits fromneighbouringmetallicities
ov
4.5 arenotablyworse,asareneighbouringδ values.Ourfitisnot
ov
perfectmostlybecausewedonotfindagoodfittothesecondary,
asshownin the middle-rightpanelofFigure2. Oursolutionis
(cid:12)
R/R 4 in goodagreementwith Polsetal. (1997), whofoundclearev-
idenceforasuperiorsolutionwhenovershootingwasincluded,
butnotedthatthe bestfitmetallicity layoutside themetallicity
3.5 range(theirmaximumZ was 0.033)of theirisochrones.Claret
(2007) attempted to fit this system with models at Z=0.02 and
wasunabletoobtainasatisfactorysolution.
3
9500 9000 8500 8000
3.6.V364Lac,WXCep,AQSer
T /K
eff
Forthreeofoursystems,weareunabletoobtainreasonablefits
Fig.6.Evolutionarytracksintheradius-T planeforthecom- atall.ForV364Lac,WX Cep andAQ Ser,thebest-fitP-value
eff
ponentsofSZCen.Theprimaryisdenotedbythedarkgreyline isbelow0.1.
and the secondary by the light grey line. Crosses represent the Ofthethreesystems,WXCepistheleastworstfittingsys-
best fit models for each track and the errorbars denote the ob- tem. We find a best fit value of P = 0.025 at Z=0.02 and δ
ov
servedsystem. = 0.12. The main difficulty with this system is that the best
fits to the componentsdo not occur at the same age. The fit to
the primary alone is at a maximum of about 0.75 at an age of
6
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
System BestfitZ Bestfitδ P
3 ov
V364Lac 0.02 0.09 6.914×10−4
AIHya 0.04 0.15 0.4922
2.8 EICep 0.02 0.15 0.9066
TZFor 0.03 0.24 0.7036
WXCep 0.02 0.12 2.549×10−2
2.6 V1031Ori 0.03 0.21(4) 0.9297
(cid:12)
R SZCen 0.01 0.12 0.8767
R/ AYCam 0.02 0.09 0.9267
2.4 AQSer 0.02 0.24 8.501×10−55
V2291Oph 0.03 0.24 0.4777
CVVel 0.02 0.09 0.8632
2.2
V539Ara 0.02 0.15 0.9267
Table3.BestfitpropertiesforeachofoursystemsusingonlyT
2 andR.
7000 6800 6600 6400 6200 6000
T /K
eff
Fig.7.Evolutionarytracksintheradius-Teff planeforthecom- height plot, as these systems are evolved beyond the main se-
ponentsofAQ Ser,assumingamodelwith Z= 0.01andδov = quenceand we cannotdefine this quantityif there is no longer
0.18.Theprimaryisdenotedbythedarkgreylineandthesec- a convectivecore. For all of our systems, we require some de-
ondarybythelightgreyline.Theerrorbarsdenotetheobserved greeofovershootingandinagreementwithpreviousworksthe
system. amount required is only moderate. The extent of overshooting
ranges from δ = 0.06 to 0.24, though the highest values are
ov
found only in our most evolved and most problematic systems
5.37×108yr, but the fit to the secondary reachesa peak value
(AQ Ser, TZ For and V2291 Oph). Most systems have δ be-
of0.92atjust5.22×108yr.Thereislittle overlapbetweenthe ov
tween 0.09 and 0.15. This is consistent with the 0.1-0.3 pres-
twofitdistributions,resultinginapoorfitoverall,ascanbeseen
sure scale heights of overshooting found by other authors. We
inFig.2.Ourbest-fitmetallicity(andhencealsotheageofthe
see no reason to suggest that δ is a function of mass (unlike
ov
system)agreeswiththatofClaret(2007),whofindsabest-fitso-
Ribasetal.2000),thoughwe notethatthe majorityofoursys-
lutionfor0-0.2pressurescaleheightsofovershooting.Polsetal.
temsareclusteredaround2M ,withjusttwosystemsataround
⊙
(1997)alsofavoura modelwithovershootingandametallicity
6M .Thereisalsonoevidenceforametallicitydependenceto
⊙
ofZ=0.02.
the extentof overshooting,thoughthis may simply be because
V364 Lac is a peculiar case. We are able to obtain almost we lack enough systems across a range of metallicities. While
perfect fits to the individual components of this system (with our three Z=0.03 systems do have higher δ values, we stress
ov
best-fit values of almost unity) at Z=0.02 and with δov = 0.09. that two of these are very poor fits hence we cannot provide a
However,thesesolutionshaveverydifferentages.Theprimary definiteconclusionregardingapossiblemetallicitydependence.
isbestfitatanageof6.7×108yr,whilethesecondaryisbestfit
Theextentofovershootingwefindisconsistentwithvalues
atanageof5.9×108yr.Wearenotawareofanyotherattempts
obtainedfromasteroseismic determinations.Aerts(2015) finds
tofitthissystemintheliterature.
valuesbetween0-0.5pressurescaleheights,alsowithconsider-
AQ Ser hasalready beennoted as a problematicsystem by
able star-to-starvariations.She also reportsthatthereisno ob-
Torresetal. (2014). These authors note that the more massive
viousrelationbetweenextentofovershootingandstellar mass.
star appearssystematically youngerthan its less massivecoun-
Neineretal.(2012)find0.3-0.35pressurescaleheightsofover-
terpart.TheP-valueforourbestfittoAQ Serissovanishingly
shootingintheirtwolateBestars,HD181231andHD175869.
smallthatitmaywellbeconsideredzero.We canobtainarea-
Againourresultsareconsistentwiththesevalues.
sonablefittotheprimaryalonewithZ=0.01andδov=0.18(see To complement our stars models, we have also calculated
Fig. 7), but it is clear from the evolutionarytrack that our sec- bestfit modelsfor each ofour systems usingmesa. The details
ondary is not massive enough. It is possible that a better fit to
ofthesefitsaregivenintheappendix.Wefindgeneralagreement
the system wouldbe obtainedif the masses of the two compo-
betweenthetwocodes.Bothpredictthesamemetallicityforall
nents were closer together, given how similar the radii and ef-
butoneofthe systems(theexceptionis AIHya).Thereis also
fective temperatures of the two stars are. As a test of this, we
agreementin the trendof the overshootingparameterrequired:
ran a single calculation with Z=0.01 and δov = 0.18, in which for systems where stars requires a high δ , we also find that
ov
wesettheprimarymasstobe1.395M⊙andthesecondarymass mesarequiresahigh f .Theextentofovershootinginthemesa
0
to be 1.370M which are the minimum and maximum masses
⊙ modelsisabout0.15-0.4pressurescaleheights.ComputingaP-
(respectively)allowedbytheerrors.Inthiscase,weobtainafit
weighted average of the overshootingparameter f results in a
ofP=0.101,whichisadramaticimprovementoverouroriginal 0
valueof0.031.
calculations.
Ideally,onewouldliketobeabletoselectasinglevaluefor
theovershootingparameterthatwouldworkreasonablywellfor
all systems. This is particularlyrelevantfor future work in our
4. Discussionandconclusions
project,wherewe aim to create gridsof low-massstellar mod-
We summarise the best fits to each of our systems in Table 3. els for use in bonnsai. If we neglect the systems with P-values
In Fig. 8 we have plotted the extentof overshootingas a func- less than 0.1 and take an average of the overshooting param-
tion of stellar mass, showingthe overshootingboth in termsof eter for the remaining systems, in which we weight each sys-
δ and as a fractionof the pressure scale height.Note that we tembyitsrelativeP-value,weobtainδ =0.156.Adoptingthis
ov ov
have not plotted TZ For and V2291 Oph on the pressure scale value,wewouldstillobtaingoodfitsforthesystemsCVVeland
7
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
System BestfitZ Bestfit f P
0
V364Lac 0.02 0.030 2.244×10−3
AIHya 0.03 0.030 1.604×10−3
EICep 0.02 0.040 0.3987
TZFor 0.03 0.050 0.01072
WXCep 0.02 0.030 0.05469
V1031Ori 0.03 0.045 0.9474
SZCen 0.01 0.025 0.01779
AYCam 0.02 0.020 0.5296
AQSer - -
V2291Oph 0.03 0.045 0.09719
CVVel 0.02 0.030 0.9427
V539Ara 0.02 0.035 0.9808
TableA.1.Bestfitpropertiesforeachofoursystemsusingonly
T and R as determined using the mesa code. No reasonable fit
couldbefoundtoAQSer(P<10−96).
von Humboldt Foundation. FNRS is funded via the Bonn Cologne Graduate
School.JCPalsothanksNorbertLangerforhissupport.
References
Aerts, C. 2015, in Proceedings of IAU symposium 307, ed. J. G. .
P. S. G. Meynet, C. Georgy (Cambridge University Press), in press
(arXiv:1407.6479)
Andersen,J.1991,A&ARev.,3,91
Brott,I.,deMink,S.E.,Cantiello,M.,etal.2011,A&A,530,A115
Casagrande,L.,Scho¨nrich,R.,Asplund,M.,etal.2011,A&A,530,A138
Claret,A.2007,A&A,475,1019
Claret,A.2009,A&A,507,377
Eggleton,P.P.1971,MNRAS,151,351
Eggleton,P.P.1972,MNRAS,156,361
Guenther,D.B.,Demarque,P.,&Gruberbauer,M.2014,ApJ,787,164
Fig.8.Extentofovershootingasafunctionofmass.Theupper Herwig,F.2000,A&A,360,952
paneldisplaystheovershootingintermsofδ ,whilethelower Ko¨hler,K.,Langer,N.,deKoter,A.,etal.2015,A&A,573,A71
ov
panelshowsthesamequantityintermsofafractionofthepres- Maeder,A.&Meynet,G.1991,A&AS,89,451
Marshall,K.P.1996,MNRAS,280,977
surescaleheight.Symbolsdenotethebestfitmetallicityforthe
Meng,Y.&Zhang,Q.S.2014,ApJ,787,127
systems:Z=0.01(triangles),Z=0.02(squares),Z=0.03(circles)
Montalba´n,J.,Miglio,A.,Noels,A.,etal.2013,ApJ,766,118
andZ=0.04(pentagons). Neiner,C.,Mathis,S.,Saio,H.,etal.2012,A&A,539,A90
Paxton,B.,Bildsten,L.,Dotter,A.,etal.2011,ApJS,192,3
Pols,O.R.,Tout,C.A.,Eggleton,P.P.,&Han,Z.1995,MNRAS,274,964
V1031Ori(whichgiveP=0.7551and0.7243respectivelyforδ Pols,O.R.,Tout,C.A.,Schro¨der,K.-P.,Eggleton,P.P.,&Manners,J.1997,
ov
MNRAS,289,869
= 0.15).AY Cam andV2291Ophwouldbemoderatefits(P=
Ribas,I.,Jordi,C.,&Gime´nez,A´.2000,MNRAS,318,L55
0.2733and 0.2240respectively),thoughthe best fit metallicity
Schneider,F.R.N.,Langer,N.,deKoter,A.,etal.2014,A&A,570,A66
forthelattersystemwoulddroptoZ=0.02.ForthesystemsSZ Schro¨der,K.-P.,Pols,O.R.,&Eggleton,P.P.1997,MNRAS,285,696
CenandTZForweareunabletoobtainanyfitatδ =0.15.Of Stancliffe,R.J.2006,MNRAS,370,1817
ov
theninesystemsweareabletofitwhenourchoiceofZandδ Stancliffe,R.J.&Eldridge,J.J.2009,MNRAS,396,1699
ov Tkachenko,A.,Aerts,C.,Pavlovski,K.,etal.2014,MNRAS,442,616
isfree,we wouldonlyfitsevenofthesesystemsif we adopted
Torres,G.,Andersen,J.,&Gime´nez,A.2010,A&ARev.,18,67
ouraveragevalueofδov=0.156. Torres,G.,Vaz,L.P.R.,SandbergLacy,C.H.,&Claret,A.2014,AJ,147,36
Itwouldbeusefultobeabletoextendthisworktoawider VandenBerg,D.A.,Bergbusch,P.A.,&Dowler,P.D.2006,ApJS,162,375
sample of binaries, particularlyas many of our systems cluster
around2M . Additionaldata forstars of between3 and5 M
⊙ ⊙
(and above6M⊙) would be particularlydesirable. However,as AppendixA: mesamodelfits
canbeseenfromFig.1,fewsystemsintheTorresetal.(2010)
datasetlieinthismassrangeevenbeforewetakeintoaccount In addition to EI Cep, we have also computed models for all
ourrequirementsthatthesystemsshouldbefairlywideandcon- our systems using mesa. The best fit parameters are given in
taincomponentsthatareevolvedawayfromtheZAMS.Nature, TableA.1.Fig.A.1showsthisdata,togetherwiththeextentof
it seems, has not been kind to us in this regard. One can hope theovershootingregionasafunctionofthepressurescaleheight
thatfuturesurveymissions,suchas Gaia3,mayhelpto fillthis at the top of the convective region. Calculating an average f
0
deficiency. weightedbytheP-values,weobtain f =0.031.
0
Acknowledgements. TheauthorsthankG.Torresforusefuldiscussionregard-
ing the parameters of SZ Cen and the referee for her/his useful remarks.
RJS is the recipient of a Sofja Kovalevskaja Award from the Alexander von
HumboldtFoundation. LFandJCPacknowledge funding fromtheAlexander
3 sci.esa.int/gaia/
8
R.J.Stancliffeetal.:CUSP–calibratingconvectiveovershooting
Fig.A.1. Extent of overshooting as a function of mass for the
mesamodels.Theupperpaneldisplaystheovershootinginterms
of δ , while the lower panelshows the same quantityin terms
ov
of a fraction of the pressure scale height. Symbols denote the
best fit metallicity for the systems: Z=0.01 (triangles), Z=0.02
(squares),Z=0.03(circles).
9