Confronting uncertainties in stellar physics II. exploring differences in main-sequence stellar evolution tracks

0.59 MB·

by  R. J. Stancliffe

Astronomy&Astrophysicsmanuscriptno.CUSPII˙v3 (cid:13)c ESO2016
January14,2016
Confronting uncertainties in stellar physics
II. exploring differences in main-sequence stellar evolution tracks
R.J.Stancliffe1,L.Fossati2,1,J.-C.Passy1,andF.R.N.Schneider1,3
1 Argelander-Institutfu¨rAstronomie,UniversityofBonn,AufdemHu¨gel71,D-53121Bonn,Germany
2 SpaceResearchInstitute,AustrianAcademyofSciences,Schmiedlstrasse6,A-8042Graz,Austria
3 DepartmentofPhysics,UniversityofOxford,DenysWilkinsonBuilding,KebleRoad,OxfordOX13RH,U.K.
6
1 ABSTRACT
0
2 Weassess thesystematic uncertainties in stellar evolutionary calculations for low- to intermediate-mass, main-sequence stars. We
comparepublishedstellartracksfromseveraldifferentevolutioncodeswithourowntrackscomputedusingthestellarcodesstars
n andmesa.Inparticular, wefocusontracksof1and3M atsolar metallicity.Wefindthatthespreadintheavailable1M tracks
⊙ ⊙
a (computedbeforetherecentsolarcompositionrevisionbyAsplundetal.)canbecoveredbytracksbetween0.97-1.01M computed
⊙
J withthestarscode.Weassesssomepossiblecausesoftheoriginofthisuncertainty,includinghowthechoiceofinputphysicsand

2 thesolarconstraintsusedtoperformthesolarcalibrationaffectthetracks.Wefindthatfora1M track,uncertaintiesofaround10%
⊙
1 intheinitialhydrogenabundanceandinitialmetallicityproducearounda2%errorinmass.Forthe3M tracks,thereisverylittle
⊙
differencebetweenthetracksfromthevariousdifferentstellarcodes.Themaindifferencecomesintheextentofthemainsequence,

] whichwebelieveresultsfromthedifferentchoicesoftheimplementationofconvectiveovershootinginthecore.Uncertaintiesinthe
R
initialabundancesleadtoa1-2%errorinthemassdetermination.Theseuncertaintiescoveronlypartofthetotalerrorbudget,which
S shouldalsoincludeuncertaintiesintheinputphysics(e.g.,reactionrates,opacities,convectivemodels)andanymissingphysics(e.g.,
. radiativelevitation,rotation,magneticfields).Uncertaintiesinstellarsurfacepropertiessuchasluminosityandeffectivetemperature
h
willfurtherreducetheaccuracyofanypotentialmassdeterminations.
p
- Keywords.stars:evolution,stars:interiors,stars:low-mass
o
r
t
s
a 1. Introduction to dramatically wrong conclusions: a planet presumed to be a
[ super-Earthin the habitablezonemay instead byan uninhabit-
Stellar evolution codes are a lot like religions: there are many
able gaseousplanet.Thisis particularlyrelevantbecausemany
1 of them to choose from, they possess many similarities, and it
planet-finding facilities, which are about to begin their opera-
v
isnotobviouswhichofthem(ifany)iscorrect.Theoutputsof
tions (e.g., NGTS [Wheatleyetal. 2013], TESS [Rickeretal.
4
these codes, in the form of evolutionarytracksand isochrones,
5 2014], andCHEOPS[Broegetal. 2013]), aredesignedto have
0 areusedinseveralfieldsofastrophysics,rangingfromthestudy their maximum efficiency in finding mini-Neptunesand super-
of extra-solar planets (exoplanets)to stellar populations.Many
3 Earthsorbitingnearbylate-typestars.
0 grids of tracks and isochrones, computed with different stellar With the “Confronting Uncertainties in Stellar Physics”
. evolutioncodes,arefreelyavailableandareusedtoderivefun- (CUSP)projectwe intendtoestimatethesystematicuncertain-
1
damental properties of stars (e.g., mass, radius, age) and star
0 tiesinvolvedinthecalculationofstellarevolutiontracks,forex-
clusters (e.g., age, distance) on the basis of observables (e.g.,
6 ampletoevaluatetheirimpactintermsofplanetparameters.For
1 magnitudes, effective temperature, surface gravity, luminosity, thistask,wewillusebonnsai1(Schneideretal.2014),which,to
: chemical composition). When comparing observations to stel- thebestofourknowledge,istheonlypubliclyavailabletoolal-
v lar evolution tracks one often considers only the observational
lowingonetoderivestellarparameters(e.g.,mass,radius,age)
i
X uncertainties in the error budget. This neglects the systematic from a set of observational parameters (e.g., effective temper-
theoretical uncertainties in the stellar codes (see below), some
r ature, surface gravity, luminosity), using Bayesian statistics to
a ofwhicharedifficulttoquantify. properlyaccountfor the observationaluncertainties. In its cur-
In the exoplanet field, evolutionary tracks are commonly rentstatus,bonnsaicontainsevolutionarytracksonlyformassive
used to derive the mass and radius of planet-hosting stars on stars (M>5M ; Brottetal. 2011; Ko¨hleretal. 2015). We wish
⊙
the basis of properties measured from spectra and transit light to extend the model database to lower mass stars from 0.8 to
curves(e.g.,effectivetemperature,surface gravity,andaverage 10M onthebasisofgridscalculatedusingthestars(Eggleton
⊙
density). The stellar parameters are then used to directly de- 1971;Stancliffe&Eldridge2009)andmesa(Paxtonetal.2011,
rive the planet mass and radius, and age of the system; as a 2013,2015)stellarevolutioncodes.
consequence, systematic uncertainties in the stellar parameters In the first paper of this series (Stancliffeetal. 2015), we
propagate to the planet parameters as well. It is important to useda setof12eclipsingbinaries,withcomponentmassesbe-
note that small systematic uncertainties in the stellar parame- tween 1.3 and 6.2M , to constrain the convective core over-
⊙
ters may be strongly significant in terms of planet properties, shootingparameterto beadoptedforthe modelgrids.We con-
for example for planets in the 5–10M transition region be-
⊕
tween gaseous and rocky planets. For these planets underes- 1 The bonnsai web-service is available at
timating the uncertainty of their mass and/or radius may lead www.astro.uni-bonn.de/stars/bonnsai.
1
R.J.Stancliffeetal.:CUSPII–stellartracks
cluded that the majority of the analysed systems can be well – BASTI.TracksfromBASTIaretakenfromtheBASTIweb-
describedwithamass-independentovershootingparametercor- site4. Details of the code are described in Pietrinfernietal.
respondingto an extensionof the mixed region abovethe core (2004).
of about 0.1–0.3 pressure scale heights. In this work, we aim – Padova.ThePadovatrackshavebeentakenfromtheirweb-
atincreasingtheawarenessofusers(observersinparticular)of site5. Details of the code and the models are described in
gridsofstellarevolutionmodelsandisochronesonthepresence Bertellietal.(2008).
of theoretical uncertainties and systematics. We also highlight – Dartmouth. The Darmouth stellar evolution code is de-
caveatspresentwhenconvertingstellaratmosphericparameters scribedinBjork&Chaboyer(2006)andDotteretal.(2007).
(i.e.,effectivetemperature,surfacegravity,andmetallicity)into The models used here come from the work of Dotteretal.
fundamentalparameters(i.e.,mass,radius,andage).Wedothis (2008)andareavailableonline6.
here by investigating the uncertainties in main-sequence, low- – Geneva. These models are taken from the work of
mass stellar evolution tracks, based on comparisons between Mowlavietal. (2012) and are available from the Geneva
tracksfor1and3M models(comparisonsforhigh-massstars website7.
⊙
have recently be carried out by Martins&Palacios 2013 and – PARSEC.ThisisanupdatedversionofthePadovacodeand
Jonesetal.2015).We limitourselftothemainsequencephase isdescribedinBressanetal.(2012).Tracksmaybeobtained
ofevolution,becauseitistheonewhichismostintensivelycon- online8.
sideredintermsofstellarspectroscopicanalysisandconversion – Victoria-Regina.ThiscodeisdescribedinVandenBergetal.
oftheatmosphericparametersintostellarmass,radius,andage (2012)andreferencestherein.Thetrackspresentherecome
(e.g.,thevastmajorityoftheknownplanet-hostingstarsareon fromtheworkofVandenBergetal.(2014),andareavailable
themainsequence).Wewillconsideruncertaintiesinlaterstages fordownload9.
ofevolutioninfuturework.
Manyofthesecodesandgridsaremassivelyused,mostlybyob-
Uncertainties in stellar modelling can be divided into two servers,toconvertspectroscopicallyderivedstellaratmospheric
groups:physicaluncertaintiesandcomputational/numericalun- parametersintofundamentalparameters.Thisconversionisof-
certainties.Theformergroupincludesuncertaintiesinmeasured tendonewithoutconsideringthephysicsincludedbythediffer-
quantitiesusedasinputsforcomputations,suchasreactionrates ent grids, whether the adopted input parameters (e.g., reaction
and opacities. It also includes our ignorance of physical pro- rates) are up to date, the calibration procedures, and the true
cesses such as the efficiency of mixing processes and the na- meaning of some of the basic parameters (e.g., present-day vs
tureofmixingatconvectiveboundaries(Vialletetal.2015).The initialmetallicity).Giventhelargeamountandqualityofobser-
second group relates to how the equations of stellar structure vationaldataalreadyavailable,forexamplefromlargespectro-
and evolution are solved. It includes issues related to spatial scopicsurveys,andsoontocome,forexamplefromGAIA,itis
and temporal resolution in models, the use of simultaneous or imperativethatthedifferentcommunitiesarefullyawareofthe
non-simultaneous solution of the equations of stellar structure presenceandsizeofthesystematicuncertaintiesassociatedwith
(Stancliffe 2006), the choice of solution scheme (Meynetetal. stellarevolutioncalculations.
2004), and the algorithmic treatment of mixing at convective- It should be stressed that this paper is not intended to rep-
radiativeboundaries(Gabrieletal. 2014) to name a few exam- resentacodecomparison,thatistosay,atestofthemechanics
ples. of the various stellar evolution codes. Such comparisons have
Wehavecollectedstellartracksfromvariouspubliclyavail- beenperformedbymanygroups(e.g.atTheAarhusRedGiants
ablesources,aswellascomputingtrackswithtwoothercodes. Workshop10). The mostimportantaspectof a codecomparison
Thisis byno meansan exhaustivelist ofwhatis available,but istocompletelystandardizetheinputphysicsandthencompare
we believe it provides a representative range of current stellar theoutputs,andthusassesswhetherthemachineryofthestellar
evolutioncalculations.Thecodesusedare: codesworksthesame way.However,forthe variouspublished
tracksproducedbydifferentgroupsthechoicesofinputphysics
isnotthesame.Atsomepoint,choiceshavetobemade(forex-
– mesa. This code is described extensively by Paxtonetal. ample,fortheinclusionofdiffusionaswillbediscussedbelow)
(2011),Paxtonetal.(2013)andPaxtonetal.(2015).Weuse and it is these choices that lead to the differences between the
revision7503ofthecode. calculations made by different groups. Although we are mak-
– stars.Thestarsstellarevolutioncodewasoriginallydevel-
ing no assertions in this work as to which choices are correct,
opedbyEggleton(1971)andhasbeenupdatedbymanyau-
usersofpublishedstellarmodelsshouldcriticallyexaminehow
thors(e.g.Polsetal.1995).Thecodeisfreelyavailablefor
welltheavailablegridssatisfyempiricalconstraints(e.g.theso-
download2. This code solves the equations of stellar struc-
lar calibration, binary stars) and the extent to which they take
tureandchemicalevolutioninafullysimultaneousmanner,
intoaccountrecentadvancesinstellarphysics(e.g.updatedre-
iterating on all variables at the same time in order to con-
actionrates,opacities).Thisrepresents(someof)thetheoretical
vergeamodel(seeStancliffe2006,foradetaileddiscussion).
uncertaintiesinherentinstellarmodelling.Ouraimistoputour-
The version employed here is that of Stancliffe&Eldridge
selves in the perspective of someone who needs to derive e.g.,
(2009) which was developed for doing binary stellar evo-
themassofastaronthebasisofspectroscopically-derivedatmo-
lution. The code treats all forms of mixing by means of a
sphericparameters,andtoestimatetheuncertaintiesinvolvedin
diffusiveformalism(Eggleton1972).
– Y2.TracksfromtheYonsei-Yalecollaborationareavailable 4 http://basti.oa-teramo.inaf.it/index.html
online3.Detailsofthecodeanditscalibrationcanbefound 5 http://stev.oapd.inaf.it/YZVAR/
inYietal.(2001). 6 http://stellar.dartmouth.edu/models/index.html
7 http://obswww.unige.ch/Recherche/evol/-Database-
8 http://stev.oapd.inaf.it/cgi-bin/cmd
2 http://www.ast.cam.ac.uk/∼stars 9 http://www.canfar.phys.uvic.ca/vosui/#/VRmodels
3 http://www.astro.yale.edu/demarque/yystar.html 10 http://users-phys.au.dk/victor/rgwork/
2
R.J.Stancliffeetal.:CUSPII–stellartracks
thisprocesscausedsimplybychoosingonesetoftracksinstead
ofanother.We willaddresstheissueofuncertaintiesinagesin
aseparatepaper.
2. Comparisonof1M tracks
⊙
In Fig. 1, we plot evolutionary tracks for a 1M model com-
⊙
putedwiththeevolutioncodeslistedabove.We haveseparated
thetracksintotwogroupsbasedontheirmetallicity:thoseplot-
tedwithan‘old’solarmetallicityareshownintheupperpanel,
while those with the Asplundetal. (2009) metallicity are plot-
tedinthelowerpanel.Asashorthand,wewillusetheterm‘old’
torefertopre-Asplundetal.(2009)solarmodels,and‘new’to
refer to modelscomputedwith the Asplundetal. (2009) abun-
dances.PerfectagreementwiththeSunshouldnotbeexpected
asweare plotting1M evolutionarytrackscomputedforgrids
⊙
ofstellarmodelswhicharenotnecessarilythesolarcalibration
tracks themselves. The exceptions here are stars and mesa, as
these are our own solar-calibration tracks. In some cases, e.g.
BASTI,thesolarcalibrationisperformedwithdiffusion,butthe
publiclyavailabletrackiscomputedwithout.Forthe‘old’Sun,
mostofthecodesgiveeffectivetemperatureswithin20-30Kof
the solar value of 5777 K (Christensen-Dalsgaardetal. 1996).
Inallcases,luminosityvariationsareextremelysmall.Thebox
drawninFig.1showswhatweconsidertobetypicalpost-GAIA
uncertainties.The errorbaron the luminosityhasbeen derived
assumingthatthewholeuncertaintyisinthebolometriccorrec-
tion (0.05mag; see e.g., Flower 1996). The uncertainty on the
effectivetemperatureistheaverageonetypicallyexpectedfrom
state-of-the-artanalysis of high quality, high resolutionspectra
of solar-like stars (70K; see e.g. Ryabchikovaetal. 2016). All
these tracks would lie comfortably within the estimated post-
GAIA errors.Thespreadin temperaturesbetweenthe tracksat
thesolarageiscomparabletotheinitialspreadintemperatures
at500Myr,whichisaround80K.Atthe10Gyrpoint,thespread
intemperaturesismuchlarger,around150Kwhichisaboutthe
size of the whole error box. The spread in luminosity remains
smallataround0.03dex.Thespreadinthestellartracksrepre- Fig.1.Evolutionarytracksfor a 1M⊙ modelfor variousstellar
sentstheuncertaintyinthecalculationofa1M star.Thehottest evolution codes. The box represents the estimated post-GAIA
⊙
trackisroughlyreproducedbya1.01M starsmodel,whilethe uncertainties of the luminosity and effective temperature.Top:
⊙
coolestrequiresa0.97M model.Thusifweweretoobservea Models comuputed with a metallicity of Z≈ 0.02. The inset
⊙
starwiththeSun’sluminosityandtemperature(andmetallicity), showsazoomedinregionaroundthesolarluminosityandtem-
theerroronthemasswedeterminebyfittinga1M tracktoit perature, with the box displaying the uncertainties of the solar
⊙
wouldbearound±0.02M . luminosityandeffectivetemperature.Bottom:Modelscomput-
⊙
Much of the spread between the various ‘old’ solar tracks ing using the abundancesof Asplundetal. (2009). The Padova
comesfromthevarietyofinputphysics.Welistsomeofthesein track, which is intermediate in metallicity is included in both
Table1.NoteveryonecalibratestheSunusingthesamephysics. panels.
Thereareavarietyofchoicesofpresent-daysolarabundances,
whichaffectsthebulkmetallicityandinitialhydrogenmassfrac-
tion.Gravitationalsettling/diffusion11ofelementsisconsidered a process that should be included in stellar calculations.
by many, but not all codes. On top of this, not everyone uses Christensen-Dalsgaardetal.(1993)showedthattheinclusionof
thesamesetofconstraintsforthecalibration.Inadditiontous- helium settling has a significant effect on the structure and os-
ingthecurrentsolarluminosityandradius,somecodesalsouse cillationfrequenciesoftheSun.However,diffusionaloneistoo
thedepthofthesolarconvectionzone.We shalldiscusstheef- efficientatdepletingmaterialsandanadditionalmixingprocess
fects of these different choices below. In the case of the ‘new’ seems to be necessary. Brunetal. (1999) showed that for the
solarmodel,theagreementbetweencodesisfarbetter(see the Sun, turbulentmixingcan inhibitsettling by 25%and thatthis
lower panel of Fig. 1) presumably as a result of a much more leadstoasignificantimprovementinthesolarsoundspeedpro-
homogenoussetofinputphysics:allthecodesuseverysimilar file. At low metallicity, diffusion together with turbulent mix-
abundancesandallincludetheeffectsofdiffusion,althoughthe ing may present an explanationfor the uniform Li abundances
Victoria-Reginacodeonlyappliesdiffusiontohelium. observedinunevolvedstars(theso-calledSpiteplateau)which
The inclusion of diffusion in stellar codes presents some solvesthecosmiclithiumproblemwithouttheneedforanyex-
issues. From a purely physical perspective, diffusion is oticphysicsduringBigBangnucleosynthesis(seeRichardetal.
2005,andreferencestherein).Asimilarconclusionisreachedby
11 Weshallusethesetwotermsinterchangeably. Kornetal.(2006)fromtheobservationoflithiuminthemetal-
3
R.J.Stancliffeetal.:CUSPII–stellartracks
Code solar Z X α diffusion X X˜ σ Reference
0 0 MLT X
abundance logL/L 0.0 4×10−4 Fro¨hlich&Lean(2004)
⊙
stars GN93 0.0184 0.7012 2.090 yes T 5777K 5K Scottetal.(2015)
eff
A09 0.0142 0.7293 2.025 yes Z/X 0.0231or0.0181 0.002 GS98,A09
mesa GS98 0.0185 0.7101 1.877 yes He 0.2485 0.0034 Basu&Antia(2004)
A09 0.0149 0.7193 1.783 yes R /R 0.713 10−3 Basu&Antia(1997)
cz ⊙
Y2 GNS96 0.0181 0.7149 1.7432 Heonly δc 0.0 10−4 Rhodesetal.(1997)
s
BASTI GN93 0.198 0.7068 1.913 yes
Padova GS98 0.017 0.723 1.68 no Table 2. Observed values used for the mesa calibration. The
Dartmouth GS98 0.0188 0.7063 1.938 yes
value of Z/X dependson the use of the GS98 (0.0231)or A09
Geneva A05 0.014 0.7200 1.6467 yes
PARSEC GS98+C11 0.0177 0.7027 1.74 yes (0.0181)abundances.
Victoria- A09 0.0133 0.7367 2.007 Heonly
Regina
2.1.Theeffectsofthechoiceofsolarcalibration
Table 1. Details of the parameters for the solar calibra-
tions of the various codes included in this study. The ab- To calibrate stellar evolutionarytracks, it is typical to calibrate
breviations for the choice of solar abundances are: GN93, acode’smixinglengthparameterusingasinglereferencepoint,
Grevesse&Noels (1993); GNS96, Grevesseetal. (1996); namely the Sun. It is well known that with the current avail-
GS98, Grevesse&Sauval (1998); A05, Asplundetal. (2005); able input physics and the Asplundetal. (2009) abundances a
A09,Asplundetal.(2009);C11,Caffauetal.(2011).Notethat 1M star at the currentage of the Solar System does not give
⊙
the quoted values for mesa are based on the calibration with a structureconsistentwith helioseismicmeasurements(see e.g.
L,Teff andZ/Xalone. Villanteetal. 2014, and references therein). Specifically, one
cannotproduceamodelthatfitsthecurrentradiusandluminos-
ityoftheSunwhichalsohasthecorrectdepthoftheconvective
envelope.
To test the effects of the choice of calibration constraints,
we perform the following tests using the mesa code. We first
poorglobularclusterNGC6397.Thebehaviourofheavierele-
calibrate the solar model using only the solar luminosity, tem-
mentsinthissameclusterisalsoconsistentwithstellarmodels
perature and Z/X. We then repeat the calibration adding addi-
including diffusion and turbulent mixing (Lindetal. 2008). In
tionalconstraints.First,weaddthedepthofthesolarconvection
additionto these surface effects, theinclusionof diffusionmay
zoneR ,followedbytheheliumabundanceinthesolarconvec-
reduce the age of a low-mass star at a given turn-off luminos- cz
tionzone.Lastly,weconsidertheRMSdeviationofthemodel’s
ity (Proffitt&Vandenberg 1991) and is therefore an important
sound speed profile δc from that deduced from helioseismol-
s
considerationifonewishestodeterminestellarages.
ogy(Rhodesetal.1997).ThevaluesusedaregiveninTable2.
Itthereforeseemsthatoneshouldpreferstellartrackscom- Ineachcasea χ2 minimisationisusedtodeterminethebestfit
puted with the inclusion of diffusion. However,there is an im- model,i.e.
portantcaveat. First, the above evidence points to the need for
more than justdiffusion alone.An additionalmixingprocesses 1 X¯ −X˜ 2
χ2 = i i (1)
isrequiredtoinhibittheeffectsofdiffusion.Nophysicalcause n σ !
forthismixinghasbeenidentifiedandsotheprescriptionsused Xi Xi
are entirely ad hoc. Additionally, most of the codes presented where n is the number of observables, X is a given observ-
i
heredonotemploysuchturbulentmixing.BoththeDartmouth able, X¯ its value for the model, X˜ its observedvalue, and σ
and PARSEC codes inhibit the action of diffusion in an ad the obsierved uncertainty. This is eivaluated at the solar age oXfi
hoc way, but with different prescriptions. The Victoria-Regina 4.57Gyrs (Bahcalletal. 1995). We use the simplex algorithm
code adds a turbulent diffusion coefficient which is similar in (Nelder&Mead1965)implementedinmesatominimizetheχ2
formtotheRichardetal.(2005)prescription(VandenBergetal. andstopthecalibrationoncechangesintheinitialparametersno
2012). None of the codes describe here includes the effects of longeraffecttheobtainedχ2.Thecodeisfreetochangetheini-
radiativelevitation,whichisthecounterpartofgravitationalset- tialmetallicity,hydrogenabundanceandmixinglengthparame-
tling, though it typically only influences the heavier elements ter.WeperformthiscalibrationforboththeGrevesse&Sauval
(Richardetal. 2002). There is clearly more to the issue of set- (1998)andtheAsplundetal.(2009)solarabundances,usingOP
tling than we currently understand. For this reason, we do not opacities (Badnelletal. 2005) and including gravitational set-
dismisscodesthatdonotincludeit.Weshallseebelowthepo- tling.TheresultsofthiscalibrationareshowninFig.2andthe
tentialerrorthatmaybeintroducedbynotincludingdiffusion. detailsofthebestfitmodelsarepresentedintableA.1.
Diffusionalsoprovidestheobserverwithanotherheadache: FortheGS98abundances,thecalibrationsallgiveextremely
which composition should the stellar tracks used for compari- similarresults.Thisisnotparticularlysurprising–itiswelles-
son have? Theoristslabel their tracks by initial abundance,but tablished that the ‘old’ solar abundances were extremely well
anobserverseesonlythepresent-daysurfaceabundance.Ifset- reconciledwithhelioseismicmeasurements.Theexactchoiceof
tlinghassubstantiallyalteredthelatter,itnolongerreflectsthe calibration should not be crucial when using ‘old’ solar abun-
former.FortheSun,thecurrentsurfacemetallicityisthoughtto dances.Forthe‘new’Sun,thechoiceofcalibrationconstraints
bearound10%lowerthantheinitialmetallicity(Asplundetal. hasa noticeableeffect. Whenthe depthof the solar convection
2009). We are able to estimate this because we know the solar zoneisadded,thetracknolongerhasaluminosityandeffective
ageandhenceweknowtheSun’sevolutionarystate.Foragiven temperaturethat fit the Sun:the modelis 20K too hot and has
field star, this information is not available and we have to live logL/L =−4×10−4(i.e.about0.1%belowthesolarluminos-
⊙
withtheconsequencesofourignorance. ity).Whenweincludethesoundspeedprofileinthecalibration,
4
R.J.Stancliffeetal.:CUSPII–stellartracks
0.4 0.4
0.3 0.3
0.2 0.2
(cid:12) (cid:12)
L L
L/ L/
g 10 0.1 g 10 0.1
o o
l l
0 0
-0.1 -0.1
-0.2 -0.2
5900 5850 5800 5750 5700 5650 5600 5900 5850 5800 5750 5700 5650 5600
T (K) T (K)
eff eff
0.4 Fig.3. Calibrated evolutionary tracks for the ‘old’ (red) and
‘new’ (black) Sun. The solid lines were computed using grav-
0.3 itationalsettlingwhilethedashedlineswerecomputedwithout.
Plusesmarkagesof500Myr,4.57Gyrand10Gyr.Theredcir-
0.2 clerepresentsthelocationoftheSun.
(cid:12)
L
L/
og 10 0.1 Oncethecalibrationpointispassed,thesituationisreversedand
l
0 themodelswithoutsettlingarehotterthanthosewithsettling.
One additional point to consider here is the effect of using
-0.1 an α value that has been calibrated using diffusion and then
applied to a grid where diffusion is not considered. The use
ofgravitationalsettlingbecomesproblematicathighermasses.
-0.2
5900 5850 5800 5750 5700 5650 5600 As the size of the convective envelope decreases, the surface
Teff (K) layers can more readily be depleted, leading to models whose
abundancesare clearly inconsistent with observations. For this
Fig.2. The effect of different calibrations for the Sun using reason,the Genevagridproducedby Mowlavietal. (2012) ne-
MESA. The solid line indicates the track for calibration per- glectssettlingforstarsabove1.1M⊙,whiletheBASTIgriddoes
formedusingjustthesolarluminosity,temperatureandZ/X.The not use settling even though it is included in the solar calibra-
dashedlineadditionallyincludesthedepthofthesolarconvec- tion(Pietrinfernietal.2004).InFig.4weshowtracksforboth
tionzone.Thedottedline includesallconstraintsincludingthe the ‘old’ and ‘new’ Sun, whose calibrations have been made
RMSsoundspeed.TheredpointrepresentstheSun.Theupper withgravitationalsettlingincluded.Alongsidethese,weinclude
panelshowstrackscomputedwiththeGS98abundances,while tracks for the same initial conditions, but without settling in-
theloweroneshowstrackscomputedwiththeA09abundances. cluded.Atthesolarage,themodelswithoutsettlingarearound
30Khotterthanthosewithsettling.By10Gyrthespreadintem-
perature has increased to around 100K (which is still smaller
the fit in the HR diagram is much improved. However, this is thanourestimatedGAIAerrorbox)12.Thispresumablyexplains
achievedattheexpenseofthesurfaceZ/X,whichis0.0246and whytheBASTItrackissomewhathotterthanothermodels.
thus clearly incompatible with the Asplundetal. (2009) abun-
dances. This model cannot be considered a reasonable calibra-
2.2.Uncertaintiesininputphysics
tion.
We now turn our attention to how small changes in the input
physics can affect the stellar tracks. We use the stars model
2.1.1. Settling
with A09 abundances and settling included as a baseline. We
Hereweillustratethedifferencebetweenmodelscalibratedwith will first deal with the effectsof changesin the assumed abun-
gravitationalsettlingandthosewithout.InFig.3,weshow1M⊙ dances. When one applies a stellar track to a given observa-
tracks for both the ‘old’ and ‘new’ Sun computed with and tion,anyuncertaintyintheobject’scompositionwillimpactthe
without settling. These tracks have been calculated using the fit of the stellar model. In Fig. 5, we show evolutionary tracks
starscode.Thecalibrationforthemodelswithoutdiffusion(Z , wherewehavealteredtheinitialhydrogenabundanceby±0.01.
0
X , α ) give (0.0132, 0.7279, 1.900) for the ‘new’ Sun and Decreasingthehydrogenabundanceby0.01(andthusincreasing
0 MLT
(0.0171,0.7000,1.965)forthe‘old’Sun.Thecalibrationshave theheliumabundancebythesameamount)makesthestarhotter
more of an effect on temperature than they do on luminosity. andmoreluminous.Forcomparison,wehavealsoplottedtracks
ClosetotheZAMS,theeffectivetemperaturesofthefourmod- for stars of 0.98M and 1.02M . These tracks are extremely
⊙ ⊙
els differ by about 20K. By 10Gyr, this spread has grown to similartoourtrackswithmodifiedinitialabundances.Thissug-
around 80K. At the hottest point, the spread in temperature is geststhattheuseofastellartrackwhosehydrogenabundanceis
around30K.Priortothesolarcalibrationpoint,modelswithout
gravitationalsettlingare coolerthanthosethatincludesettling. 12 Wefindsimilarresultswhenweperformthistestusingmesa.
5
R.J.Stancliffeetal.:CUSPII–stellartracks
0.4 0.4
0.3 0.3
0.2 0.2
(cid:12) (cid:12)
L L
L/ L/
g 10 0.1 g 10 0.1
o o
l l
0 0
-0.1 -0.1
-0.2 -0.2
5900 5850 5800 5750 5700 5650 5600 5900 5850 5800 5750 5700 5650 5600
T (K) T (K)
eff eff
Fig.4.Evolutionarytracksforthe‘old’(red)and‘new’(black) Fig.6.Evolutionarytracksfora1M modelshowingtheeffect
⊙
Sun for the initial conditions listed in Table 1. The solid lines of a small change in the total metallicity, Z. The solid line in-
were computed using gravitational settling while the dashed dicates the calibrated model,while the dotted and dashed lines
lines were computed without. The red circle represents the lo- showmodelswhereZhasbeendecreasedandincreasedby10%
cationoftheSun. respectively.Plusesmarkagesof500Myr,4.57Gyrand10Gyr.
TheredcirclerepresentsthelocationoftheSun.Redlinesmark
stellartrackswithmassesof0.98M and1.02M .
0.4 ⊙ ⊙
0.3 0.4
0.2 0.3
(cid:12)
L
L/
g 10 0.1 (cid:12) 0.2
lo L
0 g L/10 0.1
o
l
-0.1 0
-0.2 -0.1
5900 5850 5800 5750 5700 5650 5600
T (K)
eff
-0.2
5900 5850 5800 5750 5700 5650 5600
Fig.5. Evolutionary tracks for a 1M⊙ model showing the ef- Teff (K)
fect of a small change in the hydrogen abundance. The solid
lineindicatesthecalibratedmodel,whilethedottedanddashed Fig.7.Evolutionarytracksfora1M modelshowingtheeffect
⊙
lines show models where the initial hydrogen abundances has
of a smallchangein the mixinglengthparameter,α. The solid
been increased and decreased by 0.01 (a change of 14% from
lineindicatesthecalibratedmodel,whilethedottedanddashed
thecalibratedvalue)respectively.Plusesmarkagesof500Myr,
linesshowmodelswhereαhasbeensetto2.00and2.05respec-
4.57Gyrand10Gyr.Theredcirclerepresentsthelocationofthe
tively.Plusesshowmarkagesof500Myr,4.57Gyrand10Gyr.
Sun. Red lines mark stellar trackswith masses of 0.98M and
⊙ TheredcirclerepresentsthelocationoftheSun.
1.02M .
⊙
offby0.01(about14%)canleadtoa2%errorinthedetermina- 0.98M⊙ and1.02M⊙ (showningreyinthefigure)suggestthat
tionofthestellarmass. a 10% error in Z amounts to an error of over 2% in the mass
In Fig. 6 we show the effectsof a 10%variationin the ini- determination.
tialmetallicity.Noteforthesemodelsthatwehavenotincluded InFig.7,weshowtheeffectsofsmallchangesinthemixing
anyscalingoftheheliumabundancewithZasistypicallydone lengthcoefficientα.WhiletherangeofαgiveninTable1varies
whenconstructinggridsofmodelsoverlargemetallicityranges. from 1.65 to 2.09, such a large variation in α would produce
Becausewehavekepttheinitialhydrogenabundanceconstant, tracksthatareclearlyinconsistentwiththesolarcalibration.We
our 10% change in metallicity corresponds to less than a 1% adoptachangeoftheorderof0.05torepresentanotunreason-
changeintheinitialheliumabundance.Theeffectofmetallicity ableerrorinthesolarcalibration.Theluminosityofeachofthe
changesis slightly more pronouncedthat changesin the initial threeagemarkersonourtracksisscarcelychangedbythe≈1%
hydrogenabundance.Forourthreeagemarkers,theluminosity changein α and theyvaryat the levelof≈ 0.05%.Changesin
variesbyabout7-15%whilethetemperaturediffersbynomore thetemperatureofthese3timepointsarealsosmall,atthelevel
than 2%. Comparison of the modified metallicity tracks to our of≈0.1%.
6
R.J.Stancliffeetal.:CUSPII–stellartracks
2.3
2.25
2.2
2.15
(cid:12)
L/L 2.1
g 10 2.05
o
l
2
1.95
1.9
1.85
13000 12500 12000 11500 11000 10500 10000 9500 9000 8500
T (K)
eff
Fig.9.Evolutionarytracksfora3M modelshowingtheeffects
⊙
ofasmallchangeintheovershootingparameter,δ .Thesolid
ov
Fig.8. Evolutionarytracks for a 3M modelof Z ≈ 0.02.The lineindicatesthebasemodelwithδ = 0.15,whilethedotted
⊙ ov
boxrepresentstheestimatedpost-GAIAerroronluminosityand anddashedlineshaveδ = 0.12and0.18respectively.Pluses
ov
effectivetemperature. markagesof30,200and300Myr.
tivestability(Schro¨deretal.1997),suchthataregionisdefined
3. 3M⊙ models tobeconvectiveunstableif▽rad >▽ad−δ,where
In Fig. 8 we compare evolutionary tracks for 3M models of δ
⊙ δ= ov (2)
solar metallicity. BASTI’s canonical tracks do not go up to 2.5+20ζ+16ζ2
this mass, so we have instead plotted a 3M track from the
⊙
Binary Evolution Code (BEC, see Brottetal. 2011 and refer- and ζ is the ratio of radiation to gas pressure and δov is a con-
encesthereinforadescriptionofthiscode).Thetracksfallinto stantthatmustbe determined.Despitethe differentformalisms
two groups:the majorityofcodesshowextremelyclose agree- (aswellasthedifferentmethodstocalibratethefreeparameters
mentformostof the mainsequence.The Dartmouthandstars thatsuchformalismsnecessitate),allthecodesgiveverysimilar
tracksarehotterandmoreluminousthanthisgroup,thoughthe extentstothemainsequence.
two tracks agree extremely well with each other. We currently Aswithour1M⊙ models,wealsoshowtheeffectsofsmall
have no explanation for this difference. While the mesa track variationsin both the initial hydrogenabundance(Fig. 10) and
initiallybeginsclosetothefirstgroup,itssubsequentevolution thetotalmetallicity(Fig.11).Again,thesemodelsarecomputed
becomesclosetothestarsandDartmouthtracks,thoughtheul- with the stars code. For a change of 0.01 in X, the luminosity
timateextentofitsmainsequenceissomewhatlonger.Thesim- atourthreeagemarkers(30,200and300Myr)variesbetween
ilaritytothestarstrackisnotsurprising,givenourrecentrecal- 3 and 8% while the effective temperaturesdo not varyat these
ibrationoftheovershootinginbothcodesusingthesamesetof pointsbymorethan1%.Thevariationsinhydrogenabundance
eclipsing binaries(Stancliffeetal. 2015). The CESAM track is arecomparabletochangingthemassby1-2%.Similarvariations
anoutlierbecauseitisnotcomputedwiththeinclusionofcon- arefoundwhenwemodifytheinitialmetallicityby10%.Unlike
vective overshooting. Considering points of fixed age, the dif- the 1M⊙ models,the 3M⊙ modelsare extremelyinsensitive to
ferencesbetweenthetracksfallwithintheexpectedpost-GAIA variationsin the mixing length: tracks computed with α = 1.5
errorboxestimatedwiththesameassumptionsadoptedinSect. are indistinguishable from those computedwith α = 2.025 for
2withuncertaintiesof0.1maginthebolometriccorrection(see the whole of the main sequence through the Hertzsprung Gap.
e.g.Landstreetetal.2007)andof300Kineffectivetemperature This is because these stars do not have deep convective en-
(seee.g.Fossatietal.2009,forabestcasescenario). velopeswhileonthemainsequence.Notethatasthestarsevolve
Differences between the codes start to appear once the star tothegiantbranchonewouldexpecttoseedifferencesoncethe
surfaceconvectiveregionbecomesmoreextensive.
approachestheendofthemainsequence.Thisismostlikelydue
to the treatment of convective overshooting, which affects the
sizeoftheconvectivecoreandhencetheavailabilityofhydrogen
4. Discussion
for fusion. More extensive overshooting raises the star’s lumi-
nosityandthemainsequenceisextended(seeFig.9).Different Three physical inputs that we have not so far discussed and
codesapplyovershootingindifferentways.Somesimplyextend whichalsocontributetotheuncertaintiesinstellar calculations
thesizeoftheconvectiveregionbyafixedfractionofthelocal are:reactionrates,opacitiesandtheequationofstate.Reaction
pressurescaleheight(e.g.Dartmouth,Y2).mesaadoptsanexpo- ratesareperhapsoftheleastconcernasasourceoferror.Theen-
nentiallydecayingmixingcoefficientbasedonconditionsclose ergeticallyimportantreactionsfor hydrogenburning(i.e. those
totheformalconvectiveboundary(seeFreytagetal.1996,fora that will affect the stellar structure) are (mostly) all well de-
discussionofthisformalism).ThePadovagroupsetsthebound- termined. As an example, the CNO cycle bottleneck reaction
ary of the convectivecore at the location where convectiveve- 14N(p,γ)15O asmeasuredby Imbrianietal. (2005) (therecom-
locitiesgotozero,basedonthemethodofBressanetal.(1981). mended rate given in both the JINA REACLIB [Cyburtetal.
Finally,starsappliesamodificationtothecriterionforconvec- 2010] and Starlib [Sallaskaetal. 2013] reaction rate libraries)
7
R.J.Stancliffeetal.:CUSPII–stellartracks
Above a temperature of around 104K two sets of opac-
2.3
ities are commonly used in stellar evolution computations:
2.25
the OPAL data (Iglesias&Rogers 1996) and the OP data
2.2 (Badnelletal. 2005). At lower temperatures, the tables of
Alexander&Ferguson (1994) or Fergusonetal. (2005) are
2.15
(cid:12) used. The latter are to be preferred as they are an updated and
L/L 2.1 expandedversionoftheformer.Thechoiceoflowtemperature
g 10 2.05 opacitytableswillaffecttheeffectivetemperaturepredictedfor
lo stars on the red giant branch,or for very low-mass stars at the
2 beginningofthemainsequence.TheOPALandOPdata,which
1.95 arerelevantforthemajorityofstellarinteriorsconsideredhere,
generallydifferatthelevelof5-10%(Badnelletal.2005).This
1.9
risestoaround30%inthehightemperaturewingoftheZ-bump,
1.85 because the OP data places this bump at a slightly highertem-
13000 12500 12000 11500 11000 10500 10000 9500 9000 8500 perature (Badnelletal. 2005). We have used mesa to compute
T (K)
eff calibrated1M tracksusingbothOPandOPALopacitydatafor
⊙
solarcompositionofGrevesse&Sauval(1998).TheOPopacity
Fig.10. Evolutionary tracks for a 3M⊙ model showing the ef- isusedbydefaultinthecalculationspresentedabove.Whenthe
fects of a small change in the initial hydrogenabundance.The OPALdataisused,theparametersrequiredareX =0.7095,Z
0 0
dashedanddottedlinesindicatemodelsinwhichtheinitialhy- =0.0186andα = 1.8917(cf.thevaluesinTable1).Theevolu-
drogenabundanceshasbeenreducedandincreasedby0.01,re- tionarytracksarealmostindistinguishable,showingadifference
spectively.Evolutionarytracksformodelsof2.94and3.06M⊙ ofno morethan 3K atthe hottestpoint,with the OPAL model
areshowninred.Plusesmarkagesof30,200and300Myr. beingthecoolerofthetwo.
Thefinalpieceofinputmicrophysicsthatremainstobedis-
2.3 cussedisthechoiceofequationofstate(EoS).Mostofthecodes
listed employ the OPAL equation of state (Rogersetal. 1996),
2.25 whereas stars employs the Eggleton,Faulkner,&Flannery
2.2 (1973) EoS (hereinafter EFF), with additional physics added
byPolsetal.(1995),andtheDartmouthcodeusesanidealgas
2.15
L/L (cid:12) 2.1 E(CohSawboityherth&eDKeimbye1-9H9u¨5c).keDla¨cpoprreenct(i1o9n9f2o)rmcoamsspeasreadbotvhee0O.8PMAL⊙
g 10 2.05 andEFFEoSsforconditionsrelevanttothesolarhydrogenand
lo heliumionisation zones,findingdifferencesof the orderof 1%
2
(see also Rogersetal. 1996). These differenceswere attributed
1.95 to the lack of a Coulomb-pressureterm (Da¨ppen 1992), which
was subsequently added by Polsetal. (1995). The differences
1.9
should presumably now be much smaller, however we cannot
1.85 confirmthisasweknowofnorecentcomparisonsofthevarious
13000 12500 12000 11500 11000 10500 10000 9500 9000 8500
availableEoSs.Totesttheeffectsofusingdifferentequationsof
T (K)
eff state,wehaveusedmesatocomputetwo3M modelsequences,
⊙
oneusingtheOPALEoS,andoneusingFreeEoS14.At30Myr,
Fig.11. Evolutionary tracks for a 3M⊙ model showing the ef- the differencein effectivetemperaturebetween the two models
fectsofasmallchangeintheinitialmetallicity.Thedashedand isabout80K,withtheOPALmodelbeingthehotterofthetwo.
dottedlines indicatemodelsin which the initial Z has beenre-
The treatment of convection is perhaps still the Achilles’
duced and increased by 10%, respectively. Evolutionarytracks
Heelofstellarevolutioncalculations.Themixing-lengththeory
for modelsof 2.94 and 3.06M are shown in red. Pluses mark
⊙ (Bo¨hm-Vitense1958)isstillthemostwidelyusedofallthecon-
agesof30,200and300Myr.
vectivetheoriescurrentlyavailable (andindeedit is used in all
thecodeswehavelookedathere).Fromthecomputationalview-
point, it has the advantage of being a completely local theory
has an uncertainty13 of around 10% in the range of tempera-
sothatoneonlyneedsto knowabouttheconditionsatthecur-
turespertinenttomainsequencestars.Varyingthisratebetween
rentmeshpoint.Ithasalsoworkedextremelywell(onceprop-
the upper and lower limits produces no discernible change in
erlycalibrated)fora verylongtime.However,MLT doeshave
the evolutionary track of our 3M model computed with the
⊙ more‘free’parametersthanjustthemixinglengthasinthefor-
stars code.Note thatrevisionsto energeticallyimportantreac-
mulation of MLT one must make choices aboutsuch things as
tionratescanhaveasignificantimpactonstellarages.Revisions
the energy exchange a blob makes with its surroundings. Not
of the 14N(p,γ)15O rate by Formicolaetal. (2004) led to an
all formulations of MLT choose the same parameters and one
increase in the predicted age of the oldest globular clusters
can legitimately wonder what effect these choices may have.
(Imbrianietal. 2004), in addition to altering the lifetimes and
Salaris&Cassisi(2008)comparedtheeffectsofusingthestan-
luminositiesofadvancedevolutionaryphasesoflow-massstars
dard treatment of MLT used in most stellar evolution codes to
(Pietrinfernietal. 2010). When utilising a given set of tracks,
the treatment commonly used in modelling white dwarf atmo-
oneshouldalwayscheckthatthe reactionratesused areup-to-
date.
14 The FreeEoS fortran libraries are available from
13 SeeSallaskaetal.(2013)foradetaileddiscussionconcerningthe http://freeeos.sourceforge.net/. This equation of state is
meaninganddeterminationofreactionrateuncertainties. asubstantiallymodifiedversionoftheEFFEoS.
8
R.J.Stancliffeetal.:CUSPII–stellartracks
spheres.Theyshowedthatwithappropriatecalibration,thetwo puted with the stars code. 3M tracks from seven different
⊙
treatments give extremely similar results. This confirmsearlier codeshavealsobeencompared.Here,exceptionalagreementis
workbyPedersenetal.(1990),whoalsofoundthatwithappro- found between four of the codes, while two codes agree with
priate calibration evolutionary tracks are uniquely defined, re- each other very well despite being different from the majority
gardlessoftheMLTimplementationused.Recently,Arnettetal. group. Variations in the initial Z or X of around 10% lead to
(2010)proposedamodifiedMLT,inwhichthefreeparameters shifts in the stellar tracks at aroundthe 2% level. These differ-
of MLT are replaced by comparison of the relevant quantities ences amountto a changein the initial mass of 1-2%. Thus, if
with hydrodynamicalsimulations of convection.To the best of thecompositionofa star isnotknownto betterthan10%,a fit
our knowledge, this has never been employed in a stellar evo- ofanevolutionarytracktothestar’spositionintheHRdiagram
lutioncalculationthoughWood&Arnett(2011)didapplyitto cannotgivethestar’smasstobetterthan1-2%,evenifitslumi-
modelspulsationinasymptoticgiantbranchstars. nosityandeffectivetemperatureareknownexactly.
Itshouldbenotedthatoneofthe assumptionsofMLTthe-
oryisthatthemixinglengthobtainedbycalibrationtotheSun Acknowledgements. Wethank the anonymous referee forher/his constructive
criticismofthemanuscript.RJSistherecipientofaSofjaKovalevskajaAward
is applicable to all convective situations, regardless of stellar
fromtheAlexandervonHumboldtFoundation.LFandJCPacknowledgefund-
mass, metallicity or evolutionarystate. There is some observa- ing from the Alexander von Humboldt Foundation. JCP also thanks Norbert
tional evidence to suggest that this may not be the case (e.g. Langerforhissupport.
McSaveneyetal. 2007, require high mixing lengths to repro-
duce the coloursof their asymptotic giantbranch stars). In ad-
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AppendixA: mesasolarcalibration
IntableA.1,wepresentthedetailsofthevarioussolarcalibra-
tionscarriedoutwithmesa.
10