MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the construction of stellar isochrones

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by  Aaron Dotter

DRAFTVERSIONJANUARY21,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
MESAISOCHRONESANDSTELLARTRACKS(MIST)0:
METHODSFORTHECONSTRUCTIONOFSTELLARISOCHRONES
AARONDOTTER
ResearchSchoolofAstronomyandAstrophysics
AustralianNationalUniversity
Canberra,ACT
Australia
DraftversionJanuary21,2016
ABSTRACT
6
I describe a method to transform a set of stellar evolution tracks onto a uniform basis and then interpolate
1
withinthatbasistoconstructstellarisochrones. Themethodaccommodateabroadrangeofstellartypes,from
0
substellarobjectstohigh-massstars,andphasesofevolution,fromthepre-mainsequencetothewhitedwarf
2
coolingsequence.Idiscusssituationsinwhichstellarphysicsleadstodeparturesfromtheotherwisemonotonic

n relation between initial stellar mass and lifetime and how these may be dealt with in isochrone construction.
a I close with convergence tests and recommendations for the number of points in the uniform basis and the
J
mass between tracks in the original grid required in order to achieve a certain level accuracy in the resulting

0 isochrones. Theprogramsthatimplementthesemethodsarefreeandopen-source;theymaybeobtainedfrom
2 theprojectwebpage.a

Subjectheadings:methods: numerical—stars: evolution
]
R
S
1. INTRODUCTION a denser grid of tracks than is shown here for clarity. The
.
h thin lines running diagonally through Figure 1 identify three
To borrow a line from the poet John Godfrey Saxe,
p different phases of stellar evolution (see 2.2): the pre-main
isochrones, like sausages, cease to inspire respect in propor-
- sequence(‘Pre-MS’),thezeroagemains§equence(‘ZAMS’),
o tion as we know how they are made. Nevertheless, the in-
and the terminal age main sequence (‘TAMS’). The shaded
r tentofthispaperistoexplainonemethodofisochronecon-
t structionandshowtheresultsofthecodesthatimplementthe regionabovetheTAMSlineindicatesthepost-MSevolution.
s
Theprocessofisochroneconstructionistrivialifthestellar
a method.
[ A stellar evolution code produces output at fixed points in modelsallhavelifetimesfarexceedingthedesiredisochrone
age. Inthiscase,allthatisrequiredisasimpleinterpolation
time—timesteps—thatallowonetofollowtheevolutionofa
1 within each stellar evolution track to the desired age. This
model star over some portion of its lifetime. Timesteps are
v is evident in Figure 1 where multiple tracks intersect with a
chosen to meet various numerical tolerances and may vary
4 given isochrone before and during the main sequence (MS).
byordersofmagnitudeoverthespanoftheevolutionaryse-
4
An example of such are the Lyon models of very-low-mass
1 quence. The resulting data per timestep constitutes a stellar
stars(Baraffeetal.1997,1998). Theproblembecomesmuch
5 evolution track, which is a fundamental tool in the study of
harderwhenwewishtocapturethelatephasesofstellarevo-
0 stellar evolution. The primary input parameters of a stellar
lution. Referring again to Figure 1, the entirety of the post-
1. evolutiontrackareitsinitialmass(Minit)andchemicalcom- MS may lie between two stellar tracks in the model grid,
position but there are a host of other details relating to the
0 see the gray shaded region in the figure. In this case, we
physicsandnumericsassumedinanymodel.
6 encounter a situation in which neighboring stellar evolution
Anisochroneisanother, complimentarytoolthatisuseful
1 tracks in the model grid are in completely different phases
whenthepropertiesofastellarpopulation—ratherthanasin-
: ofevolutionandasophisticatedapproachisrequiredtocon-
v glestar—areofinterest. Anisochroneisderivedfromasetof
i stellarevolutiontracksspanningarangeofM butwiththe structisochronesthatfaithfullyreproducethemorphologyof
X init thetracks. Manyoftheisochronelibrariesinusetoday,such
same initial chemical composition. The underlying assump-
r tioninthecreationofanisochroneisthatallstarsareformed as BaSTI (Pietrinferni et al. 2004), Dartmouth (Dotter et al.
a 2008), PARSEC (Bressan et al. 2012), Pisa (Tognelli et al.
simultaneouslyfromahomogeneousgascloud. (Whetheror
2011), Victoria-Regina (VandenBerg et al. 2006), and Yale-
not these assumptions are ever actually met is beyond the
Yonsei (Yi et al. 2001), are based on the more-sophisticated
scope of this paper.) The goal of isochrone construction is
approach, one implementation of which is described in the
to change the independent variable from M in the set of
init following sections. The literature describing isochrone con-
stellarevolutiontrackstotimeintheisochrones,asillustrated
struction is sparse considering their widespread use in astro-
inFigure1.
physics;themostnotableexampleistheworkofP.Bergbusch
A representation of stellar evolution tracks and isochrones
and D. VandenBerg (Bergbusch & VandenBerg 1992, 2001;
inthesameplaneisshowninFigure1. Theisochrones(hor-
VandenBergetal.2006,2012).
izontal lines) have been generated from the tracks (vertical
A common question: Isn’t it possible to make isochrones
lines) using the methods described in this paper. It should
from a sufficiently-finely-sampled grid of stellar evolution
be noted, however, that the isochrones were generated from
tracks? Then isochrone construction requires a trivial inter-
polation to ‘connect the dots’ for a given age. The short an-
[email protected] swer is that such an approach is both inefficient and inele-
ahttps://github.com/dotbot2000/iso
2 A.Dotter
10
Isochrones
9
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a
8 Po r
s T
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-
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y 7 T t
[ A S
M
S
e
g
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g Z
o A
l M
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4
P
r
e-
M
3 S
0.5 0.0 0.5 1.0 1.5 2.0
−
log(M /M )
init
(cid:12)
Figure1. SchematicviewofstellarevolutiontracksandisochronesintheMinit-ageplane. Thediagonallinesandcorrespondinglabelsmarkthepositionsof
evolutionaryphasesuptoandincludingthemainsequence.Thegrayshadedregionshowsallpost-mainsequenceevolution.
gant. Inefficientbecausethenumberoftracksrequiredisor- were evolved to the white dwarf cooling sequence (WDCS)
dersofmagnitudegreaterthanistypicallyseen(104-105com- and the isochrones show the full range of evolution in the
paredtohundreds). Inelegantbecausethereisagreatdealof tracks.
similaritybetweentwostellarevolutiontrackswithcompara- Theagreementbetweenthesimpleandthesophisticatedin
ble M ; a sophisticated approach to isochrone construction Figure2isdecentalongthemainsequencebutbeginstobreak
init
willexploitthesimilarity. AnexampleisprovidedinFigure down on the red giant branch (RGB). The simple approach
2 where isochrones were constructed using this simple ap- skipsfromapointontheRGBtothecoreHe-burningphase,
proachaswellasthemore-sophisticatedapproachdescribed bypassingtheupperRGB,andthenfromcoreHe-burningto
laterinthispaperfromagridofstellarevolutiontrackswith near the bottom of the WDCS. Put another away, the differ-
mass sampling ∆M=0.001 M(cid:12) between 0.85 and 2.15 M(cid:12) ence in Minit between the early AGB and the WDCS is only
(1301tracksintotal).1 Thestellarevolutiontracksinthisgrid
lutionlibraries,where0.05M(cid:12)samplingistypicalforthisrangeofstellar
1Thismasssamplingismuchfinerthanistypicallyfoundinstellarevo- masses.
Isochroneconstruction 3
4
3
2
)
(cid:12)
L
/
L
( 1
g
o
l
0
1
Age = 10 Gyr
−
5.0 4.5 4.0 3.5
log(T [K])
eff
Figure2. Thefilledcirclesshowthepointsatwhichagridofstellarevolutiontracksreach10Gyr. ThedifferenceinMinitbetweeneachsuccessivepointis
0.001M(cid:12).Thegreenlineshowsthesimpleapproachtoisochroneconstructioninwhichagivenageislocatedineachtrackandthenthesepointsareconnected
bylinesegments.Thegraylineshowsanisochronemadeusingthesophisticatedapproachdescribedinthispaper.
0.001 M ! Yet this interval covers a great deal of distance Thesimpleapproachoflocatingthepointatwhicheachtrack
(cid:12)
in the H-R diagram and includes the brightest stars in the hasanageof13Gyrand‘connectingthedots’willresultin
isochrone. anisochronethatjumpsdirectlyfromtheMStotheWDCS,
Asamorein-depthexample,considerthatwewishtocon- skipping over all the intervening evolutionary phases. If in-
structa13Gyrisochronefromasetofstellarevolutiontracks steadweusetheapproachoutlinedinthefollowingsections,
with a mass interval of 0.05 M as visualized in Figure 3. thenweobtaintheisochroneshowninFigure3,whichfaith-
(cid:12)
The 1.0 M track has a MS lifetime of 10 Gyr and a total fullyreproducesalloftheevolutionaryphases.Theisochrone
(cid:12)
lifetimeof14Gyr. The0.95M trackhasaMSlifetimeof showninFigure3closelyfollowsthe1M trackfromthered
(cid:12) (cid:12)
12.3Gyrandatotallifetimeof16.5Gyr. The0.90M track giant branch through to the end because the later phases are
(cid:12)
hasaMSlifetimeof15.1Gyrandatotallifetimeof20Gyr. relativelyshort-lived.
At13Gyrthe0.9M modelisstillontheMS,the0.95M Thestellarevolutiontracksusedthroughoutthispaperare
(cid:12) (cid:12)
model is a subgiant, and the 1 M model is on the WDCS. takenfromMESAIsochronesandStellarTracks(MIST;Choi
(cid:12)
4 A.Dotter
4 3.0
M=0.90
3 2.5
M=0.95
M=1.00
2 2.0
13Gyr
) 1 1.5
(cid:12)
L
/
L 0 1.0
(
g
o
l
1 0.5
−
2 0.0
−
3 0.5
− −
5.0 4.5 4.0 3.5 3.80 3.75 3.70 3.65 3.60 3.55
log(T [K]) log(T [K])
eff eff
Figure3. A13GyrisochroneandthreestellarevolutiontrackswithMinit=0.90,0.95,1.00M(cid:12). Theleftpanelshowstheentirespanofa13Gyrisochrone
(black)andthe1M(cid:12)track(green). The0.90(red)and0.95M(cid:12)(blue)tracksareonlyplottedfromthepre-mainsequencetotheendoftheRGBphasefor
clarity. Therightpanelfocusesonthemainsequence,subgiantbranch,andredgiantbranch. Theisochroneisplottedasasolidlineforallpointsthatsatisfy
Minit<0.95M(cid:12)andasadashedlineelsewhere.Finally,thepointatwhicheachtrackhasanageof13Gyrisshownasafilledcircle;forthe1M(cid:12)trackthis
pointisonlyvisibleintheleftpanel.
et al., ApJS, submitted) and were computed with the stellar The complexities of stellar structure and evolution imply
evolution code MESA star, part of the MESA code library thatstellarevolutiontracksspanningarangeofM arelikely
init
(Paxton et al. 2011, 2013, 2015).2 The programs that im- to have (perhaps vastly) different lifetimes and numbers of
plement the methods described in the following sections are timesteps. Themodelstarsmayexperienceentirelydifferent
writteninFortrananduseMESAmodules,primarilyforinter- evolutionaryphases.Thissituationisfarfromidealifthegoal
polation. AlthoughoriginallydesignedtoingestMESAstar istointerpolateamongstthetrackstoconstructisochrones.
historyfiles,itiscertainlypossibletoincorporatetracksfrom Thetaskofinterpolatingamongstasetofstellarevolution
otherstellarevolutioncodes. Itisonlyamatterofloadingthe tracks can be greatly simplified by transforming the original
tracksintothedatastructuresusedinthecodes. tracks,asoutputbythestellarevolutioncode,ontoauniform
Interpolationplaysakeyroleinisochroneconstruction. It basis. Theuniformbasisisdesignedinsuchawaythateach
isthereforeworthwhiletodescribethekeyfeaturesofthein- phaseofstellarevolutionisrepresentedbyafixednumberof
terpolationmethodusedthroughoutthefollowingsections. A points and that the nth point in one track has a comparable
cubicinterpolationschemeprovidesagoodbalancebetween interpretationinanothertrack. Thisisaccomplishedbyintro-
smoothness,includingcontinuousfirstderivatives,andarea- ducingtheconceptofequivalentevolutionaryphases(EEPs),
sonablysmallnumberofneighboringpointsrequiredtocon- a series of points that can be identified in all stellar evolu-
struct the interpolating function. One important considera- tiontracks.3 EEPsserveastheuniformbasistodescribethe
tion is that the interpolation method preserve monotonicity evolutionofallstarsandaredividedintotwocategories. Pri-
throughouttheinterpolationinterval. Thisisparticularlyim- mary EEPs identify a relatively small number of physically-
portantinisochroneconstructionbecausethestandardproce- motivated phases in the tracks. Secondary EEPs provide a
dure hinges on a monotonic relation between M and age. uniformspacingbetweentheprimaryEEPsineachtrack.
init
Thepiecewise-monotoniccubicinterpolationmethodofStef-
fen(1990), asimplementedintheMESAinterp_1dmod- 2.1. PrimaryEEPs
ule,meetsbothofthesecriteriaandisusedbydefaultinthese PrimaryEEPsareexplicitlydefinedforeachtrackandhave
codes. Allreferencestointerpolationthroughouttheremain- thesamephysicalinterpretationacrossdifferenttracks. How-
derofthispaperrefertotheSteffen(1990)method. ever,sincethegoalofthecodesdescribedhereistobeappli-
Units in this paper are generally cgs, except as explicitly cabletoallstars,thereisonebranchpoint(seeEEP8inthe
noted,andagesarealwaysgiveninyears. Thefollowingsec- list below). The distinction between low- and intermediate-
tionspresentthetheoryandrelevantdetailsofhowthecodes mass stars on one hand and high-mass stars on the other is
are implemented; a practical guide to using the programs is determined by whether or not the central temperature of the
distributedalongwiththecodesthemselves. modelattheendofitsevolutionarytrackislessthanthecen-
traltemperatureaftertheendofcoreHe-burning. TheT cri-
c
2. IDENTIFYINGEQUIVALENTEVOLUTIONARYPHASES
3 AdescriptionofisochroneconstructionusingcentralHmassfraction
asatimeanalogbySimpsonetal.(1970)is,tomyknowledge,theearliest
2http://mesa.sourceforge.net discussionintheliteratureofsomethingliketheEEPformalism.
Isochroneconstruction 5
1.0 4
TAMS
0.5 PreMS
3
0.0
IAMS
2
) ZAMS
) ]c
(cid:12) 0.5 c
L − /
/ g 1
L [
(g 1.0 (ρc
o − TAMS g
l o
l 0
1.5
−
1
2.0 IAMS
− ZAMS − M = 0.3M
PreMS init
(cid:12)
2.5
− 3.64 3.62 3.60 3.58 3.56 3.54 3.52 3.50 5.5 6.0 6.5 7.0 7.5
log(T [K]) log(T [K])
eff c
Figure4. A0.3M(cid:12)stellarevolutiontrackintheH-Rdiagram(right)andtheTc-ρcdiagram(left). Theoriginaltrackisshownasthesolidline. Theprimary
EEPsdescribedin§2.1arelabeledandmarkedbylargedots;secondaryEEPsdescribedin§2.2areshownassmalldotsalongthetrack.
8
4
WDCS
PostAGB PostAGB
RGBTip 6 RGBTip TP-AGB
3 TP-AGB
ZACHeB
) 4 TACHeB
) 2 TACHeB ]c
(cid:12) PreMS c TAMS
L ZACHeB /
/ g 2 IAMS
(L 1 [ρc ZAMS
g (
o g
l TAMS o
0 l 0
IAMS
ZAMS
1 2
− − M = 1M
WDCS PreMS init
(cid:12)
2 4
−5.5 5.0 4.5 4.0 3.5 3.0 −5.5 6.0 6.5 7.0 7.5 8.0 8.5
log(T [K]) log(T [K])
eff c
Figure5. EquivalenttoFigure4fora1M(cid:12)stellarevolutiontrack.
terionisanindicationofwhetherthemodelsstarwillevolve central H mass fraction. Examples of 0.3, 1, 5, and 20 M
(cid:12)
toacoolingWD(inwhichcaseT islowerattheend)orpro- stellartrackswithprimaryEEPsidentifiedareshowninFig-
c
ceedontolaterstagesofnuclearburning(inwhichcaseT is ures4,5,6,and7,respectively.NotethatthenumberofEEPs
c
higherattheend). Furthermore,substellarobjects,forwhich showninFigures4-7hasbeenreducedbyafactorof10from
substantialnuclearburningdoesnotoccur,aretreateddiffer- the recommended numbers so that the individual EEPs may
ently. beclearlyseen.
ThissectionlistsalloftheprimaryEEPsandexplainshow
1. The pre-MS EEP (PreMS) is chosen to identify the
each is defined. In what follows X andY refer to the mass
point at which the central temperature (T) rises above
fractions of H and He, respectively, and subscript c denotes c
acertainvalue(lowerthannecessaryforsustainednu-
the property at the center of the star. For example, X is the
c clearreactions).Bydefaultthisissetatlog(T)=5.0but
c
6 A.Dotter
5.0 8
WDCS
7
4.5 PostAGB
PostAGB
TP-AGB
TP-AGB 6
4.0
)
) ]c 5
(cid:12) c
L /
RGBTip
/ 3.5 g 4 TACHeB
L [
(og TAMS TACHeB ZRAGCBHTiepB (ρgc 3 ZACHeB
l 3.0 IAMS PreMS lo TAMS
2
ZAMS
2.5 ZAMS
1 IAMS
M = 5M
init
(cid:12)
2.0 0
4.4 4.2 4.0 3.8 3.6 3.4 7.0 7.5 8.0 8.5 9.0
log(T [K]) log(T [K])
eff c
Figure6. EquivalenttoFigure5fora5M(cid:12)model.NotallprimaryEEPsareshowninbothpanels.
5.4 7
CBurn
CBurn
5.2
RGBTip 6
TAMS TACHeB
5.0 ZACHeB
5
) 4.8 IAMS ])c
(cid:12) c 4
L /
/ 4.6 g
(gL ZAMS ([ρc 3 RGBTip TACHeB
lo 4.4 og TAMS ZACHeB
l
2
4.2 PreMS
1
4.0 ZAMS
M = 20M
IAMS init
(cid:12)
3.8 0
4.6 4.4 4.2 4.0 3.8 3.6 3.4 7.0 7.5 8.0 8.5 9.0 9.5
log(T [K]) log(T [K])
eff c
Figure7. EquivalenttoFigure5fora20M(cid:12)model. NotallprimaryEEPsareshowninbothpanels. NotethatintheTc-ρcplanetwopairsofprimaryEEPs
arenearlycoincident:ZAMSandIAMS;RGBTipandZACHeB.TheRGBTipprimaryEEPdoesnothavethesamesignificanceinsuchhigh-massstarsbutis
maintainedforconsistency,asnotedinthetext.
maybesettoadifferentvalue. Itisrecommendedthat mass fraction has fallen by below its initial value by
thepre-MSpointbechosenasearlyaspossible. Ifthe 0.0015. In the substellar case where neither of these
firstpointinthestellarevolutiontrackisalreadyabove criteria is met, the ZAMS point is taken as the maxi-
this threshold, we simply use the first point. It is im- muminT alongtheevolutionarytrack.
c
plicitlyassumedthatevensubstellarobjects,nominally
M(cid:46)0.08M ,willmeetthePreMST condition. 3.-4. TowardstheendofcoreH-burning2primaryEEPsare
(cid:12) c defined at X =0.3 (intermediate age main sequence:
c
2. The zero-age main sequence (ZAMS) EEP is taken as IAMS) and X = 10−12 (terminal age main sequence:
c
the first point after the H-burning luminosity exceeds TAMS).Twopointsareusedtoisolatetheportionofthe
99.9%ofthetotalluminosityand beforethecentralH MSthatmayormaynotbeinfluencedbythepresence
Isochroneconstruction 7
1.5
1.0
)
(cid:12)
L
0.5
/
L
(
g
o
l
0.0
0.5
−
3.80 3.75 3.70 3.65 3.60
log(T [K])
eff
Figure8. ThisfigureshowshowtheprimaryandsecondaryEEPsaredistributedalongthreestellarevolutiontrackswithMinit=0.95,1,and1.05M(cid:12). The
originaltracksareplottedassolidlines;thesecondaryEEPsareshownassmalldotsandtheprimaryEEPsareshownaslargedots. Thisshouldgivesome
indicationthattheEEP-basedtracksconstituteauniformbasis.
of a convective core. The practical choice of placing 6. ThezeroagecoreHeburning(ZACHeB)EEPdenotes
twoprimaryEEPsneartheendoftheMSgreatlysim- the onset of sustained core He burning. The point is
plifiesthetreatmentofthe‘convectivehook’featurein identified as the T minimum that occurs after the on-
c
the H-R diagram. In the substellar case, for which no setofHe-burning(RGBTip)whileY >Y −0.03.
c c,RGBTip
central H is consumed, the final point in the track is This temperature minimum is readily identifiable in
chosenastheTAMSprovidedthattheageatthatpoint lower-massstars(M <2M )becausetheT-ρ evo-
init (cid:12) c c
isgreaterthansomeminimum(e.g.,20Gyr). lution in this interval has a particular shape due to the
off-centerignitionofHeburningunderdegeneratecon-
ditions (Paxton et al. 2011). The same feature is less
5. The RGB tip (RGBTip) EEP identifies the point at obviousinhigher-massstarswithnon-degeneratecores
which the stellar luminosity reaches a maximum—or butstillidentifiable.
thestellarT reachesaminimum—aftercoreHburn-
eff
ing is complete but before core He burning has pro- 7. One primary EEP is identified at the end of core He
gressedsignificantly.ThisEEPhasarecognizableloca- burning(terminalagecoreHeburning: TACHeB)cor-
tionontheH-Rdiagramoflow-andintermediatemass respondingY =10−4.
c
stars, hence the name, but the point defined here can
alsobelocatedinhigh-massstellartracksthatdonotgo 8a. The EEP marking the onset of the thermally-pulsing
througha‘redgiant’phase. Thisisachievedbytaking AGB(TP-AGB)isidentifiedasthepointaftercoreHe
the point at which the luminosity reaches a maximum burning (Y < 10−6) when the difference in mass be-
c
or the T reaches a minimum, whichever comes first, tween the H-burning and He-burning shell is less than
eff
before the center He mass fraction (Y ) is significantly 0.1M . ThisisthesamecriterionusedinMESAstar
c (cid:12)
reducedbyHeburning:Y Y −0.01. toidentifytheonsetofthermalpulsations.
c c,TAMS
≥
8 A.Dotter
8b. For stellar models that are massive enough to bypass ForthesecondaryEEPstobe‘equally-spaced’wemustde-
theTP-AGBandproceedtolaterphasesofcoreburn- fine a metric along the evolutionary track. The only (mathe-
ing, the final EEP is set at the end of core C burning matical)constraintonthemetricisthatitmustbepositivedef-
(CBurn), whenthe central C massfraction falls below inite.TraditionallythemetrichasbeendefinedasaEuclidean
10−4. ThismarkstheendoftheprimaryEEPsformas- distancealongthestellarevolutiontrackintheHertzsprung-
sivestars. TheremainingprimaryEEPsareonlyappli- Russell (H-R) diagram. However, it can be useful to in-
cabletolow-andintermediate-massstars. cludeanagetermwithappropriateweight(seethediscussion
in 4 of VandenBerg et al. 2012). The terms are weighted
9. Apost-AGB(PostAGB)EEPisidentifiedonlyinstellar (str§etched)suchthatthelogarithmicrangesspannedbyastel-
modelsthatwillgoontoformaWD.Itismeanttolo- larevolutiontrackinluminosityandT contributetothemet-
eff
cate the point at which the TP-AGB phase has ended ricdistanceinroughlyequalamounts. Additionalterms,with
and the star has begun to cross the H-R diagram at arbitraryweights,maybeaddedastheuserwishes. Themet-
nearly constant luminosity. The PostAGB EEP is de- ricdistanceDalongthetrackisdefinedsuchthatD =0and
0
fined as the point at which the H-rich stellar envelope thedistancebetweenanytwopointsiandi+1intheoriginal
fallsbelow20%ofthecurrentstellarmass. evolutionarytrackaregivenby
(cid:118)
10. TheWDCSEEP,whichfollowsthePost-AGB,isbased (cid:117) N
(cid:117)(cid:88)
onthecentralvalueoftheCoulombcouplingparameter Di+1=Di+(cid:116) wj(xj,i+1−xj,i)2. (1)
Γ, with a default upper limit of 100. WDCS is only
j=1
consideredformodelsthathaveaPostAGBEEP.
(cid:80)
Then the total distance along the track is D but, in prac-
i i
The code processes each primary EEP in order using the tice,thequantityofinterestisthedistancebetweentwoadja-
location of the previous EEP as starting point and searching centprimaryEEPs. Thewj inEquation1arearbitraryweight
for the conditions needed to identify the next EEP through factorsappliedtotheindividualterms. Thexj arecolumnsof
to the end of the track. The process ends either when the datafromthestellarevolutiontrack. Itisconvenient,butnot
full list of primary EEPs have been identified or at the first strictlynecessary,touselogarithmicquantitiesforthexj be-
failuretoidentifyaprimaryEEP.Thelistpresentedaboveis causemanyofthequantitiesusedspananorderofmagnitude
thecompletelistofprimaryEEPsinthecode. However,itis ormore. Inprinciple,anyquantityfromthestellarevolution
notnecessarytouseallofthem. Forexample,onecouldskip track may be used in Equation 1. Experiments with central
thePreMSprimaryEEPforagridofstellarevolutiontracks temperatureanddensityindicatethatbothcanbeusedeffec-
thatbeginfromtheZAMS. tively in the metric funcion. It should be noted that the sin-
gularpurposeofthemetricistoplacesecondaryEEPsalong
2.2. SecondaryEEPs theevolutionarytrackbetweentwoprimaryEEPs;themetric
isnotusedexplicitlyinanylatersteps.
Secondary EEPs serve the purpose of faithfully capturing
themorphologyofeachsegmentoftheevolutionarytrackthat 2.4. Summary
lies between two adjacent primary EEPs. After the primary
Section2describesatwo-stepprocesstoassignEEPsand
EEPshavebeenidentified,eachsegmentispopulatedwitha
create uniform, EEP-based stellar evolution tracks that can
number of equally-spaced secondary EEPs. In order to con-
laterbeusedforinterpolationofothertracksandisochrones.
struct a uniform basis of EEP-based tracks for interpolation,
The first step is to cycle through the original stellar evolu-
the number of secondary EEPs between a given pair of pri-
tiontrackandidentifiesthepointsthatarecoincidentwiththe
mary EEPs is held constant over the set of stellar evolution
primary EEPs. The second step is to cycle through the orig-
tracks.Figures4through7showbothprimary(aslargerdots)
inal track again and locate the desired number of secondary
andsecondaryEEPs(assmallerdots). Furthermore,Figure8
EEPs between each pair of primary EEPs. It is important to
shows 3 neighboring tracks from a model grid with the pri-
note that although primary and secondary EEPs are distinct
maryandsecondaryEEPsshown;inthiscaseitiseasytosee
in the way that they are identified in the original stellar evo-
howtheEEPsinonetrackcorrespondtothoseineachofthe
lution tracks there is no difference in the way they are used
othersand,thus,formasuitablebasisforinterpolation.
forinterpolationpurposes. Thetotalnumberofpointsinthe
OncetheprimaryEEPshavebeenidentifiedthestellarevo-
EEP-basedtrackwillequalthesumofthenumberofprimary
lutiontrackisreadytobeconvertedfromitsoriginalformto
EEPs and the number of secondary EEPs between each pair
thenewformconsistingonlyofEEPs. Thenew, EEP-based
ofprimaries.
trackconsistsofalltheprimaryEEPsthatwereidentifiedin
the original track plus a larger number of secondary EEPs. 3. CONSTRUCTINGNEWTRACKSANDISOCHRONES
Using a metric function calculated along the original track
Once a set of stellar evolution tracks has been processed
( 2.3), eachintervalbetween2primaryEEPsisdividedinto
§ ontoauniformgridofEEPs,itisstraightforwardtointerpo-
afixednumberofequally-spacedsecondaryEEPs;theinfor-
latenewstellarevolutiontracksforM valuesnotincluded
mation to be included in the EEP-based track is interpolated init
intheoriginalgridandtogenerateisochrones; extrapolation
fromtheoriginaltrackontotothesecondaryEEPs. Process-
isnotpermitted.
ing stellar evolution tracks in this way results in a reduction
inthesizeoftheevolutionarytracksbyafactorof 10with- 3.1. Creatinganewstellarevolutiontrack
∼
out significant loss of information for a suitable number of
Theprocessofgeneratinganewstellarevolutiontrackisas
secondaryEEPs(see 3.5).
§ simpleasidentifyingasubsetof2ormore(4forcubicinter-
polation)tracksintheoriginalgridwhoseM valuesenvelop
2.3. Themetricfunction init
that of the new track. Once a set of tracks with appropriate
Isochroneconstruction 9
M -valueshavebeenindentified,anewtrackcanbecreated have a higher M than the one before it. However, stellar
init init
byloopingoverallEEPs,interpolatingamongstthosetracks physics is not always so conformant and there are instances
atfixedEEPnumberusingM astheindependentvariable. where, over a small interval in M , the M -age relation
init init init
TheresultingtrackhasthesamenumberofEEPsasthosethat is non-monotonic. The mathematical formalism described
wentintoconstructingit. above does not work in such cases. In fact, it is possible to
ConsidertheexampleshowninFigure9,whereanewstel- missthesignatureofsuchtransitionsifthegridofstellarevo-
larevolutiontrackwithM =1.2M ,createdbycubicinter- lutiontracksissufficientlycoarse.Thefinerthegrid,themore
init (cid:12)
polationfrom4trackswithM =1.10, 1.15, 1.25, and1.30 pronouncedtheeffect(Girardietal.2013).
init
M , is compared with the actual M = 1.2 M track from Even with careful choice of primary EEPs ( 2.1) and dis-
(cid:12) init (cid:12)
§
the existing set of models. Figure 9 shows both the H-R di- tancemetric( 2.3),changesinstellarphysicscanleadtonon-
§
agram(toprow)andthecentraltemperature-densitydiagram monotonic behavior in the M -age relation. There are two
init
(bottom row). Panels on the left show the full evolutionary well-known examples of this: the first is the appearance of
tracks,frompre-MStoacoolingwhitedwarf(WD),whileon theconvectivecoreduringcoreH-burninginstarsmoremas-
therightareshowntheevolutionfromZAMStoRGBTip. sivethan1M .Thesecondisthetransitionbetweentheonset
(cid:12)
ofcoreHe-burningunderdegenerateandnon-degeneratecon-
3.2. Creatingisochrones ditionsaround1.8-2M (Girardietal.2013).5 Inbothcases,
(cid:12)
The process of generating an isochrone is more involved the change in stellar physics leads to a marked difference in
than that of generating a new track but it is, nevertheless, stellarlifetimethattemporarilyreversesthetrendofdecreas-
straightforward on a uniform grid of EEPs. For a given age, inglifetimewithincreasingMinit.
anisochroneisconstructedbyloopingoverthecompleteset Figure10givesanexampleofsuchnon-monotonicbehav-
ofEEPs,identifyingwhicharevalidforthatage,andthenper- ior towards the end of the MS due to the appearance of a
formingthenecessaryinterpolationstoarriveatthestellarpa- convectivecore(leftpanel)andthecasepresentedbyGirardi
rametersforthatEEP.Inthiscontext,anEEPis‘valid’ifmore etal.(2013,rightpanel).ShownintheleftpanelaretheMinit-
thanoneevolutionarytrackexistsatthatageforthatEEP.For agerelationsforMSEEPs.ShownintherightpanelareMinit-
example, an ancient isochrone (age > 10 Gyr) will have no age relations for core He-burning EEPs. In either case, the
validEEPsonthepre-MSwhileaveryyoungisochrone(age intermediateEEPisnoticeablynon-monotonic. Anisochrone
<5Myr)willhavenoTP-AGBorWDCS. for a particular range of ages (suggested by the dashed lines
The construction of an isochrone for a given age is per- inFigure10)willhavetwoMinitvaluesthatcorrespondtothe
formedasaloopoverallEEPs: sameageandthesameEEP.EachpointplottedinFigure10
istakenfromoneEEP-basedtrackfromthedensegridmen-
1. Make a list of the stellar evolution tracks that include tionedearlierinthepaper. Inbothcases,evenataresolution
each EEP at that age. If none exist, then cycle to the of0.001M thetransitionismarkedbyadiscontinuity.
(cid:12)
nextEEP. One way to avoid this problem is to break the non-
monotonicM -agerelationintotwomonotonicrelationsand
2. ForthatEEPandlistoftracks,anorderedM -agere- init
init then choose one or the other to represent that EEP. This ap-
lationisconstructed.
proachislessthanidealbecauseitrequiresanarbitrarychoice
to use one and neglect the other. Another approach is to ar-
3. Usingtheinputage, obtaintheM -valueappropriate
init
tificially smooth the M -age relation before interpolating a
for that age and EEP by interpolation in the M -age init
init
newM .
relation(seeFigure10). init
4. With M in hand, all stellar parameters for that EEP 3.4. Embracingnon-monotonicbehaviorintheM -age
init init
are obtained by another round of interpolations using relation
M astheindependentvariable.
init The discussion in 3.3 suggests that if we relax the stan-
§
dardmathematicalformalismforconstructingisochronesde-
Thewholeprocessisthenrepeatedforotherages.
scribedabove, thenwewillbeabletostudyphenomenathat
Instances, primarily during the TP-AGB phase, can occur
we otherwise could not. The most obvious case is that in
when interpolation and even finite numerical precision can
whichtheM -agerelationofagivenEEPisnon-monotonic
lead to non-monotonic M values along a given isochrone. init
init
duetosometransitioninstellarphysicsoverasmallrangeof
Such instances are numerical rather than physical and, thus,
M .
distinct from those discussed in 3.3 and 3.4. In this case it init
§ Asimpleapproach,perhapsthesimplestapproach,istoal-
ispossibletoenforcemonotonicityviathe‘pooladjacentvi-
olators’(PAV)algorithm.4 ThePAValgorithmisaniterative low one EEP to represent as many Minit values as the Minit-
age relation allows. In the case of a multi-valued M -age
procedurethatenforcesmonotonicitybysearchingthroughan init
relation we simply divide it into as many monotonic inter-
array and replacing a non-monotonic value with a weighted
vals as are present and proceed as usual. An isochrone con-
averageofneighboringpoints.
structedinsuchamannerwillhavethesamenumberofEEPs
3.3. Mitigatingnon-monotonicbehaviorintheMinit-age but a greater number of Minit values than one constructed
relation in the standard way. Following the example of Figure 10, a
multivaluedisochronewouldhave2M valuesforthenon-
Thestandardassumptionemployedinisochroneconstruc- init
monotonicEEPshown.
tion is that, for a given EEP, the M -age relation will be
init
monotonic. Thus for each age and EEP a unique M value
init 5Athird,weakercaseisthetransitionfromfully-convectivestarstostars
isobtained. EachEEPinagivenisochroneshould,therefore,
withradiativecores,whichtakesplacearound0.35M(cid:12). Indeed,theremay
beothercasesbutthetworeferredtointhetextarethemostpronounced
4http://stat.wikia.com/wiki/Isotonic_regression examples.
10 A.Dotter
4
M = 1.2 M
3 init
(cid:12)
3
)
(cid:12)
L 2 2
/
L
(
g
o 1
l 1
0
0
3.8 3.7 3.6 3.5 5.2 4.6 4.0 3.4
log(T [K]) log(T [K])
eff eff
6
6
5 4
)
]
c
c
/
g 2
4
[
c
ρ
(
g 0
o
3
l
Original
2
− Interpolated
2
7.1 7.3 7.5 7.7 7.9 5.5 6.5 7.5 8.5
log(T [K]) log(T [K])
c c
Figure9. Comparisonofa1.20M(cid:12)stellarevolutiontrackcreatedbyinterpolationwithonecomputeddirectlybyMESAstar.
One challenge with this approach is that the resulting from original tracks to EEPs and from EEPs to isochrones.
isochroneisitselfmulti-valuedand,therefore,cannotbeused Hereweconsiderbothinturn.
in the same way as a canonical isochrone. For example, in- Thefirstisthetransformationofstellarevolutiontracksto
tegrating an initial mass function over a canonical isochrone EEP-basedtracksasdescribedin 2.Theresolutionisentirely
§
is straightforward but the same integral over a multi-valued controlled by the number of secondary EEPs since the num-
isochroneisnot. Thisalternativeapproachtoisochronecon- ber of primary EEP is fixed (and few). The most important
structionwillrequiresomechangesinthewayisochronesare consideration in setting the number of secondary EEPs be-
usedbutisneverthelessaworthwhileavenuetoexplore. tweenanygivenpairofprimaryEEPsisthatthesecondaries
properlyreproducethemorphologyoftheoriginaltracks. In
practice,from50to200secondaryEEPsbetweenanypairof
3.5. ConvergenceTests
primaryEEPsshouldbesufficienttocoversmoothphasesof
It is informative to consider what level of resolution is
stellar evolution. Here smooth is loosely defined as lacking
neededtofaithfullyreproducethepredictionsoftheunderly-
largederivativesand/oroscillatorybehaviorofthequantities
inggridofstellarevolutiontracksinboththetransformation