A new generation of PARSEC-COLIBRI stellar isochrones including the TP-AGB phase

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by  Paola Marigo

To appear soon in ApJ
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
A NEW GENERATION OF PARSEC–COLIBRI STELLAR ISOCHRONES INCLUDING THE TP-AGB PHASE
Paola Marigo1, Le´o Girardi2, Alessandro Bressan3, Philip Rosenfield4, Bernhard Aringer1, Yang Chen1,
Marco Dussin1, Ambra Nanni1, Giada Pastorelli1, Tha´ıse S. Rodrigues1,2, Michele Trabucchi1, Sara Bladh1,
Julianne Dalcanton5, Martin A.T. Groenewegen6, Josefina Montalba´n1, Peter R. Wood7
To appear soon in ApJ
ABSTRACT
We introduce a new generation of PARSEC–COLIBRI stellar isochrones that include a detailed treat-
ment of the thermally-pulsing asymptotic giant branch (TP-AGB) phase, and covering a wide range
7 of initial metallicities (0.0001 < Z < 0.06). Compared to previous releases, the main novelties and
i
1
improvements are: use of new TP-AGB tracks and related atmosphere models and spectra for M and
0
C-type stars; inclusion of the surface H+He+CNO abundances in the isochrone tables, accounting
2
for the effects of diffusion, dredge-up episodes and hot-bottom burning; inclusion of complete ther-

n mal pulse cycles, with a complete description of the in-cycle changes in the stellar parameters; new
a pulsation models to describe the long-period variability in the fundamental and first overtone modes;
J new dust models that follow the growth of the grains during the AGB evolution, in combination

0 with radiative transfer calculations for the reprocessing of the photospheric emission. Overall, these
3 improvements are expected to lead to a more consistent and detailed description of properties of TP-
AGB stars expected in resolved stellar populations, especially in regard to their mean photometric

] properties from optical to mid-infrared wavelengths. We illustrate the expected numbers of TP-AGB
R
stars of different types in stellar populations covering a wide range of ages and initial metallicities,
S providingfurtherdetailsonthe“C-starisland”thatappearsatintermediatevaluesofageandmetal-
. licity, and about the AGB-boosting effect that occurs at ages close to 1.6-Gyr for populations of all
h
metallicities. The isochrones are available through a new dedicated web server.
p
- Subject headings: stars: general
o
r
t
s 1. INTRODUCTION This need for more information is particularly impor-
a tant when we refer to thermally-pulsing asymptotic gi-
[ Theoretical stellar isochrones are remarkably useful
antbranch(TP-AGB)stars. Theycontributetoasizable
datasets in astrophysics. Historically, they have been
1 commonly used to attribute approximated ages and dis- fractionoftheintegratedlightofstellarpopulations,and
v tances to star clusters observed in at least two filters, are responsible for a significant fraction of the chemical
0 in the traditional process of isochrone fitting (since e.g. enrichmentanddustproductioningalaxies. Yet,theTP-
1 Demarque & McClure 1977). In the last decades, their AGB phase is still the most uncertain among the main
5 evolutionary phases of single stars, since its evolution is
use has expanded in many different ways. Since Charlot
8 determined by a series of complex and interconnected
& Bruzual (1991), they are at the basis of the isochrone
0 processes that still cannot be simply modeled from first
method of the spectrophotometric evolutionary popula-
. principles–likethethirddredge-upepisodes,long-period
1 tion synthesis, which has revolutionized the interpreta-
pulsationandmassloss,etc.(seeFrost&Lattanzio1996;
0 tion of the spectra and photometry of distant galaxies.
7 They have also been extensively used in the more com- Mowlavi 1999; Herwig 2005; Marigo 2015). The calibra-
1 plexmethodsofCMDfittingofnearbygalaxiesandstar tion of this phase requires the construction of extended
: clusters(e.g.Dolphin2002),whichaimatderivingquan- sets of isochrones in which all the relevant processes are
v
consideredandalltherelevantobservablesaretabulated.
i titative measurements of their star formation and chem-
X ical enrichment histories. Only in this way we can enable a quantitative compari-
son between the model predictions and the properties of
r While the most traditional applications of isochrones
a refer to the interpretation of spectrophotometric data TP-AGB populations observed in nearby galaxies.
Indeed, in Marigo et al. (2008) we made a significant
only, more recent applications regard other quantities
step in this direction, by providing the first isochrones
as well. For instance, the advent of large microlensing
in which the properties of TP-AGB stars were consid-
(Paczynski 1996) and spectroscopic surveys (York et al.
ered in more detail, and including crucial processes like
2000) in the nineties has opened access to the variabil-
third dredge-up, hot-bottom burning, low-temperature
ity and the chemical composition information for huge
opacitychanges,thedistinctspectraofcarbon(C)stars,
populations of stars in the Milky Way and its satellite
the reprocessing of radiation by circumstellar dust in
galaxies,whiletheonsetofspace-basedasteroseismology
phases of increased mass-loss, and the expected pulsa-
now gives access to fundamental quantities such as the
tion periods. In this paper, we provide new isochrones
masses and radii even for stars located tens of kilopar-
derived from the stellar evolutionary tracks computed
secs away (Chaplin & Miglio 2013). The interpretation
with the more recent PARSEC (Bressan et al. 2012) and
of such data in terms of stellar populations and ages re-
COLIBRI (Marigo et al. 2013) codes. They include more
quires that, besides the photometry, additional intrinsic
details than the previous isochrones, and are specifically
stellar properties are provided along the isochrones.
designed to allow us to advance in the process of cali-
2 Marigo et al.
brationoftheTP-AGBevolution. Someaspectsofthese the numerical solution of the atmosphere and complete
isochrones, such as the inclusion of chemical abundance envelope model, down to the bottom of the H-burning
information, will be of interest to many other isochrone shell. The integration strategy is detailed Marigo et al.
usersaswell. Theinputdataandmethodsaredescribed (2013), whereseveralaccuracytestswithrespectstofull
inSect.2. Somenewpropertiesofthenewisochronesare stellar models are also presented. COLIBRI is charac-
illustrated in Sect. 3. Data retrieval and ongoing work terised by a high computational speed and a robust nu-
are briefly described in Sect. 4. merical stability, allowing us to promptly compute large
sets of TP-AGB tracks any time we aim at exploring
2. DATAANDMETHODS the impact of a different model prescription. Such a fea-
2.1. PARSEC tracks ture is really essential to perform the demanding TP-
AGB calibration cycle. At the same time, COLIBRI in-
PARSEC (the PAdova and tRieste Stellar Evolutionary
corporates many revisions in the input physics, some of
Code) represents a deeply-revised version of the Padova
which (e.g. the nuclear reaction rates) are also common
code used in many popular sets of isochrones (e.g. in
to PARSEC. More importantly, COLIBRI is the first code
Bertelli et al. 1994; Girardi et al. 2000, 2010; Marigo
to fully include the on-the-fly computation of the equa-
et al. 2008). The main features in PARSEC are described
tion of state and Rosseland mean opacities by suitably
in Bressan et al. (2012, 2013), and include the updating
calling,ateachtimestep,theOpacity Projectroutines
oftheinputphysics(equationofstate,opacities,nuclear
(forT >12.000K;Seaton2005),andtheÆSOPUSroutines
reaction rates, solar reference abundances) and of mix-
(for1.500≤T ≤12.000K;Marigo&Aringer2009)infull
ing processes, in particular with the addition of micro-
consistencywiththeactualcompositionofthestellaren-
scopic diffusion in low-mass stars. Further updates re-
velope. Therefore, composition-changing processes such
gard the treatment of boundary conditions in low-mass
as the third-dredge up (3DU) and the hot-bottom burn-
stars (Chen et al. 2014), and envelope overshooting and
ing (HBB) at the base of the convective envelope, pro-
mass-loss in intermediate- and high-mass stars (Tang
duce an immediate effect in the stellar structure and on
et al. 2014; Chen et al. 2015). As it is clear from these
stellar properties such as the effective temperature, T ,
papers, PARSEC is a rapidly-evolving code, with further eff
hencealsochangingtheefficiencyofotherprocessessuch
revisions being underway.
as the mass-loss rate, M˙ , pulsation periods, etc. Simi-
In the present work, we use a subset of the PARSEC
lareffectsoffeedbackbetweenchemicalcompositionand
V1.2S evolutionary tracks as a reference. They include
stellar structure were also included in Marigo & Girardi
grids of tracks for 15 values of initial metal content,
(2007) models, but in a much simpler way (see Marigo
Z, between 0.0001 and 0.06. The helium initial con-
i 2002). Moreover, COLIBRI improves in the description
tent, Y, follows the initial metal content according to
i of other effects, such as the stellar sphericity, and the
the Y = 1.78×Z +0.2485 relation, so as to reproduce
i i integration of extended nuclear networks for all burning
both the primordial Y value by Komatsu et al. (2011),
processes – including the HBB which requires a detailed
andthechemicalcompositionofthepresentSun(namely
consideration of both nuclear and convective timescales.
Z = 0.01524, Y = 0.2485, see Bressan et al. 2013, for
(cid:12) (cid:12) The occurrence and efficiency of the 3DU process in
details). The reference solar-scaled composition is taken
TP-AGB stars is notoriously uncertain and sensitive
from Caffau et al. (2011). Adopting the simple approxi-
to numerical details (Frost & Lattanzio 1996; Mowlavi
mationof[M/H]=log(Z/Z )−log(X/X ),themetal-
i (cid:12) i (cid:12) 1999). As in Marigo & Girardi (2007) models, this pro-
licity [M/H] ranges from −2.19 to +0.70 dex.
cess is parameterised also in COLIBRI. This causes the
TherangeofmassescomputedwithPARSECisalsovery
model to have a few efficiency parameters (for 3DU and
wide,generallyincluding∼120differentmassvaluesdis-
mass-loss) that can be tuned so as to reproduce the ob-
tributedintherangefrom0.1to350M ,foreachmetal-
(cid:12) served properties of populations of AGB stars.
licity. Themassrangemorerelevantforthispaperisthe
The COLIBRI models illustrated in this work are the
onefrom∼0.5M uptothemaximummassofstarsde-
(cid:12) sameonesasdescribedinRosenfieldetal.(2016), which
velopingtheTP-AGBphase,whichislocatedsomewhere
are shown to reproduce the AGB star numbers in a set
between 5 and 6.4 M , depending on metallicity. This
(cid:12) of nearby dwarf galaxies. They include the complete
mass–metallicity range is presented in Fig. 1, together
range of masses and metallicities required to comple-
with a summary of the evolutionary phases computed in
mentthePARSECV1.2Sgridoftracks,asshowninFig.1.
every case.
About 60 TP-AGB tracks are computed for each metal-
licity. TheinitialconditionsofeveryCOLIBRItrack(core
2.2. COLIBRI tracks
mass, luminosity, envelope composition, etc.) are taken
The TP-AGB evolutionary tracks from the first ther- taken from the corresponding PARSEC track just before
malpulseuptothecompleteejectionoftheenvelopeare the first significant thermal pulse; therefore the subse-
computed with the COLIBRI code developed by Marigo quent section of the PARSEC track is simply replaced by
et al. (2013), and including the updates in the mass- the COLIBRI one. As illustrated by Marigo et al. (2013),
loss described by Rosenfield et al. (2014, 2016). While thisgivesorigintoanalmostcontinuoustrack, withjust
many details of COLIBRI are described in the aforemen- a small jump in the T (typically less that 50 K, see
eff
tioned works, let us highlight here the main differences their figure 6) at the PARSEC–COLIBRI junction.
with respect to the previous TP-AGB tracks computed Figure 2 shows two of the basic properties of these
by Marigo & Girardi (2007), which were used in the tracks, namely the number of TPCs and the total TP-
stellar isochrones of Marigo et al. (2008). We recall AGB lifetime, as a function of initial mass and metallic-
that COLIBRI couples a synthetic module (which con- ity.
tains the main free parameters to be calibrated) with
PARSEC–COLIBRI isochrones 3
Figure 1. Thesetoftracksinvolvedintheconstructionof PARSEC-COLIBRIisochrones,inthe[M/H]vs.Miplane. TracksfromPARSECare
dividedintothreebroadclassestaggedasRGB(low-masstracksfromthepre-MSuptotheHe-flashatthetipoftheRGB),HB(low-mass
tracksfromthestartofquiescentcoreHe-burninguptothefirstTPContheTP-AGB)andINT(intermediate-massandmassivetracks
fromthepre-MSuptoeitherafirstTPC,orC-ignition). ForallcaseswhereafirstTPCwasdetected,theTP-AGBevolution(astagged)
wasfollowedwiththeCOLIBRIcode. Inadditiontothetracksherepresented,thePARSECdatabaseincludesverylow-masstracksdownto
0.1M(cid:12) evolveduptoanagelargerthantheHubbletime,andmassivetracksupto350M(cid:12) evolveduptoCignition;theyarenotshown
intheplotbutarealsoincludedintheavailableisochrones.
2.3. Equivalent evolutionary points and interpolation interpolations turn out to be smooth not only as a func-
method tion of initial mass and age, but also of metallicity.
According to this scheme, the interpolated tracks can
The grids of stellar evolutionary tracks describe how
be built with any arbitrary density in initial mass co-
the stellar properties – here generically denoted by p –
ordinate; the higher the density, the larger the number
varyasafunctionofstellarage,t,foragivensetofinitial
of points in the resulting isochrones. In order to limit
masses,M. Buildingisochronesisessentiallytheprocess
i these numbers, we use a dynamical interpolation algo-
of interpolating inside this grid to produce a sequence of
rithmthat, foreverytracksectionbeingconsidered, cre-
such properties as a function of M, for a given set of t,
i ates as many interpolated tracks (and hence isochrone
that is,
points) as necessary to obtain a given resolution in the
Tracks Isochrones logL versus logTeff space. In the present release, this
p(t)| ⇒ p(M)| . (1) resolutionissettoaminimumof∆logL=0.04dexand
Mi=const. i t=const. ∆logT = 0.01 dex, for all sections of PARSEC tracks;
eff
For doing so, we use the classic method of interpolat- thisresultsinisochronescontainingafewhundredpoints
ingbetweentracksusingpairsofequivalentevolutionary before the first TPC. The TP-AGB resolution is defined
points (EEP) as a reference. Essentially, once a pair of in a different way, as we specify below.
EEPs is identified in two adjacent tracks, the whole evo- The algorithm we developed to build the isochrones
lutionary sequence between these EEPs is assumed to is very general, even allowing us to identify the pres-
beequivalentandinterpolatedinallquantities, byusing ence of multiple sequences of evolutionary stages along
M and age as the independent variables. After build- the same isochrone, as those illustrated in Girardi et al.
i
ing a dense grid of interpolated tracks of masses M, the (2013); more specifically, in that work we have identified
i
simple selection of points with desired age t allows us to thepresenceofdoubleortripleTP-AGBsequencescaus-
draw a complete isochrone. Moreover, by following an ing a marked “AGB-boosting” effect in isochrones (and
algorithm similar to Bertelli et al. (2008), the available star clusters) of ages close to 1.6-Gyr.
grids of tracks are also interpolated for any intermediate
2.4. Mass loss on the RGB
valueofinitialmetallicity. Tracksthatsharesimilarevo-
lutionary properties – e.g. tracks with an extended red Another important detail is that we can apply a mod-
giant branch (RGB), or tracks with a convective core on est amount of mass loss between the tip of the RGB
themainsequence–aregroupedtogetherandhavetheir and the zero-age core-helium burning (CHeB) stage of
EEPsselectedwithexactlythesamecriteria,sothatthe low-mass stars, in order to simulate the effect of mass-
4 Marigo et al.
Figure 2. NumberofTPCsineveryCOLIBRITP-AGBtrack(toppanel),anditstotallifetime(bottompanel).
lossalongtheRGB.Justasinpreviousreleases(Girardi 2.5. Thermal pulse cycle L variations
et al. 2000; Marigo et al. 2008), this is done in an ap-
Thermal pulse cycle (TPC) L and T variations are a
proximative way: we first estimate the amount of mass eff
basic feature of TP-AGB evolutionary tracks. They are
loss expected from a given mass-loss formula, ∆M, by
quasi-periodic changes in L and T caused by the onset
integrating the mass-loss rate along the RGB section of eff
of He-shell flashes and the subsequent cycle of envelope
theevolutionarytracks,i.e.∆M =(cid:82) M˙ dt. Then,the expansion,switchingoffoftheHe-shell,contraction,and
RGB
function ∆M(Mi) is used to assign every RGB track of gradualexpansionasthequiescentH-shellburningtakes
mass Mi to a given CHeB+AGB evolutionary sequence over(seee.g.Boothroyd&Sackmann1988;Vassiliadis&
of mass Mi −∆M(Mi). The CHeB+AGB sequence is Wood 1993; Wagenhuber & Groenewegen 1998). These
created by interpolation in the existing grid, and then basic features of the tracks, however, are very hard to
attachedtotheRGBtrackwhileensuringthecontinuity obtain in the isochrones. The main problem is that the
of the stellar age along the entire sequence. The modi- number of TPCs in general varies from track to track,
fiedRGB+CHeB+AGBtracksarethenusedtobuildthe even if they are separated by small intervals of initial
isochrones. Thisisarealisticapproximationforlow-mass mass and metallicity, e.g. 0.05 M and 0.1 dex, respec-
(cid:12)
stars, inwhichtheRGBmasslossneitheraffecttheevo- tively. Thisvariationcanbeappreciatedinthetoppanel
lution of the stellar core, nor the shape of evolutionary of Fig. 2.
tracks in a significant way. Being the numbers of TPC different, it is practically
By default, we apply the classical Reimers (1975)’s impossible to obtain TPC-looking features from the di-
mass loss formula with a multiplicative coefficient of rect interpolation between adjacent TP-AGB evolution-
ηR =0.2 (Miglio et al. 2012), for stars of masses smaller ary tracks. A typical situation is illustrated in panel
than 1 M(cid:12). Then, an additional multiplicative factor is (a) of Fig. 3, where COLIBRI tracks of masses 1.60 and
assumed in order to decrease the mass loss predicted by 1.65 M , and Z = 0.001, are plotted in absolute age
(cid:12) i
this formula gradually as the initial mass increases from versus logL/L . Although the tracks look similar in
(cid:12)
1.0to1.7M(cid:12). Inthisway,starsofmassesMi (cid:38)1.5M(cid:12), slope and span comparable age and logL/L(cid:12) intervals,
for which the present algorithm might provide inconsis- the 1.65 M track has 9 TPCs (out of which, just 8 are
(cid:12)
tent results, are little affected by mass loss. Stars of really complete), whereas the 1.60 M has 7 TPCs.
(cid:12)
masses Mi (cid:38) 1.7 M(cid:12) are assumed not to lose mass be- Tosolvethisproblem, weapplyaschemethatwasde-
fore the TP-AGB. vised during the first implementation of TP-AGB tracks
in the population synthesis code TRILEGAL (Girardi &
PARSEC–COLIBRI isochrones 5
Figure 3. The sequence of operations involved in building a TP-AGB section of the isochrones, limited to the stellar property logL
only. Left panels refer to evolutionary tracks, while right panels refer to the derived isochrones. The split panel (a) shows a couple of
COLIBRItracksofsamemetallicityandcloseininitialmass(withMi=1.65and1.60M(cid:12),fromlefttoright),intheabsoluteagetversus
logLplot. ThecontinuousgraylinesarethedetailedTP-AGBevolutionproducedby COLIBRI,whereasthebluedotsmarkthepre-flash
quiescentstages,orLq. Panel(b)showsthesametwotracksinamuchsimplifiedway,containingjustthepre-flashquiescentstages. They
draw simple sequences in the t versus logL plane, which can be easily split into a series of equivalent sections, as illustrated in this case
by splitting each track into 6 sections of equal ∆t (delimited by the red crosses). Linear interpolations among these couple of equivalent
points allow us to derive the logL for any arbitrary mass or age located between these tracks, as illustrated here by producing a series
of intermediate points at a fixed age of 1.52 Gyr (black plus signs). Panel (c) shows the same situation as panel (b), but now in the Mi
versus logL plane, showing the initial mass Mi of every point produced at an age of 1.52 Gyr; they define a TP-AGB isochrone section
containingonlyquiescentstages. Theplotshowsafewadditionalgreenpoints,whichrepresentthesomewhatdensergridofinterpolated
quiescentstagesthatisproduced,bydefault,byourcode. Panel(d)showsthefinal1.52Gyrisochroneindetail,afterthedetailedTPCs
arere-introducedbetweenthequiescentTP-AGBstages.
Marigo 2007): First, the complex TP-AGB tracks from pre-TP-AGB tracks; since they draw smooth lines in
COLIBRI are converted into simplified ones containing the Hertzsprung-Russell (HR) diagram, the derived
justthepre-flashquiescentstages1. Acoupleofsuchsim- isochrone sections will also contain a smooth and well-
plified tracks are illustrated in Fig. 3b. They look much behavedsequenceofquiescentpre-flashstages,ascanbe
smootherthantheoriginaltracks,anditiseasytomake seen in the interpolated isochrone section of Fig. 3c.
interpolations along them to derive versions containing InadditiontotheL,otherquantitiescharacterizingthe
more or less evolutionary points. In the illustrative case quiescent stages are stored and interpolated while creat-
of Fig. 3b, we have generated simplified tracks contain- ing the TP-AGB isochrone sections: they include T ,
eff
ing 6 equally-lasting TP-AGB sections. This number of the core mass M , and the surface chemical compo-
core
sub-intervals can be set at any arbitrary value; in gen- sition. These quantities suffice to describe the complete
eral, however, we set it at being equal to the maximum luminosityprofileL(φ)/L asafunctionofthephasedur-
q
number of TPCs found in the mass interval being taken ing the TPC, φ, normalized to the pre-flash luminosity
into consideration. maximum L , using the formulas provided by Wagenhu-
q
Quantities between these simplified tracks are in- ber(1996)andWagenhuber&Groenewegen(1998)(just
terpolated using the same EEP scheme used for the asdoneinCOLIBRIbyMarigoetal.2013). Therefore,we
first produce the isochrone TP-AGB sections containing
1 Pre-flash quiescent stages are those in which the H-burning just the quiescent luminosities (Fig. 3c), and then we in-
shellprovidesmostofthestellarluminosity,immediatelypriorthe sert a number of points n between them, following
ignitionoftheHe-shellinaflash;seeWagenhuber&Groenewegen inTPC
the detailed TPC luminosity evolution. This last step is
(1998)foraprecisedefinition.
6 Marigo et al.
Figure 4. Examplesofisochrones,zoomingonafewoftheirTPCsinthelogL,logTeff,andperiodvs.Mi plots(fromtoptobottom).
In thetop andmiddle panels, thedots alongthe isochronesare thosederivedfor ninTPC =25. Theinset axisillustrates thevariation of
phaseφalongoneoftheinsertedTPCs. TheleftpanelsshowthecompleteTP-AGBsectionofanintermediate-ageisochrone,wherethird
dredge-up is operating and the C/O ratio steadily increases with Mi. This isochrone describes the sequence from M to S to C-type stars
(i.e.fromC/O<1toC/O∼1toC/O>1,cf.thecolorscale). Therightpanelsshowsamuchyoungerisochrone,whereHBBisoperating
in addition to the third dredge up. In this case HBB becomes gradually weaker for stars of higher Mi, since they correspond to stars in
more advanced stages of the TP-AGB, inwhichthere is asignificantreduction ofthe envelope (andtotal) masses. This situationresults
intothesequencefromMtoStoCtypestooccurinreverseorderinside individual TPCs. Finally,thebottompanelsshowthevariation
ofLPVperiodsalongtheseisochronesections,forthefundamentalmodeandfirstovertonemodes(thegreylineswithlongerandshorter
periods,respectively). Thecoloreddotsinthiscasesignaltheexpecteddominantperiod.
PARSEC–COLIBRI isochrones 7
illustrated in Fig. 3d. In this example we have adopted constructed along the TPCs in the isochrones, in par-
n =15,whichsufficestorecoverthemaindetailsof ticular the pulsation periods and chemical abundances.
inTPC
the TPCs. Thanks to our choices for the number of age These are discussed later in Sects. 3.4 to 3.5.
intervalsinthesimplifiedTP-AGBtracks,thenumberof
quiescent points on every isochrone (and hence of TPCs 2.7. Further computational details
being inserted) turns out similar to the number found in Although the scheme described and illustrate above
the track with a mass equal to the turn-off one. is quite general, a couple of additional details help us
At this point, we have isochrones with detailed L vari- to produce isochrone TP-AGB sections more closely re-
ations along the TPCs, ∆logL, as further illustrated in sembling those of the original COLIBRI tracks. The first
the upper panels of Fig. 4 for two particular isochrone one is to adopt logM as the independent variable in all
i
sections: one intermediate-age isochrone at around the interpolations involving mass, instead of M; this choice
i
point in which third dredge-up drives the transition be- producessmootherisochroneswheneverthegridscontain
tween O-rich and C-rich phases, and a young isochrone tracks widely spaced in M, but in reality it makes lit-
i
at around the point in which mass-loss weakens the ef- tle difference for the present, dense grids of tracks. The
fect of HBB allowing the same transition to occur. The second detail is that, whenever two adjacent TP-AGB
remarkable difference in the TPC shape between these tracks contain a C-star phase, those tracks are split into
cases is caused mainly by their very different core and twoequivalentsections. Inotherwords,weplaceanaddi-
envelope masses. tionalEEPattheO-toC-richtransition,henceimposing
that the interpolations between tracks occur largely in-
2.6. Thermal pulse cycle T variations
eff ternallytotheirmainO-richandC-richsections. Inthis
HavingtheLvariationsbeingre-constructed,thenext way we avoid that the main change in ∆logT /∆logL
eff
problem is to insert the detailed Teff variations in the slope between C- and O-rich stars, causes artificial fea-
isochrones. FormostTPCsintheoriginalsetof COLIBRI tures in the shape of the derived isochrones2.
tracks, these Teff variations closely follow the L varia- Moreover, we note that the TPC points are not
tions, as the star goes up and down along its Hayashi inserted at evenly spaced intervals of M along the
i
line in the HR diagram. Indeed, we verify that every isochrones. Instead, they are more closely spaced at the
single TPC develop along lines of nearly constant slope beginning of the TPCs, which present the largest varia-
∆logTeff/∆logL. We recall that this behavior is de- tions in logL (see Fig. 4).
rived from detailed envelope integrations in COLIBRI,
which take into account the opacity variations deriv- 2.8. Spectral libraries for cool giants
ing from changes in atomic and molecular concentra-
The PARSEC isochrones are transformed into absolute
tions. ∆logT /∆logL is generally well-behaved along
eff magnitudes via a series of bolometric correction (BC)
theO-richsectionoftheTP-AGBtracks(dependingpri-
tables which are suitably interpolated in the logg ver-
marily on the initial metallicity), but its value changes
sus logT plane, and taking into account the surface
abruptly from about −0.12 to −0.16 whenever a transi- eff
chemical abundances, i.e.
tion between O-rich and C-rich star occurs – following
the large variations in molecular opacities in the stellar M =M −BC(X ,logg,logT ) , (2)
λ bol i eff
atmospheres (Marigo 2002). In addition to this general
where M =−2.5 log(L/L )+4.77.
behavior, we have the changes in T due to substan- bol (cid:12)
eff The formalism to derive the BC tables starting from
tial mass loss occurring at the last TPCs, in which while
librariesofsyntheticspectraisfullydescribedinGirardi
the luminosity increases mildly, T decreases dramati-
eff et al. (2002, 2010).
cally due to large mass loss which reduces the envelope
For most stars, the X interpolation of Eq. 2 means
mass. Thus, the star moves between Hayashi lines of i
simply a linear interpolation in the variable [M/H] –
different masses. In this case, it is not appropriate to
hencea3DlinearinterpolationinlogT ×logg×[M/H]
model the T variations simply with ∆logT /∆logL eff
eff eff space. The main exception to this rule regards exactly
slopes. Therefore, in each TPC we fit T as a func-
eff the most luminoust TP-AGB stars, which are treated
tion of L and the envelope mass M , simultaneously,
env according to a different scheme, reflecting the more ex-
as logT =a +a logL+a M . The envelope mass
eff 0 1 2 env tendedsetsofatmosphericmodelsbuilttodescribethem:
M is computed through the current mass M and the
env
core mass M , as M = M − M . Besides the
core env core 2.8.1. Carbon-rich giants
quantities at the quiescent stages, we also store the lin-
ear fitting parameters within each TPC of the original C-rich giants (with a surface carbon-to-oxygen ratio
COLIBRI set of tracks, and interpolate them while going larger than 1, C/O > 1) have their atmospheres rich
fromthe quiescenttrackstothe isochrones. Theseinter- in carbon-bearing molecules like CN, C2, C3, HCN and
polated fitting parameters allow us to easily convert the C2H2, presenting spectra significantly different than O-
TPC L and M variations into T variations. rich giants of similar parameters. For this reason, a sep-
env eff
T variations computed in this way are illustrated in arate database of C-rich spectra is needed. The Marigo
eff
the middle panels of Fig. 4. They clearly show the main et al. (2008) isochrones distributed since 2009 have used
change in the behaviour of T that occurs at the O- to either the Loidl et al. (2001) or the Aringer et al. (2009)
eff
C-rich transition. In the case of the younger isochrone
(right panels), this event is further complicated by the 2Lateronintheevolution,asthestarlosesmostofitsenvelope
andstartstoheatinitswaytowardstheplanetarynebulaestage,
strong mass loss that is associated to every one of its
the ∆logT /∆logL slopes change appreciably again, becoming
eff
TP-AGB points. evenpositive. Thislatechangeisalsonaturallypresentinthefinal
Other stellar properties are interpolated and/or re- sectionoftheisochrones.
8 Marigo et al.
Figure 5. Map of the 2MASS J−Ks colours attributed to O-rich giants of metallicities [M/H] = −0.5, 0, and +0.5 (top, middle and
bottompanels,respectively),asafunctionoflogT andlogg,fortwocases: thepreviousspectrallibraryusedinMarigoetal.(2008)(left
eff
panels),andthepresentonewhichincorporatestheAringeretal.(2016)results(rightpanels). Acoupleofisochroneswithagesof0.1and
10 Gyr and initial Zi =0.00483 (top panels), Zi =0.01471 (middle panels) and Zi =0.04149 (bottom panels) are shown for comparison,
displayinginparticulartheTP-AGBpart–recognizablebythezig-zagduringthermal-pulsecycles,atlogg(cid:46)0.5. OnlytheO-richsection
of these isochrones is plotted. We note in particular that with the present prescriptions (right panels) the colours of the coolest giants
dependonlogg and[M/H],whilewiththeformerprescriptions(leftpanels),onlythedependencewithT wasbeingconsidered.
eff
databases of synthetic C-rich spectra. For the present etal.(2009): inshort,thespectralpropertiesareinterpo-
work, this library has been partially replaced: for solar lated inside the multidimensional grid with primary pa-
metallicities (i.e. for the same content of heavy met- rametersbeingZ,T ,logg,andC/O.Then,asmallcor-
eff
als, excluding C and O) we implement a new set of rection due to the spectral variations with mass (which,
spectral calculations provided by Aringer et al. (2016), for a fixed T and logg, corresponds to adopting an at-
eff
based on the latest version of the COMARCS hydrostatic mospheric model with a different sphericity) is applied.
atmosphere models. This set comprises over 950 models In practice, the latest update in the C-star library af-
with parameters in the ranges 4000 < T /K < 2500, fects all isochrones containing C-type stars at metallic-
eff
2 < logg < −1, C/O = [1.01,1.05,1.1,1.4,2.0], and ities larger than [M/H] = −0.5. For these stars, the
masses of M/M = [1,1.5,2,3]. Most of the computed bolometriccorrectionswillbebasednotonlyonthenew
(cid:12)
spectra, however, correspond to stars of low masses (1 models, but also benefit from the better interpolation
and 2 M ), with logg <0 and T <3400 K, which are inside a richer grid of models. In this context it should
(cid:12) eff
theintervalsofparametersmorerelevanttodescribethe be noted that the new database also covers objects with
TP-AGB C stars observed in nearby galaxies. C/O ratios as low as 1.01, while the smallest value in
The interpolation of bolometric corrections inside the Aringer et al. (2009) was 1.05.
C-rich grid follows the procedure outlined in Aringer
PARSEC–COLIBRI isochrones 9
2.8.2. Oxygen-rich giants Fig.5),O-richsectionsaregenerallyhotterthanthisT
eff
limit. Moreover, atsmallermetallicitiesa largerfraction
Regarding the “normal” O-rich stars (with C/O<1), of the TP-AGB appears as C-rich (see Sect. 3.6 below).
in previous releases the spectral database was either the It is also interesting to note that the spectral library
ATLAS9-based from Castelli & Kurucz (2003) or the from Aringer et al. (2016), by including C/O ratios up
PHOENIXfromAllardetal.(2012, seeChenetal.2014 to 0.97, approaches the region of parameters covered by
fordetails),replacedbyFluksetal.(1994)forcoolMgi- S-type stars, in which the scarcity of free C and O and
ants. Aringer et al. (2016) has recently published a huge the enrichment of s-process elements give place for the
database of O-rich spectra derived from the COMARCS appearance of molecular features of species such as ZrO
code, especially suited to describe the spectra of cool gi- andYO.Moreextendedcomputations,fullycoveringthe
ants. Followingtheindicationsfromthislatterpaper,we transition between C-, S- and M-type spectra, will be
now adopt a smooth transition between the BCs derived provided in Aringer et al. (in prep.)
from the previous databases, and those from Aringer One can also notice in Fig. 5 that the present
etal.(2016),intheT intervalbetween4000and5000K isochrones reach smaller logg and cooler T than the
eff eff
(4.6 < log(T /K) < 4.7). This interval roughly corre- logg < −1, T > 2600 K limits of the Aringer et al.
eff eff
spond to the CHeB phase (including the low-mass red (2016) spectral library – especially at young ages (TP-
clump) in solar-metallicity isochrones, and affects only AGB stars with HBB) or for very old and metal-rich
the brightest RGB, TP-AGB and RSG stars of metal- isochrones. There is no easy solution for this problem,
licities [M/H] (cid:46) −1. In most optical and infrared pass- since the hydrostatic model atmospheres from COMARCS
bands, such a transition will appear completely smooth. areboth(a)hardtoconvergeforstarsoflowT ,and(b)
eff
Some artifacts might appear at UV wavelengths, where not realistic given the rise of high-amplitude pulsation
the different spectral libraries are provided with a very and of huge convective cells at the stellar photosphere.
differentsamplinginλ. However,havingpreciseUVpho- Notice however that such very cool stars are rare, and
tometry of cool stars is somewhat unlikely, so that this thattheirobservedpropertiesareseverelyaffectedbycir-
aspect might be less of a problem. cumstellardust(cf. nextsubsection),hencelittlereflect-
One of the most important consequences of adopt- ing the detailed photospheric properties. However, for
ing the Aringer et al. (2016) spectral library is that practical purposes these stars also need to be attributed
of having a more physically-sound description of how aphotosphericmagnitudeandcolour;wedosobysimply
the molecular lines vary as a function of stellar param- extrapolatingthebolometriccorrectiontableslinearlyin
eters. Especially important is the behaviour of water the logg×logT plane.
eff
lines which heavily determine the near-infrared colours
2.9. Circumstellar dust
of O-rich giants; depending on their detailed behaviour,
near-infrared colours such as J−H and J−K may not Ontopofthephotosphericspectra,ourmodelsinclude
s
increase monotonically with decreasing T , as could the effect of light reprocessing by circumstellar dust in
eff
be naively expected. This effect can be appreciated theextendedenvelopesofmass-losingstarsforwhichwe
in Fig. 5, which illustrates the changes we have in the computedtheradiativetransfer(RT).Thenoveltyinthe
2MASS J−K colours of metal-rich O-rich giants, as a present isochrones is the inclusion of a self-consistent
s
function of logT and logg, for metallicities going from treatment of dust growth, fully calculated as a func-
eff
a third of solar to three times solar ([M/H] = −0.5, 0 tion of the input stellar parameters, i.e. luminosity, ac-
and +0.5), and comparing present prescriptions based tual mass, effective temperature, mass-loss and elemen-
on Aringer et al. (2016) with those adopted in the pre- talabundancesintheatmosphere,asdescribedinNanni
vious Marigo et al. (2008) isochrones. As can be appre- etal.(2013,2014). Ourdustgrowthdescriptionprovides
ciated, J−K tends to reach an almost-constant value the dust mixture as a function of the stellar parameters
s
for the coolest giants, with T (cid:46) 3000 K. The exact as well as the optical depth at λ=1µm. The dust code
eff
color of this “saturation”, and the T at which it oc- is coupled with a RT one (MoD by Groenewegen 2012a),
eff
curs, depends heavily on the total metallicity – which basedonDUSTY(Ivezic&Elitzur1997). Themostrecent
is crucial in determining the efficiency of water forma- update of our dust growth model (Nanni et al. 2016) re-
tion – and mildly also on the chosen linelist (see Aringer gards the production of carbon dust in circumstellar en-
et al. 2016, for details). Moreover, the J−K colors de- velopesofC-stars,whichareparticularlyrelevantforthe
s
pend also on the exact logg reached by the giants; for interpretationofthecolorsoftheso-calledextreme-AGB
themorecompactgiants(generallycorrespondingtothe starsobservedininfraredsurveysofnearbygalaxies(e.g.
starsoflowerinitialmassstarsfoundinolderisochrones) Boyer et al. 2011). Since in Circumstellar Envelopes of
thereiseventhepossibilitythattheJ−K ×T relation C-starsthebulkofthedustproducedismainlycomposed
s eff
reverses, with the coolest giants becoming slightly bluer byamorphouscarbon,thedusttemperatureattheinner
thantheT ∼3000Kones. Fig.5showsthatthiscom- boundary of the dust zone is assumed to be the one of
eff
plexbehaviourispresentinthenewisochroneswhenthe carbondust, evenifwealsoincludesiliconcarbide(SiC)
Aringer et al. (2016) spectral library is adopted. and iron dust in our calculations. For the same reason,
All isochrones with O-rich sequences cool enough to for M-stars the dust temperature at the inner boundary
enter in the T (cid:46) 3000 K range will be affected by of the dust zone is assumed to be the one of the first
eff
this change in the spectral library of O-rich giants. silicate dust condensed (either pyroxene or olivine). For
This regards especially isochrones of solar and super- M-stars the calculations also take into account the for-
solar metallicity, like those illustrated in the middle and mation of Al O , quartz (SiO ), periclase (MgO) and
2 3 2
bottom panels of Fig. 5. In isochrones of metallicities iron.
[M/H] (cid:46) −0.5 dex (e.g. those in the upper panels of Therefore, the approach adopted allows us to com-
10 Marigo et al.
LPVpropertieswereintroducedinisochronesforthefirst
Table 1
time in Marigo et al. (2008), using a series of fitting re-
CoefficientsfortheperiodsinEq.3.
lations derived from Fox & Wood (1982), Wood et al.
m T a b c d (1983) and Ostlie & Cox (1986) to describe periods in
0 O −0.683 1.874 −0.109 −1.957 thefundamentalandfirstovertonemodes(P andP re-
0 1
0 C −0.757 2.018 −0.121 −2.282
spectively), as well as luminosity at which the dominant
1 O −0.585 1.620 0.083 −1.641
period changes between these modes. These relations
1 C −0.499 1.515 0.107 −1.406
are presently being revised with the aid of extensive cal-
putebolometriccorrectionsself-consistentlyasafunction culations of pulsation models for a wide grid of input
of stellar parameters, and provides the corresponding parameters, computed with linear non-adiabatic radial
change in the broad-band absolute magnitudes and col- oscillationcodedescribedinWood&Olivier(2014). For
ors. A large grid of such models was computed, covering the present work, we update the fitting formula for the
2 values of masses (0.8 and 2 M(cid:12)), 5 of mass loss (from predicted P0 and P1. They are of the form:
10−7 to10−5 M(cid:12)yr−1 atlogarithmicspacedintervals),3 log(PT/days)=alog(M/M )+blog(R/R )+
luminosities (log(L/L )=3.25, 3.75 and 4.25), five T m (cid:12) (cid:12)
(cid:12) eff
clog(M /M)+d (3)
(from 2600 to 3400 K), 3 metallicities (Z =0.001, 0.004, env
and 0.008), and 6 values of carbon excess for C stars
where m stands for the mode (0/1 for fundamental/first
(with8.0<log(n −n )−log(n )+12<8.8,wheren ,
C O H C overtone) and T for the chemical type (O- or C-rich).
n , and n are the surface number fractions of carbon,
O H The fitting coefficients are presented in Table 1. These
oxygen,andhydrogenrespectively). Thebolometriccor-
relations where derived from models along COLIBRI evo-
rection applied to every star in the isochrones is deter-
lutionary tracks covering the Z interval from 0.002 to
i
mined via a multi-dimensional interpolation in this grid.
0.017, and for masses between 0.8 to 2.6 M . The fit-
(cid:12)
FortheO-richstars,themodelsadoptthesameparame-
ting relations provide periods accurate to within a few
tersandopticaldataasinNannietal.(2013,2014). For
per cent. Further updates and a thorough discussion
theC-richstars,twogridsarepresentlyprovided. Oneis
will be presented in Trabucchi et al. (in prep.). The
computedusingtheRouleau&Martin(1991)setofopti-
likely transition luminosity between the first overtone to
cal data with typical size of dust grains ∼0.1 µm, while
fundamental mode was computed in the same way as in
the other is computed by employing Jager et al. (1998)’s
Marigo et al. (2008), based on Ostlie & Cox (1986).
dataset produced at a temperature of 400 K with grains
Importantly, P and P pulsation periods are com-
0 1
of size ∼ 0.06 µm. As detailed in Nanni et al. (2016),
puted along the isochrones but the periodic changes
among several possible choices of optical data and pa-
in the photometry are not taken into account. There-
rameters, thesetwoaretheonesthatbestreproducethe
fore, the photometric properties we provide should be
infrared colors of C stars in the SMC. However, larger
regarded as mean properties over the LPV pulsation pe-
deviations for the Rouleau & Martin (1991) optical data
riods, rather than instantaneous values.
set are expected for the reddest colors, for which smaller
grains (∼0.06 µm) are better in reproducing the data. 3. RESULTS
The treatment of dust and radiative transfer adopted
Here, we give just a quick overview of the main novel-
in this new release of isochrones represents a remarkable
ties in the present isochrones:
improvement with respect to the previous prescriptions
assumedinMarigoetal.(2008). There,theinterpolation
3.1. Surface chemical abundances along isochrones
wasperformedfortablesofspectrapre-computedforfew
As already commented, with this isochrone release
dust mixtures (Bressan et al. 1998; Groenewegen 2006),
we start providing detailed surface abundances {X } –
since a full model for the growth and destruction of dust i
presently consisting of the mass fractions X, Y, Z, X ,
was still not developed. C
X , and X – along the isochrones. These values are
N O
2.10. Interstellar dust allinterpolatedlinearly(alwaysusingtheEEPsasaref-
erence) from those given along the evolutionary tracks.
In addition, the effect of heterochromatic interstellar
Their values change as a result of microscopic diffusion
dust extinction is considered in the isochrones as in Gi-
(especially during the main sequence of low-mass stars),
rardi et al. (2010). This means applying, to the points
of dredge-up events, and of HBB at the most massive
alongtheisochrones,theextinctioncoefficientswhichare
TP-AGB stars. Fig. 6 illustrates the regions of the HR
afunctionnotonlyofthepassbandbeingconsidered,but
diagram which are most affected by these changes, for a
also of the stellar T , of the total extinction in the V
eff set of given initial metallicity, by showing the changes in
band, A , and of the ratio between selective and abso-
V the total metal content by mass, Z. The main changes
lute extinction R , as defined in Cardelli et al. (1989)
V in Z occur at the main sequence of low-mass stars due
and O’Donnell (1994)’s generalized extinction curve.
to microscopic diffusion (the purple area), and at the
TP-AGB due to third dredge-up events (the yellow-to-
2.11. Long period variability
red sections at the top). In addition, the isochrones also
As demonstrated by extensive microlensing surveys of contain the variations in Z due to the second dredge-up
the Magellanic Clouds and Milky Way Bulge, TP-AGB inintermediate-massstars,andthealmost-imperceptible
stars are frequently observed as long-period-variables variations along the RGB due to protons locked up in
(LPV), pulsating in at least one of several pulsation CNO nuclei, which appear after the first dredge-up.
modes between the fourth overtone and the fundamen- Figure 7 illustrates one of the many possible applica-
tal(Miras)mode(Lattanzio&Wood2004;Wood2015). tions of the new abundance information: its left panel