The effects of the overshooting of the convective core on main-sequence turnoffs of young- and intermediate-age star clusters

2.2 MB·

by  Wuming Yang

The effects of the overshooting of the convective core on
main-sequence turnoffs of young- and intermediate-age star
clusters
7
1
0 Wuming Yang1, Zhijia Tian2
2 1Department of Astronomy, Beijing Normal University,Beijing 100875, China

n [email protected]
a 2Department of Astronomy, Peking University,Beijing 100871, China
J

1
2

]
R ABSTRACT
S
Recent investigations have shown that the extended main-sequence turnoffs (eMSTOs) are a
.
h common feature of intermediate-age star clusters in the Magellanic Clouds. The eMSTOs are
p also found in the color-magnitude diagram (CMD) of young-age star clusters. The origin of
-
o the eMSTOs is still an open question. Moreover, asteroseismology shows that the value of the
r overshootingparameterδov oftheconvectivecoreisnotfixedforthestarswithanapproximatelly
t
s equalmass. ThustheMSTOofstarclustersmaybeaffectedbytheovershootingoftheconvective
a
core (OVCC). We calculated the effects of the OVCC with different δov on the MSTO of young-
[
and intermediate-age star clusters. If δov varies between stars in a cluster, the observed

1 eMSTOs of young- and intermediate-age star clusters can be explained well by the effects. The
v
equivalent age spreads of MSTO caused by the OVCC are relatedto the age of star clusters and
3
are in good agreement with observed results of many clusters. Moreover, the observed eMSTOs
6
9 of NGC 1856 are reproduced by the coeval populations with different δov. The eMSTOs of star
5 clusters may be relevant to the effects of the OVCC. The effects of the OVCC are similar to
0 that of rotation in some respects. But the effects cannot result in a significant split
.
1 of main sequence of young star clusters at mU .21. The presence of a rapid rotationcan
0 make the split of main sequence of young star clusters more significant.
7
1 Subject headings: stars: evolution — convection — globular clusters: general — Magellanic Clouds
:
v
1. INTRODUCTION ingofstarclustersbeingsimplestellarpopulations
i
X (SSPs).
Thedoubleorextendedmain-sequenceturnoffs
r The eMSTOs were also discovered in young
a (eMSTOs)werediscoveredinthecolor-magnitude
clustersNGC1856(Correnti et al2015;Milone et al.
diagram (CMD) of intermediate-age massive star
2015),NGC1755(Milone et al.2016a),NGC1850
clustersintheMagellanicClouds(Mackey & Broby Nielsen
(Bastian et al.2016),andNGC1866(Milone et al.
2007;Mackey et al.2008;Glatt et al.2008;Milone et al.
2016b). Moreover, NGC 1856, NGC 1755, and
2009; Girardi et al. 2009; Goudfrooij et al. 2009,
NGC 1866 are found to exhibit dual main se-
2011; Keller et al. 2012; Piatti 2013). One in-
quences (MS) below their MSTO (Milone et al.
terpretation of the eMSTOs is that the clus-
2015, 2016a,b).
ters have experienced an extended star-formation
Analternativeinterpretationofthe eMSTOsis
histories (eSFH) with a duration of ∼ 100 −
the effects of star rotation (Bastian & de Mink
700 Myr (Mackey et al. 2008; Glatt et al. 2008;
2009; Yang et al. 2013; Brandt & Huang 2015;
Milone et al.2009;Girardi et al.2009;Rubele et al.
D’Antona et al. 2015; Niederhofer et al. 2015a).
2010; Goudfrooij et al. 2011, 2014; Correnti et al
Yang et al. (2013) show that the extension of
2014), which disagrees with classical understand-
1
MSTO caused by star rotations is related to the (2011, 2014) and Correnti et al (2014) argued
rotation rate of stars, the efficiency of rotational that the eMSTOs can occur only in clusters with
mixing, and the age of star clusters. A relatively masses larger than a threshold of about 105 M⊙
high rotation rate and a high efficient rotational and with escape velocity greater than 12 − 15
mixing are required to explain the eMSTOs of km s−1. However, the eMSTOs of NGC 1755
young clusters [see Figure 8 in Yang et al. (2013, (Milone et al. 2016a) and NGC 411 (Li et al.
but see Niederhofer et al. 2015a and D’Antona 2016a) would represent a challenge for this sce-
et al. 2015)]. Niederhofer et al. (2015a) claimed nario. Furthermore, the observation that
that the eMSTO of NGC 1856 can be explained there exists a strong correlation between
by a rotation of 0.5 times the Keplerian rotation cluster age and the inferred age spread as
rate (Ωcr). But in order to explain the dual MSs found by Niederhofer et al. (2015a) also
ofclustersNGC 1856andNGC1755,therotation rules out an actual age spread being the
rate of 0.9 Ωcr is required (D’Antona et al. 2015; origin of the eMSTO.
Milone et al. 2016a). A large number of rapid Li et al. (2014) analyzed the sub-giant branch
rotatingstarshavebeenfoundinNGC1850 (SGB) of NGC 1651 harbouring an eMSTO and
and NGC 1856 by Bastian et al. (2017). found that the SGB is narrower and offsets from
However, neither stellar models with dif- what would be inferred from the eMSTO region
ferent ages only, nor rapid rotating mod- if the large age spreads would be present within
els with different rotation rates, properly the cluster. Similar results were found in NGC
reproduce the observed split MS and eM- 1806andNGC 1846(Bastian& Niederhofer 2015)
STO of NGC 1866 (Milone et al. 2016b). and NGC 411 (Li et al. 2016a). Hence, they
The populations with both different ages concluded that age spreads are not likely to be
and different rotation rates are needed to the cause of the eMSTO phenomenon. How-
explain NGC 1866 (Milone et al. 2016b). ever,Goudfrooij et al.(2015)foundthatthecross-
Moreover,D’Antona et al.(2015)statedthattheir SGB profiles of NGC 1651, NGC 1806, and NGC
rotating models fail to reproduce the stars “after 1846 are consistent with their cross-MSTO pro-
the end of the central H-burning phase” of NGC files when the latter are interpreted as age distri-
1856. However, these stars might be stars butions. Conversely, their SGB morphologies are
with decretion disks (Bastian et al. 2017) inconsistent with those of simulated SSPs. The
seen nearly edge on, so they suffer from ex- origin of the eMSTOs is still an open question.
tinctionwhichpushesthemintothisregion. Theovershootingoftheconvectivecore(OVCC)
Another coeval interpretation of the eMSTOs can bring more hydrogen-rich material into H-
is interacting binaries (Yang et al. 2011; Li et al. burningcore,whichsignificantlyprolongsthe life-
2012,2015, 2016). Yang et al. (2011) showedthat timeoftheburningofcorehydrogenandenhances
interacting binaries including merged binary sys- the He-core mass left behind. The distance of the
tems and the binaries with mass transfer canlead overshooting of a convection is defined as δovHp,
to both the eMSTOs and the dual red-clumps. where δov is a free parameter and Hp is the lo-
The effects of the interacting binaries on the cal pressure scale-height. Recently, Yang (2016b)
CMDs of some clusters should not be neglected, developed a method to determine the size of the
although the number of the interacting binaries convective core including the overshooting region
in a cluster could be not enough to explain the fromobservedoscillationfrequenciesoflow-degree
eMSTOs alone. p-modes. It was found that the value of δov is
One of the important predictions of the eSFH variable for stars with an approximatelly equal
scenariois thatthe ongoingstar-formationshould mass. For example, the value ofδov is 0.2 forKIC
be observed in young star clusters with an age 9812850 with M ≃ 1.48 M⊙ (Yang 2016b), ∼ 1.4
of a few hundred Myr. However, up to now, for KIC 2837475 with M ≃ 1.50 M⊙ (Yang et al.
the expected ongoing star-formation is not ob- 2015), 0.9−1.5 for Procyon with M ≃ 1.50 M⊙
served in young clusters with age beyond 10 Myr (Guenther et al. 2015; Bond et al2015), ∼0.6 for
(Bastian et al.2013,2016;Cabrera-Ziriet al2014, HD 49933 with M ≃ 1.28 M⊙ (Liu et al. 2014),
2016; Niederhofer et al. 2015b). Goudfrooij et al. and1.7−1.8forKIC11081729withM ≃1.27M⊙
2
(Yang 2015). The typical errors of the value = 0 is 12.5 Myr, which is 4.5% of its MS lifetime.
of δov are 0.1−0.2. If a variable overshooting But the lifetime of the SGB of the model with
exists in stars with masses larger than 1.50 M⊙, M =3.0M⊙ andδov =0.4isonly0.7Myr,which
the MSTO of young- and intermediate-age star isabout0.2%ofitsMSlifetime. TheHertzsprung
clusters would be affected by the overshooting. gapofthemodelwithalargeδov isobviouslynar-
In this work, we mainly focus on whether the rower than that of the model without the OVCC.
eMSTOsofyoung-andintermediate-agestarclus- ThevaluesofthecentralhydrogenX atpoints
c
ters can be explained by the effects of the OVCC. B, E, and G in Figure 1 are about 0.05, 0.025,
The paper is organized as follows: we show our and 0.016, respectively, while the values of X at
c
calculation results in Section 2, and the results points C, F, and H are around 0.0005, 8×10−6,
arecomparedwithobservationsinSection3,then and 8×10−12, respectively. Compared with the
wediscussandsummarizetheresultsinSection4. whole MS lifetime of stars with a large δov, the
timescale of their MS hook is very short. Thus
2. CALCULATION RESULTS we define the points B, E, and G as the end of
central H-burning phase. Due to the fact that
2.1. Evolutionary tracks
more hydrogen-rich material is brought into H-
burning core by the OVCC, the stars with the
In order to study the effects of overshooting
OVCC produce more nuclear energy. Part of the
of the convective core on the MSTO of star clus-
energymakes the starsmore luminous. The other
ters, we computed a grid of evolutionary mod-
els with the initial metallicity Zi = 0.008, δov is transformed into internal energy, which leads
in the range of 0.0 − 0.6 with a resolution of to the fact that the stars expand obviously, i.e.,
0.2, supplemented by 0.7, and masses between theradiusofthestarsincreasessignificantly. Asa
0.9 and 6.0 M⊙. The resolution of mass varies consequence,theeffectivetemperatureofthestars
from 0.005 to 0.03 M⊙. We used the Yale Rota- decreases at the end of central H-burning phase.
tionEvolutionCode (YREC)(Pinsonneault et al. And the starwith alargerδov is brighterandred-
1989; Yang & Bi 2007; Yang 2016a) to con- der than the star with a small δov at the end of
centralH-burningphase. Thus,theMSwidthand
struct the models in its non-rotation configura-
MSTO of a star cluster must be extended by the
tion. The OPAL equation-of-statetable EOS2005
(Rogers & Nayfonov 2002) and OPAL opacity ta- presence of the OVCC with different δov.
ble GN93 (Iglesias & Rogers 1996) were adopted, Moreover, due to the fact that the timescale
supplementedbytheAlexander & Ferguson(1994) of the MS hook of the model with a large δov is
opacity tables at low temperature. The diffusion significantly shrunk, the structure of MS hook of
and settling of both helium and heavy elements ayoungstarclusterincludingmanystarswiththe
are computed by using the diffusion coefficients large δov is hard to be exhibited in the observed
of Thoul et al. (1994) for models with a mass less CMD of the cluster.
than 1.3 M⊙. Convection is treated according to
the standard mixing-length theory. The value of 2.2. Isochrones
the mixing-length parameter is fixed at 1.75. The
In order to understand the effects of an OVCC
fullmixingofmaterialisassumedintheovershoot-
ontheMSTOofyoung-andintermediate-agestar
ing region. All models are evolved from zero-age
clusters, we calculated isochrones with different
MS to the red giant branch (RGB).
δov. To obtain the CMDs of the isochrones, the
Figure1 showsthe evolutionarytracksofmod- theoretical properties ([Fe/H], Teff, logg, logL)
els with M = 3.0 M⊙ but with different δov. The of models are transformed into colors and mag-
evolutionary tracks are obviously affected by the nitudes using the color transformation tables of
OVCC.Themodelwithalargeδov mainlyexhibits Lejeune et al. (1998). A distance modulus of 18.6
to be bluer than the model with a smaller δov at is adopted, and the value of 0.12 is adopted for
a givenageandthe lifetime ofthe burning ofcore E(B −V) in the calculation. Figure 2 shows the
hydrogen is significantly prolonged, but the life- CMDs of isochrones of models with different δov.
time of SGB is shrunk. For example, the lifetime Comparing the isochrones of models with a large
oftheSGBofthemodelwithM =3.0M⊙andδov δov to those of models with a smaller δov, one can
3
find that the upper MS and the MSTO of young- to overlap with the overshooting isochrone with
and intermediate-age star clusters are extended δov = 0.4 and age = 400 Myr. Thus, the equiv-
by the OVCC when the different OVCCs exist in alent age spread caused by the overshooting with
clusters. However, the lower MS of the clusters δov = 0.4 is 120 Myr when the age of the cluster
are almost not affected. For the isochrones with is 400 Myr. The equivalent age spreads for other
the age of 400 Myr, the separation between the isochroneswithdifferentδov andagesareobtained
isochrone of models with δov = 0.4 and that of in the similar way. The “points C and D” on the
models with δov = 0.0 is about 0.1 mag in color isochrone with age =400 Myr and δov =0.4 can-
U-VatmU =19.5,but itisonly around0.02mag notbematchedbythoseofanyisochronewithδov
atm =21. However,theseparationisabout0.03 = 0. Thus, we cannot obtain the equivalent age
U
mag in color V-I at m = 19.5. The eMSTO of spread through comparing the points C or D of
V
youngclusters is better visible in U-V thanin V-I isochrones with different δov.
(see the panel b of Figure 2 and Figure 3). These Due to the fact that the models with a large
characteristics are similar to those found in NGC δov are obviously brighter and redder than the
1856 and NGC 1755 (Milone et al. 2015, 2016a) models with a smaller δov at the end of central
and are also expected in the rotational sce- H-burning phase, the end point of the central H-
nario. burning phase of an isochrone with δov= 0.4 can-
not be reproduced by that of the isochrones with
3. COMPARISON WITH OBSERVA- δov= 0 (see Figure 4). Thus, when one uses the
TIONAL RESULTS isochrones with a small δov to fit the CMD of a
clusterincludingmanystarswithalargerδov,one
3.1. The equivalent age spread caused by
couldfindthattherearemanystarsthatcannotbe
the OVCC
reproducedafter the end of the centralH-burning
Weassumedthatadistributionofδov suchasa phase.
bimodal of 0 and 0.4 exists in a star cluster. The In order to understand whether the effects of
isochrones of models with δov = 0 are chosen as the OVCC can explain the observed eMSTOs of
thestandardone. Theisochronesofmodelswitha star clusters, the equivalent age spreads caused
largeδov arecomparedtothoseofmodelswithδov by the OVCC are extracted as mentioned above
=0to obtainthe equivalentagespreadcausedby and compared to observed ones. We do a
the OVCC. Hereafter, the beginning and the end similar experiment as Niederhofer et al.
pointsofMShookofanisochronearelabeledasB (2015a). Figure5showsthecomparisonbetween
and C, respectively. The local bluest point (local the equivalent age spreads and those observed
minimum U-V) of MS is marked as point A, and by Goudfrooij et al. (2014), Niederhofer et al.
the brightest point (minimum mU) after point C (2015b), Correnti et al (2014, 2015), Milone et al.
is labeled as point D (see Figure 4). The values (2015, 2016a), and Bastian et al. (2016). The
of the centralhydrogenXc of models at points A, values of the age spreads of Goudfrooij sample
B,C,andD onthe isochronewithage=280Myr adopted here are the values of W20MSTO, which
are about 0.27, 0.05, 0.0005, and 0, respectively. are considered to cannot be explained by rotating
According to the initial mass function, the more models (Goudfrooij et al. 2014, but see Brandt
massivethestars,thelessthenumberofthestars. & Huang 2015 and Niederhofer et al. 2015a).
Moreover, the more massive the stars, the faster The equivalent age spreads increase with an in-
their evolutions. The stars between point C and crease in age of star clusters and δov, and are in
D(Xc betweenabout0.0005and0forδov =0but good agreement with the observed ones, which
betweenabout8×10−6and0forδov =0.4)should shows that the observed eMSTOs of young- and
be rare in a young star cluster. Thus we compare intermediate-age star cluster can be interpreted
the non-overshooting isochrone between point A by the effects of the OVCC. The inferred age
and B to the isochrone with an overshooting to spreads of a given δov shown in Figure 5 are
obtain the equivalent age spread. For example, relative to the same age isochrone with δov
inspected by eyes, the non-overshootingisochrone =0, i.e. theextensionofMSTOcausedby a
between point A and B in Figure 4 is considered δov atagiven ageisrelativetothe isochrone
4
with the same age but with δov = 0. Keep Gyr, the MS of models with δov =0 is very
in mind that a single value of δov cannot produce close to that of models with δov = 0.4 in
any eMSTO in the CMD of a cluster. And the the CMD; the separation between the δov
trend between the age of star clusters and =0.4isochroneand theδov =0.2isochroneis
the inferred age spread is also expected in about 0.04mag at m =21.8mag. The max-
V
the rotational scenario. imum separation between the MS of the δov
Figure 5 shows that the equivalent age spreads =0.2 or 0.4 isochrone and that of the δov =0
caused by the OVCC increase with an increase in isochrone in color V-I at age = 1.4 Gyr is
ageofstarclusters. However,whentheageofstar about 1.3 times as large as that at age =0.8
clusters is larger than 2 Gyr, the equivalent age Gyr, and is around 2 times as large as that
spreads caused by δov = 0.2 almost not increase at age = 2.5 Gyr. The eMSTO caused by
withanincreaseinageofclusters. Thisisbecause a large δov should be present in the CMD
theconvectivecoreoflowmassstarsshrinkduring of star clusters with ages > 2.0 Gyr un-
their MS. But for more massive stars, the convec- less the fraction of stars with the large δov
tivecoreincreasesinsizeduringtheinitialstageof is too small. But it is not as evident as
MSofthesestarsbeforethe corebeginsto shrink. that in intermediate-age star clusters. This
The larger the δov, the later the core begins to trend is not reflected by the equivalent age
shrink. In the late stage of the MS of stars, when spreads.
the core shrinks to a certain degree, the chemical
3.2. Comparison with NGC 1856
compositionofthecoreincludingtheovershooting
region is mainly helium. At this time, the effects 3.2.1. The eSFH scenario
of the overshooting on the evolution of stars are
not significant. The smaller the value of the δov, NGC 1856 exhibits the extended MSTOs.
the earlier this scenario appears. Therefore, the Niederhofer et al. (2015a) claimed that the eM-
equivalent age spreads caused by the OVCC can- STOsofNGC1856canbeexplainedbytheeffects
not increase continuously with an increase in age of rotationwith initial rotationrate Ωi =0.5 Ωcr.
of star clusters. As a consequence, the equivalent InFigure 7, we comparethe data ofNGC 1856
agespreadscausedbyδov =0.2reachamaximum tomodelswithdifferentδov,whichshowsthatthe
of about 330 Myr at the age of about 2.2 Gyr. starsattheendofthecentralH-burningphaseand
Niederhofer et al. (2016) have shown the subgiants can be reproduced by the models
that observationally there is a peak in the with δov = 0.4 and 0.6 but cannot be reproduced
inferred age spread at age ∼1.7 Gyr, which by the models with δov = 0 or 0.2. The age of
is consistent with that expected by rotation a star cluster obtained from isochrones increases
models of Yang et al. (2013). The equiva- with an increase in δov. For example, when one
lentagespreadisstillpresentatage>2Gyr uses the isochrones of models with δov = 0 to fit
in OVCC scenario. But the width of eM- theCMDofNGC1856,onecouldobtainanageof
STO caused by the OVCC reaches a max- ∼300Myr;however,whenoneusestheisochrones
imum in color V-I of CMD of star clusters ofmodelswithδov =0.4tofittheCMD,anageof
at age ∼ 1.4 Gyr, and then decreases with ∼ 500 Myr would be obtained. Moreover, Figure
an increase or decrease in age (see panels 7 also shows that one would obtain a smaller age
c and d of Figure 2 and Figure 6). The spread if one uses the isochrones with a small δov
equivalent age spread caused by δov = 0.2 tofittheeMSTOsofNGC1856. PanelcofFigure
is about 300 Myr at age = 3.2 Gyr. But 7 shows that the eMSTOs of NGC 1856 can be
the separation between the MS of models explainedbyanagespreadnomorethan300Myr
with δov = 0.2 and that of models with δov of models with δov =0.4.
= 0 is only 0.026 mag at m = 21.9 mag, NGC 1856 also exhibits the dual MSs.
V
which is comparable with the observational The dual MSs of NGC 1856 are separated
error. The separation is around 0.044 mag by ∼ 0.1 mag in color U-V at m ∼ 20.0
U
at m = 21.6 mag (see panel c of Figure and merge together ∼ 1.5 magnitudes be-
V
6). When the age of clusters is about 3.7 low (Milone et al. 2015). In order to ex-
5
plain the split MS, the rotation as high as density of the region, i.e. a decrease in radius
0.9 Ω is required (D’Antona et al. 2015). of stars, which results in the fact that rotating
cr
Moreover, the simulated populations of models have a higher effective temperature at the
D’Antona et al. (2015) cannot reproduce endofcentralH-burningphase. Asaconsequence,
the stars near the end of the central H- rotating overshooting models are slightly brighter
burning phase (see their figure 4). But this and bluer than non-rotating overshooting models
might be due to stars with decretion disks at the end of the central H-burning phase.
seen nearly edge on (Bastian et al. 2017). Inpanelsa,b,andcofFigure9,wecomparero-
The separationbetween the isochrone with age tating and non-rotating models with different δov
= 700 Myr and that with age = 400 Myr is ∼ to the data of NGC 1856,which show that the ef-
0.1 mag in color U-V at m ∼ 20.0, which is in fects of the rotation of 0.5 Ω are not enough to
U cr
good agreement with observation of Milone et al. explain the eMSTOs of NGC 1856 in our calcula-
(2015), and is only ∼ 0.02 mag at m ∼ 21.5. tions. Due to the fact that the centrifugal effect
U
Recently, Li et al. (2016c) analyzed the data of plays a dominant role in the early stage of MS
NGC 1856 and speculated that the rapid stellar phase of stars whose envelope is radiative (with
rotationscenariois the favoredexplanationof the masses & 1.3 M⊙) and leads to the fact that ro-
multiplesequencesofNGC1856ratherthaneSFH tating models are fainter and redder than non-
scenario. rotating models, the MS of rotating models is al-
mostparallelwiththatofnon-rotatingmodelsbe-
3.2.2. Rapid rotation scenario tween m ∼ 19.5 and m ∼ 22. The separation
U U
between the MS of rotating models and that of
We computed rapid rotation models with Ω =
i
non-rotatingmodelsisabout0.016mag. Ahigher
0.5 Ω and with the rotational mixing efficiency
cr
rotation rate will lead to a larger separation. But
for the Sun (Pinsonneault et al. 1989; Yang & Bi
the characteristic of the parallel separation seems
2007). In Figure 8, we compare the evolutionary
to be not in good agreement with that of NGC
tracks of our models with those of Geneva mod-
1856.
els (Georgy et al. 2013), which shows that our ro-
tating and non-rotating models with δov = 0 are If there exist a large δov and a rapid ro-
comparable with Geneva models. Figure 8 shows tation in stars, the elements produced in
that the effects of the OVCC with δov = 0.4 on the core can be easily brought to the sur-
the end point of central H-burning phase cannot face of the stars. For the model with δov
be mimicked by the effects of the rapid rotation. = 0.6 and Ωi = 0.5 in young clusters, the
When both the OVCC and the rapid rotation ex- surface nitrogen abundance at the end of
ist in a star, the effects of the OVCC will play MS is about 1.5 times as large as the ini-
a dominant role because the OVCC can mix all tial abundance. But for the models with δov
material in the overshooting region with that in = 0.2 and Ωi = 0.5 and the models with δov
theconvectivecorebuttherotationalmixingonly ≤0.6 without rotation, the surface nitrogen
brings a small part of material in the shell on the abundanceisalmostinvariableinMSphase.
top of the overshooting region into the convective If there is a nitrogen spread in MSTO stars
core. that would reflect the coexistence of rapid
Due to the centrifugal effect, rotating mod- rotation and large δov. Due to the mixing
effect of deep convective envelope, the sur-
els are fainter and redder than non-rotating over-
face nitrogen abundance in RGB is about
shooting models in the early stage of MS. How-
1.5 − 3 times as large as the initial abun-
ever, in the late stage of MS, due to the fact that
dance. But for the cluster with an age of
slightly more H-rich material is brought into H-
550 Myr, the envelopes of RGB stars with
burning core by rotational mixing, rotating mod-
els produce slightly more nuclear energy. Thus, δov =0.7 and Ωi =0 are radiative. Their ni-
trogen abundance cannot be enhanced. For
they exhibit slightly brighter than non-rotating
RGB stars with an age >4 Gyr, the surface
overshooting models (see the panel b of Figure
8). However, the rotational mixing in the radia- nitrogen abundance ofmodelswith δov =0.6
is about 1.4 times as large as that of mod-
tive region also leads to an increase in the mean
6
els with δov =0 in non-rotation case, but is standwhytherearemanystarswithU-Vbetween
about 1.7 times in rotation case. about 0.6 and 1.2 in the CMD of NGC 1856, we
calculated a binary-star stellar population follow-
3.2.3. Variable overshooting scenario ing Yang et al. (2011) by using the Hurley rapid
binary evolution codes (Hurley et al. 2002). The
In Figure 10, we compare overshooting models
calculation shows that the effects of binary stars
with the same age but with different δov to the
are not enough to explain the eMSTOs of NGC
data of NGC 1856. Panel a of Figure 10 shows
1856. However,the calculation clearly shows that
that the eMSTOs of NGC 1856 can be explained
there aremany interactingbinary starswith core-
well by the effects of δov between 0 and 0.7. The
heliumburning(CHeB)whoseluminosity andthe
separation between the isochrone with δov = 0.7
effectivetemperatureareveryclosetothatofsub-
and that with δov = 0 is about 0.1 mag at mU
giants (see Figure 11). The subgiant branch of
≃20.0 and is around 0.014 mag at m ≃21.5.
U
NGC 1856 could be polluted by the CHeB inter-
In order to understand whether or not over-
acting binary stars. Many of stars of NGC 1856
shooting models can produce the eMSTOs and
withU-Vbetweenabout0.6and1.2andwithm
U
the split MSs of NGC 1856, we performed a stel-
around 18.5 mag might be the CHeB interacting
lar population synthesis by way of Monte Carlo
binaries. Moreover,theCHeBinteractingbinaries
simulations following the initial mass function of
couldmakeusmisunderstandthatthe“subgiants”
Salpeter (1955). Milone et al. (2015) found that
of NGC 1856 mainly concentrate towards the end
about 70% and 30% of stars belong to the blue-
of the blue-MS expected from the MSTOs of the
and red-MS in NGC 1856, respectively. We as-
cluster.
sumed 10% of population with δov = 0.7, 20% of
When both the effects of a rapid rotation and
populationwithδov =0.6,40%ofpopulationwith
that of different OVCCs exist in the star cluster
δov = 0.4, 20% of population with δov = 0.2, and
NGC1856,Figure12clearlyshowsthattheeffects
10%ofpopulationwithδov=0. Inthesynthesized
oftherapidrotationof0.5Ω ontheeMSTOsare
population,weincludedobservationalerrorstaken cr
negligible compared with the effects of the differ-
to be a Gaussian distribution with a standardde-
viation of 0.03in magnitude andcolor. Panel b of entδov. TheeMSTOsoftheisochronesmainlyre-
sultfromthe effectsoftheOVCC.Theseparation
Figure 10 shows that the eMSTO of NGC 1856 is
reproduced well by the simulated coeval popula- between the isochrone with δov = 0.7 and Ωi = 0
tion. and that with δov = 0.2 and Ωi = 0.5 is about
0.09 mag atm ≃20.0and is around0.028 mag
U
Moreover, panel a of Figure 10 shows that the
at m ≃ 21.5. The centrifugal effect increases
U
bottom of the RGB of our variable δov models is with an increase in rotation rate. A higher ro-
located at U-V ∼ 1.2 and m ∼ 20.7. Both the
U tation rate will lead to the more significant split
observationaland the simulated results show that
at m ∼ 21, which cannot be mimicked by the
U
therearemanystars. Thesestarscannotappearin
effects of OVCC. Thus, the determinations of ro-
thesimulationsofeSFHmodelsandrapidrotation
tation rates of stars and the split of MS of young
models.
clustersatm .21 aidin determiningthe effects
U
The intermediate-age star clusters have a of rotation.
relatively tight SGB (Li et al. 2014, 2016a).
The split MS of NGC 1856 was not re-
The subgiantsofNGC 1856seemto be notrepro-
produced by OVCC models. This could not
duced well by the simulation. The larger the δov, ruleoutOVCCscenariobecausevariableδov
the shorter the timescale of subgiants. Moreover,
and rapid rotation could coexist in a star
thenumberofstarswithδov =0.6and0.7areless cluster. Moreover, the blue MS could be
than that of stars with δov =0.4. Thus, it is hard affected by interacting binaries (see Figure
for the subgiants with δov = 0.6 and 0.7 to occur 11).
in the CMD. The number of stars with δov = 0
are much less than that of stars with δov = 0.4.
Therefore, the number of subgiants obtained by
the simulation are also rare. In order to under-
7
3.2.4. A prediction of OVCC models differentstarswithanapproximatellyequalmass.
It has not revealed the fraction of stars with a
There is a pulsation constant
large δov. Why there are different δov for stars
M R is anotheropen question for asteroseismologyand
Q=Π( )1/2( )−3/2 (1)
M R stellar physicists.
⊙ ⊙
The OVCC is more efficient than rotation at
for pulsating stars, where Π is a period of
bringing H-rich material into the H-burning core.
pulsating stars, or a large frequency sepa-
The effects of the OVCC with δov = 0.4 on the
ration
MSTO of star clusters is hard to be mimicked by
M R the effects of a rapid rotation. In order to repro-
∆ν =134.9×( )1/2( )−3/2µHz (2)
M R ducethestarsofNGC1856attheendofcentralH-
⊙ ⊙
burning phase, an overshooting of around 0.4 H
p
for stars with solar-like oscillations. In a is requiredforour models. Whenboththe OVCC
star cluster, the radii of MSTO stars with and a rapid ratation exist in stars, the MSTO is
a large δov are much larger than those of mainly affected by the effects of the OVCC. How-
MSTO stars with a small δov. Thus, the ever, the centrifugal effect of the rapid rotation
periods of the former could be about 2−4 playing a role in the MS stars whose envelope is
times as long as those of the latter. For ex- radiative leads to the fact that the split of MS is
ample, at the end of the central H-burning more significant.
phase of a star with M = 3.0 M⊙, the pe- The points C and D of the rotating isochrones
riod of the model with δov = 0.6 is about with Ω = 0.5 can be matched by those of non-
i
4.4 times as long as that of the model with
rotating isochrones with the same δov but with
δov = 0. In other words, if the variable different ages. The equivalent age spread ob-
δov exists in a star cluster, long-periodic tained by comparing points C and D of a rotat-
pulsating/variable stars (in bright MSTO)
ing isochrone to those of non-rotating isochrone
and short-periodic pulsating stars (in faint
withthesameδov istwicemorethanthatobtained
MSTO) would be observed in the star clus-
through comparing points A and B of the rotat-
ter. This case cannot occur in rotation sce-
ing isochrone to those of non-rotating isochrone.
nario and eSFH scenario. For example, a
Niederhofer et al. (2015b) might overestimate the
difference of 300 Myr in age can only lead
equivalent age spreads caused by rotation. More-
to a change of about 10%, depending on the
over, the higher the efficiency of the rotational
age, in frequencies or periods of pulsation
mixing, the more the H-rich material brought
of MSTO stars. Gravity darkening caused
into the H-burning core, the larger the luminos-
by the angle of view cannot result in any
ity of the rotatingmodels atthe end ofcentralH-
change in the frequencies or periods. Thus,
burningphase,whichcanleadtoanincreaseinthe
asteroseismical observation for the young-
equivalent age spread caused by rotation. These
andintermediate-agestarclusterswoulddi-
may be the reason why the equivalent age spread
rectly rule out or confirm OVCC scenario,
of Niederhofer et al. (2015b) rotating models can
and show what plays a dominant role in the
match that of NGC 1856 but that of our rotating
origin of the eMSTO.
models cannot. However, a high efficiently rota-
tional mixing in the radiative region of stars can
4. DISCUSSION AND SUMMARY
hinder stellar expansion and lead to the fact that
the evolutionary tracks of the stars move towards
Figures5and10showthattheOVCCcanlead
theupperleftoftheHertzsprung-Russelldiagram.
totheobservedeMSTOs. ThevariableOVCCisa
good potential mechanism for explaining the eM- Thus, the effects of a large δov are different from
that of rotation.
STOsofstarclusters. However,itisdependenton
the fractionof stars with a large δov that whether Theequivalentagespreadbetweentheisochrone
or not the OVCC is responsible for the observed with δov = 0.4 and that with δov = 0 in panel a
eMSTOs of star clusters. Asteroseismology only of Figure 10 is 170 Myr, and the spread between
tells us that the value of δov can be different for the isochrone with δov = 0.6 and that with δov
8
=0.2 is 180 Myr, which are close to the spreadof tation models. If eMSTO does not exist in
150 Myr determined by Milone et al. (2015). The star clusters with ages between 2.0 and 2.5
equivalent age spread between the isochrone with Gyr that is hard to be explained by OVCC
δov =0.6andthatwithδov =0 is230Myr,which scenario. Perhaps, that is due to the frac-
is larger than that determined by Milone et al. tion of stars with a large δov being small in
(2015). However, Figure 10 shows that the width the star clusters.
betweentheMSTOoftheisochronewithδov =0.6 At the end of the central H-burning
andthatwithδov=0doesnotexceedthatofNGC phase of rapid rotation models, the sur-
1856. face nitrogen abundance can be enriched by
Duetothefactthatthelifetimes oftheSGBof 25−50% in young star clusters for a δov be-
the stars with a large δov are much shorter than tween 0.4 and 0.6. This does not happen in
those of the stars without overshooting or with a the rotating models with δov = 0.2 and the
small δov, the population of SGB is not only de- non-rotating models. Moreover, the pulsat-
pendentonthenumberofstarsbutalsodependent ing periods of bright MSTO stars would be
on δov. Moreover, the SGB of young star clusters longer than those of faint MSTO stars in
could be polluted by the CHeB interacting bina- OVCC scenario because the former has a
ries. Thus, the SGB of young star clusters cannot low mean density, which also does not oc-
be used to study the origin of the eMSTOs. cur in rotation scenario and eSFH scenario.
The effects of OVCC on the MSTO of These characteristics could aid us in differ-
young- and intermediate-age star clusters entiating OVCC scenario and rotation sce-
are similar to that of rotation. The equiv- nario.
alent age spread caused by rotation has a Asteroseismology has shown that the value of
peak at age ∼ 1.7 Gyr (Yang et al. 2013), overshooting parameter of the convective core is
which is dependent on the efficiency of ro- variable for stars with an approximatelly equal
tational mixing. The observationally in- mass. We calculated the effects of the OVCC on
ferred age spread also has a peak at age the MSTO of young- and intermediate-age star
∼1.7 Gyr (Niederhofer et al. 2016; Li et al. clusters, and found that the observed eMSTOs of
2016b). This characteristic does not occur young- and intermediate-age star clusters can be
in OVCC models. The eMSTO caused by explained well by the effects of the OVCC with
the OVCC is still present in star clusters different δov. The equivalent age spread caused
with age >2.0 Gyr. The eMSTO is hard to by the OVCC increases with an increase in age of
be exactly described by the equivalent age starclustersandanincreaseinδov. Duetothefact
spreads relative to the same age isochrone thattheconvectivecoreofstarscanshrinkduring
withδov =0, especiallyforstarclusterswith the late stage of MS, the equivalent age spread
age & 3.0 Gyr. Our models show that the cannot increase continuously with an increase in
width of the eMSTO caused by a δov of 0.2 age of star clusters. The eMSTOs of NGC 1856
or 0.4 at age ∼2.5 Gyr is about 0.5 times as can be reproduced well by the coeval populations
wide as that at age ∼ 1.4 Gyr. The width with different δov. The effects of a rotation with
of the eMSTO caused by the OVCC has a Ω = 0.5 and the rotational mixing efficiency for
i
peak in color V-I of CMDof star clusters at the Sun on the MSTO of a cluster with an age of
age ∼ 1.4 Gyr, and then decreases with an 550 Myr are negligible compared with the effects
increaseordecreaseinage, whichcannotbe of the OVCC. However, the presence of the rota-
reflected by the equivalent age spreads. In tion can result in the fact that the split of MS of
rotation models of Yang et al. (2013), the the cluster is more significant. The eMSTOs of
eMSTO can occurs in star clusters with age star clusters may be related to the effects of the
larger than 2.0 Gyr because rotational mix- OVCC with different δov. But keep in mind
ing exists in stars like the Sun (Yang & Bi that a single value of δov cannot produce
2007; Yang 2016a). Thus the eMSTO of any eMSTO in the CMD of clusters. As-
star clusters with age > 2.0 Gyr cannot be teroseismical observation would confirm or
used to differentiate OVCC models and ro- rule out OVCC scenario.
9
We thank the anonymous referee for helpful Goudfrooij,P.,Girardi,L.,Kozhurina-Platais,V.,
comments that helped the authors improve this et al. 2014, ApJ, 797, 35
work,A.P.Milone forprovidingthe data ofNGC
Goudfrooij, P., Girardi, L., Rosenfield, P., et al.
1856, and F. Niederhofer for the data of Nieder-
2015,MNRAS, 450, 1693
hofer sample, as well as the support from the
NSFC 11273012.
Goudfrooij, P., Puzia, T. H., Chandar, R., &
Kozhurina-Platais,V. 2011, ApJ, 737, 4
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