Rapidly rotating second-generation progenitors for the blue hook stars of {\omega} Cen

Abstract

Horizontal branch stars belong to an advanced stage in the evolution of the oldest stellar galactic population, occurring either as field halo stars or grouped in globular clusters. The discovery of multiple populations in clusters that were previously believed to have single populations gave rise to the currently accepted theory that the hottest horizontal branch members (the 'blue hook' stars, which had late helium-core flash ignition, followed by deep mixing) are the progeny of a helium-rich 'second generation' of stars. It is not known why such a supposedly rare event (a late flash followed by mixing) is so common that the blue hook of ω Centauri contains approximately 30 per cent of the horizontal branch stars in the cluster, or why the blue hook luminosity range in this massive cluster cannot be reproduced by models. Here we report that the presence of helium core masses up to about 0.04 solar masses larger than the core mass resulting from evolution is required to solve the luminosity range problem. We model this by taking into account the dispersion in rotation rates achieved by the progenitors, whose pre-main-sequence accretion disk suffered an early disruption in the dense environment of the cluster's central regions, where second-generation stars form. Rotation may also account for frequent late-flash-mixing events in massive globular clusters.

Full text

Rapidly rotating second-generation progenitors for the
blue hook stars of ωCen
Marco Tailo1,2, Francesca D’Antona1, Enrico Vesperini3, Marcella Di Criscienzo1, Paolo Ventura1,
Antonino P. Milone4, Andrea Bellini5, Aaron Dotter4, Thibaut Decressin1, Annibale D’Ercole6,
Vittoria Caloi7, Roberto Capuzzo-Dolcetta2
1INAF- Osservatorio Astronomico di Roma, I-00040 Monte Porzio (Roma), Italy.
2Dipartimento di Fisica, Universit´
a degli Studi di Roma, La Sapienza, Roma, Italy.
3Department of Astronomy, Indiana University, Bloomington, IN (USA)
4Research School of Astronomy & Astrophysics, Australian National University, Canberra ACT
2611, Australia
5Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
6INAF- Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy
7INAF, IAPS, Roma, via Fosso del Cavaliere 100, I-00133 Roma, Italy
Horizontal Branch stars belong to an advanced stage in the evolution of the oldest stellar
galactic population, occurring either as field halo stars or grouped in globular clusters.
The discovery of multiple populations in these clusters1, 2, that were previously believed to
have single populations gave rise to the currently accepted theory that the hottest horizontal
branch members (the blue hook stars, which had late helium-core flash ignition3, followed
by deep mixing4, 5) are the progeny of a helium-rich “second generation” of stars6, 7. It is not
known why such a supposedly rare event8, 9(a late flash followed by mixing) is so common that
1
arXiv:1506.07463v1 [astro-ph.SR] 24 Jun 2015

the blue hook of ωCen contains ∼30% of horizontal branch stars10, or why the blue hook
luminosity range in this massive cluster cannot be reproduced by models. Here we report
that the presence of helium core masses up to ∼0.04 solar masses larger than the core mass
resulting from evolution is required to solve the luminosity range problem. We model this by
taking into account the dispersion in rotation rates achieved by the progenitors, whose pre-
main sequence accretion disc suffered an early disruption in the dense environment of the
cluster’s central regions where second-generation stars form11. Rotation may also account
for frequent late-flash-mixing events in massive globular clusters.
In the colourmagnitude diagrams of globular clusters, the horizontal branch is the locus of
core-helium-burning structures that are the progeny of red giant stars. For each such structure, the
helium flash ignition has occurred either at the tip of the red giant branch, or anywhere along the
evolutionary path that moves the star into the white dwarf stage5. In general no mixing occurs,
and the result is a standard horizontal branch structure: the smaller its hydrogen-rich envelope is,
the hotter is the location of effective temperature (Tef f ) in the model in the HertzsprungRussell
diagram (that is, along the horizontal branch), up to about 32,000 K.
A very late helium ignition, along the white dwarf cooling pathway, can cause the helium
core to mix with the small hydrogen-rich envelope5,12, 13 (late-flash-mixing), resulting in a slightly
smaller helium-burning core plus a helium-rich, or a helium-dominated, envelope. Such “blue
hook” (BHk) structures attain a lower luminosity, and a higher Teff than do stars in the extreme
hottest horizontal branch standard locus, owing to the smaller opacity of the helium-rich atmo-
2

sphere; they have been found in the ωCen14 and in a few other massive clusters9.
The blue hook in ωCen is particularly strikingis particularly striking because it contains
about 30% of the horizontal branch stars10, and it is extremely well defined in the Hubble Space
Telescope (HST) observations. The optical colour-magnitude diagram10 displays a blue hook that
extends approximately 1.0 magnitudes in the F625W band, with a redder side much less popu-
lated (Extended Data Fig. 1a). A smaller sample of blue and UV data15 (Fig. 1a) displays a strong
peak on the blue side of the colour distribution, where the helium-richer late-flashmixed stars are
located16, 17.
The models we adopt (see Methods) assume the helium core masses implied by the evolutionary
process, and different initial envelope hydrogen abundances (Xenv−in), together with an efficiency
of helium settling calibrated to reproduce as well as possible the values of He/H versus Teff data
from the literature17 (Fig. 2). Tracks (that is, evolutionary sequences) with very low Xenv−in values
span only about 0.4 mag in F225W, but an extra 0.35 mag or so are obtained in models starting
with larger Xenv−in, as a result of the change in atmospheric composition from helium-dominated
to hydrogen-rich (Extended Data Fig. 2 and Extended Data Table 1). Considering the whole range
of Xenv−in values explored, from 0.007 to 0.41, we can obtain a global extension of 0.9 mag in the
F225W band (6th column of Extended Data Table 1), close to the range observed, but the tracks
with Xenv−in>0.06 do not match the colour-magnitude diagram well (Extended Data Fig. 4) nor
the He/H versus Teff data (Fig. 2), and tend to merge with the extreme horizontal branch. For
Xenv−in≤0.06, the tracks cover a maximum of 0.69 mag and the total extension is 0.79 mag.
3

Making the currently accepted assumption that the blue hook contains the progeny of the very-
helium-rich second-generation stars, the existence of the blue hook stars can be attributed to late-
flashmixing events in models having an initial helium mass fraction of Y = 0.37. Simulations
including only objects having up toXenv−in=0.06 show the discrepancy with the observed luminos-
ity range (Fig. 1b)
Modifications in model inputs have not much effect on the covered magnitude range:(1) standard
flash-mixed models occur in a small core mass range (δMcore <0.008 M)8, 9, corresponding to
a negligible magnitude difference in the tracks extension9; (2) the magnitude range can not be
extended by increasing the adopted core overshooting parameters. 3) we also have good reason
to reject the possibility that the blue hook contains both the progeny of the standard and of the
helium-rich populations (see discussion in Methods and Extended Data Fig. 3).
The high-luminosity portion of the blue hook, at the correct colour location, can be covered by
tracks having larger helium core masses Mc, as we show in Fig. 1a and Extended Data Fig. 1a
for the track having δMc=+0.04 M, and Y=0.37, where Mis the solar mass. The helium flash
ignites in more massive cores if very rapid core rotation delays the attainment of the flash temper-
atures. Of the (few) models available, we select a first-order approximation18 of models starting
from solid-body rotation on the main sequence, and preserving angular-momentum in shells, to es-
timate the increase in the helium-flash core mass, δMc, as a function of the initial angular velocity
ω. These models provide upper limits to δMc,because they do not allow for angular momentum
transport and losses during main-sequence and post-main-sequence evolution. In this approxima-
tion, about half of the break-up main-sequence rotation rate (at which the centrifugal force at the
4

equator becomes equal to the gravitational force, which is about 7×10−4s−1) provides a huge
δMc=0.06 M, leaving room for different Mc(ω)laws when available.
We propose that high rotation rates may be a consequence of the star-formation history and early
dynamics in very massive clusters. Specifically, in the model based on the formation of second-
generation stars from the ejecta of asymptotic giant branch stars, a cooling flow collects such
ejecta in the central regions of the first-generation cluster and produces a centrally concentrated
second-generation subcluster11, 19. Observations showing that the helium-rich blue main-sequence
starsthe progenitors of the blue hook starsare spatially more concentrated than the rest of the main-
sequence stars20 and retain some ’memory’ of their initial spatial segregation,provide evidence that
indeed such progenitors formed segregated in the innermost regions of the cluster, as predicted11
and assumed in this study.
The contracting pre-main-sequence low-mass objects in the galactic disc (that is, stars like
the variable star T Tauri)rotate with periods in the range from 2 days to 12 days. The rotation
rates of classical T Tauri stars do not increase with stellar age, because of magnetic disk-locking
between the star and the disk21, which keeps the stellar rotation constant until the disk is lost. Stars
in young clusters (such as αPersei) show a wide distribution of rotation velocities, owing to differ-
ences in the time at which different objects break their magnetic coupling with the wind22. Rapidly
rotating stars result from an early breaking of the of the coupling. Afterwards, during the main
sequence, stellar winds slow down the rotation of the convective stellar envelopes, and old stars all
appear to rotate slowly, but the inner core rotation is still fast despite the slow angular momentum
5

transfer from the core to the envelope. We note that this model implies the presence of a crowded
second generation.
Figure 3a shows the fast decrease of the momentum of inertia of first- and second-generation stars
in globular clusters (the latter ones have no initial deuterium, because the gas from which they
formed was nuclearly processed at high temperature11). Figure 3b describes the stellar angular
velocity which can be obtained at the main sequence (age exceeding 2×107yr), as a function of
the time at which the disk-star coupling is destroyed and evolution proceeds at constant angular
momentum. The moment of inertia evolution of the case of a mass of M=0.7 M, Y=0.35 is as-
sumed,for rotation periods at detachment taken in the range 2-12 days. We see that there is ample
model space to reach high angular velocities, provided that the second-generation (the progenitors
of the blue hook stars) lose their disc at young ages, from 105yr (for the longer initial periods) to
about 3×106yr (for the short periods).
We model the dynamical encounters that destroy the disk, assuming that three encounters are nec-
essary (see details in Methods), and use the δMcresulting from the timings of encounters as direct
inputs for the simulation. The resulting distribution of the blue hook is shown in Fig. 1c and Ex-
tended Data Fig. 1b. We remark that fast rotation will favour deep mixing in the giant envelopes
during the last phases of evolution, enhancing the chemical anomalies of the most extreme second
generation23, 24 and probably also favouring mixing at the onset of the flash: the late-flash-mixing
event, although very different from what is described in one-dimensional models, will be more
probable, thanks to the reduced entropy barrier between the core and the envelope, justifying the
existence of the blue hook itself. In this scheme, the non-mixed extreme horizontal branch stars
6

should have been slowly rotating, and in fact standard models match this group well. In the sim-
ulation we technically model them by assuming that the cool side of the hot horizontal branch is
populated by the progeny of scarcely rotating (rotation rate ω < 10−6s−1) stars (red squares in
Fig. 1 and Extended Data Figs 1, 3 and 4). Mcfor this side of the simulation is the mass of the
non-rotating core, and in fact standard tracks match the extreme horizontal branch well.
The presence of rapidly rotating stars among the second-generation population should not be con-
fined to modelling the blue hook in ωCen. Two conditions are at the basis of the success of the
dynamical model used here: first, we must deal with second-generation stars, which form in a
cooling flow at stellar densi- ties much higher than that of the first-generation stars; and second,
the population must be abundant, so it requires the presence of very massive clusters to start with.
Of the most massive clusters, M54 (NGC 6715) shows a blue hook very similar in shape to that
of ωCen25, and NGC 2419 also has an extended blue hook26. NGC 6388 and NGC 6441 have
peculiarly extended horizontal branches, unlike other metal-rich clusters. It is certainly possible
that their thick red horizontal branch (the red clump), modelled by assuming a large helium spread
in the second generation27, is also caused, in part, by the presence of larger core masses due to
fast rotation. The reproduction of the horizontal branch morphology would then require, in the
second generation, a smaller helium content increase than predicted by standard models27, and this
would be more consistent with the helium spread derived by the main-sequence colour thickness
in NGC 644128. We note that a large rotation spread in this case does not produce a prominent blue
hook, but an anomalous red clump. NGC 2808, just a bit less massive, could be a borderline case,
in which the presence of fast rotators is uncertain.
7

Smaller but important rotation ranges may be present in other less-massive clusters as well, and
may have been due to the same process of early disk destruction in second-generation stars: in
many clusters the mass loss necessary to explain the location of second-generation stars has to be
slightly larger than in the first generation, possibly because of their rotation rate29, 30.
The connection between the magnitude extension of the blue hook stars in ωCen and the range
of rotation rates of their progenitors might be another manifestation of the interplay between the
formation of multiple populations in clusters, their dynamical behaviour and the properties of the
component stars. At the same time, the model developed in this work for the blue hook supports
the model for the dynamical and chemical formation of multiple populations11 based on the con-
tribution of asymptotic giant branch stars and adopted in this study.
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Acknowledgements A.P.M. acknowledges support by the Australian Research Council through
Discovery Early Career Researcher Award DE150101816. E.V. acknowledges support from grant
NASANNX13AF45G. P.V. and F.D’A. acknowledge support from PRIN INAF 2011 ’Multiple pop-
ulations in globular clusters: their role in the Galaxy assembly”(principal investigator E. Carretta),
and P.V. acknowledges support from PRIN MIUR 2010-2011, project ”The Chemical and Dynamical
Evolution of the Milky Way and Local Group Galaxies” (principal investigator F. Matteucci). T.D. ac-
knowledges support from the UE Program (FP7/20072013) under grant agreement number 267251 of
Astronomy Fellowships in Italy (ASTROFit). M.D.C. acknowledges support from INAF-OAR. A.B.
acknowledges support from STScI grant AR-12656.
Author Contributions M.T., F.D’A., E.V. and M.D.C. jointly designed and coordinated this study.
F.D’A. proposed and designed the rotational evolution model. E.V. designed and computed the dynami-
cal simulation. M.T. and P.V. computed the new evolutionary models. A.D. computed synthetic colours
with an internally consistent treatment of extinction across all bandpasses. M.T. and M.D.C. performed
the simulations and the analysis. A.B. performed the data reduction and calibration for the WFC3/UVIS
exposures. A.M. dealt with the optical data and the comparison with spectroscopic data. T.D. dealt with
the problems connected to the modelling of stellar rotation. V.C. contributed to the discussion and to
the writing of the text. A.DE. and R.C.-D. provided insight on the dynamical aspects. All authors read,
commented on and approved submission of this article.
Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to Francesca
D’Antona (email: francesca.dantona@oa-roma.inaf.it or franca.dantona@gmail.com).
12

FIGURES
13

Figure 1 - Simulations for the ultraviolet data.
Grey dots are the observeds tars in ωCen. Tracks plotted on the blue hook in aare: for
M= 0.466 Mand Y=0.37 (black); for 0.489 M, Y=0.25, standard Mc(green); and for
Mcincreased by 0.04 M(magenta). Models have Xenv−in=0.06. Also shown are the zero-age
horizontal branch for Y=0.37 (dashed balck) and the 0.471 Mtrack (dotted black). Labels
indicate bolometric distance modulus (m−M)0and reddening E(B−V)adopted for the
comparison. b, Simulation with a late-flash mixture of Xenv−instars for Y= 0.37 (see text
for the details of the chemistry labels at the top). Comparison with data are shown in the
histograms on the left (blue hook, blue triangles) and right (extreme horizontal branch, red
squares) side. On the histograms, 1σPoisson errors are shown with respect to the counts N. c,
Simulation resulting from coupling dynamics and evolution (same mixture of Xenv−in). Colour
code and comparisons as in b.
14

Figure 2 - Calibration of H–He diffusion
15

Grey squares show the He/H versus Teffdata17 with error bars. The evolution of three models
is shown for different values of Xenv−in. Diffusion is computed assuming that the envelope
mass below which we assume diffusion is operating, Mturb, is 10−6M. The orange tracks
show the evolution of the smallest non mixed tracks. Triangles mark each 107yr interval, for
Mturb=10−7. The red dots represent 26 stars randomly extracted from the simulation in Fig. 1c.
Along the magenta curve, no star with Xenv−in=0.21 are extracted, they would be cooler than
the observed sample.
16

Figure 3 - Rotational evolution from disc–detachment to the main sequence
17

a, Time evolution of moment of inertia (Inorm, in units of momentum at t=2×107yr), for
different masses, values of Y and deuterium abundance XD.b, For M=0.7 Mand Y=0.35, we
show the rotation rate ωat age t=2×107yr, as a function of the age at which the star detaches
from the proto-stellar disk, and conservation of angular momentum begins. The lines represent
different assumptions for the rotation period the disc is lost, from left to right: Pin=12 days,
10 days, 8 days, 6 days, 4 days, 3 days and 2 days.
Methods
The data sets. We analyse optical10 and ultraviolet15 Hubble Space Telescope data for the
hot horizontal branch stars of ωCen. Photometric errors are about 0.015 mag in filters F435W
and F625W of ACS/WFC (Wide Field Channel of the Advanced Camera for Surveys), and
0.010 mag in the filters F225W and F438W of the WFC3. In the comparisons with optical
data, we adopt distance moduli and reddening that fit the luminous part of the horizontal
branch in these same data (M.T., PhD thesis, in preparation). The values correspond to
E(B−V)=0.156 and (m−M)0=13.63, when using extinction ratios AF435W /AV=1.362
and AF625W/AV=0.868, explicitly computed for this work, and appropriate for stars of
Teff >20000K, using standard extinction curves31,32 , and AV=3.1×E(B−V). The F225W
and F438W data15 have been recalibrated. Distance modulus and reddening are chosen by
adjustment of the simulations results, and are labelled in Fig. 1. Using AF225W/AV=2.670 and
AF438W/AV=1.356, the bolometric distance modulus results to be ∼0.08 mag smaller than the
modulus derived from optical data.
The prominent peak shown by number counts on the blue side (histogram at the bottom of Fig.
18

1a), followed by a sharp decline, and by a second broader peak at redder colours, corresponds
to a change in the spectroscopic abundances: the cool side of the distribution shows very
helium-poor spectra16, a feature ascribed to the effect of helium-diffusion in hydrogen-rich
envelopes (the hottest standard horizontal branch stars). An almost abrupt transition to
hydrogen-helium intermediate composition occurs at Teff '35000K16, 17, where the sharp
peak occurs. The contemporary presence of helium and carbon7, 17, 33 points out that the stars
at Teff &35000K are the result of late-flash-mixing5, so they must be blue hook stars. Standard
one-dimensional simulations of late-flash-mixing end up with very small hydrogen abundance
left in the envelope (X∼4×10−4, see ref. 13). A shallow mixing, leaving some percentage
of hydrogen abundance, occurs only for a metal mass fraction that is ten times larger(Z≥0.01
ref. 8) than the relevant metallicity in our sample. Nevertheless, the comparison with observed
values of N(He)/N(H) requires that many blue hook stars are the result of a shallow mixing.
We hand-draw a possible dividing line between blue hook and extreme, non mixed, horizontal
branch stars. With this subdivision, in the optical sample the blue hook stars are ∼320 and the
cooler stars are ∼150. In the UV sample we get 130 blue hook stars plus 60 cooler stars.
Standard models. Horizontal branch models are computed with the ATON code34. Some
useful results are given in Extended Data Tables 1 and 2.
Most models start from zero age horizontal branch, where the helium core mass is fixed by
previous evolution up to the helium flash, and the rest of the mass is in the hydrogen rich
envelope. We also use as guidelines the results of some full late-flash evolutions, computed
19

following the standard methodology3,35 of evolving the mass corresponding to an age of
12 billion years (Gyr) along the red giant branch, for increasing wind mass loss rates. Our
results substantially confirm the previous findings5,8, 13 . The mass range of late flashers can be
estimated in ∼0.02–0.03 M, and our range of late flash mixed models is .0.005–0.006 M
(the largest mass range found in the literature is .0.008 M; ref.8).
Models including sedimentation Guided by the evolutionary results, we built up sets of mod-
els, characterized by three main parameters: core mass Mc, envelope mass Menv, and initial
hydrogen mass abundance in the envelope Xenv−in, and followed their evolution, starting from
the zero age horizontal branch. As there are both helium-dominated and hydrogen-richer spec-
tra among the BHk stars17, 33, models need to include helium diffusion, to correctly derive the
ultraviolet magnitudes, which are strongly dependent on the Teff of the models. An increase
in the flux is expected, due to the shift in Teff that the star suffers when diffusion changes
the helium carbon dominated atmosphere into a hydrogen-rich one. The speed of diffusion, a
byproduct of the residual turbulence in the outer envelope36, 37, must be calibrated to be com-
patible with the observed values of hydrogen abundance in the blue hook stars17, 33. In the
models we calibrate the parameter Mturb. Mass loss can play a concomitant role, and, when
included, Mturb must be larger to remain consistent with the observations. We show our choice
among computations in Extended Data Fig. 2. The theoretical (Extended Data Fig. 2b) and ob-
servational (Extended Data Fig. 2a) Hertzsprung-Russell diagram evolution of the tracks hav-
ing Xenv−in from 0.007 to 0.41 are plotted for two settings of diffusion efficiency. The F225W
20

extension of the tracks increases from about 0.4 mag up to cover almost the full extension
of the blue hook(histogram in Extended Data Fig. 2a), by increasing Xenv−inup to the maxi-
mum value. Extended Data Fig. 2c shows the time evolution of Xenv−in for the two choices of
Mturb. A better representation of spectroscopic observations requires use of the models with
slower diffusion. Figure 2 shows the comparison of tracks and the location of simulated points
(red dots) for a mixture of stars with different initial abundances, Xenv−in= 0.007 (10%), 0.03
(45%) and 0.06 (45%). The Teff location and the He/H observed abundances are instead less
compatible if we include a percentage of tracks having Xenv−in>0.06 (see also the simulation
comparisons).
Assumptions for the rotating models. In the blue hook progeny of rotating stars the core
mass at the He–flash, Mc(ω), is larger than the core mass of late-flash-mixing models com-
puted by standard stellar evolution. From models of evolution starting from solid-body rotation
on the main sequence and angular-momentum conservation in shells18, we derive a parabolic
expression for the helium-flash core-mass increase δ(Mcore)as a function of the angular ve-
locity ω(in units of per second):
δ(Mcore/M) = 3.86 ×105·ω2(1)
A self-consistent approach needs computation of rotating models starting from the zero age
main sequence (where we can safely assume a rotation rate simply acquired by angular
momentum conservation) and including all possible mechanisms of transfer of angular
momentum from the core to the envelope, plus the loss of momentum of the envelope due
to magnetic wind. In the models available in the literature38,39 , the parameters necessary to
21

model these mechanisms have been calibrated from the atmospheric abundance variations
induced by the associated chemical mixing, but most of the sampled stars were slowly rotating
from the beginning, given that the fraction of young main-sequence stars which are fast
rotating is quite small40. The coexistence of a fast-rotating core and slow-rotating envelope
will lead to strong differential rotation that can be responsible for strong chemical mixing
along the red giant branch, compared to the mixing possible for initially slower-rotating stars,
an outcome that may explain extreme abundance anomalies in some clusters24,41 , but it is not
known how much these events may break down the inner fast core rotation. So we adopt the
simple Mc(ω) relation18 which does not take into account the core–envelope interactions.
The most important parameter in our investigation is the main sequence lifetime of these fast
rotating stars, which is longer than the lifetime of non-rotating stars of the same mass. If the
age is fixed, the rotating evolving mass Mev(ω)will be larger than the non rotating mass. If a
fraction of the rotating stars evolves through the helium flash, otherwise there would be no
blue hook, we have to assume that its mass satisfies the relation:
Mev(ω)&Mc(ω)+∆MRGB(ω)
where ∆MRGB(ω)is the total mass lost. The flash luminosity in our models increases by
δLflash/δMc'1.9×104(where L and Mcin solar units). Assuming Reimers’ mass loss rate,
we expect an extra mass loss of 0.010-0.015 Mfor each extra 0.01 Mincrease in the core
mass. If for example we require δMc=0.04 M, corresponding to ω'3.5×10−4s−1, the
evolving mass must be ∼0.08–0.10 Mlarger than the non-rotating mass, if it has to ignite the
22

helium-flash. If we assume a stronger dependence of mass loss on L than in Reimers’, we need
larger Mev(ω). Such an increase is compatible with the scarce existing estimates18 but a large
computational effort is needed to solve the problem. A spread in the evolving mass Mev(ω)is
also useful to meet, at least in a range of ω, the strict requirements of the late-flash-mixing
conditions and let the stars populate the blue hook: those outside the correct range will evolve
into the helium-core white-dwarf remnants whose existence has been recently established15.
The core mass vs. luminosity relation shows that the tip of the RGB is extended by
δlog L/L'0.14 (0.35 bolometric mag) for the core mass increase of ∼0.04 M. If the
model developed here is correct, the brightest giants should belong to the He–rich population,
contrary to standard models (the core flash occurs at slightly smaller luminosities in the high
Y, non rotating, models).
Model transformations: from the theoretical to the observational plane We use bolomet-
ric corrections for Z=0.0005 and [α/Fe]=0.2 ([Fe/H]=–1.74) available for atmosphere models
with standard hydrogen abundance42. A correction is necessary, as these spectra do not rep-
resent accurately the peculiar atmospheres of late-flash-mixed stars5, with strongly enhanced
helium and carbon abundances at the surface. At a given Teff , we correct the helium dominated
model magnitudes making them fainter by 0.057mag in the F225W band, an estimate based on
the comparison of fluxes in helium-and-carbon-rich and in hydrogen-rich model atmospheres9.
We keep this correction until the helium surface abundance remains above Y=0.9, then we
switch to the direct table correlations. This choice contributes to stretch in magnitude exten-
23

sion the tracks with 0.03≤Xenv−in≤0.06.
Standard Simulations Synthetic models for the simulations shown in Fig. 1b follow
standard guidelines43. For the cases employing standard tracks, we assume that the red giant
mass MRG at a fixed age is a function of helium, at assumed metallicity. The mass on the
horizontal branch is MHB =MRG (Y)-∆M, where ∆M is the mass lost during the red giant
phase. In our case, assuming Y=0.37 and 12 Gyr we derive MRG=0.658 M. The observed
ratio of stars between late flash mixed and “normal” extreme horizontal branch stars is
reproduced by assuming that ∆M has a Gaussian dispersion σ=0.008 around an average value
∆M=0.19 M. With this choice, the resulting 60 extreme horizontal branch stars that did
not suffer mixing are shown in Fig. 1b as red squares, and the 130 blue triangles are the late
flash mixed stars. When we extract a mass smaller than the late flash mixed masses, the star
is assumed to evolve into the helium-core white-dwarf. With the chosen parameters we find
110 helium-core white dwarfs. Comparison is shown by histograms of counts as a function of
color and magnitude, given separately for extreme horizontal branch and blue hook stars. The
used bins are 0.03mag and 0.08mag for colours and magnitudes respectively. The error bars
are the result of individual count Poisson error (√N). The observational histograms are grey,
the simulated ones are blue and red for the colour (scale on the right) and magnitude (scale on
the top) distributions.
This modelling shows the inherent difficulty in building the blue hook: the mass range
of extreme horizontal branch models (defined as the models at Teff >20000K) is of a few
hundredths of M(from 0.03 to 0.07 M, larger for a larger Y), while late flash mixing
24

covers a mass range of only .0.008 M. The most recent modelling of horizontal branches
of GCs shows that the mass loss spread of each component of multiple populations must be
very small, <0.01 M. If the blue hook standard models are truly describing the blue hook
stars in globular clusters, the mass lost by the red giant progenitor must be finely tuned, so
that it covers precisely this tiny mass range. If the blue hook is made up by more than one
of the globula cluster populations (groups of stars with different helium and metal content)
such a fine tuning must have been successfully met twice. This is a key issue to interpret the
simulations.
First we assume that all the extreme horizontal branch stars have Y=0.37. We use this Y
value because it is consistent with the requirements of the asymptotic giant branch model for
the formation of the extreme populations in clusters44. We use different sets of flash mixed
models, with Xenv−in=0.007, 0.03, 0.06, 0.21 and 0.41. The simulations with Xenv−in≤0.06
are unsatisfactory, as they do not cover the whole BHk extension (Fig. 1b). Including a
fraction of stars with Xenv−in=0.21, the magnitude range is marginally better covered, but the
colour agreement is worse (Extended Data Fig. 4b). The magnitude range is not extended
significantly, because already the track having Xenv−in=0.06 reaches high atmospheric
hydrogen content along the evolution including hydrogen versus helium diffusion (Extended
Data Fig. 2c). Including even stars with Xenv−in=0.41, these latter merge with the extreme
horizontal branch (Extended Data Fig. 4c).
Extended Data Fig. 3 shows the simulation according to which the BHk is made up by
25

superposition of two different late flash mixing populations, with Y=0.25 and Y=0.37,
respectively45.∆M’s and σvalues used are reported in the figure. The mass losses necessary
to obtain masses in the late flash mixing range differ by ∼0.13 Mfor the two different Y
groups. While it is already very difficult to accept that the mass-loss of a unique population
is so well constrained that the blue hook is populated at all, this double constraint is an even
more extreme assumption. In spite of this, the whole extension of the hook is not yet fully
covered.
Coupled dynamical and evolutionary simulations To estimate the distribution of rotation
rates we use a simple semi-analytical model to calculate the distribution of times, Tenc100,
needed for second-generation star-disk systems to have a close encounter with another star
at a distance d < 100 astronomical units (AU) (the estimated dimension of the protostellar
disk) and assume one - or more - such collisions to be able to break the magnetic disc-locking
(in this case Tenc100 is the detachment time). In order to calculate Tenc100 we have followed
the orbits of 50,000 particles distributed as a King model with a concentration c'1.7and
integrated the collision rate (see Eq. 7.194 in ref. 46) along the orbit to estimate Tenc100. The
system we are modelling is meant to represent the dense second-generation subcluster. For our
calculation we assume its total mass to be equal to 105.5Mand half-mass radius of about one
parsec. From the distribution of Tenc100 (Extended Data Fig. 5a) we directly derive the rotation
rates assuming the conservation of angular momentum from Fig. 3, and the corresponding
increase in the core-mass from Eq. 1 in Methods (Extended Data Fig. 5c). The initial periods
26

Pin at detachment from the disk are randomly extracted in the range 3<P(days)<12, using a
normal distribution centered at P=6 days, with standard deviation σ=2 days. The distribution
of the stellar rotation periods result of the simulation at detachment from the disc are given in
Extended Data Fig. 5b. The distribution is not Gaussian, as it also depends on the limitations
we have imposed on the maximum δMc.
For the photometric simulation, we then assume the dispersion in core-masses inferred by the
described simulation (Extended Data Fig. 5c). We are implicitly assuming that the increase in
initial evolving mass due to rotation is enough to accommodate the increase both in core mass
and mass loss in such a way that the stars are able to arrive at late-flash-mixing ignition for
rotation rates ω > 10−6s−1. We also impose that a tail of slower rotating stars populate the
extreme horizontal branch of non–flash mixed Y=0.37 models.
Code availability For the stellar evolution calculations we used ATON (not pub-
licly available). For the N-body simulations we used Starlab (publicly available at
http://www.sns.ias.edu/,starlab/). For the synthetic populations, we used a program written
explicitly for this work (not publicly available).
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ultraviolet extinction. Astrophys. J 345, 245–256 (1989).
32. O’Donnell, J. E. Rnu-dependent optical and near-ultraviolet extinction. Astrophys. J 422,
158–163 (1994).
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33. Moehler, S. et al. The hot horizontal-branch stars in ωCentauri. Astron. Astrophys. 526, A136
(2011).
34. Ventura, P., D’Antona, F. & Mazzitelli, I. The ATON 3.1 stellar evolutionary code. A version
for asteroseismology. Astrophys. Space Sci. 316, 93–98 (2008).
35. Castellani, M. & Castellani, V. Mass loss in globular cluster red giants - an evolutionary
investigation. Astrophys. J 407, 649–656 (1993).
36. Michaud, G., Richer, J. & Richard, O. Horizontal Branch Evolution and Atomic Diffusion.
Astrophys. J 670, 1178–1187 (2007).
37. Michaud, G., Richer, J. & Richard, O. Abundance Anomalies in Horizontal Branch Stars and
Atomic Diffusion. Astrophys. J 675, 1223–1232 (2008).
38. Gallet, F. & Bouvier, J. Improved angular momentum evolution model for solar-like stars.
Astron. Astrophys. 556, A36 (2013).
39. Palacios, A., Talon, S., Charbonnel, C. & Forestini, M. Rotational mixing in low-mass stars. I
Effect of the mu-gradients in main sequence and subgiant Pop I stars. Astron. Astrophys. 399,
603–616 (2003).
40. Spada, F., Lanzafame, A. C., Lanza, A. F., Messina, S. & Collier Cameron, A. Modelling the
rotational evolution of solar-like stars: the rotational coupling time-scale. Mon. Not. R. As-
tron. Soc. 416, 447–456 (2011).
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41. Charbonnel, C. & Talon, S. Influence of Gravity Waves on the Internal Rotation and Li Abun-
dance of Solar-Type Stars. Science 309, 2189–2191 (2005).
42. Castelli, F. & Kurucz, R. L. New Grids of ATLAS9 Model Atmospheres. ArXiv Astrophysics
e-prints (2004).
43. D’Antona, F. & Caloi, V. The fraction of second generation stars in globular clusters from the
analysis of the horizontal branch. Mon. Not. R. Astron. Soc. 390, 693–705 (2008).
44. Ventura, P. & D’Antona, F. Hot bottom burning in the envelope of super asymptotic giant
branch stars. Mon. Not. R. Astron. Soc. 410, 2760–2766 (2011).
45. Cassisi, S. et al. Hot Horizontal Branch Stars in ωCentauri: Clues about their Origin from the
Cluster Color Magnitude Diagram. Astrophys. J 702, 1530–1535 (2009).
46. Binney, J. & Tremaine, S. Galactic dynamics, Second Edition Princeton University Press
(2008).
ED Tables
29

Xenv−in F225W F 225WF225W δF225W δF225W (F225W−F438W)0Teff
ZAHB Peak TAHB (track) total ZAHB ZAHB
0.007 +2.29 +1.86 +1.97 0.43 0.43 -2.734 37660
0.030 +2.21 +1.61 +1.69 0.60 0.68 -2.731 37200
0.060 +2.19 +1.50 +1.56 0.69 0.79 -2.726 36700
0.210 +2.10 +1.43 +1.46 0.67 0.83 -2.693 34500
0.410 +2.03 +1.38 +1.39 0.65 0.90 -2.603 31600
∆Mcore = +0.04M
0.007 +1.96 +1.58 +1.76 0.38 0.71 -2.700 39400
0.030 +1.93 +1.38 +1.71 0.55 0.91 -2.697 38900
0.060 +1.91 +1.27 +1.35 0.64 1.02 -2.693 38400
Extended Data Table 1 - Range in magnitude F225W for M=0.466 M, Mcore=0.463 M
For each initial envelope starting H-abundance (col. 1), we give F225W band values of late flash
mixed models at the zero age horizontal branch (col. 2), at the maximum luminosity (col. 3) and
at the terminal point (col. 4) (defined as the model in which the core helium abundance becomes
zero), and the magnitude interval spanned by each model (col. 5) and by a mixture (col. 6). The
mass of the model is 0.466 M, with an envelope mass of 0.003 M, thus the model core mass is
0.463M.
30

YMTip MHBmin Mcore MLF M Mcore LF M ∆MLF
0.25 0.812 0.500 0.496 0.489 0.486 0.323
0.37 0.657 0.471 0.469 0.466 0.463 0.191
0.40 0.620 0.466 0.465 0.462 0.459 0.158
Extended Data Table 2 - Data at 12 Gyr for models of Z=0.0005, [α/Fe]=0.2
We list some important quantities from our isochrones of 12 Gyr, Metallicity Z=0.0005 and [α/Fe]=0.2.
Col. 1: helium mass fraction; col.2: evolving mass at the tip of the red giant branch (no mass loss);
col.3: minimum “standard” horizontal branch mass; col. 4: core mass in M, when the He–flash
is ignited at the red giant branch tip; col. 5: mass for which we have late flash mixing; col. 6: core
mass for the late flash mixed model; col 7: mass loss needed to achieve late flash mixing.
ED Figures
31

32

Extended Data Figure 1 Comparison simulation vs. optical data
The observed data10 are shown as grey dots. The diagonal line suggests the division between
blue hook and cooler extreme horizontal branch stars. a: Tracks shown are for M=0.466 M,
Y=0.37 and evolutionary Mc=0.463 M(black solid line), plus the corresponding track in which
Mcis increased by 0.04 M(magenta solid line). b: Result of the coupled dynamical–evolutionary
simulation (detailed explanation in Fig. 1).
33

34

Extended Data Figure 2 Models with He vs. H diffusion
aand bshow a comparison of blue hook tracks from Xenv−in=0.007 to Xenv−in=0.41, from left to
right; Mturb=10−6M(full lines) and 10−7M(dashed lines). The bolometric luminosity (b) and
the F225W magnitude (a) are shown as function of Teff . The hot side of the zero age horizontal
branch for Y=0.37 is shown (solid black line with dots). c: Evolution with time of the surface
N(He)/N(H) due to diffusion. The solar ratio is shown as a horizontal grey dotted line. The
observations of blue hook stars show that the slower diffusion is a better description (Fig. 2).
35

36

Extended Data Figure 3 Simulation vs. data, including first and second generation stars
As in Fig. 1 and Extended Data Fig. 1, but the ultraviolet data are compared with a simulation
assuming that 50% of blue hook stars are progeny of the Y=0.37 population (squares), and 50%
are progeny of the Y=0.25 population (triangles). The error bars are the result of individual one
sigma Poisson error on the stellar counts.
37

38

Extended Data Figure 4 Tracks and simulation vs. data including models having Xenv−in>0.06
As in Fig. 1. a: tracks with different Xenv−in and the end of the zero age horizontal branch with
the location of the minimum non-mixed track for Y=0.37. b: simulation including Xenv−in up to
0.21, for the listed percentages of stars. c: simulation including also stars with Xenv−in =0.41. In
the latter simulation, the main aim is to show that the most hydrogen rich stars are located on the
extreme horizontal branch, and not on the blue hook.
39

40

Extended Data Figure 5 The dynamical model. Two different cases of dynamical interactions
are compared: either we make the hypothesis that the magnetic coupling of the disc is destroyed by
one encounter with another star, at a distance entering the accretion disc (<100a.u), or that three
of such encounters are necessary. a: histogram of encounters versus time of detachment. b: initial
rotation period distribution employed in the two cases; c: distribution of the core mass increase
which enters in the photometric simulation.
41