Rapidly rotating second-generation progenitors for the blue hook stars of ω Cen
Marco Tailo, Francesca D'Antona, Enrico Vesperini, Marcella Di Criscienzo, Paolo Ventura, Antonino Milone, Andrea Bellini, Aaron Dotter, Thibaut Decressin, Annibale D'Ercole, Vittoria Caloi, Roberto Capuzzo Dolcetta
RRapidly rotating second-generation progenitors for theblue hook stars of ω Cen
Marco Tailo , , Francesca D’Antona , Enrico Vesperini , Marcella Di Criscienzo , Paolo Ventura ,Antonino P. Milone , Andrea Bellini , Aaron Dotter , Thibaut Decressin , Annibale D’Ercole ,Vittoria Caloi , Roberto Capuzzo-Dolcetta INAF- Osservatorio Astronomico di Roma, I-00040 Monte Porzio (Roma), Italy. Dipartimento di Fisica, Universit´a degli Studi di Roma, La Sapienza, Roma, Italy. Department of Astronomy, Indiana University, Bloomington, IN (USA) Research School of Astronomy & Astrophysics, Australian National University, Canberra ACT2611, Australia Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA INAF- Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy INAF, 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 stellargalactic population, occurring either as field halo stars or grouped in globular clusters.The discovery of multiple populations in these clusters
1, 2 , that were previously believed tohave single populations gave rise to the currently accepted theory that the hottest horizontalbranch members (the blue hook stars, which had late helium-core flash ignition , followedby deep mixing
4, 5 ) are the progeny of a helium-rich “second generation” of stars
6, 7 . It is notknown why such a supposedly rare event
8, 9 (a late flash followed by mixing) is so common that a r X i v : . [ a s t r o - ph . S R ] J un he blue hook of ω Cen contains ∼
30% of horizontal branch stars , or why the blue hookluminosity range in this massive cluster cannot be reproduced by models. Here we reportthat the presence of helium core masses up to ∼ . Rotation may also accountfor frequent late-flash-mixing events in massive globular clusters. In the colourmagnitude diagrams of globular clusters, the horizontal branch is the locus ofcore-helium-burning structures that are the progeny of red giant stars. For each such structure, thehelium flash ignition has occurred either at the tip of the red giant branch, or anywhere along theevolutionary path that moves the star into the white dwarf stage . 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 (T eff ) in the model in the HertzsprungRusselldiagram (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 heliumcore to mix with the small hydrogen-rich envelope
5, 12, 13 (late-flash-mixing), resulting in a slightlysmaller helium-burning core plus a helium-rich, or a helium-dominated, envelope. Such “bluehook” (BHk) structures attain a lower luminosity, and a higher T eff than do stars in the extremehottest horizontal branch standard locus, owing to the smaller opacity of the helium-rich atmo-2phere; they have been found in the ω Cen and in a few other massive clusters .The blue hook in ω Cen is particularly strikingis particularly striking because it containsabout 30% of the horizontal branch stars , and it is extremely well defined in the Hubble SpaceTelescope (HST) observations. The optical colour-magnitude diagram displays a blue hook thatextends 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 data (Fig. 1a) displays a strongpeak on the blue side of the colour distribution, where the helium-richer late-flashmixed stars arelocated
16, 17 .The models we adopt (see Methods) assume the helium core masses implied by the evolutionaryprocess, and different initial envelope hydrogen abundances (X env − in ), together with an efficiencyof helium settling calibrated to reproduce as well as possible the values of He/H versus T eff datafrom the literature (Fig. 2). Tracks (that is, evolutionary sequences) with very low X env − in valuesspan only about 0.4 mag in F225W, but an extra 0.35 mag or so are obtained in models startingwith larger X env − in , as a result of the change in atmospheric composition from helium-dominatedto hydrogen-rich (Extended Data Fig. 2 and Extended Data Table 1). Considering the whole rangeof X env − in values explored, from 0.007 to 0.41, we can obtain a global extension of 0.9 mag in theF225W band (6th column of Extended Data Table 1), close to the range observed, but the trackswith X env − in > . do not match the colour-magnitude diagram well (Extended Data Fig. 4) northe He/H versus T eff data (Fig. 2), and tend to merge with the extreme horizontal branch. ForX env − in ≤ env − 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) standardflash-mixed models occur in a small core mass range ( δ M core < . M (cid:12) )
8, 9 , corresponding toa negligible magnitude difference in the tracks extension ; (2) the magnitude range can not beextended by increasing the adopted core overshooting parameters. 3) we also have good reasonto reject the possibility that the blue hook contains both the progeny of the standard and of thehelium-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 bytracks having larger helium core masses M c , as we show in Fig. 1a and Extended Data Fig. 1afor the track having δ M c =+0.04 M (cid:12) , and Y=0.37, where M (cid:12) is the solar mass. The helium flashignites 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 approximation of models startingfrom solid-body rotation on the main sequence, and preserving angular-momentum in shells, to es-timate the increase in the helium-flash core mass, δ M c , as a function of the initial angular velocity ω . These models provide upper limits to δ M c ,because they do not allow for angular momentumtransport 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 the4quator becomes equal to the gravitational force, which is about × − s − ) provides a huge δ M c =0.06 M (cid:12) , leaving room for different M c ( ω ) laws when available.We propose that high rotation rates may be a consequence of the star-formation history and earlydynamics 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 suchejecta in the central regions of the first-generation cluster and produces a centrally concentratedsecond-generation subcluster
11, 19 . Observations showing that the helium-rich blue main-sequencestarsthe progenitors of the blue hook starsare spatially more concentrated than the rest of the main-sequence stars and retain some ’memory’ of their initial spatial segregation,provide evidence thatindeed such progenitors formed segregated in the innermost regions of the cluster, as predicted and assumed in this study.The contracting pre-main-sequence low-mass objects in the galactic disc (that is, stars likethe variable star T Tauri)rotate with periods in the range from 2 days to 12 days. The rotationrates of classical T Tauri stars do not increase with stellar age, because of magnetic disk-lockingbetween the star and the disk , which keeps the stellar rotation constant until the disk is lost. Starsin young clusters (such as α Persei) show a wide distribution of rotation velocities, owing to differ-ences in the time at which different objects break their magnetic coupling with the wind . Rapidlyrotating stars result from an early breaking of the of the coupling. Afterwards, during the mainsequence, stellar winds slow down the rotation of the convective stellar envelopes, and old stars allappear to rotate slowly, but the inner core rotation is still fast despite the slow angular momentum5ransfer from the core to the envelope. We note that this model implies the presence of a crowdedsecond generation.Figure 3a shows the fast decrease of the momentum of inertia of first- and second-generation starsin globular clusters (the latter ones have no initial deuterium, because the gas from which theyformed was nuclearly processed at high temperature ). Figure 3b describes the stellar angularvelocity which can be obtained at the main sequence (age exceeding × yr), as a function ofthe time at which the disk-star coupling is destroyed and evolution proceeds at constant angularmomentum. The moment of inertia evolution of the case of a mass of M=0.7 M (cid:12) , Y=0.35 is as-sumed,for rotation periods at detachment taken in the range 2-12 days. We see that there is amplemodel space to reach high angular velocities, provided that the second-generation (the progenitorsof the blue hook stars) lose their disc at young ages, from 10 yr (for the longer initial periods) toabout × yr (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 δ M c resulting from the timings of encounters as directinputs 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 envelopesduring the last phases of evolution, enhancing the chemical anomalies of the most extreme secondgeneration
23, 24 and probably also favouring mixing at the onset of the flash: the late-flash-mixingevent, although very different from what is described in one-dimensional models, will be moreprobable, thanks to the reduced entropy barrier between the core and the envelope, justifying theexistence of the blue hook itself. In this scheme, the non-mixed extreme horizontal branch stars6hould 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 ispopulated by the progeny of scarcely rotating (rotation rate ω < − s − ) stars (red squares inFig. 1 and Extended Data Figs 1, 3 and 4). M c for this side of the simulation is the mass of thenon-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 ω Cen. Two conditions are at the basis of the success of thedynamical model used here: first, we must deal with second-generation stars, which form in acooling 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 thatof ω Cen , and NGC 2419 also has an extended blue hook . NGC 6388 and NGC 6441 havepeculiarly extended horizontal branches, unlike other metal-rich clusters. It is certainly possiblethat their thick red horizontal branch (the red clump), modelled by assuming a large helium spreadin the second generation , is also caused, in part, by the presence of larger core masses due tofast rotation. The reproduction of the horizontal branch morphology would then require, in thesecond generation, a smaller helium content increase than predicted by standard models , and thiswould be more consistent with the helium spread derived by the main-sequence colour thicknessin NGC 6441 . We note that a large rotation spread in this case does not produce a prominent bluehook, 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.7maller but important rotation ranges may be present in other less-massive clusters as well, andmay have been due to the same process of early disk destruction in second-generation stars: inmany clusters the mass loss necessary to explain the location of second-generation stars has to beslightly larger than in the first generation, possibly because of their rotation rate
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A.P.M. acknowledges support by the Australian Research Council throughDiscovery Early Career Researcher Award DE150101816. E.V. acknowledges support from grantNASANNX13AF45G. 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 DynamicalEvolution 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 ofAstronomy 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 colourswith an internally consistent treatment of extinction across all bandpasses. M.T. and M.D.C. performedthe simulations and the analysis. A.B. performed the data reduction and calibration for the WFC3/UVISexposures. A.M. dealt with the optical data and the comparison with spectroscopic data. T.D. dealt withthe problems connected to the modelling of stellar rotation. V.C. contributed to the discussion and tothe 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 FrancescaD’Antona (email: [email protected] or [email protected]). IGURES igure 1 - Simulations for the ultraviolet data. Grey dots are the observeds tars in ω Cen. Tracks plotted on the blue hook in a are: for M = 0 . M (cid:12) and Y=0.37 (black); for 0.489 M (cid:12) , Y=0.25, standard M c (green); and forM c increased by 0.04 M (cid:12) (magenta). Models have X env − in =0.06. Also shown are the zero-agehorizontal branch for Y=0.37 (dashed balck) and the 0.471 M (cid:12) track (dotted black). Labelsindicate bolometric distance modulus ( m − M ) and reddening E ( B − V ) adopted for thecomparison. b , Simulation with a late-flash mixture of X env − in stars for Y = 0 . (see textfor the details of the chemistry labels at the top). Comparison with data are shown in thehistograms on the left (blue hook, blue triangles) and right (extreme horizontal branch, redsquares) side. On the histograms, σ Poisson errors are shown with respect to the counts N. c ,Simulation resulting from coupling dynamics and evolution (same mixture of X env − in ). Colourcode and comparisons as in b . 14 igure 2 - Calibration of H–He diffusion eff data with error bars. The evolution of three modelsis shown for different values of X env − in . Diffusion is computed assuming that the envelopemass below which we assume diffusion is operating, M turb , is 10 − M (cid:12) . The orange tracksshow the evolution of the smallest non mixed tracks. Triangles mark each 10 yr interval, forM turb =10 − . The red dots represent 26 stars randomly extracted from the simulation in Fig. 1c.Along the magenta curve, no star with X env − in =0.21 are extracted, they would be cooler thanthe observed sample. 16 igure 3 - Rotational evolution from disc–detachment to the main sequence , Time evolution of moment of inertia (I norm , in units of momentum at t= × yr), fordifferent masses, values of Y and deuterium abundance X D . b , For M=0.7 M (cid:12) and Y=0.35, weshow the rotation rate ω at age t=2 × yr, as a function of the age at which the star detachesfrom the proto-stellar disk, and conservation of angular momentum begins. The lines representdifferent assumptions for the rotation period the disc is lost, from left to right: P in =12 days,10 days, 8 days, 6 days, 4 days, 3 days and 2 days. MethodsThe data sets.
We analyse optical and ultraviolet Hubble Space Telescope data for thehot horizontal branch stars of ω Cen. Photometric errors are about 0.015 mag in filters F435Wand F625W of ACS/WFC (Wide Field Channel of the Advanced Camera for Surveys), and0.010 mag in the filters F225W and F438W of the WFC3. In the comparisons with opticaldata, we adopt distance moduli and reddening that fit the luminous part of the horizontalbranch in these same data (M.T., PhD thesis, in preparation). The values correspond to E ( B − V ) =0.156 and ( m − M ) =13.63, when using extinction ratios A F435W /A V =1.362and A F625W /A V =0.868, explicitly computed for this work, and appropriate for stars ofT eff >
31, 32 , and A V =3.1 × E ( B − V ) . The F225Wand F438W data have been recalibrated. Distance modulus and reddening are chosen byadjustment of the simulations results, and are labelled in Fig. 1. Using A F225W /A V =2.670 andA F438W /A V =1.356, the bolometric distance modulus results to be ∼ , a feature ascribed to the effect of helium-diffusion in hydrogen-richenvelopes (the hottest standard horizontal branch stars). An almost abrupt transition tohydrogen-helium intermediate composition occurs at T eff (cid:39) K
16, 17 , where the sharppeak occurs. The contemporary presence of helium and carbon
7, 17, 33 points out that the starsat T eff (cid:38) , so they must be blue hook stars. Standardone-dimensional simulations of late-flash-mixing end up with very small hydrogen abundanceleft in the envelope (X ∼ × − , see ref. 13). A shallow mixing, leaving some percentageof hydrogen abundance, occurs only for a metal mass fraction that is ten times larger(Z ≥ ∼
320 and thecooler stars are ∼ Standard models.
Horizontal branch models are computed with the ATON code . Someuseful 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 byprevious evolution up to the helium flash, and the rest of the mass is in the hydrogen richenvelope. We also use as guidelines the results of some full late-flash evolutions, computed19ollowing the standard methodology
3, 35 of evolving the mass corresponding to an age of12 billion years (Gyr) along the red giant branch, for increasing wind mass loss rates. Ourresults substantially confirm the previous findings
5, 8, 13 . The mass range of late flashers can beestimated in ∼ M (cid:12) , and our range of late flash mixed models is (cid:46) M (cid:12) (the largest mass range found in the literature is (cid:46) M (cid:12) ; ref.8). Models including sedimentation
Guided by the evolutionary results, we built up sets of mod-els, characterized by three main parameters: core mass M c , envelope mass M env , and initialhydrogen mass abundance in the envelope X env − in , and followed their evolution, starting fromthe zero age horizontal branch. As there are both helium-dominated and hydrogen-richer spec-tra among the BHk stars
17, 33 , models need to include helium diffusion, to correctly derive theultraviolet magnitudes, which are strongly dependent on the T eff of the models. An increasein the flux is expected, due to the shift in T eff that the star suffers when diffusion changesthe helium carbon dominated atmosphere into a hydrogen-rich one. The speed of diffusion, abyproduct of the residual turbulence in the outer envelope
36, 37 , must be calibrated to be com-patible with the observed values of hydrogen abundance in the blue hook stars
17, 33 . In themodels we calibrate the parameter M turb . Mass loss can play a concomitant role, and, whenincluded, M turb must be larger to remain consistent with the observations. We show our choiceamong 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 X env − in from 0.007 to 0.41 are plotted for two settings of diffusion efficiency. The F225W20xtension of the tracks increases from about 0.4 mag up to cover almost the full extensionof the blue hook(histogram in Extended Data Fig. 2a), by increasing X env − in up to the maxi-mum value. Extended Data Fig. 2c shows the time evolution of X env − in for the two choices ofM turb . A better representation of spectroscopic observations requires use of the models withslower 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, X env − in = 0.007 (10%), 0.03(45%) and 0.06 (45%). The T eff location and the He/H observed abundances are instead lesscompatible if we include a percentage of tracks having X env − in > Assumptions for the rotating models.
In the blue hook progeny of rotating stars the coremass at the He–flash, M c ( 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 rotationon the main sequence and angular-momentum conservation in shells , we derive a parabolicexpression for the helium-flash core-mass increase δ ( M core ) as a function of the angular ve-locity ω (in units of per second): δ ( M core /M (cid:12) ) = 3 . × · ω (1)A self-consistent approach needs computation of rotating models starting from the zero agemain sequence (where we can safely assume a rotation rate simply acquired by angularmomentum conservation) and including all possible mechanisms of transfer of angularmomentum from the core to the envelope, plus the loss of momentum of the envelope dueto magnetic wind. In the models available in the literature
38, 39 , the parameters necessary to21odel these mechanisms have been calibrated from the atmospheric abundance variationsinduced by the associated chemical mixing, but most of the sampled stars were slowly rotatingfrom the beginning, given that the fraction of young main-sequence stars which are fastrotating is quite small . The coexistence of a fast-rotating core and slow-rotating envelopewill lead to strong differential rotation that can be responsible for strong chemical mixingalong the red giant branch, compared to the mixing possible for initially slower-rotating stars,an outcome that may explain extreme abundance anomalies in some clusters
24, 41 , but it is notknown how much these events may break down the inner fast core rotation. So we adopt thesimple M c ( ω ) relation which does not take into account the core–envelope interactions.The most important parameter in our investigation is the main sequence lifetime of these fastrotating stars, which is longer than the lifetime of non-rotating stars of the same mass. If theage is fixed, the rotating evolving mass M ev ( ω ) will be larger than the non rotating mass. If afraction of the rotating stars evolves through the helium flash, otherwise there would be noblue hook, we have to assume that its mass satisfies the relation:M ev ( ω ) (cid:38) M c ( ω ) + ∆ M RGB ( ω ) where ∆ M RGB ( ω ) is the total mass lost. The flash luminosity in our models increases by δ L flash /δ M c (cid:39) . × (where L and M c in solar units). Assuming Reimers’ mass loss rate,we expect an extra mass loss of 0.010-0.015 M (cid:12) for each extra 0.01 M (cid:12) increase in the coremass. If for example we require δ M c =0.04 M (cid:12) , corresponding to ω (cid:39) . × − s − , theevolving mass must be ∼ M (cid:12) larger than the non-rotating mass, if it has to ignite the22elium-flash. If we assume a stronger dependence of mass loss on L than in Reimers’, we needlarger M ev ( ω ) . Such an increase is compatible with the scarce existing estimates but a largecomputational effort is needed to solve the problem. A spread in the evolving mass M ev ( ω ) isalso useful to meet, at least in a range of ω , the strict requirements of the late-flash-mixingconditions and let the stars populate the blue hook: those outside the correct range will evolveinto the helium-core white-dwarf remnants whose existence has been recently established .The core mass vs. luminosity relation shows that the tip of the RGB is extended by δ log L/L (cid:12) (cid:39) . (0.35 bolometric mag) for the core mass increase of ∼ M (cid:12) . If themodel 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 highY, 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 modelswith standard hydrogen abundance . A correction is necessary, as these spectra do not rep-resent accurately the peculiar atmospheres of late-flash-mixed stars , with strongly enhancedhelium and carbon abundances at the surface. At a given T eff , we correct the helium dominatedmodel magnitudes making them fainter by 0.057mag in the F225W band, an estimate based onthe comparison of fluxes in helium-and-carbon-rich and in hydrogen-rich model atmospheres .We keep this correction until the helium surface abundance remains above Y=0.9, then weswitch to the direct table correlations. This choice contributes to stretch in magnitude exten-23ion the tracks with 0.03 ≤ X env − in ≤ . . Standard Simulations
Synthetic models for the simulations shown in Fig. 1b followstandard guidelines . For the cases employing standard tracks, we assume that the red giantmass M RG at a fixed age is a function of helium, at assumed metallicity. The mass on thehorizontal branch is M HB =M RG (Y)- ∆ M, where ∆ M is the mass lost during the red giantphase. In our case, assuming Y=0.37 and 12 Gyr we derive M RG =0.658 M (cid:12) . The observedratio of stars between late flash mixed and “normal” extreme horizontal branch stars isreproduced by assuming that ∆ M has a Gaussian dispersion σ =0.008 around an average value ∆ M=0.19 M (cid:12) . With this choice, the resulting 60 extreme horizontal branch stars that didnot suffer mixing are shown in Fig. 1b as red squares, and the 130 blue triangles are the lateflash mixed stars. When we extract a mass smaller than the late flash mixed masses, the staris assumed to evolve into the helium-core white-dwarf. With the chosen parameters we find110 helium-core white dwarfs. Comparison is shown by histograms of counts as a function ofcolor and magnitude, given separately for extreme horizontal branch and blue hook stars. Theused bins are 0.03mag and 0.08mag for colours and magnitudes respectively. The error barsare 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 onthe top) distributions.This modelling shows the inherent difficulty in building the blue hook: the mass rangeof extreme horizontal branch models (defined as the models at T eff > M (cid:12) (from 0.03 to 0.07 M (cid:12) , larger for a larger Y), while late flash mixing24overs a mass range of only (cid:46) M (cid:12) . The most recent modelling of horizontal branchesof GCs shows that the mass loss spread of each component of multiple populations must bevery small, < . M (cid:12) . If the blue hook standard models are truly describing the blue hookstars in globular clusters, the mass lost by the red giant progenitor must be finely tuned, sothat it covers precisely this tiny mass range. If the blue hook is made up by more than oneof 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 thesimulations.First we assume that all the extreme horizontal branch stars have Y=0.37. We use this Yvalue because it is consistent with the requirements of the asymptotic giant branch model forthe formation of the extreme populations in clusters . We use different sets of flash mixedmodels, with X env − in =0.007, 0.03, 0.06, 0.21 and 0.41. The simulations with X env − in ≤ . are unsatisfactory, as they do not cover the whole BHk extension (Fig. 1b). Including afraction of stars with X env − in =0.21, the magnitude range is marginally better covered, but thecolour agreement is worse (Extended Data Fig. 4b). The magnitude range is not extendedsignificantly, because already the track having X env − in =0.06 reaches high atmospherichydrogen content along the evolution including hydrogen versus helium diffusion (ExtendedData Fig. 2c). Including even stars with X env − in =0.41, these latter merge with the extremehorizontal branch (Extended Data Fig. 4c).Extended Data Fig. 3 shows the simulation according to which the BHk is made up by25uperposition of two different late flash mixing populations, with Y=0.25 and Y=0.37,respectively . ∆ M’s and σ values used are reported in the figure. The mass losses necessaryto obtain masses in the late flash mixing range differ by ∼ M (cid:12) for the two different Ygroups. While it is already very difficult to accept that the mass-loss of a unique populationis so well constrained that the blue hook is populated at all, this double constraint is an evenmore extreme assumption. In spite of this, the whole extension of the hook is not yet fullycovered.
Coupled dynamical and evolutionary simulations
To estimate the distribution of rotationrates we use a simple semi-analytical model to calculate the distribution of times, T enc100 ,needed for second-generation star-disk systems to have a close encounter with another starat a distance d < astronomical units (AU) (the estimated dimension of the protostellardisk) and assume one - or more - such collisions to be able to break the magnetic disc-locking(in this case T enc100 is the detachment time). In order to calculate T enc100 we have followedthe orbits of 50,000 particles distributed as a King model with a concentration c (cid:39) . andintegrated the collision rate (see Eq. 7.194 in ref. 46) along the orbit to estimate T enc100 . Thesystem we are modelling is meant to represent the dense second-generation subcluster. For ourcalculation we assume its total mass to be equal to . M (cid:12) and half-mass radius of about oneparsec. From the distribution of T enc100 (Extended Data Fig. 5a) we directly derive the rotationrates assuming the conservation of angular momentum from Fig. 3, and the correspondingincrease in the core-mass from Eq. 1 in Methods (Extended Data Fig. 5c). The initial periods26 in at detachment from the disk are randomly extracted in the range 3 < P(days) <
12, using anormal distribution centered at P=6 days, with standard deviation σ =2 days. The distributionof the stellar rotation periods result of the simulation at detachment from the disc are given inExtended Data Fig. 5b. The distribution is not Gaussian, as it also depends on the limitationswe have imposed on the maximum δ M c .For the photometric simulation, we then assume the dispersion in core-masses inferred by thedescribed simulation (Extended Data Fig. 5c). We are implicitly assuming that the increase ininitial evolving mass due to rotation is enough to accommodate the increase both in core massand mass loss in such a way that the stars are able to arrive at late-flash-mixing ignition forrotation rates ω > − s − . We also impose that a tail of slower rotating stars populate theextreme horizontal branch of non–flash mixed Y=0.37 models. Code availability
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ED Tables env − in F W F W F225W δ F225W δ F225W ( F W − F W ) T eff ZAHB Peak TAHB (track) total ZAHB ZAHB . +2.29 +1.86 +1.97 0.43 0.43 -2.734 37660 . +2.21 +1.61 +1.69 0.60 0.68 -2.731 37200 . +2.19 +1.50 +1.56 0.69 0.79 -2.726 36700 . +2.10 +1.43 +1.46 0.67 0.83 -2.693 34500 . +2.03 +1.38 +1.39 0.65 0.90 -2.603 31600 ∆ M core = +0 . M (cid:12) . +1.96 +1.58 +1.76 0.38 0.71 -2.700 39400 . +1.93 +1.38 +1.71 0.55 0.91 -2.697 38900 . +1.91 +1.27 +1.35 0.64 1.02 -2.693 38400Extended Data Table 1 - Range in magnitude F225W for M=0.466 M (cid:12) , M core =0.463 M (cid:12) For each initial envelope starting H-abundance (col. 1), we give F225W band values of late flashmixed models at the zero age horizontal branch (col. 2), at the maximum luminosity (col. 3) andat the terminal point (col. 4) (defined as the model in which the core helium abundance becomeszero), and the magnitude interval spanned by each model (col. 5) and by a mixture (col. 6). Themass of the model is 0.466 M (cid:12) , with an envelope mass of 0.003 M (cid:12) , thus the model core mass is . M (cid:12) . 30 M Tip M HB min M core M LF M M core LF M ∆ MLF
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 (cid:12) , when the He–flashis ignited at the red giant branch tip; col. 5: mass for which we have late flash mixing; col. 6: coremass for the late flash mixed model; col 7: mass loss needed to achieve late flash mixing. ED Figures
Comparison simulation vs. optical data
The observed data are shown as grey dots. The diagonal line suggests the division betweenblue hook and cooler extreme horizontal branch stars. a : Tracks shown are for M=0.466 M (cid:12) ,Y=0.37 and evolutionary M c =0.463 M (cid:12) (black solid line), plus the corresponding track in whichM c is increased by 0.04 M (cid:12) (magenta solid line). b : Result of the coupled dynamical–evolutionarysimulation (detailed explanation in Fig. 1). 334xtended Data Figure 2 Models with He vs. H diffusiona and b show a comparison of blue hook tracks from X env − in =0.007 to X env − in =0.41, from left toright; M turb = − M (cid:12) (full lines) and − M (cid:12) (dashed lines). The bolometric luminosity ( b ) andthe F225W magnitude ( a ) are shown as function of T eff . The hot side of the zero age horizontalbranch for Y=0.37 is shown (solid black line with dots). c : Evolution with time of the surfaceN(He)/N(H) due to diffusion. The solar ratio is shown as a horizontal grey dotted line. Theobservations of blue hook stars show that the slower diffusion is a better description (Fig. 2).356xtended 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 simulationassuming 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 onesigma Poisson error on the stellar counts. 378xtended Data Figure 4
Tracks and simulation vs. data including models having X env − in > As in Fig. 1. a : tracks with different X env − in and the end of the zero age horizontal branch withthe location of the minimum non-mixed track for Y=0.37. b : simulation including X env − in up to0.21, for the listed percentages of stars. c : simulation including also stars with X env − in =0.41. Inthe latter simulation, the main aim is to show that the most hydrogen rich stars are located on theextreme horizontal branch, and not on the blue hook.390xtended Data Figure 5 The dynamical model.
Two different cases of dynamical interactionsare compared: either we make the hypothesis that the magnetic coupling of the disc is destroyed byone encounter with another star, at a distance entering the accretion disc ( < a : histogram of encounters versus time of detachment. b : initialrotation period distribution employed in the two cases; cc