The Origin of the Two Populations of Blue Stragglers In

arXiv:1811.00058v1 [astro-ph.SR] 31 Oct 2018 lesrglri h rdc famre ewe two mass between (with merger star a single of a a into that product merged hypothesis the that the is stars adopt straggler We blue simulations. binaries merger contact lar UMa it case W which (2017). of in al. composed 2015), et Jiang al. mainly et attributed be (Xin been of transfer should has collapse mass %) (60 core binary ago population the to red by Gyr The triggered 1-3 con- cluster. burst only the (2009) short formed relatively al. population a et blue Ferraro in the cluster. that the jecture of radius half-mass all and glers oemmn ntime in moment some sbihe yaot07 a.Bt ouain r cen- are populations of that Both (90%) population mag. majority The (red) 0.75 call concentrated. second trally about (2009) by a al. brighter and the et is Ferraro population) along (which blue positioned the sequence is populations that main distinct population than zero-age one in Hertzsprung-Russe brighter two the but are in into 1953) diagram, point populations turn-off cluster split Both (Sandage current the 2009). be stragglers al. et to blue (Ferraro appear of M30 population The Introduction 1. rtclprpciew eemn h oiino blue evolving a by of diagram position Hertzsprung-Russell the the the- in determine a straggler we From straggler’s diagram. perspective blue Hertzsprung-Russell oretical the the with in consistent position is a that in product our result merger in both distinction as a A scenarios, make straggler. type two blue not single of these do phase between we unstable 1992; 2009), models (Bailyn an al. transfer et from mass Knigge 1967) or Weigert evolution & cluster dynamical (Kippenhahn the star during the collision of direct a from sult edopitrqet to requests offprint Send srnm Astrophysics & Astronomy oebr2 2018 2, November h rgno h w ouain fbu tagesi M30 in stragglers blue of populations two the of origin The ets hs yohssb odcigasre fstel- of series a conducting by hypotheses these test We h oeto egri eemndb niga finding by determined is merger of moment The hog iayeouin nta aeaothl h binari the half for words. about constantly Key case the that whereas In Gyr, 9.8 evolution. about binary of through age an at cluster tages lg ln h eoaemi-eunewherea main-sequence zero-age t the by brighter along are stragg stragglers align blue blue stragglers) of of populations populations two Both the diagram. of position the analyze We Accepted / Received 1 ∼ bu 0 ftebu tagesi 0 sfre taconstan a at formed is 30, M in stragglers blue the of 40% about e Gyr per ednOsraoy ednUiest,P o 53 30RA 2300 9513, Box PO University, Leiden Observatory, Leiden 10 ∼ y.Tebu ouaini omdi us htstarted that burst a in formed is population blue The Gyr. 0 . 75 − red 1 a.Bsdo tla vlto n egrsmltosw ar we simulations merger and evolution stellar on Based mag. iha -odn iesaeof scale time e-folding an with Sas)bu tages—(aay)goua lses ge clusters: globular (Galaxy:) — stragglers blue (Stars:) lesrglr r ihnteprojected the within are stragglers blue .PreisZwart Portegies S. : t mrg uhamre a ihrre- either can merger a Such . aucitn.bss_M30 no. manuscript blue 0 . 93 lestrag- blue y.W pclt httebrtrsle rmtecr colla core the from resulted burst the that speculate We Gyr. .PreisZwart Portegies S. M tot ABSTRACT at ) l a h lse’ uno,btoepplto tebu blue blue (the population one but turn-off, cluster’s the han h rd ouaini lvtdi rgtes(rcolour) (or brightness in elevated is population (red) the s si h lse ffcieyrsl nabu straggler. blue a in result effectively cluster the in es esi h lblrcutrM0i h Hertzsprung-Russell the in M30 cluster globular the in lers n h uno aso 0.85 of mass turn-off the and eaotdthe adopted We setup experimental The 2. birth. the at to stars insensitive two rather as the but well of as merger, masses product the is individual merger of diagram the moment of Hertzsprung-Russell the mass to the position total the in the Harris section, to straggler to sensitive following blue according the the a in Gyr (13 for of explain cluster we product As the merger 1996). of the evolve age to remaining continue and lation rmsm diinlms os o emt eutin al. result McMillan et & (Sills to Zwart Portegies product seem of (2018)). 5.3.3 merger not Chapter also the except see loss, 2001, in do, mass differences collisions in additional off-centre qualitative described a some because then such tentatively (1997) from but collisions, We al. et head-on automated. (2002), Sills to completely al. ourselves the et is limit to Lombardi 2013; comparable procedure in is our al. analysis et described Zwart Our Portegies method 2013). al. 2018; (AMUSE, et Pelupessy al. Environment et Zwart Software Portegies Multipurpose nomical M30. cluster the of age w tr oacranage certain a to stars two Pxo ta.21)t oe h vlto ftesaswith stars the of evolution the model [ to 2011) al. et (Paxton trrsligfo egrbtentosas fe this using After product stars. merger two the evolve between to merger continue th we a of from structure the resulting calculate star which to principle 2008), Archimedes’ al. uses et (Gaburov Make-Me-A-Massive-star ing ohsasaeiiilzda h eoaemi-eunean 2009 main-sequence zero-age al. to the et evolved at Carretta initialized are metallicity stars cluster’s Both the with sistent e ouaini h euto astase n mergers and transfer mass of result the is population med e/H F eiiilz rdo rmr assbten0.5 between masses primary of grid a initialize We Astro- the with realized is setup numerical The edn h Netherlands The Leiden, , aeof rate t ∼ = ] ea tr:eouin—Mtos numerical Methods: — evolution Stars: — neral 3 1 . 2 t − u htterdpplto,wihcomposes which population, red the that gue y g tapa aeof rate peak a at ago Gyr mrg 2 ∼ . tta oetw eg h w tr us- stars two the merge we moment that At . 33 2 . 8 wihacrigt hi a.1i con- is 1 Tab. their to according (which lesrglr e y vrtelast the over Gyr per stragglers blue MESA eyyselreouincode evolution stellar Henyey t mrg M efr h egrcalcu- merger the perform , ⊙ ril ubr ae1o 5 of 1 page number, Article nseso 0.05 of steps in 30 lestragglers blue s fthe of pse

c MESA S 2018 ESO M ⊙ - othe to and M ). ⊙ d e A&A proofs: manuscript no. bss_M30

Fig. 1. Hertzsprung-Russell diagram of the M30 blue stragglers. The original data is from Ferraro et al. (2015) was convoluted Fig. 2. Same as Fig. 1, except for the time since collision, which to the temperature-luminosity plane. With effective temperature is color coded here. and luminosity from Ferraro et al. (2009). The blue and red blue stragglers are indicated as such. We fitted both distributions with a constant blue strag- gler formation rate combined with a burst and exponen- secondary masses between 0.2 M⊙ with the same upper tial decay. The best fits are obtained using the Nelder- limit in steps of 0.005 M⊙. The merger time is chosen be- Mead simplex optimization (Nelder & Mead 1965) to find tween 0.1 Gyr and the age of the cluster with steps of the minimum Kolmogorov–Smirnov (KS) statistic over the 0.98 Gyr. The evolutionary state of the merger product at free parameters tmrg, and e-folding time scale τ, in combi- any time after the collision is predominantly determined by nation with a line describing the constant formation rate. the total mass of the merger product M . Small variations tot The best fit (with KS statistics D = 0.10, p =0.24) to in the mass lost during the collision therefore have little ef- the blue blue stragglers is obtained for t =9.8 Gyr, τ = fect on our determination of the merger time, because the mrg 0.93 Gyr with a peak formation rate of 30 blue stragglers per location in the Hertzsprung-Russell diagram then depends Gyr and an additional constant formation rate of 1.8 ± 0.6 on the total mass of the merger product and the moment per Gyr. of collision, rather than on the masses of the two stars that participate in the merger. Fitting the red blue straggler formation rate with the In Appendix A we present the AMUSE script to reproduce same set of functions (a constant rate plus a power-law) did the calculations in this paper. not result in a satisfactory fit, but a single linear formation rate did produce the KS statistic of D = 0.19 (p = 0.23) with a constant formation rate of only 2.8 ± 0.5 per Gyr 3. Results between an age of 3Gyr to 10Gyr. It is interesting to note that the formation rate for the red population levels off The Hertzsprung-Russell diagram of the blue stragglers is when the blue population reaches its maximum rate. presented in Fig. 1. Overplotted in colour, is the total mass of the merger products that remain on the main-sequence until an age of 13 Gyr. Information about the masses of the two stars is largely 4. Interpretation lost in the merger process, and can hardly be used for diag- nostics (see also Lombardi et al. 2002). We, therefore, use The majority of blue stragglers in star clusters are the total blue straggler mass and the merger time as a di- thought to originate from either stellar collisions (Leonard agnostic tool. 1989) or from mass transfer in a close binary system In fig. 2 we present the same data as in fig. 1, but now (Collier & Jenkins 1984). We will argue here that the two overplotted in color is the time since collision. The light- distinct populations found in M30 can be attributed to est shades indicate the most recent mergers. The blue blue these different formation channels (see also Ferraro et al. stragglers tend to cluster around a time since collision be- (2009)). We argue that the red population is consistent with tween 1 Gyr and 3 Gyr ago (in light-green), whereas the red being formed continuously and through mass transfer and blue stragglers span a much wider range of merger times. mergers in binary systems, whereas the blue population is We quantify this statement in fig. 3, where we present the mainly the result of collisions during the core collapse of cumulative distribution of merger times for the blue and the star cluster. In that perspective, we attribute the burst red blue-stragglers together (colours) and separately (solid population to the collision scenario, whereas the continu- curves). ously formed population is the result of binary evolution.

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4.2. The continuously formed blue stragglers

Mass transfer in binary systems are less likely to depend strongly on the cluster core density because binaries are present in the halo as well as in the cluster centre, which causes them to be more homogeneously distributed across the cluster (Hut et al. 1992), whereas direct stellar colli- sions are predominantly occurring at the very centre of the cluster Portegies Zwart et al. (1997b). The binary merger rate is also not expected to be particularly affected by the cluster density profile. We, therefore, argue that the con- stant rate is a result of binary mass transfer and coales- cence.

We can constrain the underlying binary semi-major axis distribution and mass ratio distribution that produces a constant blue straggler formation rate (or a constant rate of binaries that engage in a phase of mass transfer). Mass transfer in a binary system is typically initiated by the pri- mary star, which overfills its Roche lobe when it either as- cend the giant branch or, for very tight binaries, along the main sequence. Since the timescale between the terminal- Fig. 3. The cumulative distribution of merger times (tmrg) for all age main-sequence and the post-AGB phase is only a small < the blue stragglers (blue plus red as the colour shaded area where fraction ( ∼ 0.15) of the main-sequence lifetime, we adopt the color corresponds to that in Fig. 2). The solid blue and solid the main-sequence lifetime as the limiting factor between red curves give the cumulative distribution for the blue and red zero age and the start of Roche-lobe overflow. blue stragglers, respectively. The dashed and dotted blue curves give the fit to the blue blue stragglers (the dotted curve gives The lifetime of a main-sequence star scales as t ∝ the linear component and dashes give the sum of the exponential ms m2.5 Spitzer (1962). A primary mass distribution of f(m) ∝ and the linear fits). The red dashed curve gives the linear fit to −2.35 the red blue stragglers. The color coding is identical to that used m Salpeter (1955) then produces a roughly constant in Fig. 2, here indicating the time since the merger occurred. rate at which stars leave the main sequence, consistent with the observed constant rate of blue-straggler formation.

4.1. The burst population of blue stragglers M30 has a binary fraction of about 3% (Romani & Weinberg 1991; Milone et al. 2012), so with According to our analysis about one third (15) of the blue 1.6 · 105 stars the cluster has 4800 binaries. A standard stragglers in M30 are formed in a rather short burst that Salpeter mass function has about 5.8% of the stars between started at 9.8 Gyr with power-law decay with a character- 0.5 M⊙ and ∼ 0.85 M⊙. A 0.5 M⊙ star requires an equal istic time scale of 0.9 Gyr. At the peak the blue stragglers mass secondary star to evolve into a blue straggler in in the burst formed at a rate of about 30 blue stragglers an unstable phase of mass transfer, whereas a 0.85 M⊙ per Gyr. But due to the exponential, we adopted (and sat- > star only requires a companion with a mass of 0.1 M⊙ isfactorily fitted) this burst lasts only a short while, long ∼ (Portegies Zwart et al. 1997b). On average about half the enough to produce some 20 blue stragglers. binaries in the appropriate mass range then produce blue We estimate the expected formation rate through stellar stragglers, totalling a potential number of 280. Because mergers during core collapse. This is realized by calculating the mass-ratio distribution in cluster binaries tends to the collision rate be flat Kouwenhoven et al. (2007), roughly half of these Γmrg = nσv. (1) binaries have a total mass that upon a merger results in a blue straggler. The binaries with small mass ratio do not Here n is the stellar number density, v the velocity dis- form as a blue straggler directly upon the merger because persion, and the approximate gravitational-focused cross- the total mass of the merger product does not exceed the section σ is turn-off mass, but these stars pop-up later when their σ = rν2. (2) rejuvenation causes them to stay behind in their evolution (Portegies Zwart et al. 1997a). The orbital separations of Here ν ≡ v/v∞ is the stellar velocity dispersion as fraction primordial binaries range from a few R⊙ and a maximum of the stellar escape speed (Binney & Tremaine 1987). of ∼ 104 AU at the Heggie (1975) limit for hard-soft Davies et al. (2004) derived a formation rate of blue binaries. Roche-lobe overflow on the main-sequence is stragglers for a star cluster through direct stellar collisions, most favourable for the formation of blue straggler. This using the above arguments. We can adopt their eq. 4 to process is effective for binaries with an orbital separation of < calculate the expected number of blue stragglers formed ∼ 10 R⊙. With a flat distribution in the logarithm of the through collisions. By adopting the current observed clus- semi-major axis (Zinnecker et al. 2004; Kouwenhoven et al. ter parameters (n ≃ 3.8 · 105pc−3, N = 1.6 · 105 stars, 2007), only about one in four binaries will be effectively rcore ≃ 0.2 pc and adopting a mean stellar mass of 0.5 M⊙ producing a blue straggler (Chen & Han 2009). The from Harris (1996)) we arrive at the current average blue- entire binary reservoir then produces ∼ 35 blue stragglers straggler production-rate through collisions of 20 Gyr−1. through mass transfer or coalescence.

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5. Discussion import numpy from amuse.lab import ∗ The 13 Gyr–old globular cluster M30 has a rich population of blue stragglers, which appear to be distributed bimodally def merge_stars(Mprim=0.8|units .MSun, in the Hertzsprung-Russell diagram (Ferraro et al. 2009). Msec=0.6|units .MSun, We tested the hypothesis that all these blue stragglers are t_coll=11|units .Gyr): the result of a merger due to an unstable phase of mass transfer in a binary system or a stellar collision. We simulate code = MESA( ) the current population of blue stragglers that could have p = Particles (mass=Mprim) resulted from the coalescence of two stars at some time primary = code. particles .add_particle(p) tmrg with a total mass of mtot. The stellar merger product s = Particles(mass=Msec) was subsequently evolved to the current age of the cluster of secondary = code. particles .add_particle(s) 13 Gyr. For each point in the Hertzsprung-Russell diagram code.evolve_model(t_coll) we then obtain a unique solution for the mass of the blue stars = code.particles straggler and the moment of merger tmrg. The two masses print "Pre merger:\n", stars of the stars that merge are not well discriminated in the results, because the memory of the two stellar masses is n_zones = stars .get_number_of_zones() lost in the merger process due to the mixing in the merger code. merge_colliding(primary.copy() , process (Benz & Hills 1987, and much later literature). secondary.copy() , The merger time-distribution for the blue blue strag- MakeMeAMassiveStar , glers is best described by a peak of formation of ∼ 30 blue dict(), stragglers per Gyr at tmrg ≃ 9.8 Gyr and an e-folding time dict (target_n_shells_mixing=max(n_zones)) , scale of 0.93 Gyr superposed with an additional constant return_merge_products=["se "]) formation rate of 1.8 per Gyr between tmrg ≃ 8 Gyr and code.evolve_model(13| units .Gyr) the age of the cluster. This is consistent with the conjec- print "Post merger stellar parameters:",\ ture by Ferraro et al. (2009) that these blue stragglers ware "T=", stars .temperature ,\ born in a burst during the core-collapse phase of the cluster "L=", stars.luminosity.in_(units .LSun) some 2–3Gyr ago. The population of red blue stragglers is − best described with a constant formation rate of 2.8Gyr 1 code.stop() between an age of tmrg ≃ 3 Gyr and 11Gyr (between 11 and 2 Gyr ago). About 10% of the blue and red blue stragglers ap- pear to be missed in the observational data. We interpret Acknowledgments this bimodality of blue stragglers with two distinct chan- nels through which they form, much in the same way as It is a great pleasure to thank Alex Rimoldi for the pre- Ferraro et al. (2009) argued based on the observations. The liminary analysis and making the figures for this paper and continuously formed population is consistent with originat- Tjibaria Pijloo for reconstructing the cluster’s moment of ing from mass transfer in primordial binaries. In that case, core collapse. This work was in part done at the Canadian about 10-15% (∼ 35/280) of any binary leads to the forma- Institute for Theoretical Astronomy and I am grateful for tion of a blue straggler at a constant rate. The burst with their support, in particular to Norm Murray who made this an exponential decay in the formation of blue stragglers is possible. the result of direct stellar mergers during the core collapse of the star cluster. We attribute the start of the blue straggler formation Appendix: minimal AMUSE script for the runs burst to the moment of core collapse in the star cluster, at an age of ∼ 9.8 Gyr. This is consistent with the inverse In the listing we present an AMUSE cluster-evolution analysis by (Pijloo et al. 2015, for the de- (Portegies Zwart & McMillan 2018; Portegies Zwart et al. tails of the analysis, but the results adopted here were pre- 2018) script to calculate the evolution of two stars that sented at the IAU conference in 2015), which leads to a core underwent a merger, and that is continued to evolve to collapse at 9.5 ± 0.4 Gyr and which is consistent with the some late time. This script is tuned for M30, to limit start of the blue-straggler formation burst. the number of input parameters. The script starts by We conclude that the core collapse of the cluster was initializing the MESA stellar evolution code (Paxton et al. associated with a burst in the formation of blue strag- 2010, 2011), declare the two stars and submit them to the glers. The exponential decay is a result of the relatively stellar evolution code. In the subsequent block of lines, extended period during which the cluster remains in a col- the stars are evolved to an age of 9 Gyr, and the resulting lapsed –or post-collapsed– state following the primary col- stellar models are merged using MakeMeAMassiveStar lapse (Heggie & Ramamani 1989). This could indicate a (Gaburov et al. 2008; Lombardi & Warren 2014). In the prolonged period of gravothermal oscillations following the last block of lines, the merger product is submitted again to primary collapse of the cluster core (Heggie et al. 1994). We the stellar evolution code and continued to evolve to an age argue that the post-collapsed phase lasted for about 1 Gyr. of 13 Gyr. For a more thorough explanation of the script we The current cluster has a relatively low density consistent refer to the AMUSE book (Portegies Zwart & McMillan with the late stages of post-collapse Bettwieser & Sugimoto 2018) Chapter 4 (List.4.7) or see amusecode.org for the (1984). complete source.

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