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+MNRAS 000, 1–29 (2021)
+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Accretion of sub-stellar companions as the origin of chemical
+abundance inhomogeneities in globular clusters
+Andrew J. Winter★1,2 and Cathie J. Clarke3
+1Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, 06300 Nice, France
+2Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
+Accepted 2023 January 25. Received 2023 January 08; in original form 2022 August 05
+ABSTRACT
+Globular clusters exhibit abundance variations, defining ‘multiple populations’, which have
+prompted a protracted search for their origin. Properties requiring explanation include: the
+high fraction of polluted stars (∼ 40−90 percent, correlated with cluster mass), the absence
+of pollution in young clusters and the lower pollution rate with binarity and distance from
+the cluster centre. We present a novel mechanism for late delivery of pollutants into stars via
+accretion of sub-stellar companions. In this scenario, stars move through a medium polluted
+with AGB and massive star ejecta, accreting material to produce companions with typical
+mass ratio 𝑞 ∼ 0.1. These companions undergo eccentricity excitation due to dynamical
+perturbations by passing stars, culminating in a merger with their host star. The accretion
+of the companion alters surface abundances via injected pollutant. Alongside other self-
+enrichment models, the companion accretion model can explain the dilution of pollutant and
+correlation with intra-cluster location. The model also explains the ubiquity and discreteness of
+the populations and correlations of enrichment rates with cluster mass, cluster age and stellar
+binarity. Abundance variations in some clusters can be broadly reproduced using AGB and
+massive binary ejecta abundances from the literature. In other clusters, some high companion
+mass ratios (𝑞 ≳ 1) are required. In these cases, the available mass budget necessitates a
+variable degree of mixing of the polluted material with the primary star, deviations from
+model ejecta abundances or mixing of internal burning products. We highlight the avenues of
+further investigation which are required to explore some of the key processes invoked in this
+model.
+Key words: globular clusters: general – binaries: general – stars: abundances – planets and
+satellites: formation, dynamical evolution and stability
+1
+INTRODUCTION
+Ubiquitous, discretised and complex structure in the abundance dis-
+tributions of stars in old, massive globular clusters have remained
+a challenging puzzle for many decades (e.g. Cohen 1978; Dick-
+ens et al. 1979; Kraft 1994; Gratton et al. 2004; Bastian & Lardo
+2018). While the specific abundance variations vary significantly
+from cluster to cluster, a common trend in the abundance varia-
+tions relative to the primordial stellar population is N enhancement,
+which combines with O and C depletion to yield approximately
+constant C+N+O abundances (e.g. Dickens et al. 1991). These vari-
+ations are also correlated or anti-correlated with He, Na, Al and
+Mg abundances (see review by Gratton et al. 2004). The occurrence
+of these discretised variations is correlated with numerous cluster
+and stellar properties, including (but not limited to) the cluster mass
+(e.g. Milone et al. 2017) and age (Martocchia et al. 2018a), and the
+binarity of a given star (e.g. D’Orazi et al. 2015).
+Despite numerous mechanisms put forward for the origin of
+★ andrew.winter@oca.eu
+multiple populations in globular clusters, to date all fail to explain
+numerous properties without violating other empirical constraints
+(for a review, see Bastian & Lardo 2018). One of the most stringent
+of these constraints is known as the mass budget problem (see
+Section 2.3), which is that the polluted population comprises up to
+𝑓II ∼ 90 percent of the total mass of the present day cluster. Many
+explanations for abundance variation hinge on the formation of a
+secondary population from the ejecta of massive or asymptotic giant
+branch (AGB) stars. However, the maximum mass available from
+these stars to form this second population is ≲ 10 percent of the
+accompanying mass of low mass stars, long-lived stars. This leads to
+a challenging problem in producing the observed population, which
+cannot be explained by many current models.
+In this work, we propose an origin for the abundance variations
+in globular clusters that helps solve this mass budget problem, as
+well as explaining the origin for many of the empirical correlations
+associated with multiple populations. This mechanism takes inspi-
+ration from recent results suggesting that sub-stellar companions to
+pulsating red giant stars are the origin of the long secondary period
+variability (Soszyński et al. 2021, see also Beck et al. 2014). Assum-
+© 2021 The Authors
+arXiv:2301.11412v1  [astro-ph.GA]  26 Jan 2023
+
+2
+Winter & Clarke
+ing this explains all such variability, such sub-stellar companions
+would also seem to exist in globular clusters (Percy & Gupta 2021).
+Here we make use of our recently developed analytic expressions
+describing how such companions evolve dynamically in dense envi-
+ronments (Winter et al. 2022a), in order to connect these apparently
+unrelated phenomena: sub-stellar companions and multiple stellar
+populations.
+In brief, our proposed scenario proceeds in the following way.
+Sub-stellar companions with mass ratio 𝑞 ∼ 0.1 form from AGB or
+massive star ejecta via tidal capture, disc sweep-up or Bondi-Hoyle-
+Lyttleton accretion. They are then subject to extreme eccentricity
+excitation resulting from many distant encounters in the dense core
+of the stellar cluster. Eventually, this eccentricity evolution leads to
+a collision with the host star. This collision induces deep (rotational)
+mixing, allowing pollution from the companion (and possibly inter-
+nal fusion products) to produce surface abundance variations in the
+primary.
+In the following manuscript, we explore this companion accre-
+tion model quantitatively and qualitatively in terms of the observa-
+tional constraints. In doing so, we demonstrate both the feasibility
+of the model and the advantages over many existing models. In Sec-
+tion 2 we discuss the observational requirements of any successful
+model for producing multiple populations in more detail. We explain
+the predictions and requirements of our model in Section 3, making
+comparisons with the existing observational constraints. We draw
+our conclusions in Section 4, outlining possible future directions
+that may further test the companion accretion model.
+2
+OBSERVATIONAL REQUIREMENTS
+2.1
+Discrete elemental abundance variations
+The defining feature of the multiple populations found in globular
+clusters is their elemental abundance variations. These variations
+are nearly ubiquitous in all massive and old clusters. In general, the
+polluted stars are characterised by enhancement of He, N and Na
+and depletion of O and C, although the specific variations are unique
+to each globular cluster. We do not review the specific variations in
+detail here, but refer the reader to the review by Bastian & Lardo
+(2018). However, we highlight that abundance spreads are limited to
+light elements, with little star-to-star Fe or heavy element variation.
+This suggests a unique chemical processing mechanism applies only
+in dense cluster environments.
+While the abundance variations in each cluster are unique,
+some prevailing correlations and anti-correlations appear to be com-
+mon. Enhancements in N are associated with depletion in C and O,
+such that the overall C+N+O abundance remains approximately
+constant (e.g. Dickens et al. 1991). Meanwhile, enhanced N abun-
+dance (depleted C, O) is positively correlated with Na (e.g. Sneden
+et al. 1992). A weak anti-correlation may also be apparent between
+enhanced Al and depleted Mg (see review by Gratton et al. 2004).
+Changes in the total He abundance Δ𝑌 are most reliably inferred
+from MS isochrone fitting (Cassisi et al. 2017). Enrichment of He
+exhibits a positive correlation with cluster mass, and typical spreads
+in abundance Δ𝑌 ∼ 0.1−0.2 from the pristine 𝑌 = 0.24 (e.g. King
+et al. 2012; Milone et al. 2015). Finally and significantly, Li deple-
+tion that is expected from the hot H burning that produces N, Na
+and Al is found in some instances (D’Orazi et al. 2015), but appears
+not to be universal (e.g. Mucciarelli et al. 2011). In clusters for
+which Li abundances are approximately constant through the MS,
+this would suggest that a large quantity of pristine material to dilute
+the pollutant is necessary.
+An interesting property of the chemical enrichment is its ap-
+parent bimodality (or multi-modality). Multi-modality in the abun-
+dance distribution is particularly seen in the C and N abundances,
+for which sub-populations are apparent almost ubiquitously (where
+errors on CN measurements are small enough – e.g. Norris 1987).
+Discrete sub-divisions can also be seen in high resolution observa-
+tions constraining abundances of O, Na and Al (e.g. Marino et al.
+2008; Lind et al. 2011). Thus, any pollution must proceed in a dis-
+crete way, which may be a problem for many proposed scenarios,
+such as early disc accretion (Bastian et al. 2013a).
+Another important property of the variations in the chemical
+abundances is that pollutants are mixed through the majority of
+the star. This can be inferred from the near constant abundances
+observed through the main sequence (MS), main sequence turn-off
+(MSTO) and red giant branch (RGB) phases (e.g. Briley et al. 2004).
+Such stars all have radically different convective zones, varying
+from ∼ 1 percent of the mass for MSTO stars to ∼ 70 percent of
+the mass for RGB stars (see Figure 1 of Briley et al. 2004). Thus,
+surface pollution is ruled out as the origin, and this has lead to the
+conclusion for many authors that the stars must form directly out of
+the polluted material as part of a second generation. Assuming that
+each of the populations represent different generations comes with
+its own problems, including the pervasive mass budget problem
+(Section 2.3).
+2.2
+Enrichment fraction and cluster properties
+The typical fraction of chemically enriched stars in globular clusters
+is 𝑓II ∼ 0.4−0.9, and is most robustly positively correlated with the
+total mass of the cluster (e.g. Milone et al. 2017). Meanwhile, the
+polluted star fraction 𝑓II is neither strongly correlated with galacto-
+centric distance nor metallicity (Bastian & Lardo 2015). The former
+strongly suggests that enrichment comes from an internal source,
+while the latter suggests that the fraction is not dependent on stellar
+evolution but rather on some dynamical process.
+Particularly problematic for most models, the presence of mul-
+tiple populations in clusters is also associated with age. The youngest
+globular cluster found to contain an enriched population is the
+1.9 ± 0.1 Gyr old NGC 1978 (Mucciarelli et al. 2007; Martoc-
+chia et al. 2018a; Saracino et al. 2020). By contrast, the globular
+cluster NGC 419 contains no evidence of an enriched population.
+The latter has stellar mass 𝑀c ≈ 8 · 104 𝑀⊙ (Song et al. 2021),
+half-mass radius 𝑅hm ≈ 8.1 pc (Glatt et al. 2009) and an age of
+𝑡 = 1.5 Gyr (Glatt et al. 2008). This is close to the 𝑀c ≈ 2×105 𝑀⊙
+and 𝑅hm = 8.7 pc of NGC 1978 (Milone et al. 2017, and references
+therein). Meanwhile, the age spread between populations within
+individual clusters is small (< 20 Myr, Martocchia et al. 2018b),
+indicating that the multiple populations initially form nearly simul-
+taneously. Combined with the absence of multiple populations in
+young clusters, this provides a difficult challenge for enrichment
+models. Since few young clusters are very massive and dense, one
+possible solution is that the absence of polluted stars at young ages
+is in fact related to a strict mass/radius requirement. In this instance,
+given the properties of the clusters in which multiple populations
+have not been found, then enrichment can only occur at all when
+𝑀c > 105 𝑀⊙ or 𝑀c/𝑅c > 104 𝑀⊙ pc−1. However, multiple popu-
+lations in older clusters with lower masses and larger radii than these
+thresholds have been found, such that one needs to appeal to much
+more massive initial conditions for these clusters. Alternatively, the
+above findings may be reconciled if the pristine population becomes
+the polluted population over Gyr timescales.
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+3
+2.3
+Mass budget problem
+Another severe constraint on any mechanism proposed to produce
+multiple populations is that it produces enough mass to make the
+second population. Since the mass of the second constitutes up
+to ∼ 90 percent of the total stellar population, this is a particularly
+difficult requirement for self-enrichment models invoking a standard
+initial mass function (e.g. Prantzos & Charbonnel 2006). The total
+mass of AGB stars is ∼ 8 percent of the initial stellar mass, while
+the mass in the low mass stars (𝑚∗ < 0.8 𝑀⊙) is ∼ 40 percent.
+Hence, if all of the AGB star mass (an extreme assumption) went
+into forming a second generation with a similar initial mass function
+IMF, then 93 percent of the total mass now should be primordial
+(or 83 percent if all second generation stars are low mass), while
+8 percent (17 percent) of the low mass IMF would have undergone
+processing. Even for globular clusters with a relatively large present
+day fraction of pristine stars 𝑓I ∼ 1/3, we would therefore need
+to increase the number of second generation stars by a factor ∼
+10 or decrease the number of first generation stars by 95 percent.
+Alternatively, a hard minimum of a 90 percent reduction in mass of
+the first generation stars corresponds to the very extreme assumption
+that 100 percent of the mass within AGB stars goes into low mass
+second generation stars.
+Some mitigation in this regard may be expected from the fact
+that dilution is needed for the observed chemical abundances (e.g.
+D’Ercole et al. 2010). To avoid too much iron from supernovae in
+the material forming the second generation, models have invoked
+the removal of the primordial gas by these supernovae (although
+long tails in Fe content are observed in some globular clusters –
+Carretta et al. 2010). However, then the diluting material must be
+re-accreted onto the globular cluster from the surrounding envi-
+ronment after 20−30 Myr (D’Ercole et al. 2016). This secondary
+accretion process is expected to be inefficient and requires a large
+reservoir of remaining gas (Conroy & Spergel 2011). Even in this
+case, the time difference between the two populations would be
+in tension with the apparently small age spreads between multiple
+populations (Martocchia et al. 2018b). The fresh material accreted
+from the surrounding galaxy also needs to match the abundances
+in the pristine medium; it is unclear that this would be consistent
+with the observed Fe spreads (e.g. Bailin & von Klar 2022). Fi-
+nally, and most importantly, the quantity of pristine gas must not
+greatly exceed the pollutant in order to produce the observed abun-
+dance variations (D’Ercole et al. 2011), so this dilution process in
+isolation does not solve the mass budget problem.
+We might also consider the ejection of first generation stars over
+the life-time of the cluster. The required depletion factor is 𝑓dep =
+𝑀c/𝑀c,0 ≲ 0.1 (more than 90 percent mass-loss). This might be
+increased to 𝑓dep ≈ 0.25 if we take the very extreme assumptions
+stated above (100 percent AGB stars to low mass second generation
+stars). This assumes that practically all of the second generation are
+retained (Conroy 2012; Cabrera-Ziri et al. 2015). Instead, 𝑓dep ∼ 0.5
+for typical present day globular clusters is expected theoretically
+(Kruijssen 2015) and from their current demographics (Webb &
+Leigh 2015). A tidally disrupted cluster is required in order to reach
+such small 𝑓dep (Vesperini et al. 2010). Even then, Larsen et al.
+(2012) showed that this is also inconsistent with the globular cluster
+stars in the Fornax Dwarf galaxy, which make up 20−25 percent of
+the total low metallicity stars including those in the field. This means
+that even if all of the field stars formed in globular clusters, these
+clusters could only have been a factor ≲ 4−5 more massive at their
+birth. Further, even if this rapid depletion could be responsible, it
+seems implausible that the degree of depletion would increase with
+increasing cluster mass. If anything we would expect the opposite
+trend given that the dispersal timescale for clusters evolving in a
+tidal field increases with cluster mass (e.g. Lamers et al. 2005).
+Therefore dynamical depletion alone is not sufficient to resolve the
+mass budget problem.
+2.4
+Correlations with binarity and intra-cluster location
+The pristine and enriched stellar populations often appear not to
+have the same spatial distribution within their host cluster. Most
+studies indicate a greater degree of central concentration in the
+enriched population than the pristine one (e.g. Bellini et al. 2009;
+Lardo et al. 2011; Simioni et al. 2016), although in some cases they
+appear to be well-mixed (e.g. Dalessandro et al. 2014; Vanderbeke
+et al. 2015; Miholics et al. 2015) or even less concentrated than the
+pristine population (e.g. Larsen et al. 2015). The pristine population
+exhibits an enhanced binary fraction with respect to the polluted
+population (D’Orazi et al. 2010; Lucatello et al. 2015).
+3
+MODEL
+3.1
+Overall picture
+In this section we present a novel model to explain the observed
+chemical enrichment for populations of stars in globular clusters.
+Our model is in some sense a combination of the ‘early disc ac-
+cretion’ scenario, put forward by Bastian et al. (2013a), and the
+second generation models (e.g. Decressin et al. 2007; D’Ercole
+et al. 2008). In the early disc accretion scenario, mass is added to
+the accretion disc as the young star-disc system moves through a
+the interstellar medium that is enriched by massive rapidly rotating
+stars and/or binaries. This polluted material is then accreted onto
+the host star, leading to inhomogeneous chemistry in the stellar pop-
+ulation. However, this scenario suffers a number of issues, perhaps
+most prominently that the accretion must occur extremely rapidly
+while the star is fully convective to explain abundance variations
+that remain constant across MS, MSTO and RGB (e.g. Briley et al.
+2004).
+The scenario we explore in this work is the ‘companion accre-
+tion model’. Instead of requiring that all the mass accreted onto the
+disc in the very early stages of cluster evolution, this mechanism
+invokes a later delivery of polluted material. The chronology of this
+scenario can be summarised as follows:
+(i) The first generation of stars forms, while material from the
+already formed AGB stars and massive binaries continues to be
+ejected. This material remains bound in the core of the cluster due
+to their low velocities (∼ 10−30 km s−1 – Loup et al. 1993).
+(ii) Primordial, non-polluted stars move through the interstellar
+medium that is polluted, as in the early disc accretion scenario
+(Bastian et al. 2013a). This material is accreted to produce discs
+around lower mass stars by tidal cloud capture, disc sweep-up or
+Bondi-Hoyle-Lyttleton-like accretion, but is not efficiently accreted
+onto the star. This process can occur simultaneously with (i) as long
+as the gas has not yet cooled to collapse and form new stars.
+(iii) This capture process can proceed more quickly than star
+formation, particularly if the gas reservoir is heated by the first
+generation of stars (Conroy & Spergel 2011).
+(iv) Companions form from the captured gas, and undergo ec-
+centricity excitation due to numerous hyperbolic stellar encounters
+that result in accretion of the companion (Hamers & Tremaine 2017;
+Winter et al. 2022a).
+MNRAS 000, 1–29 (2021)
+
+4
+Winter & Clarke
+(v) The merging of the two bodies results in angular momentum
+injection and mixing of the pollutant in the primary. This yields
+variations in stellar abundances.
+(vi) The primary returns to the main sequence on a thermal
+timescale (of order 10 Myr). If and when it becomes an RGB star, it
+still retains similar surface abundances due to the fact that material is
+already mixed through the convective zone by the previous accretion
+event.
+If feasible, this model immediately has several obvious advan-
+tages over the early accretion scenario. First, the polluted star no
+longer has to accrete a large quantity of mass via the disc while it
+is still fully convective, which would require extremely rapid and
+efficient accretion of material on timescales shorter than a typical
+disc life-time. Secondly, and related to the first point, this allows
+stars with longer main sequence life-times (such as AGB stars) to
+contribute to the accreted material. Finally, it predicts a dearth of
+polluted stars at young ages, in line with observations. We further
+explore the merits and shortcomings of this model as follows.
+3.2
+Origin of the contaminants
+The origin of the pollutant is important for quantifying the time and
+amount of polluted material that can be delivered to the interstellar
+medium. For the model we suggest, our primary interest is in pro-
+ducing the requisite contamination with a sufficiently low mass ratio
+𝑞 such that the contaminated companion is not easily detectable for
+the majority of stars (i.e. 𝑞 ≪ 1). Given that brown dwarf-mass
+companions to RGB stars appear to be common (Soszyński et al.
+2021; Percy & Gupta 2021), a mass ratio of 𝑞 ∼ 0.1 is a reasonable
+expectation. It is possible to achieve the required chemical signature
+using arbitrarily low mass ratios if the contamination is limited to
+the surface layers of a star. However, abundances of N change little
+along the MS and MSTO for polluted stars (Briley et al. 2004),
+and are comparable to the polluted RGB abundances (Briley 1997),
+suggesting that much of the convective zone of stars at the tip of
+the RGB in 47 Tuc (∼ 70 percent of the stellar mass) is initially
+contaminated (although see discussion in Section 3.2.5). For sim-
+plicity, we will assume a uniform mixing of the companion material
+through a fraction 𝑓mix of the outer stellar remnant of the merger.
+When observed, this results in an apparent pristine mass fraction
+𝑓pr at the stellar surface. The ratio of polluted to unpolluted mass
+in the well mixed portion of the star is thus 𝑞/ 𝑓mix, from which we
+can write:
+𝑞 = 𝑓mix(1 − 𝑓pr)
+𝑓pr
+.
+(1)
+Figure 1 shows this parameterisation schematically.
+In the following, we ask whether possible enrichment sources
+can yield observed elemental abundance variations when mixed at
+this ratio. There are numerous potential sources for enrichment (see
+review by Bastian & Lardo 2018), however we consider only a few
+relevant mechanisms here.
+3.2.1
+Rapidly rotating massive stars and binaries
+Rapidly rotating massive stars and binaries are an attractive way
+to pollute a star early during its evolution. Massive main sequence
+(MS) stars produce many of the observed chemical abundance corre-
+lations (Maeder & Meynet 2006). The problem in the first instance
+comes in allowing the fusion products to reach the surface. De-
+cressin et al. (2007) put forward mixing induced by rapid rotation
+Mass coordinate: 
+m(r)/m*
+fmix
+fpr =
+1
+1 + q/fmix
+Pristine 
+core
+fconv
+Convective zone
+fpr = 1
+Polluted 
+region 
+Figure 1. Stellar pollution schematic showing our simplified model for the
+pollution of the star as a result of a merger with a companion of mass ratio
+𝑞. The radial coordinate is the enclosed mass 𝑚(𝑟) as a function of radius 𝑟
+in the star, normalised by the total stellar mass 𝑚∗ of the remnant. In order
+to parameterise the concentration of the pollutant, we assume the mixing
+is uniform over a fraction 𝑓mix of the outer layers of the star. Generally,
+𝑓mix ≥ 𝑓conv, where 𝑓conv is the mass fraction of the convective zone.
+as a solution. The slow wind may then be ejected into the primordial
+gas (allowing the required dilution) to produce a new generation of
+stars. However, in-of-itself it is not clear how this scenario could
+produce discrete populations, Mg abundance changes or overcome
+the mass budget problem (see Section 2.3).
+A related mechanism that may produce similar chemical sig-
+natures is the interaction between massive stars in a close binary
+system (de Mink et al. 2013), which are common (Sana et al. 2012)
+and have short life-times that are comparable to that of the primor-
+dial circumstellar disc (e.g. Haisch et al. 2001). As an example of the
+abundances expected in the massive binary scenario, de Mink et al.
+(2009) computed the evolution of a 20 𝑀⊙ star in a close 15 𝑀⊙
+companion with an orbital period of 12 days. The authors show that
+the system sheds about 10 𝑀⊙ of material enriched in He, N, Na,
+and Al and depleted in C and O. The most extreme abundances for
+the ejecta the authors obtain are: 𝑌 = 0.64 for He (absolute abun-
+dance) and [𝑋] = 1.63, 1.44, 0.49, −1.07, −1.13 for Na, N, Al,
+C and O respectively. Here, [𝑋] = log 𝑋/𝑋i where 𝑋 is the mass
+fraction and 𝑋i is the initial mass fraction. The Mg abundance does
+not change significantly, indicating that variations in its abundances
+(and anti-correlation with Al – Carretta et al. 2012) must originate
+from another source.
+Here we are invoking late stage accretion, rather than early
+disc accretion (Bastian et al. 2013a), to produce the companions
+(see Section 3.2). We are therefore no longer limited to the massive
+binaries that rapidly eject material on timescales < 10 Myr, or even
+shorter timescales that enable the material to become convectively
+mixed in the pre-main sequence star (see discussion by Bastian &
+Lardo 2018). We are now able to revisit enrichment by other means
+over longer periods, such as via AGB winds that occur over several
+10 Myr timescales.
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+5
+3.2.2
+AGB stars
+When less massive stars (𝑚∗ ∼ 4−9 𝑀⊙) reach the AGB phase, their
+ejecta can also enrich the surrounding ISM. D’Ercole et al. (2008,
+see also D’Ercole et al. 2010) envisioned a picture in which the AGB
+ejecta sink to core of the young globular cluster, achieve high density
+and undergo collapse into a second generation of stars. Variations on
+this mechanism are among the oldest and most popular explanations
+for producing the observed enrichments (Cottrell & Da Costa 1981).
+One reason for this popularity is that AGB stars burn H at higher
+temperatures than main sequence massive stars (dependent on mass
+and metallicity Prantzos et al. 2007), allowing them to activate the
+Al-Mg burning chain, depleting Mg and increasing Al. This goes
+some way to explaining the shortcomings in the binary model above.
+For example, Ventura & D’Antona (2009) find an [Al/Fe] ≳ 1, for
+stellar masses 𝑚∗ ≳ 5 𝑀⊙, bringing down the quantity of material
+required to produce the most extreme observed Al enhancements to
+𝑞 ∼ 0.3.
+However, the composition of AGB ejecta is highly uncer-
+tain, and depends on the complex interplay between the secondary
+dredge-up (SDU), tertiary dredge-up (TDU) and hot bottom burning
+(HBB; e.g. Karakas & Lattanzio 2014). The ejecta must come from
+stars with mass 𝑚∗ ≳ 3.5 𝑀⊙ to conserve the C+N+O content (we
+discuss in Section 3.3 that the sweep-up time of contaminants into
+companions occurs over ∼ 200 Myr, such that the lower mass AGB
+stars do not contribute in t scenario). HBB may produce O deple-
+tion while TDU would increase the C+N+O on the surface, however
+the balance of these two processes alters the yields of other light
+elements such that it is challenging to reproduce all of the observed
+abundance variations. Although this might be solved by appealing
+to deviations in yields from theoretical predictions (Renzini 2013),
+the uncertainties in parameters required in AGB ejecta modelling
+make it challenging to produce quantitative predictions (e.g. Ven-
+tura & D’Antona 2005a,b). Nonetheless, AGB enrichment models
+require significant dilution of ejecta in pristine material, both for
+mass budget and abundance reasons. While the quantity of diluting
+material is poorly constrained, if all of the ∼ 8 percent of the total
+stellar mass in AGB stars is accreted into companions around low
+mass stars that make up 40 percent of the total IMF mass, then the
+typical mass ratio is ⟨𝑞⟩ ∼ 0.2. While the assumption of 100 per-
+cent accretion efficiency is extreme, such a mass ratio would be
+reasonable for a binary in the context of our model.
+Apart from abundances in ejecta, a number of issues are asso-
+ciated with this traditional AGB enrichment scenario. In the first in-
+stance, this scenario suffers from the requirement that the gas cools
+sufficiently to collapse, which is only possible after ≳ 100 Myr
+(Conroy & Spergel 2011). This delay is a severe problem for the
+AGB scenario, as the C+N+O abundance would not be conserved
+for AGB stars at this mass, contradicting observations (e.g. Dickens
+et al. 1991). This delay also leads to problems in retaining enough
+pristine gas in order for the cluster to have the necessary primordial
+gas for dilution, which is required for the observed chemical abun-
+dance correlations (for example, a lack of variation in Li abundances
+– e.g. Mucciarelli et al. 2011). Related to this, without dilution the
+amount of mass required to produce the second generation of stars
+with a mass of ∼ 0.4−0.9 times the mass of the initial population is
+far higher than the mass fraction that can be supplied by the AGB
+population, which is the mass budget problem that is common to
+many enrichment scenarios (see Section 2.3).
+In the context of the companion accretion model, none of the
+above issues are necessarily a problem. Rather than forming a new
+population from collapsing gas, we require only capture of the gas
+(Section 3.3). Thus the gas does not need to immediately cool to
+produce the next generation of stars. Dilution is naturally achieved
+by the mixing of the material in the polluted companion with the
+host star, without the need to appeal to a significant retention of
+primordial gas or re-accretion of fresh surrounding gas. The total
+quantity of mass needed is no longer determined by the total mass
+of the enriched stars, but the amount of mass needed to pollute
+a pristine star. The latter is dependent on the abundance yields
+from AGB stars that are highly uncertain, but given the wider time
+window available for pollution we can now appeal to material that
+originates in multiple types of progenitors (see also discussion in
+Section 3.3).
+3.2.3
+Internal mixing
+Early in the study of multiple populations in globular clusters, evo-
+lutionary mixing was suggested as the origin of the chemical in-
+homogeneities in RGB stars (Denisenkov & Denisenkova 1990).
+Stellar models for RGB stars with rotational mixing are well-known
+to be capable of reproducing abundance anomalies for C, N, O
+and Na (Langer et al. 1993; Charbonnel 1995; Denissenkov & Tout
+2000, and review by Salaris et al. 2002). However, this hypothesis
+was effectively dismissed with the finding that the MS and MSTO
+stars exhibit similar abundance patterns (Cannon et al. 1998; Bri-
+ley et al. 2004). Since MS stars close to the turn off have small
+convective zones (encompassing a mass fraction ∼ 10 percent for
+𝑚∗ ≈ 0.8 𝑀⊙), enrichment could not be the product of convective
+mixing of inner H burning products. In addition, while the CNO
+cycle occurs in low mass stars, and may result in enhancements in
+N and depletion of C and O, variations in Na, Al, and Mg cannot
+by produced. Their temperatures are too low to activate the NeNa-
+and MgAl-chains (e.g Prantzos et al. 2007, 2017).
+However, in the scenario we put forward, a significant fraction
+of the evolved MS star undergoes mixing as a result of collision
+with the companion (Section 3.5). Subsequently, the high angular
+momentum companion material induces rapid rotation inside the
+combined star, which may result in rotational mixing similar to
+those suggested to contribute to RGB surface abundances. Thus it
+is possible that mixing does indeed contribute to changes in surface
+abundances. During companion accretion, the presence of exter-
+nally produced elements (e.g. Mg, Na and Al) is naturally correlated
+with mixing of internal burning products, because a star needs to
+have merged with its companion to have produced the deep mixing
+and CNO variations. However, the heavier elements have their ori-
+gin in the AGB and/or massive star ejecta. Thus, a combination of
+deep mixing and external origins becomes plausible (Briley et al.
+2002). As commented by Briley et al. (2004) on internal dredge-up
+versus high mass star origins for abundance variations: ‘we find
+ourselves now facing a wealth of evidence that suggests not one
+origin or the other but rather both’. Indeed, the authors point to cor-
+relations between C depletion with decreasing absolute magnitude
+in several low-metallicity clusters (Bellman et al. 2001) and the C
+isotopic variation with stellar mass (Shetrone 2003) as strong evi-
+dence that deep dredge-up does contribute to abundance variations.
+The scenario we present in this work has the capacity to facilitate
+both internal mixing and massive/AGB star origins for abundance
+variations.
+3.2.4
+Absent supernovae ejecta
+The small dispersion in iron content in most globular clusters indi-
+cates that the fraction of mass ejected by core collapse supernovae
+MNRAS 000, 1–29 (2021)
+
+6
+Winter & Clarke
+that is retained in the cluster cannot generally exceed a few percent
+(Renzini 2013; Marino et al. 2019). This has led some authors to
+conclude that the pollution must originate either before or well af-
+ter the onset of these supernovae (Renzini et al. 2015). Scenarios
+in which the supernovae deplete the available gas reservoir while
+the second generation is forming may exacerbate the mass budget
+problem (see discussion by Renzini et al. 2022). However, the mass
+budget problem is less severe in the companion accretion model,
+while in a highly structured medium supernovae ejecta may follow
+low density channels to escape efficiently without removing all the
+existing gas (e.g. Rogers & Pittard 2013; Krause et al. 2013). In this
+work, we consider a scenario in Section 3.9 in which all massive
+binary ejecta are removed by core collapse supernovae, then AGB
+stars eject material afterwards.
+3.2.5
+Outlook for producing observed abundance variations
+We show the dilution tracks derived from the extreme abundances
+of the massive binary simulation of de Mink et al. (2009) and the
+AGB ejecta as adopted by D’Ercole et al. (2010) in Figure 2a. We
+have chosen pristine abundances [O/Fe] = 0.38, [Na/Fe] = 0.0,
+[Mg/Fe] = 0.35 and [Al/Fe] = 0.11 in order to compare with
+the sample of abundances for RGB stars as reported by Carretta
+et al. (2010), also shown in Figure 2. Dilution tracks start where the
+ejecta abundance lines converge, at which point the composition is
+100 percent pristine – i.e. the fraction of pollutant fraction 𝑓poll = 0.
+As the fraction of pristine material 𝑓pr = 1 − 𝑓poll decreases, the
+material becomes more concentrated until it is composed entirely
+of the ejecta material ( 𝑓poll = 1, 𝑓pr = 0). We show increments of
+0.1 in 𝑓poll ( 𝑓pr) as triangles in Figure 2a. We also show the pristine
+mass fraction 𝑓pr as a function of the effective mixing fraction 𝑓mix
+and companion mass ratio 𝑞 in Figure 2b.
+We see in Figure 2a that it is challenging to produce some of the
+lowest O abundances. These cases imply very high concentrations
+of the pollutant for the most extreme variations, requiring some
+combination of large 𝑞 or small 𝑓mix. We suggest that this may
+be somewhat mitigated by a combination of a number of possible
+factors, including:
+(i) A small number of companions with large 𝑞.
+(ii) Variable/inhomogeneous mixing of contaminants.
+(iii) Late collapse of the gas reservoir to form a (small) secondary
+population.
+(iv) Some surface enhancements of internal burning products
+via rotational mixing.
+(v) Uncertainties in the model ejecta abundances.
+For example, we indicate in Figure 2a how the first three points (i–
+iii) may shape the observed population of M54. The most extreme
+population can result from residual star formation, while a tail of
+highly polluted stars may result from a range of 𝑓mix and 𝑞 (see
+Figure 2b).
+In Figure 3 we show the distribution of the mass-ratios 𝑞 re-
+quired to produce the inferred pristine mass fractions, assuming
+fixed mixing fractions 𝑓mix. Here we exclude those stars composed
+entirely of pollutant ( 𝑓pr < 0.05), which we assume are the results
+of star formation directly from the ejecta. We include all remaining
+stars, although we highlight that within the companion accretion
+model we expect some fraction (∼ 20 percent) of companions to
+have been ionised rather than accreted. We therefore expect to some-
+what underestimate the median 𝑞 when including all of the RGB
+stars. With this caveat, the median 𝑞 varies between 0.03 and 0.2 for
+𝑓mix between 0.1 and 0.7. These values would be consistent with the
+available mass from massive and/or AGB stars, which contribute
+up to 25 percent and 20 percent of the mass of the low mass stellar
+population respectively. However, if 𝑓mix = 0.7, we also apparently
+require a tail of high mass ratio companions up to 𝑞 ∼ 4, which
+may be challenging to produce via the companion accretion model.
+We highlight that the exact 𝑓pr in the low 𝑓pr limit strongly depends
+on the AGB or massive binary model abundances, which remain
+uncertain. However, if the ejecta abundances are accurate, we may
+still produce the low 𝑓pr stars by adopting small 𝑓mix.
+Some empirical constraints limit the degree of variation in
+the mixing fraction 𝑓mix. Empirically, in 47 Tuc and M71 Briley
+et al. (2004) find similar nitrogen and carbon abundances along the
+MS and MSTO stars, which have relatively thin convective zones,
+spanning a range in mass fraction 𝑓conv ∼ 0.01 − 0.1. This would
+be consistent with any mixing fraction 𝑓mix ≳ max( 𝑓conv) ∼ 0.1. A
+more stringent constraint follows that the abundance variations are
+also approximately constant among polluted RGB stars, as surveyed
+by Briley (1997). In this sample, 𝑓conv varies up to ∼ 70 percent,
+suggesting that 𝑓mix is not less than an order of magnitude lower
+than this value across the sample. However, some variation of a
+factor few in 𝑓mix over this range is not ruled out (see discussion in
+Section 4.3 of Briley 1997).
+Residual star formation from the pollutant in some clusters
+also remains a viable way to produce the highly polluted popula-
+tion, as in previous models for self-enrichment (e.g. D’Ercole et al.
+2010). In this scenario, the second population would be present
+early in the cluster lifetime. This apparently contradicts the absence
+of multiple populations in young clusters (e.g. Martocchia et al.
+2019) and the low age spreads in slightly older clusters with mul-
+tiple populations (e.g. Martocchia et al. 2018b). However, such a
+mechanism would be contingent on retaining a sufficient quantity
+of gas at the time star formation is instigated. This may explain
+why clusters with large mass-radius ratio 𝑀c/𝑅c, such as M54 and
+NGC 2808 (𝑀c/𝑅c = 2.4×105 𝑀⊙ pc−1 and 3.4×105 𝑀⊙ pc−1 re-
+spectively) exhibit extreme abundance variations not seen in 47 Tuc
+(𝑀c/𝑅c = 1.3 × 105 𝑀⊙ pc−1, see discussion in Section 3.9). The
+young clusters surveyed for multiple populations are not so massive
+or dense as the most massive (old) globular clusters, particularly
+when factoring in dynamical mass-loss. If clusters less concen-
+trated than 47 Tuc cannot form a population composed purely of
+AGB ejecta then this scenario appears consistent with observations.
+However, this still requires that in some clusters that ∼ 20 percent of
+the present day cluster mass forms from pure ejecta, and the same in
+companions. If massive binary and AGB ejecta can both contribute
+to this budget, then a maximum of 45 percent of the present day
+low mass population can be reached. This is sufficient, and can be
+effectively enhanced by the concentration of the ejecta material into
+the core of the cluster, where it survives the subsequent dynamical
+ejection stars over Gyr timescales.
+There remain some further concerns in appealing to self-
+enrichment by massive progenitors. In particular, some authors
+predict that AGB stars do not give Na variations, which would
+rule them out as candidates for these variations (e.g. Doherty et al.
+2014). Predicted yields from all enrichment sources may struggle
+to reproduce observed variations when more than two elements are
+included (Bastian et al. 2015). For example, He abundance varia-
+tions are often over produced when sufficient material is injected
+to produce the observed O depletion and Na enhancement. We will
+discuss this issue more quantitatively the role of the pollutant in
+producing observed abundance variations in Section 3.9.
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+7
+°1.0
+°0.5
+0.0
+0.5
+[O/Fe]
+°0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+[Na/Fe]
+4.0 MØ
+5.0 MØ
+6.0 MØ
+7.0 MØ
+9.0 MØ
+Massive binary
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pristine mass fraction: fpr
+x
+Residual star formation
+x
+Primordial
+(a)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mass-ratio of the companion: q
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Mixing fraction: fmix
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Pristine mass fraction: fpr
+(b)
+Figure 2. Dilution tracks for O and Na abundances (Figure 2a) and the pristine mass fraction 𝑓pr as a function of companion mass ratio 𝑞 and mixing fraction
+𝑓mix (Figure 2b). We adopt the AGB star abundances adopted as in D’Ercole et al. (2010, coloured lines) and the average abundances for the massive binary
+explored by de Mink et al. (2009, black lines). AGB stars of mass 𝑚AGB ∼ 4−9 𝑀⊙ are included because they have conserved C+N+O abundances, undergo
+recurrent mixing events (Karakas & Lattanzio 2014), and have life-times short enough to pollute their environment before cooling and collapse of the medium
+(Conroy & Spergel 2011). The triangle markers show increments of 0.1 𝑓pr, where 𝑓pr = 1 − 𝑓poll is the fraction of pristine material compared to contiminated
+material. We have assumed pristine abundances [O/Fe] = 0.33, [Na/Fe] = 0.0, [Mg/Fe] = 0.37 and [Al/Fe] = 0.11, in order to compare with the abundances
+in M54 (NGC 6715) as reported by Carretta et al. (2010), shown here as circles with error bars. Upper limits are shown as arrows, points without error bars
+are those for which no uncertainty is listed by Carretta et al. (2010). The colours of the observational data points show an estimate of the surface pristine
+mass fraction 𝑓pr by minimizing the distance between this point and an interpolated AGB ejecta contour using the scipy package optimize.minimize. These
+values of 𝑓pr can be mapped to a corresponding 𝑓mix and 𝑞 using Figure 2b.
+3.3
+Companion formation from contaminants
+3.3.1
+General principle
+We now consider how companions with mass ratio 𝑞 ∼ 0.1 may
+form around stars in the globular cluster. Unlike the early disc sce-
+nario (Bastian et al. 2013a), we can now consider a slower accumula-
+tion of material over 100 Myr timescales, rather than the ≪ 10 Myr
+needed to pollute a star while it is still convective. The collision of
+the companion with the host will supply the mixing (see discussion
+in Section 3.5). We first consider the properties of the gas reservoir,
+and then some different mechanisms that may allow a star to gain a
+companion composed of massive binary or AGB ejecta.
+In all scenarios, we consider an ISM that is the product of slow
+stellar winds that are unable to escape the potential of the dense
+cluster. These winds may occupy the core of the cluster (D’Ercole
+et al. 2008), but do not undergo rapid collapse to produce a second
+generation (Conroy & Spergel 2011). The latter authors argue that
+material does not collapse while the Lyman-Werner photon flux
+remains sufficient to prevent the formation of molecular hydrogen
+at typical interstellar distances within dense clusters (i.e. a few
+100 Myr), maintaining a temperature of ≳ 100 K. However, we will
+show in Section 3.3.7 that, even if star formation proceeds similarly
+to normal molecular clouds, companions can feasibly be produced
+on shorter timescales. The density for the contaminant reservoir can
+be simply estimated as:
+𝜌gas = 𝑓wind · 3𝑀c,0
+4𝜋𝑅3gas
+= 3𝑀gas
+4𝜋𝑅3gas
+(2)
+where 𝑓wind ∼ 0.1 is the fraction of material re-injected into the
+ISM by the AGB and massive stars, 𝑀c,0 ∼ 106 𝑀⊙ is the initial
+stellar mass of the cluster, and 𝑅gas ∼ 1 pc is the radius in which
+the gas is contained (of order the core radius). These numbers yield
+a density 𝜌gas ∼ 3 · 10−18 g cm−3.
+In the framework we present in this work there is no inherent
+constraint on whether the pollutant is from AGB or massive (binary)
+stars. If the first wave of ejecta from massive stars is very dense
+(e.g. Wünsch et al. 2017) it may result in rapid accretion onto
+discs/companions around young stars or it may survive the feedback
+due to preferential clearing of the gas along low density regions
+between filaments (e.g. Dale et al. 2014). We now consider different
+mechanisms by which the ejecta can be captured onto the low mass
+stellar population.
+3.3.2
+Formation of companion in a disc
+We generally consider a scenario in which the first generation of
+stars capture the gas to produce a circumstellar disc. If the cap-
+tured gas cannot cool immediately, it will form a structure that is
+initially supported by pressure and rotation. This disc is gravitation-
+ally unstable if 𝑄 ≲ 1, where 𝑄 is the Toomre (1964) parameter:
+MNRAS 000, 1–29 (2021)
+
+8
+Winter & Clarke
+10−2
+10−1
+100
+101
+Mass-ratio: q
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cumulative distribution function
+fmix = 0.1
+fmix = 0.3
+fmix = 0.7
+Figure 3. The mass-ratio 𝑞 distribution required to produce the distribution
+of RGB star abundances in M54 for those stars that are not composed purely
+of pollutants, where purely pollutant is defined to be 𝑓pr < 0.05. We show
+the distribution for fixed assumed mixing fractions 𝑓mix = 0.1 (dotted), 0.3
+(solid) and 0.7 (dashed), with corresponding medians shown as vertical lines.
+An important caveat is that the precise shape of the cumulative distribution
+function at the low 𝑞 end is dependent on the assumed primordial abundances
+and observational uncertainties. The shape of the function at the high 𝑞 end
+is similarly dependent on AGB model ejecta abundances and observational
+uncertainties.
+𝑄 = 𝜅𝑐s
+𝜋𝐺Σ ≈ 𝑞−1 𝑐s
+𝑣K
+,
+(3)
+where 𝜅 ≈ ΩK = 𝑅disc𝑣K is the epicyclic frequency, similar to
+the Keplerian frequency ΩK, Σ ≈ 𝑀disc/𝜋𝑅2
+disc is the mass of the
+accreted contaminant, and 𝑐s is the sound speed to the accreted gas
+when it fragments. We have adopted 𝑞 = 𝑀disc/𝑚∗ in the RHS of
+equation 3. If the gas is able to cool to 𝑐s = 1 km s−1 (temperature
+𝑇 ∼ 200 K), then for 𝑣K ∼ 10 km s−1 and 𝑞 ≳ 𝑞crit = 0.1 the
+gas is gravitationally unstable and collapses to form a (sub-)stellar
+companion, or companions.
+If a large quantity of gas is captured at once (for example,
+via tidal capture), then the gravitational collapse may initiate im-
+mediately without the need for long-lived disc survival. However,
+a replenished gaseous disc composed of ejecta material may also
+survive longer than typical protoplanetary discs (3−10 Myr – e.g.
+Haisch et al. 2001). For a low mass main sequence star, the X-ray
+and EUV luminosity of the host star drops rapidly over time (Tu
+et al. 2015), such that the internal photoevaporation of the disc be-
+comes less efficient (e.g. Alexander et al. 2006; Owen et al. 2010).
+External photoevaporation of protoplanetary discs by neighbour-
+ing stars is dominated by massive stars of mass ≳ 30 𝑀⊙, absent
+after several Myr (e.g. Johnstone et al. 1998, see Winter & Ha-
+worth 2022 for a review). Accretion may be suppressed for the disc
+around a main sequence star when the spherical Alfven radius ex-
+ceeds the co-rotation radius (Königl 1991; Armitage 1995; Clarke
+et al. 1995). This suppression may be efficient for stars that rotate
+rapidly, particularly if they lose their primordial discs early due to
+external photoevaporation (e.g. Clarke & Bouvier 2000; Roquette
+et al. 2021). Thus the disc lifetime need not be prohibitive for mass
+accumulation.
+3.3.3
+Bondi-Hoyle-Lyttleton accretion
+Material may accrete slowly via BHL style accretion in a scenario
+similar to that considered by Throop & Bally (2008) and Moeckel &
+Throop (2009). When the gas medium is not homogeneous, we ex-
+pect the stagnation point to be offset from the axis that is coincident
+with the velocity vector of the star. Material accreted in this way
+retains angular momentum, and so forms a disc that is not accreted
+immediately onto the star. We can estimate the average rate at which
+a star captures material in such a scenario using the Bondi (1952)
+accretion rate (see also Shima et al. 1985):
+�𝑀BHL ≈
+∫ ∞
+0
+𝜋𝑅2
+BHL𝜌gas(𝜆𝑐2
+s + 𝑣2
+∗)1/2 𝑔(𝑣∗) d𝑣∗,
+(4)
+where 𝜆 = 𝑒3/2/4 ≈ 1.12 in the isothermal limit, 𝑔(𝑣∗) is the
+probability that a star is moving with velocity 𝑣∗ through the gas
+and
+𝑅BHL = 2𝐺𝑚∗
+𝑐2s + 𝑣2∗
+(5)
+is the BHL radius for a star of mass 𝑚∗, and the gas in the interstellar
+medium has sound speed 𝑐s. Since 𝑔 ≈ 𝑣2∗/2√𝜋𝜎3���� where 𝑣∗ < 𝜎𝑣:
+�𝑀BHL ≈ 2√𝜋𝐺2𝑚2∗𝜌gas
+𝜎3𝑣
+[I(𝜎𝑣) − I(0)]
+(6)
+where the term in square brackets on the RHS is defined by the
+function:
+I(𝑥) = −
+𝑥
+√︃
+𝑐2s𝜆 + 𝑥2
+2(𝑐2s + 𝑥2)
++
+ln
+�√︃
+𝑐2s𝜆 + 𝑥2 + 𝑥
+�
++
+(𝜆 − 2) tan−1
+�
+𝑥
+√
+𝜆−1
+√
+𝑐2s 𝜆+𝑥2
+�
+2
+√
+𝜆 − 1
+.
+(7)
+We can adopt the sound speed 𝑐s ∼ 1 km s−1, 𝑣∗ ∼ 10 km s−1 and
+𝑚∗ = 1 𝑀⊙, we find 𝑅BHL ≈ 18 au. With a similar 𝜎𝑣 = 10 km s−1,
+then we obtain an average ⟨ �𝑀BHL⟩ ∼ 6·10−9 𝑀⊙ yr−1. This results
+in an average change in the mass ratio of the contaminant to the
+pristine material in the whole system �𝑞 ∼ 6 · 10−3 Myr−1. A mass
+ratio 𝑞 ∼ 0.1 is therefore reached (globally averaged) on a timescale
+of tens of Myr, shorter than the several 100 Myr that may be required
+for the Lyman-Werner flux density to drop to a level to allow the
+ejecta to cool and rapidly form a second population of stars (Conroy
+& Spergel 2011). However, we revisit this rate in Section 3.3.7,
+showing that BHL accretion is generally less efficient than other gas
+capture mechanisms.
+3.3.4
+Tidal cloud capture
+In a highly sub-structured medium populated by dense cloudlets,
+gas can be accreted onto pre-existing stars via the mechanism of
+tidal capture and disruption of passing cloudlets (see Dullemond
+et al. 2019; Kuffmeier et al. 2020). This might work similarly to the
+tidal compression (and disruption) between the black hole and an
+infalling cloud in the galactic centre, subject both to the pressure
+from the low density ambient medium and the tidal forces of the
+compact object as modelled by a number of authors (Burkert et al.
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+9
+2012; Lucas et al. 2013; Steinberg et al. 2018). The disruption of the
+cloud and capture of gas in this way may be responsible for bursts of
+star formation in the galactic centre (Bonnell & Rice 2008), possibly
+enhanced by convergent flows (Hobbs & Nayakshin 2009).
+To attempt a quantitative estimate at the rate of tidal capture,
+here we speculate that this capture may operate similarly to the tidal
+exchange of orbital energy during close passages between stars and
+other stars (e.g. Robe 1968; Fabian et al. 1975; Bonnell et al. 2003;
+Winter et al. 2022b) or protoplanetary discs (Ostriker 1994; Winter
+et al. 2018a). If we assume that the cloud capture is physically
+similar to stellar capture, then the capture rate in the hyperbolic
+regime is (Winter et al. 2022b):
+Γcapt ∼ 0.013
+� 𝑛clouds
+105 pc−3
+� �� 𝑅cloud
+10 au
+�5 � 𝑀cloud
+1 𝑀⊙
+�
+×
+�
+𝜎𝑣
+10 kms−1
+� 𝑚∗(𝑀cloud + 𝑚∗)
+𝑀2
+cloud
+�1/3
+Myr−1.
+(8)
+Here 𝑛clouds = 𝜌gas/𝑀cloud is the number density of clouds. In
+the hyperbolic (high velocity) regime, the capture cross-section is
+insensitive to the internal equation of state (Lee & Ostriker 1986),
+such that equation 8 might be a reasonable approximation for tidal
+capture by clouds.
+The assumption of an idealised geometry as well as uncertain-
+ties in cloud properties mean that equation 8 only offers an order
+of magnitude estimate. In addition, tidal encounters involving ‘cap-
+ture’ of a gas cloud probably result in tidal disruption of the cloud
+rather than capture of the entire mass. Therefore, to estimate a mass
+accretion rate using equation 8, we multiply the amount of mass
+expected to be gained in each capture encounter:
+�𝑀capt = 𝑓captΓcapt𝑀cloud.
+(9)
+All of the complex physics is hidden in the factor 𝑓capt. For illus-
+trative purposes, we will here assume that the capture process can
+yield a maximum efficiency of 𝑓capt = min{0.1𝑚∗/𝑀cloud, 1}, such
+that a maximum of 𝑞 = 0.1 is produced in one encounter.
+3.3.5
+Disc sweeping
+Early disc sweeping is the scenario for surface pollution envisioned
+by Bastian et al. (2013a). This mechanism may still operate in the
+context of the companion accretion model. If the primordial disc
+(or a subsequently formed one) is continuously supplied with gas,
+then this can enhance the encounter cross-section and lead to rapid
+sweep-up of material. Integrating over the relative velocities, the
+accretion rate for disc of radius 𝑅disc and mass ratio 𝑞 is (Binney &
+Tremaine 2008):
+�𝑀sweep = 16√𝜋𝜌gas𝜎𝑣 𝑅2
+disc
+�
+1 + 𝐺𝑚∗(1 + 𝑞)
+2𝜎𝑣 𝑅disc
+�
+= 1.63 × 10−8𝜒grav
+� 𝑀gas
+105 𝑀⊙
+�
+×
+� 𝑅gas
+1 pc
+�−3 �
+𝜎𝑣
+10 km s−1
+� � 𝑅disc
+10 au
+�2
+𝑀⊙ yr−1.
+(10)
+Here we define the gravitational focusing parameter:
+𝜒grav = 1 + 0.44
+� 𝑚∗(1 + 𝑞)
+1 𝑀⊙
+� �
+𝜎𝑣
+10 kms−1
+�−1 � 𝑅disc
+10 au
+�−1
+(11)
+We have adopted the average density of the gas reservoir 𝜌gas =
+3𝑀gas/4𝜋𝑅3gas, where 𝑀gas is the total mass and 𝑅gas is the radial
+100
+101
+Cluster scale radius: astars [pc]
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Relative accretion rate: ˙q · τff
+BHL
+Capture
+Sweep-up
+astars/Rgas = 1
+astars/Rgas = 2
+astars/Rgas = 3
+Figure 4. The relative accretion rate (change of mass ratio multiplied by
+the free-fall time) according to BHL style accretion (dotted lines), tidal
+cloud capture (dashed lines) and disc sweep-up (solid lines) for a solar mass
+star as a function of the scale radius of the stellar population 𝑎stars. We
+adopt a total initial mass 𝑀c,0 = 107 𝑀⊙, of which 40 percent remains in
+the low mass stellar population and 10 percent goes into the gas reservoir
+(𝑀gas = 106 𝑀⊙). The gas radius 𝑅gas that is equal to (blue), one half
+(blue) and one third (yellow) of 𝑎stars. The red horizontal line shows the
+threshold above which we expect companion formation to yield 𝑞 ∼ 0.1
+before significant star formation has occurred, assuming the star formation
+efficiency per free-fall time is 𝜖SF = 0.01.
+extent. We see that the disc does not need to be very extended
+in order to yield rapid accretion if the density is large. In dense
+environments the initial disc extent may shrink due to star-disc
+encounters and external heating (e.g. Winter et al. 2018b). Perhaps
+more importantly in a dense interstellar medium, such a disc is also
+subject to ram pressure stripping. Here we adopt 𝑅disc = 10 au, for
+which a disc survives face-on accretion in a medium with density
+∼ 10−18 g cm−3 moving at a mutual velocity of ∼ 10 km s−1
+(Wijnen et al. 2017).
+3.3.6
+Accretion after companion formation
+Once a companion is formed, either from primordial or ejecta ma-
+terial, some further accretion may occur due to angular momentum
+exchanges between the gas and the companion. Whether residual
+infalling material is more likely to accrete on to the host star or the
+companion depends on the specific angular momentum and temper-
+ature of the gas (e.g Artymowicz 1983; Bate 1997; Bate & Bonnell
+1997), though this issue has not been investigated in the context of
+BHL accretion. If a circumbinary disc can form in this way, for low
+𝑞 we might expect the majority of this disc mass to accrete onto the
+secondary (Duffell et al. 2020), unless the disc collapses to form
+another companion.
+MNRAS 000, 1–29 (2021)
+
+10
+Winter & Clarke
+3.3.7
+Overall rates and outlook for companion formation
+We now estimate the timescale of gas capture relative to the free-fall
+timescale of the gaseous reservoir:
+𝜏ff =
+√︄
+3𝜋
+32𝐺𝜌gas
+,
+(12)
+for gas density 𝜌gas. Here, for simplicity, we assume a Plummer
+density profile for the stars:
+𝜌stars(𝑟) = 3𝑀stars
+4𝜋𝑎3
+stars
+�
+1 +
+𝑟2
+𝑎2
+stars
+�−5/2
+(13)
+as a function of radius 𝑟 within the cluster. The three dimensional
+velocity dispersion is then:
+𝜎𝑣 (𝑟) =
+√︁
+𝜙/2,
+(14)
+where
+𝜙(𝑟) = 𝐺𝑀stars
+𝑎stars
+�
+1 + 𝑟2
+𝑎2
+�−1/2
+.
+(15)
+We assume a uniform gas density profile inside a sphere of radius
+𝑅gas, which we consider to be proportional to 𝑎stars. We consider
+scenarios in which the ejecta are more concentrated than the stellar
+population, which might plausibly arise when the high mass pro-
+genitors are mass segregated (as discussed by Bastian et al. 2013a).
+Where appropriate, we assume that the ejecta is composed of clumps
+of gas with radius 𝑅cloud = 10 au and mass 𝑀cloud = 0.1 𝑀⊙, which
+is marginally Jeans stable at temperature 𝑇 = 200 K.
+In Figure 4 we show �𝑞·𝜏ff, the product of the free-fall timescale
+and the rate of change of the mass ratio 𝑞 for a disc around a solar
+mass star undergoing BHL, tidal cloud capture or disc sweep-up at
+the centre of a cluster with scale radius 𝑎stars. If we require 𝑞 to
+reach 0.1 over the timescale for star formation:
+𝜏SF = 𝜖−1
+SF𝜏ff,
+(16)
+where 𝜖SF is the star formation efficiency per free-fall time. The
+value of 𝜖SF is generally estimated to be ≲ 1 percent (see discussion
+by Krumholz et al. 2019), in which case configurations above the
+red horizontal line in Figure 4 would produce sufficient companion
+mass before significant star formation. If 𝜖SF is smaller due to the
+increased temperature of the gas (Conroy & Spergel 2011), then
+this red line would move downwards. In general, tidal capture and
+sweep-up are promising mechanisms by which stars can capture the
+requisite pollutant to produce a companion, while BHL accretion
+may be too slow.
+3.4
+Survival of the gas
+For a significant fraction of polluted material to end up in compan-
+ions that are subsequently accreted, we require that the gas reservoir
+in the core of the cluster is maintained until the gas can be captured.
+We now investigate whether this is consistent with the absence of
+evidence of cold dust and gas in young massive clusters (Bastian &
+Strader 2014; Cabrera-Ziri et al. 2015), nor significant quantities of
+fully ionised hydrogen indicating ongoing star formation (Bastian
+et al. 2013b). This absence has been attributed to the removal of
+gas via ram pressure stripping as a young globular cluster passes
+through a dense medium (Chantereau et al. 2020). As discussed by
+Longmore (2015), the absence of observational signatures of resid-
+ual gas in young massive clusters represents a significant problem
+for traditional models of second generation formation.
+3.4.1
+Gas mass evolution
+We can estimate the evolution of the mass of the gas reservoir in the
+cluster over time by comparing the rate of accretion onto the circum-
+stellar discs (we assume disc sweep-up here, using equation 10) and
+the approximate rate at which material is ejected into the medium:
+�𝑀eject =
+� 𝑀c,0
+⟨𝑚∗⟩ | �𝑚TO|𝜉(𝑚TO)Δ𝑚TO
+𝑚TO > 𝑚min
+0
+𝑚TO < 𝑚min
+.
+(17)
+Here Δ𝑚TO = 𝑚TO − 𝑚rem is the mass of the stars at main se-
+quence turn-off 𝑚TO minus the remnant mass 𝑚rem. We will assume
+𝑚rem ≪ 𝑚TO to yield maximal mass ejection. We have introduced
+𝜉 which is the mass function, for which in the high mass limit we
+will assume 𝜉 ∝ 𝑚−2.35, normalised to yield 8 percent of the total
+stellar mass in the AGB stars in the mass range 4−9 𝑀⊙, where
+the minimum mass of a star contributing to the gas reservoir is
+𝑚min = 4 𝑀⊙. The normalisation is given by the number of stars in
+the cluster or the total initial stellar mass 𝑀c,0 divided by the aver-
+age stellar mass ⟨𝑚∗⟩ ≈ 0.5 𝑀⊙. We assume that the main sequence
+life-time for a star of mass 𝑚∗ is
+𝜏MS = 20
+� 𝑚∗
+9 𝑀⊙
+�−2.5
+Myr,
+(18)
+such that
+�𝑚TO = −0.18
+�
+𝑡
+20 Myr
+�−7/5
+𝑀⊙ Myr−1.
+(19)
+We adopt a maximum mass 𝑚max = 50 𝑀⊙ that contributes to the
+ejecta, although our results are not sensitive to this choice.
+Using the above, the rate of change of the total mass of gas in
+the reservoir is then:
+�𝑀gas = �𝑀eject − 𝑓lm𝑀c,0⟨ �𝑞sweep⟩ − �𝑀SF.
+(20)
+Here we estimate 𝑓lm ≈ 0.4, the fraction of the initial stellar mass
+in the low mass stars (𝑚∗ ≲ 1 𝑀⊙) that capture the gas (with an
+IMF following Kroupa 2001). We will here assume that the total
+star formation rate �𝑀SF = 0. The rate of change of the average disc
+mass-ratio is:
+⟨ �𝑞⟩ = ⟨ �𝑞sweep⟩ = �𝑀sweep/⟨𝑚∗⟩.
+(21)
+The only further information needed is the total initial stellar
+mass of the cluster and the scale radius of the stars, 𝑎stars, and
+radius of the gas reservoir 𝑅gas. Hereafter, we assume a Plummer
+sphere density profile of stars, and a uniform gas density such that
+𝑅gas = 𝑎stars. We will assume that the half-mass of the present day
+cluster is equal to the half-mass of the initial cluster 𝑅hm, such that
+𝑎stars ≈ 𝑅hm/1.3. For the total stellar mass, unless otherwise stated
+we will assume the present day low mass star population has been
+dynamical depleted by 50 percent, while the total mass in low mass
+stars is 40 percent. Hence the initial total mass is 𝑀c,0 = 5𝑀c,
+where 𝑀c is the present day cluster mass.
+In order to demonstrate the evolution of the gas content and
+disc mass ratio, we consider parameters appropriate for NGC 2808.
+We adopt 𝑀c = 8.6 × 105 and 𝑅hm = 3.9 pc (Hilker et al. 2020).
+We integrate the coupled ordinary differential equations 20 and 21
+using a fourth order Runge-Kutta scheme. The results are shown in
+Figure 5. We find that in this case, the mass in the gas remains <
+1 percent of the total stellar mass at time 𝑡 ≳ 30 Myr. This is similar
+to the constraint quoted by Bastian & Strader (2014) for several
+clusters aged ∼ 30−300 Myr using Spitzer 160 𝜇m observations.
+While the upper limit for the dust mass is more stringent if the dust
+is hot, we also expect a high optical depth at these wavelengths,
+which we revisit below.
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+11
+101
+102
+Time: t [Myr]
+10−3
+10−2
+10−1
+100
+Mass-ratios
+Mgas/Mc,0
+⟨q⟩
+tmin = τMS(mmin)
+Figure 5. Evolution of the fraction of the cluster mass remaining in ejected
+material that has not been swept up by discs 𝑀gas/𝑀c,0, together with the
+averaged disc mass ratio ⟨𝑞⟩. We adopt a simple model for NGC 2808:
+𝑀c,0 = 4.3 × 106 𝑀⊙, 𝑎stars = 𝑅gas = 3.7 pc (see text for details). We show
+the fraction of the total mass ejected from massive binaries (left of red line)
+and AGB stars (right of red line) as a solid black line. We assume accretion
+onto a disc with 𝑅disc = 10 au around a star with 𝑚∗ = 0.5 𝑀⊙ according
+to equation 10.
+3.4.2
+Extinction and dust
+Even if the accretion of the gas reservoir is slower than the above
+estimate, the properties of the medium may further hide observa-
+tional signatures of a massive gas reservoir. With regards to the
+cold medium, the bright sub-mm emission and internal extinction
+is geometry dependent. Bastian & Strader (2014) and Longmore
+(2015) argue that the lack of FIR dust emission and the lack of (op-
+tical) colour gradients in the stellar population point to an absence
+of dust in the interstellar medium. However, this absence may be
+explained by the turbulent properties of the medium. If this medium
+is composed of cloudlets of radius 𝑅cloud and masses 𝑀cloud, then
+the optical depth of the dust in a cloudlet is:
+𝜏 ≈ 𝜅𝜈Σdust
+(22)
+≈ 90
+�
+𝜅𝜈
+10 cm2 g−1
+� � 𝑀cloud
+0.1 𝑀⊙
+� �
+𝜌dust
+10−2 · 𝜌gas
+� � 𝑅cloud
+10 au
+�−2
+,
+(23)
+where the dust-to-gas ratio is 𝜌dust/𝜌gas and we estimate the opacity:
+𝜅𝜈 = 𝜅0
+� 𝜈
+𝜈0
+�𝛽
+,
+(24)
+where 𝛽 ≈ 1 and 𝜅𝜈 ∼ 10 cm2 g−1 at 103 GHz (Hildebrand 1983;
+Beckwith et al. 1990). Clearly, at 160 𝜇m, the longest wavelength
+observed with Spitzer by Bastian & Strader (2014), this yields a large
+optical depth. For cloudlets with radius 𝑅cloud ∼ 𝑅BHL ∼ 10 au and
+mass 𝑀cloud ∼ 0.1 𝑀⊙, this results in an underestimate by a factor
+> 100 in the observed total gas mass at 160 𝜇m. The total flux of the
+warm dust is therefore dependent on the amount of mass in small
+cloudlets.
+In the case of a clumpy medium, Longmore (2015) also pre-
+dicts a patchy extinction pattern. However, this is contingent on the
+effective (two dimensional) filling factor of the gas clouds. This can
+be approximated:
+F ≈
+𝑀gas
+𝑀cloud
+� 𝑅cloud
+𝑅gas
+�2
+(25)
+if F ≪ 1. Factoring in three dimensional structure, the average
+fraction of obscured stars in the cluster is always < F . If we assume
+𝑀gas = 105 𝑀⊙ and 𝑀cloud = 0.1 𝑀⊙, with 𝑅cloud = 10 au and
+𝑅gas = 1 pc, we obtain a small F ≈ 2 × 10−3. This is a conservative
+estimate in terms of the reservoir properties; the maximum 𝑀gas
+in our example for NGC 2808 older than 20 Myr is much smaller
+than this, while the scale radius is also larger (𝑅gas = 3.7 pc). The
+properties of the gas cloudlets remains uncertain, but as discussed
+above we expect clouds that are this size and mass to be Jeans stable
+at temperature 200 K. Such clouds would also be optically thick, so
+that obscured stars may be undetected rather than detected as red-
+dened stars. In reality we expect some range of cloud masses and
+radii, but this illustrative example demonstrates that if enough ma-
+terial is in small clouds then this reduces the influence of extinction
+on the stellar population.
+3.4.3
+Constraints on the gas content
+In terms of signatures of gas in the cluster, Cabrera-Ziri et al. (2015)
+inferred an upper limit of ≲ 9 percent of the total cluster mass from
+an ALMA search for CO 𝐽 = 3−2 transitions in three young massive
+clusters in the Antennae, aged 50 − 200 Myr. This upper limit is
+not strongly restrictive for the companion accretion model, since
+the gas reservoir (Figure 5) never approaches this limit. Further,
+in the scenario we explore in this work the heating source is the
+Lyman-Werner photons, which photodissociate the CO over several
+100 Myr timescales (Conroy & Spergel 2011). Even if the CO is not
+fully dissociated, the 3−2 transition may in some cases be optically
+thick (Polychroni et al. 2012; Salak et al. 2014; Fukui et al. 2015),
+particularly if gas is concentrated into small cloudlets as discussed
+above.
+Since there is little or no ongoing star formation, the fully
+ionised assumption adopted by Bastian et al. (2013b) need not hold,
+such that an absence of strong H𝛼 signatures is expected. More
+restrictive are the searches for HI gas in 13 LMC and SMC young
+clusters (30 − 300 Myr old) by Bastian & Strader (2014), who in-
+ferred an upper limit of 1 percent of the total cluster mass. While
+this is comparable to the maximum gas mass in the model assuming
+rapid sweep-up accretion of the contaminants, it remains an uncom-
+fortable constraint if accretion is slower. Once again, this constraint
+may be mitigated in some cases where HI can be optically thick
+(Fukui et al. 2015; Seifried et al. 2022). Following the discussion
+in Appendix B of Seifried et al. (2022), this can occur when the
+column density of HI is ≳ 10−18 g cm−2, depending on the tem-
+perature. The gas in small clouds with mass 𝑀cloud ∼ 0.1 𝑀⊙ and
+radii 𝑅cloud ∼ 10 au would have much larger column density than
+this threshold.
+3.4.4
+Survival of gas against stripping
+Finally, the clumpiness of the gas reservoir may have important
+consequences for the ram pressure stripping as the globular clus-
+ter moves through the external medium. Chantereau et al. (2020)
+assumed a smooth density distribution when showing that a gas
+MNRAS 000, 1–29 (2021)
+
+12
+Winter & Clarke
+reservoir could be stripped efficiently from a globular cluster mov-
+ing through a dense gaseous environment. However, for a clumpy
+reservoir the external, high velocity medium may pass through low
+density channels, similar that seen in simulations of stellar winds
+during the formation of stellar clusters (e.g. Rogers & Pittard 2013).
+This would reduce the efficiency of stripping, allowing the ejecta
+products to remain bound to the cluster for longer.
+In summary, a sub-structured gas reservoir may survive long
+enough to be captured by the stellar population. A combination
+of relatively rapid accretion of the gas, photodissociation and high
+optical depth resulting from substructure may contribute to the non-
+detection of such a gas reservoir in young massive clusters.
+3.5
+Mixing of companion material with host star
+3.5.1
+Binary collisional mixing
+In order to produce the observed abundance variations, mixing of
+the secondary material in the majority of the host star is required.
+This is in order to produce the same chemical signatures in MS and
+MSTO as in RGB stars (e.g. Briley et al. 1996, 2004, although we
+expect some variation in the degree of mixing – see discussion in
+Section 3.2.5). RGB stars have convective envelopes that encompass
+∼ 70 percent of the total mass, while for MSTO stars at 10 Gyr this
+is closer to ∼ 1 percent. Hence, we must ask whether it is plausible
+to expect mixing of the pollutant in the scenario we consider.
+Some authors have performed smoothed particle hydrodynam-
+ics simulations of such collisions, primarily for their influence on
+blue straggler properties (e.g. Benz & Hills 1992; Lombardi et al.
+1996, 2002). It is not clear how accurate such models are for comput-
+ing the degree of shock heating and convective mixing induced by
+a collision. Nonetheless, the studies agree that the degree of mixing
+is dependent on whether the encounter is ‘direct’ (i.e. with closest
+approach distance 𝑥min → 0) or ‘grazing’ (i.e. with 𝑥min ∼ 𝑅∗,
+for 𝑅∗ the stellar radius). In our case, the dynamical history of
+the companion prior to collision with the host star is driven by an
+accumulation of small eccentricity changes due to successive per-
+turbations by neighbouring stars (Winter et al. 2022a). We therefore
+expect the collisional encounter to be grazing in nature. This case
+is depicted as Case W in the bottom panels of Figure 5 of Lombardi
+et al. (2002). The authors define the entropic parameter 𝐴 ≡ 𝑃/𝜌𝛾,
+for 𝑃 pressure, 𝜌 density and 𝛾 the adiabatic index. For radius 𝑟
+within the star, 𝐴 must satisfy the requirement d𝐴/d𝑟 > 0 in ther-
+mal equilibrium. The value of 𝐴 is significantly increased due to
+shock heating compared to the initial conditions (Figure 1 in that
+work) in the majority of the primary and throughout the secondary.
+This suggests that the contaminant brought in by the secondary may
+be well mixed. As discussed by Lombardi et al. (1996), the star is
+initially very far from thermal equilibrium, and may undergo sig-
+nificant convective mixing before it returns to the MS on a thermal
+timescale (∼ 1−10 Myr).
+More recently, in their hydrodynamic simulations of a merger
+(or extremely grazing encounter) between a brown dwarf and a
+solar mass star, Cabezón et al. (2022) showed that approximately
+40 percent of the brown dwarf mass stays in the outer 30 percent
+in radius of the primary, indicating a mixing fraction similar those
+required for the companion accretion model. We highlight that the
+aforementioned models, as well as those of other authors following
+post-collisional evolution of MS stars (e.g. Glebbeek & Pols 2008),
+are not necessarily sensitive to the later stage rotational mixing of
+the remnant.
+We conclude that the collision of the companion with the star
+and the resultant merged star’s subsequent evolution plausibly re-
+sults in deep mixing. Unless otherwise stated, we will proceed on
+the assumption that sufficient mixing occurs to evenly distribute the
+contaminants through 70 percent of the star, while only initiating a
+deviation from the MS that is much shorter than the star’s life-time.
+3.5.2
+Lithium survival
+Lithium may survive accretion of a companion with mass ratio
+𝑞 ∼ 0.1 companion, as in the simulations of Lombardi et al. (2002)
+and Cabezón et al. (2022). This may explain why there is no strong
+correlation in observed between other abundance variations and
+lithium abundances (e.g. Dobrovolskas et al. 2014). However, some
+degree of variation should be expected simply due to the different
+composition of the secondary, which would presumably be Li poor.
+One mitigating factor may be that the two individual stars of mass
+𝑚1 and 𝑚2 are lower mass than the collisional product 𝑚1 + 𝑚2,
+and therefore deplete their lithium more slowly than a star that has
+always had the mass 𝑚∗ = 𝑚1+𝑚2 (e.g. Bildsten et al. 1997). This is
+unlikely to significantly influence lithium evolution in the primary
+for small 𝑞, but may allow any residual lithium to survive on the
+low mass secondary.
+3.5.3
+Hertzsprung-Russell diagram
+Another requirement for the merger is that it does not produce
+strong features in the Hertzsprung-Russell diagram that could be
+interpreted as large age disparity with unpolluted stars. This is be-
+cause Martocchia et al. (2018b) found that the multiple populations
+NGC 1978 (∼ 2 Gyr old) clusters are consistent with being coeval
+within ∼ 20 Myr when using optical CMDs of SGB stars (also in
+NGC 2121 – see Saracino et al. 2020) or within ∼ 65 Myr when
+using the MSTO width. As noted by Martocchia et al. (2018b), this
+finding ‘suggest[s] that multiple populations may be due to a stellar
+evolutionary effect not yet recognized in standard evolution models.
+This effect would need to only efficiently operate in stars within
+massive/dense stellar clusters’. Interestingly, other young clusters
+do exhibit extended or bimodal MSTOs (e.g. Milone et al. 2009),
+although this is now generally considered to be unrelated to stellar
+age (Cordoni et al. 2022). The breadth of these extended MSTO may
+be attributed to stellar rotation (Bastian & de Mink 2009; D’Antona
+et al. 2015), which mimics the age spreads proportional to the age
+of the cluster (Niederhofer et al. 2015). Magnetic braking would
+then suppress this later in the stellar life-time, explaining the lack
+of extended MSTOs after a few Gyr.
+Wang et al. (2020b) used theoretical binary evolution models
+to show that the MSTO width can also be extended when massive
+binaries interact. This may produce an ersatz age spread that is
+proportional to the age of the cluster, similar to the stellar rotation
+hypothesis and the empirical constraints (Bastian et al. 2016). If
+the merger of binaries in the companion accretion model would
+produce an extended MS in the F814W band used by Martocchia
+et al. (2018b) to constrain age differences between populations, then
+this would contradict the model. However, there are two reasons why
+this is not necessarily the case:
+(i) Wang et al. (2020b) studied massive binaries with MS lifetime
+up to ∼ 40 Myr. In the context of the companion accretion model, we
+are interested in mergers with lower mass stars (𝑚∗ ∼ 1 𝑀⊙), with
+companions that have typical mass-ratios 𝑞 ∼ 0.1−0.3. Both abso-
+lute mass and companion mass-ratio are important in the evolution
+of rotation, magnetic fields, and H-R diagram position. The results
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+13
+therefore do not necessarily generalise trivially to the distributions
+of stellar masses and mass-ratios explored in this work.
+(ii) Mergers extend the MS in part because of their influence
+on the stellar rotation, which produces something similar to the
+observed extended MS in numerous clusters. This extension is less
+pronounced at longer (optical) wavelengths (Brandt & Huang 2015).
+It is unclear how the Wang et al. (2020b) results would appear in
+the F814W band used by Martocchia et al. (2018b) to constrain age
+differences between populations.
+We therefore consider this possibility as an important future test of
+the companion accretion model, but do not conclude that the results
+of Wang et al. (2020b) in conjunction with the observational results
+of Martocchia et al. (2018b) rule out mergers at present.
+3.6
+Fraction of accreted companions
+We now turn our attention to quantifying the rate at which sub-stellar
+companions can be accreted onto the primary star. We base these
+estimates on the analytic prescription presented by Winter et al.
+(2022a), which is developed from the theoretical cross-sections for
+eccentricity excitation in dense clusters (Heggie & Rasio 1996),
+and has been benchmarked against the numerical simulations of
+Hamers & Tremaine (2017). In brief, numerous hyperbolic encoun-
+ters in high velocity dispersion environments result in a diffusive
+eccentricity evolution for a binary system. When a high eccentricity
+is reached, a companion can either circularise to produce a short
+period companion (i.e. a hot Jupiter if 𝑞 is small), or it can become
+disrupted and/or merge with the host star. For sufficiently high den-
+sity environments, rapid eccentricity fluctuations effectively forbid
+circularisation. For the typical semi-major axes and stellar densities
+we consider here, we are always in the regime where collisions are
+more probable than circularisation. We will therefore assume that
+no companions survive circularisation, but are instead accreted onto
+their host star. Over sufficient time, all companions will eventually
+either become ionised by a close dynamical perturbation or they
+will merge. The overall number of collisions is therefore dependent
+on the relative rates of these two outcomes among the population.
+The rate of collision of the companion with the host star can
+be estimated (Winter et al. 2022a):
+Γcoll ≈
+𝛾0𝑒0
+√︃
+1 − 𝑒2
+0
+2(1 − 𝑒0)
+(26)
+where 𝑒0 is the initial eccentricity and we have defined the factor
+𝛾0 ≡ 4.6
+√︁
+1 + 𝑞M (coll)
+∗
+𝑛∗
+105 pc−3
+� 𝑚∗
+1 𝑀⊙
+�1/2 � 𝑎0
+5 au
+�3/2
+Gyr−1,
+(27)
+for
+M (coll)
+∗
+=
+∫ ∞
+0
+d𝑚pert 𝑞pert 𝜉(𝑚pert).
+(28)
+In the above equations, 𝑛∗ is the local stellar density, 𝜎𝑣 is the three
+dimensional velocity dispersion, 𝑎0 is the initial semi-major axis of
+the companion, 𝑞pert = 𝑚∗/𝑚pert is the mass-ratio of a perturber
+with mass 𝑚pert and
+𝜉(𝑚∗) ∝
+����
+����
+𝑚−𝛼1
+∗
+𝑚min ≤ 𝑚∗ < 𝑚br
+𝑚−𝛼2
+∗
+𝑚br ≤ 𝑚∗ ≤ 𝑚max
+0
+𝑚∗ > 𝑚max or 𝑚∗ < 𝑚min
+(29)
+is the local mass function. We adopt 𝛼1 = 0.4, 𝛼2 = 2.8, 𝑚br =
+0.8 𝑀⊙, 𝑚min = 0.08 𝑀⊙ and truncate above 𝑚max = 1.3 𝑀⊙ to
+approximately replicate the mass function at 5 Gyr. However, this
+choice does not strongly influence our results because the overall
+dominant perturbations are due to stars below this mass.
+The initial eccentricity 𝑒0 in this context is not immediately
+obvious. We expect the binary formation via gravitational instability
+in a more compact disc than for hierarchical star formation. In
+the field, wide binary orbits are generally not circular, but exhibit
+a wide range of eccentricities (e.g. Duquennoy & Mayor 1991).
+In fact, binaries at separations ≲ 100 au, that may have formed
+through gravitational instability in a disc, exhibit an approximately
+uniform distribution of eccentricities Hwang et al. (2022). We will
+therefore adopt 𝑒0 = 0.5 as the mean of the uniform eccentricity
+distribution. However, we highlight that if eccentricities are small
+then equation 26 implies the collision rate is also small (and vanishes
+as 𝑒0 → 0). This is because the dominant terms in the dynamical
+cross sections vanish at 𝑒0 = 0, such that only close encounters play
+an important role in the initial changes to eccentricity evolution
+(Ostriker 1994; Winter et al. 2018a). In this case, the collision rate
+may be initially slower, but the early eccentricity excitation must be
+treated with an alternative prescription that factors in higher order
+terms.
+From equation 27, we see that the rate of enrichment by col-
+lision with the companion is independent of the local velocity dis-
+persion, but linearly dependent on the local density. This approx-
+imation does not necessarily apply when the orbital frequency of
+the companion becomes comparable to the encounter rate (Kaib &
+Raymond 2014), or more stringently becomes less accurate when
+the majority of encounters are not ‘slow’ (i.e. when 𝜎v ≫ 𝑣orb, for
+the orbital velocity 𝑣orb ∼ 10 km s−1; see discussion in Appendix
+A of Winter et al. 2022a). However, since 𝜎𝑣 is never more than a
+factor few larger than the orbital velocity, we are satisfied that our
+estimate should be a reasonable approximation.
+By comparison, the rate at which a star-companion system
+undergoes a resonant or ionising encounter can be approximated
+(Hut & Bahcall 1983):
+Γion = 2.8M (ion)
+∗
+�
+𝜎𝑣
+10 km s−1
+�−1
+𝑚∗
+1 𝑀⊙
+×
+× 𝑎0
+5 au
+𝑛∗
+105 pc−3 Gyr−1.
+(30)
+Here we have defined:
+M (ion)
+∗
+=
+∫ ∞
+0
+d𝑚pert (1 + 𝑞pert + 𝑞)𝑞1/3
+pert𝜉(𝑚pert).
+(31)
+We consider all strong dynamical perturbations of this kind to be
+ionising as a necessary simplification. Strong encounters may in
+reality alter the energy of the companion, which then may subse-
+quently be ionised or accreted as before (albeit with a different semi-
+major axis). Equation 30 therefore may moderately overestimate the
+role of ionisation relative to collisions. However, we immediately in-
+fer from a comparison between equation 30 and equation 27 that for
+sufficiently large velocity dispersions, accretion of the companion
+dominates over ionisation.
+With these expressions, we can estimate the relative probability
+of a star having experienced enrichment by the accretion of the
+companion at time 𝑡:
+𝑃coll = Γcoll
+Γtot
+{1 − exp [−Γtot𝑡]} ,
+(32)
+where Γtot = Γcoll + Γion. In the absence of any further dynamical
+effects (see Section 3.7), this simple equation gives the overall frac-
+tion of stars that are contaminated with pollutants. To first order, at
+MNRAS 000, 1–29 (2021)
+
+14
+Winter & Clarke
+0
+2
+4
+6
+8
+10
+12
+14
+Age [Gyr]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction or probability
+fI,dep
+fII,dep
+fdep
+Pcoll
+Mc = 5 · 104 M⊙
+Mc = 106 M⊙
+Figure 6. The assumed scaling of the depletion factors for a fixed cluster
+mass 𝑀c = 106 𝑀⊙ (solid lines) and 5 × 104 𝑀⊙ (dashed lines) over time.
+The red lines show the depletion of the non-polluted population of stars,
+while the blue lines show the equivalent for the polluted stars. The black
+lines show the overall assumed depletion factor 𝑓dep = 𝑀c/𝑀c,0, and the
+cyan lines shows the probability of a star having been enriched due to
+accretion of the sub-stellar companion. All clusters are assumed to have a
+current half-mass radius 𝑅hm = 10 pc. Note, this figure does not show the
+evolution of a specific cluster, but the associated fractions for a fixed mass at
+the specified age (the initial mass 𝑀c,0 = 𝑀c/ 𝑓dep increases with decreasing
+𝑓dep).
+𝑡 ≫ 1/Γtot, we see that 𝑃coll ∝ 𝜎𝑣 ∝
+√︁
+𝑀c/𝑅c for total cluster mass
+𝑀c and radius 𝑅c. Given the lack of clear correlation between 𝑀c
+and 𝑅c among globular clusters (Krumholz et al. 2019), this sug-
+gests 𝑃coll ∝ 𝑀1/2
+c
+. This positive correlation is in broad agreement
+with what is observed (see Bastian & Lardo 2018, and references
+therein), however we return to this in more detail in Section 3.8.
+When we compute the rate of capture and ionisation we will
+generally adopt the central properties of the cluster within the scale
+radius 𝑎c = 𝑅hm/1.305 in a Plummer sphere. We choose the central
+values because this is where the most rapid encounters occur, and
+the velocity dispersion in this region is therefore the most important
+factor in setting the ratio of enrichment to ionisation. Indeed, if stars
+outside of the core are more likely to pass beyond the tidal radius and
+be lost from the cluster, then the stars that survive to the present day
+should also be biased to those that occupied the cluster core earlier
+in the cluster evolution. Meanwhile, the assumption of a Plummer
+sphere may not always be the most accurate profile, however it
+allows us to homogeneously compare across different clusters with
+simple scaling relations while adopting the half-mass radius from
+observations. The latter is a comparatively robust quantity, that
+does not depend on density profile definitions and should not vary
+rapidly with cluster evolution. It is probable that deviations in the
+present day true central density compared to this assumption are less
+important than the variations over the dynamical history of a given
+cluster. We leave these considerations of more accurate dynamical
+modelling to future work.
+3.7
+Removal of pristine stars
+The removal of the pristine population over time is a necessary in-
+gredient of all enrichment models to enhance the overall enrichment
+fraction. The loss of some fraction of the total mass of a cluster over
+time is naturally expected due to the two-body relaxation of the clus-
+ter, as well as by other sources of dynamical heating such as tidal
+shocks. In models that invoke a second generation of stars (not com-
+panions), a second population may be expected to form in the core
+region of the cluster (e.g. D’Ercole et al. 2008), so that the assump-
+tion that the pristine population is preferentially removed is to some
+extent justified. However, the fraction of the pristine population that
+must be removed in multi-generation models is usually extremely
+high (> 90 percent), unless some mechanism for significant dilution
+can be invoked (e.g. Conroy & Spergel 2011).
+In the context of the mechanism we present here, we also natu-
+rally expect the contaminated population to survive in much higher
+numbers than the pristine population. This is partly because the
+stars in the core, where accretion of the companion is most proba-
+ble due to higher density and 𝜎𝑣, are also the most gravitationally
+bound. In addition, we might expect some correlation between stars
+that are ejected and those that remain pristine through dynamical
+arguments. The energy input required to ionise the star-companion
+system is:
+Δ𝐸ion = 𝐺𝑚2∗𝑞
+𝑎0
+.
+(33)
+This is generally smaller than the energy input required to unbind
+the star from the cluster:
+Δ𝐸unbind = −𝐸i ∼ −1
+2𝑚∗𝑣2
+∗ + 𝐺𝑀c𝑚∗
+𝑎c
+�
+1 + 𝑟2∗
+𝑎2c
+�−1/2
+,
+(34)
+where 𝐸i is the initial energy of the stellar orbit in the cluster and
+the RHS of equation 34 is an approximation adopting a Plummer
+density profile with scale radius 𝑎c, where a star is at radius 𝑟∗ from
+the centre and moving with velocity 𝑣∗. For the sake of illustration,
+plugging in 𝑣∗ = 10 km s−1, 𝑀c = 105 𝑀⊙, 𝑟∗ = 𝑎c = 2 pc, 𝑎0 =
+5 au, 𝑚∗ = 1 𝑀⊙ and 𝑞 = 0.1, we obtain Δ𝐸unbind/Δ𝐸ion = 5.75.
+Thus only a small fraction of an encounter that unbinds the star needs
+to go into the binary orbit in order to result in ionisation. However,
+this is only the case if stars become unbound due to individual
+encounters that impart large kinetic energy changes, and it is unclear
+whether this is the case in practice. Preferential loss from the cluster
+of binaries that have been ionised would lead to an increased binary
+fraction in the cluster core. However, it is not straight forward to
+interpret this from simulations that consider evolving binaries where
+𝑞 ∼ 1 (e.g. Hong et al. 2019), since other effects such as mass
+segregation also play a role. We will here assume preferential loss
+of the pristine population, but the degree to which this is true needs
+to be investigated further in future simulations.
+The consequences of a depletion in stellar mass by a factor
+𝑓dep ≡ 𝑀c/𝑀c,0 for the overall fraction of polluted stars in the
+companion accretion model is two-fold. In the first instance, the
+fact that a cluster has been depleted means that its initial mass
+was larger. We will generally assume the half-mass radius does not
+change substantially. Thus the increase in mass yields an increase
+in initial density and velocity dispersion, somewhat altering the
+pollution fraction via the local density and relative ionisation and
+collision rates (see Section 3.6). More importantly, the depletion
+can reduce the number of pristine stars remaining in the cluster. The
+total depletion factor can be written:
+𝑓dep = (1 − 𝑃coll) 𝑓I,dep + 𝑃coll 𝑓II,dep
+(35)
+where 𝑓I,dep and 𝑓II,dep are the depletion factors for the population I
+(pristine) and population II (polluted) stars respectively. With these
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+15
+definitions, we can determine
+𝑓I =
+𝑓I,dep(1 − 𝑃coll)
+𝑓I,dep(1 − 𝑃coll) + 𝑓II,dep𝑃coll
+,
+(36)
+which is the fraction of unpolluted stars remaining in the clus-
+ter. We discuss sensible functional forms of 𝑓I,dep and 𝑓II,dep in
+Appendix A. We expect differential depletion between the two pop-
+ulations because the collisions occur preferentially when a star is in
+the core of the cluster. This is where the density is largest and the
+timescale for encounters is shortest. If 𝑃coll (the collision probabil-
+ity) is low, then this should mean nearly all collisions have happened
+in the core, where they are most frequent. These stars should also
+therefore be those should take the longest to be ejected by two-body
+encounters. On the other hand, as 𝑃coll grows, eventually stars that
+have undergone collision are distributed throughout the cluster. This
+is because there is a relatively large chance of having had a collision
+even for stars outside the centre.
+Examples of the assumed scaling for the depletion factors of
+the polluted and pristine populations are shown in Figure 6. To
+generate the depletion factors in a given cluster as a function of
+time, we have assumed that:
+𝑓dep = 1 + ( 𝑓dep,0 − 1) 𝑡
+𝑡0
+,
+(37)
+with 𝑓dep,0 = 0.5 at time 𝑡 = 𝑡0 = 10 Gyr. This corresponds to a
+simple, linear decrease in the stellar mass that very approximately
+mimics the influence of two body encounters over time. The nor-
+malisation is chosen such that 𝑓dep ∼ 0.3−0.5 at the typical ages
+of globular clusters (e.g. Kruijssen 2015; Webb & Leigh 2015).
+For the two different fixed cluster masses are adopted at each time,
+𝑀c = 106 𝑀⊙ and 𝑀c = 5·104 𝑀⊙, both with half-mass radii 𝑅hm.
+The two regimes of the depletion behaviour for each population
+can be seen at later times in the higher mass case. As discussed
+above, the depletion of the polluted (II) stars becomes less efficient
+relative to the pristine (I) stars when a large fraction of the stars in
+the cluster, including those outside of the core, become polluted. In
+the lower mass model, 𝑃coll stays low for all ages, meaning that the
+polluted stars are preferentially in the core. The polluted population
+is therefore never large enough to become depleted in this case.
+We emphasise that this is not a unique solution, but is a physically
+motivated toy model as discussed in Appendix A. Broadly, our toy
+depletion model reproduces the kind of factors we might expect, and
+completes the model to be compared to observations in Section 3.8.
+3.8
+Trends with globular cluster properties
+In the following we will consider how our models reproduce the
+observational trends in the enrichment fractions in various globular
+clusters. For simplicity, we will always assume that the initial semi-
+major axis of the companion has 𝑎0 = 5 au (approximately half the
+BHL radius) and the companion eccentricity is initially 𝑒0 = 0.5, the
+average thermalised value. All other assumptions and calculations
+are as outlined in the preceding section. Although our models are
+highly simplified, we have deliberately kept our model simple and
+fixed parameters in order to avoid fine-tuning of the model to the
+data. Our concern is mostly on the trends predicted by the model
+rather than the exact fit of the data, although the parameters choices
+are all physically or empirically motivated.
+3.8.1
+Presence of multiple populations
+In our statistical comparison between the model we have presented
+in the preceding section and the rate of stellar enrichment in glob-
+ular clusters, we first consider the regions for which evidence of
+multiple populations has been found compared with those that have
+not. Within the companion accretion model, the age and velocity
+dispersion are the two most important parameters in determining the
+fraction of polluted stars. The virial velocity dispersion scales with
+√︁
+𝑀c/𝑅hm, and we therefore consider globular clusters in terms
+of 𝑀c/𝑅hm and age. In our model, we assume that the total clus-
+ter population depletion follows equation 37, with 𝑓dep = 0.5 at
+𝑡 = 10 Gyr.
+In Figure 7 we show the distribution of cluster properties di-
+vided by whether or not they show evidence of harboring multiple
+populations (Krause et al. 2016). We have further included/modified
+the clusters surveyed by Martocchia et al. (2018a) and Martocchia
+et al. (2019), with cluster parameters estimated by Song et al. (2021)
+except for Lindsay 113 and Lindsay 38 for which we use an approxi-
+mate half-mass radius based on the findings of Bica & Dutra (2000).
+When compared with the predicted fraction of pristine (unpolluted)
+stars in the model (using equation 32 combined with equation 36),
+we find generally good agreement in terms of clusters in which we
+expect to find multiple populations. In particular, we predict a non-
+negligble polluted population in the young NGC 1978, which is only
+2 Gyr old and has two, coeval populations (Martocchia et al. 2018a;
+Saracino et al. 2020). Using the actual properties of the cluster in
+our estimate (𝑀c = 1.9 × 105 𝑀⊙, 𝑅hm = 8.7 pc and 𝑡 = 1.9 Gyr),
+we expect a moderate fraction ( 𝑓II = 5 percent assuming 𝑓dep = 1,
+or 𝑓II = 21 percent if 𝑓dep = 0.5) of polluted stars. Based on Figure
+5 in the study Martocchia et al. (2018a), this fraction seems rea-
+sonable. This polluted population can therefore be explained by our
+model.
+In three instances, NGC 2121, NGC 2155 and Lindsay 113,
+Martocchia et al. (2019) do not find evidence of two populations
+in their Gaussian mixture models, but do find N spreads that are
+broader than can be accounted for by observational error alone
+(empty green triangles in Figure 7). Companion accretion may fea-
+sibly contribute to this spread. For example, in the case of NGC
+2155, the half-mass radius is 𝑅hm ≈ 6.5 pc (Song et al. 2021),
+which means that the cluster is higher density than our fiducial esti-
+mate. Therefore both ionisations and collisions occur more rapidly.
+When substituting 𝑅hm = 6.5 pc, 𝑀c = 6 × 103 𝑀⊙, 𝑡 = 3 Gyr
+into our model, we obtain 𝑓II = 0.01 for 𝑓dep = 1 or 𝑓II = 0.03
+for 𝑓dep = 0.5, such that a small fraction of stars may have been
+enriched by companion accretion. Similarly, a few percent of the
+stars in NGC 2121 may be polluted via companion accretion.
+The marginal cases where no conclusive evidence of multi-
+ple populations have been found among small sample sizes, Terzan
+7 and Palomar 12, sit close to the border where a non-negligible
+fraction of polluted stars are expected. The most challenging case
+is that of Ruprecht 106. In this case, 9 stars have been searched
+for evidence of Na-O abundance variations (Villanova et al. 2013).
+Assuming 𝑓dep = 0.5, our model with the properties of Ruprecht
+106 (𝑅hm = 11 pc, 𝑀c = 8.3 × 104 𝑀⊙, 𝑡 = 12 Gyr) predicts a
+82 percent pristine fraction, which gives a 17 percent probability
+of finding 9 pristine stars randomly, which is not statistically sig-
+nificant. Further, for a small amount of depletion ( 𝑓dep = 1), this
+increases to 95 percent pristine fraction (63 percent chance).
+There remains a curious case in the cluster NGC 6791, which
+has a mass of 𝑀c ≈ 5 × 104 𝑀⊙ and radius 𝑅hm ≈ 5 pc and age
+𝑡 ≈ 7.5 Gyr, for which our model would yield 𝑓I = 88 percent if
+MNRAS 000, 1–29 (2021)
+
+16
+Winter & Clarke
+0
+2
+4
+6
+8
+10
+12
+14
+Age [Gyr]
+10−4
+10−3
+10−2
+10−1
+100
+101
+Mass-radius ratio: (Mc/105 M⊙)/(Rhm/1 pc)
+NGC 419
+(8.1 pc)
+Ruprecht 106
+(11.0 pc)
+Terzan 7
+(8.7 pc)
+Palomar 12
+(16.0 pc)
+Berkley 39
+(11.0 pc)
+NGC 6791
+(5.0 pc)
+NGC 1978
+(8.7 pc)
+Kron 3
+(11.5 pc)
+Lindsay 1
+(19.1 pc)
+NGC 416
+(5.0 pc)
+NGC 339
+(12.7 pc)
+NGC 2121
+(10.8 pc)
+NGC 2155
+(6.5 pc)
+Lindsay 113
+(1.7 pc)
+Lindsay 38
+(7.8 pc)
+MPs
+No MPs
+Uncertain
+Spread
+0.00
+0.15
+0.30
+0.45
+0.60
+0.75
+0.90
+Predicted pristine star fraction (Rhm = 10 pc)
+Figure 7. The distribution of observed clusters that have been searched for evidence of multiple populations (MPs), as compiled by Krause et al. (2016). We
+include only globular clusters with age estimates in that dataset. We have added a number of clusters from the recent works of (Martocchia et al. 2017, 2018a,
+2019; Saracino et al. 2020) and Palomar 12, Terzan 7, NGC 6791 and Ruprecht 106 are designated ‘uncertain’, mostly due to the low sample sizes (see Bastian
+& Lardo 2018, and references therein). From Martocchia et al. (2019), we have denoted Lindsay 113, NGC 2121 and NGC 2155 with empty triangles. These
+clusters exhibit a larger spread in abundances than expected from observational uncertainties, but Gaussian mixture models do not demonstrate the presence of
+discrete populations. The colour bar is a guide as to the enrichment fraction with an assumed present day half-mass radius 𝑅hm = 10 pc and a depletion factor
+𝑓dep that varies such that (1 − 𝑓dep) = 0.5 · 10 Gyr/t for age 𝑡. The names of some notable clusters are indicated, with half-mass radii shown in brackets. Note
+that where the half-mass radius 𝑅hm < 10 pc this can result in reduced pristine fractions at early times with respect to the colourbar estimate (and vice versa,
+see text for details).
+𝑓dep = 1 and 67 percent if 𝑓dep = 0.5. As dicussed by Villanova
+et al. (2018), the nature of NGC 6791 is a controversial topic, it
+being an old cluster with a very high metallicity (Cunha et al. 2015,
+and references therein). Different authors have characterised it both
+as an open cluster, globular cluster, or remnant dwarf galaxy and
+a member of the thick disc, thin disc, or the galactic bulge. While
+Geisler et al. (2012) found evidence of inhomogeneities and a Na-O
+anti-correlation, subsequent studies have not reproduced this find-
+ing. In particular, Villanova et al. (2018) did not find evidence
+of inhomogeneities, despite choosing the same sample as Geisler
+et al. (2012). This was a sample of 17 stars, which with 𝑓dep = 1
+would give a 11 percent chance of a null result, but 0.1 percent if
+𝑓dep = 0.5. Given the peculiarity of this cluster, we do not con-
+sider this a strong counterexample for our model. We conclude that
+the predictions from the companion accretion model for the con-
+ditions under which multiple populations are found remain largely
+consistent with current constraints.
+Finally, we highlight that all of the studies of young clusters
+shown as squares in Figure 7 were performed on post-main sequence
+stars. By contrast, Cadelano et al. (2022) find hints of a N spread in
+the main sequence stars of the 1.5 Gyr old NGC 1783. The absence
+of this signature for evolved stars may be due to first dredge up
+mixing wiping out apparent N differences between their surface
+abundances (Salaris et al. 2020). In the context of the companion
+accretion model, some abundance spread among young MS stars
+may result from surface contamination in a similar manner to the
+early disc accretion scenario (Bastian et al. 2013a).
+3.8.2
+Scaling with cluster mass
+We will now consider how the enrichment fraction correlates with
+the (initial) cluster mass. For this purpose, we use the enrichment
+fractions compiled by Milone et al. (2017). Since the globular clus-
+ters are all old, we here assume that they have all have the same
+depletion factor 𝑓dep = 0.5 rather than compounding uncertainties
+in the dynamical timescales, tidal histories and ages to estimate the
+depletion factor for each cluster individually.
+We show these data in Figure 8, where we also show the
+expected fraction of pristine (unpolluted) stars expected from our
+models, assuming an age 𝑡 = 10 Gyr and either a depletion factor
+𝑓dep = 1 or 0.5 and a half-mass radius 𝑅hm = 10 pc or 1 pc.
+We do not introduce a mass-radius relation since any such trends
+for globular clusters is weak, washed out by scatter (see Figure 9 of
+Krumholz et al. 2019). We find that if no depletion is considered then
+our model still predicts a similar trend with the initial cluster mass,
+however it overestimates the number of unpolluted stars remaining.
+When we adopt a modest 𝑓dep = 0.5, the fraction of unpolluted
+stars decreases due to the preferential removal of these stars. These
+models are in quantitative agreement with observed pristine fraction
+in Milone et al. (2017) dataset. Qualitatively, the fraction of pristine
+stars exhibits a ‘knee’ at the transition from the assumption that all
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+17
+104
+105
+106
+107
+Initial mass of cluster: Mc,0 [M⊙]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fraction of pristine stars
+fdep = 1, t = 10 Gyr, Rhm = 10 pc
+fdep = 0.5, t = 10 Gyr, Rhm = 10 pc
+fdep = 0.5, t = 10 Gyr, Rhm = 1 pc
+Observed, fdep = 0.5
+Figure 8. The number of unpolluted stars based on the compilation of Milone et al. (2017), shown as crosses with assumed depletion factor of 𝑓dep = 0.5 to
+yield the initial masses. The lines represent three different illustrative examples for the outcome of the enrichment model put forward in this work. The solid
+line includes no depletion 𝑓dep = 1, with an assumed age of 10 Gyr and cluster half-mass 𝑅hm = 10 pc. The dashed line is the same, but with a more reasonable
+depletion factor 𝑓dep = 0.5. The dotted line has the same depletion factor 𝑓dep = 0.5 and a smaller 𝑅hm = 1 pc. All models assume an initial semi-major axis
+for the companion of 𝑎0 = 5 au.
+stars that are dynamically ejected are pristine, to a mix of the two
+(see Appendix A). This naturally produces a greater spread in the
+fraction of polluted stars at lower cluster masses, similar to what is
+found empirically. Quantitatively and qualitatively, our model is a
+good description of the enrichment trends with cluster mass.
+3.8.3
+Individual cluster enrichment fractions
+We now consider how well the model we have put forward re-
+produces the enrichment fraction for each cluster based on their
+specific dynamical properties. Specifically, we take the masses and
+half-mass radii of the globular clusters in the compilation by Milone
+et al. (2017) from the database of Hilker et al. (2020), and the ages
+as compiled by (Krause et al. 2016). We then use our model to
+predict, for a fixed depletion factor 𝑓dep = 0.5, what the observed
+fraction of unpolluted stars should be.
+The outcome of this exercise is shown in Figure 9. Immediately,
+we find a clear correlation between the predicted and observed
+pristine star fraction (Spearmann rank correlation 𝑝-value < 10−11).
+To better quantify the agreement, we perform a fitting procedure
+using the linmix package (Kelly 2007). The best fit relationship
+between the model predictions 𝑓model and the observed fractions
+𝑓obs is assumed to take the form:
+𝑓model = 𝑚 𝑓obs + 𝑐,
+(38)
+where 𝑚 = 1 and 𝑐 = 0 correspond to perfect agreement. We obtain
+𝑚 = 1.24 ± 0.13 and 𝑐 = −0.01 ± 0.05, which is remarkably good
+2 https://github.com/jmeyers314/linmix
+agreement given the simplicity of the model. Further excluding
+the clusters NGC 5053 and NGC 5466 that have both undergone
+tidal disruption, as evident from their extended tidal tails (Lauchner
+et al. 2006; Belokurov et al. 2006), we obtain an even better fit
+with 𝑚 = 1.11 ± 0.12 and 𝑐 = 0.03 ± 0.05. The minor tension
+between model and observations could originate from variations
+in the depletion rates, dynamical timescale, or initial semi-major
+axes (for example, via variations in the temperature of the polluted
+medium; see Section 3.3) among other possibilities.
+We can also turn our model around to ask ‘what is the depletion
+factor needed to obtain the observed enrichment rate?’ In order to
+do this, we apply the MCMC implementation emcee3 Foreman-
+Mackey et al. (2013). We adopt a log-likelihood:
+ln L = − ( 𝑓I,mod − 𝑓I,obs)2
+2𝜎2err
+,
+(39)
+where 𝑓I,mod and 𝑓I,obs are the model and observed pristine fractions
+respectively and 𝜎2err = 𝜎2
+mod+𝜎2
+obs is the combined uncertainty. We
+adopt a uniform prior between 10−3 < 𝑓dep < 1, and the resulting
+estimates are listed in Table 1. We find that for the majority of
+clusters 𝑓dep ∼ 0.2−0.6 are sufficient to reproduce the observed
+fractions, with some exceptions.
+In some cases, much smaller 𝑓dep ∼ 0.01 are required under
+our assumed model in order to reproduce the observed pristine star
+fractions. This may be due to extreme depletion in these cases,
+however we also highlight that our model struggles to produce the
+3 https://github.com/dfm/emcee
+MNRAS 000, 1–29 (2021)
+
+18
+Winter & Clarke
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Pristine fraction, observed
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Pristine fraction, model with fdep = 0.5
+linmix fit
+(m = 1.24 ± 0.13, c = −0.01 ± 0.05)
+w/o dis. GCs
+(m = 1.11 ± 0.12, c = 0.03 ± 0.05)
+1:1 agreement
+Non-disrupted GCs
+NGC 5053 and NGC 5466
+Figure 9. The number of unpolluted stars in globular clusters as compiled by Milone et al. (2017), compared with the prediction of our model using the
+dynamical data of Hilker et al. (2020) with a depletion factor 𝑓dep = 0.5 and initial semi-major axes 𝑎0 = 5 au. The uncertainties in the y-axis (model) are
+obtained using the uncertainty in the total cluster mass in the database by Hilker et al. (2020). The red line is the 1 : 1 line, along which the model agrees
+completely with the data. The yellow lines have been obtained using the linmix2fitting procedure (Kelly 2007), the solid line including all datapoints while
+the dashed line excludes the two red points that correspond to the tidally disrupted clusters NGC 5053 and NGC 5466 (Belokurov et al. 2006; Lauchner et al.
+2006).
+lowest pristine fractions 𝑓I ≲ 0.25 due to assumptions in our de-
+pletion model. In particular, for small 𝑓I the depleted population
+transitions from being 100 percent unpolluted stars ( 𝑓I,dep = 𝑓 (1)
+I,min,
+equation A1) to also including enriched stars ( 𝑓I,dep = 𝑓 (2)
+I,min, equa-
+tion A2). By instead setting 𝑓 (2)
+I,min = 0, we can obtain the maximal
+depletion of the pristine population to yield the factor 𝑓dep,max (i.e.
+no polluted stars are lost until no pristine stars are left). Once we
+do this, we obtain the results shown again in Table 1, which now
+show practically all clusters with 𝑓dep,max > 0.25, consistent with
+inferred depletion (Kruijssen 2015; Webb & Leigh 2015). The two
+exceptions are the disrupted NGC 5053 and NGC 5466 (Lauch-
+ner et al. 2006; Belokurov et al. 2006). We therefore conclude that
+the depletion factors required to reproduce the observed fraction of
+unpolluted stars are consistent with empirical constraints.
+3.9
+Abundance distribution synthesis
+3.9.1
+Synthesis procedure
+In this section, we consider the elemental abundance variations
+within individual globular clusters. We first need to estimate the
+composition of the companions that form via accretion of the pol-
+luted ISM. We will assume that accretion is dominated by sweep-up
+by a long-lived disc (see discussion in Section 3.2). In order to esti-
+mate the evolution of the ejecta abundances, we follow an approach
+similar to that of D’Ercole et al. (2010). As in Section 3.4.1, we
+consider the quantity of mass added to the gas reservoir due to the
+evolution of massive stars. We consider the evolution of equation 20
+with the star formation rate:
+�𝑀SF = 𝑀c,0 · 𝜖SF
+𝜏ff
+,
+(40)
+for free-fall time 𝜏ff. The star formation efficiency per free-fall time
+𝜖SF is a free parameter. We will always assume 𝜖SF = 0 initially, but
+‘switch-on’ star formation at a time 𝑡SF defined as the MS lifetime
+for a star of mass 𝑚SF. We then draw new stars at times obtained
+from a probability distribution function proportional to �𝑀SF, with
+a composition of pure ejecta material defined by the composition of
+the ISM at the time they form.
+In order to simulate the evolution of the ISM composition, we
+keep track of the individual elements:
+�𝑀el,𝑘 = 𝛼𝑘 (𝑡) · �𝑀eject − 𝛽𝑘
+� 𝑓lm𝑀c,0⟨ �𝑞sweep⟩ + �𝑀SF
+� ,
+(41)
+where 𝑘 denotes an index referring to a specific chemical element,
+𝛽𝑘 is the current mass fraction of the element 𝑘 in the reservoir, and
+𝛼𝑘 (𝑡) is the mass fraction of element 𝑘 being ejected by the first
+generation stars. We estimate the specific yield of each element by
+interpolating between the model ejecta abundances of AGB ejecta
+(as for D’Ercole et al. 2010). For stars of mass 𝑚∗ > 9 𝑀⊙, we
+use the massive binary ejecta abundances found by de Mink et al.
+(2009).
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+19
+Cluster
+Mass
+Half-mass radius
+Tidal radius
+Age
+Unpol. frac. (obs)
+Unpol. 𝑓dep = 0.5
+Dep. factor
+Maximum dep. factor
+𝑀c [105 𝑀⊙]
+𝑅hm [pc]
+𝑅t [pc]
+𝑡 [Gyr]
+𝑓I,obs
+𝑓I,mod
+𝑓dep
+𝑓dep,max
+NGC 104
+8.95 ± 0.06
+6.30
+126.80
+12.80
+0.175 ± 0.009
+0.25
+0.046+0.010
+−0.008
+0.589+0.005
+−0.005
+NGC 288
+0.93 ± 0.03
+8.37
+94.86
+12.20
+0.542 ± 0.031
+0.59
+0.063+0.413
+−0.025
+0.470+0.024
+−0.023
+NGC 362
+2.84 ± 0.04
+3.79
+91.96
+10.00
+0.279 ± 0.015
+0.24
+0.445+0.127
+−0.197
+0.570+0.010
+−0.010
+NGC 1261
+1.82 ± 0.03
+5.23
+140.39
+10.24
+0.359 ± 0.016
+0.33
+0.516+0.012
+−0.347
+0.519+0.011
+−0.010
+NGC 1851
+3.18 ± 0.04
+2.90
+127.09
+7.64
+0.264 ± 0.015
+0.25
+0.413+0.143
+−0.147
+0.611+0.011
+−0.010
+NGC 2298
+0.56 ± 0.08
+3.31
+74.71
+12.40
+0.370 ± 0.037
+0.48
+0.439+0.035
+−0.035
+0.435+0.038
+−0.034
+NGC 2808
+8.64 ± 0.06
+3.89
+165.91
+11.20
+0.232 ± 0.014
+0.25
+0.380+0.195
+−0.140
+0.681+0.011
+−0.010
+NGC 3201
+1.60 ± 0.03
+6.78
+77.53
+11.10
+0.436 ± 0.036
+0.41
+0.513+0.029
+−0.036
+0.517+0.026
+−0.024
+NGC 4590
+1.22 ± 0.09
+7.58
+77.43
+12.70
+0.381 ± 0.024
+0.49
+0.434+0.022
+−0.021
+0.432+0.023
+−0.022
+NGC 4833
+2.06 ± 0.10
+4.76
+80.15
+12.50
+0.362 ± 0.025
+0.28
+0.549+0.022
+−0.028
+0.552+0.022
+−0.021
+NGC 5024
+4.55 ± 0.32
+10.18
+177.72
+12.70
+0.328 ± 0.020
+0.28
+0.529+0.022
+−0.022
+0.529+0.023
+−0.022
+NGC 5053
+0.74 ± 0.16
+17.31
+97.05
+12.30
+0.544 ± 0.062
+0.91
+0.218+0.042
+−0.037
+0.209+0.046
+−0.047
+NGC 5272
+4.06 ± 0.17
+6.34
+127.28
+11.40
+0.305 ± 0.014
+0.24
+0.565+0.014
+−0.014
+0.566+0.014
+−0.014
+NGC 5286
+3.53 ± 0.12
+3.79
+93.89
+12.50
+0.342 ± 0.015
+0.25
+0.643+0.015
+−0.015
+0.642+0.016
+−0.015
+NGC 5466
+0.60 ± 0.10
+14.03
+79.88
+13.60
+0.467 ± 0.063
+0.87
+0.238+0.035
+−0.030
+0.237+0.039
+−0.039
+NGC 5897
+1.57 ± 0.14
+10.99
+69.58
+12.30
+0.547 ± 0.042
+0.59
+0.473+0.045
+−0.041
+0.474+0.044
+−0.040
+NGC 5904
+3.94 ± 0.06
+5.68
+81.73
+11.50
+0.235 ± 0.013
+0.24
+0.322+0.171
+−0.201
+0.535+0.008
+−0.008
+NGC 5986
+3.34 ± 0.23
+4.25
+65.86
+12.20
+0.246 ± 0.012
+0.25
+0.353+0.160
+−0.174
+0.556+0.019
+−0.020
+NGC 6093
+3.38 ± 0.09
+2.62
+61.98
+12.50
+0.351 ± 0.029
+0.25
+0.697+0.028
+−0.026
+0.698+0.028
+−0.025
+NGC 6101
+1.78 ± 0.32
+14.06
+89.11
+12.50
+0.654 ± 0.032
+0.70
+0.469+0.079
+−0.074
+0.462+0.076
+−0.074
+NGC 6121
+0.87 ± 0.01
+3.69
+53.43
+13.10
+0.285 ± 0.037
+0.41
+0.209+0.215
+−0.125
+0.435+0.018
+−0.016
+NGC 6144
+0.79 ± 0.14
+4.91
+28.18
+13.80
+0.444 ± 0.037
+0.49
+0.475+0.049
+−0.047
+0.471+0.051
+−0.049
+NGC 6171
+0.75 ± 0.04
+3.94
+36.17
+13.40
+0.397 ± 0.031
+0.45
+0.462+0.026
+−0.036
+0.467+0.023
+−0.022
+NGC 6205
+5.45 ± 0.21
+5.26
+134.25
+11.70
+0.184 ± 0.013
+0.25
+0.044+0.016
+−0.011
+0.557+0.011
+−0.012
+NGC 6218
+1.07 ± 0.03
+4.05
+45.27
+13.40
+0.400 ± 0.029
+0.39
+0.508+0.022
+−0.022
+0.510+0.020
+−0.020
+NGC 6254
+2.05 ± 0.04
+4.81
+54.06
+12.40
+0.364 ± 0.028
+0.28
+0.550+0.021
+−0.019
+0.550+0.020
+−0.019
+NGC 6341
+3.52 ± 0.04
+4.49
+120.87
+13.20
+0.304 ± 0.015
+0.25
+0.592+0.011
+−0.011
+0.592+0.011
+−0.010
+NGC 6352
+0.65 ± 0.02
+4.56
+33.58
+12.70
+0.474 ± 0.035
+0.51
+0.475+0.028
+−0.029
+0.476+0.027
+−0.024
+NGC 6362
+1.27 ± 0.03
+7.23
+51.84
+13.60
+0.574 ± 0.035
+0.47
+0.583+0.041
+−0.039
+0.585+0.038
+−0.033
+NGC 6366
+0.38 ± 0.02
+5.56
+34.82
+13.30
+0.418 ± 0.045
+0.64
+0.355+0.027
+−0.288
+0.359+0.025
+−0.022
+NGC 6388
+12.50 ± 0.11
+4.34
+100.55
+11.70
+0.245 ± 0.010
+0.24
+0.553+0.124
+−0.139
+0.726+0.009
+���0.008
+NGC 6397
+0.97 ± 0.01
+3.90
+52.44
+13.40
+0.345 ± 0.036
+0.40
+0.466+0.023
+−0.029
+0.471+0.021
+−0.019
+NGC 6496
+0.69 ± 0.07
+6.42
+28.78
+12.40
+0.674 ± 0.035
+0.57
+0.619+0.068
+−0.059
+0.619+0.068
+−0.060
+NGC 6535
+0.22 ± 0.04
+3.65
+25.75
+10.50
+0.536 ± 0.081
+0.65
+0.427+0.084
+−0.064
+0.424+0.085
+−0.063
+NGC 6541
+2.93 ± 0.09
+4.34
+38.94
+12.90
+0.396 ± 0.020
+0.24
+0.636+0.020
+−0.017
+0.638+0.019
+−0.018
+NGC 6584
+1.02 ± 0.17
+5.37
+55.84
+11.30
+0.451 ± 0.026
+0.45
+0.498+0.049
+−0.048
+0.498+0.050
+−0.050
+NGC 6624
+1.56 ± 0.04
+3.69
+19.45
+12.50
+0.279 ± 0.020
+0.28
+0.263+0.232
+−0.131
+0.499+0.012
+−0.012
+NGC 6637
+1.55 ± 0.18
+3.69
+25.34
+13.10
+0.425 ± 0.017
+0.28
+0.591+0.036
+−0.037
+0.590+0.038
+−0.036
+NGC 6652
+0.48 ± 0.07
+1.96
+18.84
+12.00
+0.344 ± 0.026
+0.40
+0.470+0.037
+−0.032
+0.467+0.038
+−0.037
+NGC 6656
+4.76 ± 0.05
+5.29
+77.20
+12.70
+0.274 ± 0.020
+0.25
+0.376+0.179
+−0.152
+0.591+0.013
+−0.013
+NGC 6681
+1.16 ± 0.02
+2.89
+27.72
+12.80
+0.234 ± 0.019
+0.29
+0.282+0.141
+−0.184
+0.471+0.009
+−0.009
+NGC 6715
+17.80 ± 0.30
+5.20
+283.30
+10.80
+0.267 ± 0.012
+0.24
+0.742+0.031
+−0.145
+0.767+0.012
+−0.011
+NGC 6717
+0.36 ± 0.08
+4.23
+19.80
+13.20
+0.637 ± 0.039
+0.60
+0.539+0.088
+−0.080
+0.541+0.089
+−0.079
+NGC 6723
+1.77 ± 0.11
+5.06
+37.16
+13.10
+0.363 ± 0.017
+0.33
+0.515+0.023
+−0.339
+0.522+0.020
+−0.019
+NGC 6752
+2.76 ± 0.04
+5.27
+67.43
+13.80
+0.294 ± 0.023
+0.24
+0.387+0.150
+−0.207
+0.534+0.014
+−0.013
+NGC 6779
+1.86 ± 0.18
+4.51
+96.87
+13.70
+0.469 ± 0.041
+0.29
+0.629+0.051
+−0.046
+0.629+0.050
+−0.045
+NGC 6809
+1.93 ± 0.08
+6.95
+51.52
+13.80
+0.311 ± 0.029
+0.38
+0.454+0.025
+−0.299
+0.465+0.019
+−0.018
+NGC 6838
+0.66 ± 0.03
+6.57
+43.17
+12.70
+0.622 ± 0.038
+0.58
+0.540+0.046
+−0.037
+0.543+0.047
+−0.040
+NGC 6934
+1.36 ± 0.19
+5.16
+86.40
+11.10
+0.326 ± 0.020
+0.39
+0.470+0.033
+−0.029
+0.466+0.036
+−0.036
+NGC 6981
+0.69 ± 0.12
+5.96
+49.82
+10.90
+0.542 ± 0.027
+0.55
+0.495+0.057
+−0.054
+0.494+0.057
+−0.055
+NGC 7078
+6.33 ± 0.07
+4.30
+140.75
+13.60
+0.399 ± 0.019
+0.25
+0.763+0.020
+−0.019
+0.763+0.020
+−0.019
+NGC 7089
+6.20 ± 0.11
+4.77
+111.48
+11.80
+0.224 ± 0.014
+0.25
+0.199+0.245
+−0.083
+0.607+0.010
+−0.009
+Table 1. Properties of globular clusters as listed by Hilker et al. (2020) and ages listed by (Krause et al. 2016), with empirical unpolluted star fractions as
+listed by Milone et al. (2017). The last three columns are the fraction of these pristine stars in the model presented in this work assuming a total depletion
+factor 𝑓dep = 𝑀c/𝑀c,0 = 0.5, the 𝑓dep inferred from an MCMC exploration with our model, and 𝑓dep,max that is the result of the same MCMC exploration but
+assuming only unpolluted stars are removed from the cluster.
+The cluster properties are taken from Table 1, while we as-
+sume that the initial cluster mass 𝑀c,0 = 5𝑀c, where 𝑀c is the
+present day mass. The scale radius in a Plummer sphere where
+𝑎stars = 𝑅hm/1.3 is fixed and we assume the gas is uniformly dis-
+tributed with 𝑅gas = 𝑎stars. This also determines the central velocity
+dispersion. We then draw a sample of 400 companions that share the
+abundances of the well-mixed ejecta at the time of their formation
+𝑡 = 𝑡form. The formation time is drawn from a probability density
+function which is proportional to ⟨ �𝑞(𝑡)⟩, the mean rate of change
+of the companion mass ratio. We initially fix the mixing fraction
+MNRAS 000, 1–29 (2021)
+
+20
+Winter & Clarke
+101
+102
+Time: t [Myr]
+−1.0
+−0.5
+0.0
+0.5
+1.0
+Abundance: [X/Fe] [dex]
+O
+Na
+Mg
+Al
+Primordial
+Ejecta solution
+AGB transition
+Figure 10. The abundances of elements in the gas reservoir of our model for
+47 Tuc (solid lines) compared to the pristine abundances (horizontal dashed
+lines) as a function of time. Up to 20 Myr we assume gas is expelled from the
+cluster. Afterwards we interpolate over the grid of ejecta used by D’Ercole
+et al. (2010) for AGB stars. All elements are assumed to be instantaneously
+mixed with the existing gas reservoir at each time. The minimum stellar
+mass that contributes is 𝑚min = 6 𝑀⊙. After a star of 𝑚min reaches the end
+of the MS abundances remain fixed.
+𝑓mix = 0.7 when modifying the primary star abundance in order to
+compare to observations, varying only the companion mass-ratio 𝑞.
+However, as per the discussion in Section 3.2.5, the required 𝑞 may
+be significantly smaller if 𝑓mix is smaller by a factor of a few. We
+demonstrate how this works by two different approaches to mod-
+elling M54 (Section 3.9.3) and NGC 2808 (Section 3.9.4). In the
+former, we assume a broad range of 𝑞, while in the latter we assume
+a narrow range of 𝑞 and vary 𝑓mix.
+We do not produce a full model-fitting procedure. This is be-
+cause, as discussed in Section 3.9.5, there are numerous uncertain-
+ties and caveats in the model. This renders any outcomes of a fitting
+exercise difficult to interpret. However, we can vary some choice
+parameters in order to try to qualitatively fit some observed abun-
+dance distributions. In the following, we compare the outcome of
+our simple synthesis model to observed abundances in 47 Tuc, M54
+and NGC 2808. For all these clusters we assume that a fraction
+𝑓I = 0.25 of the present day population had companions that un-
+derwent ionisation rather than accretion, consistent with our model
+predictions.
+3.9.2
+Population synthesis in 47 Tuc
+In 47 Tuc, we will find that we do not need to appeal to residual star
+formation from pure pollutant in order to reproduce the observed
+abundance variations. Nor do we require the contribution of both
+massive binaries and AGB ejecta. We therefore set 𝜖SF = 0 through-
+out, and only include ejecta after 20 Myr, the assumed MS lifetime
+of an AGB star of mass 𝑚max = 9 𝑀⊙. The minimum mass for the
+AGB stars contributing to the ejecta is 𝑚min = 6 𝑀⊙. The resultant
+abundance variations of the ISM and the total mass in the reservoir
+and companions are shown in Figures 10 and 11 respectively. In the
+101
+102
+Time: t [Myr]
+10−3
+10−2
+10−1
+100
+Mass-ratios
+Mgas/Mc,0
+⟨q⟩
+tmin = τMS(mmin)
+Figure 11. Total mass of ejecta in our model for 47 Tuc. The solid line is the
+total mass of ejecta material in the gas reservoir, the dashed line represents
+the average mass-ratio of companions. The dashed red vertical line shows
+when the ejecta contributions to the medium are ‘switched-off’.
+latter, we see that the total gas reservoir never exceeds a few percent
+of the total initial cluster mass, while the average mass ratio of the
+companions eventually reaches 𝑞 ≈ 0.1.
+In order to estimate the abundance variations expected from
+our model, we draw 400 companions with a mass ratio 𝑞 from a
+log-normal distribution with median mass ratio 𝑞1/2 = 0.1 and
+0.5 dex dispersion. We assume pristine abundances [O/Fe] = 0.45,
+Na/Fe] = −0.14, [Al/Fe] = 0.52 and [Mg/Fe] = 0.52, and that
+a merger occurs for 75 percent of the stars. We then compare to
+the abundances of O and Na inferred by Dobrovolskas et al. (2014)
+and Mg and Al by Carretta et al. (2009). We assume observational
+errors of 0.1 dex for O and Na abundances (abudance errors are
+not listed for these elements by Dobrovolskas et al. 2014), and for
+Al and Mg abundances we take the median errors from Carretta
+et al. (2009). We then offset the abundances by amount drawn from
+a Gaussian distribution with standard deviation determined by this
+observational error.
+The companion accretion model is somewhat successful in
+reproducing the abundance variations in 47 Tuc, as shown in Fig-
+ure 12. The outcome is in reasonable agreement to the observed
+distributions, both for O and Na (Figure 12a) and Mg and Al (Fig-
+ure 12b). Some discrepancies between the observed abundances and
+our model are clear. The synthetic O abundances exhibit a disper-
+sion of 𝜎mod = 0.13 dex, only slighly smaller than 𝜎obs = 0.17 dex
+in the Dobrovolskas et al. (2014) sample. For Na, the observed
+and model dispersions agree at 0.15 dex. The scatter in the Mg
+abundance in the model 𝜎mod = 0.12 dex exceeds the observed dis-
+persion 𝜎obs = 0.03, however this is due to the quoted observational
+uncertainties rather than the physical model. The Al dispersion in
+the model (𝜎mod = 0.1 dex) is slightly lower than the observed
+(𝜎obs = 0.16 dex), with a single high Al data point that could be an
+outlier in this case.
+We also show the abundances of He for our model for 47
+Tuc in Figure 13. The maximum dispersion in the He abundances
+inferred from horizontal branch stars is Δ𝑌 ∼ 0.03 (di Criscienzo
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+21
+−0.25
+0.00
+0.25
+0.50
+0.75
+[O/Fe]
+−0.4
+−0.2
+0.0
+0.2
+0.4
+[Na/Fe]
+47 Tuc, Dobrovolskas et al. 2014
+Model
+101
+102
+Time of formation: tform [Myr]
+(a)
+0.2
+0.4
+0.6
+0.8
+[Mg/Fe]
+0.2
+0.4
+0.6
+0.8
+1.0
+[Al/Fe]
+47 Tuc, Carretta 2009
+Model
+101
+102
+Time of formation: tform [Myr]
+(b)
+Figure 12. Synthesised chemical enrichment of 47 Tuc stars according to our
+simple model as described in Section 3.9. We show the assumed primordial
+abundances as black circles. Magenta circles show the abundances inferred
+by Dobrovolskas et al. (2014) for Na and O (Figure 12a) and Carretta et al.
+(2009) for Mg and Al (Figure 12b). A sample of 400 Monte Carlo drawings
+from our model are shown as small circles coloured by the time at which
+the companion formed from the ejecta of massive binaries of AGB ejecta.
+Note that although all stars are assumed to form a companion, 25 percent of
+these companions are ionised rather than accreted. The Na and O errors are
+not listed by Dobrovolskas et al. (2014), so we have assumed observational
+errors of 0.1 dex in order to replicate observational scatter in our model
+sample. For the Al and Mg distributions, we adopt the median errors from
+the Carretta et al. (2009) sample.
+et al. 2010; Gratton et al. 2013). As pointed out by Bastian et al.
+(2015), the largest O depletion values are inconsistent with this
+change in He abundances. Indeed, in Figure 13 we see a tail of stars
+with larger He abundance variations. However, the dispersion in the
+model sample actually remains consistent with the small variation
+inferred from observations. We interpret this success cautiously,
+since some other elements have a somewhat smaller dispersion
+than the observed abundances. However, the upper limit in the He
+abundance dispersion of 0.03 is only in slight tension with the
+D’Ercole et al. (2010) ejecta abundances.
+0.225
+0.250
+0.275
+0.300
+0.325
+0.350
+0.375
+Helium fraction: Y
+0
+10
+20
+30
+40
+50
+60
+Stellar probability density function
+47 Tuc model
+Primordial
+1 σ range
+Max. dispersion obs.
+Figure 13. He distribution obtained from our model for the pollution in 47
+Tuc (as in Figure 12). We show the assumed primordial 𝑌 = 0.248 as a
+vertical solid blue line and maximal dispersion Δ𝑌 = 0.03 as a vertical red
+line inferred from constraints in the spread in the horizontal branch stars (di
+Criscienzo et al. 2010; Gratton et al. 2013). The range of the 1 𝜎 dispersion
+in the model is shown by vertical dashed lines.
+101
+102
+Time: t [Myr]
+−0.6
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Abundance: [X/Fe] [dex]
+O
+Na
+Mg
+Al
+Primordial
+Ejecta solution
+AGB transition
+Figure 14. The abundances of elements in the gas reservoir of our model
+for M54. Line colors and styles have similar meanings as in Figure 10.
+3.9.3
+Population synthesis in M54
+We also apply our simple model to the the RGB star abundances
+in the older, more massive cluster M54 (Carretta et al. 2010). In
+this case, as discussed in Section 3.2.5, we must appeal to residual
+star formation, and possibly some contribution from massive bina-
+ries to the available mass budget in order to produce the observed
+abundances. We here adopt 𝑚max = 11 𝑀⊙, 𝑚min = 4.9 𝑀⊙ and
+𝑚SF = 7.9 𝑀⊙, with a star formation efficiency per free-fall time
+𝜖SF = 4 × 10−4. We assume pristine abundances [O/Fe] = 0.38,
+Na/Fe] = −0.06, [Al/Fe] = 0.11 and [Mg/Fe] = 0.35.
+The abundance evolution of the model ISM is shown in Fig-
+MNRAS 000, 1–29 (2021)
+
+22
+Winter & Clarke
+101
+102
+Time: t [Myr]
+10−3
+10−2
+10−1
+100
+Mass-ratios
+Mgas/Mc,0
+⟨q⟩
+MSF/Mc,0
+tSF = τMS(mSF)
+tmin = τMS(mmin)
+Figure 15. The relative mass in the gas reservoir (solid line) in second
+generation star formation (dotted line) and the average mass ratio (dashed)
+line in the model for M54. The solid red vertical line marks the time where
+star formation is switched on (stars of mass 𝑚SF = 8 𝑀⊙ reach the end of
+the MS), while the dashed red line marks the time where ejecta pollution of
+the environment is switched off (stars of mass 𝑚min = 5 𝑀⊙ reach the end
+of the MS). We assume a star formation efficiency constant 𝜖SF = 4 · 10−4.
+ure 14, while the mass evolution of the components of the system
+is shown in Figure 15. The latter indicates that the final ⟨𝑞⟩ slightly
+exceeds 0.1 and the total mass of stars formed from pollutant, 𝑀SF,
+is a few percent of the initial total cluster mass 𝑀c,0. If these stars
+are all low mass and do not get dynamically ejected from the cluster
+over time (due to formation in the core), then the fraction of them
+at the present day would be:
+𝑓SF =
+𝑀SF
+𝑀SF + 𝑓dep · 𝑓lm · 𝑀c,0
+.
+(42)
+We will here fix the depletion factor 𝑓dep = 0.5 and the fraction
+of low mass stars 𝑓lm = 0.4. In the case of our M54 model, this
+yields approximately 13 percent of present day stars composed of
+pure ejecta.
+Motivated by our results in Section 3.2.5, we adopt a log-
+uniform distribution between log 𝑞min = −1.3 and log 𝑞max = 0.3.
+This distribution of 𝑞 contains too many large 𝑞 values to be consis-
+tent with the mass budget. However, this may be mitigated in several
+ways. As discussed in Section 3.2.5, a somewhat reduced 𝑓mix may
+explain the largest 𝑞 > 1, which we may not form frequently in
+the companion accretion model. When we model the population in
+NGC 2808 in Section 3.9.4, we will use a more moderate range
+of 𝑞 and vary 𝑓mix to demonstrate how this might work. In addi-
+tion, the gas reservoir for forming companions may be centrally
+concentrated, leading to higher 𝑞 companions in the central cluster
+stars that remain bound over Gyr timescales. Finally, the required
+mass-ratio for the highly polluted stars is strongly dependent on the
+model abundances of the pollutants; for example, slightly more Na
+and less O can significantly reduce the maximum required 𝑞.
+We show the abundance variations in the model for M54 in
+Figure 16. We obtain qualitative agreement with the O and Na abun-
+dance distributions as shown in Figure 16a. In particular, the gap
+between the pure ejecta stars and those which have accreted a com-
+panion explains the substructure in the high Na, low O population.
+We obtain a model O dispersion 𝜎mod = 0.24, moderately underes-
+timating the observed 𝜎obs = 0.34. For Na, we obtain 𝜎mod = 0.28,
+close to the observed 𝜎obs = 0.31. In Figure 16b, although the Mg
+distribution is reasonable (𝜎mode = 0.14, 𝜎obs = 0.17), we find
+problems reproducing the Al distribution which is not (strongly)
+bimodal in our model.
+We conclude that aspects of the abundance variations in M54
+can be well-produced by our model, while others suffer from the
+same problems as those of other self-enrichment scenarios. In gen-
+eral, our model can help to alleviate the mass budget problem by
+appealing to both massive binary and AGB ejecta and a variable
+𝑓mix. If 𝑓mix ∼ 0.1−0.3 for the subset of apparently high 𝑞 stars,
+then the observed abundance variations may be reproduced without
+requiring additional mass. However, even in this case, we are not be
+able to reproduce the bimodial Al distribution, or the most extreme
+O depletion in our model. This is not an issue of the mass budget,
+but must require additional physics – for example, deviations from
+the theoretical model ejecta abundances.
+3.9.4
+Population synthesis in NGC 2808
+Our final population synthesis effort is for NGC 2808, in which
+self-enrichment models have particular trouble to explain observed
+abundances (see discussion by D’Ercole et al. 2010). We com-
+pare our model with the O, Na and Mg abundances obtained by
+Carretta (2015) and Al abundances from UVES spectra obtained
+with FLAMES/GIRAFFE (Carretta et al. 2018). The massive bi-
+nary ejecta yield too much Na and O compared with the observed
+abundances in the RGB population of NGC 2808. We therefore
+consider a scenario where little of the massive binary ejecta is re-
+tained, adopting 𝑚max = 9.2 𝑀⊙. We also take 𝑚SF = 8.9 𝑀⊙
+with 𝜖SF = 2.5 × 10−4 and 𝑚min = 5.5 𝑀⊙. We assume pristine
+abundances [O/Fe] = 0.26, [Na/Fe] = 0.0, [Mg/Fe] = 0.37 and
+[Al/Fe] = −0.22. The evolution of the elemental abundances is
+shown in Figure 17, while the mass evolution of the gas reservoir,
+average mass-ratio and residual star formation is shown in Figure 18.
+In this case, due to the smaller mass range of the ejecta contributors,
+we obtain a slightly lower mass budget, with ⟨𝑞⟩ ≈ 0.07.
+We adopt a log-normal distribution in 𝑞 with a median mass-
+ratio 𝑞1/2 = 0.2 that is moderately larger than that obtained directly
+from the integration, this time with 0.2 dex dispersion. In order to
+obtain the strongly polluted stars, we will here adopt a range of 𝑓mix
+rather than a broad range of 𝑞. In order to do this simply, we draw
+𝑓mix uniformly between 0.1 and 0.7, although highlight that this
+choice is arbitrary and intended only to illustrate the influence of
+𝑓mix.
+The results are shown in Figure 19. As in M54, we can qualita-
+tively reproduce some of the features seen in the abundance distri-
+butions. In both the O-Na case and Mg-Al cases, we reproduce the
+overall shape of the distribution. Here we also recover a multi-modal
+distribution in Al due to the clear distinction between the polluted
+and non-polluted stars. However, there is not the same large jump in
+Al abundances for low-Mg stars seen in the observed population in
+Figure 19b. Possible origins for this discrepancy may be that mixing
+of the ejecta in the gas reservoir is inefficient, or if Al production
+is more efficient in massive binaries or AGB stars than predicted.
+Quantitatively, the model abundance dispersions are 𝜎mod = 0.18,
+0.22, 0.12 and 0.36 dex for Na, O, Mg and Al respectively. Com-
+pared with the observed dispersions, 𝜎obs = 0.22, 0.37, 0.15 and
+0.57 dex, Al is the only element that is more than 0.2 dex underesti-
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+23
+−0.5
+0.0
+0.5
+[O/Fe]
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+[Na/Fe]
+M54, Carretta et al. 2010
+Upper limit
+Model
+101
+2 × 101
+3 × 101
+4 × 101
+6 × 101
+Time of formation: tform [Myr]
+(a)
+−0.5
+0.0
+0.5
+[Mg/Fe]
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+[Al/Fe]
+M54, Carretta et al. 2010
+Model
+101
+2 × 101
+3 × 101
+4 × 101
+6 × 101
+Time of formation: tform [Myr]
+(b)
+Figure 16. As in Figures 12, but for a model of M54 as described in the text. Here the observational data points are from Carretta et al. (2010).
+101
+102
+Time: t [Myr]
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Abundance: [X/Fe] [dex]
+O
+Na
+Mg
+Al
+Primordial
+Ejecta solution
+AGB transition
+Figure 17. The abundances of elements in the gas reservoir of our model for
+NGC 2808. Line colors and styles have similar meanings as in Figure 10.
+mated in the model. The need for more Al in the ejecta is common
+to all three clusters we investigate in this work. In order to produce
+this Al, we may appeal to an additional progenitor, such as binary
+mergers for stars with masses lower than 9 𝑀⊙ (see e.g. Wang et al.
+2020a, and discussion Section 3.9.5), to activate the appropriate
+reaction chains.
+Finally, we show the He distribution in Figure 20. This is in
+broad agreement with the observed abundances in the triple main
+sequence in NGC 2808, which has discrete populations with inferred
+𝑌 ≈ 0.248, 0.3 and 0.35−0.40 (Piotto et al. 2007). The exact relative
+sizes and distinctness of these three populations is dependent on the
+star formation efficiency, the companion mass-ratio distribution and
+mixing fractions.
+101
+102
+Time: t [Myr]
+10−3
+10−2
+10−1
+100
+Mass-ratios
+Mgas/Mc,0
+⟨q⟩
+MSF/Mc,0
+tSF = τMS(mSF)
+tmin = τMS(mmin)
+Figure 18. As in Figures 11 and 15 but for a model of NGC 2808. Free
+parameters are: 𝑚max = 9.2 𝑀⊙, 𝑚SF = 8.9 𝑀⊙, 𝑚min = 5.5 𝑀⊙, 𝜖SF =
+2.5 · 10−4.
+3.9.5
+Caveats and the origins of the model short-comings
+The very simple model for chemical enrichment comes with nu-
+merous caveats, which may help or hinder the reproduction of the
+observed abundances. Some such considerations are as follows:
+• AGB (and massive binary) ejecta abundances are model de-
+pendent (e.g. Ventura & D’Antona 2005a,b; Karakas & Lattanzio
+2007). For example, Ventura & D’Antona (2009) predict lower Mg
+and higher Al yields, which would help in the case of the compar-
+isons to the M54 and NGC 2808 abundances. On the other hand,
+the hotter temperatures in massive stars or binary interactions may
+be needed to activate the MgAl chain (for example, see Prantzos &
+Charbonnel 2006; Kobayashi et al. 2020, and references therein).
+Frequent early binary mergers, possibly driven by cluster dynamics
+MNRAS 000, 1–29 (2021)
+
+24
+Winter & Clarke
+−1.0
+−0.5
+0.0
+0.5
+[O/Fe]
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+[Na/Fe]
+NGC 2808, Carretta et al. 2015
+Upper limit
+Model
+101
+2 × 101
+3 × 101
+4 × 101
+6 × 101
+Time of formation: tform [Myr]
+(a)
+−0.2
+0.0
+0.2
+0.4
+[Mg/Fe]
+0.0
+0.5
+1.0
+1.5
+[Al/Fe]
+NGC 2808, Carretta 2015
+Carretta et al. 2018
+Model
+101
+2 × 101
+3 × 101
+4 × 101
+6 × 101
+Time of formation: tform [Myr]
+(b)
+Figure 19. As in Figures 12 and 16, but for a model of NGC 2808 as described in the text.
+0.225
+0.250
+0.275
+0.300
+0.325
+0.350
+0.375
+Helium fraction: Y
+0
+10
+20
+30
+40
+Stellar probability density function
+NGC 2808 model
+Obs. pop. 1
+Obs. pop. 2
+Obs. pop. 3
+Figure 20. The helium abundance distribution obtained from our model
+for NGC 2808. The primordial abundance is assumed to be 𝑌 = 0.248.
+We show the approximate abundances of the three populations identified by
+Piotto et al. (2007) as coloured vertical lines.
+(Wang et al. 2020a), may further contribute to the pollutants, in-
+creasing both Al content and mass budget from which to form com-
+panions. Overall, the predictive power of AGB ejecta abundances is
+limited, and this problem is inherent in all self-enrichment models.
+• The internal sources of pollution via the merging, rotational
+mixing and later dredge-up processes are not accounted for in the
+model. Any combination of these processes may further alter the
+abundances in the collision product from the pristine values, as
+supported by a number of observational constraints discussed in
+Section 3.2.3.
+• Further to the above, if we assume that in some cases a mi-
+nority of stars can have massive polluted companions 𝑞 ∼ 1, then
+the internal burning products of both stars, not just the host, can
+contribute to surface abundance variations.
+3.9.6
+Outlook for producing abundance variations by companion
+accretion
+We have demonstrated that under feasible assumptions, the compan-
+ion accretion model can reproduce many aspects of the observed
+abundance distributions in specific stellar clusters. However, in the
+cases of the massive and old clusters M54 and NGC 2808, we have
+needed to assume some moderate enhancement of the mass ratios
+of the companions, as well as variable mixing and residual star
+formation. There remain uncertainties about the degree of feasible
+mixing and the maximum possible mass ratio of a companion, while
+Al abundances in particular requires appealing to uncertainties in
+AGB ejecta composition. A number of possible avenues exist for
+more detailed study of the evolution of the collision product post-
+accretion. Overall, we conclude that the abundances we obtain from
+our models are promising, but not yet conclusive.
+3.10
+Stellar property trends
+3.10.1
+Binarity correlations
+We expect the mechanism we have described to naturally give rise
+to the observed absence of enrichment among binary stars (e.g.
+D’Orazi et al. 2015). This is because the triple system will prefer-
+entially eject the lower mass component once a high eccentricity is
+excited (e.g. Anosova 1986), and hence lose the potentially polluting
+companion.
+3.10.2
+Central concentration
+In the companion accretion model, we expect that the polluted com-
+panions are accreted most frequently at high velocity dispersions in
+the central regions of a cluster. This would be in line with several
+observed regions (e.g. Simioni et al. 2016). In general, we expect
+the dynamical mixing over time to dilute the polluted population,
+although since our model allows the pollution to occur much later
+than other models, this should be the optimum scenario for produc-
+ing spatial segregation of the two populations. Clusters that do not
+show spatially segregated populations may still be consistent with
+dynamical mixing (e.g. Vanderbeke et al. 2015). It is less obvious
+whether some circumstances may precipitate an outcome in some
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+25
+clusters where the polluted population is actually less centrally con-
+centrated than the pristine stars (e.g. Larsen et al. 2012). The rate
+of companion capture is dependent on the local density and veloc-
+ity dispersion of the stars and gas reservoir and the localised star
+formation rate. Hence it is possible that in some birth environments
+the polluted material is accreted most efficiently outside of the high
+velocity central regions. However, the nature of this scenario is de-
+pendent on the (turbulent) properties of the wind ejecta, and we do
+not explore this further here.
+3.11
+Summary and caveats
+3.11.1
+Successes of the companion accretion model
+We have presented a model for producing multiple stellar popula-
+tions in globular clusters that hinges on the accretion of polluted
+sub-stellar companions due to eccentricity excitation by stellar en-
+counters. This mechanism allows us to make quantitative predic-
+tions for the rate of enrichment, which agree with those observed.
+Our model offers the following successes, many of which are not
+achieved by existing models:
+• Presence of absence of multiple populations: The rate of ac-
+cretion of companions is a predictable consequence of averaged
+dynamical encounters. In general our model can accurately predict
+which clusters exhibit evidence of multiple populations. In a small
+number of clusters there is some minor in tension between our model
+and the presence or absence of multiple populations. However, this
+is only in clusters where the sample sizes are small (Ruprecht 106)
+or where the populations are not unambiguously identified (NGC
+2155).
+• Age correlation: The enrichment in the companion accretion
+model typically occurs on Gyr timescales, such that we expect an
+absence of polluted stars for young clusters that is consistent with
+observations (Martocchia et al. 2018a).
+• Scaling of pollution fraction with cluster properties: Our model
+is in excellent agreement with the observed fractions of polluted
+populations, reproducing similar scalings with the cluster properties
+for an assumed depletion factor 𝑓dep = 0.5. Conversely, the observed
+fractions can be reproduced by appealing to moderate depletion
+factors, in contrast to many multi-generational models (e.g. Conroy
+& Spergel 2011).
+• Binarity: Our model predicts lower rates of polution among
+binary stars (e.g. D’Orazi et al. 2015), since a lower mass companion
+in an unstable configuration would be dynamically ejected rather
+than accreted.
+• Discreteness: We have shown in Section 3.9 that whether or
+not the companion merges with its host, and different ejecta mate-
+rial and mixing fractions can produce complexity in the elemental
+‘chomosome’ maps (Milone et al. 2017) and the multi-modal He
+distribution as inferred from the multiple main sequences in some
+clusters (e.g. Piotto et al. 2007).
+There are further possible advantages of our model that remain
+more speculative:
+• Dilution and mass budget: In the scenario we explore in this
+work, the dilution of the heavier elements occurs within the pristine
+star itself, long after all the ejecta from massive and AGB stars in
+the cluster has been expelled or accreted. We have shown that a
+mass ratio 𝑞 ∼ 0.1 of a companion composed of polluted mate-
+rial mixed through 𝑓mix = 70 percent of the primary is sufficient
+to produce typical abundance variations in some clusters (47 Tuc
+is explored here). However, late collapse of the gas reservoir and
+variable mixing fractions are required to explain the most extreme
+abundance variations while maintaining a small 𝑞. In the old and
+massive clusters M54 and NGC 2808, similar abundance variations
+to those observed can be reproduced in this way, but we must also
+appeal to order unity enhancements in the typical mass-ratios of
+companions. These may be possible if the gaseous reservoir is cen-
+trally concentrated. Alternatively additional physics may contribute
+to surface abundance variations, such as mixing internal burning
+products in the primary into the convective region as a result of the
+merger.
+• Dynamical removal of pristine stars: Two-body encounters that
+would ionise the companion from its host star may be correlated to
+those that impart large amounts of kinetic energy into the trajectory
+of a star. It is possible that this offers a mechanism for preferential
+depletion of the pristine population, on top of the central concen-
+tration of the stars that are expected to undergo collisions with their
+companion. However, this possibility must be investigated with de-
+tailed dynamical models.
+3.11.2
+Uncertainties and caveats
+While the accretion rate of companions at a given stellar density
+and velocity dispersion is well-quantified as a function of time in
+this work, the feasibility of a number of aspects of the model require
+more detailed examination in future. The most serious of these are:
+• Formation of the sub-stellar companion(s): We have invoked a
+number of possible mechanisms to produce sub-stellar companions,
+but have particularly adopted the rate at which a long-lived, compact
+disc sweeps-up material (cf. Bastian et al. 2013a). However, the gas
+capture scenarios we discuss require follow-up with hydrodynamics
+simulations. This formation process may also be in competition with
+ram pressure stripping of the (structured) gas reservoir (Chantereau
+et al. 2020).
+• Stellar mixing: The chemical signatures in our model are de-
+pendent on the degree of mixing. We have appealed to simulations
+of grazing, high mass ratio mergers and mixing performed by Lom-
+bardi et al. (2002) and Cabezón et al. (2022). These simulations
+suggest that deep mixing through the majority of the star is possi-
+ble, but depends on the nature of the collision. Further quantification
+of the degree and homogeneity of the rotational mixing post-merger,
+and the subsequent return to the MS are required to fully test the
+expected changes to the surface abundances.
+• Lithium survival: Linked to the above, it is unclear whether
+we should expect lithium to survive after a merger with a low mass
+companion. The models of Lombardi et al. (2002) and Cabezón
+et al. (2022) show survival of lithium on the stellar surface during
+the merger. However, these models have not been performed for an
+initially Li depleted companion, while the long term survival is also
+dependent on subsequent mixing.
+• Ejecta content: We have assumed that it is possible to produce
+the required pollutants out of some combination of the massive
+and AGB ejecta. In general, the wind abundances, particularly for
+AGB stars, are highly uncertain (e.g. Ventura & D’Antona 2005a,b;
+Renzini 2013). This remains true for any scenario that invokes first
+generation stars within the cluster as a source for abundance varia-
+tions.
+• Initial eccentricity of the companion: We have assumed an ini-
+tial eccentricity of the companion 𝑒0 = 0.5, which the rate at which
+collisions occur in the cluster. We make this choice as the mean of
+a uniform eccentricity distribution, which appears to characterise
+binaries of separation ∼ 100 au (Hwang et al. 2022). Such binaries
+MNRAS 000, 1–29 (2021)
+
+26
+Winter & Clarke
+may have formed via a similar process of fragmentation of a disc.
+However, it is unclear if such a process is similar for the formation
+process invoked in the context of this work.
+3.11.3
+Future tests of the companion accretion model
+While the companion accretion model is feasibly consistent with
+current observation constraints, future empirical tests may falsify
+the companion accretion model:
+• Companions in stellar clusters: By far the most direct test of
+the model is the search for 𝑞 ∼ 0.1 companions in surveys of mas-
+sive clusters aged ∼ 1−4 Gyr, before most have been ionised or
+accreted. The main problem with this is that long baseline mea-
+surements are required to detect companions at ∼ 5 au separations
+in radial velocity surveys. Unambiguously identifying companions
+becomes particularly problematic for high orbital eccentricities in
+dense clusters, when dynamical interactions with neighbours may
+also accelerate the star. However, such an investment may have
+numerous benefits beyond a test of the companion accretion model.
+• Chemistry of RGB ‘companions’: As discussed in the introduc-
+tion, some RGB stars appear to host companions that may be under-
+going disruption or exchange of material with their host (Soszyński
+et al. 2021). These companions appear to also exist in globular clus-
+ters (Percy & Gupta 2021). If these examples represent the leftover
+survivors of companion accretion, then studying these stars and
+their companions for abundance variations may offer a window into
+this process.
+• Companion merger simulations: Further testing of the com-
+panion accretion model may be based on predictions from detailed
+simulations for the mergers, including the post-merger evolution
+of the remnant. These simulations may be compared quantitatively
+with observational constraints from Hertzsprung-Russell contours,
+for example (cf. the early interactions of massive close binaries by
+Wang et al. 2020b).
+4
+CONCLUSIONS
+In this work, we have explored a novel mechanism for producing
+multiple stellar populations in globular clusters: ‘the companion
+accretion model’. In this scenario, the enriched wind ejecta from
+massive and AGB stars forms a bound gaseous reservoir, which re-
+mains unable to form stars efficiently for ∼ 100 Myr timescales (e.g.
+Conroy & Spergel 2011). Stars moving through this medium may
+accrete gas from this reservoir by a combination of disc-sweeping,
+tidal cloud capture and Bond-Hoyle-Lyttleton accretion. This ma-
+terial may subsequently cool and collapse to produce a sub-stellar
+companion(s) of mass ratio 𝑞 ∼ 0.1. Over Gyr timescales, compan-
+ions undergo eccentricity excitations due to dynamical encounters
+that can result in a grazing collision with the host star (Kaib &
+Raymond 2014; Hamers & Tremaine 2017; Winter et al. 2022a).
+Such a collision may inject the pollutant in the secondary (Lombardi
+et al. 2002; Cabezón et al. 2022), which can broadly produce typi-
+cal abundance variations in globular clusters. Extreme abundances
+may be produced by the leftover pollutant that remains bound and
+is allowed to cool and collapse to form a small second population
+from pure pollutant in some clusters.
+The fraction of pristine stars in this model can be calculated,
+in a given environment, by considering the ratio in which stellar en-
+counters induce ionisations compared with collisions, also factoring
+in the effect of dynamical depletion of ionised (hence pristine) sys-
+tems from the cluster core. We find that the predicted fraction of
+pristine stars as a function of environment is in good agreement
+with observations, requiring only moderate (factor two) depletion
+of the stellar mass at the present day.
+This model has numerous benefits over existing models. It
+provides a natural way to produce ubiquitous multiple populations
+in dense and old clusters (e.g. Milone et al. 2017) and a dearth
+of multiple populations in young clusters (e.g. Martocchia et al.
+2018a). Companion accretion offers a comparatively simple dilution
+mechanism, without needing to retain primordial gas (e.g. D’Ercole
+et al. 2011). Most notably, the companion accretion model predicts
+a correlation of the enrichment fraction with cluster mass, or more
+precisely the internal velocity dispersion, since ionisation of the
+companions is less efficient for high velocity encounters.
+There remain some important assumptions and possible
+sources of tension between our model and the observational con-
+straints. For example, resolving the mass budget problem requires
+a variable mixing fraction, residual star formation and moderate
+enhancements to the average 𝑞 for stars that survive in a globular
+cluster to the present day. Particular future attention may be given
+to an exploration of the initial accretion and gravitational collapse
+of the companion from the polluted ejecta in the turbulent medium
+(e.g. Kuffmeier et al. 2020), the mixing of pollutant in the primary
+(e.g. Lombardi et al. 2002; Cabezón et al. 2022), the uncertainties
+in the yield of elements in massive star and AGB ejecta (e.g. Ven-
+tura & D’Antona 2005a,b; Renzini 2013), and the stripping of the
+structured gas reservoir (Chantereau et al. 2020). Each constitutes a
+complex process, for which the conclusions of this manuscript offer
+motivation for further consideration.
+DATA AVAILABILITY
+The data used in this manuscript is all publicly available. Any scripts
+used for making figures shown are available from the corresponding
+author upon reasonable request.
+ACKNOWLEDGEMENTS
+We sincerely thank the anonymous referee for an extremely useful
+referee report, which had a substantial impact on aspects of the
+model presented in this work as well the clarity of the manuscript.
+AJW thanks the Institute of Astronomy, Cambridge for the funding
+for the visit in which this manuscript was written. We further thank
+Chris Tout and Arnab Sarkar for useful discussion regarding stellar
+mixing. In addition, AJW thanks Kees Dullemond, Maria Berge-
+mann and Ivan Cabrera-Ziri for discussions on aspects of the topics
+covered in this manuscript. This project has received funding from
+the European Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation programme (PROTOPLAN-
+ETS, grant agreement No. 101002188).
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+APPENDIX A: PRISTINE STAR DYNAMICAL
+DEPLETION
+We wish to quantify the relative depletion of the pristine and polluted
+population. Unfortunately, without more information, equation 35
+is the only constraint available to us linking 𝑓I,dep and 𝑓II,dep, so that
+to go further we must make a choice on the form of these functions.
+A minimum for 𝑓I,dep can be found by enforcing 𝑓II,dep ≤ 1,
+such that:
+𝑓I,dep ≥ 𝑓 (1)
+I,min = 𝑓dep ·
+1 − 𝑃coll/ 𝑓dep
+1 − 𝑃coll
+.
+(A1)
+When the fraction of unpolluted stars (1 − 𝑃coll) is large, adopting
+𝑓I,dep = 𝑓I,min (i.e. all lost stars are not polluted) makes sense, since
+the vast majority of lost stars should anyway be unpolluted. This is
+because polluted stars will preferentially occupy the core region of
+the cluster, where they are most likely to remain bound, as discussed
+in Section 3.7.
+As 𝑃coll grows, possibly exceeding 𝑓dep to yield 𝑓I,dep < 0,
+then this choice must eventually underestimate the loss of the pol-
+luted population II stars. Thus, we impose another minimum:
+𝑓I,dep ≥ 𝑓 (2)
+I,min = 𝑓dep · 𝑃coll.
+(A2)
+We choose this minimum because it remains greater than zero for
+all 𝑓dep > 0 and 𝑃coll > 0, as well as scaling with, but is always
+smaller than, the overall 𝑓dep (as we expect). This minimum also
+MNRAS 000, 1–29 (2021)
+
+Multiple populations via sub-stellar accretion
+29
+scales with the overall fraction of polluted stars, which means that
+as the enrichment fraction grows the unpolluted stars become com-
+paratively less likely to be ejected. This is what we expect, given
+that a large 𝑃coll should result in less central concentration of the
+polluted stars.
+In practice, we take the maximal of these two minima as 𝑓I,dep,
+from which we can obtain the corresponding 𝑓II,dep for any given
+𝑓dep and 𝑃coll. The scaling of many of our results do not depend
+strongly on this choice, however we highlight that the form of 𝑓I,dep
+and 𝑓II,dep should be the study of further work. They are uncertain
+here as they are for other models for multiple populations.
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–29 (2021)
+