A Comparison of Rotating and Binary Stellar Evolution Models: Effects on Massive Star Populations
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A Comparison of Rotating and Binary Stellar Evolution Models: Effects on Massive Star Populations
Trevor Z. Dorn-Wallenstein and Emily M. Levesque University of Washington Astronomy DepartmentPhysics and Astronomy Building, 3910 15th Ave NESeattle, WA 98105, USA (Received; Revised; Accepted)
Submitted to ApJABSTRACTBoth rotation and interactions with binary companions can significantly affect massive star evolution,altering interior and surface abundances, mass loss rates and mechanisms, observed temperatures andluminosities, and their ultimate core-collapse fates. The Geneva and BPASS stellar evolution codesinclude detailed treatments of rotation and binary evolutionary effects, respectively, and can illustratethe impact of these phenomena on massive stars and stellar populations. However, a direct comparisonof these two widely-used codes is vital if we hope to use their predictions for interpreting observations.In particular, rotating and binary models will predict different young stellar populations, impactingthe outputs of stellar population synthesis (SPS) and the resulting interpretation of large massivestar samples based on commonly-used tools such as star count ratios. Here we compare the Genevaand BPASS evolutionary models, using an interpolated SPS scheme introduced in our previous workand a novel Bayesian framework to present the first in-depth direct comparison of massive stellarpopulations produced from single, rotating, and binary non-rotating evolution models. We calculateboth models’ predicted values of star count ratios and compare the results to observations of massivestars in Westerlund 1, h + χ Persei, and both Magellanic Clouds. We also consider the limitations ofboth the observations and the models, and how to quantitatively include observational completenesslimits in SPS models. We demonstrate that the methods presented here, when combined with robuststellar evolutionary models, offer a potential means of estimating the physical properties of massivestars in large stellar populations.
Keywords: binaries: general, stars: rotation, stars: statistics, stars: massive, galaxies: stellar content INTRODUCTIONRotation is a ubiquitous property of stars, and has asignificant effect on the physical properties of massive( M ini (cid:38) M (cid:12) ) stars. Rotationally induced mixing viadiffusion and meridional circulation (Zahn 1992) altersstellar interiors enhances surface element abundances,centrifugal forces alter stellar shapes (von Zeipel 1924),and both radiative (Maeder & Meynet 2000a) and me-chanical (Georgy 2010) mass loss are boosted by rota-tion. The Geneva code (Ekstr¨om et al. 2012; Georgyet al. 2013; Groh et al. 2019) is the current state of the Corresponding author: Trevor Z. [email protected] art implementation of rotating stellar evolutionary mod-els, and has been used to study the impact of rotationon the radiative output (Levesque et al. 2012), chemicalyields (Hirschi et al. 2005), and final fates (Meynet et al.2015) of massive stars.Simultaneously, many massive stars are born intobinary and higher-order systems (Sana et al. 2012;Duchˆene & Kraus 2013; Sana et al. 2014; Moe & DiStefano 2017), and binary systems with dwarf, super-giant, Wolf-Rayet, and compact components are com-monly observed (Sana et al. 2013; Neugent & Massey2014; Neugent et al. 2018a, 2019). Interactions in bi-nary systems through both tides (Hurley et al. 2002)and mass transfer can completely disrupt the evolutionof both stars, creating otherwise-impossible evolution-ary states. Both the detailed makeups and integrated a r X i v : . [ a s t r o - ph . S R ] A p r Dorn-Wallenstein & Levesque spectral energy distributions of populations of massivestars are affected, especially at low metallicity (Stanwayet al. 2016). The Binary Population and Spectral Syn-thesis code (BPASS, Eldridge et al. 2017; Stanway &Eldridge 2018) is the current state of the art rigorousimplementation of these effects.Despite the importance of binary interactions and ro-tation, the theory of single, nonrotating stars has thus-far been successful at predicting the evolution of starsof varying compositions and initial masses (the “Contiscenario”, Conti et al. 1983). The addition of the effectsof rotation has further refined single-star evolution to re-produce observed surface abundance enhancements andmass loss rates (e.g., Meynet & Maeder 2000). Thesesuccesses are often used as evidence that binary inter-actions are a secondary effect, or can be approximatedby simple methods (e.g., by instantaneously removinga model star’s H envelope after it leaves the main se-quence; for a review of recent stellar evolution modelsfeaturing simple implementations of binary interactions,see Eldridge et al. 2017, and citations therein). Indeed,at first glance some of the effects predicted by binarystellar evolution models (such as enhanced surface abun-dances, stronger mass loss, broader mass and age rangesfor Wolf-Rayet progenitors, and harder ionizing spec-tra produced by stellar populations) are identical to thepredictions from rotating stellar models (e.g. Levesqueet al. 2012). Regardless, massive binary systems do ex-ist, necessitating detailed modeling of their evolution aswell as observations of the frequency of binary systemsto understand their impact on stellar populations.Finding binary systems via radial velocity variability(e.g., Sana et al. 2012; Sana et al. 2013) requires longtime baselines, high spectral resolution, and long inte-gration times, and is currently infeasible for stars be-yond the Magellanic Clouds (MCs), or for some starssuch as red supergiant binaries (Neugent et al. 2019).For more-distant systems, we are left studying (semi-)resolved stellar populations. Simultaneously, measur-ing rotation periods requires high-cadence time-seriesphotometry (Blomme et al. 2011; Buysschaert et al.2015; Balona et al. 2015; Balona 2016; Johnston et al.2017; Ramiaramanantsoa et al. 2018; Pedersen et al.2019). Spectroscopic measurements yield projected ro-tational velocities, with the caveat that the inclinationof the rotational axis to the line of sight is unknown (e.g.,Huang et al. 2010). In either case, both methods againrequire high quality observations of nearby stars. Fortu-nately, massive stars are luminous, and can be resolvedin galaxies around the Local Group and beyond. Previ-ously, we used the BPASS models to construct grids ofsynthetic populations with varying metallicity ( Z ), star formation history (SFH), and the natal binary fraction( f bin ), and predicted the frequency of various evolution-ary phases (Dorn-Wallenstein & Levesque 2018, Paper Ihereafter). Here, we incorporate stellar evolution mod-els that include rotation (Ekstr¨om et al. 2012; Georgyet al. 2013; Groh et al. 2019), and perform a detailedcomparison between the two model sets.In §
2, we compare the two evolutionary codes, fromthe BPASS and Geneva groups, used to generate binaryand rotating stellar tracks respectively. We also detailthe population synthesis method we employ to gener-ate theoretical rotating and binary stellar populationsand subsequent predictions for the frequency of variousspectral types in these populations. In § § CREATING THEORETICAL POPULATIONS2.1.
The Models
In Paper I, we created synthetic populations usingBPASS version 2.2.1, which incorporates the effects ofboth tides and mass transfer to predict the evolution ofsingle and binary stars on a dense grid of initial primaryand secondary masses ( M and M ), initial periods P ,and mass ratios ( q ≡ M /M
1) at 13 metallicities. Weexpress the metallicity as a mass fraction Z , and BPASSadopts metallicities in the range 10 − ≤ Z ≤ .
04. Notethat for the duration of this paper, we assume solarmetallicity Z (cid:12) = 0 .
014 (Asplund et al. 2009). Binaryinteractions are modeled as enhanced mass loss/gainfrom/onto its model stars via Roche Lobe Overflow(RLOF). The orbital energy is tracked throughout, al-lowing BPASS to model a broad range of simulated evo-lutionary scenarios.Here, we also incorporate the Geneva evolutionarytracks. Stellar evolution is modeled at two differentinitial rotation rates (nonrotating, with v ini /v crit = 0,and rotating, with v ini /v crit = 0 .
4) and at three dif-ferent metallicities: Z = 0 .
014 (Ekstr¨om et al. 2012), Z = 0 .
002 (Georgy et al. 2013), and Z = 0 . nterpreting Star Count Ratios in Stellar Populations Table 1.
Model parameters used to label model timesteps with an evolutionary phase, and other variables introducedin the text.
Parameter Description Unitlog( L ) Logarithm of the luminosity L (cid:12) log( T eff ) Logarithm of the effective temperature Klog( g ) Logarithm of the surface gravity cm s − X Hydrogen Surface Mass Fraction - Y Helium Surface Mass Fraction - C Carbon Surface Mass Fraction (Sum of C and C for Geneva tracks) - N Nitrogen Surface Mass Fraction - O Oxygen Surface Mass Fraction -log t Logarithm of time yr Z Mass fraction metals, Z (cid:12) = 0 .
014 - f bin Binary fraction - f rot Rotating fraction - f Generic term for either f bin or f rot - n S Observed frequency of an arbitrary spectral type S - R S /S Observed ratio of the frequency of two spectral types, S and S -ˆ n S Intrinsic frequency of an arbitrary spectral type S -ˆ R S /S Intrinsic ratio of the frequency of two spectral types, defined as ˆ R S /S ≡ ˆ n S / ˆ n S -bined following Chaboyer & Zahn (1992). Angular mo-mentum transport is included, and angular momentumis conserved following Georgy (2010). Finally, rotation-enhanced radiative (Maeder & Meynet 2000a) and me-chanical (Georgy 2010) mass loss are also implemented;for a detailed disucssion see (Ekstr¨om et al. 2012).2.2. Population Synthesis
Evolutionary tracks in hand, we synthesize a popula-tion by weighing each track with the initial mass func-tion, Φ( M ). We adopt the default form of Φ in BPASSv2.2.1, which is a broken power law with slope -1.3 be-low 0.5 M (cid:12) , and a slope of -2.35 for higher masses, witha minimum mass of 0.1 M (cid:12) and a maximum mass of 300 M (cid:12) , normalized so the total stellar mass is 10 M (cid:12) . Bi-nary models are also weighted according to the distribu-tions of the fundamental natal period P and mass ratio q from Moe & Di Stefano (2017). Because the Genevatracks are sampled on a much coarser grid of initial massthan the BPASS single star tracks, we linearly interpo-late the available Geneva tracks onto the BPASS sin-gle star initial mass grid following Georgy et al. (2014),and adopt the IMF weighting from the BPASS v2.2.1 inputs . Additionally, both rotating and nonrotatingtracks at all three metallicities are only available be-tween 1.7 and 120 M (cid:12) . We choose not to introduce anycorrection factors to the IMF weights — e.g., boost-ing the weight of the 120 M (cid:12) model to represent allstars with initial masses between 120 and 300 M (cid:12) —to ensure that tracks with identical masses are weightedidentically. Because no single star below the 1.7 M (cid:12) threshold would become any of the stellar types consid-ered here, only the exclusion of these very massive starswould affect our results. However, these stars are sorare, and their lifetimes so short, that their impact onour synthetic populations is minimal.We create four sets of synthetic populations: one com-posed entirely of single, nonrotating stars using the in-put files provided in the BPASS v2.2 data release ( f bin =0), one composed entirely of binary stars ( f bin = 1)using custom input files provided by the BPASS team(J. J. Eldridge 2018, private communication), one com-posed of single, nonrotating stars from the Geneva mod-els (hereafter referred to as the population with “rotat-ing fraction” f rot = 0), and one composed entirely of Details on the mass grids, parameter values and more can befound in the BPASS v2.2 User Manual, currently hosted online atbpass.auckland.ac.nz
Dorn-Wallenstein & Levesque rotating stars ( f rot = 1). We note that there are nosingle stars in the custom f bin = 1 population, thoughthe distribution of periods and mass ratios is identicalto the default BPASS v2.2 binary population, which isdrawn from Moe & Di Stefano (2017).2.3. Number Counts vs. Time
Photometric surveys of nearby massive stars (e.g., theLocal Group Galaxy Survey, LGGS; Massey et al. 2006,2007), can yield fairly complete catalogs after filteringfor foreground contaminants (e.g., Massey et al. 2009),and follow-up narrow-band surveys can be used to findevolved emission line stars (e.g., Neugent & Massey 2011and Neugent et al. 2012, 2018b). Even without follow-up spectroscopy, photometric measurements can then beused to categorize stars into broad spectral types. Thusit is useful to classify all timesteps of a given evolution-ary track into one of a number of coarse spectral types,using the position of the model timestep on the HR di-agram, as well as its surface composition, based on themodel parameters listed in Table 1; we largely adapt theclassification scheme from Eldridge et al. (2017), shownin Table 2. Georgy et al. (2013) use a similar classifica-tion scheme, with slightly different temperature or com-position thresholds. All labels refer explicitly to spectraltypes. WNH corresponds to Hydrogen-rich Wolf-Rayetstars; O f stars are O stars with particularly strongwinds and He II emission (Brinchmann et al. 2008). TheBSG, YSG, RSG, and WR numbers are computed bysumming the numbers of the indicated species, and ap-plying the relevant luminosity threshold, as describedbelow.Where they are different, we adopt the Geneva crite-ria to classify the Geneva tracks, and use the BPASScriteria in cases where the BPASS classification is morespecific than the Geneva classification (e.g., BPASS dis-tinguishes between K and M stars). This choice is moti-vated by two facts. Firstly, in some cases, these criteriaare used within the individual evolutionary tracks to dis-tinguish between different prescriptions for, e.g., mass-loss. Secondly, the coupling between the outermost layerof a model star and a model stellar atmosphere in orderto produce a synthetic spectrum is non-trivial. Indeed,the criteria for classifying WR stars have nothing to dowith the mass-loss rate, which might be observed fromsuch a spectrum. In lieu of synthesizing a spectrum foreach timestep of each model, we defer to the creatorsof each code in how to best classify their models. How-ever, it should be noted that the exact choice of thevalues presented in Table 2 do not drastically affect theresults (Meynet & Maeder 2003). When we attemptedto classify the Geneva tracks using the BPASS criteria, we found little substantive changes, except the rotatingGeneva models produce more WN stars (still less thanhalf the WNs produced by BPASS).We then use the classification, age, and weight of eachtrack to find the frequency of each spectral type in Ta-ble 2 as a function of time, binned to 51 time bins thatare logarithmically-spaced between 10 and 10 yearsin 0.1 dex increments as described in Paper I. We alsoassign each model to one of 31 luminosity bins with 0.1dex width between log( L ) = 3 and 6. Thus we can ap-ply coarse luminosity thresholds to mimic observationalcompleteness limits. This allows us to account for unre-solved binaries by making the assumption that all starsbelow a given luminosity, including secondaries, are notdetected, while all stars above this threshold are. Whilea somewhat simplistic assumption, the evolved stagesof massive star evolution are so short-lived that mostevolved massive stars have main sequence companions(e.g., Neugent et al. 2018a, 2019). Of course, binarieswith two evolved components do exist (e.g., WR+WRbinaries), but are usually detectable via their wind in-teractions. In the case that spectral observations are ofinsufficient SNR or resolution to classify the componentsof such a binary, the WC/WN ratio would be unreliable.We encourage observational efforts dedicated to makinga census of the massive component of stellar populationsto discuss their insensitivity to unresolved binaries.Each column in Figure 1 shows the number, ˆ n , of thespectral types in Table 2. For clarity, values on the y-axis are not shown, but both lines in a given panel areplotted on the same, linear scale. Line styles are usedto indicate the four different populations, with dash-dotted lines for f bin = 0, dashed for f bin = 1, dotted for f rot = 0, and solid for f rot = 1. The top row correspondsto the single, nonrotating populations from both evolu-tionary codes, the second row shows the f bin = 0 and f bin = 1 populations, the third row shows the f rot = 0and f rot = 1 populations, and the bottom row shows the f bin = 1 and f rot = 1 populations. Note that, as in Pa-per I, we do not include Luminous Blue Variables in ouranalysis. This is due to the present uncertainty in theevolutionary status of LBVs (Smith & Tombleson 2015;Humphreys et al. 2016; Aadland et al. 2018), and thelack of a clear consensus in how to observationally clas-sify a statistically significant number of them withoutlong-term monitoring.2.4. Diagnostic Ratios
Because the calculated number counts are the fre-quency of each subtype per 10 M (cid:12) of stars formed,direct application to observed populations requires anestimate of the stellar mass of a population, M ∗ . Such nterpreting Star Count Ratios in Stellar Populations O f b i n = f r o t = BB S G Y S G R S G W R W N W C f b i n = f b i n = f r o t = f r o t = f b i n = f r o t = l og t ˆ n Figure 1.
Number of various stellar subtypes at Z = 0 .
014 between 10 and 10 . years. The top row compares the f bin = 0population (dash-dotted) with the f rot = 0 population (dotted), the second row compares the f bin = 1 (dashed) and f bin = 0(dash-dotted) populations, the third row compares the f rot = 1 (solid) and f rot = 0 (dotted) populations, and the bottom rowcompares the f bin = 1 (dashed) population with the f rot = 1 (solid) population. Both lines in a given panel are plotted on thesame linear y-scale to allow for comparison. Dorn-Wallenstein & Levesque
Table 2.
Criteria used to classify evolution tracks. Adapted from Table 3 of Eldridgeet al. (2017). We specify where the Geneva tracks are classified with different criteriathan the BPASS tracks.
Label BPASS Criteria Geneva CriteriaWNH log( T eff ) ≥ . X ≤ . T eff ) ≥ . X ≤ . T eff ) ≥ . X ≤ − ( C + O )) /Y ≤ .
03 log( T eff ) ≥ . X ≤ − N > C
WC log( T eff ) ≥ . X ≤ − ( C + O )) /Y > .
03 log( T eff ) ≥ . X ≤ − N ≤ C O log( T eff ) ≥ .
48 log( T eff ) ≥ . f log( T eff ) ≥ . g ) > .
676 log( T eff ) + 13 . . ≤ log( T eff ) < .
48 4 . ≤ log( T eff ) < .
5A 3 . ≤ log( T eff ) < .
041 3 . ≤ log( T eff ) < . . ≤ log( T eff ) < . . ≤ log( T eff ) < .
8K 3 . ≤ log( T eff ) < .
66M log( T eff ) < . f + B + Alog( L ) ≥ . L ) ≥ . L ) ≥ . L ) ≥ . nterpreting Star Count Ratios in Stellar Populations − − B/R − − − − WR/RSG − WC/WN − − − − − WR/O − − − O/BSG . . . . . − − . . . . − − − − . . . . . . − . . . . . − − − − − . . . . − − − . . . f bin . . . f rot log t l ogˆ R Figure 2.
Predicted values of five diagnostic ratios (where
B/R is shorthand for the ratio of blue to red supergiants), at solarmetallicity, for the BPASS (top) and Geneva (bottom) populations as a function of time, for values of f between 0 and 1, asindicated by the colorbar. measurements are often model dependent, and are basedon inferences of the sometimes-undetected low-mass endof the population. Instead, ratios of the frequency ofthese types (hereafter “number count ratios”) are inde-pendent of the stellar mass, while remaining sensitive toboth rotation and binary interactions.We first construct the predicted number counts forsubtypes in a population with a given f bin or f rot (gener-ically f hereafter). All notation used is summarized inTable 1. We calculate ˆ n S , the frequency of a subtype Sat time t and metallicity Z asˆ n S ( t, f, Z ) = f ˆ n S f =1 ( t, Z ) + (1 − f )ˆ n S f =0 ( t, Z ) (1)where ˆ n S f =0 and ˆ n S f =1 are the frequencies in the f = 0and f = 1 populations respectively. We note that, forthe BPASS populations, the naive interpretation of f bin is rather straightforward: f bin is the fraction of binarystars in the population . However, for the Geneva pop-ulations, f rot would then correspond to the fraction ofstars in the population born with v ini /v crit = 0 . f rot to correspond to the average initial rotation rate (i.e., (cid:104) v ini /v crit (cid:105) = 0 . f rot , see, for example, Levesque et al. There is a secondary factor here, in that f bin is mass-dependent (Duchˆene & Kraus 2013). However, all of the evo-lutionary phases here are mostly descended from O and early Bstars (at least in the single-star paradigm), for which observedsamples are too small to determine any mass dependence. f is simply a factorused to linearly combine the output number counts fromeach population, and does not necessarily correspondto the fraction of binary/rotating star models that areused, a fact that the reader should be aware of wheninterpreting our results.We then calculate number count ratios usingˆ R S /S ( t, f, Z ) = ˆ n S ( t, f, Z ) / ˆ n S ( t, f, z ) (2)for two subtypes S and S .In Paper I we described four number count ratios fre-quently found in the literature, as well as a novel numbercount ratio, O/BSG. Here we briefly describe these ra-tios, and the physical effects that they probe. We notethat, in theory, our model populations would allow us toperform a search for the ratios and completeness limitsthat would best differentiate between different channelsof stellar evolution. However the existing data tendsto focus only on individual species, making the avail-able space of ratios that can be measured quite small.In future work, we plan to perform this search in orderto guide observers to evolutionary species for which anaccurate census is most useful for constraining stellarevolution. • B/R : The ratio of the number of BSGs to RSGs(B/R) is among the most frequently used ratios,and has a long history as a metallicity diagnos-tic (Walker 1964; van den Bergh 1968; Langer &Maeder 1995; Massey 2003). It is sensitive tothe physics governing the timescale of rightward
Dorn-Wallenstein & Levesque evolution of stars on the HR diagram — rota-tional/convective mixing — as well as the inter-ruption of a star’s expansion by RLOF. • WR/RSG : In the single star paradigm, theWR/RSG ratio probes the boundary betweenstars that experience only redward evolution andstars that lose enough mass to evolve bluewardat the end of their lives (Conti et al. 1983), andis therefore also sensitive to metallicity (Maederet al. 1980). Mass loss via binary channels servesto artificially boost this ratio by decreasing thenumber of RSGs, and commensurately increasingthe number of WRs. • WC/WN : The WC/WN ratio probes the evolutionof stars that have already lost enough mass to be-come WRs. Thus it is mostly insensitive to thebinary fraction, and is a very sensitive diagnosticof radiative mass loss in the WR phase (Vanbev-eren & Conti 1980; Hellings & Vanbeveren 1981). • WR/O : The WR/O ratio probes the largestswatch of the mass spectrum considered here. Asthe only ratio in the literature with main sequencecomponents, it is the most subject to contamina-tion by unresolved O+O binaries (Maeder 1991)that is difficult to address via our simple imple-mentation of completeness limits. • O/BSG : In Paper I, we introduced the O/BSGratio. Both species are recovered by photometriccensuses of bright blue stars in stellar populations,and don’t require narrow band imaging or spec-troscopic follow-up to detect. While, in theory,it is mostly sensitive to the main sequence life-time (and thus rotational and convective mixing),some main sequence O stars are luminous enoughto be classified as BSGs in our scheme. As wedemonstrated, O/BSG is mostly insensitive to thebinary fraction, except in a narrow window aroundlog t = 7. This is due to lower mass stars losingenough of their envelopes via RLOF to evolve blue-ward, without losing enough Hydrogen to be la-belled as WR stars — observationally these mightbe classified as sdB/O stars, which are not countedseparately from O and B stars in our classificationscheme (see Section 3.2 and Figure 10 of Eldridgeet al. 2017). This effect boosts the O/BSG ratiorelative to the single star population until all ofthe O stars have evolved. Thereafter, only smallnumbers of models in the binary population residewithin the Luminosity-Temperature-Compositionboundary of O stars or BSGs at different times, causing rapid changes to O/BSG (for a particu-larly drastic example of a star rapidly enteringand leaving the O and BSG regimes, see Figure1 of Dorn-Wallenstein & Levesque 2018).Figure 2 shows the values of five different ratios vs.time for solar metallicity BPASS (top row) and Geneva(bottom row) populations with 0 ≤ f ≤ f . RESULTS3.1.
Comparing Rotating and Binary ModelPopulations
The differences between the single/nonrotating O andB stars in the top row of Figure 1 are minimal, where thedotted line shows the nonrotating Geneva population,and the dash-dotted line shows the BPASS single-starpopulation. We warn the reader that, in general, theBPASS user manual cautions against using the f bin = 0population in isolation; only the f bin > f bin =0 results for completeness, and discuss the differencesbetween the two f = 0 populations in detail here andthroughout the text.Significant differences arise in the yellow supergiantphase, where the Geneva models predict the existence offar more YSGs. However, theoretical uncertainty in thisvery short-lived phase has long stymied our understand-ing of massive star evolution (Kippenhahn & Weigert1990); for this reasons we do not use the YSG phasein our subsequent ratio diagnostics and caution againstusing it as a diagnostic of stellar population propertiesuntil it is better understood. The single BPASS modelsproduce approximately twice as many RSGs as the non-rotating Geneva models. This is largely due to the factthat the BPASS models cross the HR diagram quicker(reflected in the significantly smaller number of YSGscompared to the Geneva models), increasing the amountof time the stars spend as RSGs before ending their lives.However, the two model sets also adopt slightly differ-ent mass loss prescriptions during the RSG phase: theGeneva tracks use mass loss rates from Reimers (1975,1977) for the models less massive than 12 M (cid:12) during theRSG phase, and a combination of mass loss rates from deJager et al. (1988), Sylvester et al. (1998), and van Loonet al. (1999) for more massive models, while BPASS onlyuses rates from de Jager et al. (1988) that are higher onaverage. This serves to modulate the increased numbers nterpreting Star Count Ratios in Stellar Populations f bin = 1 population at late times. Rotation prolongs thelength of the early evolutionary phases, serving to delaythe onset of the RSG and YSG phases. Rotating modelsalso produce higher mass loss rates thanks to luminosity-dependent mass loss prescriptions and the higher lumi-nosities of rotating stars, beginning at the terminal agemain sequence and persisting through their post-main-sequence evolution (Maeder & Meynet 2000b; Ekstr¨omet al. 2012), as well as contributions from mechanicalmass loss (e.g., mass loss via the stellar equator frommatter rotating above the critical velocity). We also seean increase in the number of WR stars formed in therotating Geneva populations; this boost primarily man-ifests as an increase of WNH stars, with slight decreases in the number of WN and WC stars. This is the resultof a longer lifetime for the WNH phase and subsequentshorter lifetimes for the WN and WC phase (a conse-quence of rotational mixing), as well as efficient massloss producing lower-mass WNH stars (for more discus-sion of rotation effects of WR subtypes see Georgy et al.2012). Finally, the bottom panel serves as a comparisonbetween the binary BPASS population (dashed line) andthe rotating Geneva population (solid line) as shown inthe above rows. In summary: rotation causes a delay inthe appearance of evolved supergiants and an increasein the number of WNH stars, while binary interactionseffectively trade evolved supergiants for WR stars. How-ever, the exact timing and degree of these effects as afunction of initial mass/luminosity is more complicated,and this effect manifests in the observed stellar popula-tions as we will demonstrate.The predictions of the number count ratios in Fig-ure 2 are fairly similar between the Geneva and BPASSpopulations at log t (cid:46) . −
7. Both binary interac-tions and rotation introduce similar effects, especiallyin B/R and WR/RSG. The Geneva models predict ahigher overall value of WC/WN before log t ∼ .
75 (dueto the increase/decrease of WNH/WN stars respectivelyin the Geneva models), and a higher local maximum ofWR/O at log t ∼ .
8. At this time, the WR compo- nent of the Geneva populations becomes increasinglydominated by WC and WNH stars at increasing f rot ,while the BPASS WRs are largely WN type, which is re-flected in the multiple orders of magnitude difference inthe prediction for WC/WN. At increasingly later times,more stars in binaries become stripped. Models that loseenough Hydrogen increase the number of WRs well af-ter log t ∼ .
8, when the Geneva populations predict thelast WRs have died, while models with only moderatemass transfer/loss become O stars/BSGs, depending ontheir luminosity — this is reflected in the rapid changesin the O/BSG ratio at late times. Overall, depending onthe ratio chosen and the approximate age of the popu-lation being analyzed, different ratios are most sensitiveto age, f bin , f rot or all three. For example, WC/WN isincredibly sensitive to f rot in moderately evolved popu-lations, while B/R is a relatively powerful age indicatorin the earliest populations.Below solar metallicity, the dominant differences arethat all WR stars in the Geneva populations are H-rich(WNH in our labelling scheme), and binary interactionsbecome increasingly important for producing WRs inthe BPASS populations. The former is consistent withGeorgy et al. (2013) and Groh et al. (2019), and is aknown feature of the Geneva models (Leitherer et al.2014). While this might indicate that all H-deficientWRs at low metallicity are formed by binary interac-tions, other possibilities and evolutionary pathways ex-ist, which we defer to work focused more specifically onWR populations.3.2. Comparisons with Real Data
Ensuring Self-Consistency
Two important effects must be considered before di-rectly comparing some observed number count ratio toa theoretical prediction of this ratio, to ensure that thequantities being compared are identical: • The value (and corresponding uncertainty) re-ported by an observer must be an estimate ofthe underlying number count ratio, ˆ R (an intrinsiccharacteristic of the stellar population belongingto the set of real numbers), rather than the rawobserved ratio, R (which is a characteristic of thedata belonging to the set of rational numbers dueto the integer nature of the measurement). • The theoretical population must approximate theobserved population, and reflect the completenessof the catalog of massive stars (which may varywith spectral type).We first consider how to estimate number count ratiosand confidence intervals from observed data. Here we0
Dorn-Wallenstein & Levesque present a novel framework for making point-estimates(with corresponding uncertainties) for an intrinsic num-ber count ratio. Much like photons in the low-countregime, the frequency of finding a given spectral typeis determined by Poisson statistics (this is especiallytrue for counting massive stars, where “shot noise” isthe dominant source of uncertainty). In particular, themeasurement uncertainty of the frequency of two sub-types, n and n , can be approximated as σ n = √ n and σ n = √ n respectively. Past studies that calcu-late the number count ratio (e.g., Massey & Olsen 2003;Neugent et al. 2012), apply traditional propagation ofuncertainties, and report the observed ratio R = n /n ,with corresponding uncertainty σ R = R (cid:113) n − + n − .As discussed by Neugent et al. (2012), this approach isproblematic, and more sophisticated corrections can bemade. However, an additional problem exists in that,with few enough stars (e.g., the Wolf-Rayet populationof the SMC, where 1 WC star is known; Neugent et al.2018b), or where the true underlying ratio is large, yetfinite, n = 0 is well-within a “3 σ ” error bar, and ameasurement of R = ∞ could have been made.This is a well-studied problem in the X-ray astronomycommunity, with a tractable solution within a Bayesianframework. Park et al. (2006) derive the posterior prob-ability distribution of colors and hardness ratios for X-ray sources, in the limit of few (or no) photon counts.The problem here corresponds to a special case wherethe background is guaranteed to be 0, which simplifiesthe calculations somewhat; unknown sample contami-nation by foreground stars can be accounted for withminimal added complexity.Say we measure a ratio R = n /n . Both n and n are assumed to be Poisson variables with (unknown) ex-pectation values ˆ n and ˆ n . What we wish to reportis an estimate of the true ratio, ˆ R ≡ ˆ n / ˆ n , which isa property of the underlying stellar population — in-deed, it is the exact quantity plotted in Figure 2. FromBayes’ theorem, the probability distribution for ˆ n givena measurement of n is given by p (ˆ n | n ) ∝ p (ˆ n ) p ( n | ˆ n ) (3)and similarly for p (ˆ n | n ), where p (ˆ n ) reflects ourprior knowledge on the value of ˆ n , and p ( n | ˆ n ) is thelikelihood of drawing n from a Poisson distribution withexpectation value ˆ n . For a prior, we adopt p (ˆ n ) ∝ ˆ n φ − . As discussed in Park et al. (2006) and van Dyket al. (2001), this is a special case of a γ -prior: p (ˆ n , α, β ) = 1Γ( α ) β α ˆ n α − e − β ˆ n (4) with α = φ and β →
0. This choice of prior en-sures that the posterior probability function takes thesame parametric form as the likelihood function. Parket al. (2006) found that, in Monte Carlo simulations, thechoice of φ only has a moderate impact on the coverage(the percentage of simulations where the ground truthvalue of ˆ R is within a 95% confidence interval). Wechoose φ = 1 /
2, which generally provides the best cov-erage for the observed number counts reported in typicalextragalactic surveys.Assuming ˆ n and ˆ n are independent (i.e., no stars oftype 1 would also be counted as type 2 ), the joint poste-rior distribution is p (ˆ n , ˆ n | n , n ) = p (ˆ n | n ) p (ˆ n | n ).Transforming ˆ n = ˆ R ˆ n and marginalizing over ˆ n , p ( ˆ R | n , n ) d ˆ R = d ˆ R (cid:90) ˆ n d ˆ n ˆ n p ( ˆ R ˆ n , ˆ n | n , n ) (5)Utilizing Eq. (3), and substituting in the prior andlikelihood functions, p ( ˆ R | n , n ) ∝ (cid:90) ˆ n d ˆ n ˆ R φ − ˆ n φ − ˆ R n ˆ n ( n + n )2 e − ˆ n ( ˆ R +1) n ! n ! (6)We use emcee (Foreman-Mackey et al. 2013), a MarkovChain Monte Carlo package, to sample the joint pos-terior probability distribution for ˆ R and ˆ n with 100walkers initialized around the observed value R , 500burn-in steps that are discarded, and an additional 3000steps to explore the stationary distribution of walkers.In cases with R → ∞ or R →
0, we force R to bein the range [10 − , ] when initializing the walkers.We estimate the value of both ˆ R and ˆ n , as well as a68% (1 σ ) confidence interval, using the 16th-, 50th-, and84th-percentile values of the samples. The key advan-tages of this method are that the estimated quantitycan be directly compared to the model predictions, thatthe reported errorbars correspond to the actual poste-rior probability distribution (and can be assymetric),and that the estimate of ˆ R is meaningful even if n or n are 0.The challenge of accounting for complete samples isdiscussed in depth in Paper I. Here we reiterate that ac-counting for incompleteness in the observed samples —here defined as the lowest luminosity to which all starsof a given subtype have been found — is critical, andincorrectly handling or ignoring this effect can result in In one example below, n is subset of n . There, a simpletransformation can be made, but handling more complex situa-tions is nontrivial. Software for performing these calculations, as well as re-producing all of the results in this work, is available online athttps://github.com/tzdwi/Diagnostics/ nterpreting Star Count Ratios in Stellar Populations ∼ . L cut , that can betuned for each subtype under consideration, and doesnot include models with L < L cut . Thus, even if thesample is assembled spectroscopically, or has a limitingmagnitude in some photometric band that correspondsto different luminosity thresholds depending on the ef-fective temperatures of the different subtypes, the modelpopulations can be adjusted accordingly.We note that assuming an observed sample is 100%complete above L cut , and no stars are detected below itis a somewhat simplistic assumption. Below we compareour models to actual observed samples. In the two starclusters that we focus on, the data mostly come from fo-cused studies that are designed to detect a given species.These stars are the brightest objects in a given partof the color magnitude diagram, have been followed upspectroscopically to remove contaminants, and the sam-ple should be complete above the lowest luminosity star.In the Magellanic Clouds, the current existing samplesof WRs and RSGs claim to be mostly-complete censusesof both species down to quite stringent magnitude lim-its (Neugent et al. 2018b, 2012). Some confusion arisesin the counting of BSGs; however, the topic of brightblue stars in the Magellanic Clouds is currently beingdebated. We defer here to authors with more exper-tise (Aadland et al. 2018). Finally, observational bias,particularly in spectroscopic searches for WR stars, islikely to lead to missed weak-lined WR stars or WRstars with less evolved companions when not carefullyaccounted for, as discussed by Neugent et al. (2018b).3.2.2. Starburst Comparisons
We now wish to test our models in an environmentwhere both sets of populations produce roughly iden-tical predictions in a simple stellar population. FromFigure 2, the best examples are young ( <
10 Myr) starclusters, where we can assume that all of the stars be-long to a single burst of star formation (see caveats inGossage et al. 2018). At these young ages, most WRsformed by binary interactions are evolved from progeni-tors that were massive enough to become WRs anyway.There are very few such clusters with enough confirmedmembers to adequately sample the IMF. With a mass of M ∗ ≈ × M (cid:12) cluster (Andersen et al. 2017), and awell-studied cohort of evolved massive stars (Clark et al.2005; Crowther et al. 2006), including a large number ofBSGs, and an appreciable amount of WRs and RSGs,Westerlund 1 (Wd 1) is perhaps the best Galactic testbench for our model populations. In Paper I, we demon-strated that, when including binary effects and account- ing for completeness, we can use two number count ra-tios, to estimate an age consistent with Crowther et al.(2006), who use a single diagnostic ratio and did not ac-count for binarity or completeness. We can now applyour updated proscription for estimating number countratios, as well as the rotating populations.We first apply the Monte Carlo method describedabove to estimate the value of two ratios, O/BSG andWR/RSG. Using data from Clark et al. (2005) andCrowther et al. (2006), we count n O = 22, n BSG = 29, n W R = 24, and n RSG = 3. As an illustration, the sam-ples of the joint posterior distribution for ˆ R W R/RSG andˆ n RSG are shown in the bottom-left panel of Figure 3,along with the marginalized posterior distributions foreach parameter and accompanying point estimates (solidblue vertical line) and 68% confidence intervals (dashedblack vertical lines). Note that because n RSG is so low,the distribution of ˆ R W R/RSG is very skewed, and the97.5th-percentile upper limit is much higher than thereported 84th-percentile. All of the O stars in our sam-ple are also blue supergiants, and so our assumptionof independent variables no longer holds. Instead, weestimate ˆ R O/ ( BSG − O ) where BSG-O refers to all BSGsthat are not O stars. We then transform each poste-rior sample of O/(BSG-O) into a sample of O/BSG esti-mates. With this method we measure intrinsic values ofˆ R W R/RSG = 7 . +6 . − . and ˆ R O/BSG = 0 . +0 . − . .
20 40 60 80 ˆ R W R/RSG ˆ n R S G ˆ n RSG
Figure 3.
Posterior distribution samples of the estimate ofWR/RSG for Wd 1. The 1-D histograms show the marginal-ized posterior distribution for the true ratio ˆ R , and the truenumber of RSGs, ˆ n RSG , as well as the point estimates (inblue vertical lines) and 68% confidence intervals (in dashedvertical lines) Dorn-Wallenstein & Levesque
We now wish to estimate age and f from thedata. Figure 4 shows the predictions for WR/RSGvs. O/BSG, calculated on a grid of Geneva (left) andBPASS (center) populations with varying age (with6 . ≤ log t ≤ .
9) and f . Lines of constant age and f are shown; the inset plot can be used to translatefrom the ratio-space into f and log age. The mod-els incorporate completeness limits consistent with thelowest luminosities of each subtype reported by Clarket al. (2005) and Crowther et al. (2006) (specificially, L cut = 4 . , . , . , . f = 0 pop-ulations (shown in purple in all three panels). The datafor Wd 1, plotted in blue, are consistent with an age oflog t = 6 . f bin and f rot .In the example above focused on Wd 1, most of thestars in the sample were OB dwarfs or supergiants,where the differences between binary and rotating sce-narios are small. For intermediate age clusters (log t (cid:38) L = 4 . f = 0 case fromboth model sets, where ˆ R W R/RSG →
0. Both grids re-flect this, as they asymptote off the bottom left of theplot.There exists only one stellar population that is mas-sive enough to test our synthetic populations in this ageregime: h + χ Persei. Photometric and spectroscopicstudies of the members of h + χ Per have determined ages of 13-14 Myr (log( t ) ≈ .
1) for both clusters (Slesnicket al. 2002; Currie et al. 2010). While these studies haverevealed a population of O stars, BSGs, and RSGs, noobvious WRs have been found, and thus we were unableto compare the data with the models in Paper I (seethat work for a detailed discussion of detecting low-massWRs). Now, we can use the data from Table 3 in Currieet al. (2010) to count 1 O star (HD 14434), 29 BSGs,and 7 RSGs, and estimate ˆ R W R/RSG = 0 . +0 . . and ˆ R O /BSG = 0 . +0 . . , assuming 0 observed WRs.These values are plotted in blue in Figure 5. Note thatthe samples O stars and BSGs are independent , andwe do not need to perform the same transformation asabove. Because Currie et al. (2010) study the main se-quence down to G0 dwarfs, the luminosity cutoffs ap-plied here are consistent with the data. The point es-timates for both ratios are consistent with the BPASSmodels, though the Geneva models are not completelyexcluded. Interestingly, this difference is driven primar-ily by our measurement of O/BSG, and not WR/RSG(which we estimate without observing any WR stars).Using the BPASS model grid, the data corresponds toan age of ∼
10 Myr, consistent with previous age esti-mates. 3.2.3.
Constant Star Formation Histories
While these simple stellar populations in our Galaxyare useful, the 68% confidence intervals on our numbercount ratio estimates are too wide to accurately deter-mine age and f bin / f rot . To obtain a larger sample ofevolved massive stars, we turn to galaxies in the Lo-cal Group. Not only do these galaxies contain moremassive stars, they also sample a broad range of metal-licities. However, these populations have complex starformation histories (SFHs), which we describe in PaperI. Here, we only consider populations that are constantlyforming stars, following Eldridge et al. (2017). Despiteincluding stars from all age bins, all of the evolved typesof massive stars considered here are only sensitive to, atmost, 50-100 Myr of star formation, after which the pop-ulations of massive stars reach an equilibrium. Figure 6shows WR/O, B/R, WR/RSG, and WC/WN as a func-tion of metallicity and f rot for galaxies constantly form-ing stars. A minimum luminosity of log( L ) = 4 . Z (cid:12) are presented instead as individual points. Con-sistent with past models and existing observations (vanden Bergh 1973; Humphreys & Davidson 1979; Maeder nterpreting Star Count Ratios in Stellar Populations − ˆ R O/BSG ˆ R W R / R S G Westerlund 1 . . . t . . . f r o t − ˆ R O/BSG ˆ R W R / R S G Westerlund 1 . . . t . . . f b i n − ˆ R O/BSG ˆ R W R / R S G Westerlund 1
Figure 4. ˆ R WR/RSG vs ˆ R O/BSG for the Geneva (left) and BPASS (center) populations, calculated on a grid of 6 . ≤ log t ≤ . ≤ f ≤
1, as shown by the inset panels. The rightmost panel shows both grids overlain on top of each other, with identicalcolor coding. An estimate for ˆ R WR/RSG and ˆ R O/BSG in Westerlund 1, as well as corresponding 68% confidence intervals, iscalculated using data from Clark et al. (2005) and Crowther et al. (2006) and shown in blue. − − − − − ˆ R O/BSG − − − − ˆ R W R / R S G h + χ Per . . t . . . f r o t − − − − − ˆ R O/BSG − − − − ˆ R W R / R S G h + χ Per . . t . . . f b i n − − − − − ˆ R O/BSG − − − − ˆ R W R / R S G h + χ Persei
Figure 5.
Similar to Figure 4: ˆ R WR/RSG vs ˆ R O/BSG for the Geneva (left) and BPASS (center) populations, now calculatedon a grid of 6 . ≤ log t ≤ .
4. An estimate for ˆ R WR/RSG and ˆ R O/BSG in h + χ Persei, as well as corresponding 68% confidenceintervals, is calculated using data from Currie et al. (2010) and shown in blue. et al. 1980), B/R is the least sensitive ratio to f rot , and isa good indicator of metallicity, while WR/RSG is largelyindependent of metallicity (a fact that has long been intension with observations, see section 6.4 of Levesque2017, and citations therein).Recent measurements of the frequency of all threesubtypes are only available for the Magellanic Clouds(MCs). Figure 7 shows ˆ R W R/RSG vs. ˆ R B/R using theGeneva (left), and BPASS models (center), with bothgrids overlaid (right), with lines of constant f rot , f bin and Z , as indicated with the inset grids. We note thatthe f = 0 case from both grids do not agree; BPASSpredicts approximately an order of magnitude smallervalues in both ratios. This is largely due to the signif-icant difference in the number of RSGs predicted (seethe top row of Figure 1), and thus, the difference in themass loss prescription used by each code. However, bothgrids reside in largely the same part of this ratio space.Furthermore, consistent with Figure 6, B/R remains auseful proxy for metallicity, and WR/RSG is a usefuldiagnostic of either f bin or f rot .For both the Large Magellanic Cloud (LMC, plottedin orange) and Small Magellanic Cloud (SMC, plotted in blue), we estimate ˆ R B/R using data from Massey& Olsen (2003) and ˆ R W R/RSG from Massey & Olsen(2003), Neugent & Massey (2011), and Neugent et al.(2012), and find ˆ R B/R = 13 . +0 . . , ˆ R W R/RSG =0 . +0 . . in the LMC, and ˆ R B/R = 16 . +1 . . ,ˆ R W R/RSG = 0 . +0 . . in the SMC. The Geneva mod-els fail to reproduce the observations, while the BPASSmodels underpredict the metallicity of both clouds ( Z ≈ . / .
002 for the SMC/LMC respectively). In bothcases, the data indicate that the initial f bin or f rot islarger in the higher metallicity LMC than in the lowermetallicity SMC, consistent with the results in Dorn-Wallenstein & Levesque (2018). We stress here thatthese results refer to the natal values of these parame-ters. Indeed, the exact opposite trend is predicted andobserved in the frequency of X-ray binaries (Belczynskiet al. 2004); however, this is a reflection the metallicity-dependent angular momentum evolution of binary sys-tems. Measurements of the frequency of O star binaries(which in theory have not undergone mass transfer in-teractions in our model populations, and should serve asa decent proxy for the initial value of f bin ) reveal that f bin does indeed increase from the LMC ( f bin ∼ . Dorn-Wallenstein & Levesque ˆ R B / R ˆ R W R / R S G − − Z2030406010 ˆ R W C / W N − − Z 10 − − − − ˆ R W R / O . . . f rot Figure 6.
Predicted values of B/R (top left), WR/RSG (topright), WC/WN (bottom left), and WR/O (bottom right)for Geneva populations with constant star formation, andvalues of f rot between 0 and 1, as indicated by the color bar.A minimum luminosity of log( L ) = 4 . Sana et al. 2013) to the Galaxy ( f bin ∼ .
7, Sana et al.2012). Meanwhile, the observed rotation rates of metal-poor stars are expected to be higher (i.e., f rot decreaseswith metallicity), due to stars being more compact atlow Z (Chiappini et al. 2006). However, the agreementbetween the overall trends in the data and those pre-dicted by both model sets imply that number count ra-tios may be the best way to infer f bin or f rot in galaxieswhere direct measurements are currently infeasible. DISCUSSION AND CONCLUSIONBefore discussing our results, we note two importantcaveats. Our results here assume that both stellar evolu-tion codes accurately describe the evolution of the mod-elled stars. If this were true, then the f = 0 cases fromeach population should be identical, as both popula-tions contain only single, nonrotating stars. Of course,both stellar evolution codes make different assumptions;while we refer the reader to the papers describing bothcodes for more details, we note that the different massloss prescriptions in particular can result in the differingnumbers of predicted cool supergiants.We are furthermore assuming that both codes arecompletely accurate descriptions of rotation or binaryinteractions. Indeed, both codes are now seen as theindustry standard for modelling their respective effectsin massive stars. However, both codes do exhibit short-comings. For example, the Geneva group only provides models that rotate at 0 . v crit . More rapidly rotatingmodels are available, but only in a limited mass win-dow below 15 M (cid:12) . With rapidly rotating models athigher mass incorporated into our model populations,we would be able to probe stars with significantly en-hanced rotational mixing and enhanced mass loss —perhaps increasing the number of WNs while decreas-ing the number of WNHs.Meanwhile, the BPASS team is the only group thatexplicitly models RLOF in binary systems and makestheir results publicly available. However, as noted inEldridge et al. (2017), BPASS makes a number of sim-plifying assumptions in its treatment of circular orbits,rotation, and common envelope evolution. Furthermore,BPASS does not model systems with initial orbital pe-riods shorter than one day. Regardless, BPASS is stillsuccessful at producing all classes of observed binary sys-tems; in theory, some merger products are not modelled,but those products are either the result of very short-period systems — which merge very early, and evolve assingle stars (J. Eldridge 2018, private communication),which may make result in an effectively top-heavier IMF— or are incredibly rare objects (e.g., Thorne- ˙ZytkowObjects, Thorne & Zytkow 1975, 1977).These caveats aside, it is still important to examinehow these two widely-used codes compare to each otherand to observations when predicting the evolution andpopulations of massive stars, and to consider this com-parison when interpreting the use and application ofthese models in future work. Our main results are asfollows:1. While rotation and binary interactions predictqualitatively similar effects on stellar populations,the predictions for the detailed makeup of simu-lated simple and complex stellar populations showdramatic differences between the two evolution-ary scenarios. While some of these can be at-tributed to fundamental differences in the Genevaand BPASS codes (as discussed in Section 2.1), itis also clear that rotation and binarity have quan-titatively different effects on the evolution of mas-sive star populations (in particular, the diagnosticnumber count ratios discussed in Section 3.1, andshown in Figure 2).2. We introduced a novel Bayesian method of es-timating both the value and error of diagnosticratios in the low-number count regime typical ofsamples of massive stars. This method, combinedwith our implementation of completeness limits inthe model populations, make our comparisons be-tween models and data more accurate. nterpreting Star Count Ratios in Stellar Populations ˆ R B/R − − ˆ R W R / R S G SMCLMC − − − − Z . . . f r o t ˆ R B/R − − ˆ R W R / R S G SMCLMC − − − Z . . . f b i n ˆ R B/R − − ˆ R W R / R S G SMCLMC
Figure 7.
Similar to Figure 4: ˆ R WR/RSG vs ˆ R B/R for the Geneva (left) and BPASS (center) populations, assuming constantstar formation, and calculated on a grid of 10 − ≤ Z ≤ .
04. Estimates for ˆ R WR/RSG and ˆ R B/R in the LMC and SMC, as wellas corresponding 68% confidence intervals, are shown in orange and blue, respectively. Data are from Massey & Olsen (2003),Neugent & Massey (2011), and Neugent et al. (2012).
3. The data from observed Galactic populationsagree with the grids of simulated SSPs, but sufferfrom poor signal-to-noise. The increased sam-ple sizes in Local Group galaxies allows us tomake higher-precision estimates of diagnostic ra-tios, and show that measurements of the natalbinary fraction or rotation rate of stellar popula-tions beyond the Magellanic Clouds may be possi-ble with currently obtainable data. In the comingdecades, JWST and WFIRST are scheduled tolaunch, giving us immediate access to red/optical-mid IR photometry, as well as sparsely sampledlightcurves for massive stars well-beyond the Lo-cal Group. Using comparable existing data inthe Galaxy and Magellanic Clouds, it is possibleto accurately fit for coarse spectral types akin tothose used here (Dorn-Wallenstein et al. in prep).4. Figures 4, 5, and 7 show a comparison betweenintrinsic number count ratios derived from gridsof model stellar populations and inferred from ob-served data. For the starburst populations, weshow that the ages consistent with the observeddata are reasonably close to already published ageestimates after correcting for completeness. How-ever, the true power of this technique lies in itsability to derive the values and uncertainties ofunknown parameters that are critically importantfor stellar evolution — e.g. f bin or f rot — in en-vironments where traditional means of measuringthese quantities are expensive or impossible given current technology. In this case, deriving the like-lihood of obtaining a given set of star count ra-tios given a set of values for these parametersis nontrivial. Future work will focus on apply-ing Approximate Bayesian Computation (ABC,Sunn˚aker et al. 2013) to derive constraints on f bin and compare them to existing values found in theliterature (Sana et al. 2012, 2014).5. Finally, we stress that rotating stars are found inbinary (and higher order) systems. A completemodel of stellar evolution should not designate oneor the other effect as secondary. Rather, as wedemonstrate here, binary interactions and rotationand produce both similar and contradictory effectsin stellar populations, and a rigorous simultaneoustreatment of both is necessary.TZDW acknowledges J. Eldridge and C. Georgy fortheir advice and help. This research was supported byNSF grant AST 1714285 awarded to EML. This workmade use of v2.2.1 of the Binary Population and Spec-tral Synthesis (BPASS) models as described in Eldridgeet al. (2017) and Stanway & Eldridge (2018).This work made use of the following software: Software:
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