Northwestern University, 2145 Sheridan Rd., Evanston, IL ... · Aaron M. Geller1 ,2 † ∗, Emily M. Leiner3, Sourav Chatterjee1, Nathan W. C.Leigh4, Robert D. Mathieu3, Alison Sills

arX

iv1

703

1017

2v2

[as

tro-

phS

R]

17

May

201

7Draft version July 9 2018Preprint typeset using LATEX style emulateapj v 121611

ON THE ORIGIN OF SUB-SUBGIANT STARS III FORMATION FREQUENCIES

Aaron M Geller12daggerlowast Emily M Leiner3 Sourav Chatterjee1 Nathan W C Leigh4 Robert D Mathieu3Alison Sills5

1Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics amp AstronomyNorthwestern University 2145 Sheridan Rd Evanston IL 60201 USA

2Adler Planetarium Department of Astronomy 1300 S Lake Shore Drive Chicago IL 60605 USA3Department of Astronomy University of WisconsinndashMadison 475 North Charter Street Madison WI 53706 USA

4Department of Astrophysics American Museum of Natural History Central Park West and 79th Street New York NY 10024 USA5Department of Physics and Astronomy McMaster University Hamilton ON L8S 4M1 Canada

Draft version July 9 2018

ABSTRACT

Sub-subgiants are a new class of stars that are optically redder than normal main-sequence stars andfainter than normal subgiant stars Sub-subgiants and the possibly related red stragglers (which fallto the red of the giant branch) occupy a region of the color-magnitude diagram that is predicted tobe devoid of stars by standard stellar evolution theory In previous papers we presented the observeddemographics of these sources and defined possible theoretical formation channels through isolatedbinary evolution the rapid stripping of a subgiantrsquos envelope and stellar collisions Sub-subgiantsoffer key tests for single- and binary-star evolution and stellar collision models In this paper wesynthesize these findings to discuss the formation frequencies through each of these channels Theempirical data our analytic formation rate calculations and analyses of sub-subgiants in a largegrid of Monte Carlo globular cluster models suggest that the binary evolution channels may be themost prevalent though all channels appear to be viable routes to sub-subgiant creation (especially inhigher-mass globular clusters) Multiple formation channels may operate simultaneously to producethe observed sub-subgiant population Finally many of these formation pathways can produce stars inboth the sub-subgiant and red straggler (and blue straggler) regions of the color-magnitude diagramin some cases as different stages along the same evolutionary sequenceSubject headings open clusters and associations ndash globular clusters ndash binaries (including multiple)

close ndash blue stragglers ndash stars evolution ndash stars variables general

1 INTRODUCTION

This is the third paper in a series investigating theorigins of a relatively new class of stars known as sub-subgiants (SSGs) In Geller et al (2017 hereafter PaperI) we gather the available observations for these sourcesand find that SSGs share the following important empir-ical characteristics

1 SSGs occupy a unique location on a color-magnitude diagram (CMD) redward of the nor-mal MS stars but fainter than the subgiant branchwhere normal single-star evolution does not predictstars (see Figure 1)

2 More than half of the SSGs are observed to beX-ray sources with typical luminosities of order1030minus31 erg sminus1 consistent with active binaries

3 At least one third of the SSGs exhibit Hα emission(an indicator of chromospheric activity)

4 At least two thirds of the SSGs are photometricandor radial-velocity variables with typical peri-ods of 15 days

5 At least three quarters of the variable SSGs areradial-velocity binaries

lowastNSF Astronomy and Astrophysics Postdoctoral FellowElectronic address daggera-gellernorthwesternedu

6 The specific frequency of SSGs increases towardlower-mass star clusters

The fractions of sources given in items 2-5 above are alllower limits because not all sources have the necessaryobservations to investigate each characteristicIn Leiner et al (2017 hereafter Paper II) we study

three specific formation channels in detail that can pro-duce stars in the SSG region of the color-magnitude di-agram (CMD) namely (i) ongoing mass-transfer from asubgiant donor ldquoSG MTrdquo (ii) a reduced convective effi-ciency likely related to increased magnetic activity ldquoSGMagrdquo and (iii) rapid and partial stripping of a subgiantsrsquoenvelope ldquoSG Striprdquo Paper II also briefly considers afourth channel (iv) a main-sequence ndash main-sequencestellar collision where the collision product is observedwhile settling back down onto the normal main-sequenceldquoMS Collrdquo We provide details and models for thesemechanisms in Paper II (and references therein) and de-scribe them qualitatively in Section 2 below For refer-ence in Figure 1 we show example evolutionary tracksfor each of these mechanism plotted over an isochroneat the age of M67 Within this same figure we also definethe SSG region on a CMD with the dark-gray shading(see also Figure 1 in Paper I)The ldquored stragglerrdquo (RS) stars which are possibly re-

lated to the SSGs are found in the region with the light-gray shading in Figure 1 We note here (and in PapersI and II) that there has been some confusion in the lit-erature with the naming convention of these two types

2 Geller et al

of stars We urge readers to adopt the convention thatwe set forth in this series of papers to identify SSG andRS stars There are far fewer RS stars than SSGs butdespite their different location on the CMD their empir-ical characteristics appear to be very similar to the SSGstars Some of the SSG formation channels discussedin Paper II predict an evolutionary relationship betweenstars in the SSG and RS regions (where one is the pre-cursor to the other) and furthermore at least two of theformation channels (ldquoSG MTrdquo and ldquoMS Collrdquo) can leadto the formation of a blue straggler star (BSS)In this paper we investigate the formation rates of

SSGs through these four mechanisms through analyticcalculations Our goal is to identify if indeed all of thesemechanisms are viable or if one or more clearly domi-nates the production of SSGs After providing a qual-itative description of the four formation mechanisms inSection 2 we then discuss the probabilities of observingSSGs from each of these theoretical formation channelsin Sections 3 and 4 We investigate SSGs created in N -body and Monte Carlo star cluster models in Section 5Finally in Section 6 we provide a brief discussion andconclusions

2 SUMMARY OF THEORETICAL FORMATIONCHANNELS

In Paper II we study SSG formation channels in de-tail primarily through in-depth analyses of MESA mod-els (Paxton et al 2015) In this section we provide abrief summary of these formation channels specificallyfor SSGs formed through ongoing binary mass transfer(Section 21) increased magnetic activity leading to in-hibited convection (Section 22) rapid loss of an envelope(Section 23) and MS ndash MS collisions (Section 24) Werefer the reader to Paper II for further details about thephysics behind the first three mechanism (and a wider ex-ploration of parameter space) MS stellar collisions havebeen modeled previously in detail (eg Sills et al 19972001 2002 2005) in the context of BSS formation

21 Ongoing Binary Mass Transfer Involving aSubgiant Star (ldquoSG MTrdquo)

If a binary containing a MS star has a short enoughorbital period (sim15 days for a circular binary with aprimary star at the turnoff in M67) this MS star canoverfill its Roche lobe shortly after evolving off of theMS As mass transfer begins the now subgiant star losesmass becomes fainter and moves into the SSG regionThe MESA model in Figure 1 shows a 10 day binary

with a 13 M⊙ primary and a 07 M⊙ secondary Masstransfer begins when the primary overflows its Rochelobe on the subgiant branch Stable mass transfer pro-ceeds with an efficiency of 50 In this model the binaryremains in the SSG region for sim400 Myr which is com-parable to the duration of the subgiant phase of a normalsim13 M⊙ star (of 600 Myr in MESA)In this scenario the subgiant must be the brighter star

in the binary in the optical for it to appear in the SSGregion The accretor could be a MS star or a compactobject However if a MS accretor is massive enoughinitially it may gain enough mass to become a BSSand dominate the combined light before the subgiantbecomes sub-luminous enough to enter the SSG region(unless the mass transfer is extremely nonconservative)

04 06 08 10 12 14 16MB - MV

6

5

4

3

2

1

MV

SG

Str

ip SG

Mag

SG MTMS Coll

5 10

5

10

15

N

Fig 1mdash Color-magnitude diagram showing the theoretical SSGformation channels (colored lines) and our definition of the SSGregion (dark gray shaded area) and RS region (light gray shadedarea) with respect to a PARSEC isochrone (Bressan et al 2012)for M67 (solid line) In the main panel the dashed line showsthe equal-mass binary sequence for M67 We show evolutionarytracks from MESA (Paxton et al 2015) for a subgiant undergo-ing stable mass transfer (ldquoSG MTrdquo purple line) a subgiant thathas been stripped of much of its envelope (ie after ejecting acommon-envelope or after a grazing collision ldquoSG Striprdquo dark-redline) and a star that has a reduced convective mixing length co-efficient (and increased magnetic activity ldquoSG Magrdquo green line)We also show the result of a collision between two 07 M⊙ MSstars (from Sills et al 2002 ldquoMS Collrdquo yellow line) For all coloredlines the arrows indicate the direction of time along the evolu-tionary sequence In the two sub-panels we show histograms ofthe observed distribution of SSG and RS stars from the open andglobular clusters studied in Paper I in MV (right) and MB minusMV

(top) The black-filled histograms show the contribution from theglobular cluster sources alone with the additional white-filled re-gion (up to the solid lines) coming from the open cluster sourcesWe refer the reader to Figure 1 in Paper 1 for CMDs of these starsin the individual clusters

Here one would expect to observe a short-period bi-nary likely with a rapidly rotating primary (subgiant)star Photometric variability could potentially arise fromellipsoidal variations on the subgiant spot activity oreclipses X-rays (and Hα emission) could be producedby chromospheric activity on the rapidly rotating sub-giant andor hot spots in the accretion stream onto acompact object (likely requiring a neutron star or blackhole accretor to reach X-ray temperatures)In many cases where the accretor is a MS star the

evolution will lead to the coalescence of the two starsand interestingly in this scenario the merger productmay be observed eventually as a BSS The mass-transfermodel shown in Figure 1 eventually creates a BSS witha WD companion Indeed binary mass transfer is one ofthe primary BSS formation mechanisms (McCrea 1964

Formation Frequencies of Sub-subgiant Stars 3

Tian et al 2006 Chen amp Han 2008 Geller amp Mathieu2011) especially in low-density (ρc 103Msunpc3) en-vironments (Chatterjee et al 2013a) We will come backto this potential connection between SSGs and BSS inSection 6

22 Increased Magnetic Activity in a Subgiant Star(ldquoSG Magrdquo)

While the effects of magnetic fields on stellar evolu-tion are in general not well known there is evidence thatmagnetic fields may alter the temperature and radii ofstars by lowering the efficiency of convection For ex-ample low-mass eclipsing binaries are found to be largerand cooler than model predictions which has been at-tributed to magnetic activity (eg Chabrier et al 2007Clausen et al 2009) A similar mechanism may be atwork in SSGs (Paper II) Chabrier et al (2007) lower themixing length coefficient in their models to mimic low-ered convective efficiency Our preliminary MESA mod-els where we reduce the convective mixing length coeffi-cient to α = 12 (see green line in Figure 1 and note thatMESArsquos standard mixing length coefficient is α = 20)suggest that this mechanism may produce SSGs primar-ily after the downturn on the subgiant branch and on thelower red-giant branch (RGB)

23 Rapid Loss of a Subgiant Starrsquos Envelope (ldquoSGStriprdquo)

If the envelope of a subgiant star is rapidly strippedaway it will become fainter while losing mass1 Oncemass loss stops the star will begin to evolve toward thesubgiant and giant branches as before but now alonga path appropriate for its new lower mass If enoughmass is lost the star will be fainter than the clusterrsquossubgiant branch and eventually redder than the giantbranch moving through the observed SSG regionFor the ldquoSG Striprdquo track shown in Figure 1 we use

MESA to evolve a 13 M⊙ star and we remove 045 M⊙

from the envelope soon after the star begins to evolveoff of the MS and at a rate of 10minus5 M⊙ yrminus1 At thisrate the star is driven out of thermal equilibrium initiallybut quickly returns to an equilibrium position once theremoval of mass is complete (This is the largest masstransfer rate we are able to reliably model in MESA forthis stellar mass and evolutionary state at larger rateshydrodynamical effects become important)One possible method to induce this rapid mass loss is a

grazing collision between a subgiant and some more com-pact star (perhaps a compact object or MS star) withan impact parameter small enough to strip the subgiantrsquosenvelope but large enough that the two stars donrsquot di-rectly merge We will refer to this pathway as ldquoSG CollrdquoA second potential method is through the ejection of acommon-envelope we will refer to this pathway as ldquoSGCErdquo For our purposes here we do not consider whathappens to the mass lost from the subgiant (whether itcan be accreted by the other star or lost from the system

1 Stripping the envelope of a red-giant star has only a very minoraffect on its luminosity (Leigh et al 2016a) because the luminosityof a red giant is controlled almost entirely by the He core Thusdespite the larger physical size of a red giant and therefore thelarger collision rate this stripping mechanism may be most easilyobserved for subgiants

entirely) Both of these mechanisms require further de-tailed modeling here we will simply assume that they areboth possible and focus on the possibility of observingthe product of such an eventAfter both processes a tight binary companion could

remain (for the ldquoSG Collrdquo scenario this could be akinto a tidal capture see eg Fabian et al 1975 andPress amp Teukolsky 1977) The subgiant may be spunup in this process If the stripped subgiant is rotat-ing rapidly then one may expect to observe photometricvariability and X-ray emission due to chromospheric ac-tivity and spots

24 Collision of Two MS Stars (ldquoMS Collrdquo)

In Figure 1 we show a collision product from Sills et al(1997) resulting from two 07 M⊙ stars Immediately af-ter a collision between two MS stars the collision prod-uct will become brighter (due primarily to the kineticenergy input from the motion of the stars leading up tocollision) by a factor of about 10 to 50 (in luminosity) forthe mass range of interest here Afterwards the star willsettle back into thermal equilibrium by contracting andreleasing gravitational potential energy along analogoustracks to pre-MS stars Through this contraction phasethe star becomes fainter and eventually settles back nearthe normal MS stars but before reaching the MS thecollision product may reside in the SSG region The con-traction phase occurs over roughly a thermal timescalewhich is between about 1-15 Myr for the masses of inter-est hereIf the collision is off axis the product will likely be

very rapidly rotating (Sills et al 2005) which could leadto similar photometric variability and X-ray emission as(particularly if a magnetic field can be maintained) asobserved for some SSGs Scattering experiments and N -body star cluster simulations suggest that it would bedifficult for the collision product to retain a binary com-panion at the short periods that are observed for manySSGs (ie of order 10 days) directly after a collision(eg Fregeau et al 2004 Leigh amp Sills 2011 Geller et al2013) Subsequent exchanges or tidal capture encounterscould become more likely with the increased mass (andtemporary increase in radius) of the collision productFurther scattering experiments and N -body models arenecessary to better understand the likelihood for creat-ing a short-period binary containing a collision productwithin such a short timescale after the collision (as wouldbe required to produce SSGs in binaries with periods oforder 10 days)Though we show one specific collision model a wide

range of component masses can produce collision prod-ucts in the SSG region Furthermore for certain combi-nations of MS stars the collision product may be ldquobornrdquoin the RS region and contract through the SSG region asit settles back into thermal equilibrium This mechanismhas also been invoked to explain BSS (eg Hills amp Day1976 Leonard 1989 Sills et al 2009) A collision prod-uct that could be observed as a SSG may later be ob-served as a BSS after the normal stars of similar massevolve toward the subgiant and giant branches

4 Geller et al

3 PROBABILITIES OF OBSERVING THEPRODUCTS OF EACH FORMATION CHANNEL

Each of these theoretical formation channels can pro-duce products that have characteristics consistent withat least a subset of the observed SSGs Many of theseproducts are predicted to be relatively short-lived in re-lation to the age of the clusters that have SSGs Weinvestigate here the probability of observing at least oneSSG from each mechanism respectively in different starclusters both over a range in cluster masses (eg Fig-ure 2) and for the observed parameters of the specificclusters that have SSGs (eg Table 1 and Figure 3)We follow the same framework in our calculations for

each mechanism based on the cumulative Poisson prob-ability

Ψ(t τ) = 1minus eminus(tτ)nminus1sum

x=0

(tτ)x

x (1)

where t is the time interval of interest (here the durationthat the star remains in the SSG region) τ is the meantime in between events and n is the number of eventsEquation 1 gives the probability of observing n or moreevents over the time interval t when the mean numberof events is expected to be tτ We discuss our estimatesfor t and τ for each respective formation channel belowand in all cases we attempt to take the most optimisticassumptionsFirst our timescale calculations depend on the clus-

ter age mass (Mcl) metallicity ([FeH]) binary frac-tion (fb) central velocity dispersion (σ0) central den-sity (ρ0) core radius (rc) andor half-mass radius (rhm)We describe how we obtain these values in Section 4In general for our study of the specific clusters (Sec-tion 42) we obtain values from the literature (Table 1)For our general calculations (Section 41 and also as es-timates for cluster specific values that are unavailablein the literature) we assume a Plummer (1911) modeland also use the semi-analytic cluster evolution codeEMACSS (Alexander amp Gieles 2012 Gieles et al 2014Alexander et al 2014)To start we use the rapid Single Star Evolution code

SSE (Hurley et al 2000) to determine the mass of a starthat would reside at the base of the giant branch for agiven cluster age and metallicity We take the evolution-ary states for stars in these calculations directly fromSSE We will refer to this star as S1 below We thenuse SSE to determined the mass radius and luminosityof this star when it was on the zero-age main sequence(ZAMS) the terminal-age MS (TAMS) and at the baseof the RGB for a given metallicityFor many of the scenarios we also require the number

of subgiants (or the fraction of stars that are subgiantsfSG) expected to be in a given cluster To estimate thisvalue we first determine an appropriate mass functionof a cluster of a given age and mass using the methodof Webb amp Leigh (2015) which accounts for the changeto a Kroupa (2001) IMF due to dynamical evolution andmass loss from the cluster2 This method requires an es-

2 The true cluster mass function depends on many uncertain fac-tors (eg the IMF initial Jacobi filling factor remnant retentionfractions etc) which are neglected in the simplified Webb amp Leigh(2015) relation However this simplified relation is sufficient for the

timate of the initial cluster mass which we derive byiteratively modeling clusters of different initial massesusing EMACSS until reproducing the observed presentday cluster mass (at either the solar Galactocentric dis-tance for Section 41 or the true Galactocentric distanceof the given cluster for Section 42) We then use SSE toestimate the masses of stars that would evolve off the MSat +- 1 Gyr from the cluster age These masses com-bined with the mass function provides a rate at whichstars evolve off the MS at the given cluster age and metal-licity Γev This rate multiplied by the lifetime of S1 onthe subgiant branch yields an estimate of the number ofsubgiant stars in a given cluster (and a similar methodcan provide the number of MS stars in the cluster)This theoretical estimate for the number of subgiants

is consistent with observed values For instance in theopen clusters studied in Paper II we count 20-30 sub-giant stars in M67 and about 100 subgiant stars in NGC6791 (These numbers of course depend on where onedefines the end of the MS and the base of the RGB whichcan be somewhat subjective on a CMD) Following thetheoretical procedure above we predict 32 subgiants inM67 and 120 in NGC 6791 both consistent with the ob-served valuesGiven the mass function we can also estimate the

mean single-star mass in the cluster 〈ms〉 For somecalculations we also desire the mean mass of an object(single or binary) We estimate this value as 〈m〉 =(1minus fb) 〈ms〉 + fb 〈mb〉 where fb is the cluster binaryfraction 〈mb〉 is the mean binary mass and we assumea mean binary mass ratio of 05 (a reasonable guess foran approximately uniform mass ratio distribution asis observed for solar-type binaries in the Galactic fieldand globular clusters see eg Raghavan et al 2010 andMilone et al 2012) such that 〈mb〉 = 15 〈ms〉For our general calculations discussed in Section 41

we obtain the binary frequency fb for globular clustersfrom the empirical study of Leigh et al (2013) For openclusters we estimate fb by first assuming that prior todynamical disruptions the binaries would follow the fieldsolar-type stars with a 50 binary frequency and a log-normal binary period distribution (Raghavan et al 2010with a mean of log(P [days]) = 503 and σ = 228)Then we truncated the period distribution at the hard-soft boundary

Phs =πGradic2

(

m1 〈ms〉〈m〉

)32

(m1 + 〈ms〉)minus12σminus30 (2)

derived using the virial theorem to relate the mean bi-nary binding energy to the local mean kinetic energyof a colliding star where m1 is the initial mass of S1and σ0 is the three-dimensional velocity dispersion inthe core (and we assume a Plummer (1911) model and

that σ0 =radic3σ01D) We calculate the cluster binary fre-

quency as the ratio of the area under the truncated pe-riod distribution to that of the full distribution times the50 solar-type field binary frequency This assumes thatthe cluster has lived through sufficient relaxation timesthat all binaries have cycled through the core which isreasonable for the open clusters known to contain SSGs(A more detailed calculation might account for the time

approximate calculations performed here


and radial dependence of the hard-soft boundary butthat is beyond the scope of this paper) This producesbinary fractions consistent with open cluster observa-tions (eg Geller amp Mathieu 2012 Geller et al 2015)In practice this method for open clusters requires aniterative derivation of fb 〈m〉 and Phs For our cluster-specific calculations discussed in Section 42 we take theobserved binary fractions (where available)In the following we describe our derivation of the

timescale τ from Equation 1 for each specific formationmechanism For the MSndashMS collision channel we alsoderive t while for all others we simply take t equal tothe lifetime of S1 on the subgiant branch Again ourassumption for t represents the most optimistic scenariofor the duration of each mechanism

31 Ongoing Binary Mass Transfer Involving aSubgiant Star

We calculate τ here as the mean time between starsin appropriate binaries evolving off of the MS Only bi-naries with orbital periods large enough to avoid Rochelobe overflow (RLOF) on the MS and small enough toundergo RLOF on the subgiant branch are of interestwhich defines a fraction of the binary population by pe-riod fP Here we use the Roche radius equation fromEggleton (1983)

rLa

=049qminus23

06qminus23 + ln(

1 + qminus13) (3)

where we set q = 〈ms〉 m1 a is the binaryrsquos semi-major axis and we assume circular orbits (a standardassumption given the expectation of tidal circulariza-tion and sufficient for these approximate calculations)Likewise only binaries expected to undergo stable masstransfer are of interest We impose a critical mass ra-tio of qcrit = maccretormdonor = 13 below which weassume that the system undergoes a common envelopeand is not included in this particular mechanism Thevalue of 13 is similar to values used in binary populationsynthesis codes for such stars (eg Hurley et al 2002Belczynski et al 2008 and see also Geller et al 2013 andEggleton 2006) Assuming a uniform mass-ratio distri-bution this critical mass ratio allows only 23 of thebinaries to potentially undergo stable mass transfer andthereby provides a factor of fq = 23 below These fac-tors multiplied by the rate at which stars evolve off theMS at the given cluster age and metallicity (Γev see Sec-tion 3) yield

τSG MT = (ΓevfbfPfq)minus1

(4)

32 Increased Magnetic Activity in a Subgiant Star

To calculate τ we follow a similar method as in Sec-tion 31 to estimate the mean time between stars in ap-propriate binaries evolving off of the MS Here for fPwe set the short-period limit to be that at the Rocheradius (see Equation 3 thereby excluding any binariesincluded in Section 31) and the long-period limit to thebinary circularization period of the cluster We estimatethe circularization period of a cluster of a given age fromthe results of Geller et al (2013 dotted line in their Fig-ure 2 that matches the observed binary circularizationperiods from Meibom amp Mathieu 2005 ) The fraction of

binaries with these short periods defines fP We allowall mass ratios hereHowever not all short-period binaries containing a

subgiant star must become SSGs A sample of the openclusters (NGC 188 NGC 2682 NGC 6819 and NGC6791) have sufficient time-series radial-velocity andorphotometric observations to count the known binarieswith orbital periods less than 15 days amongst the SSGsand subgiants as a rough estimate of the efficiency ofSSG formation through this mechanism Within theseclusters we find four normal subgiants and nine SSGsrespectively in binaries with periodslt15 days We applythis fraction of α = 913 to our calculation

τSG Mag = (αΓevfbfP)minus1

(5)

Finally as noted above here we again simply take t asthe lifetime of S1 on the subgiant branch It is possiblethat such stars can remain in the SSG region also duringthe early evolution of the red-giant phase Adding thisto t would increase our probabilties of observing a SSGfrom ldquoSG Magrdquo

33 Rapid Mass Loss from a Subgiant Star

Here we investigate two stripping mechanismsthrough (i) common-envelope or (ii) a grazing collisionFor the common-envelope case ldquoSG CErdquo we use nearlythe same calculations as for the ldquoSG MTrdquo channel (Sec-tion 31) but here we set fq = 13 in Equation 4This optimistic scenario assumes that every subgiantthat undergoes a common-envelope will have its enve-lope stripped in such a way as to produce a SSGFor the grazing collision case ldquoSG Collrdquo τ is the mean

time between collisions involving the stars of interest

τSG Coll (a) = [fSG (2Γ11 + 3fc12Γ12 (a) + 4fc22Γ22 (a))]minus1

(6)

where Γ11 Γ12 and Γ22 are the single-singlesingle-binary and binary-binary encounter rates fromLeigh amp Sills (2011)3 (and τ = 1Γ) except here wemultiply each rate by a factor (NfSG) to account forthe requirement that at least one of the stars involvedmust be a subgiant where N = 234 is the numberof stars in the encounter and fSG is the fraction ofstars in the cluster that are expected to be subgiants(as explained above) fc12 and fc22 are the fractionsof 1+2 and 2+2 encounters respectively that result indirect collisions taken from the grid of scattering ex-periments of Geller amp Leigh (2015) for a given clustermass and half-mass radius As these scattering exper-iments only include MS stars we multiply these fac-tors by the ratio of the gravitionally-focused cross sec-tion for S1 to that of a MS star at the turnoff (ie(MS1RS1) (MMSTORMSTO) Leonard 1989)Γ12 and Γ22 both depend on the binary semi-major

axis a (or orbital period) and we allow binaries fromthe Roche limit of S1 on the ZAMS up to the hard-soft boundary (thereby excluding encounters with softbinaries) To calculate τSG for Equation 1 we take the

3 The encounter rates depend on the binary fraction core ra-dius stellar density velocity dispersion mean stellar mass andthe physical size of the object (ie the stellar radius for a 1+1encounter and the semi-major axis for the 1+2 or 2+2 encounter)We describe how we estimate these values in Section 3

6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag

log10(Mcl[M⊙])Fig 2mdash Poisson probabilities of observing SSGs resulting from the formation channels discussed in Section 2 ldquoSG MTrdquo (blue) is

the probability of observing a binary in the process of mass transfer from a subgiant donor (Sections 21 and 31) ldquoMS Collrdquo (yellow) isthe probability of observing a MS-MS collision product before it settles back to the ZAMS (Sections 24 and 34) ldquoSG Collrdquo and ldquoSGCErdquo (red) are the probabilities of observing a subgiant after having its envelope rapidly stripped (Sections 23 and 33) either througha grazing collision (dashed) or a common-envelope ejection (dotted) ldquoSG Magrdquo (green) is the probability of observing a subgiant witha reduced convective mixing length from enhanced magnetic activity (Sections 22 and 32) Each region shows the Poisson probabilitiesderived from the weighted average timescales (t and τ from Equation 1) over our grid of models weighted by the observed distributions ofages half-mass radii and metallicities for open clusters (left) and globular clusters (right) as described in Sections 3 and 41 The widthsshow one (weighted) standard deviation above and below the weighted mean Additionally we plot predictions from globular cluster MonteCarlo models for the probability of observing SSGs created through each channel (see Section 5) points show the weighted means verticalerror bars show the standard errors of the mean and horizontal bars show the widths of each mass bin (Mass bins are the same for eachchannel for the lowest-mass bin of the ldquoSG MTrdquo and ldquoSG Collrdquo channels we shift the points slightly for readability)

average of τSG Coll (a) weighted by the log-normal pe-riod distribution (within the appropriate Roche limit andhard-soft boundary)We assume here that each collision results in sufficient

stripping to produce a SSG This is likely an overestimateof the true SSG production rate through this mechanismAgain we aim for the most optimistic assumptions in ourcalculations hereFinally as mentioned above we set tSG Strip equal to

the lifetime of S1 on the subgiant branch In our ex-ploratory MESA modeling in Paper II we see that fordifferent amounts of stripping and for different assump-tions about the time the stripping occurs the productcan have a lifetime in the SSG region that is somewhatgreater than or less than the subgiant lifetime of S1Accounting for this level of detail is beyond the scope ofthis paper but may warrant future investigation

34 Collision of two MS Stars

To estimate t here we start with the mean timeof all collision products in Sills et al (1997) to evolvefrom immediately after the collision back to the ZAMStc0 = 674 Myr The mean increase in luminosity for allcollision products in Sills et al (1997) from immediatelyafter the collision until settling back to the MS is a fac-tor of 1015 and we assume this increase for all collisionproducts in our calculations We then make the simplify-ing assumption that the productrsquos luminosity decreases

linearly in time Finally we step through bins in stellarmass and calculate a weighted average of the time thata MS-MS collision product is estimated to remain in theSSG region for a given cluster

tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)

where w(m) weights by the mass function at the massm mf is the ZAMS mass of S1 and m0 is the mass ofa MS star with a luminosity that is 1015 times smallerthan mf (from SSE) The factor f(m) is an estimate ofthe fraction of the time from collision to ZAMS that theproduct is expected to remain in the SSG region thisfactor follows from our assumption that the luminosity ofthe product immediately after the collision increases by afactor of 1015 then decreases back to the ZAMS linearlywith time and may pass through the SSG region thatextends from the magnitude of the main-sequence turnoffdown to 15 magnitudes fainter (approximately coveringthe region of observed SSGs see Figure 1) Certainly amore detailed treatment of this factor is desirable but isbeyond the scope of this paperWe follow the same approach to calculate τMS Coll

as in Section 23 but take fc12 and fc22 directly fromGeller amp Leigh (2015) and use the fraction of MS starswith masses between m0 and mf in place of the fraction


of subgiant stars (fSG) in the cluster

4 COMPARISON OF THE PROBABILITIES OFOBSERVING EACH PRODUCT

We use two methods to compare the probabilities ofobserving at least one product of each respective forma-tion channel (given the two timescales for each channeldiscussed above) one general and averaged over all ob-served open and globular clusters as a function of clustermass (Section 41 and Figure 2) and the other specificto each cluster with observed SSGs (Section 42 Table 1and Figure 3)

41 General

We begin by producing a grid of timescales (t and τfrom Equation 1) for each mechanism covering the rangeof relevant cluster ages (from 2 to 13 Gyr in steps of 1Gyr) masses (from log(Mcl [M⊙ ]) = 3 to 6 in steps of001) half-mass radii (from rhm = 1 to 10 pc in steps of1 pc) and metallicities (from [FeH] = -23 to 02 withsteps of 05 for [FeH] between -2 and 0 the metallicityrange possible in SSE is Z = [00001 003] which corre-sponds to [FeH]sim[-23 02]) for observed open and glob-ular clusters We use a Plummer model and EMACSSwhere necessary and the assumptions discussed in Sec-tion 2We then compile all available observed values of

age rhm and [FeH] for open (Salaris et al 2004van den Bergh 2006)4 and globular (Marın-Franch et al2009 Harris 1996 2010) clusters Then for each of thesetwo samples we take a weighted average of our calcu-lated grid of timescales for each respective mechanismweighted by the fraction of open or globular clusterswithin each bin of age rhm and [FeH] Finally we usethese weighted average timescales to calculate the Pois-son probabilities of observing at least one SSG within acluster of the given mass We divide our results at a massof 104 M⊙ which separates our sample at roughly thetransition mass between open and globular cluster massThe resulting probabilities for each SSG formation

mechanism are shown in Figure 2 in the different coloredregions with widths equal to one (weighted) standard de-viation from the weighted mean value In general taking1Ψ gives the number of clusters that should be observedin order to expect to detect at least one SSG from thegiven mechanism Our calculations predict that roughlyone in every few open clusters and nearly every globularcluster should host at least one SSG This is in reason-able agreement with the current state of observations (seeFigure 4 and Section 42) though no systematic surveyfor SSGs exists (in open or globular clusters) As wersquovetaken optimistic assumptions in our calculations theseprobabilities may be interpreted as upper limitsOur calculations predict that the probability of observ-

ing SSGs from all mechanisms will increase with increas-ing cluster mass This is simply due to the larger numberof stars More importantly for clusters of all masses wepredict that isolated binary evolution mechanisms are

4 We note that a larger catalog for these parameters exists inPiskunov et al (2008) and Kharchenko et al (2013) but here weare more interested in the older open clusters like those observedto have SSGs which were more carefully analyzed and provided inthe given references

SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67

Fig 3mdash Percent of total SSGs predicted from each formationmechanism (see Sections 2 and 3) in all the observed clusters inTable 1

dominant The other mechanisms follow at lower prob-abilities though toward the highest-mass globular clus-ters it becomes equally likely to observe at least one SSGfrom all mechanismsAlthough we show in Figure 2 the probabilities of ob-

serving SSGs as a function of cluster mass cluster den-sity (and encounter rate) is also important For a givencluster mass the rate of SSG formation through the col-lision channels increases with increasing density whilethe rate of SSG formation through the binary evolutionmechanisms is nearly independent of density (within therange of parameters relevant to observed open and glob-ular clusters) The only dynamical mechanism that canaffect the binary evolution channels in these calculationsis the truncation of the binary orbital period distributionat the hard-soft boundary which for clusters of interestis at longer periods than the synchronization period (andthe period at Roche lobe overflow) Again these are op-timistic assumptions meant to provide an upper limit onSSG formation rates As we discuss below more sub-tle dynamical effects like perturbations and exchangeswithin hard binaries may decrease the true SSG produc-tion rate through the binary evolution channels for themost massive clusters

42 Cluster specific

In addition to the general calculation described abovewe also perform specific calculations of the respectiveprobabilities to observe at least one product of each ofthe formation channels for each cluster with a SSG candi-date Here we compile all available data for each clusterthat would serve as an input into our probability cal-culations described in Section 3 and provide these inTable 1 As described above our calculations requirethe age mass metallicity and either the core or half-mass radius Where available we provide the additionalempirical input to our calculations of the observed bi-nary frequency (fb) central density (ρ0) core radius(rc) half-mass radius (rhm) and circularization period(Pcirc) All other necessary values that are unavailable

8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3

Fig 4mdash Number (top) and specific frequency (bottom num-ber of SSGs NSSG divided by the cluster mass Mcl) of SSGs asa function of the cluster mass Observed openglobular clustersfrom Paper I are plotted in openfilled symbols As in Paper Iwe show only those observed SSGs with the highest-likelihood ofcluster membership and within the same radial completeness limitof lt 33 core radii (see Paper I for details) Error bars show thestandard Poisson uncertainties on NSSG (and we truncate the lowererror bars for cases with NSSG = 1) The gray-filled region showsthe predicted number of SSGs from our calculations in Section 41through all mechanisms combined and the hatched region showsthe predicted number of SSGs for the collision mechanisms alone(ie ldquoSG Collrdquo and ldquoMS Collrdquo) Note that our Poisson calcula-tions are not limited in radius from the center of the cluster (asare the observations) and rely on optimistic assumptions thesecalculations show upper limits

in the literature are inferred using the same assumptionsas aboveWe use these empirical values to determine t and τ in

Equation 1 as described in Section 3 and provide theprobabilities of observing at least one SSG from the givenmechanism in each cluster in Table 1 We also providethe combined Poisson probabilities of observing the ob-served number of SSGs (nSSG) in each cluster5 from anyformation channel (calculated by summing the tτ valuesfrom each mechanism and using this in Equation 1 andonly given for clusters with SSGs) For ease of readingwe do not include uncertainties on these probabilities inthe table however we do follow the uncertainties on eachinput parameter through our calculations for each prob-ability If a parameter does not have uncertainties in theliterature (and therefore no error is given in the table)we assume a 10 uncertainty for our calculations Thenumber of digits provided in the Table shows the order of

5 The number of SSGs is taken from Paper I where we selectstars that reside in the SSG region of the CMD in at least one avail-able color-magnitude combination and have a lt 10 probabilityof being a field star

magnitude of the inferred range in probabilities resultingfrom the uncertainties in input values We round anyprobability gt 099 up to 1In Paper II we investigate the SSGs in two of these

clusters NGC 6791 and M67 in depth and perform morecareful calculations of their formation (involving moredetailed empirical input and using a slightly differentmethod) Our results here agree very well with thosefrom Paper II which provides further confidence in ourcalculations here Specifically in Paper II we find a prob-ability of observing at least one SSG from the ldquoSG MTrdquomechanism in M67 of 4 and in NGC 6791 of 14 wherehere we find 5 and 9 respectively In Paper II wefind a probability of 42 and 94 of observing at leastone SSG from the ldquoSG Magrdquo in M67 and NGC 6791respectively as compared to 47 and 82 here For theldquoSG Collrdquo scenario in Paper II we find a probability ofsim3 that we would observe at least one in M67 com-pared to 20 here Though this particular probabilityvalue appears higher here (due to our more optimisticassumptions) the uncertainty on this probability is ofthe same order as the value itselfAdditionally we show the results graphically in Fig-

ure 3 where we plot the percentage of SSGs predictedover all clusters in Table 1 to come from each mech-anism To construct this plot we sum the number ofpredicted SSGs for a given mechanism over the observedclusters and divide by the total number of SSGs pre-dicted for all clusters from all mechanisms For instanceour calculations predict that 67 of sub-subgiants inthese observed clusters may come from the ldquoSG MagrdquomechanismNonetheless if we sum the probabilities for each mech-

anism given in Table 1 we expect to observe at least oneSSG from each mechanism when considering all clustersFor nearly all of the globular clusters our calculationssuggest that these formation channels are sufficient toexplain all observed SSGs (ie ΨnSSG sim 1 for these clus-ters) In the open cluster regime the number of SSGspredicted for clusters in this mass range is in rough agree-ment with the observations (Figure 4) though the spe-cific ΨnSSG values for the observed open clusters are be-low one in Table 1 This may indicate that we haveoverlooked viable formation channels in the open clusterregime or that we have underestimated values in our cal-culations primarily for open clusters and we return tothis in Section 6In Figure 4 we show the number of SSGs predicted

by our model as a function of cluster mass comparedto that of the observed clusters (see Paper I Figure 7)The gray band combines all formation channels whilethe hatched region shows only the collision channels Ourmodel agrees with the general trend in the observationsof decreasing specific frequency of SSGs toward increas-ing cluster mass However toward the high-mass end ourmodel begins to over-predict the number of SSGs Thismay imply that there are more SSGs to be discoveredin these clusters (which indeed is expected see Paper I)This discrepancy may also be tied at least in part toour simplified treatment of how dynamics affects the bi-nary evolution channels Perhaps more subtle dynamicaleffects (such as perturbations or exchanges not includedin these calculations) inhibit the binary evolution chan-nels significantly in clusters with high encounter rates


(like the massive observed clusters in our sample) Weinvestigate this further in the following section Indeedfor the most massive clusters in our sample our modelpredicts that the collision mechanisms alone can nearlyproduce the observed numbers of SSGs

5 SUB-SUBGIANTS IN STAR CLUSTER N -BODYMODELS

Our Poisson probability calculations make simplifyingassumptions about SSG formation and provide upperlimits for SSG formation rates N -body star clustermodels can alleviate some of these simplifications andin particular can allow us to study the effects from morecomplex dynamical encounters and subtle perturbationsthat we do not consider in our analytic calculations

51 Direct N -body Models

To our knowledge the Hurley et al (2005) N -bodymodel of M67 is the only star cluster model that specif-ically discusses the creation of a SSG star They usedthe NBODY4 code (Aarseth 1999) which utilizes BSE(Hurley et al 2002) for binary-star evolution The onlypathway available for SSG formation in these modelsis through binary evolution the other mechanisms dis-cussed here are not yet implemented in the N -body codefor SSG formation (though some are implemented to pro-duce BSS)This specific binary first went through a stage of con-

servative stable mass transfer where the subgiant pri-mary transferred mass onto its MS companion This thenled to a common-envelope merger event that created theSSG single star seen at the age of M67 (We refer thereader to Hurley et al 2005 for a more detailed descrip-tion of this starrsquos history) This mechanism is similarin part to our ldquoSG MTrdquo pathway (Section 21) and isformally included in the ldquoSG MTrdquo rate calculations de-scribed in Sections 3 and 4 (because the system startswith stable mass transfer) Unlike our mechanism how-ever the Hurley et al (2005) star is more massive thanthe normal giants in the cluster at the age of M67 butwith a lower core mass than the normal giants Theyattribute the lower luminosity of the object to this lowercore mass Through our extensive BSE modeling (seePaper II) we do not see common-envelope merger prod-ucts as a dominant SSG formation channel within themass-transfer mechanism though we have likely not cov-ered the entire parameter space leading to SSG formationin BSE (and common-envelope evolution remains poorlyunderstood and only approximated within BSE) Fur-thermore as most of the observed SSGs in open clustersappear to be in short-period binaries this specific path-way may not produce SSGs similar to the majority ofthose observed

52 Monte Carlo Models

We also investigated a grid of Monte Carloglobular cluster models from the Northwest-ern group (Joshi et al 2000 2001 Fregeau et al2003 Fregeau amp Rasio 2007 Chatterjee et al 2010Umbreit et al 2012) Specifically we use a supersetof the simulations presented in Chatterjee et al (20102013ab) which includes 327 models that cover theparameter space of the observed globular clusters in

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3

log10(ρ [M pcminus3])

2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag

Fig 5mdash Comparison of the number of SSGs from the ldquoSG MTrdquo(NSG MT) and ldquoSG Magrdquo (NSG Mag) channels created in a gridof Monte Carlo globular cluster models that have the given totalnumbers of stars (Nstars) binaries (Nb) and blue straggler stars(NBSS) core radius (rc) central density (log10(ρ)) ratio of thehalf-mass and core radii (rhrc) core collision rate (Γc) and corebinary frequency (fb) These parameters are all calculated theo-retically at the same snapshot times as we use to identify the SSGsand some may be slightly different from what an observer wouldmeasure (Chatterjee et al 2013b) We show network diagrams foreach channel (top ldquoSG MTrdquo bottom ldquoSG Magrdquo) where eachpath around the plot defines a specific cluster model crossing theaxes at the given cluster parameters and colored by the number ofSSGs created by that channel (see color bars at left of each plot)

our Galaxy (though all at a metallicity of Z=0001)We examine snapshots from these models between 9and 12 Gyr We used two methods to identify SSGsin these models (i) we selected SSGs based on thelocation in the H-R diagram (as in Figure 1) and (ii)we identified other stars that may be observed as SSGsin a real cluster but were not found in the SSG regionof the simulated H-R diagram due to limitations of BSE

10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS

ρ = minus0497ρ = minus0727

0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981

Fig 6mdash Comparison of the number of SSGs from the ldquoSG MTrdquo (NSG MT blue circles) and ldquoSG Magrdquo (NSG Mag green triangles)channels created in a grid of Monte Carlo globular cluster models and showing the same parameters as in Figure 5 Here we plot thenumber of SSGs as a function of each of these parameters respectively showing only models that produced at least one SSG Small pointsshow the raw values from the grid and larger points show the mean values in bins with vertical error bars equal to the standard errors ofthe mean and horizontal lines showing the bin sizes (which are smaller than the symbols in some cases) For reference we also include therespective Pearson correlation statistics (ρ) calculated for the mean values in each panel

(which is used in both the NBODY4 and Monte Carlomodels)Method (i) discovers all SSGs produced through the

ldquoSG MTrdquo channel this is the only mechanism availableto producing SSGs within BSE We identified over 1100ldquoSG MTrdquo SSGs in these models 99 of these simu-lated SSGs are currently in binaries and the remainderwere previously in binaries 98 of the SSGs in bina-ries are currently undergoing RLOF Of the few that aredetached sim80 contain an evolved star that had pre-viously lost ge01M⊙ presumably from a recently com-pleted period of mass transfer (a subpopulation that wealso briefly discuss in Paper II) Importantly only sim10

of these SSGs suffered strong encounters or direct colli-sions prior to becoming a SSG (though weak fly-bys arenot tracked in these models as this is part of the re-laxation process) The vast majority of ldquoSG MTrdquo SSGsin these models avoided strong encounters for the entirelifetime of the globular clusterTo investigate predictions for the other formation chan-

nels we follow similar assumptions as in Section 3 Morespecifically we identify ldquoSG Magrdquo SSGs as binaries inthe models with orbital periods P lt Pcirc that contain asubgiant (and then multiply the number identified by ourempirical fraction of 913 see Section 32) We identifyldquoMS Collrdquo SSGs as the products of collisions involving


two main-sequence stars that occurred close enough intime to the model snapshot output time and have a prod-uct bright enough to reside in the SSG region (using thesame assumptions as Section 34) Finally we identifyldquoSG Collrdquo SSGs as the products of collisions involvingat least one subgiant star that occurred close enough intime to the model snapshot output time (Likely not allof these collisions would create SSGs but this will pro-vide an upper limit) Through this method we identifymore than 12000 additional SSGs6 primarily from theldquoSG Magrdquo channelWe plot the Poisson probabilities of observing at least

one SSG from these models in bins of cluster mass withinFigure 2 For the ldquoSG MTrdquo and ldquoSG Magrdquo points wefirst apply a correction factor to the number of SSGsin each model to account for a different assumed binaryorbital period (or semi-major axis) distribution we as-sume a log-normal period distribution in Section 3 whilethe Monte Carlo models use a distribution that is flat inthe log For a given binary frequency a flat distributioncreates a factor of about 25 more short-period binaries(eg that can undergo RLOF on the subgiant branch)than does the log-normal distribution For all channelswe then take the average number of SSGs in each massbin weighted by the observed distributions of half-massradii and cluster age (in a similar manner as described inSection 41) We then set tτ from Equation 1 equal tothis weighted average number of SSGs from the modelsin each mass bin to calculate the Poisson probabilitiesThe predictions from the Monte Carlo models agree wellwith those from our analytic upper limits from Figure 2even given the different assumptions that go into eachmethod The Monte Carlo models predict a factor ofa few less ldquoMS Collrdquo SSGs than predicted analyticallylikely due to our implicit assumptions in Section 34 ofall encounters occurring directly at the cluster centerand with zero impact parameter (neither of which arerequired in the Monte Carlo model) Nonetheless theagreement with this (relatively) independent method ofderiving Ψ for all channels supports the results of ourmore simplified analytic calculationsAs a further step we also investigate the grid of Monte

Carlo models for predictions of the type of clusters thatshould harbor the most SSGs The collision channelsbehave as expected where more SSGs are produced inclusters with larger collision rates However the vastmajority of the SSGs produced in all these Monte Carlomodels (gt 99) derive from the binary evolution chan-nels Furthermore these models (plus our assumptionsin identifying SSGs therein) predict on average about fivetimes more ldquoSG Magrdquo than ldquoSG MTrdquo SSGsWe focus on these ldquoSG MTrdquo and ldquoSG Magrdquo mecha-

nisms here and show detailed comparisons of these twochannels in Figures 5 and 6 Here we do not apply anycorrection to the number of SSGs from each model basedon the input binary period distribution (as we did above)Some of these Monte Carlo models contain very largenumbers of SSGs inconsistent with the (much smaller)number of SSGs observed in the clusters wersquove studied

6 Collisions are tracked continuously within these models whilefull snapshot output occurs roughly every Gyr common-envelopeevents are not tracked continuously and therefore we cannot in-vestigate ldquoSG CErdquo here

This likely results from a combination of initial condi-tion choices (some of which produce clusters that donrsquotmatch those wersquove studied) and also the details of binaryevolution in BSE However here we are not interested inthe raw number of SSGs produced instead we investi-gate for trends in number of SSGs versus various clusterparameters predicted for these modelsIn Figure 5 we show network diagrams to visualize

how all of the parameters from a given model relate tothe number of SSGs created In this diagram one arcaround the figure corresponds to one model hitting theaxes at the appropriate values for the model and with acolor defined by the number of SSGs In Figure 6 we plotthe number of SSGs against various (mostly observable)cluster parametersFor both channels we see correlations of increasing

number of SSGs with increasing number of stars (Nstars)number of binaries (Nb) and binary frequency (fb)These correlations are expected as nearly any popula-tion of stars that involve binaries (exotic or otherwise)should behave this way Plotting the relative numberof SSGs with respect to Nstars and Nb (second row ofFigure 6) shows no significant correlationThe more interesting result from this comparison is

that the number of SSGs produced through both bi-nary channels increases toward decreasing central density(log10(ρ)) increasing core radii (rc) and a decreasing ra-tio of the half-mass to core radii (rhrc) In other wordsthese model predicts that diffuse clusters are most effi-cient at producing SSGs through binary channels Fur-thermore these trends are far more dramatic for SSGsproduced through ongoing mass transfer (ldquoSG MTrdquo)While our analytic calculations from Section 3 only ac-count for disruptions of soft binaries the Monte Carlomodel predicts that even these hard binaries can be sub-jected to perturbations exchanges etc that can stop bi-naries from forming SSGs Apparently the mass transferchannel is particularly vulnerable to these dynamical in-terruptions (see also Leigh et al 2016b)We also investigate the relation between the number

of SSGs and the core collision rate (Γc here we calcu-late the combined rate for 1+2 and 2+2 encounters fora binary semi-major axis equal to the Roche radius ofa 10 Gyr star at the end of the subgiant phase with a045M⊙ MS star companion roughly the expected meanMS mass) For both the ldquoSG MTrdquo and ldquoSG Magrdquo chan-nels the number of SSGs rises toward modest Γc values(sim003 Myrminus1) The ldquoSG MTrdquo channel then decreasesagain toward high Γc values while the ldquoSGMagrdquo channelremains roughly constantGenerally as Γc increases the more frequent dy-

namical encounters become more efficient at hardening(ie shrinking the semi-major-axis of) hard binaries inthis case to potentially create SSGs through both bi-nary channels Additionally as Γc increases dynamicalexchanges that insert subgiants into sufficiently short-period binaries becomes more likely This may accountfor the increase in the number of SSGs in both binarychannels up to modest Γc valuesOn the other hand toward higher Γc values encoun-

ters may be energetic and frequent enough to perturbbinaries away from producing SSGs (eg through in-ducing binary coalescence or otherwise inhibiting masstransfer) This may at least partly explain the decrease

12 Geller et al

in NSG MT and the flattening in NSG Mag toward higherΓc values Though we also believe that initial conditionchoices may contribute to this trendSome additional insight into this relation between Γc

and the number of SSGs can be found by comparingagainst the number of BSS NBSS BSS are producedin the Monte Carlo model through both collisions andbinary evolution and here we include both channelsin NBSS For the few models that produce gt150 BSS(beyond the peak in the relation between NBSS andNSG MT) the mean encounter rate 〈Γc〉 sim 019 as com-pared to 〈Γc〉 sim 006 for models with lt150 BSS At thelow NBSS and low Γc end both the SSGs and BSS areproduced primarily through binary evolution and there-fore the number of SSGs increases with increasing num-ber of BSS However the models with high Γc produceBSS primarily through collisions due to higher encounterrates Encounters can also perturb the ldquoproto - SG MTrdquobinaries away from producing SSGs through mass trans-fer which results in a peaked distribution of NBSS andNSG MT On the other hand we see again that the ldquoSGMagrdquo channel is less affected by dynamics and NSG Mag

simply continues to increase with NBSSFor both the ldquoSG Magrdquo and ldquoSG MTrdquo channels we

see the relative number of SSGs with respect to NBSS

decreases toward larger NBSS Again the models thatproduce the most BSS do so primarily through collisionsthus the most interesting portion of this panel is towardthe low-NBSS end where the BSS are produced moreoften through binary evolution (like the SSGs here) Themodels predict that for some clusters with low encounterrates the number of SSGs may be comparable (to withina factor of a few) to the number of BSSIn summary the prediction from these Monte Carlo

models is that the binary evolution channels dominatethe production of SSGs Furthermore the largest num-ber of SSGs produced through the binary evolution chan-nels should be found in massive diffuse clusters withhigh binary frequencies and modest encounter rates Atpresent the observed data are too sparse to search for atrend in number of SSGs with encounter rate Nonethe-less this result from the Monte Carlo models aligns withour suggestion in Paper I that dynamical disruptionsperturbations and other alterations to ldquoproto-SSGrdquo bi-naries could explain the empirical trend of decreasingspecific SSG frequency with increasing cluster mass (Fig-ure 4) These dynamical effects inhibit the binary evo-lution channels and particularly the ldquoSG MTrdquo chan-nel in clusters with higher encounter rates (like thosein our observed sample of globular clusters) Clusterswith the highest encounter rates may begin to produceSSGs through the collision mechanisms at a similar orperhaps higher rate than the binary mechanisms

6 DISCUSSION AND CONCLUSIONS

In Paper I we identify from the literature a sample of65 SSG and RS stars in 16 star clusters including bothopen and globular clusters and we summarize their em-pirical demographics within this paper in Section 1 InPaper II we discuss in detail three potential formationchannels for SSGs The mechanisms within these chan-nels involve isolated subgiant binary evolution rapid par-

tial stripping of a subgiants envelope (for which we en-vision two mechanisms one through common-envelopeevolution and another through dynamical encounters)or reduced luminosity due to magnetic fields that inhibitconvection In addition Paper II briefly considers a for-mation channel through collisions of two main-sequencestars during a binary encounter which we elaborate uponhereWith isolated binaries SSGs may be produced through

ongoing binary mass transfer involving a subgiant star(Section 21 ldquoSG MTrdquo) reduced convective efficiency ona rapidly rotating magnetically active subgiant likely ina tidally locked binary (Section 22 ldquoSG Magrdquo) or rapidstripping of a subgiantrsquos envelope during a common-envelope phase (Section 23 ldquoSG CErdquo) Invoking stel-lar collisions (most likely involving at least one binaryLeigh amp Geller 2012 2013) SSGs can be created througha collision and subsequent merger of two MS stars ob-served while contracting back onto the MS (Section 24ldquoMS Collrdquo) or a grazing collision involving a subgiantthat rapidly strips much of its envelope (Section 23 ldquoSGCollrdquo) The binary evolution channels can happen in iso-lation while the collision channels require the dynamicalenvironment of a star cluster Yet all of these channelsare catalyzed by binary starsOur analytic Poisson probability calculations (Sec-

tions 3 and 4 which are upper limits) and our analysisof a large grid of Monte Carlo models (Section 5) suggestthat the binary evolution channels are dominant In par-ticular both of these methods predict that we are mostlikely to observe SSGs that originate from magneticallyactive subgiants with reduced convective efficiency (seeFigures 2 and 3)This result is based on the SSG formation rates alone

without any constraint on the expected binarity of theproduct Observationally we know that the SSGs areprimarily in short-period active binaries (Paper I andsee Section 1 here) At least two thirds of the SSGshave photometric andor radial-velocity periods of 15days and at least three quarters of these variables areconfirmed to be radial-velocity binaries These short or-bital periods are consistent with tidally locked binaries(eg Meibom amp Mathieu 2005) as expected for the ldquoSGMagrdquo mechanism The SSGs with the shortest-periodvariability may be in binaries currently (or very recently)undergoing mass transfer Indeed there are a few WUMa contact binaries amongst the SSGs in our sample(in NGC 188 ω Centauri and NGC 6397) which supportthe ldquoSG MTrdquo mechanism In short the ldquoSG MTrdquo andldquoSG Magrdquo mechanisms naturally explain the binarityAdditional empirical evidence supporting SSG forma-

tion through isolated binary evolution may be found inthe nearly 10000 stars in the ldquoNo-Manrsquos-Landrdquo from Ke-pler (Batalha et al 2013 Huber et al 2014) which maybe field SSGs These stars are important targets for fu-ture observations and we will investigate them in moredetail within a future paperConversely producing SSGs through collisions may

only be relevant in very dense star clusters Further-more encounters that lead to the ldquoMS Collrdquo mechanismgenerally produce collision products in wider binaries (orwithout companions) sometimes with periods that areorders of magnitude larger than observed for the SSGs(Leigh et al 2011 Geller et al 2013) When also consid-


ering the low Poisson probabilities calculated here for theldquoMS Collrdquo channel and the even lower number predictedby the Monte Carlo models (see Figure 2 and Section 5)we conclude that in most clusters observing a SSG fromthe ldquoMS Collrdquo channel is unlikely especially for SSGsfound in a short-period binary The few globular clus-ters studied in Paper I with very high encounter ratesmay be the best places to find SSGs produced throughthis mechanism (see Section 42 and Table 1)Observing a SSG resulting from the rapid loss of a

subgiantrsquos envelope (ldquoSG Striprdquo) through either mecha-nism explored here is also relatively unlikely given ourPoisson probability calculations and our analysis of theMonte Carlo models The expected binarity of the prod-uct for ldquoSG Striprdquo is less clear than for the other mecha-nisms It may be possible that a grazing encounter thatstrips a subgiantrsquos envelope can leave a bound compan-ion in a short-period binary (akin to a tidal capture bi-nary) but further study is required to confirm if this isindeed possible Likewise stripping in common-envelopeevolution is highly uncertain and it is unclear what thebinarity of the product would beOther efficient mechanisms may also exist that we have

not identified which could explain why our Ψ(nSSG)Poisson probabilities do not reach unity for some clus-ters (and particularly the open clusters) in Table 1where nSSG SSGs are in fact observed For instancethere may be other ldquoSG Striprdquo mechanisms that we havenot investigated Perhaps SSGs can be created if stablemass transfer is interrupted dynamically as discussed inLeigh et al (2016b) In addition very close companionsto neutron stars can be evaporated as in the well-knownldquoblack widowrdquo pulsars (eg Fruchter et al 1990) Per-haps companions in the early stages of being evaporatedwould appear as SSGs as may be the case for SSG U12in NGC 6397 (DrsquoAmico et al 2001 Ferraro et al 2003)Massive and diffuse globular clusters may be the most

promising targets for future observations aimed at identi-fying additional SSGs The Monte Carlo globular clustermodels (Section 5) predict that such clusters should havethe largest frequency of SSGs created through the binaryevolution channels The Monte Carlo models also pre-dict that the binary evolution channels may be inhibitedfor the densest clusters with high encounter rates whichis consistent with the current observations (Figure 4though note that the observations are incomplete seePaper I) It is clear that in some clusters multiple mech-anisms likely operate simultaneously to produce SSGs(eg see Table 1)Many of these observed and predicted trends in num-

ber of SSGs are also seen for BSS For instance thefrequency of BSS in globular clusters is observed to beanticorrelated with the absolute luminosity (mass) ofthe cluster (Piotto et al 2004 Leigh et al 2007) butcorrelated with the binary fraction (Sollima et al 2008Milone et al 2012) These observations point to bina-ries as a critical ingredient for BSS formation in globularclusters (Knigge et al 2009) The correlations seen inglobular cluster observations have been interpreted the-oretically to indicate that binary evolution is an impor-

tant and sometimes dominant BSS production mecha-nism (Leigh et al 2011) though binary-mediated colli-sions may also be important at high densities (Sills et al2013 Chatterjee et al 2013a) The reduced survival ofbinaries (ie BSS and SSG progenitors) in high density(and high velocity dispersion) environments likely alsocontributes to these observed correlations (Davies et al2004 Sollima 2008) as does the preferential retention ofbinary stars compared to the less massive single stars inclusters that experience significant mass loss (as may bethe case for the lower-mass clusters in our observed SSGsample) Binaries are also critical for BSS (and likelyalso SSG) formation in open clusters (Mathieu amp Geller2009) and the field (Carney et al 2005) The discus-sion from this body of literature may help to explain theobserved decreasing trend in specific frequency of SSGswith increasing cluster mass shown in Figure 4Though we focus on the SSGs throughout the major-

ity of the paper the RS stars (ie stars that occupy thelighter gray regions in Figure 1) have very similar em-pirical characteristics (Paper I) As shown in Figure 1RS and SSG stars may be produced through the samemechanisms and in some cases one can be the evolu-tionary precursor to the other Furthermore at leasttwo of these mechanisms that form SSGs mass trans-fer and MS ndash MS collisions are also invoked to ex-plain the origins of BSS and yellow stragglersgiants(McCrea 1964 Mathieu amp Latham 1986 Leonard 1989Chen amp Han 2008 Leigh et al 2011 Chatterjee et al2013a Sills et al 2013 Gosnell et al 2015 Leiner et al2016) Some fraction of these stars may have been bornthrough the same (or similar) formation channels andperhaps in some cases these stars may represent differentstages along the same evolutionary sequence Comparingthe frequencies and binary characteristics of these stel-lar populations across multiple star clusters could revealimportant insights into their formation mechanism(s)and provide important guidance for detailed evolution-ary models of binary mass transfer and the products ofstellar collisions

AMG acknowledges support from NASA throughHST grant AR-13910 and a National Science Foun-dation Astronomy and Astrophysics Postdoctoral Fel-lowship Award No AST-1302765 SC acknowledgessupport from NASA through HST grant HST-AR-12829004-A Support for Programs AR-13910 and HST-AR-12829004-A were provided by NASA through agrant from the Space Telescope Science Institute whichis operated by the Association of Universities for Re-search in Astronomy Incorporated under NASA con-tract NAS5-26555 This research was supported in partthrough the computational resources and staff contribu-tions provided for the Quest high performance comput-ing facility at Northwestern University which is jointlysupported by the Office of the Provost the Office for Re-search and Northwestern University Information Tech-nology

REFERENCES

Aarseth S J 1999 PASP 111 1333Alexander P E R amp Gieles M 2012 MNRAS 422 3415

Alexander P E R Gieles M Lamers H J G L M ampBaumgardt H 2014 MNRAS 442 1265

14 Geller et al

Batalha N M Rowe J F Bryson S T et al 2013 ApJS204 24

Belczynski K Kalogera V Rasio F A et al 2008 ApJS 174223

Bressan A Marigo P Girardi L et al 2012 MNRAS 427 127Carney B W Lee J-W amp Dodson B 2005 AJ 129 656Carraro G Girardi L amp Marigo P 2002 MNRAS 332 705Chabrier G Gallardo J amp Baraffe I 2007 AampA 472 L17Chatterjee S Fregeau J M Umbreit S amp Rasio F A 2010

ApJ 719 915Chatterjee S Rasio F A Sills A amp Glebbeek E 2013a ApJ

777 106Chatterjee S Umbreit S Fregeau J M amp Rasio F A 2013b

MNRAS 429 2881Chen X amp Han Z 2008 Mon Not R Astron Soc 387 1416Chumak Y O Platais I McLaughlin D E Rastorguev A S

amp Chumak O V 2010 MNRAS 402 1841Clausen J V Bruntt H Claret A et al 2009 AampA 502 253DrsquoAmico N Possenti A Manchester R N et al 2001 ApJ

561 L89Davies M B Piotto G amp de Angeli F 2004 MNRAS 349 129Di Cecco A Bono G Prada Moroni P G et al 2015 AJ

150 51Eggleton P 2006 Evolutionary Processes in Binary and Multiple

Stars ed Eggleton PEggleton P P 1983 ApJ 268 368Fabian A C Pringle J E amp Rees M J 1975 MNRAS 172

15PFerraro F R Sabbi E Gratton R et al 2003 ApJ 584 L13Fregeau J M Cheung P Portegies Zwart S F amp Rasio F A

2004 MNRAS 352 1Fregeau J M Gurkan M A Joshi K J amp Rasio F A 2003

ApJ 593 772Fregeau J M amp Rasio F A 2007 ApJ 658 1047Fruchter A S Berman G Bower G et al 1990 ApJ 351 642Geller A M Hurley J R amp Mathieu R D 2013 AJ 145 8Geller A M Latham D W amp Mathieu R D 2015 AJ 150

97Geller A M amp Leigh N W C 2015 ApJ 808 L25Geller A M amp Mathieu R D 2011 Nature 478 356mdash 2012 AJ 144 54Geller A M Mathieu R D Harris H C amp McClure R D

2008 AJ 135 2264Geller A M Leiner E M Bellini A et al 2017 ArXiv

e-prints arXiv170310167Gieles M Alexander P E R Lamers H J G L M amp

Baumgardt H 2014 MNRAS 437 916Gosnell N M Mathieu R D Geller A M et al 2015 ApJ

814 163Harris W E 1996 AJ 112 1487mdash 2010 ArXiv e-prints arXiv10123224Hills J G amp Day C A 1976 Astrophys Lett 17 87Hole K T Geller A M Mathieu R D et al 2009 AJ 138

159Huber D Silva Aguirre V Matthews J M et al 2014 ApJS

211 2Hurley J R Pols O R Aarseth S J amp Tout C A 2005

MNRAS 363 293Hurley J R Pols O R amp Tout C A 2000 MNRAS 315 543Hurley J R Tout C A amp Pols O R 2002 MNRAS 329 897Joshi K J Nave C P amp Rasio F A 2001 ApJ 550 691Joshi K J Rasio F A amp Portegies Zwart S 2000 ApJ 540

969Kalirai J S Richer H B Fahlman G G et al 2001 AJ 122

266Kharchenko N V Piskunov A E Schilbach E Roser S amp

Scholz R-D 2013 AampA 558 A53Knigge C Leigh N amp Sills A 2009 Nature 457 288

Kroupa P 2001 MNRAS 322 231Leigh N amp Geller A M 2012 MNRAS 425 2369Leigh N Knigge C Sills A et al 2013 MNRAS 428 897Leigh N amp Sills A 2011 MNRAS 410 2370Leigh N Sills A amp Knigge C 2007 ApJ 661 210mdash 2011 MNRAS 416 1410Leigh N W C Antonini F Stone N C Shara M M amp

Merritt D 2016a MNRAS 463 1605Leigh N W C amp Geller A M 2013 MNRAS 432 2474Leigh N W C Geller A M amp Toonen S 2016b ApJ 818 21Leiner E Mathieu R D amp Geller A M 2017 ArXiv e-prints

arXiv170310181Leiner E Mathieu R D Stello D Vanderburg A amp

Sandquist E 2016 ApJ 832 L13Leonard P J T 1989 AJ 98 217Marın-Franch A Aparicio A Piotto G et al 2009 ApJ 694

1498Mathieu R D amp Geller A M 2009 Nature 462 1032Mathieu R D amp Latham D W 1986 AJ 92 1364

McCrea W H 1964 Mon Not R Astron Soc 128 147Meibom S amp Mathieu R D 2005 ApJ 620 970Meibom S Grundahl F Clausen J V et al 2009 AJ 137

5086Milliman K E Mathieu R D Geller A M et al 2014 AJ

148 38Milone A P Piotto G Bedin L R et al 2012 AampA 540

A16Paxton B Marchant P Schwab J et al 2015 ApJS 220 15Piotto G De Angeli F King I R et al 2004 ApJ 604 L109Piskunov A E Schilbach E Kharchenko N V Roser S amp

Scholz R-D 2008 AampA 477 165Platais I Cudworth K M Kozhurina-Platais V et al 2011

ApJ 733 L1Plummer H C 1911 MNRAS 71 460Press W H amp Teukolsky S A 1977 ApJ 213 183Raghavan D McAlister H A Henry T J et al 2010 ApJS

190 1Salaris M Weiss A amp Percival S M 2004 AampA 414 163Sandquist E L Shetrone M Serio A W amp Orosz J 2013

AJ 146 40Sarajedini A von Hippel T Kozhurina-Platais V amp

Demarque P 1999 AJ 118 2894Sills A Adams T amp Davies M B 2005 MNRAS 358 716Sills A Adams T Davies M B amp Bate M R 2002

MNRAS 332 49Sills A Faber J A Lombardi Jr J C Rasio F A amp

Warren A R 2001 ApJ 548 323Sills A Glebbeek E Chatterjee S amp Rasio F A 2013 ApJ

777 105Sills A Karakas A amp Lattanzio J 2009 ApJ 692 1411Sills A Lombardi Jr J C Bailyn C D et al 1997 ApJ

487 290Sollima A 2008 MNRAS 388 307Sollima A Lanzoni B Beccari G Ferraro F R amp Fusi

Pecci F 2008 AampA 481 701Straizys V Maskoliunas M Boyle R P et al 2014 MNRAS

437 1628Thompson I B Kaluzny J Rucinski S M et al 2010 AJ

139 329Tian B Deng L Han Z amp Zhang X B 2006 AampA 455 247Tofflemire B M Gosnell N M Mathieu R D amp Platais I

2014 AJ 148 61Umbreit S Fregeau J M Chatterjee S amp Rasio F A 2012

ApJ 750 31van den Bergh S 2006 AJ 131 1559Webb J J amp Leigh N W C 2015 MNRAS 453 3278

Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15

TABLE 1Sub-subgiant Formation Probabilities

Cluster age [FeH] Mcl fb σ0 log(ρ0) rc rhm Pcirc nSSG ΨSG MT ΨMS Coll ΨSG Coll ΨSG CE ΨSG Mag Ψ(nSSG)[Gyr] [M⊙] [km sminus1] [M⊙pc3] [pc] [pc] [day]

Open Clusters

NGC 188 62 00 1500plusmn400 05plusmn005 041plusmn004 middot middot middot 21 40 145plusmn18 3 0042 0003 002 002 05 004NGC 2158 2 -06 15000 middot middot middot middot middot middot middot middot middot 323 middot middot middot middot middot middot 1 005 0006 0001 0026 027 033NGC 2682 4 00 2100plusmn600 057plusmn004 059plusmn007 middot middot middot 1 middot middot middot 121plusmn13 2 005 005 01 00 047 02NGC 6791 8 04 4600plusmn1500 middot middot middot 062plusmn01 middot middot middot 34 middot middot middot middot middot middot 5 0086 0004 003 004 082 004NGC 6819 24 00 2600 04plusmn002 middot middot middot middot middot middot 175 middot middot middot 62plusmn11 1 0016 0005 0001 0008 012 015NGC 7142 36 01 500 middot middot middot middot middot middot middot middot middot 31 middot middot middot middot middot middot 0 0009 000013 00003 00045 01 middot middot middot

Globular Clusters

NGC 104 131 -072 10times106 002plusmn001 11plusmn03 518 047 415 middot middot middot 8 071 1 1 046 1 1NGC 5139 115 -153 22times106 middot middot middot 168plusmn03 345 359 756 middot middot middot 15 099 04 04 09 1 1NGC 6121 125 -116 13times105 01plusmn001 4plusmn02 394 074 277 middot middot middot 2 04 073 085 025 1 1NGC 6218 127 -137 14times105 006plusmn001 45plusmn04 353 110 247 middot middot middot 1 038 03 04 02 1 1NGC 6366 133 -059 48times105 011plusmn003 13plusmn05 270 221 298 middot middot middot 1 06 02 04 04 1 1NGC 6397 127 -202 77times104 002plusmn001 45plusmn02 606 003 194 middot middot middot 3 007 05 03 00 05 03NGC 6652 129 -081 79times104 01plusmn001 middot middot middot 478 029 140 middot middot middot 0 089 089 1 068 1 middot middot middot

NGC 6752 118 -154 21times105 001plusmn001 49plusmn04 534 020 222 middot middot middot 0 01 097 097 005 06 middot middot middot

NGC 6809 123 -194 18times105 middot middot middot 4plusmn03 252 283 445 middot middot middot 2 043 007 004 025 099 098NGC 6838 120 -078 30times104 022plusmn002 23plusmn02 313 073 194 middot middot middot 2 033 013 04 02 1 1

Note References for the values in this table other than the probabilities are as follows For the open clusters NGC 188 We take the age from Meibom et al (2009) and the adpoted [FeH] from Sarajedini et al (1999)

Mcl rc rhm from Chumak et al (2010) fb from Geller et al (2013) σ0 from Geller et al (2008) and Pcirc from Meibom amp Mathieu (2005) NGC 2158 We take the age [FeH] Mcl from Carraro et al (2002) and rc

from Kharchenko et al (2013) NGC 2682 We take the age [FeH] Mcl fb σ0 rc from Geller et al (2015 and references therein) and Pcirc from Meibom amp Mathieu (2005) NGC 6791 We take the age [FeH] from

Carney et al (2005) Mcl σ0 from Tofflemire et al (2014) and rc from Platais et al (2011) NGC 6819 We adopt the age [FeH] from Hole et al (2009 and references therein) take Mcl rc from Kalirai et al (2001)

and fb (scaled here to full period distribution using method from Geller et al 2015) Pcirc from Milliman et al (2014) NGC 7142 We take the age [FeH] from Sandquist et al (2013 and references therein) estimate Mcl

from Straizys et al (2014) and take rc from Kharchenko et al (2013) For the globular clusters we take the age from Marın-Franch et al (2009 using the ldquoG00CGrdquo values and normalized using the age of 47 Tuc from

Thompson et al 2010) [FeH] σ0 (where available) Mcl and log ρ0 (both calculated assuming a mass-to-light ratio of 2) rc rhm from Harris (1996 2010) and fb (where available) from Milone et al (2012) For NGC 6366

we calculate the mass from σ0 assuming a Plummer model Finally for NGC 6838 we take the age from Di Cecco et al (2015)

2 Geller et al

of stars We urge readers to adopt the convention thatwe set forth in this series of papers to identify SSG andRS stars There are far fewer RS stars than SSGs butdespite their different location on the CMD their empir-ical characteristics appear to be very similar to the SSGstars Some of the SSG formation channels discussedin Paper II predict an evolutionary relationship betweenstars in the SSG and RS regions (where one is the pre-cursor to the other) and furthermore at least two of theformation channels (ldquoSG MTrdquo and ldquoMS Collrdquo) can leadto the formation of a blue straggler star (BSS)In this paper we investigate the formation rates of

SSGs through these four mechanisms through analyticcalculations Our goal is to identify if indeed all of thesemechanisms are viable or if one or more clearly domi-nates the production of SSGs After providing a qual-itative description of the four formation mechanisms inSection 2 we then discuss the probabilities of observingSSGs from each of these theoretical formation channelsin Sections 3 and 4 We investigate SSGs created in N -body and Monte Carlo star cluster models in Section 5Finally in Section 6 we provide a brief discussion andconclusions

2 SUMMARY OF THEORETICAL FORMATIONCHANNELS

In Paper II we study SSG formation channels in de-tail primarily through in-depth analyses of MESA mod-els (Paxton et al 2015) In this section we provide abrief summary of these formation channels specificallyfor SSGs formed through ongoing binary mass transfer(Section 21) increased magnetic activity leading to in-hibited convection (Section 22) rapid loss of an envelope(Section 23) and MS ndash MS collisions (Section 24) Werefer the reader to Paper II for further details about thephysics behind the first three mechanism (and a wider ex-ploration of parameter space) MS stellar collisions havebeen modeled previously in detail (eg Sills et al 19972001 2002 2005) in the context of BSS formation

21 Ongoing Binary Mass Transfer Involving aSubgiant Star (ldquoSG MTrdquo)

If a binary containing a MS star has a short enoughorbital period (sim15 days for a circular binary with aprimary star at the turnoff in M67) this MS star canoverfill its Roche lobe shortly after evolving off of theMS As mass transfer begins the now subgiant star losesmass becomes fainter and moves into the SSG regionThe MESA model in Figure 1 shows a 10 day binary

with a 13 M⊙ primary and a 07 M⊙ secondary Masstransfer begins when the primary overflows its Rochelobe on the subgiant branch Stable mass transfer pro-ceeds with an efficiency of 50 In this model the binaryremains in the SSG region for sim400 Myr which is com-parable to the duration of the subgiant phase of a normalsim13 M⊙ star (of 600 Myr in MESA)In this scenario the subgiant must be the brighter star

in the binary in the optical for it to appear in the SSGregion The accretor could be a MS star or a compactobject However if a MS accretor is massive enoughinitially it may gain enough mass to become a BSSand dominate the combined light before the subgiantbecomes sub-luminous enough to enter the SSG region(unless the mass transfer is extremely nonconservative)

04 06 08 10 12 14 16MB - MV

6

5

4

3

2

1

MV

SG

Str

ip SG

Mag

SG MTMS Coll

5 10

5

10

15

N

Fig 1mdash Color-magnitude diagram showing the theoretical SSGformation channels (colored lines) and our definition of the SSGregion (dark gray shaded area) and RS region (light gray shadedarea) with respect to a PARSEC isochrone (Bressan et al 2012)for M67 (solid line) In the main panel the dashed line showsthe equal-mass binary sequence for M67 We show evolutionarytracks from MESA (Paxton et al 2015) for a subgiant undergo-ing stable mass transfer (ldquoSG MTrdquo purple line) a subgiant thathas been stripped of much of its envelope (ie after ejecting acommon-envelope or after a grazing collision ldquoSG Striprdquo dark-redline) and a star that has a reduced convective mixing length co-efficient (and increased magnetic activity ldquoSG Magrdquo green line)We also show the result of a collision between two 07 M⊙ MSstars (from Sills et al 2002 ldquoMS Collrdquo yellow line) For all coloredlines the arrows indicate the direction of time along the evolu-tionary sequence In the two sub-panels we show histograms ofthe observed distribution of SSG and RS stars from the open andglobular clusters studied in Paper I in MV (right) and MB minusMV

(top) The black-filled histograms show the contribution from theglobular cluster sources alone with the additional white-filled re-gion (up to the solid lines) coming from the open cluster sourcesWe refer the reader to Figure 1 in Paper 1 for CMDs of these starsin the individual clusters

Here one would expect to observe a short-period bi-nary likely with a rapidly rotating primary (subgiant)star Photometric variability could potentially arise fromellipsoidal variations on the subgiant spot activity oreclipses X-rays (and Hα emission) could be producedby chromospheric activity on the rapidly rotating sub-giant andor hot spots in the accretion stream onto acompact object (likely requiring a neutron star or blackhole accretor to reach X-ray temperatures)In many cases where the accretor is a MS star the

evolution will lead to the coalescence of the two starsand interestingly in this scenario the merger productmay be observed eventually as a BSS The mass-transfermodel shown in Figure 1 eventually creates a BSS witha WD companion Indeed binary mass transfer is one ofthe primary BSS formation mechanisms (McCrea 1964

















4 Geller et al





x=0

(tτ)x

x (1)










Phs =πGradic2

(

m1 〈ms〉〈m〉

)32











rLa

=049qminus23

06qminus23 + ln(

1 + qminus13) (3)



(4)






(5)






(6)




6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag









tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)







41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters




























4 Geller et al





x=0

(tτ)x

x (1)










Phs =πGradic2

(

m1 〈ms〉〈m〉

)32











rLa

=049qminus23

06qminus23 + ln(

1 + qminus13) (3)



(4)






(5)






(6)




6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag









tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)







41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












4 Geller et al





x=0

(tτ)x

x (1)










Phs =πGradic2

(

m1 〈ms〉〈m〉

)32











rLa

=049qminus23

06qminus23 + ln(

1 + qminus13) (3)



(4)






(5)






(6)




6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag









tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)







41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters

















rLa

=049qminus23

06qminus23 + ln(

1 + qminus13) (3)



(4)






(5)






(6)




6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag









tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)







41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












6 Geller et al

30 35 40 4510-3

10-2

10-1

100Ψ

50 55 60

SG MT

MS Coll

SG Coll

SG CE

SG Mag









tMS Coll =

summf

m=m0

(

674[Myr]

)

f(m)w(m)summf

m=m0w(m)

(7)







41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters
















41 General






SG MT

5MS Coll

8

SG Coll 18

SG CE2

SG Mag67




42 Cluster specific


8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












8 Geller et al

01

10

100

1000

NS

SG

01

10

100

1000

103 104 105 106

Mcl (M )

10minus6

10minus5

10minus4

10minus3

NS

SG

Mcl (

M minus

1 )

103 104 105 10610minus6

10minus5

10minus4

10minus3



















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters





















fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

10

20

30

40

50

60708088

NSG MT

fb

0

030

Nstars

0

106

Nb

0

3x105

NBSS0350

rc (pc)

0

3


2

6

log10(rhrc)

0

1

Γc (Myrminus1)0 07

0

25

50

75

100

125

150

175

200225242

NSG Mag



10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












10 Geller et al

3x105 6x105 8x105 1x106

Nstars

0

5

10

15

20

25N

SG

MT

0

20

40

60

80

NS

G M

ag

ρ = 0996ρ = 0994

0 1x105 2x105 3x105

Nb

0

10

20

30

40

50

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0963ρ = 0997

0 100 200 300NBSS

0

15

30

45

60

NS

G M

T

0

50

100

150

NS

G M

ag

ρ = 0632ρ = 0983

3x105 6x105 8x105 1x106

Nstars

0

5

10

NS

G M

T

Nst

ars

x 10

5

0

5

10

15

20

NS

G M

ag

Nst

ars

x 10

5ρ = 0874ρ = 0781

0 1x105 2x105 3x105

Nb

0

1

2

3

4

NS

G M

T

Nb

x 10

4

0

2

4

6

8

NS

G M

ag

Nb

x 10

4

ρ = 0155ρ = 0799

0 100 200 300NBSS

00

05

10

NS

G M

T

NB

SS

00

05

10

15

20

25

NS

G M

ag

NB

SS


0 1 2 3rc (pc)

0

10

20

30

40

NS

G M

T

0

40

80

120

NS

G M

ag

ρ = 0908ρ = 0853

25 30 35 40 45 50log10(ρ [M pcminus3])

0

5

10

15

20

25

NS

G M

T

0

20

40

60

80

NS

G M

ag


04 06 08 10 12log10(rh rc)

0

10

20

30

NS

G M

T

0

20

40

60

80

NS

G M

ag


000 002 004 006 008 010Γc (Myrminus1)

0

15

30

45

60

NS

G M

T

0

40

80

120

NS

G M

ag


00 01 02 03fb

0

10

20

30

40

NS

G M

T

0

30

60

90

120

150

NS

G M

ag

ρ = 0954ρ = 0981



















12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters

























12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












12 Geller et al
























REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters





















REFERENCES



14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












14 Geller et al
















































Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












Form

atio

nFreq

uencies

ofSub-su

bgiantStars

15



Open Clusters


Globular Clusters












Northwestern University, 2145 Sheridan Rd., Evanston, IL ... · Aaron M. Geller1 ,2 † ∗, Emily M. Leiner3, Sourav Chatterjee1, Nathan W. C.Leigh4, Robert D. Mathieu3, Alison Sills

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