A relation between the radial velocity dispersion of young clusters and their age: Evidence for hardening as the formation scenario of massive close binaries
M.C. Ramírez-Tannus, F. Backs, A. de Koter, H. Sana, H. Beuther, A. Bik, W. Brandner, L. Kaper, H. Linz, Th. Henning, J. Poorta
AAstronomy & Astrophysics manuscript no. MassiveBinaryFormation © ESO 2021January 6, 2021 L etter to the E ditor A relation between the radial velocity dispersion of young clustersand their age:
Evidence for hardening as the formation scenario of massive close binaries
M.C. Ramírez-Tannus , F. Backs , A. de Koter , , H. Sana , H. Beuther , A. Bik , W. Brandner , L. Kaper , H. Linz ,Th. Henning , and J. Poorta Max Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany;e-mail: [email protected] Astronomical Institute "Anton Pannekoek", University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Institute of Astronomy, KU Leuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium Department of Astronomy, Stockholm University, Oskar Klein Center, SE-106 91 Stockholm, SwedenJanuary 6, 2021
ABSTRACT
The majority of massive stars ( > (cid:12) ) in OB associations are found in close binary systems. Nonetheless, the formation mechanismof these close massive binaries is not understood yet. Using literature data, we measured the radial-velocity dispersion ( σ ) as aproxy for the close binary fraction in ten OB associations in the Galaxy and the Large Magellanic Cloud, spanning an age range from1 to 6 Myrs. We find a positive trend of this dispersion with the cluster’s age, which is consistent with binary hardening. Assuminga universal binary fraction of f bin = σ behavior to an evolution of the minimum orbital period P cuto ff from ∼ ∼ ∼ Key words.
Stars: binaries (close) – Stars: formation – Stars: early-type – (Galaxy:) Open clusters and associations
1. Introduction
It is well established that the vast majority of massive stars( M > (cid:12) ) come in pairs or as higher-order multiples (e.g., Ma-son et al. 2009; Chini et al. 2012; Peter et al. 2012; Kiminki &Kobulnicky 2012; Kobulnicky et al. 2014; Sana et al. 2014; Dun-stall et al. 2015). A large fraction of these binaries have orbitalperiods on the order of 2 months or shorter (Sana & Evans 2011;Sana et al. 2012; Kiminki & Kobulnicky 2012; Almeida et al.2017; Barbá et al. 2017). These binaries are e ffi ciently detectedwith spectroscopic techniques measuring periodic Doppler shiftsof the photospheric lines. Massive binaries produce a variety ofexotic products later in their evolution such as X-ray binaries,rare types of supernovae (Ibc, IIn, super-luminous SNe, Whe-lan & Iben 1973; Yoon et al. 2010; Langer 2012), gamma-raybursts (Woosley et al. 1993; Cantiello et al. 2007), and, eventu-ally, gravitational wave sources (e.g., Ivanova et al. 2013; Man-del & de Mink 2016; de Mink & Mandel 2016; Eldridge &Stanway 2016). However, the origin of massive close binariesremains unknown.The first e ff ort to characterize the binarity properties of asample of O stars in compact H ii regions was performed by Apaiet al. (2007). They did a multi-epoch (two to three epochs) radialvelocity (RV) study of a sample of 16 embedded O stars in sevenmassive star-forming regions. They identified two close binarystars based on their RV variations ( ∼
90 km s − ) and measured an RV dispersion ( σ ) of 35 km s − for the whole sample and25 km s − when excluding the two close binaries.After pioneering studies to spectroscopically characterizesingle massive young stellar objects (mYSOs) such as those car-ried out by Bik et al. (2006, 2012), Ochsendorf et al. (2011),and Ellerbroek et al. (2013), Ramírez-Tannus et al. (2017) per-formed a single-epoch VLT / X-shooter spectroscopic study ofa sample of eleven candidate mYSOs in the very young gi-ant H ii region M17 ( (cid:46) −
25 M (cid:12) which is the mass range that dominates the samplesfrom which multiplicity characteristics of 2-4 Myr old main-sequence OB stars are derived (Sana et al. 2012; Kobulnickyet al. 2014). The measured radial-velocity dispersion of thesemYSOs is σ = . ± . − . In low density clusters, suchas M17, σ of a single epoch is strongly dominated by the or-bital properties of the binary population. For example, if a givencluster has several close binaries of similar masses, one wouldexpect the individual radial velocities of the stars to di ff er sig-nificantly from each other and, therefore, for σ to be large.For 2-4 Myr clusters, with binary fractions > . ∼ . − is typi-cal (e.g., Kouwenhoven et al. 2007; Sana et al. 2008, 2012; Sotaet al. 2014; Kobulnicky et al. 2014). The latter is in stark contrastwith our observation of M17, suggesting a lack of close massivebinaries in this region. Article number, page 1 of 7 a r X i v : . [ a s t r o - ph . S R ] J a n & A proofs: manuscript no. MassiveBinaryFormation
In Sana et al. (2017), two scenarios are explored that may ex-plain the low σ observed in M17: a small binary fraction f bin and / or a lack of short-period binaries. They conclude that theobserved dispersion can be explained either if f bin = . + . − . orif the minimum orbital period P cuto ff > f bin > .
42 or P cuto ff <
47 days can be rejected atthe 95% significance level. Since it is unlikely that the binaryfraction for M17 would be so far below that of other clusters,this very interesting result suggests that massive binaries formin wide orbits that migrate inward over the course of a few mil-lion years. In this letter, we refer to the generic mechanism ofshrinking binary periods as the migration scenario. One strongtest for this scenario is to compare the velocity dispersion ob-served in clusters spanning a range of ages. If the binary orbitsharden with time, one would expect σ to increase as the clusterage increases.In Ramírez-Tannus et al. (2020), VLT / KMOS spectra ofaround 200 stars in three very young clusters (M8, NGC 6357,and G333.6-0.2) were obtained. Introducing an automaticmethod to classify the spectra, the e ff ective temperatures, andluminosities of the observed stars were characterized in orderto place them in the Hertzsprung-Russell diagram (HRD). Theage and mass range of the observed populations was constrainedby comparison to MESA evolutionary tracks obtained from theMIST project (Paxton et al. 2011, 2013, 2015; Dotter 2016; Choiet al. 2016). The main sequence stars in M8 have masses be-tween ∼ ∼
70 M (cid:12) and the age of this cluster is between 1and 3 Myr. In G333.6-0.2, the main sequence population rangesin mass between ∼ ∼
35 M (cid:12) and the estimated age of thisregion is < ∼
10 and ∼
100 M (cid:12) and their ages range from0 . − σ , and P cuto ff , to constrain atimescale for binary hardening assuming a universal binary frac-tion ( f bin = .
7; Sana et al. 2012). We base our analysis on clus-ters younger than 6 Myr to ensure that neither secular evolutionnor the e ff ect of binary interactions (Wellstein & Langer 1999;de Mink et al. 2007) a ff ect our results significantly. In Section 2we measure the radial velocities of the high-mass stars in M8and NGC 6357 and calculate their σ . Next, we compare ourfindings with those presented by Sana et al. (2012) for Galac-tic clusters of 2-4 Myr, with those from Zeidler et al. (2018)for Westerlund 2 (Wd2), with those from Hénault-Brunet et al.(2012) for R136 in the Large Magellanic Cloud, and with thosefrom Ramírez-Tannus et al. (2017) for the very young massive-star forming region M17. This reveals a temporal behavior of σ (Section 3) that is converted into an evolution of the mini-mum binary period (Section 4), as binary motion is dominatingthe velocity dispersion of young massive clusters. In Section 5we discuss and conclude this work.
2. Observations
The sample studied in this paper consists of the OB stars in M8and NGC 6357. The data acquisition and reduction are describedin detail in Ramírez-Tannus et al. (2020). In short, we obtainedaround 200 H and K -band intermediate resolution spectra (withspectral resolution power, λ/ ∆ λ , between 6700 and 8500, i.e.,30 < ∆ v <
40 km s − ) of stars in the abovementioned giant H ii regions with VLT / KMOS (Sharples et al. 2013). The final sam-ples of massive stars consist of 16 stars in M8, 22 in NGC 6357,and four in G333.6-0.2. We discarded G333.6-0.2 from our anal- ysis because there are not enough stars with RV measurementsto calculate σ . A description of the age and mass range de-termination can be found in Ramírez-Tannus et al. (2020), andAppendix A presents a detailed discussion about the accuracy ofthe age determination.The radial velocity (RV) of the intermediate to high-massstars was obtained by measuring the Doppler shifts of a suitableset of photospheric lines. Tables B.1 and B.2 list the RV obtainedfor each star together with its error and the spectral lines used inour analysis.The RV-fitting approach is similar to the one adopted by Sanaet al. (2017). First, for the profile fitting, we adopted Gaussianprofiles. Second, we clipped the core of diagnostic lines thatwere still contaminated by residuals of the nebular emission.Third, we simultaneously fit all spectral lines available, therebyassuming that the Doppler shift is the same for all lines (see Sec-tion 2 and Appendix B of Sana et al. 2013).Figure C.1 shows the radial-velocity distribution for the tworegions. We calculated the errors of the histogram bins by ran-domly drawing RV values from a Gaussian centered at each mea-sured RV and with a sigma corresponding to the measurementerror; we repeated that process 10 times. The value shown foreach bin is the mean of all the RVs in that bin’s measurementsand the error bar corresponds to the standard deviation. We ob-tained σ by calculating the weighted standard deviation ofthe measured RVs. The weighted mean and standard deviationare listed in the top-left corner of each histogram in Fig. C.1.The measured σ for M8 and NGC 6357 are 32 . ± . . ± . − , respectively.
3. Velocity dispersion versus cluster age
Based on single-epoch radial-velocity measurements of youngmassive stars in M17, Sana et al. (2017) conclude that this young( ∼ σ , increases with time.We plotted σ of the clusters studied in this paper andcompare it with the results by Sana et al. (2012, NGC 6231,IC 2944, IC 1805, IC 1848, NGC 6611), Zeidler et al. (2018,Wd2), Hénault-Brunet et al. (2012, R136), and Sana et al. (2017,M17). R136 and Wd2 are very relevant for our study given theirrelatively young age (1-2 Myr) which is in between the agemeasured for M17 (Ramírez-Tannus et al. 2017) and that of thesomewhat older clusters (Sana et al. 2012). In order to comparethe multi-epoch RV data provided by Sana et al. (2012) with thesingle epoch data of M17, M8, NGC 6357, and Wd2, we drewthe RV measured for each star in a given cluster in a randomepoch and we computed the RV dispersion. We repeated this pro-cedure 10 times and then calculated the most probable σ andits standard deviation. The σ obtained for each cluster is listedin the third column of Table 1. The second and seventh columnsshow the age of the clusters and the respective references.In Figure 1 we show σ versus age of the clusters. Weperformed an orthogonal distance regression (ODR; Churchwell1990) to the data and find a positive correlation between the ageof the clusters and σ . The solid black line represents the bestfit to the data and the gray area represents the 1- σ error on thefit. The Pearson coe ffi cient for the observed relation is 0.7, which Article number, page 2 of 7.C. Ramírez-Tannus et al.: A relation between the radial velocity dispersion of young clusters and their age:
Table 1.
Age, radial-velocity dispersion, number of stars, and massrange for our sample of young clusters hosting massive stars.
Cluster Age σ D N Mass P min AgeMyr km s − stars M (cid:12) days ref.IC1805 1.6 – 3.5 65 . ± . . + . . ± . . + . . ± . . + . − . . ± . . + . . ± . . + . − . . ± . . + . − . . ± . − . ± . . + . − . . ± . . + . − . . ± . . + . − . References. (1) Sung et al. (2017); (2) Lim et al. (2014); (3) Baumeet al. (2014); (4) van der Meij et al. (2020), in prep. ; (5) Gvaramadze& Bomans (2008); (6) Zeidler et al. (2018); (7) Ramírez-Tannus et al.(2017); (8) Ramírez-Tannus et al. (2020); (9) Hénault-Brunet et al.(2012) t (Myr) D ( k m s ) IC1805 IC1848IC2944 NGC6231NGC6611Wd2M17M8NGC6357R136 D = (13.3 ± ± Fig. 1.
Radial-velocity dispersion ( σ ) versus age of the clusters. Thepurple data points show the data from Sana et al. (2012), the magentapoints show Wd2 and R136 (Zeidler et al. 2018; Hénault-Brunet et al.2012), and the green data points show the clusters studied in Ramírez-Tannus et al. (2020) and Sana et al. (2017). The solid black line repre-sents the linear fit to the data and the gray area shows the 1- σ errors onthe fit. indicates a strong positive correlation. Nevertheless, this coe ffi -cient does not take into account the errors in the parameters. Totest the validity of our results we performed two Monte Carlotests whose results are shown in Figure 2. The left panel showsthe distribution of Pearson coe ffi cients obtained from drawingrandom points centered on our data (age, σ ) with a standarddeviation equal to our error bars. The right panel shows the prob- Pearson r F r e q u e n c y Pearson r Fig. 2.
Distribution of Pearson coe ffi cients after randomly drawing 10 samples from the data shown in Figure 1. Left:
2D Gaussians centeredat our data points and with σ equal to our error bars. The green, red,and blue areas show the 68, 95, and 99% confidence intervals, respec-tively. The black line shows the observed coe ffi cient. Right:
2D Gaus-sians centered at random locations of the parameter space (age between0 and 7 Myr and σ between 0 and 80 km s − ) and with σ equal to ourerror bars. The blue shaded area represents the probability (2%) that theobserved coe ffi cient is caused by a random distribution. ability (2%) that a random distribution causes the observed co-e ffi cient.Even though the scatter is substantial, we conclude that forthe present data there is a positive correlation between σ andthe age of the clusters. Although the number of clusters consid-ered here remains limited and there are sizable uncertainties asto the individual age determinations, our results indicate an in-crease in the fraction of close binary systems as the clusters getolder. M17 seems to be a unique cluster given its very young ageand extremely low σ . Trumpler 14 in this sense seems compa-rable, being ∼ σ .
4. Physical parameters
Our aim is to characterize the multiplicity properties of observedclusters based on the observed σ . This section presents the re-sults of Monte Carlo population syntheses computed with di ff er-ent underlying multiplicity properties. We focus on the e ff ect that the binary fraction, f bin , and minimumorbital period, P cuto ff , have on the observed σ . The methodol-ogy is similar to that used in Sana et al. (2017). A parent pop-ulation of stars is generated with a certain f bin and P cuto ff . Thebinary star systems in this population are described by their or-bital properties. These are the primary mass, M , mass ratio, q ,period, P , and eccentricity, e . For the multiplicity properties thatwe do not vary, we adopt the values from Sana et al. (2012) forGalactic young clusters as the basic properties for our popula-tion. For the primary star, we adopt a Kroupa mass distribu-tion (Kroupa 2001). The mass ratio distribution is uniform with0 . < q <
1. The probability density function of the period isdescribed as pdf( P ) ∝ (log P ) − . , with the period in days andlog P cuto ff < log P < .
5. The eccentricity distribution dependson the period of the binary system. For P < Article number, page 3 of 7 & A proofs: manuscript no. MassiveBinaryFormation ities are sampled from pdf( e ) ∝ e − . , with 0 ≤ e < .
5; for pe-riods longer than 6 days the same distribution is used, but with0 ≤ e < . i , lon-gitude of periastron, ω , and eccentric anomaly, E . The latter isdetermined by generating a random mean anomaly and numeri-cally solving Kepler’s equation to find the corresponding eccen-tric anomaly using Brent’s method (Brent 1973) . This gives allof the information required to calculate the binary component ofthe radial velocity of the primary star through v r , b = K e cos( ω ) + cos (cid:114) + e − e tan (cid:32) E (cid:33) + ω , (1)with K being the amplitude of the orbital velocity of the pri-mary star. The binary component of the radial velocity is addedto the velocity due to cluster dynamics, σ dyn . This results in apopulation of stars with radial velocities based on either clusterdynamics only (in the case of f bin =
0) or both cluster dynam-ics and binary orbits. The secondary stars are assumed to beundetected. We adopt a cluster velocity dispersion for all clus-ters of 2 km s − , which corresponds to the typical value foundby Kuhn et al. (2019) who measured σ dyn for several massiveclusters in the Milky Way. Adopting σ dyn = σ dyn = − has no significant e ff ect on our results. Sana et al. (2012) alsoconclude that the dynamical dispersion of young clusters is neg-ligible with respect to the dispersion due to binary motions. EachMonte Carlo run consists of a generated parent population of 10 stars, which is sampled 10 times to simulate the observation ofa cluster. The mass distribution of the stars in the parent distribu-tion are matched to the mass distribution in the observed cluster.The samples contain the same number of stars and measurementaccuracy as the observed sample. We then constructed densitydistributions of σ for each parent population. Each grid pointin Figure 3 shows the median σ obtained for a simulated clus-ter of 20 stars sampled from a parent population with massesbetween 15 and 60 M (cid:12) , where we varied f bin , and P cuto ff . Thecolor bar corresponds to σ in km s − . The dashed lines showthe σ contours for 5.5, 15, 25.3, 30.9, 50.3, and 66 km s − ,which correspond to the σ measured for our clusters. As thenumber of stars and the mass ranges are di ff erent for each clus-ter, the contours do not represent the exact way in which wemeasured P cuto ff but are meant to show examples of the trendsthat a certain σ follows in this diagram. A comparison of ourresults with Sana et al. (2017) is shown in Appendix D. Assuming that the binary fraction of massive stars is consistentwith that observed in OB stars in young open clusters ( f bin = . P cuto ff which best represents the observed σ for each cluster. For each cluster, we kept the distribution of or-bital properties fixed and we adjusted the sample size and massranges in accordance to those of the observed samples (see Ta-ble 1). This allowed us to make a similar plot as in Figure 1 butin terms of P cuto ff . Figure 4 shows the estimated P cuto ff for eachcluster as a function of cluster age. We determined that, assum-ing f bin = .
7, the most likely P cuto ff to explain the observed σ https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.brentq.html Fig. 3.
Median σ obtained for distributions with di ff erent combina-tions of f bin and P cuto ff . For this set of simulations, we used a sampleof 20 stars ranging in masses from 15 to 60 M (cid:12) . The contours showthe trends followed by σ values of 5.5, 15, 25.3, 30.9, 50.3, and 66km s − . of M17 is around 3500 days and that a P cuto ff of 665 days and126 days can be rejected at the 68% and 95% confidence levels,respectively. For the clusters with the largest σ in our sample,IC 1805, IC 1848, and NGC 6231, the cuto ff period that best ex-plains the observed σ is 1.4 days, which is the P cuto ff adoptedby Sana et al. (2012).In order to get a first estimate of the binary hardeningtimescale, we fit a function of the form P ( t ) = P e − t / t + c to thedata in Figure 4, where P is the minimum period at the momentof binary formation and t corresponds to the e-folding time. As P is very uncertain, we assumed a typical value of 10 days,corresponding to an initial separation of ∼
100 AU for a pair of10 M (cid:12) stars. In order to find the range of t , we varied P from10 to 10 days . The resulting fit and corresponding parame-ter ranges are shown with the blue line and shaded area in Fig-ure 4. We obtain a typical e-folding time of t = . + . − . Myr.To harden an orbit from 3500 to 1.4 days, ∼ ∼ (cid:12) stars,this would imply a mean hardening rate of ∼ . − . ForM17, we discarded periods lower than 665 d with a 68% con-fidence level. Following the same argument as before, the typi-cal timescale to harden an orbit from 665 to 1.4 days would be ∼ ∼ − .
5. Discussion and conclusions
In this paper we present observational evidence that the 1D ve-locity dispersion of massive stars in young clusters ( σ ) in-creases as they get older (Figure 1). Additionally, we performedMonte Carlo simulations which allowed us to convert the mea-sured values of σ to physical parameters. Assuming that starsare born with the binary fraction representative of OB stars in2 − f bin = ff periodthat would correspond to each of the observed populations. FromFigure 4 we can conclude that the orbits would harden in 1- Article number, page 4 of 7.C. Ramírez-Tannus et al.: A relation between the radial velocity dispersion of young clusters and their age: t (Myr) P c u t o ff ( d a y s ) NGC6357 NGC6231 I C IC1805R136 NGC6611M17 IC2944M8 IC1805Wd2 P cutoff (t)= (5±1) exp ( t /(0.19 +0.060.04 )) + 1.40 Fig. 4. P cuto ff versus age of the clusters. The error bars represent the P cuto ff corresponding to the distributions that represent each σ withintheir 68% confidence range. The blue line and shaded region show thefit to the data and its 1- σ error bars. (cid:12) stars, a total angular momen-tum of ∼ . × kg m s − ( ∼ . × kg m s − ) needsto be transferred for the binary to harden from 3500 (665) to 1.4days. Rose et al. (2019) explored the possibility of the orbit hard-ening via the Eccentric Kozai-Lidov (EKL) mechanism, where athird companion perturbs the orbit of a binary system. They findthat, beginning with a cuto ff period of 9 months the EKL mech-anism is insu ffi cient to reproduce the population of short periodbinaries observed by Sana et al. (2012). They suggest that typeII migration (Lin & Papaloizou 1986) might explain the tighten-ing of binaries in such a time period. Moe & Kratter (2018) findthat the main mechanism to harden binaries should be the dy-namical disruption of coplanar triples that initially fragmentedin the disk in combination with energy dissipation within thisdisk. Other mechanisms include the combination of EKL oscil-lations with tidal friction both during the pre-main sequence andthe main sequence (Bate et al. 2002; Bate 2009). Given the densi-ties of the clusters studied, stellar encounters are uncommon and,therefore, mechanisms such as binary-binary or single-binary in-teractions should not play a significant role in the formation ofclose binaries (Fujii & Portegies Zwart 2011).Our findings a ff ect predictions of binary population synthe-sis models that follow the evolution of an ensemble of binarysystems subject to, among others, a distribution function for thezero-age orbital periods. Such models (e.g., Sana et al. 2012;Schneider et al. 2015) predict that a small fraction of systems in-teract within a timescale of 3 Myr, almost invariably resulting ina merging of the two components thus creating a blue straggler. Specifically, Schneider et al. (2015) report that of the popula-tion within two magnitudes in brightness of the main-sequenceturno ff , only 1%, 2%, and 10% is a blue straggler after 1, 2, or 3Myr. Therefore, unless one is specifically interested in early bluestraggler formation, the temporal evolution of orbital propertiesin the first few million years reported here should not a ff ect pre-dictions made by binary population synthesis models that relyon initial conditions defined at the zero-age main sequence only.Except for the caveat mentioned, adopting initial conditions forthe massive star population as reported on by Sana et al. (2012),for example, remains a justified approach. Acknowledgements.
Based on observations collected at the European Organi-sation for Astronomical Research in the Southern Hemisphere under ESO pro-gram 095.C-0048. This research made use of Astropy, a community-developedcore Python package for Astronomy (The Astropy Collaboration et al. 2018),NASA’s Astrophysics Data System Bibliographic Services (ADS), the SIMBADdatabase, operated at CDS, Strasbourg, France (Wenger et al. 2000), and thecross-match service provided by CDS, Strasbourg. The research leading to theseresults has received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation programme (grantagreement numbers 772225: MULTIPLES).
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Appendix A: Accuracy of the age determination
The e ff ective temperature and bolometric correction of the lumi-nosity class V stars are taken from Pecaut & Mamajek (2013);for the luminosity class III stars, we used the calibrations fromAlonso et al. (1999) for spectral types F0-F9, and for G4-M5stars we used the calibration of Cox (2000). In order to calcu-late the luminosity of the observed targets, we used the absolute K -band magnitude assuming the Indebetouw et al. (2005) ex-tinction law to derive the K -band extinction A K . We then ob-tained the absolute magnitudes by scaling the apparent mag-nitudes to the distance of the clusters (1336 + − pc for M8 and1770 + − pc for NGC 6357; Kuhn et al. 2019). The luminosi-ties were obtained by using the bolometric correction and theabsolute magnitudes. Finally, we determined the age of the re-gions by comparing the position of the observed stars to MESAisochrones obtained from the MIST project (Paxton et al. 2011,2013, 2015; Dotter 2016; Choi et al. 2016). The isochronesand tracks used have a solar metallicity and an initial rotationalvelocity of v ini = . v crit , the age obtained using tracks with v ini = ff ect the agedetermination and compare our age estimates with those pub-lished in the literature. i) Binarity could a ff ect the luminositydetermination by making stars look brighter than they actuallyare. Given the spacing of the isochrones at the young age of M8and NGC 6357, this does not have a significant e ff ect on our agedetermination. ii) The use of di ff erent evolutionary tracks couldlead to di ff erent age estimates; for example, Martins & Palacios(2013) show that the main-sequence age for di ff erent modelscould vary by ∼ . (cid:12) isochrone. In our case, thisuncertainty is likely smaller than that caused by the uncertaintyon the spectral type. iii) The age determination may depend onthe adopted extinction law. By using the Nishiyama et al. (2009) extinction law, we obtain lower A K values and, therefore, slightlyfainter objects. Nevertheless, given that the extinction in thesethree regions is relatively modest ( A K , max ∼ . ff erencein luminosity produced by the various extinction laws does notsignificantly a ff ect the age determination. iv) The classificationof the earliest spectral type (O3) is normally degenerate (see e.g.,Wu et al. 2014, 2016). In the case of NGC 6357, the most mas-sive star is of spectral type O3.5. The fact that its spectral typeis uncertain could a ff ect the lower age limit. To avoid this, wedetermined the age based on the whole population, including thelow-mass PMS stars, instead of focusing only on the massivestars.Arias et al. (2007) estimated the age of M8 to be < ∼ . − . − − ∼ . − Appendix B: Lines used to measure the radialvelocity
Table B.1.
Radial velocities and lines used to measure the radial veloc-ity for each star in M8.
Name RV (km s − ) Br-12 Br-11 HeI Br-10 HeI Br γ − ± − ± − ± ± − ± − ± ± ± − ± ± ± − ± − ± ± − ± ± Article number, page 6 of 7.C. Ramírez-Tannus et al.: A relation between the radial velocity dispersion of young clusters and their age:
Table B.2.
Radial velocities and lines used to measure the radial veloc-ity for each star in NGC6357.
Name RV (km s − ) Br-12 Br-11 HeI Br-10 HeI Br γ
107 9.5 ± ± ± ± ± ± − ± ± ± ± ± ± − ± − ± − ± ± ± − ± ± ± ± ± Appendix C: Distribution of radial velocities of theOB stars in M8 and NGC 6357 (a) M8
140 70 0 70 140Radial velocity (km/s)0246 N = -2.5 ± ± (b) NGC 6357
140 70 0 70 140Radial velocity (km/s)02468 N = 16.6 ± ± Fig. C.1.
Distribution of radial velocities for the massive stars in M8and NGC 6357.
Appendix D: Comparison to previous work
We compare the σ distribution obtained with the distributionpresented in Sana et al. (2017). Therefore, we sampled clus-ters of 12 stars with P cuto ff = . f bin = . σ = . + . − . km s − for the method described above. Sanaet al. (2017) found a lower value of σ = . + . − . km s − . Theconfidence intervals of the two values overlap due to the widthof the distribution.We assessed the correctness of our method by generatinga parent population using numerically calculated two body or-bits. The binary star systems were generated in a similar way asdescribed above. Each system is described by a primary mass,mass ratio, period, and eccentricity using identical distributionsas before. The orbits of the binaries were simulated in a two di-mensional plane. Only a single orbit starting at periastron wassimulated numerically using a fourth order Runge-Kutta method(Dormand & Prince 1980). The systems were tested on their sta-bility by integrating for a large number of orbits. The velocitywas obtained by selecting a random time step from the orbit. Thetwo dimensional x and y velocities were converted to a radial ve-locity using a randomly generated inclination and longitude ofperiastron. Finally, the e ff ect of cluster dynamics was added tothe radial velocity.The radial velocities calculated from the numerical two bodyorbits were combined with the radial velocities of single stars togenerate a new parent population of 10 stars. This parent popu-lation can be sampled as described above to generate a σ dis-tribution. Using the same cluster properties, we produced a σ distribution for M17 with f bin = . P cuto ff = . σ = . + . − . km s − , which agrees closely with thiswork. No significant di ff erence with this work was found for anybinary fraction or cuto ff period.period.