A self-consistent, absolute isochronal age scale for young moving groups in the solar neighbourhood
MMon. Not. R. Astron. Soc. , 1–24 (2014) Printed 26 August 2015 (MN L A TEX style file v2.2)
A self-consistent, absolute isochronal age scale for youngmoving groups in the solar neighbourhood
Cameron P. M. Bell (cid:63) , Eric E. Mamajek and Tim Naylor Department of Physics & Astronomy, University of Rochester, Rochester, NY 14627, USA School of Physics, University of Exeter, Exeter EX4 4QL, UK
Accepted ?, Received ?; in original form ?
ABSTRACT
We present a self-consistent, absolute isochronal age scale for young ( (cid:46)
200 Myr),nearby ( (cid:46)
100 pc) moving groups in the solar neighbourhood based on homogeneousfitting of semi-empirical pre-main-sequence model isochrones using the τ maximum-likelihood fitting statistic of Naylor & Jeffries in the M V , V − J colour-magnitudediagram. The final adopted ages for the groups are: 149 +51 − Myr for the AB Dor movinggroup, 24 ± β Pic moving group (BPMG), 45 +11 − Myr for the Carinaassociation, 42 +6 − Myr for the Columba association, 11 ± η Cha cluster,45 ± ± +4 − Myr for the 32 Ori group. At this stage we are uncomfortableassigning a final, unambiguous age to the Argus association as our membership listfor the association appears to suffer from a high level of contamination, and thereforeit remains unclear whether these stars represent a single population of coeval stars.Our isochronal ages for both the BPMG and Tuc-Hor are consistent with recentlithium depletion boundary (LDB) ages, which unlike isochronal ages, are relativelyinsensitive to the choice of low-mass evolutionary models. This consistency betweenthe isochronal and LDB ages instills confidence that our self-consistent, absolute agescale for young, nearby moving groups is robust, and hence we suggest that these agesbe adopted for future studies of these groups.Software implementing the methods described in this study is available from . Key words: stars: evolution – stars: formation – stars: pre-main-sequence – stars:fundamental parameters – techniques: photometric – solar neighbourhood – open clus-ters and associations: general – Hertzsprung-Russell and colour-magnitude diagrams
Over the past couple of decades, hundreds of young low- andintermediate-mass stars have been discovered in close prox-imity to the Sun. These stars are not uniformly dispersedacross the sky, but instead comprise sparse, (mostly) gravita-tionally unbound stellar associations within which the mem-bers share a common space motion. Approximately 10 suchmoving groups, with ages of between (cid:39)
10 and 200 Myr, havebeen identified within a distance of 100 pc from the Sun (seee.g. Zuckerman & Song 2004; Torres et al. 2008; Mamajek2015). Given their proximity to Earth, the members of thesegroups therefore play a crucial role in our understanding ofthe early evolution of low- and intermediate-mass stars. Fur-thermore, such stars provide ideal targets for direct imag- (cid:63)
E-mail: [email protected] (CPMB) ing and other measurements of dusty debris discs, substellarobjects and, of course, extrasolar planets (e.g. Janson et al.2013; Chauvin et al. 2015; MacGregor et al. 2015).Whilst the relative ages of these young associations arewell known e.g. the TW Hya association (TWA) is youngerthan the β Pic moving group (BPMG), which in turn isyounger than the AB Dor moving group, the absolute agesof these groups are still under-constrained. There are sev-eral methods for deriving the absolute age of a given younggroup of stars (see the review of Soderblom et al. 2014),however there are known problems with almost every one.Arguably the most common age-dating technique, using the-oretical model isochrones, still suffers from a high level ofmodel dependency which arises from the differences in thetreatment of various physical aspects, as well as the val-ues of adopted parameters; most importantly the treatmentof convection, the sources of opacity, the handling of the c (cid:13) a r X i v : . [ a s t r o - ph . S R ] A ug C. P. M. Bell et al. stellar interior/atmospheric boundary conditions and theinitial chemical composition (see e.g. Dahm 2005; Hillen-brand, Bauermeister, & White 2008). Even model indepen-dent methods, such as using kinematic information to in-fer the expansion rate of a given group or estimating thetime at which said group occupied the smallest volume inspace, suffer from issues of subjectivity with regard to thestars which are included/excluded (e.g. Song, Zuckerman, &Bessell 2003; de la Reza, Jilinski, & Ortega 2006; Soderblomet al. 2014). Furthermore, these kinematic methods have alsobeen shown to be unreproducible when similar analyses havebeen performed using improved astrometric data (e.g. Mur-phy, Lawson, & Bessell 2013; Mamajek & Bell 2014).Despite the apparent problems with deriving absoluteages for young associations, recent age estimates basedon the lithium depletion boundary (LDB) technique havestarted to instill confidence that we have both precise and accurate ages (to within just a few Myr) for at least two ofthese groups – the BPMG and Tucana-Horologium movinggroup (Tuc-Hor). The LDB technique works by identifyingthe lowest luminosity within a given (presumed coeval) pop-ulation of stars at which the resonant lithium (Li) feature at6708 ˚A shows that Li remains unburned. Although the LDBtechnique relies on the same theoretical models of stellar evo-lution from which (model-dependent) isochrones are created,the luminosity at which the transition from burned to un-burned Li occurs is remarkably insensitive to the inputs andassumptions noted above (see e.g. Burke, Pinsonneault, &Sills 2004; Tognelli, Prada Moroni, & Degl’Innocenti 2015).Given this high level of model insensitivity, Soderblomet al. (2014) propose that LDB ages therefore provide thebest means of establishing a reliable and robust age scale foryoung ( (cid:46)
200 Myr) stellar populations. Unfortunately, thereis a lower age limit to the applicability of the LDB tech-nique ( (cid:39)
20 Myr) which arises due to a much higher level ofmodel dependency between the various evolutionary mod-els in this age regime. Furthermore, for many of the younggroups studied here, the censuses of low-mass members is farfrom complete and therefore calculating a robust LDB ageis not possible. Therefore, if we are to establish an absoluteage scale for young moving groups in the solar neighbour-hood, we must use different age-dating techniques, howeverwe must also ensure that the resultant ages are consistentwith those calculated using the LDB technique.In a series of papers (Bell et al. 2012, 2013, 2014) we dis-cussed some of the main issues with using pre-main-sequence(pre-MS) model isochrones to derive ages for young clustersin colour-magnitude diagrams (CMDs), and in particular,the inability of these models to reproduce the observed lociin CMD space for clusters with well-constrained ages anddistances. In these papers we introduced a method of cre-ating semi-empirical pre-MS model isochrones using the ob-served colours of stars in the Pleiades to derive empiricalcorrections to the theoretical colour-effective temperature( T eff ) relations and bolometric corrections (BCs; hereafterreferred to together as BC- T eff relations) predicted by at-mospheric models. Of the clusters studied in the Bell et al.studies, Jeffries et al. (2013) recently identified the LDB inNGC 1960 and derived an age of 22 ± ± (cid:39) We focus on the following young ( (cid:46)
200 Myr), nearby ( (cid:46)
100 pc) moving groups: the AB Dor moving group, Argusassociation, BPMG, Carina association, Columba associa-tion, η Cha cluster, Tuc-Hor, TWA, and 32 Ori group. As-signing members to young moving groups is typically basedon a combination of kinematic diagnostics (e.g. proper mo-tions, radial velocities, etc.) in conjunction with youth indi-cators such as H α and X-ray emission, and Li absorption.Having established a list of members and high-probabilitycandidate members (hereafter simply referred to together as‘members’) for each group, we then compile broadband pho-tometric measurements and assign distances to these starsso that we can then place them in a CMD before fittingwith model isochrones. Note that in our adopted member-ships for each young group we exclude known brown dwarfs.The reason for this is that the low-mass cut-off of the semi-empirical model isochrones we use occurs at (cid:39) . (cid:12) (seeSection 4.1). Table 1 details our list of members for eachgroup, along with the compiled V J photometry, spectraltypes and distances. c (cid:13) , 1–24 olar neighbourhood age scale T a b l e . V J ph o t o m e t r y , s p ec t r a l t y p e s a ndd i s t a n ce s f o r m e m b e r s t a r s i n o u r s a m p l e o f y o un g m o v i n gg r o up s . T h e fin a l c o l u m n i nd i c a t e s w h e t h e r t h e g i v e n s t a r h a s a p o s i t i o n i n C M D s p a ce w h i c h i s c o mm e n s u r a t e w i t h g r o up m e m b e r s h i pb a s e d o n t h e b e s t - fi tt i n g m o d e l a s c a l c u l a t e du s i n g t h e τ fi tt i n g s t a t i s t i c ( s ee S ec t i o n . f o r d e t a il s ) . T h e f u ll t a b l e i s a v a il a b l e a s Supp o r t i n g I n f o r m a t i o n w i t h t h e o n li n e v e r s i o n o f t h e p a p e r a nd i n c l ud e s a ll m e m b e r s o f t h e A B D o r m o v i n gg r o up , A r g u s , B P M G , C a r i n a , C o l u m b a , η C h a , T W A , T u c - H o r a nd O r i. S t a r G r o upSp . T . R e f . V σ V R e f . V − J a σ V − J D i s t . σ D i s t . R e f . M V σ M V τ m e m b e r ( m ag )( m ag )( m ag )( m ag )( p c )( p c )( m ag )( m ag ) H R A BB I V n . . - . b . . . - . . Y H R A BB V s . . - . b . . . . . Y H R A BB V n . . - . b . . . . . Y H R A B A V p . . . c . . . . . Y H R ∗ A B A V a + n . . . c . . . . . Y H R A B A V . . . . . . . . Y H R A B A V . . . . . . . . Y H D A B F V . . . . . . . . Y H D ∗ A B F V . . . . . . . . Y H R A B F V . . . b . . . . . Y N o t e s : G r o up s : ( A B ) A B D o r m o v i n gg r o up ; ( A r ) A r g u s ; ( B P ) B P M G ; ( C a ) C a r i n a ; ( C o ) C o l u m b a ; ( E C ) η C h a ; ( T H ) T u c - H o r ; ( T W ) T W A ; ( ) O r i ∗ d e n o t e s d o ub l e o r m u l t i p l e s t a r w i t hun r e s o l v e d s p ec t r a l t y p e a J - b a ndph o t o m e t r y i s f r o m t h e M A SS P S C ( C u t r i e t a l. ; S k r u t s k i ee t a l. ) , un l e sss t a t e d o t h e r w i s e . b V − J c o l o u r e s t i m a t e d f r o m t h e B − V c o l o u r du e t o p oo r M A SS P S C ph o t o m e t r y ( s ee S ec t i o n . ) . c V − J c o l o u r e s t i m a t e d f r o m t h e V − K s c o l o u r du e t o p oo r J - b a nd M A SS P S C ph o t o m e t r y ( s ee S ec t i o n . ) . d D i s t a n ce b a s e d o n w e i g h t e d a v e r ag e o f s t a r s w i t h i n t h e g r o up w i t h t r i go n o m e t r i c d i s t a n ce s ( s ee S ec t i o n . ) . R e f e r e n c e s f o r s p e c t r a l t y p e s , ph o t o m e t r y a ndd i s t a n c e s : ( ) G a rr i s o n & G r a y ( ) ; ( ) M e r m illi o d ( ) ; ( ) T r i go n o m e t r i c d i s t a n ce f r o m v a n L ee u w e n ( ) ; ( ) A b t & M o rr e ll ( ) ; ( ) G r a y & G a rr i s o n ( ) ; ( ) H o u k & C o w l e y ( ) ; ( ) T y c h o - V T ph o t o m e t r y c o n v e r t e d t o J o hn s o n V f o ll o w i n g M a m a j e k e t a l. ( ) ; ( ) C o w l e y e t a l. ( ) ; ( ) S c h li e d e r , L ´ e p i n e , & S i m o n ( ) ; ( ) M oo r e & P a dd o c k ( ) ; ( ) H a u c k & M e r m illi o d ( ) ; ( ) G r a y e t a l. ( ) ; ( ) T o rr e s e t a l. ( ) ; ( ) H o u k & S w i f t( ) ; ( ) G r a y e t a l. ( ) ; ( ) K h a r c h e n k o & R o e s e r ( ) ; ( ) A b t( ) ; ( ) M o n t e s e t a l. ( ) ; ( ) H o u k & S m i t h - M oo r e ( ) ; ( ) O p o l s k i ( ) ; ( ) Y o ss ( ) ; ( ) K i n e m a t i c d i s t a n ce d e r i v e du s i n g t h e ‘ c o n v e r g e n t p o i n t ’ m e t h o d f o ll o w i n g M a m a j e k ( , s ee S ec t i o n . ) ; ( ) Z a c h a r i a s e t a l. ( ) ; ( ) K i n e m a t i c d i s t a n ce f r o m M a l o e t a l. ( ) ; ( ) S t e ph e n s o n ( ) ; ( ) V y ss o t s ky ( ) ; ( ) E lli o tt e t a l. ( ) ; ( ) R i a z , G i z i s , & H a r v i n ( ) ; ( ) T r i go n o m e t r i c d i s t a n ce f r o m W a hh a j e t a l. ( ) ; ( ) Sp ec t r a l t y p e r o und e d t o t h e n e a r e s t h a l f s ub - c l a ss f r o m Sh k o l n i k e t a l. ( ) ; ( ) G i r a r d e t a l. ( ) ; ( ) K i n e m a t i c d i s t a n ce f r o m M a l o e t a l. ( ) ; ( ) D a e m g e n e t a l. ( ) ; ( ) H e n r y , K i r k p a t r i c k , & S i m o n s ( ) ; ( ) B o w l e r e t a l. ( ) ; ( ) T r i go n o m e t r i c d i s t a n ce f r o m Sh k o l n i k e t a l. ( ) ; ( ) Z a c h a r i a s e t a l. ( ) ; ( ) P h o t o m e t r i c m e a s u r e m e n t a nd t r i go n o m e t r i c d i s t a n ce f r o m R i e d e l e t a l. ( ) ; ( ) C h a u v i n e t a l. ( ) ; ( ) G r a y & G a rr i s o n ( ) ; ( ) L ´ e p i n e & G a i d o s ( ) ; ( ) T r i go n o m e t r i c d i s t a n ce f r o m R i e d e l e t a l. ( ) ; ( ) L o w r a n cee t a l. ( ) ; ( ) C o r b a ll y ( ) ; ( ) G r a y ( ) ; ( ) P ec a u t & M a m a j e k ( ) ; ( ) H o u k ( ) ; ( ) H o u k ( ) ; ( ) N e uh ¨a u s e r e t a l. ( ) ; ( ) N e f s e t a l. ( ) ; ( ) P o j m a´ n s k i ( ) ; ( ) Z u c k e r m a n & S o n g ( ) ; ( ) W e i s ( ) ; ( ) K i n e m a t i c d i s t a n ce f r o m B i n k s & J e ff r i e s ( ) ; ( ) S c h r ¨o d e r & S c h m i tt( ) ; ( ) D u c o u r a n t e t a l. ( ) ; ( ) G r a y & C o r b a ll y ( ) ; ( ) H a r l a n ( ) ; ( ) L a w s o n e t a l. ( ) ; ( ) L uh m a n & S t ee g h s ( ) ; ( ) L a w s o n e t a l. ( ) ; ( ) L y o e t a l. ( ) ; ( ) H il t n e r , G a rr i s o n , & S c h il d ( ) ; ( ) M a s o n e t a l. ( ) ; ( ) Sp ec t r a l t y p e b a s e d o n a c o m b i n a t i o n o f i nd i v i du a l M V m ag n i t ud e s a nd B − V c o l o u r s ( s ee A pp e nd i x A . ) ; ( ) D e - c o n s t r u c t e d c o l o u r a nd m ag n i t ud ec a l c u l a t e du s i n g t h e t ec hn i q u e o f M e r m illi o d e t a l. ( , s ee A pp e nd i x A ) ; ( ) L e v a t o ( ) ; ( ) Sp ec t r a l t y p e ( r o und e d t o t h e n e a r e s t h a l f s ub - c l a ss ) a nd k i n e m a t i c d i s t a n ce ( a d o p t i n ga5 p e r ce n t un ce r t a i n t y ) f r o m K r a u s e t a l. ( ) ; ( ) H a w l e y , G i z i s , & R e i d ( ) ’ ( ) C o s t a e t a l. ( ) ; ( ) W e i nb e r g e r e t a l. ( ) ; ( ) W h i t e & H ill e nb r a nd ( ) ; ( ) W e bb e t a l. ( ) ; ( ) B a rr a d o y N a v a s c u ´ e s ( ) ; ( ) K a s t n e r , Z u c k e r m a n , & B e ss e ll ( ) ; ( ) L oo p e r e t a l. ( ) ; ( ) S c hn e i d e r e t a l. ( b ) ; ( ) A b t & L e v a t o ( ) ; ( ) Sh v o n s k i e t a l. ( i np r e p a r a t i o n ) . c (cid:13) , 1–24 C. P. M. Bell et al.
Our list of member stars was assembled from literature mem-bership lists and includes the following studies: Mamajek,Lawson, & Feigelson (1999), Luhman & Steeghs (2004), Lyoet al. (2004), Zuckerman & Song (2004), Mamajek (2005),L´opez-Santiago et al. (2006), Torres et al. (2006), Mamajek(2007), L´epine & Simon (2009), Schlieder, L´epine, & Simon(2010), Kiss et al. (2011), Rodriguez et al. (2011), Zuck-erman et al. (2011), Schlieder, L´epine, & Simon (2012a),Schlieder, L´epine, & Simon (2012b), Shkolnik et al. (2012),Binks & Jeffries (2014), Ducourant et al. (2014), Kraus et al.(2014), Riedel et al. (2014), and Shvonski et al. (in prepa-ration). In addition we include the ‘bona fide’ and high-probability ( ≥
90 per cent) candidate members as definedby Malo et al. (2013), Malo et al. (2014a) and Malo et al.(2014b).The series of papers by Malo et al. have identified sev-eral hundred high-probability candidate members for sev-eral of the groups included in this study, however many ofthese still require additional measurements (e.g. Li equiva-lent width, radial velocity, trigonometric parallax measure-ment) to unambiguously assign final membership to a givengroup. Given that our aim here is to derive the ‘best’ rep-resentative isochronal ages for young moving groups in thesolar neighbourhood, it is therefore critical that we minimisethe number of contaminating interlopers i.e. candidate mem-bers which still have a questionable status. Therefore, for theinclusion of Malo et al. high-probability candidate members,we require that such stars must have a measured radial ve-locity and/or trigonometric parallax which are/is consistentwith membership for a given group.Not accounting for unresolved multiples, our sample in-cludes a total of 89 members of the AB Dor moving group, 27members of Argus, 97 members of the BPMG, 12 membersof Carina, 50 members of Columba, 18 members of η Cha,189 members of Tuc-Hor, 30 members of TWA, and 14 mem-bers of 32 Ori.
The stars in our list are spread over a large area on the sky,and although there have been dedicated searches for low-mass members in some of the groups studied here (see e.g.Torres et al. 2006), there is a significant dearth of homoge-neous optical photometric coverage of these moving groups(especially in the M dwarf regime). This is not the casein the near-infrared (near-IR), however, where all memberstars have counterparts in the Two-Micron All-Sky Survey(2MASS; Skrutskie et al. 2006) Point Source Catalog (PSC;Cutri et al. 2003).Although ages can be derived using near-IR CMDs, theloci become vertical with colours J − K s (cid:39) . (cid:46) T eff (cid:46) (cid:46)
10 Myr), observations are further complicated bythe presence of circumstellar material. Therefore, to deriveages for our sample of groups we supplement the near-IRphotometry with V -band data. Note that due to increasedcontamination in the K s -band photometry as a result of cir- cumstellar material (especially for η Cha and TWA), wederive ages using the M V , V − J CMD.Whilst many of the stars in our list of members have
Hipparcos entries, this catalogue, unfortunately, does notquote explicit uncertainties on the V -band magnitude. Wehave therefore assembled V -band photometry from thefollowing sources: Vyssotsky (1956), Hauck & Mermilliod(1998), Tycho-2 (Høg et al. 2000) V T transformed into John-son V using the relation of Mamajek, Meyer, & Liebert(2006), Lawson et al. (2001), Lawson et al. (2002), theAll-Sky Automated Survey (ASAS; Pojma´nski 2002), Lyoet al. (2004), Zuckerman & Song (2004), the Naval Ob-servatory Merged Astrometric Dataset (NOMAD; Zachariaset al. 2005), Barrado y Navascu´es (2006), Costa et al. (2006),Mermilliod (2006), the Search for Associations ContainingYoung stars (SACY) sample (Torres et al. 2006), the All-Sky Compiled Catalogue of 2.5 million stars (ASCC-2.5;Kharchenko & Roeser 2009), Chauvin et al. (2010), theSouthern Proper Motion Catalog 4 (SPM4; Girard et al.2011), L´epine & Gaidos (2011), the USNO CCD AstrographCatalog 4 (UCAC4; Zacharias et al. 2013), and Riedel et al.(2014).Whilst the majority of these studies provide associateduncertainties on the V -band magnitude, stars for which wehave adopted NOMAD and SPM4 photometry, as well asphotometry taken from the studies of Zuckerman & Song(2004), Barrado y Navascu´es (2006), Chauvin et al. (2010),and L´epine & Gaidos (2011) do not. In such cases we adopta conservative V -band uncertainty of σ V = 0 . V = 12 mag we adopt an uncertainty of σ V = 0 .
01 mag,whereas for fainter sources we again adopt the rather con-servative σ V = 0 . V - and J -bands, forothers, the optical and/or near-IR photometry is unresolved.If the two components are resolved in both the V - and J -bands, then we plot each component separately in the CMD.However, given that we have heterogeneous information con-cerning the binary population of all the co-moving groups(e.g. the lack of information concerning the swathe of newTuc-Hor members from Kraus et al. 2014) coupled with thefact that the model isochrones we use to derive ages includethe effects of binary stars (see Section 4.2), then if eitherthe V - and/or J -band photometry is unresolved we insteadopt to plot the combined system measurement in the CMD.Note that the model isochrones do not account for higher or-der multiples (e.g. triples or quadruples), and so if we haveinformation concerning the number of components for suchsystems (e.g. TWA 4 is a quadruple system) we then attemptto derive individual component colours and magnitudes fol-lowing the technique of Mermilliod et al. (1992) based onthe system colour and magnitude as well as the magnitudedifference between the components in a given photometricbandpass. This process is explained fully in Appendix A andspecifies the systems for which this technique was adopted.In the case of higher order multiple systems for which wedo not have the required information to derive individual c (cid:13) , 1–24 olar neighbourhood age scale component colours and magnitudes (e.g. the magnitude dif-ference between the various components) we simply plot thecombined system measurement in the CMD.Secondly, to ensure we are using the best possible pho-tometry from the 2MASS PSC, we only use those objects forwhich the associated Qflg is ‘A’. For many of the brighter,early-type stars, however, the associated Qflg is ‘C’, ‘D’ or‘E’ i.e. either a magnitude was extracted but the associateduncertainty is prohibitively large or there were serious is-sues extracting a magnitude at all. For several cases, whilstthe K s -band photometry has an associated Qflg of ‘A’, the J -band photometry does not, and so we therefore use themain-sequence relation of Pecaut & Mamajek (2013) to in-fer a V − J colour for the star based on its V − K s colour. Fora smaller number of cases all three 2MASS PSC bands haveassociated Qflgs which are not ‘A’, and in such instances weinstead use the associated B − V colour (typically from Mer-milliod 2006) to infer the V − J colour again using the Pecaut& Mamajek (2013) relation. The stars which are affectedas such are expected to have settled on the main-sequence(based on their spectral type) and so we can be confidentthat such a technique will yield colours which can be usedto reliably place stars in the CMD. A full list of the starsfor which we have adopted this boot-strapping of colours,in addition to objects with questionable memberships andthose we have excluded from our age analysis, can be foundin Appendix B. We preferentially adopt trigonometric parallax measure-ments for assigning distances to each star, however in caseswhere these are not available or the associated uncertaintiesare prohibitively large, we adopt either a kinematic distancefrom the literature or derive one using the so-called ‘con-vergent point’ method (de Bruijne 1999; Mamajek 2005). Incertain cases, the trigonometric parallaxes are possibly er-roneous (e.g. TWA 9A and TWA 9B where the
Hipparcos parallax may be in error at the ∼ σ level; see Pecaut & Ma-majek 2013), and therefore we adopt a kinematic distanceinstead. Of the young groups in our sample we require kine-matic distances for several members of the AB Dor movinggroup, BPMG, Tuc-Hor and TWA, for which we adopt theconvergent point solutions of Barenfeld et al. (2013), Mama-jek & Bell (2014), Kraus et al. (2014) and Weinberger et al.(2013) respectively.For the two most distant groups in our sample – η Cha and 32 Ori – there are only a small number of starswith trigonometric parallax measurements (RECX 2 [ η Cha]and RECX 8 [RS Cha] in η Cha and 32 Ori, HR 1807 andHD 35714 in 32 Ori). As opposed to deriving individual kine-matic distances for the other members in these groups, weinstead adopt distances of 94 . ± .
18 pc and 91 . ± .
42 pcfor stars in η Cha and 32 Ori respectively based on theweighted average of stars within these clusters with trigono-metric parallaxes from the revised
Hipparcos reduction ofvan Leeuwen (2007).
In this study we adopt four sets of semi-empirical modelisochrones based on different interior models, namely theDotter et al. (2008), Tognelli et al. (2011), Bressan et al.(2012) and Baraffe et al. (2015) models (hereafter referred toas Dartmouth, Pisa, PARSEC and BHAC15 respectively) .Note that the Pisa models represent a customised set ofinterior models which span a much greater mass and agerange than those available via the Pisa webpages (see Bellet al. 2014 for details). In addition, we note that the semi-empirical PARSEC models are not the recent v1.2S modelsby Chen et al. (2014). Instead, they are based on the v1.1interior models (computed assuming a solar composition of Z = 0 . T eff relations, the aforementioned discrepancy between themodels and the data still exists, especially in optical band-passes (see e.g. Bell et al. 2014). Hence, in Bell et al. (2013,2014) we described a method of calculating additional em-pirical corrections to the theoretical BC- T eff relations pre-dicted by the BT-Settl atmospheric models (Allard et al.2011) so as to match the observed colours of low-mass starsin the Pleiades assuming an age of 130 Myr, distance mod-ulus of 5 .
63 mag and reddening of E ( B − V ) = 0 .
04 mag.These corrections were then combined with the theoreticaldependence of the BCs on surface gravity from the atmo-spheric models to derive semi-empirical BC- T eff relations.In Baraffe et al. (2015) the authors suggest that the newBHAC15 models now match the shape of the observed lo-cus, in the sense that the optical colours for M dwarfs are nolonger too blue. In Fig. 1 we show the V, V − J CMD of thePleiades and overlay 130 Myr model isochrones from differ-ent ‘generations’ of the Baraffe et al. models (assuming thesame distance modulus and reddening as noted in the previ-ous paragraph). From Fig. 1 it is clear that whilst progresshas been made since the original Baraffe et al. (1998) modelswere first published, there are still unresolved problems withthe latest BHAC15 in the optical and/or near-IR bandpassesat low T eff . Repeating the binary analysis of Bell et al. (2012)in which we compared the theoretical system magnitudespredicted by the theoretical models with the measured sys-tem magnitudes in the V I c JHK s bandpasses, we find that,for the BHAC15 models, the observed spread in the opticalbandpasses is smaller than for the other models, however it isstill appreciably larger than in the near-IR bandpasses. Thusdespite the fact that the discrepancy between observed lociand the models is smaller when using the BHAC15 models(compared with previous ‘generations’), it is still necessaryto calculate additional empirical corrections to the theoret-ical BC- T eff relations before using these to derive absoluteages for young stellar populations using CMDs. The semi-empirical Dartmouth and Pisa models we use hereare available via the Cluster Collaboration isochrone server c (cid:13) , 1–24 C. P. M. Bell et al.
Figure 1.
V, V − J CMD of the Pleiades. Overlaid are different‘generations’ of the Baraffe et al. model isochrones with ages of130 Myr, each of which has been reddened by the equivalent of E ( B − V ) = 0 .
04 mag and shifted vertically by a distance modulus dm = 5 .
63 mag. The original Baraffe et al. (1998) models (with asolar-calibrated mixing-length parameter α = 1 .
9) coupled withthe NextGen atmospheric models are shown as the green line. Thesame interior models, but coupled with the BT-Settl atmosphericmodels (computed using the Asplund et al. 2009 abundances)are represented by the blue line. The latest Baraffe et al. (2015)interior models coupled with the BT-Settl models (both computedusing the Caffau et al. 2011 abundances) are denoted by the redline. It is clear that whilst the discrepancy between the modelsand the data has decreased with time, there are still unresolvedproblems in the optical and/or near-IR bandpasses at low T eff . We use the maximum-likelihood τ fitting statistic of Naylor& Jeffries (2006) and Naylor (2009) to derive ages from the M V , V − J CMDs of our sample of young moving groups.The τ fitting statistic can be viewed as a generalisation ofthe χ statistic to two dimensions in which both the modelisochrone and the uncertainties for each datapoint are two-dimensional distributions. Not only does the use of the τ fitting statistic remove the issue of objectivity introducedthrough ‘by-eye’ fitting of model isochrones, but it also al-lows us to include the effects of binarity in our model dis-tribution, provides reliable uncertainties on the derived pa-rameters and allows us to test whether the model providesa good fit to the data. Similarly to χ , the best-fit model isfound by minimising τ . The two-dimensional model distributions are generated us-ing a Monte Carlo method to simulate 10 stars over a givenmass range. To populate our models we adopt the canonicalbroken power law mass function of Dabringhausen, Hilker, & Kroupa (2008). An important feature of our fitting methodis that we include the effects of binary stars, thereby modify-ing the model isochrone at a given age from a curve in CMDspace to a two-dimensional probability distribution. To in-clude binary stars in our fits we assume a binary fractionof 50 per cent and a uniform secondary distribution rangingfrom zero to the mass of the primary.Our sample of member stars ranges from early B-typestars to late M-type dwarfs, and given that across this rangethere is an obvious trend for the observed binary fraction todecrease as a function of decreasing spectral type (or mass;see e.g. Duchˆene & Kraus 2013), our choice of a uniform bi-nary fraction of 50 per cent for all spectral types is somewhatidealised. The effect of varying the adopted binary fractionin model distributions has previously been investigated byNaylor & Jeffries (2006) who demonstrated that even at un-realistic values (e.g. 80 per cent across all spectral types) thebest-fit age is affected at the <
10 per cent level. The modelgrids we calculate are dense and cover a large age range[log(age) = 6 . − . .
01 dex], andso given this insensitivity to the adopted binary fraction,we adopt a uniform fraction of 50 per cent for the ease ofcomputing them.We note that due to the processes involved in creat-ing the semi-empirical BC- T eff relations (see Section 3), thelower T eff limit of the BCs may be reached before the lowermass limit of the interior models. Effectively this means thatthe semi-empirical pre-MS model isochrones only extenddown to masses of (cid:39) . (cid:12) . As a result, simulated sec-ondary stars with masses corresponding to T eff values belowthis T eff threshold are assumed to make a negligible contri-bution to the system flux i.e. this is equivalent to placing thebinary on the single-star sequence. Essentially, this meansthat we begin to lose the binary population before the single-star population in our two-dimensional model distributionsand results in a ‘binary wedge’ of zero probability betweenthe binary and single-star sequences at low masses (see e.g.Fig. 2). Whilst we have made every effort to include only high-probability ( ≥
90 per cent) candidate members in our sam-ple of young moving groups, it is likely that a fraction ofthese stars are in fact non-members. We have further im-proved the τ fitting statistic since the modifications of Nay-lor (2009) and the reader is referred to Appendix C in whichwe introduce an updated method (cf. Bell et al. 2013) todeal with non-member contamination by assuming a uni-form non-member distribution. We choose such a concep-tual framework, in part because it is better suited to theproblem in hand, but also because it allows us to calculatea goodness-of-fit parameter, a step which was missing fromour earlier soft-clipping technique used in Bell et al. (2013).Prior to deriving ages from the M V , V − J CMDs for oursample of young groups, we must first address whether weneed to account for local levels of interstellar extinction andreddening in the solar neighbourhood. All of our memberstars lie within the so-called Local Bubble ( (cid:46)
100 pc). A re-cent study by Reis et al. (2011) performed an analysis of theinterstellar reddening in the Local Bubble using Str¨omgrenphotometry and provided estimates of how E ( b − y ) varies as c (cid:13) , 1–24 olar neighbourhood age scale Figure 2.
Best-fitting M V , V − J CMDs of the AB Dor moving group. The red circles represent fitted data, whereas the blue squaresdenote objects which are removed prior to fitting (see Section 4.2 for details).
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa. The dashed line in each panel represents the ZAMS relation for that specific set of model isochrones. a function of distance. To calculate how this local reddeningcould affect the colours and magnitudes of our member stars,we assume that A V = 4 . × E ( b − y ) and A J /A V = 0 . A V (cid:39) .
03 mag and E ( V − J ) (cid:39) .
02 mag respectively.Compared to the combined photometric and distance uncer-tainties, such levels of extinction and reddening are insignif-icant and therefore we can leave the semi-empirical modelsin the absolute magnitude-intrinsic colour plane to deriveages from the M V , V − J CMDs.To illustrate our fitting procedure we will use the ABDor moving group as an example, for which the best-fitting M V , V − J CMDs are shown in Fig. 2. Prior membershipprobabilities for the individual stars are taken from the anal-yses of Malo et al. (2013, 2014a,b), and for this dataset themean prior probability of a given star being a member of thecluster P ( M c ) is greater than 0.99 (see Appendix C1.2 for anexplanation of how the τ fitting statistic can be used to in-clude prior membership probabilities for individual sources).Therefore, with 89 stars in our catalogue, this would thenimply that, at most, one object does not originate from themodel sequence. Note that for member stars in our sampleof young moving groups which are not included in the anal-yses of Malo et al. we adopt a uniform prior probability of0.9. An examination of Fig. 2, however, suggests that sucha low non-member fraction is unlikely. There is a group offour objects which lie significantly above the equal-mass binary envelope of the model distribution between 1 . (cid:46) V − J (cid:46) . M V , also lie above the upper envelope of the best-fitmodel. It is worth noting that none of these objects are sin-gle stars, but are either resolved components of binary sys-tems or unresolved binaries, all of which have separations of < M V (cid:39) . V − J (cid:39) . (cid:39) . P ( M c ). This conclusion is justified by examiningthe distribution of the τ values for each individual data-point. The top left panel of Fig. 3 shows the cumulativedistribution for these values, along with the predicted dis-tribution calculated from the same 1000 simulated clusters(in this case based on the Dartmouth model distributions)used to calculate the goodness-of-fit parameter [Pr( τ ); seeAppendix C3.1]. There is an obvious tail of points with val-ues much higher than the prediction from the simulations.To overcome this problem we therefore set a maximumprior membership probability ( P max ) and multiply all theprior membership probabilities by this factor. We assessedthe correct value for P max by adjusting it until the total of c (cid:13) , 1–24 C. P. M. Bell et al.
Figure 3.
Comparing the real and expected τ distributions (us-ing the Dartmouth models) for the AB Dor moving group. Theblack histograms show the fraction of stars with individual τ values less than a given value in the fit to the data. The redcurves represent the expected distributions of the probability ofobtaining a given τ calculated for the simulated observation (us-ing the best-fit model). Each panel is for a different value of themaximum allowed prior membership probability P max (see text). the prior and posterior memberships roughly matched. Thisprocess is illustrated in Fig. 3. The drop at the end of thedistribution is the stars which are apparent non-members,and this is fairly closely matched at P max = 0 .
94 which isthe value where the total prior and posterior membershipsare best matched. For the AB Dor moving group we findthat, regardless of which set of semi-empirical pre-MS mod-els is adopted, the maximum prior membership probabilityis P max ≥ .
9, and that this results in goodness-of-fit valuesof Pr( τ ) ≥ . . − . . (cid:12) , which means that any ‘handle’ on the age providedby higher mass stars (specifically those evolving between thezero-age main-sequence [ZAMS] and the terminal-age main-sequence [TAMS]) is effectively ignored by these models,which is not the case for the other models which have uppermass cut-offs of ≥ (cid:12) . Although we have tried to excludeknown brown dwarfs from our membership lists, there arestill some stars which occupy regions of the CMD not cov-ered by the models (at both high- and low-masses; see e.g.the blue squares in the upper right panel of Fig. 9), whichmust be removed from the fit before calculating either of theaforementioned parameters.A further consequence of our fitting technique is thatour best-fit model returns posterior membership probabili-ties. For a given group in our sample we expect a negligibleage spread (or equivalently luminosity spread), and there-fore we can use the posteriors to identify stars which appearto be non-members (based solely on CMD position) in aneffort to further refine the membership lists of these youngmoving groups. Such a technique is far from unassailable. For example, we need only think of the obscuring effectsof edge-on discs or uncertainties in the distances to objects(see e.g. TWA 9 in Section 2.3) which will act to shift starsaway from the observed locus of the group. Identification ofsuch stars, however, represents a sensible ‘first-order sanitycheck’ which should be used in combination with other met-rics (e.g. Li equivalent widths, radial velocities, etc.) to es-tablish whether a given star is a high-probability candidatemember of a given moving group. For example, the anal-yses of Malo et al. have identified several high-probabilitycandidate members in the BPMG, however an examinationof Fig. 5 clearly shows that the inclusion of these stars re-sults in a luminosity spread of (cid:39) M V at a colour of V − J (cid:39) . (cid:38)
20 Myr, isincomprehensibly vast and the fact that the faintest stars inthis sample lie below the computed ZAMS lends confidenceto our assertion that our membership list still contains likelynon-members. This is also borne out by the calculated pos-terior probabilities for these stars which are all less than 0.1i.e. indicating that these are likely non-members.Figs. 4–11 show the best-fitting M V , V − J CMDs forthe remainder of our sample of young groups. Of the re-maining groups, we find that, except for Argus, the BPMGand η Cha, the maximum prior membership probability is P max (cid:38) . τ ) are (cid:38) . P max values required forthe BPMG and η Cha (of between 0.7 and 0.8) are a re-sult of contamination by non-members in the former (seeabove) and astrophysical phenomena affecting the CMD po-sitions of (cid:39)
10 per cent of the members in the latter (seeSection 5.5) in conjunction with our adoption of a well-constrained distance ( < P max values for Argus are all (cid:38) .
85, the corresponding goodness-of-fit values Pr( τ ) are (cid:46) . (cid:39) −
15 per cent level contaminationin our membership list for this association (see Section 5.3).Table 2 shows the best-fit age for each group accordingto the four sets of semi-empirical models, in addition to ourfinal adopted age. For the final age we adopt the average ofi) the median, ii) the Chauvenet clipped mean (Bevington& Robinson 1992), and iii) the probit mean (Lutz & Upgren1980) of the individual best-fit ages. The quoted uncertain-ties on the final ages represent the statistical and system-atic uncertainties added in quadrature. Given the asymmet-ric statistical uncertainties on several of the individual agesshown in Table 2, we calculate both an upper and lower sta-tistical uncertainty by taking the median of the four individ-ual upper and lower statistical uncertainties. Our estimateof the systematic uncertainty arising from the use of differ-ent model isochrones is based on the average of i) the 68 percent confidence levels and ii) the probit standard deviation(Lutz & Upgren 1980) of the individual best-fit ages.
In Section 4.2 we used semi-empirical models to derive aself-consistent, absolute isochronal age scale for nine young,nearby moving groups within 100 pc of the Sun. To assessthe reliability of these ages, we must first compare these to c (cid:13) , 1–24 olar neighbourhood age scale Figure 4.
Best-fitting M V , V − J CMDs of Argus. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.
Figure 5.
Best-fitting M V , V − J CMDs of the BPMG. The coloured symbols and dashed lines are the same as those in Fig. 2.
Topleft:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.c (cid:13) , 1–24 C. P. M. Bell et al.
Figure 6.
Best-fitting M V , V − J CMDs of Carina. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.
Figure 7.
Best-fitting M V , V − J CMDs of Columba. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa. c (cid:13)000
Pisa. c (cid:13)000 , 1–24 olar neighbourhood age scale Figure 8.
Best-fitting M V , V − J CMDs of η Cha. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.
Figure 9.
Best-fitting M V , V − J CMDs of Tuc-Hor. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.c (cid:13) , 1–24 C. P. M. Bell et al.
Figure 10.
Best-fitting M V , V − J CMDs of TWA. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa.
Figure 11.
Best-fitting M V , V − J CMDs of 32 Ori. The coloured symbols and dashed lines are the same as those in Fig. 2.
Top left:
BHAC15.
Top right:
Dartmouth.
Bottom left:
PARSEC.
Bottom right:
Pisa. c (cid:13)000
Pisa. c (cid:13)000 , 1–24 olar neighbourhood age scale Table 2.
Ages for the young groups studied in this paper. Individual ages are shown for each set of semi-empirical pre-MS modelisochroness and have been derived using the τ fitting statistic for which we have set P max so that the total prior and posteriormemberships are roughly equal (see Section 4.2 for details). The penultimate row lists our final adopted age for each group on whichthe quoted uncertainties represent the statistical and systematic uncertainties added in quadrature. The final row shows literature LDBages for the BPMG and Tuc-Hor (see Section 5.1 for references) which highlights the consistency between the two age diagnostics.Model Group age (Myr)AB Dor Argus a BPMG Carina Columba η Cha Tuc-Hor TWA 32 OriBHAC15 145 +889 − +19 − ± +16 − +8 − ± +1 − ± +3 − Dartmouth 135 +15 − +64 − ± +13 − +8 − ± +1 − +2 − +5 − PARSEC 151 +24 − +268 +1 − +7 − +3 − ± ± ± +2 − Pisa 166 +74 − +235 − +1 − +8 − ± ± +1 − ± ± Adopted 149 + − – 24 ± + − + − ± ± ± + − LDB age – – 24 ± ± Notes: a We do not provide a final adopted age for Argus as it remains unclear whether the stars in our list of members are representative of asingle population of coeval stars (see Section 5.3 for details). Note also that the best-fit age for Argus using the PARSEC models onlyprovides an upper limit on the age of the association and hence we do not include a lower age uncertainty in the table. what are considered to be well-constrained ages for the samegroups.
As discussed in the Introduction, LDB ages are arguably themost reliable age diagnostic – in terms of calculating abso-lute ages – currently available for stellar populations withages of between ∼
20 and 200 Myr. Note that although LDBages have been advocated as our best hope of establishing areliable age scale in this age range (primarily due to the factthat LDB ages have been shown to be relatively insensitiveto variations in the physical inputs adopted in the evolution-ary models and the fact that between different models thereis excellent agreement; see e.g. Soderblom et al. 2014), recentstudies have demonstrated that by not accounting for the ef-fects of starspots in the evolution of pre-MS stars, the LDBage scale may in fact only be good to (cid:39) −
20 per cent (seee.g. Jackson & Jeffries 2014; Somers & Pinsonneault 2015).Binks & Jeffries (2014) and Malo et al. (2014b) haveboth identified the LDB in the BPMG, deriving ages of 21 ± ± ± ± ± M V , B − V CMD analysis of the A-, F- and G-type membersof the group. Whilst such consistency lends confidence toour absolute age scale for the young moving groups in thesolar neighbourhood, there are (currently) no other LDBages for our sample with which to compare the isochronal ages against. Hence, we must place the isochronal ages forthe seven remaining groups in context by comparing themto other age-dating techniques adopted in the literature.
Literature ages for the AB Dor moving group vary fromthe relatively young ( (cid:39) −
70 Myr; see e.g. Zuckerman,Song, & Bessell 2004; Torres et al. 2008; da Silva et al. 2009)to essentially coeval with the Pleiades ( (cid:39)
130 Myr; see e.g.Luhman, Stauffer, & Mamajek 2005). A more recent analysisby Barenfeld et al. (2013) provided a strong constraint onthe age of the group by identifying the main-sequence turn-on for AB Dor nucleus stars in the M V , V − K s CMD. Theydemonstrated that the late K-type stars have already settledonto the ZAMS, and used this to place a firm lower limit onthe age of 110 Myr.Interestingly, the recent discovery of a Li-rich M8high-probability brown dwarf candidate member (2MASSJ00192626+4614078; Gagn´e et al. 2014) provides us witha strong upper limit on the age of the group of 196 Myr(Binks, private communication). Additional candidate mem-bers with spectral types in the range M4-M8 have been re-ported by Gagn´e et al. (2015) and thus spectroscopic follow-up of these objects could place stronger constraints on theabsolute age of the AB Dor moving group. Our isochronalage therefore lies almost exactly midway between the lowerand upper age limits provided by the main-sequence turn-on and the lower edge of the tentative identification of theLDB.From Table 2, it is clear that the main ‘handle’ on theage of the AB Dor moving group comes from the high-massstars which are evolving between the ZAMS and TAMS.The BHAC15 models demonstrate that if these stars areomitted from the fit then, although the best-fit age is ingeneral agreement with those from the other models, the as-sociated upper uncertainty on the age is prohibitively large.The reason why the low-mass population of the group can-not provide a precise age is due simply to the fact that at c (cid:13) , 1–24 C. P. M. Bell et al. ages of (cid:38)
100 Myr the model isochrones occupy essentiallythe same position in CMD space (except for very-low-massobjects with V − J (cid:38) . τ space across a widerange of ages. The kinematics of Argus are very similar to those of thenearby young cluster IC 2391 (Torres et al. 2003), for whichBarrado y Navascu´es, Stauffer, & Jayawardhana (2004) de-rived an LDB age of 50 ± ∼
40 Myr) was proposed by Torres et al. (2008) on the basisof Li equivalent widths and positions in the
V, V − I c CMD,and therefore our isochronal ages of ∼
60 Myr disagree withthe current literature age for this association. Furthermore,whilst the different sets of semi-empirical models tend toagree upon an age of ∼
60 Myr for Argus, the uncertain-ties on these individual ages are extremely large and implythat based on our current membership list, it is not valid toassign a unique, unambiguous age to Argus (see Table 2).We believe that the smaller uncertainties on the BHAC15best-fit age stem from the upper-mass cut-off of these mod-els which neglect any age information from the higher massmembers. Excluding the BHAC15 models for the moment,the other ages in Table 2 clearly demonstrate that althoughthere is a minimum τ value within the grid, the τ spaceis also relatively flat over a wide range of ages (see e.g. thePARSEC models which can only give an upper age limit of328 Myr).Our membership list for Argus stems primarily from theBayesian analyses of Malo et al. (2013, 2014a,b) and com-paring it with the membership list of Torres et al. (2008,their Table 12) there are two notable differences i) only threestars in common appear in both lists (BW Phe [HD 5578],HD 84075 and NY Aps [HD 133813]) and ii) the mediandistance of members in our list is 32 pc whereas for the Tor-res et al. list it is 96 pc. Furthermore, looking also at thedistribution of distances (from the nearest to the farthest)in these two membership lists, those stars in our list cover arange of (cid:39)
65 pc, whereas those in Torres et al. (2008) covera range of more than twice this at (cid:39)
150 pc (cf. (cid:39)
50 pcfor the supposedly coeval Tuc-Hor). Given this spread indistances for the Torres et al. members, one can reasonablyask whether we expect these objects to have formed togetherand therefore be coeval in the first place.Fig. 4 shows the best-fitting CMDs of Argus and thereare two points worth mentioning. First, there is a group of5 A-type stars (all of which were proposed as members byZuckerman et al. 2011 and none of which are unresolved bi-naries [according to the Washington Visual Double Star Cat-alog; Mason et al. 2001]), which we would expect to definethe ZAMS of the association (as is the case with the othergroups in our sample). None of these, however, actually lieon the ZAMS (4 are over-luminous and 1 is under-luminouswith respect to the ZAMS), which suggests that these starsmay not be coeval, but rather represent stars at differentevolutionary stages. Second, of the K- and M-type stars inthe association, a significant fraction of these appear to liebelow the best-fitting model in all panels of Fig. 4. Whilstthese stars all appear to be young and active, as evidenced by a combination of high R X = L X /L bol values and largeH α equivalent widths (see e.g. Riaz, Gizis, & Harvin 2006),they all lack Li measurements which may help us discrimi-nate between young active stars belonging to a given groupand slightly older active ZAMS stars which do not (at leastfor the mid to late M-type objects). Given these ambigui-ties, we are therefore reluctant (at present) to assign a final,unambiguous age to Argus as it appears as though our mem-bership list suffers from a high level of contamination, andhence it remains unclear whether this list represents a single,coeval population of stars, or even whether the associationis in fact physical. Carina is an extremely sparse association whose Galacticspace motion is, to within the uncertainties, statistically in-distinguishable from that of Columba (see e.g. Malo et al.2014a), although spatially the two associations are ratherdistinct. Both associations were identified by Torres et al.(2008) who further demonstrated that they share a simi-lar age of ∼
40 Myr through a combination of Li equivalentwidths and positions in the
V, V − I c CMD. Our isochronalages for both Carina and Columba are consistent with agesof ∼
40 Myr and, to within the uncertainties, appear to becoeval (as well as share a common age with Tuc-Hor; seeTorres et al. 2008).Fig. 7 demonstrates that our best-fit models appear tofollow the observed locus of Columba (tracing both the up-per and lower envelopes of the relatively well-populated as-sociation). For the much sparser Carina association, how-ever, such agreement between the models and the datais not so obvious [see Fig. 6; this is also reflected in thesystematically lower Pr( τ ) values for Carina compared toColumba]. Although we only have 12 stars in our mem-bership list for Carina, there is a case to be made thatthere are two apparent outliers which appear to lie belowthe observed locus. These stars (namely 2MASS J04082685-7844471 and 2MASS J09032434-6348330) will act to ‘drag’the best-fit model to older ages, the result of which, is that2MASS J10140807-7636327 (the reddest star in our list)is assigned a posterior membership probability which indi-cates non-member status, despite occupying a position inCMD space which is commensurate with its binary statusi.e. (cid:39) . (cid:39)
36 Myr (cf. 45 Myr). It remains unclear whether the kine-matic distance estimates for these two stars are erroneousor whether they are older than the other stars in our list.To better constrain the age of Carina, it is therefore clearthat we have to determine whether these stars are genuinemembers of Carina or whether they are simply older ac-tive stars (akin to what we previously discussed in Argus;see Section 5.3). Given the spectral types of these two stars(early M-type) it is unclear whether further spectroscopicinformation (e.g. Li equivalent widths) would allow us todifferentiate between these two options, and so this may bea case of having to wait until
Gaia provides us with thenecessary kinematic information to unambiguously do so. c (cid:13) , 1–24 olar neighbourhood age scale η Cha cluster
The use of the LDB technique to derive ages is restricted topopulations with ages of (cid:38)
20 Myr and thus η Cha is tooyoung for the adoption of such a technique. There are, how-ever, model-independent methods of deriving an age whichcan then be used as an independent diagnostic to compareagainst isochronal ages. One such method involves calculat-ing the time of minimum separation between stellar groupsin the past on the assumption that the groups share a com-mon origin. As with other kinematic methods this approachhas led to contradictory conclusions. For example, Jilinski,Ortega, & de la Reza (2005) performed a kinematic trace-back of the η Cha and (cid:15)
Cha clusters assuming a Galacticpotential and found that the smallest separation of only afew pc was ∼ η Cha afew Myr older than (cid:15)
Cha, but also that there is little ev-idence that the two clusters were appreciably closer thantheir current ∼
30 pc in the past.The literature isochronal ages for η Cha imply an ageof 5 − − T eff and L bol for bothcomponents by Alecian et al. (2007) and Gennaro, PradaMoroni, & Tognelli (2012) imply an age of (cid:39) (cid:39)
11 Myr for the cluster. Furthermore, Fig. 8 showsthat our semi-empirical pre-MS models provide a good fitto the data, tracing both the lower and upper envelope ofthe observed locus, except for the two outliers RECX 13 andRECX 15.Luhman & Steeghs (2004) demonstrated that unlike theother members of η Cha, which exhibit negligible levels ofinterstellar extinction, RECX 13 has a measured extinctionof A V = 0 . E ( V − J ) = 0 .
29 magbased on the A J /A V = 0 .
282 relation of Rieke & Lebofsky1985; see Section 4.2]. De-reddening RECX 13 using thesevalues would place the star much closer to the single-starsequence of the model distribution. RECX 15, on the otherhand, has a large IR excess at wavelengths of (cid:38) µ m whichis indicative of circumstellar material (see e.g. Megeath et al.2005). The inclination of this disc with respect to our line-of-sight remains ill-constrained, however high inclinationsfor discs around other η Cha members have been reported( (cid:38) o ; Lawson, Lyo, & Muzerolle 2004). If the disc aroundRECX 15 is so inclined, then we would expect significantdimming in the optical wavelengths as a result of observingthe star through it’s disc, and this could therefore explainwhy RECX 15 appears fainter than the other stars which represent the lower envelope of the cluster locus in CMDspace. TWA was the first of the young moving groups in our sam-ple to be identified in the literature (see e.g. de la Reza et al.1989; Kastner et al. 1997) and as such many age estimatesare now available for this association. Several authors haveattempted to use kinematic information to derive a model-independent age for the TWA, however, they are either con-tradictory or cover a prohibitively large range to provide astrong constraint on the age. For example, Mamajek (2005)noted that whilst the data are consistent with expansion, thecorresponding expansion age of 20 +25 − Myr has such large as-sociated uncertainties that it is of limited use.More recent analyses by Weinberger, Anglada-Escud´e,& Boss (2013) and Ducourant et al. (2014), which have mea-sured a larger number of parallaxes and proper motions thanin the Mamajek (2005) study, have come to completely op-posite conclusions. Weinberger et al. (2013) find that thespace motions of TWA members are essentially parallel anddo not indicate convergence at any time in the past 15 Myr,whereas Ducourant et al. (2014) find that a subset of 16members occupied the smallest volume of space ∼ . ∼
10 Myr (see e.g.Webb et al. 1999; Barrado y Navascu´es 2006; Weinbergeret al. 2013). Whilst our isochronal age is in excellent agree-ment with previous isochronal age estimates, Fig. 10 showsthat the Dartmouth models (in particular) do not provide agood fit to the data. Whilst it is apparent that these mod-els have an enlarged ‘binary wedge’ when compared to theother models (presumably stemming from the lower-masscut-off in the interior models and the slightly different T eff scale) which could play a role in this discrepancy, it is alsonotable that there is a mismatch between the observed slopeof the locus and that of the best-fitting model. Furthermore,even the other models appear to lie slightly above the lowerenvelope of the TWA locus. Hence, it is possible that ourisochronal age could be underestimated and that a morerepresentative age would be closer to ∼
15 Myr.Even when using models which include the effects of bi-nary stars, it is clear that we are unable to fit both the lowerand upper envelopes of the observed locus in the CMD ofthe TWA. One reason for this increased luminosity spreadcould be due to obscuration effects arising from discs aroundthe stars which define the lower envelope (namely TWA 6,TWA 9B, TWA 21 and 2MASS J10252092-4241539) which,depending on their orientation, may act to make them ap-pear dimmer in the CMD (see e.g. RECX 15 in Section 5.5).A recent analysis of IR excess disc emission by Schneider,Melis, & Song (2012a) suggests that none of these stars have c (cid:13) , 1–24 C. P. M. Bell et al. appreciable excess emission and therefore this is unlikely tobe the cause of the apparent luminosity spread.An alternative possibility is that the apparent luminos-ity spread could be a consequence of a real age spread withinthe association. Weinberger et al. (2013) propose that an agespread is probable given that the width of the age distribu-tion for TWA members exceeds that which would be ex-pected for a population of stars formed in a single burst. Toplace stricter constraints on whether the data are consistentwith a real age spread or not, a more comprehensive study,such as that of Reggiani et al. (2011) in the Orion NebulaCluster which accounted for uncertainties in the distance,spectral type, unresolved binarity, accretion and photomet-ric variability, would have to be performed.
The 32 Ori group is fairly new and has not been as well-characterised as the other groups in this study. As such,our isochronal age represents the first definitive publishedage for the group. The group is located in northern Orion,centred near 32 Ori and Bellatrix (although not containingthe latter), with a distinctly high proper motion ( µ α , µ δ (cid:39) +7 , −
33 mas yr − ) and radial velocity ( v r (cid:39) − . − )compared to the much more distant Ori OB1 young stel-lar population. It was first reported by Mamajek (2007),and a Spitzer
IR survey of the group reported by Shvon-ski et al. (2010) found several members to have dusty de-bris discs cooler than ∼
200 K, namely HD 35499 (F4V),HD 36338 (F4.5V), HR 1807 (A0Vn), TYC 112-917-1 (K4),and TYC 112-1486-1 (K3). Kharchenko et al. (2013) re-covered the cluster and estimated it to to be at 95 pc,with mean proper motion µ α , µ δ (cid:39) +10 . , − . − ( ± . − ), a core radius of ∼ . ± In this study we present a self-consistent, absolute age scalefor eight young ( (cid:46)
200 Myr), nearby ( (cid:46)
100 pc) movinggroups in the solar neighbourhood. We use previous liter-ature assessments to compile a list of member and high-probability candidate members for each of the young groupsin our sample. Creating four sets of semi-empirical pre-MSmodel isochrones based on the observed colours of youngstars in the Pleiades, in conjunction with theoretical cor-rections for the dependence on log g , we combine these withthe τ maximum-likelihood fitting statistic to derive ages foreach group using the M V , V − J CMD. Our final adoptedages for each group in our sample are: 149 +51 − Myr for the ABDor moving group, 24 ± β Pic moving group(BPMG), 45 +11 − Myr for the Carina association, 42 +6 − Myrfor the Columba association, 11 ± η Cha cluster,45 ± ± +4 − Myrfor the 32 Ori group. At this stage, we are uncomfortableassigning a final, unambiguous age to Argus as it appears asthough the stars in our membership list for the associationsuffer from a high level of contamination and hence maynot represent a single, coeval population. Comparing ourisochronal ages to literature LDB ages, which are currentlyonly available for the BPMG and Tuc-Hor, we find consis-tency between these two age diagnostics for both groups.This consistency instills confidence that our self-consistent,absolute age scale for young, nearby moving groups is robustand hence we suggest that these ages be adopted for futurestudies of these groups.
ACKNOWLEDGEMENTS
CPMB and EEM acknowledge support from the Universityof Rochester School of Arts and Sciences. EEM also acknowl-edges support from NSF grant AST-1313029. We thank Li-son Malo for discussions regarding the Bayesian analysis ofmembership probabilities for the young moving groups inthis study and Alex Binks for his tentative upper age limitfor the AB Dor moving group. This research has made use ofarchival data products from the Two-Micron All-Sky Survey(2MASS), which is a joint project of the University of Mas-sachusetts and the Infrared Processing and Analysis Center,funded by the National Aeronautics and Space Adminis-tration (NASA) and the National Science Foundation. Thisresearch has also made extensive use of the VizieR and SIM-BAD services provided by CDS as well as the WashingtonDouble Star Catalog maintained at the U.S. Naval Observa-tory and the Tool for OPerations on Catalogues And Tables(TOPCAT) software package (Taylor 2005).
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APPENDIX A: BETWEEN A ROCK AND AHARD PLACE: DE-CONSTRUCTINGPHOTOMETRY OF HIGHER-ORDERMULTIPLE SYSTEMSA1 Methodology
To de-construct photometry of higher-order multiple sys-tems into individual components measurements we adoptthe technique of Mermilliod et al. (1992) which, in our case,uses the system V -band magnitude, system V − J colourand ∆ V -band magnitude between the two components.We illustrate this technique using the TWA memberTWA 4 as an example. TWA 4 is a quadruple system (un-resolved spectral type of K6IVe; Pecaut & Mamajek 2013)comprising two visual components (separation < V -band magnitude of V AB = 8 . ± .
053 mag based on 601 observations by theASAS. The 2MASS PSC gives a system J -band magnitudeof J AB = 6 . ± .
020 mag, yielding a system V − J colour c (cid:13) , 1–24 C. P. M. Bell et al. of ( V − J ) AB = 2 . ± .
057 mag. For the difference in V -band magnitude between the A and B components weadopt ∆ V A , B = 0 . ± .
030 mag from Soderblom et al.(1998) based on two separate V -band measurements.Whilst the Mermilliod et al. (1992) formalism is demon-strated for splitting BV photometry, this can be generalisedto any combination of photometric bandpasses so long asthe relationship between the colour and the magnitude isknown (in our case the trend between M V and V − J ). Wetherefore fit an empirical (linear) slope to the resolved TWAmembers in the M V , V − J CMD, for which we calculate aslope a = 1 . V − J ) A = ( V − J ) AB +2 . − . m )(1+ a ) /a − . m (A1)and V A = V AB + 2 . − . m ) (A2)where ∆ m is the difference in the V -band magnitude be-tween the two components. The colour and magnitude ofthe secondary component is then simply( V − J ) B = ( V − J ) A + ∆ m/a (A3)and V B = V A + ∆ m. (A4)Hence for TWA 4 we derive, for the two visual compo-nents A and B, colours and magnitudes of ( V − J ) A =2 . ± .
057 mag, V A = 9 . ± .
061 mag, ( V − J ) B =2 . ± .
057 mag, and V B = 9 . ± .
061 mag. For the un-certainties on the individual component V − J colours weassume the same value as that for the system V − J colour,whereas for the individual V -band measurements we add theuncertainties on the system V -band and ∆ V -band measure-ments in quadrature. A2 Other cases
A2.1 HD 20121
HD 20121 is a member of Tuc-Hor and consists of atriple system (A0V+F7III+F5V according to the Wash-ington Double Star Catalog; Mason et al. 2001) in whichthe A+B components are separated by less than 1 arcsecand the C component lies a further 3 arcsec away. Weadopt the combined system V -band magnitude of V ABC =5 . ± .
005 mag from Mermilliod (2006) and a system J -band magnitude of J ABC = 5 . ± .
030 mag from the2MASS PSC, thereby defining a system V − J colour of( V − J ) ABC = 0 . ± .
030 mag. Tycho-2 provides resolvedphotometry for the combined A+B component and C com-ponent, from which we derive (using the relation of Ma-majek et al. 2006 to transform V T photometry into John-son V ) a difference between these two visual componentsof ∆ V AB , C = 2 . ± .
036 mag. Given the early spectraltypes of the components, in conjunction with the age ofTuc-Hor, we expect that all three components are on the main-sequence. Therefore, to split the photometry we fit anempirical (linear) fit to the A/F/G-type main-sequence re-lation of Pecaut & Mamajek (2013, slope a = 3 . V − J ) AB = 0 . ± .
030 mag, ( V − J ) C = 1 . ± .
030 mag, V AB = 6 . ± .
036 mag, and V C = 8 . ± .
036 mag.The reader will note that the spectral types we pro-vide in Table 1 for the three components of this system dif-fer from those given in the Washington Double Star Cata-log. The reason for this is simply that the spectral typesquoted in the Catalog are inconsistent with the derivedcolours and magnitudes of the individual components. Høget al. (2000) and Fabricius et al. (2002) provide resolvedTycho-2 ( BV ) T photometry for the individual componentsof this system. We use the conversion of Mamajek et al.(2006) and the trigonometric parallax measurement of vanLeeuwen (2007) to calculate individual absolute M V magni-tudes and Johnson B − V colours, and after comparing theseto the main-sequence relation of Pecaut & Mamajek (2013),find that both metrics are consistent with spectral types ofF4V+F9V+G9V. A2.2 GJ 3322
GJ 3322 is a member of the BPMG and is described byRiedel et al. (2014) as a triple system (unresolved spec-tral type of M4IVe) in which the A+C components forman unresolved binary and the B component is (cid:39) V -band magnitudeof V ABC = 11 . ± .
030 mag from UCAC4 and J -bandmagnitude of J ABC = 7 . ± .
023 mag from the 2MASSPSC, we derive a system V − J colour of ( V − J ) ABC =4 . ± .
038 mag. Riedel et al. (2014) quote a difference inmagnitude between the combined A+C component and theB component of ∆ K AC , B = 1 .
03 mag, where the K -bandused is that of the CIT system. To estimate what the cor-responding difference between the two components in the V -band is, we first assume that ∆ K CIT (cid:39) ∆ K s and thenfit an empirical (linear) fit to the BPMG members in the M V , V − K s CMD (for which we find a slope of a = 1 . V AC , B = 2 .
579 mag (for which we assume an un-certainty of 0.05 mag). To split the photometry we fit an em-pirical (linear) fit to the BPMG members (slope a = 1 . V − J ) AC = 4 . ± .
038 mag, ( V − J ) B = 5 . ± .
038 mag, V AC = 11 . ± .
058 mag, and V B = 14 . ± .
058 mag.The recent study of Riedel et al. (2014) also includes de-blended photometric measurements for the individual com-ponents, yielding V A = V C = 12 .
46 mag, V B = 13 .
56 mag,( V − J ) A = ( V − J ) C = 4 .
16 mag, and ( V − J ) B = 4 .
91 mag.Riedel et al. assume that the A and C components are ofequal-mass, which is clearly not true as Delfosse et al. (1999)provides radial velocity amplitudes. On the other hand, ourmethod assumes that we are disentangling unresolved bi-naries, and therefore if one of the components is itself anunresolved binary, then our method is insufficiently robustto deal with the fact that one of the sources is unnaturallybrighter because it is a multiple. As we do not have any in-formation regarding the photometric difference between theA and C components, we opt to simply adopt the aforemen-tioned values of Riedel et al. (2014) for the three individualcomponents of the system. c (cid:13) , 1–24 olar neighbourhood age scale APPENDIX B: BOOT-STRAPPED COLOURSAND QUESTIONABLE MEMBERSHIPSB1 Boot-strapped colours
In Section 2.2 we discussed cases in which we were forced toinfer V − J colours for member stars from either the V − K s or B − V colour. We used the V − K s colour when onlythe associated 2MASS PSC J -band Qflg was not ‘A’. Tocalculate the uncertainty on the inferred V − J colour, weadopted the 2MASS PSC K s -band uncertainty and addedthis in quadrature with the V -band uncertainty. When theassociated Qflgs of all three 2MASS PSC bandpasses werenot ‘A’ we used the B − V colour to estimate V − J . In suchcases, given that the 2MASS PSC photometry is unreliable,to assign an uncertainty on the V − J colour, we assumedan uncertainty of 0.03 mag combined in quadrature with the V -band uncertainty.The following stars have V − J colours inferred fromtheir V − K s colours: 32 Ori, AK Pic, HD 31647A,HIP 88726, HR 136, HR 789, HR 1190, HR 1189, HR 1474,HR 1621, HR 2466, HR 6070, HR 7329, HR 7736, HR 8352,HR 8911, HR 9016, and HR 9062.The following stars have V − J colours inferred fromtheir B − V colours: HD 31647B, HR 126, HR 127, HR 674,HR 806, HR 838, HR 1249, HR 2020, HR 4023, HR 4534,HR 7012, HR 7348, HR 7590, HR 7790, and HR 8425. B2 Questionable membership
In this Section we discuss those stars for which membershipto a given moving group is either questionable or unlikely.
B2.1 AB Dor moving group
HR 7214 was designated a member of the AB Dor movinggroup by Zuckerman et al. (2011), however Malo et al. (2013)derives a membership probability (including radial velocityand parallax information) of P v + π = 80 per cent for thisobject due to its seemingly anomalous space motion U incomparison to the bulk motion of the group as defined bybonafide members. We retain HR 7214 for our age analysison the basis that there is no categoric evidence that is shouldbe discarded. B2.2 BPMG
HR 789 is a tight spectroscopic binary which also has amore distance (25 arcsec) co-moving M dwarf companion(2MASS J02394829-4253049). This system was suggested asa member of Columba by Zuckerman et al. (2011), howeverMalo et al. (2013) find it is more likely a BPMG member( P v + π = 96 per cent). We therefore include HR 789 and2MASS J02394829-4253049 as BPMG members for our ageanalysis.GJ 9303 (HIP 47133) was proposed as a BPMG memberby Schlieder et al. (2012a), however the Bayesian analysis ofMalo et al. (2013) suggests that it is more likely associatedwith the field ( P v + π = 99 per cent). The main reason forthis assignment is that its Galactic position in Z is far ( ∼
40 pc) from the centre of bonafide BPMG members. With no available Li diagnostic to infer its youth, we discard GJ 9303from our age analysis.HR 7329 was classified as a BPMG member by Zucker-man et al. (2001), however Malo et al. (2013) demonstratesthat the radial velocity is discrepant (∆ v (cid:39)
14 km s − ) withregards to that predicted for the BPMG, and that if it is in-cluded in the prior the probability P v + π becomes zero. Maloet al. (2013) suggest that either the close-by low-mass com-panion may affect the systematic velocity or that the radialvelocity is erroneous due to the fast rotation of the primary( v sini = 330 km s − ). With no observations currently avail-able to categorically rule out its association with the BPMG,we retain HR 7329 in the age analysis. B2.3 Carina
AB Pic (HD 44627) has an ambiguous membership havingpreviously been assigned to Tuc-Hor by Zuckerman & Song(2004) only to be revised to Carina by Torres et al. (2008).The Bayesian analysis of Malo et al. (2013) suggests that itis more likely a member of the latter ( P v + π = 71 per centfor Carina compared with only 29 per cent for Columbaand 0 per cent for Tuc-Hor). The higher probability that itbelongs to Carina is a result of its Galactic positions Y Z which are more akin to that of Carina than Tuc-Hor. Fur-thermore, the reason it has a higher probability of beinga Columba member (as compared to Tuc-Hor) is becausethere is a difference of only 1 km s − between the predictedradial velocities for Columba and Carina (Malo et al. 2013).Until additional radial velocity measurements are made tocategorically demonstrate that it is not a member of Carina,we retain AB Pic as a member for our age analysis. B2.4 Columba
HD 15115 was previously believed to be a member of theBPMG (see e.g. Mo´or et al. 2006), however Malo et al. (2013)suggests that it is more likely to be a member of Columba( P v + π = 87 per cent) and therefore we retain this as a mem-ber of Columba for the age determination. Similarly, whilstHD 23524 was originally designated a Tuc-Hor member byZuckerman et al. (2011), Malo et al. (2013) derive a proba-bility of it being a Columba member of P v + π = 98 per cent,and thus we include it in our age determination of Columba.In addition, we also include V1358 Ori (BD-03 1386) andDK Leo (GJ 2079; originally assigned memberships in Tuc-Hor and the BPMG respectively) in our age analysis ofColumba, as Malo et al. (2013) derive membership prob-abilities of P v + π = 85 and 93 per cent respectively.The membership status of AS Col (HD 35114) is am-biguous primarily as a result of a poorly constrained ra-dial velocity (see e.g. Bobylev, Goncharov, & Bajkova 2006and Bobylev & Bajkova 2007 who derive values of 15 . ± . − and 23 . ± . − respectively). The analy-sis of Malo et al. (2013) finds that AS Col belongs to ei-ther Columba or Tuc-Hor ( P v + π = 99 per cent in bothcases) depending on which radial velocity is adopted. Us-ing the Bayesian Analysis for Nearby Young AssociatioNs(BANYAN; see Malo et al. 2013) for AS Col, but remov-ing the radial velocity measurement as a prior, we find thatthe star appears to be a bona fide member of Columba. We c (cid:13) , 1–24 C. P. M. Bell et al. therefore retain AS Col as a Columba member for our ageanalysis of the association. κ And (HR 8976) is designated a bona fide member ofColumba by Malo et al. (2013) with a probability of P v + π =95 per cent. Recent evidence, however, suggests that κ Andis in fact much older ( (cid:38)
100 Myr depending on assumedcomposition; see e.g. Hinkley et al. 2013; Brandt & Huang2015). Given this evidence we exclude κ And from the ageanalysis of Columba.
B2.5 TWA
We discard three stars (TWA 14, TWA 18 and TWA 31)from the age analysis of the TWA. The mean distance es-timate of the group is ∼
57 pc with a 1 σ scatter of roughly11 −
12 pc (Mamajek 2005), however both TWA 14 andTWA 18 have distances (trigonometric and kinematic re-spectively) of (cid:38)
100 pc and thus if they were members, wouldrepresent 3 σ outliers. In addition, we do not include TWA 31in our age analysis due to its designation as a non-memberby Ducourant et al. (2014). B2.6 Tuc-Hor
GJ 3054 (HIP 3556), HD 12894, HD 200798, and BS Ind(HD 202947) are all designated members of Tuc-Hor byZuckerman & Song (2004), however Malo et al. (2013)demonstrated that if their radial velocities are included aspriors, their respective probabilities of belonging to Tuc-Horsignificantly decrease due to a difference of (cid:39) − − be-tween the measured and predicted radial velocities. Withoutadditional, higher precision, radial velocity measurementsfor these stars, we retain all four in our age determinationof Tuc-Hor.HR 943 was classified as a Tuc-Hor member by Zuck-erman et al. (2011), however Malo et al. (2013) find thatit is equally likely to be a member of Tuc-Hor or Columba( P v + π = 50 and 49 per cent respectively). Until further mea-surements are able to discriminate between which of themoving groups it belongs to, we HR 943 as a member ofTuc-Hor for our age analysis.HR 6351 and V857 Ara (HD 155915) are given as Tuc-Hor members by Zuckerman et al. (2011), and although ex-hibiting signs of youth (e.g. circumstellar material and Liabsorption), the analysis of Malo et al. (2013) suggests thatboth are field objects. Without categoric evidence that thesestars are either field dwarfs or young interlopers belongingto another co-moving group, we retain both stars as Tuc-Hormembers in our age analysis. B2.7 Duplicate members
Given that we have collated memberships from numer-ous different literature sources, it is possible that starswhich have been assigned membership to one movinggroup in a particular study, may be classified as a mem-ber of a different group by another study. We identi-fied a total of 6 duplicate stars, 5 of which are classi-fied as members of Tuc-Hor and Columba by Kraus et al.(2014) and Malo et al. (2014a) respectively, namely CD-44 753 (2MASS J02303239-4342232), 2MASS J03050976- 3725058, 2MASS J04240094-5512223, 2MASS J04515303-4647309 and 2MASS J05111098-4903597. In each case thestar has been assigned membership solely on the basis ofthe measured radial velocity, and for 4 out of the 5 cases(excepting CD-44 753) the two independent radial velocitymeasurements of Kraus et al. (2014) and Malo et al. (2014a)agree to within the quoted uncertainties. Given the addi-tional prior information included in the Bayesian analysis ofMalo et al., we retain these stars as members of Columba,but note that the inclusion/exclusion of these stars for eithergroup has a negligible effect on the best-fit age.TYC 5853-1318-1 (2MASS J01071194-1935359) wasalso found to be a duplicate, first suggested as a BPMGmember by Kiss et al. (2011), but more recently advo-cated as a Tuc-Hor member by Kraus et al. (2014). TheBayesian analysis of Malo et al. (2013) is unable to cat-egorically assign membership to only one of the movinggroups, instead suggesting it could belong to any of theBPMG, Tuc-Hor or Columba. Interestingly, Kraus et al.(2014) only assigns membership to Tuc-Hor on the basisof strong Li absorption i.e. it is young. Its measured ra-dial velocity ( v = 9 . ± . − ) is somewhat discrepantwith respect to that of the bulk of the other Tuc-Hor mem-bers (∆ v = 8 .
25 km s − ), and on the basis of this shouldbe considered a non-member (as stated in Kraus et al.2014). The measured radial velocity of Kiss et al. (2011, v = 11 . ± . − ) is consistent with that of Krauset al. (2014) to within the uncertainties. Furthermore, bothof these velocities are consistent with that predicted by Maloet al. (2013) if it is in fact a member of the BPMG. Based onthis we prefer to assign membership of TYC 5853-1318-1 tothe BPMG and include it in our age analysis of the group. APPENDIX C: UPDATED τ MODEL FORDEALING WITH NON-MEMBERCONTAMINATIONC1 Fitting datasets with non-members
C1.1 Background
Non-members whose positions in the CMD lie outside thearea of the cluster sequence can have an overwhelming effecton the fitted parameters, distorting them far away from thetrue values, or make the fit fail entirely. The reason is thatthe total likelihood (the logarithm of which is proportionalto τ ) is the product of the likelihoods that the individualdata points originate from the cluster sequence. Thus onedatapoint with a likelihood close to zero for an otherwisegood fit will drag the model towards it, or even give a prob-ability of zero for the whole fit.Conceptually the most straightforward solution is tohave a model of the non-member contamination in CMDspace. Perhaps surprisingly it turns out that a very crudemodel is effective; in Bell et al. (2013) we used a uniformdistribution over the area delineated by the maximum andminimum colours and magnitudes in the dataset (see alsovan Dyk et al. 2009). In the same paper we showed howthe uniformly distributed non-member model was formallyequivalent to a soft-clipping scheme, and in fact actuallyused the latter in the fitting. However, here we will use theconceptual framework of a uniform non-member distribu- c (cid:13) , 1–24 olar neighbourhood age scale tion. In part because it is better suited to the problem inhand, but also because it allows us to calculate a goodness-of-fit parameter, a step which was missing from our earliersoft-clipping technique.Finally, before embarking on the formalism of thismethod, we should remark that whilst we will talk aboutnon-members, in fact we should really refer to stars whichdo not fit our cluster model. For example, equal-mass tripleswill be very slightly above our equal-mass binary sequences,and hence in a region of zero probability because our modeldoes not include higher order multiples. Therefore, such ob-jects will be treated by our method as non-members, whenin fact it is the model which is at fault. In this sense ouruniform distribution is in part a Jaynes’ fire extinguisher,a hypothesis which remains in abeyance unless needed bydata which have a low probability of originating from ourmain hypothesis (see Section 4.4.1 of Jaynes & Bretthorst2003). C1.2 Formalism and implementation
As in Naylor & Jeffries (2006) we define the function to beminimised as τ = − (cid:88) i =1 , N ln (cid:90) (cid:90) U i ( c − c i , m − m i ) ρ ( c, m ) d c d m (C1)(see also the elegant proof in Walmswell et al. 2013). Themodel of the expected density of stars in the CMD isgiven by ρ ( c, m ) and U i represents the uncertainties for thedata point i . In terms of Bayes’ theorem, the integral is P ( M ) P ( D | M ), and so if we have two competing models thisshould be replaced by P ( M n ) P ( D | M n ) + P ( M c ) P ( D | M c )where the subscripts c and n refer to cluster members andnon-members respectively. If the probability that any givenstar is a member is given by F i , then P ( M n ) = 1 − F i and P ( M c ) = F i . Furthermore, P ( D | M n ) is the integral of U i ρ n and P ( D | M c ) the integral of U i ρ c , where ρ c is our usual clus-ter model and ρ n is the model of the non-members. If thearea delineated by the maximum and minimum colours andmagnitudes in the dataset is A , then where ρ n is non-zero ρ n = 1 /A since it must integrate to one (see Naylor 2009).Hence τ = − (cid:88) i =1 , N ln (cid:20) (1 − F i ) ρ n (cid:90) (cid:90) U i d c d m + F i (cid:90) (cid:90) U i ρ c d c d m (cid:21) , (C2)and thus τ = − (cid:88) i =1 , N ln (cid:20) − F i A + F i (cid:90) (cid:90) U i ρ c d c d m (cid:21) , (C3)where we have used the facts that where ρ n is non-zero inthe CMD it is constant, and that U i integrates to one. Usingthis formula directly (rather than adding a constant to ρ c )is very straightforward to implement, since one simply cal-culates the likelihood in the normal way, adjusts it for theprobability of membership and adds a constant. This has theadvantage over our soft-clipping procedure that individualstars can be given different membership probabilities.A further advantage of this formalism is that we can calculate how the position of a star in the CMD modifiesour estimate of how likely it is to be a member. If we applyBayes’ theorem to the hypothesis M c , that a star is a mem-ber of the cluster, which we are testing against a dataset D then P ( M c | D ) = P ( M c ) P ( D | M c ) P ( D ) (C4)= P ( M c ) P ( D | M c ) P ( M c ) P ( D | M c ) + P ( M n ) P ( D | M n ) . Using the same expressions for the probabilities and likeli-hoods we used for Eqn. C3 we obtain P i ( M c | D ) = F i (cid:82)(cid:82) U i ρ c d c d m − F i A + F i (cid:82)(cid:82) U i ρ c d c d m , (C5)where in the denominator we have used the same simplifica-tions as between Eqns. C2 and C3. The intuitive interpre-tation of this equation is that at any point in the CMD theratio of the model densities for members and non-membersgives the membership probability for a star at that position,were its uncertainties in colour and magnitude infinitelysmall. Allowance for uncertainties is made by convolving thedensities with the uncertainty function. C2 Testing the prior membership probabilities
If the sum of the prior and posterior membership probabil-ities are very different, this could be an indication that thepriors were incorrect. A second way of exploring this is de-scribed in Section 4.2 where we examined the distributionof the individual values of τ . The cumulative plots in Fig. 3show how mis-classified members have values of τ far ex-ceeding those predicted by the fitting process. If we lowerthe maximum prior membership probability, then both thepredicted and measured values of τ develop a pedestal atthe value of τ corresponding to the maximum membershipprobability (again shown in Fig. 3). The simplest way toachieve this is to multiply all the priors by a factor, whichwe call P max as it then corresponds to the maximum mem-bership probability any star can have. We found that ad-justing P max until either the pedestals in Fig. 3 matched,or the sums of the prior and posterior membership proba-bilities were roughly in agreement gave very similar answersfor the best value of P max . Given the number of fits we hadto perform, we took the latter course as straightforward toimplement in an automated procedure.We emphasise that P max should not be adjusted to ob-tain a reasonable value for the goodness-of-fit [Pr( τ )], as itis possible have a reasonable value of Pr( τ ) but still havesystematic residuals. This can be seen by examining the cu-mulative distribution plots for the individual values of τ ,which seem to be a good space in which to examine the qual-ity of a fit, rather like the residual plots in more conventionalfitting. Hence there are two metrics for a good fit, the nu-merical one of the value of Pr( τ ), and the qualitative oneof the match between the predicted and model distributionsof τ . It is possible to adjust P max to achieve a good valueof Pr( τ ) , but if this does not achieve a change in shapeof the τ distribution that brings data and prediction intoagreement, the fit is probably incorrect. c (cid:13) , 1–24 C. P. M. Bell et al.
C3 Calculation of
Pr( τ )Naylor & Jeffries (2006) showed how to assess whether agiven model was a good fit to the data [Pr( τ )] by calculat-ing the probability that a random dataset drawn from themodel, when fitted to the model would have a value of τ which exceeded that for the observed dataset. The methodpresented for calculating this involved convolving the dis-tributions of τ for each individual datapoint with the dis-tributions for all other data points. This procedure has theadvantage of being a numerical equivalent to the way theexpression for Pr( χ ) is derived, but for the data presentedhere there are two significant problems. First the methodwas unacceptably slow for some of the data, this is becausesome of the uncertainties are very large leading to slow con-volutions. Second the method has to be modified to allowfor non-members. C3.1 The new method
We therefore elected to use a more direct method to calcu-late Pr( τ ) where we simply simulated a thousand observeddatasets using the parameters of the best-fitting model, andcalculated τ for each of them. For the simulation of clustermembers, the simulated stars should in principle be drawnrandomly from the stellar mass function multiplied by anyselection effects, such as magnitude limits. In practice the se-lection effects are often poorly understood, and so the func-tion from which the stars are drawn is best defined from themagnitude distribution of the real dataset. This problem isnot unique to this method of calculating Pr( τ ), an analo-gous problem exists for the method described in Naylor &Jeffries (2006).To calculate Pr( τ ) we began by simulating a cluster ofa million stars with the parameters of the best-fitting model(practically one can use the CMD of ρ created for the fit-ting process) and grouped the stars into a number of mag-nitude bins equal to the number of observed data points.The boundaries between these bins were set at the mid-magnitudes between each observed datapoint. We then sim-ulated a data point from each magnitude bin by first usingthe prior probability of membership to assign it to either thecluster or the field. Non-members were simulated by plac-ing the star randomly within the rectangular area definedby the maximum and minimum colours and magnitudes ofthe real dataset (and thus there is a small chance it may beplaced within the cluster sequence). Cluster members weresimulated taking their colours and magnitudes from a simu-lated star randomly chosen from within the magnitude bin.Since the bins are more closely spaced where there are manyreal data points, this means the distribution of data pointsin the simulated dataset broadly follows the magnitude dis-tribution of real data points. Each simulated star was thendisplaced from its model colour and magnitude using the un-certainties of the corresponding real data point. Followingthis procedure for each magnitude bin resulted in a simu-lated observation, for which we could calculate τ in thenormal way. The distribution of 1000 values of τ calculatedin this way could then be used to calculate Pr( τ ) for thereal observation.Finally, we must allow for the effect of free parameterson the expected values of τ , since they will be lower for the Figure C1.
The probability of a model exceeding a given valueof τ calculated from ten different simulated observations of 30stars from a model cluster of a million stars. best-fitting model if there are more free parameters than inthe case where a single model is fitted. For the reasons dis-cussed in Naylor (2009) we correct for this by scaling thevalues of τ associated with a particular Pr( τ ). We firstsubtract the expectation value of τ from each of the val-ues, multiply them by (cid:112) N − n ), where n here refers tothe number of free parameters and N to the number of datapoints, and then add back the the expectation value less n . Note the scaling factor was incorrectly stated in Nay-lor (2009), and was also incorrect in earlier versions of thecode, though the corrections are so small that it does notsignificantly affect earlier results. C3.2 The accuracy of
Pr( τ )In principle the cumulative distribution function for the ex-pected values of Pr( τ ) is a function only of the best-fittingmodel and the way in which we select those stars which areobserved. If we have two different datasets consisting of dif-ferent stars in the same cluster, provided those stars wereselected using the same random process, Pr( τ ) would bethe same for both datasets. Unfortunately, the fact we donot know the selection criteria for the stars means that, asdescribed above (see Appendix C3.1) we have to rely on theobservations themselves to give us the distribution in magni-tudes expected for the sample, and so the value we calculatefor Pr( τ ) depends on our sample. This produces an uncer-tainly in our value of Pr( τ ), which it is important to quan-tify. We performed an example simulation of 30 stars in a10 Myr-old cluster spread over 4 magnitudes in an M V , V − J CMD. Fig. C1 shows ten of the resulting distributions of τ which show that whilst the shape and the width of the dis-tribution remain largely constant, the τ associated with agiven probability varies with an RMS (calculated from 100models) of (cid:39)
5. For good fits where Pr( τ ) ∼ . τ , and so one would almost always conclude the fit is agood one. However, in the low-probability tail the test willshow the fit is probably not a good one, but cannot give auseful answer as to how poor it is. c (cid:13)000