The velocity distribution of nearby stars from Hipparcos data II. The nature of the low-velocity moving groups
aa r X i v : . [ a s t r o - ph . GA ] J un The velocity distribution of nearby stars from
Hipparcos dataII. The nature of the low-velocity moving groups
Jo Bovy1 , , ABSTRACT
The velocity distribution of nearby stars ( .
100 pc) contains many overden-sities or “moving groups”, clumps of comoving stars, that are inconsistent withthe standard assumption of an axisymmetric, time-independent, and steady-stateGalaxy. We study the age and metallicity properties of the low-velocity movinggroups based on the reconstruction of the local velocity distribution in Paper Iof this series. We perform stringent, conservative hypothesis testing to establishfor each of these moving groups whether it could conceivably consist of a coevalpopulation of stars. We conclude that they do not: the moving groups are nottrivially associated with their eponymous open clusters nor with any other inho-mogeneous star formation event. Concerning a possible dynamical origin of themoving groups, we test whether any of the moving groups has a higher or lowermetallicity than the background population of thin disk stars, as would generi-cally be the case if the moving groups are associated with resonances of the baror spiral structure. We find clear evidence that the Hyades moving group hashigher than average metallicity and weak evidence that the Sirius moving grouphas lower than average metallicity, which could indicate that these two groups arerelated to the inner Lindblad resonance of the spiral structure. Further we findweak evidence that the Hercules moving group has higher than average metallic-ity, as would be the case if it is associated with the bar’s outer Lindblad resonance.The Pleiades moving group shows no clear metallicity anomaly, arguing against acommon dynamical origin for the Hyades and Pleiades groups. Overall, however,the moving groups are barely distinguishable from the background population ofstars, raising the likelihood that the moving groups are associated with transientperturbations. Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 WashingtonPlace, New York, NY 10003, USA To whom correspondence should be addressed: [email protected] Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117 Heidelberg, Germany
Subject headings:
Galaxy: fundamental parameters — Galaxy: kinematics anddynamics — Galaxy: structure — methods: statistical — Solar neighborhood —stars: kinematics
1. Introduction
Moving groups—clumps of stars in the Solar neighborhood sharing the same spacevelocity—have been known for over a century (M¨adler 1846; Proctor 1869) and their inter-pretation has touched on some of the most basic facts about our Galaxy and the Universe.From the location of the center of the Milky Way (M¨adler 1847) over the age and dy-namical state of the Universe (Jeans 1915, 1935; Bok 1946), presently, the moving groupsare used to constrain the dynamical properties of the Galactic disk (e.g., Dehnen 2000;Quillen & Minchev 2005). However, in order to quantitatively constrain the fundamentalproperties of the Galaxy using the presence of structure in the local velocity distribution,the nature of the moving groups needs to be clarified. At present, the evidence that themoving groups are not unmixed structure in phase space consisting of the ghosts of paststar-formation events, but are instead dynamical effects arising from non-axisymmetric com-ponents of the Galaxy’s mass distribution, is by and large circumstantial. Currently, anyconstraint on Galaxy dynamics arising from the moving groups’ existence or properties issubject to the large uncertainty as to what the actual origin of the moving groups is.The structure of the local velocity distribution has received much attention during thelast century. While the simplest assumption is that the distribution of velocities is a sim-ple Gaussian distribution (Schwarzschild 1907), this assumption was untenable in the lightof observations that showed the presence of multiple “streams” in the velocity distribution(Kapteyn 1905; Eddington 1910). That these streams are very prominent and make up alarge part of the full distribution is clear from the fact that their existence was so readilyestablished. Until the
Hipparcos mission, the actual contribution of substructure in thevelocity distribution was only poorly characterized, but the rich
Hipparcos data set con-clusively showed that a large fraction of the local velocity distribution is in the form ofclumps (Dehnen 1998; Skuljan, Hearnshaw, & Cottrell 1999); a quantitative analysis showsthat about 40 percent of the stars in the Solar neighborhood ( .
100 pc) is part of a smallnumber of moving groups (Bovy, Hogg, & Roweis 2009a). The velocity distribution with themoving groups indicated is shown in Figure 1.The nature and origin of the moving groups has remained elusive all this time, although 3 –considerable effort has been made both observationally and theoretically to explain andinterpret the existence of the moving groups. For much of the last century the consensusview was that the moving groups are the remnants of past star formation events, coevalpopulations of stars that were once closely associated in position as well as velocity but thathave now dispersed and spread out over vast regions of space into the loose associations ofstars that still retain a common motion. This view of a dynamically unrelaxed Galaxy wasfirst expressed by Jeans (Jeans 1915) and its most vociferous proponent during the secondhalf of the century was Eggen (e.g., Eggen 1996). The Hyades and Ursa Major moving groupsseemed to fit into this framework as disrupting clusters in a differentially rotating disk (Bok1934, 1936, 1946). The inspection of the properties of likely Hyades members showed thatthese followed a similar color-luminosity relation as the Hyades and Praesepe open clusters(Eggen 1958), which seemed to vindicate the view of moving groups as disrupting clusters.This explanation of the moving groups’ origins was contested, however, (e.g., Breger 1968;Wielen 1971; Williams 1971; Soderblom & Clemens 1987; Boesgaard & Budge 1988) andstarted to fall out of favor by the end of the century as observational evidence started toappear that moving group members were a much more varied population of stars than theopen clusters with which they were believed to be associated: Eggen (1993) found thatthe Hyades moving group has a different luminosity function than the Hyades open cluster;Dehnen (1998) found that moving groups are present in various color subsamples of
Hipparcos stars and that therefore, using color as a proxy for mean age, moving groups contain starsof a wide range of ages. Nevertheless, the evaporating cluster narrative still holds sway for(parts of) some moving groups (Asiain, Figueras, & Torra 1999), in particular for the smallHR 1614 moving group (Feltzing & Holmberg 2000; De Silva et al. 2007), which we do notstudy here because it does not stand out as a kinematic overdensity in the overall velocitydistribution. In §§ §
7, in which we test various of these dynamical scenarios.Most of the non-axisymmetric perturbations that have been proposed to create themoving groups are associated with stable, long-lived perturbations, e.g., a long-lived den-sity wave (Lin & Shu 1964). However, several pieces of evidence indicate that spiral struc-ture might be only short-lived and/or transient: spiral structure gradually heats the disk(Carlberg & Sellwood 1985) such that it eventually becomes stable against non-axisymmetricperturbations in the absence of a cooling mechanism (Sellwood & Carlberg 1984); spiraldensity waves tend to dissipate within a few galactic revolutions if fresh waves are notcontinuously created (Toomre 1969); spiral structure is more common in high density envi-ronments than in the field (Elmegreen & Elmegreen 1982, 1983) where interactions betweengalaxies that could induce transient spiral structure are more common; and nearby galax-ies show strong variations of the pattern speed with galactocentric radius, which stronglyconstrains the lifetime of grand-design spiral structure (Merrifield, Rand, & Meidt 2006;Meidt, Rand, & Merrifield 2009). The velocity distribution inferred from
Hipparcos dataitself, with its large amount of substructure, shows that spiral structure does not operateon a smooth phase-space density and that spiral instabilities that grow because of featuresin the phase-space distribution (e.g., Sellwood & Lin 1989; Sellwood & Kahn 1991) shouldtherefore be expected to be present.One such instability driven by features in the angular-momentum distribution suchas grooves or ridges is the scenario proposed by Sellwood & Kahn (1991) (see alsoLovelace & Hohlfeld 1978). In this model for the growth of spiral modes, an initialnarrow groove in the angular-momentum density grows into a well-defined large-scale spiralpattern that dies off again after a few galactic rotations (at corotation, which lies nearthe groove center). Since stars are scattered at the inner Lindblad resonance (ILR of thespiral pattern, an underdensity of stars in energy–angular-momentum space forms at theLindblad resonance, which could spur a new cycle of growth of a spiral instability, albeitwith a corotation radius near the ILR of the previous pattern. Since the corotation radiiof subsequent spiral patterns move steadily inward, this recurrent cycle stops at a certainpoint. In §
8, we ask whether any of the moving groups is a manifestation of this scenario. 5 –Although the
Hipparcos data allowed the velocity distribution in the Solar neighborhoodto be studied in detail for the first time using complete samples of stars, and theoretical workon the origin of the moving groups has blossomed in recent years, little progress has beenmade observationally to elucidate the nature of the moving groups. In this paper, we use largesamples of
Hipparcos stars—an order of magnitude improvement over previous studies—to investigate the origin of the kinematical substructures seen in Figure 1. We use thereconstruction of the local velocity distribution from Bovy, Hogg, & Roweis (2009a) to assignmoving-group membership probabilities to stars. We propagate the membership uncertaintythrough all of the analyses of the properties of the moving-group member stars. This avoidsall of the biases that result from making hard cuts on membership in investigations of thiskind and allows us to perform comprehensive tests to establish the origin of each individualmoving group.Before we continue, it is worth pointing out that OB associations—spatially localizedassociations of young stars (e.g., de Zeeuw et al. 1999)—are also sometimes referred to asmoving groups. The following does not concern these OB associations.The main parts of this paper are the following. In § §
6, we extend this result to showthat the moving groups are not associated with any single episode of star formation; in § §
8, we look at whether the moving groups areassociated with the recurrent spiral structure scenario of Sellwood & Kahn (1991).
2. Data
We use the standard Galactic velocity coordinate system, with the directions x , y ,and z (and associated unit vectors ˆx , ˆy , and ˆz ) pointing toward the Galactic center, inthe direction of circular orbital motion, and toward the north Galactic Pole, respectively.Vectors are everywhere taken to be column vectors. The components of the velocity vector, ˆx ⊤ v , ˆy ⊤ v , and ˆz ⊤ v , are conventionally referred to as U , V , and W , respectively, but we willrefer to them as v x , v y , and v z . We follow the procedure of Dehnen & Binney (1998) and Aumer & Binney (2009) toselect a magnitude-limited, kinematically unbiased sample of single main-sequence stars with 6 –accurate astrometry from the
Hipparcos catalog. We start by determining the magnitude towhich the
Hipparcos catalog is complete in 16 × ×
10 equal width bins in sin b , l , and color B T − V T , the latter measured in the Tycho passbands in the interval (-0.3,1.5), by finding the V T magnitude of the second brightest star that is included in the Tycho catalog (Høg et al.2000a,b), but absent in the Hipparcos catalog. We then select in each bin all stars from theoriginal
Hipparcos catalog (ESA 1997) brighter in V T than the limiting magnitude in that bin.From this sample of stars we select single stars by using the “Solution type” isol n <
10 in thenew reduction of the
Hipparcos data (van Leeuwen 2007), stars with accurate astrometry byselecting stars with relative parallax uncertainties smaller than 10 percent (using the formalerror on the parallax in the new
Hipparcos catalog). Main-sequence stars are selected byusing the color–magnitude cuts from Aumer & Binney (2009)M
Hip < . × ( B − V ) − . , B − V ≤ . Hip < . × ( B − V ) − . , . ≤ B − V ≤ . Hip < . × ( B − V ) + 1 . , . ≤ B − V M Hip > . × ( B − V ) + 2 . , B − V ≤ . Hip > . × ( B − V ) + 1 . , . ≤ B − V ≤ . Hip > . × ( B − V ) + 4 . , . ≤ B − V ≤ . Hip > . × ( B − V ) + 0 . , . ≤ B − V (1)where M Hip is the absolute magnitude in
Hipparcos ’ own passband.This procedure selects 19,631 stars from the
Hipparcos catalog, 15,023 of which aremain-sequence stars. The color–magnitude diagram of the full sample of 19,631 stars isshown in Figure 2; the cuts defining the main-sequence are also shown in this figure.We refer the reader to Bovy, Hogg, & Roweis (2009a, hereafter BHR) for a detailedexplanation of how three-dimensional velocities are projected onto the two-dimensional tan-gential plane observed by
Hipparcos —since the
Hipparcos mission did not measure radialvelocities, this third velocity component is missing for all of the stars in the sample—andhow the uncertainties given in the
Hipparcos catalog are propagated to the uncertaintiesin the tangential velocity components. In what follows w i will represent the observed tan-gential velocity of star i , v i its (unobserved) three-dimensional velocity, R i the projectionmatrix onto the tangential plane for star i —i.e., R i v i = w i —and S i the two-dimensionalobservational-uncertainty variance matrix in the tangential-velocity plane. 7 – BHR reconstructed the velocity distribution of nearby stars by deconvolving the ob-served tangential velocity distribution of a kinematically unbiased sample of 11,865
Hipparcos stars. The deconvolution algorithm (Bovy, Hogg, & Roweis 2009b) represents the underly-ing velocity distribution as a sum, or mixture, of Gaussian components, and can properlyhandle arbitrary uncertainties, including missing data, provided that there are no significantstar–to–star correlations. These are believed to be insignificant in the most recent releaseof the
Hipparcos data (van Leeuwen 2007). Model selection, most notably the selection ofthe “right’ number of components in the mixture, was based on predicting the radial veloc-ities in the
Geneva-Copenhagen Survey (GCS; Nordstr¨om et al. 2004). BHR found that theunderlying three-dimensional velocity distribution was best represented by a ten-componentmixture of Gaussians and found the 99 best-fit parameters of this decomposition.Although in Gaussian-mixture deconvolution the individual components do not neces-sarily have any meaningful interpretation—the Gaussians are simply basis functions of anexpansion—many of the Gaussian components in the best-fit mixture could be identifiedunambiguously with peaks in the velocity distribution, most of which correspond to knownmoving groups. For the purposes of this paper we will use the representation of the velocitydistribution as a mixture of 10 Gaussian components with parameters given in Table 1 ofBHR; we will come back to this choice in the discussion in §
9. We will identify the mainmoving groups in the velocity distribution with components in Table 1 of BHR as follows:component 2 corresponds to the NGC 1901 moving group; component 4 to the Herculesmoving group; component 5 to the Sirius moving group; components 6 and 7 to the Pleiadesmoving group; component 8 to the Hyades moving group; and component 10 to the Arcturusmoving group.We can now probabilistically assign stars to Gaussian components or moving groups.For each star i we calculate the probability that it is associated with component j of theGaussian mixture model for the local velocity distribution p ij = α j N ( w i | R i m j , R i V j R ⊤ i + S i ) P k α k N ( w i | R i m k , R i V k R ⊤ i + S i ) , (2)where α j , m j , V j are the amplitude, mean, and variance of the j th Gaussian component,which are given in Table 1 of BHR; see Bovy, Hogg, & Roweis (2009b) for a derivation ofthis formula. For the Pleiades moving group for each star i we add up the probabilitiesof it being associated with component 6 or 7, i.e., p i, Pleiades = p i + p i , since two of thecomponents of the mixture are associated with the Pleiades moving group (see BHR for anextended discussion of this). 8 –
3. A first look: Are the low-velocity moving groups associated with theireponymous open clusters?
To get a first idea about the properties of the moving groups we can look at “probabilis-tic” color–magnitude diagrams of the groups, which will form the basis for everything we doin the remainder of this paper. Using the probabilities p ij for each star i to be part of movinggroup j , we can create color–magnitude diagrams for the different moving groups that areweighted by the probabilities of each star to be part of that particular moving group. Suchcolor–magnitude diagrams are shown in Figure 3 for the six moving groups unambiguouslydetected by BHR. In these color–magnitude diagrams each star is plotted as a dot with thegrayscale of that dot proportional to the probability of the star to be part of the movinggroup. For clarity only those stars with p ij > . are plotted for each of the moving groups corresponding to the pro-posed ages and metallicities for the associated open clusters found in the literature; see thecaption of Figure 3 for the details on each open cluster. It is clear from this figure that theisochrones of the open clusters do not represent the color–magnitude relation of their associ-ated moving groups well, although a caveat remains about the effect of parallax uncertaintiesand the effect of low-probability moving groups members, which is hard to gauge from thisfigure. To make the comparison between the open clusters’ and the moving groups’ ageand metallicity more quantitative, we show in Figure 4 a comparison between the observedparallaxes of the moving group members and the predicted parallaxes based on the openclusters’ isochrone and the observed photometry for each main-sequence star in the sample; Retrieved using the Web interface provided by Leo Girardi at the Astronomical Observatory of Padua http://stev.oapd.inaf.it/cgi-bin/cmd_2.2 M V versus B − V relation corresponding to theisochrone of the associated open cluster we predict the absolute magnitude of the star andconvert this to a model parallax using the observed apparent V magnitude. Conservatively,we do not consider any star for which we cannot obtain a photometric parallax in this way,for example because its color is inconsistent with the age and metallicity of the cluster; if anysuch stars is a high probability member of the moving group, this star alone rules out theopen-cluster origin of the moving group. In order to compute the photometric parallax weuse, for each moving group, the first of the isochrones mentioned in the caption of Figure 3;the results for the second set of isochrones are very similar to the ones presented below.In each of the histograms all of the main-sequence stars of the sample are plotted; theircontributions to each histogram are weighted by their probabilities of being members of themoving group in question, as calculated in § ∼ Hipparcos
10 –catalog from Table 2 of Perryman et al. (1998): We only select those stars that have afinal membership entry ‘1’, that are single, and that lie within 10 pc of the center of theHyades cluster. This procedure selects 61 Hyades members. The color–magnitude diagramof these stars is shown in Figure 5 with the 625 Myr, Z = 0 .
019 isochrone overlaid. TheHyades members hug the isochrone closely, especially in the range 0.4 mag < B − V < B − V > . p ij = 0 . B − V = 1 . p ij < .
1) and/or thered ( B − V > .
4. Strategy for the second part of this paper
Even though we have shown that the moving groups cannot be considered to be theevaporating parts of their associated open clusters, the question still remains whether theycan be considered to be some single-burst stellar population, perhaps originating from anopen cluster that has completely evaporated and thus has no presently identifiable core. Itmight be that it is merely a coincidence that the kinematically defined moving groups’ spacevelocities roughly coincide with those of prominent open clusters, while the moving groupsare actually remnants of older open clusters that are hard to identify at the present day. Wehave also failed to explain the origin of the Hercules moving group in the previous sectionbecause of the lack of an associated open cluster. Therefore, in what follows we will testthe hypothesis that the low-velocity moving groups each comprise some single-burst stellarpopulation, with an a priori unknown age and metallicity. If the moving groups fail to liveup to this hypothesis, we can confidently say that they are not remnants of inhomogeneousstar formation, but instead most likely have a dynamical origin.Hypothesis testing or model selection is at its strongest when two mutually exclusivehypotheses can be pitted against each other as opposed to merely testing whether a particularhypothesis provides a good fit to the data. Fortunately, we are in a situation here in whichthere is a well-specified background hypothesis: this hypothesis is simply that the stars inthe moving groups are nothing more than a sparse sampling of the full locally observed diskpopulation, that is, that there is nothing special about their age and chemical compositionto distinguish them from local disk stars as a whole. We are also lucky to have a non-parametric model at our disposal for this background hypothesis: this model is nothingmore than the observed local population of disk stars. Thus, we can test whether themoving groups’ photometric properties are better described by the model in which each 12 –contains just a single-burst stellar population or by the model in which each contains justthe same population as the background stars. The single-burst stellar population model canmake very tight predictions for the photometric properties of the stars while the backgroundmodel can only make very broad statements about the moving groups’ member properties.If the tight predictions play out, this will lead to clear evidence of the evaporating-clusternature of the moving groups because the photometric properties of the member stars will bemuch more probable than they are under the background hypothesis. However, if the single-burst stellar population hypothesis fails to predict the photometric properties of the movinggroup members, then the background model will be preferred. This conceptual view of themodel selection procedure which we will use in the second part of this paper is illustrated inFigure 7.Coupled with an initial mass function, the age and metallicity of a single-burst stel-lar population imply a density, or distribution, in the color–magnitude plane, and testingwhether a population of stars consists of a single stellar population is equivalent to checkingwhether the observed distribution of stars in the color–magnitude plane is consistent withthis density. This is a very strict test of the coeval hypothesis that depends on choosing, orinferring, the right initial mass function and having complete samples of stars at one’s dis-posal. A more conservative approach, which does not rely on these two assumptions, wouldbe to test whether the relation between color and absolute magnitude predicted for a coevalpopulation of a certain age and metallicity is observed in the sample. That is, rather thantesting whether the predicted density is observed in the color–magnitude plane of a sampleof stars, we test whether the predicted regression M V ( B − V ) is consistent with the data. Inpractice, we use the predicted M V ( B − V ) relation combined with the observed photometryof a star to predict a photometric parallax for the star, in exactly the way described onthe previous section. This photometric parallax is then compared to the observed parallax,taking into account the observational uncertainty on the parallax.For the hypothesis that we are interested in testing this is advisable because masssegregation and selective evaporation have been shown to affect the luminosity func-tions of open clusters, both in simulations (Aarseth & Woolf 1972; Terlevich 1987;de la Fuente Marcos 1995; Bonnell & Davies 1998)—whether primordial (Bonnell & Davies1998; Hillenbrand & Hartmann 1998) or dynamical (McMillan, Vesperini, & Portegies Zwart2007; Moeckel & Bonnell 2009; Allison et al. 2009)—as well as observationally in some ofthe open clusters associated with the moving groups studied here (Hyades: Reid 1992;Perryman et al. 1998; Reid & Hawley 1999; Dobbie et al. 2002; Bouvier et al. 2008; Pleiades:Bouvier et al. 1998; Hambly et al. 1999; Adams et al. 2001; NGC 1901: Carraro et al. 2007).There is some debate about whether mass segregation has actually been observed in massiveopen clusters (Ascenso, Alves, & Lago 2009). We can expect low-mass stars to be prefer- 13 –entially ejected from open clusters, although quantitative estimates of this effect are stillhighly uncertain. It would be hard to predict the complete two-dimensional model densityin the color–magnitude plane. However, whether or not selective evaporation plays a largerole in the evolution and evaporation of open clusters, the relation M V ( B − V ) shouldalways hold if the sample of stars originated from a single star-formation event and themodel selection test based on it will not be affected.The test will hinge on the existence of a background model that states that the stars inthe moving groups are similar to the local disk population as a whole. In the next section wewill refine this background model and put it in such a form that we can use it quantitativelyin the model selection test. That is, we will turn the bulk photometric properties of thelocal disk stars in Hipparcos into a photometric parallax relation—predicted model parallaxplus model scatter—which can be compared to the photometric parallax obtained for asingle-burst stellar population for each star.
5. The background model
Given that we have at our disposal a large number of stars with accurate photometry andparallaxes to estimate a one-dimensional photometric parallax relation, it is unlikely that anyparametric model could capture the observed relation and its scatter in all of its details. It is,therefore, advisable to use a non-parametric approach to estimate the photometric parallaxrelation and its intrinsic scatter for the background population. Principled probabilisticapproaches to this exist (e.g., using Gaussian Process regression: Rasmussen and Williams2006) but given the low-dimensional nature of the problem and the large amount of data—from Figure 2 it is clear that at most points there are at least dozens of stars with which toestimate the local relation—we can expect simpler procedures to perform adequately.To constrain the background model we use all of the 15,023 stars in our
Hipparcos sample. Strictly speaking, we are testing whether one or more of the moving groups isdistinct from the local disk population of stars in that it consists solely of stars of a narrowage and metallicity range, and therefore, including these moving group members in thebackground model mixes into the background model the stellar populations of the movinggroups. This could complicate model selection, since it will be harder to distinguish betweenthe background and the foreground models (for the purposes of this section and the next, theforeground model for each moving group is that it is a single-burst stellar population) whenthe background model is more like the foreground model than it should be. In principle weshould test each combination of moving-group/not-a-moving-group for each of the movinggroups and build background models out of stars that are not believed to be part of a single- 14 –burst stellar population in that particular model selection test. This would be impractical,not in the least because few of the stars can be confidently assigned to a specific movinggroup or even background, and making subsamples would necessarily involve making hardcuts on membership probabilities, with all of the biases that would result from that. Wetherefore use all of the stars to construct the background model and investigate each movinggroup separately in the following by testing against this background model. Given that morethan 60 percent of the stars are believed to be part of the background (see BHR) and thatthe population of moving groups taken together would presumably span some range of ageand metallicity, the effect of including real moving group members in the background sampleshould be small. It is important to note, however, that even if the moving groups significantlyaffect the background model, this will only bring the foreground and the background modelcloser together, but the foreground model should still be preferred over the background modelwhen the moving group is a single-burst stellar population.From our sample of 15,023 main-sequence stars, we construct a non-parametric photo-metric parallax relation: For each star i we take the stars in our Hipparcos sample in a smallcolor bin (see below) around star i ’s color and consider the absolute magnitudes of the starsin this color bin to represent the complete set of absolute magnitudes that a star in this colorbin could have, that is, the probability of the absolute magnitude of star i is given by p ( M V,i | ( B − V ) i ) = X j ( B − V ) j ≈ ( B − V ) i δ ( M V,i − M V,j ) , (3)where δ ( · ) is the Dirac delta function. The exact meaning and implementation of ( B − V ) j ≈ ( B − V ) i are discussed in detail below. Given this finite set of possible absolute magnitudesfor star i , we use the observed V magnitude of star i to derive a probability estimate for itsparallax π i , that is, p ( π i | V i , M V ) = δ ( π i − π [ V i , M V ]) , (4) π [ V, M V ] = 10 [( M V − V ) / , (5) p ( π i | V i , ( B − V ) i ) = X j ( B − V ) j ≈ ( B − V ) i δ ( π i − π [ V i , M V,j ]) , (6)where [ π ] = mas. To define the notion of “nearness” that is the implementation of ( B − V ) j ≈ ( B − V ) i in the expressions above we use the concept of a kernel (in this sense the methoddescribed here is similar to that of a linear smoother in non-parametric statistics; Wasserman2005). Using a kernel w ( · ; λ ) we define a distance between two colors x i ≡ ( B − V ) i and x j ≡ ( B − V ) j as w ( x i − x j ; λ ), where λ is a width parameter of the kernel, and we use thisnotion of distance to weight the contributions of the various stars in the background sample. 15 –These weights are inserted into equation (3) as follows p ( M V,i | ( B − V ) i ) = 1 W X j w ( x i − x j ; λ ) δ ( M V,i − M V,j ) , (7)where x i and x j are the colors of the stars and W is a normalization factor equal to P j w ( x i − x j ; λ ). To compare this photometric parallax with the observed trigonometric parallax π obs ,i we convolve this distribution with the observational uncertainty σ π,i , assumed Gaussian: p ( π obs ,i | σ π,i , V i , ( B − V ) i ) = 1 W X j w ( x i − x j ; λ ) N ( π obs ,i | π [ V i , M V,j ] , σ π,i ) , (8)where N ( · ) is the normalized, one-dimensional Gaussian distribution and π [ V, M V ] is givenby equation (5). The probability distributions for the observed parallax obtained in this wayare shown for a random sample of stars in Figure 8 together with the actual observed valueof the trigonometric parallax (the kernel and its width used in this figure are the optimalones for the background model as discussed below).Several considerations play a role in choosing a kernel function w ( · ; λ ). On the onehand one wants a kernel that is as smooth as possible, smoothly going from giving highweights to points that are close in color space to low weights for stars on the other side ofthe main sequence. However, it is computationally advantageous to use a kernel that hasfinite support such that in constructing the photometric parallax prediction in equation (8)only a subset of the 15,023 in the whole sample are used. For this reason, a Gaussian kernel w ( u ; λ ) = exp (cid:18) − u λ (cid:19) , (9)while smooth, is unwieldy. Therefore, we have considered the following finite-range kernels:the Tricube kernel w ( u ; λ ) = (cid:18) − (cid:16) uλ (cid:17) (cid:19) , u ≤ λ , (10)and the Epanechnikov kernel w ( u ; λ ) = (1 − (cid:16) uλ (cid:17) ) , u ≤ λ . (11)Of these, the Tricube kernel is everywhere differentiable; it combines the best of both worlds.Each of the kernels has a width parameter λ that is unknown a priori . We need to trainthe background model, i.e., establish a good value of λ . We train the background modelusing leave-one-out cross-validation (Stone 1974): for each choice of the width parameter λ on a logarithmically spaced grid in λ , we compute the probability of each of the observed 16 –parallaxes in our sample using as the training { B − V, M V } -set all of the other stars in oursample. We then multiply the probabilities thus obtained for all of the stars and take thelogarithm of this product to compute the “score” for that value of λ ; this quantity is alsoknown as the “pseudo-likelihood”. The value of λ with the highest score is the preferredvalue of λ .We computed the cross-validation score for a range of values of λ for each of the kernels;these are shown in Figure 9. It is clear that all three kernels agree on the best value of λ (keeping in mind that the Gaussian kernel has infinite range and only approaches zero for u > λ ). As expected, the resulting cross-validation score curve for the Gaussian kernel ismuch smoother than the corresponding curves for the Tricube and Epanechnikov kernels andthe maximum score for the Gaussian kernel is somewhat higher than that for the Tricubekernel, but because computations with the Gaussian kernel are much slower and the gainin performance is small, we choose the Tricube kernel for the background model. This is,again, a conservative choice, since a slightly worse background fit can only make it easierfor the foreground model to be preferred. All three kernels agree that the optimal width isapproximately λ = 0.05 mag and this is the value used in the background model.To test whether the background model with the chosen kernel parameters actually pro-vides a good fit to the data or whether it is merely the best possible fit among the possibilitiesexplored (note that we do not expect this to be the case as this is a non-parametric model),we have checked whether the background-model parallax probability distribution in equa-tion (8) is a consistent description of the parallax distribution in that all of the quantiles ofthe distribution are correct. Figure 10 shows the distribution of the quantiles of the parallaxdistribution at which the observed, trigonometric parallaxes are found. If the backgroundmodel is a good description of the observed parallaxes, then this distribution should beuniform. That is, if equation (8) correctly predicts the 95 percent confidence interval, the90 percent interval, and so on, then the background model is a good fit to the data. Thedistribution in Figure 10 is flat over most of the range between zero and one, with the onlymajor deviations near the edges of this interval, and we can therefore say that the back-ground model provides a good fit to the bulk of the data. That the background model failsfor stars at the edges of the parallax distribution is not surprising as these are rare: nearbyand faint or distant and bright stars are sparsely sampled regions of the color–magnitudediagram in a magnitude-limited sample as is clear from Figure 2. 17 –
6. The moving groups are not single-burst stellar populations
The goal of this section is to establish whether the moving groups could conceivablyarise from an evaporating cluster, or whether their stellar content is inconsistent with beingproduced by a single burst of star formation. We will fit a model of a single-burst stellarpopulation to each of the five moving groups and test whether this model is a better fit tothe moving-groups data than the background model described in the previous section. Theforeground hypothesis for the purpose of this section is therefore that the moving group ischaracterized by a single age and metallicity.Like the background model, the foreground model defines a photometric parallax rela-tion. While that defined by the background sample is a broad model, roughly consisting ofa mean photometric parallax relation and a large amount of scatter around this mean, theforeground model’s photometric parallax relation is very narrow, or informative, in that itis given by the single isochrone corresponding to an assumed age and metallicity (single inthe sense of being the unique isochrone in the Padova database), smoothed by the observa-tional uncertainty. The probability of an observed, trigonometric parallax π obs ,i assuming acertain age and metallicity Z and given the star’s color ( B − V ) i , apparent magnitude V i ,and observational uncertainty σ π,i is given by p ( π obs ,i | Age , Z, σ π,i , V i , ( B − V ) i ) = N (cid:0) π obs ,i | π [ V i , ( B − V ) i , Age , Z ] , σ π,i (cid:1) , (12)where the photometric parallax π [ V i , ( B − V ) i , Age , Z ] is derived from the isochrone absolutemagnitude as in equation (5). The absolute magnitude is derived from the isochrone byreading off the absolute magnitude along the isochrone corresponding to the assumed ageand metallicity at the star’s observed color ( B − V ) i .Equation (12) is not the whole story. First, Figures 5 and 6 show that even an opencluster itself is not perfectly fit by a single isochrone, that is, the right histogram in Figure 6is much broader than the unit variance Gaussian distribution. We find that there is 0.2 magof root variance in absolute magnitude with respect to the isochrone locus of the stars inFigure 5. We propagate this to a variance in the parallaxes of the Hyades cluster membersand add it in quadrature to the observational parallax uncertainty. The resulting photometricparallax–observed parallax comparison is also shown in Figure 6 as the dashed histogram.This distribution has close to unit variance; now the open-cluster scenario provides a good fitto the data (this procedure is somewhat equivalent to adding a small amount of unmodelednoise or “jitter”).Second, the assumption of a certain age and metallicity for a moving group is too easilyfalsified. When we observe a star that is a member of a moving group, but that has acolor that is inconsistent with that age and metallicity, e.g., because the star is too young 18 –to still be on the main sequence of an old population of stars, this combination of age andmetallicity is ruled out by the existence of this single star alone. As useful as the idea offalsification has been in epistemology, and as helpful as it could be in this case if we hadhigh probability members of the moving groups in our sample, the ease of falsification isactually problematic, since we cannot confidently assign any of the stars in our sample tomoving groups and we need to take the odds that a star is in fact a background interloperinto account. The proper way to take this interloper probability into account is to dividethe probability of a star’s properties among the foreground and background hypotheses in away that is proportional to the probability that the star is part of the moving group or not.Thus, we write the probability of the observed parallax of each star as p ( π obs ,i ) = p ( π obs ,i | foreground) p ij p ( π obs ,i | background) − p ij , (13)where p ij is the probability that star i is a member of moving group j ; see equation (2).A low-probability member of a moving group, one that is most likely not a member, hasthe bad property that it can rule out a certain age and metallicity due to its color beinginconsistent with it, since the first factor in equation (13) will be zero for any non-zero p ij and the probability of an age and metallicity of a moving group is given by Bayes’s theorem p (Age , Z |{ π obs ,i } ) ∝ p (Age , Z ) Y i p ( π obs ,i | Age , Z ) , (14)where we have implicitly assumed the other observational properties of the star (i.e., its color,apparent magnitude, and observational parallax uncertainty) in the conditional probabilities.A single star with a color that is inconsistent with the age and metallicity under investigationfor the moving group therefore rules out this age and metallicity, as it factors in a zeroprobability in the product in equation (14).Instead of just using the isochrone prediction in evaluating the probability of an observedparallax under the foreground model in equation (12), we add a small contribution from thebackground into the probability, such that the first factor in equation (13) becomes p ( π obs ,i | foreground) = (1 − α ) p ( π obs ,i | Age , Z ) + α p ( π obs ,i | background) , (15)where the background probability is given by equation (8). The parameter α is, in general,a free parameter and is a measure of the amount of background contamination. The totalforeground probability is obtained by substituting this equation into equation (13). A starwhose color is inconsistent with an assumed age and metallicity will now automatically resortto its background probability, since then p ( π obs ,i | Age , Z ) is zero and p ( π obs ,i | foreground) = [(1 − α ) p ( π obs ,i | Age , Z ) + α p ( π obs ,i | background)] p ij × p ( π obs ,i | background) − p ij = α p ij p ( π obs ,i | background) . (16) 19 –Low probability members have p ij ≈ α p ij ≈
1. This expression shows thatwhen α is a free parameter, it will be advantageous to make it large when high-probabilitymembers are inconsistent with the assumed age and metallicity, to make the fit at least asgood as the background fit.When α is allowed to take any value between zero and one, it is clear that if the fitprefers a value of α that is close to one, this will be an indication that the single-burst stellarpopulation model is not a very good fit to the moving-group data. But what value of α dowe expect if the data are consistent with the moving group having originated from a singleburst of star formation? In order to answer this question, we look at the overall propertiesof the velocity distribution. We look at the fraction of stars in one of the moving groups as afunction of a hard cut on the membership probabilities p ij to assign moving-group members.We find that a hard cut of p ij > α are given in Table 1 on the first line for each movinggroup.If we allow the fit to vary the background contamination level α and we find that thefit prefers values of α that are much larger than the value of α estimated for each movinggroup from the global analysis described above, that is a strong indication that the movinggroups are not single-burst stellar populations. This does not rule out that certain parts ofthe moving group are consistent with being created in a single burst—a preferred value of α that is close to but not equal to unity could suggest this. Therefore, we perform two fits:one in which we fix α at the value determined in the last paragraph and the other in whichwe allow α to take on any value between zero and one. In both cases we vary the age andmetallicity of the underlying isochrone on a grid in log age and metallicity space. The bestfit is then given by the combination of age, metallicity, and—if left free— α that maximizesthe probability of the foreground model in p (log Age , Z ( , α )) ∝ Y i p ( π obs ,i | log Age , Z ( , α )) , (17)where the individual conditional probabilities are given by equation (13) and the parenthesesaround α indicate that we either fix α or vary it. This is a maximum-likelihood fit or, equiv-alently but relevant in what follows, the maximum of the posterior probability distributionfor age, metallicity, and α with uniform priors on log age, metallicity, and α . The latter 20 –attitude permits marginalization over subsets of the parameters. In performing the fit, thelast factor in equation (13) is irrelevant, as it does not depend on any of the fit parameters,but in the model selection it does need to be taken into account.The results of this fit when fixing α are shown in Figure 11 for each of the five movinggroups. The logarithm of the expression in equation (17) (up to an arbitrary normalizationterm) is shown and the best-fit value of age and metallicity is indicated. These best-fitvalues are given in Table 1 on the first line for each moving group. For the moving groupswith an associated open cluster the best-fit ages are similar to those of the open clusters,but the metallicities are very different. This confirms the result from § α to be fit as well, the best-fitages and metallicities are similar to those obtained for fixed α , but the parameter α is drawnto values close to unity. The posterior distribution for the age and the metallicity, marginal-ized over α using a uniform prior on α , is shown in Figure 12; the posterior distributionfor α , likewise marginalized over the logarithm of the age and the metallicity, is shown inFigure 13. The best-fit values are listed in Table 1 on the second line for each moving group.It is clear from these results that the best-fit level of background contamination is very highfor each of the moving groups; for Hercules, the best-fit value of α is actually equal to unity.Especially in the marginalized distributions for α —our degree of belief concerning α giventhat we believe that part of the moving group was produced in a single burst of star forma-tion without caring about the age and metallicity of that event—the peak of the distributionis at large values of α and even at α = 1 for the NGC 1901 and Hyades moving groups, andin all cases much higher than the expected level of background contamination indicated bythe vertical line. This tells us that most of each moving group, if not all of it, is better fitby the background than by any single-burst stellar population.Although it is telling that the background contamination level in each moving group,if left as a free parameter in the fit, is drawn to high levels of contamination, we will takeour hypothesis testing one step further by examining which of the two hypotheses for eachmoving group, i.e., that it is an evaporating cluster or that it is merely a sparse samplingof the background population of stars, is better at predicting the properties of an externaldata set. As this external data set we use the stars in the GCS sample (Nordstr¨om et al. 21 –2004), which consists of a subset of the Hipparcos data set with additional radial velocities.We select stars that are not suspected to be giants or to be part of a binary in exactly thesame way as described in § Hipparcos catalog (van Leeuwen 2007)for all of the other properties of these stars. This sample of stars contains stars that are inthe basic
Hipparcos sample that we used before to fit the age and metallicity of the movinggroups and that we used to construct the background model as well. The trigonometricparallaxes are therefore not an entirely independent sample of parallaxes. But the GCSsample is completely external in the following sense: we can use the radial velocities fromthe GCS catalog to calculate the membership probabilities p ij for all of the stars in theGCS sample in a similar way as in equation (2) but with R i now the projection onto theline-of-sight direction. This way of assigning membership probabilities is independent ofthe way we assigned membership probabilities before, since those were calculated using thetangential velocities. It is in this sense that the GCS data set is external; in what followswe will determine whether the foreground model trained on the basic Hipparcos sampleusing tangential velocities predicts the properties of the moving group members in the GCSsample, assigned using radial velocities, better than the background model.The background model predicts the distribution of the observed parallax in equation (8).For the foreground model, specified by an age, a metallicity, and optionally a value of thebackground contamination level, the predicted distribution is given by equation (13), wherethe first factor is given by equation (15) and the membership probabilities p ij are calculatedusing the radial velocities.In Figure 14 we show figures similar in spirit to Figure 7. For a few specially selectedstars (high probability members of the Sirius moving group) we have calculated the back-ground prediction for the parallax (left panel in each row), the foreground prediction whenfixing α at the value determined from the global contamination analysis (middle panel ineach row), and the foreground prediction when fitting the background contamination (rightpanel in each row); in making these figures we chose the best-fit parameters for the Siriusmoving group from Table 1. The two stars in this figure are chosen to illustrate the modelselection and do not reflect the general trend. The first row shows an example where theforeground model performs well: the foreground model with fixed α makes a good predic-tion for the parallax of this star and, by virtue of being narrow and informative, gives ahigher probability to the observed parallax than the background model—note the differencein scale on the y -axes. The second row shows the much more common situation in which theforeground model misses completely and the observed parallax is found in the tails of thepredicted distribution; the background model performs better by virtue of being broader. 22 –We repeat this for all of the stars in the GCS sample. We only consider the 7,577 starswith colors B − V < α do matter now since these provide the integration measureon the space of properties through which we can integrate over these properties.A first thing to note is that the foreground model when fixing α , both using only thebest-fit values for the parameters as when marginalizing over the posterior distribution for theparameters of the foreground model, predicts the GCS parallaxes worse than the backgroundmodel, except in the case of the Hercules moving group. That the Hercules moving groupcould be considered a single-burst stellar population is somewhat surprising, as it is generallyregarded as the best established example of a moving group with a dynamical origin. Thepreference for the foreground model is only slight and the fact that, if left free, the backgroundcontamination parameter runs to α = 1 is strong evidence against it being an evaporatingcluster. When we let α be a free parameter, all of the foreground models perform at leastas well as the background model, although only slightly for most groups. There might bea subsample of stars in each of the moving groups that is the remnant of a cluster of stars.However, at the best-fit background contamination levels in Table 1, hardly any stars areassigned to the moving groups when using the relevant hard cut on membership probability.In the case of the Hyades, even though the foreground model only performs as well as thebackground hypothesis, the best-fit foreground model is very similar to the Hyades cluster’sproperties, such that a subset of stars in the Hyades moving group may have originatedfrom the open cluster. This is not entirely unexpected, as there must be some stars thathave already been lost to the open cluster but that still share its space motion. However,this fraction is not simply equal to the difference between the best-fit values of α in Table 1and one. In the analysis above, we did not remove open-cluster members from our sample,and, for example, 28 of the stars in our sample are confirmed Hyades members—they arepart of the sample from Perryman et al. (1998) described in §
3. These 28 stars makeup 11 percent of the expected 261 Hyades for this sample—they are all high-probabilitymembers of the Hyades moving group—comparable to the 14 percent of non-backgroundfound in the best fit to the Hyades moving group. These 28 stars were selected using astringent membership criterion and therefore we can expect the actual number of Hyades-open-cluster members present in our sample to be even higher. Thus, we find that only avery insignificant fraction—at most a few percent—of a moving group can be explained bythe evaporation of a single open cluster, in disagreement with the 15 to 40 percent, for low-and intermediate-mass stars respectively, found by Famaey et al. (2007). 23 –
7. A resonant dynamical origin of the low-velocity moving groups
Now that we have firmly established that none of the moving groups can be entirelyinterpreted as being an evaporating open cluster, we can turn to investigate possible dynam-ical origins of the moving groups. If not an evaporating moving cluster, the next a-priorimost likely explanation of the moving groups is that they are generated by one of the non-axisymmetric perturbations to the Galactic potential, e.g., by the bar or spiral arms. This isnot to say that there are no other possible explanations of the moving groups’ existence—e.g.,projection effects of partially mixed phase-space structure (Tremaine 1999)—but theoreticalwork has suggested that moving groups naturally arise in various non-axisymmetric scenar-ios. As mentioned in the Introduction, the evidence for the dynamical origin of the movinggroups has been largely circumstantial, amounting to little more than finding that the mov-ing groups display some variety of ages and metallicities. The purpose of this section is totest the hypothesis of a dynamical origin in a more specific, albeit generic, manner.We can broadly distinguish between two classes of dynamical origin for the movinggroups: those in which the moving groups are generated through steady-state non-axisymmetric perturbations, and those in which they are due to transient perturbations.This section will mostly test the former category. Steady-state perturbations such as thoseassociated with the bar or spiral structure are characterized by a pattern speed, which couldpotentially vary although this is not the case in any of the dynamical scenarios consideredin the literature so far. Since orbits have associated natural frequencies—the radial andazimuthal frequencies in the plane, or the epicycle and angular frequencies in the epicycleapproximation—strong interactions between the non-axisymmetric perturbations and thestars occur when these two sets of frequencies are commensurate, that is, when the differencebetween the perturbation’s frequency and the angular or azimuthal frequency of the orbitis commensurate with the radial frequency. This gives rise to the co-rotation resonance,where the period of the perturbation is equal to the angular period of the orbit, and theLindblad resonances, which are associated with closed orbits in the rotating frame of theperturbation that do not cross themselves (e.g., Binney & Tremaine 2008). The influenceof a weak non-axisymmetric perturbation to the overall potential is therefore most stronglyfelt at these resonances (e.g., Lynden-Bell & Kalnajs 1972). If the moving groups’ originlies in steady non-axisymmetric perturbations, we would expect the Sun’s present locationto be near one of the resonances of the non-axisymmetric structure to account for its stronginfluence on the velocity distribution.Simulations confirm this basic picture. Several simulations have shown that moving-group-like structures form near the resonances associated with the bar (e.g., Dehnen 2000;Fux 2001) or spiral structure (Quillen & Minchev 2005) or at the overlap of resonances of 24 –these two (Quillen 2003). Even though spiral arms are observed to start near the end ofthe bar in many galaxies, the pattern speeds of these two features are probably not stronglyrelated in a dynamical sense, i.e., their resonances would generically be independent of eachother because their pattern speeds are in general very different (Sellwood & Sparke 1988).Note that in order to explain the Hercules moving group as being due to the outer Lindbladresonance (OLR) of the bar and the lower velocity moving groups as being due to resonancesof the spiral structure, the Sun would have to be in a rather special location in the Galaxyto be at exactly the right spot with respect to all of these. More observational evidence foreither of these scenarios is thus needed to check that the velocity distribution is not beingoverfit.It is instructive to see what happens to the orbits of stars that are near one of theresonances to understand what properties we expect the moving-group members to have ifthey are associated with a resonance of dynamical origin. Generically, in the neighborhood ofa resonance we expect to see a bifurcation of the orbits into two families (Contopoulos 1975;Weinberg 1994; Kalnajs 1991). This bifurcation could be such that one of the families is onnearly circular orbits and the other significantly lagging with respect to the local standardof rest, as is the case near the OLR of the bar (Dehnen 2000), or it could be such that thereare no stars on nearly circular orbits any longer and one family lags with respect to the localstandard of rest, while the other family moves faster than purely circular location (in bothcases, this is at the present location of the Sun in the successful dynamical scenarios). Atazimuths where these two families cross, we expect to see streams in pairs in the velocitydistribution.Under the hypothesis of an OLR origin for the Hercules moving group, there is a familyof orbits that are anti-aligned with the bar and spend most of their time inside the OLR, andthere is a family of orbits that are aligned with the bar and spend most of its time outside theOLR (Contopoulos & Grosbøl 1989). When invoking steady-state spiral arms to explain theexistence of the Hyades/Pleiades moving groups and the Coma Berenices (which we do notstudy in this paper because it did not show up at high significance in the reconstruction of thevelocity distribution in BHR) or Sirius moving group, the Sun’s present location is near the4:1 ILR and there is one family of orbits that is elongated, square-shaped, and that spendsmost of the orbit outside of the ILR, and another family of orbits that is also elongated,square-shaped, and is more typically found inside the ILR (Contopoulos & Grosbøl 1986).Varying the parameters of the spiral structure, moving-group-like structure also forms forother types of orbits, but generically one family of orbits’ mean radius is inside the resonanceand the other family’s mean radius is outside the resonance. This is even somewhat the casewhen the moving groups are created by transient behavior of the bar—e.g., recent bar growth(Minchev et al. 2009)—although the situation is a lot messier in these cases because of the 25 –time-dependent nature of the problem.Thus, if the moving groups are particular manifestations of dynamics near the resonancesof the bar or spiral structure, then the previous argument shows that the orbits of stars inthe moving groups concern different and mostly non-overlapping regions of the Galaxy: Starsthat are part of moving groups that on average lag the local circular motion are near theirapogalacticon, so their orbits will be mostly confined to the inner Galaxy. Stars in movinggroups that are ahead of circular motion on average are near their perigalacticon, so thesestars spend most of their orbits in the outer Galaxy. Therefore, we can expect the stellarpopulations of moving groups to be different depending on their position in the v x − v y plane,as these populations of stars are born in different physical conditions. It is this hypothesisthat we will test in this section.Specifically, the hypothesis we set out to test is the following. If moving groups areassociated with a family of orbits that either spend most of the orbit inside of the Solarcircle or outside of it, then, since stars reflect the conditions of the regions in which they areborn, stars in moving groups will either have a higher than average metallicity, or lower thanaverage metallicity, because there is a metallicity gradient in the Galaxy, declining outwardfrom the Galactic Center (e.g., Shaver et al. 1983; Afflerbach, Churchwell, & Werner 1997;Nordstr¨om et al. 2004; Rudolph et al. 2006). For each moving group we can estimate thisexpected metallicity difference by calculating the mean metallicity at the mean radius of eachmoving group if it was moving in a simple axisymmetric potential. We can approximate thismean radius by the radius of the circular orbit that has the same angular momentum asthe center of each moving group. The mean radii found by assuming a flat rotation curvewith a circular velocity of 235 km s − and a distance to the Galactic center of 8.2 kpc(Bovy, Hogg, & Rix 2009) are listed in Table 3. Assuming a metallicity gradient in [Fe/H]of -0.1 dex kpc − (e.g., Mayor 1976; Nordstr¨om et al. 2004) these mean radii translate inthe expected metallicity differences of a few hundreds of a dex to a tenth of a dex forHercules in Table 3. In reality, we can expect these differences to be larger, since theresonance-trapped orbits will make larger excursions inward or outward than in this simpleaxisymmetric argument.There will be some spread around this mean value, but this spread will certainly besmaller than the width of the full thin-disk metallicity distribution, which is, as we will showbelow, about 0.14 dex. We can estimate the expected width of a moving group’s metallicitydistribution based on its velocity width by using the same procedure that we used in thelast paragraph. Using the velocity widths from BHR we find expected widths of a fewhundreds of a dex or less; these are given in Table 3. These expected widths are smallerthan the expected metallicity offset for each moving group except for NGC 1901. Therefore, 26 –we expect each moving groups’ metallicity distribution to be largely contained in either thehigher than average or lower than average part of the local thin disk metallicity distributionand this effect should be detectable.Thus, we ask for each of the moving groups whether it is better fit by a model witha higher or lower metallicity than the background model, which reflects the full metallicitydistribution in the Solar neighborhood. Since the Hipparcos sample that we have beenusing throughout this paper does not include spectroscopic metallicity information, we usea sample selected from the GCS catalog instead. We use less conservative cuts on thebinary and giant contamination of this sample to maximize the number of stars in thesample. Giant contamination is in fact very small in this sample of F and G dwarfs, and thepresence of binaries is not really an issue since the multiple radial velocity epochs availablefor all stars in the GCS data allow for an accurate determination of the mean motion,although the photometric parallax technique that we use will be slightly biased by thepresence of unresolved binaries. This affects both background and foreground models—foreground models in this section are those with low/high metallicities—and is taken care ofin the non-parametric photometric parallax relation that we will again establish (all modelseffectively use a noise-in, noise-out approach as far as unresolved binaries are concerned).As before we will train a non-parametric model to represent the background hypothesis,a non-parametric photometric parallax relation that we will establish for the GCS stars inexactly the same way that we trained the background model in §
5. We cannot re-use theprevious background model, as the GCS data are a much finer sampling in a narrow colorrange of the rich color–magnitude diagram than our previous
Hipparcos sample. Ratherthan using a parametric model for the foreground hypothesis, we will build a non-parametricmodel similar to the background model, by training it on stars that have higher or lowermetallicity than average.We construct the GCS sample used in this section as follows: From the GCS catalogwe take all of the stars that have a
Hipparcos counterpart and take their radial veloci-ties with uncertainties and their metallicities from the GCS catalog (the latest reduction;Holmberg, Nordstr¨om, & Andersen 2007, 2009). We take the rest of the spatial, kinematic,and photometric data from the
Hipparcos catalog (ESA 1997; van Leeuwen 2007). Fromthis sample we select those stars with accurate parallaxes ( π/σ π ≥ σ/ √ f thin N , where σ = 0 .
16 dex is the width of the thin diskdistribution convolved with the typical uncertainty in the GCS [Fe/H] values, N =9,575,and f thin = 0 .
9; the uncertainty in the mean is therefore about 0.002 dex. Of course, thisuncertainty does not include the uncertainty in the thin-thick disk decomposition, but thisis expected to be small.To get a first sense of the metallicities of the various moving groups, we have computedthe average metallicity of the stars in the GCS sample described in the previous section(before the color cut, but the results are the same after the color cut) by weighting the indi-vidual metallicities by the probability that the star is part of the moving group in question,i.e., h [Fe / H] i j = P i p ij [Fe / H] i P i p ij . (18)In the same way we can calculate the second moment of the metallicity distribution ofeach moving group. These average metallicities and widths are given in Table 3. All ofthe moving groups except for Sirius have higher metallicities than the average thin diskmetallicity, which we established above to be -0.13 dex. The Hyades moving group has adistinctively higher metallicity than average (see also Famaey et al. 2007, who find about thesame value from a simple cut in velocity space); for the other moving groups the differenceis smaller and it is not clear what the significance of this result is. The fact that the secondmoment of each moving group’s metallicity distribution is comparable to that of the fulllocal metallicity distribution indicates that the moving groups’ metallicity distributions areall very similar to that of the background. To test the significance of the non-zero offsetsfrom the average metallicity, we perform a simple hypothesis test to see whether the movinggroups’ metallicities are significantly different from that of the general thin disk population.We create two subsamples from the full sample of 9,575 stars by taking stars withmetallicities larger than the average thin disk metallicity, and stars with metallicities lowerthan the average; these samples contain 4,593 and 4,737 stars, respectively. That the lattersuffer from some contamination from thick disk stars does not matter for our purposes aswe are merely interested in creating a model with lower metallicities than the average thin 28 –disk.We now fit a non-parametric photometric parallax relation to each of these samples—the background model consisting of all of the stars and the two foreground models consistingonly of the low/high metallicity subsamples of the full sample—in exactly the same way as in §
5. In order to avoid an excessively spiky non-parametric model, we focus on the color region0.35 mag < B − V < p ( π obs ,i | foreground) = p ( π obs ,i | high / low Z ) p ij p ( π obs ,i | background) − p ij , (19)where we now make use of the full kinematical information for the GCS stars, since we haveall three components of the velocity to assign moving-group membership for this sample.The logarithm of the total probability under both foreground models and the backgroundmodel thus calculated is tabulated in Table 3. If a moving group shows clear signs of ahigher or lower metallicity, and thus of a resonant origin in a steady-state non-axisymmetricpotential, we would expect the moving group’s properties to be better fit by the higher/lowermetallicity subsample than by the background model. As is clear from Table 3, no movinggroup shows convincing evidence that this is the case.The only moving group that shows weak evidence that it has a different metallicitythan the background of Solar-neighborhood stars is the Hyades moving group, confirmingthe result for the Hyades found above by calculating a weighted average of the metallicitiesof Hyades members. That the Hyades moving group has a slight preference for a highermetallicity could indicate that it is associated with a family of orbits whose mean radii arewithin the Solar circle, although the evidence is very weak. This may seem like a largefactor, but one needs to consider that this is for a sample α Hyades N = 0 . × , ≈
8. Hints of recurrent spiral structure
If the solar neighborhood is currently near the ILR of the current cycle in the recurrentspiral-structure scenario of Sellwood & Kahn (1991) scenario described in §
1, we wouldexpect to see a feature in the local energy-angular-momentum distribution correspondingto stars being scattered at the ILR. Some tentative signs of this have been detected inthe distribution of
Hipparcos stars (Sellwood 2000), although this analysis made use of thereconstruction of the local velocity distribution derived from tangential velocities alone byDehnen (1998). With the full kinematical information in the GCS catalog, we can constructthe energy–angular-momentum distribution without making any symmetry assumptions,and we can ask whether any of the moving groups are actually a manifestation of the groovefeature in the angular-momentum distribution that drives spiral structure.In order to calculate the integrals of the motion of the stars in the GCS sample we 31 –need to assume a Galactic disk potential to convert positions and velocities into energy andangular momentum. We use a simple model for the disk potential, a Mestel disk (Mestel1963; Binney & Tremaine 2008), which has a flat rotation curve and is uniquely characterizedby the circular velocity; we assume a circular velocity at the Sun of V c = 235 km s − andcalculate Galactocentric distances using R = 8 . p ij > .
5, where p ij is again calculatedusing the full three-dimensional velocity vector and the level of the hard cut is set to thevalue that gives an overall fraction of stars in moving groups of about 40 percent. Theredoes not seem to be a clear scattering feature in this distribution. The Hercules movinggroup is, unsurprisingly, the only moving group that could potentially be associated with ascattering feature, but since it lies very close to the selection cutoff, it is hard to tell whetherthe Hercules moving group corresponds to a genuine scattering feature in this diagram orwhether this is just the selection cutoff.Recently, Sellwood (2010) has argued that the Hyades moving group rather than theHercules moving group corresponds to the inner-Lindblad scattering feature. This feature isnot apparent in Figure 17, since it concerns stars with an order of magnitude less randomenergy. For ease of comparison with Sellwood (2010), Figure 18 shows the GCS stars fromFigure 17 with the smallest random motions, as well as the Hyades moving-group members.It is clear from this figure that the Hyades members do indeed correspond to the weakfeature apparent in the top panel, confirming that the Hyades moving group might be atelltale of the recurrent nature of the Milky Way’s spiral structure. This explanation doesleave a few questions unanswered. The other low-velocity moving groups do not stand outin the energy–angular-momentum space. Ignoring the Hercules moving group, which canpotentially be explained by the bar, how are the other moving groups formed if they are notthe result of inhomogeneous star formation? Since the recurrent spiral structure is supposedto move inward, with the next spiral pattern’s corotation radius near the inner Lindbladradius of the previous pattern, it is unlikely that the other moving groups are the result ofscattering features associated with previous patterns since these features should be at largervalues of the angular momentum and random energy. The result that the Hyades movinggroup is created by the scattering of stars at the ILR is also slightly at odds with the highermetallicity preference for the Hyades moving group found in §
7: Since stars are scattered 32 –inward at the ILR, the Hyades stars originate at greater Galactocentric radii and shouldtherefore be, if anything, less metal-rich than average.
9. Discussion and future work
The tests and discussions above have all focused on determining the nature of the movinggroups identified in Figure 1, and we have been able to rule out and provide support forsome possible scenarios through which these moving groups may have formed. However, thegroups shown in Figure 1 have been determined as Gaussian components in a deconvolutionof the observed velocity distribution using 10 Gaussians. BHR found that the best-fittingmixture-of-Gaussians model contained only 10 components: When using more components,the velocity distribution was overfit as it became clear by testing its predictions of the externalGCS radial-velocity data set. This does not , however, constitute an endorsement that theindividual components have any physical interpretation: only the mixture itself, that is, thefull distribution, can be considered real, the individual components are just positioned insuch a way as to best describe the overall velocity distribution. It is therefore fair to askwhether the results in this paper have not been unduly influenced by our identification ofmoving groups with individual components of the mixture.In BHR, we argued that moving groups can be associated with individual componentsof the mixture for a few different reasons. The overall reconstructed velocity distributioncontains a number of distinct peaks (see Figure 1). These peaks can be unambiguouslyidentified with specific components of the mixture, and therefore we can cross-correlatestructures in the velocity distribution with the Gaussian components. Peaks, or overdensities,in the velocity distribution are what are generally called moving groups. Thus, since peaks inthe velocity distribution are what define moving groups, and these peaks can be identified asindividual components in the mixture, we can associate individual components with movinggroups. Furthermore, the peaks in the velocity distribution compare favorably with thefiducial locations of the classical moving groups that are studied in this paper, althoughthere are some small differences, such as that the Pleiades is resolved as two components,and that the Hercules moving group is both more smoothly connected to the bulk of thedistribution than is generally thought to be the case and is located at slightly lower velocitiesthan usual.The generally accepted kinematic properties of the moving groups amount to not muchmore than a rough location and an even rougher estimate of the size and orientation of themoving group. The shape of the moving groups in the direction out of the plane is rarelydiscussed, although all of the moving groups’ vertical velocities are presumably as well mixed 33 –as those of the general background population, because of the efficiency of phase mixing inthe vertical direction. Similarly, until BHR, the weight of the individual moving groups inthe velocity distribution, or even the total weight of substructure in the distribution hadnever been quantitatively determined. It is hard to make quantitative estimates of groupmembership for individual stars, especially if not all of the velocity components of the starsare measured. The locations, shapes, and relative importance that we used in this paperallow for an objective way to estimate membership probabilities for a large sample of starsfor all of the moving groups. While one can argue over whether these locations, shapes,and relative weights are exactly right for the moving groups, the objective, probabilisticprocedure that we followed in this paper should be preferred over ad hoc choices on whichto base membership assignments.We also do not expect small biases in the parameters of the moving groups to affect theconclusions of this paper very much. If the moving groups are actually located at slightlydifferent locations in velocity space, if their profiles deviate from Gaussians in the wings, orif their relative weights are slightly higher or slightly lower than that which was assumedhere, the computed membership probabilities will be somewhat wrong, but not by largefactors. That is, high probability members based on the parameters that we assumed for themoving groups will remain high probability members even for slightly different parameters.If the moving groups had shown a clear preference for an explanation of their existenceover the others in the previous sections, e.g., if they were much better fit by a single-burststellar population than by the background distribution, this conclusion would have stoodout at high significance even if we computed membership probabilities slightly wrong. Thus,the main conclusion of this paper—that no moving group shows clear evidence of havingoriginated through one of the scenarios discussed here—holds whatever you believe aboutour parameterization. The more tentative conclusions reached here, however, should beinterpreted with care.Another caveat has to do with the possibility of radial mixing playing an important rolein the chemical evolution of the Galactic disk. Radial mixing (Sellwood & Binney 2002) is theprocess in which stars can migrate radially from their birthplaces over large distances whileremaining on nearly circular orbits. Such mixing causes a wider range of birth radii to bepresent at any Galactocentric radius and can therefore weaken expected correlations between,for example, metallicity and Galactocentric radius or metallicity and age (e.g., Roˇskar et al.2008). Radial mixing occurs naturally in galactic disks with transient spiral structure—onlystars scattered at corotation can be scattered without increasing their random motion, so alarge range of frequencies needs to be present for radial mixing to occur throughout the disk—but recently it has been shown that the coupling between a steady-state bar and steady-statespiral arms can also lead to significant radial migration (Minchev & Famaey 2009). In this 34 –scenario, stars from a wide range of birth radii and metallicities can migrate radially and betrapped into the bar’s and spiral structure’s resonances, leading to a potentially significantdilution of the metallicity-offset effect we searched for in §
7. More work is necessary totest whether the resonance-overlap radial mixing is consistent with observations of the Solarneighborhood (cf. Scho¨enrich & Binney 2009) and whether the metallicity distributions ofthe moving groups created by the resonances are consistent with the results from § several open clusters. All of these alternative explanations provide a priori reasonableexplanations of the moving groups’ existence and should therefore be tested. Testing theseexplanations will be harder because the stellar content of the moving groups will have to bedetermined in greater detail than what has been done here. Theoretical work and simulationswill also have to establish the nature of the moving groups in the scenarios where they aredue to transient perturbations to allow the data on the stellar content of the moving groupsto be interpreted in terms of these models.Future work to elucidate the origin of the moving groups could go beyond thesimple tests performed here by fitting more complicated models for the chemical com-position and star-formation history of each moving group. This “chemical tagging”(Freeman & Bland-Hawthorn 2002) could lead to greater insight into the kind of stars ororbits that make up the moving groups. Fitting these more general models will be con-siderably more complicated than what has been done here. Nevertheless, the probabilisticapproach followed here in which all stars in the sample are carried through the analysis ofeach moving group with appropriate membership-probability weights—a weak cut could bedone for computational efficiency—will be essential in these more sophisticated analyses tostudy the kinematic structures that are the moving groups.
10. Conclusions
A summary of our results is the following: 35 – • We use large samples of stars extracted from the
Hipparcos and GCS catalogs tostudy the properties of the five most prominent low-velocity moving groups: the NGC 1901group, the Sirius group, the Pleiades group, the Hyades group, and the Hercules group.Using membership probabilities calculated in a probabilistic manner based on the tangentialvelocities of the stars, the radial velocities, or both, and by propagating these membershipprobabilities through our whole analysis, we are able to use the maximum number of starsin the study of each moving group—an order of magnitude improvement for most of themoving groups—and avoid any possible biases that could result from making hard cuts onmembership probabilities in analyses of this kind. • For the four moving groups in our sample with an associated open cluster, we askedwhether the moving groups could consist of stars that have evaporated from these openclusters. By comparing the parallaxes of the stars that we predict if the stars in the movinggroups have the same age and metallicity as the open cluster that the moving group isassociated with the observed trigonometric parallax, we establish that a large part of eachmoving group is poorly fit by the assumption that it has the same stellar population as theopen cluster. This establishes beyond any reasonable doubt that the moving groups are not fundamentally associated with their eponymous open clusters. • Next we studied whether each moving group could conceivably be associated with any open cluster, not necessarily the one normally associated with it. We constructed a back-ground model in which the moving group is nothing more than a sparse sampling of the localdisk population of stars and single-burst stellar population foreground models parameterizedby an age, a metallicity, and a level of background contamination. For reasonable values ofthe background contamination we find that only the Hercules moving group displays marginalevidence that it could be a remnant of a past star formation event. However, letting thelevel of background contamination run free, all of the moving groups prefer very large valuesof the contamination, reaching values close to complete contamination by the background,especially in the case of the Hercules moving group. Therefore, we can confidently concludethat none of the moving groups is a remnant of a single open cluster. • To test scenarios in which moving groups are formed as a consequence of resonancesassociated with the bar and/or spiral structure, we asked whether the moving groups arebetter fit by a model with higher than average—or lower than average—metallicity, such aswould generically be the case in resonant models for the moving groups. We find that ofall the moving groups only the Hyades moving group shows a metallicity preference, towardhigher metallicity. All of the other moving groups are best represented by the backgroundpopulation of stars, although the Sirius moving group prefers a lower than average metallicityover higher than average, which, together with the higher than average metallicity of the 36 –Hyades could be an indication of a spiral-structure-associated resonance origin for the Hyadesand Sirius moving groups. The Pleiades moving group is preferably fit by a lower than averagemetallicity rather than a higher than average metallicity, arguing against a common origin forthe Hyades and Pleiades moving groups. The Hercules moving group has a preference towardhigher metallicity, consistent with it being associated with the OLR of the bar. We stressthat all of this evidence is very tentative and the background model is the preferred modelin most cases, raising the likelihood of transient non-axisymmetric perturubation scenariosfor the origin of the moving groups. • We confirm the result of Sellwood (2010) that the Hyades moving groups might beassociated with features—grooves—in the angular momentum distribution as would be ex-pected in some models of recurrent spiral structure.It is a pleasure to thank the anonymous referee for valuable comments and MichaelAumer, Mike Blanton, Iain Murray and Sam Roweis for helpful discussions and assistance.Financial support for this project was provided by the National Aeronautics and SpaceAdministration (grant NNX08AJ48G) and the National Science Foundation (grant AST-0908357). DWH is a research fellow of the Alexander von Humboldt Foundation.
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This preprint was prepared with the AAS L A TEX macros v5.2.
43 –Table 1. Best fit single-stellar-population models for the low-velocity moving groupsGroup Age
Z α a(Myr)NGC1901........ 180 0.030 0.41NGC1901........ 56 0.030 0.98Sirius............... 350 0.026 0.53Sirius............... 413 0.023 0.90Pleiades........... 67 0.030 0.57Pleiades........... 67 0.030 0.90Hyades............. 488 0.029 0.58Hyades............. 679 0.027 0.86Hercules........... 180 0.030 0.83Hercules........... 180 0.030 1.00Note. — The first line for each grouplists the best-fit age and metallicity forthe fixed value for α in the last column—this value was obtained from a globalcontamination analysis (see the text)—the second line lists the overall best fitage, metallicity, and α . a Background contamination level. 44 –Table 2. Model selection using the GCS sample: is the single-stellar-population model forthe moving groups preferred?
Group Best-fit SSP, Marginalized SSP, Best fit SSP, Marginalized SSP,fixed α fixed α free α free α NGC1901......... -262 -262 17 17Sirius............... -61 -61 0 1Pleiades........... -70 -70 1 5Hyades............. -8 -8 0 0Hercules........... 2 2 0 15Note. — The difference between the logarithm of the probability of the parallaxes of the7,577 stars in the GCS sample used in § B − V < α . Table 3. High/low metallicity model selection: do the moving groups have higher or lower metallicities than thebackground disk population?
Group R c ( L )a Expected ∆[Fe/H]b Expected σ [Fe/H]b h ∆[Fe/H] i c σ [Fe/H]c High Metallicity Low Metallicity(kpc) (dex) (dex) (dex) (dex)NGC1901......... 8.0 0.02 0.025 0.02 0.16 -203 -144Sirius............... 8.5 -0.03 0.015 -0.03 0.15 -108 -8Pleiades........... 7.6 0.05 0.015 0.02 0.16 -43 -37Hyades............. 7.6 0.05 0.003 0.11 0.14 3 -14Hercules........... 7.2 0.10 0.040 0.01 0.17 -40 -106Note. — The difference between the logarithm of the probability of the parallaxes of the 9,330 stars in the GCS sample (0.35mag
46 –Fig. 1.— Velocity distribution in the Solar neighborhood (from Bovy, Hogg, & Roweis2009a) in the Galactic plane with the moving groups studied in this work indicated. Thedensity grayscale is linear and contours contain, from the inside outward, 2, 6, 12, 21, 33, 50,68, 80, 90, and 95 percent of the distribution. The first five of these contours are white andsomewhat blended together; 50 percent of the distribution is contained within the innermostdark contour. The origin in each of these plots is at the Solar velocity; the velocity of theLocal Standard of Rest (Hogg et al. 2005) is indicated by a triangle. 47 –Fig. 2.— Color–magnitude diagram of the full
Hipparcos sample of 19,631 stars, selected tobe kinematically unbiased and consist of single stars with relative parallax uncertainties .
10 percent. The 15,023 main-sequence stars that we use in the hypothesis tests in §§
3, 5, and6 lie between the gray lines. M
Hip is the absolute magnitude in
Hipparcos ’ own passband. 48 –Fig. 3.— Color–magnitude diagrams of the six moving groups detected in BHR. The pointsare grayscale-coded with the probability of each star to be part of the moving group (see thetext); only stars that have a probability larger than 0.1 of being part of the moving group areplotted. For those moving groups potentially associated with an open cluster, theoreticalisochrones (Marigo et al. 2008; Bertelli et al. 1994) for the open cluster are overlaid: the400 Myr, Z = 0 .
016 (Carraro et al. 2007) and the 600 Myr, Z = 0 .
016 (Pavani et al. 2001)isochrone for the NGC 1901 cluster; the 300 Myr, Z = 0 .
016 (Soderblom & Mayor 1993) andthe 500 Myr, Z = 0 .
016 (King et al. 2003) isochrone for the Ursa Major (Sirius) cluster; the100 Myr, Z = 0 .
018 (Boesgaard & Friel 1990; Gratton 2000) and the 100 Myr, Z = 0 . Z = 0 .
026 isochrone (Perryman et al. 1998) and the 625 Myr, Z = 0 .
019 isochrone for theHyades cluster. The main-sequence cuts from Figure 2 are indicated in gray. 49 –Fig. 4.— Observed parallaxes vs. model parallaxes assuming a single-burst stellar popu-lation identical to that of the associated cluster of the moving groups: comparison of thedistribution of observed parallaxes ( dashed lines ) with that of the model parallaxes ( solidlines ) in the left figure of each panel; histogram of the normalized difference between modeland observed parallax in the right figure. Each star is weighted by its probability of beingpart of the moving group in question. The isochrone used in this figure corresponds to thefirst age and metallicity pair mentioned in the caption of Figure 3 for each open cluster. 50 –Fig. 5.— Color–magnitude diagram of the Hyades cluster with the 625 Myr, Z = 0 . y -axis represents the probability of measuring the valueon the x -axis for a foreground model ( thin, black curve ) and a background model ( thick,gray curve ). The foreground model makes very informative predictions while the broaderbackground model makes less informative predictions. Therefore, when both the foregroundmodel and the background model predict the right observed value ( vertical line ) the observedvalue has a larger probability for the foreground model ( left panel ); when the foregroundmodel fails to predict the observed value, the observed value is more probable under thebackground model right panel ). 53 –Fig. 8.— Background model predictions for the parallax of 9 random stars in the basic Hipparcos sample. The background model consists of a linear smoother with a Tricubekernel with width parameter λ = 0.05. In each panel the background model has beenconvolved with the observational parallax uncertainty. The observed parallax ( thick, blackline ) as well as 95 percent confidence regions ( thin, gray lines ) are indicated. 54 –Fig. 9.— Selection of the width parameter λ of the kernel used in the kernel-regressionbackground model. 55 –Fig. 10.— Distribution of the quantiles at which the observed parallax is found of thebackground-model predictive distribution for the parallax. This curve should be flat forperfectly consistent predictive distributions—meaning that they correctly predict all of thequantiles of the distribution. 56 –Fig. 11.— Logarithm of the likelihood of different single-burst stellar population modelscharacterized by an age and metallicity Z for the low-velocity moving groups, with thebackground contamination level α for each group set to the value obtained from a globalcontamination analysis (see the text). The best-fit model is indicated by a white cross. 57 –Fig. 12.— Logarithm of the likelihood of different single stellar population models character-ized by an age and metallicity Z for the low-velocity moving groups, marginalized over thebackground contamination level α with a uniform prior on α . The best-fit model is indicatedby a white cross. 58 –Fig. 13.— Posterior distribution for the background contamination level α for each of themoving groups, marginalized over age and metallicity of the foreground model with uniformpriors on the metallicity and the logarithm of the age. Total contamination— α = 1—ispreferred in most cases. The value of α obtained from a global contamination analysis—thevalue used in Figure 11—is indicated by the vertical line. 59 –Fig. 14.— Model selection using the GCS sample: the background model prediction for twoindividual stellar parallaxes in the GCS sample is contrasted with the best-fit foregroundsingle-burst stellar population model for a fixed value of the background contamination α and the best-fit value for α for the Sirius moving group (see Table 1 for the details ofthese best fit single-burst stellar populations). The foreground models are trained usingprobabilistic moving-group assignments from the Hipparcos tangential velocities, while theGCS radial velocity is used to probabilistically assign the two GCS stars featured in thisfigure to moving groups. The top row shows an example where the informative foregroundprediction does better than the broad background model prediction; the bottom row shows anexample where the narrow foreground prediction is wrong and the uninformative backgroundpredictions performs better. In each panel the probability of the parallax π is conditional onthe star’s positional, kinematic, and photometric data (except for the observed trigonometricparallax). 60 –Fig. 15.— Color–magnitude diagram of the magnitude-limited GCS sample of 9,575 starsused in §§ .
10 percent. 61 –Fig. 16.— Metallicity distribution in the Solar neighborhood: the distribution of metallicitiesof 9,575 in the GCS sample. The best-fit two-Gaussian decomposition is overlaid: the twocomponents as the dashed lines (the “thin” disk component has been scaled down for clarity)and the resulting distribution as the full line. The parameters for the best fit two-Gaussiandistribution are given in the top-left corner as mean ± standard deviation of each component. 62 –Fig. 17.— Lindblad diagram: Distribution of the GCS stars in energy–angular momentumspace assuming a Mestel disk model for the Galaxy with circular velocity of 235 km s − and R = 8 . E c ≡ E c ( L ) is the energy of a circular orbit with angular momentum L , L c ( R ) is the angular momentum of the circular orbit going through the Sun’s presentlocation. The location of high probability ( p ij > ..