473 THE KINEMATICS OF MAIN-SEQUENCE STARS FROM HIPPARCOS DATA 1 1 2 3 4 J.J. Binney , W. Dehnen , N. Houk , C.A. Murray , M.J. Penston 1 Theoretical Physics, Keble Road, Oxford OX1 3NP,UK 2 Dept. of Astronomy, UniversityofMichigan, 1041 Dennison Building, Ann Arb or, MI 48109-1090, USA 3 12 Derwent Road, Eastb ourne, Sussex BN20 7PH, UK 4 Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 0EZ, UK ABSTRACT We analyze a kinematically unbiased sample of 5610 stars around the south celestial p ole that i haveMK sp ectral typ es in the Michigan catalogues with lumi- nosity class V and ii had photometric parallaxes that placed them within 80 pc of the Sun. We bin the stars by B V and determine for each bin the so- lar motion from prop er motions alone. As exp ected, the U and W comp onents of the derived solar mo- tions do not vary signi cantly from bin to bin, while the V comp onentvaries systematically. As the clas- sic Stromb erg relation predicts, V is a linear function 2 of the variance S within each bin around the solar 2 motion. Extrap olating V S toS =0 we determine the solar motion with resp ect to the LSR, obtaining a signi cantly smaller value of V than is usually em- ployed. Parenago's discontinuityin the dep endence 2 of S on sp ectral typ e emerges with exceptional clar- ity. Figure1. The distribution with respect to distance of the 5610 objects in the sample. 2384 of these have a distance > 80 pc. Key words: Stars: main-sequence; Galactic Struc- ture: solar motion. The Hipparcos satellite provides an imp ortant opp or- tunity to re-examine the fundamental data of solar- 1. INTRODUCTION neighb ourho o d kinematics by providing the rst all- sky catalogue of absolute parallaxes and prop er mo- The kinematics of stars near the Sun has long b een tions. From the Hipparcos catalogue we can, more- known to provide crucial information regarding b oth over, extract samples of ob jects that are completely the structure and the evolution of the Milky Way. free of the kinematic biases that have plagued simi- Karl Schwarzschild 1908 already interpreted the lar studies in the past. The present pap er rep orts the distribution of random velo cities as forming a tri- analysis of one such kinematically unbiased sample. axial `velo city ellipsoid', which Oort, B. Lindblad Other asp ects are analysed by Murray et al 1997 and Stromb erg were able to relate to the large-scale and Houk et al. 1997. structure of the disk. In the early 1950s Pare- nago 1950, Nancy Roman 1950, 1952 and oth- ers p ointed out that stellar kinematics varies system- 2. THE SAMPLE atically with stellar typ e, in the sense that groups of stars that are on average younger have smaller velo city disp ersions and larger mean Galactic rota- tion velo cities than older stellar groups. Spitzer & The sample is based on the rst three volumes of Schwarzschild 1953, Barbanis & Woltjer 1967 and the Michigan Sp ectral Catalogue Houk & Cowley Wielen 1977 explained these correlations in terms 1975, Houk 1978, Houk 1982, which provide two- of the di usion of stars through phase space as the dimensional MK classi cations of stars in the Henry Galactic disk ages|for recent studies of these pro- Drap er Catalogue with declinations < 26 . All cesses see Binney & Lacey 1988 and Jenkins 1992. stars of luminosity class IV-V and V were identi ed in 474 randomly chosen star has a radial velo city increases strongly with its prop er motion b ecause high-prop er motion stars are preferentially put on radial-velo city programmes. In the absence of precise knowledge of how stars with radial velo cities have b een selected, it is dangerous to employ any radial velo cities in a study such as this. Thus, we feel obliged to discard the radial velo cities that wedohave and to work with prop er motions alone. The unsatisfactoriness of this state of a airs cannot b e to o strongly emphasized. 4. THE SOLAR MOTION Let the direction to the k th star b e given by the unit ^ vector r and let the distance and prop er motion of k this star b e denoted by d and , resp ectively. Then k k is related to the space velo cities v and v of the k k k th star and the Sun by: Figure 2. The distribution with respect to magnitude of proper motion of the 5610 stars in the sample upper his- d = A v v 1 k k k k togram and of the 1072 sample stars for which radial velocities are available shaded. ^ ^ where the matrix A I r r pro jects velo cities k k k onto the plane of the sky. The solar velo city relative to any given group of stars is de ned to b e the value the Michigan Catalogue that were judged, from their of v that minimizes: apparent magnitudes and sp ectral classi cations, to E E D D lie within 80 pc of the Sun. The great ma jority of 2 2 2 d + A v 2 A v = S these stars were brighter than m = 10 see Figure 2 of Houk et al. 1997, and at m = 9 Hipparcos p osi- tions havea typical error of 1 mas. Hence Hipparcos where hi denotes sample average. Indeed, in this could determine distances with negligible error for frame we have hA v i = 0, i.e. the stars have zero any programme star that actually lies within 80 p c. mean velo city. The solar motion in this frame and 1 its variance can b e estimated as: Hipparcos measured the parallaxes and prop er mo- tions of the 6840 programme stars and showed that 1 v = hAi hdi 3 only 3091 of these stars actually lie within 80 pc| 1 1 2 errors in photometric distance mo duli resulted in V v =N hAi S 4 roughly half the programme stars lying beyond the 80 pc cuto Murray et al. 1997. All but two of 2 where S Equation 2 is evaluated at v from 3. the 5612 stars that have distances accurate to b et- 2 The variance of S is estimated as: ter than 20 per cent b elong to luminosity class V. In this study we analyse the kinematics of these 5610 E E D D 2 2 4 2 1 class V stars. Figure 1 shows the distribution of their d+Av V S =N d+Av distances. 5 again using v from 3. 3. RADIAL VELOCITIES 2 The minimum value of S is a measure of the velo city disp ersion of the group. In fact, in the frame de ned by Equation 3: For many studies it is of the greatest imp ortance to have space velo cities rather than merely the two com- 2 2 2 ^ 6 v r S = jv j p onents of a star's velo city on the sky. For this rea- son it was early on hop ed that the Hipparcos mis- ^ so if the direction of the line of sight, r , did not sion would b e accompanied byaninternational pro- k 2 gramme to determine the radial velo cities of stars in vary signi cantly from star to star, S would b e the the Hipparcos catalogue and weawait with interest mean-square random velo city of the stars less the ^ the publication of the results of the two survey pro- mean-square velo city parallel to r . Consider the k grammes with the Coravel sp ectrometers Mayor et case in which the velo city ellipsoid of the stars is al. 1997. For this work wewere able to nd a radial aligned with the frame in which the x axis p oints velo city for only one in ve of our stars in either the towards the Galactic centre, the y axis p oints in the Hipparcos Input Catalogue Turon et al. 1993 or the direction of Galactic rotation and the z axis p oints catalogues of Barbier-Brossat et al. 1994 and Du- towards the north Galactic p ole. Then at the cen- 2 ot et al. 1995 that are available in the CDS archive ^ tre of our eld, v r has an exp ectation value of in Strasb ourg. Figure 2 shows the distribution over 1 prop er motion of b oth stars in our sample upp er The contribution of uncertainties in distance and prop er histogram and the 1072 stars with radial velo cities motion of the individual stars in the sample is neglible com- pared to the Poisson noise due to the nite sample size. shaded histogram. Clearly the probability that a 475 Figure3. The variation with colour B{V of the components of solar motion U; V ; W and of the dispersion S . 2 2 2 2 0:23 +0:55 +0:21 , where is the mean- for V . Parenago's discontinuity is thought to arise x y z i from the fact that the mean age of stars decreases square comp onentof random velo city along the ith as one moves blueward of the discontinuity, while it axis. Substituting this into Equation 6 wehave: is indep endent of colour redward of the discontinu- 2 2 2 2 ity: scattering pro cesses cause the random velo cities S =0:77 +0:45 +0:79 7 x y z of stars to increase steadily with age e.g.