1 1 2 3 4

J.J. Binney , W. Dehnen , N. Houk , C.A. Murray , M.J. Penston


Theoretical Physics, Keble Road, Oxford OX1 3NP,UK


Dept. of , UniversityofMichigan, 1041 Dennison Building, Ann Arb or, MI 48109-1090, USA


12 Derwent Road, Eastb ourne, Sussex BN20 7PH, UK


Royal Greenwich Observatory, Madingley Road, Cambridge, CB3 0EZ, UK


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

that placed them within 80 pc of the . 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


of the variance S within each bin around the solar


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


of S on sp ectral typ e emerges with exceptional clar-


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-


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 . 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 varies system-


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  < 26 . All

cesses see Binney & Lacey 1988 and Jenkins 1992.

stars of class IV-V and V were identi ed in


randomly chosen 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.


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


this star b e denoted by d and  , resp ectively. Then



 is related to the space velo cities v and v of the



k th star and the Sun by:

Figure 2. The distribution with respect to of

of the 5610 stars in the sample upper his-

d  = A  v v 1

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


lie within 80 pc of the Sun. The great ma jority of

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


its variance can b e estimated as:

Hipparcos measured the parallaxes and prop er mo-

tions of the 6840 programme stars and showed that


v = hAi hdi 3

only 3091 of these stars actually lie within 80 pc|


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


where S Equation 2 is evaluated at v from 3.

the 5612 stars that have distances accurate to b et-


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



2 4

2 1

class V stars. Figure 1 shows the distribution of their

d+Av V S =N d+Av



again using v from 3.



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-



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


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-


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


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


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


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. Jenkins

Since our eld has nite extent and the velo city ellip-

1992. Hence velo city disp ersion re ects age, and de-

soid is probably not p erfectly aligned with the x; y ; z

creases as one moves blueward from the discontinuity


through ever younger stellar groups, while remaining

frame, S will in reality b e a slightly di erent linear

constant with mean age redward of the discontinuity.

combination of velo city moments. But Equation 7


The discontinuity itself should o ccur at the colour for

gives a useful idea of what S is measuring.

which the main-sequence lifetime of a star equals the

age of the Galactic disk. However, since stars change

Before applying our metho d, we have corrected the

colour during their life on the , detailed

observed prop er motions for galactic rotation using

mo delling of stellar p opulations is necessary, to infer

Feast & Whitelo ck's 1997 values for Oort's con-

the age of the stellar disk from this datum.

stants. In the solution for v Equation 3 wehave

excluded outlying stars by an iterative pro cess: at

each solution we excluded any star that would have

2 2

contributed to S more than  times the previously


obtained value of S , where  ranged from 2.5 to 4.

5. THE SUN'S VELOCITY W.R.T. THE LSR The results obtained prove to be only weakly sen-

sitive to the value of . Below we cite results for

 = 4, when 12 stars of 5610 are excluded against an

exp ected numb er of 0.35 for a Gaussian distribution


Figure 4 is a plot of U , V and W versus S . This


of contributions to S .


clearly shows the linear dep endence of V on S that

is predicted by Stromb erg's asymmetric drift equa-

Figure 3 shows the values of S and the comp onents

tion e.g. Equation 4-34 of Binney & Tremaine 1987.

^ ^ ^

U v  x, V v  y and W v  z of the so-


That is, V increases systematically with S b ecause

lar motion for our stars when they are group ed by

the larger a stellar group's velo city disp ersion is, the

B V colour. U and W do not vary signi cantly

more slowly it rotates ab out the Galactic centre and

between groups, while b oth V and S increase sys-

the faster the Sun moves with resp ect to its lagging

tematically from early to late sp ectral typ es. The

frame. The velo cityU ;V ;W  of the Sun with re-

0 0 0

p oints for S displayvery b eautifully Parenago's dis-

sp ect to the lo cal standard of rest the velo city of

continuity: around B V ' 0:62 there is an abrupt

the closed orbit in the plane that passes through the

change in gradient from a strongly p ositive value

lo cation of the Sun may be read o from Figure 4

to ab out zero Parenago 1950. The same discon-


by extrap olating any trends of U , V and W with S tinuity is visible, though less clearly, in the data



Figure4. The dependence of U; V ; W on S . The dotted lines correspond to the linear relation tted V  or the mean

values U and W .

stars that lie south of  = 26 . When backto S =0. We nd:

the stars are group ed by B V , the random velo cities


U =11:00:6 km s

within each group increase linearly from B V ' 0:4



to B V ' 0:62 and then are essentially indep endent


V = 5:3  1:7 km s



of B V redward of B V ' 0:62. Thus these data

W = 7:0  0:6 km s


displayvery cleanly the classical Parenago disconti-

nuity, which arises b ecause the random velo cities of

The quoted errors on U and W are smaller than

0 0

stars increase steadily with time.

that on V , b ecause we have assumed that U and


W are indep endent of B V such that no binning

in colour was made. The signs of the comp onents

are such that the Sun is moving towards the Galactic

The values U and W of the solar motion relativeto

centre, towards the North Galactic p ole and currently

stars of di erent colours show no signi cant trends.

lies inside the guiding centre of its epicyclic orbit.

Our mean value of W is in go o d agreement with ear-

lier determinations of the Sun's velo city with resp ect

The solar velo city Equation 8 relative to the LSR

to the Lo cal Standard of Rest, while our mean value

may b e compared with the classical value of Delhaye

of U is marginally larger than Delhaye's 1965 mean

1965 that is used in many studies: U ;V ;W =

value. Our values of V increase linearly with the

0 0 0



9; 12; 7 km s . Clearly our value W do es not dif-

mean-square random velo cities S of the groups, just


fer signi cantly from Delhaye's. Our values of U

as Stromb erg's asymmetric drift equation predicts.


and V di er from Delhaye's values by3 and 4 , re-

Extrap olating the dep endence of V up on S to S =0


sp ectively. Delhaye's result may have b een a ected

we deriveavalue of the Sun's tangential velo city with


by biases towards high prop er motions in the cata-

resp ect to the LSR, V = 5:2  1:7 km s that is


logues he employed|such biases tend to exaggerate

smaller than Delhaye's classical 1965 value but in

V . Several more recent determinations have, like us,


b etter agreement with several recent studies.

derived smaller values of V Mayor 1974, G omez &


Mennessier 1977, Delhaye 1982, Oblak 1983, Bien-

aym e & Sechaud 1997.

Our analysis is based on the very small subset of Hip-

parcos Catalogue to which we had privileged early

access. When the catalogue is published in mid 1997


it will b e p ossible to p erform a much more satisfac-

tory analysis. In particular, with all-sky coverage it

will b e p ossible to determine the shap e and, p ossibly,

the orientation of the velo city ellipsoid. We exp ect We have used prop er motions and parallaxes from

to rep ort the results of such an analysis in the near the Hipparcos satellite to study the kinematics of

future. a kinematically unbiased sample of main-sequence



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