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. 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
2
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
2
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 main sequence, 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
2
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
2
Figure 4 is a plot of U , V and W versus S . This
2
of contributions to S .
2
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-
2
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 city U ;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-
2
by extrap olating any trends of U , V and W with S tinuity is visible, though less clearly, in the data
476
2
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 declination = 26 . When backto S =0. We nd:
the stars are group ed by B V , the random velo cities
1
U =11:00:6 km s
within each group increase linearly from B V ' 0:4
0
1
to B V ' 0:62 and then are essentially indep endent
8
V = 5:3 1:7 km s
0
1
of B V redward of B V ' 0:62. Thus these data
W = 7:0 0:6 km s
0
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
0
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
1
2
9; 12; 7 km s . Clearly our value W do es not dif-
mean-square random velo cities S of the groups, just
0
fer signi cantly from Delhaye's. Our values of U
as Stromb erg's asymmetric drift equation predicts.
0
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
0
sp ectively. Delhaye's result may have b een a ected
we deriveavalue of the Sun's tangential velo city with
1
by biases towards high prop er motions in the cata-
resp ect to the LSR, V = 5:2 1:7 km s that is
0
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,
0
b etter agreement with several recent studies.
derived smaller values of V Mayor 1974, G omez &
0
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
6. CONCLUSIONS
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
477
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