and Cosmology

2. The Milky Way as a Galaxy
38
ric parallax could be extended to stars up to distances
of ∼ 300 pc. The satellite GAIA, the successor mission
to HIPPARCOS, is scheduled to be launched in
2012. GAIA will compile a catalog of ∼ 10 9 stars up
to V ≈ 20 in four broad-band and eleven narrow-band
filters. It will measure parallaxes for these stars with
an accuracy of ∼ 2 × 10 −4 arcsec, with the accuracy
for brighter stars even being considerably better. GAIA
will thus determine the distances for ∼ 2 × 10 8 stars
with a precision of 10%, and tangential velocities (see
next section) with a precision of better than 3 km/s.
The trigonometric parallax method forms the basis
of nearly all distance determinations owing to its purely
geometrical nature. For example, using this method the
distances to nearby stars have been determined, allowing
the production of the Hertzsprung–Russell diagram
(see Appendix B.2). Hence, all distance measures that
are based on the properties of stars, such as will be
described below, are calibrated by the trigonometric
parallax.
2.2.2 Proper Motions
Stars are moving relative to us or, more precisely, relative
to the Sun. To study the kinematics of the Milky
Way we need to be able to measure the velocities of
stars. The radial component v r of the velocity is easily
obtained from the Doppler shift of spectral lines,
v r = Δλ
λ 0
c , (2.5)
where λ 0 is the rest-frame wavelength of an atomic
transition and Δλ = λ obs − λ 0 the Doppler shift of the
wavelength due to the radial velocity of the source. The
sign of the radial velocity is defined such that v r > 0
corresponds to a motion away from us, i.e., to a redshift
of spectral lines.
In contrast, the determination of the other two velocity
components is much more difficult. The tangential
component, v t , of the velocity can be obtained from the
proper motion of an object. In addition to the motion
caused by the parallax, stars also change their positions
on the sphere as a function of time because of
the transverse component of their velocity relative to
the Sun. The proper motion μ is thus an angular velocity,
e.g., measured in milliarcseconds per year (mas/yr).
This angular velocity is linked to the tangential velocity
component via
( )( )
v t
D μ
v t = Dμ or
km/s = 4.74 1pc 1 ′′ .
/yr
(2.6)
Therefore, one can calculate the tangential velocity from
the proper motion and the distance. If the latter is derived
from the trigonometric parallax, (2.6) and (2.4) can be
combined to yield
( )
v t
μ ( p
) −1
km/s = 4.74 . (2.7)
1 ′′ /yr 1 ′′
HIPPARCOS measured proper motions for ∼ 10 5 stars
with an accuracy of up to a few mas/yr; however, they
can be translated into physical velocities only if their
distance is known.
Of course, the proper motion has two components,
corresponding to the absolute value of the angular velocity
and its direction on the sphere. Together with v r
this determines the three-dimensional velocity vector.
Correcting for the known velocity of the Earth around
the Sun, one can then compute the velocity vector v
of the star relative to the Sun, called the heliocentric
velocity.
2.2.3 Moving Cluster Parallax
The stars in an (open) star cluster all have a very similar
spatial velocity. This implies that their proper motion
vectors should be similar. To what extent the proper
motions are aligned depends on the angular extent of the
star cluster on the sphere. Like two railway tracks that
run parallel but do not appear parallel to us, the vectors
of proper motions in a star cluster also do not appear
parallel. They are directed towards a convergence point,
as depicted in Fig. 2.4. We shall demonstrate next how
to use this effect to determine the distance to a star
cluster.
We consider a star cluster and assume that all stars
have the same spatial velocity v. The position of the i-th
star as a function of time is then described by
r i (t) = r i + vt , (2.8)
where r i is the current position if we identify the origin
of time, t = 0, with “today”. The direction of a star