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which is in accord with our discussion prior to (4.8) wherein the parsec was defined as the distance at which the parallax is 1.00". Traditionally, astronomers define "parallax" n = 0par (arcsec) so that n(arcsec) = [r(pc)]-1. Thus if one has n = 0.10", the distance is r = 10 pc.

Traditional trigonometric parallax with ground-based telescopes has good precision out to a distance of ~300 LY. Here the angular displacements become so small

(0.01" at 100 pc) that the errors become significant. The centroids of stellar images of size ~ 1" must be measured to these precisions. Typically ~40 images taken over a period of ~5 years will be measured. Satellite observatories do much better. The Hipparcos satellite (launched 1989) was dedicated solely to astrometry and yielded ~105 stellar positions more precise than 1 milliarcsec (mas). The Hubble Space Telescope carries out limited astrometry to accuracies of ~0.5 mas. At 1 mas, one reaches to about 3000 LY.

Note that the stellar distances are obtained in units of the sun-earth distance (1 AU). Thus the determination of the magnitude of the astronomical unit (see above) could be called the first step on the distance ladder, and the distances to other stars with trigonometric parallax would be the second step.

Distances to open clusters

The distribution of stars on the sky shows a number of open clusters; these are groups of stars that presumably formed together from the same collapsing cloud at a given location within the Galaxy at some time in the past. A typical open cluster might contain ~100 cataloged stars and be ~20 LY in diameter. Such groups eventually disperse due to the random motions of the individual stars. The open clusters now in the sky are significantly younger than the Galaxy; they have not yet had time to disperse. It turns out that a number of these clusters are moving at a significant speed through the Galaxy, relative to the sun. It is this feature that makes it possible to obtain a distance to a cluster by purely geometric means called the moving cluster method, which we now describe.

Convergence

The motion of the cluster as a whole typically turns out to be much greater than the motions of individual stars within the cluster. Thus one can visualize a cluster of individual stars moving through space as a group, each with the velocity of the cluster. If the radial portion of the motion is such that the cluster is receding from us, the cluster will become smaller and smaller in angular size as time progresses. The azimuthal and radial motions of the cluster cause the tracks of the individual stars on the sky (as seen by the observer, Fig. 3a) to eventually converge to a single point, just as two rails of a railroad track converge toward a single point at great distances. This is strictly true only for our ideal picture of stars motionless with respect to the cluster.

The celestial locations of selected stars of the Hyades cluster at time t1 are at the tails of arrows in Fig. 3a; star A is an example. The arrows indicate the apparent directions and angular speeds of motion d9/dt. This motion, the proper motion (Section 4.3), is observed from the comparison of sky images taken at

(a) Observer's view of the sky; time 11

Hyades cluster

Star A propei- time t1 motion 1 / d6/di 1

Star A time t2

Star A time t3

(a) Observer's view of the sky; time 11

Hyades cluster

Star A propei- time t1 motion 1 / d6/di 1

Star A time t2

Star A time t3

Right Ascension a

Right Ascension a

(b) Views normal to line of sight and to star vector velocity v

Time t1

Direction of 6 velocity vector \ ^ \

Time t2 v

Time t1

Star A

Star A

Figure 9.3. The moving cluster method of obtaining the distance to an open cluster. The motion of the stars relative to the center of mass of the cluster is assumed to be small. (a) The individual stars of the Hyades cluster plotted on the sky with the directions and relative magnitudes of their proper motion shown (arrows) as derived from data from 1908 to 1954. All of the stars appear to be headed toward the convergence point at a « 6 h and 8 « +9°. The angle 01 for star A at time t1 is shown. Anticipated positions of star A at times t2 and t3 are also shown. (b) Sketches in plane of view direction and velocity vector of star showing velocity of star A at three times. The angular size of the cluster and the angle 0 both decrease as time progresses. [(a) After J. A. Pearce, PASP 67, 23 (1955) and in O. Struve, B. Lynds, and H. Pillans, Elementary Astronomy, © Oxford University Press, 1959]

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