Adapted from an exercise for Astronomy 200, University of Victoria, Victoria, BC, Canada.
Summary
The student finds the distance to the Hyades star cluster using the convergent point method.

Background and Theory
The zero-point for the entire cosmic distance scale, out to the farthest reaches of the observable universe, has been based upon the distance to Hyades star cluster. It is a bit unsettling to think that even over the past decade, distance determinations to this cluster have varied by as much as 15%. The problem? From ground-based telescopes, reliable parallaxes for individual stars can be measured out to about 40 parsecs. At 40-50 parsecs, the Hyades cluster is just beyond this distance. As a result, other (often ingenious) methods have been required to find the distance to this star cluster.

Most distance determinations used for the Hyades have either been based upon the convergent point method or have been judged according to their agreement or disagreement with it. The geometry used is relatively simple, but before we discuss how to use the method, we must define and explain some important concepts.

The positions of stars are not really fixed. They only seem that way because of our crude measuring instruments or our limited observational time period (a few hundred years, at most). We can determine their motion in space by measuring two quantities: the proper motion and the radial velocity.

1. Proper Motion: Proper motion (usually denoted by the Greek letter "mu": ) is the angular change in position of a star across our line of sight, measured in arc seconds per year. For stars visible to the naked eye, the average proper motion is around 0.1 arcsec per year (remember how small an arc second is? A tennis ball 8 miles away is an arcsecond in diameter). The largest proper motion award goes to Barnard's star: a whopping 10.27 arcsec per year. Proper motion is generally measured by taking photographs several years apart and measuring the movement of the image of a star with respect to more distant background stars over that time period. Usually decades must elapse between successive photographs before a reliable measurement can be made. The proper motion of a star results from its "transverse velocity", or the velocity across the sky (see Fig. 1 below).
2. Radial Velocity: The absorption lines in a stellar spectrum will be shifted in wavelength if the star is moving towards or away from us (see Fig. 1). This shift in wavelength (or frequency) is called the Doppler shift, and is analogous to the pitch of sound waves coming from a rapidly approaching and then receding siren. The radial velocity of a star is remarkably easy to measure: we just take a spectrogram of a star and measure the displacement, , of a spectral line from its expected wavelength, . The radial velocity, v is given by:

where c is the speed of light.

Ultimately, what we want to determine is the distance to the star. We can measure the proper motion and radial velocity of that star. But the proper motion is not an actual transverse velocity, but is instead an angular velocity. Just as the velocity of a plane in the sky can not be determined unless we know the distance to the plane, so proper motion can be converted to transverse velocity only if the distance to the star is known. But the distance is what we wish to find! We need a "trick".

For stars that are too far away to measure a reliable parallax, we can apply a different geometrical method if those stars belong to a close cluster. In this case, we turn the problem inside out, and solve for the distance from an actual velocity and a proper motion.

If a group of stars is moving together (as happens in a cluster), we can sketch the motion of each star in space as shown in Figure 2. Just as two parallel railroad tracks appear to converge in the distance, so also will parallel star paths. This point of convergence is determined on a chart of the sky by simply extending the lines of proper motion of each star, and finding their point of intersection. The angle of sight between a star and the convergent point is measured on the chart, or in the sky. This angle is denoted by the greek letter "theta":

The important point is that this angle, , is equal to the angle between the true space velocity of the star, v, and its radial velocity Vr. We measure the radial velocity, vr, directly. We can solve the velocity right triangle for the transverse velocity:

The geometry of this method is shown in Figure 3. Make sure you understand the geometry and why this works.

We now have the transverse velocity, vt and the proper motion, m, and can calculate the distance D (in parsecs) by using the equation:

If this procedure is carried out for many stars in a cluster, an average of the distances calculated will be a good indication of the actual distance to the cluster. It is not expected that all stars in the cluster will all be at the same distance, as the cluster has some depth (see Fig. 5 at the end of this exercise). Also, even though the stars are moving together through space in the Galaxy, they also orbit a common center of mass. This method also assumes that the cluster is neither expanding, contracting, nor rotating.

Procedure

Print out the worksheet.

Table 1 lists the radial velocity and proper motion for each of the 10 stars indicated on this copy of the Hyades cluster (Fig. 4):
1. The image on the worksheet shows the member stars and vectors (arrows) depicting the direction and magnitude of the proper motion for those stars.
2. Tape another page to the left-hand side of this image. With a ruler, extend the x-axis scale onto the side sheet. (Note that this scale gives the right ascension of the stars in degrees. This will help you determine the angle q easily.)
3. With a ruler, carefully extend the vectors for about 10 - 12 stars (don't do too many!) about 25 cm in the direction of the arrows. Because this cluster of stars is travelling through space as a unit, the lines will seem to converge. But, since within the cluster the stars are moving around the center of mass in addition to progressing along with the general cluster, there will not be exact convergence. In addition, there are errors in the determination of each proper motion, and as you can see, some crowding of the data. These inaccuracies are great enough that we ignore the fact that we are working with a 3-dimensional cluster on a 2-dimensional page -- in fact, these extensions should be slightly curved.
4. Decide where the density of the lines is greatest and circle that point. (You may wish to use a coin to draw your circle: if the convergent point is tight, use a dime; if it is hard to determine, use a quarter; if it is extremely imprecise, use a jar lid!) This circle gives you some indication of how precise your determination is.
5. Complete the data in Table 1 and find the distance to the Hyades cluster by averaging the distance to each of the 10 stars.
1. Find the transverse velocity for each of the 10 stars marked in Fig. 4
• measure the angle q between each of the 10 stars and the convergent point, using the right ascension, marked in degrees of arc instead of hours of time, along the x-axis to do this.
• you may wish to devise your own measuring tool or determine a scale to use with a ruler.
• use the given radial velocity and the appropriate equation given above.
2. Find the distance to each star using the transverse velocity and the proper motion for each of the 10 stars, using the appropriate equation given above.
3. Determine an approximate error in your measurement of the distance based upon the size of your "error" circle. That is, for you determination, what is the farthest and what is the nearest the cluster can be?

To get a better feel for the actual dimensions of this cluster, take a look at Fig. 5, a 3-D image of the Hyades.

The Hipparcos results for the Hyades are truly remarkable and have spawned a number of papers on various topics. All of the Hipparcos results are on-line, and you are strongly encouraged to check out this site for more information, animations, and other goodies.