This method also assumes that the cluster is neither expanding, contracting, nor rotating. Also, even though the stars are moving together through space in the Galaxy, they also orbit a common center of mass. It is not expected that all stars in the cluster will all be at the same distance, as the cluster has some depth (seeįig. 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. We now have the transverse velocity, v t and the proper motion, m, and can calculate the distance D (in parsecs) by using the equation: Make sure you understand the geometry and why this works. The geometry of this method is shown in Figure 3. We can solve the velocity right triangle for the transverse velocity: We measure the radial velocity, v r, directly. 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 V r. This angle is denoted by the greek letter "theta": The angle of sight between a star and the convergent point is measured on the chart, or in the sky. 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. Just as two parallel railroad tracks appear to converge in the distance, so also will parallel star paths. 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. In this case, we turn the problem inside out, and solve for the distance from an actual velocity and a proper motion. But the distance is what we wish to find! We need a "trick".įor 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. 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 proper motion is not an actual transverse velocity, but is instead an angular velocity. We can measure the proper motion and radial velocity of that star. Ultimately, what we want to determine is the distance to the star. , of a spectral line from its expected wavelength. The radial velocity of a star is remarkably easy to measure: we just take a spectrogram of a star and measure the displacement, 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.
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