The fixed stars, part III
Seeing the forest instead of the trees
As we’ve noted, the “fixed stars” aren’t actually fixed in place. They move around at quite respectable speeds. It’s just that they’re so far away, and so far between, that it takes a very long time for any motion to show up. On human scales of time, that means taking very careful measurements and being patient.
By the late eighteenth century, the movements of many stars relative to each other had been measured. These are called “proper motions” to distinguish them from the general change of coordinates due to Earth’s precession, and are different for each star. They are also tiny, measured in seconds of arc per year or (mostly) less. (A second of arc is the size of a dime seen from two miles away, which won’t actually mean much to people who aren’t in the habit of looking at pocket change in the next town. Instead, it’s probably best to think of a second of arc as the dividing line between what can be seen by any competent observer in a good telescope on a decent night, and what takes much skill and steady air to measure.) In a remarkable calculation, Sir William Herschel found from these proper motions that the Sun is moving relative to the average of these stars, heading toward a point in Hercules. The interesting part is that the Sun’s reflex motion is a small part of any star’s proper motion. It’s only by looking carefully at the mass of data that the average comes out.
Herschel also noted that many stars seem to occur in pairs, many more than one would expect from random placement on the sky. By statistics alone, some must be connected to each other. By the mid-nineteenth century the motions of several relative to each other had been worked out: elliptical orbits, just as planets orbit the Sun. So Newton’s laws of motion and gravitation apply everywhere! Well, at least they work far beyond the region where they’d been used before.
But beyond double stars the ability to calculate motion breaks down. With three or more objects the nice conic sections (ellipse, hyperbola, etc.) no longer work. Indeed, except in special cases there is no closed-form solution to the equations of motion. And there are many more than three stars in the sky! Johann Mädler, a German astronomer, worked out that the Milky Way was rotating about a central point, by assuming something like the motion of the planets around the Sun. But John Herschel (son of William) expressed the view of most nineteenth-century astronomers when he admitted himself baffled by the problem of the dynamics of a cluster of stars.
Then, in the twentieth century, Sir James Jeans had an idea. Calculating the motion of thousands of separate stars is, at best, impractical. So how about treating the stars in a galaxy as a fluid? Smooth out the bumps and apply the mathematics developed for air and water flow (with the addition of gravity). It’s actually a very good approximation for a galaxy with, say, 1011 stars and enormous spaces between them.
Well, here in the twenty-first century there are fewer limitations on calculations. A computer can actually work out the motion of a cluster of a thousand or more stars. And the Gaia mission is producing proper motions for over a billion stars. We are back to the star-by-star universe. But it’s still useful, most times, to take one’s suns a cup at a time and apply Jeans’ Equations of Stellar Dynamics.
