The nearness of the UMa cluster, like the famous Hyades, allows us to determine the true space motion of member stars. A star's distance is determined through basic trigonometry by measuring its position from opposite sides of the Earth's orbit, giving the so-called parallax. For even the nearest stars this is a bit less than one arc second, and the technique is good only to maybe 200 light years; beyond that, distances are arrived at spectroscopically. By taking a high-resolution spectrogram, a star's motion toward or away from us is revealed by the doppler-shifting of the absorption lines. This component of motion is called the radial velocity, and is given as km/sec. The other important component of motion, the proper motion, takes years or decades (and often longer) to accurately fix. This is because we have to observe the very small apparent drift across the line of sight, against the background of very distant stars. For all but a few stars this amounts to a fraction of an arc second per year. With the position, distance, and radial and tangential components of motion in hand, a little spherical trigonometry will yield the star's velocity vector in space.
The core of the Ursa Major cluster is marked by the five middle stars of the Big Dipper, beta, gamma, delta, epsilon and zeta. (The end stars, alpha and eta, are not members; their proper motions are almost opposite that of the cluster). These are early A-type main sequence stars of about fifty times the Sun's luminosity, appearing at second and third magnitude. Our sun at their distance would be magnitude 6.7 -- barely visible to a keen-eyed observer on the darkest night. The brightest is Alioth (epsilon), which has actually evolved to the subgiant stage and is a small amplitude variable of the alpha CVn type. The most-observed star of the group is Mizar (zeta), a well-known double for small telescopes. Its naked eye companion, Alcor (80 UMa), is not orbiting Mizar, but is a member of the cluster. There are four other members easily visible without optical aid: 37 UMa, 78 UMa and 21 LMi (all late A to early F types of fourth and fifth magnitude); and Alphecca (alpha CrB), an early A-type subgiant. 21 LMi and Alphecca appear rather far from the cluster centre -- about 30 and 40 degrees, respectively; Alphecca may actually be a stream member (more later).
Rounding out the cluster are a half-dozen sixth to eighth magnitude type F to K stars of lower mass. Two are about 15 degrees to the northeast in Draco's tail, and the other four are near the Dipper's handle. Visually we are presented with a very sparse, flattened group, the main concentration of which is about 20 degrees long, with the outliers spanning almost 50 degrees. In all, then, we find 16 star systems (a few stars are double or multiple), the main concentration of which occupies a roughly ellipsoidal volume of 18 by 30 light years. A rich open cluster could pack 200 stars in the same volume! Another way to appreciate its sparseness is to realize that at 78 light years' distance our sun and Alpha Centauri would appear three degrees apart, or about half the distance between any two neighbouring Dipper stars -- actually less than half when we consider the extra separation of cluster stars resulting from the difference in distances.
It was first realized that the Dipper is a real cluster before the turn of the century, when the stars were found to have similar proper motions -- almost due east, or to the left, on the sky. This annual drift amounts to about 0.11 seconds of arc per year, or the equivalent of the Moon's apparent diameter in 16,000 years. In addition, the spectroscope shows the cluster to be approaching us at 11 to 12 km/sec, although this varies depending on the position of the member stars on the sky. When the cluster nears the Keystone of Hercules in a little under a million years from now, it will be at its closest to us and two-thirds its current distance.
The true space velocity with respect to the Sun is 16 km/sec, with the cluster appearing to approach from a point on the sky at roughly 8h 20m, +35d -- about 10 degrees northeast of Castor and Pollux in Gemini. This point is called the divergent, and is analogous to a meteor shower radiant. On the opposite side the sky is the divergent, located in extreme eastern Sagittarius at 20h 20m, -35d. If the stars were moving exactly parallel, there would be one divergent and one convergent for all members. Even neglecting measurement errors, the effects of gravity among cluster stars and from material in the galaxy conspire against this, and we see some scatter among the stars' individual convergents -- a reflection of their instantaneous velocity vectors.
Many other stars scattered all over the sky share a common space motion with the UMa cluster. And like the cluster the most massive, hence brightest, stars have evolved to the subgiant and even giant phases. One hundred or so stars have been accepted as members of the UMa stream, as it is called. Spanning a few hundred light years of space, it's much too big to be a cluster proper. As these stars are seen in all directions, we are obviously passing through the stream. The spectral types give stellar ages similar to that of the cluster, and this together with the space motions suggests a common origin about 150 million years ago -- just over half a galactic rotation. This age is about midway, ratio-wise, between the youngest and oldest star clusters (one million to 10 billion years).
In the literature I've seen only a few stream stars identified. Using a Basic program I wrote, I went through the bright star list in the RASC Observer's Handbook and culled a dozen or so likely candidates. As these are among the brightest visible stars, I wasn't surprised to see that most were evolving subgiants and giants. Going to fainter magnitudes would net more and more run-of-the-mill main sequence members. Here are the 17 I have so far, in order of R.A.: gamma Cet, beta Eri, beta Lep?, beta Aur, gamma Gem, Sirius, delta Vel, gamma Leo, delta Leo, beta Ser, eta Her, alpha Oph, g Sco, zeta Sgr, beta Pav, zeta Cyg? and iota Cep. The computed true space velocity of these stars averages nearly 22 km/sec, or six km/sec faster than the cluster. Keep in mind that while the stream is nearly 40% faster than the cluster with respect to the Sun, everything is whirling about the galaxy at more like 250 km/sec, so the difference is really about 2%.
Whereas the cluster member convergents mostly fit within a 10-degree ellipse on the sky, the stream members' convergents are scattered across an area of 25 by 45 degrees, perpendicular to the galactic plane (the cluster member convergent pattern is curiously aligned in the plane containing the cluster, Earth and convergents). The cause of this out-of-plane dispersion could be due to the gravitational gradient of the galaxy being stronger perpendicular to the disk -- where much material lies in a heavily stratified form -- than within the plane of the disk. This effect is causing the solar system to oscillate up and down through the disk about 3.5 times per galactic rotation (we are currently above the disk). The spread in the pattern is so large because we and the stars are moving more or less together, so that the effect of small departures from the mean motion is exaggerated.
Note that the cluster and much of the stream appear to be heading 30 degrees below the galactic plane. This is actually a consequence of our sun's motion upward (toward galactic long. 56d, lat. +23d near Vega). If the Sun's orbit remained in the galactic plane, the stars' convergents would straddle the galactic equator, telling us that those stars are now at their highest point above the plane, and will fall back through the midplane millions of years in the future. These stars also appear to be heading almost toward the galactic centre, across the general flow which is toward long. 90d. Once again the Sun's anomalous motion adds its effect. We are moving faster than the general flow by about 20 km/sec -- if we subtract this to get our speed back to normal (250 km/sec), the cluster/stream would exhibit a more forward component of motion toward long. 35d -- still quite a bit inward. Additionally, the resulting increase in relative velocity would tighten up the convergents' dispersion.
Finally, let's consider the motion within a more absolute frame of reference, from a point at rest with respect to the galactic nucleus. This would be like observing multi-lane highway traffic from an overpass. If the stars were westbound vehicles, the galactic centre is to the north. At first glance everything seems to be heading in exactly the same direction and at the same speed, but a closer look reveals cars changing lanes -- their actual direction of travel (and often speed) is slightly different. To a truck driver watching a car in front of him cross from the passing lane to the inside lane, it would seem as if the car were moving left to right at nearly right angles to his truck's path. Our observer on the overpass would record the true difference in direction as being just a few degrees. Even at highway speeds (the general motion of the galactic disk) small differences in speed and direction are significant when you're moving with the traffic (observing from the Earth).
The fact that we see the inward-moving cluster/stream crossing our already inward line of travel shows that the stars' orbits are more eccentric than the Sun's, and like the Sun are heading toward perigalacticon (closest point to the galactic nucleus). But don't envision comet orbit-like ellipses; the orbits are still fairly circular. The Sun's true motion in the galaxy is toward long. 87.8d, lat. +1.6d, and the cluster/stream is toward long. 84d, lat. 0d (a circular orbit would be directed toward long. 90d). The group's somewhat higher speed indicates higher orbital energy, therefore a larger orbit, but not by much. Whereas the Sun has gone around the galaxy almost 20 times since its birth, the UMa cluster has not yet completed one orbit. Theory predicts young(er) stars to have fairly circular orbits, so why is the cluster's orbit more eccentric? Maybe the cloud of material it formed from was jostled out of a more circular orbit, either by a previous period of energetic star formation or perhaps by the gravitational effects of density variations in that spiral arm region. It's interesting to contemplate that, given the elapsed time and relative velocities involved, the birthplace of the group was likely 5,000 to 8,000 l.y. from the Sun -- on the order of the spacing between spiral arms.
The included chart [109K GIF] encompasses almost one half of the entire sky. The galactic co-ordinate system is used, as it best imparts a sense of the dynamics. The galactic equator is the solid line running along the milky way, and the lines of longitude and latitude are spaced every ten degrees. On the equator line, longitudes 0 and 90 degrees are indicated, in Sagittarius and Cygnus respectively. The galactic north pole is toward the top, in Coma Berenices. Sixteen cluster and stream member stars are shown with direction vectors converging on the largest of the three patterns, which contains the apparent convergent points of the stars I've identified. The dense hatching shows the area toward which most of the UMa cluster stars are heading; the lighter hatching includes the rest. The large unshaded area includes all stream member convergents.
The smaller, somewhat similar pattern on the galactic equator represents
the convergent point pattern as it would appear if our sun were orbiting
the galaxy in a purely circular orbit at the nominal 250 km/sec. Its single
hatched area includes all cluster members. The small hatched area at longitude
86 degrees in Cygnus includes all cluster and stream convergents,
and shows where those stars would appear to recede toward (at around 285
km/sec) if viewed from a point at rest with respect to the galaxy. The
elongation in the galactic plane is the result of the spread in space velocities
among those stars and the consequent effect on the individual orbits.