Stars traveling through space

After Herschel, other astronomers tackled the problem of the detailed layout of the stars in space. To solve this problem requires the measurement of distances to stars. But that's the one of the hardest measurements to make in astronomy! You have already seen how astronomers use heliocentric parallaxes to find the distances to nearby stars. With a Hertzsprung-Russell diagram in hand, we can estimate spectroscopic distances to more distant stars. We can also estimate the distances to stars by observing their motions. We will see how to do this, but first let's look at stellar motions in general.

Space Velocity A star's motion in space has two components (Fig.23.10): a transverse velocity across the line of sight and a radial velocity along the line of sight. The combination of these component velocities across and along the line of sight fix the star's space velocity, its total speed and direction in a reference frame at rest with respect to the sun. Though the space velocity has both direction and magnitude, we will use the term to refer to the magnitude only, the speed of the star through space.

The motion of a star in space. The part of a star's motion across our line of sight is its tranverse velocity, proportional to its angular speed, called the star's proper motion; the part along the line of sight is its radial velocity.

Proper Motion Early in 1718 Edmund Halley compared the positions in the sky he had found for Arcturus and Sirius with those given by Ptolemy. Halley found that these stars had moved a considerable amount in 1500 years; the "fixed" stars were not in fact fixed, but moved about in space. In one year the change in a star's position on the sky relative to other stars amounts to a very small angular movement. But over many years-such as the time between Ptolemy and Halley-the changes add up and are easier to observe. This change in position on the sky of a star with respect to other stars because of its motion in space is called its proper motion. (The rather archaic term arose historically to differentiate this real motion from the apparent motion of rising and setting and that due to precession.)

This makes sense: for a given transverse velocity, the stars that have large proper motions, crossing the sky at high angular speed, must be nearby (a rule of thumb to help remember this relation is: swiftness means nearness or slowness means far-ness). And to provide a given proper motion, a star with a high transverse velocity must be farther away.

We could calculate the distance of a star if we knew its proper motion and transverse velocity. The trouble is, we don't know the transverse velocity of an individual star. First, although the average space velocity of stars is 25 km/s, some move faster and some move slower. Second, the space velocity may be in any direction. Two stars that have the same space velocity may have quite different transverse velocities if one is moving mainly in the transverse direction and the other mainly in the radial direction. So there isn't any way we can get the distance for an individual star from its proper motion.

What we can do is pick a large number of stars of similar type, which we think are at about the same distance (because they have the same apparent brightness, or flux at the earth). We then measure the spectra of these stars and find their average radial velocity. If we then assume that the space velocities are in random directions, we can say that the average transverse velocity will be the same as the average radial velocity. We also measure the average proper motion of the stars. Then we can calculate the average distance of the group from the average transverse velocity and the average proper motion, using the relation given above. We can of course translate this distance into a parallax. A parallax determined in this way, because of the use of averages and reliance on the assumption of random directions for the space velocities, is called a statistical parallax.

The Sun's Motion There is a complication in using the method of motions of only thirteen stars, Herschel deduced statistical parallaxes, however: some of a star's that the direction in which the sun moves in space, proper motion may arise from the motion of the called the apex of the sun's motion, lies in the sun itself relative to the center of mass of the stars constellation Hercules. How? He noted in the solar neighborhood, a reference frame called that, in general, the direction of the proper mo-the local standard of rest. In 1783 Herschel noted that the sun's motion should be reflected in a part of the proper motions of stars, varying in a regular way with position on the sky. They converged on a point in the opposite part of the sky, the antapex, in the conway with position on the sky. stellation of Columba. He suggested that this effect was caused by the motion of our own star through space.

Imagine this situation with the following analogy. Consider standing in a snowfall with no wind blowing. Then the snowflakes fall in their usual shifting patterns. Drive a car through the snowstorm. Peering through the front windshield you note that the snowflakes appear to fan out from a point directly in front of the car. If you use the rear-view mirror to look out the back window, the snowflakes appear to converge at a point behind it. Their apparent motions only reflect the car's motion through the swarm of falling snowflakes.

Similarly, the sun's motion at about 20 km/s through the local stars creates the effect of an apex and antapex of solar motion. How do we know how fast were moving? By using the Doppler shift. In general, stars in Hercules have blueshifted spectra; those in Columba show redshifts. If we measure the blueshifts for a number of stars in the Hercules region and find an average, it comes out to about 20 km/s. Secular Parallaxes Once the motion of the sun through space is known, we can separate the observed proper motions of stars into a part due to the sun's motion and a part due to the stars' motions with respect to the local standard of rest. The statistical parallax method can be applied to the latter for determining stellar distances. But the solar motion itself also helps us determine distances of stars. Let's see how.

For the moment, let's ignore the random motion of the stars themselves, and think of them as fixed in space. The sun moves at 20 km/s through space with respect to the average position of these nearby stars. Since all motion is relative, we can think of the solar motion in terms of the sun being at rest, and all the stars moving at 20 km/s with respect to the sun, but in the opposite direction. Suppose a star is in a direction making an angle (3 with respect to the direction of the solar motion.

Recall that the method of trigonometric parallaxes can be used only to distances of about 100 pc, because the angular shift then gets too small to measure. The methods of statistical parallax and of secular parallax greatly extend the range of distance determinations, out to about 500 pc or more. That's because the baselines for these methods are not limited to 2 AU. If we are willing to wait long enough, the sun, or the distant star, will move many AU, and the angular shift, from which we get the proper motion, can become as large as we like. On the other hand, both these methods, because of their statistical nature, can be used only to get the average distance for a group of stars, not for the distance of any particular individual star. Nevertheless, this information provides a crucial leap beyond the direct measure of distances to the nearest stars-a leap necessary to discover the structure of the Galaxy.