PROPER MOTION We've seen that diurnal motion of stars is due to the rotation of the earth on its axis. Annual motion of stars is due to the revolution of the earth around the sun. And the motion of planets is due to the actual orbits of the planets around the sun. It was finally discovered, in the eigtheenth century, that the stars do have real motion, called proper motion. We see this as a very small angular displacement, over time, relative to other stars.
In telescope it would look as Figure 1 below, with one star displaced after, say, a year relative to a pattern among other stars.
This figure looks similar to the one we saw earlier showing the wandering of a planet. The difference is that the angular shift of a star is very much less, typically about 1/1000 of a degree per year (whereas for a planet it might be 90 degrees or 180 degrees in a year). What is happening in space is shown in Figure 2.
Seen against a background of 7 distant stars, the near star is first at point P, aligned with star 2, then at later time (years later) at point Q, aligned with star 6. The stars that exhibit proper motion are always those that are relatively near to us, and their motion seen relative to distant stars. "Near" in this context means from a few light years out to a few tens of light years. The distant stars may be millions of light years away.
Great patience and diligence is required to study this motion. One must be assured that the telescope is in exactly the same position and orientation consistently over a period of years, even decades.
BINARY STARS
The first binary, or double, star was discovered in 1650 by Jean Baptiste Riccioli, who observed two stars separated by about 1/1000 of a degree. He did not observe motion, but in 1805 William Herschel observed a famous binary pair, one star bright and the other faint, and the faint one moved about 1/7000 of a degree in one year. After a few years he saw that the motion was not a straight line. Given Newton's laws of motion, he had to conclude that the faint star was feeling a force. And the only reasonable assumption was that this was the gravitational force exerted by the bright star.
Eventually it was seen that the dim star orbitted the bright star. The bright star would also feel gravitational force due to the dim star, but it is more massive (usually brightness correlates with mass) and so its motion would have been less -- too little to be detectable. Following the method we have used in connection with the solar system, we can deduce the mass of the bright star from the orbital parameters of the dim star. This is in fact the first determination of the mass of a star, and it is the main technique we have today for determining stellar masses. From observation of many binary stars that we have gotten a good idea of the range of star masses. Among other things we find that the sun's mass falls in this range, which adds to our conviction that the sun is a star.
Figure 3 below is a logarithmic scale which shows the range of star masses, and where the solar system planets fit in.
A logarithmic scale is one in which a fixed spacing on the line corresponds to multiplying by a constant factor (the factor here is 10). Each mass is given relative to the sun. Note the large "empty space" between the smallest stars and Jupiter, the largest planet.
TYPES OF PROPER MOTION
Figure 4 shows several possible types of proper motion:
FINDING A PLANET
In the case of an observation like (e)(i), the invisible companion could be a planet. Planets don't produce light themselves, but, like the moon, they become visible by reflecting light from the star. If the wobbly path represents the motion of a planet, P, feeling the gravitational force of a star, S, the law of gravitation tell us that from the motion of S we can deduce the mass of P. Note here that we are explicitly using the universality of the law of gravitation -- not only in the formula for the force in terms of the masses and the distance, but in the value of the constant, G. Discovered by Cavendish in an earth-bound laboratory, we now assume the same number applies to stars many light years away.
If we can determine the mass of the invisible companion, we can see where it fits in Figure 3. Is it among the solar system planets, or is it among the stars? Observationally this is a very difficult game. The amount of the wobble is very small, of the order of thousandths of a degree. If the mass of the companion is large -- close to the masses of the small stars or between the stars and Jupiter, the wobbling is relatively greater and easier to detect. If the mass of the companion is small, like that of earth or Mars, the wobbling is much more difficult to detect.