surface. The larger body, having four times the surface of the smaller, will then emit four times as much light. The volumes being proportional to the cubes of their diameters, it will have eight times its volume. The densities being supposed equal, it will have eight times the mass. Suppose that each has a satellite revolving around it, and that the orbit of the satellite of the larger body is twice the radius of that of the smaller one.
Calling the radius of the nearer satellite 1, that of the more distant one will be 2. The cube of this number is 8. It follows from the extension of Kepler's third law, which we have cited, that the times of revolution of the two satellites will be the same. Thus the two bodies, A and B, with their satellites, C and C, form two binary systems whose proportions and whose periods are the same, only the linear dimensions of B are all double those of A. In other words, we shall have a pair of binary systems which may look alike in every respect, but of which one will have double the dimensions and eight times the mass of the other.
Now let us suppose the larger system to be placed at twice the distance of the smaller. The two will then appear of the same size, and, if stars, will appear of the same brightness, while the two orbits will have the same apparent dimensions. In a word, the two systems will appear alike when examined with the telescope, and the periodic times will be equal.
Near the end of the second chapter we have given a little table showing the magnitude that the Sun would appear to us to have were it placed at different distances among the stars. The parallaxes we have there given are simply the apparent angle which would have to be subtended by the radius of the earth's orbit at different distances. It follows that, were the stars all of similar constitution to the Sun, the numbers given in the last column of the table referred to would, in all cases, express the apparent distance from the star of a companion, having a time of revolution of one year. From this we may easily show what would be the time of revolution of any binary system of which the companions were separated by 1", if the stars were of the same constitution as the Sun.
Periods of binary systems whose components are separated by 1" and whose constitution is the same as that of the Sun.