199. ORBITS OF DOUBLE STARS.--The sun's nearest neighbor, α Centauri, is such a double star, whose position angle and distance have been measured by successive generations of astronomers for more than a century, and Fig. 127 shows the result of plotting their observations. Each black dot that lies on or near the circumference of the long ellipse stands for an observed direction and distance of the fainter of the two stars from the brighter one, which is represented by the small circle at the intersection of the lines inside the ellipse. It appears from the figure that during this time the one star has gone completely around the other, as a planet goes around the sun, and the true orbit must therefore be an ellipse having one of its foci at the center of gravity of the two stars. The other star moves in an ellipse of precisely similar shape, but probably smaller size, since the dimensions of the two orbits are inversely proportional to the masses of the two bodies, but it is customary to neglect this motion of the larger star and to give to the smaller one an orbit whose diameter is equal to the sum of the diameters of the two real orbits. This practice, which has been followed in Fig. 127, gives correctly the relative positions of the two stars, and makes one orbit do the work of two. [Illustration: FIG. 127.--The orbit of α Centauri.--SEE.] In Fig. 127 the bright star does not fall anywhere near the focus of the ellipse marked out by the smaller one, and from this we infer that the figure does not show the true shape of the orbit, which is certainly distorted, foreshortened, by the fact that we look obliquely down upon its plane. It is possible, however, by mathematical analysis, to find just how much and in what direction that plane should be turned in order to bring the focus of the ellipse up to the position of the principal star, and thus give the true shape and size of the orbit. See Fig. 128 for a case in which the true orbit is turned exactly edgewise toward the earth, and the small star, which really moves in an ellipse like that shown in the figure, appears to oscillate to and fro along a straight line drawn through the principal star, as shown at the left of the figure. In the case of α Centauri the true orbit proves to have a major axis 47 times, and a minor axis 40 times, as great as the distance of the earth from the sun. The orbit, in fact, is intermediate in size between the orbits of Uranus and Neptune, and the periodic time of the star in this orbit is 81 years, a little less than the period of Uranus. [Illustration: FIG. 128.--Apparent orbit and real orbit of the double star 42 Comæ Berenicis.--SEE.] 200. MASSES OF DOUBLE STARS.--If we apply to this orbit Kepler's Third Law in the form given it at page 179, we shall find-- a^3 / T^2 = (23.5)^3 / (81)^2 = k (M + m),