II. WE are in possession of numerous methods of computing double star orbits. Sir John Herschel gave orie of the first solutions of this problem, and his method has been used more than any other up to this, and so far from becoming obsolete, it is yearly gaining ground at the cost of the methods that have been proposed elsewhere. It starts with the construction of the orbit, which the companion appears to describe round the main star. It is clear that as the planes of the orbits may be inclined in every direction in spaced we see only the projection of the real orbits on the heavens, but this, as well as every other projection of an ellipse on a plane surface, is another ellipse, though the main star does no longer appear situated in the focus. Five points determine an ellipse, if we therefore possess five complete observations, we can determine the apparent ellipse. Now the observations are not perfectly accurate, but the calculus of probabilities furnishes us with means to ascertain the most probable ellipse from a great number of observations, to which different weight may be attributed, according to their reliability, as far as known. But at Herschel's time, though the angles had been fairly observed, the measurement of these minute distances was still in its infancy. He, in consequence, threw them away, and computed distances by aid of the Keplerian law referred to above, from the angular velocities, concluded from a comparison of observations separated by moderate intervals. He improved the angles in the following way:—On a paper neatly divided into squares, he lays down a point for every observed angle of position, the epoch in years and decimals being measured as an abscissa along the horizontal lines, and the angle in degrees as an ordinate along the vertical ones. A series of points are thus obtained, which, if the observations were exact, would necessarily admit of a regular curve being drawn through them, whose nature is of course determined by the laws of elliptic motion, and one of whose essential characters is to have within those limits of the abscissa, which correspond to a whole period of revolution (that is, to a difference of 360 units in the ordinates), in some cases two, in some four, points of contrary flexure, but never more than the latter, nor fewer than the former, and to have, moreover, in all its points, a peculiarly graceful and flowing outline. The errors of observation, however, prevent the drawing of such a curve through all the points. It must be drawn with a free but careful hand, not through, but among the points, and so that it shall deviate less from every point, according as it is more or less reliable. Now after Herschel's time the accuracy of the observed distances has wonderfully improved, and we are therefore able to draw another curve representing the distances as ordinates, which then ought to agree with those deduced from the angles, and the angles ought to agree with those deducible by aid of integral calculus from the distances. The curves must be varied till they thus mutually support each other, and then we may construct any number of points of the apparent orbit by reading off the angles and distances for the corresponding epochs on the curves, and if we find the are described sufficiently extensive, the apparent ellipse is simply drawn as nearly as possible through them. From the apparent orbit the elements of the real orbit, described in space, are then determined. These are seven in number:—
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Jul 31, 2015
G Perović + 1
Open Access
