162. Elements of a comet's orbit.—There is a theorem of geometry to the effect that through any three points not in the same straight line one circle, and only one, can be drawn. Corresponding to this there is a theorem of celestial mechanics, that through any three positions of a comet one conic section, and only one, can be passed along which the comet can move in accordance with the law of gravitation. This conic section is, of course, its orbit, and at the discovery of a comet astronomers always hasten to observe its position in the sky on different nights in order to obtain the three positions (right ascensions and declinations) necessary for determining the particular orbit in which it moves. The circle, to which reference was made above, is completely ascertained and defined when we know its radius and the position of its center. A parabola is not so simply defined, and five numbers, called the elements of its orbit, are required to fix accurately a comet's path around the sun. Two of these relate to the position of the line of nodes and the angle which the orbit plane makes with the plane of the ecliptic; a third fixes the direction of the axis of the orbit in its plane, and the remaining two, which are of more interest to us, are the date at which the comet makes its nearest approach to the sun (perihelion passage) and its distance from the sun at that date (perihelion distance). The date, September 17th, placed near the center of [Fig. 108], is the former of these elements, while the latter, which is too small to be accurately measured here, may be found from [Fig. 109] to be 0.82 of the sun's diameter, or, in terms of the earth's distance from the sun, 0.008.
Fig. 108.—Orbits of the earth and the Great Comet of 1882.
[Fig. 109] shows on a large scale the shape of that part of the orbit near the sun and gives the successive positions of the comet, at intervals of 2/10 of a day, on September 16th and 17th, showing that in less than 10 hours—17.0 to 17.4—the comet swung around the sun through an angle of more than 240°. When at its perihelion it was moving with a velocity of 300 miles per second! This very unusual velocity was due to the comet's extraordinarily close approach to the sun. The earth's velocity in its orbit is only 19 miles per second, and the velocity of any comet at any distance from the sun, provided its orbit is a parabola, may be found by dividing this number by the square root of half the comet's distance—e. g., 300 miles per second equals 19 ÷ √ 0.004.
Fig. 109.—Motion of the Great Comet of 1883 in passing around the sun.
Most of the visible comets have their perihelion distances included between 1/3 and 4/3 of the earth's distance from the sun, but occasionally one is found, like the second comet of 1885, whose nearest approach to the sun lies far outside the earth's orbit, in this case half-way out to the orbit of Jupiter; but such a comet must be a very large one in order to be seen at all from the earth. There is, however, some reason for believing that the number of comets which move around the sun without ever coming inside the orbit of Jupiter, or even that of Saturn, is much larger than the number of those which come close enough to be discovered from the earth. In any case we are reminded of Kepler's saying, that comets in the sky are as plentiful as fishes in the sea, which seems to be very little exaggerated when we consider that, according to Kleiber, out of all the comets which enter the solar system probably not more than 2 or 3 per cent are ever discovered.