1911 Encyclopædia Britannica/Orbit
ORBIT (from Lat. orbita, a track, orbis, a wheel), in astronomy, the path of any body, and especially of a heavenly body, revolving round an attracting centre. If the law of attraction is that of gravitation, the orbit is a conic section — ellipse, parabola or hyperbola — having the centre of attraction in one of its foci; and the motion takes place in accordance with Kepler's laws (see Astronomy). But unless the orbit is an ellipse the body will never complete a revolution, but will recede indefinitely from the centre of motion. Elliptic orbits, and a parabolic orbit considered as the special case when the eccentricity of the ellipse is 1, are almost the only ones the astronomer has to consider, and our attention will therefore be confined to them in the present article. If the attraction of a central body is not the only force acting on the moving body, the orbit will deviate from the form of a conic section in a degree depending on the amount of the extraneous force; and the curve described may not be a re-entering curve at all, but one winding around so as to form an indefinite succession of spires. In all the cases which have yet arisen in astronomy the extraneous forces are so small compared with the gravitation of the central body that the orbit is approximately an ellipse, and the preliminary computations, as well as all determinations in which a high degree of precision is not necessary, are made on the hypothesis of elliptic orbits. Below are set forth the methods of determining and dealing with such orbits.
The true anomaly, AFP, is commonly determined through the mean anomaly conceived thus: Describe a circle of radius aCA around F, and let a fictitious planet start from K at the same moment that the actual planet passes A, and let it move with a uniform speed such that it shall complete its revolution in the same time T as the actual planet. From the law of angular motion of the latter its radius vector will run ahead of PQ near A, PQ will overtake and pass it at apocentre, and the two will again coincide at pericentre when the revolution is completed. The anomaly AFQ of Q at any moment is called the mean anomaly, and the angle QFP by which the true anomaly exceeds it at that moment is the equation of the centre.
Two elements define the position of the plane passing through the attracting centre in which the orbit lies. One of these is the position of the line MN through the sun at F in which the plane of the orbit cuts some fundamental plane of reference, commonly the ecliptic. This is called the line of nodes, and its position is specified by the angle which it makes with some fixed line FX in the fundamental plane. At one of the nodes, say N, the body passes from the south to the north side of the fundamental plane; this is called the ascending node. The other element is the inclination of the plane of the orbit to the fundamental plane, called the inclination simply. A fifth element is the position of the pericentre which may be expressed by its angular distance XFN from the ascending node. A sixth is the position of the planet in the orbit at a given moment, for which may be substituted the moment at which it passed the pericentre. Another element is the time of revolution of the body in its orbit, called its period. Instead of theperiod it is common in astronomical practice to use the mean angular speed, called the mean motion of the body. This is defined as the speed of revolution of the fictitious body already described, revolving with a uniform angular motion and the same periodic time as the planet. It follows that putting n for the mean motion and T for the period of revolution we shall have in degrees nT360°.
The process by which the position of a planet at any time is determined from its elements may now be conceived as follows:—The epoch of passage through pericentre being given, let t be the interval of time between this epoch and that for which the position of the body is required. Representing by P this position, it follows that the area of that portion of the ellipse contained between the radiiWhen a new celestial body, say a planet or a comet is discovered, the astronomer meets with the problem of determining the orbit from several observed positions of the body. To form a conception of this problem it is to be noted that since the position of the body in space can be computed from the six elements of the orbit at any time we may ideally conceive the coordinates of the body to be algebraically expressed as functions of the six elements and of the time. Since the distance of a body from the observer cannot be observed directly, but only the right ascension and declination, calling these a and δ we conceive ideal equations of the form FB and FP will bear the same ratio to the whole area of the ellipse that t does to T, the time of revolution. The problem of finding a radius vector satisfying this condition is one which can be solved only by successive approximations, or tentatively. Its discussion may be found in any work on theoretical astronomy. The solution may be worked out directly or through the determination of the equation of the centre which, being added to the mean anomaly, gives the true anomaly. The angle from the pericentre to the actual radius vector, and the length of the latter being found, the angular distance of the planet from the node in the plane of the orbit is found by adding to the true anomaly the distance from the node to the pericentre. This, and the inclination of the orbit being given, we have all the geometrical data necessary to compute the coordinates of the planet itself. The coordinates thus found will in the case of a body moving around the sun be heliocentric. The reduction to the earth's centre is a problem of pure geometry.
The problem of determining an orbit may be regarded as coeval with Hipparchus, who, it is supposed, found the moving positions of the apogee and perigee of the moon's orbit. The problem of determining a heliocentric orbit first presented itself to Kepler, who actually determined that of Mars. The modern method of determining orbits from three or four observations was first developed by C. F. Gauss in his celebrated work Theoria Motus Corporum Coelestium. This classical work is still a favourite among students, the improvements on its methods made since its publication being rather in details than in general principles.
Authorities.—Among recent works on the determination of orbits, J. C. Watson's Theoretical Astronomy is the most complete in the English language. The most complete existing work, an encyclopedia of the subject in fact, is T. von Oppolzer's Lehrbuch zur Bahnbestimmung der Kometen und Planeten (2 vols.), which contains voluminous tables, formulae, and instructions for the computation of orbits in the many special cases that arise. More recent and better adapted to study is Bauschinger's Bahnbestimmung der Himmelskorper (i vol., Leipzig, 1906), which, alone of the three, treats orbits of satellites and double stars. (S.N.)