Page:EB1911 - Volume 20.djvu/191

Rh 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.

We begin by considering the laws of motion in the orbit itself, regardless of the position of the latter.

Let the curve represent an elliptic orbit, AB being the major axis, DE the minor axis, and F the focus in which the centre of attraction is situated, which centre we shall call the sun. From the properties of the ellipse, A is the pericentre or nearest point of the orbit to the centre of attraction and B the apocentre or most distant point. The semi-major axis, CA or CB, is called the mean distance, and is represented by the symbol a. We put e for the eccentricity of the ellipse, represented by the ratio CF:CA. P is the position of the planet at any time, and we call r the radius vector FP. The angle AFP between the pericentre and the position P of the planet is the anomaly called v. By Kepler’s second law the radius vector, FP, sweeps over equal areas in equal times. To do this the actual speed in the orbit, and in a yet higher degree the angular speed around F, must be greatest at pericentre, and continually diminish till the apocentre is reached. Let P, P′ be two consecutive positions of the radius vector. Since the area of the triangle FPP′ is one half the product of FP into the perpendicular p from P on FP′, it follows that if these perpendiculars were equal all round the orbit, the areas described during the infinitesimal time would be smallest at the pericentre and continually increase during the passage of the body to B. It follows that p must be greatest at pericentre, where its distance from F is least. By geometrical consideration it can be shown that the angle subtended by p, as seen from F, must be inversely as the square of its distance r. We therefore have the fundamental theorem that the angular velocity of the body around the centre of attraction varies inversely as the square of its distance, and is therefore at every point proportional to the gravitation of the sun. Another curious theorem proposed by Bouilland in 1625 as a substitute for Kepler's second law is that the angular motion of the body as measured around the empty focus F′ is (approximately) uniform. That is to an eye at F′, the planet would seem to move around the sky with a nearly uniform speed.

The true anomaly, AFP, is commonly determined through the mean anomaly conceived thus: Describe a circle of radius a＝CA 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 the period 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 nT＝360°.

It is shown in the article (Celestial Mechanics) that the mean distance and mean motion or time of revolution of a planet are so related by Kepler’s third law that, when one of these elements is given, the other can be found. Hence the number of independent elements assigned to a planet or other body moving around the sun is commonly six. But the same relation does not hold of a satellite the mass of whose primary is not regarded as an absolutely known quantity, or of a binary star. In these cases therefore the mean distance and mean motion are regarded as different elements, and the whole number of the latter is seven.

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 radii vectores 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.

When 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 we conceive ideal equations of the form $\displaystyle a=f(a, b, c, e, f, g, t)$ and $\displaystyle\delta = \phi(a, b, c, e, f, g, t)$, the symbols a, b,. . . t, representing the six elements and the time. If the values of a and, defining the position of the body on the celestial sphere, are observed at three different times, we may conceive six equations like the above, one for each of the three observed values of a and. Then by solving these equations, regarding the six elements as unknown quantities, the values of the latter may be computed. The actual process of solution is vastly more complex than is indicated by this description of it. Instead of the six ideal equations just described we have to combine a number of equations of various forms containing other quantities than the elements. But the logical framework of the process is that which we have set forth.

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.

.—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 encyclopaedia 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 Himmelskörper (1 vol., Leipzig, 1906), which, alone of the three, treats orbits of satellites and double stars.

 ORCAGNA (c. 1308–c. 1368 ), Italian painter, sculptor and architect, whose full name was, called