Elliptic orbit

“ Euler extended this idea so as to represent any perturbation of a planet, whether experienced in the course of one revolution or in a longer time, by means of changes in an elliptic orbit. For wherever a planet may be, and whatever (within certain limits [11] ) be its speed or direction of motion some ellipse can be found, having the sun in one focus, such that the planet can be regarded as moving in it for a short time. Hence as the planet describes a perturbed orbit it can be regarded as moving at any instant ”

“ According to Einstein the path would also be elliptical, but before a revolution would be quite completed, the planet would start along a slightly advanced line, forming a new ellipse slightly in advance of the first. The elliptic orbit slowly turns in the direction in which the planet is moving. After many years—centuries—the orbit will be in a different direction.
The rapidity of the orbit’s change of direction depends on the velocity of the planet. Mercury [69] moving at the rate of 30 miles a second is the fastest among the planets. ”

“ It follows that, if we go on computing the elements a, b, c, d from the actual values of x, y, x′ and y′, at each moment when the planet is subject to the attraction of another body, they will no longer be invariable, but will slowly vary from day to day and year to year. These ever varying elements represent an ever varying elliptic orbit,—not an orbit which the planet actually describes through its whole course, but an ideal one in which it is moving at each instant, and which continually adjusts itself to the actual motion of the planet at the instant. ”

“ The method of treating the elements of the elliptic orbits as variable is specially suitable for secular inequalities; but for periodic inequalities it is generally better to treat the body as being disturbed from an elliptic path, and to study these deviations.
"The simplest way of regarding these various perturbations consists in imagining a planet moving in accordance with the laws of elliptic motion, on an ellipse the elements of which vary by insensible degrees ”

“ Imagine a shell travelling in an elliptic orbit round the [Pg 301] earth to suddenly explode: the centre of gravity of all its fragments would continue moving along precisely the same path as had been traversed by the centre of the shell before explosion, and would complete its orbit quite undisturbed. Each fragment would describe an orbit of its own, because it would be affected by a different initial velocity; but every orbit would be a simple ellipse, and consequently every piece would in time return through its starting-point—viz. ”

“ But if the motions of a planet at two widely different epochs are compared, though on each occasion the path described is very nearly an ellipse, the ellipses differ in some respects. For example, between the time of Ptolemy (A.D. 150) and that of Euler the direction of the apse line of the earth's orbit altered by about 5°, and some of the other elements also varied slightly. Hence in dealing with the motion of a planet through a long period of time it is convenient to introduce the idea of an elliptic path which is gradually changing its position and possibly also its size and shape. ”

“ The orbits of the planets cannot be regarded as true circles. They are ovals, or, to speak in technical language, "ellipses." Their ovalness or "ellipticity" is, however, in each case not by any means of the same degree. Some orbits—for instance, that of the earth—differ only slightly from circles; while others—those of Mars or Mercury, for example—are markedly elliptic. The orbit of the tiny planet Eros is, however, far and away the most elliptic of all, as we shall see when we come to deal with that little planet in detail. ”

“ IN Bohr’s theory, the electron always moves round the hydrogen nucleus in a circle. But according to Newtonian principles, the electron ought also to be able to move in an ellipse, and the generalized quantum-principle can be applied to elliptic orbits as well as to those that are circular. It is natural to inquire whether it is possible to work out a theory that allows for elliptic orbits, and, if so, whether it will fit the facts better or worse than Bohr’s original theory. It is found that, as regards the broad facts, it makes no difference whether we admit or reject elliptic orbits ”

“ The determination of an orbit from three complete observations by the solution of the equations which represent elliptic motion presents so great difficulties in the general case, that in the first solution of the problem we must generally limit ourselves to the case in which the intervals between the observations are not very long. ”

“ If the spiral nebulae have been formed in accordance with Chamberlin and Moulton's hypothesis, the secondary nuclei in them must revolve in a great variety of elliptic orbits. The orbits would intersect, and in the course of long ages the separate masses would collide and combine and the number of separate masses would constantly grow smaller. ”

“ At the same time in any single revolution a planet may for most practical purposes be said to be actually moving in an ellipse. As, however, time goes on, the ellipse gradually varies. It alters its shape, it alters its plane, and it alters its position in that plane. If, therefore, we want to study the movements of the planets, when great intervals of time are concerned, it is necessary to have the means of learning the nature of the movement of the orbit in consequence of the disturbances it has experienced. ”

“ On the other hand, under the influence of a force acting centripulsively in the inverse ratio of the square roots of the distances, we find the mean effect to be as the minor axis of the ellipse directly, so that two planets in orbits of different eccentricity, but having the same major axis, experience a different amount from the action of this radial stream, the least eccentric orbit being that which receives the greatest mean effect. ”

“ One of the achievements by which Sir John Herschel is best known is his invention of a method by which the orbits of binary stars could be determined. It will be observed that when one star revolves around another in consequence of the law of gravitation, the orbit described must be an ellipse. This ellipse, however, generally speaking, appears to us more or less foreshortened, for it is easily seen that only under highly exceptional circumstances would the plane in which the stars move happen to be directly square to the line of view. ”

“ To appreciate the significance of the discovery of the elliptical orbit of the planets, it is necessary to understand the complicated confusion that prevailed in the conception of planetary motions. The primal thought was that the motions of the planets were uniform and circular. This intuition of circular orbits was a happy one, and was, perhaps, necessitated by the very structure of the human mind. The sweeping and centrifugal soul, darting manifold rays of equal reach, realizes the conception of the circle, that is, a figure all of whose radii are equidistant from a central point. ”

“ Is not motion in an ellipse common enough? Does not the earth revolve in an ellipse round the sun? And do not the planets also revolve in ellipses?
It is this very fact that elliptic motion is so common in the planets of the solar system which renders its discovery in binary stars of such importance. From what does the elliptic [Pg 437] motion in the solar system arise? Is it not due to the law of attraction, discovered by Newton, which states that every mass attracts every other mass with a force which varies inversely as the square of the distance? ”

“ In virtue of the gravitative attraction or gravitation field between the two, they will mutually attract one another in their motion, and each will, in consequence, be deflected more or less out of that orbital path which it would normally pursue in the absence of the other. This attraction will naturally be greatest when the planets are in the closest proximity; the planet having the widest orbit will then be drawn inwards towards the central axis, the other will be drawn outwards. ”

“ When motion is required to be given them, the globes of light are sometimes attached to a car that is made to run on a single rail elliptical track, which may be suspended at any elevation and reduced to a minimum weight by proper adjustments of its gravitation, the light globe being either suspended from the car or floating above it. The elliptical orbit is inclined enough to enable gravity to propel the car. An automatic shunt turns on repulsion when the car reaches the lowest part of the orbit and it is then forwarded on the up grade portion, shunted again at the top and so on perpetually. ”

“ The orbits of the planets change not only their position, but even their magnitude and their form: the longer axis of each has a slow angular motion; and, though its length remains fixed, the shorter axis increases and diminishes, so that the form of the orbit approaches to that of a circle, and recedes from it by turns. ”

“ It is slowly distorted by the attractions of the other planets, for ages it may be nearly circular, for ages it is more or less elliptical. As the ellipse becomes most nearly circular, then the focus becomes most nearly the centre. When the orbit becomes most elliptical, then the position of the sun becomes most remote from the middle or, to use the astronomer’s phrase, most eccentric. ”

“ In the first place, we must note that the ellipticity of the earth's orbit is not developed on fixed lines, but is endlessly varied, as we can readily imagine it would be for the reason that its form depends upon the wandering of the outer planetary spheres which pull the earth about. ”

“ Fourthly, a body in motion will move in a straight line, unless diverted out of that line by some external force; and the body will resume its straight-forward motion, whenever the force that turns it aside is withdrawn. Every body that is revolving in an orbit, like the moon around the earth, or the earth around the sun, tends to move in a straight line which is a tangent [7] to its orbit. Thus, if A B C, Fig. 28, represents the orbit of the moon around the earth, were it not for the [130] constant action of some force that draws her towards the earth, she would move off in a straight line. ”

“ When a star is near enough to be measured by this trigonometric method it seems to describe a minute ellipse "like a reflection of the ellipse of the earth's orbit." If it lies so far in space that it does not [Pg 100] seem to change its position even when viewed from stations on either side of the earth's orbit, its distance may never be measured by using the diameter of the earth's orbit as a baseline.
Even the near stars are so very distant that their displacement is almost imperceptible. ”

“ But this circumstance must greatly affect the rotation of 269 the resulting planet. In a decidedly hoop-shaped nebulous ring, the differences of velocity between the inner and outer surfaces will be very small; and such a ring, aggregating into a mass whose greatest diameter is at right angles to the plane of the orbit, will almost certainly give to this mass a predominant tendency to rotate in a direction at right angles to the plane of the orbit. ”

“ In the case of a planetary body, there is a real limit to the extent of the transformation of its orbital energy of motion under the influence of the gravitation field. As the orbit of the planet widens, and its mean distance from the primary becomes greater, its velocity in its orbital path must correspondingly decrease. ”

“ The profound and long-continued researches of the illustrious geometer completely established the perpetual variability of the planetary ellipses. He demonstrated that the extremities of their major axes make the circuit of the heavens; that independent of oscillation, the planes of their orbits undergo displacements by which their intersections with the plane of the terrestrial orbit are each year directed toward different stars. ”

“ Instead of a complete revolution, we may only be concerned with part of one orbit. Let’s say that this part lies between the two positions P1 and P2 that the satellite occupies at the two times t1 and t2 (see Figure 6) . Then another of Kepler’s laws says that the ratio between the time difference t2 - t1 and the period T equals the ratio between the sector of the ellipse OP1P2 and the area of the entire ellipse. ”