Kepler's laws of planetary motion

“ Not only, however, is the first law true of planetary motion, it is equally true of the motions of all satellites moving round their primary planets. I wish, however, to point out, and prove in an indisputable manner, that Kepler's First Law does not sufficiently explain and determine the exact orbit of any satellite as it revolves around its primary planet, or even of any planet as it revolves around the sun. ”

“ But the simple law of universal gravitation, combined, of course, with the laws of motion, not only leads to Kepler's laws as a very close approximation to the actual motions, but also accounts for those slight changes which have just been mentioned as necessary to make Kepler's laws fit observation exactly. We are inevitably led to regard the attraction of gravitation as the cause which keeps the planets in their orbits. ”

“ The planets do not revolve in circles but in ellipses, (see page 126,) and therefore, the distance of each planet from the sun undergoes considerable alteration. Kepler's second law of planetary motion was this: that if we draw a line from the sun to a planet, that line passes over equal areas in every successive hour; that area not being the same for different planets, but being constantly the same for the same planet ”

“ If this law were departed from, then I do not say that it would be impossible for the planet to revolve around the sun in some fashion, but the motion would not be performed in an ellipse described around the sun in the focus.
You see how very instructive are the laws which Kepler discovered. From the first of them we were able to infer that the sun attracts the planets; from the second, we have learned how the magnitude of the attracting force varies. ”

“ Perturbations of the Planets.—Kepler’s laws do not completely express the motion of a planet around a central body, except when no force but the mutual attraction of the two bodies comes into play. When one or more other bodies form a part of the system, their action produces deviations from the elliptic motion, which are called perturbations. The problem of determining the perturbations of the 805 heavenly bodies is perhaps the most complicated with which the mathematical astronomer has to grapple ”

“ All these laws, it will be observed, take for granted the fact that the sun is the centre of the planetary orbits. It must be understood, too, that the earth is constantly regarded, in accordance with the Copernican system, as being itself a member of the planetary system, subject to precisely the same laws as the other planets. Long familiarity has made these wonderful laws of Kepler seem such a matter of course that it is difficult now to appreciate them at their full value. ”

“ The relation between the periodic time of a planet and its mean distance, approximately expressed by Kepler’s third law, follows very simply from the laws of centrifugal force. It is an elementary principle of mechanics that this force varies directly as the product of the distance of the moving body from the centre of motion into the square of its angular velocity. ”

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Then followed a contribution from the professor, Don Jeronimo de la Fuente; it was a serious, violent attack on Kepler's three great laws of the motions of the planetary bodies, or rather on two of them, for he preserved silence on the first, which treats of the elliptical orbit of the planets. ”

“ Besides these, we must have given the position of the planet in the orbit at some specified moment. Having these data, the position of the planet at any other time may be geometrically constructed by Kepler’s laws. The third law enables us to compute the time taken by the radius vector to sweep over the entire area of the orbit, which is identical with the time of revolution. The problem of constructing successive radii vectores, the angles of which are measured off from the radius vector of the body at the original given position, is then a geometric one, known as Kepler’s problem. ”

“ A planet, for example, moves nearly as if no other body but the sun existed, and the moon’s motion relative to the earth is roughly the same as if the other bodies of the solar system were non-existent.
The problem of the motion of two mutually gravitating spheres was completely solved by Newton, and was shewn to lead to Kepler’s first two laws. Hence each body of the solar system could be regarded as moving nearly in an ellipse round some one body, but as slightly disturbed by the action of others. ”

“ They afford yet another great advantage in that the derivation of the results requires only the analytic operations of the infinitesimal calculus.
The power and spirit of the analytic method will be appreciated by showing how it expresses the relations of motion as they were conceived geometrically by Newton and Kepler. It is quite evident that Kepler’s laws do not in themselves enable us to determine the actual motion of the planets. We must have, in addition, in the case of each special planet, certain specific facts, viz. ”

“ We must, however, proceed to the subject of Mars, which was, as before noted, the first important investigation entrusted to Kepler on his arrival at Prague.
The time taken from one opposition of Mars to the next is decidedly unequal at different parts of his orbit, so that many oppositions must be used to determine the mean motion. The ancients had noticed that what was called the “second inequality,” due as we now know to the orbital motion of the earth, only vanished when earth, sun, and planet were in line, i.e. at the planet’s opposition ”

“ Whereas the Ptolemaic system required a number of motions round mere geometrical points, centres of epicycles or eccentrics, equants, etc., unoccupied by any real body, and many such motions were still required by Coppernicus, Kepler’s scheme of the solar system placed a real body, the sun, at the most important point connected with the path of each planet, and dealt similarly with the moon’s motion round the earth and with that of the four satellites round Jupiter. ”

“ He corrects definitely the order of the planets outwards from the sun, a matter which had been in dispute. A notable defect is due to the idea that a body can only revolve about another body or a point, as if rigidly connected with it, so that, in order to keep the earth’s axis in a constant direction in space, he has to invent a third motion. ”

“ We know that the laws of Kepler are a consequence of the laws of gravitation. We know that the planet moves in an elliptic path around the sun, in virtue of the sun's attraction, and we know that the ellipse will be preserved without the minutest alteration if the sun and the planet be left to their mutual attractions, and if no other force intervene. We can also calculate the influence of each of the known planets on the form and position of the orbit. ”

“ In making any hypothesis as to the physical cause of Kepler's Laws, if it can be shown that the same aetherial medium that gives rise to the centrifugal force, also gives rise to the centripetal force, and that the same medium by its rotatory motions also fulfils the three laws of motion, and gives a satisfactory physical explanation of all Kepler's Laws; then, according to our three Rules of Philosophy, we shall have found a physical medium which, by its motions and pressures and tensions, can give rise to all the phenomena exhibited in the celestial mechanism. ”

“ It was reserved for Newton to establish the Laws of Motion, to find the law of force that would constrain a planet to obey Kepler’s first and second Laws, and to prove that it must therefore also obey the third. ”

“ It is a good thing it is so: accurate astronomy would be impossible if we had to take into account the perturbations caused by a crowd of invisible bodies. Astronomy would then approach in complexity some of the problems of physics.
But though we may be convinced from the facts of gravitation that these meteoric stones, and all other bodies flying through space near our solar system, must be constrained by the sun to obey Kepler's laws, and fly round it in some regular elliptic or hyperbolic orbit, what chance have we of determining that orbit? ”

“ Newton's law of gravitation is said to furnish the principle which is involved in Kepler's formula of details; because once Newton's law is admitted as governing planetary motion, what Kepler observed of the movements of the planets, can be deduced by calculation. It would be perhaps more philosophical, in the present state of our knowledge, to regard even the most apparently elementary and fundamental law as only empirical, and the ultimate principle as lying deeper than any known law. ”

“ If we could imagine a planet to be stopped to have its velocity reversed, and then to be started in a precisely opposite direction, it would still continue to describe precisely the same path; it would still obey Kepler’s laws with unfailing accuracy, so far as our present argument is concerned, and the velocity which it would have at each point of the track would be quite the same whether the planet were going one way or whether it was going the other. ”

“ Every science-worker, be his efforts ever so humble, will be able to sympathise with the enthusiastic delight of Kepler when at last, after years of toil, the glorious light broke forth, and that which he considered to be the greatest of his astonishing laws first dawned upon him. Kepler rightly judged that the number of days which a planet required to perform its voyage round the sun must be connected in some manner with the distance from the planet to the sun ”