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The Elliptic Orbits of the Planets the Law of the Inverse Square

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The Elliptic Orbits of the Planets the Law of the Inverse Square The Elliptic Orbits of the Planets Before proceeding with the subject of orbit dynamics, wc shall briefly describe the remarkable achievement of Johannes Kepler in establishing the proposition that the planetary orbits are not compounded circles but simple ellipses. This great discovery was based almost entirely on the analysis of the motion of a single planet—Mars. The behavior of Mars had puzzled and exasperated astronomers for a very long time, because the apparent irregularities in its motion were greater than those of any other planet and defied any easy analysis in terms of combinations of uniform circular motions. To appreciate the development of Kepler's discovery one must constantly keep in mind the fact that the primary data of observational astronomy (and this was especially true in the days before the telescope) are directions rather than distances. Although it was well understood that variations in the apparent brightness of the planets were linked to variations of distance from the earth, the precise data were only of angular positions. The whole theoretical machinery of superposed circular motions was primarily a means of reproducing the observed angular position of each planet as a function of time. Kepler began his study of Mars at the direction of the great observational astronomer Tycho Brahe, whom he joined as an assistant in 1600. Kepler's task was to construct the actual path of Mars in space from the accumulation of original observations; it took him 6 years, and many false scents, before he arrived at the picture that is now familiar to us. Kepler fully accepted a heliocentric model of the solar system, and (unlike Copernicus himself) he consistently held to the idea that the path of a planet must be a smooth, continuous curve of some kind around the sun. His problem was to find this curve on the basis of observations made from a laboratory—the earth— which was itself orbiting the sun in a nonuniform way. A first task was therefore to establish the path of the earth itself. Kepler attacked this problem in several ways. The one most directly based on observation was brilliant. Kepler published the full story of his labors—the many failures as well as the final successes—in a book, The New Astronomy (Astronomia Nova), published in 1609. It is a classic of scientific discovery. The law of the inverse square Newton’s law of gravitational attraction states that two particles of masses , at a distance r apart, attract one another with equal and opposite forces of magnitude , (1) where G is the gravitational constant. Coulomb’s law of electrostatic attraction states that two particles carrying electric charges (in electrostatic units), at a distance r apart, repel one another with equal and opposite forces of magnitude . (2) If and have opposite signs, this force is a force of attraction. Here we have two examples of the law of the inverse square. The law (1) governs astronomical phenomena – in particular, the motion of a planet round the sun. The law (2) governs atomic phenomena – in particular, the motion of an electron in an atom about the central nucleus. In this case, of course, the charges have opposite signs, so that the force is one of attraction, as in the gravitational case. It is remarkable that the same form for the law of attraction should hold on such different scales. The expressions (1) and (2), combined with Newton’s law of motion, constitute two hypothesis regarding phenomena in gravitational and electrostatic fields. For a long time, they were accepted as completely valid from a physical point of view, but that is no longer the case. The modern astronomer knows that gravitational attraction should be discussed in terms of the general theory of relativity, and the physicist insists that problems on the atomic scale belong to quantum mechanics. It would, however, create a completely false impression if we were to say that the law of the inverse square has disappeared from modern science. Nearly all the calculations of astronomers are still bases on (1) and give results in excellent agreement with observation. Moreover, the physicist often falls back on simple atomic picture based on (2) and Newton’s law of motion. In what follows, we shall discuss the motion of a planet attracted by the sun. Obviously, be a mere change of constant, the same reasoning will apply to the motion of an electron in an atom. Determination of the orbit The sun and a planet are regarded as particles, of masses M and m, respectively. The attraction of the sun on the planet, given by (1), produces an acceleration of the sun and treats it as if it were at rest. We consider then the case of a particle attracted toward a fixed centre by a force per unit mass, where , (3) being some positive constant. The polar differential equation (the Path Equation (6) of Mechanics-3) for the orbit now becomes . (4) The general solution is , (5) where C and are constants of integration. This is, in polar coordinates, the equation of the most general orbit described under a central force varying as the inverse square of the distance. The potential energy per unit mass (given by equation (1b) of Mechanics-3) is , (6) the constant of integration being chosen to make V vanish at infinity. Let us now substitute from (5) in equation (7) of Mechanics-3 i.e. , the equation of energy, in order to express the constant C in terms of E and h (the total energy and angular momentum per unit mass). We get , So that (7) By rotating the base line , we can make and in (5); this we shall suppose done. Then the equation (5) for the orbit reads . (8) From the focus-directrix property of conic, we know that its equation in polar coordinates may be written as , (9) Where l is the semi-latus-rectum (i.e. half the focal chord parallel to the directrix) and e the eccentricity; is measured from the perpendicular dropped from the focus on the directrix. The conic may be of any of the following types : ellipse , parabola , hyperbola . In the case of the hyperbola, (9) gives only the branch adjacent to the focus. Comparing (8) and (9), we note that it is always possible to bring the equations into complete agreement by choosing for and the values . (10) Accordingly, we may say: The orbit described by a particle, attracted to a fixed centre by a force varying as the inverse square of the distance, is a conic having the centre of force for focus. The semi-latus-rectum and the eccentricity are given by (10) in terms of the angular momentum and energy per unit mass. The orbit may be of the following types: ellipse , parabola , hyperbola . The fact that orbits may be classified thus in terms of the total energy is remarkable. Possible orbits under an inverse square attractive central conservative force The most important orbits in astronomy (those of the planets) are ellipses. Recurring comets describe orbits which are elongated ellipses, approximating to parabolas. A body with a parabolic or hyperbolic orbit would pass out from the solar system, never to return. Constants of the elliptical orbit Let us now confine our attention to the elliptical orbit. Since the orbits of the planets are of this type, a great wealth of technical detail has been developed about the elliptical orbit. We shall discuss here in brief. It is evident from (9) that the shape and size of an orbit (but not its orientation in space) are determined by the two constants l, e. These are related to the constants E, h by (10). Thus, of the various constants which appear in our equations, we are to regard (the intensity of the force centre) as given once for all, whereas the constants l, e, E, h take different values for different orbits. On account of (10) only two of these constants are independent. We may use as an independent pair any two which prove convenient. Instead of using (l, e) as fundamental constants, it is better to use (a, e), where a is the semiaxis major of the orbit. Now, , (11) B being the semiaxis minor. We shall refer to (a, e) as the geometrical constants of an orbit and (E, h) as its dynamical constants. The formulas of transformation from one set to the other are as follows : (12) Velocity at any point There is a simple formula giving the speed at any point of the orbit in terms of the radius vector. By the equation of energy (equation (1) of Mechanics-3) , and (equation (1b) of Mechanics-3) we have by (3) . (since , therefore ) Substituting for E from (12), we obtain (13) The periodic time We now ask : How long does the particle take to describe the elliptical orbit ? This time is called the periodic time ( ). We seek an expression for in terms of the fundamental constants. For obtaining periodic time we refer to the equation (16) of Mechanics-3, which gives for the areal velocity of a particle moving along a plane curve . Figure : The rate of increase of A is constant If F is the focus at which the centre of force is situated, it follows at once that the particle describes an arc VP, starting from the vertex V nearer to F, in a time , where is the area of the sector subtended at F by this arc (see figure above). The vertex V is called perihelion – the point closest to the sun – the other vertex being called aphelion – the point away from the sun.
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