Hence the time of describing, about the focus, an angle measured from the nearer apse is, in the ellipse, 2 that is, of the sectorial area ASP (figure to § 160); and, h 2 h √(e+1) cos√(e-1) sin 2 Ꮎ √(e+1)cos2+√(e−1)sin that is, of the sectorial area ASP of the hyperbola. Hence these expressions for the time through any area of an elliptic or hyperbolic orbit about a focus might have been written down from the known expressions for the area of an elliptic or hyperbolic sector. 153. In the parabola, if d be the apsidal distance, the integral becomes {since e = · 1, a (1 − e) = d, a (1 − e2) = 2d}, 154. From the result for the ellipse we see that the O periodic time is 2′′ This might also have been found from the consideration of equable description of areas by the radius vector. where n, which is called the Mean Motion, is R 155. By laborious calculation from an immense series of observations of the planets, and of Mars in particular, Kepler was led to enuntiate the following as the laws of the planetary motions about the Sun. I The planets describe, relatively to the Sun, Ellipses of which the Sun occupies a focus. The radius vector of each planet traces out equal areas in equal times. III. The squares of the periodic times of any two planets are as the cubes of the major axes of their orbits. 156. From the second of these laws we conclude that the planets are retained in their orbits by an attraction tending to the Sun. For, If the radius vector of a particle moving in a plane describe equal areas in equal times about a point in that plane, the resultant attraction on the particle tends to that point. Take the point as origin, and let x, y be the co-ordinates of the particle at time t; X, Y the component accelerations due to the attraction acting on it, resolved parallel to the axes; the equations of motion are But by hypothesis, if A be the area traced out by the dA radius vector, is constant. Hence, dt Differentiating, x and by the parallelogram of forces (§ 67) the resultant of X and Y passes through the origin. 157. From the first of these laws it follows that the law of the intensity of the attraction is that of the inverse square of the distance. The polar equation of an Ellipse referred to its focus is and therefore the attraction to the focus requisite for the description of the ellipse is (§ 135) Hence, if the orbit be an ellipse, described about a centre of attraction at the focus, the law of intensity is that of the inverse square of the distance. 158. From the third it follows that the attraction of the Sun (supposed fixed) which acts on unit of mass of each of the planets is the same for each planet at the same distance. For, in the formula in § 154, T2 will not vary as a3 unless μ be constant, i.e. unless the strength of attraction of the Sun be the same for all the planets. We shall find afterwards that for more reasons than one Kepler's laws are only approximate, but their enuntiation was sufficient to enable Newton to propound the doctrine of Universal Gravitation; viz. that every particle of matter in R R the universe attracts every other with an attraction whose direction is that of the line joining them and whose magnitude is as the product of the masses directly, and as the square of the distance inversely; or according to Maxwell's "Matter and Motion," between every pair of particles there is a stress of the nature of a tension, proportional to the product of the masses of the particles divided by the square of their distance. On this hypothesis, neglecting the mutual attractions of the planets, Kepler's third law should be stated (Chap. XI.): The cubes of the major axes of the orbits are as the squares of the periodic times and the sums of the masses of the Sun and the planet. 159. Suppose APA' to be an elliptic orbit described about a centre of attraction in the focus S. Also suppose P to be the position of the particle at any time t. Draw PM perpendicular to the major axis ACA', and produce it to cut the auxiliary circle in the point Q. Let C be the common centre of the curves. Join CQ. When the moving particle is at A, the nearest point of the orbit to S, it is said to be in Perihelion. The angle ASP, or the excess of the particle's longitude over that of the perihelion, is called the True Anomaly. Let us denote it by 0. The angle ACQ is called the Excentric Anomaly, and is 2π generally denoted by u. And if be the time of a complete n revolution, nt is the circular measure of an imaginary angle called the Mean Anomaly; it would evidently be the true anomaly if the particle's angular velocity about S were constant. 160. It is easy from known properties of the ellipse to deduce relations between the mean and excentric, and also between the true and excentric, anomalies; this we proceed to do. |