The Heavens Above: A Popular Handbook of Astronomy

by J. A. Gillet; W. J. Rolfe

Fig. 57.

45. Kepler's Second Law.—Kepler next discovered that a planet's rate of motion in the various parts of its orbit is such that a line drawn from the planet to the sun would always sweep over equal areas in equal times. Thus, in Fig. 57, suppose the planet would move from P to P¹ in the same time that it would move from P² to P³, or from P⁴ to P⁵; then the dark spaces, which would be swept over by a line joining the sun and the planet, in these equal times, would all be equal.

A line drawn from the sun to a planet is called the radius vector of the planet. The radius vector of a planet is shortest when the planet is nearest the sun, or at perihelion, and longest when the planet is farthest from the sun, or at aphelion: hence, in order to have the areas equal, it follows that a planet must move fastest when at perihelion, and slowest at aphelion.

Kepler's Second Law of planetary motion is usually stated as follows: The radius vector of a planet describes equal areas in equal times in every part of the planet's orbit.

46. Kepler's Third Law.—Kepler finally discovered that the periodic times of the planets bear the following relation to the distances of the planets from the sun: The squares of the periodic times of the planets are to each other as the cubes of their mean distances from the sun. This is known as Kepler's Third Law of planetary motion. By periodic time is meant the time it takes a planet to revolve around the sun.

These three laws of Kepler's are the foundation of modern physical astronomy.

The Newtonian System.

47. Newton's Discovery.—Newton followed Kepler, and by means of his three laws of planetary motion made his own immortal discovery of the law of gravitation. This law is as follows: Every portion of matter in the universe attracts every other portion with a force varying directly as the product of the masses acted upon, and inversely as the square of the distances between them.

48. The Conic Sections.—The conic sections are the figures formed by the various plane sections of a right cone. There are four classes of figures formed by these sections, according to the angle which the plane of the section makes with the axis of the cone.

OPQ, Fig. 58, is a right cone, and ON is its axis. Any section, AB, of this cone, whose plane is perpendicular to the axis of the cone, is a circle.