Tycho Brahe’s largest and most accurate collections of planetary observations helped Kepler to present the first ever natural laws First law: Two planets P1 and P2 move on separate ellipses with the Sun at a common focus. Second law: For planet P1 the line SP1 sweeps out equal area in equal intervals of time. Third law: The square of the time taken by P1 and P2 to complete their respective orbits are in proportion to the cube of their respective semi-major axes. First law indicates clearly what path the orbiting planets follow. Second law explains about the speed of a planet i.e. when the planet is nearest to the Sun (focus) speed is less e.g. at perihelion. When planet is farthest to the Sun e.g. at the aphelion the speed is more, so that at equal intervals of time the line joining the Sun and planet sweeps out equal areas. Third law is purely qualitative and compares the orbital motion of the different planets i.e. the ratio of square of period of revolution to the cube of semi-major axes is constant. If T1 and T2 are periods of two planets P1 and P2 and a1 and a2 are their respective semi-major axes then T12a13 = T22a23 =constant
The Kepler’s laws are a-posteriori i.e. derived by experience. Hence none of the laws state why planets follow elliptical orbit why not straight lines. In those days one of the theories proposed was that the planets went around the elliptical orbits because behind them were the invisible angels beating their wings and driving the planets forward. Newton freed nature from an all embracing god-relatedness of Kepler’s era by making postulates which allow the determination of concepts such as mass, inertia, force, cause, space, time and motion. Newton had become the first systematizer of modern science. Before Newton Galileo discovered a very remarkable fact about motion, that is the principle of inertia- if something is moving with nothing touching it and completely undisturbed, it will go on for-ever, coasting at a uniform speed in a straight line. Newton modified his idea saying that the only way to change the motion of a body is to use force. If the body speed up, a force has been applied sideways, i.e. a force is needed to change the speed or the direction of motion of a body.
2. Inverse square law of gravitation
The fact that the planets does not move in a straight line with uniform speed indicates that there is a force acting on it and it is proportional to the acceleration according to first and second laws of Newton. Newton found the acceleration of a planet by two methods: the first conventional method he used was the method of geometry, the second method was the use of new technique of mathematics, which Newton himself was invented, the calculus.
Figure 1.Illustration of Kepler’s laws. Two planets P1 and P2 are shown to move on separate ellipses with the Sun S as a common focus. For planet P1, for example, the second focus is at S’. The line joining S to S’ and extended in both directions meets the orbit of P1 at A and A’. The point A is closest to the Sun and is called Perihelion. While A’ is the farthest point and is called aphelion. The length AA’ is called the major axis of the ellipse and is denoted by 2a. The ratio of lengths SS’ to AA’ is called the eccentricity of the ellipse.
Refer to fig-1 and denote the distance SP1 by r and the angle P1SA’ by θ. Then according to Kepler’s first law P1 moves along an ellipse with focus at S, its equation is given by
1r = 1l1-ecosθ
Where l is constant known as semi-latus rectum...