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Planetary Motion

On the basis of exhaustive data recorded by Tycho Brahe on the motion of planets, Johannes Kepler in 1609, after sixteen years of intense effort solved the mystery of planetary motion. Newton then discovered his law of universal gravitation from the observations of Galileo regarding motion of objects on earth and those of Kepler regarding motion of planets. We shall now, going backwards, deduce Kepler’s laws of planetary motion on the basis of Newton’s law of gravitation.

According to Newton’s law of gravitation, any two point masses M₁ and M₂ separated by distance r (at any instant) attract each other by an instantaneous force of gravitation given by,

where G is universal gravitational constant.

In respect of planetary motion, Newton showed that each planet moves around the Sun under the influence of gravitational force exerted by Sun on the planet. That is, the motion of a planet is equivalent to motion of a particle of mass m moving in the central force of gravitational produced by Sun as the fixed center of force:

where M_s is the mass of Sun. There are two points to note about the above statement:

1. Planetary problem is essentially a two body problem. However, if one mass is very large as compared to other (M_s >> m), then the heavy mass (M_s) can be taken as the fixed center of force.

2. The sun and planets are not point particles. However, as we shall show that the gravitational force behaves as if the entire mass of a sphere is concentrated as its center. Hence, r refers to the distance between the centersof Sun and the plant (both being regarded as spheres).