Kepler Laws of Planetary Motion:
After carrying out the analysis of the whole astronomical data on the basis of Copernicus theory, Kepler in 1608 obtained three laws for the orbital motion of the planets. In terms of modern terminology, the three laws are as follows-
(I) Kepler’s First Law or Law of Orbits- The orbits of the planets are elliptical with the sun at one focus. Kepler first law reaffirms the Copernicus picture of the heliocentric solar system and at the same time makes a significant modification regarding the shape of the orbits. Copernicus treated orbits as simple circles but Kepler pointed out that although the orbits of planets are very nearly circular but actually they are elliptical.
(II) Kepler’s Second Law or Law of Areas- A line joining from the sun to the planet sweeps out equal areas in equal time. If a planet P moves from A to B in a certain interval of time and again moves from A’ to B’ in the same interval of time, then according to Kepler second law the area ASB and A’SB’ must be equal where S represents the position of the sun. In other words, the areal velocity of the planet remains constant. The linear speed, however, changes from place to place, being maximum where the planet is nearest to the sun. The nearest position of the planet from the sun is called perihelion and the farthest position is aphelion.
(III) Kepler’s Third Law or Law of Periods- The squares of the periods of revolution of the planets are proportional to the cubes of the semi-major axis of the elliptical orbit.
If the period of revolution is ‘T’ and the semi-major axis of the orbit is ‘a’, then
|T2 ∝ a3|
or T2 = Ka3
Where ‘K’ is a constant. For circular orbits since a = b = r.
Where ‘r’ is the radius of the orbit,
|T2 = Kr3|
Significance of Kepler Laws:
Kepler laws serve to describe very accurately the orbits of the planets, however, they do not give directly the underlying mechanics to which they must somehow be related.
Kepler laws of planetary motion provided a very exciting challenge to the philosophers of the 17th century. Why does a planet move in an elliptical path? What can be the fundamental principle of mechanics which may lead to the derivation of Kepler’s laws? In order to examine these questions, let us examine the Kepler laws critically.
The first law of Kepler clearly suggests that there must be some force of attraction between a planet and the sun. The force of attraction between two material bodies is known as gravitational force. Now the question arises regarding the direction and magnitude of this gravitational force.
The second law of Kepler gives information regarding the direction of the gravitational force that should act between the planet and the sun. On the basis of Kepler’s second law, it can be proved that the angular momentum of the planet around the sun remains constant.
In other words, the gravitational force acting on the planet is such that it has no perpendicular component i.e., it acts along the line joining the planet with the sun.
The third law of Kepler can be utilized to know the factors on which the magnitude of this force depends. Assuming that a planet of mass ‘m’ is revolving in an orbit of radius ‘r’ with a velocity ‘ν’, we find that as the gravitational force must provide the centripetal force, it must be given by-
|F = mν2/r = m/r(2πr/T)2 = 4π2mr/T2|
Where ‘T’ is the period of revolution. But according to Kepler’s third law, T2 = Kr3, therefore replacing T2 by Kr3, we get from the above relations-
|F = 4π2mr/Kr3 = (4π2/K)m/r2|
or F ∝ m/r2
The gravitational force depends directly on the mass of the planet and inversely on the square of the distance between the planet and the sun. Since the force is mutual and acts on the sun as well, it is not unreasonable to suppose that it is also proportional to the mass ‘M’ of the sun. Therefore, we have
| F ∝ Mm/r2 |
or F = G Mm/r2
Where G is a constant.