## Gravitation and Keplers Laws

Kepler's three laws, although predating Newton's theory, result from the law of universal gravitation, which postulates that 2 masses mi and m2, separated by a distance R, exert on one another an attractive force F, parallel to the radius vector R that joins their respective centres of mass (this force being described as 'central'). This force is expressed by the relationship:

where G is the universal constant of gravitation (= 6.67 x 10~n N.m2.kg~2).

Kepler's laws are always valid for a system consisting of two bodies, but remain valid for multiple systems (several planets) when the approximation is made that the planets are of negligible mass relative to the central star. The laws were advanced as applying to the Solar System, but may be generalized to apply to any planetary system.

The first law, known as the law of orbits (1605) states that in the heliocentric reference frame, the orbit of each planet is an ellipse, of which one focus is occupied by the Sun.

The second law, known as the law of areas (1604) states that the motion of each planet is such that the section of the straight line joining the Sun and the planet (the radius vector) sweeps out equal areas in equal times.

The third law, known as the law of periods (1618) states that for all the planets, the ratio of the cube of the semi-major axis (a) of the orbit and the square of the orbital period (T ) is constant, and is expressed as:

=Cte = G(mstar + mplanet) _ Gmstar T2 = = 4 n2 ~ 4n2

where mstar and mplanet are the masses of the star and planet, respectively.