KEPLER’S LAWS

Kepler’s first law or law of orbits:

It states “all the planets moves around the sun in an elliptical orbit with sun at one of the focus of ellipse”. The point when the planet is nearest to the sun is termed as point when the planet is nearest to the sun is termed as perihelion and the farthest one is known as aphelion.

Kepler’s second law or law of areas:

An imaginary line drawn from the centre of the sun to the centre of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)

Hence, ΔA = 1/2r (rΔθ) = 1/2r2Δθ

You can write, ΔA/Δt = 1/2r2 Δθ/Δt

⇒limΔt→0 ΔAΔt =12r2.  limΔt→0 ΔθΔt  Taking limits both sides as Δt → 0

=> dA/dt = 1/2r2ω

=> dA/dt = l/2m

Now, by conservation of angular momentum, L is constant,

Thus, dA/dt = Constant

Third Law (Law of Periods):

The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

By Kepler’s 3rd law can be derived from newton’s inverse square law of gravitation where gravitation force acts as centripetal force for the planet

Therefore mv2/r = GmM/r2

Where, m – mass of planet

M - Mass of sun

v2 = GM/r

Where v = rω

v2 = r2ω2

v2r = GM

r2ω2 = GM

ω2 = GM/r3

(2π)2/T2 = GM/r3

r3(2π)2 = (GM) T2

T2 α r3