Question

A planet of mass m moves around the sun of mass M in an elliptical orbit .the maximum and minimum distances of the planet from the sun are r1 and r2 respectively .the time period of the planet is proportional to

Answer 1

Given the perihelion = r2 = shortest distance of a planet from Sun
                 aphelion = r1 = longest distance of planet from Sun
                 mass of planet = m.
                 mass of Sun = M

So the length of the major axis of the Elliptical orbit of the planet
    = r1 + r2.

Semimajor axis = R = (r1 + r2)/2.

According to Kepler's laws:

    The square of time period T of a planet revolving around Sun is proportional to the cube of semi major axis R of the elliptical orbit of the planet.

    T² ∞ R³
    T ∞  R³/²
    T  ∞ [r1 + r2] ³/²     Answer.

Actual value of the time period is given by:

$T=2 \pi \sqrt{\frac{R^3}{GM}} = 2 \pi \sqrt{\frac{(r_1+r_2)^3}{8G M}}$

Answer 2

semi major axis= (r1+r2)/2.

T2 proportional to r3

T proportional to [(r1+r2)/2]power 3/2