Question

3. A Renaissance astronomer, Johannes Kepler, observed that the amount of time that a planet takes to travel around the Sun T is proportional to the 3/2 power of the long axis of its orbit. We are not equipped to analyze this for general orbits. However, we can calculate this for a circular orbit. Consider a planet of mass m orbiting a star of mass M in a circular orbit of radius r. Find the relationship between the time T it takes to travel around the Sun and the radius r of its orbit. (Your answer will also involve the gravitational constant and the mass of the star.) Show that your value for the time it takes to go around the Sun is proportional to the 3/2 power of the radius. Hint: Think about the relationship between T and w.

Answer 1

In a circle, the speed stays constant as v

so,

we can write

F = ma = -GMm / r²

cancel 'm' on both sides, we get

a = - GM / r² --------- (1)

Note that object moving in circular motion is always accelerating due to change in direction.

The object experience the centripetal acceleration given as

a_c = - w² r -------- (2)

where

w = 2$\pi$ / T

put (1) in (2)

- (2$\pi$ / T)² r = - GM / r²

4$\pi$² / T² = GM / r³

rearrange, we can write as

T² = ( 4$\pi$² / GM) r³

as ( 4$\pi$² / GM) is constant for a given star, we can also write as

T² = r³

T = r^3/2