Kepler's 3rd Law problem?

Question

A planet moves in an elliptical orbit around the sun. The mass of the sun is M. The minimum and maximum distances of the planet from the sun are R1 and R2, respectively.

Using Kepler's 3rd law and Newton's law of universal gravitation, find the period of revolution P of the planet as it moves around the sun. Assume that the mass of the planet is much smaller than the mass of the sun.

Use G for the gravitational constant.

Express the period in terms of G, M, R1, and R2.

Thanks for any help!!

Answer 1

Kepler's third law states the square of the period is proportional to the cube of the average distance. The rest of the equation is just some constants - of course, you need the constants if you want to attach a number to the period instead of just a relationship.

The average distance is equal to half the long axis (the semi-major axis). The minimum and maximum distance occur at opposite sides of the orbit, meaning their sum is equal to the long axis. Divide the sum by 2 to get the semi-major axis.

That gives you a formula of:

T = sqrt { 4 * PI^2 /(GM) * [(R1 + R2)/2]^3 }

(The constants were grouped together to focus on the relationship of the variables.)