# Gravitation - Period of revolution of planet

1. Jul 9, 2013

### Pranav-Arora

1. The problem statement, all variables and given/known data
A planet of mass $M$ moves around the Sun along an ellipse so that its minimum distance from the Sun is equal to $r$ and the maximum distance is $R$. Making use of Kepler's laws, find its period of revolution.

(Ans: $\pi \sqrt{(r+R)^3/(2GM)}$)

2. Relevant equations
Kepler's laws:
1. The orbit of every planet is an ellipse with the Sun at one of the two foci.
2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

3. The attempt at a solution
From the third law, $T^2 \propto R^3$ but according to the answer there should be a $(r+R)^3$ and also, I don't know how would I determine the constants here.

Any help is appreciated. Thanks!

2. Jul 9, 2013

### TSny

Does $R$ represent the semi-major axis?

3. Jul 9, 2013

### Pranav-Arora

Silly me, it is $(r+R)/2$, correct now?

4. Jul 9, 2013

### TSny

Yes.

Not sure what you are asking here.

5. Jul 9, 2013

### Pranav-Arora

We have $T^2 \propto (r+R)^3/8 \Rightarrow T^2=k(r+R)^3/8$. How do I determine k here?

6. Jul 9, 2013

### TSny

Note that the formula for the period does not depend on the eccentricity of the ellipse when the period is expressed in terms of the semi-major axis. So, the constant factor will be the same for all elliptical orbits. Pick a value of eccentricity that would make the analysis simple.

7. Jul 9, 2013

### Pranav-Arora

How about eccentricity be zero? :P

Thank you TSny! I have reached the correct answer. :)

8. Jul 9, 2013

### haruspex

Where M is the mass of the planet? Doesn't sound right. If you doubled the mass of the planet, wouldn't it follow the same path and have the same period?

9. Jul 9, 2013

### Pranav-Arora

I think you are right, shouldn't that be the mass of Sun?

10. Jul 10, 2013

### ehild

It should be the mass of the Sun.

ehild

11. Jul 10, 2013

### TSny

Yes, Thanks haruspex and ehild. I didn't even notice that M was given as the mass of the planet.