Gravitation - Period of revolution of planet

[Saitama]

Homework Statement

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

Homework 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.

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!

 

[TSny]

Kepler's laws:
...
3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

The attempt at a solution[/b]
From the third law, $T^2 \propto R^3$

Does $R$ represent the semi-major axis?

 

[Saitama]

Does $R$ represent the semi-major axis?

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

What about the constants?

 

[TSny]

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

Yes.

What about the constants?

Not sure what you are asking here.

 

[Saitama]

Not sure what you are asking here.

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

 

[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.

 

[Saitama]

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.

How about eccentricity be zero? :P

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

 

[haruspex]

Homework Statement

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

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?

 

[Saitama]

Where M is the mass of the planet? Doesn't sound right.

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

 

[ehild]

It should be the mass of the Sun.

ehild

 

[TSny]

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