Gravitation - Period of revolution of planet

In summary, a planet of mass ##M## moves around the Sun along an ellipse with minimum distance ##r## and maximum distance ##R##. Using Kepler's laws, the period of revolution can be calculated to be ##\pi \sqrt{(r+R)^3/(2Gm)}##, where ##m## is the mass of the Sun.
  • #1
Saitama
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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!
 
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  • #2
Pranav-Arora said:
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?
 
  • #3
TSny said:
Does ##R## represent the semi-major axis?

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

What about the constants? :confused:
 
  • #4
Pranav-Arora said:
Silly me, it is ##(r+R)/2##, correct now?
Yes.

What about the constants? :confused:

Not sure what you are asking here.
 
  • #5
TSny said:
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?
 
  • #6
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.
 
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  • #7
TSny said:
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. :)
 
  • #8
Pranav-Arora said:

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?
 
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  • #9
haruspex said:
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? :confused:
 
  • #10
It should be the mass of the Sun.

ehild
 
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  • #11
Yes, Thanks haruspex and ehild. I didn't even notice that M was given as the mass of the planet.
 

Related to Gravitation - Period of revolution of planet

1. What is the period of revolution of a planet?

The period of revolution of a planet refers to the time it takes for a planet to complete one full orbit around its star. It is also known as the orbital period.

2. How is the period of revolution of a planet calculated?

The period of revolution can be calculated using Kepler's Third Law, which states that the square of a planet's orbital period is proportional to the cube of its semi-major axis. This means that the farther a planet is from its star, the longer its period of revolution will be.

3. What affects the period of revolution of a planet?

The period of revolution of a planet is primarily affected by the mass of the planet and the distance between the planet and its star. Other factors, such as the presence of other planets in the system, can also have an impact.

4. How does the period of revolution relate to a planet's speed?

The period of revolution and a planet's speed are inversely related. This means that as the period of revolution increases, the planet's speed decreases. This is because the planet has a greater distance to travel in the same amount of time.

5. Can the period of revolution change over time?

Yes, the period of revolution of a planet can change over time due to gravitational interactions with other celestial bodies. For example, if a planet's orbit is affected by the gravitational pull of a nearby planet, its period of revolution may change slightly.

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