Homework Help: Changing the Seasons by Shifting Center of Mass

1. Mar 19, 2017

Seth Newman

1. The problem statement, all variables and given/known data
On Earth, seasons occur due to the tilt of our planet. In the SSDS (A Dyson Sphere built around our sun), we can cause seasons to occur by having the Dyson sphere be not perfectly centered around the sun. If you want the sun to be 10% closer to the one side of the Dyson sphere in “Summer” and 10% further than average in “Winter”, what will be the length of a “year”?

2. Relevant equations
None given, but we are all pretty familiar with angular kinematics, gravitation, and most other ideas in introductory physics.

3. The attempt at a solution
This is a group challenge problem for the last section of intro calc-physics. We are having trouble deciding how to interpret the problem. First, we tried to solve the problem using properties of angular velocity, but it didn't end up making a whole lot of sense. We then made the assumption that the Dyson sphere is similar to the earth-sun system, so we adjusted "R" to be +/- 10% and solved for the period. We got an answer (366.8 days) but conceptually this makes no sense.

Any thoughts/ideas or bumps in the right direction would be greatly appreciated. Thanks!

Last edited: Mar 19, 2017
2. Mar 19, 2017

John Park

I'm not sure what you did here. What do you mean by "adjusted 'R'"--the radius of the Dyson sphere? Are you referring to two different orbits or an eccentric orbit? (By Kepler's laws, how does the period vary with eccentricity?)

3. Mar 19, 2017

Staff: Mentor

If the Dyson sphere (a shell, really) is of uniform density and perfectly symmetric, then what's its gravitational influence on anything inside it? And by symmetry, what's the gravitational influence of anything inside it (a Sun, perhaps) on the Dyson sphere?

4. Mar 19, 2017

John Park

Ah--true. (And obvious.)

5. Mar 19, 2017

Seth Newman

Okay. So if we use the spherical shell theorem, then no net gravitational force on the inside has a net gravitational force on anything outside. So the forces are equal and opposite? Does this mean the sun exerts no net force on the Dyson sphere?

6. Mar 19, 2017

John Park

Yes. As I should have remembered. (For a Niven ring, if you've come across that, it's worse--the situation is unstable and the ring will fall towards the star.)

7. Mar 19, 2017

Staff: Mentor

That's not what the shell theorem states. You might want to review it.
Right. There will be no net force between the Sun and shell (or between the shell and anything else inside the shell). What can you do with this information?

8. Mar 19, 2017

Seth Newman

Ahh. Okay. In a previous problem we calculated the angular velocity needed in order to simulate Earth's gravity within the Dyson sphere (there's supposedly people living inside the sphere) so we should be able to use the angular velocity to calculate the period of one rotation (which is the length of a year)?

9. Mar 19, 2017

Staff: Mentor

Sure.

If the Sun is at the exact center then there would be nothing to distinguish seasons anywhere on the inside surface since the distance would be always the same everywhere. I suppose your job is to place the Sun at a location that allows for seasons as desired. Your rotation rate depends on other factors as you've already determined.

10. Mar 19, 2017

Seth Newman

Great, thanks a bunch. We were stuck in a conceptual hell trying to solve this. I feel bad that it was so obvious. Appreciate it!

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