Changing the Seasons by Shifting Center of Mass

Homework Statement

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”?

Homework Equations

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

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!

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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.

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?)

gneill
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?

Ah--true. (And obvious.)

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?

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?

Does this mean the sun exerts no net force on the Dyson sphere?
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.)

• Seth Newman
gneill
Mentor
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.
That's not what the shell theorem states. You might want to review it.
So the forces are equal and opposite? Does this mean the sun exerts no net force on the Dyson sphere?
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?

• Seth Newman
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?

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)?

gneill
Mentor
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)?
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.

• Seth Newman
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.

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!