Two Planets Orbiting an Empty Spot

[SHISHKABOB]

Okay so I just had this problem on my astronomy midterm and I was sort of stumped by it. I hope I can remember all of it.

Basically there are two planets orbiting around what appears to be a blank area of space. From observations we can see that they are not in a binary orbit with each other and that there is an object of large mass in the blank spot that does not emit any light. Its mass is much greater than that of each of the two planets. In this problem, the orbital period of the inner planet is defined as a "new year" or y_n and the orbital radius is defined as r_o. M_d is defined as a unit of mass. The gravitational constant is defined as 4\pi²y_n^-2r₀^-3M_d^-1

so basically I saw that I was supposed to use Kepler's third law here, where P² = \frac{4\pi^{2}}{GM}a³

The first planet has y_n = 1 and r_o = 1
The second planet has y_n = 1 and r_o = 3

in each case I needed to solve for the mass of the "invisible" object and I got 1 M_d and 3 M_d respectively.

The problem then asks me to reconcile the difference between these two numbers and... this is where I had no idea what to do. It tells me that the observations can be made many times without a large error and that the gravitational potential energy is such that no correction from general relativity needs to be made.

I simply had no idea what to do so I jokingly answered that there was a sphere of dark matter beyond the orbit of the second planet.

There's *obviously* something wrong with this system, right? The mass of the central body should be the same in both cases. But I have no idea where I should have gone from there.

 

[Mike Pemulis]

I'll just leave this here.

 

[SHISHKABOB]

I'll just leave this here.

augh, my only recourse was not even a viable recourse

 

[Mike Pemulis]

Yes, but you had the right idea. Read the article again and think.

 

[gulfcoastfella]

Yeah, this is the problem that astronomers face with galaxies, right? When they measure the velocities of stars in the galaxy, it doesn't match up with what is expected from the gravity of visible matter. The only thing that makes sense is for dark matter to be distributed throughout the galaxy. Is that right?

 

[SHISHKABOB]

the only other thing I could think of is another large body beyond the second planet that is also orbiting the dark body :/

 

[Mike Pemulis]

That wouldn't work, right? An third planet might disturb the second planet's orbit, deforming it away from a perfect circle, but it wouldn't really make it go faster. And that's what we need to explain.

Let's try again.

We need to explain why the outer planet orbits faster than we would expect, given the orbit of the second planet. We need to explain this by invoking gravity only; it is no fair to attach a giant rocket to the outer planet to push it around faster.

So there needs to be some large mass which affects the outer planet but not the inner planet. How is that possible? Read the Wikipedia article again.

 

[SHISHKABOB]

a sphere of mass between the orbits? Or would even just a ring work? A ring in the same plane as both the planets' orbits

 

[Mike Pemulis]

Yup, a sphere between the orbits. If I had to guess, I would say a ring would work too, but I'm not positive. A sphere would definitely work.

 

[SHISHKABOB]

well, I guess that makes sense... thanks for pointing me in an interesting direction though. I guess I'll find out tomorrow or sometime this week when he's got the tests all done. Maybe it was an open ended question... lol

though of course, then'd I'd still be wrong, heh