Period of Planets orbiting a Star

AI Thread Summary
The discussion centers on the orbital periods of two planets orbiting different stars with the same radius, where Planet 1 has a longer period than Planet 2. The key point is that the mass of the stars affects the orbital speed and period; specifically, if Planet 1 orbits a less massive star (Star 1), it will have a longer orbital period compared to Planet 2 orbiting a more massive star (Star 2). The equations used in the analysis clarify that the mass of the planets is negligible compared to the stars, leading to the conclusion that the correct answer is that Star 1 has less mass than Star 2. The conversation also highlights the importance of correctly identifying variables in gravitational equations to avoid confusion. Ultimately, understanding the relationship between mass and orbital speed is crucial for determining the periods of the planets.
Soniteflash
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Homework Statement


Planet 1 orbits Star 1 and Planet 2 orbits Star 2 in circular orbits of the same radius. However, the orbital period of Planet 1 is longer than the orbital period of Planet 2. What could explain this?

A) Star 1 has less mass than Star2.
B) Star 1 has more mass than Star 2
C) Planet 1 has less mass than Planet 2
D )Planet 1 has more mass than Planet 2.
E) The masses of the planet are much less than the masses of the stars.

Homework Equations



F=(G m1 x m2 ) / (r2)
ac = mv2 / r
(2π x r ) / T = V

The Attempt at a Solution


I think it is C.
I used F=(G m1 x m2 ) / (r2) and set it equal to mv2 / r
I didn't see anything related to Period so I remembered that circumference divided by period equals V.
I solved for V in the first equation and got : v = √[(Gm)/r]. Mass is the only variable that could cause the speed and therefore the Period to change. So I thought that increasing the mass of Planet would increase the speed and make the Period longer.
I probably messed up ...
 
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You sure the mass in your expression is the planet mass and not the star mass ? In short: sort out which is which in m1, m2 and m and which one divides out !
 
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BvU said:
You sure the mass in your expression is the planet mass and not the star mass ? In short: sort out which is which in m1, m2 and m and which one divides out !

Ohhhhh, dumb me...
I assigned m1 as the Planet and m2 as the Star.
Since the m in mv2 / r is referring to the orbiting mass, the mass of the planet cancels out and leaves me with the star so it should be B right?
 
How do you deduce that from your equations ?
 
I canceled m1 out.
 
I mean how do you deduce that it's B and not A
 
So if V increases that means the period would increase. Oh wait. T is in the denominator of 2πr / T.. so that means if I increase V that would mean that Period would go down. I assume that's my mistake. So it has to be that T increases when V decreases meaning that the mass of the star has to be less. So A.
 
Something like that. If you work out T2 you get something with .../(GM..)
 
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BvU said:
Something like that. If you work out T2 you get something with .../(GM..)
How would your approach look like? I am wondering how I would arrive at T2
 
  • #10
  1. F=(G m1 x m2 ) / r2
  2. not ac but Fc= m2v2 / r
  3. (2π x r ) / T = v
3: ##\ \ \displaystyle {1\over T^2} = \left ({v\over 2\pi r} \right )^2\ \ ##. Now equate 1 and 2:
$${v^2\over r} = {Gm_1 \over r^2} \quad \Rightarrow \quad 1/T^2 = \left ({v\over 2\pi r} \right )^2 = {1\over (2\pi)^2} {Gm_1 \over r^3} \quad \Rightarrow \\ T = 2\pi\sqrt {r^3\over Gm_1} $$

as in wikipedia (here you can ignore the planet mass m wrt the M of the star)
 
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  • #11
why did this get so complicated... [V][/orb]=√GM/r
And since the problem indicates that r is the same for both systems, it isn't going to affect the speed of the orbital, G is also a constant so it isn't going to affect the system, however, a change in M can affect the speed of the orbital, for example: If you increase M the V orbital is going to speed up, and if you decrease M its going to slow down. Knowing that, we can see that the planet is taking longer to orbit because its slower than the other planet, so V orbital is slower, and since M is the only changing variable, we can see that M for planet 1 is less than planet 2.
 
  • #12
Hello MiaPow, :welcome:

Don't hijack another thread -- Start a new thread and state the problem that YOU have when working this out. Where do you get stuck ?

What kind of assistnce can be brought to bear on a question like 'Why did this get so complicated ... ?'
I don't think it's all that complicated at all !
 
  • #13
Oh I apologize for my poor wording. I had no problem following through your solution, just an alternative method of answering.
 
  • #14
No problem. The OP probably needed some specific guidance and things got intricate before they became (almost) straightforward. Not everyone 'sees through' an exercise straightaway :rolleyes: (not even helpers...)
And, eh: :welcome:
 
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