# Period of Planets orbiting a Star

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1. May 4, 2015

### Soniteflash

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

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

3. 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 .....

2. May 4, 2015

### BvU

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 !

3. May 4, 2015

### Soniteflash

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?

4. May 4, 2015

### BvU

How do you deduce that from your equations ?

5. May 4, 2015

### Soniteflash

I cancelled m1 out.

6. May 4, 2015

### BvU

I mean how do you deduce that it's B and not A

7. May 4, 2015

### Soniteflash

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.

8. May 4, 2015

### BvU

Something like that. If you work out T2 you get something with .../(GM..)

9. May 4, 2015

### Soniteflash

How would your approach look like? I am wondering how I would arrive at T2

10. May 5, 2015

### BvU

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)