Low altitude satellites orbiting around planet with twice radius

[totallyclone]

Homework Statement

Two remote planets consist of identical material, but one has a radius twice as large as the other. IF the shortest possible period for a low altitude satellite orbiting the smaller planet is 40 minutes, what is the shortest possible period for a similar low altitude satellite orbiting the larger one? Answer in minutes.

Homework Equations

volume of sphere=4/3 ∏r³
speed=distance/time
E_k=-1/2 E_g

The Attempt at a Solution

r_B=2r_A

time_A=2∏r/V_A
40=2∏r/V_A

Don't really know how to move on from here? I know I have to find the time taken for the satellite to orbit planet B.

T_B=2∏(2r_A)/V_B

I don't know V_B, how would I find that?

 

[PeterO]

Homework Statement

Two remote planets consist of identical material, but one has a radius twice as large as the other. IF the shortest possible period for a low altitude satellite orbiting the smaller planet is 40 minutes, what is the shortest possible period for a similar low altitude satellite orbiting the larger one? Answer in minutes.

Homework Equations

volume of sphere=4/3 ∏r³
speed=distance/time
E_k=-1/2 E_g

The Attempt at a Solution

r_B=2r_A

time_A=2∏r/V_A
40=2∏r/V_A

Don't really know how to move on from here? I know I have to find the time taken for the satellite to orbit planet B.

T_B=2∏(2r_A)/V_B

I don't know V_B, how would I find that?

The larger planet will have a much larger mass [same material presumably means same density]

Radius of orbit around the larger planet is larger [just a little larger than the planet itself]

I would be looking at the effect of doubling the radius of orbit of the satellite around the smaller planet, then the effect of having a larger mass attracting the satellite.

 

[totallyclone]

The larger planet will have a much larger mass [same material presumably means same density]

Radius of orbit around the larger planet is larger [just a little larger than the planet itself]

I would be looking at the effect of doubling the radius of orbit of the satellite around the smaller planet, then the effect of having a larger mass attracting the satellite.

So I doubled the radius. I didn't notice their densities would be the same since they're made of the same material. Anyways:

E_KA=-1/2 E_GA
1/2 mv_A²=-1/2(-GmM_A/r_A)
v_A²=GM_A/r_A

and

E_KB=-1/2 E_GB
1/2 mv_B²=-1/2(-GmM_B/2r_A)
v_B²=GM_B/2r_A

 

[PeterO]

So I doubled the radius. I didn't notice their densities would be the same since they're made of the same material. Anyways:

E_KA=-1/2 E_GA
1/2 mv_A²=-1/2(-GmM_A/r_A)
v_A²=GM_A/r_A

and

E_KB=-1/2 E_GB
1/2 mv_B²=-1/2(-GmM_B/2r_A)
v_B²=GM_B/2r_A

I was looking for an answer like: If you double the radius (while retaining the planet mass), you double/halve/quadruple/quarter the period.

You would then look at: When you double/treble/quadruple/etc the mass of the planet (independent of the radius), you (some change) the Period.

You then combine those effects:

eg if one change means halving, and the other means increasing by a factor of 12; then net result is an increase by a factor of 6.

 

[totallyclone]

Volume of B=4/3 ∏r³
=4/3∏(2r_A)³
=4/3∏(8r_A³)
=8(4/3∏r_A³)
=8 volume of A

So, planet B's volume is 8 times of A's...

 

[PeterO]

So I doubled the radius. I didn't notice their densities would be the same since they're made of the same material. Anyways:

E_KA=-1/2 E_GA
1/2 mv_A²=-1/2(-GmM_A/r_A)
v_A²=GM_A/r_A

and

E_KB=-1/2 E_GB
1/2 mv_B²=-1/2(-GmM_B/2r_A)
v_B²=GM_B/2r_A

I would be addressing the expression a = 4π².R / T²
one of the more useful expressions relating to the acceleration of a body moving in a circle.

 

[PeterO]

Volume of B=4/3 ∏r³
=4/3∏(2r_A)³
=4/3∏(8r_A³)
=8(4/3∏r_A³)
=8 volume of A

So, planet B's volume is 8 times of A's...

Given they are made of the same material (imagine if you were using two steel balls to model the situation) what effect would that have on the mass.