Finding Orbital Period of Unknown Planet

[tristanslater]

Homework Statement

A satellite is in circular orbit at an altitude of 800 km above the surface of a nonrotating planet with an orbital speed of 3.7 km/s. The minimum speed needed to escape from the surface of the planet is 9.8 km/s, and G = 6.67 × 10-11 N · m2/kg2. The orbital period of the satellite is closest to

Homework Equations

This question is supposed to be related to Kepler's law, so I imagine it has something to do with:

$k = \frac{T^2}{R^3}$

Another period equation I've tried:

$T = \frac{2\pi\sqrt{R}^3}{\sqrt{GM}}$

I think energy is going to factor in somehow, so:

$K = \frac{1}{2}mv^2$ and $U = \frac{GMm}{R}$

And escape velocity is given, so maybe:

$v_e = \sqrt{\frac{2GM}{R}}$

The Attempt at a Solution

Everything I try, it seems like there is not enough information.

I tried starting with:

$T = \frac{2\pi\sqrt{R}^3}{\sqrt{GM}}$

Then I solved the escape velocity formula for $\sqrt{GM}$:

$\sqrt{GM} = v_e\sqrt{\frac{R}{2}}$

This way I can sub it into the perio equation:

$T = \frac{2\pi\sqrt{r}^3}{v_e\sqrt{\frac{R}{2}}}$

Simplifying I get:

$T = \frac{2\pi R\sqrt{2}}{v_e}$

This gets rid of most of the unknowns, but it still contains R, which we don't know. This is where I get stuck. How can we find a substitution for R?

Thanks.

 

[Orodruin]

You have not used the information on the orbital velocity. Without it there would indeed be too little information.

 

[tristanslater]

You have not used the information on the orbital velocity. Without it there would indeed be too little information.

I would love to use the velocity, but I can't seem to find a way to incorporate both velocities. I could use $T = \frac{2\pi R}{v}$, but I'm still stuck with $R$. I've tried working around other equations, but it always seems that without either the radius of the planet or the mass of the planet, I reach a dead end. This kind of makes intuitive sense as well. If the size of the planet went down, the satellite would be closer to the center of gravity, but if the mass also went down, the speed could stay the same. Same with escape velocity. We don't know where the surface is, so it doesn't give us much information. It seems like one of those factors is crucial. Or is there some relationship between the escape velocity and orbital velocity that tells us something?

Thanks.

 

[Orodruin]

So you have two equations and two unknowns. This is a solvable system.

Also note that the escape velocity is not given at the same radius as the satellite.

 

[tristanslater]

So you have two equations and two unknowns. This is a solvable system.

Also note that the escape velocity is not given at the same radius as the satellite.

So you have two equations and two unknowns. This is a solvable system.

Also note that the escape velocity is not given at the same radius as the satellite.

Oops, you're right, I overlooked the difference in radius. The escape velocity is at $R$, and the orbital velocity is at $R+h$, and we know $h$, so that maintains the number of unknowns. I'm not sure we can use those two equations as a system though, because they are the same relationship. I think we need another relationship. I tried $a = \frac{v^2}{R}$, but that just introduces $a$.

Can someone give me a little guidance?

Thank you.

 

[Orodruin]

But you already have two relations relating T and R in terms of known variables! Why do you want more?