# Kepler's Laws and orbiting satellite

1. Nov 11, 2009

### G-reg

1. The problem statement, all variables and given/known data
A orbiting satellite stays over a certain spot on the equator of (rotating) Mercury. What is the altitude of the orbit (called a "synchronous orbit")?

Mercury mass = 3.3022 e23 kg
Mercury Radius = 2439.7 km

2. Relevant equations

$$T^2$$ = (4$$\Pi^2$$/GM) * r^3

and that's all I can think of which is why I'm stuck

3. The attempt at a solution

I'm sorry that I have no attempt, I'm really lost on this one and would greatly appreciate any help!

2. Nov 11, 2009

### mgb_phys

Do you know what that equation means?

What T are you looking for?

3. Nov 11, 2009

### G-reg

Yes, the equation is telling you the period of the motion squared.

But we're not looking for the period or the mass or the radius..we're looking for the altitude but i can't find an equation in my text book that would help me out with that.

4. Nov 11, 2009

### mgb_phys

What's 'r' in that equation?

5. Nov 11, 2009

### G-reg

The radius..
Are you hinting that the period is the same thing as the altitude?

6. Nov 11, 2009

### ideasrule

T is the time it takes for the satellite to make one orbit. r isn't the radius; it's the distance from the center of Mercury to the satellite. So to use that equation, you'll need to find Mercury's rotation period.

7. Nov 12, 2009

### G-reg

Oh ok, I think I understand now. So we are looking for 'r'.
And we're using the "law of periods" to solve for 'r'.
Correct?

8. Nov 12, 2009

### G-reg

Well I guess this guess is wrong because I got a wrong answer..
So where should I go from here?

9. Nov 12, 2009

### buffordboy23

What is the definition of altitude? In other words, where is the reference point for which we determine the altitude?

10. Nov 12, 2009

### G-reg

So it should be..

GM * (T/2pi)^2 = x

x - radius of mercury = answer?

11. Nov 12, 2009

### buffordboy23

Hi G-reg,

You are really lost. You don't have enough data. A geosynchronous orbit simply means that the satellite remains fixed over a certain point on the surface of the planet. In other words, it's period is the same as the planet's period (length of Mercury's day). You need to obtain or look up this data for Mercury.

The second point to understand is in regards to your last post. You want to use "x", which represents Mercury's radius. What your saying here is that the satellite's is position is on the surface of Mercury, and this doesn't correspond to the satellite being in orbit. The satellite has some altitude; i.e., it is some distance above Mercury's surface. How can write an expression for the altitude h in terms of Mercury's radius R and the satellite's radial distance r from Mercury's center of mass?

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