- #1

- 78

- 0

**[SOLVED] Density of a planet using orbit of a satellite**

## Homework Statement

A satellite is in a circular orbit about a planet of radius R. If the altitude of the satellite is h and its period is T, show that the density of the planet is [tex]\rho=\frac{3\pi}{GT^2}(1+\frac{h}{R})^3[/tex]

## The Attempt at a Solution

I feel that I am mostly doing this problem correctly, but I think I'm leaving out something I need. I am able to derive the entire formula except for the part in the parentheses. I use the fact that the density equals M\V. To find the mass, I solved for M in the equation [tex]T^2=(\frac{4\pi^2}{GM})(R+h)^3[/tex]. I used R=H to account for the radius plus the altitude of the satellite. I do some rearranging and get [tex]M=\frac{4\pi^2(R+h)^3}{T^2G}[/tex]

I then use [tex]V=\frac{4\piR^2}{3}[/tex]. When I place M over V, I get that the density is equal to [tex]\frac{3\pi}{GT^2}(R+h)^3[/tex]

I have absolutely no clue where the 1 + h/R comes from that I must fine. I must have done something wrong in my deriving or rearranging...could someone point out a mistake? Thanks so much!