Satellite motion¶
For an object to become an permanent satellite around earth ith has to travel at a certain speed.
\[\begin{split}
F_g = ma \\
\frac{Gmm_E}{r^2} = \frac{mv^2}{r} \\
v = \frac{Gm_E}{r}
\end{split}\]
\(a = \frac{v^2}{r}\) since we assume that the object ravels in a circular trajectory.
Here we can see that we cannot choose v and r independently, but the mass of an object plays no role.
Period¶
We can also determine the period of one revolution. First we assume that:
\[
v = \frac{2 \pi r}{T}
\]
T is the time it takes to make one revolution
If we put this together we get:
\[\begin{split}
T = \frac{2 \pi r}{v} = 2 \pi r \sqrt{\frac{r}{Gm_E}} = \frac{2\pi r^{3/2}}{\sqrt{Gm_E}}\\
T^2 = \frac{4\pi^2r^3}{Gm_E}
\end{split}\]
From this we see that larger orbits require lower speed and take longer.