I have been looking at satellite orbits around the earth, or any object around any planet in fact, and am trying to find the orbital radius, or semi major length of a given satellite.
Kepler's third law gives the equation $P^2 = a^3$ where $P$ is the period of orbit and $a$ the distance.
I have a table of satellites currently orbiting the earth, as well as their altitude in the sky on their geosynchronous trajectory. One in particular is 99.9 and has an altitude of 705.
By solving the equation for $a$, I get $a = (P^2)^{1/3}$.
When I plug in the numbers, they don't correspond.
So my questions are:
- Are there unit standards I need for both $P$ and $a$? Currently $P$ is in minutes, $a$ in kilometres.
- Am I missing something, like Newton's universal gravitational constant? I get a page deriving Kepler's third law using this constant.
Best Answer
that equality should be a proportional to sign. In particular, in SI, the squared period has units of seconds squared, and the semi-major radius of of the orbit cubed is in meters cubed, so they can't be equal.
Instead, I'd be checking whether $T^{2}/a^{3}$ is constant for different satellites orbiting the same object (Like the ISS and the moon, for example)