Mryntu

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:

1. Are there unit standards I need for both $P$ and $a$? Currently $P$ is in minutes, $a$ in kilometres.


2. Am I missing something, like Newton's universal gravitational constant? I get a page deriving Kepler's third law using this constant.

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)

The general form of Kepler's period law is $T^2 = frac4pi^2G(M+m)a^3$. Often, we make the simplifying assumption that $M>>m$, so that $M+m approx M$.

Kepler's period law only takes the form $T^2 = a^3$ (forgetting about the units) when you use certain quantities- in this case, $M$ being solar mass, $T$ being an Earth year, and $a$ being an astronomical unit.

Try plugging into the equation for the mass of earth (and don't bother with the satellite mass) and use units of meters and seconds. See if you get the right result!

Kepler's third law claims that $p^2 propto a^3$. The equality sign you use is incorrect.