Circular Orbit

Velocity of Circular Orbit

We're going to look at circular orbit. Imagine a satellite with mass m orbiting a planet, which has mass M. It is at a distance r from the centre of the planet. It is moving at velocity v, at right angles to the planet, and this velocity allows the outward centrifugal force F_cent to just about balance the gravitational force F_grav. Thus

F_grav = F_cent

$$ \frac{GMm}{r^2} = \frac{mv^2}{r} $$

Rearranging for v:

$$ v = \sqrt{\frac{GM}{r}} $$

This is the speed the satellite needs to be travelling in order to maintain circular orbit.

Orbital Period

The orbital period T is the time taken for the satellite to complete a single orbit. For a circular orbit, multiplying the velocity by the time taken equals the circumference of the circular orbit: $$ vT = 2 \pi r $$

Rearranging for T, and substituting the equation for velocity:

$$ T = \frac{2 \pi r }{v} = 2 \pi r \sqrt{\frac{r}{GM}} $$

This is the equation for the orbital period of a satellite orbiting a body of mass M at a distance r from its centre:

$$ T = \frac{2 \pi }{\sqrt{GM}}r^{\frac{3}{2}} $$

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