Circular orbit of a satellite: speed, period, energy

Setting up the equation

A satellite of mass m in a circular orbit of radius r around a planet of mass M is subject only to the gravitational force, which acts as the centripetal force: GMm/r^2 = m*v^2/r

Orbital speed

From this we deduce the circular speed: v = sqrt(G*M/r)

The closer the satellite is to the planet, the greater its speed. Counter-intuitive result: raising a satellite to a higher orbit slows it down.

Orbital period

T = 2pir/v = 2pisqrt(r^3/(G*M))

Mechanical energy

Kinetic energy: Ec = (1/2)mv^2 = GMm/(2r) Gravitational potential energy (zero reference at infinity): Ep = -GMm/r Total mechanical energy: Em = Ec + Ep = -GMm/(2r)

Em is negative: the satellite is in a bound state. A positive or zero mechanical energy would correspond to an escape trajectory (hyperbola or parabola, not closed).

Table of quantities

Quantity	Expression
Speed	v = sqrt(G*M/r)
Period	T = 2pisqrt(r^3/(G*M))
Mechanical energy	Em = -GMm/(2*r)

Classic pitfall

Do not confuse Ec and Em: we have Ec = -Em (and not Ec = Em). This is a direct consequence of the virial theorem for a 1/r potential, and a frequent source of sign errors in energy calculations.