Gravitational Fields: Orbits of Planets and Satellites

Kepler's Third Law

the square of the orbital period (T) is directly proportional to the cube of the radius
T^2 ∝ r^3

T^2 ∝ r^3 derivation step 1

when an object orbits a mass, it experiences a gravitational force towards the centre of the mass
this acts as a centripetal force, due to the circular motion of orbits
this means we can equate gravitational force to centripetal force
mv^2/r = GMm/r^2

T^2 ∝ r^3 derivation step 2

rearrange the equation to make v^2 the subject
v^2 = GM/r

T^2 ∝ r^3 derivation step 3

velocity is the rate of change of displacement
this means you can find v in terms of radius (r) and orbital period (T):
v = 2πr/T ——> v^2 = 4π^2r^2/T^2
(because the diameter of a circle is 2πr and the object will travel this distance in one orbital period

T^2 ∝ r^3 derivation step 4

substitute the equation for v^2 in terms of r and T into the original equation (from step 2)
4π^2r^2/T^2 = GM/r

T^2 ∝ r^3 derivation step 5

rearrange to make T^2 = 4π^2/GM x r^3

as 4π^2/GM is a constant, this shows that T^2 ∝ r^3

what is the total energy of an orbiting satellite made up of?

its kinetic and potential energy

• it's constant

total energy of a satellite

kinetic energy + potential energy

escape velocity

the minimum velocity an object must travel at in order to escape the gravitational field at the surface of a mass