• Since most planets and satellites have a near circular orbit, the gravitational force?F_G?between the sun or another planet provides the?centripetal force?needed to stay in an orbit
• Both the gravitational force and centripetal force are?perpendicular?to the direction of travel of the planet
• Consider a satellite with mass?m?orbiting Earth with mass?M?at a distance?r?from the centre travelling with linear speed?v

• Equating the gravitational force to the centripetal force for a planet or satellite in orbit gives:

• The mass of the satellite?m?will cancel out on both sides to give:

• This means that all satellites, whatever their mass, will travel at the same speed?v?in a particular orbit radius?r
• Recall that since the direction of a planet orbiting in circular motion is constantly changing, it has?centripetal acceleration

A satellite in orbit around the Earth travels in circular motion

Kepler’s Third Law of Planetary Motion

• For the orbital time period?T?to travel the circumference of the orbit 2πr, the linear speed?v?can be written as

• This is a result of the well-known equation, speed = distance / time
• Substituting the value of the linear speed?v?into the above equation:

• Rearranging leads to Kepler’s third law equation:

• The equation shows that the orbital period?T?is related to the radius?r?of the orbit. This is known as Kepler’s third law:

For planets or satellites in a circular orbit about the same central body, the square of the time period is proportional to the cube of the radius of the orbit

• Kepler’s third law can be summarised as:

Maths Tip

• The ∝ symbol means ‘proportional to’
• Find out more about proportional relationships between two variables in the “proportional relationships?” section of the A Level Maths revision notes

(1) The angular speed of ? of M₁

Step 1: Equating the centripetal force of mass M₁?to the gravitational force between M₁?and M₂

Step 2: M₁?cancels on both sides

Step 3: Rearrange for angular velocity ?

Step 4: Square root both sides

(2)? The time period T for each star in terms of angular speed??

Step 1: Angular speed equation with time period T

Step 2:?Rearrange for T

Step 3: Substitute in ?