- Since most planets and satellites have a near circular orbit, the gravitational force FG 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
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:
- 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:
- 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
Worked example: Circular orbits in gravitational fields
A binary star system constant of two stars orbiting about a fixed point B.
The star of mass M1 has a circular orbit of radius R1 and mass M2 has a radius of R2. Both have linear speed v and an angular speed ⍵ about B.
State the following formula, in terms of G, M2, R1 and R2.
- The angular speed ⍵ of M1
- The time period T for each star in terms of angular speed ⍵
(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 ⍵
Many of the calculations in the Gravitation questions depend on the equations for circular motion. Be sure to revisit these and understand how to use them!