Orbital Motion
Newton's Law of Gravitation & Orbits
- 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
- This centripetal force is perpendicular to the velocity 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 from Newton's Law of Gravitation to the centripetal force for a planet or satellite in orbit gives:
FG = Fcentripetal
- The mass of the satellite m will cancel out on both sides to give:
- Where:
- v = linear speed of the mass in orbit (m s–1)
- G = Newton's Gravitational Constant (N m2 kg–2)
- M = mass of the object being orbited (kg)
- r = orbital radius (m)
- This means that all satellites, whatever their mass, will travel at the same speed v in a particular orbital radius r
Newton's Laws of Motion & Orbits
- Newton's first law of motion states that a body remain at rest or at constant velocity unless a resultant force acts on it
- Bodies in orbit do have a resultant force acting on them
- This is the gravitational force due to the mass M being orbited
- This force is a centripetal force because it acts towards the centre of M, perpendicular to the velocity of the planet or satellite
- Therefore, since the direction of a planet or satellite orbiting in circular motion is constantly changing, it must be accelerating
- This is called centripetal acceleration
A satellite in orbit around the Earth travels in circular motion
Time Period & Orbital Radius Relation
- A planet or a satellite orbits in circular motion
- Therefore, its orbital time period T is the time taken to travel the circumference of the orbit 2πr
- This means the linear speed, or orbital speed v is:
- This is a result of the well-known equation speed = distance / time
- Equating the two equations for orbital speed gives:
- Squaring out the brackets and rearranging for T2 gives the equation relating the time period T and orbital radius r:
- Where:
- T = time period of the orbit (s)
- r = orbital radius (m)
- G = Newton's Gravitational Constant (N m2 kg–2)
- M = mass of the object being orbited (kg)
- The equation shows the relationship between the orbital period and the orbital radius for any planet or satellite in orbit
- It is summarised mathematically as:
Worked example
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
(i) The angular speed ⍵ of M1
(ii) The time period T for each star in terms of angular speed ⍵
Exam Tip
This worked example helps you practise two crucial techniques that are often examined:
- The centripetal force is expressed as
or equivalently as 
. In our case the angular speed is given in the question, so it is best to use the latter expression when equating the centripetal force to the gravitational force. - The gravitational force is given as
but note that the distance r in this question is given as a sum, R1 + R2. You should remember that r is defined as the distance between the centre of masses of M and m, therefore, r = R1 + R2 and so r2 = (R1 + R2)2.
