• Since most planets and satellites have a near circular orbit, the gravitational force?FG?between the Sun and another planet provides the?centripetal force?needed to stay in an orbit
• The gravitational force is centripetal, therefore it is?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

FG?=?Fcentripetal

• 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:

• Where:
  • v?= linear speed of the mass in orbit (m s-1)
  • G?=?Newton's Gravitational Constant
  • 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 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

Time Period & Orbital Radius Relation

• Since a planet or a satellite is travelling in circular motion when in orbit, its orbital time period?T?to travel the circumference of the orbit 2πr, the linear speed?v?is:

• This is a result of the well-known equation, speed = distance / time and first introduced in the circular motion topic
• Substituting the value of the linear speed?v?from equating the gravitational and centripetal force into the above equation 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
  • M?= mass of the object being orbited (kg)

   

• The equation shows that the orbital period?T?is related to the radius?r?of the orbit. This is also 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:

Graphical Representation of T2?∝ r3

• The relationship between?T?and?r?can be shown using a logarithmic plot
• The graph of log(T) in years against log(r) in AU (astronomical units) for the planets in our solar system is a straight line graph:

• The graph does not go through the origin since it has a negative y-intercept
• Only the log of both?T?and?r?will produce a straight line graph

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