7.3.1 Circular Orbits in Gravitational Fields

• Since most planets and satellites have a near circular orbit, the gravitational force F_G 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

F_G = F_centripetal

• 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