5.8.2 Circular Orbits in Gravitational Fields

• Typically, most planets and satellites have a near circular orbit
  • Therefore, the gravitational force F between the Sun and another planet provides the centripetal force needed to stay in an orbit

• Since the gravitational force is centripetal, it is perpendicular to the direction of travel of the planet

• Consider a satellite with mass m orbiting the Earth, with mass M, at a distance r from its centre and travelling with linear speed v
• The gravitational force F on the satellite is centripetal, therefore:

F = 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:

• This can be reduced to give an equation for the orbital speed v, as:

• Where:
  • v = orbital 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
  • 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

• Assuming a planet or a satellite is travelling in circular motion when in orbit, its orbital speed v is given by the distance travelled divided by the orbital period T:

• This is a result of the well-known equation, speed = distance / time
  • Where the distance is equal to the circumference of a circle = 2πr
• Recall, orbital speed v is also given by: 

• Therefore:

• Expanding the brackets gives:

• Rearranging for T² gives the mathematical expression of Kepler's Third Law which relates the time period T and orbital radius r as:

• Where:
  • T = time period of the orbit, or "orbital period" (s)
  • r = orbital radius (m)
  • G = Newton's Gravitational Constant
  • M = mass of the object being orbited (kg)