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: