Kepler's Laws of Planetary Motion (SL IB Physics)

Kepler's First Law

• Kepler's First Law describes the shape of planetary orbits
• It states: 

The orbit of a planet is an ellipse, with the Sun at one of the two foci

The orbit of all planets are elliptical, and with the Sun at one focus

• An ellipse is just a 'squashed' circle
  • Some planets, like Pluto, have highly elliptical orbits around the Sun
  • Other planets, like Earth, have near circular orbits around the Sun

Kepler's Second Law

• Kepler's Second Law describes the motion of all planets around the Sun
• It states: 

A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals

• The consequence of Kepler's Second Law is that planets move faster nearer the Sun and slower further away from it

Kepler's Third Law

• Kepler's Third Law states

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

• This law describes the relationship between the time of an orbit and its radius

• Where: 
  • T = orbital time period (s)
  • r = mean orbital radius (m)

Time Period & Orbital Radius Relation

• Since a planet or a satellite is travelling in circular motion when in order, 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 = Gravitational Constant
  • M = mass of the object being orbited (kg)

• 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 logarithmic graph of log T against log r gives a straight line

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