Kepler's Third Law (College Board AP® Physics 1: Algebra-Based)

Kepler's third law

• For a satellite in circular orbit around a central body, the satellite’s centripetal acceleration is caused only by gravitational attraction

• 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: 

  • = orbital time period, measured in

  • = mean orbital radius, measured in

• The period and radius of the circular orbit are related to the mass of the central body using the equation for Kepler's third law

• Where:

  • time period of the orbit, measured in

  • orbital radius, measured in

  • Gravitational Constant

  • mass of the object being orbited, measured in

Derived equation

• The period and radius of the circular orbit are related to the mass of the central body using the equation for Kepler's third law

Derivation:

Step 1: Identify the fundamental principles

• Uniform circular motion equation:

• Where:

  • = time period, measured in

  • = radius of orbit, measured in

  • = tangential velocity of object in orbit, measured in

• Newton's law of gravitation:

  

• Where:

  • magnitude of the gravitational force between the two objects, measured in

  • universal gravitational constant =

  • mass of object 1, measured in

  • mass of object 2, measured in

  • the distance between the center of mass of the two objects, measured in

• Centripetal force equation:

• Where:

  • = centripetal force, measured in

  • = mass of orbiting object, measured in

  • = tangential speed of orbiting object, measured in

  • = orbital radius, measured in

Step 2: Combine the equations for centripetal and gravitational force

• For an object in orbit in a uniform gravitational field around a larger mass, the centripetal force is created by the gravitational force

  • Where object 1 is the larger mass, and object 2 the smaller mass

Step 3: Rearrange the equation to make the subject

Step 4: Substitute for from the uniform circular motion equation

Step 5: Expand the brackets, rearrange to make the subject and simplify