6.6 Motion of Orbiting Satellites

Describe the motions of a system consisting of two objects or systems interacting only via gravitational forces.

• In a system consisting only of a massive central object and an orbiting satellite with mass that is negligible in comparison to the central object’s mass, the motion of the central object itself is negligible.
• The motion of satellites in orbits is constrained by conservation laws.
  • i. In circular orbits, the system’s total mechanical energy, the system’s gravitational potential energy, and the satellite’s angular momentum and kinetic energy are constant.
  • ii. In elliptical orbits, the system’s total mechanical energy and the satellite’s angular momentum are constant, but the system’s gravitational potential energy and the satellite’s kinetic energy can each change.
  • iii. The gravitational potential energy of a system consisting of a satellite and a massive central object is defined to be zero when the satellite is an infinite distance from the central object. Relevant equation: UG mm r g 12=−
• The total energy of a system consisting of a satellite orbiting a central object in a circular path can be written in terms of the gravitational potential energy of that system or the kinetic energy of the satellite. Derived equations: KU 1 2=− EU GMm r 1 22 total == −
• The escape velocity of a satellite is the satellite’s velocity such that the mechanical energy of the satellite–central-object system is equal to zero.
  • i. When the only force exerted on a satellite is gravity from a central object, a satellite that reaches escape velocity will move away from the central body until its speed reaches zero at an infinite distance from the central body.
  • ii. The escape velocity of a satellite from a central body of mass M can be derived using conservation of energy laws. Derived equation: v GM r 2 esc =