Two-Body Problem

Motion of two point masses under their mutual gravity: the starting point for everything in the course. See also ENAE 441: Two Body Problem.

Law of Universal Gravitation

• Two-body motion: two point masses orbiting about their common center of mass
• One mass is much greater than the other the barycenter is (essentially) the center of mass of that big body
  • e.g. Earth and a satellite: we follow the satellite relative to the Earth’s center

Kepler’s Laws

1. The orbit of a planet is an ellipse, with the Sun at the focus
   • see Conic Sections
2. The line from the Sun to the planet sweeps out equal areas in equal times
   • proved from conservation of angular momentum in Kepler’s second law
3. For any two planets:
   • is the orbital period, the semimajor axis; the constant is found in Kepler’s third law

Newton’s Law of Gravitation

• Force on the satellite (mass ) from the Earth (mass ):
  • : distance from the Earth’s center to the satellite
  • : unit vector from the Earth’s center toward the satellite
• Gravitational parameter of the Earth:
• Properties of this force:
  1. Independent of velocity and time
  2. A central force, i.e., directed along the radial line connecting the bodies
  3. Inverse square dependence on distance
• These properties give three conservation laws:
  1. Conservation of energy: Specific Mechanical Energy
  2. Conservation of angular momentum: Specific Angular Momentum
  3. Conservation of the eccentricity vector: Eccentricity Vector

Equation of Motion

• Newton’s second law on the satellite:
• The mass cancels:

Two-Body Equation of Motion

• Equivalently , since and