Newtonian orbits and the equivalent one-dimensional problem

• Unit mass is 2-body system's reduced mass m.
• Unit length is the position of the minimum of the effective potential energy in the attractive case (i.e., the radius of the circular orbit): r₀ = L²/m|k|.
• Time unit is L³/mk².
• In these unit we have m = L = |k| = 1, the orbit equation is 1/r = ε cos(φ-φ₀) + η with η = |k|/k. The eccentricity is ε = (1+2E)^1/2, the effective potential energy -η/r+1/2r² and its minimum is at point (1,-1/2) for attractive forces.

• If Orbit is selected, along with the orbit one can see the conserved Laplace-Runge-Lenz vector, which goes through the pericentron.
• To get information on one element, put over it the mouse pointer to see the corresponding tooltip.
  

Activities

• On the left one can see the effective potential energy corresponding to the equivalent one-dimensional problem (in red).
  • Use the mouse (or the corresponding numerical entries) to change the mechanical energy E and the initial value of the polar distance r: the polar angle φ (phi in the simulation) will be set automatically.
  • Discuss the values of the eccentricity ε (e) and the form of the orbit for different values of E.
  • Uncheck Attracting and discuss what happens with repulsive forces.
  • Change orbit's spatial orientation by choosing the pericentron position φ₀ (called per in the simulation) or use the following procedure.
• The plane relative motion is displayed on the right. Use the mouse to select the initial position and the orbit orientation.
• With the simultaneous evolutions in the double display you can check that the return points of the one-dimensional problem correspond in the full problem to apses, where the velocity does not vanish, but is perpendicular to the radius.