Center of mass and relative motion

• 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 V_e(r) = -η/r+1/2r² and its minimum is at point (1,-1/2) for attractive forces.

• If Orbits 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

• Use the mouse on the right figure or the corresponding numerical entries to change the mechanical energy E and the initial value of the polar distance r and polar angle φ.
• Discuss the relationship (for different energies and in the cases of both attracting and repulsive forces) between the orbits around the center of mass (displayed on the left figure) and the relative motion of one particle around the other, as displayed on the right.
• Discuss the influence of the quotient of masses m₁/m₂ = m1/m2.
• Now, let us assume the motion is analyzed from another inertial reference frame in which the center of mass moves with a velocity whose components Vx and Vy you have to choose. Which kind of orbits do you get now?