Orbital Gravity (2D orbit plane)

The planet feels a = -G M r_hat / (r^2 + epsilon) (the 3D lab uses G = 1, epsilon = 0.35 to keep close passes finite; here epsilon = 0 gives exact Newtonian gravity). With epsilon = 0 the motion is a conic: circle at v_c = sqrt(GM/r), ellipse for v between 0 and escape speed sqrt(2) v_c, parabola at escape, hyperbola above. Kepler's laws follow: equal areas swept in equal times (angular momentum L = r x v is constant for any central force), and T^2 = 4 pi^2 a^3 / (G M) with a = -GM / (2E).

With softening the force is still central, so L is conserved, and the energy uses the potential Phi = (GM / sqrt(epsilon)) atan(r / sqrt(epsilon)); but the orbit is no longer a closed ellipse and Kepler's third law only holds approximately.

Simplifications: a single light planet (the star does not move), no planet-planet attraction, no damping or relativity. The 3D lab includes planet-planet interaction and a tiny drag, so its orbits slowly differ from these ideal ones.