Conic sections and orbital shape determination
An orbiting object traces one of four fundamental curves defined by geometry: a circle, ellipse, parabola, or hyperbola. The shape is determined entirely by the object's speed relative to escape velocity at a given point. A satellite with insufficient speed falls back to Earth on an elliptical arc. One with exactly the right speed maintains a circular path. Higher speeds extend the ellipse, while speeds at or exceeding escape velocity trace parabolic or hyperbolic paths that do not return. This mathematical relationship means that knowing a single speed measurement uniquely determines the orbital path.
Energy balance in closed and open orbits
Circular and elliptical orbits represent bound systems where the orbiting body retains total negative mechanical energy, ensuring it never escapes. Parabolic and hyperbolic paths correspond to positive or zero total energy, resulting in objects departing to infinity. At the moment of encounter, the object's total energy (kinetic plus gravitational potential) dictates whether it will stay, swing by, or escape. This energy perspective unifies orbital mechanics across wildly different scales, from electron clouds around nuclei to galaxies orbiting galactic centers.