What defines a trajectory as hyperbolic in celestial mechanics?

Energy is positive. In celestial mechanics, a trajectory is considered hyperbolic when the total mechanical energy of the system is positive. This occurs when the kinetic energy of the object exceeds the potential energy due to gravity, allowing the object to escape the gravitational influence of the larger body. In the case of a hyperbolic trajectory, the object follows a path that does not form a closed orbit but instead extends to infinity, as the object possesses sufficient energy to overcome the gravitational attraction and move away from the central body without returning. Hyperbolic trajectories are important in space exploration and astronomy, as they can be observed in scenarios such as comets passing through our solar system or spacecraft performing flyby maneuvers around planets.