What paths can a comet follow under Newton's law of gravitation?

Under Sir Isaac Newton's law of gravitation, the path a body—such as a comet—follows depends on its total energy and initial velocity relative to a central mass. While Johannes Kepler established that planets move in elliptical orbits, Newton demonstrated that gravitational motion can also follow other open conic sections.12

The possible paths are defined by the orbital eccentricity ($e$):

• Elliptical orbits ($e<1$): These are closed, bound orbits. Most objects in the solar system, including planets and many comets, follow these paths.12
• Parabolic orbits ($e=1$): These represent the boundary between bound and unbound motion. An object on a parabolic path has just enough energy to escape the gravitational pull of the central body.12
• Hyperbolic orbits ($e>1$): These are open, unbound paths. An object with sufficient energy, such as a comet entering the solar system at high velocity, can follow a hyperbolic trajectory, passing the Sun and leaving the system without returning.12