Bound and Unbound Trajectories

An object 'm' in the gravitational field of another object 'M' capital M can move on a bound (orbit) or unbound trajectory. Bound and unbound trajectories can be described by conic sections, and calculated using the concept of conservation of energy.

Conic Sections

In physics, the paths that can be taken by an object under gravitational attraction are called the conic sections. In mathematics, these curves are obtained by taking a slice from a cone at different angles. These shapes are the circle, the ellipse, the parabola and the hyperbola. The circle and ellipse are bound orbits (like planets around the sun), and the parabola and hyperbola are unbound (like a deflecting rocketship on a slingshot orbit).

The image of four conic sections of a transparent cone. The first section is an orange circle placed horizontally, a bit above the tip of the cone. The second section, ellipse, is placed below the circle. The third parabola section has a base at the base of the cone and a highest point at one of the walls of the cone. The last section, hyperbola, is similarly placed as the parabola, but taking a smaller section, closer to the edge of the base of the cone.

Figure 1: Illustration of the four different conic sections.

Calculating Bound and Unbound Trajectories

To determine whether an object with mass 'm' will follow a bound or unbound trajectory, it is useful to apply conservation of energy and calculate the kinetic and gravitational potential energy of the object in the gravitational field of mass 'M' capital M.

Ekinetic > Epotential : Object 'm' has enough energy to escape the gravitational pull of object 'M' capital M and follows an unbound orbit, escaping to infinity.

Ekinetic < Epotential : The velocity of object 'm' is too small to escape the gravitational attraction of object 'M' and is stuck in freefall around object 'M'. capital M

The kinetic energy Ekinetic capital E kinetic and the gravitational potential energy Epotential capital E potential of an object with mass 'm' in the gravitational field of mass 'M' capital 'M'can be described by equations in the escape velocity theory page.