In astronomy, a parabolic trajectory is the path an object takes when it moves under gravity along a curve that is geometrically a parabola. The term is often used interchangeably with parabolic orbit or parabolic escape trajectory. In the ideal two-body problem this motion occurs exactly when the orbit's eccentricity equals 1, which makes it the transition case between bound and unbound motion.
According to Newtonian gravity and Kepler's laws, a parabolic trajectory corresponds to zero specific orbital energy: the object has just enough energy to escape the central body but not to recede with excess speed. Real small bodies — for example, an asteroid or a comet arriving from very far away — can follow paths that are very close to parabolic. In practice, however, planetary perturbations, non-gravitational forces, and measurement uncertainties usually make the measured eccentricity differ slightly from exactly 1.
Mathematical and observational notes
Mathematically, parabolic motion is one of the conic-section solutions to the inverse-square law of attraction. It can be described in polar coordinates with the focus at the origin by an equation using the semi-latus rectum; dynamically it is characterised by angular momentum and zero total energy. Observationally, distinguishing a true parabola from a near-parabolic hyperbola or a very elongated ellipse often requires precise long-arc measurements.
Relation to other orbital types
- circular orbit — a special case of an ellipse with eccentricity 0
- elliptical orbit — bound trajectories with eccentricity less than 1
- hyperbolic trajectory — unbound paths with eccentricity greater than 1
In practical astronomy, exact parabolic trajectories are uncommon because no real system is perfectly isolated and measurements are finite. Nevertheless, the parabolic solution remains important as the limiting case between elliptical and hyperbolic motion and as a useful approximation for some long-period comets and near-interstellar flybys.