For an attractive Kepler orbit with gravitational parameter , velocity and specific angular momentum , the eccentricity vector isIts magnitude is the orbital eccentricity, and it points toward periapsis for a noncircular orbit. It is a normalized attractive Laplace-Runge-Lenz vector; it remains constant under the pure inverse-square force.
If the relative acceleration is , with constant masses, then the specific orbital energy , specific angular momentum and eccentricity vector obeyThe inverse-square-force terms cancel in each derivative. These identities express perturbing work and torque, and track the instantaneous change of the ellipse's shape and orientation.
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The eccentricity vector, often denoted as **e**, is a vector that describes the shape and orientation of an orbit in celestial mechanics. It is particularly relevant in the context of conic sections, which are used to describe orbits of celestial bodies (like planets, comets, and satellites) around other massive bodies.