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The specific angular momentum (J) of an orbiting body is angular momentum associated with its orbit divided by its mass, i.e., the angular momentum per unit mass.
J = L/m
Each of the two bodies of an orbit has a specific angular momentum. In the simple case (which includes no interference from a third body), the orbiting body's specific angular momentum is conserved, a fact useful for calculating orbits.
Angular momentum is a "momentum" associated with rotation and is represented as a vector along the axis of rotation, its vector magnitude representing the amount. For a point circling an axis, the magnitude is:
L = rmv
Conventionally, the choice of directions along the axis uses the right hand rule. Following this convention allows the vectors to be added to calculate the angular momentum of multiple points fixed to each other as a rigid structure. The "right handedness" in its definition was an arbitrary choice, but adhering to the choice makes things consistent.
Angular momentum itself is conserved; if you begin spinning yourself, the Earth's rotation is very slightly modified in the opposite direction such that the total angular momentum including you and the Earth has not changed.