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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-sized mass circling an axis, the magnitude is:
L = rmv
Conventionally, the choice between the two possible directions along the axis that is used to specify the "direction" of the angular momentum follows the right hand rule. Adherence to this convention allows such angular-momentum vectors to be added, e.g., to calculate the angular momentum of multiple points fixed to each other as a rigid structure. The "right handedness" in this convention was an arbitrary choice, but adhering to such a convention keeps interpretation and calculation methods 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.