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The reduced mass is a function of two masses that can be used to simplify the calculations of the mechanics of two bodies interacting by force (e.g., gravity). It is used in calculations of electron/nucleus orbits as well as the orbits of astronomical bodies. The function produces a "mass" smaller than either of the two masses, just slightly smaller than the smaller of the two, if the two masses differ substantially. Formula:
mreduced = ( m1 m2 ) / ( m1 + m2 )
or equivalently:
1 / mreduced = 1 / m1 + 1 / m2
The motion of a pair of bodies (primarily) interacting only with each other, is in relation to their center of mass (barycenter), e.g., the Moon and Earth orbit the center of mass of the pair (a point between their individual centers of mass, which in this case happens to be within the Earth, given its much larger mass, but still some distance from the Earth's center). The motion of the Moon in relation to the Earth itself (rather than in relation to their center of mass) could be calculated by using some slightly smaller mass for the Moon and giving Earth some slightly larger mass. This would describe their motion in a frame of reference centered on the Earth's own center of mass, a frame-of-reference that "wobbles", i.e., moves in circles around the center of mass of the pair, but the calculation could then assume the Earth was held still, a much easier and shorter calculation. The masses that accurately enable such a calculation are the reduced mass for the Moon and the sum of the two masses for the Earth.
Since the motions are relative to each other (e.g., on Earth, you see the Moon orbiting you while on the Moon, you would see the Earth orbiting you at the same distance and rate), the calculation could be done giving the Moon the larger (sum of the two masses) and the Earth the smaller (the same reduced mass), yielding the correct motion of the Earth around the Moon using the Moon's center as the frame-of-reference. In either case, to find their motions relative to surroundings, the result would need adjustment so their mutual center of mass is the fixed point.