### reduced mass

(the reciprocal of the sum of the reciprocals of the masses of two objects)

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 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 greatly.

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 not at the Earth's center).
The motion of the Moon in relation to the Earth itself
(rather than 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
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, 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.

Equivalent formulae:

m_{reduced} = ( m_{1} m_{2} ) / ( m_{1} + m_{2} )

or

m_{reduced} = 1 / ( 1 / m_{1} + 1 / m_{2} )

(*physics,mass,mechanics*)
http://en.wikipedia.org/wiki/Reduced_mass

**Referenced by:**

chirp mass (M_{c})

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