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Reduced mass is the "effective" inertial mass appearing in the two-body problem
Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....
 of Newtonian mechanics. This is a quantity with the unit
Units of measurement

The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day....
 of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not reduced. In the computation one mass can be replaced by the reduced mass, if this is compensated by replacing the other mass by the sum of both masses.

Given two bodies, one with mass and the other with mass , they will orbit the barycenter of the two bodies.






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Reduced mass is the "effective" inertial mass appearing in the two-body problem
Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....
 of Newtonian mechanics. This is a quantity with the unit
Units of measurement

The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day....
 of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not reduced. In the computation one mass can be replaced by the reduced mass, if this is compensated by replacing the other mass by the sum of both masses.

Given two bodies, one with mass and the other with mass , they will orbit the barycenter of the two bodies. The equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass



where the force on this mass is given by the gravitational force between the two bodies. This is just half the harmonic mean
Harmonic mean

In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of Rate s is desired....
 of the two masses.

This can be proven easily. Use Newton's second law, the force exerted by body 2 on body 1 is

The force exerted by body 1 on body 2 is

According to Newton's third law, for every action there is an equal and opposite reaction:

Therefore,

and

The relative acceleration between the two bodies is given by


So we conclude that body 1 moves with respect to the position of body 2 as a body of mass equal to the reduced mass.

The reduced mass is frequently denoted by the Greek letter

Applying the gravitational formula we get that the position of the first body with respect to the second is governed by the same differential equation as the position of a very small body orbiting a body with a mass equal to the sum of the two masses, because

The reduced mass is always less than or equal to the mass of each body.

"Reduced mass" may also refer more generally to an algebraic
Algebraic

Algebraic may refer to:* Algebraic chess notation — a method used to record and describe the play of chess games.* Algebraic data types....
 term of the form

that simplifies an equation of the form

The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. This relationship is determined by the physical properties of the elements as well as the continuity equation
Continuity equation

A continuity equation is a differential equation that describes the conservative transport of some kind of quantity. Since mass, energy, momentum, and other natural quantities are conserved, a vast variety of physics may be described with continuity equations....
 linking them.

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