Definitions

# Reduced mass

Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit 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 $m_\left\{1\right\}!,$ and the other with mass $m_\left\{2\right\}!,$, 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

$m_text\left\{red\right\} = mu = cfrac\left\{1\right\}\left\{cfrac\left\{1\right\}\left\{m_1\right\}+cfrac\left\{1\right\}\left\{m_2\right\}\right\} = cfrac\left\{m_1 m_2\right\}\left\{m_1 + m_2\right\},!,$

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

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

$F_\left\{12\right\} = m_1 a_1. !,$

The force exerted by body 1 on body 2 is

$F_\left\{21\right\} = m_2 a_2. !,$

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

$F_\left\{12\right\} = - F_\left\{21\right\}.!,$

Therefore,

$m_1 a_1 = - m_2 a_2. !,$

and

$a_2=-\left\{m_1 over m_2\right\} a_1. !,$

The relative acceleration between the two bodies is given by

$a= a_1-a_2 = left\left(\left\{1+\left\{m_1 over m_2\right\}\right\}right\right) a_1 = \left\{\left\{m_2+m_1\right\}over\left\{m_1 m_2\right\}\right\} m_1 a_1 = \left\{F_\left\{12\right\} over m_text\left\{red\right\}\right\}.$

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 $mu.!,$

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

$\left\{m_1 m_2 over m_text\left\{red\right\}\right\} = m_1+m_2.!,$

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 term of the form

$x_text\left\{red\right\} = \left\{1 over \left\{1 over x_1\right\} + \left\{1 over x_2\right\}\right\} = \left\{x_1 x_2 over x_1 + x_2\right\}!,$

that simplifies an equation of the form

$\left\{1over x_text\left\{eq\right\}\right\} = sum_\left\{i=1\right\}^n \left\{1over x_i\right\} = \left\{1over x_1\right\} + \left\{1over x_2\right\} + cdots+ \left\{1over x_n\right\}.!,$

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 linking them.