Reduced mass

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Reduced mass is a concept that allows one to solve the two-body problem of mechanics as if it were a one body problem. Given two bodies, one with mass $m 1$ and the other with mass $m 2$ , 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 inertial mass

$m_{red} = {1 \over {{1 \over m_1} + {1 \over m_2}}} = {{m_1 m_2} \over {m_1 + m_2}}$

with the force the actual one.

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 ${{m_1 m_2} \over {m_{red}}}=m_1+m_2$ .