Reduced mass

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 ${\displaystyle m_{1}}$ and the other with mass ${\displaystyle m_{2}}$, 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
${\displaystyle m_{red}\equiv {1 \over {{1 \over m_{1}}+{1 \over m_{2}}}}={{m_{1}m_{2}} \over {m_{1}+m_{2}}}}$
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 ${\displaystyle {{m_{1}m_{2}} \over {m_{red}}}=m_{1}+m_{2}}$.