## Reduced Mass

\[\mu\]

.The educed for masses

\[m, \: M\]

is labelled \[\mu\]

and satisfies\[\frac{1}= \frac{1}{m} + \frac{1}{M} \rightarrow \mu = \frac{mM}{m+M}\]

The reduced mass is less that the total mass

\[M=m\]

and in fact less than each individual mass.Or ital equations and equations using Newton's Second Law of motion can be rewritten in terms of the reduced mass.

Example: Suppose planets 1 and 2 with masses

\[M\]

and m \[m\]

are in orbit. 1 will be attracted to 2 with a force

\[\mathbf{F}_{12}= \frac{GmM}{r^2} \mathbf{e}_{12} \]

1 will therefore experience an acceleration

\[\mathbf{a}_{12} = \frac{\mathbf{F}_{12}}{M}= \frac{Gm}{r^2} \mathbf{e}_{12} \]

2 will be attracted to 1 with a an equal and opposite force

\[- \mathbf{F}_{12}=- \frac{GmM}{r^2} \mathbf{e}_{12} \]

1 will therefore experience an acceleration

\[\mathbf{a}_{21} =- \frac{\mathbf{F}_{12}}{M}= \frac{Gm}{r^2} \mathbf{e}_{12} \]

Subtracting these gives a relative acceleration.

\[\mathbf{a} = \mathbf{a}_{21} - \mathbf{a}_{12}= -\frac{Gm}{r^2} \mathbf{e}_{12} - \frac{GM}{r^2} \mathbf{e}_{12} = - \frac{G \mu Mm}{r^2} \mathbf{e}_{12}\]