## Reduced Mass

In planetary science the reduced mass is very important. For two bodies in orbit, if the mass of one is much greater than the mass of the other, we often assume the smaller to orbit the larger. This is an approximation, and sometimes not accurate enough. In fact both bodies orbit their common centre of mass and we introduce a concept called the '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.

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