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9.3: Some Functions of the Masses

Last updated

Mar 5, 2022
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- 9.2: Kepler's Second Law from Conservation of Angular Momentum
- 9.4: Kepler's First and Third Laws from Newton's Law of Gravitation

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In section 9.5 I am going to consider the motion of two masses, $M$ and $m$ around their mutual centre of mass under the influence of their gravitational attraction. I shall probably want to make use of several functions of the masses, which I shall define here, as follows:

Total mass of the system:

$\textbf{M} = M + m. \label{9.4.1} \tag{9.4.1}$

"Reduced mass" $\text{m} = \frac{Mm}{M + m} . \label{9.4.2} \tag{9.4.2}$

"Mass function": $\mathfrak{M} = \frac{M^3}{(M+m)^2} . \label{9.4.3} \tag{9.4.3}$

No particular name: $m_+ = m \left( 1 + \frac{m}{M} \right) . \label{9.4.4} \tag{9.4.4}$

Mass ratio: $q = m/M . \label{9.4.5} \tag{9.4.5}$

Mass fraction: $\mu = m/(M+m) . \label{9.4.6} \tag{9.4.6}$

The first four are of dimension M; the last two are dimensionless. When $m << M$ , $\text{m} → m$ , $\mathfrak{M} → M$ and $m_+ → m$ .

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