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9.3: Some Functions of the Masses

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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 \(m_+ → m\).