3 The time period $T$ of a satellite in circular orbit around a planet satisfies the equation\\
$G M T ^ { 2 } = 4 \pi ^ { 2 } R ^ { 3 }$,\\
where

\begin{itemize}
  \item $G$ is the universal gravitational constant,
  \item $M$ is the mass of the planet,
  \item $\quad R$ is the radius of the orbital circle.
\begin{enumerate}[label=(\alph*)]
\item Find the dimensions of $G$.
\end{itemize}

A student suggests the following formula to model the approach speed between two orbiting bodies.\\
$v = k G { } ^ { \alpha } { } ^ { \beta } { } _ { r } \gamma _ { m _ { 1 } } m _ { 2 } \left( m _ { 1 } + m _ { 2 } \right)$,\\
where

\begin{itemize}
  \item $\quad v$ is the approach speed of the two bodies,
  \item $k$ is a dimensionless constant,
  \item $\quad c$ is the speed of light,
  \item $\quad r$ is the distance between the two bodies,
  \item $\quad m _ { 1 }$ and $m _ { 2 }$ are the masses of the bodies.
\item Use dimensional analysis to determine the values of $\alpha , \beta$ and $\gamma$.
\item Calculate, according to the student's model, how many times greater the approach speed is between a pair of stars which are 6.13 light-years apart and the same pair of stars if they were 8.64 light-years apart. (A light-year is a unit of distance.)
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Mechanics A AS 2023 Q3 [9]}}