Gravitational force: Moon causing a tidal force on the Earth's ocean

Question:
The gravitational force between the moon and Earth creates a tidal force. From the figure, $a$ is the distance between the moon and the Earth. $M$ and $m$ are the masses of Earth and moon, respectively. $r$ denotes the radius of Earth. Find the differential tidal acceleration.

Answer:
The tidal force is obtained by the difference of gravitational fields between C (center of mass) and S (place to get tidal force). This can be associated with the differential tidal acceleration. Let us write down each gravitational acceleration.
\begin{eqnarray}
g_C &=& \frac{Gm}{a^2}  \\
g_S &=& \frac{Gm}{(a+r)^2}
\end{eqnarray}
The difference of them is the tidal acceleration.
\begin{eqnarray}
g_C -g_S &=& \frac{Gm}{a^2} - \frac{Gm}{(a+r)^2}  \\
   &=& \frac{Gm}{a^2}\left(1-\frac{a^2}{(a+r)^2}\right)  \\
   &=& \frac{Gm}{a^2}\left(1-\frac{1}{(1+\frac{r}{a})^2}\right)  \\
   &\sim& \frac{Gm}{a^2}\left(1-\left\{1-2\frac{r}{a}\right\}\right)
\end{eqnarray}
The above uses approximation. Hence, we have
\[
g_{\mathrm{tidal}} = \frac{2Gmr}{a^3}
\]

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