\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad \rho
 & = & \frac{m}{V} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad \rho
 & = & \frac{m}{V} \\
m
 & = & \rho V \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad \mbox{d}m
 & = & \rho \, \mbox{d}V \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad \mbox{d}m
 & = & \rho \, \mbox{d}V \\
& = & 4 \pi \rho r^{2} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad \int\limits_{r = r_{A}}^{r = \infty} \mbox{d}m
 & = & \int\limits_{r = r_{A}}^{r = \infty} 4 \pi \rho r^{2} \, \mbox{d}r \\
m
 & = & 4 \pi \int\limits_{r = r_{A}}^{r = \infty} \rho r^{2} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad \rho
 & = & \rho_{0} e^{- (r - r_{A})/\tau} \\
& = & \rho_{0} e^{- r/\tau + r_{A}/\tau} \\
& = & \rho_{0} e^{r_{A}/\tau} e^{- r/\tau} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(7)} \qquad m
 & = & 4 \pi \int\limits_{r = r_{A}}^{r = \infty} \rho r^{2} \, \mbox{d}r \\
& = & 4 \pi \int\limits_{r = r_{A}}^{r = \infty} \rho_{0} e^{r_{A}/\tau} e^{- r/\tau} r^{2} \, \mbox{d}r \\
& = & 4 \pi \rho_{0} e^{r_{A}/\tau} \int\limits_{r = r_{A}}^{r = \infty} e^{- r/\tau} r^{2} \, \mbox{d}r \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(8)} \qquad \int x^{2} e^{ax} \, \mbox{d}x
 & = & \frac{1}{a^{3}} e^{ax} (a^{2} x^{2} - 2ax + 2) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(9)} \qquad m
 & = & 4 \pi \rho_{0} e^{r_{A}/\tau} \int\limits_{r = r_{A}}^{r = \infty} e^{- r/\tau} r^{2} \, \mbox{d}r \\
& = & 4 \pi \rho_{0} e^{r_{A}/\tau}
 \left [ - \tau^{3} e^{- r/\tau} \left ( \frac{1}{\tau^{2}} r^{2} + \frac{2}{\tau} r + 2 \right ) \right ]_{r = r_{A}}^{r = \infty} \\
& = & 4 \pi \rho_{0}
 \left [ - \tau^{3} e^{- (r - r_{A})/\tau} \left ( \frac{1}{\tau^{2}} r^{2} + \frac{2}{\tau} r + 2 \right ) \right ]_{r = r_{A}}^{r = \infty} \\
& = & 4 \pi \rho_{0} \left (
 \left ( - \tau^{3} e^{- (\infty - r_{A})/\tau} \left ( \frac{1}{\tau^{2}} \infty^{2} + \frac{2}{\tau} \infty + 2 \right ) \right )
 - \left ( - \tau^{3} e^{- (r_{A} - r_{A})/\tau} \left ( \frac{1}{\tau^{2}} r_{A}^{2} + \frac{2}{\tau} r_{A} + 2 \right ) \right ) \right ) \\
& = & 4 \pi \rho_{0} \left ( 0 + \tau^{3} \left ( \frac{1}{\tau^{2}} r_{A}^{2} + \frac{2}{\tau} r_{A} + 2 \right ) \right ) \\
& = & 4 \pi \rho_{0} \left ( \tau r_{A}^{2} + 2 \tau^{2} r_{A} + 2 \tau^{3} \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(10)} \qquad m
 & = & 4 \pi \rho_{0} \left ( \tau r_{A}^{2} + 2 \tau^{2} r_{A} + 2 \tau^{3} \right ) \\
& \approx & 4 \pi \rho_{0} \tau r_{A}^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(11)} \qquad p_{0}
 & = & \frac{F}{A} \\
& = & \frac{m g}{4 \pi r_{A}^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(12)} \qquad g
 & = & \frac{G M_{A}}{r^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(13)} \qquad p_{0}
 & = & \frac{m g}{4 \pi r_{A}^{2}} \\
& = & 4 \pi \rho_{0} e^{r_{A}/\tau} \int\limits_{r = r_{A}}^{r = \infty} e^{- r/\tau} r^{2} \frac{G M_{A}}{r^{2}} \frac{1}{4 \pi r_{A}^{2}} \, \mbox{d}r \\
& = & \frac{G M_{A} \rho_{0}}{r_{A}^{2}} e^{r_{A}/\tau} \int\limits_{r = r_{A}}^{r = \infty} e^{- r/\tau} \, \mbox{d}r \\
& = & \frac{G M_{A} \rho_{0}}{r_{A}^{2}} e^{r_{A}/\tau} \left [ - \tau e^{- r/\tau} \right ]_{r = r_{A}}^{r = \infty} \\
& = & \frac{G M_{A} \rho_{0}}{r_{A}^{2}} \left [ - \tau e^{- (r - r_{A})/\tau} \right ]_{r = r_{A}}^{r = \infty} \\
& = & \frac{G M_{A} \rho_{0}}{r_{A}^{2}} (- \tau e^{- (\infty - r_{A})/\tau} + \tau e^{- (r_{A} - r_{A})/\tau}) \\
& = & \frac{G M_{A} \rho_{0}}{r_{A}^{2}} (0 + \tau) \\
& = & \frac{G M_{A} \rho_{0} \tau}{r_{A}^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(14)} \qquad p_{0}
 & = & \frac{G M_{A} \rho_{0} \tau}{r_{A}^{2}} \\
\tau
 & = & \frac{p_{0} r_{A}^{2}}{G M_{A} \rho_{0}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(15)} \qquad m
 & = & 4 \pi \rho_{0} \tau r_{A}^{2} \\
& = & 4 \pi \rho_{0} \frac{p_{0} r_{A}^{2}}{G M_{A} \rho_{0}} r_{A}^{2} \\
& = & \frac{4 \pi p_{0} r_{A}^{4}}{G M_{A}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(16)} \qquad m
 & = & \frac{4 \pi p_{0} r_{A}^{4}}{G M_{A}} \\
& = & 5.2847 \cdot 10^{18} \mbox{ kg} \\
\end{eqnarray*}
\end{document}
