\begin{table}%t4
\caption{\label{sinoptico}Synopsis of the basic equations that characterize the studied gas
  media and the propagation of a shock wave within them$^{a}$. }
\par
%\centerline
{
\begin{tabular}{l l l c }
\hline \hline 
 $\rho (Z)=$  &    $\frac{\rho (z)}{\rho_{0}}= F(z)=$   &  Solutions for
 the shock-wave motion &  Blow-out  time $t_{b}$  \\ 
\hline
 $\rho_{\rm c} \exp~ (-Z/{H})$  &   $\exp~ (-z/{H})$              &  $r= 2
 {H} \arctan \left( \frac{t_{\star} \cos \varphi}{1-t_{\star} \sin
    \varphi} \right)$  &   1.0  \\
       &                  &  $z= - 2 {H} \ln \left(\sqrt{ 1-2 t_{\star} \sin \varphi +
    t_{\star}^2}\right)$ &       \\
 for $ 0\leq Z \leq \infty$  & for $ -Z_{0}\leq z \leq \infty$  & for $
 -Z_{0}\leq z \leq \infty$   &                       \\
\hline
$\frac{\rho_{\rm c}}{ (1+\frac{Z}{{H}})}$ & $\frac{1}{
  (1+\frac{z}{{H_{\star}}})}$ 
  & 
  $r = 2 {H_{\star}}  \big \{  t_{\star} \cos \varphi+ \frac{\cos 2 \varphi}{(2
    \cos \varphi)^{2} } \left[ \sin (2
t_{\star} \cos \varphi)- 2
t_{\star} \cos \varphi)\right ] + $
     & 
$\infty $  \\
 &  where ${H_{\star}=H} +Z_{0}$ &   $ \frac{\sin \varphi}{2 \cos \varphi}  \left [1- \cos (2
    t_{\star} \cos \varphi ) \right ] \big \}$  &  \\
              &   &  $ z = 2 {H_{\star}}  \big \{  \frac{\sin 2 \varphi}{2
    \cos \varphi } \sin (2
t_{\star} \cos \varphi)+ 
     \frac{\cos 2 \varphi}{(2 \cos \varphi)^{2}} \big [\cos (2
t_{\star} \cos \varphi)-1\big ] \big \}. $ & \\
for $0 \leq Z \leq \infty$ & for $-Z_{0} \leq z \leq \infty$ & for $-Z_{0}
  \leq z \leq \infty$ & \\
\hline
$\frac{\rho_{\rm c}}{ (1+\frac{Z}{{H}})^{2}}$ & $\frac{1}{
  (1+\frac{z}{\rm{H_{\star}}})^{2}}$ & $r= {H_{\star}} \frac{\cos
  \varphi}{\coth 2 t_{\star} -  \sin~ \varphi}$ & $\infty $  \\
              & where ${H_{\star}=H} +Z_{0}$ &  $z= {H_{\star}} \frac{2
  \sin
  \varphi - \cos^{2}\varphi ~~ \sinh 2 t_{\star} }{\coth~ t_{\star} -
   \sin \varphi +   \cos^{2}\varphi  ~~ \sinh 2 t_{\star} } $ & \\
for $0 \leq Z \leq \infty$ & for $-Z_{0} \leq z \leq \infty$ & for $-Z_{0}
  \leq z \leq \infty$ & \\
\hline
  $\rho_{\rm c}~ {\rm sech }^{2} ~(Z/ {H})$  & $ [\cosh~ (z/{H})+ \alpha \sinh~
  (z/{H})]^{-2}$  & $r= {H} ~\arctan \frac{
  (p + q) \cos
   \varphi}{1 - p~ q - (p - q) \sin\varphi}  $ &  0.5--0.78   \\ 
&  where $\alpha= \tanh~ (Z_{0}/{H})$ & $ z=  {H}~   \ln \sqrt{\frac{1+ 2 q \sin \varphi
      + q^{2}}{1- 2 p \sin \varphi + p^{2}} }.$   &          \\
    &    &   where $p=(1+\alpha)~ t_{\star}+{\sum_{n=2}^{\infty} (-1+\alpha)^{n-1}
 (1+\alpha)^{n}} B_{2n} {\frac{(-4^{n})(1-4^{n})}{(2
      n)!}}~t_{\star}^{2n-1}$ & \\
& & and,  $q= -(-1+\alpha)~ t_{\star}-{\sum_{n=2}^{\infty} (-1+\alpha)^{n}
 (1+\alpha)^{n-1}} B_{2n}{ \frac{(-4^{n})(1-4^{n})}{(2
      n)!}}~t_{\star}^{2n-1}.$ & \\ 
 for $-\infty\leq Z \leq \infty$ &  for $ -\infty\leq z \leq \infty $ & for $ -\infty\leq z \leq \infty $  &   \\
\hline
\end{tabular}}\medskip
\par
$^a$  Density
  distribution of the  plane-parallel stratified medium, with respect to  a
  $Z$ axis (Col.~1) and to a $z$ axis (Col.~2). The $z$ and $Z$ axes are
  along the line that passes through the explosion point and  is  perpendicular to the
  stratification plane. The origin of $Z$-axis lies on the plane of maximum
  gas density $\rho_{\rm c}$ or symmetry plane, e.g. the Galactic plane. Both
  positive axes point in the same sense, toward decreasing densities. We denote the $Z$-position
  (or altitude) of
  the explosion point by $Z_{0}$, origin of the $z$ axis. The positions $(r, z)$ of the wave, given by
  the parametric equations (Col.~3),  are referred to the cylindrical
  coordinate system  $(r, z)$ with origin
  in  the explosion
  point and symmetry  around the $z$-axis. The valid range for each  function is given.  
\end{table}