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Home All issues Volume 506 / No 3 (November II 2009) A&A, 506 3 (2009) 1215-1228 Full HTML
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Table 4:

Synopsis of the basic equations that characterize the studied gas media and the propagation of a shock wave within thema.
$\rho (Z)=$ $\frac{\rho (z)}{\rho_{0}}= F(z)=$ Solutions for the shock-wave motion Blow-out time tb
$\rho_{\rm c} \exp~ (-Z/ {H})$ $\exp~ (-z/{H})$ $r= 2 {H} \arctan \left( \frac{t_{\star} \cos \varphi}{1-t_{\star} \sin \varphi} \right)$ 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$  
$\frac{\rho_{\rm c}}{ (1+\frac{Z}{{H}})}$ $\frac{1}{ (1+\frac{z}{{H_{\star}}})}$ $r = 2 {H_{\star}} \big \{ t_{\star} \cos \varphi+ \frac{\cos 2 \varphi}{(2 \co... ... } \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_{\... ...}{(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$  
$\frac{\rho_{\rm c}}{(1+\frac{Z}{H})^{2}}$ $\frac{1}{ (1+\frac{z}{\rm {H_{\star}}})^{2}}$ $r= {H_{\star}} \frac{\cos \varphi}{\coth 2 t_{\star} - \sin~ \varphi}$ $\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$  
$\rho_{\rm c}~ {\rm sech }^{2} ~(Z/ {H})$ $ [\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 $  

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 Z0, 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.


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