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					@ -45,7 +45,7 @@ include-headers: | | 
				
			
			
		
	
		
			
				
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					# Introduction | 
				
			
			
		
	
		
			
				
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					In the \emph{nucleated instability\/} (also called core | 
				
			
			
		
	
		
			
				
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					In the *nucleated instability* (also called core | 
				
			
			
		
	
		
			
				
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					instability) hypothesis of giant planet | 
				
			
			
		
	
		
			
				
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					formation, a critical mass for static core  envelope | 
				
			
			
		
	
		
			
				
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					protoplanets has been found. \citet{langley00} determined | 
				
			
			
		
	
	
		
			
				
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					@ -106,18 +106,45 @@ spherical gas clouds with the following properties: | 
				
			
			
		
	
		
			
				
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					* thermal equilibrium, | 
				
			
			
		
	
		
			
				
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					* energy transport by grey radiation diffusion. | 
				
			
			
		
	
		
			
				
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					For equations, we can use several environment: | 
				
			
			
		
	
		
			
				
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					~~~ | 
				
			
			
		
	
		
			
				
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					\begin{equation} | 
				
			
			
		
	
		
			
				
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					\begin{eqnarray} | 
				
			
			
		
	
		
			
				
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					\begin{array} | 
				
			
			
		
	
		
			
				
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					\begin{displaymath} | 
				
			
			
		
	
		
			
				
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					\begin{align} | 
				
			
			
		
	
		
			
				
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					~~~ | 
				
			
			
		
	
		
			
				
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					This is an example: | 
				
			
			
		
	
		
			
				
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					~~~ | 
				
			
			
		
	
		
			
				
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					\begin{align} | 
				
			
			
		
	
		
			
				
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					\nabla \cdot  \vec{E} &= \rho \nonumber \\ | 
				
			
			
		
	
		
			
				
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					\nabla \cdot  \vec{B} &= 0    \nonumber \\ | 
				
			
			
		
	
		
			
				
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					\nabla \times \vec{E} &= -\frac{\vec{B}}{t} | 
				
			
			
		
	
		
			
				
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					\end{align} | 
				
			
			
		
	
		
			
				
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					~~~ | 
				
			
			
		
	
		
			
				
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					\begin{align} | 
				
			
			
		
	
		
			
				
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					\nabla \cdot  \vec{E} &= \rho \nonumber \\ | 
				
			
			
		
	
		
			
				
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					\nabla \cdot  \vec{B} &= 0    \nonumber \\ | 
				
			
			
		
	
		
			
				
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					\nabla \times \vec{E} &= -\frac{\vec{B}}{t} | 
				
			
			
		
	
		
			
				
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					\end{align} | 
				
			
			
		
	
		
			
				
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					For the one-zone-model Baker obtains necessary conditions | 
				
			
			
		
	
		
			
				
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					for dynamical, secular and vibrational (or pulsational) | 
				
			
			
		
	
		
			
				
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					stability (Eqs.\ (34a,\,b,\,c) in Baker \citeyear{DudaHart2nd}). Using Baker's | 
				
			
			
		
	
		
			
				
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					stability (Eqs. (34a,b,c) in Baker \citeyear{DudaHart2nd}). Using Baker's | 
				
			
			
		
	
		
			
				
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					notation: | 
				
			
			
		
	
		
			
				
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					\noindent | 
				
			
			
		
	
		
			
				
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					   and with the definitions of the \emph{local cooling time\/} | 
				
			
			
		
	
		
			
				
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					   and with the definitions of the *local cooling time* | 
				
			
			
		
	
		
			
				
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					   (see == [@fig:FigGam] == Fig. \ref{fig:FigGam}) | 
				
			
			
		
	
		
			
				
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					   \begin{equation} | 
				
			
			
		
	
		
			
				
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					      \tau_{\mathrm{co}} = \frac{E_{\mathrm{th}}}{L_{r0}} \,, | 
				
			
			
		
	
		
			
				
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					   \end{equation} | 
				
			
			
		
	
		
			
				
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					   and the \emph{local free-fall time} | 
				
			
			
		
	
		
			
				
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					   and the *local free-fall time* | 
				
			
			
		
	
		
			
				
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					   \begin{equation} | 
				
			
			
		
	
		
			
				
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					      \tau_{\mathrm{ff}} = | 
				
			
			
		
	
		
			
				
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					         \sqrt{ \frac{3 \pi}{32 G} \frac{4\pi r_0^3}{3 M_{\mathrm{r}}} | 
				
			
			
		
	
	
		
			
				
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					@ -164,7 +191,7 @@ notation: | 
				
			
			
		
	
		
			
				
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					                             { \partial\ln T} \right)_{T} | 
				
			
			
		
	
		
			
				
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					   \end{displaymath} | 
				
			
			
		
	
		
			
				
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					   one obtains, after some pages of algebra, the conditions for | 
				
			
			
		
	
		
			
				
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					   \emph{stability\/} given | 
				
			
			
		
	
		
			
				
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					   *stability* given | 
				
			
			
		
	
		
			
				
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					   below: | 
				
			
			
		
	
		
			
				
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					   \begin{eqnarray} | 
				
			
			
		
	
		
			
				
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					      \frac{\pi^2}{8} \frac{1}{\tau_{\mathrm{ff}}^2} | 
				
			
			
		
	
	
		
			
				
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					@ -195,31 +222,31 @@ notation: | 
				
			
			
		
	
		
			
				
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					   We observe that these criteria for dynamical, secular and | 
				
			
			
		
	
		
			
				
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					   vibrational stability, respectively, can be factorized into | 
				
			
			
		
	
		
			
				
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					   \begin{enumerate} | 
				
			
			
		
	
		
			
				
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					      \item a factor containing local timescales only, | 
				
			
			
		
	
		
			
				
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					      \item a factor containing only constitutive relations and | 
				
			
			
		
	
		
			
				
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					         their derivatives. | 
				
			
			
		
	
		
			
				
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					   \end{enumerate} | 
				
			
			
		
	
		
			
				
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					   The first factors, depending on only timescales, are positive | 
				
			
			
		
	
		
			
				
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					   by definition. The signs of the left hand sides of the | 
				
			
			
		
	
		
			
				
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					   inequalities (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) | 
				
			
			
		
	
		
			
				
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					   therefore depend exclusively on the second factors containing | 
				
			
			
		
	
		
			
				
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					   the constitutive relations. Since they depend only | 
				
			
			
		
	
		
			
				
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					   on state variables, the stability criteria themselves are \emph{ | 
				
			
			
		
	
		
			
				
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					   functions of the thermodynamic state in the local zone}. The | 
				
			
			
		
	
		
			
				
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					   one-zone stability can therefore be determined | 
				
			
			
		
	
		
			
				
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					   from a simple equation of state, given for example, as a function | 
				
			
			
		
	
		
			
				
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					   of density and | 
				
			
			
		
	
		
			
				
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					   temperature. Once the microphysics, i.e.\ the thermodynamics | 
				
			
			
		
	
		
			
				
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					   and opacities (see Table \ref{KapSou}), are specified (in practice | 
				
			
			
		
	
		
			
				
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					   by specifying a chemical composition) the one-zone stability can | 
				
			
			
		
	
		
			
				
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					   be inferred if the thermodynamic state is specified. | 
				
			
			
		
	
		
			
				
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					   The zone -- or in | 
				
			
			
		
	
		
			
				
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					   other words the layer -- will be stable or unstable in | 
				
			
			
		
	
		
			
				
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					   whatever object it is imbedded as long as it satisfies the | 
				
			
			
		
	
		
			
				
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					   one-zone-model assumptions. Only the specific growth rates | 
				
			
			
		
	
		
			
				
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					   (depending upon the time scales) will be different for layers | 
				
			
			
		
	
		
			
				
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					   in different objects. | 
				
			
			
		
	
		
			
				
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					1. a factor containing local timescales only, | 
				
			
			
		
	
		
			
				
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					2. a factor containing only constitutive relations and | 
				
			
			
		
	
		
			
				
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					   their derivatives. | 
				
			
			
		
	
		
			
				
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					3. To make a numered list, make sure that: | 
				
			
			
		
	
		
			
				
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					    1. it stands on its own paragraph (blank line above and below the list), | 
				
			
			
		
	
		
			
				
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					    2. the number starts at first column on each line. | 
				
			
			
		
	
		
			
				
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					The first factors, depending on only timescales, are positive | 
				
			
			
		
	
		
			
				
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					by definition. The signs of the left hand sides of the | 
				
			
			
		
	
		
			
				
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					inequalities (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) | 
				
			
			
		
	
		
			
				
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					therefore depend exclusively on the second factors containing | 
				
			
			
		
	
		
			
				
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					the constitutive relations. Since they depend only | 
				
			
			
		
	
		
			
				
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					on state variables, the stability criteria themselves are *functions of the thermodynamic state in the local zone*. The | 
				
			
			
		
	
		
			
				
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					one-zone stability can therefore be determined | 
				
			
			
		
	
		
			
				
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					from a simple equation of state, given for example, as a function | 
				
			
			
		
	
		
			
				
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					of density and temperature. Once the microphysics, i.e. the thermodynamics | 
				
			
			
		
	
		
			
				
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					and opacities (see Table \ref{KapSou}), are specified (in practice | 
				
			
			
		
	
		
			
				
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					by specifying a chemical composition) the one-zone stability can | 
				
			
			
		
	
		
			
				
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					be inferred if the thermodynamic state is specified. | 
				
			
			
		
	
		
			
				
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					The zone -- or in other words the layer -- will be stable or unstable in | 
				
			
			
		
	
		
			
				
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					whatever object it is imbedded as long as it satisfies the | 
				
			
			
		
	
		
			
				
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					one-zone-model assumptions. Only the specific growth rates | 
				
			
			
		
	
		
			
				
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					(depending upon the time scales) will be different for layers | 
				
			
			
		
	
		
			
				
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					in different objects. | 
				
			
			
		
	
		
			
				
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					@ -238,10 +265,10 @@ Table: Simple table using default markdown table. Currently not working in two-c | 
				
			
			
		
	
		
			
				
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					|             |          Grouping           || | 
				
			
			
		
	
		
			
				
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					First Header  | Second Header | Third Header | | 
				
			
			
		
	
		
			
				
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					 ------------ | :-----------: | -----------: | | 
				
			
			
		
	
		
			
				
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					Content       |          *Long Cell*        || | 
				
			
			
		
	
		
			
				
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					Content       |   **Cell**    |         Cell | | 
				
			
			
		
	
		
			
				
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					Content       |          *Long Cell*          || | 
				
			
			
		
	
		
			
				
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					Content       |   **Cell**        |         Cell | | 
				
			
			
		
	
		
			
				
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					New section   |     More      |         Data | | 
				
			
			
		
	
		
			
				
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					And more      | With an escaped '\|'         || | 
				
			
			
		
	
		
			
				
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					And more      | With an escaped '\|'        || | 
				
			
			
		
	
		
			
				
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					[More complicated table can be done using multimarkdown in .multiTable Code Block. You have to use this format for all table as default.] | 
				
			
			
		
	
		
			
				
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					~~~ | 
				
			
			
		
	
		
			
				
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					@ -318,11 +345,10 @@ Lp. & Miejscowość | 
				
			
			
		
	
		
			
				
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					   We will now write down the sign (and therefore stability) | 
				
			
			
		
	
		
			
				
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					   determining parts of the left-hand sides of the inequalities | 
				
			
			
		
	
		
			
				
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					   (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) and thereby | 
				
			
			
		
	
		
			
				
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					   obtain \emph{stability equations of state}. | 
				
			
			
		
	
		
			
				
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					   obtain *stability equations of state*. | 
				
			
			
		
	
		
			
				
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					   The sign determining part of inequality (\ref{ZSDynSta}) is | 
				
			
			
		
	
		
			
				
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					   $3\Gamma_1 - 4$ and it reduces to the | 
				
			
			
		
	
		
			
				
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					   criterion for dynamical stability | 
				
			
			
		
	
		
			
				
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					   $3\Gamma_1 - 4$ and it reduces to the criterion for dynamical stability | 
				
			
			
		
	
		
			
				
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					   \begin{equation} | 
				
			
			
		
	
		
			
				
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					     \Gamma_1 > \frac{4}{3}\,\cdot | 
				
			
			
		
	
		
			
				
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					   \end{equation} | 
				
			
			
		
	
	
		
			
				
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					@ -344,7 +370,7 @@ Lp. & Miejscowość | 
				
			
			
		
	
		
			
				
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					   we find the sign determining terms in inequalities (\ref{ZSSecSta}) | 
				
			
			
		
	
		
			
				
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					   and (\ref{ZSVibSta}) respectively and obtain the following form | 
				
			
			
		
	
		
			
				
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					   of the criteria for dynamical, secular and vibrational | 
				
			
			
		
	
		
			
				
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					   \emph{stability}, respectively: | 
				
			
			
		
	
		
			
				
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					   *stability*, respectively: | 
				
			
			
		
	
		
			
				
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					   \begin{eqnarray} | 
				
			
			
		
	
		
			
				
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					      3 \Gamma_1 - 4 =: S_{\mathrm{dyn}}      > & 0 & \label{DynSta}  \\ | 
				
			
			
		
	
		
			
				
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					      \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 ) | 
				
			
			
		
	
	
		
			
				
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					@ -403,7 +429,7 @@ create label for \ref{FigVibStab} using #FigVibStab | 
				
			
			
		
	
		
			
				
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					         indicated by vibrational instability. These regions | 
				
			
			
		
	
		
			
				
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					         of instability are much larger in extent and degree of | 
				
			
			
		
	
		
			
				
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					         instability than the second He ionization zone | 
				
			
			
		
	
		
			
				
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					         that drives the Cephe{\"\i}d pulsations. | 
				
			
			
		
	
		
			
				
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					         that drives the Cephe\text{\"\i}d pulsations. | 
				
			
			
		
	
		
			
				
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					   \end{enumerate} | 
				
			
			
		
	
		
			
				
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