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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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<!-- |
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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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