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---
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title: "Paperlighter Template Implementation Example"
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author:
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- number: 1
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name: "Author One"
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correspond: true
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affiliation: "My City University"
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address: "Orenomachi, Orenoshi, Orenoken, Japan"
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email: "one@myuni.ac.jp"
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- number: 2
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name: "Author Two"
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affiliation: "My Other City University"
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address: "Hokanomachi, Orenoshi, Orenoken, Japan"
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email: "two@myuni.ac.jp"
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email: "xxx@myuni.ac.jp"
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titleshort: "Paperlighter Example"
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authorshort: "Author One et.al."
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linkDir:
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- Figure
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- Output
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appendix:
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- appendix/1
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- appendix/2
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abstractTex:
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\abstract{To investigate the physical nature of the `nuc\-leated instability' of
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proto giant planets, the stability of layers
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in static, radiative gas spheres is analysed on the basis of Baker's
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standard one-zone model.}
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{To investigate the physical nature of the `nuc\-leated instability' of
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proto giant planets, the stability of layers
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in static, radiative gas spheres is analysed on the basis of Baker's
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standard one-zone model.}
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{It is shown that stability depends only upon the equations of state, the opacities and the local
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thermodynamic state in the layer. Stability and instability can
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therefore be expressed in the form of stability equations of state
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which are universal for a given composition.}
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{The stability equations of state are
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calculated for solar composition and are displayed in the domain
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$-14 \leq \lg \rho / \mathrm{[g\, cm^{-3}]} \leq 0 $,
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$ 8.8 \leq \lg e / \mathrm{[erg\, g^{-1}]} \leq 17.7$. These displays
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may be
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used to determine the one-zone stability of layers in stellar
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or planetary structure models by directly reading off the value of
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the stability equations for the thermodynamic state of these layers,
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specified
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by state quantities as density $\rho$, temperature $T$ or
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specific internal energy $e$.
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Regions of instability in the $(\rho,e)$-plane are described
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and related to the underlying microphysical processes.}
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{Vibrational instability is found to be a common phenomenon
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at temperatures lower than the second He ionisation
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zone. The $\kappa$-mechanism is widespread under `cool'
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conditions.}
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{}
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keywords: giant planet formation -- $\kappa$-mechanism -- stability of gas spheres
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acknowledgements:
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Part of this work was supported by the German
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\emph{Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} project
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number Ts~17/2--1.
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---
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# Introduction
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In the \emph{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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the critical mass of the core to be about $12 \,M_\oplus$
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($M_\oplus=5.975 \times 10^{27}\,\mathrm{g}$ is the Earth mass), which
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is independent of the outer boundary
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conditions and therefore independent of the location in the
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solar nebula. This critical value for the core mass corresponds
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closely to the cores of today's giant planets.
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Although no hydrodynamical study has been available many workers
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conjectured that a collapse or rapid contraction will ensue
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after accumulating the critical mass. The main motivation for
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this article
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is to investigate the stability of the static envelope at the
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critical mass. With this aim the local, linear stability of static
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radiative gas spheres is investigated on the basis of Baker's
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(\citeyear{mitchell80}) standard one-zone model.
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Phenomena similar to the ones described above for giant planet
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formation have been found in hydrodynamical models concerning
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star formation where protostellar cores explode
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(Tscharnuter \citeyear{kearns89}, Balluch \citeyear{MachineLearningI}),
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whereas earlier studies found quasi-steady collapse flows. The
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similarities in the (micro)physics, i.e., constitutive relations of
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protostellar cores and protogiant planets serve as a further
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motivation for this study.
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# Baker's standard one-zone model
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\begin{figure*}
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\centering
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\caption{Adiabatic exponent $\Gamma_1$.
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$\Gamma_1$ is plotted as a function of
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$\lg$ internal energy $\mathrm{[erg\,g^{-1}]}$ and $\lg$
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density $\mathrm{[g\,cm^{-3}]}$.}
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\label{FigGam}%
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\end{figure*}
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In this section the one-zone model of \citet{DudaHart2nd},
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originally used to study the Cephe{\"{\i}}d pulsation mechanism, will
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be briefly reviewed. The resulting stability criteria will be
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rewritten in terms of local state variables, local timescales and
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constitutive relations.
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\citet{DudaHart2nd} investigates the stability of thin layers in
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self-gravitating,
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spherical gas clouds with the following properties:
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\begin{itemize}
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\item hydrostatic equilibrium,
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\item thermal equilibrium,
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\item energy transport by grey radiation diffusion.
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\end{itemize}
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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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notation:
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\noindent
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and with the definitions of the \emph{local cooling time\/}
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(see Fig.~\ref{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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\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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}\,,
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\end{equation}
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Baker's $K$ and $\sigma_0$ have the following form:
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\begin{eqnarray}
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\sigma_0 & = & \frac{\pi}{\sqrt{8}}
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\frac{1}{ \tau_{\mathrm{ff}}} \\
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K & = & \frac{\sqrt{32}}{\pi} \frac{1}{\delta}
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\frac{ \tau_{\mathrm{ff}} }
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{ \tau_{\mathrm{co}} }\,;
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\end{eqnarray}
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where $E_{\mathrm{th}} \approx m (P_0/{\rho_0})$ has been used and
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\begin{equation}
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\begin{array}{l}
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\delta = - \left(
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\frac{ \partial \ln \rho }{ \partial \ln T }
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\right)_P \\
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e=mc^2
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\end{array}
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\end{equation}
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is a thermodynamical quantity which is of order $1$ and equal to $1$
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for nonreacting mixtures of classical perfect gases. The physical
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meaning of $\sigma_0$ and $K$ is clearly visible in the equations
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above. $\sigma_0$ represents a frequency of the order one per
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free-fall time. $K$ is proportional to the ratio of the free-fall
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time and the cooling time. Substituting into Baker's criteria, using
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thermodynamic identities and definitions of thermodynamic quantities,
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\begin{displaymath}
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\Gamma_1 = \left( \frac{ \partial \ln P}{ \partial\ln \rho}
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\right)_{S} \, , \;
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\chi^{}_\rho = \left( \frac{ \partial \ln P}{ \partial\ln \rho}
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\right)_{T} \, , \;
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\kappa^{}_{P} = \left( \frac{ \partial \ln \kappa}{ \partial\ln P}
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\right)_{T}
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\end{displaymath}
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\begin{displaymath}
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\nabla_{\mathrm{ad}} = \left( \frac{ \partial \ln T}
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{ \partial\ln P} \right)_{S} \, , \;
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\chi^{}_T = \left( \frac{ \partial \ln P}
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{ \partial\ln T} \right)_{\rho} \, , \;
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\kappa^{}_{T} = \left( \frac{ \partial \ln \kappa}
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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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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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( 3 \Gamma_1 - 4 )
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& > & 0 \label{ZSDynSta} \\
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\frac{\pi^2}{\tau_{\mathrm{co}}
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\tau_{\mathrm{ff}}^2}
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\Gamma_1 \nabla_{\mathrm{ad}}
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\left[ \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T }
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( \kappa^{}_T - 4 )
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+ \kappa^{}_P + 1
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\right]
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& > & 0 \label{ZSSecSta} \\
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\frac{\pi^2}{4} \frac{3}{\tau_{ \mathrm{co} }
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\tau_{ \mathrm{ff} }^2
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}
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\Gamma_1^2 \, \nabla_{\mathrm{ad}} \left[
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4 \nabla_{\mathrm{ad}}
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- ( \nabla_{\mathrm{ad}} \kappa^{}_T
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+ \kappa^{}_P
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)
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- \frac{4}{3 \Gamma_1}
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\right]
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& > & 0 \label{ZSVibSta}
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\end{eqnarray}
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For a physical discussion of the stability criteria see \citet{DudaHart2nd} or \citet{anonymous}.
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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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\begin{table}
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\caption[]{Opacity sources.}
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\label{KapSou}
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$$
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\begin{array}{p{0.5\linewidth}l}
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\hline
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\noalign{\smallskip}
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Source & T / {[\mathrm{K}]} \\
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\noalign{\smallskip}
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\hline
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\noalign{\smallskip}
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Yorke 1979, Yorke 1980a & \leq 1700^{\mathrm{a}} \\
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Kr\"ugel 1971 & 1700 \leq T \leq 5000 \\
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Cox \& Stewart 1969 & 5000 \leq \\
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\noalign{\smallskip}
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\hline
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\end{array}
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$$
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\end{table}
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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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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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\begin{equation}
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\Gamma_1 > \frac{4}{3}\,\cdot
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\end{equation}
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Stability of the thermodynamical equilibrium demands
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\begin{equation}
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\chi^{}_\rho > 0, \;\; c_v > 0\, ,
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\end{equation}
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and
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\begin{equation}
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\chi^{}_T > 0
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\end{equation}
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holds for a wide range of physical situations.
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With
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\begin{eqnarray}
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\Gamma_3 - 1 = \frac{P}{\rho T} \frac{\chi^{}_T}{c_v}&>&0\\
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\Gamma_1 = \chi_\rho^{} + \chi_T^{} (\Gamma_3 -1)&>&0\\
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\nabla_{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0
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\end{eqnarray}
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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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\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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+ \kappa^{}_P + 1 =: S_{\mathrm{sec}} > & 0 & \label{SecSta} \\
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4 \nabla_{\mathrm{ad}} - (\nabla_{\mathrm{ad}} \kappa^{}_T
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+ \kappa^{}_P)
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- \frac{4}{3 \Gamma_1} =: S_{\mathrm{vib}}
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> & 0\,.& \label{VibSta}
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\end{eqnarray}
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The constitutive relations are to be evaluated for the
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unperturbed thermodynamic state (say $(\rho_0, T_0)$) of the zone.
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We see that the one-zone stability of the layer depends only on
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the constitutive relations $\Gamma_1$,
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$\nabla_{\mathrm{ad}}$, $\chi_T^{},\,\chi_\rho^{}$,
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$\kappa_P^{},\,\kappa_T^{}$.
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These depend only on the unperturbed
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thermodynamical state of the layer. Therefore the above relations
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define the one-zone-stability equations of state
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$S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
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and $S_{\mathrm{vib}}$. See Fig.~\ref{FigVibStab} for a picture of
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$S_{\mathrm{vib}}$. Regions of secular instability are
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listed in Table~1.
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\begin{figure}
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\centering
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\caption{Vibrational stability equation of state
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$S_{\mathrm{vib}}(\lg e, \lg \rho)$.
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$>0$ means vibrational stability.
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}
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\label{FigVibStab}
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\end{figure}
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# Conclusions
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\begin{enumerate}
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\item The conditions for the stability of static, radiative
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layers in gas spheres, as described by Baker's (\citeyear{DudaHart2nd})
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standard one-zone model, can be expressed as stability
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equations of state. These stability equations of state depend
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only on the local thermodynamic state of the layer.
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\item If the constitutive relations -- equations of state and
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Rosseland mean opacities -- are specified, the stability
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equations of state can be evaluated without specifying
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properties of the layer.
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\item For solar composition gas the $\kappa$-mechanism is
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working in the regions of the ice and dust features
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in the opacities, the $\mathrm{H}_2$ dissociation and the
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combined H, first He ionization zone, as
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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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\end{enumerate}
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