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354 lines
14 KiB
--- |
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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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|
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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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<!-- we need figure* for make it full screen, so it must be in latex code as follows |
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--> |
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\begin{figure*} |
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\centering |
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\includegraphics{Figure/icml_numpapers.eps} |
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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\text{\"{\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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|
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* hydrostatic equilibrium, |
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* thermal equilibrium, |
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* energy transport by grey radiation diffusion. |
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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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<!-- calculate height and width of picture manually in inch, using evince -> Properties |
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create label for \ref{FigVibStab} using #FigVibStab |
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--> |
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![Vibrational stability equation of state $S_{\mathrm{vib}}(\lg e, \lg \rho)$. $>0$ means vibrational stability.](Figure/icml_numpapers.eps){#FigVibStab width=3.43in height=2.71in} |
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<!-- |
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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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--> |
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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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