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335 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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processDate: |
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received: "March 11, 2022" |
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accepted: "April 11, 2022" |
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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 in static, radiative gas spheres is analysed on the basis of Baker's 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 in static, radiative gas spheres is analysed on the basis of Baker's 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 thermodynamic state in the layer. Stability and instability can therefore be expressed in the form of stability equations of state which are universal for a given composition.} |
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{The stability equations of state are calculated for solar composition and are displayed in the domain $-14 \leq \lg \rho / \mathrm{[g\, cm^{-3}]} \leq 0 $, $ 8.8 \leq \lg e / \mathrm{[erg\, g^{-1}]} \leq 17.7$. These displays may be used to determine the one-zone stability of layers in stellar or planetary structure models by directly reading off the value of the stability equations for the thermodynamic state of these layers, specified by state quantities as density $\rho$, temperature $T$ or specific internal energy $e$. Regions of instability in the $(\rho,e)$-plane are described and related to the underlying microphysical processes.} |
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{Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone. The $\kappa$-mechanism is widespread under `cool' 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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*Deutsche Forschungsgemeinschaft, 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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<!-- 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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* 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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{#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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