#This file was created by Mon Nov 24 22:57:48 1997 #LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team \lyxformat 2.15 \textclass aapaper \language default \inputencoding latin1 \fontscheme default \graphics default \paperfontsize default \spacing single \papersize Default \paperpackage a4 \use_geometry 0 \use_amsmath 0 \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle default \layout Thesaurus 06(03.11.1;16.06.1;19.06.1;19.37.1;19.53.1;19.63.1) \layout Title Hydrodynamics of giant planet formation \layout Subtitle I. Overviewing the \begin_inset Formula \( \kappa \) \end_inset -mechanism \layout Author G. Wuchterl \layout Address Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna \layout Offprint G. Wuchterl \layout Email wuchterl@amok.ast.univie.ac.at \layout Date Received September 15, 1996 / Accepted March 16, 1997 \layout Abstract To investigate the physical nature of the `nuc\SpecialChar \- leated instability' of proto giant planets (Mizuno \begin_inset LatexCommand \cite{mizuno} \end_inset ), the stability of layers in static, radiative gas spheres is analysed on the basis of Baker's \begin_inset LatexCommand \cite{baker} \end_inset standard one-zone model. 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. \layout Abstract The stability equations of state are calculated for solar composition and are displayed in the domain \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \) \end_inset , \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \) \end_inset . 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 \begin_inset Formula \( \rho \) \end_inset , temperature \begin_inset Formula \( T \) \end_inset or specific internal energy \begin_inset Formula \( e \) \end_inset . Regions of instability in the \begin_inset Formula \( (\rho \, e) \) \end_inset -plane are described and related to the underlying microphysical processes. Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone. The \begin_inset Formula \( \kappa \) \end_inset -mechanism is widespread under `cool' conditions. \layout Abstract \latex latex \backslash keywords{ \latex default giant planet formation -- \begin_inset Formula \( \kappa \) \end_inset -mechanism -- stability of gas spheres \latex latex } \layout Section Introduction \layout Standard In the \emph on nucleated instability \emph default (also called core instability) hypothesis of giant planet formation, a critical mass for static core envelope protoplanets has been found. Mizuno ( \begin_inset LatexCommand \cite{mizuno} \end_inset ) determined the critical mass of the core to be about \begin_inset Formula \( 12\, M_{\oplus } \) \end_inset ( \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \) \end_inset is the Earth mass), which is independent of the outer boundary conditions and therefore independent of the location in the solar nebula. This critical value for the core mass corresponds closely to the cores of today's giant planets. \layout Standard Although no hydrodynamical study has been available many workers conjectured that a collapse or rapid contraction will ensue after accumulating the critical mass. The main motivation for this article is to investigate the stability of the static envelope at the critical mass. With this aim the local, linear stability of static radiative gas spheres is investigated on the basis of Baker's ( \begin_inset LatexCommand \cite{baker} \end_inset ) standard one-zone model. The nonlinear, hydrodynamic evolution of the protogiant planet beyond the critical mass, as calculated by Wuchterl ( \begin_inset LatexCommand \cite{wuchterl} \end_inset ), will be described in a forthcoming article. \layout Standard The fact that Wuchterl ( \begin_inset LatexCommand \cite{wuchterl} \end_inset ) found the excitation of hydrodynamical waves in his models raises considerable interest on the transition from static to dynamic evolutionary phases of the protogiant planet at the critical mass. The waves play a crucial role in the development of the so-called nucleated instability in the nucleated instability hypothesis. They lead to the formation of shock waves and massive outflow phenomena. The protoplanet evolves into a new quasi-equilibrium structure with a \emph on pulsating \emph default envelope, after the mass loss phase has declined. \layout Standard Phenomena similar to the ones described above for giant planet formation have been found in hydrodynamical models concerning star formation where protostellar cores explode (Tscharnuter \begin_inset LatexCommand \cite{tscarnuter} \end_inset , Balluch \begin_inset LatexCommand \cite{balluch} \end_inset ), whereas earlier studies found quasi-steady collapse flows. The similarities in the (micro)physics, i.e., constitutive relations of protostel lar cores and protogiant planets serve as a further motivation for this study. \layout Section Baker's standard one-zone model \layout Standard \begin_float wide-fig \layout Standard \latex latex \backslash rule{0.4pt}{4cm} \hfill \backslash parbox[b]{55mm}{ \layout Caption Adiabatic exponent \begin_inset Formula \( \Gamma \) \end_inset . \begin_inset Formula \( \Gamma _{1} \) \end_inset is plotted as a function of \begin_inset Formula \( \lg \) \end_inset internal energy \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \) \end_inset and \begin_inset Formula \( \lg \) \end_inset density \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \) \end_inset \begin_inset LatexCommand \label{FigGam} \end_inset \latex latex } \end_float In this section the one-zone model of Baker ( \begin_inset LatexCommand \cite{baker} \end_inset ), originally used to study the Cepheïd pulsation mechanism, will be briefly reviewed. The resulting stability criteria will be rewritten in terms of local state variables, local timescales and constitutive relations. \layout Standard Baker ( \begin_inset LatexCommand \cite{baker} \end_inset ) investigates the stability of thin layers in self-gravitating, spherical gas clouds with the following properties: \layout Itemize hydrostatic equilibrium, \layout Itemize thermal equilibrium, \layout Itemize energy transport by grey radiation diffusion. \layout Standard For the one-zone-model Baker obtains necessary conditions for dynamical, secular and vibrational (or pulsational) stability [Eqs. \protected_separator (34a, \latex latex \backslash , \latex default b, \latex latex \backslash , \latex default c) in Baker \begin_inset LatexCommand \cite{baker} \end_inset ]. Using Baker's notation: \begin_inset Formula \begin{eqnarray*} M_{\mathrm{r}} & & \mathrm{mass}\, \mathrm{internal}\, \mathrm{to}\, \mathrm{the}\, \mathrm{radius}\, r\\ m & & \mathrm{mass}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}\\ r_{0} & & \mathrm{unperturbed}\, \mathrm{zone}\, \mathrm{radius}\\ \rho _{0} & & \mathrm{unperturbed}\, \mathrm{density}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\ T_{0} & & \mathrm{unperturbed}\, \mathrm{temperature}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\ L_{r0} & & \mathrm{unperturbed}\, \mathrm{luminosity}\\ E_{\mathrm{th}} & & \mathrm{thermal}\, \mathrm{energy}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone} \end{eqnarray*} \end_inset and with the definitions of the \emph on local cooling time \emph default (see Fig. \protected_separator \begin_inset LatexCommand \ref{FigGam} \end_inset ) \layout Standard \begin_inset Formula \begin{equation} \label{} \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, , \end{equation} \end_inset and the \emph on local free-fall time \layout Standard \begin_inset Formula \begin{equation} \label{} \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}\, ,} \end{equation} \end_inset Baker's \begin_inset Formula \( K \) \end_inset and \begin_inset Formula \( \sigma _{0} \) \end_inset have the following form: \begin_inset Formula \begin{eqnarray} \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\ K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ; \end{eqnarray} \end_inset where \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/\rho _{0}) \) \end_inset has been used and \layout Standard \begin_inset Formula \begin{equation} \label{} \begin{array}{l} \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\ e=mc^{2} \end{array} \end{equation} \end_inset is a thermodynamical quantity which is of order \begin_inset Formula \( 1 \) \end_inset and equal to \begin_inset Formula \( 1 \) \end_inset for nonreacting mixtures of classical perfect gases. The physical meaning of \begin_inset Formula \( \sigma _{0} \) \end_inset and \begin_inset Formula \( K \) \end_inset is clearly visible in the equations above. \begin_inset Formula \( \sigma _{0} \) \end_inset represents a frequency of the order one per free-fall time. \begin_inset Formula \( K \) \end_inset is proportional to the ratio of the free-fall time and the cooling time. Substituting into Baker's criteria, using thermodynamic identities and definitions of thermodynamic quantities, \begin_inset Formula \[ \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\: \chi _{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\: \kappa _{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\] \end_inset \layout Standard \begin_inset Formula \[ \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\: \chi _{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\: \kappa _{T}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\] \end_inset one obtains, after some pages of algebra, the conditions for \emph on stability \emph default given below: \layout Standard \begin_inset Formula \begin{eqnarray} \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\ \frac{\pi ^{2}}{\tau _{\mathrm{co}}\tau _{\mathrm{ff}}^{2}}\Gamma _{1}\nabla _{\mathrm{ad}}\left[ \frac{1-3/4\chi _{\rho }}{\chi _{T}}(\kappa _{T}-4)+\kappa _{P}+1\right] & > & 0\label{ZSSecSta} \\ \frac{\pi ^{2}}{4}\frac{3}{\tau _{\mathrm{co}}\tau _{\mathrm{ff}}^{2}}\Gamma _{1}^{2}\nabla _{\mathrm{ad}}\left[ 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa _{T}+\kappa _{P})-\frac{4}{3\Gamma _{1}}\right] & > & 0\label{ZSVibSta} \end{eqnarray} \end_inset For a physical discussion of the stability criteria see Baker ( \begin_inset LatexCommand \cite{baker} \end_inset ) or Cox ( \begin_inset LatexCommand \cite{cox} \end_inset ). \layout Standard We observe that these criteria for dynamical, secular and vibrational stability, respectively, can be factorized into \layout Enumerate a factor containing local timescales only, \layout Enumerate a factor containing only constitutive relations and their derivatives. \layout Standard The first factors, depending on only timescales, are positive by definition. The signs of the left hand sides of the inequalities \protected_separator ( \begin_inset LatexCommand \ref{ZSDynSta} \end_inset ), ( \begin_inset LatexCommand \ref{ZSSecSta} \end_inset ) and ( \begin_inset LatexCommand \ref{ZSVibSta} \end_inset ) therefore depend exclusively on the second factors containing the constitutive relations. Since they depend only on state variables, the stability criteria themselves are \emph on functions of the thermodynamic state in the local zone \emph default . The one-zone stability can therefore be determined from a simple equation of state, given for example, as a function of density and temperature. Once the microphysics, i.e. the thermodynamics and opacities (see Table \protected_separator \begin_inset LatexCommand \ref{KapSou} \end_inset ), are specified (in practice by specifying a chemical composition) the one-zone stability can be inferred if the thermodynamic state is specified. The zone -- or in other words the layer -- will be stable or unstable in whatever object it is imbedded as long as it satisfies the one-zone-model assumptions. Only the specific growth rates (depending upon the time scales) will be different for layers in different objects. \layout Standard \begin_float tab \layout Caption Opacity sources \begin_inset LatexCommand \label{KapSou} \end_inset \layout Standard \align center \LyXTable multicol4 4 2 0 0 -1 -1 -1 -1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 2 0 0 2 0 0 0 2 1 0 0 0 0 0 2 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0 0 2 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0 Source \newline T/[K] \newline Yorke 1979, Yorke 1980a \newline \begin_inset Formula \( \leq 1700^{\mathrm{a}} \) \end_inset \newline Krügel 1971 \newline \begin_inset Formula \( 1700\leq T\leq 5000 \) \end_inset \newline Cox & Stewart 1969 \newline \begin_inset Formula \( 5000\leq \) \end_inset \layout Standard \begin_inset Formula \( \mathrm{a} \) \end_inset This is footnote a \end_float \begin_float wide-tab \layout Caption Regions of secular instability \begin_inset LatexCommand \label{TabSecInst} \end_inset \layout Standard \latex latex \backslash vspace{4cm} \end_float We will now write down the sign (and therefore stability) determining parts of the left-hand sides of the inequalities ( \begin_inset LatexCommand \ref{ZSDynSta} \end_inset ), ( \begin_inset LatexCommand \ref{ZSSecSta} \end_inset ) and ( \begin_inset LatexCommand \ref{ZSVibSta} \end_inset ) and thereby obtain \emph on stability equations of state \emph default . \layout Standard The sign determining part of inequality \protected_separator ( \begin_inset LatexCommand \ref{ZSDynSta} \end_inset ) is \begin_inset Formula \( 3\Gamma _{1}-4 \) \end_inset and it reduces to the criterion for dynamical stability \layout Standard \begin_inset Formula \begin{equation} \label{} \Gamma _{1}>\frac{4}{3} \end{equation} \end_inset Stability of the thermodynamical equilibrium demands \begin_inset Formula \begin{equation} \label{} \chi _{\rho }>0,\: \: c_{v}>0\, , \end{equation} \end_inset and \layout Standard \begin_inset Formula \begin{equation} \label{} \chi _{T}>0 \end{equation} \end_inset holds for a wide range of physical situations. With \layout Standard \begin_inset Formula \begin{eqnarray} \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi _{T}}{c_{v}} & > & 0\\ \Gamma _{1}=\chi _{\rho }+\chi _{T}(\Gamma _{3}-1) & > & 0\\ \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0 \end{eqnarray} \end_inset we find the sign determining terms in inequalities \protected_separator ( \begin_inset LatexCommand \ref{ZSSecSta} \end_inset ) and ( \begin_inset LatexCommand \ref{ZSVibSta} \end_inset ) respectively and obtain the following form of the criteria for dynamical, secular and vibrational \emph on stability \emph default , respectively: \layout Standard \begin_inset Formula \begin{eqnarray} 3\Gamma _{1}-4=:\, S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\ \frac{1-3/4\chi _{\rho }}{\chi _{T}}(\kappa _{T}-4)+\kappa _{P}+1=:\, S_{\mathrm{sec}}> & 0 & \label{SecSta} \\ 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa _{T}+\kappa _{P}-\frac{4}{3\Gamma _{1}}=:\, S_{\mathrm{vib}}> & 0 & \label{VibSta} \end{eqnarray} \end_inset The constitutive relations are to be evaluated for the unperturbed thermodynamic state (say \begin_inset Formula \( (\rho _{0},T_{0}) \) \end_inset ) of the zone. We see that the one-zone stability of the layer depends only on the constitutiv e relations \begin_inset Formula \( \Gamma _{1} \) \end_inset , \begin_inset Formula \( \nabla _{\mathrm{ad}} \) \end_inset , \begin_inset Formula \( \chi _{T},\, \chi _{\rho } \) \end_inset , \begin_inset Formula \( \kappa _{P},\, \kappa _{T} \) \end_inset . These depend only on the unperturbed thermodynamical state of the layer. Therefore the above relations define the one-zone-stability equations of state \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \) \end_inset and \begin_inset Formula \( S_{\mathrm{vib}} \) \end_inset . See Fig. \protected_separator \begin_inset LatexCommand \ref{FigVibStab} \end_inset for a picture of \begin_inset Formula \( S_{\mathrm{vib}} \) \end_inset . Regions of secular instability are listed in Table \protected_separator \begin_inset LatexCommand \ref{TabSecInst} \end_inset . \layout Standard \begin_float fig \layout Standard \latex latex \backslash vspace{5cm} \layout Caption Vibrational stability equation of state \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \) \end_inset . \begin_inset Formula \( >0 \) \end_inset means vibrational stability. \begin_inset LatexCommand \label{FigVibStab} \end_inset \end_float \layout Section Conclusions \layout Enumerate The conditions for the stability of static, radiative layers in gas spheres, as described by Baker's ( \begin_inset LatexCommand \cite{baker} \end_inset ) standard one-zone model, can be expressed as stability equations of state. These stability equations of state depend only on the local thermodynamic state of the layer. \layout Enumerate If the constitutive relations -- equations of state and Rosseland mean opacities -- are specified, the stability equations of state can be evaluated without specifying properties of the layer. \layout Enumerate For solar composition gas the \begin_inset Formula \( \kappa \) \end_inset -mechanism is working in the regions of the ice and dust features in the opacities, the \begin_inset Formula \( \mathrm{H}_{2} \) \end_inset dissociation and the combined H, first He ionization zone, as indicated by vibrational instability. These regions of instability are much larger in extent and degree of instabilit y than the second He ionization zone that drives the Cepheïd pulsations. \layout Acknowledgement Part of this work was supported by the German \emph on Deut\SpecialChar \- sche For\SpecialChar \- schungs\SpecialChar \- ge\SpecialChar \- mein\SpecialChar \- schaft, DFG \emph default project number Ts \protected_separator 17/2--1. \layout Bibliography \bibitem [1966]{baker} Baker N., 1966, in: Stellar Evolution, eds. \protected_separator R. F. Stein, A. G. W. Cameron, Plenum, New York, p. \protected_separator 333 \layout Bibliography \bibitem [1988]{balluch} Balluch M., 1988, A&A 200, 58 \layout Bibliography \bibitem [1980]{cox} Cox J. P., 1980, Theory of Stellar Pulsation, Princeton University Press, Princeton, p. \protected_separator 165 \layout Bibliography \bibitem [1969]{cox69} Cox A. N., Stewart J. N., 1969, Academia Nauk, Scientific Information 15, 1 \layout Bibliography \bibitem [1971]{kruegel} Krügel E., 1971, Der Rosselandsche Mittelwert bei tiefen Temperaturen, Diplom--Th esis, Univ. \protected_separator Göttingen \layout Bibliography \bibitem [1980]{mizuno} Mizuno H., 1980, Prog. Theor. Phys. 64, 544 \layout Bibliography \bibitem [1987]{tscarnuter} Tscharnuter W. M., 1987, A&A 188, 55 \layout Bibliography \bibitem [1989]{wuchterl} Wuchterl G., 1989, Zur Entstehung der Gasplaneten. Ku\SpecialChar \- gel\SpecialChar \- sym\SpecialChar \- me\SpecialChar \- tri\SpecialChar \- sche Gas\SpecialChar \- strö\SpecialChar \- mun\SpecialChar \- gen auf Pro\SpecialChar \- to\SpecialChar \- pla\SpecialChar \- ne\SpecialChar \- ten, Dissertation, Univ. Wien \layout Bibliography \bibitem [1979]{yorke79} Yorke H. W., 1979, A&A 80, 215 \layout Bibliography \bibitem [1980a]{yorke80a} Yorke H. W., 1980a, A&A 86, 286 \the_end