#LyX 1.4.1 created this file. For more info see http://www.lyx.org/ \lyxformat 245 \begin_document \begin_header \textclass aa \begin_preamble \usepackage{graphicx} % \end_preamble \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize default \spacing single \papersize default \use_geometry false \use_amsmath 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 2 \papersides 2 \paperpagestyle default \tracking_changes false \output_changes true \end_header \begin_body \begin_layout Title Hydrodynamics of giant planet formation \end_layout \begin_layout Subtitle I. Overviewing the \begin_inset Formula \( \kappa \) \end_inset -mechanism \end_layout \begin_layout Author G. Wuchterl \begin_inset ERT status collapsed \begin_layout Standard \backslash inst{1} \backslash and \end_layout \begin_layout Standard \end_layout \end_inset C. Ptolemy \begin_inset ERT status collapsed \begin_layout Standard \backslash inst{2} \backslash fnmsep \end_layout \end_inset \begin_inset Foot status collapsed \begin_layout Standard Just to show the usage of the elements in the author field \end_layout \end_inset \end_layout \begin_layout Offprint G. Wuchterl \end_layout \begin_layout Address Institute for Astronomy (IfA), University of Vienna, T\i \"{u} rkenschanzstrasse 17, A-1180 Vienna \newline \begin_inset ERT status collapsed \begin_layout Standard \backslash email{wuchterl@amok.ast.univie.ac.at} \backslash and \end_layout \begin_layout Standard \end_layout \end_inset University of Alexandria, Department of Geography, ... \newline \begin_inset ERT status collapsed \begin_layout Standard \backslash email{c.ptolemy@hipparch.uheaven.space} \end_layout \end_inset \begin_inset Foot status collapsed \begin_layout Standard The university of heaven temporarily does not accept e-mails \end_layout \end_inset \end_layout \begin_layout Date Received September 15, 1996; accepted March 16, 1997 \end_layout \begin_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. 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. \begin_inset ERT status collapsed \begin_layout Standard \end_layout \begin_layout Standard \backslash keywords{giant planet formation -- \backslash ( \backslash kappa \backslash )-mechanism -- stability of gas spheres } \end_layout \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard In the \emph on nucleated instability \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \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. \end_layout \begin_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. \end_layout \begin_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{tscharnuter} \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. \end_layout \begin_layout Section Baker's standard one-zone model \end_layout \begin_layout Standard \begin_inset Float figure wide true sideways false status open \begin_layout Caption Adiabatic exponent \begin_inset Formula \( \Gamma _{1} \) \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 \end_layout \begin_layout Standard \begin_inset LatexCommand \label{FigGam} \end_inset \end_layout \end_inset In this section the one-zone model of Baker ( \begin_inset LatexCommand \cite{baker} \end_inset ), originally used to study the Cephe\i \"{\i} d pulsation mechanism, will be briefly reviewed. The resulting stability criteria will be rewritten in terms of local state variables, local timescales and constitutive relations. \end_layout \begin_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: \end_layout \begin_layout Itemize hydrostatic equilibrium, \end_layout \begin_layout Itemize thermal equilibrium, \end_layout \begin_layout Itemize energy transport by grey radiation diffusion. \end_layout \begin_layout Standard \noindent For the one-zone-model Baker obtains necessary conditions for dynamical, secular and vibrational (or pulsational) stability (Eqs. \begin_inset ERT status collapsed \begin_layout Standard \backslash \end_layout \end_inset (34a, \begin_inset ERT status collapsed \begin_layout Standard \backslash , \end_layout \end_inset b, \begin_inset ERT status collapsed \begin_layout Standard \backslash , \end_layout \end_inset c) in Baker \begin_inset LatexCommand \cite{baker} \end_inset ). Using Baker's notation: \end_layout \begin_layout Standard \align left \begin_inset Formula \begin{eqnarray*} M_{r} & & \textrm{mass internal to the radius }r\\ m & & \textrm{mass of the zone}\\ r_{0} & & \textrm{unperturbed zone radius}\\ \rho _{0} & & \textrm{unperturbed density in the zone}\\ T_{0} & & \textrm{unperturbed temperature in the zone}\\ L_{r0} & & \textrm{unperturbed luminosity}\\ E_{\textrm{th}} & & \textrm{thermal energy of the zone} \end{eqnarray*} \end_inset \end_layout \begin_layout Standard \noindent and with the definitions of the \emph on local cooling time \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \emph default (see Fig.\InsetSpace ~ \begin_inset LatexCommand \ref{FigGam} \end_inset ) \begin_inset Formula \begin{equation} \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, , \end{equation} \end_inset and the \emph on local free-fall time \emph default \begin_inset Formula \begin{equation} \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 \begin_inset Formula \begin{equation} \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 \begin_inset Formula \[ \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\] \end_inset one obtains, after some pages of algebra, the conditions for \emph on stability \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \emph default given below: \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 ). \end_layout \begin_layout Standard We observe that these criteria for dynamical, secular and vibrational stability, respectively, can be factorized into \end_layout \begin_layout Enumerate a factor containing local timescales only, \end_layout \begin_layout Enumerate a factor containing only constitutive relations and their derivatives. \end_layout \begin_layout Standard The first factors, depending on only timescales, are positive by definition. The signs of the left hand sides of the inequalities\InsetSpace ~ ( \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. \begin_inset ERT status collapsed \begin_layout Standard \backslash \end_layout \end_inset the thermodynamics and opacities (see Table\InsetSpace ~ \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. \end_layout \begin_layout Standard \begin_inset Float table wide false sideways false status open \begin_layout Caption \begin_inset LatexCommand \label{KapSou} \end_inset Opacity sources \end_layout \begin_layout Standard \begin_inset Tabular \begin_inset Text \begin_layout Standard Source \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula \( T/[\textrm{K}] \) \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard Yorke 1979, Yorke 1980a \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula \( \leq 1700^{\textrm{a}} \) \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard Krügel 1971 \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula \( 1700\leq T\leq 5000 \) \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Standard Cox & Stewart 1969 \end_layout \end_inset \begin_inset Text \begin_layout Standard \begin_inset Formula \( 5000\leq \) \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \begin_inset Formula \( ^{\textrm{a}} \) \end_inset This is footnote a \end_layout \end_inset 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 . \end_layout \begin_layout Standard The sign determining part of inequality\InsetSpace ~ ( \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 \begin_inset Formula \begin{equation} \Gamma _{1}>\frac{4}{3}\, \cdot \end{equation} \end_inset Stability of the thermodynamical equilibrium demands \begin_inset Formula \begin{equation} \chi ^{}_{\rho }>0,\; \; c_{v}>0\, , \end{equation} \end_inset and \begin_inset Formula \begin{equation} \chi ^{}_{T}>0 \end{equation} \end_inset holds for a wide range of physical situations. With \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\InsetSpace ~ ( \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: \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 thermodynami c 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.\InsetSpace ~ \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\InsetSpace ~ 1. \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_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 \end_layout \begin_layout Standard \begin_inset LatexCommand \label{FigVibStab} \end_inset \end_layout \end_inset \end_layout \begin_layout Section Conclusions \end_layout \begin_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. \end_layout \begin_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. \end_layout \begin_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 Cephe\i \"{\i} d pulsations. \end_layout \begin_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 \begin_inset ERT status collapsed \begin_layout Standard \backslash /{} \end_layout \end_inset \emph default project number Ts\InsetSpace ~ 17/2--1. \end_layout \begin_layout Bibliography \bibitem [1966]{baker} Baker, N. 1966, in Stellar Evolution, ed. \begin_inset ERT status collapsed \begin_layout Standard \backslash \end_layout \end_inset R. F. Stein,& A. G. W. Cameron (Plenum, New York) 333 \end_layout \begin_layout Bibliography \bibitem [1988]{balluch} Balluch, M. 1988, A&A, 200, 58 \end_layout \begin_layout Bibliography \bibitem [1980]{cox} Cox, J. P. 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton) 165 \end_layout \begin_layout Bibliography \bibitem [1969]{cox69} Cox, A. N.,& Stewart, J. N. 1969, Academia Nauk, Scientific Information 15, 1 \end_layout \begin_layout Bibliography \bibitem [1980]{mizuno} Mizuno H. 1980, Prog. Theor. Phys., 64, 544 \end_layout \begin_layout Bibliography \bibitem [1987]{tscharnuter} Tscharnuter W. M. 1987, A&A, 188, 55 \end_layout \begin_layout Bibliography \bibitem [1992]{terlevich} Terlevich, R. 1992, in ASP Conf. Ser. 31, Relationships between Active Galactic Nuclei and Starburst Galaxies, ed. A. V. Filippenko, 13 \end_layout \begin_layout Bibliography \bibitem [1980a]{yorke80a} Yorke, H. W. 1980a, A&A, 86, 286 \end_layout \begin_layout Bibliography \bibitem [1997]{zheng} Zheng, W., Davidsen, A. F., Tytler, D. & Kriss, G. A. 1997, preprint \end_layout \end_body \end_document