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+% laa.dem version 1.1 as of 25-Feb-91
+%
+% This is LAA.DEM, the demonstration file of the
+% LaTeX style file from Springer-Verlag for the
+% Astronomy and Astrophysics Main Journal
+%
+% It is for use with LaTeX version 2.09
+%
+% Please report all errors via e-mail to SPRINGER@DHDSPRI6.bitnet
+% or to the address mentioned on page 2 of the documentation
+%
+%\documentstyle{laamt} % LaTeX A&A Monotype Times Fonts
+\documentstyle{laa} % LaTeX A&A Standard Fonts
+
+%_____________________________________ `Thermodynamical' derivatives.
+%
+\newcommand{\DXDYCZ}[3]{\left( \frac{ \partial #1 }{ \partial #2 }
+ \right)_{#3}
+ }
+
+\begin{document}
+
+ \thesaurus{06 % A&A Section 6: Form. struct. and evolut. of stars
+ (03.11.1; % Cosmogony,
+ 16.06.1; % Planets and satellites: general,
+ 19.06.1; % Solar system: general,
+ 19.37.1; % Stars: formation of,
+ 19.53.1; % Stars: oscillations of,
+ 19.63.1) % Stars: structure of.
+ }
+%
+ \title{Hydrodynamics of giant planet formation}
+
+ \subtitle{I. Overviewing the $\kappa$-mechanism}
+
+ \author{G. Wuchterl
+% \inst{1}
+ }
+
+% \offprints{G. Wuchterl}
+
+ \institute{Institut f\"ur Theoretische
+ Astrophysik der Universit\"at Heidelberg, Im Neuenheimer
+ Feld 561,\\
+ W--6900 Heidelberg, Federal Republic of Germany\\
+ EARN: WCAH at DS0RUS1I
+ }
+
+ \date{Received September 15, 1989; accepted March 16, 1990}
+
+ \maketitle
+
+ \begin{abstract}
+%______________________________________ Do not leave a blank line here!
+%
+% 14.Sep.'90: Demo-Vs.
+%_____________________________________ Do not leave a blank line here!
+ To investigate the physical nature of the `nucleated instability'
+ of proto giant planets (Mizuno 1980), the stability of layers
+ in static,
+ radiative gas spheres is analysed on the basis of Baker's
+ 1966 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
+ $-14 \leq \lg \rho / {\rm [g\, cm^{-3}]} \leq 0 $,
+ $ 8.8 \leq \lg e / {\rm [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.
+ 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.
+
+ \keywords{giant planet formation --
+ $\kappa$-mechanism --
+ stability of gas spheres
+ }
+ \end{abstract}
+
+%
+% 14.Sep.'90: Demo-Vs.
+%________________________________________________________________
+
+\section{Introduction}
+
+ In the {\em nucleated instability\/} (also called core
+ instability) hypothesis of giant planet
+ formation, a critical mass for static core envelope
+ protoplanets has been found. Mizuno (1980) determined
+ the critical mass of the core to be about $12 \,M_\oplus$
+ ($M_\oplus=5.975 \, 10^{27}\rm \,g$ 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.
+
+ Although no hydrodynamical study was 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
+ (1966) standard one-zone model. The nonlinear, hydrodynamic
+ evolution of the protogiant planet
+ beyond the critical mass, as calculated by Wuchterl
+ (1989), will be described in a forthcoming article.
+
+ The fact that Wuchterl (1989) 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 {\em pulsating} envelope, after the mass loss phase
+ has declined.
+
+ 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 1987, Balluch 1988),
+ whereas earlier studies found quasi-steady collapse flows. The
+ similarities in the
+ (micro)physics, i.e., constitutive relations of protostellar cores and
+ protogiant planets serve as a further motivation for this study.
+
+%
+% 14.Sep.'90: Demo-Vs.
+%__________________________________________________________________
+
+\section{Baker's standard one-zone model}
+
+% Two column figure (place early!)
+%______________________________________________ Gamma_1 (lg rho, lg e)
+ \begin{figure*}
+ \picplace{4cm}
+ \caption{Adiabatic exponent $\Gamma_1$.
+ $\Gamma_1$ is plotted as a function of
+ $\lg$ internal energy $\rm [erg\,g^{-1}]$ and $\lg$ density
+ $\rm [g\,cm^{-3}]$
+ }
+ \label{FigGam}
+ \end{figure*}
+%
+ In this section the one-zone model of Baker (1966), originally
+ used to study the Cephe{\"{\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.
+
+ Baker (1966) investigates the stability of thin layers in
+ self-gravitating,
+ spherical gas clouds with the following properties:
+ \begin{itemize}
+ \item hydrostatic equilibrium,
+ \item thermal equilibrium,
+ \item energy transport by grey radiation diffusion.
+ \end{itemize}
+ For the one-zone-model Baker obtains necessary conditions
+ for dynamical, secular and vibrational (or pulsational)
+ stability [Eqs.\ (34a,\,b,\,c) in Baker 1966]. Using Baker's
+ notation:
+ \[
+ \begin{array}{lp{0.8\linewidth}}
+ M_{\rm r} & mass internal to the radius $r$ \\
+ m & mass of the zone \\
+ r_0 & unperturbed zone radius \\
+ \rho_0 & unperturbed density in the zone \\
+ T_0 & unperturbed temperature in the zone \\
+ L_{r0} & unperturbed luminosity \\
+ E_{\rm th} & thermal energy of the zone
+ \end{array}
+ \]
+\noindent
+ and with the definitions of the {\em local cooling time\/}
+ (see Fig.~\ref{FigGam})
+ \begin{equation}
+ \tau_{\rm co} = \frac{E_{\rm th}}{L_{r0}} \,,
+ \end{equation}
+ and the {\em local free-fall time\/}
+ \begin{equation}
+ \tau_{\rm ff} =
+ \sqrt{ \frac{3 \pi}{32 G} \frac{4\pi r_0^3}{3 M_{\rm r}} }\,,
+ \end{equation}
+ Baker's $K$ and $\sigma_0$ have the following form:
+ \begin{eqnarray}
+ \sigma_0 & = & \frac{\pi}{\sqrt{8}}
+ \frac{1}{ \tau_{\rm ff} } \\
+ K & = & \frac{\sqrt{32}}{\pi} \frac{1}{\delta}
+ \frac{ \tau_{\rm ff} }
+ { \tau_{\rm co} }\,;
+ \end{eqnarray}
+ where $ E_{\rm th} \approx m (P_0/{\rho_0})$ has been used and
+ \begin{equation}
+ \delta = - \left(
+ \frac{ \partial \ln \rho }{ \partial \ln T }
+ \right)_P
+ \end{equation}
+ is a thermodynamical quantity which is
+ %of order $1$ and
+ equal to $1$ for nonreacting mixtures of classical perfect
+ gases.
+ The physical meaning of $ \sigma_0 $ and $K$ is clearly visible in
+ the equations above. $\sigma_0$ represents a frequency of the order one
+ per free-fall time. $K$ 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{equation}
+ \Gamma_1 = \DXDYCZ{\ln P}{\ln \rho}{S} \; , \;
+ \chi^{}_\rho = \DXDYCZ{\ln P}{\ln \rho}{T} \; , \;
+ \kappa^{}_{P} = \DXDYCZ{\ln \kappa}{\ln P}{T} \, ,
+ \end{equation}
+ \begin{equation}
+ \nabla_{\rm ad} = \DXDYCZ{\ln T}{\ln P}{S} \; , \;
+ \chi^{}_T = \DXDYCZ{\ln P}{\ln T}{\rho} \; , \;
+ \kappa^{}_{T} = \DXDYCZ{\ln \kappa}{\ln T}{T} \, ,
+ \end{equation}
+ one obtains, after some pages of algebra, the conditions for
+ {\em stability} given
+ below:
+ \begin{eqnarray}
+ \frac{\pi^2}{8} \frac{1}{\tau_{\rm ff}^2}
+ ( 3 \Gamma_1 - 4 )
+ & > & 0 \label{ZSDynSta} \\
+ \frac{\pi^2}{\tau_{\rm co}
+ \tau_{\rm ff}^2}
+ \Gamma_1 \nabla_{\rm 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_{ \rm co }
+ \tau_{ \rm ff }^2
+ }
+ \Gamma_1^2 \, \nabla_{\rm ad} \left[
+ 4 \nabla_{\rm ad}
+ - ( \nabla_{\rm ad} \kappa^{}_T
+ + \kappa^{}_P
+ )
+ - \frac{4}{3 \Gamma_1}
+ \right]
+ & > & 0 \label{ZSVibSta}
+ \end{eqnarray}
+%
+ For a physical discussion of the stability criteria see Baker (1966)
+ or Cox (1980).
+
+ We observe that these criteria for dynamical, secular and
+ vibrational stability, respectively, can be factorized into
+ \begin{enumerate}
+ \item a factor containing local timescales only,
+ \item a factor containing only constitutive relations and
+ their derivatives.
+ \end{enumerate}
+ The first factors, depending on only timescales, are positive
+ by definition. The signs of the left hand sides of the
+ inequalities~(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta})
+ therefore depend exclusively on the second factors containing
+ the constitutive relations. Since they depend only
+ on state variables, the stability criteria themselves are {\em
+ functions of the thermodynamic state in the local zone}. 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~\ref{KapSou}), 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.
+
+%__________________________________________________ One column table
+ \begin{table}
+ \caption{Opacity sources}
+ \label{KapSou}
+ \[
+ \begin{array}{p{0.5\linewidth}l}
+ \hline
+ \noalign{\smallskip}
+ Source & T / {[\rm K]} \\
+ \noalign{\smallskip}
+ \hline
+ \noalign{\smallskip}
+ Yorke 1979, Yorke 1980a & \leq 1700 \\
+ Kr\"ugel 1971 & 1700 \leq T \leq 5000 \\
+ Cox \& Stewart 1969 & 5000 \leq \\
+ \noalign{\smallskip}
+ \hline
+ \end{array}
+ \]
+ \end{table}
+%
+%
+%___________________________________ Two column table (place early!)
+ \begin{table*}
+ \caption{Regions of secular instability}
+ \label{TabSecInst}
+ \picplace{4cm}
+ \end{table*}
+ We will now write down the sign (and therefore stability)
+ determining parts of the left-hand sides of the inequalities
+ (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) and thereby
+ obtain {\em stability equations of state}.
+
+ The sign determining part of inequality~(\ref{ZSDynSta}) is
+ $3\Gamma_1 - 4$ and it reduces to the
+ criterion for dynamical stability
+ \begin{equation}
+ \Gamma_1 > \frac{4}{3}\,.
+ \end{equation}
+ Stability of the thermodynamical equilibrium demands
+ \begin{equation}
+ \chi^{}_\rho > 0, \;\; c_v > 0\, ,
+ \end{equation}
+ and
+ \begin{equation}
+ \chi^{}_T > 0
+ \end{equation}
+ holds for a wide range of physical situations.
+ With
+ \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_{\rm ad} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0
+ \end{eqnarray}
+ we find the sign determining terms in inequalities~(\ref{ZSSecSta})
+ and (\ref{ZSVibSta}) respectively and obtain the following form
+ of the criteria for dynamical, secular and vibrational
+ {\em stability}, respectively:
+ \begin{eqnarray}
+ 3 \Gamma_1 - 4 =: S_{\rm dyn} > & 0 & \label{DynSta} \\
+%
+ \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 )
+ + \kappa^{}_P + 1 =: S_{\rm sec} > & 0 & \label{SecSta} \\
+%
+ 4 \nabla_{\rm ad} - ( \nabla_{\rm ad} \kappa^{}_T + \kappa^{}_P )
+ - \frac{4}{3 \Gamma_1} =: S_{\rm vib}
+ > & 0\,.& \label{VibSta}
+ \end{eqnarray}
+ The constitutive relations are to be evaluated for the
+ unperturbed thermodynamic state (say $(\rho_0, T_0)$) of the zone.
+ We see that the one-zone stability of the layer depends only on
+ the constitutive relations $\Gamma_1$,
+ $\nabla_{\rm ad}$, $\chi_T^{},\,\chi_\rho^{}$,
+ $\kappa_P^{},\,\kappa_T^{}$.
+ These depend only on the unperturbed
+ thermodynamical state of the layer. Therefore the above relations
+ define the one-zone-stability equations of state
+ $S_{\rm dyn},\,S_{\rm sec}$
+ and $S_{\rm vib}$. See Fig.~\ref{FigVibStab} for a picture of
+ $S_{\rm vib}$. Regions of secular instability are
+ listed in Table~\ref{TabSecInst}.
+
+%
+% One column figure
+%----------------------------------------------------------- S_vib
+ \begin{figure}[htbp]
+ \picplace{5cm}
+ \caption{Vibrational stability equation of state
+ $S_{\rm vib}(\lg e, \lg \rho)$.
+ $>0$ means vibrational stability
+ }
+ \label{FigVibStab}
+ \end{figure}
+%
+%
+% 14.Sep.'90: Demo Vs.
+%______________________________________________________________
+
+\section{Conclusions}
+
+ \begin{enumerate}
+ \item The conditions for the stability of static, radiative
+ layers in gas spheres, as described by Baker's (1966) 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.
+ \item 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.
+ \item For solar composition gas the $\kappa$-mechanism is
+ working in the regions of the ice and dust features
+ in the opacities, the $\rm H_2$ 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
+ instability than the second He ionization zone
+ that drives the Cephe{\"\i}d pulsations.
+ \end{enumerate}
+
+ \acknowledgements
+%________________________________________ Do not leave a blank line here!
+ Part of this work was supported by the German
+ {\em Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} project
+ number Ts~17/2--1.
+
+%
+% 14.Sep.'90: Demo-Vs.
+%_____________________________________________________________________
+
+\begin{thebibliography}{}
+
+ \bibitem{} Baker N., 1966,
+ in: Stellar Evolution,
+ eds.\ R. F. Stein, A. G. W. Cameron,
+ Plenum, New York, p.\ 333
+
+ \bibitem{} Balluch M., 1988,
+ A\&A 200, 58
+
+ \bibitem{} Cox J. P., 1980,
+ Theory of Stellar Pulsation,
+ Princeton University Press, Princeton, p.\ 165
+
+ \bibitem{} Cox A. N., Stewart J. N., 1969,
+ Academia Nauk, Scientific Information 15, 1
+
+ \bibitem{} Kr\"ugel E., 1971,
+ Der Rosselandsche Mittelwert bei tiefen Temperaturen,
+ Diplom--Thesis, Univ.\ G\"ottingen
+
+ \bibitem{} Mizuno H., 1980,
+ Prog. Theor. Phys. 64, 544
+
+ \bibitem{} Tscharnuter W. M., 1987,
+ A\&A 188, 55
+
+ \bibitem{} Wuchterl G., 1989,
+ Zur Entstehung der Gasplaneten.\ Ku\-gel\-sym\-me\-tri\-sche
+ Gas\-str\"o\-mun\-gen auf Pro\-to\-pla\-ne\-ten,
+ Dissertation, Univ.\ Wien
+
+ \bibitem{} Yorke H. W., 1979,
+ A\&A 80, 215
+
+ \bibitem{} Yorke H. W., 1980a,
+ A\&A 86, 286
+
+\end{thebibliography}
+
+\end{document}
+ \ No newline at end of file