+2001-06-12 Peter Suetterlin <P.Suetterlin@astro.uu.nl>
+
+ * examples/aa_head.lyx:
+ * examples/aa_paper.lyx: removed
+
+ * doc/LaTeXConfig.lyx.in:
+ * examples/aa_sample.lyx:
+ * layouts/aa.layout:
+ * layouts/aapaper.inc:
+ * layouts/aapaper.layout:
+ * templates/aa.lyx: aa.layout is for the new version of the
+ A&A document class, while aapaper.layout is for the older (and
+ slightly incompatible) version.
+
2001-06-07 Jean-Marc Lasgouttes <Jean-Marc.Lasgouttes@inria.fr>
* doc/LaTeXConfig.lyx.in:
2001-05-22 Adrien Rebollo <rebollo@iaf.cnrs-gif.fr>
- * lib/kbd/european.kmap:
- * lib/kbd/francais.kmap:
- * lib/kbd/iso8859-15.cdef:
+ * kbd/european.kmap:
+ * kbd/francais.kmap:
+ * kbd/iso8859-15.cdef:
* encodings: add iso8859-15 support.
- * lib/kbd/iso8859-1.cdef: cleanup
+ * kbd/iso8859-1.cdef: cleanup
2001-05-22 Peter Suetterlin <P.Suetterlin@astro.uu.nl>
aa
\layout Description
-Found: @chk_aapaper@
+Found: @chk_aa@
\layout Description
-CTAN: N/A (available from Springer's ftp site
+CTAN: N/A (available from ftp site
\family typewriter
- ftp.springer.de
+ ftp.edpsciences.org
\family default
in directory
\family typewriter
-/pub/tex/latex/aa
+/pub/aa/
\family default
)
\layout Description
\family sans
aa
\family default
- can be used to write articles for submission to the scientific journal
+ (Version 5.01) can be used to write articles for submission to the
+ scientific journal
\emph on
Astronomy and Astrophysics
\emph default
- and the accompanying
-\emph on
-Supplement Series
-\emph default
-published by Springer -Verlag.
+published by EDP Sciences.
\layout Subsection
aastex
+++ /dev/null
-#This file was created by <pit> Tue Nov 25 23:06:48 1997
-#LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team
-\lyxformat 2.15
-\textclass aa
-\language english
-\inputencoding latin1
-\fontscheme times
-\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 2
-\papersides 2
-\paperpagestyle default
-
-\layout Thesaurus
-
-Sorry, no thesaurus for that
-\layout Title
-
-The Use of LyX in Astronomy
-\layout Subtitle
-
-Looking towards a bright future
-\layout Author
-
-P.
- Sütterlin
-\latex latex
-
-\backslash
-inst{
-\latex default
-1
-\latex latex
-}
-\begin_float footnote
-\layout Standard
-
-
-\series medium
-\emph on
-Present Address:
-\emph default
- Universitäts-Sternwarte, Geismarlandstr.
- 11, D-37083 Göttingen
-\end_float
-
-\latex latex
-
-\backslash
-and
-\latex default
- A.N.
- Coauthor
-\latex latex
-
-\backslash
-inst{
-\latex default
-2
-\latex latex
-}
-\layout Address
-
-Kiepenheuer-Institut für Sonnenphysik, Schöneckstr.
- 6, D-79104 Freiburg
-\latex latex
-
-\backslash
-and
-\latex default
- Somewhere-in-the-Country
-\layout Offprint
-
-P.
- Sütterlin
-\layout Email
-
-pit@uni-sw.gwdg.de
-\layout Date
-
-Received: maybe / Accepted: who knows
-\layout Abstract
-
-In this paper we show that LyX can be used to write papers for puplication
- in the journal Astronomy and Astrophysics.
-
-\layout Abstract
-
-
-\latex latex
-
-\backslash
-keywords{
-\latex default
-LaTeX - Wordprocessing - Publication
-\latex latex
-}
-\layout Section
-
-Intoduction
-\layout Standard
-
-In 1997, Springer Verlag decided to accept papers for publication in their
- journals
-\emph on
-Astronomy and Astrophysics
-\emph default
- and the acompanying
-\emph on
-Supplement Series
-\emph default
- only in form of LaTeX.
- At the same time, they switched the layout macros to LaTeX2
-\begin_inset Formula \( \epsilon \)
-\end_inset
-
- (Lamport
-\begin_inset LatexCommand \cite{kopka}
-
-\end_inset
-
-).
- This made it easy to adapt a corresponding LyX style to produce the correct
- output for direct submission to A&A.
-\layout Section
-
-Discussion
-\layout Standard
-
-What is there to discuss about? The existence of LyX with this layout makes
- it as easy as one could wish to write scientific papers.
-\layout Bibliography
-\bibitem [1994]{lamport}
-
-L.
- Lamport, 1994:
-\emph on
-LaTeX: A Document Preparation System
-\emph default
-, Addison-Wesley Publishing Company, 2nd edition
-\the_end
+++ /dev/null
-#LyX 1.1 created this file. For more info see http://www.lyx.org/
-\lyxformat 218
-\textclass aa
-\language english
-\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 2
-\papersides 2
-\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.\SpecialChar ~
-(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.\SpecialChar ~
-
-\begin_inset LatexCommand \ref{FigGam}
-
-\end_inset
-
-)
-\layout Standard
-
-
-\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
-\layout Standard
-
-
-\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
-\layout Standard
-
-
-\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
-
-
-\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\SpecialChar ~
-(
-\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\SpecialChar ~
-
-\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
-
-
-\begin_inset Tabular
-<lyxtabular version="2" rows="4" columns="2">
-<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
-<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
-<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
-<row topline="true" bottomline="true" newpage="false">
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-Source
-\end_inset
-</cell>
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-T/[K]
-\end_inset
-</cell>
-</row>
-<row topline="false" bottomline="false" newpage="false">
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-Yorke 1979, Yorke 1980a
-\end_inset
-</cell>
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-
-\begin_inset Formula \( \leq 1700^{\mathrm{a}} \)
-\end_inset
-
-
-\end_inset
-</cell>
-</row>
-<row topline="false" bottomline="false" newpage="false">
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-Krügel 1971
-\end_inset
-</cell>
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-
-\begin_inset Formula \( 1700\leq T\leq 5000 \)
-\end_inset
-
-
-\end_inset
-</cell>
-</row>
-<row topline="false" bottomline="true" newpage="false">
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-Cox & Stewart 1969
-\end_inset
-</cell>
-<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
-\begin_inset Text
-
-\layout Standard
-
-
-\begin_inset Formula \( 5000\leq \)
-\end_inset
-
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\layout Standard
-\added_space_top medskip*
-
-\begin_inset Formula \( ^{\textrm{a}} \)
-\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\SpecialChar ~
-(
-\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}
-\Gamma _{1}>\frac{4}{3}
-\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
-\layout Standard
-
-
-\begin_inset Formula \begin{equation}
-\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\SpecialChar ~
-(
-\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.\SpecialChar ~
-
-\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\SpecialChar ~
-
-\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\SpecialChar ~
-17/2--1.
-\layout Bibliography
-\bibitem [1966]{baker}
-
-Baker N., 1966, in: Stellar Evolution, eds.\SpecialChar ~
-R.
- F.
- Stein, A.
- G.
- W.
- Cameron, Plenum, New York, p.\SpecialChar ~
-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.\SpecialChar ~
-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.\SpecialChar ~
- 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
--- /dev/null
+#LyX 1.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 218
+\textclass aa
+\begin_preamble
+\usepackage{graphicx}
+%
+\end_preamble
+\language english
+\inputencoding auto
+\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 2
+\papersides 2
+\paperpagestyle default
+
+\layout Title
+
+Hydrodynamics of giant planet formation
+\layout Subtitle
+
+I.
+ Overviewing the
+\begin_inset Formula \( \kappa \)
+\end_inset
+
+-mechanism
+\layout Author
+
+G.
+ Wuchterl
+\latex latex
+
+\backslash
+inst{1}
+\backslash
+and
+\newline
+
+\latex default
+C.
+ Ptolemy
+\latex latex
+
+\backslash
+inst{2}
+\backslash
+fnmsep
+\begin_float footnote
+\layout Standard
+
+Just to show the usage of the elements in the author field
+\end_float
+
+\layout Offprint
+
+G.
+ Wuchterl
+\layout Address
+
+Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
+rkenschanzstrasse
+ 17, A-1180 Vienna
+\newline
+
+\latex latex
+
+\backslash
+email{wuchterl@amok.ast.univie.ac.at}
+\backslash
+and
+\newline
+
+\latex default
+University of Alexandria, Department of Geography, ...
+\newline
+
+\latex latex
+
+\backslash
+email{c.ptolemy@hipparch.uheaven.space}
+\latex default
+
+\begin_float footnote
+\layout Standard
+
+The university of heaven temporarily does not accept e-mails
+\end_float
+
+\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.
+ 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.
+\latex latex
+
+\newline
+
+\backslash
+keywords{giant planet formation --
+\backslash
+(
+\backslash
+kappa
+\backslash
+)-mechanism -- stability of gas spheres }
+\latex default
+
+\layout Section
+
+Introduction
+\layout Standard
+
+In the
+\emph on
+nucleated instability
+\latex latex
+
+\backslash
+/{}
+\emph default
+\latex 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.
+\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.
+\layout Section
+
+Baker's standard one-zone model
+\layout Standard
+
+\begin_float wide-fig
+\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
+
+
+\layout Standard
+
+
+\begin_inset LatexCommand \label{FigGam}
+
+\end_inset
+
+
+\end_float
+ 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.
+\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
+\noindent
+For the one-zone-model Baker obtains necessary conditions for dynamical,
+ secular and vibrational (or pulsational) stability (Eqs.
+\latex latex
+
+\backslash
+
+\latex default
+(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:
+\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
+
+
+\layout Standard
+\noindent
+and with the definitions of the
+\emph on
+local cooling time
+\latex latex
+
+\backslash
+/{}
+\emph default
+\latex default
+ (see Fig.\SpecialChar ~
+
+\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
+\latex latex
+
+\backslash
+/{}
+\emph default
+\latex 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
+
+).
+\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\SpecialChar ~
+(
+\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.
+\latex latex
+
+\backslash
+
+\latex default
+the thermodynamics and opacities (see Table\SpecialChar ~
+
+\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
+
+
+\begin_inset LatexCommand \label{KapSou}
+
+\end_inset
+
+Opacity sources
+\layout Standard
+
+
+\begin_inset Tabular
+<lyxtabular version="2" rows="4" columns="2">
+<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
+<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
+<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
+<row topline="true" bottomline="false" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Source
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+
+\begin_inset Formula \( T/[\textrm{K}] \)
+\end_inset
+
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="false" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Yorke 1979, Yorke 1980a
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+
+\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
+\end_inset
+
+
+\end_inset
+</cell>
+</row>
+<row topline="false" bottomline="false" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Krügel 1971
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+
+\begin_inset Formula \( 1700\leq T\leq 5000 \)
+\end_inset
+
+
+\end_inset
+</cell>
+</row>
+<row topline="false" bottomline="true" newpage="false">
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+Cox & Stewart 1969
+\end_inset
+</cell>
+<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
+\begin_inset Text
+
+\layout Standard
+
+
+\begin_inset Formula \( 5000\leq \)
+\end_inset
+
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\layout Standard
+
+
+\begin_inset Formula \( ^{\textrm{a}} \)
+\end_inset
+
+This is footnote a
+\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\SpecialChar ~
+(
+\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\SpecialChar ~
+(
+\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.\SpecialChar ~
+
+\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\SpecialChar ~
+1.
+\layout Standard
+
+\begin_float fig
+\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
+\layout Standard
+
+
+\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 Cephe\i \"{\i}
+d 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
+\latex latex
+
+\backslash
+/{}
+\emph default
+\latex default
+ project number Ts\SpecialChar ~
+17/2--1.
+
+\layout Bibliography
+\bibitem [1966]{baker}
+
+ Baker, N.
+ 1966, in Stellar Evolution, ed.
+\latex latex
+
+\backslash
+
+\latex default
+R.
+ F.
+ Stein,& A.
+ G.
+ W.
+ Cameron (Plenum, New York) 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)
+ 165
+\layout Bibliography
+\bibitem [1969]{cox69}
+
+ Cox, A.
+ N.,& Stewart, J.
+ N.
+ 1969, Academia Nauk, Scientific Information 15, 1
+\layout Bibliography
+\bibitem [1980]{mizuno}
+
+ Mizuno H.
+ 1980, Prog.
+ Theor.
+ Phys., 64, 544
+\layout Bibliography
+\bibitem [1987]{tscharnuter}
+
+ Tscharnuter W.
+ M.
+ 1987, A&A, 188, 55
+\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
+\layout Bibliography
+\bibitem [1980a]{yorke80a}
+
+ Yorke, H.
+ W.
+ 1980a, A&A, 86, 286
+\layout Bibliography
+\bibitem [1997]{zheng}
+
+Zheng, W., Davidsen, A.
+ F., Tytler, D.
+ & Kriss, G.
+ A.
+ 1997, preprint
+\the_end
#
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+# This file contains additional style definitions for the
+# A&A paper style not found in the standard include files.
+# It is Input by aapaper.layout
+#
+# Author: Peter Sütterlin <pit@uni-sw.gwdg.de>
+
+# Subitle style definition
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+
+
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+#% Do not delete the line below; configure depends on this
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+#
+# Author: Peter Sütterlin <pit@uni-sw.gwdg.de>
+
+# This is for the old Springer layout of A&A (aa.cls version 4.x)
+# If you still have this, please upgrade to version 5 (see Extended.lyx)
+#
+# If you want to keep the layout for old papers I'd suggest to rename
+# the old style file to aa-old and change the second line in this file to
+#
+# \DeclareLaTeXClass[aa-old]{article (A&A V4)}
+#
+# and reconfigure LyX (after running texhash, of course).
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+
+# I want to keep the entries in a logical order.
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+
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+
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+
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+
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+
-#This file was created by <pit> Mon Nov 24 23:26:21 1997
-#LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team
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+#LyX 1.1 created this file. For more info see http://www.lyx.org/
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projects / features.git / commitdiff