-#LyX 1.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 221
+#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/
+\lyxformat 245
+\begin_document
+\begin_header
\textclass aa
\begin_preamble
\usepackage{graphicx}
\graphics default
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\spacing single
-\papersize Default
-\paperpackage a4
-\use_geometry 0
+\papersize default
+\use_geometry false
\use_amsmath 0
-\use_natbib 0
-\use_numerical_citations 0
+\cite_engine basic
+\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
-\quotes_times 2
\papercolumns 2
\papersides 2
\paperpagestyle default
+\tracking_changes false
+\output_changes true
+\end_header
-\layout Title
+\begin_body
+
+\begin_layout Title
Hydrodynamics of giant planet formation
-\layout Subtitle
+\end_layout
+
+\begin_layout Subtitle
I.
Overviewing the
\begin_inset Formula \( \kappa \)
-\end_inset
+\end_inset
-mechanism
-\layout Author
+\end_layout
+
+\begin_layout Author
G.
Wuchterl
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
inst{1}
-\backslash
+\backslash
and
-\newline
+\end_layout
+
+\begin_layout Standard
-\end_inset
+\end_layout
+
+\end_inset
C.
Ptolemy
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
inst{2}
-\backslash
+\backslash
fnmsep
-\end_inset
+\end_layout
+
+\end_inset
\begin_inset Foot
-collapsed true
+status collapsed
-\layout Standard
+\begin_layout Standard
Just to show the usage of the elements in the author field
-\end_inset
+\end_layout
+
+\end_inset
-\layout Offprint
+\end_layout
+
+\begin_layout Offprint
G.
Wuchterl
-\layout Address
+\end_layout
+
+\begin_layout Address
Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
rkenschanzstrasse
17, A-1180 Vienna
-\newline
+\newline
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
email{wuchterl@amok.ast.univie.ac.at}
-\backslash
+\backslash
and
-\newline
+\end_layout
+
+\begin_layout Standard
-\end_inset
+\end_layout
+
+\end_inset
University of Alexandria, Department of Geography, ...
-\newline
+\newline
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
email{c.ptolemy@hipparch.uheaven.space}
-\end_inset
+\end_layout
+
+\end_inset
\begin_inset Foot
-collapsed true
+status collapsed
-\layout Standard
+\begin_layout Standard
The university of heaven temporarily does not accept e-mails
-\end_inset
+\end_layout
+
+\end_inset
-\layout Date
+\end_layout
+
+\begin_layout Date
Received September 15, 1996; accepted March 16, 1997
-\layout Abstract
+\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
+\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
+\end_inset
standard one-zone model.
It is shown that stability depends only upon the equations of state, the
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
+\end_inset
,
\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
-\end_inset
+\end_inset
.
These displays may be used to determine the one-zone stability of layers
of the stability equations for the thermodynamic state of these layers,
specified by state quantities as density
\begin_inset Formula \( \rho \)
-\end_inset
+\end_inset
, temperature
\begin_inset Formula \( T \)
-\end_inset
+\end_inset
or specific internal energy
\begin_inset Formula \( e \)
-\end_inset
+\end_inset
.
Regions of instability in the
\begin_inset Formula \( (\rho ,e) \)
-\end_inset
+\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
+\end_inset
-mechanism is widespread under `cool' conditions.
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\newline
+\end_layout
-\backslash
+\begin_layout Standard
+
+\backslash
keywords{giant planet formation --
-\backslash
+\backslash
(
-\backslash
+\backslash
kappa
-\backslash
+\backslash
)-mechanism -- stability of gas spheres }
-\end_inset
+\end_layout
+
+\end_inset
-\layout Section
+\end_layout
+
+\begin_layout Section
Introduction
-\layout Standard
+\end_layout
+
+\begin_layout Standard
In the
-\emph on
+\emph on
nucleated instability
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
/{}
-\end_inset
+\end_layout
+\end_inset
-\emph default
+
+\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
+\end_inset
) determined the critical mass of the core to be about
\begin_inset Formula \( 12\, M_{\oplus } \)
-\end_inset
+\end_inset
(
\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
-\end_inset
+\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
+\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
is investigated on the basis of Baker's (
\begin_inset LatexCommand \cite{baker}
-\end_inset
+\end_inset
) standard one-zone model.
-\layout Standard
+\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
+\end_inset
, Balluch
\begin_inset LatexCommand \cite{balluch}
-\end_inset
+\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
+\end_layout
+
+\begin_layout Section
Baker's standard one-zone model
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset Float figure
wide true
-collapsed false
+sideways false
+status open
-\layout Caption
+\begin_layout Caption
Adiabatic exponent
\begin_inset Formula \( \Gamma _{1} \)
-\end_inset
+\end_inset
.
\begin_inset Formula \( \Gamma _{1} \)
-\end_inset
+\end_inset
is plotted as a function of
\begin_inset Formula \( \lg \)
-\end_inset
+\end_inset
internal energy
\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
-\end_inset
+\end_inset
and
\begin_inset Formula \( \lg \)
-\end_inset
+\end_inset
density
\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset LatexCommand \label{FigGam}
-\end_inset
+\end_inset
+
+\end_layout
-\end_inset
+\end_inset
In this section the one-zone model of Baker (
\begin_inset LatexCommand \cite{baker}
-\end_inset
+\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
+\end_layout
+
+\begin_layout Standard
Baker (
\begin_inset LatexCommand \cite{baker}
-\end_inset
+\end_inset
) investigates the stability of thin layers in self-gravitating, spherical
gas clouds with the following properties:
-\layout Itemize
+\end_layout
+
+\begin_layout Itemize
hydrostatic equilibrium,
-\layout Itemize
+\end_layout
+
+\begin_layout Itemize
thermal equilibrium,
-\layout Itemize
+\end_layout
+
+\begin_layout Itemize
energy transport by grey radiation diffusion.
-\layout Standard
-\noindent
+\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
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
-\end_inset
+\end_layout
+
+\end_inset
(34a,
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
,
-\end_inset
+\end_layout
+
+\end_inset
b,
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
,
-\end_inset
+\end_layout
+
+\end_inset
c) in Baker
\begin_inset LatexCommand \cite{baker}
-\end_inset
+\end_inset
).
Using Baker's notation:
-\layout Standard
-\align left
+\end_layout
+
+\begin_layout Standard
+\align left
\begin_inset Formula \begin{eqnarray*}
M_{r} & & \textrm{mass internal to the radius }r\\
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
-\noindent
+\end_layout
+
+\begin_layout Standard
+\noindent
and with the definitions of the
-\emph on
+\emph on
local cooling time
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
/{}
-\end_inset
+\end_layout
+\end_inset
-\emph default
- (see Fig.\SpecialChar ~
+
+\emph default
+ (see Fig.\InsetSpace ~
\begin_inset LatexCommand \ref{FigGam}
-\end_inset
+\end_inset
)
\begin_inset Formula \begin{equation}
\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
\end{equation}
-\end_inset
+\end_inset
and the
-\emph on
+\emph on
local free-fall time
-\emph default
+\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
+\end_inset
Baker's
\begin_inset Formula \( K \)
-\end_inset
+\end_inset
and
\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\end_inset
have the following form:
\begin_inset Formula \begin{eqnarray}
K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
\end{eqnarray}
-\end_inset
+\end_inset
where
\begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
-\end_inset
+\end_inset
has been used and
\begin_inset Formula \begin{equation}
\end{array}
\end{equation}
-\end_inset
+\end_inset
is a thermodynamical quantity which is of order
\begin_inset Formula \( 1 \)
-\end_inset
+\end_inset
and equal to
\begin_inset Formula \( 1 \)
-\end_inset
+\end_inset
for nonreacting mixtures of classical perfect gases.
The physical meaning of
\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\end_inset
and
\begin_inset Formula \( K \)
-\end_inset
+\end_inset
is clearly visible in the equations above.
\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\end_inset
represents a frequency of the order one per free-fall time.
\begin_inset Formula \( K \)
-\end_inset
+\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
\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
+\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
+\end_inset
one obtains, after some pages of algebra, the conditions for
-\emph on
+\emph on
stability
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
/{}
-\end_inset
+\end_layout
+
+\end_inset
-\emph default
+\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}
+\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
+\end_inset
For a physical discussion of the stability criteria see Baker (
\begin_inset LatexCommand \cite{baker}
-\end_inset
+\end_inset
) or Cox (
\begin_inset LatexCommand \cite{cox}
-\end_inset
+\end_inset
).
-\layout Standard
+\end_layout
+
+\begin_layout Standard
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
-\layout Enumerate
+\end_layout
+
+\begin_layout Enumerate
a factor containing local timescales only,
-\layout Enumerate
+\end_layout
+
+\begin_layout Enumerate
a factor containing only constitutive relations and their derivatives.
-\layout Standard
+\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\SpecialChar ~
+ The signs of the left hand sides of the inequalities\InsetSpace ~
(
\begin_inset LatexCommand \ref{ZSDynSta}
-\end_inset
+\end_inset
), (
\begin_inset LatexCommand \ref{ZSSecSta}
-\end_inset
+\end_inset
) and (
\begin_inset LatexCommand \ref{ZSVibSta}
-\end_inset
+\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
+\emph on
functions of the thermodynamic state in the local zone
-\emph default
+\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
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
-\end_inset
+\end_layout
+
+\end_inset
-the thermodynamics and opacities (see Table\SpecialChar ~
+the thermodynamics and opacities (see Table\InsetSpace ~
\begin_inset LatexCommand \ref{KapSou}
-\end_inset
+\end_inset
), are specified (in practice by specifying a chemical composition) the
one-zone stability can be inferred if the thermodynamic state is specified.
assumptions.
Only the specific growth rates (depending upon the time scales) will be
different for layers in different objects.
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset Float table
wide false
-collapsed false
+sideways false
+status open
-\layout Caption
+\begin_layout Caption
\begin_inset LatexCommand \label{KapSou}
-\end_inset
+\end_inset
Opacity sources
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset Tabular
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
Source
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
\begin_inset Formula \( T/[\textrm{K}] \)
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
Yorke 1979, Yorke 1980a
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
Krügel 1971
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
\begin_inset Formula \( 1700\leq T\leq 5000 \)
-\end_inset
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
<row bottomline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
Cox & Stewart 1969
-\end_inset
+\end_layout
+
+\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Standard
\begin_inset Formula \( 5000\leq \)
-\end_inset
+\end_inset
+
+\end_layout
-\end_inset
+\end_inset
</cell>
</row>
</lyxtabular>
-\end_inset
+\end_inset
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset Formula \( ^{\textrm{a}} \)
-\end_inset
+\end_inset
This is footnote a
-\end_inset
+\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
+\end_inset
), (
\begin_inset LatexCommand \ref{ZSSecSta}
-\end_inset
+\end_inset
) and (
\begin_inset LatexCommand \ref{ZSVibSta}
-\end_inset
+\end_inset
) and thereby obtain
-\emph on
+\emph on
stability equations of state
-\emph default
+\emph default
.
-\layout Standard
+\end_layout
-The sign determining part of inequality\SpecialChar ~
+\begin_layout Standard
+
+The sign determining part of inequality\InsetSpace ~
(
\begin_inset LatexCommand \ref{ZSDynSta}
-\end_inset
+\end_inset
) is
\begin_inset Formula \( 3\Gamma _{1}-4 \)
-\end_inset
+\end_inset
and it reduces to the criterion for dynamical stability
\begin_inset Formula \begin{equation}
-\Gamma _{1}>\frac{4}{3}\, \cdot
+\Gamma _{1}>\frac{4}{3}\, \cdot
\end{equation}
-\end_inset
+\end_inset
Stability of the thermodynamical equilibrium demands
\begin_inset Formula \begin{equation}
\chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
\end{equation}
-\end_inset
+\end_inset
and
\begin_inset Formula \begin{equation}
\chi ^{}_{T}>0
\end{equation}
-\end_inset
+\end_inset
holds for a wide range of physical situations.
With
\nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
\end{eqnarray}
-\end_inset
+\end_inset
- we find the sign determining terms in inequalities\SpecialChar ~
+ we find the sign determining terms in inequalities\InsetSpace ~
(
\begin_inset LatexCommand \ref{ZSSecSta}
-\end_inset
+\end_inset
) and (
\begin_inset LatexCommand \ref{ZSVibSta}
-\end_inset
+\end_inset
) respectively and obtain the following form of the criteria for dynamical,
secular and vibrational
-\emph on
+\emph on
stability
-\emph default
+\emph default
, respectively:
\begin_inset Formula \begin{eqnarray}
3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
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
+\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
+\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
+\end_inset
,
\begin_inset Formula \( \nabla _{\mathrm{ad}} \)
-\end_inset
+\end_inset
,
\begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
-\end_inset
+\end_inset
,
\begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
-\end_inset
+\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
+\end_inset
and
\begin_inset Formula \( S_{\mathrm{vib}} \)
-\end_inset
+\end_inset
.
- See Fig.\SpecialChar ~
+ See Fig.\InsetSpace ~
\begin_inset LatexCommand \ref{FigVibStab}
-\end_inset
+\end_inset
for a picture of
\begin_inset Formula \( S_{\mathrm{vib}} \)
-\end_inset
+\end_inset
.
- Regions of secular instability are listed in Table\SpecialChar ~
+ Regions of secular instability are listed in Table\InsetSpace ~
1.
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset Float figure
wide false
-collapsed false
+sideways false
+status open
-\layout Caption
+\begin_layout Caption
Vibrational stability equation of state
\begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
-\end_inset
+\end_inset
.
\begin_inset Formula \( >0 \)
-\end_inset
+\end_inset
means vibrational stability
-\layout Standard
+\end_layout
+
+\begin_layout Standard
\begin_inset LatexCommand \label{FigVibStab}
-\end_inset
+\end_inset
+
+
+\end_layout
+\end_inset
-\end_inset
+\end_layout
-\layout Section
+\begin_layout Section
Conclusions
-\layout Enumerate
+\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
+\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
+\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.
-\layout Enumerate
+\end_layout
+
+\begin_layout Enumerate
For solar composition gas the
\begin_inset Formula \( \kappa \)
-\end_inset
+\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
+\end_inset
dissociation and the combined H, first He ionization zone, as indicated
by vibrational instability.
y than the second He ionization zone that drives the Cephe\i \"{\i}
d pulsations.
-\layout Acknowledgement
+\end_layout
+
+\begin_layout Acknowledgement
Part of this work was supported by the German
-\emph on
+\emph on
Deut\SpecialChar \-
sche For\SpecialChar \-
schungs\SpecialChar \-
mein\SpecialChar \-
schaft, DFG
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
/{}
-\end_inset
+\end_layout
+
+\end_inset
-\emph default
- project number Ts\SpecialChar ~
+\emph default
+ project number Ts\InsetSpace ~
17/2--1.
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1966]{baker}
Baker, N.
1966, in Stellar Evolution, ed.
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Standard
-\backslash
+\backslash
-\end_inset
+\end_layout
+
+\end_inset
R.
F.
G.
W.
Cameron (Plenum, New York) 333
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1988]{balluch}
Balluch, M.
1988, A&A, 200, 58
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1980]{cox}
Cox, J.
P.
1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
165
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1969]{cox69}
Cox, A.
N.,& Stewart, J.
N.
1969, Academia Nauk, Scientific Information 15, 1
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1980]{mizuno}
Mizuno H.
1980, Prog.
Theor.
Phys., 64, 544
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1987]{tscharnuter}
Tscharnuter W.
M.
1987, A&A, 188, 55
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1992]{terlevich}
Terlevich, R.
A.
V.
Filippenko, 13
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1980a]{yorke80a}
Yorke, H.
W.
1980a, A&A, 86, 286
-\layout Bibliography
+\end_layout
+
+\begin_layout Bibliography
\bibitem [1997]{zheng}
Zheng, W., Davidsen, A.
& Kriss, G.
A.
1997, preprint
-\the_end
+\end_layout
+
+\end_body
+\end_document
projects / lyx.git / blobdiff