-#LyX 1.4.0cvs created this file. For more info see http://www.lyx.org/
-\lyxformat 243
+#LyX 1.6.0svn created this file. For more info see http://www.lyx.org/
+\lyxformat 341
\begin_document
\begin_header
\textclass aa
\end_preamble
\language english
\inputencoding auto
-\fontscheme default
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
\graphics default
\paperfontsize default
\spacing single
+\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 0
+\use_esint 0
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\paragraph_separation indent
\defskip medskip
\quotes_language english
-\quotes_times 2
\papercolumns 2
\papersides 2
\paperpagestyle default
\tracking_changes false
-\output_changes true
+\output_changes false
+\author ""
\end_header
\begin_body
\begin_layout Title
-
Hydrodynamics of giant planet formation
\end_layout
\begin_layout Subtitle
-
I.
Overviewing the
-\begin_inset Formula \( \kappa \)
+\begin_inset Formula $\kappa$
\end_inset
-mechanism
\end_layout
\begin_layout Author
-
G.
Wuchterl
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
inst{1}
and
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
+
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
inst{2}
\begin_inset Foot
status collapsed
-\begin_layout Standard
-
+\begin_layout Plain Layout
Just to show the usage of the elements in the author field
\end_layout
\end_inset
-
\end_layout
\begin_layout Offprint
-
G.
Wuchterl
\end_layout
\begin_layout Address
-
-Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
-rkenschanzstrasse
+Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
17, A-1180 Vienna
-\newline
+\begin_inset Newline newline
+\end_inset
+
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
email{wuchterl@amok.ast.univie.ac.at}
and
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
\end_inset
University of Alexandria, Department of Geography, ...
-\newline
+\begin_inset Newline newline
+\end_inset
+
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
email{c.ptolemy@hipparch.uheaven.space}
\begin_inset Foot
status collapsed
-\begin_layout Standard
-
+\begin_layout Plain Layout
The university of heaven temporarily does not accept e-mails
\end_layout
\end_inset
-
\end_layout
\begin_layout Date
-
Received September 15, 1996; accepted March 16, 1997
\end_layout
\begin_layout Abstract
-
To investigate the physical nature of the `nuc\SpecialChar \-
leated instability' of proto
giant planets (Mizuno
-\begin_inset LatexCommand \cite{mizuno}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
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 \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
+\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
\end_inset
,
-\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
+\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
\end_inset
.
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 \( \rho \)
+\begin_inset Formula $\rho$
\end_inset
, temperature
-\begin_inset Formula \( T \)
+\begin_inset Formula $T$
\end_inset
or specific internal energy
-\begin_inset Formula \( e \)
+\begin_inset Formula $e$
\end_inset
.
Regions of instability in the
-\begin_inset Formula \( (\rho ,e) \)
+\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 \( \kappa \)
+\begin_inset Formula $\kappa$
\end_inset
-mechanism is widespread under `cool' conditions.
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
keywords{giant planet formation --
\end_layout
\begin_layout Section
-
Introduction
\end_layout
\begin_layout Standard
-
In the
\emph on
nucleated instability
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
/{}
(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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "mizuno"
\end_inset
) determined the critical mass of the core to be about
-\begin_inset Formula \( 12\, M_{\oplus } \)
+\begin_inset Formula $12\, M_{\oplus}$
\end_inset
(
-\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
+\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
\end_layout
\begin_layout Standard
-
Although no hydrodynamical study has been available many workers conjectured
that a collapse or rapid contraction will ensue after accumulating the
critical mass.
the 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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
\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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "tscharnuter"
\end_inset
, Balluch
-\begin_inset LatexCommand \cite{balluch}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "balluch"
\end_inset
\end_layout
\begin_layout Section
-
Baker's standard one-zone model
\end_layout
\begin_layout Standard
-
\begin_inset Float figure
wide true
sideways false
status open
-\begin_layout Caption
+\begin_layout Plain Layout
+\begin_inset Caption
+\begin_layout Plain Layout
Adiabatic exponent
-\begin_inset Formula \( \Gamma _{1} \)
+\begin_inset Formula $\Gamma_{1}$
\end_inset
.
-\begin_inset Formula \( \Gamma _{1} \)
+\begin_inset Formula $\Gamma_{1}$
\end_inset
is plotted as a function of
-\begin_inset Formula \( \lg \)
+\begin_inset Formula $\lg$
\end_inset
internal energy
-\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
+\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
\end_inset
and
-\begin_inset Formula \( \lg \)
+\begin_inset Formula $\lg$
\end_inset
density
-\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
+\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
\end_inset
\end_layout
-\begin_layout Standard
+\end_inset
-\begin_inset LatexCommand \label{FigGam}
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "FigGam"
\end_inset
\end_inset
In this section the one-zone model of Baker (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
-), originally used to study the Cephe\i \"{\i}
-d pulsation mechanism, will be briefly
+), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
reviewed.
The resulting stability criteria will be rewritten in terms of local state
variables, local timescales and constitutive relations.
\end_layout
\begin_layout Standard
-
Baker (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
\end_layout
\begin_layout Itemize
-
hydrostatic equilibrium,
\end_layout
\begin_layout Itemize
-
thermal equilibrium,
\end_layout
\begin_layout Itemize
-
energy transport by grey radiation diffusion.
\end_layout
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
,
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
,
\end_inset
c) in Baker
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
\begin_layout Standard
\align left
-
\begin_inset Formula \begin{eqnarray*}
M_{r} & & \textrm{mass internal to the radius }r\\
m & & \textrm{mass of the zone}\\
r_{0} & & \textrm{unperturbed zone radius}\\
-\rho _{0} & & \textrm{unperturbed density in the zone}\\
+\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*}
+E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
\end_inset
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
/{}
\emph default
- (see Fig.\InsetSpace ~
+ (see Fig.
+\begin_inset space ~
+\end_inset
-\begin_inset LatexCommand \ref{FigGam}
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "FigGam"
\end_inset
)
\begin_inset Formula \begin{equation}
-\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
-\end{equation}
+\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
\end_inset
\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}
+\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 \( K \)
+\begin_inset Formula $K$
\end_inset
and
-\begin_inset Formula \( \sigma _{0} \)
+\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}
+\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 \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
+\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}
+\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 \( 1 \)
+\begin_inset Formula $1$
\end_inset
and equal to
-\begin_inset Formula \( 1 \)
+\begin_inset Formula $1$
\end_inset
for nonreacting mixtures of classical perfect gases.
The physical meaning of
-\begin_inset Formula \( \sigma _{0} \)
+\begin_inset Formula $\sigma_{0}$
\end_inset
and
-\begin_inset Formula \( K \)
+\begin_inset Formula $K$
\end_inset
is clearly visible in the equations above.
-\begin_inset Formula \( \sigma _{0} \)
+\begin_inset Formula $\sigma_{0}$
\end_inset
represents a frequency of the order one per free-fall time.
-\begin_inset Formula \( K \)
+\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 \[
-\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
+\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 \[
-\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
+\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
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
/{}
\emph default
given below:
\begin_inset Formula \begin{eqnarray}
-\frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
-\frac{\pi ^{2}}{\tau _{\mathrm{co}}\tau _{\mathrm{ff}}^{2}}\Gamma _{1}\nabla _{\mathrm{ad}}\left[ \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1\right] & > & 0\label{ZSSecSta} \\
-\frac{\pi ^{2}}{4}\frac{3}{\tau _{\mathrm{co}}\tau _{\mathrm{ff}}^{2}}\Gamma _{1}^{2}\, \nabla _{\mathrm{ad}}\left[ 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}\right] & > & 0\label{ZSVibSta}
-\end{eqnarray}
+\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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
) or Cox (
-\begin_inset LatexCommand \cite{cox}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "cox"
\end_inset
\end_layout
\begin_layout Standard
-
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
\end_layout
\begin_layout Enumerate
-
a factor containing local timescales only,
\end_layout
\begin_layout Enumerate
-
a factor containing only constitutive relations and their derivatives.
\end_layout
\begin_layout Standard
-
The first factors, depending on only timescales, are positive by definition.
- The signs of the left hand sides of the inequalities\InsetSpace ~
+ The signs of the left hand sides of the inequalities
+\begin_inset space ~
+\end_inset
+
(
-\begin_inset LatexCommand \ref{ZSDynSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
\end_inset
), (
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
\end_inset
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
\end_inset
-the thermodynamics and opacities (see Table\InsetSpace ~
+the thermodynamics and opacities (see Table
+\begin_inset space ~
+\end_inset
+
-\begin_inset LatexCommand \ref{KapSou}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "KapSou"
\end_inset
\end_layout
\begin_layout Standard
-
\begin_inset Float table
wide false
sideways false
status open
-\begin_layout Caption
-
+\begin_layout Plain Layout
+\begin_inset Caption
-\begin_inset LatexCommand \label{KapSou}
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "KapSou"
\end_inset
Opacity sources
\end_layout
-\begin_layout Standard
+\end_inset
-\begin_inset Tabular
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="2">
<features>
<column alignment="left" valignment="top" width="0pt">
<column alignment="left" valignment="top" width="0pt">
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
+\begin_layout Plain Layout
Source
\end_layout
\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
-
-\begin_inset Formula \( T/[\textrm{K}] \)
+\begin_layout Plain Layout
+\begin_inset Formula $T/[\textrm{K}]$
\end_inset
\end_inset
</cell>
</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
+\begin_layout Plain Layout
Yorke 1979, Yorke 1980a
\end_layout
\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
-
-\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
+\begin_layout Plain Layout
+\begin_inset Formula $\leq1700^{\textrm{a}}$
\end_inset
</cell>
</row>
<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
-Krügel 1971
+\begin_layout Plain Layout
+Krügel 1971
\end_layout
\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
-
-\begin_inset Formula \( 1700\leq T\leq 5000 \)
+\begin_layout Plain Layout
+\begin_inset Formula $1700\leq T\leq5000$
\end_inset
\end_inset
</cell>
</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
+\begin_layout Plain Layout
Cox & Stewart 1969
\end_layout
\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
-\begin_layout Standard
-
-
-\begin_inset Formula \( 5000\leq \)
+\begin_layout Plain Layout
+\begin_inset Formula $5000\leq$
\end_inset
\end_layout
-\begin_layout Standard
-
-
-\begin_inset Formula \( ^{\textrm{a}} \)
+\begin_layout Plain Layout
+\begin_inset Formula $^{\textrm{a}}$
\end_inset
This is footnote a
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}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
\end_inset
), (
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
\end_inset
\end_layout
\begin_layout Standard
+The sign determining part of inequality
+\begin_inset space ~
+\end_inset
-The sign determining part of inequality\InsetSpace ~
(
-\begin_inset LatexCommand \ref{ZSDynSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
\end_inset
) is
-\begin_inset Formula \( 3\Gamma _{1}-4 \)
+\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}
+\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}
+\chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
\end_inset
and
\begin_inset Formula \begin{equation}
-\chi ^{}_{T}>0
-\end{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}
+\Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
+\Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
+\nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
\end_inset
- we find the sign determining terms in inequalities\InsetSpace ~
+ we find the sign determining terms in inequalities
+\begin_inset space ~
+\end_inset
+
(
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
\end_inset
\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}
+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 \( (\rho _{0},T_{0}) \)
+\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 \( \Gamma _{1} \)
+\begin_inset Formula $\Gamma_{1}$
\end_inset
,
-\begin_inset Formula \( \nabla _{\mathrm{ad}} \)
+\begin_inset Formula $\nabla_{\mathrm{ad}}$
\end_inset
,
-\begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
+\begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
\end_inset
,
-\begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
+\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 \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
+\begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
\end_inset
and
-\begin_inset Formula \( S_{\mathrm{vib}} \)
+\begin_inset Formula $S_{\mathrm{vib}}$
\end_inset
.
- See Fig.\InsetSpace ~
+ See Fig.
+\begin_inset space ~
+\end_inset
-\begin_inset LatexCommand \ref{FigVibStab}
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "FigVibStab"
\end_inset
for a picture of
-\begin_inset Formula \( S_{\mathrm{vib}} \)
+\begin_inset Formula $S_{\mathrm{vib}}$
\end_inset
.
- Regions of secular instability are listed in Table\InsetSpace ~
+ Regions of secular instability are listed in Table
+\begin_inset space ~
+\end_inset
+
1.
\end_layout
\begin_layout Standard
-
\begin_inset Float figure
wide false
sideways false
status open
-\begin_layout Caption
+\begin_layout Plain Layout
+\begin_inset Caption
+\begin_layout Plain Layout
Vibrational stability equation of state
-\begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
+\begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
\end_inset
.
-\begin_inset Formula \( >0 \)
+\begin_inset Formula $>0$
\end_inset
means vibrational stability
\end_layout
-\begin_layout Standard
+\end_inset
-\begin_inset LatexCommand \label{FigVibStab}
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "FigVibStab"
\end_inset
\end_inset
+
\end_layout
\begin_layout Section
-
Conclusions
\end_layout
\begin_layout Enumerate
-
The conditions for the stability of static, radiative layers in gas spheres,
as described by Baker's (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
\end_inset
\end_layout
\begin_layout Enumerate
-
If the constitutive relations -- equations of state and Rosseland mean opacities
-- are specified, the stability equations of state can be evaluated without
specifying properties of the layer.
\end_layout
\begin_layout Enumerate
-
For solar composition gas the
-\begin_inset Formula \( \kappa \)
+\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 \( \mathrm{H}_{2} \)
+\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.
+y than the second He ionization zone that drives the Cepheı̈d pulsations.
\end_layout
\begin_layout Acknowledgement
-
Part of this work was supported by the German
\emph on
Deut\SpecialChar \-
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
/{}
\emph default
- project number Ts\InsetSpace ~
+ project number Ts
+\begin_inset space ~
+\end_inset
+
17/2--1.
\end_layout
\begin_layout Bibliography
-\bibitem [1966]{baker}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1966"
+key "baker"
+
+\end_inset
Baker, N.
1966, in Stellar Evolution, ed.
\begin_inset ERT
status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
+
\backslash
\end_layout
\begin_layout Bibliography
-\bibitem [1988]{balluch}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1988"
+key "balluch"
+
+\end_inset
Balluch, M.
1988, A&A, 200, 58
\end_layout
\begin_layout Bibliography
-\bibitem [1980]{cox}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980"
+key "cox"
+
+\end_inset
Cox, J.
P.
\end_layout
\begin_layout Bibliography
-\bibitem [1969]{cox69}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1969"
+key "cox69"
+
+\end_inset
Cox, A.
N.,& Stewart, J.
\end_layout
\begin_layout Bibliography
-\bibitem [1980]{mizuno}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980"
+key "mizuno"
+
+\end_inset
Mizuno H.
1980, Prog.
\end_layout
\begin_layout Bibliography
-\bibitem [1987]{tscharnuter}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1987"
+key "tscharnuter"
+
+\end_inset
Tscharnuter W.
M.
\end_layout
\begin_layout Bibliography
-\bibitem [1992]{terlevich}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1992"
+key "terlevich"
+
+\end_inset
Terlevich, R.
1992, in ASP Conf.
\end_layout
\begin_layout Bibliography
-\bibitem [1980a]{yorke80a}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980a"
+key "yorke80a"
+
+\end_inset
Yorke, H.
W.
\end_layout
\begin_layout Bibliography
-\bibitem [1997]{zheng}
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1997"
+key "zheng"
+
+\end_inset
Zheng, W., Davidsen, A.
F., Tytler, D.
projects / lyx.git / blobdiff