+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+This is an example LyX file for articles to be submitted to the Journal
+ of Astronomy & Astrophysicssing (A&A).
+ How to install the A&A LaTeX class to your LaTeX system is explained in
+
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
- 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.
+ giant planets, the stability of layers in static, radiative gas spheres
+ is analysed on the basis of Baker's 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
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
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
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
\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
\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
\emph default
(also called core instability) hypothesis of giant planet formation, a
critical mass for static core envelope protoplanets has been found.
Mizuno (
\emph default
(also called core instability) hypothesis of giant planet formation, a
critical mass for static core envelope protoplanets has been found.
Mizuno (
Although no hydrodynamical study has been available many workers conjectured
that a collapse or rapid contraction will ensue after accumulating the
critical mass.
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 (
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 (
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
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
- 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.
+ The similarities in the (micro)physics, i.
+\begin_inset space \thinspace{}
+\end_inset
+
+g.
+\begin_inset space \space{}
+\end_inset
+
+constitutive relations of protostellar cores and protogiant planets serve
+ as a further motivation for this study.
-), 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
reviewed.
The resulting stability criteria will be rewritten in terms of local state
variables, local timescales and constitutive relations.
\end_layout
\begin_layout Standard
\noindent
For the one-zone-model Baker obtains necessary conditions for dynamical,
secular and vibrational (or pulsational) stability (Eqs.
\noindent
For the one-zone-model Baker obtains necessary conditions for dynamical,
secular and vibrational (or pulsational) stability (Eqs.
M_{r} & & \textrm{mass internal to the radius }r\\
m & & \textrm{mass of the zone}\\
r_{0} & & \textrm{unperturbed zone radius}\\
M_{r} & & \textrm{mass internal to the radius }r\\
m & & \textrm{mass of the zone}\\
r_{0} & & \textrm{unperturbed zone radius}\\
T_{0} & & \textrm{unperturbed temperature in the zone}\\
L_{r0} & & \textrm{unperturbed luminosity}\\
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
T_{0} & & \textrm{unperturbed temperature in the zone}\\
L_{r0} & & \textrm{unperturbed luminosity}\\
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
-\begin_inset Formula \begin{equation}
-\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
+\begin_inset Formula
+\begin{equation}
+\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
-\begin_inset Formula \begin{equation}
-\tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
+\begin_inset Formula
+\begin{equation}
+\tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
-\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}}}\, ;
+\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_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,
\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 \[
-\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
+\begin_inset Formula
+\[
+\Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}
+\]
-\begin_inset Formula \[
-\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
+\begin_inset Formula
+\[
+\nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}
+\]
-\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}
+\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}
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
\end_layout
\begin_layout Enumerate
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
\end_layout
\begin_layout Enumerate
.
The one-zone stability can therefore be determined from a simple equation
of state, given for example, as a function of density and temperature.
.
The one-zone stability can therefore be determined from a simple equation
of state, given for example, as a function of density and temperature.
<column alignment="left" valignment="top" width="0pt">
<column alignment="left" valignment="top" width="0pt">
<column alignment="left" valignment="top" width="0pt">
<column alignment="left" valignment="top" width="0pt">
We will now write down the sign (and therefore stability) determining parts
of the left-hand sides of the inequalities (
We will now write down the sign (and therefore stability) determining parts
of the left-hand sides of the inequalities (
-\begin_inset Formula \begin{equation}
-\chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
+\begin_inset Formula
+\begin{equation}
+\chi_{\rho}^{}>0,\;\; c_{v}>0\,,
-\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
+\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
-\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}
+\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_inset
.
These depend only on the unperturbed thermodynamical state of the layer.
Therefore the above relations define the one-zone-stability equations of
state
\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
The conditions for the stability of static, radiative layers in gas spheres,
as described by Baker's (
The conditions for the stability of static, radiative layers in gas spheres,
as described by Baker's (
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.
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_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
\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