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+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
+
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+status open
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+http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
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- 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
.
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
.
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
-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
-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
(also called core instability) hypothesis of giant planet formation, a
critical mass for static core envelope protoplanets has been found.
Mizuno (
(also called core instability) hypothesis of giant planet formation, a
critical mass for static core envelope protoplanets has been found.
Mizuno (
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.
is the Earth mass), which is independent of the outer boundary conditions
and therefore independent of the location in the solar nebula.
This critical value for the core mass corresponds closely to the cores
of today's giant planets.
Although no hydrodynamical study 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.
-\layout Section
+ The similarities in the (micro)physics, i.
+\begin_inset space \thinspace{}
+\end_inset
+
+g.
+\begin_inset space \space{}
+\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.
reviewed.
The resulting stability criteria will be rewritten in terms of local state
variables, local timescales and constitutive relations.
) investigates the stability of thin layers in self-gravitating, spherical
gas clouds with the following properties:
) investigates the stability of thin layers in self-gravitating, spherical
gas clouds with the following properties:
For the one-zone-model Baker obtains necessary conditions for dynamical,
secular and vibrational (or pulsational) stability (Eqs.
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}
\end{eqnarray*}
T_{0} & & \textrm{unperturbed temperature in the zone}\\
L_{r0} & & \textrm{unperturbed luminosity}\\
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
\end{eqnarray*}
-\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}}}\,;
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,
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
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
) therefore depend exclusively on the second factors containing the constitutive
relations.
Since they depend only on state variables, the stability criteria themselves
are
) therefore depend exclusively on the second factors containing the constitutive
relations.
Since they depend only on state variables, the stability criteria themselves
are
.
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.
), are specified (in practice by specifying a chemical composition) the
one-zone stability can be inferred if the thermodynamic state is specified.
), 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.
assumptions.
Only the specific growth rates (depending upon the time scales) will be
different for layers in different objects.
<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}
.
These depend only on the unperturbed thermodynamical state of the layer.
Therefore the above relations define the one-zone-stability equations of
state
.
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 (
) 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.
) 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.
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.
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
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
Cox, A.
N.,& Stewart, J.
N.
1969, Academia Nauk, Scientific Information 15, 1
Cox, A.
N.,& Stewart, J.
N.
1969, Academia Nauk, Scientific Information 15, 1