mirror of
https://git.lyx.org/repos/lyx.git
synced 2024-12-02 05:55:38 +00:00
27de1486ca
git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@140 a592a061-630c-0410-9148-cb99ea01b6c8
940 lines
20 KiB
Plaintext
940 lines
20 KiB
Plaintext
#This file was created by <pit> Mon Nov 24 22:57:48 1997
|
|
#LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team
|
|
\lyxformat 2.15
|
|
\textclass aapaper
|
|
\language default
|
|
\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 1
|
|
\papersides 1
|
|
\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.
|
|
\protected_separator
|
|
(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.
|
|
\protected_separator
|
|
|
|
\begin_inset LatexCommand \ref{FigGam}
|
|
|
|
\end_inset
|
|
|
|
)
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\label{}
|
|
\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}
|
|
\label{}
|
|
\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}
|
|
\label{}
|
|
\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
|
|
\protected_separator
|
|
(
|
|
\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
|
|
\protected_separator
|
|
|
|
\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
|
|
\align center \LyXTable
|
|
multicol4
|
|
4 2 0 0 -1 -1 -1 -1
|
|
1 0 0 0
|
|
1 0 0 0
|
|
0 0 0 0
|
|
0 1 0 0
|
|
2 0 0
|
|
2 0 0
|
|
0 2 1 0 0 0 0
|
|
0 2 1 0 0 0 0
|
|
0 8 1 0 0 0 0
|
|
0 8 1 0 0 0 0
|
|
0 2 1 0 0 0 0
|
|
0 8 1 0 0 0 0
|
|
0 8 1 0 0 0 0
|
|
0 8 1 0 0 0 0
|
|
|
|
Source
|
|
\newline
|
|
T/[K]
|
|
\newline
|
|
Yorke 1979, Yorke 1980a
|
|
\newline
|
|
|
|
\begin_inset Formula \( \leq 1700^{\mathrm{a}} \)
|
|
\end_inset
|
|
|
|
|
|
\newline
|
|
Krügel 1971
|
|
\newline
|
|
|
|
\begin_inset Formula \( 1700\leq T\leq 5000 \)
|
|
\end_inset
|
|
|
|
|
|
\newline
|
|
Cox & Stewart 1969
|
|
\newline
|
|
|
|
\begin_inset Formula \( 5000\leq \)
|
|
\end_inset
|
|
|
|
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula \( \mathrm{a} \)
|
|
\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
|
|
\protected_separator
|
|
(
|
|
\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}
|
|
\label{}
|
|
\Gamma _{1}>\frac{4}{3}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
Stability of the thermodynamical equilibrium demands
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\label{}
|
|
\chi _{\rho }>0,\: \: c_{v}>0\, ,
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
and
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
\label{}
|
|
\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
|
|
\protected_separator
|
|
(
|
|
\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.
|
|
\protected_separator
|
|
|
|
\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
|
|
\protected_separator
|
|
|
|
\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
|
|
\protected_separator
|
|
17/2--1.
|
|
\layout Bibliography
|
|
\bibitem [1966]{baker}
|
|
|
|
Baker N., 1966, in: Stellar Evolution, eds.
|
|
\protected_separator
|
|
R.
|
|
F.
|
|
Stein, A.
|
|
G.
|
|
W.
|
|
Cameron, Plenum, New York, p.
|
|
\protected_separator
|
|
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.
|
|
\protected_separator
|
|
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.
|
|
\protected_separator
|
|
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
|