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964 lines
21 KiB
Plaintext
964 lines
21 KiB
Plaintext
#LyX 1.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 218
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\textclass aa
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\begin_preamble
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\usepackage{graphicx}
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%
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\end_preamble
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\language english
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\inputencoding auto
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\fontscheme default
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\graphics default
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\paperfontsize default
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\spacing single
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\papersize Default
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\paperpackage a4
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\use_geometry 0
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\use_amsmath 0
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\paperorientation portrait
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\defskip medskip
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\quotes_language english
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\quotes_times 2
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\papercolumns 2
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\papersides 2
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\paperpagestyle default
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\layout Title
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Hydrodynamics of giant planet formation
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\layout Subtitle
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I.
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Overviewing the
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\begin_inset Formula \( \kappa \)
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\end_inset
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-mechanism
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\layout Author
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G.
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Wuchterl
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\latex latex
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\backslash
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inst{1}
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\backslash
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and
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\newline
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\latex default
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C.
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Ptolemy
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\latex latex
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\backslash
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inst{2}
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\backslash
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fnmsep
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\begin_float footnote
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\layout Standard
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Just to show the usage of the elements in the author field
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\end_float
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\layout Offprint
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G.
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Wuchterl
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\layout Address
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Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
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rkenschanzstrasse
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17, A-1180 Vienna
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\newline
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\latex latex
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\backslash
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email{wuchterl@amok.ast.univie.ac.at}
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\backslash
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and
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\newline
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\latex default
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University of Alexandria, Department of Geography, ...
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\newline
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\latex latex
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\backslash
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email{c.ptolemy@hipparch.uheaven.space}
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\latex default
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\begin_float footnote
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\layout Standard
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The university of heaven temporarily does not accept e-mails
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\end_float
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\layout Date
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Received September 15, 1996; accepted March 16, 1997
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\layout Abstract
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To investigate the physical nature of the `nuc\SpecialChar \-
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leated instability' of proto
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giant planets (Mizuno
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\begin_inset LatexCommand \cite{mizuno}
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\end_inset
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), the stability of layers in static, radiative gas spheres is analysed
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on the basis of Baker's
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\begin_inset LatexCommand \cite{baker}
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\end_inset
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standard one-zone model.
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It is shown that stability depends only upon the equations of state, the
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opacities and the local thermodynamic state in the layer.
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Stability and instability can therefore be expressed in the form of stability
|
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equations of state which are universal for a given composition.
|
||
The stability equations of state are calculated for solar composition and
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||
are displayed in the domain
|
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\begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
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\end_inset
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||
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,
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\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
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\end_inset
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||
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.
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These displays may be used to determine the one-zone stability of layers
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in stellar or planetary structure models by directly reading off the value
|
||
of the stability equations for the thermodynamic state of these layers,
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specified by state quantities as density
|
||
\begin_inset Formula \( \rho \)
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||
\end_inset
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||
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, temperature
|
||
\begin_inset Formula \( T \)
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||
\end_inset
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||
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or specific internal energy
|
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\begin_inset Formula \( e \)
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\end_inset
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.
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Regions of instability in the
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||
\begin_inset Formula \( (\rho ,e) \)
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\end_inset
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||
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-plane are described and related to the underlying microphysical processes.
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Vibrational instability is found to be a common phenomenon at temperatures
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lower than the second He ionisation zone.
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The
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\begin_inset Formula \( \kappa \)
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\end_inset
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||
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-mechanism is widespread under `cool' conditions.
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\latex latex
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|
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\newline
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\backslash
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keywords{giant planet formation --
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\backslash
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(
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\backslash
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||
kappa
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\backslash
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||
)-mechanism -- stability of gas spheres }
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||
\latex default
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\layout Section
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||
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||
Introduction
|
||
\layout Standard
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||
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||
In the
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||
\emph on
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||
nucleated instability
|
||
\latex latex
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||
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||
\backslash
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||
/{}
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\emph default
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||
\latex default
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||
(also called core instability) hypothesis of giant planet formation, a
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||
critical mass for static core envelope protoplanets has been found.
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||
Mizuno (
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\begin_inset LatexCommand \cite{mizuno}
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||
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||
\end_inset
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||
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||
) determined the critical mass of the core to be about
|
||
\begin_inset Formula \( 12\, M_{\oplus } \)
|
||
\end_inset
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||
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||
(
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||
\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
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||
\end_inset
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||
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||
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
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||
of today's giant planets.
|
||
\layout Standard
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||
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||
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}
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||
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||
\end_inset
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||
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||
) standard one-zone model.
|
||
\layout Standard
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||
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||
Phenomena similar to the ones described above for giant planet formation
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||
have been found in hydrodynamical models concerning star formation where
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||
protostellar cores explode (Tscharnuter
|
||
\begin_inset LatexCommand \cite{tscharnuter}
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||
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||
\end_inset
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||
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||
, Balluch
|
||
\begin_inset LatexCommand \cite{balluch}
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||
\end_inset
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||
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||
), whereas earlier studies found quasi-steady collapse flows.
|
||
The similarities in the (micro)physics, i.e., constitutive relations of protostel
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||
lar cores and protogiant planets serve as a further motivation for this
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||
study.
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||
\layout Section
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||
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||
Baker's standard one-zone model
|
||
\layout Standard
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||
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||
\begin_float wide-fig
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||
\layout Caption
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||
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||
Adiabatic exponent
|
||
\begin_inset Formula \( \Gamma _{1} \)
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||
\end_inset
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||
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||
.
|
||
|
||
\begin_inset Formula \( \Gamma _{1} \)
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||
\end_inset
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||
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||
is plotted as a function of
|
||
\begin_inset Formula \( \lg \)
|
||
\end_inset
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||
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||
internal energy
|
||
\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
|
||
\end_inset
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||
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||
and
|
||
\begin_inset Formula \( \lg \)
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||
\end_inset
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||
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||
density
|
||
\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
|
||
\end_inset
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||
|
||
|
||
\layout Standard
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||
|
||
|
||
\begin_inset LatexCommand \label{FigGam}
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||
|
||
\end_inset
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||
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||
|
||
\end_float
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||
In this section the one-zone model of Baker (
|
||
\begin_inset LatexCommand \cite{baker}
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||
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||
\end_inset
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||
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||
), originally used to study the Cephe\i \"{\i}
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d pulsation mechanism, will be briefly
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reviewed.
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The resulting stability criteria will be rewritten in terms of local state
|
||
variables, local timescales and constitutive relations.
|
||
\layout Standard
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||
|
||
Baker (
|
||
\begin_inset LatexCommand \cite{baker}
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||
|
||
\end_inset
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||
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||
) investigates the stability of thin layers in self-gravitating, spherical
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||
gas clouds with the following properties:
|
||
\layout Itemize
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||
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||
hydrostatic equilibrium,
|
||
\layout Itemize
|
||
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||
thermal equilibrium,
|
||
\layout Itemize
|
||
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||
energy transport by grey radiation diffusion.
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||
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||
\layout Standard
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||
\noindent
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||
For the one-zone-model Baker obtains necessary conditions for dynamical,
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||
secular and vibrational (or pulsational) stability (Eqs.
|
||
\latex latex
|
||
|
||
\backslash
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||
|
||
\latex default
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||
(34a,
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||
\latex latex
|
||
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||
\backslash
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||
,
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||
\latex default
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||
b,
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||
\latex latex
|
||
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||
\backslash
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||
,
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||
\latex default
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||
c) in Baker
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||
\begin_inset LatexCommand \cite{baker}
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||
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||
\end_inset
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||
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||
).
|
||
Using Baker's notation:
|
||
\layout Standard
|
||
\align left
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||
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||
\begin_inset Formula \begin{eqnarray*}
|
||
M_{r} & & \textrm{mass internal to the radius }r\\
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||
m & & \textrm{mass of the zone}\\
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||
r_{0} & & \textrm{unperturbed zone radius}\\
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||
\rho _{0} & & \textrm{unperturbed density in the zone}\\
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||
T_{0} & & \textrm{unperturbed temperature in the zone}\\
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||
L_{r0} & & \textrm{unperturbed luminosity}\\
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||
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
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||
\end{eqnarray*}
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||
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||
\end_inset
|
||
|
||
|
||
\layout Standard
|
||
\noindent
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||
and with the definitions of the
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\emph on
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||
local cooling time
|
||
\latex latex
|
||
|
||
\backslash
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||
/{}
|
||
\emph default
|
||
\latex default
|
||
(see Fig.\SpecialChar ~
|
||
|
||
\begin_inset LatexCommand \ref{FigGam}
|
||
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||
\end_inset
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||
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||
)
|
||
\begin_inset Formula \begin{equation}
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||
\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
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||
\end{equation}
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||
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||
\end_inset
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||
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||
and the
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||
\emph on
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||
local free-fall time
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||
\emph default
|
||
|
||
\begin_inset Formula \begin{equation}
|
||
\tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
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||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
Baker's
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||
\begin_inset Formula \( K \)
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||
\end_inset
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||
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||
and
|
||
\begin_inset Formula \( \sigma _{0} \)
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||
\end_inset
|
||
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||
have the following form:
|
||
\begin_inset Formula \begin{eqnarray}
|
||
\sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
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||
K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
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||
\end{eqnarray}
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||
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||
\end_inset
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||
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||
where
|
||
\begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
|
||
\end_inset
|
||
|
||
has been used and
|
||
\begin_inset Formula \begin{equation}
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||
\begin{array}{l}
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||
\delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
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||
e=mc^{2}
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||
\end{array}
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||
\end{equation}
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||
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||
\end_inset
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||
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||
is a thermodynamical quantity which is of order
|
||
\begin_inset Formula \( 1 \)
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||
\end_inset
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||
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||
and equal to
|
||
\begin_inset Formula \( 1 \)
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||
\end_inset
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||
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||
for nonreacting mixtures of classical perfect gases.
|
||
The physical meaning of
|
||
\begin_inset Formula \( \sigma _{0} \)
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||
\end_inset
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||
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||
and
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||
\begin_inset Formula \( K \)
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||
\end_inset
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||
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||
is clearly visible in the equations above.
|
||
|
||
\begin_inset Formula \( \sigma _{0} \)
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||
\end_inset
|
||
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||
represents a frequency of the order one per free-fall time.
|
||
|
||
\begin_inset Formula \( K \)
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||
\end_inset
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||
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||
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 \[
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||
\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
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||
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||
\end_inset
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||
|
||
|
||
\begin_inset Formula \[
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||
\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
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||
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||
\end_inset
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||
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||
one obtains, after some pages of algebra, the conditions for
|
||
\emph on
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||
stability
|
||
\latex latex
|
||
|
||
\backslash
|
||
/{}
|
||
\emph default
|
||
\latex 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} \\
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||
\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\SpecialChar ~
|
||
(
|
||
\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.
|
||
\latex latex
|
||
|
||
\backslash
|
||
|
||
\latex default
|
||
the thermodynamics and opacities (see Table\SpecialChar ~
|
||
|
||
\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
|
||
|
||
|
||
\begin_inset LatexCommand \label{KapSou}
|
||
|
||
\end_inset
|
||
|
||
Opacity sources
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Tabular
|
||
<lyxtabular version="2" rows="4" columns="2">
|
||
<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
|
||
<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
|
||
<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
|
||
<row topline="true" bottomline="false" newpage="false">
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
Source
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Formula \( T/[\textrm{K}] \)
|
||
\end_inset
|
||
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row topline="true" bottomline="false" newpage="false">
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
Yorke 1979, Yorke 1980a
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
|
||
\end_inset
|
||
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row topline="false" bottomline="false" newpage="false">
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
Kr<EFBFBD>gel 1971
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Formula \( 1700\leq T\leq 5000 \)
|
||
\end_inset
|
||
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row topline="false" bottomline="true" newpage="false">
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
Cox & Stewart 1969
|
||
\end_inset
|
||
</cell>
|
||
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
|
||
\begin_inset Text
|
||
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Formula \( 5000\leq \)
|
||
\end_inset
|
||
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\layout Standard
|
||
|
||
|
||
\begin_inset Formula \( ^{\textrm{a}} \)
|
||
\end_inset
|
||
|
||
This is footnote a
|
||
\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\SpecialChar ~
|
||
(
|
||
\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
|
||
\begin_inset Formula \begin{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}
|
||
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula \begin{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}
|
||
|
||
\end_inset
|
||
|
||
we find the sign determining terms in inequalities\SpecialChar ~
|
||
(
|
||
\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:
|
||
\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 thermodynami
|
||
c 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.\SpecialChar ~
|
||
|
||
\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\SpecialChar ~
|
||
1.
|
||
\layout Standard
|
||
|
||
\begin_float fig
|
||
\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
|
||
\layout Standard
|
||
|
||
|
||
\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 Cephe\i \"{\i}
|
||
d 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
|
||
\latex latex
|
||
|
||
\backslash
|
||
/{}
|
||
\emph default
|
||
\latex default
|
||
project number Ts\SpecialChar ~
|
||
17/2--1.
|
||
|
||
\layout Bibliography
|
||
\bibitem [1966]{baker}
|
||
|
||
Baker, N.
|
||
1966, in Stellar Evolution, ed.
|
||
\latex latex
|
||
|
||
\backslash
|
||
|
||
\latex default
|
||
R.
|
||
F.
|
||
Stein,& A.
|
||
G.
|
||
W.
|
||
Cameron (Plenum, New York) 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)
|
||
165
|
||
\layout Bibliography
|
||
\bibitem [1969]{cox69}
|
||
|
||
Cox, A.
|
||
N.,& Stewart, J.
|
||
N.
|
||
1969, Academia Nauk, Scientific Information 15, 1
|
||
\layout Bibliography
|
||
\bibitem [1980]{mizuno}
|
||
|
||
Mizuno H.
|
||
1980, Prog.
|
||
Theor.
|
||
Phys., 64, 544
|
||
\layout Bibliography
|
||
\bibitem [1987]{tscharnuter}
|
||
|
||
Tscharnuter W.
|
||
M.
|
||
1987, A&A, 188, 55
|
||
\layout Bibliography
|
||
\bibitem [1992]{terlevich}
|
||
|
||
Terlevich, R.
|
||
1992, in ASP Conf.
|
||
Ser.
|
||
31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
|
||
ed.
|
||
A.
|
||
V.
|
||
Filippenko, 13
|
||
\layout Bibliography
|
||
\bibitem [1980a]{yorke80a}
|
||
|
||
Yorke, H.
|
||
W.
|
||
1980a, A&A, 86, 286
|
||
\layout Bibliography
|
||
\bibitem [1997]{zheng}
|
||
|
||
Zheng, W., Davidsen, A.
|
||
F., Tytler, D.
|
||
& Kriss, G.
|
||
A.
|
||
1997, preprint
|
||
\the_end
|