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958 lines
22 KiB
Plaintext
958 lines
22 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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\language english
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\inputencoding latin1
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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 Thesaurus
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06(03.11.1;16.06.1;19.06.1;19.37.1;19.53.1;19.63.1)
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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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\layout Address
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Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
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17, A-1180 Vienna
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\layout Offprint
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G.
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Wuchterl
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\layout Email
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wuchterl@amok.ast.univie.ac.at
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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.
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\layout Abstract
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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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\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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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
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of the stability equations for the thermodynamic state of these layers,
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specified by state quantities as density
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\begin_inset Formula \( \rho \)
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\end_inset
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, temperature
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\begin_inset Formula \( T \)
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\end_inset
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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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-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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-mechanism is widespread under `cool' conditions.
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\layout Abstract
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\latex latex
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\backslash
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keywords{
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\latex default
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giant planet formation --
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\begin_inset Formula \( \kappa \)
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\end_inset
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-mechanism -- stability of gas spheres
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\latex latex
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}
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\layout Section
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Introduction
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\layout Standard
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In the
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\emph on
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nucleated instability
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\emph 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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\end_inset
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) determined the critical mass of the core to be about
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\begin_inset Formula \( 12\, M_{\oplus } \)
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\end_inset
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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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is the Earth mass), which is independent of the outer boundary conditions
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and therefore independent of the location in the solar nebula.
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This critical value for the core mass corresponds closely to the cores
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of today's giant planets.
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\layout Standard
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Although no hydrodynamical study has been available many workers conjectured
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that a collapse or rapid contraction will ensue after accumulating the
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critical mass.
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The main motivation for this article is to investigate the stability of
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the static envelope at the critical mass.
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With this aim the local, linear stability of static radiative gas spheres
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is investigated 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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The nonlinear, hydrodynamic evolution of the protogiant planet beyond the
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critical mass, as calculated by Wuchterl (
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\begin_inset LatexCommand \cite{wuchterl}
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\end_inset
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), will be described in a forthcoming article.
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\layout Standard
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The fact that Wuchterl (
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\begin_inset LatexCommand \cite{wuchterl}
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\end_inset
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) found the excitation of hydrodynamical waves in his models raises considerable
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interest on the transition from static to dynamic evolutionary phases of
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the protogiant planet at the critical mass.
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The waves play a crucial role in the development of the so-called nucleated
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instability in the nucleated instability hypothesis.
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They lead to the formation of shock waves and massive outflow phenomena.
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The protoplanet evolves into a new quasi-equilibrium structure with a
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\emph on
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pulsating
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\emph default
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envelope, after the mass loss phase has declined.
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\layout Standard
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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
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\begin_inset LatexCommand \cite{tscarnuter}
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\end_inset
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, Balluch
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\begin_inset LatexCommand \cite{balluch}
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\end_inset
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), whereas earlier studies found quasi-steady collapse flows.
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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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Baker's standard one-zone model
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\layout Standard
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\begin_float wide-fig
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\layout Standard
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\latex latex
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\backslash
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rule{0.4pt}{4cm}
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\hfill
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\backslash
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parbox[b]{55mm}{
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\layout Caption
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Adiabatic exponent
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\begin_inset Formula \( \Gamma \)
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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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is plotted as a function of
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\begin_inset Formula \( \lg \)
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\end_inset
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internal energy
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\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
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\end_inset
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and
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\begin_inset Formula \( \lg \)
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\end_inset
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density
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\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
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\end_inset
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\begin_inset LatexCommand \label{FigGam}
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\end_inset
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\latex latex
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}
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\end_float
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In this section the one-zone model of Baker (
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\begin_inset LatexCommand \cite{baker}
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\end_inset
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), originally used to study the Cepheïd 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
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variables, local timescales and constitutive relations.
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\layout Standard
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Baker (
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\begin_inset LatexCommand \cite{baker}
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\end_inset
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) investigates the stability of thin layers in self-gravitating, spherical
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gas clouds with the following properties:
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\layout Itemize
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hydrostatic equilibrium,
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\layout Itemize
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thermal equilibrium,
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\layout Itemize
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energy transport by grey radiation diffusion.
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\layout Standard
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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.\SpecialChar ~
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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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\end_inset
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].
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Using Baker's notation:
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\begin_inset Formula \begin{eqnarray*}
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M_{\mathrm{r}} & & \mathrm{mass}\, \mathrm{internal}\, \mathrm{to}\, \mathrm{the}\, \mathrm{radius}\, r\\
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m & & \mathrm{mass}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}\\
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r_{0} & & \mathrm{unperturbed}\, \mathrm{zone}\, \mathrm{radius}\\
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\rho _{0} & & \mathrm{unperturbed}\, \mathrm{density}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\
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T_{0} & & \mathrm{unperturbed}\, \mathrm{temperature}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\
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L_{r0} & & \mathrm{unperturbed}\, \mathrm{luminosity}\\
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E_{\mathrm{th}} & & \mathrm{thermal}\, \mathrm{energy}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}
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\end{eqnarray*}
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\end_inset
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and with the definitions of the
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\emph on
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local cooling time
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\emph default
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(see Fig.\SpecialChar ~
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\begin_inset LatexCommand \ref{FigGam}
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\end_inset
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)
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\layout Standard
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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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\end_inset
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and the
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\emph on
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local free-fall time
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\layout Standard
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\begin_inset Formula \begin{equation}
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\tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}\, ,}
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\end{equation}
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\end_inset
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Baker's
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\begin_inset Formula \( K \)
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\end_inset
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and
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\begin_inset Formula \( \sigma _{0} \)
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\end_inset
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have the following form:
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\begin_inset Formula \begin{eqnarray}
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\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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\end_inset
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where
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\begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/\rho _{0}) \)
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\end_inset
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has been used and
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\layout Standard
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\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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\end_inset
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is a thermodynamical quantity which is of order
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\begin_inset Formula \( 1 \)
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\end_inset
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and equal to
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\begin_inset Formula \( 1 \)
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\end_inset
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for nonreacting mixtures of classical perfect gases.
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The physical meaning of
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\begin_inset Formula \( \sigma _{0} \)
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\end_inset
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and
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\begin_inset Formula \( K \)
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\end_inset
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is clearly visible in the equations above.
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\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.
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\begin_inset Formula \( K \)
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\end_inset
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is proportional to the ratio of the free-fall time and the cooling time.
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Substituting into Baker's criteria, using thermodynamic identities and
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definitions of thermodynamic quantities,
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\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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\end_inset
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\layout Standard
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\begin_inset Formula \[
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\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\: \chi _{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\: \kappa _{T}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
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\end_inset
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one obtains, after some pages of algebra, the conditions for
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\emph on
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stability
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\emph default
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given below:
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\layout Standard
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\begin_inset Formula \begin{eqnarray}
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\frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
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\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}
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\end{eqnarray}
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\end_inset
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For a physical discussion of the stability criteria see Baker (
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\begin_inset LatexCommand \cite{baker}
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\end_inset
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) or Cox (
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\begin_inset LatexCommand \cite{cox}
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\end_inset
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).
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\layout Standard
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We observe that these criteria for dynamical, secular and vibrational stability,
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respectively, can be factorized into
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\layout Enumerate
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a factor containing local timescales only,
|
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\layout Enumerate
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a factor containing only constitutive relations and their derivatives.
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\layout Standard
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The first factors, depending on only timescales, are positive by definition.
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The signs of the left hand sides of the inequalities\SpecialChar ~
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(
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\begin_inset LatexCommand \ref{ZSDynSta}
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\end_inset
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), (
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\begin_inset LatexCommand \ref{ZSSecSta}
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\end_inset
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) and (
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\begin_inset LatexCommand \ref{ZSVibSta}
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\end_inset
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) therefore depend exclusively on the second factors containing the constitutive
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relations.
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Since they depend only on state variables, the stability criteria themselves
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are
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\emph on
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functions of the thermodynamic state in the local zone
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\emph default
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.
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The one-zone stability can therefore be determined from a simple equation
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of state, given for example, as a function of density and temperature.
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Once the microphysics, i.e.
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the thermodynamics and opacities (see Table\SpecialChar ~
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|
|
\begin_inset LatexCommand \ref{KapSou}
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\end_inset
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|
), are specified (in practice by specifying a chemical composition) the
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one-zone stability can be inferred if the thermodynamic state is specified.
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The zone -- or in other words the layer -- will be stable or unstable in
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whatever object it is imbedded as long as it satisfies the one-zone-model
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assumptions.
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Only the specific growth rates (depending upon the time scales) will be
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different for layers in different objects.
|
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\layout Standard
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\begin_float tab
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\layout Caption
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Opacity sources
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\begin_inset LatexCommand \label{KapSou}
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\end_inset
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|
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\layout Standard
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|
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\begin_inset Tabular
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<lyxtabular version="2" rows="4" columns="2">
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<features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
|
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<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
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|
<column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
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<row topline="true" bottomline="true" newpage="false">
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<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
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\begin_inset Text
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\layout Standard
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Source
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\end_inset
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</cell>
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<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
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\begin_inset Text
|
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\layout Standard
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T/[K]
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\end_inset
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</cell>
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</row>
|
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<row topline="false" bottomline="false" newpage="false">
|
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<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
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\begin_inset Text
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\layout Standard
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Yorke 1979, Yorke 1980a
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\end_inset
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</cell>
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<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
|
|
\begin_inset Text
|
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\layout Standard
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|
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|
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|
\begin_inset Formula \( \leq 1700^{\mathrm{a}} \)
|
|
\end_inset
|
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|
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\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
|
|
|
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\layout Standard
|
|
|
|
Krügel 1971
|
|
\end_inset
|
|
</cell>
|
|
<cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="true" rotate="false" usebox="none" width="" special="">
|
|
\begin_inset Text
|
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\layout Standard
|
|
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|
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|
\begin_inset Formula \( 1700\leq T\leq 5000 \)
|
|
\end_inset
|
|
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
<row topline="false" bottomline="true" newpage="false">
|
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<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="true" rotate="false" usebox="none" width="" special="">
|
|
\begin_inset Text
|
|
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula \( 5000\leq \)
|
|
\end_inset
|
|
|
|
|
|
\end_inset
|
|
</cell>
|
|
</row>
|
|
</lyxtabular>
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|
|
|
\end_inset
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|
|
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\layout Standard
|
|
\added_space_top medskip*
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|
|
|
\begin_inset Formula \( ^{\textrm{a}} \)
|
|
\end_inset
|
|
|
|
This is footnote a
|
|
\end_float
|
|
\begin_float wide-tab
|
|
\layout Caption
|
|
|
|
Regions of secular instability
|
|
\begin_inset LatexCommand \label{TabSecInst}
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|
|
|
\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}
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|
|
|
\end_inset
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|
), (
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|
\begin_inset LatexCommand \ref{ZSSecSta}
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|
|
|
\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
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula \begin{equation}
|
|
\Gamma _{1}>\frac{4}{3}
|
|
\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
|
|
\layout Standard
|
|
|
|
|
|
\begin_inset Formula \begin{equation}
|
|
\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\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:
|
|
\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.\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 ~
|
|
|
|
\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\SpecialChar ~
|
|
17/2--1.
|
|
\layout Bibliography
|
|
\bibitem [1966]{baker}
|
|
|
|
Baker N., 1966, in: Stellar Evolution, eds.\SpecialChar ~
|
|
R.
|
|
F.
|
|
Stein, A.
|
|
G.
|
|
W.
|
|
Cameron, Plenum, New York, p.\SpecialChar ~
|
|
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.\SpecialChar ~
|
|
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.\SpecialChar ~
|
|
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
|