-#LyX 1.3 created this file. For more info see http://www.lyx.org/
-\lyxformat 221
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
+\begin_document
+\begin_header
\textclass aa
\begin_preamble
\usepackage{graphicx}
%
\end_preamble
+\options traditabstract
+\use_default_options false
+\maintain_unincluded_children false
\language english
+\language_package default
\inputencoding auto
-\fontscheme default
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
\paperfontsize default
\spacing single
-\papersize Default
-\paperpackage a4
-\use_geometry 0
+\use_hyperref false
+\papersize default
+\use_geometry false
\use_amsmath 0
-\use_natbib 0
-\use_numerical_citations 0
+\use_esint 0
+\use_mhchem 1
+\use_mathdots 1
+\cite_engine basic
+\use_bibtopic false
+\use_indices false
\paperorientation portrait
+\suppress_date false
+\use_refstyle 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
-\defskip medskip
+\paragraph_indentation default
\quotes_language english
-\quotes_times 2
\papercolumns 2
\papersides 2
\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+This is an example LyX file for articles to be submitted to the Journal
+ of Astronomy & Astrophysicssing (A&A).
+ How to install the A&A LaTeX class to your LaTeX system is explained in
+
+\begin_inset Flex URL
+status open
+
+\begin_layout Plain Layout
+
+http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
-\layout Title
+\begin_layout Standard
+\begin_inset Note Note
+status open
+\begin_layout Plain Layout
+Depending on the submission state and the abstract layout, you need to use
+ different document class options that are listed in the aa manual
+\family sans
+aadoc.pdf
+\family default
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Title
Hydrodynamics of giant planet formation
-\layout Subtitle
+\end_layout
+\begin_layout Subtitle
I.
Overviewing the
-\begin_inset Formula \( \kappa \)
-\end_inset
+\begin_inset Formula $\kappa$
+\end_inset
-mechanism
-\layout Author
+\end_layout
+\begin_layout Author
G.
- Wuchterl
+ Wuchterl
+\begin_inset Flex institutemark
+status open
+
+\begin_layout Plain Layout
+1
+\end_layout
+
+\end_inset
+
+
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
+
-\layout Standard
+\backslash
+and
+\end_layout
-\backslash
-inst{1}
-\backslash
-and
-\newline
-
-\end_inset
+\end_inset
-C.
+ C.
Ptolemy
+\begin_inset Flex institutemark
+status collapsed
+
+\begin_layout Plain Layout
+2
+\end_layout
+
+\end_inset
+
+
\begin_inset ERT
-status Collapsed
+status collapsed
-\layout Standard
+\begin_layout Plain Layout
-\backslash
-inst{2}
-\backslash
-fnmsep
-\end_inset
+\backslash
+fnmsep
+\end_layout
+
+\end_inset
-\begin_inset Foot
-collapsed true
-\layout Standard
+\begin_inset Foot
+status collapsed
+\begin_layout Plain Layout
Just to show the usage of the elements in the author field
-\end_inset
+\end_layout
+\end_inset
-\layout Offprint
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+
+\backslash
+fnmsep is only needed for more than one consecutive notes/marks
+\end_layout
+
+\end_inset
+
+\end_layout
+
+\begin_layout Offprint
G.
Wuchterl
-\layout Address
+\end_layout
-Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
-rkenschanzstrasse
+\begin_layout Address
+Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
17, A-1180 Vienna
-\newline
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Flex Email
+status open
+
+\begin_layout Plain Layout
+wuchterl@amok.ast.univie.ac.at
+\end_layout
+
+\end_inset
+
\begin_inset ERT
-status Collapsed
+status collapsed
+
+\begin_layout Plain Layout
-\layout Standard
-\backslash
-email{wuchterl@amok.ast.univie.ac.at}
-\backslash
-and
-\newline
+\backslash
+and
+\end_layout
-\end_inset
+\end_inset
University of Alexandria, Department of Geography, ...
-\newline
+\begin_inset Newline newline
+\end_inset
-\begin_inset ERT
-status Collapsed
-\layout Standard
+\begin_inset Flex Email
+status collapsed
-\backslash
-email{c.ptolemy@hipparch.uheaven.space}
-\end_inset
+\begin_layout Plain Layout
+c.ptolemy@hipparch.uheaven.space
+\end_layout
+
+\end_inset
-
-\begin_inset Foot
-collapsed true
-\layout Standard
+\begin_inset Foot
+status collapsed
+\begin_layout Plain Layout
The university of heaven temporarily does not accept e-mails
-\end_inset
+\end_layout
+\end_inset
-\layout Date
+\end_layout
+\begin_layout Date
Received September 15, 1996; accepted March 16, 1997
-\layout Abstract
+\end_layout
+\begin_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.
+ giant planets, the stability of layers in static, radiative gas spheres
+ is analysed on the basis of Baker's standard one-zone model.
It is shown that stability depends only upon the equations of state, the
opacities and the local thermodynamic state in the layer.
Stability and instability can therefore be expressed in the form of stability
equations of state which are universal for a given composition.
The stability equations of state are calculated for solar composition and
are displayed in the domain
-\begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
-\end_inset
+\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
+\end_inset
,
-\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
-\end_inset
+\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
+\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
+\begin_inset Formula $\rho$
+\end_inset
, temperature
-\begin_inset Formula \( T \)
-\end_inset
+\begin_inset Formula $T$
+\end_inset
or specific internal energy
-\begin_inset Formula \( e \)
-\end_inset
+\begin_inset Formula $e$
+\end_inset
.
Regions of instability in the
-\begin_inset Formula \( (\rho ,e) \)
-\end_inset
+\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
+\begin_inset Formula $\kappa$
+\end_inset
-mechanism is widespread under `cool' conditions.
-\begin_inset ERT
-status Collapsed
+\begin_inset Note Note
+status open
-\layout Standard
+\begin_layout Plain Layout
+Citations are not allowed in A&A abstracts.
+\end_layout
-\newline
+\end_inset
-\backslash
-keywords{giant planet formation --
-\backslash
-(
-\backslash
-kappa
-\backslash
-)-mechanism -- stability of gas spheres }
-\end_inset
-
-\layout Section
+\begin_inset Note Note
+status open
-Introduction
-\layout Standard
+\begin_layout Plain Layout
+This is the unstructured abstract type, an example for the structured abstract
+ is in the
+\family sans
+aa.lyx
+\family default
+ template file that comes with LyX.
+\end_layout
+
+\end_inset
-In the
-\emph on
-nucleated instability
-\begin_inset ERT
-status Collapsed
-\layout Standard
+\end_layout
-\backslash
-/{}
-\end_inset
+\begin_layout Keywords
+giant planet formation --
+\begin_inset Formula $\kappa$
+\end_inset
+-mechanism -- stability of gas spheres
+\end_layout
-\emph default
+\begin_layout Section
+Introduction
+\end_layout
+
+\begin_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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "mizuno"
-\end_inset
+\end_inset
) determined the critical mass of the core to be about
-\begin_inset Formula \( 12\, M_{\oplus } \)
-\end_inset
+\begin_inset Formula $12\, M_{\oplus}$
+\end_inset
(
-\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
-\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
+\end_layout
+\begin_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 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}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\end_inset
) standard one-zone model.
-\layout Standard
+\end_layout
+\begin_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{tscharnuter}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "tscharnuter"
-\end_inset
+\end_inset
, Balluch
-\begin_inset LatexCommand \cite{balluch}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "balluch"
-\end_inset
+\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
+ The similarities in the (micro)physics, i.
+\begin_inset space \thinspace{}
+\end_inset
+
+g.
+\begin_inset space \space{}
+\end_inset
+constitutive relations of protostellar cores and protogiant planets serve
+ as a further motivation for this study.
+\end_layout
+
+\begin_layout Section
Baker's standard one-zone model
-\layout Standard
+\end_layout
+\begin_layout Standard
\begin_inset Float figure
wide true
-collapsed false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Caption
-\layout Caption
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:FigGam"
+
+\end_inset
Adiabatic exponent
-\begin_inset Formula \( \Gamma _{1} \)
-\end_inset
+\begin_inset Formula $\Gamma_{1}$
+\end_inset
.
-\begin_inset Formula \( \Gamma _{1} \)
-\end_inset
+\begin_inset Formula $\Gamma_{1}$
+\end_inset
is plotted as a function of
-\begin_inset Formula \( \lg \)
-\end_inset
+\begin_inset Formula $\lg$
+\end_inset
internal energy
-\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
-\end_inset
+\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
+\end_inset
and
-\begin_inset Formula \( \lg \)
-\end_inset
+\begin_inset Formula $\lg$
+\end_inset
density
-\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
-\end_inset
-
+\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
+\end_inset
-\layout Standard
+\end_layout
-\begin_inset LatexCommand \label{FigGam}
+\end_inset
-\end_inset
+\end_layout
-\end_inset
+\end_inset
In this section the one-zone model of Baker (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\end_inset
-), originally used to study the Cephe\i \"{\i}
-d pulsation mechanism, will be briefly
+), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
reviewed.
The resulting stability criteria will be rewritten in terms of local state
variables, local timescales and constitutive relations.
-\layout Standard
+\end_layout
+\begin_layout Standard
Baker (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\end_inset
) investigates the stability of thin layers in self-gravitating, spherical
gas clouds with the following properties:
-\layout Itemize
+\end_layout
+\begin_layout Itemize
hydrostatic equilibrium,
-\layout Itemize
+\end_layout
+\begin_layout Itemize
thermal equilibrium,
-\layout Itemize
+\end_layout
+\begin_layout Itemize
energy transport by grey radiation diffusion.
-\layout Standard
-\noindent
+\end_layout
+
+\begin_layout Standard
+\noindent
For the one-zone-model Baker obtains necessary conditions for dynamical,
secular and vibrational (or pulsational) stability (Eqs.
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-
-\backslash
-
-\end_inset
+\begin_inset space \space{}
+\end_inset
(34a,
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-
-\backslash
-,
-\end_inset
+\begin_inset space \thinspace{}
+\end_inset
b,
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-
-\backslash
-,
-\end_inset
+\begin_inset space \thinspace{}
+\end_inset
c) in Baker
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\end_inset
).
Using Baker's notation:
-\layout Standard
-\align left
+\end_layout
-\begin_inset Formula \begin{eqnarray*}
+\begin_layout Standard
+\align left
+\begin_inset Formula
+\begin{eqnarray*}
M_{r} & & \textrm{mass internal to the radius }r\\
m & & \textrm{mass of the zone}\\
r_{0} & & \textrm{unperturbed zone radius}\\
-\rho _{0} & & \textrm{unperturbed density in the zone}\\
+\rho_{0} & & \textrm{unperturbed density in the zone}\\
T_{0} & & \textrm{unperturbed temperature in the zone}\\
L_{r0} & & \textrm{unperturbed luminosity}\\
E_{\textrm{th}} & & \textrm{thermal energy of the zone}
\end{eqnarray*}
-\end_inset
+\end_inset
-\layout Standard
-\noindent
+\end_layout
+
+\begin_layout Standard
+\noindent
and with the definitions of the
-\emph on
+\emph on
local cooling time
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-
-\backslash
-/{}
-\end_inset
-
+\emph default
+ (see Fig.
+\begin_inset space ~
+\end_inset
-\emph default
- (see Fig.\SpecialChar ~
-\begin_inset LatexCommand \ref{FigGam}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:FigGam"
-\end_inset
+\end_inset
)
-\begin_inset Formula \begin{equation}
-\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
+\begin_inset Formula
+\begin{equation}
+\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
\end{equation}
-\end_inset
+\end_inset
and the
-\emph on
+\emph on
local free-fall time
-\emph default
+\emph default
-\begin_inset Formula \begin{equation}
-\tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
+\begin_inset Formula
+\begin{equation}
+\tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
\end{equation}
-\end_inset
+\end_inset
Baker's
-\begin_inset Formula \( K \)
-\end_inset
+\begin_inset Formula $K$
+\end_inset
and
-\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\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}}}\, ;
+\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
+\end_inset
where
-\begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
-\end_inset
+\begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
+\end_inset
has been used and
-\begin_inset Formula \begin{equation}
+\begin_inset Formula
+\begin{equation}
\begin{array}{l}
-\delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
+\delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
e=mc^{2}
\end{array}
\end{equation}
-\end_inset
+\end_inset
is a thermodynamical quantity which is of order
-\begin_inset Formula \( 1 \)
-\end_inset
+\begin_inset Formula $1$
+\end_inset
and equal to
-\begin_inset Formula \( 1 \)
-\end_inset
+\begin_inset Formula $1$
+\end_inset
for nonreacting mixtures of classical perfect gases.
The physical meaning of
-\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\begin_inset Formula $\sigma_{0}$
+\end_inset
and
-\begin_inset Formula \( K \)
-\end_inset
+\begin_inset Formula $K$
+\end_inset
is clearly visible in the equations above.
-\begin_inset Formula \( \sigma _{0} \)
-\end_inset
+\begin_inset Formula $\sigma_{0}$
+\end_inset
represents a frequency of the order one per free-fall time.
-\begin_inset Formula \( K \)
-\end_inset
+\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 \[
-\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
+\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
+\end_inset
-\begin_inset Formula \[
-\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
+\begin_inset Formula
+\[
+\nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}
+\]
-\end_inset
+\end_inset
one obtains, after some pages of algebra, the conditions for
-\emph on
+\emph on
stability
-\begin_inset ERT
-status Collapsed
-
-\layout Standard
-
-\backslash
-/{}
-\end_inset
-
-
-\emph default
+\emph 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} \\
-\frac{\pi ^{2}}{4}\frac{3}{\tau _{\mathrm{co}}\tau _{\mathrm{ff}}^{2}}\Gamma _{1}^{2}\, \nabla _{\mathrm{ad}}\left[ 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}\right] & > & 0\label{ZSVibSta}
+\begin_inset Formula
+\begin{eqnarray}
+\frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
+\frac{\pi^{2}}{\tau_{\mathrm{co}}\tau_{\mathrm{ff}}^{2}}\Gamma_{1}\nabla_{\mathrm{ad}}\left[\frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1\right] & > & 0\label{ZSSecSta}\\
+\frac{\pi^{2}}{4}\frac{3}{\tau_{\mathrm{co}}\tau_{\mathrm{ff}}^{2}}\Gamma_{1}^{2}\,\nabla_{\mathrm{ad}}\left[4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}\right] & > & 0\label{ZSVibSta}
\end{eqnarray}
-\end_inset
+\end_inset
For a physical discussion of the stability criteria see Baker (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\end_inset
) or Cox (
-\begin_inset LatexCommand \cite{cox}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "cox"
-\end_inset
+\end_inset
).
-\layout Standard
+\end_layout
+\begin_layout Standard
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
a factor containing local timescales only,
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
a factor containing only constitutive relations and their derivatives.
-\layout Standard
+\end_layout
+\begin_layout Standard
The first factors, depending on only timescales, are positive by definition.
- The signs of the left hand sides of the inequalities\SpecialChar ~
+ The signs of the left hand sides of the inequalities
+\begin_inset space ~
+\end_inset
+
(
-\begin_inset LatexCommand \ref{ZSDynSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
-\end_inset
+\end_inset
), (
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
-\end_inset
+\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
-\end_inset
+\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
+\emph on
functions of the thermodynamic state in the local zone
-\emph default
+\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.
-\begin_inset ERT
-status Collapsed
+ Once the microphysics, i.
+\begin_inset space \thinspace{}
+\end_inset
-\layout Standard
+g.
+\begin_inset space \space{}
+\end_inset
-\backslash
-
-\end_inset
+the thermodynamics and opacities (see Table
+\begin_inset space ~
+\end_inset
-the thermodynamics and opacities (see Table\SpecialChar ~
-\begin_inset LatexCommand \ref{KapSou}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:KapSou"
-\end_inset
+\end_inset
), are specified (in practice by specifying a chemical composition) the
one-zone stability can be inferred if the thermodynamic state is specified.
assumptions.
Only the specific growth rates (depending upon the time scales) will be
different for layers in different objects.
-\layout Standard
+\end_layout
+\begin_layout Standard
\begin_inset Float table
wide false
-collapsed false
-
-\layout Caption
+sideways false
+status open
+\begin_layout Plain Layout
+\begin_inset Caption
-\begin_inset LatexCommand \label{KapSou}
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:KapSou"
-\end_inset
+\end_inset
Opacity sources
-\layout Standard
+\end_layout
+\end_inset
-\begin_inset Tabular
+
+\end_layout
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="2">
-<features>
+<features tabularvalignment="middle">
<column alignment="left" valignment="top" width="0pt">
<column alignment="left" valignment="top" width="0pt">
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
+\begin_layout Plain Layout
Source
-\end_inset
+\end_layout
+
+\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Plain Layout
+\begin_inset Formula $T/[\textrm{K}]$
+\end_inset
-\begin_inset Formula \( T/[\textrm{K}] \)
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
+\begin_layout Plain Layout
Yorke 1979, Yorke 1980a
-\end_inset
+\end_layout
+
+\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" topline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Plain Layout
+\begin_inset Formula $\leq1700^{\textrm{a}}$
+\end_inset
-\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Plain Layout
+Krügel 1971
+\end_layout
-Krügel 1971
-\end_inset
+\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" usebox="none">
\begin_inset Text
-\layout Standard
-
-
-\begin_inset Formula \( 1700\leq T\leq 5000 \)
-\end_inset
+\begin_layout Plain Layout
+\begin_inset Formula $1700\leq T\leq5000$
+\end_inset
-\end_inset
+\end_layout
+
+\end_inset
</cell>
</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<row>
+<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
-\layout Standard
-
+\begin_layout Plain Layout
Cox & Stewart 1969
-\end_inset
+\end_layout
+
+\end_inset
</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+<cell alignment="center" valignment="top" bottomline="true" usebox="none">
\begin_inset Text
-\layout Standard
+\begin_layout Plain Layout
+\begin_inset Formula $5000\leq$
+\end_inset
-\begin_inset Formula \( 5000\leq \)
-\end_inset
+\end_layout
-
-\end_inset
+\end_inset
</cell>
</row>
</lyxtabular>
-\end_inset
-
+\end_inset
-\layout Standard
+\end_layout
-\begin_inset Formula \( ^{\textrm{a}} \)
-\end_inset
+\begin_layout Plain Layout
+\begin_inset Formula $^{\textrm{a}}$
+\end_inset
This is footnote a
-\end_inset
+\end_layout
+
+\end_inset
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}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
-\end_inset
+\end_inset
), (
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
-\end_inset
+\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
-\end_inset
+\end_inset
) and thereby obtain
-\emph on
+\emph on
stability equations of state
-\emph default
+\emph default
.
-\layout Standard
+\end_layout
+
+\begin_layout Standard
+The sign determining part of inequality
+\begin_inset space ~
+\end_inset
-The sign determining part of inequality\SpecialChar ~
(
-\begin_inset LatexCommand \ref{ZSDynSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSDynSta"
-\end_inset
+\end_inset
) is
-\begin_inset Formula \( 3\Gamma _{1}-4 \)
-\end_inset
+\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
+\begin_inset Formula
+\begin{equation}
+\Gamma_{1}>\frac{4}{3}\,\cdot
\end{equation}
-\end_inset
+\end_inset
Stability of the thermodynamical equilibrium demands
-\begin_inset Formula \begin{equation}
-\chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
+\begin_inset Formula
+\begin{equation}
+\chi_{\rho}^{}>0,\;\; c_{v}>0\,,
\end{equation}
-\end_inset
+\end_inset
and
-\begin_inset Formula \begin{equation}
-\chi ^{}_{T}>0
+\begin_inset Formula
+\begin{equation}
+\chi_{T}^{}>0
\end{equation}
-\end_inset
+\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
+\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
+\end_inset
+
+ we find the sign determining terms in inequalities
+\begin_inset space ~
+\end_inset
- we find the sign determining terms in inequalities\SpecialChar ~
(
-\begin_inset LatexCommand \ref{ZSSecSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSSecSta"
-\end_inset
+\end_inset
) and (
-\begin_inset LatexCommand \ref{ZSVibSta}
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "ZSVibSta"
-\end_inset
+\end_inset
) respectively and obtain the following form of the criteria for dynamical,
secular and vibrational
-\emph on
+\emph on
stability
-\emph default
+\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}
+\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
+\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
+\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 $\Gamma_{1}$
+\end_inset
,
-\begin_inset Formula \( \nabla _{\mathrm{ad}} \)
-\end_inset
+\begin_inset Formula $\nabla_{\mathrm{ad}}$
+\end_inset
,
-\begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
-\end_inset
+\begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
+\end_inset
,
-\begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
-\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
+\begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
+\end_inset
and
-\begin_inset Formula \( S_{\mathrm{vib}} \)
-\end_inset
+\begin_inset Formula $S_{\mathrm{vib}}$
+\end_inset
.
- See Fig.\SpecialChar ~
+ See Fig.
+\begin_inset space ~
+\end_inset
-\begin_inset LatexCommand \ref{FigVibStab}
-\end_inset
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:VibStabEquation"
+
+\end_inset
for a picture of
-\begin_inset Formula \( S_{\mathrm{vib}} \)
-\end_inset
+\begin_inset Formula $S_{\mathrm{vib}}$
+\end_inset
.
- Regions of secular instability are listed in Table\SpecialChar ~
+ Regions of secular instability are listed in Table
+\begin_inset space ~
+\end_inset
+
1.
-\layout Standard
+\end_layout
+\begin_layout Standard
\begin_inset Float figure
wide false
-collapsed false
+sideways false
+status open
-\layout Caption
+\begin_layout Plain Layout
+\begin_inset Caption
+
+\begin_layout Plain Layout
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:VibStabEquation"
+
+\end_inset
Vibrational stability equation of state
-\begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
-\end_inset
+\begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
+\end_inset
.
-\begin_inset Formula \( >0 \)
-\end_inset
+\begin_inset Formula $>0$
+\end_inset
means vibrational stability
-\layout Standard
+\end_layout
+\end_inset
-\begin_inset LatexCommand \label{FigVibStab}
-\end_inset
+\end_layout
+\end_inset
-\end_inset
-\layout Section
+\end_layout
+\begin_layout Section
Conclusions
-\layout Enumerate
+\end_layout
+\begin_layout Enumerate
The conditions for the stability of static, radiative layers in gas spheres,
as described by Baker's (
-\begin_inset LatexCommand \cite{baker}
+\begin_inset CommandInset citation
+LatexCommand cite
+key "baker"
-\end_inset
+\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
+\end_layout
+\begin_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
+\end_layout
+\begin_layout Enumerate
For solar composition gas the
-\begin_inset Formula \( \kappa \)
-\end_inset
+\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
+\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.
+y than the second He ionization zone that drives the Cepheı̈d pulsations.
-\layout Acknowledgement
+\end_layout
+\begin_layout Acknowledgement
Part of this work was supported by the German
-\emph on
+\emph on
Deut\SpecialChar \-
sche For\SpecialChar \-
schungs\SpecialChar \-
ge\SpecialChar \-
mein\SpecialChar \-
schaft, DFG
-\begin_inset ERT
-status Collapsed
+\emph default
+ project number Ts
+\begin_inset space ~
+\end_inset
+
+17/2--1.
+\end_layout
-\layout Standard
+\begin_layout Standard
+\begin_inset Note Note
+status open
-\backslash
-/{}
-\end_inset
+\begin_layout Plain Layout
+You can alternatively use BibTeX.
+ You must then use the BibTeX style
+\family sans
+aa.bst
+\family default
+ that is part of the A&A LaTeX-package.
+\end_layout
+\end_inset
-\emph default
- project number Ts\SpecialChar ~
-17/2--1.
-
-\layout Bibliography
-\bibitem [1966]{baker}
- Baker, N.
- 1966, in Stellar Evolution, ed.
-\begin_inset ERT
-status Collapsed
+\end_layout
-\layout Standard
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1966"
+key "baker"
-\backslash
-
-\end_inset
+\end_inset
+
+ Baker, N.
+ 1966, in Stellar Evolution, ed.
+\begin_inset space \space{}
+\end_inset
R.
F.
G.
W.
Cameron (Plenum, New York) 333
-\layout Bibliography
-\bibitem [1988]{balluch}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1988"
+key "balluch"
+
+\end_inset
Balluch, M.
1988, A&A, 200, 58
-\layout Bibliography
-\bibitem [1980]{cox}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980"
+key "cox"
+
+\end_inset
Cox, J.
P.
1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
165
-\layout Bibliography
-\bibitem [1969]{cox69}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1969"
+key "cox69"
+
+\end_inset
Cox, A.
N.,& Stewart, J.
N.
1969, Academia Nauk, Scientific Information 15, 1
-\layout Bibliography
-\bibitem [1980]{mizuno}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980"
+key "mizuno"
+
+\end_inset
Mizuno H.
1980, Prog.
Theor.
Phys., 64, 544
-\layout Bibliography
-\bibitem [1987]{tscharnuter}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1987"
+key "tscharnuter"
+
+\end_inset
Tscharnuter W.
M.
1987, A&A, 188, 55
-\layout Bibliography
-\bibitem [1992]{terlevich}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1992"
+key "terlevich"
+
+\end_inset
Terlevich, R.
1992, in ASP Conf.
A.
V.
Filippenko, 13
-\layout Bibliography
-\bibitem [1980a]{yorke80a}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1980a"
+key "yorke80a"
+
+\end_inset
Yorke, H.
W.
1980a, A&A, 86, 286
-\layout Bibliography
-\bibitem [1997]{zheng}
+\end_layout
+
+\begin_layout Bibliography
+\labelwidthstring References
+\begin_inset CommandInset bibitem
+LatexCommand bibitem
+label "1997"
+key "zheng"
+
+\end_inset
Zheng, W., Davidsen, A.
F., Tytler, D.
& Kriss, G.
A.
1997, preprint
-\the_end
+\end_layout
+
+\end_body
+\end_document
projects / lyx.git / blobdiff