-#This file was created by <pit> Mon Nov 24 22:57:48 1997
-#LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team
-\lyxformat 2.15
-\textclass aapaper
+#LyX 1.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 218
+\textclass aa
\language english
\inputencoding latin1
\fontscheme default
\defskip medskip
\quotes_language english
\quotes_times 2
-\papercolumns 1
-\papersides 1
+\papercolumns 2
+\papersides 2
\paperpagestyle default
\layout Thesaurus
\layout Standard
For the one-zone-model Baker obtains necessary conditions for dynamical,
- secular and vibrational (or pulsational) stability [Eqs.
-\protected_separator
+ secular and vibrational (or pulsational) stability [Eqs.\SpecialChar ~
(34a,
\latex latex
].
Using Baker's notation:
-\begin_inset Formula
-\begin{eqnarray*}
+\begin_inset Formula \begin{eqnarray*}
M_{\mathrm{r}} & & \mathrm{mass}\, \mathrm{internal}\, \mathrm{to}\, \mathrm{the}\, \mathrm{radius}\, r\\
m & & \mathrm{mass}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}\\
r_{0} & & \mathrm{unperturbed}\, \mathrm{zone}\, \mathrm{radius}\\
\emph on
local cooling time
\emph default
- (see Fig.
-\protected_separator
+ (see Fig.\SpecialChar ~
\begin_inset LatexCommand \ref{FigGam}
\layout Standard
-\begin_inset Formula
-\begin{equation}
-\label{}
+\begin_inset Formula \begin{equation}
\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
\end{equation}
\layout Standard
-\begin_inset Formula
-\begin{equation}
-\label{}
+\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
have the following form:
-\begin_inset Formula
-\begin{eqnarray}
+\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}
\layout Standard
-\begin_inset Formula
-\begin{equation}
-\label{}
+\begin_inset Formula \begin{equation}
\begin{array}{l}
\delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
e=mc^{2}
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
-\[
+\begin_inset Formula \[
\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\: \chi _{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\: \kappa _{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
\end_inset
\layout Standard
-\begin_inset Formula
-\[
+\begin_inset Formula \[
\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\: \chi _{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\: \kappa _{T}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
\end_inset
\layout Standard
-\begin_inset Formula
-\begin{eqnarray}
+\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}
\layout Standard
The first factors, depending on only timescales, are positive by definition.
- The signs of the left hand sides of the inequalities
-\protected_separator
+ The signs of the left hand sides of the inequalities\SpecialChar ~
(
\begin_inset LatexCommand \ref{ZSDynSta}
The one-zone stability can therefore be determined from a simple equation
of state, given for example, as a function of density and temperature.
Once the microphysics, i.e.
- the thermodynamics and opacities (see Table
-\protected_separator
+ the thermodynamics and opacities (see Table\SpecialChar ~
\begin_inset LatexCommand \ref{KapSou}
\layout Standard
-\align center \LyXTable
-multicol4
-4 2 0 0 -1 -1 -1 -1
-1 0 0 0
-1 0 0 0
-0 0 0 0
-0 1 0 0
-2 0 0
-2 0 0
-0 2 1 0 0 0 0
-0 2 1 0 0 0 0
-0 8 1 0 0 0 0
-0 8 1 0 0 0 0
-0 2 1 0 0 0 0
-0 8 1 0 0 0 0
-0 8 1 0 0 0 0
-0 8 1 0 0 0 0
+
+
+\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="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
Source
-\newline
+\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
+
T/[K]
-\newline
+\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
+
Yorke 1979, Yorke 1980a
-\newline
+\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 \( \leq 1700^{\mathrm{a}} \)
\end_inset
-\newline
+\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
+
Krügel 1971
-\newline
+\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 \( 1700\leq T\leq 5000 \)
\end_inset
-\newline
+\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
-\newline
+\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
-\layout Standard
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+\end_inset
+
+
+\layout Standard
+\added_space_top medskip*
-\begin_inset Formula \( \mathrm{a} \)
+\begin_inset Formula \( ^{\textrm{a}} \)
\end_inset
This is footnote a
.
\layout Standard
-The sign determining part of inequality
-\protected_separator
+The sign determining part of inequality\SpecialChar ~
(
\begin_inset LatexCommand \ref{ZSDynSta}
\layout Standard
-\begin_inset Formula
-\begin{equation}
-\label{}
+\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}
-\label{}
+\begin_inset Formula \begin{equation}
\chi _{\rho }>0,\: \: c_{v}>0\, ,
\end{equation}
\layout Standard
-\begin_inset Formula
-\begin{equation}
-\label{}
+\begin_inset Formula \begin{equation}
\chi _{T}>0
\end{equation}
\layout Standard
-\begin_inset Formula
-\begin{eqnarray}
+\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_inset
-we find the sign determining terms in inequalities
-\protected_separator
+we find the sign determining terms in inequalities\SpecialChar ~
(
\begin_inset LatexCommand \ref{ZSSecSta}
\layout Standard
-\begin_inset Formula
-\begin{eqnarray}
+\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_inset
.
- See Fig.
-\protected_separator
+ See Fig.\SpecialChar ~
\begin_inset LatexCommand \ref{FigVibStab}
\end_inset
.
- Regions of secular instability are listed in Table
-\protected_separator
+ Regions of secular instability are listed in Table\SpecialChar ~
\begin_inset LatexCommand \ref{TabSecInst}
mein\SpecialChar \-
schaft, DFG
\emph default
- project number Ts
-\protected_separator
+ project number Ts\SpecialChar ~
17/2--1.
\layout Bibliography
\bibitem [1966]{baker}
-Baker N., 1966, in: Stellar Evolution, eds.
-\protected_separator
+Baker N., 1966, in: Stellar Evolution, eds.\SpecialChar ~
R.
F.
Stein, A.
G.
W.
- Cameron, Plenum, New York, p.
-\protected_separator
+ Cameron, Plenum, New York, p.\SpecialChar ~
333
\layout Bibliography
\bibitem [1988]{balluch}
Cox J.
P., 1980, Theory of Stellar Pulsation, Princeton University Press, Princeton,
- p.
-\protected_separator
+ p.\SpecialChar ~
165
\layout Bibliography
\bibitem [1969]{cox69}
\bibitem [1971]{kruegel}
Krügel E., 1971, Der Rosselandsche Mittelwert bei tiefen Temperaturen, Diplom--Th
-esis, Univ.
-\protected_separator
+esis, Univ.\SpecialChar ~
Göttingen
\layout Bibliography
\bibitem [1980]{mizuno}
--- /dev/null
+#% Do not delete the line below; configure depends on this
+# \DeclareLaTeXClass[aa]{article (A&A)}
+#
+# Author: Peter Sütterlin <pit@uni-sw.gwdg.de>
+#
+# 2001-05-17 take out include of stdlayouts; merge aapaper.inc
+# remove LaTeX style, rename to aa
+
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projects / features.git / commitdiff