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lyx_mirror/lib/examples/aa_sample.lyx
José Matox 0ed05b32be Update lyx examples to latest file format (for 1.5.0 release) 
git-svn-id: svn://svn.lyx.org/lyx/lyx-devel/trunk@19192 a592a061-630c-0410-9148-cb99ea01b6c8
2007-07-24 17:45:46 +00:00

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#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
\textclass aa
\begin_preamble
\usepackage{graphicx}
%
\end_preamble
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\paperfontsize default
\spacing single
\papersize default
\use_geometry false
\use_amsmath 0
\use_esint 0
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 2
\papersides 2
\paperpagestyle default
\tracking_changes false
\output_changes false
\end_header
\begin_body
\begin_layout Title
Hydrodynamics of giant planet formation
\end_layout
\begin_layout Subtitle
I.
Overviewing the
\begin_inset Formula \( \kappa \)
\end_inset
-mechanism
\end_layout
\begin_layout Author
G.
Wuchterl
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
inst{1}
\backslash
and
\end_layout
\begin_layout Standard
\end_layout
\end_inset
C.
Ptolemy
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
inst{2}
\backslash
fnmsep
\end_layout
\end_inset
\begin_inset Foot
status collapsed
\begin_layout Standard
Just to show the usage of the elements in the author field
\end_layout
\end_inset
\end_layout
\begin_layout Offprint
G.
Wuchterl
\end_layout
\begin_layout Address
Institute for Astronomy (IfA), University of Vienna, Tü
rkenschanzstrasse
17, A-1180 Vienna
\newline
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
email{wuchterl@amok.ast.univie.ac.at}
\backslash
and
\end_layout
\begin_layout Standard
\end_layout
\end_inset
University of Alexandria, Department of Geography, ...
\newline
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
email{c.ptolemy@hipparch.uheaven.space}
\end_layout
\end_inset
\begin_inset Foot
status collapsed
\begin_layout Standard
The university of heaven temporarily does not accept e-mails
\end_layout
\end_inset
\end_layout
\begin_layout Date
Received September 15, 1996; accepted March 16, 1997
\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
key "mizuno"
\end_inset
), the stability of layers in static, radiative gas spheres is analysed
on the basis of Baker's
\begin_inset LatexCommand cite
key "baker"
\end_inset
standard one-zone model.
It is shown that stability depends only upon the equations of state, the
opacities and the local thermodynamic state in the layer.
Stability and instability can therefore be expressed in the form of stability
equations of state which are universal for a given composition.
The stability equations of state are calculated for solar composition and
are displayed in the domain
\begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
\end_inset
,
\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
\end_inset
.
These displays may be used to determine the one-zone stability of layers
in stellar or planetary structure models by directly reading off the value
of the stability equations for the thermodynamic state of these layers,
specified by state quantities as density
\begin_inset Formula \( \rho \)
\end_inset
, temperature
\begin_inset Formula \( T \)
\end_inset
or specific internal energy
\begin_inset Formula \( e \)
\end_inset
.
Regions of instability in the
\begin_inset Formula \( (\rho ,e) \)
\end_inset
-plane are described and related to the underlying microphysical processes.
Vibrational instability is found to be a common phenomenon at temperatures
lower than the second He ionisation zone.
The
\begin_inset Formula \( \kappa \)
\end_inset
-mechanism is widespread under `cool' conditions.
\begin_inset ERT
status collapsed
\begin_layout Standard
\end_layout
\begin_layout Standard
\backslash
keywords{giant planet formation --
\backslash
(
\backslash
kappa
\backslash
)-mechanism -- stability of gas spheres }
\end_layout
\end_inset
\end_layout
\begin_layout Section
Introduction
\end_layout
\begin_layout Standard
In the
\emph default
\emph on
nucleated instability
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
/{}
\end_layout
\end_inset
\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
key "mizuno"
\end_inset
) determined the critical mass of the core to be about
\begin_inset Formula \( 12\, M_{\oplus } \)
\end_inset
(
\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
\end_inset
is the Earth mass), which is independent of the outer boundary conditions
and therefore independent of the location in the solar nebula.
This critical value for the core mass corresponds closely to the cores
of today's giant planets.
\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 main motivation for this article is to investigate the stability of
the static envelope at the critical mass.
With this aim the local, linear stability of static radiative gas spheres
is investigated on the basis of Baker's (
\begin_inset LatexCommand cite
key "baker"
\end_inset
) standard one-zone model.
\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
key "tscharnuter"
\end_inset
, Balluch
\begin_inset LatexCommand cite
key "balluch"
\end_inset
), whereas earlier studies found quasi-steady collapse flows.
The similarities in the (micro)physics, i.e., constitutive relations of protostel
lar cores and protogiant planets serve as a further motivation for this
study.
\end_layout
\begin_layout Section
Baker's standard one-zone model
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide true
sideways false
status open
\begin_layout
\begin_inset Caption
\begin_layout
Adiabatic exponent
\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
internal energy
\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
\end_inset
and
\begin_inset Formula \( \lg \)
\end_inset
density
\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset LatexCommand label
name "FigGam"
\end_inset
\end_layout
\end_inset
In this section the one-zone model of Baker (
\begin_inset LatexCommand cite
key "baker"
\end_inset
), originally used to study the Cepheı̈
d pulsation mechanism, will be briefly
reviewed.
The resulting stability criteria will be rewritten in terms of local state
variables, local timescales and constitutive relations.
\end_layout
\begin_layout Standard
Baker (
\begin_inset LatexCommand cite
key "baker"
\end_inset
) investigates the stability of thin layers in self-gravitating, spherical
gas clouds with the following properties:
\end_layout
\begin_layout Itemize
hydrostatic equilibrium,
\end_layout
\begin_layout Itemize
thermal equilibrium,
\end_layout
\begin_layout Itemize
energy transport by grey radiation diffusion.
\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
\begin_layout Standard
\backslash
\end_layout
\end_inset
(34a,
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
,
\end_layout
\end_inset
b,
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
,
\end_layout
\end_inset
c) in Baker
\begin_inset LatexCommand cite
key "baker"
\end_inset
).
Using Baker's notation:
\end_layout
\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}\\
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_layout
\begin_layout Standard
\noindent
and with the definitions of the
\emph default
\emph on
local cooling time
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
/{}
\end_layout
\end_inset
\emph default
(see Fig.\InsetSpace ~
\begin_inset LatexCommand ref
reference "FigGam"
\end_inset
)
\begin_inset Formula \begin{equation}
\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
\end{equation}
\end_inset
and the
\emph default
\emph on
local free-fall time
\emph default
\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
Baker's
\begin_inset Formula \( K \)
\end_inset
and
\begin_inset Formula \( \sigma _{0} \)
\end_inset
have the following form:
\begin_inset Formula \begin{eqnarray}
\sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
\end{eqnarray}
\end_inset
where
\begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
\end_inset
has been used and
\begin_inset Formula \begin{equation}
\begin{array}{l}
\delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
e=mc^{2}
\end{array}
\end{equation}
\end_inset
is a thermodynamical quantity which is of order
\begin_inset Formula \( 1 \)
\end_inset
and equal to
\begin_inset Formula \( 1 \)
\end_inset
for nonreacting mixtures of classical perfect gases.
The physical meaning of
\begin_inset Formula \( \sigma _{0} \)
\end_inset
and
\begin_inset Formula \( K \)
\end_inset
is clearly visible in the equations above.
\begin_inset Formula \( \sigma _{0} \)
\end_inset
represents a frequency of the order one per free-fall time.
\begin_inset Formula \( K \)
\end_inset
is proportional to the ratio of the free-fall time and the cooling time.
Substituting into Baker's criteria, using thermodynamic identities and
definitions of thermodynamic quantities,
\begin_inset Formula \[
\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
\end_inset
\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
one obtains, after some pages of algebra, the conditions for
\emph default
\emph on
stability
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
/{}
\end_layout
\end_inset
\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}
\end{eqnarray}
\end_inset
For a physical discussion of the stability criteria see Baker (
\begin_inset LatexCommand cite
key "baker"
\end_inset
) or Cox (
\begin_inset LatexCommand cite
key "cox"
\end_inset
).
\end_layout
\begin_layout Standard
We observe that these criteria for dynamical, secular and vibrational stability,
respectively, can be factorized into
\end_layout
\begin_layout Enumerate
a factor containing local timescales only,
\end_layout
\begin_layout Enumerate
a factor containing only constitutive relations and their derivatives.
\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\InsetSpace ~
(
\begin_inset LatexCommand ref
reference "ZSDynSta"
\end_inset
), (
\begin_inset LatexCommand ref
reference "ZSSecSta"
\end_inset
) and (
\begin_inset LatexCommand ref
reference "ZSVibSta"
\end_inset
) therefore depend exclusively on the second factors containing the constitutive
relations.
Since they depend only on state variables, the stability criteria themselves
are
\emph default
\emph on
functions of the thermodynamic state in the local zone
\emph default
.
The one-zone stability can therefore be determined from a simple equation
of state, given for example, as a function of density and temperature.
Once the microphysics, i.e.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
\end_layout
\end_inset
the thermodynamics and opacities (see Table\InsetSpace ~
\begin_inset LatexCommand ref
reference "KapSou"
\end_inset
), are specified (in practice by specifying a chemical composition) the
one-zone stability can be inferred if the thermodynamic state is specified.
The zone -- or in other words the layer -- will be stable or unstable in
whatever object it is imbedded as long as it satisfies the one-zone-model
assumptions.
Only the specific growth rates (depending upon the time scales) will be
different for layers in different objects.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout
\begin_inset Caption
\begin_layout
\begin_inset LatexCommand label
name "KapSou"
\end_inset
Opacity sources
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="2">
<features>
<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">
\begin_inset Text
\begin_layout Standard
Source
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\begin_inset Formula \( T/[\textrm{K}] \)
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row topline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
Yorke 1979, Yorke 1980a
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\begin_inset Formula \( \leq 1700^{\textrm{a}} \)
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
Krügel 1971
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\begin_inset Formula \( 1700\leq T\leq 5000 \)
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row bottomline="true">
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
Cox & Stewart 1969
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Standard
\begin_inset Formula \( 5000\leq \)
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula \( ^{\textrm{a}} \)
\end_inset
This is footnote a
\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
reference "ZSDynSta"
\end_inset
), (
\begin_inset LatexCommand ref
reference "ZSSecSta"
\end_inset
) and (
\begin_inset LatexCommand ref
reference "ZSVibSta"
\end_inset
) and thereby obtain
\emph default
\emph on
stability equations of state
\emph default
.
\end_layout
\begin_layout Standard
The sign determining part of inequality\InsetSpace ~
(
\begin_inset LatexCommand ref
reference "ZSDynSta"
\end_inset
) is
\begin_inset Formula \( 3\Gamma _{1}-4 \)
\end_inset
and it reduces to the criterion for dynamical stability
\begin_inset Formula \begin{equation}
\Gamma _{1}>\frac{4}{3}\, \cdot
\end{equation}
\end_inset
Stability of the thermodynamical equilibrium demands
\begin_inset Formula \begin{equation}
\chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
\end{equation}
\end_inset
and
\begin_inset Formula \begin{equation}
\chi ^{}_{T}>0
\end{equation}
\end_inset
holds for a wide range of physical situations.
With
\begin_inset Formula \begin{eqnarray}
\Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
\Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
\nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
\end{eqnarray}
\end_inset
we find the sign determining terms in inequalities\InsetSpace ~
(
\begin_inset LatexCommand ref
reference "ZSSecSta"
\end_inset
) and (
\begin_inset LatexCommand ref
reference "ZSVibSta"
\end_inset
) respectively and obtain the following form of the criteria for dynamical,
secular and vibrational
\emph default
\emph on
stability
\emph default
, respectively:
\begin_inset Formula \begin{eqnarray}
3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
\frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
\end{eqnarray}
\end_inset
The constitutive relations are to be evaluated for the unperturbed thermodynami
c state (say
\begin_inset Formula \( (\rho _{0},T_{0}) \)
\end_inset
) of the zone.
We see that the one-zone stability of the layer depends only on the constitutiv
e relations
\begin_inset Formula \( \Gamma _{1} \)
\end_inset
,
\begin_inset Formula \( \nabla _{\mathrm{ad}} \)
\end_inset
,
\begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
\end_inset
,
\begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
\end_inset
.
These depend only on the unperturbed thermodynamical state of the layer.
Therefore the above relations define the one-zone-stability equations of
state
\begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
\end_inset
and
\begin_inset Formula \( S_{\mathrm{vib}} \)
\end_inset
.
See Fig.\InsetSpace ~
\begin_inset LatexCommand ref
reference "FigVibStab"
\end_inset
for a picture of
\begin_inset Formula \( S_{\mathrm{vib}} \)
\end_inset
.
Regions of secular instability are listed in Table\InsetSpace ~
1.
\end_layout
\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open
\begin_layout
\begin_inset Caption
\begin_layout
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
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset LatexCommand label
name "FigVibStab"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Section
Conclusions
\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
key "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.
\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.
\end_layout
\begin_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 Cepheı̈
d pulsations.
\end_layout
\begin_layout Acknowledgement
Part of this work was supported by the German
\emph default
\emph on
Deut\SpecialChar \-
sche For\SpecialChar \-
schungs\SpecialChar \-
ge\SpecialChar \-
mein\SpecialChar \-
schaft, DFG
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
/{}
\end_layout
\end_inset
\emph default
project number Ts\InsetSpace ~
17/2--1.
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1966"
key "baker"
\end_inset
Baker, N.
1966, in Stellar Evolution, ed.
\begin_inset ERT
status collapsed
\begin_layout Standard
\backslash
\end_layout
\end_inset
R.
F.
Stein,& A.
G.
W.
Cameron (Plenum, New York) 333
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1988"
key "balluch"
\end_inset
Balluch, M.
1988, A&A, 200, 58
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1980"
key "cox"
\end_inset
Cox, J.
P.
1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
165
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1969"
key "cox69"
\end_inset
Cox, A.
N.,& Stewart, J.
N.
1969, Academia Nauk, Scientific Information 15, 1
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1980"
key "mizuno"
\end_inset
Mizuno H.
1980, Prog.
Theor.
Phys., 64, 544
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1987"
key "tscharnuter"
\end_inset
Tscharnuter W.
M.
1987, A&A, 188, 55
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1992"
key "terlevich"
\end_inset
Terlevich, R.
1992, in ASP Conf.
Ser.
31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
ed.
A.
V.
Filippenko, 13
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1980a"
key "yorke80a"
\end_inset
Yorke, H.
W.
1980a, A&A, 86, 286
\end_layout
\begin_layout Bibliography
\begin_inset LatexCommand bibitem
label "1997"
key "zheng"
\end_inset
Zheng, W., Davidsen, A.
F., Tytler, D.
& Kriss, G.
A.
1997, preprint
\end_layout
\end_body
\end_document
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