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-\lyxformat 341
+#LyX 2.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 413
\begin_document
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\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
+
+.
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+
+
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+
+\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
\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
\backslash
-inst{1}
-\backslash
-and
+and
\end_layout
-\begin_layout Plain Layout
+\end_inset
-
+ C.
+ Ptolemy
+\begin_inset Flex institutemark
+status collapsed
+
+\begin_layout Plain Layout
+2
\end_layout
\end_inset
-C.
- Ptolemy
+
\begin_inset ERT
status collapsed
\backslash
-inst{2}
-\backslash
-fnmsep
+fnmsep
\end_layout
\end_inset
\end_inset
+\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
\end_inset
-\begin_inset ERT
-status collapsed
+\begin_inset Flex Email
+status open
\begin_layout Plain Layout
+wuchterl@amok.ast.univie.ac.at
+\end_layout
+\end_inset
-\backslash
-email{wuchterl@amok.ast.univie.ac.at}
-\backslash
-and
-\end_layout
+
+\begin_inset ERT
+status collapsed
\begin_layout Plain Layout
+
+\backslash
+and
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\end_inset
\end_inset
-\begin_inset ERT
+\begin_inset Flex Email
status collapsed
\begin_layout Plain Layout
-
-
-\backslash
-email{c.ptolemy@hipparch.uheaven.space}
+c.ptolemy@hipparch.uheaven.space
\end_layout
\end_inset
-
+
\begin_inset Foot
status collapsed
\begin_layout Abstract
To investigate the physical nature of the `nuc\SpecialChar \-
leated instability' of proto
- giant planets (Mizuno
-\begin_inset CommandInset citation
-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 CommandInset citation
-LatexCommand cite
-key "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
\end_inset
-mechanism is widespread under `cool' conditions.
-\begin_inset ERT
-status collapsed
+\begin_inset Note Note
+status open
\begin_layout Plain Layout
-
+Citations are not allowed in A&A abstracts.
\end_layout
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
\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
-\backslash
-keywords{giant planet formation --
-\backslash
-(
-\backslash
-kappa
-\backslash
-)-mechanism -- stability of gas spheres }
\end_layout
+\begin_layout Keywords
+giant planet formation --
+\begin_inset Formula $\kappa$
\end_inset
-
+-mechanism -- stability of gas spheres
\end_layout
\begin_layout Section
In the
\emph on
nucleated instability
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\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.
\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.
+ 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
\begin_inset 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
\end_inset
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "FigGam"
-
-\end_inset
-
-
\end_layout
\end_inset
\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 Plain Layout
-
-
-\backslash
-
-\end_layout
-
+\begin_inset space \space{}
\end_inset
(34a,
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-,
-\end_layout
-
+\begin_inset space \thinspace{}
\end_inset
b,
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-,
-\end_layout
-
+\begin_inset space \thinspace{}
\end_inset
c) in Baker
\begin_layout Standard
\align left
-\begin_inset Formula \begin{eqnarray*}
+\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*}
+E_{\textrm{th}} & & \textrm{thermal energy of the zone}
+\end{eqnarray*}
\end_inset
and with the definitions of the
\emph on
local cooling time
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-/{}
-\end_layout
-
-\end_inset
-
-
\emph default
(see Fig.
\begin_inset space ~
\begin_inset CommandInset ref
LatexCommand ref
-reference "FigGam"
+reference "fig:FigGam"
\end_inset
)
-\begin_inset Formula \begin{equation}
-\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
+\begin_inset Formula
+\begin{equation}
+\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
+\end{equation}
\end_inset
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}
+\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
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}
+K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
+\end{eqnarray}
\end_inset
\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}\\
-e=mc^{2}\end{array}\end{equation}
+e=mc^{2}
+\end{array}
+\end{equation}
\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
-\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
one obtains, after some pages of algebra, the conditions for
\emph on
stability
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-/{}
-\end_layout
-
-\end_inset
-
-
\emph default
given below:
-\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}\end{eqnarray}
+\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
.
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 Plain Layout
-
-
-\backslash
-
-\end_layout
+ Once the microphysics, i.
+\begin_inset space \thinspace{}
+\end_inset
+g.
+\begin_inset space \space{}
\end_inset
the thermodynamics and opacities (see Table
\begin_inset CommandInset ref
LatexCommand ref
-reference "KapSou"
+reference "tab:KapSou"
\end_inset
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
-name "KapSou"
+name "tab:KapSou"
\end_inset
\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>
\end_inset
and it reduces to the criterion for dynamical stability
-\begin_inset Formula \begin{equation}
-\Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
+\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}
+\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}
+\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}
+\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}
+\nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
+\end{eqnarray}
\end_inset
stability
\emph default
, respectively:
-\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{eqnarray}
+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
\begin_inset CommandInset ref
LatexCommand ref
-reference "FigVibStab"
+reference "fig:VibStabEquation"
\end_inset
\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
\end_inset
-\end_layout
-
-\begin_layout Plain Layout
-\begin_inset CommandInset label
-LatexCommand label
-name "FigVibStab"
-
-\end_inset
-
-
\end_layout
\end_inset
ge\SpecialChar \-
mein\SpecialChar \-
schaft, DFG
-\begin_inset ERT
-status collapsed
+\emph default
+ project number Ts
+\begin_inset space ~
+\end_inset
-\begin_layout Plain Layout
+17/2--1.
+\end_layout
+\begin_layout Standard
+\begin_inset Note Note
+status open
-\backslash
-/{}
+\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
-\begin_inset space ~
-\end_inset
-
-17/2--1.
-
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1966"
Baker, N.
1966, in Stellar Evolution, ed.
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-
-\end_layout
-
+\begin_inset space \space{}
\end_inset
R.
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1988"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1980"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1969"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1980"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1987"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1992"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1980a"
\end_layout
\begin_layout Bibliography
+\labelwidthstring References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "1997"