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Hasan al Rasyid 3 years ago
parent 7c02b79255
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      manuscript.md

@ -26,40 +26,18 @@ appendix:
- appendix/2
abstractTex:
\abstract{To investigate the physical nature of the `nuc\-leated instability' of
proto giant planets, the stability of layers
in static, radiative gas spheres is analysed on the basis of Baker's
standard one-zone model.}
proto giant planets, the stability of layers in static, radiative gas spheres is analysed on the basis of Baker's standard one-zone model.}
{To investigate the physical nature of the `nuc\-leated instability' of
proto 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
$-14 \leq \lg \rho / \mathrm{[g\, cm^{-3}]} \leq 0 $,
$ 8.8 \leq \lg e / \mathrm{[erg\, g^{-1}]} \leq 17.7$. 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 $\rho$, temperature $T$ or
specific internal energy $e$.
Regions of instability in the $(\rho,e)$-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 $\kappa$-mechanism is widespread under `cool'
conditions.}
proto 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 $-14 \leq \lg \rho / \mathrm{[g\, cm^{-3}]} \leq 0 $, $ 8.8 \leq \lg e / \mathrm{[erg\, g^{-1}]} \leq 17.7$. 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 $\rho$, temperature $T$ or specific internal energy $e$. Regions of instability in the $(\rho,e)$-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 $\kappa$-mechanism is widespread under `cool' conditions.}
{}
keywords: giant planet formation -- $\kappa$-mechanism -- stability of gas spheres
acknowledgements:
Part of this work was supported by the German
\emph{Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} project
number Ts~17/2--1.
*Deutsche Forschungsgemeinschaft, DFG* project
number Ts 17/2--1.
---
@ -222,7 +200,7 @@ notation:
\end{enumerate}
The first factors, depending on only timescales, are positive
by definition. The signs of the left hand sides of the
inequalities~(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta})
inequalities (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta})
therefore depend exclusively on the second factors containing
the constitutive relations. Since they depend only
on state variables, the stability criteria themselves are \emph{
@ -267,7 +245,7 @@ notation:
(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) and thereby
obtain \emph{stability equations of state}.
The sign determining part of inequality~(\ref{ZSDynSta}) is
The sign determining part of inequality (\ref{ZSDynSta}) is
$3\Gamma_1 - 4$ and it reduces to the
criterion for dynamical stability
\begin{equation}
@ -288,7 +266,7 @@ notation:
\Gamma_1 = \chi_\rho^{} + \chi_T^{} (\Gamma_3 -1)&>&0\\
\nabla_{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0
\end{eqnarray}
we find the sign determining terms in inequalities~(\ref{ZSSecSta})
we find the sign determining terms in inequalities (\ref{ZSSecSta})
and (\ref{ZSVibSta}) respectively and obtain the following form
of the criteria for dynamical, secular and vibrational
\emph{stability}, respectively:

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