@ -62,69 +62,75 @@ acknowledgements:
# Introduction
In the \emph{nucleated instability\/} (also called core
instability) hypothesis of giant planet
formation, a critical mass for static core envelope
protoplanets has been found. \citet{langley00} determined
the critical mass of the core to be about $12 \,M_\oplus$
($M_\oplus=5.975 \times 10^{27}\,\mathrm{g}$ 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.
In the \emph{nucleated instability\/} (also called core
instability) hypothesis of giant planet
formation, a critical mass for static core envelope
protoplanets has been found. \citet{langley00} determined
the critical mass of the core to be about $12 \,M_\oplus$
($M_\oplus=5.975 \times 10^{27}\,\mathrm{g}$ 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.
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
(\citeyear{mitchell80}) standard one-zone model.
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
(\citeyear{mitchell80}) standard one-zone model.
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 \citeyear{kearns89}, Balluch \citeyear{MachineLearningI}),
whereas earlier studies found quasi-steady collapse flows. The
similarities in the (micro)physics, i.e., constitutive relations of
protostellar cores and protogiant planets serve as a further
motivation for this study.
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 \citeyear{kearns89}, Balluch \citeyear{MachineLearningI}),
whereas earlier studies found quasi-steady collapse flows. The
similarities in the (micro)physics, i.e., constitutive relations of
protostellar cores and protogiant planets serve as a further
motivation for this study.
# Baker's standard one-zone model
\begin{figure*}
\centering
\caption{Adiabatic exponent $\Gamma_1$.
$\Gamma_1$ is plotted as a function of
$\lg$ internal energy $\mathrm{[erg\,g^{-1}]}$ and $\lg$
density $\mathrm{[g\,cm^{-3}]}$.}
\label{FigGam}%
\end{figure*}
In this section the one-zone model of \citet{DudaHart2nd},
originally used to study the Cephe{\"{\i}}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.
\citet{DudaHart2nd} investigates the stability of thin layers in
self-gravitating,
spherical gas clouds with the following properties:
\begin{itemize}
\item hydrostatic equilibrium,
\item thermal equilibrium,
\item energy transport by grey radiation diffusion.
\end{itemize}
For the one-zone-model Baker obtains necessary conditions
for dynamical, secular and vibrational (or pulsational)
stability (Eqs.\ (34a,\,b,\,c) in Baker \citeyear{DudaHart2nd}). Using Baker's
notation:
<!-- we need figure* for make it full screen, so it must be in latex code as follows
-->
\begin{figure*}
\centering
\includegraphics{Figure/icml_numpapers.eps}
\caption{Adiabatic exponent $\Gamma_1$.
$\Gamma_1$ is plotted as a function of
$\lg$ internal energy $\mathrm{[erg\,g^{-1}]}$ and $\lg$
density $\mathrm{[g\,cm^{-3}]}$.}
\label{FigGam}%
\end{figure*}
In this section the one-zone model of \citet{DudaHart2nd},
originally used to study the Cephe\text{\"{\i}}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.
\citet{DudaHart2nd} investigates the stability of thin layers in
self-gravitating,
spherical gas clouds with the following properties:
* hydrostatic equilibrium,
* thermal equilibrium,
* energy transport by grey radiation diffusion.
For the one-zone-model Baker obtains necessary conditions
for dynamical, secular and vibrational (or pulsational)
stability (Eqs.\ (34a,\,b,\,c) in Baker \citeyear{DudaHart2nd}). Using Baker's
notation:
\noindent
and with the definitions of the \emph{local cooling time\/}
(see Fig.~\ref{FigGam})
(see Fig. \ref{FigGam})
\begin{equation}
\tau_{\mathrm{co}} = \frac{E_{\mathrm{th}}}{L_{r0}} \,,
\end{equation}
@ -222,7 +228,7 @@ acknowledgements:
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~ \ref{KapSou}), are specified (in practice
and opacities (see Table \ref{KapSou}), 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
@ -302,10 +308,16 @@ acknowledgements:
thermodynamical state of the layer. Therefore the above relations
define the one-zone-stability equations of state
$S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
and $S_{\mathrm{vib}}$. See Fig.~ \ref{FigVibStab} for a picture of
and $S_{\mathrm{vib}}$. See Fig. \ref{FigVibStab} for a picture of
$S_{\mathrm{vib}}$. Regions of secular instability are
listed in Table~1.
listed in Table 1.
<!-- calculate height and width of picture manually in inch, using evince - > Properties
create label for \ref{FigVibStab} using #FigVibStab
-->
{#FigVibStab width=3.43in height=2.71in}
<!--
\begin{figure}
\centering
\caption{Vibrational stability equation of state
@ -314,6 +326,7 @@ acknowledgements:
}
\label{FigVibStab}
\end{figure}
-->
# Conclusions
Hasan al Rasyid
3 years ago