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1262 lines
24 KiB
Plaintext
#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\lyxformat 544
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\begin_document
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\begin_header
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\origin /systemlyxdir/examples/
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status open
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\begin_layout Plain Layout
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\family roman
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This is an example \SpecialChar LyX
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file for articles to be submitted to the Journal of
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Astronomy & Astrophysics (A&A).
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How to install the A&A \SpecialChar LaTeX
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class to your \SpecialChar LaTeX
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system is explained in
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\begin_inset Flex URL
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status open
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\begin_layout Plain Layout
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https://wiki.lyx.org/Layouts/Astronomy-Astrophysics
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\end_layout
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\end_inset
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.
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\begin_inset Newline newline
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\end_inset
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Depending on the submission state and the abstract layout, you need to use
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different document class options that are listed in the aa manual.
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\family default
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\begin_inset Newline newline
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\end_inset
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\family roman
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\series default
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Note:
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\series medium
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If you use accented characters in your document, you must use the predefined
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document class option
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\series default
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latin9
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\series medium
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in the document settings.
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\end_inset
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\end_layout
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\begin_layout Title
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Hydrodynamics of giant planet formation
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\end_layout
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\begin_layout Subtitle
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I.
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Overviewing the
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\begin_inset Formula $\kappa$
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\end_inset
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-mechanism
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\end_layout
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\begin_layout Author
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G.
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Wuchterl
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\begin_inset Flex institutemark
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status open
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\begin_layout Plain Layout
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1
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\begin_inset ERT
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status collapsed
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\begin_layout Plain Layout
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\backslash
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and
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\end_layout
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\end_inset
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C.
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Ptolemy
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\begin_inset Flex institutemark
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status collapsed
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\begin_layout Plain Layout
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2
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\begin_inset ERT
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status collapsed
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\backslash
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fnmsep
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\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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Just to show the usage of the elements in the author field
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\end_layout
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\end_inset
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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\backslash
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fnmsep is only needed for more than one consecutive notes/marks
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Offprint
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G.
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Wuchterl
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\end_layout
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\begin_layout Address
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Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
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17, A-1180 Vienna
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\begin_inset Newline newline
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\end_inset
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\begin_inset Flex Email
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status open
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\begin_layout Plain Layout
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wuchterl@amok.ast.univie.ac.at
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\begin_inset ERT
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status collapsed
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\begin_layout Plain Layout
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\backslash
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and
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\end_layout
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\end_inset
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University of Alexandria, Department of Geography, ...
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\begin_inset Newline newline
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\end_inset
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\begin_inset Flex Email
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status collapsed
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\begin_layout Plain Layout
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c.ptolemy@hipparch.uheaven.space
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\end_layout
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\end_inset
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\begin_inset Foot
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status collapsed
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\begin_layout Plain Layout
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The university of heaven temporarily does not accept e-mails
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Date
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Received September 15, 1996; accepted March 16, 1997
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\end_layout
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\begin_layout Abstract (unstructured)
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To investigate the physical nature of the `nuc\SpecialChar softhyphen
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leated instability' of proto
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giant planets, the stability of layers in static, radiative gas spheres
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is analysed on the basis of Baker's standard one-zone model.
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It is shown that stability depends only upon the equations of state, the
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opacities and the local thermodynamic state in the layer.
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Stability and instability can therefore be expressed in the form of stability
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||
equations of state which are universal for a given composition.
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The stability equations of state are calculated for solar composition and
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are displayed in the domain
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||
\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
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\end_inset
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,
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\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
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\end_inset
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.
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These displays may be used to determine the one-zone stability of layers
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||
in stellar or planetary structure models by directly reading off the value
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||
of the stability equations for the thermodynamic state of these layers,
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||
specified by state quantities as density
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||
\begin_inset Formula $\rho$
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\end_inset
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, temperature
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\begin_inset Formula $T$
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\end_inset
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or specific internal energy
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\begin_inset Formula $e$
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\end_inset
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.
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Regions of instability in the
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\begin_inset Formula $(\rho,e)$
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\end_inset
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-plane are described and related to the underlying microphysical processes.
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Vibrational instability is found to be a common phenomenon at temperatures
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||
lower than the second He ionisation zone.
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||
The
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\begin_inset Formula $\kappa$
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\end_inset
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-mechanism is widespread under `cool' conditions.
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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Citations are not allowed in A&A abstracts.
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\end_layout
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\end_inset
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\begin_inset Note Note
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status open
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|
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\begin_layout Plain Layout
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This is the unstructured abstract type, an example for the structured abstract
|
||
is in the
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||
\family sans
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||
aa.lyx
|
||
\family default
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||
template file that comes with \SpecialChar LyX
|
||
.
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||
\end_layout
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||
\end_inset
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||
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||
\end_layout
|
||
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||
\begin_layout Keywords
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||
giant planet formation –
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\begin_inset Formula $\kappa$
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\end_inset
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-mechanism – stability of gas spheres
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\end_layout
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\begin_layout Section
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||
Introduction
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||
\end_layout
|
||
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||
\begin_layout Standard
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||
In the
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||
\emph on
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||
nucleated instability
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||
\emph default
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||
(also called core instability) hypothesis of giant planet formation, a
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||
critical mass for static core envelope protoplanets has been found.
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||
Mizuno (
|
||
\begin_inset CommandInset citation
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||
LatexCommand cite
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||
key "Eisenstein2005"
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||
literal "true"
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||
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||
\end_inset
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||
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||
) determined the critical mass of the core to be about
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||
\begin_inset Formula $12\,M_{\oplus}$
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||
\end_inset
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||
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||
(
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||
\begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
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||
\end_inset
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||
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||
is the Earth mass), which is independent of the outer boundary conditions
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||
and therefore independent of the location in the solar nebula.
|
||
This critical value for the core mass corresponds closely to the cores
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||
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
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||
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
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||
is investigated on the basis of Baker's (
|
||
\begin_inset CommandInset citation
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||
LatexCommand cite
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||
key "Abernethy2003"
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||
literal "true"
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||
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||
\end_inset
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||
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||
) standard one-zone model.
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||
\end_layout
|
||
|
||
\begin_layout Standard
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||
Phenomena similar to the ones described above for giant planet formation
|
||
have been found in hydrodynamical models concerning star formation where
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||
protostellar cores explode (Tscharnuter
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "Cotton1999"
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||
literal "true"
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||
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||
\end_inset
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||
|
||
, Balluch
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||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "Mena2000"
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||
literal "true"
|
||
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||
\end_inset
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||
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||
), whereas earlier studies found quasi-steady collapse flows.
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||
The similarities in the (micro)physics, i.
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||
\begin_inset space \thinspace{}
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||
\end_inset
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||
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||
g.
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||
\begin_inset space \space{}
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||
\end_inset
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||
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||
constitutive relations of protostellar cores and protogiant planets serve
|
||
as a further motivation for this study.
|
||
\end_layout
|
||
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||
\begin_layout Section
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||
Baker's standard one-zone model
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||
\end_layout
|
||
|
||
\begin_layout Standard
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||
\begin_inset Float figure
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||
wide true
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||
sideways false
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status open
|
||
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||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
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||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
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||
LatexCommand label
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name "fig:FigGam"
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||
\end_inset
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||
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||
Adiabatic exponent
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||
\begin_inset Formula $\Gamma_{1}$
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||
\end_inset
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||
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||
.
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||
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||
\begin_inset Formula $\Gamma_{1}$
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||
\end_inset
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||
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||
is plotted as a function of
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||
\begin_inset Formula $\lg$
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||
\end_inset
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||
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||
internal energy
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||
\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
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||
\end_inset
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||
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||
and
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||
\begin_inset Formula $\lg$
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||
\end_inset
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||
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density
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\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
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\end_inset
|
||
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||
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||
\end_layout
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||
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||
\end_inset
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||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
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||
|
||
In this section the one-zone model of Baker (
|
||
\begin_inset CommandInset citation
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||
LatexCommand cite
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||
key "Abernethy2003"
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||
literal "true"
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||
|
||
\end_inset
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||
|
||
), originally used to study the Cepheïd pulsation mechanism, will be briefly
|
||
reviewed.
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||
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 CommandInset citation
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||
LatexCommand cite
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||
key "Abernethy2003"
|
||
literal "true"
|
||
|
||
\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 space \space{}
|
||
\end_inset
|
||
|
||
(34a,
|
||
\begin_inset space \thinspace{}
|
||
\end_inset
|
||
|
||
b,
|
||
\begin_inset space \thinspace{}
|
||
\end_inset
|
||
|
||
c) in Baker
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "Abernethy2003"
|
||
literal "true"
|
||
|
||
\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 on
|
||
local cooling time
|
||
\emph default
|
||
(see Fig.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:FigGam"
|
||
|
||
\end_inset
|
||
|
||
)
|
||
\begin_inset Formula
|
||
\begin{equation}
|
||
\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
|
||
\end{equation}
|
||
|
||
\end_inset
|
||
|
||
and the
|
||
\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 on
|
||
stability
|
||
\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 CommandInset citation
|
||
LatexCommand cite
|
||
key "Abernethy2003"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
) or Cox (
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Parkin2005"
|
||
literal "true"
|
||
|
||
\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
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSDynSta"
|
||
|
||
\end_inset
|
||
|
||
), (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSSecSta"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
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 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.
|
||
\begin_inset space \thinspace{}
|
||
\end_inset
|
||
|
||
g.
|
||
\begin_inset space \space{}
|
||
\end_inset
|
||
|
||
the thermodynamics and opacities (see Table
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "tab: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 Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset CommandInset label
|
||
LatexCommand label
|
||
name "tab:KapSou"
|
||
|
||
\end_inset
|
||
|
||
Opacity sources
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\align center
|
||
\begin_inset Tabular
|
||
<lyxtabular version="3" rows="4" columns="2">
|
||
<features tabularvalignment="middle">
|
||
<column alignment="left" valignment="top" width="0pt">
|
||
<column alignment="left" valignment="top" width="0pt">
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Source
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $T/[\textrm{K}]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Yorke 1979, Yorke 1980a
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" topline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $\leq1700^{\textrm{a}}$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Krügel 1971
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $1700\leq T\leq5000$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
<row>
|
||
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
Cox & Stewart 1969
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
||
\begin_inset Text
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Formula $5000\leq$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
</cell>
|
||
</row>
|
||
</lyxtabular>
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Plain Layout
|
||
\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 CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSDynSta"
|
||
|
||
\end_inset
|
||
|
||
), (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSSecSta"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSVibSta"
|
||
|
||
\end_inset
|
||
|
||
) and thereby obtain
|
||
\emph on
|
||
stability equations of state
|
||
\emph default
|
||
.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
The sign determining part of inequality
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset CommandInset ref
|
||
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
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSSecSta"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSVibSta"
|
||
|
||
\end_inset
|
||
|
||
) respectively and obtain the following form of the criteria for dynamical,
|
||
secular and vibrational
|
||
\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.
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "fig:VibStabEquation"
|
||
|
||
\end_inset
|
||
|
||
for a picture of
|
||
\begin_inset Formula $S_{\mathrm{vib}}$
|
||
\end_inset
|
||
|
||
.
|
||
Regions of secular instability are listed in Table
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
1.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
|
||
wide false
|
||
sideways false
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
\begin_inset Caption Standard
|
||
|
||
\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
|
||
|
||
.
|
||
|
||
\begin_inset Formula $>0$
|
||
\end_inset
|
||
|
||
means vibrational stability
|
||
\end_layout
|
||
|
||
\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 CommandInset citation
|
||
LatexCommand cite
|
||
key "Abernethy2003"
|
||
literal "true"
|
||
|
||
\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 Cepheïd pulsations.
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Acknowledgement
|
||
Part of this work was supported by the German
|
||
\emph on
|
||
Deut\SpecialChar softhyphen
|
||
sche For\SpecialChar softhyphen
|
||
schungs\SpecialChar softhyphen
|
||
ge\SpecialChar softhyphen
|
||
mein\SpecialChar softhyphen
|
||
schaft, DFG
|
||
\emph default
|
||
project number Ts
|
||
\begin_inset space ~
|
||
\end_inset
|
||
|
||
17/2–1.
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset CommandInset bibtex
|
||
LatexCommand bibtex
|
||
btprint "btPrintAll"
|
||
bibfiles "biblioExample"
|
||
options "aa"
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\series bold
|
||
Note:
|
||
\series default
|
||
If you cannot see the bibliography in the output, assure that you have
|
||
given the full path to the Bib\SpecialChar TeX
|
||
style file
|
||
\family sans
|
||
aa.bst
|
||
\family default
|
||
that is part of the A&A \SpecialChar LaTeX
|
||
-package.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_body
|
||
\end_document
|