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#LyX 2.2 created this file. For more info see http://www.lyx.org/
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2016-01-12 21:30:37 +00:00
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\lyxformat 504
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2005-07-18 09:42:29 +00:00
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\begin_document
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\begin_header
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\save_transient_properties true
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\origin /systemlyxdir/examples/
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\textclass aa
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\language english
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\index_command default
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\paperfontsize default
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\use_hyperref false
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\papersize default
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\cite_engine basic
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\cite_engine_type default
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\biblio_style plain
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\use_bibtopic false
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\use_indices false
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\paperorientation portrait
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\suppress_date false
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\justification true
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\use_refstyle 0
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\secnumdepth 3
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\tocdepth 3
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\quotes_language english
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\papercolumns 2
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\papersides 2
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\paperpagestyle default
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\tracking_changes false
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\output_changes false
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\html_math_output 0
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\html_be_strict false
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\end_header
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\begin_body
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\begin_layout Title
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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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\family roman
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\series medium
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\size normal
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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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http://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_layout
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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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\end_layout
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\end_inset
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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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\end_layout
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\end_inset
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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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fnmsep
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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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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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\end_layout
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\end_inset
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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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\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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\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_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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1999-09-27 18:44:28 +00:00
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leated instability' of proto
|
2008-10-28 22:29:02 +00:00
|
|
|
|
giant planets, the stability of layers in static, radiative gas spheres
|
|
|
|
|
is analysed on the basis of Baker's standard one-zone model.
|
1999-09-27 18:44:28 +00:00
|
|
|
|
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.
|
2001-06-13 13:45:51 +00:00
|
|
|
|
The stability equations of state are calculated for solar composition and
|
1999-09-27 18:44:28 +00:00
|
|
|
|
are displayed in the domain
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
,
|
2008-10-01 20:26:35 +00:00
|
|
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|
\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
|
2005-07-18 09:42:29 +00:00
|
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|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
|
|
|
|
|
|
.
|
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|
|
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
|
2008-10-01 20:26:35 +00:00
|
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|
\begin_inset Formula $\rho$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
|
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, temperature
|
2008-10-01 20:26:35 +00:00
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\begin_inset Formula $T$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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or specific internal energy
|
2008-10-01 20:26:35 +00:00
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|
\begin_inset Formula $e$
|
2005-07-18 09:42:29 +00:00
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
|
.
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Regions of instability in the
|
2008-10-01 20:26:35 +00:00
|
|
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|
\begin_inset Formula $(\rho,e)$
|
2005-07-18 09:42:29 +00:00
|
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|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
-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.
|
|
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|
|
The
|
2008-10-01 20:26:35 +00:00
|
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|
|
\begin_inset Formula $\kappa$
|
2005-07-18 09:42:29 +00:00
|
|
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
|
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|
|
-mechanism is widespread under `cool' conditions.
|
2008-10-28 22:29:02 +00:00
|
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|
\begin_inset Note Note
|
2008-10-25 22:46:25 +00:00
|
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|
status open
|
2001-06-13 13:45:51 +00:00
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|
2008-10-01 20:26:35 +00:00
|
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|
|
\begin_layout Plain Layout
|
2008-10-28 22:29:02 +00:00
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|
Citations are not allowed in A&A abstracts.
|
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|
\end_layout
|
2008-10-01 20:26:35 +00:00
|
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|
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|
2008-10-28 22:29:02 +00:00
|
|
|
|
\end_inset
|
2005-07-18 09:42:29 +00:00
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|
2008-10-28 22:29:02 +00:00
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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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This is the unstructured abstract type, an example for the structured abstract
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|
is in the
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|
\family sans
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aa.lyx
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\family default
|
2016-01-13 01:51:42 +00:00
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template file that comes with \SpecialChar LyX
|
2015-11-13 18:14:45 +00:00
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.
|
2005-07-18 09:42:29 +00:00
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\end_layout
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\end_inset
|
2003-01-17 13:50:11 +00:00
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2008-10-28 22:29:02 +00:00
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\end_layout
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\begin_layout Keywords
|
2016-01-13 01:51:42 +00:00
|
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|
giant planet formation –
|
2008-10-28 22:29:02 +00:00
|
|
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|
\begin_inset Formula $\kappa$
|
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|
\end_inset
|
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|
2016-01-13 01:51:42 +00:00
|
|
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|
-mechanism – stability of gas spheres
|
2005-07-18 09:42:29 +00:00
|
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|
\end_layout
|
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|
\begin_layout Section
|
1999-09-27 18:44:28 +00:00
|
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Introduction
|
2005-07-18 09:42:29 +00:00
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|
\end_layout
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\begin_layout Standard
|
2008-10-01 20:26:35 +00:00
|
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|
In the
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph on
|
1999-09-27 18:44:28 +00:00
|
|
|
|
nucleated instability
|
2005-07-18 09:42:29 +00:00
|
|
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|
\emph default
|
1999-09-27 18:44:28 +00:00
|
|
|
|
(also called core instability) hypothesis of giant planet formation, a
|
|
|
|
|
critical mass for static core envelope protoplanets has been found.
|
|
|
|
|
Mizuno (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
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|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Eisenstein2005"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) determined the critical mass of the core to be about
|
2016-01-13 01:51:42 +00:00
|
|
|
|
\begin_inset Formula $12\,M_{\oplus}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
(
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
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.
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
1999-09-27 18:44:28 +00:00
|
|
|
|
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 (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Abernethy2003"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) standard one-zone model.
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
1999-09-27 18:44:28 +00:00
|
|
|
|
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
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Cotton1999"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
, Balluch
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Mena2000"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
), whereas earlier studies found quasi-steady collapse flows.
|
2008-10-28 23:39:29 +00:00
|
|
|
|
The similarities in the (micro)physics, i.
|
|
|
|
|
\begin_inset space \thinspace{}
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
g.
|
2008-10-25 22:46:25 +00:00
|
|
|
|
\begin_inset space \space{}
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
constitutive relations of protostellar cores and protogiant planets serve
|
|
|
|
|
as a further motivation for this study.
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Section
|
1999-09-27 18:44:28 +00:00
|
|
|
|
Baker's standard one-zone model
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
2003-01-17 13:50:11 +00:00
|
|
|
|
\begin_inset Float figure
|
|
|
|
|
wide true
|
2005-07-18 09:42:29 +00:00
|
|
|
|
sideways false
|
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|
|
|
status open
|
2003-01-17 13:50:11 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_layout Plain Layout
|
2013-04-08 06:58:33 +00:00
|
|
|
|
\begin_inset Caption Standard
|
2007-07-24 17:45:46 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_layout Plain Layout
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset CommandInset label
|
|
|
|
|
LatexCommand label
|
|
|
|
|
name "fig:FigGam"
|
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|
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|
|
|
|
\end_inset
|
|
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|
|
1999-09-27 18:44:28 +00:00
|
|
|
|
Adiabatic exponent
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $\Gamma_{1}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
.
|
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|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $\Gamma_{1}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
is plotted as a function of
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $\lg$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
internal energy
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
and
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $\lg$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
density
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
2005-07-18 09:42:29 +00:00
|
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|
\end_layout
|
|
|
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|
|
2007-07-24 17:45:46 +00:00
|
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|
\end_inset
|
|
|
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|
2001-06-13 13:45:51 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2003-01-17 13:50:11 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
In this section the one-zone model of Baker (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Abernethy2003"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
|
1999-09-27 18:44:28 +00:00
|
|
|
|
reviewed.
|
|
|
|
|
The resulting stability criteria will be rewritten in terms of local state
|
|
|
|
|
variables, local timescales and constitutive relations.
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
1999-09-27 18:44:28 +00:00
|
|
|
|
Baker (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Abernethy2003"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) investigates the stability of thin layers in self-gravitating, spherical
|
2001-06-13 13:45:51 +00:00
|
|
|
|
gas clouds with the following properties:
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Itemize
|
2001-06-13 13:45:51 +00:00
|
|
|
|
hydrostatic equilibrium,
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Itemize
|
2001-06-13 13:45:51 +00:00
|
|
|
|
thermal equilibrium,
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Itemize
|
1999-09-27 18:44:28 +00:00
|
|
|
|
energy transport by grey radiation diffusion.
|
2001-06-13 13:45:51 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Standard
|
|
|
|
|
\noindent
|
1999-09-27 18:44:28 +00:00
|
|
|
|
For the one-zone-model Baker obtains necessary conditions for dynamical,
|
2001-06-13 13:45:51 +00:00
|
|
|
|
secular and vibrational (or pulsational) stability (Eqs.
|
2008-10-25 22:46:25 +00:00
|
|
|
|
\begin_inset space \space{}
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2003-01-17 13:50:11 +00:00
|
|
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|
1999-09-27 18:44:28 +00:00
|
|
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|
(34a,
|
2008-10-25 22:46:25 +00:00
|
|
|
|
\begin_inset space \thinspace{}
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2003-01-17 13:50:11 +00:00
|
|
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1999-09-27 18:44:28 +00:00
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b,
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2008-10-25 22:46:25 +00:00
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\begin_inset space \thinspace{}
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2005-07-18 09:42:29 +00:00
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2003-01-17 13:50:11 +00:00
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1999-09-27 18:44:28 +00:00
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c) in Baker
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2008-10-01 20:26:35 +00:00
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\begin_inset CommandInset citation
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LatexCommand cite
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2013-05-26 15:55:10 +00:00
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key "Abernethy2003"
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2008-10-01 20:26:35 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
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).
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1999-09-27 18:44:28 +00:00
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Using Baker's notation:
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2015-11-13 18:14:45 +00:00
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\begin_inset Separator parbreak
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\end_inset
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2016-01-13 01:51:42 +00:00
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\end_layout
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\begin_layout Standard
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\align left
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2011-06-15 01:02:15 +00:00
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\begin_inset Formula
|
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\begin{eqnarray*}
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2001-06-13 13:45:51 +00:00
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M_{r} & & \textrm{mass internal to the radius }r\\
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m & & \textrm{mass of the zone}\\
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r_{0} & & \textrm{unperturbed zone radius}\\
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2008-10-01 20:26:35 +00:00
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\rho_{0} & & \textrm{unperturbed density in the zone}\\
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2001-06-13 13:45:51 +00:00
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T_{0} & & \textrm{unperturbed temperature in the zone}\\
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L_{r0} & & \textrm{unperturbed luminosity}\\
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2011-06-15 01:02:15 +00:00
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E_{\textrm{th}} & & \textrm{thermal energy of the zone}
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\end{eqnarray*}
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1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\begin_layout Standard
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\noindent
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2008-10-01 20:26:35 +00:00
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and with the definitions of the
|
2005-07-18 09:42:29 +00:00
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\emph on
|
1999-09-27 18:44:28 +00:00
|
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local cooling time
|
2005-07-18 09:42:29 +00:00
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\emph default
|
2008-10-01 20:26:35 +00:00
|
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(see Fig.
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\begin_inset space ~
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\end_inset
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1999-09-27 18:44:28 +00:00
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2008-10-01 20:26:35 +00:00
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\begin_inset CommandInset ref
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LatexCommand ref
|
2011-06-15 01:02:15 +00:00
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reference "fig:FigGam"
|
2008-10-01 20:26:35 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
|
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|
)
|
2011-06-15 01:02:15 +00:00
|
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|
\begin_inset Formula
|
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|
\begin{equation}
|
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|
\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
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\end{equation}
|
1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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2008-10-01 20:26:35 +00:00
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and the
|
2005-07-18 09:42:29 +00:00
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\emph on
|
1999-09-27 18:44:28 +00:00
|
|
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local free-fall time
|
2005-07-18 09:42:29 +00:00
|
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\emph default
|
1999-09-27 18:44:28 +00:00
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|
2011-06-15 01:02:15 +00:00
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\begin_inset Formula
|
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\begin{equation}
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\tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
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|
\end{equation}
|
1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
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Baker's
|
2008-10-01 20:26:35 +00:00
|
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\begin_inset Formula $K$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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and
|
2008-10-01 20:26:35 +00:00
|
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|
\begin_inset Formula $\sigma_{0}$
|
2005-07-18 09:42:29 +00:00
|
|
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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2001-06-13 13:45:51 +00:00
|
|
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|
have the following form:
|
2011-06-15 01:02:15 +00:00
|
|
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|
\begin_inset Formula
|
|
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|
|
\begin{eqnarray}
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
|
2011-06-15 01:02:15 +00:00
|
|
|
|
K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
|
|
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|
|
\end{eqnarray}
|
1999-09-27 18:44:28 +00:00
|
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|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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|
2001-06-13 13:45:51 +00:00
|
|
|
|
where
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
|
2005-07-18 09:42:29 +00:00
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
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|
has been used and
|
2011-06-15 01:02:15 +00:00
|
|
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|
\begin_inset Formula
|
|
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|
\begin{equation}
|
1999-09-27 18:44:28 +00:00
|
|
|
|
\begin{array}{l}
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
|
2011-06-15 01:02:15 +00:00
|
|
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|
e=mc^{2}
|
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\end{array}
|
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|
\end{equation}
|
1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
2001-06-13 13:45:51 +00:00
|
|
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|
is a thermodynamical quantity which is of order
|
2008-10-01 20:26:35 +00:00
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|
\begin_inset Formula $1$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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and equal to
|
2008-10-01 20:26:35 +00:00
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\begin_inset Formula $1$
|
2005-07-18 09:42:29 +00:00
|
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\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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for nonreacting mixtures of classical perfect gases.
|
|
|
|
|
The physical meaning of
|
2008-10-01 20:26:35 +00:00
|
|
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|
\begin_inset Formula $\sigma_{0}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
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and
|
2008-10-01 20:26:35 +00:00
|
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\begin_inset Formula $K$
|
2005-07-18 09:42:29 +00:00
|
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
is clearly visible in the equations above.
|
|
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|
2008-10-01 20:26:35 +00:00
|
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|
\begin_inset Formula $\sigma_{0}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
represents a frequency of the order one per free-fall time.
|
|
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|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $K$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
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|
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|
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,
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
\Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}
|
|
|
|
|
\]
|
1999-09-27 18:44:28 +00:00
|
|
|
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|
2005-07-18 09:42:29 +00:00
|
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
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|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\[
|
|
|
|
|
\nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}
|
|
|
|
|
\]
|
1999-09-27 18:44:28 +00:00
|
|
|
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|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
one obtains, after some pages of algebra, the conditions for
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph on
|
1999-09-27 18:44:28 +00:00
|
|
|
|
stability
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph default
|
2001-06-13 13:45:51 +00:00
|
|
|
|
given below:
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\begin{eqnarray}
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\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}\\
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\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}
|
1999-09-27 18:44:28 +00:00
|
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2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
For a physical discussion of the stability criteria see Baker (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Abernethy2003"
|
2008-10-01 20:26:35 +00:00
|
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2005-07-18 09:42:29 +00:00
|
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
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|
) or Cox (
|
2008-10-01 20:26:35 +00:00
|
|
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|
\begin_inset CommandInset citation
|
|
|
|
|
LatexCommand cite
|
2013-05-26 15:55:10 +00:00
|
|
|
|
key "Parkin2005"
|
2008-10-01 20:26:35 +00:00
|
|
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
|
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|
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).
|
2005-07-18 09:42:29 +00:00
|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Standard
|
1999-09-27 18:44:28 +00:00
|
|
|
|
We observe that these criteria for dynamical, secular and vibrational stability,
|
2001-06-13 13:45:51 +00:00
|
|
|
|
respectively, can be factorized into
|
2005-07-18 09:42:29 +00:00
|
|
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|
\end_layout
|
|
|
|
|
|
|
|
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|
\begin_layout Enumerate
|
2001-06-13 13:45:51 +00:00
|
|
|
|
a factor containing local timescales only,
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
|
|
|
|
|
|
\begin_layout Enumerate
|
1999-09-27 18:44:28 +00:00
|
|
|
|
a factor containing only constitutive relations and their derivatives.
|
2001-06-13 13:45:51 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
|
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|
|
|
|
|
\begin_layout Standard
|
1999-09-27 18:44:28 +00:00
|
|
|
|
The first factors, depending on only timescales, are positive by definition.
|
2008-10-01 20:26:35 +00:00
|
|
|
|
The signs of the left hand sides of the inequalities
|
|
|
|
|
\begin_inset space ~
|
|
|
|
|
\end_inset
|
|
|
|
|
|
1999-09-27 18:44:28 +00:00
|
|
|
|
(
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSDynSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
), (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSSecSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) and (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSVibSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) therefore depend exclusively on the second factors containing the constitutive
|
|
|
|
|
relations.
|
|
|
|
|
Since they depend only on state variables, the stability criteria themselves
|
2008-10-01 20:26:35 +00:00
|
|
|
|
are
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph on
|
2001-06-13 13:45:51 +00:00
|
|
|
|
functions of the thermodynamic state in the local zone
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph default
|
1999-09-27 18:44:28 +00:00
|
|
|
|
.
|
|
|
|
|
The one-zone stability can therefore be determined from a simple equation
|
|
|
|
|
of state, given for example, as a function of density and temperature.
|
2008-10-28 23:39:29 +00:00
|
|
|
|
Once the microphysics, i.
|
|
|
|
|
\begin_inset space \thinspace{}
|
|
|
|
|
\end_inset
|
|
|
|
|
|
|
|
|
|
g.
|
2008-10-25 22:46:25 +00:00
|
|
|
|
\begin_inset space \space{}
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2003-01-17 13:50:11 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
the thermodynamics and opacities (see Table
|
|
|
|
|
\begin_inset space ~
|
|
|
|
|
\end_inset
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1999-09-27 18:44:28 +00:00
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2008-10-01 20:26:35 +00:00
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\begin_inset CommandInset ref
|
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LatexCommand ref
|
2011-06-15 01:02:15 +00:00
|
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reference "tab:KapSou"
|
2008-10-01 20:26:35 +00:00
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|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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), are specified (in practice by specifying a chemical composition) the
|
|
|
|
|
one-zone stability can be inferred if the thermodynamic state is specified.
|
2016-01-13 01:51:42 +00:00
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|
|
The zone – or in other words the layer – will be stable or unstable in
|
1999-09-27 18:44:28 +00:00
|
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|
whatever object it is imbedded as long as it satisfies the one-zone-model
|
|
|
|
|
assumptions.
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|
|
Only the specific growth rates (depending upon the time scales) will be
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|
|
different for layers in different objects.
|
2005-07-18 09:42:29 +00:00
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|
\end_layout
|
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\begin_layout Standard
|
2003-01-17 13:50:11 +00:00
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\begin_inset Float table
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wide false
|
2005-07-18 09:42:29 +00:00
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sideways false
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status open
|
2003-01-17 13:50:11 +00:00
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|
2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
|
2013-04-08 06:58:33 +00:00
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\begin_inset Caption Standard
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1999-09-27 18:44:28 +00:00
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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LatexCommand label
|
2011-06-15 01:02:15 +00:00
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|
name "tab:KapSou"
|
2008-10-01 20:26:35 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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2001-06-13 13:45:51 +00:00
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Opacity sources
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2005-07-18 09:42:29 +00:00
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\end_layout
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2007-07-24 17:45:46 +00:00
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\end_inset
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\end_layout
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
|
2011-06-15 01:02:15 +00:00
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\align center
|
2008-10-01 20:26:35 +00:00
|
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|
\begin_inset Tabular
|
2003-01-17 13:50:11 +00:00
|
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|
<lyxtabular version="3" rows="4" columns="2">
|
2016-01-13 01:51:42 +00:00
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|
<features tabularvalignment="middle">
|
2003-01-17 13:50:11 +00:00
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<column alignment="left" valignment="top" width="0pt">
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<column alignment="left" valignment="top" width="0pt">
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2008-10-01 20:26:35 +00:00
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<row>
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<cell alignment="center" valignment="top" topline="true" usebox="none">
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2001-05-22 13:10:12 +00:00
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\begin_inset Text
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
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1999-09-27 18:44:28 +00:00
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Source
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2005-07-18 09:42:29 +00:00
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\end_layout
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\end_inset
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2001-05-22 13:10:12 +00:00
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</cell>
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2008-10-01 20:26:35 +00:00
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<cell alignment="center" valignment="top" topline="true" usebox="none">
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2001-05-22 13:10:12 +00:00
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\begin_inset Text
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
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\begin_inset Formula $T/[\textrm{K}]$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\end_inset
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2001-05-22 13:10:12 +00:00
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</cell>
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</row>
|
2008-10-01 20:26:35 +00:00
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<row>
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<cell alignment="center" valignment="top" topline="true" usebox="none">
|
2001-05-22 13:10:12 +00:00
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\begin_inset Text
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
|
1999-09-27 18:44:28 +00:00
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Yorke 1979, Yorke 1980a
|
2005-07-18 09:42:29 +00:00
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\end_layout
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\end_inset
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2001-05-22 13:10:12 +00:00
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</cell>
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2008-10-01 20:26:35 +00:00
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<cell alignment="center" valignment="top" topline="true" usebox="none">
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2001-05-22 13:10:12 +00:00
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\begin_inset Text
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|
2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
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\begin_inset Formula $\leq1700^{\textrm{a}}$
|
2005-07-18 09:42:29 +00:00
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|
\end_inset
|
1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\end_inset
|
2001-05-22 13:10:12 +00:00
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</cell>
|
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|
|
|
</row>
|
2003-01-17 13:50:11 +00:00
|
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|
|
<row>
|
2008-10-01 20:26:35 +00:00
|
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|
<cell alignment="center" valignment="top" usebox="none">
|
2001-05-22 13:10:12 +00:00
|
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|
\begin_inset Text
|
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|
2008-10-01 20:26:35 +00:00
|
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|
\begin_layout Plain Layout
|
2007-07-24 17:45:46 +00:00
|
|
|
|
Krügel 1971
|
2005-07-18 09:42:29 +00:00
|
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|
|
\end_layout
|
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\end_inset
|
2001-05-22 13:10:12 +00:00
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|
</cell>
|
2008-10-01 20:26:35 +00:00
|
|
|
|
<cell alignment="center" valignment="top" usebox="none">
|
2001-05-22 13:10:12 +00:00
|
|
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|
\begin_inset Text
|
|
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|
2008-10-01 20:26:35 +00:00
|
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|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $1700\leq T\leq5000$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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|
\end_layout
|
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|
\end_inset
|
2001-05-22 13:10:12 +00:00
|
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|
</cell>
|
|
|
|
|
</row>
|
2008-10-01 20:26:35 +00:00
|
|
|
|
<row>
|
|
|
|
|
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
2001-05-22 13:10:12 +00:00
|
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|
\begin_inset Text
|
|
|
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|
2008-10-01 20:26:35 +00:00
|
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|
\begin_layout Plain Layout
|
1999-09-27 18:44:28 +00:00
|
|
|
|
Cox & Stewart 1969
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
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|
\end_inset
|
2001-05-22 13:10:12 +00:00
|
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|
</cell>
|
2008-10-01 20:26:35 +00:00
|
|
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|
<cell alignment="center" valignment="top" bottomline="true" usebox="none">
|
2001-05-22 13:10:12 +00:00
|
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|
\begin_inset Text
|
|
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|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $5000\leq$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
|
|
|
|
|
1999-09-27 18:44:28 +00:00
|
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|
2005-07-18 09:42:29 +00:00
|
|
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|
\end_layout
|
2001-06-13 13:45:51 +00:00
|
|
|
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|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2001-05-22 13:10:12 +00:00
|
|
|
|
</cell>
|
|
|
|
|
</row>
|
|
|
|
|
</lyxtabular>
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
2001-05-22 13:10:12 +00:00
|
|
|
|
|
|
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|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
|
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|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_layout Plain Layout
|
|
|
|
|
\begin_inset Formula $^{\textrm{a}}$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
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|
2001-06-13 13:45:51 +00:00
|
|
|
|
This is footnote a
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
|
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|
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|
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|
\end_inset
|
2003-01-17 13:50:11 +00:00
|
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|
2001-06-13 13:45:51 +00:00
|
|
|
|
We will now write down the sign (and therefore stability) determining parts
|
1999-09-27 18:44:28 +00:00
|
|
|
|
of the left-hand sides of the inequalities (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSDynSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
), (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSSecSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) and (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSVibSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
) and thereby obtain
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph on
|
1999-09-27 18:44:28 +00:00
|
|
|
|
stability equations of state
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph default
|
1999-09-27 18:44:28 +00:00
|
|
|
|
.
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_layout
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\begin_layout Standard
|
2008-10-01 20:26:35 +00:00
|
|
|
|
The sign determining part of inequality
|
|
|
|
|
\begin_inset space ~
|
|
|
|
|
\end_inset
|
2005-07-18 09:42:29 +00:00
|
|
|
|
|
1999-09-27 18:44:28 +00:00
|
|
|
|
(
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSDynSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) is
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset Formula $3\Gamma_{1}-4$
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
and it reduces to the criterion for dynamical stability
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\Gamma_{1}>\frac{4}{3}\,\cdot
|
|
|
|
|
\end{equation}
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
Stability of the thermodynamical equilibrium demands
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\begin{equation}
|
2016-01-13 01:51:42 +00:00
|
|
|
|
\chi_{\rho}^{}>0,\;\;c_{v}>0\,,
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\end{equation}
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
and
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\begin{equation}
|
|
|
|
|
\chi_{T}^{}>0
|
|
|
|
|
\end{equation}
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2001-06-13 13:45:51 +00:00
|
|
|
|
holds for a wide range of physical situations.
|
|
|
|
|
With
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\begin_inset Formula
|
|
|
|
|
\begin{eqnarray}
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
|
|
|
|
|
\Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
|
2011-06-15 01:02:15 +00:00
|
|
|
|
\nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
|
|
|
|
|
\end{eqnarray}
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
2008-10-01 20:26:35 +00:00
|
|
|
|
we find the sign determining terms in inequalities
|
|
|
|
|
\begin_inset space ~
|
|
|
|
|
\end_inset
|
|
|
|
|
|
1999-09-27 18:44:28 +00:00
|
|
|
|
(
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSSecSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) and (
|
2008-10-01 20:26:35 +00:00
|
|
|
|
\begin_inset CommandInset ref
|
|
|
|
|
LatexCommand ref
|
2007-07-24 17:45:46 +00:00
|
|
|
|
reference "ZSVibSta"
|
2008-10-01 20:26:35 +00:00
|
|
|
|
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\end_inset
|
1999-09-27 18:44:28 +00:00
|
|
|
|
|
|
|
|
|
) respectively and obtain the following form of the criteria for dynamical,
|
2008-10-01 20:26:35 +00:00
|
|
|
|
secular and vibrational
|
2005-07-18 09:42:29 +00:00
|
|
|
|
\emph on
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1999-09-27 18:44:28 +00:00
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stability
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\emph default
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, respectively:
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\begin_inset Formula
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\begin{eqnarray}
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2008-10-01 20:26:35 +00:00
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3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
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\frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
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2011-06-15 01:02:15 +00:00
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4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
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\end{eqnarray}
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1999-09-27 18:44:28 +00:00
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\end_inset
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2001-06-13 13:45:51 +00:00
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The constitutive relations are to be evaluated for the unperturbed thermodynami
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c state (say
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2008-10-01 20:26:35 +00:00
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\begin_inset Formula $(\rho_{0},T_{0})$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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) of the zone.
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We see that the one-zone stability of the layer depends only on the constitutiv
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e relations
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2008-10-01 20:26:35 +00:00
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\begin_inset Formula $\Gamma_{1}$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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,
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2008-10-01 20:26:35 +00:00
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\begin_inset Formula $\nabla_{\mathrm{ad}}$
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\end_inset
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1999-09-27 18:44:28 +00:00
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,
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\begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
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\end_inset
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1999-09-27 18:44:28 +00:00
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,
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2008-10-01 20:26:35 +00:00
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\begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
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\end_inset
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1999-09-27 18:44:28 +00:00
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.
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These depend only on the unperturbed thermodynamical state of the layer.
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Therefore the above relations define the one-zone-stability equations of
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state
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2016-01-13 01:51:42 +00:00
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\begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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and
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\begin_inset Formula $S_{\mathrm{vib}}$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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.
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See Fig.
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\begin_inset space ~
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\end_inset
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2008-10-01 20:26:35 +00:00
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\begin_inset CommandInset ref
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LatexCommand ref
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2011-06-15 01:02:15 +00:00
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reference "fig:VibStabEquation"
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2008-10-01 20:26:35 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
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for a picture of
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2008-10-01 20:26:35 +00:00
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\begin_inset Formula $S_{\mathrm{vib}}$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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.
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2008-10-01 20:26:35 +00:00
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Regions of secular instability are listed in Table
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\begin_inset space ~
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\end_inset
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2001-06-13 13:45:51 +00:00
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1.
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\end_layout
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\begin_layout Standard
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2003-01-17 13:50:11 +00:00
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\begin_inset Float figure
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wide false
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sideways false
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status open
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2003-01-17 13:50:11 +00:00
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\begin_layout Plain Layout
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2013-04-08 06:58:33 +00:00
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\begin_inset Caption Standard
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2007-07-24 17:45:46 +00:00
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2008-10-01 20:26:35 +00:00
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\begin_layout Plain Layout
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\begin_inset CommandInset label
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LatexCommand label
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name "fig:VibStabEquation"
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\end_inset
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1999-09-27 18:44:28 +00:00
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Vibrational stability equation of state
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\begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
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2005-07-18 09:42:29 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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.
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\begin_inset Formula $>0$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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means vibrational stability
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\end_layout
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2007-07-24 17:45:46 +00:00
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\end_inset
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2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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1999-09-27 18:44:28 +00:00
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2005-07-18 09:42:29 +00:00
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\end_inset
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2008-10-01 20:26:35 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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2003-01-17 13:50:11 +00:00
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\begin_layout Section
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Conclusions
|
2005-07-18 09:42:29 +00:00
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\end_layout
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\begin_layout Enumerate
|
1999-09-27 18:44:28 +00:00
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|
The conditions for the stability of static, radiative layers in gas spheres,
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as described by Baker's (
|
2008-10-01 20:26:35 +00:00
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\begin_inset CommandInset citation
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LatexCommand cite
|
2013-05-26 15:55:10 +00:00
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key "Abernethy2003"
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2008-10-01 20:26:35 +00:00
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\end_inset
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1999-09-27 18:44:28 +00:00
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) standard one-zone model, can be expressed as stability equations of state.
|
|
|
|
|
These stability equations of state depend only on the local thermodynamic
|
|
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|
|
state of the layer.
|
2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\begin_layout Enumerate
|
2016-01-13 01:51:42 +00:00
|
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If the constitutive relations – equations of state and Rosseland mean opacities
|
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|
|
– are specified, the stability equations of state can be evaluated without
|
1999-09-27 18:44:28 +00:00
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|
specifying properties of the layer.
|
2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\begin_layout Enumerate
|
1999-09-27 18:44:28 +00:00
|
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|
For solar composition gas the
|
2008-10-01 20:26:35 +00:00
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\begin_inset Formula $\kappa$
|
2005-07-18 09:42:29 +00:00
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\end_inset
|
1999-09-27 18:44:28 +00:00
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|
-mechanism is working in the regions of the ice and dust features in the
|
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|
opacities, the
|
2008-10-01 20:26:35 +00:00
|
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|
\begin_inset Formula $\mathrm{H}_{2}$
|
2005-07-18 09:42:29 +00:00
|
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\end_inset
|
1999-09-27 18:44:28 +00:00
|
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|
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
|
2008-10-01 20:26:35 +00:00
|
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|
y than the second He ionization zone that drives the Cepheı̈d pulsations.
|
2001-06-13 13:45:51 +00:00
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2005-07-18 09:42:29 +00:00
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\end_layout
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\begin_layout Acknowledgement
|
2008-10-01 20:26:35 +00:00
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|
Part of this work was supported by the German
|
2005-07-18 09:42:29 +00:00
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\emph on
|
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Deut\SpecialChar softhyphen
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sche For\SpecialChar softhyphen
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schungs\SpecialChar softhyphen
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ge\SpecialChar softhyphen
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mein\SpecialChar softhyphen
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1999-09-27 18:44:28 +00:00
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schaft, DFG
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\emph default
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2008-10-01 20:26:35 +00:00
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project number Ts
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\begin_inset space ~
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\end_inset
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2016-01-13 01:51:42 +00:00
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17/2–1.
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2008-10-28 22:29:02 +00:00
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\end_layout
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\begin_layout Standard
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2013-05-26 15:55:10 +00:00
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\begin_inset CommandInset bibtex
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LatexCommand bibtex
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btprint "btPrintAll"
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bibfiles "biblioExample"
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options "aa"
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\end_inset
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2008-10-28 22:29:02 +00:00
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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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2013-05-26 15:55:10 +00:00
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\series bold
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Note:
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\series default
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|
If you cannot see the bibliography in the output, assure that you have
|
2016-01-13 01:51:42 +00:00
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|
gievn the full path to the Bib\SpecialChar TeX
|
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style file
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\family sans
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aa.bst
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\family default
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that is part of the A&A \SpecialChar LaTeX
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-package.
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2008-10-28 22:29:02 +00:00
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\end_layout
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\end_inset
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2005-07-18 09:42:29 +00:00
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\end_layout
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\end_body
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\end_document
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