mirror of
https://git.lyx.org/repos/lyx.git
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a15303b938
Especially after the change to use semantic linefeeds (7b23c76b),
the diffs are large and it's hard to figure out what diff is the
result of the linefeed change and which diff is the result of an
edit.
By updating the docs, it will make the edits easier to understand
from the diff.
This commit used the LyX binary to write the new .lyx files since
lyx2lyx does not apply semantic linefeeds.
I used the following command:
./development/tools/updatedocs.py [path to 'lyx' binary]
A few documents were not updated (e.g., the command sequence used
failed because a dialog about a missing dependency was shown).
I did not update Additional.lyx for any language since it is
undergoing changes. I also didn't change anything in doc/fr since JP
is working on some of those. I also didn't change doc/ru since it
appears Yuriy already updated those.
1306 lines
25 KiB
Plaintext
1306 lines
25 KiB
Plaintext
#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\lyxformat 612
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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/Articles/
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\textclass aa
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\use_default_options true
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\bibtex_command bibtex
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\index_command default
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\paperfontsize default
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\spacing single
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\use_hyperref false
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\papersize default
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\use_geometry false
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\use_package undertilde 1
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\cite_engine natbib
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\cite_engine_type authoryear
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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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\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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\paragraph_separation indent
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\paragraph_indentation default
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\end_header
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|
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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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|
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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 Astronomy & Astrophysics (A&A).
|
||
How to install the A&A \SpecialChar LaTeX
|
||
class to your \SpecialChar LaTeX
|
||
system is explained in
|
||
\begin_inset Flex URL
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||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
|
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https://wiki.lyx.org/Layouts/Astronomy-Astrophysics
|
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\end_layout
|
||
|
||
\end_inset
|
||
|
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.
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
Depending on the submission state and the abstract layout,
|
||
you need to use different document class options that are listed in the aa manual.
|
||
\family default
|
||
|
||
\begin_inset Newline newline
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\end_inset
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|
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|
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\family roman
|
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\series default
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Note:
|
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|
||
\series medium
|
||
If you use accented characters in your document,
|
||
you must use the predefined document class option
|
||
\series default
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||
latin9
|
||
\series medium
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||
in the document settings.
|
||
\end_layout
|
||
|
||
\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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||
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\begin_layout Plain Layout
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||
|
||
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\backslash
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||
and
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||
\end_layout
|
||
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||
\end_inset
|
||
|
||
C.
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||
Ptolemy
|
||
\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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||
|
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\begin_inset ERT
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status collapsed
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|
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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
|
||
|
||
\begin_layout Plain Layout
|
||
Just to show the usage of the elements in the author field
|
||
\end_layout
|
||
|
||
\end_inset
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|
||
|
||
\begin_inset Note Note
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||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
\backslash
|
||
fnmsep is only needed for more than one consecutive notes/marks
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Offprint
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||
G.
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||
Wuchterl
|
||
\end_layout
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||
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||
\begin_layout Address
|
||
Institute for Astronomy (IfA),
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||
University of Vienna,
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||
Türkenschanzstrasse 17,
|
||
A-1180 Vienna
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Flex Email
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||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
wuchterl@amok.ast.univie.ac.at
|
||
\end_layout
|
||
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||
\end_inset
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||
|
||
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||
\begin_inset ERT
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
|
||
|
||
\backslash
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||
and
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
University of Alexandria,
|
||
Department of Geography,
|
||
...
|
||
\begin_inset Newline newline
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Flex Email
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
c.ptolemy@hipparch.uheaven.space
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Foot
|
||
status collapsed
|
||
|
||
\begin_layout Plain Layout
|
||
The university of heaven temporarily does not accept e-mails
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Date
|
||
Received September 15,
|
||
1996;
|
||
accepted March 16,
|
||
1997
|
||
\end_layout
|
||
|
||
\begin_layout Abstract (unstructured)
|
||
To investigate the physical nature of the `nuc\SpecialChar softhyphen
|
||
leated instability' of proto giant planets,
|
||
the stability of layers in static,
|
||
radiative gas spheres is analysed on the basis of Baker's standard one-zone model.
|
||
It is shown that stability depends only upon the equations of state,
|
||
the opacities and the local thermodynamic state in the layer.
|
||
Stability and instability can therefore be expressed in the form of stability equations of state which are universal for a given composition.
|
||
The stability equations of state are calculated for solar composition and are displayed in the domain
|
||
\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
|
||
\end_inset
|
||
|
||
,
|
||
|
||
\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
|
||
\end_inset
|
||
|
||
.
|
||
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
|
||
\begin_inset Formula $\rho$
|
||
\end_inset
|
||
|
||
,
|
||
temperature
|
||
\begin_inset Formula $T$
|
||
\end_inset
|
||
|
||
or specific internal energy
|
||
\begin_inset Formula $e$
|
||
\end_inset
|
||
|
||
.
|
||
Regions of instability in the
|
||
\begin_inset Formula $(\rho,e)$
|
||
\end_inset
|
||
|
||
-plane are described and related to the underlying microphysical processes.
|
||
Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone.
|
||
The
|
||
\begin_inset Formula $\kappa$
|
||
\end_inset
|
||
|
||
-mechanism is widespread under `cool' conditions.
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
Citations are not allowed in A&A abstracts.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\begin_inset Note Note
|
||
status open
|
||
|
||
\begin_layout Plain Layout
|
||
This is the unstructured abstract type,
|
||
an example for the structured abstract is in the
|
||
\family sans
|
||
aa.lyx
|
||
\family default
|
||
template file that comes with \SpecialChar LyX
|
||
.
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\begin_layout Keywords
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||
giant planet formation –
|
||
\begin_inset Formula $\kappa$
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||
\end_inset
|
||
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||
-mechanism – stability of gas spheres
|
||
\end_layout
|
||
|
||
\begin_layout Section
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||
Introduction
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
In the
|
||
\emph on
|
||
nucleated instability
|
||
\emph default
|
||
(also called core instability) hypothesis of giant planet formation,
|
||
a critical mass for static core envelope protoplanets has been found.
|
||
Mizuno (
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "Eisenstein2005"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
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||
) determined the critical mass of the core to be about
|
||
\begin_inset Formula $12\,M_{\oplus}$
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||
\end_inset
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||
|
||
(
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||
\begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
|
||
\end_inset
|
||
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||
is the Earth mass),
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||
which is independent of the outer boundary conditions and therefore independent of the location in the solar nebula.
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||
This critical value for the core mass corresponds closely to the cores 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 critical mass.
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||
The main motivation for this article is to investigate the stability of the static envelope at the critical mass.
|
||
With this aim the local,
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||
linear stability of static radiative gas spheres is investigated on the basis of Baker's (
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Abernethy2003"
|
||
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 protostellar cores explode (Tscharnuter
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Cotton1999"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
,
|
||
Balluch
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
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||
key "Mena2000"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
),
|
||
whereas earlier studies found quasi-steady collapse flows.
|
||
The similarities in the (micro)physics,
|
||
i.
|
||
\begin_inset space \thinspace{}
|
||
\end_inset
|
||
|
||
g.
|
||
\begin_inset space \space{}
|
||
\end_inset
|
||
|
||
constitutive relations of protostellar cores and protogiant planets serve as a further motivation for this study.
|
||
\end_layout
|
||
|
||
\begin_layout Section
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Baker's standard one-zone model
|
||
\end_layout
|
||
|
||
\begin_layout Standard
|
||
\begin_inset Float figure
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||
placement document
|
||
alignment document
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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
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||
\begin_inset CommandInset label
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||
LatexCommand label
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||
name "fig:FigGam"
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||
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||
\end_inset
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||
|
||
Adiabatic exponent
|
||
\begin_inset Formula $\Gamma_{1}$
|
||
\end_inset
|
||
|
||
.
|
||
|
||
\begin_inset Formula $\Gamma_{1}$
|
||
\end_inset
|
||
|
||
is plotted as a function of
|
||
\begin_inset Formula $\lg$
|
||
\end_inset
|
||
|
||
internal energy
|
||
\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
|
||
\end_inset
|
||
|
||
and
|
||
\begin_inset Formula $\lg$
|
||
\end_inset
|
||
|
||
density
|
||
\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
|
||
\end_layout
|
||
|
||
\end_inset
|
||
|
||
In this section the one-zone model of Baker (
|
||
\begin_inset CommandInset citation
|
||
LatexCommand cite
|
||
key "Abernethy2003"
|
||
literal "true"
|
||
|
||
\end_inset
|
||
|
||
),
|
||
originally used to study the Cepheïd pulsation mechanism,
|
||
will be briefly reviewed.
|
||
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
|
||
LatexCommand cite
|
||
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
|
||
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"
|
||
nolink "false"
|
||
|
||
\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"
|
||
nolink "false"
|
||
|
||
\end_inset
|
||
|
||
),
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSSecSta"
|
||
nolink "false"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSVibSta"
|
||
nolink "false"
|
||
|
||
\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"
|
||
nolink "false"
|
||
|
||
\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
|
||
placement document
|
||
alignment document
|
||
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"
|
||
nolink "false"
|
||
|
||
\end_inset
|
||
|
||
),
|
||
(
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSSecSta"
|
||
nolink "false"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSVibSta"
|
||
nolink "false"
|
||
|
||
\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"
|
||
nolink "false"
|
||
|
||
\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"
|
||
nolink "false"
|
||
|
||
\end_inset
|
||
|
||
) and (
|
||
\begin_inset CommandInset ref
|
||
LatexCommand ref
|
||
reference "ZSVibSta"
|
||
nolink "false"
|
||
|
||
\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 thermodynamic 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 constitutive 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"
|
||
nolink "false"
|
||
|
||
\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
|
||
placement document
|
||
alignment document
|
||
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 instability 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
|