X-Git-Url: https://git.lyx.org/gitweb/?a=blobdiff_plain;f=lib%2Fexamples%2Faa_sample.lyx;h=d8a4f8cd07aad7e0a8c124babb21625f51bebfbc;hb=90f7007a2e6c78ffd031e4636ff909ab1bc2ddec;hp=0c841c14725bbe7787b49431ffb9b4fcb9827c16;hpb=0ed05b32be7506ba9b21e1378dec9844e8ff3f23;p=lyx.git diff --git a/lib/examples/aa_sample.lyx b/lib/examples/aa_sample.lyx index 0c841c1472..d8a4f8cd07 100644 --- a/lib/examples/aa_sample.lyx +++ b/lib/examples/aa_sample.lyx @@ -1,5 +1,5 @@ -#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/ -\lyxformat 276 +#LyX 2.0 created this file. For more info see http://www.lyx.org/ +\lyxformat 413 \begin_document \begin_header \textclass aa @@ -7,87 +7,167 @@ \usepackage{graphicx} % \end_preamble +\options traditabstract +\use_default_options false +\maintain_unincluded_children false \language english +\language_package default \inputencoding auto +\fontencoding global \font_roman default \font_sans default \font_typewriter default \font_default_family default +\use_non_tex_fonts false \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 + \graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default \paperfontsize default \spacing single +\use_hyperref false \papersize default \use_geometry false \use_amsmath 0 \use_esint 0 +\use_mhchem 1 +\use_mathdots 1 \cite_engine basic \use_bibtopic false +\use_indices false \paperorientation portrait +\suppress_date false +\use_refstyle 0 +\index Index +\shortcut idx +\color #008000 +\end_index \secnumdepth 3 \tocdepth 3 \paragraph_separation indent -\defskip medskip +\paragraph_indentation default \quotes_language english \papercolumns 2 \papersides 2 \paperpagestyle default \tracking_changes false \output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false \end_header \begin_body -\begin_layout Title +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +This is an example LyX file for articles to be submitted to the Journal + of Astronomy & Astrophysicssing (A&A). + How to install the A&A LaTeX class to your LaTeX system is explained in + +\begin_inset Flex URL +status open + +\begin_layout Plain Layout + +http://wiki.lyx.org/Layouts/Astronomy-Astrophysics +\end_layout + +\end_inset + +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +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 sans +aadoc.pdf +\family default +. +\end_layout +\end_inset + + +\end_layout + +\begin_layout Title Hydrodynamics of giant planet formation \end_layout \begin_layout Subtitle - I. Overviewing the -\begin_inset Formula \( \kappa \) +\begin_inset Formula $\kappa$ \end_inset -mechanism \end_layout \begin_layout Author - G. - Wuchterl + Wuchterl +\begin_inset Flex institutemark +status open + +\begin_layout Plain Layout +1 +\end_layout + +\end_inset + + \begin_inset ERT status collapsed -\begin_layout Standard +\begin_layout Plain Layout + \backslash -inst{1} -\backslash -and +and \end_layout -\begin_layout Standard - +\end_inset + + C. + Ptolemy +\begin_inset Flex institutemark +status collapsed + +\begin_layout Plain Layout +2 \end_layout \end_inset -C. - Ptolemy + \begin_inset ERT status collapsed -\begin_layout Standard +\begin_layout Plain Layout + \backslash -inst{2} -\backslash -fnmsep +fnmsep \end_layout \end_inset @@ -96,108 +176,108 @@ fnmsep \begin_inset Foot status collapsed -\begin_layout Standard - +\begin_layout Plain Layout Just to show the usage of the elements in the author field \end_layout \end_inset - +\begin_inset Note Note +status collapsed + +\begin_layout Plain Layout + +\backslash +fnmsep is only needed for more than one consecutive notes/marks \end_layout -\begin_layout Offprint +\end_inset + + +\end_layout +\begin_layout Offprint G. Wuchterl \end_layout \begin_layout Address - -Institute for Astronomy (IfA), University of Vienna, Tü -rkenschanzstrasse +Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse 17, A-1180 Vienna -\newline +\begin_inset Newline newline +\end_inset -\begin_inset ERT -status collapsed -\begin_layout Standard +\begin_inset Flex Email +status open -\backslash -email{wuchterl@amok.ast.univie.ac.at} -\backslash -and +\begin_layout Plain Layout +wuchterl@amok.ast.univie.ac.at \end_layout -\begin_layout Standard +\end_inset + + +\begin_inset ERT +status collapsed +\begin_layout Plain Layout + + +\backslash +and \end_layout \end_inset University of Alexandria, Department of Geography, ... -\newline +\begin_inset Newline newline +\end_inset -\begin_inset ERT -status collapsed -\begin_layout Standard +\begin_inset Flex Email +status collapsed -\backslash -email{c.ptolemy@hipparch.uheaven.space} +\begin_layout Plain Layout +c.ptolemy@hipparch.uheaven.space \end_layout \end_inset - + \begin_inset Foot status collapsed -\begin_layout Standard - +\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 - To investigate the physical nature of the `nuc\SpecialChar \- leated instability' of proto - giant planets (Mizuno -\begin_inset LatexCommand cite -key "mizuno" -\end_inset - -), the stability of layers in static, radiative gas spheres is analysed - on the basis of Baker's -\begin_inset LatexCommand cite -key "baker" -\end_inset - - standard one-zone model. + 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 \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \) +\begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$ \end_inset , -\begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \) +\begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$ \end_inset . @@ -205,92 +285,89 @@ key "baker" 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 \( \rho \) +\begin_inset Formula $\rho$ \end_inset , temperature -\begin_inset Formula \( T \) +\begin_inset Formula $T$ \end_inset or specific internal energy -\begin_inset Formula \( e \) +\begin_inset Formula $e$ \end_inset . Regions of instability in the -\begin_inset Formula \( (\rho ,e) \) +\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 \( \kappa \) +\begin_inset Formula $\kappa$ \end_inset -mechanism is widespread under `cool' conditions. -\begin_inset ERT -status collapsed +\begin_inset Note Note +status open -\begin_layout Standard +\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 LyX. \end_layout -\begin_layout Standard +\end_inset + -\backslash -keywords{giant planet formation -- -\backslash -( -\backslash -kappa -\backslash -)-mechanism -- stability of gas spheres } \end_layout +\begin_layout Keywords +giant planet formation -- +\begin_inset Formula $\kappa$ \end_inset - +-mechanism -- stability of gas spheres \end_layout \begin_layout Section - Introduction \end_layout \begin_layout Standard - -In the -\emph default - +In the \emph on nucleated instability -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash -/{} -\end_layout - -\end_inset - - \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 LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "mizuno" + \end_inset ) determined the critical mass of the core to be about -\begin_inset Formula \( 12\, M_{\oplus } \) +\begin_inset Formula $12\, M_{\oplus}$ \end_inset ( -\begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \) +\begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$ \end_inset is the Earth mass), which is independent of the outer boundary conditions @@ -300,7 +377,6 @@ key "mizuno" \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. @@ -308,88 +384,93 @@ Although no hydrodynamical study has been available many workers conjectured 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 ( -\begin_inset LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset ) standard one-zone model. \end_layout \begin_layout Standard - 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 LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "tscharnuter" + \end_inset , Balluch -\begin_inset LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "balluch" + \end_inset ), whereas earlier studies found quasi-steady collapse flows. - The similarities in the (micro)physics, i.e., constitutive relations of protostel -lar cores and protogiant planets serve as a further motivation for this - study. + 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 - Baker's standard one-zone model \end_layout \begin_layout Standard - \begin_inset Float figure wide true sideways false status open -\begin_layout +\begin_layout Plain Layout \begin_inset Caption -\begin_layout +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:FigGam" + +\end_inset Adiabatic exponent -\begin_inset Formula \( \Gamma _{1} \) +\begin_inset Formula $\Gamma_{1}$ \end_inset . -\begin_inset Formula \( \Gamma _{1} \) +\begin_inset Formula $\Gamma_{1}$ \end_inset is plotted as a function of -\begin_inset Formula \( \lg \) +\begin_inset Formula $\lg$ \end_inset internal energy -\begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \) +\begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$ \end_inset and -\begin_inset Formula \( \lg \) +\begin_inset Formula $\lg$ \end_inset density -\begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \) -\end_inset - - -\end_layout - +\begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$ \end_inset \end_layout -\begin_layout Standard - - -\begin_inset LatexCommand label -name "FigGam" \end_inset @@ -398,22 +479,24 @@ name "FigGam" \end_inset In this section the one-zone model of Baker ( -\begin_inset LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset -), originally used to study the Cepheı̈ -d pulsation mechanism, will be briefly +), originally used to study the Cepheı̈d 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 LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset ) investigates the stability of thin layers in self-gravitating, spherical @@ -421,17 +504,14 @@ key "baker" \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 @@ -440,44 +520,22 @@ energy transport by grey radiation diffusion. \noindent For the one-zone-model Baker obtains necessary conditions for dynamical, secular and vibrational (or pulsational) stability (Eqs. -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash - -\end_layout - +\begin_inset space \space{} \end_inset (34a, -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash -, -\end_layout - +\begin_inset space \thinspace{} \end_inset b, -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash -, -\end_layout - +\begin_inset space \thinspace{} \end_inset c) in Baker -\begin_inset LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset ). @@ -486,12 +544,12 @@ key "baker" \begin_layout Standard \align left - -\begin_inset Formula \begin{eqnarray*} +\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}\\ +\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} @@ -504,74 +562,67 @@ E_{\textrm{th}} & & \textrm{thermal energy of the zone} \begin_layout Standard \noindent -and with the definitions of the -\emph default - +and with the definitions of the \emph on local cooling time -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash -/{} -\end_layout - +\emph default + (see Fig. +\begin_inset space ~ \end_inset -\emph default - (see Fig.\InsetSpace ~ +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:FigGam" -\begin_inset LatexCommand ref -reference "FigGam" \end_inset ) -\begin_inset Formula \begin{equation} -\tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, , +\begin_inset Formula +\begin{equation} +\tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,, \end{equation} \end_inset - and the -\emph default - + 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}}}}\, , +\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 \( K \) +\begin_inset Formula $K$ \end_inset and -\begin_inset Formula \( \sigma _{0} \) +\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}}}\, ; +\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 \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \) +\begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$ \end_inset has been used and -\begin_inset Formula \begin{equation} +\begin_inset Formula +\begin{equation} \begin{array}{l} -\delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\ +\delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\ e=mc^{2} \end{array} \end{equation} @@ -579,150 +630,149 @@ e=mc^{2} \end_inset is a thermodynamical quantity which is of order -\begin_inset Formula \( 1 \) +\begin_inset Formula $1$ \end_inset and equal to -\begin_inset Formula \( 1 \) +\begin_inset Formula $1$ \end_inset for nonreacting mixtures of classical perfect gases. The physical meaning of -\begin_inset Formula \( \sigma _{0} \) +\begin_inset Formula $\sigma_{0}$ \end_inset and -\begin_inset Formula \( K \) +\begin_inset Formula $K$ \end_inset is clearly visible in the equations above. -\begin_inset Formula \( \sigma _{0} \) +\begin_inset Formula $\sigma_{0}$ \end_inset represents a frequency of the order one per free-fall time. -\begin_inset Formula \( K \) +\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 \[ -\Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\] +\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 \[ -\nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\] +\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 default - + one obtains, after some pages of algebra, the conditions for \emph on stability -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash -/{} -\end_layout - -\end_inset - - \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} +\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 LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset ) or Cox ( -\begin_inset LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "cox" + \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\InsetSpace ~ + The signs of the left hand sides of the inequalities +\begin_inset space ~ +\end_inset + ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSDynSta" + \end_inset ), ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSSecSta" + \end_inset ) and ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSVibSta" + \end_inset ) therefore depend exclusively on the second factors containing the constitutive relations. Since they depend only on state variables, the stability criteria themselves - are -\emph default - + 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.e. -\begin_inset ERT -status collapsed - -\begin_layout Standard + Once the microphysics, i. +\begin_inset space \thinspace{} +\end_inset -\backslash - -\end_layout +g. +\begin_inset space \space{} +\end_inset +the thermodynamics and opacities (see Table +\begin_inset space ~ \end_inset -the thermodynamics and opacities (see Table\InsetSpace ~ -\begin_inset LatexCommand ref -reference "KapSou" +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:KapSou" + \end_inset ), are specified (in practice by specifying a chemical composition) the @@ -735,20 +785,19 @@ reference "KapSou" \end_layout \begin_layout Standard - \begin_inset Float table wide false sideways false status open -\begin_layout +\begin_layout Plain Layout \begin_inset Caption -\begin_layout - +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "tab:KapSou" -\begin_inset LatexCommand label -name "KapSou" \end_inset Opacity sources @@ -759,32 +808,28 @@ Opacity sources \end_layout -\begin_layout Standard - - -\begin_inset Tabular +\begin_layout Plain Layout +\align center +\begin_inset Tabular - + - - + + \begin_inset Text -\begin_layout Standard - +\begin_layout Plain Layout Source \end_layout \end_inset - + \begin_inset Text -\begin_layout Standard - - -\begin_inset Formula \( T/[\textrm{K}] \) +\begin_layout Plain Layout +\begin_inset Formula $T/[\textrm{K}]$ \end_inset @@ -793,24 +838,21 @@ Source \end_inset - - + + \begin_inset Text -\begin_layout Standard - +\begin_layout Plain Layout Yorke 1979, Yorke 1980a \end_layout \end_inset - + \begin_inset Text -\begin_layout Standard - - -\begin_inset Formula \( \leq 1700^{\textrm{a}} \) +\begin_layout Plain Layout +\begin_inset Formula $\leq1700^{\textrm{a}}$ \end_inset @@ -820,23 +862,20 @@ Yorke 1979, Yorke 1980a - + \begin_inset Text -\begin_layout Standard - +\begin_layout Plain Layout Krügel 1971 \end_layout \end_inset - + \begin_inset Text -\begin_layout Standard - - -\begin_inset Formula \( 1700\leq T\leq 5000 \) +\begin_layout Plain Layout +\begin_inset Formula $1700\leq T\leq5000$ \end_inset @@ -845,24 +884,21 @@ Krügel 1971 \end_inset - - + + \begin_inset Text -\begin_layout Standard - +\begin_layout Plain Layout Cox & Stewart 1969 \end_layout \end_inset - + \begin_inset Text -\begin_layout Standard - - -\begin_inset Formula \( 5000\leq \) +\begin_layout Plain Layout +\begin_inset Formula $5000\leq$ \end_inset @@ -878,10 +914,8 @@ Cox & Stewart 1969 \end_layout -\begin_layout Standard - - -\begin_inset Formula \( ^{\textrm{a}} \) +\begin_layout Plain Layout +\begin_inset Formula $^{\textrm{a}}$ \end_inset This is footnote a @@ -891,23 +925,27 @@ This is footnote a We will now write down the sign (and therefore stability) determining parts of the left-hand sides of the inequalities ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSDynSta" + \end_inset ), ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSSecSta" + \end_inset ) and ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSVibSta" + \end_inset -) and thereby obtain -\emph default - +) and thereby obtain \emph on stability equations of state \emph default @@ -915,144 +953,170 @@ stability equations of state \end_layout \begin_layout Standard +The sign determining part of inequality +\begin_inset space ~ +\end_inset -The sign determining part of inequality\InsetSpace ~ ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSDynSta" + \end_inset ) is -\begin_inset Formula \( 3\Gamma _{1}-4 \) +\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 +\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\, , +\begin_inset Formula +\begin{equation} +\chi_{\rho}^{}>0,\;\; c_{v}>0\,, \end{equation} \end_inset and -\begin_inset Formula \begin{equation} -\chi ^{}_{T}>0 +\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 +\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\InsetSpace ~ + we find the sign determining terms in inequalities +\begin_inset space ~ +\end_inset + ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSSecSta" + \end_inset ) and ( -\begin_inset LatexCommand ref +\begin_inset CommandInset ref +LatexCommand ref reference "ZSVibSta" + \end_inset ) respectively and obtain the following form of the criteria for dynamical, - secular and vibrational -\emph default - + 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} +\begin_inset Formula +\begin{eqnarray} +3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\ +\frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\ +4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta} \end{eqnarray} \end_inset The constitutive relations are to be evaluated for the unperturbed thermodynami c state (say -\begin_inset Formula \( (\rho _{0},T_{0}) \) +\begin_inset Formula $(\rho_{0},T_{0})$ \end_inset ) of the zone. We see that the one-zone stability of the layer depends only on the constitutiv e relations -\begin_inset Formula \( \Gamma _{1} \) +\begin_inset Formula $\Gamma_{1}$ \end_inset , -\begin_inset Formula \( \nabla _{\mathrm{ad}} \) +\begin_inset Formula $\nabla_{\mathrm{ad}}$ \end_inset , -\begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \) +\begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$ \end_inset , -\begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \) +\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 \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \) +\begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$ \end_inset and -\begin_inset Formula \( S_{\mathrm{vib}} \) +\begin_inset Formula $S_{\mathrm{vib}}$ \end_inset . - See Fig.\InsetSpace ~ + See Fig. +\begin_inset space ~ +\end_inset + + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:VibStabEquation" -\begin_inset LatexCommand ref -reference "FigVibStab" \end_inset for a picture of -\begin_inset Formula \( S_{\mathrm{vib}} \) +\begin_inset Formula $S_{\mathrm{vib}}$ \end_inset . - Regions of secular instability are listed in Table\InsetSpace ~ + Regions of secular instability are listed in Table +\begin_inset space ~ +\end_inset + 1. \end_layout \begin_layout Standard - \begin_inset Float figure wide false sideways false status open -\begin_layout +\begin_layout Plain Layout \begin_inset Caption -\begin_layout +\begin_layout Plain Layout +\begin_inset CommandInset label +LatexCommand label +name "fig:VibStabEquation" + +\end_inset Vibrational stability equation of state -\begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \) +\begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$ \end_inset . -\begin_inset Formula \( >0 \) +\begin_inset Formula $>0$ \end_inset means vibrational stability @@ -1063,31 +1127,22 @@ Vibrational stability equation of state \end_layout -\begin_layout Standard - - -\begin_inset LatexCommand label -name "FigVibStab" \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 LatexCommand cite +\begin_inset CommandInset citation +LatexCommand cite key "baker" + \end_inset ) standard one-zone model, can be expressed as stability equations of state. @@ -1097,7 +1152,6 @@ key "baker" \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. @@ -1105,29 +1159,24 @@ If the constitutive relations -- equations of state and Rosseland mean opacities \end_layout \begin_layout Enumerate - For solar composition gas the -\begin_inset Formula \( \kappa \) +\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 \( \mathrm{H}_{2} \) +\begin_inset Formula $\mathrm{H}_{2}$ \end_inset dissociation and the combined H, first He ionization zone, as indicated by vibrational instability. These regions of instability are much larger in extent and degree of instabilit -y than the second He ionization zone that drives the Cepheı̈ -d pulsations. +y than the second He ionization zone that drives the Cepheı̈d pulsations. \end_layout \begin_layout Acknowledgement - -Part of this work was supported by the German -\emph default - +Part of this work was supported by the German \emph on Deut\SpecialChar \- sche For\SpecialChar \- @@ -1135,26 +1184,36 @@ schungs\SpecialChar \- ge\SpecialChar \- mein\SpecialChar \- schaft, DFG -\begin_inset ERT -status collapsed +\emph default + project number Ts +\begin_inset space ~ +\end_inset + +17/2--1. +\end_layout \begin_layout Standard +\begin_inset Note Note +status open -\backslash -/{} +\begin_layout Plain Layout +You can alternatively use BibTeX. + You must then use the BibTeX style +\family sans +aa.bst +\family default + that is part of the A&A LaTeX-package. \end_layout \end_inset -\emph default - project number Ts\InsetSpace ~ -17/2--1. - \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1966" key "baker" @@ -1162,15 +1221,7 @@ key "baker" Baker, N. 1966, in Stellar Evolution, ed. -\begin_inset ERT -status collapsed - -\begin_layout Standard - -\backslash - -\end_layout - +\begin_inset space \space{} \end_inset R. @@ -1182,7 +1233,9 @@ R. \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1988" key "balluch" @@ -1193,7 +1246,9 @@ key "balluch" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1980" key "cox" @@ -1206,7 +1261,9 @@ key "cox" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1969" key "cox69" @@ -1219,7 +1276,9 @@ key "cox69" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1980" key "mizuno" @@ -1232,7 +1291,9 @@ key "mizuno" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1987" key "tscharnuter" @@ -1244,7 +1305,9 @@ key "tscharnuter" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1992" key "terlevich" @@ -1261,7 +1324,9 @@ key "terlevich" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1980a" key "yorke80a" @@ -1273,7 +1338,9 @@ key "yorke80a" \end_layout \begin_layout Bibliography -\begin_inset LatexCommand bibitem +\labelwidthstring References +\begin_inset CommandInset bibitem +LatexCommand bibitem label "1997" key "zheng"