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78 \begin_inset Note Note
81 \begin_layout Plain Layout
86 This is an example \SpecialChar LyX
87 file for articles to be submitted to the Journal of
88 Astronomy & Astrophysics (A&A).
89 How to install the A&A \SpecialChar LaTeX
90 class to your \SpecialChar LaTeX
91 system is explained in
95 \begin_layout Plain Layout
97 http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
103 \begin_inset Newline newline
106 Depending on the submission state and the abstract layout, you need to use
107 different document class options that are listed in the aa manual.
110 \begin_inset Newline newline
118 If you use accented characters in your document, you must use the predefined
119 document class option
123 in the document settings.
132 Hydrodynamics of giant planet formation
135 \begin_layout Subtitle
138 \begin_inset Formula $\kappa$
147 \begin_inset Flex institutemark
150 \begin_layout Plain Layout
160 \begin_layout Plain Layout
171 \begin_inset Flex institutemark
174 \begin_layout Plain Layout
184 \begin_layout Plain Layout
197 \begin_layout Plain Layout
198 Just to show the usage of the elements in the author field
204 \begin_inset Note Note
207 \begin_layout Plain Layout
210 fnmsep is only needed for more than one consecutive notes/marks
218 \begin_layout Offprint
223 \begin_layout Address
224 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
226 \begin_inset Newline newline
230 \begin_inset Flex Email
233 \begin_layout Plain Layout
234 wuchterl@amok.ast.univie.ac.at
243 \begin_layout Plain Layout
252 University of Alexandria, Department of Geography, ...
253 \begin_inset Newline newline
257 \begin_inset Flex Email
260 \begin_layout Plain Layout
261 c.ptolemy@hipparch.uheaven.space
270 \begin_layout Plain Layout
271 The university of heaven temporarily does not accept e-mails
280 Received September 15, 1996; accepted March 16, 1997
283 \begin_layout Abstract (unstructured)
284 To investigate the physical nature of the `nuc\SpecialChar softhyphen
285 leated instability' of proto
286 giant planets, the stability of layers in static, radiative gas spheres
287 is analysed on the basis of Baker's standard one-zone model.
288 It is shown that stability depends only upon the equations of state, the
289 opacities and the local thermodynamic state in the layer.
290 Stability and instability can therefore be expressed in the form of stability
291 equations of state which are universal for a given composition.
292 The stability equations of state are calculated for solar composition and
293 are displayed in the domain
294 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
298 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
302 These displays may be used to determine the one-zone stability of layers
303 in stellar or planetary structure models by directly reading off the value
304 of the stability equations for the thermodynamic state of these layers,
305 specified by state quantities as density
306 \begin_inset Formula $\rho$
310 \begin_inset Formula $T$
313 or specific internal energy
314 \begin_inset Formula $e$
318 Regions of instability in the
319 \begin_inset Formula $(\rho,e)$
322 -plane are described and related to the underlying microphysical processes.
323 Vibrational instability is found to be a common phenomenon at temperatures
324 lower than the second He ionisation zone.
326 \begin_inset Formula $\kappa$
329 -mechanism is widespread under `cool' conditions.
330 \begin_inset Note Note
333 \begin_layout Plain Layout
334 Citations are not allowed in A&A abstracts.
340 \begin_inset Note Note
343 \begin_layout Plain Layout
344 This is the unstructured abstract type, an example for the structured abstract
349 template file that comes with \SpecialChar LyX
358 \begin_layout Keywords
359 giant planet formation –
360 \begin_inset Formula $\kappa$
363 -mechanism – stability of gas spheres
366 \begin_layout Section
370 \begin_layout Standard
373 nucleated instability
375 (also called core instability) hypothesis of giant planet formation, a
376 critical mass for static core envelope protoplanets has been found.
378 \begin_inset CommandInset citation
385 ) determined the critical mass of the core to be about
386 \begin_inset Formula $12\,M_{\oplus}$
390 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
393 is the Earth mass), which is independent of the outer boundary conditions
394 and therefore independent of the location in the solar nebula.
395 This critical value for the core mass corresponds closely to the cores
396 of today's giant planets.
399 \begin_layout Standard
400 Although no hydrodynamical study has been available many workers conjectured
401 that a collapse or rapid contraction will ensue after accumulating the
403 The main motivation for this article is to investigate the stability of
404 the static envelope at the critical mass.
405 With this aim the local, linear stability of static radiative gas spheres
406 is investigated on the basis of Baker's (
407 \begin_inset CommandInset citation
414 ) standard one-zone model.
417 \begin_layout Standard
418 Phenomena similar to the ones described above for giant planet formation
419 have been found in hydrodynamical models concerning star formation where
420 protostellar cores explode (Tscharnuter
421 \begin_inset CommandInset citation
429 \begin_inset CommandInset citation
436 ), whereas earlier studies found quasi-steady collapse flows.
437 The similarities in the (micro)physics, i.
438 \begin_inset space \thinspace{}
442 \begin_inset space \space{}
445 constitutive relations of protostellar cores and protogiant planets serve
446 as a further motivation for this study.
449 \begin_layout Section
450 Baker's standard one-zone model
453 \begin_layout Standard
454 \begin_inset Float figure
459 \begin_layout Plain Layout
460 \begin_inset Caption Standard
462 \begin_layout Plain Layout
463 \begin_inset CommandInset label
470 \begin_inset Formula $\Gamma_{1}$
475 \begin_inset Formula $\Gamma_{1}$
478 is plotted as a function of
479 \begin_inset Formula $\lg$
483 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
487 \begin_inset Formula $\lg$
491 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
504 In this section the one-zone model of Baker (
505 \begin_inset CommandInset citation
512 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
514 The resulting stability criteria will be rewritten in terms of local state
515 variables, local timescales and constitutive relations.
518 \begin_layout Standard
520 \begin_inset CommandInset citation
527 ) investigates the stability of thin layers in self-gravitating, spherical
528 gas clouds with the following properties:
531 \begin_layout Itemize
532 hydrostatic equilibrium,
535 \begin_layout Itemize
539 \begin_layout Itemize
540 energy transport by grey radiation diffusion.
544 \begin_layout Standard
546 For the one-zone-model Baker obtains necessary conditions for dynamical,
547 secular and vibrational (or pulsational) stability (Eqs.
548 \begin_inset space \space{}
552 \begin_inset space \thinspace{}
556 \begin_inset space \thinspace{}
560 \begin_inset CommandInset citation
568 Using Baker's notation:
571 \begin_layout Standard
575 M_{r} & & \textrm{mass internal to the radius }r\\
576 m & & \textrm{mass of the zone}\\
577 r_{0} & & \textrm{unperturbed zone radius}\\
578 \rho_{0} & & \textrm{unperturbed density in the zone}\\
579 T_{0} & & \textrm{unperturbed temperature in the zone}\\
580 L_{r0} & & \textrm{unperturbed luminosity}\\
581 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
589 \begin_layout Standard
591 and with the definitions of the
600 \begin_inset CommandInset ref
602 reference "fig:FigGam"
609 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
621 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
627 \begin_inset Formula $K$
631 \begin_inset Formula $\sigma_{0}$
634 have the following form:
637 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
638 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
644 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
651 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
658 is a thermodynamical quantity which is of order
659 \begin_inset Formula $1$
663 \begin_inset Formula $1$
666 for nonreacting mixtures of classical perfect gases.
667 The physical meaning of
668 \begin_inset Formula $\sigma_{0}$
672 \begin_inset Formula $K$
675 is clearly visible in the equations above.
677 \begin_inset Formula $\sigma_{0}$
680 represents a frequency of the order one per free-fall time.
682 \begin_inset Formula $K$
685 is proportional to the ratio of the free-fall time and the cooling time.
686 Substituting into Baker's criteria, using thermodynamic identities and
687 definitions of thermodynamic quantities,
690 \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}
698 \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}
703 one obtains, after some pages of algebra, the conditions for
710 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
711 \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}\\
712 \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}
717 For a physical discussion of the stability criteria see Baker (
718 \begin_inset CommandInset citation
726 \begin_inset CommandInset citation
736 \begin_layout Standard
737 We observe that these criteria for dynamical, secular and vibrational stability,
738 respectively, can be factorized into
741 \begin_layout Enumerate
742 a factor containing local timescales only,
745 \begin_layout Enumerate
746 a factor containing only constitutive relations and their derivatives.
750 \begin_layout Standard
751 The first factors, depending on only timescales, are positive by definition.
752 The signs of the left hand sides of the inequalities
757 \begin_inset CommandInset ref
764 \begin_inset CommandInset ref
771 \begin_inset CommandInset ref
777 ) therefore depend exclusively on the second factors containing the constitutive
779 Since they depend only on state variables, the stability criteria themselves
782 functions of the thermodynamic state in the local zone
785 The one-zone stability can therefore be determined from a simple equation
786 of state, given for example, as a function of density and temperature.
787 Once the microphysics, i.
788 \begin_inset space \thinspace{}
792 \begin_inset space \space{}
795 the thermodynamics and opacities (see Table
800 \begin_inset CommandInset ref
802 reference "tab:KapSou"
806 ), are specified (in practice by specifying a chemical composition) the
807 one-zone stability can be inferred if the thermodynamic state is specified.
808 The zone – or in other words the layer – will be stable or unstable in
809 whatever object it is imbedded as long as it satisfies the one-zone-model
811 Only the specific growth rates (depending upon the time scales) will be
812 different for layers in different objects.
815 \begin_layout Standard
816 \begin_inset Float table
821 \begin_layout Plain Layout
822 \begin_inset Caption Standard
824 \begin_layout Plain Layout
825 \begin_inset CommandInset label
839 \begin_layout Plain Layout
842 <lyxtabular version="3" rows="4" columns="2">
843 <features tabularvalignment="middle">
844 <column alignment="left" valignment="top" width="0pt">
845 <column alignment="left" valignment="top" width="0pt">
847 <cell alignment="center" valignment="top" topline="true" usebox="none">
850 \begin_layout Plain Layout
856 <cell alignment="center" valignment="top" topline="true" usebox="none">
859 \begin_layout Plain Layout
860 \begin_inset Formula $T/[\textrm{K}]$
870 <cell alignment="center" valignment="top" topline="true" usebox="none">
873 \begin_layout Plain Layout
874 Yorke 1979, Yorke 1980a
879 <cell alignment="center" valignment="top" topline="true" usebox="none">
882 \begin_layout Plain Layout
883 \begin_inset Formula $\leq1700^{\textrm{a}}$
893 <cell alignment="center" valignment="top" usebox="none">
896 \begin_layout Plain Layout
902 <cell alignment="center" valignment="top" usebox="none">
905 \begin_layout Plain Layout
906 \begin_inset Formula $1700\leq T\leq5000$
916 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
919 \begin_layout Plain Layout
925 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
928 \begin_layout Plain Layout
929 \begin_inset Formula $5000\leq$
945 \begin_layout Plain Layout
946 \begin_inset Formula $^{\textrm{a}}$
954 We will now write down the sign (and therefore stability) determining parts
955 of the left-hand sides of the inequalities (
956 \begin_inset CommandInset ref
963 \begin_inset CommandInset ref
970 \begin_inset CommandInset ref
978 stability equations of state
983 \begin_layout Standard
984 The sign determining part of inequality
989 \begin_inset CommandInset ref
996 \begin_inset Formula $3\Gamma_{1}-4$
999 and it reduces to the criterion for dynamical stability
1000 \begin_inset Formula
1002 \Gamma_{1}>\frac{4}{3}\,\cdot
1007 Stability of the thermodynamical equilibrium demands
1008 \begin_inset Formula
1010 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1016 \begin_inset Formula
1023 holds for a wide range of physical situations.
1025 \begin_inset Formula
1027 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1028 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1029 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1034 we find the sign determining terms in inequalities
1035 \begin_inset space ~
1039 \begin_inset CommandInset ref
1041 reference "ZSSecSta"
1046 \begin_inset CommandInset ref
1048 reference "ZSVibSta"
1052 ) respectively and obtain the following form of the criteria for dynamical,
1053 secular and vibrational
1058 \begin_inset Formula
1060 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1061 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1062 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1067 The constitutive relations are to be evaluated for the unperturbed thermodynami
1069 \begin_inset Formula $(\rho_{0},T_{0})$
1073 We see that the one-zone stability of the layer depends only on the constitutiv
1075 \begin_inset Formula $\Gamma_{1}$
1079 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1083 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1087 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1091 These depend only on the unperturbed thermodynamical state of the layer.
1092 Therefore the above relations define the one-zone-stability equations of
1094 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1098 \begin_inset Formula $S_{\mathrm{vib}}$
1103 \begin_inset space ~
1107 \begin_inset CommandInset ref
1109 reference "fig:VibStabEquation"
1114 \begin_inset Formula $S_{\mathrm{vib}}$
1118 Regions of secular instability are listed in Table
1119 \begin_inset space ~
1125 \begin_layout Standard
1126 \begin_inset Float figure
1131 \begin_layout Plain Layout
1132 \begin_inset Caption Standard
1134 \begin_layout Plain Layout
1135 \begin_inset CommandInset label
1137 name "fig:VibStabEquation"
1141 Vibrational stability equation of state
1142 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1147 \begin_inset Formula $>0$
1150 means vibrational stability
1163 \begin_layout Section
1167 \begin_layout Enumerate
1168 The conditions for the stability of static, radiative layers in gas spheres,
1169 as described by Baker's (
1170 \begin_inset CommandInset citation
1177 ) standard one-zone model, can be expressed as stability equations of state.
1178 These stability equations of state depend only on the local thermodynamic
1183 \begin_layout Enumerate
1184 If the constitutive relations – equations of state and Rosseland mean opacities
1185 – are specified, the stability equations of state can be evaluated without
1186 specifying properties of the layer.
1190 \begin_layout Enumerate
1191 For solar composition gas the
1192 \begin_inset Formula $\kappa$
1195 -mechanism is working in the regions of the ice and dust features in the
1197 \begin_inset Formula $\mathrm{H}_{2}$
1200 dissociation and the combined H, first He ionization zone, as indicated
1201 by vibrational instability.
1202 These regions of instability are much larger in extent and degree of instabilit
1203 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1207 \begin_layout Acknowledgement
1208 Part of this work was supported by the German
1210 Deut\SpecialChar softhyphen
1211 sche For\SpecialChar softhyphen
1212 schungs\SpecialChar softhyphen
1213 ge\SpecialChar softhyphen
1214 mein\SpecialChar softhyphen
1218 \begin_inset space ~
1224 \begin_layout Standard
1225 \begin_inset CommandInset bibtex
1227 btprint "btPrintAll"
1228 bibfiles "biblioExample"
1234 \begin_inset Note Note
1237 \begin_layout Plain Layout
1242 If you cannot see the bibliography in the output, assure that you have
1243 gievn the full path to the Bib\SpecialChar TeX
1248 that is part of the A&A \SpecialChar LaTeX