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81 \begin_inset Note Note
84 \begin_layout Plain Layout
89 This is an example \SpecialChar LyX
90 file for articles to be submitted to the Journal of
91 Astronomy & Astrophysics (A&A).
92 How to install the A&A \SpecialChar LaTeX
93 class to your \SpecialChar LaTeX
94 system is explained in
98 \begin_layout Plain Layout
100 https://wiki.lyx.org/Layouts/Astronomy-Astrophysics
106 \begin_inset Newline newline
109 Depending on the submission state and the abstract layout, you need to use
110 different document class options that are listed in the aa manual.
113 \begin_inset Newline newline
121 If you use accented characters in your document, you must use the predefined
122 document class option
126 in the document settings.
135 Hydrodynamics of giant planet formation
138 \begin_layout Subtitle
141 \begin_inset Formula $\kappa$
150 \begin_inset Flex institutemark
153 \begin_layout Plain Layout
163 \begin_layout Plain Layout
174 \begin_inset Flex institutemark
177 \begin_layout Plain Layout
187 \begin_layout Plain Layout
200 \begin_layout Plain Layout
201 Just to show the usage of the elements in the author field
207 \begin_inset Note Note
210 \begin_layout Plain Layout
213 fnmsep is only needed for more than one consecutive notes/marks
221 \begin_layout Offprint
226 \begin_layout Address
227 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
229 \begin_inset Newline newline
233 \begin_inset Flex Email
236 \begin_layout Plain Layout
237 wuchterl@amok.ast.univie.ac.at
246 \begin_layout Plain Layout
255 University of Alexandria, Department of Geography, ...
256 \begin_inset Newline newline
260 \begin_inset Flex Email
263 \begin_layout Plain Layout
264 c.ptolemy@hipparch.uheaven.space
273 \begin_layout Plain Layout
274 The university of heaven temporarily does not accept e-mails
283 Received September 15, 1996; accepted March 16, 1997
286 \begin_layout Abstract (unstructured)
287 To investigate the physical nature of the `nuc\SpecialChar softhyphen
288 leated instability' of proto
289 giant planets, the stability of layers in static, radiative gas spheres
290 is analysed on the basis of Baker's standard one-zone model.
291 It is shown that stability depends only upon the equations of state, the
292 opacities and the local thermodynamic state in the layer.
293 Stability and instability can therefore be expressed in the form of stability
294 equations of state which are universal for a given composition.
295 The stability equations of state are calculated for solar composition and
296 are displayed in the domain
297 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
301 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
305 These displays may be used to determine the one-zone stability of layers
306 in stellar or planetary structure models by directly reading off the value
307 of the stability equations for the thermodynamic state of these layers,
308 specified by state quantities as density
309 \begin_inset Formula $\rho$
313 \begin_inset Formula $T$
316 or specific internal energy
317 \begin_inset Formula $e$
321 Regions of instability in the
322 \begin_inset Formula $(\rho,e)$
325 -plane are described and related to the underlying microphysical processes.
326 Vibrational instability is found to be a common phenomenon at temperatures
327 lower than the second He ionisation zone.
329 \begin_inset Formula $\kappa$
332 -mechanism is widespread under `cool' conditions.
333 \begin_inset Note Note
336 \begin_layout Plain Layout
337 Citations are not allowed in A&A abstracts.
343 \begin_inset Note Note
346 \begin_layout Plain Layout
347 This is the unstructured abstract type, an example for the structured abstract
352 template file that comes with \SpecialChar LyX
361 \begin_layout Keywords
362 giant planet formation –
363 \begin_inset Formula $\kappa$
366 -mechanism – stability of gas spheres
369 \begin_layout Section
373 \begin_layout Standard
376 nucleated instability
378 (also called core instability) hypothesis of giant planet formation, a
379 critical mass for static core envelope protoplanets has been found.
381 \begin_inset CommandInset citation
388 ) determined the critical mass of the core to be about
389 \begin_inset Formula $12\,M_{\oplus}$
393 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
396 is the Earth mass), which is independent of the outer boundary conditions
397 and therefore independent of the location in the solar nebula.
398 This critical value for the core mass corresponds closely to the cores
399 of today's giant planets.
402 \begin_layout Standard
403 Although no hydrodynamical study has been available many workers conjectured
404 that a collapse or rapid contraction will ensue after accumulating the
406 The main motivation for this article is to investigate the stability of
407 the static envelope at the critical mass.
408 With this aim the local, linear stability of static radiative gas spheres
409 is investigated on the basis of Baker's (
410 \begin_inset CommandInset citation
417 ) standard one-zone model.
420 \begin_layout Standard
421 Phenomena similar to the ones described above for giant planet formation
422 have been found in hydrodynamical models concerning star formation where
423 protostellar cores explode (Tscharnuter
424 \begin_inset CommandInset citation
432 \begin_inset CommandInset citation
439 ), whereas earlier studies found quasi-steady collapse flows.
440 The similarities in the (micro)physics, i.
441 \begin_inset space \thinspace{}
445 \begin_inset space \space{}
448 constitutive relations of protostellar cores and protogiant planets serve
449 as a further motivation for this study.
452 \begin_layout Section
453 Baker's standard one-zone model
456 \begin_layout Standard
457 \begin_inset Float figure
462 \begin_layout Plain Layout
463 \begin_inset Caption Standard
465 \begin_layout Plain Layout
466 \begin_inset CommandInset label
473 \begin_inset Formula $\Gamma_{1}$
478 \begin_inset Formula $\Gamma_{1}$
481 is plotted as a function of
482 \begin_inset Formula $\lg$
486 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
490 \begin_inset Formula $\lg$
494 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
507 In this section the one-zone model of Baker (
508 \begin_inset CommandInset citation
515 ), originally used to study the Cepheïd pulsation mechanism, will be briefly
517 The resulting stability criteria will be rewritten in terms of local state
518 variables, local timescales and constitutive relations.
521 \begin_layout Standard
523 \begin_inset CommandInset citation
530 ) investigates the stability of thin layers in self-gravitating, spherical
531 gas clouds with the following properties:
534 \begin_layout Itemize
535 hydrostatic equilibrium,
538 \begin_layout Itemize
542 \begin_layout Itemize
543 energy transport by grey radiation diffusion.
547 \begin_layout Standard
549 For the one-zone-model Baker obtains necessary conditions for dynamical,
550 secular and vibrational (or pulsational) stability (Eqs.
551 \begin_inset space \space{}
555 \begin_inset space \thinspace{}
559 \begin_inset space \thinspace{}
563 \begin_inset CommandInset citation
571 Using Baker's notation:
574 \begin_layout Standard
578 M_{r} & & \textrm{mass internal to the radius }r\\
579 m & & \textrm{mass of the zone}\\
580 r_{0} & & \textrm{unperturbed zone radius}\\
581 \rho_{0} & & \textrm{unperturbed density in the zone}\\
582 T_{0} & & \textrm{unperturbed temperature in the zone}\\
583 L_{r0} & & \textrm{unperturbed luminosity}\\
584 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
592 \begin_layout Standard
594 and with the definitions of the
603 \begin_inset CommandInset ref
605 reference "fig:FigGam"
612 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
624 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
630 \begin_inset Formula $K$
634 \begin_inset Formula $\sigma_{0}$
637 have the following form:
640 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
641 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
647 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
654 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
661 is a thermodynamical quantity which is of order
662 \begin_inset Formula $1$
666 \begin_inset Formula $1$
669 for nonreacting mixtures of classical perfect gases.
670 The physical meaning of
671 \begin_inset Formula $\sigma_{0}$
675 \begin_inset Formula $K$
678 is clearly visible in the equations above.
680 \begin_inset Formula $\sigma_{0}$
683 represents a frequency of the order one per free-fall time.
685 \begin_inset Formula $K$
688 is proportional to the ratio of the free-fall time and the cooling time.
689 Substituting into Baker's criteria, using thermodynamic identities and
690 definitions of thermodynamic quantities,
693 \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}
701 \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}
706 one obtains, after some pages of algebra, the conditions for
713 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
714 \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}\\
715 \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}
720 For a physical discussion of the stability criteria see Baker (
721 \begin_inset CommandInset citation
729 \begin_inset CommandInset citation
739 \begin_layout Standard
740 We observe that these criteria for dynamical, secular and vibrational stability,
741 respectively, can be factorized into
744 \begin_layout Enumerate
745 a factor containing local timescales only,
748 \begin_layout Enumerate
749 a factor containing only constitutive relations and their derivatives.
753 \begin_layout Standard
754 The first factors, depending on only timescales, are positive by definition.
755 The signs of the left hand sides of the inequalities
760 \begin_inset CommandInset ref
767 \begin_inset CommandInset ref
774 \begin_inset CommandInset ref
780 ) therefore depend exclusively on the second factors containing the constitutive
782 Since they depend only on state variables, the stability criteria themselves
785 functions of the thermodynamic state in the local zone
788 The one-zone stability can therefore be determined from a simple equation
789 of state, given for example, as a function of density and temperature.
790 Once the microphysics, i.
791 \begin_inset space \thinspace{}
795 \begin_inset space \space{}
798 the thermodynamics and opacities (see Table
803 \begin_inset CommandInset ref
805 reference "tab:KapSou"
809 ), are specified (in practice by specifying a chemical composition) the
810 one-zone stability can be inferred if the thermodynamic state is specified.
811 The zone – or in other words the layer – will be stable or unstable in
812 whatever object it is imbedded as long as it satisfies the one-zone-model
814 Only the specific growth rates (depending upon the time scales) will be
815 different for layers in different objects.
818 \begin_layout Standard
819 \begin_inset Float table
824 \begin_layout Plain Layout
825 \begin_inset Caption Standard
827 \begin_layout Plain Layout
828 \begin_inset CommandInset label
842 \begin_layout Plain Layout
845 <lyxtabular version="3" rows="4" columns="2">
846 <features tabularvalignment="middle">
847 <column alignment="left" valignment="top" width="0pt">
848 <column alignment="left" valignment="top" width="0pt">
850 <cell alignment="center" valignment="top" topline="true" usebox="none">
853 \begin_layout Plain Layout
859 <cell alignment="center" valignment="top" topline="true" usebox="none">
862 \begin_layout Plain Layout
863 \begin_inset Formula $T/[\textrm{K}]$
873 <cell alignment="center" valignment="top" topline="true" usebox="none">
876 \begin_layout Plain Layout
877 Yorke 1979, Yorke 1980a
882 <cell alignment="center" valignment="top" topline="true" usebox="none">
885 \begin_layout Plain Layout
886 \begin_inset Formula $\leq1700^{\textrm{a}}$
896 <cell alignment="center" valignment="top" usebox="none">
899 \begin_layout Plain Layout
905 <cell alignment="center" valignment="top" usebox="none">
908 \begin_layout Plain Layout
909 \begin_inset Formula $1700\leq T\leq5000$
919 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
922 \begin_layout Plain Layout
928 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
931 \begin_layout Plain Layout
932 \begin_inset Formula $5000\leq$
948 \begin_layout Plain Layout
949 \begin_inset Formula $^{\textrm{a}}$
957 We will now write down the sign (and therefore stability) determining parts
958 of the left-hand sides of the inequalities (
959 \begin_inset CommandInset ref
966 \begin_inset CommandInset ref
973 \begin_inset CommandInset ref
981 stability equations of state
986 \begin_layout Standard
987 The sign determining part of inequality
992 \begin_inset CommandInset ref
999 \begin_inset Formula $3\Gamma_{1}-4$
1002 and it reduces to the criterion for dynamical stability
1003 \begin_inset Formula
1005 \Gamma_{1}>\frac{4}{3}\,\cdot
1010 Stability of the thermodynamical equilibrium demands
1011 \begin_inset Formula
1013 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1019 \begin_inset Formula
1026 holds for a wide range of physical situations.
1028 \begin_inset Formula
1030 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1031 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1032 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1037 we find the sign determining terms in inequalities
1038 \begin_inset space ~
1042 \begin_inset CommandInset ref
1044 reference "ZSSecSta"
1049 \begin_inset CommandInset ref
1051 reference "ZSVibSta"
1055 ) respectively and obtain the following form of the criteria for dynamical,
1056 secular and vibrational
1061 \begin_inset Formula
1063 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1064 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1065 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1070 The constitutive relations are to be evaluated for the unperturbed thermodynami
1072 \begin_inset Formula $(\rho_{0},T_{0})$
1076 We see that the one-zone stability of the layer depends only on the constitutiv
1078 \begin_inset Formula $\Gamma_{1}$
1082 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1086 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1090 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1094 These depend only on the unperturbed thermodynamical state of the layer.
1095 Therefore the above relations define the one-zone-stability equations of
1097 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1101 \begin_inset Formula $S_{\mathrm{vib}}$
1106 \begin_inset space ~
1110 \begin_inset CommandInset ref
1112 reference "fig:VibStabEquation"
1117 \begin_inset Formula $S_{\mathrm{vib}}$
1121 Regions of secular instability are listed in Table
1122 \begin_inset space ~
1128 \begin_layout Standard
1129 \begin_inset Float figure
1134 \begin_layout Plain Layout
1135 \begin_inset Caption Standard
1137 \begin_layout Plain Layout
1138 \begin_inset CommandInset label
1140 name "fig:VibStabEquation"
1144 Vibrational stability equation of state
1145 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1150 \begin_inset Formula $>0$
1153 means vibrational stability
1166 \begin_layout Section
1170 \begin_layout Enumerate
1171 The conditions for the stability of static, radiative layers in gas spheres,
1172 as described by Baker's (
1173 \begin_inset CommandInset citation
1180 ) standard one-zone model, can be expressed as stability equations of state.
1181 These stability equations of state depend only on the local thermodynamic
1186 \begin_layout Enumerate
1187 If the constitutive relations – equations of state and Rosseland mean opacities
1188 – are specified, the stability equations of state can be evaluated without
1189 specifying properties of the layer.
1193 \begin_layout Enumerate
1194 For solar composition gas the
1195 \begin_inset Formula $\kappa$
1198 -mechanism is working in the regions of the ice and dust features in the
1200 \begin_inset Formula $\mathrm{H}_{2}$
1203 dissociation and the combined H, first He ionization zone, as indicated
1204 by vibrational instability.
1205 These regions of instability are much larger in extent and degree of instabilit
1206 y than the second He ionization zone that drives the Cepheïd pulsations.
1210 \begin_layout Acknowledgement
1211 Part of this work was supported by the German
1213 Deut\SpecialChar softhyphen
1214 sche For\SpecialChar softhyphen
1215 schungs\SpecialChar softhyphen
1216 ge\SpecialChar softhyphen
1217 mein\SpecialChar softhyphen
1221 \begin_inset space ~
1227 \begin_layout Standard
1228 \begin_inset CommandInset bibtex
1230 btprint "btPrintAll"
1231 bibfiles "biblioExample"
1237 \begin_inset Note Note
1240 \begin_layout Plain Layout
1245 If you cannot see the bibliography in the output, assure that you have
1246 given the full path to the Bib\SpecialChar TeX
1251 that is part of the A&A \SpecialChar LaTeX