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89 \begin_inset Note Note
92 \begin_layout Plain Layout
97 This is an example \SpecialChar LyX
98 file for articles to be submitted to the Journal of Astronomy & Astrophysics (A&A).
99 How to install the A&A \SpecialChar LaTeX
100 class to your \SpecialChar LaTeX
101 system is explained in
102 \begin_inset Flex URL
105 \begin_layout Plain Layout
107 https://wiki.lyx.org/Layouts/Astronomy-Astrophysics
113 \begin_inset Newline newline
116 Depending on the submission state and the abstract layout,
117 you need to use different document class options that are listed in the aa manual.
120 \begin_inset Newline newline
129 If you use accented characters in your document,
130 you must use the predefined document class option
134 in the document settings.
143 Hydrodynamics of giant planet formation
146 \begin_layout Subtitle
149 \begin_inset Formula $\kappa$
158 \begin_inset Flex institutemark
161 \begin_layout Plain Layout
171 \begin_layout Plain Layout
182 \begin_inset Flex institutemark
185 \begin_layout Plain Layout
195 \begin_layout Plain Layout
208 \begin_layout Plain Layout
209 Just to show the usage of the elements in the author field
215 \begin_inset Note Note
218 \begin_layout Plain Layout
221 fnmsep is only needed for more than one consecutive notes/marks
229 \begin_layout Offprint
234 \begin_layout Address
235 Institute for Astronomy (IfA),
236 University of Vienna,
237 Türkenschanzstrasse 17,
239 \begin_inset Newline newline
243 \begin_inset Flex Email
246 \begin_layout Plain Layout
247 wuchterl@amok.ast.univie.ac.at
256 \begin_layout Plain Layout
265 University of Alexandria,
266 Department of Geography,
268 \begin_inset Newline newline
272 \begin_inset Flex Email
275 \begin_layout Plain Layout
276 c.ptolemy@hipparch.uheaven.space
285 \begin_layout Plain Layout
286 The university of heaven temporarily does not accept e-mails
295 Received September 15,
301 \begin_layout Abstract (unstructured)
302 To investigate the physical nature of the `nuc\SpecialChar softhyphen
303 leated instability' of proto giant planets,
304 the stability of layers in static,
305 radiative gas spheres is analysed on the basis of Baker's standard one-zone model.
306 It is shown that stability depends only upon the equations of state,
307 the opacities and the local thermodynamic state in the layer.
308 Stability and instability can therefore be expressed in the form of stability equations of state which are universal for a given composition.
309 The stability equations of state are calculated for solar composition and are displayed in the domain
310 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
315 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
319 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,
320 specified by state quantities as density
321 \begin_inset Formula $\rho$
326 \begin_inset Formula $T$
329 or specific internal energy
330 \begin_inset Formula $e$
334 Regions of instability in the
335 \begin_inset Formula $(\rho,e)$
338 -plane are described and related to the underlying microphysical processes.
339 Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone.
341 \begin_inset Formula $\kappa$
344 -mechanism is widespread under `cool' conditions.
345 \begin_inset Note Note
348 \begin_layout Plain Layout
349 Citations are not allowed in A&A abstracts.
355 \begin_inset Note Note
358 \begin_layout Plain Layout
359 This is the unstructured abstract type,
360 an example for the structured abstract is in the
364 template file that comes with \SpecialChar LyX
373 \begin_layout Keywords
374 giant planet formation –
375 \begin_inset Formula $\kappa$
378 -mechanism – stability of gas spheres
381 \begin_layout Section
385 \begin_layout Standard
388 nucleated instability
390 (also called core instability) hypothesis of giant planet formation,
391 a critical mass for static core envelope protoplanets has been found.
393 \begin_inset CommandInset citation
400 ) determined the critical mass of the core to be about
401 \begin_inset Formula $12\,M_{\oplus}$
405 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
409 which is independent of the outer boundary conditions and therefore independent of the location in the solar nebula.
410 This critical value for the core mass corresponds closely to the cores of today's giant planets.
413 \begin_layout Standard
414 Although no hydrodynamical study has been available many workers conjectured that a collapse or rapid contraction will ensue after accumulating the critical mass.
415 The main motivation for this article is to investigate the stability of the static envelope at the critical mass.
416 With this aim the local,
417 linear stability of static radiative gas spheres is investigated on the basis of Baker's (
418 \begin_inset CommandInset citation
425 ) standard one-zone model.
428 \begin_layout Standard
429 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
430 \begin_inset CommandInset citation
439 \begin_inset CommandInset citation
447 whereas earlier studies found quasi-steady collapse flows.
448 The similarities in the (micro)physics,
450 \begin_inset space \thinspace{}
454 \begin_inset space \space{}
457 constitutive relations of protostellar cores and protogiant planets serve as a further motivation for this study.
460 \begin_layout Section
461 Baker's standard one-zone model
464 \begin_layout Standard
465 \begin_inset Float figure
472 \begin_layout Plain Layout
473 \begin_inset Caption Standard
475 \begin_layout Plain Layout
476 \begin_inset CommandInset label
483 \begin_inset Formula $\Gamma_{1}$
488 \begin_inset Formula $\Gamma_{1}$
491 is plotted as a function of
492 \begin_inset Formula $\lg$
496 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
500 \begin_inset Formula $\lg$
504 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
517 In this section the one-zone model of Baker (
518 \begin_inset CommandInset citation
526 originally used to study the Cepheïd pulsation mechanism,
527 will be briefly reviewed.
528 The resulting stability criteria will be rewritten in terms of local state variables,
529 local timescales and constitutive relations.
532 \begin_layout Standard
534 \begin_inset CommandInset citation
541 ) investigates the stability of thin layers in self-gravitating,
542 spherical gas clouds with the following properties:
546 \begin_layout Itemize
547 hydrostatic equilibrium,
551 \begin_layout Itemize
556 \begin_layout Itemize
557 energy transport by grey radiation diffusion.
561 \begin_layout Standard
563 For the one-zone-model Baker obtains necessary conditions for dynamical,
564 secular and vibrational (or pulsational) stability (Eqs.
565 \begin_inset space \space{}
569 \begin_inset space \thinspace{}
573 \begin_inset space \thinspace{}
577 \begin_inset CommandInset citation
585 Using Baker's notation:
588 \begin_layout Standard
592 M_{r} & & \textrm{mass internal to the radius }r\\
593 m & & \textrm{mass of the zone}\\
594 r_{0} & & \textrm{unperturbed zone radius}\\
595 \rho_{0} & & \textrm{unperturbed density in the zone}\\
596 T_{0} & & \textrm{unperturbed temperature in the zone}\\
597 L_{r0} & & \textrm{unperturbed luminosity}\\
598 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
606 \begin_layout Standard
608 and with the definitions of the
617 \begin_inset CommandInset ref
619 reference "fig:FigGam"
627 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
639 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
645 \begin_inset Formula $K$
649 \begin_inset Formula $\sigma_{0}$
652 have the following form:
656 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
657 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
663 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
670 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
677 is a thermodynamical quantity which is of order
678 \begin_inset Formula $1$
682 \begin_inset Formula $1$
685 for nonreacting mixtures of classical perfect gases.
686 The physical meaning of
687 \begin_inset Formula $\sigma_{0}$
691 \begin_inset Formula $K$
694 is clearly visible in the equations above.
696 \begin_inset Formula $\sigma_{0}$
699 represents a frequency of the order one per free-fall time.
701 \begin_inset Formula $K$
704 is proportional to the ratio of the free-fall time and the cooling time.
705 Substituting into Baker's criteria,
706 using thermodynamic identities and definitions of thermodynamic quantities,
710 \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}
718 \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}
724 after some pages of algebra,
733 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
734 \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}\\
735 \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}
740 For a physical discussion of the stability criteria see Baker (
741 \begin_inset CommandInset citation
749 \begin_inset CommandInset citation
759 \begin_layout Standard
760 We observe that these criteria for dynamical,
761 secular and vibrational stability,
763 can be factorized into
766 \begin_layout Enumerate
767 a factor containing local timescales only,
771 \begin_layout Enumerate
772 a factor containing only constitutive relations and their derivatives.
776 \begin_layout Standard
778 depending on only timescales,
779 are positive by definition.
780 The signs of the left hand sides of the inequalities
785 \begin_inset CommandInset ref
794 \begin_inset CommandInset ref
802 \begin_inset CommandInset ref
809 ) therefore depend exclusively on the second factors containing the constitutive relations.
810 Since they depend only on state variables,
811 the stability criteria themselves are
813 functions of the thermodynamic state in the local zone
816 The one-zone stability can therefore be determined from a simple equation of state,
818 as a function of density and temperature.
819 Once the microphysics,
821 \begin_inset space \thinspace{}
825 \begin_inset space \space{}
828 the thermodynamics and opacities (see Table
833 \begin_inset CommandInset ref
835 reference "tab:KapSou"
841 are specified (in practice by specifying a chemical composition) the one-zone stability can be inferred if the thermodynamic state is specified.
842 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.
843 Only the specific growth rates (depending upon the time scales) will be different for layers in different objects.
846 \begin_layout Standard
847 \begin_inset Float table
854 \begin_layout Plain Layout
855 \begin_inset Caption Standard
857 \begin_layout Plain Layout
858 \begin_inset CommandInset label
872 \begin_layout Plain Layout
875 <lyxtabular version="3" rows="4" columns="2">
876 <features tabularvalignment="middle">
877 <column alignment="left" valignment="top" width="0pt">
878 <column alignment="left" valignment="top" width="0pt">
880 <cell alignment="center" valignment="top" topline="true" usebox="none">
883 \begin_layout Plain Layout
889 <cell alignment="center" valignment="top" topline="true" usebox="none">
892 \begin_layout Plain Layout
893 \begin_inset Formula $T/[\textrm{K}]$
903 <cell alignment="center" valignment="top" topline="true" usebox="none">
906 \begin_layout Plain Layout
913 <cell alignment="center" valignment="top" topline="true" usebox="none">
916 \begin_layout Plain Layout
917 \begin_inset Formula $\leq1700^{\textrm{a}}$
927 <cell alignment="center" valignment="top" usebox="none">
930 \begin_layout Plain Layout
936 <cell alignment="center" valignment="top" usebox="none">
939 \begin_layout Plain Layout
940 \begin_inset Formula $1700\leq T\leq5000$
950 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
953 \begin_layout Plain Layout
959 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
962 \begin_layout Plain Layout
963 \begin_inset Formula $5000\leq$
979 \begin_layout Plain Layout
980 \begin_inset Formula $^{\textrm{a}}$
988 We will now write down the sign (and therefore stability) determining parts of the left-hand sides of the inequalities (
989 \begin_inset CommandInset ref
998 \begin_inset CommandInset ref
1000 reference "ZSSecSta"
1006 \begin_inset CommandInset ref
1008 reference "ZSVibSta"
1013 ) and thereby obtain
1015 stability equations of state
1020 \begin_layout Standard
1021 The sign determining part of inequality
1022 \begin_inset space ~
1026 \begin_inset CommandInset ref
1028 reference "ZSDynSta"
1034 \begin_inset Formula $3\Gamma_{1}-4$
1037 and it reduces to the criterion for dynamical stability
1038 \begin_inset Formula
1040 \Gamma_{1}>\frac{4}{3}\,\cdot
1045 Stability of the thermodynamical equilibrium demands
1046 \begin_inset Formula
1048 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1054 \begin_inset Formula
1061 holds for a wide range of physical situations.
1063 \begin_inset Formula
1065 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1066 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1067 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1072 we find the sign determining terms in inequalities
1073 \begin_inset space ~
1077 \begin_inset CommandInset ref
1079 reference "ZSSecSta"
1085 \begin_inset CommandInset ref
1087 reference "ZSVibSta"
1092 ) respectively and obtain the following form of the criteria for dynamical,
1093 secular and vibrational
1100 \begin_inset Formula
1102 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1103 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1104 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1109 The constitutive relations are to be evaluated for the unperturbed thermodynamic state (say
1110 \begin_inset Formula $(\rho_{0},T_{0})$
1114 We see that the one-zone stability of the layer depends only on the constitutive relations
1115 \begin_inset Formula $\Gamma_{1}$
1120 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1125 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1130 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1134 These depend only on the unperturbed thermodynamical state of the layer.
1135 Therefore the above relations define the one-zone-stability equations of state
1136 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1140 \begin_inset Formula $S_{\mathrm{vib}}$
1145 \begin_inset space ~
1149 \begin_inset CommandInset ref
1151 reference "fig:VibStabEquation"
1157 \begin_inset Formula $S_{\mathrm{vib}}$
1161 Regions of secular instability are listed in Table
1162 \begin_inset space ~
1168 \begin_layout Standard
1169 \begin_inset Float figure
1176 \begin_layout Plain Layout
1177 \begin_inset Caption Standard
1179 \begin_layout Plain Layout
1180 \begin_inset CommandInset label
1182 name "fig:VibStabEquation"
1186 Vibrational stability equation of state
1187 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1192 \begin_inset Formula $>0$
1195 means vibrational stability
1208 \begin_layout Section
1212 \begin_layout Enumerate
1213 The conditions for the stability of static,
1214 radiative layers in gas spheres,
1215 as described by Baker's (
1216 \begin_inset CommandInset citation
1223 ) standard one-zone model,
1224 can be expressed as stability equations of state.
1225 These stability equations of state depend only on the local thermodynamic state of the layer.
1229 \begin_layout Enumerate
1230 If the constitutive relations – equations of state and Rosseland mean opacities – are specified,
1231 the stability equations of state can be evaluated without specifying properties of the layer.
1235 \begin_layout Enumerate
1236 For solar composition gas the
1237 \begin_inset Formula $\kappa$
1240 -mechanism is working in the regions of the ice and dust features in the opacities,
1242 \begin_inset Formula $\mathrm{H}_{2}$
1245 dissociation and the combined H,
1246 first He ionization zone,
1247 as indicated by vibrational instability.
1248 These regions of instability are much larger in extent and degree of instability than the second He ionization zone that drives the Cepheïd pulsations.
1252 \begin_layout Acknowledgments
1253 Part of this work was supported by the German
1255 Deut\SpecialChar softhyphen
1256 sche For\SpecialChar softhyphen
1257 schungs\SpecialChar softhyphen
1258 ge\SpecialChar softhyphen
1259 mein\SpecialChar softhyphen
1264 \begin_inset space ~
1270 \begin_layout Standard
1271 \begin_inset CommandInset bibtex
1273 btprint "btPrintAll"
1274 bibfiles "../biblioExample"
1280 \begin_inset Note Note
1283 \begin_layout Plain Layout
1289 If you cannot see the bibliography in the output,
1290 assure that you have given the full path to the Bib\SpecialChar TeX
1295 that is part of the A&A \SpecialChar LaTeX