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80 \begin_inset Note Note
83 \begin_layout Plain Layout
88 This is an example \SpecialChar LyX
89 file for articles to be submitted to the Journal of
90 Astronomy & Astrophysics (A&A).
91 How to install the A&A \SpecialChar LaTeX
92 class to your \SpecialChar LaTeX
93 system is explained in
97 \begin_layout Plain Layout
99 http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
105 \begin_inset Newline newline
108 Depending on the submission state and the abstract layout, you need to use
109 different document class options that are listed in the aa manual.
112 \begin_inset Newline newline
120 If you use accented characters in your document, you must use the predefined
121 document class option
125 in the document settings.
134 Hydrodynamics of giant planet formation
137 \begin_layout Subtitle
140 \begin_inset Formula $\kappa$
149 \begin_inset Flex institutemark
152 \begin_layout Plain Layout
162 \begin_layout Plain Layout
173 \begin_inset Flex institutemark
176 \begin_layout Plain Layout
186 \begin_layout Plain Layout
199 \begin_layout Plain Layout
200 Just to show the usage of the elements in the author field
206 \begin_inset Note Note
209 \begin_layout Plain Layout
212 fnmsep is only needed for more than one consecutive notes/marks
220 \begin_layout Offprint
225 \begin_layout Address
226 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
228 \begin_inset Newline newline
232 \begin_inset Flex Email
235 \begin_layout Plain Layout
236 wuchterl@amok.ast.univie.ac.at
245 \begin_layout Plain Layout
254 University of Alexandria, Department of Geography, ...
255 \begin_inset Newline newline
259 \begin_inset Flex Email
262 \begin_layout Plain Layout
263 c.ptolemy@hipparch.uheaven.space
272 \begin_layout Plain Layout
273 The university of heaven temporarily does not accept e-mails
282 Received September 15, 1996; accepted March 16, 1997
285 \begin_layout Abstract (unstructured)
286 To investigate the physical nature of the `nuc\SpecialChar softhyphen
287 leated instability' of proto
288 giant planets, the stability of layers in static, radiative gas spheres
289 is analysed on the basis of Baker's standard one-zone model.
290 It is shown that stability depends only upon the equations of state, the
291 opacities and the local thermodynamic state in the layer.
292 Stability and instability can therefore be expressed in the form of stability
293 equations of state which are universal for a given composition.
294 The stability equations of state are calculated for solar composition and
295 are displayed in the domain
296 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
300 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
304 These displays may be used to determine the one-zone stability of layers
305 in stellar or planetary structure models by directly reading off the value
306 of the stability equations for the thermodynamic state of these layers,
307 specified by state quantities as density
308 \begin_inset Formula $\rho$
312 \begin_inset Formula $T$
315 or specific internal energy
316 \begin_inset Formula $e$
320 Regions of instability in the
321 \begin_inset Formula $(\rho,e)$
324 -plane are described and related to the underlying microphysical processes.
325 Vibrational instability is found to be a common phenomenon at temperatures
326 lower than the second He ionisation zone.
328 \begin_inset Formula $\kappa$
331 -mechanism is widespread under `cool' conditions.
332 \begin_inset Note Note
335 \begin_layout Plain Layout
336 Citations are not allowed in A&A abstracts.
342 \begin_inset Note Note
345 \begin_layout Plain Layout
346 This is the unstructured abstract type, an example for the structured abstract
351 template file that comes with \SpecialChar LyX
360 \begin_layout Keywords
361 giant planet formation –
362 \begin_inset Formula $\kappa$
365 -mechanism – stability of gas spheres
368 \begin_layout Section
372 \begin_layout Standard
375 nucleated instability
377 (also called core instability) hypothesis of giant planet formation, a
378 critical mass for static core envelope protoplanets has been found.
380 \begin_inset CommandInset citation
387 ) determined the critical mass of the core to be about
388 \begin_inset Formula $12\,M_{\oplus}$
392 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
395 is the Earth mass), which is independent of the outer boundary conditions
396 and therefore independent of the location in the solar nebula.
397 This critical value for the core mass corresponds closely to the cores
398 of today's giant planets.
401 \begin_layout Standard
402 Although no hydrodynamical study has been available many workers conjectured
403 that a collapse or rapid contraction will ensue after accumulating the
405 The main motivation for this article is to investigate the stability of
406 the static envelope at the critical mass.
407 With this aim the local, linear stability of static radiative gas spheres
408 is investigated on the basis of Baker's (
409 \begin_inset CommandInset citation
416 ) standard one-zone model.
419 \begin_layout Standard
420 Phenomena similar to the ones described above for giant planet formation
421 have been found in hydrodynamical models concerning star formation where
422 protostellar cores explode (Tscharnuter
423 \begin_inset CommandInset citation
431 \begin_inset CommandInset citation
438 ), whereas earlier studies found quasi-steady collapse flows.
439 The similarities in the (micro)physics, i.
440 \begin_inset space \thinspace{}
444 \begin_inset space \space{}
447 constitutive relations of protostellar cores and protogiant planets serve
448 as a further motivation for this study.
451 \begin_layout Section
452 Baker's standard one-zone model
455 \begin_layout Standard
456 \begin_inset Float figure
461 \begin_layout Plain Layout
462 \begin_inset Caption Standard
464 \begin_layout Plain Layout
465 \begin_inset CommandInset label
472 \begin_inset Formula $\Gamma_{1}$
477 \begin_inset Formula $\Gamma_{1}$
480 is plotted as a function of
481 \begin_inset Formula $\lg$
485 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
489 \begin_inset Formula $\lg$
493 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
506 In this section the one-zone model of Baker (
507 \begin_inset CommandInset citation
514 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
516 The resulting stability criteria will be rewritten in terms of local state
517 variables, local timescales and constitutive relations.
520 \begin_layout Standard
522 \begin_inset CommandInset citation
529 ) investigates the stability of thin layers in self-gravitating, spherical
530 gas clouds with the following properties:
533 \begin_layout Itemize
534 hydrostatic equilibrium,
537 \begin_layout Itemize
541 \begin_layout Itemize
542 energy transport by grey radiation diffusion.
546 \begin_layout Standard
548 For the one-zone-model Baker obtains necessary conditions for dynamical,
549 secular and vibrational (or pulsational) stability (Eqs.
550 \begin_inset space \space{}
554 \begin_inset space \thinspace{}
558 \begin_inset space \thinspace{}
562 \begin_inset CommandInset citation
570 Using Baker's notation:
573 \begin_layout Standard
577 M_{r} & & \textrm{mass internal to the radius }r\\
578 m & & \textrm{mass of the zone}\\
579 r_{0} & & \textrm{unperturbed zone radius}\\
580 \rho_{0} & & \textrm{unperturbed density in the zone}\\
581 T_{0} & & \textrm{unperturbed temperature in the zone}\\
582 L_{r0} & & \textrm{unperturbed luminosity}\\
583 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
591 \begin_layout Standard
593 and with the definitions of the
602 \begin_inset CommandInset ref
604 reference "fig:FigGam"
611 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
623 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
629 \begin_inset Formula $K$
633 \begin_inset Formula $\sigma_{0}$
636 have the following form:
639 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
640 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
646 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
653 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
660 is a thermodynamical quantity which is of order
661 \begin_inset Formula $1$
665 \begin_inset Formula $1$
668 for nonreacting mixtures of classical perfect gases.
669 The physical meaning of
670 \begin_inset Formula $\sigma_{0}$
674 \begin_inset Formula $K$
677 is clearly visible in the equations above.
679 \begin_inset Formula $\sigma_{0}$
682 represents a frequency of the order one per free-fall time.
684 \begin_inset Formula $K$
687 is proportional to the ratio of the free-fall time and the cooling time.
688 Substituting into Baker's criteria, using thermodynamic identities and
689 definitions of thermodynamic quantities,
692 \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}
700 \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}
705 one obtains, after some pages of algebra, the conditions for
712 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
713 \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}\\
714 \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}
719 For a physical discussion of the stability criteria see Baker (
720 \begin_inset CommandInset citation
728 \begin_inset CommandInset citation
738 \begin_layout Standard
739 We observe that these criteria for dynamical, secular and vibrational stability,
740 respectively, can be factorized into
743 \begin_layout Enumerate
744 a factor containing local timescales only,
747 \begin_layout Enumerate
748 a factor containing only constitutive relations and their derivatives.
752 \begin_layout Standard
753 The first factors, depending on only timescales, are positive by definition.
754 The signs of the left hand sides of the inequalities
759 \begin_inset CommandInset ref
766 \begin_inset CommandInset ref
773 \begin_inset CommandInset ref
779 ) therefore depend exclusively on the second factors containing the constitutive
781 Since they depend only on state variables, the stability criteria themselves
784 functions of the thermodynamic state in the local zone
787 The one-zone stability can therefore be determined from a simple equation
788 of state, given for example, as a function of density and temperature.
789 Once the microphysics, i.
790 \begin_inset space \thinspace{}
794 \begin_inset space \space{}
797 the thermodynamics and opacities (see Table
802 \begin_inset CommandInset ref
804 reference "tab:KapSou"
808 ), are specified (in practice by specifying a chemical composition) the
809 one-zone stability can be inferred if the thermodynamic state is specified.
810 The zone – or in other words the layer – will be stable or unstable in
811 whatever object it is imbedded as long as it satisfies the one-zone-model
813 Only the specific growth rates (depending upon the time scales) will be
814 different for layers in different objects.
817 \begin_layout Standard
818 \begin_inset Float table
823 \begin_layout Plain Layout
824 \begin_inset Caption Standard
826 \begin_layout Plain Layout
827 \begin_inset CommandInset label
841 \begin_layout Plain Layout
844 <lyxtabular version="3" rows="4" columns="2">
845 <features tabularvalignment="middle">
846 <column alignment="left" valignment="top" width="0pt">
847 <column alignment="left" valignment="top" width="0pt">
849 <cell alignment="center" valignment="top" topline="true" usebox="none">
852 \begin_layout Plain Layout
858 <cell alignment="center" valignment="top" topline="true" usebox="none">
861 \begin_layout Plain Layout
862 \begin_inset Formula $T/[\textrm{K}]$
872 <cell alignment="center" valignment="top" topline="true" usebox="none">
875 \begin_layout Plain Layout
876 Yorke 1979, Yorke 1980a
881 <cell alignment="center" valignment="top" topline="true" usebox="none">
884 \begin_layout Plain Layout
885 \begin_inset Formula $\leq1700^{\textrm{a}}$
895 <cell alignment="center" valignment="top" usebox="none">
898 \begin_layout Plain Layout
904 <cell alignment="center" valignment="top" usebox="none">
907 \begin_layout Plain Layout
908 \begin_inset Formula $1700\leq T\leq5000$
918 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
921 \begin_layout Plain Layout
927 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
930 \begin_layout Plain Layout
931 \begin_inset Formula $5000\leq$
947 \begin_layout Plain Layout
948 \begin_inset Formula $^{\textrm{a}}$
956 We will now write down the sign (and therefore stability) determining parts
957 of the left-hand sides of the inequalities (
958 \begin_inset CommandInset ref
965 \begin_inset CommandInset ref
972 \begin_inset CommandInset ref
980 stability equations of state
985 \begin_layout Standard
986 The sign determining part of inequality
991 \begin_inset CommandInset ref
998 \begin_inset Formula $3\Gamma_{1}-4$
1001 and it reduces to the criterion for dynamical stability
1002 \begin_inset Formula
1004 \Gamma_{1}>\frac{4}{3}\,\cdot
1009 Stability of the thermodynamical equilibrium demands
1010 \begin_inset Formula
1012 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1018 \begin_inset Formula
1025 holds for a wide range of physical situations.
1027 \begin_inset Formula
1029 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1030 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1031 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1036 we find the sign determining terms in inequalities
1037 \begin_inset space ~
1041 \begin_inset CommandInset ref
1043 reference "ZSSecSta"
1048 \begin_inset CommandInset ref
1050 reference "ZSVibSta"
1054 ) respectively and obtain the following form of the criteria for dynamical,
1055 secular and vibrational
1060 \begin_inset Formula
1062 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1063 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1064 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1069 The constitutive relations are to be evaluated for the unperturbed thermodynami
1071 \begin_inset Formula $(\rho_{0},T_{0})$
1075 We see that the one-zone stability of the layer depends only on the constitutiv
1077 \begin_inset Formula $\Gamma_{1}$
1081 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1085 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1089 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1093 These depend only on the unperturbed thermodynamical state of the layer.
1094 Therefore the above relations define the one-zone-stability equations of
1096 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1100 \begin_inset Formula $S_{\mathrm{vib}}$
1105 \begin_inset space ~
1109 \begin_inset CommandInset ref
1111 reference "fig:VibStabEquation"
1116 \begin_inset Formula $S_{\mathrm{vib}}$
1120 Regions of secular instability are listed in Table
1121 \begin_inset space ~
1127 \begin_layout Standard
1128 \begin_inset Float figure
1133 \begin_layout Plain Layout
1134 \begin_inset Caption Standard
1136 \begin_layout Plain Layout
1137 \begin_inset CommandInset label
1139 name "fig:VibStabEquation"
1143 Vibrational stability equation of state
1144 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1149 \begin_inset Formula $>0$
1152 means vibrational stability
1165 \begin_layout Section
1169 \begin_layout Enumerate
1170 The conditions for the stability of static, radiative layers in gas spheres,
1171 as described by Baker's (
1172 \begin_inset CommandInset citation
1179 ) standard one-zone model, can be expressed as stability equations of state.
1180 These stability equations of state depend only on the local thermodynamic
1185 \begin_layout Enumerate
1186 If the constitutive relations – equations of state and Rosseland mean opacities
1187 – are specified, the stability equations of state can be evaluated without
1188 specifying properties of the layer.
1192 \begin_layout Enumerate
1193 For solar composition gas the
1194 \begin_inset Formula $\kappa$
1197 -mechanism is working in the regions of the ice and dust features in the
1199 \begin_inset Formula $\mathrm{H}_{2}$
1202 dissociation and the combined H, first He ionization zone, as indicated
1203 by vibrational instability.
1204 These regions of instability are much larger in extent and degree of instabilit
1205 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1209 \begin_layout Acknowledgement
1210 Part of this work was supported by the German
1212 Deut\SpecialChar softhyphen
1213 sche For\SpecialChar softhyphen
1214 schungs\SpecialChar softhyphen
1215 ge\SpecialChar softhyphen
1216 mein\SpecialChar softhyphen
1220 \begin_inset space ~
1226 \begin_layout Standard
1227 \begin_inset CommandInset bibtex
1229 btprint "btPrintAll"
1230 bibfiles "biblioExample"
1236 \begin_inset Note Note
1239 \begin_layout Plain Layout
1244 If you cannot see the bibliography in the output, assure that you have
1245 gievn the full path to the Bib\SpecialChar TeX
1250 that is part of the A&A \SpecialChar LaTeX