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61 \quotes_language english
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75 \begin_inset Note Note
78 \begin_layout Plain Layout
83 This is an example \SpecialCharNoPassThru LyX
84 file for articles to be submitted to the Journal
85 of Astronomy & Astrophysics (A&A).
86 How to install the A&A \SpecialCharNoPassThru LaTeX
87 class to your \SpecialCharNoPassThru LaTeX
88 system is explained in
93 \begin_layout Plain Layout
95 http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
101 \begin_inset Newline newline
104 Depending on the submission state and the abstract layout, you need to use
105 different document class options that are listed in the aa manual.
108 \begin_inset Newline newline
116 If you use accented characters in your document, you must use the predefined
117 document class option
121 in the document settings.
130 Hydrodynamics of giant planet formation
133 \begin_layout Subtitle
136 \begin_inset Formula $\kappa$
145 \begin_inset Flex institutemark
148 \begin_layout Plain Layout
158 \begin_layout Plain Layout
169 \begin_inset Flex institutemark
172 \begin_layout Plain Layout
182 \begin_layout Plain Layout
195 \begin_layout Plain Layout
196 Just to show the usage of the elements in the author field
202 \begin_inset Note Note
205 \begin_layout Plain Layout
208 fnmsep is only needed for more than one consecutive notes/marks
216 \begin_layout Offprint
221 \begin_layout Address
222 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
224 \begin_inset Newline newline
228 \begin_inset Flex Email
231 \begin_layout Plain Layout
232 wuchterl@amok.ast.univie.ac.at
241 \begin_layout Plain Layout
250 University of Alexandria, Department of Geography, ...
251 \begin_inset Newline newline
255 \begin_inset Flex Email
258 \begin_layout Plain Layout
259 c.ptolemy@hipparch.uheaven.space
268 \begin_layout Plain Layout
269 The university of heaven temporarily does not accept e-mails
278 Received September 15, 1996; accepted March 16, 1997
281 \begin_layout Abstract (unstructured)
282 To investigate the physical nature of the `nuc\SpecialChar softhyphen
283 leated instability' of proto
284 giant planets, the stability of layers in static, radiative gas spheres
285 is analysed on the basis of Baker's standard one-zone model.
286 It is shown that stability depends only upon the equations of state, the
287 opacities and the local thermodynamic state in the layer.
288 Stability and instability can therefore be expressed in the form of stability
289 equations of state which are universal for a given composition.
290 The stability equations of state are calculated for solar composition and
291 are displayed in the domain
292 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
296 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
300 These displays may be used to determine the one-zone stability of layers
301 in stellar or planetary structure models by directly reading off the value
302 of the stability equations for the thermodynamic state of these layers,
303 specified by state quantities as density
304 \begin_inset Formula $\rho$
308 \begin_inset Formula $T$
311 or specific internal energy
312 \begin_inset Formula $e$
316 Regions of instability in the
317 \begin_inset Formula $(\rho,e)$
320 -plane are described and related to the underlying microphysical processes.
321 Vibrational instability is found to be a common phenomenon at temperatures
322 lower than the second He ionisation zone.
324 \begin_inset Formula $\kappa$
327 -mechanism is widespread under `cool' conditions.
328 \begin_inset Note Note
331 \begin_layout Plain Layout
332 Citations are not allowed in A&A abstracts.
338 \begin_inset Note Note
341 \begin_layout Plain Layout
342 This is the unstructured abstract type, an example for the structured abstract
347 template file that comes with \SpecialCharNoPassThru LyX
356 \begin_layout Keywords
357 giant planet formation \twohyphens
359 \begin_inset Formula $\kappa$
362 -mechanism \twohyphens
363 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
384 ) determined the critical mass of the core to be about
385 \begin_inset Formula $12\, M_{\oplus}$
389 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
392 is the Earth mass), which is independent of the outer boundary conditions
393 and therefore independent of the location in the solar nebula.
394 This critical value for the core mass corresponds closely to the cores
395 of today's giant planets.
398 \begin_layout Standard
399 Although no hydrodynamical study has been available many workers conjectured
400 that a collapse or rapid contraction will ensue after accumulating the
402 The main motivation for this article is to investigate the stability of
403 the static envelope at the critical mass.
404 With this aim the local, linear stability of static radiative gas spheres
405 is investigated on the basis of Baker's (
406 \begin_inset CommandInset citation
412 ) standard one-zone model.
415 \begin_layout Standard
416 Phenomena similar to the ones described above for giant planet formation
417 have been found in hydrodynamical models concerning star formation where
418 protostellar cores explode (Tscharnuter
419 \begin_inset CommandInset citation
426 \begin_inset CommandInset citation
432 ), whereas earlier studies found quasi-steady collapse flows.
433 The similarities in the (micro)physics, i.
434 \begin_inset space \thinspace{}
438 \begin_inset space \space{}
441 constitutive relations of protostellar cores and protogiant planets serve
442 as a further motivation for this study.
445 \begin_layout Section
446 Baker's standard one-zone model
449 \begin_layout Standard
450 \begin_inset Float figure
455 \begin_layout Plain Layout
456 \begin_inset Caption Standard
458 \begin_layout Plain Layout
459 \begin_inset CommandInset label
466 \begin_inset Formula $\Gamma_{1}$
471 \begin_inset Formula $\Gamma_{1}$
474 is plotted as a function of
475 \begin_inset Formula $\lg$
479 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
483 \begin_inset Formula $\lg$
487 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
500 In this section the one-zone model of Baker (
501 \begin_inset CommandInset citation
507 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
509 The resulting stability criteria will be rewritten in terms of local state
510 variables, local timescales and constitutive relations.
513 \begin_layout Standard
515 \begin_inset CommandInset citation
521 ) investigates the stability of thin layers in self-gravitating, spherical
522 gas clouds with the following properties:
525 \begin_layout Itemize
526 hydrostatic equilibrium,
529 \begin_layout Itemize
533 \begin_layout Itemize
534 energy transport by grey radiation diffusion.
538 \begin_layout Standard
540 For the one-zone-model Baker obtains necessary conditions for dynamical,
541 secular and vibrational (or pulsational) stability (Eqs.
542 \begin_inset space \space{}
546 \begin_inset space \thinspace{}
550 \begin_inset space \thinspace{}
554 \begin_inset CommandInset citation
561 Using Baker's notation:
562 \begin_inset Separator parbreak
567 \begin_layout Standard
571 M_{r} & & \textrm{mass internal to the radius }r\\
572 m & & \textrm{mass of the zone}\\
573 r_{0} & & \textrm{unperturbed zone radius}\\
574 \rho_{0} & & \textrm{unperturbed density in the zone}\\
575 T_{0} & & \textrm{unperturbed temperature in the zone}\\
576 L_{r0} & & \textrm{unperturbed luminosity}\\
577 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
585 \begin_layout Standard
587 and with the definitions of the
596 \begin_inset CommandInset ref
598 reference "fig:FigGam"
605 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
617 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
623 \begin_inset Formula $K$
627 \begin_inset Formula $\sigma_{0}$
630 have the following form:
633 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
634 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
640 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
647 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
654 is a thermodynamical quantity which is of order
655 \begin_inset Formula $1$
659 \begin_inset Formula $1$
662 for nonreacting mixtures of classical perfect gases.
663 The physical meaning of
664 \begin_inset Formula $\sigma_{0}$
668 \begin_inset Formula $K$
671 is clearly visible in the equations above.
673 \begin_inset Formula $\sigma_{0}$
676 represents a frequency of the order one per free-fall time.
678 \begin_inset Formula $K$
681 is proportional to the ratio of the free-fall time and the cooling time.
682 Substituting into Baker's criteria, using thermodynamic identities and
683 definitions of thermodynamic quantities,
686 \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}
694 \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}
699 one obtains, after some pages of algebra, the conditions for
706 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
707 \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}\\
708 \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}
713 For a physical discussion of the stability criteria see Baker (
714 \begin_inset CommandInset citation
721 \begin_inset CommandInset citation
730 \begin_layout Standard
731 We observe that these criteria for dynamical, secular and vibrational stability,
732 respectively, can be factorized into
735 \begin_layout Enumerate
736 a factor containing local timescales only,
739 \begin_layout Enumerate
740 a factor containing only constitutive relations and their derivatives.
744 \begin_layout Standard
745 The first factors, depending on only timescales, are positive by definition.
746 The signs of the left hand sides of the inequalities
751 \begin_inset CommandInset ref
758 \begin_inset CommandInset ref
765 \begin_inset CommandInset ref
771 ) therefore depend exclusively on the second factors containing the constitutive
773 Since they depend only on state variables, the stability criteria themselves
776 functions of the thermodynamic state in the local zone
779 The one-zone stability can therefore be determined from a simple equation
780 of state, given for example, as a function of density and temperature.
781 Once the microphysics, i.
782 \begin_inset space \thinspace{}
786 \begin_inset space \space{}
789 the thermodynamics and opacities (see Table
794 \begin_inset CommandInset ref
796 reference "tab:KapSou"
800 ), are specified (in practice by specifying a chemical composition) the
801 one-zone stability can be inferred if the thermodynamic state is specified.
803 or in other words the layer \twohyphens
804 will be stable or unstable in
805 whatever object it is imbedded as long as it satisfies the one-zone-model
807 Only the specific growth rates (depending upon the time scales) will be
808 different for layers in different objects.
811 \begin_layout Standard
812 \begin_inset Float table
817 \begin_layout Plain Layout
818 \begin_inset Caption Standard
820 \begin_layout Plain Layout
821 \begin_inset CommandInset label
835 \begin_layout Plain Layout
838 <lyxtabular version="3" rows="4" columns="2">
839 <features rotate="0" tabularvalignment="middle">
840 <column alignment="left" valignment="top" width="0pt">
841 <column alignment="left" valignment="top" width="0pt">
843 <cell alignment="center" valignment="top" topline="true" usebox="none">
846 \begin_layout Plain Layout
852 <cell alignment="center" valignment="top" topline="true" usebox="none">
855 \begin_layout Plain Layout
856 \begin_inset Formula $T/[\textrm{K}]$
866 <cell alignment="center" valignment="top" topline="true" usebox="none">
869 \begin_layout Plain Layout
870 Yorke 1979, Yorke 1980a
875 <cell alignment="center" valignment="top" topline="true" usebox="none">
878 \begin_layout Plain Layout
879 \begin_inset Formula $\leq1700^{\textrm{a}}$
889 <cell alignment="center" valignment="top" usebox="none">
892 \begin_layout Plain Layout
898 <cell alignment="center" valignment="top" usebox="none">
901 \begin_layout Plain Layout
902 \begin_inset Formula $1700\leq T\leq5000$
912 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
915 \begin_layout Plain Layout
921 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
924 \begin_layout Plain Layout
925 \begin_inset Formula $5000\leq$
941 \begin_layout Plain Layout
942 \begin_inset Formula $^{\textrm{a}}$
950 We will now write down the sign (and therefore stability) determining parts
951 of the left-hand sides of the inequalities (
952 \begin_inset CommandInset ref
959 \begin_inset CommandInset ref
966 \begin_inset CommandInset ref
974 stability equations of state
979 \begin_layout Standard
980 The sign determining part of inequality
985 \begin_inset CommandInset ref
992 \begin_inset Formula $3\Gamma_{1}-4$
995 and it reduces to the criterion for dynamical stability
998 \Gamma_{1}>\frac{4}{3}\,\cdot
1003 Stability of the thermodynamical equilibrium demands
1004 \begin_inset Formula
1006 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,
1012 \begin_inset Formula
1019 holds for a wide range of physical situations.
1021 \begin_inset Formula
1023 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1024 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1025 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1030 we find the sign determining terms in inequalities
1031 \begin_inset space ~
1035 \begin_inset CommandInset ref
1037 reference "ZSSecSta"
1042 \begin_inset CommandInset ref
1044 reference "ZSVibSta"
1048 ) respectively and obtain the following form of the criteria for dynamical,
1049 secular and vibrational
1054 \begin_inset Formula
1056 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1057 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1058 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1063 The constitutive relations are to be evaluated for the unperturbed thermodynami
1065 \begin_inset Formula $(\rho_{0},T_{0})$
1069 We see that the one-zone stability of the layer depends only on the constitutiv
1071 \begin_inset Formula $\Gamma_{1}$
1075 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1079 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1083 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1087 These depend only on the unperturbed thermodynamical state of the layer.
1088 Therefore the above relations define the one-zone-stability equations of
1090 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1094 \begin_inset Formula $S_{\mathrm{vib}}$
1099 \begin_inset space ~
1103 \begin_inset CommandInset ref
1105 reference "fig:VibStabEquation"
1110 \begin_inset Formula $S_{\mathrm{vib}}$
1114 Regions of secular instability are listed in Table
1115 \begin_inset space ~
1121 \begin_layout Standard
1122 \begin_inset Float figure
1127 \begin_layout Plain Layout
1128 \begin_inset Caption Standard
1130 \begin_layout Plain Layout
1131 \begin_inset CommandInset label
1133 name "fig:VibStabEquation"
1137 Vibrational stability equation of state
1138 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1143 \begin_inset Formula $>0$
1146 means vibrational stability
1159 \begin_layout Section
1163 \begin_layout Enumerate
1164 The conditions for the stability of static, radiative layers in gas spheres,
1165 as described by Baker's (
1166 \begin_inset CommandInset citation
1172 ) standard one-zone model, can be expressed as stability equations of state.
1173 These stability equations of state depend only on the local thermodynamic
1178 \begin_layout Enumerate
1179 If the constitutive relations \twohyphens
1180 equations of state and Rosseland mean opacities
1182 are specified, the stability equations of state can be evaluated without
1183 specifying properties of the layer.
1187 \begin_layout Enumerate
1188 For solar composition gas the
1189 \begin_inset Formula $\kappa$
1192 -mechanism is working in the regions of the ice and dust features in the
1194 \begin_inset Formula $\mathrm{H}_{2}$
1197 dissociation and the combined H, first He ionization zone, as indicated
1198 by vibrational instability.
1199 These regions of instability are much larger in extent and degree of instabilit
1200 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1204 \begin_layout Acknowledgement
1205 Part of this work was supported by the German
1207 Deut\SpecialChar softhyphen
1208 sche For\SpecialChar softhyphen
1209 schungs\SpecialChar softhyphen
1210 ge\SpecialChar softhyphen
1211 mein\SpecialChar softhyphen
1215 \begin_inset space ~
1222 \begin_layout Standard
1223 \begin_inset CommandInset bibtex
1225 btprint "btPrintAll"
1226 bibfiles "biblioExample"
1232 \begin_inset Note Note
1235 \begin_layout Plain Layout
1240 If you cannot see the bibliography in the output, assure that you have
1241 gievn the full path to the Bib\SpecialCharNoPassThru TeX
1246 that is part of the A&A \SpecialCharNoPassThru LaTeX