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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 \SpecialChar LyX
84 file for articles to be submitted to the Journal of
85 Astronomy & Astrophysics (A&A).
86 How to install the A&A \SpecialChar LaTeX
87 class to your \SpecialChar LaTeX
88 system is explained in
92 \begin_layout Plain Layout
94 http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
100 \begin_inset Newline newline
103 Depending on the submission state and the abstract layout, you need to use
104 different document class options that are listed in the aa manual.
107 \begin_inset Newline newline
115 If you use accented characters in your document, you must use the predefined
116 document class option
120 in the document settings.
129 Hydrodynamics of giant planet formation
132 \begin_layout Subtitle
135 \begin_inset Formula $\kappa$
144 \begin_inset Flex institutemark
147 \begin_layout Plain Layout
157 \begin_layout Plain Layout
168 \begin_inset Flex institutemark
171 \begin_layout Plain Layout
181 \begin_layout Plain Layout
194 \begin_layout Plain Layout
195 Just to show the usage of the elements in the author field
201 \begin_inset Note Note
204 \begin_layout Plain Layout
207 fnmsep is only needed for more than one consecutive notes/marks
215 \begin_layout Offprint
220 \begin_layout Address
221 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
223 \begin_inset Newline newline
227 \begin_inset Flex Email
230 \begin_layout Plain Layout
231 wuchterl@amok.ast.univie.ac.at
240 \begin_layout Plain Layout
249 University of Alexandria, Department of Geography, ...
250 \begin_inset Newline newline
254 \begin_inset Flex Email
257 \begin_layout Plain Layout
258 c.ptolemy@hipparch.uheaven.space
267 \begin_layout Plain Layout
268 The university of heaven temporarily does not accept e-mails
277 Received September 15, 1996; accepted March 16, 1997
280 \begin_layout Abstract (unstructured)
281 To investigate the physical nature of the `nuc\SpecialChar softhyphen
282 leated instability' of proto
283 giant planets, the stability of layers in static, radiative gas spheres
284 is analysed on the basis of Baker's standard one-zone model.
285 It is shown that stability depends only upon the equations of state, the
286 opacities and the local thermodynamic state in the layer.
287 Stability and instability can therefore be expressed in the form of stability
288 equations of state which are universal for a given composition.
289 The stability equations of state are calculated for solar composition and
290 are displayed in the domain
291 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
295 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
299 These displays may be used to determine the one-zone stability of layers
300 in stellar or planetary structure models by directly reading off the value
301 of the stability equations for the thermodynamic state of these layers,
302 specified by state quantities as density
303 \begin_inset Formula $\rho$
307 \begin_inset Formula $T$
310 or specific internal energy
311 \begin_inset Formula $e$
315 Regions of instability in the
316 \begin_inset Formula $(\rho,e)$
319 -plane are described and related to the underlying microphysical processes.
320 Vibrational instability is found to be a common phenomenon at temperatures
321 lower than the second He ionisation zone.
323 \begin_inset Formula $\kappa$
326 -mechanism is widespread under `cool' conditions.
327 \begin_inset Note Note
330 \begin_layout Plain Layout
331 Citations are not allowed in A&A abstracts.
337 \begin_inset Note Note
340 \begin_layout Plain Layout
341 This is the unstructured abstract type, an example for the structured abstract
346 template file that comes with \SpecialChar LyX
355 \begin_layout Keywords
356 giant planet formation –
357 \begin_inset Formula $\kappa$
360 -mechanism – stability of gas spheres
363 \begin_layout Section
367 \begin_layout Standard
370 nucleated instability
372 (also called core instability) hypothesis of giant planet formation, a
373 critical mass for static core envelope protoplanets has been found.
375 \begin_inset CommandInset citation
381 ) determined the critical mass of the core to be about
382 \begin_inset Formula $12\,M_{\oplus}$
386 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
389 is the Earth mass), which is independent of the outer boundary conditions
390 and therefore independent of the location in the solar nebula.
391 This critical value for the core mass corresponds closely to the cores
392 of today's giant planets.
395 \begin_layout Standard
396 Although no hydrodynamical study has been available many workers conjectured
397 that a collapse or rapid contraction will ensue after accumulating the
399 The main motivation for this article is to investigate the stability of
400 the static envelope at the critical mass.
401 With this aim the local, linear stability of static radiative gas spheres
402 is investigated on the basis of Baker's (
403 \begin_inset CommandInset citation
409 ) standard one-zone model.
412 \begin_layout Standard
413 Phenomena similar to the ones described above for giant planet formation
414 have been found in hydrodynamical models concerning star formation where
415 protostellar cores explode (Tscharnuter
416 \begin_inset CommandInset citation
423 \begin_inset CommandInset citation
429 ), whereas earlier studies found quasi-steady collapse flows.
430 The similarities in the (micro)physics, i.
431 \begin_inset space \thinspace{}
435 \begin_inset space \space{}
438 constitutive relations of protostellar cores and protogiant planets serve
439 as a further motivation for this study.
442 \begin_layout Section
443 Baker's standard one-zone model
446 \begin_layout Standard
447 \begin_inset Float figure
452 \begin_layout Plain Layout
453 \begin_inset Caption Standard
455 \begin_layout Plain Layout
456 \begin_inset CommandInset label
463 \begin_inset Formula $\Gamma_{1}$
468 \begin_inset Formula $\Gamma_{1}$
471 is plotted as a function of
472 \begin_inset Formula $\lg$
476 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
480 \begin_inset Formula $\lg$
484 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
497 In this section the one-zone model of Baker (
498 \begin_inset CommandInset citation
504 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
506 The resulting stability criteria will be rewritten in terms of local state
507 variables, local timescales and constitutive relations.
510 \begin_layout Standard
512 \begin_inset CommandInset citation
518 ) investigates the stability of thin layers in self-gravitating, spherical
519 gas clouds with the following properties:
522 \begin_layout Itemize
523 hydrostatic equilibrium,
526 \begin_layout Itemize
530 \begin_layout Itemize
531 energy transport by grey radiation diffusion.
535 \begin_layout Standard
537 For the one-zone-model Baker obtains necessary conditions for dynamical,
538 secular and vibrational (or pulsational) stability (Eqs.
539 \begin_inset space \space{}
543 \begin_inset space \thinspace{}
547 \begin_inset space \thinspace{}
551 \begin_inset CommandInset citation
558 Using Baker's notation:
561 \begin_layout Standard
565 M_{r} & & \textrm{mass internal to the radius }r\\
566 m & & \textrm{mass of the zone}\\
567 r_{0} & & \textrm{unperturbed zone radius}\\
568 \rho_{0} & & \textrm{unperturbed density in the zone}\\
569 T_{0} & & \textrm{unperturbed temperature in the zone}\\
570 L_{r0} & & \textrm{unperturbed luminosity}\\
571 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
579 \begin_layout Standard
581 and with the definitions of the
590 \begin_inset CommandInset ref
592 reference "fig:FigGam"
599 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
611 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
617 \begin_inset Formula $K$
621 \begin_inset Formula $\sigma_{0}$
624 have the following form:
627 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
628 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
634 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
641 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
648 is a thermodynamical quantity which is of order
649 \begin_inset Formula $1$
653 \begin_inset Formula $1$
656 for nonreacting mixtures of classical perfect gases.
657 The physical meaning of
658 \begin_inset Formula $\sigma_{0}$
662 \begin_inset Formula $K$
665 is clearly visible in the equations above.
667 \begin_inset Formula $\sigma_{0}$
670 represents a frequency of the order one per free-fall time.
672 \begin_inset Formula $K$
675 is proportional to the ratio of the free-fall time and the cooling time.
676 Substituting into Baker's criteria, using thermodynamic identities and
677 definitions of thermodynamic quantities,
680 \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}
688 \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}
693 one obtains, after some pages of algebra, the conditions for
700 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
701 \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}\\
702 \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}
707 For a physical discussion of the stability criteria see Baker (
708 \begin_inset CommandInset citation
715 \begin_inset CommandInset citation
724 \begin_layout Standard
725 We observe that these criteria for dynamical, secular and vibrational stability,
726 respectively, can be factorized into
729 \begin_layout Enumerate
730 a factor containing local timescales only,
733 \begin_layout Enumerate
734 a factor containing only constitutive relations and their derivatives.
738 \begin_layout Standard
739 The first factors, depending on only timescales, are positive by definition.
740 The signs of the left hand sides of the inequalities
745 \begin_inset CommandInset ref
752 \begin_inset CommandInset ref
759 \begin_inset CommandInset ref
765 ) therefore depend exclusively on the second factors containing the constitutive
767 Since they depend only on state variables, the stability criteria themselves
770 functions of the thermodynamic state in the local zone
773 The one-zone stability can therefore be determined from a simple equation
774 of state, given for example, as a function of density and temperature.
775 Once the microphysics, i.
776 \begin_inset space \thinspace{}
780 \begin_inset space \space{}
783 the thermodynamics and opacities (see Table
788 \begin_inset CommandInset ref
790 reference "tab:KapSou"
794 ), are specified (in practice by specifying a chemical composition) the
795 one-zone stability can be inferred if the thermodynamic state is specified.
796 The zone – or in other words the layer – will be stable or unstable in
797 whatever object it is imbedded as long as it satisfies the one-zone-model
799 Only the specific growth rates (depending upon the time scales) will be
800 different for layers in different objects.
803 \begin_layout Standard
804 \begin_inset Float table
809 \begin_layout Plain Layout
810 \begin_inset Caption Standard
812 \begin_layout Plain Layout
813 \begin_inset CommandInset label
827 \begin_layout Plain Layout
830 <lyxtabular version="3" rows="4" columns="2">
831 <features tabularvalignment="middle">
832 <column alignment="left" valignment="top" width="0pt">
833 <column alignment="left" valignment="top" width="0pt">
835 <cell alignment="center" valignment="top" topline="true" usebox="none">
838 \begin_layout Plain Layout
844 <cell alignment="center" valignment="top" topline="true" usebox="none">
847 \begin_layout Plain Layout
848 \begin_inset Formula $T/[\textrm{K}]$
858 <cell alignment="center" valignment="top" topline="true" usebox="none">
861 \begin_layout Plain Layout
862 Yorke 1979, Yorke 1980a
867 <cell alignment="center" valignment="top" topline="true" usebox="none">
870 \begin_layout Plain Layout
871 \begin_inset Formula $\leq1700^{\textrm{a}}$
881 <cell alignment="center" valignment="top" usebox="none">
884 \begin_layout Plain Layout
890 <cell alignment="center" valignment="top" usebox="none">
893 \begin_layout Plain Layout
894 \begin_inset Formula $1700\leq T\leq5000$
904 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
907 \begin_layout Plain Layout
913 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
916 \begin_layout Plain Layout
917 \begin_inset Formula $5000\leq$
933 \begin_layout Plain Layout
934 \begin_inset Formula $^{\textrm{a}}$
942 We will now write down the sign (and therefore stability) determining parts
943 of the left-hand sides of the inequalities (
944 \begin_inset CommandInset ref
951 \begin_inset CommandInset ref
958 \begin_inset CommandInset ref
966 stability equations of state
971 \begin_layout Standard
972 The sign determining part of inequality
977 \begin_inset CommandInset ref
984 \begin_inset Formula $3\Gamma_{1}-4$
987 and it reduces to the criterion for dynamical stability
990 \Gamma_{1}>\frac{4}{3}\,\cdot
995 Stability of the thermodynamical equilibrium demands
998 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1004 \begin_inset Formula
1011 holds for a wide range of physical situations.
1013 \begin_inset Formula
1015 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1016 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1017 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1022 we find the sign determining terms in inequalities
1023 \begin_inset space ~
1027 \begin_inset CommandInset ref
1029 reference "ZSSecSta"
1034 \begin_inset CommandInset ref
1036 reference "ZSVibSta"
1040 ) respectively and obtain the following form of the criteria for dynamical,
1041 secular and vibrational
1046 \begin_inset Formula
1048 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1049 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1050 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1055 The constitutive relations are to be evaluated for the unperturbed thermodynami
1057 \begin_inset Formula $(\rho_{0},T_{0})$
1061 We see that the one-zone stability of the layer depends only on the constitutiv
1063 \begin_inset Formula $\Gamma_{1}$
1067 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1071 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1075 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1079 These depend only on the unperturbed thermodynamical state of the layer.
1080 Therefore the above relations define the one-zone-stability equations of
1082 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1086 \begin_inset Formula $S_{\mathrm{vib}}$
1091 \begin_inset space ~
1095 \begin_inset CommandInset ref
1097 reference "fig:VibStabEquation"
1102 \begin_inset Formula $S_{\mathrm{vib}}$
1106 Regions of secular instability are listed in Table
1107 \begin_inset space ~
1113 \begin_layout Standard
1114 \begin_inset Float figure
1119 \begin_layout Plain Layout
1120 \begin_inset Caption Standard
1122 \begin_layout Plain Layout
1123 \begin_inset CommandInset label
1125 name "fig:VibStabEquation"
1129 Vibrational stability equation of state
1130 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1135 \begin_inset Formula $>0$
1138 means vibrational stability
1151 \begin_layout Section
1155 \begin_layout Enumerate
1156 The conditions for the stability of static, radiative layers in gas spheres,
1157 as described by Baker's (
1158 \begin_inset CommandInset citation
1164 ) standard one-zone model, can be expressed as stability equations of state.
1165 These stability equations of state depend only on the local thermodynamic
1170 \begin_layout Enumerate
1171 If the constitutive relations – equations of state and Rosseland mean opacities
1172 – are specified, the stability equations of state can be evaluated without
1173 specifying properties of the layer.
1177 \begin_layout Enumerate
1178 For solar composition gas the
1179 \begin_inset Formula $\kappa$
1182 -mechanism is working in the regions of the ice and dust features in the
1184 \begin_inset Formula $\mathrm{H}_{2}$
1187 dissociation and the combined H, first He ionization zone, as indicated
1188 by vibrational instability.
1189 These regions of instability are much larger in extent and degree of instabilit
1190 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1194 \begin_layout Acknowledgement
1195 Part of this work was supported by the German
1197 Deut\SpecialChar softhyphen
1198 sche For\SpecialChar softhyphen
1199 schungs\SpecialChar softhyphen
1200 ge\SpecialChar softhyphen
1201 mein\SpecialChar softhyphen
1205 \begin_inset space ~
1211 \begin_layout Standard
1212 \begin_inset CommandInset bibtex
1214 btprint "btPrintAll"
1215 bibfiles "biblioExample"
1221 \begin_inset Note Note
1224 \begin_layout Plain Layout
1229 If you cannot see the bibliography in the output, assure that you have
1230 gievn the full path to the Bib\SpecialChar TeX
1235 that is part of the A&A \SpecialChar LaTeX