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60 \quotes_language english
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74 \begin_inset Note Note
77 \begin_layout Plain Layout
82 This is an example \SpecialCharNoPassThru LyX
83 file for articles to be submitted to the Journal
84 of Astronomy & Astrophysics (A&A).
85 How to install the A&A \SpecialCharNoPassThru LaTeX
86 class to your \SpecialCharNoPassThru LaTeX
87 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 \SpecialCharNoPassThru LyX
355 \begin_layout Keywords
356 giant planet formation \twohyphens
358 \begin_inset Formula $\kappa$
361 -mechanism \twohyphens
362 stability of gas spheres
365 \begin_layout Section
369 \begin_layout Standard
372 nucleated instability
374 (also called core instability) hypothesis of giant planet formation, a
375 critical mass for static core envelope protoplanets has been found.
377 \begin_inset CommandInset citation
383 ) determined the critical mass of the core to be about
384 \begin_inset Formula $12\, M_{\oplus}$
388 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
391 is the Earth mass), which is independent of the outer boundary conditions
392 and therefore independent of the location in the solar nebula.
393 This critical value for the core mass corresponds closely to the cores
394 of today's giant planets.
397 \begin_layout Standard
398 Although no hydrodynamical study has been available many workers conjectured
399 that a collapse or rapid contraction will ensue after accumulating the
401 The main motivation for this article is to investigate the stability of
402 the static envelope at the critical mass.
403 With this aim the local, linear stability of static radiative gas spheres
404 is investigated on the basis of Baker's (
405 \begin_inset CommandInset citation
411 ) standard one-zone model.
414 \begin_layout Standard
415 Phenomena similar to the ones described above for giant planet formation
416 have been found in hydrodynamical models concerning star formation where
417 protostellar cores explode (Tscharnuter
418 \begin_inset CommandInset citation
425 \begin_inset CommandInset citation
431 ), whereas earlier studies found quasi-steady collapse flows.
432 The similarities in the (micro)physics, i.
433 \begin_inset space \thinspace{}
437 \begin_inset space \space{}
440 constitutive relations of protostellar cores and protogiant planets serve
441 as a further motivation for this study.
444 \begin_layout Section
445 Baker's standard one-zone model
448 \begin_layout Standard
449 \begin_inset Float figure
454 \begin_layout Plain Layout
455 \begin_inset Caption Standard
457 \begin_layout Plain Layout
458 \begin_inset CommandInset label
465 \begin_inset Formula $\Gamma_{1}$
470 \begin_inset Formula $\Gamma_{1}$
473 is plotted as a function of
474 \begin_inset Formula $\lg$
478 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
482 \begin_inset Formula $\lg$
486 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
499 In this section the one-zone model of Baker (
500 \begin_inset CommandInset citation
506 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
508 The resulting stability criteria will be rewritten in terms of local state
509 variables, local timescales and constitutive relations.
512 \begin_layout Standard
514 \begin_inset CommandInset citation
520 ) investigates the stability of thin layers in self-gravitating, spherical
521 gas clouds with the following properties:
524 \begin_layout Itemize
525 hydrostatic equilibrium,
528 \begin_layout Itemize
532 \begin_layout Itemize
533 energy transport by grey radiation diffusion.
537 \begin_layout Standard
539 For the one-zone-model Baker obtains necessary conditions for dynamical,
540 secular and vibrational (or pulsational) stability (Eqs.
541 \begin_inset space \space{}
545 \begin_inset space \thinspace{}
549 \begin_inset space \thinspace{}
553 \begin_inset CommandInset citation
560 Using Baker's notation:
561 \begin_inset Separator parbreak
566 \begin_layout Standard
570 M_{r} & & \textrm{mass internal to the radius }r\\
571 m & & \textrm{mass of the zone}\\
572 r_{0} & & \textrm{unperturbed zone radius}\\
573 \rho_{0} & & \textrm{unperturbed density in the zone}\\
574 T_{0} & & \textrm{unperturbed temperature in the zone}\\
575 L_{r0} & & \textrm{unperturbed luminosity}\\
576 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
584 \begin_layout Standard
586 and with the definitions of the
595 \begin_inset CommandInset ref
597 reference "fig:FigGam"
604 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
616 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
622 \begin_inset Formula $K$
626 \begin_inset Formula $\sigma_{0}$
629 have the following form:
632 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
633 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
639 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
646 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
653 is a thermodynamical quantity which is of order
654 \begin_inset Formula $1$
658 \begin_inset Formula $1$
661 for nonreacting mixtures of classical perfect gases.
662 The physical meaning of
663 \begin_inset Formula $\sigma_{0}$
667 \begin_inset Formula $K$
670 is clearly visible in the equations above.
672 \begin_inset Formula $\sigma_{0}$
675 represents a frequency of the order one per free-fall time.
677 \begin_inset Formula $K$
680 is proportional to the ratio of the free-fall time and the cooling time.
681 Substituting into Baker's criteria, using thermodynamic identities and
682 definitions of thermodynamic quantities,
685 \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}
693 \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}
698 one obtains, after some pages of algebra, the conditions for
705 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
706 \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}\\
707 \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}
712 For a physical discussion of the stability criteria see Baker (
713 \begin_inset CommandInset citation
720 \begin_inset CommandInset citation
729 \begin_layout Standard
730 We observe that these criteria for dynamical, secular and vibrational stability,
731 respectively, can be factorized into
734 \begin_layout Enumerate
735 a factor containing local timescales only,
738 \begin_layout Enumerate
739 a factor containing only constitutive relations and their derivatives.
743 \begin_layout Standard
744 The first factors, depending on only timescales, are positive by definition.
745 The signs of the left hand sides of the inequalities
750 \begin_inset CommandInset ref
757 \begin_inset CommandInset ref
764 \begin_inset CommandInset ref
770 ) therefore depend exclusively on the second factors containing the constitutive
772 Since they depend only on state variables, the stability criteria themselves
775 functions of the thermodynamic state in the local zone
778 The one-zone stability can therefore be determined from a simple equation
779 of state, given for example, as a function of density and temperature.
780 Once the microphysics, i.
781 \begin_inset space \thinspace{}
785 \begin_inset space \space{}
788 the thermodynamics and opacities (see Table
793 \begin_inset CommandInset ref
795 reference "tab:KapSou"
799 ), are specified (in practice by specifying a chemical composition) the
800 one-zone stability can be inferred if the thermodynamic state is specified.
802 or in other words the layer \twohyphens
803 will be stable or unstable in
804 whatever object it is imbedded as long as it satisfies the one-zone-model
806 Only the specific growth rates (depending upon the time scales) will be
807 different for layers in different objects.
810 \begin_layout Standard
811 \begin_inset Float table
816 \begin_layout Plain Layout
817 \begin_inset Caption Standard
819 \begin_layout Plain Layout
820 \begin_inset CommandInset label
834 \begin_layout Plain Layout
837 <lyxtabular version="3" rows="4" columns="2">
838 <features rotate="0" tabularvalignment="middle">
839 <column alignment="left" valignment="top" width="0pt">
840 <column alignment="left" valignment="top" width="0pt">
842 <cell alignment="center" valignment="top" topline="true" usebox="none">
845 \begin_layout Plain Layout
851 <cell alignment="center" valignment="top" topline="true" usebox="none">
854 \begin_layout Plain Layout
855 \begin_inset Formula $T/[\textrm{K}]$
865 <cell alignment="center" valignment="top" topline="true" usebox="none">
868 \begin_layout Plain Layout
869 Yorke 1979, Yorke 1980a
874 <cell alignment="center" valignment="top" topline="true" usebox="none">
877 \begin_layout Plain Layout
878 \begin_inset Formula $\leq1700^{\textrm{a}}$
888 <cell alignment="center" valignment="top" usebox="none">
891 \begin_layout Plain Layout
897 <cell alignment="center" valignment="top" usebox="none">
900 \begin_layout Plain Layout
901 \begin_inset Formula $1700\leq T\leq5000$
911 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
914 \begin_layout Plain Layout
920 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
923 \begin_layout Plain Layout
924 \begin_inset Formula $5000\leq$
940 \begin_layout Plain Layout
941 \begin_inset Formula $^{\textrm{a}}$
949 We will now write down the sign (and therefore stability) determining parts
950 of the left-hand sides of the inequalities (
951 \begin_inset CommandInset ref
958 \begin_inset CommandInset ref
965 \begin_inset CommandInset ref
973 stability equations of state
978 \begin_layout Standard
979 The sign determining part of inequality
984 \begin_inset CommandInset ref
991 \begin_inset Formula $3\Gamma_{1}-4$
994 and it reduces to the criterion for dynamical stability
997 \Gamma_{1}>\frac{4}{3}\,\cdot
1002 Stability of the thermodynamical equilibrium demands
1003 \begin_inset Formula
1005 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,
1011 \begin_inset Formula
1018 holds for a wide range of physical situations.
1020 \begin_inset Formula
1022 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1023 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1024 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1029 we find the sign determining terms in inequalities
1030 \begin_inset space ~
1034 \begin_inset CommandInset ref
1036 reference "ZSSecSta"
1041 \begin_inset CommandInset ref
1043 reference "ZSVibSta"
1047 ) respectively and obtain the following form of the criteria for dynamical,
1048 secular and vibrational
1053 \begin_inset Formula
1055 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1056 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1057 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1062 The constitutive relations are to be evaluated for the unperturbed thermodynami
1064 \begin_inset Formula $(\rho_{0},T_{0})$
1068 We see that the one-zone stability of the layer depends only on the constitutiv
1070 \begin_inset Formula $\Gamma_{1}$
1074 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1078 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1082 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1086 These depend only on the unperturbed thermodynamical state of the layer.
1087 Therefore the above relations define the one-zone-stability equations of
1089 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1093 \begin_inset Formula $S_{\mathrm{vib}}$
1098 \begin_inset space ~
1102 \begin_inset CommandInset ref
1104 reference "fig:VibStabEquation"
1109 \begin_inset Formula $S_{\mathrm{vib}}$
1113 Regions of secular instability are listed in Table
1114 \begin_inset space ~
1120 \begin_layout Standard
1121 \begin_inset Float figure
1126 \begin_layout Plain Layout
1127 \begin_inset Caption Standard
1129 \begin_layout Plain Layout
1130 \begin_inset CommandInset label
1132 name "fig:VibStabEquation"
1136 Vibrational stability equation of state
1137 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1142 \begin_inset Formula $>0$
1145 means vibrational stability
1158 \begin_layout Section
1162 \begin_layout Enumerate
1163 The conditions for the stability of static, radiative layers in gas spheres,
1164 as described by Baker's (
1165 \begin_inset CommandInset citation
1171 ) standard one-zone model, can be expressed as stability equations of state.
1172 These stability equations of state depend only on the local thermodynamic
1177 \begin_layout Enumerate
1178 If the constitutive relations \twohyphens
1179 equations of state and Rosseland mean opacities
1181 are specified, the stability equations of state can be evaluated without
1182 specifying properties of the layer.
1186 \begin_layout Enumerate
1187 For solar composition gas the
1188 \begin_inset Formula $\kappa$
1191 -mechanism is working in the regions of the ice and dust features in the
1193 \begin_inset Formula $\mathrm{H}_{2}$
1196 dissociation and the combined H, first He ionization zone, as indicated
1197 by vibrational instability.
1198 These regions of instability are much larger in extent and degree of instabilit
1199 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1203 \begin_layout Acknowledgement
1204 Part of this work was supported by the German
1206 Deut\SpecialChar softhyphen
1207 sche For\SpecialChar softhyphen
1208 schungs\SpecialChar softhyphen
1209 ge\SpecialChar softhyphen
1210 mein\SpecialChar softhyphen
1214 \begin_inset space ~
1221 \begin_layout Standard
1222 \begin_inset CommandInset bibtex
1224 btprint "btPrintAll"
1225 bibfiles "biblioExample"
1231 \begin_inset Note Note
1234 \begin_layout Plain Layout
1239 If you cannot see the bibliography in the output, assure that you have
1240 gievn the full path to the Bib\SpecialCharNoPassThru TeX
1245 that is part of the A&A \SpecialCharNoPassThru LaTeX