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114 Hydrodynamics of giant planet formation
117 \begin_layout Subtitle
120 \begin_inset Formula $\kappa$
129 \begin_inset Flex institutemark
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200 \begin_layout Offprint
205 \begin_layout Address
206 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
208 \begin_inset Newline newline
212 \begin_inset Flex Email
215 \begin_layout Plain Layout
216 wuchterl@amok.ast.univie.ac.at
225 \begin_layout Plain Layout
234 University of Alexandria, Department of Geography, ...
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239 \begin_inset Flex Email
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243 c.ptolemy@hipparch.uheaven.space
252 \begin_layout Plain Layout
253 The university of heaven temporarily does not accept e-mails
262 Received September 15, 1996; accepted March 16, 1997
265 \begin_layout Abstract
266 To investigate the physical nature of the `nuc\SpecialChar \-
267 leated instability' of proto
268 giant planets, the stability of layers in static, radiative gas spheres
269 is analysed on the basis of Baker's standard one-zone model.
270 It is shown that stability depends only upon the equations of state, the
271 opacities and the local thermodynamic state in the layer.
272 Stability and instability can therefore be expressed in the form of stability
273 equations of state which are universal for a given composition.
274 The stability equations of state are calculated for solar composition and
275 are displayed in the domain
276 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
280 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
284 These displays may be used to determine the one-zone stability of layers
285 in stellar or planetary structure models by directly reading off the value
286 of the stability equations for the thermodynamic state of these layers,
287 specified by state quantities as density
288 \begin_inset Formula $\rho$
292 \begin_inset Formula $T$
295 or specific internal energy
296 \begin_inset Formula $e$
300 Regions of instability in the
301 \begin_inset Formula $(\rho,e)$
304 -plane are described and related to the underlying microphysical processes.
305 Vibrational instability is found to be a common phenomenon at temperatures
306 lower than the second He ionisation zone.
308 \begin_inset Formula $\kappa$
311 -mechanism is widespread under `cool' conditions.
312 \begin_inset Note Note
315 \begin_layout Plain Layout
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325 \begin_layout Plain Layout
326 This is the unstructured abstract type, an example for the structured abstract
331 template file that comes with LyX.
339 \begin_layout Keywords
340 giant planet formation --
341 \begin_inset Formula $\kappa$
344 -mechanism -- stability of gas spheres
347 \begin_layout Section
351 \begin_layout Standard
354 nucleated instability
356 (also called core instability) hypothesis of giant planet formation, a
357 critical mass for static core envelope protoplanets has been found.
359 \begin_inset CommandInset citation
365 ) determined the critical mass of the core to be about
366 \begin_inset Formula $12\, M_{\oplus}$
370 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
373 is the Earth mass), which is independent of the outer boundary conditions
374 and therefore independent of the location in the solar nebula.
375 This critical value for the core mass corresponds closely to the cores
376 of today's giant planets.
379 \begin_layout Standard
380 Although no hydrodynamical study has been available many workers conjectured
381 that a collapse or rapid contraction will ensue after accumulating the
383 The main motivation for this article is to investigate the stability of
384 the static envelope at the critical mass.
385 With this aim the local, linear stability of static radiative gas spheres
386 is investigated on the basis of Baker's (
387 \begin_inset CommandInset citation
393 ) standard one-zone model.
396 \begin_layout Standard
397 Phenomena similar to the ones described above for giant planet formation
398 have been found in hydrodynamical models concerning star formation where
399 protostellar cores explode (Tscharnuter
400 \begin_inset CommandInset citation
407 \begin_inset CommandInset citation
413 ), whereas earlier studies found quasi-steady collapse flows.
414 The similarities in the (micro)physics, i.
415 \begin_inset space \thinspace{}
419 \begin_inset space \space{}
422 constitutive relations of protostellar cores and protogiant planets serve
423 as a further motivation for this study.
426 \begin_layout Section
427 Baker's standard one-zone model
430 \begin_layout Standard
431 \begin_inset Float figure
436 \begin_layout Plain Layout
439 \begin_layout Plain Layout
440 \begin_inset CommandInset label
447 \begin_inset Formula $\Gamma_{1}$
452 \begin_inset Formula $\Gamma_{1}$
455 is plotted as a function of
456 \begin_inset Formula $\lg$
460 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
464 \begin_inset Formula $\lg$
468 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
481 In this section the one-zone model of Baker (
482 \begin_inset CommandInset citation
488 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
490 The resulting stability criteria will be rewritten in terms of local state
491 variables, local timescales and constitutive relations.
494 \begin_layout Standard
496 \begin_inset CommandInset citation
502 ) investigates the stability of thin layers in self-gravitating, spherical
503 gas clouds with the following properties:
506 \begin_layout Itemize
507 hydrostatic equilibrium,
510 \begin_layout Itemize
514 \begin_layout Itemize
515 energy transport by grey radiation diffusion.
519 \begin_layout Standard
521 For the one-zone-model Baker obtains necessary conditions for dynamical,
522 secular and vibrational (or pulsational) stability (Eqs.
523 \begin_inset space \space{}
527 \begin_inset space \thinspace{}
531 \begin_inset space \thinspace{}
535 \begin_inset CommandInset citation
542 Using Baker's notation:
545 \begin_layout Standard
549 M_{r} & & \textrm{mass internal to the radius }r\\
550 m & & \textrm{mass of the zone}\\
551 r_{0} & & \textrm{unperturbed zone radius}\\
552 \rho_{0} & & \textrm{unperturbed density in the zone}\\
553 T_{0} & & \textrm{unperturbed temperature in the zone}\\
554 L_{r0} & & \textrm{unperturbed luminosity}\\
555 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
563 \begin_layout Standard
565 and with the definitions of the
574 \begin_inset CommandInset ref
576 reference "fig:FigGam"
583 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
595 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
601 \begin_inset Formula $K$
605 \begin_inset Formula $\sigma_{0}$
608 have the following form:
611 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
612 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
618 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
625 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
632 is a thermodynamical quantity which is of order
633 \begin_inset Formula $1$
637 \begin_inset Formula $1$
640 for nonreacting mixtures of classical perfect gases.
641 The physical meaning of
642 \begin_inset Formula $\sigma_{0}$
646 \begin_inset Formula $K$
649 is clearly visible in the equations above.
651 \begin_inset Formula $\sigma_{0}$
654 represents a frequency of the order one per free-fall time.
656 \begin_inset Formula $K$
659 is proportional to the ratio of the free-fall time and the cooling time.
660 Substituting into Baker's criteria, using thermodynamic identities and
661 definitions of thermodynamic quantities,
664 \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}
672 \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}
677 one obtains, after some pages of algebra, the conditions for
684 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
685 \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}\\
686 \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}
691 For a physical discussion of the stability criteria see Baker (
692 \begin_inset CommandInset citation
699 \begin_inset CommandInset citation
708 \begin_layout Standard
709 We observe that these criteria for dynamical, secular and vibrational stability,
710 respectively, can be factorized into
713 \begin_layout Enumerate
714 a factor containing local timescales only,
717 \begin_layout Enumerate
718 a factor containing only constitutive relations and their derivatives.
722 \begin_layout Standard
723 The first factors, depending on only timescales, are positive by definition.
724 The signs of the left hand sides of the inequalities
729 \begin_inset CommandInset ref
736 \begin_inset CommandInset ref
743 \begin_inset CommandInset ref
749 ) therefore depend exclusively on the second factors containing the constitutive
751 Since they depend only on state variables, the stability criteria themselves
754 functions of the thermodynamic state in the local zone
757 The one-zone stability can therefore be determined from a simple equation
758 of state, given for example, as a function of density and temperature.
759 Once the microphysics, i.
760 \begin_inset space \thinspace{}
764 \begin_inset space \space{}
767 the thermodynamics and opacities (see Table
772 \begin_inset CommandInset ref
774 reference "tab:KapSou"
778 ), are specified (in practice by specifying a chemical composition) the
779 one-zone stability can be inferred if the thermodynamic state is specified.
780 The zone -- or in other words the layer -- will be stable or unstable in
781 whatever object it is imbedded as long as it satisfies the one-zone-model
783 Only the specific growth rates (depending upon the time scales) will be
784 different for layers in different objects.
787 \begin_layout Standard
788 \begin_inset Float table
793 \begin_layout Plain Layout
796 \begin_layout Plain Layout
797 \begin_inset CommandInset label
811 \begin_layout Plain Layout
814 <lyxtabular version="3" rows="4" columns="2">
815 <features tabularvalignment="middle">
816 <column alignment="left" valignment="top" width="0pt">
817 <column alignment="left" valignment="top" width="0pt">
819 <cell alignment="center" valignment="top" topline="true" usebox="none">
822 \begin_layout Plain Layout
828 <cell alignment="center" valignment="top" topline="true" usebox="none">
831 \begin_layout Plain Layout
832 \begin_inset Formula $T/[\textrm{K}]$
842 <cell alignment="center" valignment="top" topline="true" usebox="none">
845 \begin_layout Plain Layout
846 Yorke 1979, Yorke 1980a
851 <cell alignment="center" valignment="top" topline="true" usebox="none">
854 \begin_layout Plain Layout
855 \begin_inset Formula $\leq1700^{\textrm{a}}$
865 <cell alignment="center" valignment="top" usebox="none">
868 \begin_layout Plain Layout
874 <cell alignment="center" valignment="top" usebox="none">
877 \begin_layout Plain Layout
878 \begin_inset Formula $1700\leq T\leq5000$
888 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
891 \begin_layout Plain Layout
897 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
900 \begin_layout Plain Layout
901 \begin_inset Formula $5000\leq$
917 \begin_layout Plain Layout
918 \begin_inset Formula $^{\textrm{a}}$
926 We will now write down the sign (and therefore stability) determining parts
927 of the left-hand sides of the inequalities (
928 \begin_inset CommandInset ref
935 \begin_inset CommandInset ref
942 \begin_inset CommandInset ref
950 stability equations of state
955 \begin_layout Standard
956 The sign determining part of inequality
961 \begin_inset CommandInset ref
968 \begin_inset Formula $3\Gamma_{1}-4$
971 and it reduces to the criterion for dynamical stability
974 \Gamma_{1}>\frac{4}{3}\,\cdot
979 Stability of the thermodynamical equilibrium demands
982 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,
995 holds for a wide range of physical situations.
999 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1000 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1001 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1006 we find the sign determining terms in inequalities
1007 \begin_inset space ~
1011 \begin_inset CommandInset ref
1013 reference "ZSSecSta"
1018 \begin_inset CommandInset ref
1020 reference "ZSVibSta"
1024 ) respectively and obtain the following form of the criteria for dynamical,
1025 secular and vibrational
1030 \begin_inset Formula
1032 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1033 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1034 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1039 The constitutive relations are to be evaluated for the unperturbed thermodynami
1041 \begin_inset Formula $(\rho_{0},T_{0})$
1045 We see that the one-zone stability of the layer depends only on the constitutiv
1047 \begin_inset Formula $\Gamma_{1}$
1051 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1055 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1059 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1063 These depend only on the unperturbed thermodynamical state of the layer.
1064 Therefore the above relations define the one-zone-stability equations of
1066 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1070 \begin_inset Formula $S_{\mathrm{vib}}$
1075 \begin_inset space ~
1079 \begin_inset CommandInset ref
1081 reference "fig:VibStabEquation"
1086 \begin_inset Formula $S_{\mathrm{vib}}$
1090 Regions of secular instability are listed in Table
1091 \begin_inset space ~
1097 \begin_layout Standard
1098 \begin_inset Float figure
1103 \begin_layout Plain Layout
1104 \begin_inset Caption
1106 \begin_layout Plain Layout
1107 \begin_inset CommandInset label
1109 name "fig:VibStabEquation"
1113 Vibrational stability equation of state
1114 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1119 \begin_inset Formula $>0$
1122 means vibrational stability
1135 \begin_layout Section
1139 \begin_layout Enumerate
1140 The conditions for the stability of static, radiative layers in gas spheres,
1141 as described by Baker's (
1142 \begin_inset CommandInset citation
1148 ) standard one-zone model, can be expressed as stability equations of state.
1149 These stability equations of state depend only on the local thermodynamic
1154 \begin_layout Enumerate
1155 If the constitutive relations -- equations of state and Rosseland mean opacities
1156 -- are specified, the stability equations of state can be evaluated without
1157 specifying properties of the layer.
1161 \begin_layout Enumerate
1162 For solar composition gas the
1163 \begin_inset Formula $\kappa$
1166 -mechanism is working in the regions of the ice and dust features in the
1168 \begin_inset Formula $\mathrm{H}_{2}$
1171 dissociation and the combined H, first He ionization zone, as indicated
1172 by vibrational instability.
1173 These regions of instability are much larger in extent and degree of instabilit
1174 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1178 \begin_layout Acknowledgement
1179 Part of this work was supported by the German
1182 sche For\SpecialChar \-
1183 schungs\SpecialChar \-
1189 \begin_inset space ~
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1196 \begin_inset Note Note
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1200 You can alternatively use BibTeX.
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1213 \begin_layout Bibliography
1214 \labelwidthstring References
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1216 LatexCommand bibitem
1223 1966, in Stellar Evolution, ed.
1224 \begin_inset space \space{}
1232 Cameron (Plenum, New York) 333
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1259 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1263 \begin_layout Bibliography
1264 \labelwidthstring References
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1275 1969, Academia Nauk, Scientific Information 15, 1
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1319 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1349 Zheng, W., Davidsen, A.