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48 Hydrodynamics of giant planet formation
51 \begin_layout Subtitle
54 \begin_inset Formula $\kappa$
66 \begin_layout Plain Layout
75 \begin_layout Plain Layout
87 \begin_layout Plain Layout
102 \begin_layout Plain Layout
103 Just to show the usage of the elements in the author field
111 \begin_layout Offprint
116 \begin_layout Address
117 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
119 \begin_inset Newline newline
126 \begin_layout Plain Layout
130 email{wuchterl@amok.ast.univie.ac.at}
135 \begin_layout Plain Layout
141 University of Alexandria, Department of Geography, ...
142 \begin_inset Newline newline
149 \begin_layout Plain Layout
153 email{c.ptolemy@hipparch.uheaven.space}
162 \begin_layout Plain Layout
163 The university of heaven temporarily does not accept e-mails
172 Received September 15, 1996; accepted March 16, 1997
175 \begin_layout Abstract
176 To investigate the physical nature of the `nuc\SpecialChar \-
177 leated instability' of proto
178 giant planets (Mizuno
179 \begin_inset CommandInset citation
185 ), the stability of layers in static, radiative gas spheres is analysed
186 on the basis of Baker's
187 \begin_inset CommandInset citation
193 standard one-zone model.
194 It is shown that stability depends only upon the equations of state, the
195 opacities and the local thermodynamic state in the layer.
196 Stability and instability can therefore be expressed in the form of stability
197 equations of state which are universal for a given composition.
198 The stability equations of state are calculated for solar composition and
199 are displayed in the domain
200 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
204 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
208 These displays may be used to determine the one-zone stability of layers
209 in stellar or planetary structure models by directly reading off the value
210 of the stability equations for the thermodynamic state of these layers,
211 specified by state quantities as density
212 \begin_inset Formula $\rho$
216 \begin_inset Formula $T$
219 or specific internal energy
220 \begin_inset Formula $e$
224 Regions of instability in the
225 \begin_inset Formula $(\rho,e)$
228 -plane are described and related to the underlying microphysical processes.
229 Vibrational instability is found to be a common phenomenon at temperatures
230 lower than the second He ionisation zone.
232 \begin_inset Formula $\kappa$
235 -mechanism is widespread under `cool' conditions.
239 \begin_layout Plain Layout
243 \begin_layout Plain Layout
247 keywords{giant planet formation --
253 )-mechanism -- stability of gas spheres }
261 \begin_layout Section
265 \begin_layout Standard
268 nucleated instability
272 \begin_layout Plain Layout
283 (also called core instability) hypothesis of giant planet formation, a
284 critical mass for static core envelope protoplanets has been found.
286 \begin_inset CommandInset citation
292 ) determined the critical mass of the core to be about
293 \begin_inset Formula $12\, M_{\oplus}$
297 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
300 is the Earth mass), which is independent of the outer boundary conditions
301 and therefore independent of the location in the solar nebula.
302 This critical value for the core mass corresponds closely to the cores
303 of today's giant planets.
306 \begin_layout Standard
307 Although no hydrodynamical study has been available many workers conjectured
308 that a collapse or rapid contraction will ensue after accumulating the
310 The main motivation for this article is to investigate the stability of
311 the static envelope at the critical mass.
312 With this aim the local, linear stability of static radiative gas spheres
313 is investigated on the basis of Baker's (
314 \begin_inset CommandInset citation
320 ) standard one-zone model.
323 \begin_layout Standard
324 Phenomena similar to the ones described above for giant planet formation
325 have been found in hydrodynamical models concerning star formation where
326 protostellar cores explode (Tscharnuter
327 \begin_inset CommandInset citation
334 \begin_inset CommandInset citation
340 ), whereas earlier studies found quasi-steady collapse flows.
341 The similarities in the (micro)physics, i.e., constitutive relations of protostel
342 lar cores and protogiant planets serve as a further motivation for this
346 \begin_layout Section
347 Baker's standard one-zone model
350 \begin_layout Standard
351 \begin_inset Float figure
356 \begin_layout Plain Layout
359 \begin_layout Plain Layout
361 \begin_inset Formula $\Gamma_{1}$
366 \begin_inset Formula $\Gamma_{1}$
369 is plotted as a function of
370 \begin_inset Formula $\lg$
374 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
378 \begin_inset Formula $\lg$
382 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
393 \begin_layout Plain Layout
394 \begin_inset CommandInset label
405 In this section the one-zone model of Baker (
406 \begin_inset CommandInset citation
412 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
414 The resulting stability criteria will be rewritten in terms of local state
415 variables, local timescales and constitutive relations.
418 \begin_layout Standard
420 \begin_inset CommandInset citation
426 ) investigates the stability of thin layers in self-gravitating, spherical
427 gas clouds with the following properties:
430 \begin_layout Itemize
431 hydrostatic equilibrium,
434 \begin_layout Itemize
438 \begin_layout Itemize
439 energy transport by grey radiation diffusion.
443 \begin_layout Standard
445 For the one-zone-model Baker obtains necessary conditions for dynamical,
446 secular and vibrational (or pulsational) stability (Eqs.
450 \begin_layout Plain Layout
463 \begin_layout Plain Layout
476 \begin_layout Plain Layout
486 \begin_inset CommandInset citation
493 Using Baker's notation:
496 \begin_layout Standard
498 \begin_inset Formula \begin{eqnarray*}
499 M_{r} & & \textrm{mass internal to the radius }r\\
500 m & & \textrm{mass of the zone}\\
501 r_{0} & & \textrm{unperturbed zone radius}\\
502 \rho_{0} & & \textrm{unperturbed density in the zone}\\
503 T_{0} & & \textrm{unperturbed temperature in the zone}\\
504 L_{r0} & & \textrm{unperturbed luminosity}\\
505 E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
512 \begin_layout Standard
514 and with the definitions of the
520 \begin_layout Plain Layout
536 \begin_inset CommandInset ref
543 \begin_inset Formula \begin{equation}
544 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
553 \begin_inset Formula \begin{equation}
554 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,\end{equation}
559 \begin_inset Formula $K$
563 \begin_inset Formula $\sigma_{0}$
566 have the following form:
567 \begin_inset Formula \begin{eqnarray}
568 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
569 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;\end{eqnarray}
574 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
578 \begin_inset Formula \begin{equation}
580 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
581 e=mc^{2}\end{array}\end{equation}
585 is a thermodynamical quantity which is of order
586 \begin_inset Formula $1$
590 \begin_inset Formula $1$
593 for nonreacting mixtures of classical perfect gases.
594 The physical meaning of
595 \begin_inset Formula $\sigma_{0}$
599 \begin_inset Formula $K$
602 is clearly visible in the equations above.
604 \begin_inset Formula $\sigma_{0}$
607 represents a frequency of the order one per free-fall time.
609 \begin_inset Formula $K$
612 is proportional to the ratio of the free-fall time and the cooling time.
613 Substituting into Baker's criteria, using thermodynamic identities and
614 definitions of thermodynamic quantities,
615 \begin_inset Formula \[
616 \Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}\]
621 \begin_inset Formula \[
622 \nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}\]
626 one obtains, after some pages of algebra, the conditions for
632 \begin_layout Plain Layout
644 \begin_inset Formula \begin{eqnarray}
645 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
646 \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}\\
647 \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}\end{eqnarray}
651 For a physical discussion of the stability criteria see Baker (
652 \begin_inset CommandInset citation
659 \begin_inset CommandInset citation
668 \begin_layout Standard
669 We observe that these criteria for dynamical, secular and vibrational stability,
670 respectively, can be factorized into
673 \begin_layout Enumerate
674 a factor containing local timescales only,
677 \begin_layout Enumerate
678 a factor containing only constitutive relations and their derivatives.
682 \begin_layout Standard
683 The first factors, depending on only timescales, are positive by definition.
684 The signs of the left hand sides of the inequalities
689 \begin_inset CommandInset ref
696 \begin_inset CommandInset ref
703 \begin_inset CommandInset ref
709 ) therefore depend exclusively on the second factors containing the constitutive
711 Since they depend only on state variables, the stability criteria themselves
714 functions of the thermodynamic state in the local zone
717 The one-zone stability can therefore be determined from a simple equation
718 of state, given for example, as a function of density and temperature.
719 Once the microphysics, i.e.
723 \begin_layout Plain Layout
732 the thermodynamics and opacities (see Table
737 \begin_inset CommandInset ref
743 ), are specified (in practice by specifying a chemical composition) the
744 one-zone stability can be inferred if the thermodynamic state is specified.
745 The zone -- or in other words the layer -- will be stable or unstable in
746 whatever object it is imbedded as long as it satisfies the one-zone-model
748 Only the specific growth rates (depending upon the time scales) will be
749 different for layers in different objects.
752 \begin_layout Standard
753 \begin_inset Float table
758 \begin_layout Plain Layout
761 \begin_layout Plain Layout
762 \begin_inset CommandInset label
776 \begin_layout Plain Layout
778 <lyxtabular version="3" rows="4" columns="2">
780 <column alignment="left" valignment="top" width="0pt">
781 <column alignment="left" valignment="top" width="0pt">
783 <cell alignment="center" valignment="top" topline="true" usebox="none">
786 \begin_layout Plain Layout
792 <cell alignment="center" valignment="top" topline="true" usebox="none">
795 \begin_layout Plain Layout
796 \begin_inset Formula $T/[\textrm{K}]$
806 <cell alignment="center" valignment="top" topline="true" usebox="none">
809 \begin_layout Plain Layout
810 Yorke 1979, Yorke 1980a
815 <cell alignment="center" valignment="top" topline="true" usebox="none">
818 \begin_layout Plain Layout
819 \begin_inset Formula $\leq1700^{\textrm{a}}$
829 <cell alignment="center" valignment="top" usebox="none">
832 \begin_layout Plain Layout
838 <cell alignment="center" valignment="top" usebox="none">
841 \begin_layout Plain Layout
842 \begin_inset Formula $1700\leq T\leq5000$
852 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
855 \begin_layout Plain Layout
861 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
864 \begin_layout Plain Layout
865 \begin_inset Formula $5000\leq$
881 \begin_layout Plain Layout
882 \begin_inset Formula $^{\textrm{a}}$
890 We will now write down the sign (and therefore stability) determining parts
891 of the left-hand sides of the inequalities (
892 \begin_inset CommandInset ref
899 \begin_inset CommandInset ref
906 \begin_inset CommandInset ref
914 stability equations of state
919 \begin_layout Standard
920 The sign determining part of inequality
925 \begin_inset CommandInset ref
932 \begin_inset Formula $3\Gamma_{1}-4$
935 and it reduces to the criterion for dynamical stability
936 \begin_inset Formula \begin{equation}
937 \Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
941 Stability of the thermodynamical equilibrium demands
942 \begin_inset Formula \begin{equation}
943 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
948 \begin_inset Formula \begin{equation}
949 \chi_{T}^{}>0\end{equation}
953 holds for a wide range of physical situations.
955 \begin_inset Formula \begin{eqnarray}
956 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
957 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
958 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
962 we find the sign determining terms in inequalities
967 \begin_inset CommandInset ref
974 \begin_inset CommandInset ref
980 ) respectively and obtain the following form of the criteria for dynamical,
981 secular and vibrational
986 \begin_inset Formula \begin{eqnarray}
987 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
988 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
989 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}\end{eqnarray}
993 The constitutive relations are to be evaluated for the unperturbed thermodynami
995 \begin_inset Formula $(\rho_{0},T_{0})$
999 We see that the one-zone stability of the layer depends only on the constitutiv
1001 \begin_inset Formula $\Gamma_{1}$
1005 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1009 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1013 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1017 These depend only on the unperturbed thermodynamical state of the layer.
1018 Therefore the above relations define the one-zone-stability equations of
1020 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1024 \begin_inset Formula $S_{\mathrm{vib}}$
1029 \begin_inset space ~
1033 \begin_inset CommandInset ref
1035 reference "FigVibStab"
1040 \begin_inset Formula $S_{\mathrm{vib}}$
1044 Regions of secular instability are listed in Table
1045 \begin_inset space ~
1051 \begin_layout Standard
1052 \begin_inset Float figure
1057 \begin_layout Plain Layout
1058 \begin_inset Caption
1060 \begin_layout Plain Layout
1061 Vibrational stability equation of state
1062 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1067 \begin_inset Formula $>0$
1070 means vibrational stability
1078 \begin_layout Plain Layout
1079 \begin_inset CommandInset label
1093 \begin_layout Section
1097 \begin_layout Enumerate
1098 The conditions for the stability of static, radiative layers in gas spheres,
1099 as described by Baker's (
1100 \begin_inset CommandInset citation
1106 ) standard one-zone model, can be expressed as stability equations of state.
1107 These stability equations of state depend only on the local thermodynamic
1112 \begin_layout Enumerate
1113 If the constitutive relations -- equations of state and Rosseland mean opacities
1114 -- are specified, the stability equations of state can be evaluated without
1115 specifying properties of the layer.
1119 \begin_layout Enumerate
1120 For solar composition gas the
1121 \begin_inset Formula $\kappa$
1124 -mechanism is working in the regions of the ice and dust features in the
1126 \begin_inset Formula $\mathrm{H}_{2}$
1129 dissociation and the combined H, first He ionization zone, as indicated
1130 by vibrational instability.
1131 These regions of instability are much larger in extent and degree of instabilit
1132 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1136 \begin_layout Acknowledgement
1137 Part of this work was supported by the German
1140 sche For\SpecialChar \-
1141 schungs\SpecialChar \-
1148 \begin_layout Plain Layout
1160 \begin_inset space ~
1167 \begin_layout Bibliography
1168 \begin_inset CommandInset bibitem
1169 LatexCommand bibitem
1176 1966, in Stellar Evolution, ed.
1180 \begin_layout Plain Layout
1194 Cameron (Plenum, New York) 333
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1199 LatexCommand bibitem
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1219 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
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1225 LatexCommand bibitem
1234 1969, Academia Nauk, Scientific Information 15, 1
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1275 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1303 Zheng, W., Davidsen, A.