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26 \quotes_language english
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39 Hydrodynamics of giant planet formation
42 \begin_layout Subtitle
46 \begin_inset Formula \( \kappa \)
59 \begin_layout Standard
67 \begin_layout Standard
78 \begin_layout Standard
92 \begin_layout Standard
94 Just to show the usage of the elements in the author field
103 \begin_layout Offprint
109 \begin_layout Address
111 Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
119 \begin_layout Standard
122 email{wuchterl@amok.ast.univie.ac.at}
127 \begin_layout Standard
133 University of Alexandria, Department of Geography, ...
139 \begin_layout Standard
142 email{c.ptolemy@hipparch.uheaven.space}
151 \begin_layout Standard
153 The university of heaven temporarily does not accept e-mails
164 Received September 15, 1996; accepted March 16, 1997
167 \begin_layout Abstract
169 To investigate the physical nature of the `nuc\SpecialChar \-
170 leated instability' of proto
171 giant planets (Mizuno
172 \begin_inset LatexCommand \cite{mizuno}
176 ), the stability of layers in static, radiative gas spheres is analysed
177 on the basis of Baker's
178 \begin_inset LatexCommand \cite{baker}
182 standard one-zone model.
183 It is shown that stability depends only upon the equations of state, the
184 opacities and the local thermodynamic state in the layer.
185 Stability and instability can therefore be expressed in the form of stability
186 equations of state which are universal for a given composition.
187 The stability equations of state are calculated for solar composition and
188 are displayed in the domain
189 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
193 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
197 These displays may be used to determine the one-zone stability of layers
198 in stellar or planetary structure models by directly reading off the value
199 of the stability equations for the thermodynamic state of these layers,
200 specified by state quantities as density
201 \begin_inset Formula \( \rho \)
205 \begin_inset Formula \( T \)
208 or specific internal energy
209 \begin_inset Formula \( e \)
213 Regions of instability in the
214 \begin_inset Formula \( (\rho ,e) \)
217 -plane are described and related to the underlying microphysical processes.
218 Vibrational instability is found to be a common phenomenon at temperatures
219 lower than the second He ionisation zone.
221 \begin_inset Formula \( \kappa \)
224 -mechanism is widespread under `cool' conditions.
228 \begin_layout Standard
232 \begin_layout Standard
235 keywords{giant planet formation --
241 )-mechanism -- stability of gas spheres }
249 \begin_layout Section
254 \begin_layout Standard
258 nucleated instability
262 \begin_layout Standard
272 (also called core instability) hypothesis of giant planet formation, a
273 critical mass for static core envelope protoplanets has been found.
275 \begin_inset LatexCommand \cite{mizuno}
279 ) determined the critical mass of the core to be about
280 \begin_inset Formula \( 12\, M_{\oplus } \)
284 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
287 is the Earth mass), which is independent of the outer boundary conditions
288 and therefore independent of the location in the solar nebula.
289 This critical value for the core mass corresponds closely to the cores
290 of today's giant planets.
293 \begin_layout Standard
295 Although no hydrodynamical study has been available many workers conjectured
296 that a collapse or rapid contraction will ensue after accumulating the
298 The main motivation for this article is to investigate the stability of
299 the static envelope at the critical mass.
300 With this aim the local, linear stability of static radiative gas spheres
301 is investigated on the basis of Baker's (
302 \begin_inset LatexCommand \cite{baker}
306 ) standard one-zone model.
309 \begin_layout Standard
311 Phenomena similar to the ones described above for giant planet formation
312 have been found in hydrodynamical models concerning star formation where
313 protostellar cores explode (Tscharnuter
314 \begin_inset LatexCommand \cite{tscharnuter}
319 \begin_inset LatexCommand \cite{balluch}
323 ), whereas earlier studies found quasi-steady collapse flows.
324 The similarities in the (micro)physics, i.e., constitutive relations of protostel
325 lar cores and protogiant planets serve as a further motivation for this
329 \begin_layout Section
331 Baker's standard one-zone model
334 \begin_layout Standard
336 \begin_inset Float figure
341 \begin_layout Caption
344 \begin_inset Formula \( \Gamma _{1} \)
349 \begin_inset Formula \( \Gamma _{1} \)
352 is plotted as a function of
353 \begin_inset Formula \( \lg \)
357 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
361 \begin_inset Formula \( \lg \)
365 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
371 \begin_layout Standard
374 \begin_inset LatexCommand \label{FigGam}
383 In this section the one-zone model of Baker (
384 \begin_inset LatexCommand \cite{baker}
388 ), originally used to study the Cephe\i \"{\i}
389 d pulsation mechanism, will be briefly
391 The resulting stability criteria will be rewritten in terms of local state
392 variables, local timescales and constitutive relations.
395 \begin_layout Standard
398 \begin_inset LatexCommand \cite{baker}
402 ) investigates the stability of thin layers in self-gravitating, spherical
403 gas clouds with the following properties:
406 \begin_layout Itemize
408 hydrostatic equilibrium,
411 \begin_layout Itemize
416 \begin_layout Itemize
418 energy transport by grey radiation diffusion.
422 \begin_layout Standard
424 For the one-zone-model Baker obtains necessary conditions for dynamical,
425 secular and vibrational (or pulsational) stability (Eqs.
429 \begin_layout Standard
441 \begin_layout Standard
453 \begin_layout Standard
462 \begin_inset LatexCommand \cite{baker}
467 Using Baker's notation:
470 \begin_layout Standard
473 \begin_inset Formula \begin{eqnarray*}
474 M_{r} & & \textrm{mass internal to the radius }r\\
475 m & & \textrm{mass of the zone}\\
476 r_{0} & & \textrm{unperturbed zone radius}\\
477 \rho _{0} & & \textrm{unperturbed density in the zone}\\
478 T_{0} & & \textrm{unperturbed temperature in the zone}\\
479 L_{r0} & & \textrm{unperturbed luminosity}\\
480 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
488 \begin_layout Standard
490 and with the definitions of the
496 \begin_layout Standard
506 (see Fig.\InsetSpace ~
508 \begin_inset LatexCommand \ref{FigGam}
513 \begin_inset Formula \begin{equation}
514 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
524 \begin_inset Formula \begin{equation}
525 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
531 \begin_inset Formula \( K \)
535 \begin_inset Formula \( \sigma _{0} \)
538 have the following form:
539 \begin_inset Formula \begin{eqnarray}
540 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
541 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
547 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
551 \begin_inset Formula \begin{equation}
553 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
560 is a thermodynamical quantity which is of order
561 \begin_inset Formula \( 1 \)
565 \begin_inset Formula \( 1 \)
568 for nonreacting mixtures of classical perfect gases.
569 The physical meaning of
570 \begin_inset Formula \( \sigma _{0} \)
574 \begin_inset Formula \( K \)
577 is clearly visible in the equations above.
579 \begin_inset Formula \( \sigma _{0} \)
582 represents a frequency of the order one per free-fall time.
584 \begin_inset Formula \( K \)
587 is proportional to the ratio of the free-fall time and the cooling time.
588 Substituting into Baker's criteria, using thermodynamic identities and
589 definitions of thermodynamic quantities,
590 \begin_inset Formula \[
591 \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
596 \begin_inset Formula \[
597 \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
601 one obtains, after some pages of algebra, the conditions for
607 \begin_layout Standard
618 \begin_inset Formula \begin{eqnarray}
619 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
620 \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} \\
621 \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}
626 For a physical discussion of the stability criteria see Baker (
627 \begin_inset LatexCommand \cite{baker}
632 \begin_inset LatexCommand \cite{cox}
639 \begin_layout Standard
641 We observe that these criteria for dynamical, secular and vibrational stability,
642 respectively, can be factorized into
645 \begin_layout Enumerate
647 a factor containing local timescales only,
650 \begin_layout Enumerate
652 a factor containing only constitutive relations and their derivatives.
656 \begin_layout Standard
658 The first factors, depending on only timescales, are positive by definition.
659 The signs of the left hand sides of the inequalities\InsetSpace ~
661 \begin_inset LatexCommand \ref{ZSDynSta}
666 \begin_inset LatexCommand \ref{ZSSecSta}
671 \begin_inset LatexCommand \ref{ZSVibSta}
675 ) therefore depend exclusively on the second factors containing the constitutive
677 Since they depend only on state variables, the stability criteria themselves
680 functions of the thermodynamic state in the local zone
683 The one-zone stability can therefore be determined from a simple equation
684 of state, given for example, as a function of density and temperature.
685 Once the microphysics, i.e.
689 \begin_layout Standard
697 the thermodynamics and opacities (see Table\InsetSpace ~
699 \begin_inset LatexCommand \ref{KapSou}
703 ), are specified (in practice by specifying a chemical composition) the
704 one-zone stability can be inferred if the thermodynamic state is specified.
705 The zone -- or in other words the layer -- will be stable or unstable in
706 whatever object it is imbedded as long as it satisfies the one-zone-model
708 Only the specific growth rates (depending upon the time scales) will be
709 different for layers in different objects.
712 \begin_layout Standard
714 \begin_inset Float table
719 \begin_layout Caption
722 \begin_inset LatexCommand \label{KapSou}
729 \begin_layout Standard
733 <lyxtabular version="3" rows="4" columns="2">
735 <column alignment="left" valignment="top" width="0pt">
736 <column alignment="left" valignment="top" width="0pt">
738 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
741 \begin_layout Standard
748 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
751 \begin_layout Standard
754 \begin_inset Formula \( T/[\textrm{K}] \)
764 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
767 \begin_layout Standard
769 Yorke 1979, Yorke 1980a
774 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
777 \begin_layout Standard
780 \begin_inset Formula \( \leq 1700^{\textrm{a}} \)
790 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
793 \begin_layout Standard
800 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
803 \begin_layout Standard
806 \begin_inset Formula \( 1700\leq T\leq 5000 \)
815 <row bottomline="true">
816 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
819 \begin_layout Standard
826 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
829 \begin_layout Standard
832 \begin_inset Formula \( 5000\leq \)
848 \begin_layout Standard
851 \begin_inset Formula \( ^{\textrm{a}} \)
859 We will now write down the sign (and therefore stability) determining parts
860 of the left-hand sides of the inequalities (
861 \begin_inset LatexCommand \ref{ZSDynSta}
866 \begin_inset LatexCommand \ref{ZSSecSta}
871 \begin_inset LatexCommand \ref{ZSVibSta}
877 stability equations of state
882 \begin_layout Standard
884 The sign determining part of inequality\InsetSpace ~
886 \begin_inset LatexCommand \ref{ZSDynSta}
891 \begin_inset Formula \( 3\Gamma _{1}-4 \)
894 and it reduces to the criterion for dynamical stability
895 \begin_inset Formula \begin{equation}
896 \Gamma _{1}>\frac{4}{3}\, \cdot
901 Stability of the thermodynamical equilibrium demands
902 \begin_inset Formula \begin{equation}
903 \chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
909 \begin_inset Formula \begin{equation}
915 holds for a wide range of physical situations.
917 \begin_inset Formula \begin{eqnarray}
918 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
919 \Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
920 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
925 we find the sign determining terms in inequalities\InsetSpace ~
927 \begin_inset LatexCommand \ref{ZSSecSta}
932 \begin_inset LatexCommand \ref{ZSVibSta}
936 ) respectively and obtain the following form of the criteria for dynamical,
937 secular and vibrational
942 \begin_inset Formula \begin{eqnarray}
943 3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
944 \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
945 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
950 The constitutive relations are to be evaluated for the unperturbed thermodynami
952 \begin_inset Formula \( (\rho _{0},T_{0}) \)
956 We see that the one-zone stability of the layer depends only on the constitutiv
958 \begin_inset Formula \( \Gamma _{1} \)
962 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
966 \begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
970 \begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
974 These depend only on the unperturbed thermodynamical state of the layer.
975 Therefore the above relations define the one-zone-stability equations of
977 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
981 \begin_inset Formula \( S_{\mathrm{vib}} \)
985 See Fig.\InsetSpace ~
987 \begin_inset LatexCommand \ref{FigVibStab}
992 \begin_inset Formula \( S_{\mathrm{vib}} \)
996 Regions of secular instability are listed in Table\InsetSpace ~
1000 \begin_layout Standard
1002 \begin_inset Float figure
1007 \begin_layout Caption
1009 Vibrational stability equation of state
1010 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
1015 \begin_inset Formula \( >0 \)
1018 means vibrational stability
1021 \begin_layout Standard
1024 \begin_inset LatexCommand \label{FigVibStab}
1035 \begin_layout Section
1040 \begin_layout Enumerate
1042 The conditions for the stability of static, radiative layers in gas spheres,
1043 as described by Baker's (
1044 \begin_inset LatexCommand \cite{baker}
1048 ) standard one-zone model, can be expressed as stability equations of state.
1049 These stability equations of state depend only on the local thermodynamic
1054 \begin_layout Enumerate
1056 If the constitutive relations -- equations of state and Rosseland mean opacities
1057 -- are specified, the stability equations of state can be evaluated without
1058 specifying properties of the layer.
1062 \begin_layout Enumerate
1064 For solar composition gas the
1065 \begin_inset Formula \( \kappa \)
1068 -mechanism is working in the regions of the ice and dust features in the
1070 \begin_inset Formula \( \mathrm{H}_{2} \)
1073 dissociation and the combined H, first He ionization zone, as indicated
1074 by vibrational instability.
1075 These regions of instability are much larger in extent and degree of instabilit
1076 y than the second He ionization zone that drives the Cephe\i \"{\i}
1081 \begin_layout Acknowledgement
1083 Part of this work was supported by the German
1086 sche For\SpecialChar \-
1087 schungs\SpecialChar \-
1094 \begin_layout Standard
1104 project number Ts\InsetSpace ~
1109 \begin_layout Bibliography
1110 \bibitem [1966]{baker}
1113 1966, in Stellar Evolution, ed.
1117 \begin_layout Standard
1130 Cameron (Plenum, New York) 333
1133 \begin_layout Bibliography
1134 \bibitem [1988]{balluch}
1140 \begin_layout Bibliography
1141 \bibitem [1980]{cox}
1145 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1149 \begin_layout Bibliography
1150 \bibitem [1969]{cox69}
1155 1969, Academia Nauk, Scientific Information 15, 1
1158 \begin_layout Bibliography
1159 \bibitem [1980]{mizuno}
1167 \begin_layout Bibliography
1168 \bibitem [1987]{tscharnuter}
1175 \begin_layout Bibliography
1176 \bibitem [1992]{terlevich}
1181 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
1188 \begin_layout Bibliography
1189 \bibitem [1980a]{yorke80a}
1196 \begin_layout Bibliography
1197 \bibitem [1997]{zheng}
1199 Zheng, W., Davidsen, A.