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26 \quotes_language english
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38 Hydrodynamics of giant planet formation
41 \begin_layout Subtitle
45 \begin_inset Formula \( \kappa \)
58 \begin_layout Standard
66 \begin_layout Standard
77 \begin_layout Standard
91 \begin_layout Standard
93 Just to show the usage of the elements in the author field
102 \begin_layout Offprint
108 \begin_layout Address
110 Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
118 \begin_layout Standard
121 email{wuchterl@amok.ast.univie.ac.at}
126 \begin_layout Standard
132 University of Alexandria, Department of Geography, ...
138 \begin_layout Standard
141 email{c.ptolemy@hipparch.uheaven.space}
150 \begin_layout Standard
152 The university of heaven temporarily does not accept e-mails
163 Received September 15, 1996; accepted March 16, 1997
166 \begin_layout Abstract
168 To investigate the physical nature of the `nuc\SpecialChar \-
169 leated instability' of proto
170 giant planets (Mizuno
171 \begin_inset LatexCommand \cite{mizuno}
175 ), the stability of layers in static, radiative gas spheres is analysed
176 on the basis of Baker's
177 \begin_inset LatexCommand \cite{baker}
181 standard one-zone model.
182 It is shown that stability depends only upon the equations of state, the
183 opacities and the local thermodynamic state in the layer.
184 Stability and instability can therefore be expressed in the form of stability
185 equations of state which are universal for a given composition.
186 The stability equations of state are calculated for solar composition and
187 are displayed in the domain
188 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
192 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
196 These displays may be used to determine the one-zone stability of layers
197 in stellar or planetary structure models by directly reading off the value
198 of the stability equations for the thermodynamic state of these layers,
199 specified by state quantities as density
200 \begin_inset Formula \( \rho \)
204 \begin_inset Formula \( T \)
207 or specific internal energy
208 \begin_inset Formula \( e \)
212 Regions of instability in the
213 \begin_inset Formula \( (\rho ,e) \)
216 -plane are described and related to the underlying microphysical processes.
217 Vibrational instability is found to be a common phenomenon at temperatures
218 lower than the second He ionisation zone.
220 \begin_inset Formula \( \kappa \)
223 -mechanism is widespread under `cool' conditions.
227 \begin_layout Standard
231 \begin_layout Standard
234 keywords{giant planet formation --
240 )-mechanism -- stability of gas spheres }
248 \begin_layout Section
253 \begin_layout Standard
257 nucleated instability
261 \begin_layout Standard
271 (also called core instability) hypothesis of giant planet formation, a
272 critical mass for static core envelope protoplanets has been found.
274 \begin_inset LatexCommand \cite{mizuno}
278 ) determined the critical mass of the core to be about
279 \begin_inset Formula \( 12\, M_{\oplus } \)
283 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
286 is the Earth mass), which is independent of the outer boundary conditions
287 and therefore independent of the location in the solar nebula.
288 This critical value for the core mass corresponds closely to the cores
289 of today's giant planets.
292 \begin_layout Standard
294 Although no hydrodynamical study has been available many workers conjectured
295 that a collapse or rapid contraction will ensue after accumulating the
297 The main motivation for this article is to investigate the stability of
298 the static envelope at the critical mass.
299 With this aim the local, linear stability of static radiative gas spheres
300 is investigated on the basis of Baker's (
301 \begin_inset LatexCommand \cite{baker}
305 ) standard one-zone model.
308 \begin_layout Standard
310 Phenomena similar to the ones described above for giant planet formation
311 have been found in hydrodynamical models concerning star formation where
312 protostellar cores explode (Tscharnuter
313 \begin_inset LatexCommand \cite{tscharnuter}
318 \begin_inset LatexCommand \cite{balluch}
322 ), whereas earlier studies found quasi-steady collapse flows.
323 The similarities in the (micro)physics, i.e., constitutive relations of protostel
324 lar cores and protogiant planets serve as a further motivation for this
328 \begin_layout Section
330 Baker's standard one-zone model
333 \begin_layout Standard
335 \begin_inset Float figure
340 \begin_layout Caption
343 \begin_inset Formula \( \Gamma _{1} \)
348 \begin_inset Formula \( \Gamma _{1} \)
351 is plotted as a function of
352 \begin_inset Formula \( \lg \)
356 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
360 \begin_inset Formula \( \lg \)
364 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
370 \begin_layout Standard
373 \begin_inset LatexCommand \label{FigGam}
382 In this section the one-zone model of Baker (
383 \begin_inset LatexCommand \cite{baker}
387 ), originally used to study the Cephe\i \"{\i}
388 d pulsation mechanism, will be briefly
390 The resulting stability criteria will be rewritten in terms of local state
391 variables, local timescales and constitutive relations.
394 \begin_layout Standard
397 \begin_inset LatexCommand \cite{baker}
401 ) investigates the stability of thin layers in self-gravitating, spherical
402 gas clouds with the following properties:
405 \begin_layout Itemize
407 hydrostatic equilibrium,
410 \begin_layout Itemize
415 \begin_layout Itemize
417 energy transport by grey radiation diffusion.
421 \begin_layout Standard
423 For the one-zone-model Baker obtains necessary conditions for dynamical,
424 secular and vibrational (or pulsational) stability (Eqs.
428 \begin_layout Standard
440 \begin_layout Standard
452 \begin_layout Standard
461 \begin_inset LatexCommand \cite{baker}
466 Using Baker's notation:
469 \begin_layout Standard
472 \begin_inset Formula \begin{eqnarray*}
473 M_{r} & & \textrm{mass internal to the radius }r\\
474 m & & \textrm{mass of the zone}\\
475 r_{0} & & \textrm{unperturbed zone radius}\\
476 \rho _{0} & & \textrm{unperturbed density in the zone}\\
477 T_{0} & & \textrm{unperturbed temperature in the zone}\\
478 L_{r0} & & \textrm{unperturbed luminosity}\\
479 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
487 \begin_layout Standard
489 and with the definitions of the
495 \begin_layout Standard
505 (see Fig.\InsetSpace ~
507 \begin_inset LatexCommand \ref{FigGam}
512 \begin_inset Formula \begin{equation}
513 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
523 \begin_inset Formula \begin{equation}
524 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
530 \begin_inset Formula \( K \)
534 \begin_inset Formula \( \sigma _{0} \)
537 have the following form:
538 \begin_inset Formula \begin{eqnarray}
539 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
540 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
546 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
550 \begin_inset Formula \begin{equation}
552 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
559 is a thermodynamical quantity which is of order
560 \begin_inset Formula \( 1 \)
564 \begin_inset Formula \( 1 \)
567 for nonreacting mixtures of classical perfect gases.
568 The physical meaning of
569 \begin_inset Formula \( \sigma _{0} \)
573 \begin_inset Formula \( K \)
576 is clearly visible in the equations above.
578 \begin_inset Formula \( \sigma _{0} \)
581 represents a frequency of the order one per free-fall time.
583 \begin_inset Formula \( K \)
586 is proportional to the ratio of the free-fall time and the cooling time.
587 Substituting into Baker's criteria, using thermodynamic identities and
588 definitions of thermodynamic quantities,
589 \begin_inset Formula \[
590 \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
595 \begin_inset Formula \[
596 \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
600 one obtains, after some pages of algebra, the conditions for
606 \begin_layout Standard
617 \begin_inset Formula \begin{eqnarray}
618 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
619 \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} \\
620 \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}
625 For a physical discussion of the stability criteria see Baker (
626 \begin_inset LatexCommand \cite{baker}
631 \begin_inset LatexCommand \cite{cox}
638 \begin_layout Standard
640 We observe that these criteria for dynamical, secular and vibrational stability,
641 respectively, can be factorized into
644 \begin_layout Enumerate
646 a factor containing local timescales only,
649 \begin_layout Enumerate
651 a factor containing only constitutive relations and their derivatives.
655 \begin_layout Standard
657 The first factors, depending on only timescales, are positive by definition.
658 The signs of the left hand sides of the inequalities\InsetSpace ~
660 \begin_inset LatexCommand \ref{ZSDynSta}
665 \begin_inset LatexCommand \ref{ZSSecSta}
670 \begin_inset LatexCommand \ref{ZSVibSta}
674 ) therefore depend exclusively on the second factors containing the constitutive
676 Since they depend only on state variables, the stability criteria themselves
679 functions of the thermodynamic state in the local zone
682 The one-zone stability can therefore be determined from a simple equation
683 of state, given for example, as a function of density and temperature.
684 Once the microphysics, i.e.
688 \begin_layout Standard
696 the thermodynamics and opacities (see Table\InsetSpace ~
698 \begin_inset LatexCommand \ref{KapSou}
702 ), are specified (in practice by specifying a chemical composition) the
703 one-zone stability can be inferred if the thermodynamic state is specified.
704 The zone -- or in other words the layer -- will be stable or unstable in
705 whatever object it is imbedded as long as it satisfies the one-zone-model
707 Only the specific growth rates (depending upon the time scales) will be
708 different for layers in different objects.
711 \begin_layout Standard
713 \begin_inset Float table
718 \begin_layout Caption
721 \begin_inset LatexCommand \label{KapSou}
728 \begin_layout Standard
732 <lyxtabular version="3" rows="4" columns="2">
734 <column alignment="left" valignment="top" width="0pt">
735 <column alignment="left" valignment="top" width="0pt">
737 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
740 \begin_layout Standard
747 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
750 \begin_layout Standard
753 \begin_inset Formula \( T/[\textrm{K}] \)
763 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
766 \begin_layout Standard
768 Yorke 1979, Yorke 1980a
773 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
776 \begin_layout Standard
779 \begin_inset Formula \( \leq 1700^{\textrm{a}} \)
789 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
792 \begin_layout Standard
799 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
802 \begin_layout Standard
805 \begin_inset Formula \( 1700\leq T\leq 5000 \)
814 <row bottomline="true">
815 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
818 \begin_layout Standard
825 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
828 \begin_layout Standard
831 \begin_inset Formula \( 5000\leq \)
847 \begin_layout Standard
850 \begin_inset Formula \( ^{\textrm{a}} \)
858 We will now write down the sign (and therefore stability) determining parts
859 of the left-hand sides of the inequalities (
860 \begin_inset LatexCommand \ref{ZSDynSta}
865 \begin_inset LatexCommand \ref{ZSSecSta}
870 \begin_inset LatexCommand \ref{ZSVibSta}
876 stability equations of state
881 \begin_layout Standard
883 The sign determining part of inequality\InsetSpace ~
885 \begin_inset LatexCommand \ref{ZSDynSta}
890 \begin_inset Formula \( 3\Gamma _{1}-4 \)
893 and it reduces to the criterion for dynamical stability
894 \begin_inset Formula \begin{equation}
895 \Gamma _{1}>\frac{4}{3}\, \cdot
900 Stability of the thermodynamical equilibrium demands
901 \begin_inset Formula \begin{equation}
902 \chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
908 \begin_inset Formula \begin{equation}
914 holds for a wide range of physical situations.
916 \begin_inset Formula \begin{eqnarray}
917 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
918 \Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
919 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
924 we find the sign determining terms in inequalities\InsetSpace ~
926 \begin_inset LatexCommand \ref{ZSSecSta}
931 \begin_inset LatexCommand \ref{ZSVibSta}
935 ) respectively and obtain the following form of the criteria for dynamical,
936 secular and vibrational
941 \begin_inset Formula \begin{eqnarray}
942 3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
943 \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
944 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
949 The constitutive relations are to be evaluated for the unperturbed thermodynami
951 \begin_inset Formula \( (\rho _{0},T_{0}) \)
955 We see that the one-zone stability of the layer depends only on the constitutiv
957 \begin_inset Formula \( \Gamma _{1} \)
961 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
965 \begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
969 \begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
973 These depend only on the unperturbed thermodynamical state of the layer.
974 Therefore the above relations define the one-zone-stability equations of
976 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
980 \begin_inset Formula \( S_{\mathrm{vib}} \)
984 See Fig.\InsetSpace ~
986 \begin_inset LatexCommand \ref{FigVibStab}
991 \begin_inset Formula \( S_{\mathrm{vib}} \)
995 Regions of secular instability are listed in Table\InsetSpace ~
999 \begin_layout Standard
1001 \begin_inset Float figure
1006 \begin_layout Caption
1008 Vibrational stability equation of state
1009 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
1014 \begin_inset Formula \( >0 \)
1017 means vibrational stability
1020 \begin_layout Standard
1023 \begin_inset LatexCommand \label{FigVibStab}
1034 \begin_layout Section
1039 \begin_layout Enumerate
1041 The conditions for the stability of static, radiative layers in gas spheres,
1042 as described by Baker's (
1043 \begin_inset LatexCommand \cite{baker}
1047 ) standard one-zone model, can be expressed as stability equations of state.
1048 These stability equations of state depend only on the local thermodynamic
1053 \begin_layout Enumerate
1055 If the constitutive relations -- equations of state and Rosseland mean opacities
1056 -- are specified, the stability equations of state can be evaluated without
1057 specifying properties of the layer.
1061 \begin_layout Enumerate
1063 For solar composition gas the
1064 \begin_inset Formula \( \kappa \)
1067 -mechanism is working in the regions of the ice and dust features in the
1069 \begin_inset Formula \( \mathrm{H}_{2} \)
1072 dissociation and the combined H, first He ionization zone, as indicated
1073 by vibrational instability.
1074 These regions of instability are much larger in extent and degree of instabilit
1075 y than the second He ionization zone that drives the Cephe\i \"{\i}
1080 \begin_layout Acknowledgement
1082 Part of this work was supported by the German
1085 sche For\SpecialChar \-
1086 schungs\SpecialChar \-
1093 \begin_layout Standard
1103 project number Ts\InsetSpace ~
1108 \begin_layout Bibliography
1109 \bibitem [1966]{baker}
1112 1966, in Stellar Evolution, ed.
1116 \begin_layout Standard
1129 Cameron (Plenum, New York) 333
1132 \begin_layout Bibliography
1133 \bibitem [1988]{balluch}
1139 \begin_layout Bibliography
1140 \bibitem [1980]{cox}
1144 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1148 \begin_layout Bibliography
1149 \bibitem [1969]{cox69}
1154 1969, Academia Nauk, Scientific Information 15, 1
1157 \begin_layout Bibliography
1158 \bibitem [1980]{mizuno}
1166 \begin_layout Bibliography
1167 \bibitem [1987]{tscharnuter}
1174 \begin_layout Bibliography
1175 \bibitem [1992]{terlevich}
1180 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
1187 \begin_layout Bibliography
1188 \bibitem [1980a]{yorke80a}
1195 \begin_layout Bibliography
1196 \bibitem [1997]{zheng}
1198 Zheng, W., Davidsen, A.