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46 Hydrodynamics of giant planet formation
49 \begin_layout Subtitle
53 \begin_inset Formula \( \kappa \)
66 \begin_layout Standard
74 \begin_layout Standard
85 \begin_layout Standard
99 \begin_layout Standard
101 Just to show the usage of the elements in the author field
110 \begin_layout Offprint
116 \begin_layout Address
118 Institute for Astronomy (IfA), University of Vienna, Tü
126 \begin_layout Standard
129 email{wuchterl@amok.ast.univie.ac.at}
134 \begin_layout Standard
140 University of Alexandria, Department of Geography, ...
146 \begin_layout Standard
149 email{c.ptolemy@hipparch.uheaven.space}
158 \begin_layout Standard
160 The university of heaven temporarily does not accept e-mails
171 Received September 15, 1996; accepted March 16, 1997
174 \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 LatexCommand cite
183 ), the stability of layers in static, radiative gas spheres is analysed
184 on the basis of Baker's
185 \begin_inset LatexCommand cite
189 standard one-zone model.
190 It is shown that stability depends only upon the equations of state, the
191 opacities and the local thermodynamic state in the layer.
192 Stability and instability can therefore be expressed in the form of stability
193 equations of state which are universal for a given composition.
194 The stability equations of state are calculated for solar composition and
195 are displayed in the domain
196 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
200 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
204 These displays may be used to determine the one-zone stability of layers
205 in stellar or planetary structure models by directly reading off the value
206 of the stability equations for the thermodynamic state of these layers,
207 specified by state quantities as density
208 \begin_inset Formula \( \rho \)
212 \begin_inset Formula \( T \)
215 or specific internal energy
216 \begin_inset Formula \( e \)
220 Regions of instability in the
221 \begin_inset Formula \( (\rho ,e) \)
224 -plane are described and related to the underlying microphysical processes.
225 Vibrational instability is found to be a common phenomenon at temperatures
226 lower than the second He ionisation zone.
228 \begin_inset Formula \( \kappa \)
231 -mechanism is widespread under `cool' conditions.
235 \begin_layout Standard
239 \begin_layout Standard
242 keywords{giant planet formation --
248 )-mechanism -- stability of gas spheres }
256 \begin_layout Section
261 \begin_layout Standard
267 nucleated instability
271 \begin_layout Standard
281 (also called core instability) hypothesis of giant planet formation, a
282 critical mass for static core envelope protoplanets has been found.
284 \begin_inset LatexCommand cite
288 ) determined the critical mass of the core to be about
289 \begin_inset Formula \( 12\, M_{\oplus } \)
293 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
296 is the Earth mass), which is independent of the outer boundary conditions
297 and therefore independent of the location in the solar nebula.
298 This critical value for the core mass corresponds closely to the cores
299 of today's giant planets.
302 \begin_layout Standard
304 Although no hydrodynamical study has been available many workers conjectured
305 that a collapse or rapid contraction will ensue after accumulating the
307 The main motivation for this article is to investigate the stability of
308 the static envelope at the critical mass.
309 With this aim the local, linear stability of static radiative gas spheres
310 is investigated on the basis of Baker's (
311 \begin_inset LatexCommand cite
315 ) standard one-zone model.
318 \begin_layout Standard
320 Phenomena similar to the ones described above for giant planet formation
321 have been found in hydrodynamical models concerning star formation where
322 protostellar cores explode (Tscharnuter
323 \begin_inset LatexCommand cite
328 \begin_inset LatexCommand cite
332 ), whereas earlier studies found quasi-steady collapse flows.
333 The similarities in the (micro)physics, i.e., constitutive relations of protostel
334 lar cores and protogiant planets serve as a further motivation for this
338 \begin_layout Section
340 Baker's standard one-zone model
343 \begin_layout Standard
345 \begin_inset Float figure
356 \begin_inset Formula \( \Gamma _{1} \)
361 \begin_inset Formula \( \Gamma _{1} \)
364 is plotted as a function of
365 \begin_inset Formula \( \lg \)
369 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
373 \begin_inset Formula \( \lg \)
377 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
388 \begin_layout Standard
391 \begin_inset LatexCommand label
400 In this section the one-zone model of Baker (
401 \begin_inset LatexCommand cite
405 ), originally used to study the Cepheı̈
406 d pulsation mechanism, will be briefly
408 The resulting stability criteria will be rewritten in terms of local state
409 variables, local timescales and constitutive relations.
412 \begin_layout Standard
415 \begin_inset LatexCommand cite
419 ) investigates the stability of thin layers in self-gravitating, spherical
420 gas clouds with the following properties:
423 \begin_layout Itemize
425 hydrostatic equilibrium,
428 \begin_layout Itemize
433 \begin_layout Itemize
435 energy transport by grey radiation diffusion.
439 \begin_layout Standard
441 For the one-zone-model Baker obtains necessary conditions for dynamical,
442 secular and vibrational (or pulsational) stability (Eqs.
446 \begin_layout Standard
458 \begin_layout Standard
470 \begin_layout Standard
479 \begin_inset LatexCommand cite
484 Using Baker's notation:
487 \begin_layout Standard
490 \begin_inset Formula \begin{eqnarray*}
491 M_{r} & & \textrm{mass internal to the radius }r\\
492 m & & \textrm{mass of the zone}\\
493 r_{0} & & \textrm{unperturbed zone radius}\\
494 \rho _{0} & & \textrm{unperturbed density in the zone}\\
495 T_{0} & & \textrm{unperturbed temperature in the zone}\\
496 L_{r0} & & \textrm{unperturbed luminosity}\\
497 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
505 \begin_layout Standard
507 and with the definitions of the
515 \begin_layout Standard
525 (see Fig.\InsetSpace ~
527 \begin_inset LatexCommand ref
532 \begin_inset Formula \begin{equation}
533 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
545 \begin_inset Formula \begin{equation}
546 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
552 \begin_inset Formula \( K \)
556 \begin_inset Formula \( \sigma _{0} \)
559 have the following form:
560 \begin_inset Formula \begin{eqnarray}
561 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
562 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
568 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
572 \begin_inset Formula \begin{equation}
574 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
581 is a thermodynamical quantity which is of order
582 \begin_inset Formula \( 1 \)
586 \begin_inset Formula \( 1 \)
589 for nonreacting mixtures of classical perfect gases.
590 The physical meaning of
591 \begin_inset Formula \( \sigma _{0} \)
595 \begin_inset Formula \( K \)
598 is clearly visible in the equations above.
600 \begin_inset Formula \( \sigma _{0} \)
603 represents a frequency of the order one per free-fall time.
605 \begin_inset Formula \( K \)
608 is proportional to the ratio of the free-fall time and the cooling time.
609 Substituting into Baker's criteria, using thermodynamic identities and
610 definitions of thermodynamic quantities,
611 \begin_inset Formula \[
612 \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
617 \begin_inset Formula \[
618 \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
622 one obtains, after some pages of algebra, the conditions for
630 \begin_layout Standard
641 \begin_inset Formula \begin{eqnarray}
642 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
643 \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} \\
644 \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}
649 For a physical discussion of the stability criteria see Baker (
650 \begin_inset LatexCommand cite
655 \begin_inset LatexCommand cite
662 \begin_layout Standard
664 We observe that these criteria for dynamical, secular and vibrational stability,
665 respectively, can be factorized into
668 \begin_layout Enumerate
670 a factor containing local timescales only,
673 \begin_layout Enumerate
675 a factor containing only constitutive relations and their derivatives.
679 \begin_layout Standard
681 The first factors, depending on only timescales, are positive by definition.
682 The signs of the left hand sides of the inequalities\InsetSpace ~
684 \begin_inset LatexCommand ref
689 \begin_inset LatexCommand ref
694 \begin_inset LatexCommand ref
698 ) therefore depend exclusively on the second factors containing the constitutive
700 Since they depend only on state variables, the stability criteria themselves
705 functions of the thermodynamic state in the local zone
708 The one-zone stability can therefore be determined from a simple equation
709 of state, given for example, as a function of density and temperature.
710 Once the microphysics, i.e.
714 \begin_layout Standard
722 the thermodynamics and opacities (see Table\InsetSpace ~
724 \begin_inset LatexCommand ref
728 ), are specified (in practice by specifying a chemical composition) the
729 one-zone stability can be inferred if the thermodynamic state is specified.
730 The zone -- or in other words the layer -- will be stable or unstable in
731 whatever object it is imbedded as long as it satisfies the one-zone-model
733 Only the specific growth rates (depending upon the time scales) will be
734 different for layers in different objects.
737 \begin_layout Standard
739 \begin_inset Float table
750 \begin_inset LatexCommand label
762 \begin_layout Standard
766 <lyxtabular version="3" rows="4" columns="2">
768 <column alignment="left" valignment="top" width="0pt">
769 <column alignment="left" valignment="top" width="0pt">
771 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
774 \begin_layout Standard
781 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
784 \begin_layout Standard
787 \begin_inset Formula \( T/[\textrm{K}] \)
797 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
800 \begin_layout Standard
802 Yorke 1979, Yorke 1980a
807 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
810 \begin_layout Standard
813 \begin_inset Formula \( \leq 1700^{\textrm{a}} \)
823 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
826 \begin_layout Standard
833 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
836 \begin_layout Standard
839 \begin_inset Formula \( 1700\leq T\leq 5000 \)
848 <row bottomline="true">
849 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
852 \begin_layout Standard
859 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
862 \begin_layout Standard
865 \begin_inset Formula \( 5000\leq \)
881 \begin_layout Standard
884 \begin_inset Formula \( ^{\textrm{a}} \)
892 We will now write down the sign (and therefore stability) determining parts
893 of the left-hand sides of the inequalities (
894 \begin_inset LatexCommand ref
899 \begin_inset LatexCommand ref
904 \begin_inset LatexCommand ref
912 stability equations of state
917 \begin_layout Standard
919 The sign determining part of inequality\InsetSpace ~
921 \begin_inset LatexCommand ref
926 \begin_inset Formula \( 3\Gamma _{1}-4 \)
929 and it reduces to the criterion for dynamical stability
930 \begin_inset Formula \begin{equation}
931 \Gamma _{1}>\frac{4}{3}\, \cdot
936 Stability of the thermodynamical equilibrium demands
937 \begin_inset Formula \begin{equation}
938 \chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
944 \begin_inset Formula \begin{equation}
950 holds for a wide range of physical situations.
952 \begin_inset Formula \begin{eqnarray}
953 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
954 \Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
955 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
960 we find the sign determining terms in inequalities\InsetSpace ~
962 \begin_inset LatexCommand ref
967 \begin_inset LatexCommand ref
971 ) respectively and obtain the following form of the criteria for dynamical,
972 secular and vibrational
979 \begin_inset Formula \begin{eqnarray}
980 3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
981 \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
982 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
987 The constitutive relations are to be evaluated for the unperturbed thermodynami
989 \begin_inset Formula \( (\rho _{0},T_{0}) \)
993 We see that the one-zone stability of the layer depends only on the constitutiv
995 \begin_inset Formula \( \Gamma _{1} \)
999 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
1003 \begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
1007 \begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
1011 These depend only on the unperturbed thermodynamical state of the layer.
1012 Therefore the above relations define the one-zone-stability equations of
1014 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
1018 \begin_inset Formula \( S_{\mathrm{vib}} \)
1022 See Fig.\InsetSpace ~
1024 \begin_inset LatexCommand ref
1025 reference "FigVibStab"
1029 \begin_inset Formula \( S_{\mathrm{vib}} \)
1033 Regions of secular instability are listed in Table\InsetSpace ~
1037 \begin_layout Standard
1039 \begin_inset Float figure
1045 \begin_inset Caption
1049 Vibrational stability equation of state
1050 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
1055 \begin_inset Formula \( >0 \)
1058 means vibrational stability
1066 \begin_layout Standard
1069 \begin_inset LatexCommand label
1080 \begin_layout Section
1085 \begin_layout Enumerate
1087 The conditions for the stability of static, radiative layers in gas spheres,
1088 as described by Baker's (
1089 \begin_inset LatexCommand cite
1093 ) standard one-zone model, can be expressed as stability equations of state.
1094 These stability equations of state depend only on the local thermodynamic
1099 \begin_layout Enumerate
1101 If the constitutive relations -- equations of state and Rosseland mean opacities
1102 -- are specified, the stability equations of state can be evaluated without
1103 specifying properties of the layer.
1107 \begin_layout Enumerate
1109 For solar composition gas the
1110 \begin_inset Formula \( \kappa \)
1113 -mechanism is working in the regions of the ice and dust features in the
1115 \begin_inset Formula \( \mathrm{H}_{2} \)
1118 dissociation and the combined H, first He ionization zone, as indicated
1119 by vibrational instability.
1120 These regions of instability are much larger in extent and degree of instabilit
1121 y than the second He ionization zone that drives the Cepheı̈
1126 \begin_layout Acknowledgement
1128 Part of this work was supported by the German
1133 sche For\SpecialChar \-
1134 schungs\SpecialChar \-
1141 \begin_layout Standard
1151 project number Ts\InsetSpace ~
1156 \begin_layout Bibliography
1157 \begin_inset LatexCommand bibitem
1164 1966, in Stellar Evolution, ed.
1168 \begin_layout Standard
1181 Cameron (Plenum, New York) 333
1184 \begin_layout Bibliography
1185 \begin_inset LatexCommand bibitem
1195 \begin_layout Bibliography
1196 \begin_inset LatexCommand bibitem
1204 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1208 \begin_layout Bibliography
1209 \begin_inset LatexCommand bibitem
1218 1969, Academia Nauk, Scientific Information 15, 1
1221 \begin_layout Bibliography
1222 \begin_inset LatexCommand bibitem
1234 \begin_layout Bibliography
1235 \begin_inset LatexCommand bibitem
1246 \begin_layout Bibliography
1247 \begin_inset LatexCommand bibitem
1256 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
1263 \begin_layout Bibliography
1264 \begin_inset LatexCommand bibitem
1275 \begin_layout Bibliography
1276 \begin_inset LatexCommand bibitem
1282 Zheng, W., Davidsen, A.