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50 Hydrodynamics of giant planet formation
53 \begin_layout Subtitle
56 \begin_inset Formula $\kappa$
68 \begin_layout Plain Layout
77 \begin_layout Plain Layout
89 \begin_layout Plain Layout
104 \begin_layout Plain Layout
105 Just to show the usage of the elements in the author field
113 \begin_layout Offprint
118 \begin_layout Address
119 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
121 \begin_inset Newline newline
128 \begin_layout Plain Layout
132 email{wuchterl@amok.ast.univie.ac.at}
137 \begin_layout Plain Layout
143 University of Alexandria, Department of Geography, ...
144 \begin_inset Newline newline
151 \begin_layout Plain Layout
155 email{c.ptolemy@hipparch.uheaven.space}
164 \begin_layout Plain Layout
165 The university of heaven temporarily does not accept e-mails
174 Received September 15, 1996; accepted March 16, 1997
177 \begin_layout Abstract
178 To investigate the physical nature of the `nuc\SpecialChar \-
179 leated instability' of proto
180 giant planets (Mizuno
181 \begin_inset CommandInset citation
187 ), the stability of layers in static, radiative gas spheres is analysed
188 on the basis of Baker's
189 \begin_inset CommandInset citation
195 standard one-zone model.
196 It is shown that stability depends only upon the equations of state, the
197 opacities and the local thermodynamic state in the layer.
198 Stability and instability can therefore be expressed in the form of stability
199 equations of state which are universal for a given composition.
200 The stability equations of state are calculated for solar composition and
201 are displayed in the domain
202 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
206 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
210 These displays may be used to determine the one-zone stability of layers
211 in stellar or planetary structure models by directly reading off the value
212 of the stability equations for the thermodynamic state of these layers,
213 specified by state quantities as density
214 \begin_inset Formula $\rho$
218 \begin_inset Formula $T$
221 or specific internal energy
222 \begin_inset Formula $e$
226 Regions of instability in the
227 \begin_inset Formula $(\rho,e)$
230 -plane are described and related to the underlying microphysical processes.
231 Vibrational instability is found to be a common phenomenon at temperatures
232 lower than the second He ionisation zone.
234 \begin_inset Formula $\kappa$
237 -mechanism is widespread under `cool' conditions.
241 \begin_layout Plain Layout
245 keywords{giant planet formation --
251 )-mechanism -- stability of gas spheres }
259 \begin_layout Section
263 \begin_layout Standard
266 nucleated instability
268 (also called core instability) hypothesis of giant planet formation, a
269 critical mass for static core envelope protoplanets has been found.
271 \begin_inset CommandInset citation
277 ) determined the critical mass of the core to be about
278 \begin_inset Formula $12\, M_{\oplus}$
282 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
285 is the Earth mass), which is independent of the outer boundary conditions
286 and therefore independent of the location in the solar nebula.
287 This critical value for the core mass corresponds closely to the cores
288 of today's giant planets.
291 \begin_layout Standard
292 Although no hydrodynamical study has been available many workers conjectured
293 that a collapse or rapid contraction will ensue after accumulating the
295 The main motivation for this article is to investigate the stability of
296 the static envelope at the critical mass.
297 With this aim the local, linear stability of static radiative gas spheres
298 is investigated on the basis of Baker's (
299 \begin_inset CommandInset citation
305 ) standard one-zone model.
308 \begin_layout Standard
309 Phenomena similar to the ones described above for giant planet formation
310 have been found in hydrodynamical models concerning star formation where
311 protostellar cores explode (Tscharnuter
312 \begin_inset CommandInset citation
319 \begin_inset CommandInset citation
325 ), whereas earlier studies found quasi-steady collapse flows.
326 The similarities in the (micro)physics, i.e.
327 \begin_inset space \space{}
330 constitutive relations of protostellar cores and protogiant planets serve
331 as a further motivation for this study.
334 \begin_layout Section
335 Baker's standard one-zone model
338 \begin_layout Standard
339 \begin_inset Float figure
344 \begin_layout Plain Layout
347 \begin_layout Plain Layout
349 \begin_inset Formula $\Gamma_{1}$
354 \begin_inset Formula $\Gamma_{1}$
357 is plotted as a function of
358 \begin_inset Formula $\lg$
362 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
366 \begin_inset Formula $\lg$
370 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
381 \begin_layout Plain Layout
382 \begin_inset CommandInset label
393 In this section the one-zone model of Baker (
394 \begin_inset CommandInset citation
400 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
402 The resulting stability criteria will be rewritten in terms of local state
403 variables, local timescales and constitutive relations.
406 \begin_layout Standard
408 \begin_inset CommandInset citation
414 ) investigates the stability of thin layers in self-gravitating, spherical
415 gas clouds with the following properties:
418 \begin_layout Itemize
419 hydrostatic equilibrium,
422 \begin_layout Itemize
426 \begin_layout Itemize
427 energy transport by grey radiation diffusion.
431 \begin_layout Standard
433 For the one-zone-model Baker obtains necessary conditions for dynamical,
434 secular and vibrational (or pulsational) stability (Eqs.
435 \begin_inset space \space{}
439 \begin_inset space \thinspace{}
443 \begin_inset space \thinspace{}
447 \begin_inset CommandInset citation
454 Using Baker's notation:
457 \begin_layout Standard
459 \begin_inset Formula \begin{eqnarray*}
460 M_{r} & & \textrm{mass internal to the radius }r\\
461 m & & \textrm{mass of the zone}\\
462 r_{0} & & \textrm{unperturbed zone radius}\\
463 \rho_{0} & & \textrm{unperturbed density in the zone}\\
464 T_{0} & & \textrm{unperturbed temperature in the zone}\\
465 L_{r0} & & \textrm{unperturbed luminosity}\\
466 E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
473 \begin_layout Standard
475 and with the definitions of the
484 \begin_inset CommandInset ref
491 \begin_inset Formula \begin{equation}
492 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
501 \begin_inset Formula \begin{equation}
502 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,\end{equation}
507 \begin_inset Formula $K$
511 \begin_inset Formula $\sigma_{0}$
514 have the following form:
515 \begin_inset Formula \begin{eqnarray}
516 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
517 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;\end{eqnarray}
522 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
526 \begin_inset Formula \begin{equation}
528 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
529 e=mc^{2}\end{array}\end{equation}
533 is a thermodynamical quantity which is of order
534 \begin_inset Formula $1$
538 \begin_inset Formula $1$
541 for nonreacting mixtures of classical perfect gases.
542 The physical meaning of
543 \begin_inset Formula $\sigma_{0}$
547 \begin_inset Formula $K$
550 is clearly visible in the equations above.
552 \begin_inset Formula $\sigma_{0}$
555 represents a frequency of the order one per free-fall time.
557 \begin_inset Formula $K$
560 is proportional to the ratio of the free-fall time and the cooling time.
561 Substituting into Baker's criteria, using thermodynamic identities and
562 definitions of thermodynamic quantities,
563 \begin_inset Formula \[
564 \Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}\]
569 \begin_inset Formula \[
570 \nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}\]
574 one obtains, after some pages of algebra, the conditions for
579 \begin_inset Formula \begin{eqnarray}
580 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
581 \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}\\
582 \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}
586 For a physical discussion of the stability criteria see Baker (
587 \begin_inset CommandInset citation
594 \begin_inset CommandInset citation
603 \begin_layout Standard
604 We observe that these criteria for dynamical, secular and vibrational stability,
605 respectively, can be factorized into
608 \begin_layout Enumerate
609 a factor containing local timescales only,
612 \begin_layout Enumerate
613 a factor containing only constitutive relations and their derivatives.
617 \begin_layout Standard
618 The first factors, depending on only timescales, are positive by definition.
619 The signs of the left hand sides of the inequalities
624 \begin_inset CommandInset ref
631 \begin_inset CommandInset ref
638 \begin_inset CommandInset ref
644 ) therefore depend exclusively on the second factors containing the constitutive
646 Since they depend only on state variables, the stability criteria themselves
649 functions of the thermodynamic state in the local zone
652 The one-zone stability can therefore be determined from a simple equation
653 of state, given for example, as a function of density and temperature.
654 Once the microphysics, i.e.
655 \begin_inset space \space{}
658 the thermodynamics and opacities (see Table
663 \begin_inset CommandInset ref
669 ), are specified (in practice by specifying a chemical composition) the
670 one-zone stability can be inferred if the thermodynamic state is specified.
671 The zone -- or in other words the layer -- will be stable or unstable in
672 whatever object it is imbedded as long as it satisfies the one-zone-model
674 Only the specific growth rates (depending upon the time scales) will be
675 different for layers in different objects.
678 \begin_layout Standard
679 \begin_inset Float table
684 \begin_layout Plain Layout
687 \begin_layout Plain Layout
688 \begin_inset CommandInset label
702 \begin_layout Plain Layout
704 <lyxtabular version="3" rows="4" columns="2">
706 <column alignment="left" valignment="top" width="0pt">
707 <column alignment="left" valignment="top" width="0pt">
709 <cell alignment="center" valignment="top" topline="true" usebox="none">
712 \begin_layout Plain Layout
718 <cell alignment="center" valignment="top" topline="true" usebox="none">
721 \begin_layout Plain Layout
722 \begin_inset Formula $T/[\textrm{K}]$
732 <cell alignment="center" valignment="top" topline="true" usebox="none">
735 \begin_layout Plain Layout
736 Yorke 1979, Yorke 1980a
741 <cell alignment="center" valignment="top" topline="true" usebox="none">
744 \begin_layout Plain Layout
745 \begin_inset Formula $\leq1700^{\textrm{a}}$
755 <cell alignment="center" valignment="top" usebox="none">
758 \begin_layout Plain Layout
764 <cell alignment="center" valignment="top" usebox="none">
767 \begin_layout Plain Layout
768 \begin_inset Formula $1700\leq T\leq5000$
778 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
781 \begin_layout Plain Layout
787 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
790 \begin_layout Plain Layout
791 \begin_inset Formula $5000\leq$
807 \begin_layout Plain Layout
808 \begin_inset Formula $^{\textrm{a}}$
816 We will now write down the sign (and therefore stability) determining parts
817 of the left-hand sides of the inequalities (
818 \begin_inset CommandInset ref
825 \begin_inset CommandInset ref
832 \begin_inset CommandInset ref
840 stability equations of state
845 \begin_layout Standard
846 The sign determining part of inequality
851 \begin_inset CommandInset ref
858 \begin_inset Formula $3\Gamma_{1}-4$
861 and it reduces to the criterion for dynamical stability
862 \begin_inset Formula \begin{equation}
863 \Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
867 Stability of the thermodynamical equilibrium demands
868 \begin_inset Formula \begin{equation}
869 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
874 \begin_inset Formula \begin{equation}
875 \chi_{T}^{}>0\end{equation}
879 holds for a wide range of physical situations.
881 \begin_inset Formula \begin{eqnarray}
882 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
883 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
884 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
888 we find the sign determining terms in inequalities
893 \begin_inset CommandInset ref
900 \begin_inset CommandInset ref
906 ) respectively and obtain the following form of the criteria for dynamical,
907 secular and vibrational
912 \begin_inset Formula \begin{eqnarray}
913 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
914 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
915 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}\end{eqnarray}
919 The constitutive relations are to be evaluated for the unperturbed thermodynami
921 \begin_inset Formula $(\rho_{0},T_{0})$
925 We see that the one-zone stability of the layer depends only on the constitutiv
927 \begin_inset Formula $\Gamma_{1}$
931 \begin_inset Formula $\nabla_{\mathrm{ad}}$
935 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
939 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
943 These depend only on the unperturbed thermodynamical state of the layer.
944 Therefore the above relations define the one-zone-stability equations of
946 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
950 \begin_inset Formula $S_{\mathrm{vib}}$
959 \begin_inset CommandInset ref
961 reference "FigVibStab"
966 \begin_inset Formula $S_{\mathrm{vib}}$
970 Regions of secular instability are listed in Table
977 \begin_layout Standard
978 \begin_inset Float figure
983 \begin_layout Plain Layout
986 \begin_layout Plain Layout
987 Vibrational stability equation of state
988 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
993 \begin_inset Formula $>0$
996 means vibrational stability
1004 \begin_layout Plain Layout
1005 \begin_inset CommandInset label
1019 \begin_layout Section
1023 \begin_layout Enumerate
1024 The conditions for the stability of static, radiative layers in gas spheres,
1025 as described by Baker's (
1026 \begin_inset CommandInset citation
1032 ) standard one-zone model, can be expressed as stability equations of state.
1033 These stability equations of state depend only on the local thermodynamic
1038 \begin_layout Enumerate
1039 If the constitutive relations -- equations of state and Rosseland mean opacities
1040 -- are specified, the stability equations of state can be evaluated without
1041 specifying properties of the layer.
1045 \begin_layout Enumerate
1046 For solar composition gas the
1047 \begin_inset Formula $\kappa$
1050 -mechanism is working in the regions of the ice and dust features in the
1052 \begin_inset Formula $\mathrm{H}_{2}$
1055 dissociation and the combined H, first He ionization zone, as indicated
1056 by vibrational instability.
1057 These regions of instability are much larger in extent and degree of instabilit
1058 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1062 \begin_layout Acknowledgement
1063 Part of this work was supported by the German
1066 sche For\SpecialChar \-
1067 schungs\SpecialChar \-
1073 \begin_inset space ~
1080 \begin_layout Bibliography
1081 \begin_inset CommandInset bibitem
1082 LatexCommand bibitem
1089 1966, in Stellar Evolution, ed.
1093 \begin_layout Plain Layout
1107 Cameron (Plenum, New York) 333
1110 \begin_layout Bibliography
1111 \begin_inset CommandInset bibitem
1112 LatexCommand bibitem
1122 \begin_layout Bibliography
1123 \begin_inset CommandInset bibitem
1124 LatexCommand bibitem
1132 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1136 \begin_layout Bibliography
1137 \begin_inset CommandInset bibitem
1138 LatexCommand bibitem
1147 1969, Academia Nauk, Scientific Information 15, 1
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1188 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1197 LatexCommand bibitem
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1216 Zheng, W., Davidsen, A.