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69 Hydrodynamics of giant planet formation
72 \begin_layout Subtitle
75 \begin_inset Formula $\kappa$
84 \begin_inset Flex institutemark
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160 \begin_layout Address
161 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
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167 \begin_inset Flex Email
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171 wuchterl@amok.ast.univie.ac.at
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189 University of Alexandria, Department of Geography, ...
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198 c.ptolemy@hipparch.uheaven.space
207 \begin_layout Plain Layout
208 The university of heaven temporarily does not accept e-mails
217 Received September 15, 1996; accepted March 16, 1997
220 \begin_layout Abstract
221 To investigate the physical nature of the `nuc\SpecialChar \-
222 leated instability' of proto
223 giant planets, the stability of layers in static, radiative gas spheres
224 is analysed on the basis of Baker's standard one-zone model.
225 It is shown that stability depends only upon the equations of state, the
226 opacities and the local thermodynamic state in the layer.
227 Stability and instability can therefore be expressed in the form of stability
228 equations of state which are universal for a given composition.
229 The stability equations of state are calculated for solar composition and
230 are displayed in the domain
231 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
235 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
239 These displays may be used to determine the one-zone stability of layers
240 in stellar or planetary structure models by directly reading off the value
241 of the stability equations for the thermodynamic state of these layers,
242 specified by state quantities as density
243 \begin_inset Formula $\rho$
247 \begin_inset Formula $T$
250 or specific internal energy
251 \begin_inset Formula $e$
255 Regions of instability in the
256 \begin_inset Formula $(\rho,e)$
259 -plane are described and related to the underlying microphysical processes.
260 Vibrational instability is found to be a common phenomenon at temperatures
261 lower than the second He ionisation zone.
263 \begin_inset Formula $\kappa$
266 -mechanism is widespread under `cool' conditions.
267 \begin_inset Note Note
270 \begin_layout Plain Layout
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280 \begin_layout Plain Layout
281 This is the unstructured abstract type, an example for the structured abstract
286 template file that comes with LyX.
294 \begin_layout Keywords
295 giant planet formation --
296 \begin_inset Formula $\kappa$
299 -mechanism -- stability of gas spheres
302 \begin_layout Section
306 \begin_layout Standard
309 nucleated instability
311 (also called core instability) hypothesis of giant planet formation, a
312 critical mass for static core envelope protoplanets has been found.
314 \begin_inset CommandInset citation
320 ) determined the critical mass of the core to be about
321 \begin_inset Formula $12\, M_{\oplus}$
325 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
328 is the Earth mass), which is independent of the outer boundary conditions
329 and therefore independent of the location in the solar nebula.
330 This critical value for the core mass corresponds closely to the cores
331 of today's giant planets.
334 \begin_layout Standard
335 Although no hydrodynamical study has been available many workers conjectured
336 that a collapse or rapid contraction will ensue after accumulating the
338 The main motivation for this article is to investigate the stability of
339 the static envelope at the critical mass.
340 With this aim the local, linear stability of static radiative gas spheres
341 is investigated on the basis of Baker's (
342 \begin_inset CommandInset citation
348 ) standard one-zone model.
351 \begin_layout Standard
352 Phenomena similar to the ones described above for giant planet formation
353 have been found in hydrodynamical models concerning star formation where
354 protostellar cores explode (Tscharnuter
355 \begin_inset CommandInset citation
362 \begin_inset CommandInset citation
368 ), whereas earlier studies found quasi-steady collapse flows.
369 The similarities in the (micro)physics, i.
370 \begin_inset space \thinspace{}
374 \begin_inset space \space{}
377 constitutive relations of protostellar cores and protogiant planets serve
378 as a further motivation for this study.
381 \begin_layout Section
382 Baker's standard one-zone model
385 \begin_layout Standard
386 \begin_inset Float figure
391 \begin_layout Plain Layout
394 \begin_layout Plain Layout
396 \begin_inset Formula $\Gamma_{1}$
401 \begin_inset Formula $\Gamma_{1}$
404 is plotted as a function of
405 \begin_inset Formula $\lg$
409 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
413 \begin_inset Formula $\lg$
417 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
428 \begin_layout Plain Layout
429 \begin_inset CommandInset label
440 In this section the one-zone model of Baker (
441 \begin_inset CommandInset citation
447 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
449 The resulting stability criteria will be rewritten in terms of local state
450 variables, local timescales and constitutive relations.
453 \begin_layout Standard
455 \begin_inset CommandInset citation
461 ) investigates the stability of thin layers in self-gravitating, spherical
462 gas clouds with the following properties:
465 \begin_layout Itemize
466 hydrostatic equilibrium,
469 \begin_layout Itemize
473 \begin_layout Itemize
474 energy transport by grey radiation diffusion.
478 \begin_layout Standard
480 For the one-zone-model Baker obtains necessary conditions for dynamical,
481 secular and vibrational (or pulsational) stability (Eqs.
482 \begin_inset space \space{}
486 \begin_inset space \thinspace{}
490 \begin_inset space \thinspace{}
494 \begin_inset CommandInset citation
501 Using Baker's notation:
504 \begin_layout Standard
506 \begin_inset Formula \begin{eqnarray*}
507 M_{r} & & \textrm{mass internal to the radius }r\\
508 m & & \textrm{mass of the zone}\\
509 r_{0} & & \textrm{unperturbed zone radius}\\
510 \rho_{0} & & \textrm{unperturbed density in the zone}\\
511 T_{0} & & \textrm{unperturbed temperature in the zone}\\
512 L_{r0} & & \textrm{unperturbed luminosity}\\
513 E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
520 \begin_layout Standard
522 and with the definitions of the
531 \begin_inset CommandInset ref
538 \begin_inset Formula \begin{equation}
539 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
548 \begin_inset Formula \begin{equation}
549 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,\end{equation}
554 \begin_inset Formula $K$
558 \begin_inset Formula $\sigma_{0}$
561 have the following form:
562 \begin_inset Formula \begin{eqnarray}
563 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
564 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;\end{eqnarray}
569 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
573 \begin_inset Formula \begin{equation}
575 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
576 e=mc^{2}\end{array}\end{equation}
580 is a thermodynamical quantity which is of order
581 \begin_inset Formula $1$
585 \begin_inset Formula $1$
588 for nonreacting mixtures of classical perfect gases.
589 The physical meaning of
590 \begin_inset Formula $\sigma_{0}$
594 \begin_inset Formula $K$
597 is clearly visible in the equations above.
599 \begin_inset Formula $\sigma_{0}$
602 represents a frequency of the order one per free-fall time.
604 \begin_inset Formula $K$
607 is proportional to the ratio of the free-fall time and the cooling time.
608 Substituting into Baker's criteria, using thermodynamic identities and
609 definitions of thermodynamic quantities,
610 \begin_inset Formula \[
611 \Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}\]
616 \begin_inset Formula \[
617 \nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}\]
621 one obtains, after some pages of algebra, the conditions for
626 \begin_inset Formula \begin{eqnarray}
627 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
628 \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}\\
629 \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}
633 For a physical discussion of the stability criteria see Baker (
634 \begin_inset CommandInset citation
641 \begin_inset CommandInset citation
650 \begin_layout Standard
651 We observe that these criteria for dynamical, secular and vibrational stability,
652 respectively, can be factorized into
655 \begin_layout Enumerate
656 a factor containing local timescales only,
659 \begin_layout Enumerate
660 a factor containing only constitutive relations and their derivatives.
664 \begin_layout Standard
665 The first factors, depending on only timescales, are positive by definition.
666 The signs of the left hand sides of the inequalities
671 \begin_inset CommandInset ref
678 \begin_inset CommandInset ref
685 \begin_inset CommandInset ref
691 ) therefore depend exclusively on the second factors containing the constitutive
693 Since they depend only on state variables, the stability criteria themselves
696 functions of the thermodynamic state in the local zone
699 The one-zone stability can therefore be determined from a simple equation
700 of state, given for example, as a function of density and temperature.
701 Once the microphysics, i.
702 \begin_inset space \thinspace{}
706 \begin_inset space \space{}
709 the thermodynamics and opacities (see Table
714 \begin_inset CommandInset ref
720 ), are specified (in practice by specifying a chemical composition) the
721 one-zone stability can be inferred if the thermodynamic state is specified.
722 The zone -- or in other words the layer -- will be stable or unstable in
723 whatever object it is imbedded as long as it satisfies the one-zone-model
725 Only the specific growth rates (depending upon the time scales) will be
726 different for layers in different objects.
729 \begin_layout Standard
730 \begin_inset Float table
735 \begin_layout Plain Layout
738 \begin_layout Plain Layout
739 \begin_inset CommandInset label
753 \begin_layout Plain Layout
755 <lyxtabular version="3" rows="4" columns="2">
757 <column alignment="left" valignment="top" width="0pt">
758 <column alignment="left" valignment="top" width="0pt">
760 <cell alignment="center" valignment="top" topline="true" usebox="none">
763 \begin_layout Plain Layout
769 <cell alignment="center" valignment="top" topline="true" usebox="none">
772 \begin_layout Plain Layout
773 \begin_inset Formula $T/[\textrm{K}]$
783 <cell alignment="center" valignment="top" topline="true" usebox="none">
786 \begin_layout Plain Layout
787 Yorke 1979, Yorke 1980a
792 <cell alignment="center" valignment="top" topline="true" usebox="none">
795 \begin_layout Plain Layout
796 \begin_inset Formula $\leq1700^{\textrm{a}}$
806 <cell alignment="center" valignment="top" usebox="none">
809 \begin_layout Plain Layout
815 <cell alignment="center" valignment="top" usebox="none">
818 \begin_layout Plain Layout
819 \begin_inset Formula $1700\leq T\leq5000$
829 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
832 \begin_layout Plain Layout
838 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
841 \begin_layout Plain Layout
842 \begin_inset Formula $5000\leq$
858 \begin_layout Plain Layout
859 \begin_inset Formula $^{\textrm{a}}$
867 We will now write down the sign (and therefore stability) determining parts
868 of the left-hand sides of the inequalities (
869 \begin_inset CommandInset ref
876 \begin_inset CommandInset ref
883 \begin_inset CommandInset ref
891 stability equations of state
896 \begin_layout Standard
897 The sign determining part of inequality
902 \begin_inset CommandInset ref
909 \begin_inset Formula $3\Gamma_{1}-4$
912 and it reduces to the criterion for dynamical stability
913 \begin_inset Formula \begin{equation}
914 \Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
918 Stability of the thermodynamical equilibrium demands
919 \begin_inset Formula \begin{equation}
920 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
925 \begin_inset Formula \begin{equation}
926 \chi_{T}^{}>0\end{equation}
930 holds for a wide range of physical situations.
932 \begin_inset Formula \begin{eqnarray}
933 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
934 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
935 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
939 we find the sign determining terms in inequalities
944 \begin_inset CommandInset ref
951 \begin_inset CommandInset ref
957 ) respectively and obtain the following form of the criteria for dynamical,
958 secular and vibrational
963 \begin_inset Formula \begin{eqnarray}
964 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
965 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
966 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}\end{eqnarray}
970 The constitutive relations are to be evaluated for the unperturbed thermodynami
972 \begin_inset Formula $(\rho_{0},T_{0})$
976 We see that the one-zone stability of the layer depends only on the constitutiv
978 \begin_inset Formula $\Gamma_{1}$
982 \begin_inset Formula $\nabla_{\mathrm{ad}}$
986 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
990 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
994 These depend only on the unperturbed thermodynamical state of the layer.
995 Therefore the above relations define the one-zone-stability equations of
997 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1001 \begin_inset Formula $S_{\mathrm{vib}}$
1006 \begin_inset space ~
1010 \begin_inset CommandInset ref
1012 reference "FigVibStab"
1017 \begin_inset Formula $S_{\mathrm{vib}}$
1021 Regions of secular instability are listed in Table
1022 \begin_inset space ~
1028 \begin_layout Standard
1029 \begin_inset Float figure
1034 \begin_layout Plain Layout
1035 \begin_inset Caption
1037 \begin_layout Plain Layout
1038 Vibrational stability equation of state
1039 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1044 \begin_inset Formula $>0$
1047 means vibrational stability
1055 \begin_layout Plain Layout
1056 \begin_inset CommandInset label
1070 \begin_layout Section
1074 \begin_layout Enumerate
1075 The conditions for the stability of static, radiative layers in gas spheres,
1076 as described by Baker's (
1077 \begin_inset CommandInset citation
1083 ) standard one-zone model, can be expressed as stability equations of state.
1084 These stability equations of state depend only on the local thermodynamic
1089 \begin_layout Enumerate
1090 If the constitutive relations -- equations of state and Rosseland mean opacities
1091 -- are specified, the stability equations of state can be evaluated without
1092 specifying properties of the layer.
1096 \begin_layout Enumerate
1097 For solar composition gas the
1098 \begin_inset Formula $\kappa$
1101 -mechanism is working in the regions of the ice and dust features in the
1103 \begin_inset Formula $\mathrm{H}_{2}$
1106 dissociation and the combined H, first He ionization zone, as indicated
1107 by vibrational instability.
1108 These regions of instability are much larger in extent and degree of instabilit
1109 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1113 \begin_layout Acknowledgement
1114 Part of this work was supported by the German
1117 sche For\SpecialChar \-
1118 schungs\SpecialChar \-
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1157 1966, in Stellar Evolution, ed.
1158 \begin_inset space \space{}
1166 Cameron (Plenum, New York) 333
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1191 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
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1206 1969, Academia Nauk, Scientific Information 15, 1
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1247 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1275 Zheng, W., Davidsen, A.