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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
163 \begin_inset Newline newline
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
271 Citations are not allowed in A&A abstracts.
277 \begin_inset Note Note
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.e.
370 \begin_inset space \space{}
373 constitutive relations of protostellar cores and protogiant planets serve
374 as a further motivation for this study.
377 \begin_layout Section
378 Baker's standard one-zone model
381 \begin_layout Standard
382 \begin_inset Float figure
387 \begin_layout Plain Layout
390 \begin_layout Plain Layout
392 \begin_inset Formula $\Gamma_{1}$
397 \begin_inset Formula $\Gamma_{1}$
400 is plotted as a function of
401 \begin_inset Formula $\lg$
405 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
409 \begin_inset Formula $\lg$
413 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
424 \begin_layout Plain Layout
425 \begin_inset CommandInset label
436 In this section the one-zone model of Baker (
437 \begin_inset CommandInset citation
443 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
445 The resulting stability criteria will be rewritten in terms of local state
446 variables, local timescales and constitutive relations.
449 \begin_layout Standard
451 \begin_inset CommandInset citation
457 ) investigates the stability of thin layers in self-gravitating, spherical
458 gas clouds with the following properties:
461 \begin_layout Itemize
462 hydrostatic equilibrium,
465 \begin_layout Itemize
469 \begin_layout Itemize
470 energy transport by grey radiation diffusion.
474 \begin_layout Standard
476 For the one-zone-model Baker obtains necessary conditions for dynamical,
477 secular and vibrational (or pulsational) stability (Eqs.
478 \begin_inset space \space{}
482 \begin_inset space \thinspace{}
486 \begin_inset space \thinspace{}
490 \begin_inset CommandInset citation
497 Using Baker's notation:
500 \begin_layout Standard
502 \begin_inset Formula \begin{eqnarray*}
503 M_{r} & & \textrm{mass internal to the radius }r\\
504 m & & \textrm{mass of the zone}\\
505 r_{0} & & \textrm{unperturbed zone radius}\\
506 \rho_{0} & & \textrm{unperturbed density in the zone}\\
507 T_{0} & & \textrm{unperturbed temperature in the zone}\\
508 L_{r0} & & \textrm{unperturbed luminosity}\\
509 E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
516 \begin_layout Standard
518 and with the definitions of the
527 \begin_inset CommandInset ref
534 \begin_inset Formula \begin{equation}
535 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
544 \begin_inset Formula \begin{equation}
545 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,\end{equation}
550 \begin_inset Formula $K$
554 \begin_inset Formula $\sigma_{0}$
557 have the following form:
558 \begin_inset Formula \begin{eqnarray}
559 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
560 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;\end{eqnarray}
565 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
569 \begin_inset Formula \begin{equation}
571 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
572 e=mc^{2}\end{array}\end{equation}
576 is a thermodynamical quantity which is of order
577 \begin_inset Formula $1$
581 \begin_inset Formula $1$
584 for nonreacting mixtures of classical perfect gases.
585 The physical meaning of
586 \begin_inset Formula $\sigma_{0}$
590 \begin_inset Formula $K$
593 is clearly visible in the equations above.
595 \begin_inset Formula $\sigma_{0}$
598 represents a frequency of the order one per free-fall time.
600 \begin_inset Formula $K$
603 is proportional to the ratio of the free-fall time and the cooling time.
604 Substituting into Baker's criteria, using thermodynamic identities and
605 definitions of thermodynamic quantities,
606 \begin_inset Formula \[
607 \Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}\]
612 \begin_inset Formula \[
613 \nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}\]
617 one obtains, after some pages of algebra, the conditions for
622 \begin_inset Formula \begin{eqnarray}
623 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
624 \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}\\
625 \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}
629 For a physical discussion of the stability criteria see Baker (
630 \begin_inset CommandInset citation
637 \begin_inset CommandInset citation
646 \begin_layout Standard
647 We observe that these criteria for dynamical, secular and vibrational stability,
648 respectively, can be factorized into
651 \begin_layout Enumerate
652 a factor containing local timescales only,
655 \begin_layout Enumerate
656 a factor containing only constitutive relations and their derivatives.
660 \begin_layout Standard
661 The first factors, depending on only timescales, are positive by definition.
662 The signs of the left hand sides of the inequalities
667 \begin_inset CommandInset ref
674 \begin_inset CommandInset ref
681 \begin_inset CommandInset ref
687 ) therefore depend exclusively on the second factors containing the constitutive
689 Since they depend only on state variables, the stability criteria themselves
692 functions of the thermodynamic state in the local zone
695 The one-zone stability can therefore be determined from a simple equation
696 of state, given for example, as a function of density and temperature.
697 Once the microphysics, i.e.
698 \begin_inset space \space{}
701 the thermodynamics and opacities (see Table
706 \begin_inset CommandInset ref
712 ), are specified (in practice by specifying a chemical composition) the
713 one-zone stability can be inferred if the thermodynamic state is specified.
714 The zone -- or in other words the layer -- will be stable or unstable in
715 whatever object it is imbedded as long as it satisfies the one-zone-model
717 Only the specific growth rates (depending upon the time scales) will be
718 different for layers in different objects.
721 \begin_layout Standard
722 \begin_inset Float table
727 \begin_layout Plain Layout
730 \begin_layout Plain Layout
731 \begin_inset CommandInset label
745 \begin_layout Plain Layout
747 <lyxtabular version="3" rows="4" columns="2">
749 <column alignment="left" valignment="top" width="0pt">
750 <column alignment="left" valignment="top" width="0pt">
752 <cell alignment="center" valignment="top" topline="true" usebox="none">
755 \begin_layout Plain Layout
761 <cell alignment="center" valignment="top" topline="true" usebox="none">
764 \begin_layout Plain Layout
765 \begin_inset Formula $T/[\textrm{K}]$
775 <cell alignment="center" valignment="top" topline="true" usebox="none">
778 \begin_layout Plain Layout
779 Yorke 1979, Yorke 1980a
784 <cell alignment="center" valignment="top" topline="true" usebox="none">
787 \begin_layout Plain Layout
788 \begin_inset Formula $\leq1700^{\textrm{a}}$
798 <cell alignment="center" valignment="top" usebox="none">
801 \begin_layout Plain Layout
807 <cell alignment="center" valignment="top" usebox="none">
810 \begin_layout Plain Layout
811 \begin_inset Formula $1700\leq T\leq5000$
821 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
824 \begin_layout Plain Layout
830 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
833 \begin_layout Plain Layout
834 \begin_inset Formula $5000\leq$
850 \begin_layout Plain Layout
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 CommandInset ref
868 \begin_inset CommandInset ref
875 \begin_inset CommandInset ref
883 stability equations of state
888 \begin_layout Standard
889 The sign determining part of inequality
894 \begin_inset CommandInset ref
901 \begin_inset Formula $3\Gamma_{1}-4$
904 and it reduces to the criterion for dynamical stability
905 \begin_inset Formula \begin{equation}
906 \Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
910 Stability of the thermodynamical equilibrium demands
911 \begin_inset Formula \begin{equation}
912 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
917 \begin_inset Formula \begin{equation}
918 \chi_{T}^{}>0\end{equation}
922 holds for a wide range of physical situations.
924 \begin_inset Formula \begin{eqnarray}
925 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
926 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
927 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
931 we find the sign determining terms in inequalities
936 \begin_inset CommandInset ref
943 \begin_inset CommandInset ref
949 ) respectively and obtain the following form of the criteria for dynamical,
950 secular and vibrational
955 \begin_inset Formula \begin{eqnarray}
956 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
957 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
958 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}\end{eqnarray}
962 The constitutive relations are to be evaluated for the unperturbed thermodynami
964 \begin_inset Formula $(\rho_{0},T_{0})$
968 We see that the one-zone stability of the layer depends only on the constitutiv
970 \begin_inset Formula $\Gamma_{1}$
974 \begin_inset Formula $\nabla_{\mathrm{ad}}$
978 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
982 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
986 These depend only on the unperturbed thermodynamical state of the layer.
987 Therefore the above relations define the one-zone-stability equations of
989 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
993 \begin_inset Formula $S_{\mathrm{vib}}$
1002 \begin_inset CommandInset ref
1004 reference "FigVibStab"
1009 \begin_inset Formula $S_{\mathrm{vib}}$
1013 Regions of secular instability are listed in Table
1014 \begin_inset space ~
1020 \begin_layout Standard
1021 \begin_inset Float figure
1026 \begin_layout Plain Layout
1027 \begin_inset Caption
1029 \begin_layout Plain Layout
1030 Vibrational stability equation of state
1031 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1036 \begin_inset Formula $>0$
1039 means vibrational stability
1047 \begin_layout Plain Layout
1048 \begin_inset CommandInset label
1062 \begin_layout Section
1066 \begin_layout Enumerate
1067 The conditions for the stability of static, radiative layers in gas spheres,
1068 as described by Baker's (
1069 \begin_inset CommandInset citation
1075 ) standard one-zone model, can be expressed as stability equations of state.
1076 These stability equations of state depend only on the local thermodynamic
1081 \begin_layout Enumerate
1082 If the constitutive relations -- equations of state and Rosseland mean opacities
1083 -- are specified, the stability equations of state can be evaluated without
1084 specifying properties of the layer.
1088 \begin_layout Enumerate
1089 For solar composition gas the
1090 \begin_inset Formula $\kappa$
1093 -mechanism is working in the regions of the ice and dust features in the
1095 \begin_inset Formula $\mathrm{H}_{2}$
1098 dissociation and the combined H, first He ionization zone, as indicated
1099 by vibrational instability.
1100 These regions of instability are much larger in extent and degree of instabilit
1101 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1105 \begin_layout Acknowledgement
1106 Part of this work was supported by the German
1109 sche For\SpecialChar \-
1110 schungs\SpecialChar \-
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1132 that is part of the A&A LaTeX-package.
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1149 1966, in Stellar Evolution, ed.
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1158 Cameron (Plenum, New York) 333
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1183 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
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1198 1969, Academia Nauk, Scientific Information 15, 1
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1239 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1267 Zheng, W., Davidsen, A.