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68 Hydrodynamics of giant planet formation
71 \begin_layout Subtitle
74 \begin_inset Formula $\kappa$
83 \begin_inset Flex institutemark
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154 \begin_layout Offprint
159 \begin_layout Address
160 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
162 \begin_inset Newline newline
166 \begin_inset Flex Email
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170 wuchterl@amok.ast.univie.ac.at
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188 University of Alexandria, Department of Geography, ...
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197 c.ptolemy@hipparch.uheaven.space
206 \begin_layout Plain Layout
207 The university of heaven temporarily does not accept e-mails
216 Received September 15, 1996; accepted March 16, 1997
219 \begin_layout Abstract
220 To investigate the physical nature of the `nuc\SpecialChar \-
221 leated instability' of proto
222 giant planets, the stability of layers in static, radiative gas spheres
223 is analysed on the basis of Baker's standard one-zone model.
224 It is shown that stability depends only upon the equations of state, the
225 opacities and the local thermodynamic state in the layer.
226 Stability and instability can therefore be expressed in the form of stability
227 equations of state which are universal for a given composition.
228 The stability equations of state are calculated for solar composition and
229 are displayed in the domain
230 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
234 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
238 These displays may be used to determine the one-zone stability of layers
239 in stellar or planetary structure models by directly reading off the value
240 of the stability equations for the thermodynamic state of these layers,
241 specified by state quantities as density
242 \begin_inset Formula $\rho$
246 \begin_inset Formula $T$
249 or specific internal energy
250 \begin_inset Formula $e$
254 Regions of instability in the
255 \begin_inset Formula $(\rho,e)$
258 -plane are described and related to the underlying microphysical processes.
259 Vibrational instability is found to be a common phenomenon at temperatures
260 lower than the second He ionisation zone.
262 \begin_inset Formula $\kappa$
265 -mechanism is widespread under `cool' conditions.
266 \begin_inset Note Note
269 \begin_layout Plain Layout
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279 \begin_layout Plain Layout
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285 template file that comes with LyX.
293 \begin_layout Keywords
294 giant planet formation --
295 \begin_inset Formula $\kappa$
298 -mechanism -- stability of gas spheres
301 \begin_layout Section
305 \begin_layout Standard
308 nucleated instability
310 (also called core instability) hypothesis of giant planet formation, a
311 critical mass for static core envelope protoplanets has been found.
313 \begin_inset CommandInset citation
319 ) determined the critical mass of the core to be about
320 \begin_inset Formula $12\, M_{\oplus}$
324 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
327 is the Earth mass), which is independent of the outer boundary conditions
328 and therefore independent of the location in the solar nebula.
329 This critical value for the core mass corresponds closely to the cores
330 of today's giant planets.
333 \begin_layout Standard
334 Although no hydrodynamical study has been available many workers conjectured
335 that a collapse or rapid contraction will ensue after accumulating the
337 The main motivation for this article is to investigate the stability of
338 the static envelope at the critical mass.
339 With this aim the local, linear stability of static radiative gas spheres
340 is investigated on the basis of Baker's (
341 \begin_inset CommandInset citation
347 ) standard one-zone model.
350 \begin_layout Standard
351 Phenomena similar to the ones described above for giant planet formation
352 have been found in hydrodynamical models concerning star formation where
353 protostellar cores explode (Tscharnuter
354 \begin_inset CommandInset citation
361 \begin_inset CommandInset citation
367 ), whereas earlier studies found quasi-steady collapse flows.
368 The similarities in the (micro)physics, i.
369 \begin_inset space \thinspace{}
373 \begin_inset space \space{}
376 constitutive relations of protostellar cores and protogiant planets serve
377 as a further motivation for this study.
380 \begin_layout Section
381 Baker's standard one-zone model
384 \begin_layout Standard
385 \begin_inset Float figure
390 \begin_layout Plain Layout
393 \begin_layout Plain Layout
395 \begin_inset Formula $\Gamma_{1}$
400 \begin_inset Formula $\Gamma_{1}$
403 is plotted as a function of
404 \begin_inset Formula $\lg$
408 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
412 \begin_inset Formula $\lg$
416 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
427 \begin_layout Plain Layout
428 \begin_inset CommandInset label
439 In this section the one-zone model of Baker (
440 \begin_inset CommandInset citation
446 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
448 The resulting stability criteria will be rewritten in terms of local state
449 variables, local timescales and constitutive relations.
452 \begin_layout Standard
454 \begin_inset CommandInset citation
460 ) investigates the stability of thin layers in self-gravitating, spherical
461 gas clouds with the following properties:
464 \begin_layout Itemize
465 hydrostatic equilibrium,
468 \begin_layout Itemize
472 \begin_layout Itemize
473 energy transport by grey radiation diffusion.
477 \begin_layout Standard
479 For the one-zone-model Baker obtains necessary conditions for dynamical,
480 secular and vibrational (or pulsational) stability (Eqs.
481 \begin_inset space \space{}
485 \begin_inset space \thinspace{}
489 \begin_inset space \thinspace{}
493 \begin_inset CommandInset citation
500 Using Baker's notation:
503 \begin_layout Standard
505 \begin_inset Formula \begin{eqnarray*}
506 M_{r} & & \textrm{mass internal to the radius }r\\
507 m & & \textrm{mass of the zone}\\
508 r_{0} & & \textrm{unperturbed zone radius}\\
509 \rho_{0} & & \textrm{unperturbed density in the zone}\\
510 T_{0} & & \textrm{unperturbed temperature in the zone}\\
511 L_{r0} & & \textrm{unperturbed luminosity}\\
512 E_{\textrm{th}} & & \textrm{thermal energy of the zone}\end{eqnarray*}
519 \begin_layout Standard
521 and with the definitions of the
530 \begin_inset CommandInset ref
537 \begin_inset Formula \begin{equation}
538 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,\end{equation}
547 \begin_inset Formula \begin{equation}
548 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,\end{equation}
553 \begin_inset Formula $K$
557 \begin_inset Formula $\sigma_{0}$
560 have the following form:
561 \begin_inset Formula \begin{eqnarray}
562 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
563 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;\end{eqnarray}
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}\\
575 e=mc^{2}\end{array}\end{equation}
579 is a thermodynamical quantity which is of order
580 \begin_inset Formula $1$
584 \begin_inset Formula $1$
587 for nonreacting mixtures of classical perfect gases.
588 The physical meaning of
589 \begin_inset Formula $\sigma_{0}$
593 \begin_inset Formula $K$
596 is clearly visible in the equations above.
598 \begin_inset Formula $\sigma_{0}$
601 represents a frequency of the order one per free-fall time.
603 \begin_inset Formula $K$
606 is proportional to the ratio of the free-fall time and the cooling time.
607 Substituting into Baker's criteria, using thermodynamic identities and
608 definitions of thermodynamic quantities,
609 \begin_inset Formula \[
610 \Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{S}\,,\;\chi_{\rho}^{}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{T}\,,\;\kappa_{P}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln P}\right)_{T}\]
615 \begin_inset Formula \[
616 \nabla_{\mathrm{ad}}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{S}\,,\;\chi_{T}^{}=\left(\frac{\partial\ln P}{\partial\ln T}\right)_{\rho}\,,\;\kappa_{T}^{}=\left(\frac{\partial\ln\kappa}{\partial\ln T}\right)_{T}\]
620 one obtains, after some pages of algebra, the conditions for
625 \begin_inset Formula \begin{eqnarray}
626 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
627 \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}\\
628 \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}
632 For a physical discussion of the stability criteria see Baker (
633 \begin_inset CommandInset citation
640 \begin_inset CommandInset citation
649 \begin_layout Standard
650 We observe that these criteria for dynamical, secular and vibrational stability,
651 respectively, can be factorized into
654 \begin_layout Enumerate
655 a factor containing local timescales only,
658 \begin_layout Enumerate
659 a factor containing only constitutive relations and their derivatives.
663 \begin_layout Standard
664 The first factors, depending on only timescales, are positive by definition.
665 The signs of the left hand sides of the inequalities
670 \begin_inset CommandInset ref
677 \begin_inset CommandInset ref
684 \begin_inset CommandInset ref
690 ) therefore depend exclusively on the second factors containing the constitutive
692 Since they depend only on state variables, the stability criteria themselves
695 functions of the thermodynamic state in the local zone
698 The one-zone stability can therefore be determined from a simple equation
699 of state, given for example, as a function of density and temperature.
700 Once the microphysics, i.
701 \begin_inset space \thinspace{}
705 \begin_inset space \space{}
708 the thermodynamics and opacities (see Table
713 \begin_inset CommandInset ref
719 ), are specified (in practice by specifying a chemical composition) the
720 one-zone stability can be inferred if the thermodynamic state is specified.
721 The zone -- or in other words the layer -- will be stable or unstable in
722 whatever object it is imbedded as long as it satisfies the one-zone-model
724 Only the specific growth rates (depending upon the time scales) will be
725 different for layers in different objects.
728 \begin_layout Standard
729 \begin_inset Float table
734 \begin_layout Plain Layout
737 \begin_layout Plain Layout
738 \begin_inset CommandInset label
752 \begin_layout Plain Layout
754 <lyxtabular version="3" rows="4" columns="2">
756 <column alignment="left" valignment="top" width="0pt">
757 <column alignment="left" valignment="top" width="0pt">
759 <cell alignment="center" valignment="top" topline="true" usebox="none">
762 \begin_layout Plain Layout
768 <cell alignment="center" valignment="top" topline="true" usebox="none">
771 \begin_layout Plain Layout
772 \begin_inset Formula $T/[\textrm{K}]$
782 <cell alignment="center" valignment="top" topline="true" usebox="none">
785 \begin_layout Plain Layout
786 Yorke 1979, Yorke 1980a
791 <cell alignment="center" valignment="top" topline="true" usebox="none">
794 \begin_layout Plain Layout
795 \begin_inset Formula $\leq1700^{\textrm{a}}$
805 <cell alignment="center" valignment="top" usebox="none">
808 \begin_layout Plain Layout
814 <cell alignment="center" valignment="top" usebox="none">
817 \begin_layout Plain Layout
818 \begin_inset Formula $1700\leq T\leq5000$
828 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
831 \begin_layout Plain Layout
837 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
840 \begin_layout Plain Layout
841 \begin_inset Formula $5000\leq$
857 \begin_layout Plain Layout
858 \begin_inset Formula $^{\textrm{a}}$
866 We will now write down the sign (and therefore stability) determining parts
867 of the left-hand sides of the inequalities (
868 \begin_inset CommandInset ref
875 \begin_inset CommandInset ref
882 \begin_inset CommandInset ref
890 stability equations of state
895 \begin_layout Standard
896 The sign determining part of inequality
901 \begin_inset CommandInset ref
908 \begin_inset Formula $3\Gamma_{1}-4$
911 and it reduces to the criterion for dynamical stability
912 \begin_inset Formula \begin{equation}
913 \Gamma_{1}>\frac{4}{3}\,\cdot\end{equation}
917 Stability of the thermodynamical equilibrium demands
918 \begin_inset Formula \begin{equation}
919 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,\end{equation}
924 \begin_inset Formula \begin{equation}
925 \chi_{T}^{}>0\end{equation}
929 holds for a wide range of physical situations.
931 \begin_inset Formula \begin{eqnarray}
932 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
933 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
934 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0\end{eqnarray}
938 we find the sign determining terms in inequalities
943 \begin_inset CommandInset ref
950 \begin_inset CommandInset ref
956 ) respectively and obtain the following form of the criteria for dynamical,
957 secular and vibrational
962 \begin_inset Formula \begin{eqnarray}
963 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
964 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
965 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}\end{eqnarray}
969 The constitutive relations are to be evaluated for the unperturbed thermodynami
971 \begin_inset Formula $(\rho_{0},T_{0})$
975 We see that the one-zone stability of the layer depends only on the constitutiv
977 \begin_inset Formula $\Gamma_{1}$
981 \begin_inset Formula $\nabla_{\mathrm{ad}}$
985 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
989 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
993 These depend only on the unperturbed thermodynamical state of the layer.
994 Therefore the above relations define the one-zone-stability equations of
996 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1000 \begin_inset Formula $S_{\mathrm{vib}}$
1005 \begin_inset space ~
1009 \begin_inset CommandInset ref
1011 reference "FigVibStab"
1016 \begin_inset Formula $S_{\mathrm{vib}}$
1020 Regions of secular instability are listed in Table
1021 \begin_inset space ~
1027 \begin_layout Standard
1028 \begin_inset Float figure
1033 \begin_layout Plain Layout
1034 \begin_inset Caption
1036 \begin_layout Plain Layout
1037 Vibrational stability equation of state
1038 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1043 \begin_inset Formula $>0$
1046 means vibrational stability
1054 \begin_layout Plain Layout
1055 \begin_inset CommandInset label
1069 \begin_layout Section
1073 \begin_layout Enumerate
1074 The conditions for the stability of static, radiative layers in gas spheres,
1075 as described by Baker's (
1076 \begin_inset CommandInset citation
1082 ) standard one-zone model, can be expressed as stability equations of state.
1083 These stability equations of state depend only on the local thermodynamic
1088 \begin_layout Enumerate
1089 If the constitutive relations -- equations of state and Rosseland mean opacities
1090 -- are specified, the stability equations of state can be evaluated without
1091 specifying properties of the layer.
1095 \begin_layout Enumerate
1096 For solar composition gas the
1097 \begin_inset Formula $\kappa$
1100 -mechanism is working in the regions of the ice and dust features in the
1102 \begin_inset Formula $\mathrm{H}_{2}$
1105 dissociation and the combined H, first He ionization zone, as indicated
1106 by vibrational instability.
1107 These regions of instability are much larger in extent and degree of instabilit
1108 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1112 \begin_layout Acknowledgement
1113 Part of this work was supported by the German
1116 sche For\SpecialChar \-
1117 schungs\SpecialChar \-
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1156 1966, in Stellar Evolution, ed.
1157 \begin_inset space \space{}
1165 Cameron (Plenum, New York) 333
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1190 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
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1205 1969, Academia Nauk, Scientific Information 15, 1
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1246 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
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1274 Zheng, W., Davidsen, A.