1 #This file was created by <pit> Mon Nov 24 22:57:48 1997
2 #LyX 0.11 (C) 1995-1997 Matthias Ettrich and the LyX Team
15 \paperorientation portrait
18 \paragraph_separation indent
20 \quotes_language english
24 \paperpagestyle default
28 06(03.11.1;16.06.1;19.06.1;19.37.1;19.53.1;19.63.1)
31 Hydrodynamics of giant planet formation
36 \begin_inset Formula \( \kappa \)
46 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
54 wuchterl@amok.ast.univie.ac.at
57 Received September 15, 1996 / Accepted March 16, 1997
60 To investigate the physical nature of the `nuc\SpecialChar \-
61 leated instability' of proto
63 \begin_inset LatexCommand \cite{mizuno}
67 ), the stability of layers in static, radiative gas spheres is analysed
68 on the basis of Baker's
69 \begin_inset LatexCommand \cite{baker}
73 standard one-zone model.
74 It is shown that stability depends only upon the equations of state, the
75 opacities and the local thermodynamic state in the layer.
76 Stability and instability can therefore be expressed in the form of stability
77 equations of state which are universal for a given composition.
80 The stability equations of state are calculated for solar composition and
81 are displayed in the domain
82 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
86 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
90 These displays may be used to determine the one-zone stability of layers
91 in stellar or planetary structure models by directly reading off the value
92 of the stability equations for the thermodynamic state of these layers,
93 specified by state quantities as density
94 \begin_inset Formula \( \rho \)
98 \begin_inset Formula \( T \)
101 or specific internal energy
102 \begin_inset Formula \( e \)
106 Regions of instability in the
107 \begin_inset Formula \( (\rho \, e) \)
110 -plane are described and related to the underlying microphysical processes.
111 Vibrational instability is found to be a common phenomenon at temperatures
112 lower than the second He ionisation zone.
114 \begin_inset Formula \( \kappa \)
117 -mechanism is widespread under `cool' conditions.
126 giant planet formation --
127 \begin_inset Formula \( \kappa \)
130 -mechanism -- stability of gas spheres
140 nucleated instability
142 (also called core instability) hypothesis of giant planet formation, a
143 critical mass for static core envelope protoplanets has been found.
145 \begin_inset LatexCommand \cite{mizuno}
149 ) determined the critical mass of the core to be about
150 \begin_inset Formula \( 12\, M_{\oplus } \)
154 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
157 is the Earth mass), which is independent of the outer boundary conditions
158 and therefore independent of the location in the solar nebula.
159 This critical value for the core mass corresponds closely to the cores
160 of today's giant planets.
163 Although no hydrodynamical study has been available many workers conjectured
164 that a collapse or rapid contraction will ensue after accumulating the
166 The main motivation for this article is to investigate the stability of
167 the static envelope at the critical mass.
168 With this aim the local, linear stability of static radiative gas spheres
169 is investigated on the basis of Baker's (
170 \begin_inset LatexCommand \cite{baker}
174 ) standard one-zone model.
175 The nonlinear, hydrodynamic evolution of the protogiant planet beyond the
176 critical mass, as calculated by Wuchterl (
177 \begin_inset LatexCommand \cite{wuchterl}
181 ), will be described in a forthcoming article.
184 The fact that Wuchterl (
185 \begin_inset LatexCommand \cite{wuchterl}
189 ) found the excitation of hydrodynamical waves in his models raises considerable
190 interest on the transition from static to dynamic evolutionary phases of
191 the protogiant planet at the critical mass.
192 The waves play a crucial role in the development of the so-called nucleated
193 instability in the nucleated instability hypothesis.
194 They lead to the formation of shock waves and massive outflow phenomena.
195 The protoplanet evolves into a new quasi-equilibrium structure with a
199 envelope, after the mass loss phase has declined.
202 Phenomena similar to the ones described above for giant planet formation
203 have been found in hydrodynamical models concerning star formation where
204 protostellar cores explode (Tscharnuter
205 \begin_inset LatexCommand \cite{tscarnuter}
210 \begin_inset LatexCommand \cite{balluch}
214 ), whereas earlier studies found quasi-steady collapse flows.
215 The similarities in the (micro)physics, i.e., constitutive relations of protostel
216 lar cores and protogiant planets serve as a further motivation for this
220 Baker's standard one-zone model
223 \begin_float wide-fig
238 \begin_inset Formula \( \Gamma \)
243 \begin_inset Formula \( \Gamma _{1} \)
246 is plotted as a function of
247 \begin_inset Formula \( \lg \)
251 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
255 \begin_inset Formula \( \lg \)
259 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
263 \begin_inset LatexCommand \label{FigGam}
271 In this section the one-zone model of Baker (
272 \begin_inset LatexCommand \cite{baker}
276 ), originally used to study the Cepheïd pulsation mechanism, will be briefly
278 The resulting stability criteria will be rewritten in terms of local state
279 variables, local timescales and constitutive relations.
283 \begin_inset LatexCommand \cite{baker}
287 ) investigates the stability of thin layers in self-gravitating, spherical
288 gas clouds with the following properties:
291 hydrostatic equilibrium,
297 energy transport by grey radiation diffusion.
300 For the one-zone-model Baker obtains necessary conditions for dynamical,
301 secular and vibrational (or pulsational) stability [Eqs.
316 \begin_inset LatexCommand \cite{baker}
321 Using Baker's notation:
324 M_{\mathrm{r}} & & \mathrm{mass}\, \mathrm{internal}\, \mathrm{to}\, \mathrm{the}\, \mathrm{radius}\, r\\
325 m & & \mathrm{mass}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}\\
326 r_{0} & & \mathrm{unperturbed}\, \mathrm{zone}\, \mathrm{radius}\\
327 \rho _{0} & & \mathrm{unperturbed}\, \mathrm{density}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\
328 T_{0} & & \mathrm{unperturbed}\, \mathrm{temperature}\, \mathrm{in}\, \mathrm{the}\, \mathrm{zone}\\
329 L_{r0} & & \mathrm{unperturbed}\, \mathrm{luminosity}\\
330 E_{\mathrm{th}} & & \mathrm{thermal}\, \mathrm{energy}\, \mathrm{of}\, \mathrm{the}\, \mathrm{zone}
335 and with the definitions of the
342 \begin_inset LatexCommand \ref{FigGam}
353 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
367 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}\, ,}
373 \begin_inset Formula \( K \)
377 \begin_inset Formula \( \sigma _{0} \)
380 have the following form:
383 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
384 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
390 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/\rho _{0}) \)
401 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
408 is a thermodynamical quantity which is of order
409 \begin_inset Formula \( 1 \)
413 \begin_inset Formula \( 1 \)
416 for nonreacting mixtures of classical perfect gases.
417 The physical meaning of
418 \begin_inset Formula \( \sigma _{0} \)
422 \begin_inset Formula \( K \)
425 is clearly visible in the equations above.
427 \begin_inset Formula \( \sigma _{0} \)
430 represents a frequency of the order one per free-fall time.
432 \begin_inset Formula \( K \)
435 is proportional to the ratio of the free-fall time and the cooling time.
436 Substituting into Baker's criteria, using thermodynamic identities and
437 definitions of thermodynamic quantities,
440 \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}\]
450 \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 P}\right) _{T}\]
454 one obtains, after some pages of algebra, the conditions for
464 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
465 \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} \\
466 \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}
471 For a physical discussion of the stability criteria see Baker (
472 \begin_inset LatexCommand \cite{baker}
477 \begin_inset LatexCommand \cite{cox}
484 We observe that these criteria for dynamical, secular and vibrational stability,
485 respectively, can be factorized into
488 a factor containing local timescales only,
491 a factor containing only constitutive relations and their derivatives.
494 The first factors, depending on only timescales, are positive by definition.
495 The signs of the left hand sides of the inequalities
498 \begin_inset LatexCommand \ref{ZSDynSta}
503 \begin_inset LatexCommand \ref{ZSSecSta}
508 \begin_inset LatexCommand \ref{ZSVibSta}
512 ) therefore depend exclusively on the second factors containing the constitutive
514 Since they depend only on state variables, the stability criteria themselves
517 functions of the thermodynamic state in the local zone
520 The one-zone stability can therefore be determined from a simple equation
521 of state, given for example, as a function of density and temperature.
522 Once the microphysics, i.e.
523 the thermodynamics and opacities (see Table
526 \begin_inset LatexCommand \ref{KapSou}
530 ), are specified (in practice by specifying a chemical composition) the
531 one-zone stability can be inferred if the thermodynamic state is specified.
532 The zone -- or in other words the layer -- will be stable or unstable in
533 whatever object it is imbedded as long as it satisfies the one-zone-model
535 Only the specific growth rates (depending upon the time scales) will be
536 different for layers in different objects.
543 \begin_inset LatexCommand \label{KapSou}
549 \align center \LyXTable
571 Yorke 1979, Yorke 1980a
574 \begin_inset Formula \( \leq 1700^{\mathrm{a}} \)
582 \begin_inset Formula \( 1700\leq T\leq 5000 \)
590 \begin_inset Formula \( 5000\leq \)
597 \begin_inset Formula \( \mathrm{a} \)
602 \begin_float wide-tab
605 Regions of secular instability
606 \begin_inset LatexCommand \label{TabSecInst}
619 We will now write down the sign (and therefore stability) determining parts
620 of the left-hand sides of the inequalities (
621 \begin_inset LatexCommand \ref{ZSDynSta}
626 \begin_inset LatexCommand \ref{ZSSecSta}
631 \begin_inset LatexCommand \ref{ZSVibSta}
637 stability equations of state
642 The sign determining part of inequality
645 \begin_inset LatexCommand \ref{ZSDynSta}
650 \begin_inset Formula \( 3\Gamma _{1}-4 \)
653 and it reduces to the criterion for dynamical stability
660 \Gamma _{1}>\frac{4}{3}
665 Stability of the thermodynamical equilibrium demands
669 \chi _{\rho }>0,\: \: c_{v}>0\, ,
686 holds for a wide range of physical situations.
693 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi _{T}}{c_{v}} & > & 0\\
694 \Gamma _{1}=\chi _{\rho }+\chi _{T}(\Gamma _{3}-1) & > & 0\\
695 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
700 we find the sign determining terms in inequalities
703 \begin_inset LatexCommand \ref{ZSSecSta}
708 \begin_inset LatexCommand \ref{ZSVibSta}
712 ) respectively and obtain the following form of the criteria for dynamical,
713 secular and vibrational
723 3\Gamma _{1}-4=:\, S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
724 \frac{1-3/4\chi _{\rho }}{\chi _{T}}(\kappa _{T}-4)+\kappa _{P}+1=:\, S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
725 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa _{T}+\kappa _{P}-\frac{4}{3\Gamma _{1}}=:\, S_{\mathrm{vib}}> & 0 & \label{VibSta}
730 The constitutive relations are to be evaluated for the unperturbed thermodynamic
732 \begin_inset Formula \( (\rho _{0},T_{0}) \)
736 We see that the one-zone stability of the layer depends only on the constitutiv
738 \begin_inset Formula \( \Gamma _{1} \)
742 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
746 \begin_inset Formula \( \chi _{T},\, \chi _{\rho } \)
750 \begin_inset Formula \( \kappa _{P},\, \kappa _{T} \)
754 These depend only on the unperturbed thermodynamical state of the layer.
755 Therefore the above relations define the one-zone-stability equations of
757 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
761 \begin_inset Formula \( S_{\mathrm{vib}} \)
768 \begin_inset LatexCommand \ref{FigVibStab}
773 \begin_inset Formula \( S_{\mathrm{vib}} \)
777 Regions of secular instability are listed in Table
780 \begin_inset LatexCommand \ref{TabSecInst}
797 Vibrational stability equation of state
798 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
803 \begin_inset Formula \( >0 \)
806 means vibrational stability.
807 \begin_inset LatexCommand \label{FigVibStab}
818 The conditions for the stability of static, radiative layers in gas spheres,
819 as described by Baker's (
820 \begin_inset LatexCommand \cite{baker}
824 ) standard one-zone model, can be expressed as stability equations of state.
825 These stability equations of state depend only on the local thermodynamic
829 If the constitutive relations -- equations of state and Rosseland mean opacities
830 -- are specified, the stability equations of state can be evaluated without
831 specifying properties of the layer.
834 For solar composition gas the
835 \begin_inset Formula \( \kappa \)
838 -mechanism is working in the regions of the ice and dust features in the
840 \begin_inset Formula \( \mathrm{H}_{2} \)
843 dissociation and the combined H, first He ionization zone, as indicated
844 by vibrational instability.
845 These regions of instability are much larger in extent and degree of instabilit
846 y than the second He ionization zone that drives the Cepheïd pulsations.
847 \layout Acknowledgement
849 Part of this work was supported by the German
852 sche For\SpecialChar \-
853 schungs\SpecialChar \-
862 \bibitem [1966]{baker}
864 Baker N., 1966, in: Stellar Evolution, eds.
871 Cameron, Plenum, New York, p.
875 \bibitem [1988]{balluch}
877 Balluch M., 1988, A&A 200, 58
882 P., 1980, Theory of Stellar Pulsation, Princeton University Press, Princeton,
887 \bibitem [1969]{cox69}
891 N., 1969, Academia Nauk, Scientific Information 15, 1
893 \bibitem [1971]{kruegel}
895 Krügel E., 1971, Der Rosselandsche Mittelwert bei tiefen Temperaturen, Diplom--Th
900 \bibitem [1980]{mizuno}
902 Mizuno H., 1980, Prog.
907 \bibitem [1987]{tscarnuter}
910 M., 1987, A&A 188, 55
912 \bibitem [1989]{wuchterl}
914 Wuchterl G., 1989, Zur Entstehung der Gasplaneten.
920 sche Gas\SpecialChar \-
923 gen auf Pro\SpecialChar \-
927 ten, Dissertation, Univ.
930 \bibitem [1979]{yorke79}
933 W., 1979, A&A 80, 215
935 \bibitem [1980a]{yorke80a}
938 W., 1980a, A&A 86, 286