1 #LyX 1.3 created this file. For more info see http://www.lyx.org/
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20 \paperorientation portrait
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25 \quotes_language english
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33 Hydrodynamics of giant planet formation
38 \begin_inset Formula \( \kappa \)
78 Just to show the usage of the elements in the author field
89 Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
100 email{wuchterl@amok.ast.univie.ac.at}
107 University of Alexandria, Department of Geography, ...
116 email{c.ptolemy@hipparch.uheaven.space}
125 The university of heaven temporarily does not accept e-mails
132 Received September 15, 1996; accepted March 16, 1997
135 To investigate the physical nature of the `nuc\SpecialChar \-
136 leated instability' of proto
137 giant planets (Mizuno
138 \begin_inset LatexCommand \cite{mizuno}
142 ), the stability of layers in static, radiative gas spheres is analysed
143 on the basis of Baker's
144 \begin_inset LatexCommand \cite{baker}
148 standard one-zone model.
149 It is shown that stability depends only upon the equations of state, the
150 opacities and the local thermodynamic state in the layer.
151 Stability and instability can therefore be expressed in the form of stability
152 equations of state which are universal for a given composition.
153 The stability equations of state are calculated for solar composition and
154 are displayed in the domain
155 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
159 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
163 These displays may be used to determine the one-zone stability of layers
164 in stellar or planetary structure models by directly reading off the value
165 of the stability equations for the thermodynamic state of these layers,
166 specified by state quantities as density
167 \begin_inset Formula \( \rho \)
171 \begin_inset Formula \( T \)
174 or specific internal energy
175 \begin_inset Formula \( e \)
179 Regions of instability in the
180 \begin_inset Formula \( (\rho ,e) \)
183 -plane are described and related to the underlying microphysical processes.
184 Vibrational instability is found to be a common phenomenon at temperatures
185 lower than the second He ionisation zone.
187 \begin_inset Formula \( \kappa \)
190 -mechanism is widespread under `cool' conditions.
199 keywords{giant planet formation --
205 )-mechanism -- stability of gas spheres }
216 nucleated instability
228 (also called core instability) hypothesis of giant planet formation, a
229 critical mass for static core envelope protoplanets has been found.
231 \begin_inset LatexCommand \cite{mizuno}
235 ) determined the critical mass of the core to be about
236 \begin_inset Formula \( 12\, M_{\oplus } \)
240 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
243 is the Earth mass), which is independent of the outer boundary conditions
244 and therefore independent of the location in the solar nebula.
245 This critical value for the core mass corresponds closely to the cores
246 of today's giant planets.
249 Although no hydrodynamical study has been available many workers conjectured
250 that a collapse or rapid contraction will ensue after accumulating the
252 The main motivation for this article is to investigate the stability of
253 the static envelope at the critical mass.
254 With this aim the local, linear stability of static radiative gas spheres
255 is investigated on the basis of Baker's (
256 \begin_inset LatexCommand \cite{baker}
260 ) standard one-zone model.
263 Phenomena similar to the ones described above for giant planet formation
264 have been found in hydrodynamical models concerning star formation where
265 protostellar cores explode (Tscharnuter
266 \begin_inset LatexCommand \cite{tscharnuter}
271 \begin_inset LatexCommand \cite{balluch}
275 ), whereas earlier studies found quasi-steady collapse flows.
276 The similarities in the (micro)physics, i.e., constitutive relations of protostel
277 lar cores and protogiant planets serve as a further motivation for this
281 Baker's standard one-zone model
284 \begin_inset Float figure
291 \begin_inset Formula \( \Gamma _{1} \)
296 \begin_inset Formula \( \Gamma _{1} \)
299 is plotted as a function of
300 \begin_inset Formula \( \lg \)
304 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
308 \begin_inset Formula \( \lg \)
312 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
319 \begin_inset LatexCommand \label{FigGam}
326 In this section the one-zone model of Baker (
327 \begin_inset LatexCommand \cite{baker}
331 ), originally used to study the Cephe\i \"{\i}
332 d pulsation mechanism, will be briefly
334 The resulting stability criteria will be rewritten in terms of local state
335 variables, local timescales and constitutive relations.
339 \begin_inset LatexCommand \cite{baker}
343 ) investigates the stability of thin layers in self-gravitating, spherical
344 gas clouds with the following properties:
347 hydrostatic equilibrium,
353 energy transport by grey radiation diffusion.
357 For the one-zone-model Baker obtains necessary conditions for dynamical,
358 secular and vibrational (or pulsational) stability (Eqs.
389 \begin_inset LatexCommand \cite{baker}
394 Using Baker's notation:
398 \begin_inset Formula \begin{eqnarray*}
399 M_{r} & & \textrm{mass internal to the radius }r\\
400 m & & \textrm{mass of the zone}\\
401 r_{0} & & \textrm{unperturbed zone radius}\\
402 \rho _{0} & & \textrm{unperturbed density in the zone}\\
403 T_{0} & & \textrm{unperturbed temperature in the zone}\\
404 L_{r0} & & \textrm{unperturbed luminosity}\\
405 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
413 and with the definitions of the
427 (see Fig.\SpecialChar ~
429 \begin_inset LatexCommand \ref{FigGam}
434 \begin_inset Formula \begin{equation}
435 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
445 \begin_inset Formula \begin{equation}
446 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
452 \begin_inset Formula \( K \)
456 \begin_inset Formula \( \sigma _{0} \)
459 have the following form:
460 \begin_inset Formula \begin{eqnarray}
461 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
462 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
468 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
472 \begin_inset Formula \begin{equation}
474 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
481 is a thermodynamical quantity which is of order
482 \begin_inset Formula \( 1 \)
486 \begin_inset Formula \( 1 \)
489 for nonreacting mixtures of classical perfect gases.
490 The physical meaning of
491 \begin_inset Formula \( \sigma _{0} \)
495 \begin_inset Formula \( K \)
498 is clearly visible in the equations above.
500 \begin_inset Formula \( \sigma _{0} \)
503 represents a frequency of the order one per free-fall time.
505 \begin_inset Formula \( K \)
508 is proportional to the ratio of the free-fall time and the cooling time.
509 Substituting into Baker's criteria, using thermodynamic identities and
510 definitions of thermodynamic quantities,
511 \begin_inset Formula \[
512 \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
517 \begin_inset Formula \[
518 \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
522 one obtains, after some pages of algebra, the conditions for
537 \begin_inset Formula \begin{eqnarray}
538 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
539 \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} \\
540 \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}
545 For a physical discussion of the stability criteria see Baker (
546 \begin_inset LatexCommand \cite{baker}
551 \begin_inset LatexCommand \cite{cox}
558 We observe that these criteria for dynamical, secular and vibrational stability,
559 respectively, can be factorized into
562 a factor containing local timescales only,
565 a factor containing only constitutive relations and their derivatives.
569 The first factors, depending on only timescales, are positive by definition.
570 The signs of the left hand sides of the inequalities\SpecialChar ~
572 \begin_inset LatexCommand \ref{ZSDynSta}
577 \begin_inset LatexCommand \ref{ZSSecSta}
582 \begin_inset LatexCommand \ref{ZSVibSta}
586 ) therefore depend exclusively on the second factors containing the constitutive
588 Since they depend only on state variables, the stability criteria themselves
591 functions of the thermodynamic state in the local zone
594 The one-zone stability can therefore be determined from a simple equation
595 of state, given for example, as a function of density and temperature.
596 Once the microphysics, i.e.
606 the thermodynamics and opacities (see Table\SpecialChar ~
608 \begin_inset LatexCommand \ref{KapSou}
612 ), are specified (in practice by specifying a chemical composition) the
613 one-zone stability can be inferred if the thermodynamic state is specified.
614 The zone -- or in other words the layer -- will be stable or unstable in
615 whatever object it is imbedded as long as it satisfies the one-zone-model
617 Only the specific growth rates (depending upon the time scales) will be
618 different for layers in different objects.
621 \begin_inset Float table
628 \begin_inset LatexCommand \label{KapSou}
637 <lyxtabular version="3" rows="4" columns="2">
639 <column alignment="left" valignment="top" width="0pt">
640 <column alignment="left" valignment="top" width="0pt">
642 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
650 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
656 \begin_inset Formula \( T/[\textrm{K}] \)
664 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
669 Yorke 1979, Yorke 1980a
672 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
678 \begin_inset Formula \( \leq 1700^{\textrm{a}} \)
686 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
694 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
700 \begin_inset Formula \( 1700\leq T\leq 5000 \)
707 <row bottomline="true">
708 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
716 <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
722 \begin_inset Formula \( 5000\leq \)
737 \begin_inset Formula \( ^{\textrm{a}} \)
743 We will now write down the sign (and therefore stability) determining parts
744 of the left-hand sides of the inequalities (
745 \begin_inset LatexCommand \ref{ZSDynSta}
750 \begin_inset LatexCommand \ref{ZSSecSta}
755 \begin_inset LatexCommand \ref{ZSVibSta}
761 stability equations of state
766 The sign determining part of inequality\SpecialChar ~
768 \begin_inset LatexCommand \ref{ZSDynSta}
773 \begin_inset Formula \( 3\Gamma _{1}-4 \)
776 and it reduces to the criterion for dynamical stability
777 \begin_inset Formula \begin{equation}
778 \Gamma _{1}>\frac{4}{3}\, \cdot
783 Stability of the thermodynamical equilibrium demands
784 \begin_inset Formula \begin{equation}
785 \chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
791 \begin_inset Formula \begin{equation}
797 holds for a wide range of physical situations.
799 \begin_inset Formula \begin{eqnarray}
800 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
801 \Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
802 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
807 we find the sign determining terms in inequalities\SpecialChar ~
809 \begin_inset LatexCommand \ref{ZSSecSta}
814 \begin_inset LatexCommand \ref{ZSVibSta}
818 ) respectively and obtain the following form of the criteria for dynamical,
819 secular and vibrational
824 \begin_inset Formula \begin{eqnarray}
825 3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
826 \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
827 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
832 The constitutive relations are to be evaluated for the unperturbed thermodynami
834 \begin_inset Formula \( (\rho _{0},T_{0}) \)
838 We see that the one-zone stability of the layer depends only on the constitutiv
840 \begin_inset Formula \( \Gamma _{1} \)
844 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
848 \begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
852 \begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
856 These depend only on the unperturbed thermodynamical state of the layer.
857 Therefore the above relations define the one-zone-stability equations of
859 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
863 \begin_inset Formula \( S_{\mathrm{vib}} \)
867 See Fig.\SpecialChar ~
869 \begin_inset LatexCommand \ref{FigVibStab}
874 \begin_inset Formula \( S_{\mathrm{vib}} \)
878 Regions of secular instability are listed in Table\SpecialChar ~
882 \begin_inset Float figure
888 Vibrational stability equation of state
889 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
894 \begin_inset Formula \( >0 \)
897 means vibrational stability
901 \begin_inset LatexCommand \label{FigVibStab}
913 The conditions for the stability of static, radiative layers in gas spheres,
914 as described by Baker's (
915 \begin_inset LatexCommand \cite{baker}
919 ) standard one-zone model, can be expressed as stability equations of state.
920 These stability equations of state depend only on the local thermodynamic
925 If the constitutive relations -- equations of state and Rosseland mean opacities
926 -- are specified, the stability equations of state can be evaluated without
927 specifying properties of the layer.
931 For solar composition gas the
932 \begin_inset Formula \( \kappa \)
935 -mechanism is working in the regions of the ice and dust features in the
937 \begin_inset Formula \( \mathrm{H}_{2} \)
940 dissociation and the combined H, first He ionization zone, as indicated
941 by vibrational instability.
942 These regions of instability are much larger in extent and degree of instabilit
943 y than the second He ionization zone that drives the Cephe\i \"{\i}
946 \layout Acknowledgement
948 Part of this work was supported by the German
951 sche For\SpecialChar \-
952 schungs\SpecialChar \-
967 project number Ts\SpecialChar ~
971 \bibitem [1966]{baker}
974 1966, in Stellar Evolution, ed.
989 Cameron (Plenum, New York) 333
991 \bibitem [1988]{balluch}
1000 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
1002 \layout Bibliography
1003 \bibitem [1969]{cox69}
1008 1969, Academia Nauk, Scientific Information 15, 1
1009 \layout Bibliography
1010 \bibitem [1980]{mizuno}
1016 \layout Bibliography
1017 \bibitem [1987]{tscharnuter}
1022 \layout Bibliography
1023 \bibitem [1992]{terlevich}
1028 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
1033 \layout Bibliography
1034 \bibitem [1980a]{yorke80a}
1039 \layout Bibliography
1040 \bibitem [1997]{zheng}
1042 Zheng, W., Davidsen, A.