1 #LyX 1.1 created this file. For more info see http://www.lyx.org/
12 \paperfontsize default
18 \paperorientation portrait
21 \paragraph_separation indent
23 \quotes_language english
27 \paperpagestyle default
31 Hydrodynamics of giant planet formation
36 \begin_inset Formula \( \kappa \)
64 Just to show the usage of the elements in the author field
73 Institute for Astronomy (IfA), University of Vienna, T\i \"{u}
81 email{wuchterl@amok.ast.univie.ac.at}
87 University of Alexandria, Department of Geography, ...
93 email{c.ptolemy@hipparch.uheaven.space}
99 The university of heaven temporarily does not accept e-mails
104 Received September 15, 1996; accepted March 16, 1997
107 To investigate the physical nature of the `nuc\SpecialChar \-
108 leated instability' of proto
109 giant planets (Mizuno
110 \begin_inset LatexCommand \cite{mizuno}
114 ), the stability of layers in static, radiative gas spheres is analysed
115 on the basis of Baker's
116 \begin_inset LatexCommand \cite{baker}
120 standard one-zone model.
121 It is shown that stability depends only upon the equations of state, the
122 opacities and the local thermodynamic state in the layer.
123 Stability and instability can therefore be expressed in the form of stability
124 equations of state which are universal for a given composition.
125 The stability equations of state are calculated for solar composition and
126 are displayed in the domain
127 \begin_inset Formula \( -14\leq \lg \rho /[\mathrm{g}\, \mathrm{cm}^{-3}]\leq 0 \)
131 \begin_inset Formula \( 8.8\leq \lg e/[\mathrm{erg}\, \mathrm{g}^{-1}]\leq 17.7 \)
135 These displays may be used to determine the one-zone stability of layers
136 in stellar or planetary structure models by directly reading off the value
137 of the stability equations for the thermodynamic state of these layers,
138 specified by state quantities as density
139 \begin_inset Formula \( \rho \)
143 \begin_inset Formula \( T \)
146 or specific internal energy
147 \begin_inset Formula \( e \)
151 Regions of instability in the
152 \begin_inset Formula \( (\rho ,e) \)
155 -plane are described and related to the underlying microphysical processes.
156 Vibrational instability is found to be a common phenomenon at temperatures
157 lower than the second He ionisation zone.
159 \begin_inset Formula \( \kappa \)
162 -mechanism is widespread under `cool' conditions.
168 keywords{giant planet formation --
174 )-mechanism -- stability of gas spheres }
184 nucleated instability
191 (also called core instability) hypothesis of giant planet formation, a
192 critical mass for static core envelope protoplanets has been found.
194 \begin_inset LatexCommand \cite{mizuno}
198 ) determined the critical mass of the core to be about
199 \begin_inset Formula \( 12\, M_{\oplus } \)
203 \begin_inset Formula \( M_{\oplus }=5.975\, 10^{27}\, \mathrm{g} \)
206 is the Earth mass), which is independent of the outer boundary conditions
207 and therefore independent of the location in the solar nebula.
208 This critical value for the core mass corresponds closely to the cores
209 of today's giant planets.
212 Although no hydrodynamical study has been available many workers conjectured
213 that a collapse or rapid contraction will ensue after accumulating the
215 The main motivation for this article is to investigate the stability of
216 the static envelope at the critical mass.
217 With this aim the local, linear stability of static radiative gas spheres
218 is investigated on the basis of Baker's (
219 \begin_inset LatexCommand \cite{baker}
223 ) standard one-zone model.
226 Phenomena similar to the ones described above for giant planet formation
227 have been found in hydrodynamical models concerning star formation where
228 protostellar cores explode (Tscharnuter
229 \begin_inset LatexCommand \cite{tscharnuter}
234 \begin_inset LatexCommand \cite{balluch}
238 ), whereas earlier studies found quasi-steady collapse flows.
239 The similarities in the (micro)physics, i.e., constitutive relations of protostel
240 lar cores and protogiant planets serve as a further motivation for this
244 Baker's standard one-zone model
247 \begin_float wide-fig
251 \begin_inset Formula \( \Gamma _{1} \)
256 \begin_inset Formula \( \Gamma _{1} \)
259 is plotted as a function of
260 \begin_inset Formula \( \lg \)
264 \begin_inset Formula \( [\mathrm{erg}\, \mathrm{g}^{-1}] \)
268 \begin_inset Formula \( \lg \)
272 \begin_inset Formula \( [\mathrm{g}\, \mathrm{cm}^{-3}] \)
279 \begin_inset LatexCommand \label{FigGam}
285 In this section the one-zone model of Baker (
286 \begin_inset LatexCommand \cite{baker}
290 ), originally used to study the Cephe\i \"{\i}
291 d pulsation mechanism, will be briefly
293 The resulting stability criteria will be rewritten in terms of local state
294 variables, local timescales and constitutive relations.
298 \begin_inset LatexCommand \cite{baker}
302 ) investigates the stability of thin layers in self-gravitating, spherical
303 gas clouds with the following properties:
306 hydrostatic equilibrium,
312 energy transport by grey radiation diffusion.
316 For the one-zone-model Baker obtains necessary conditions for dynamical,
317 secular and vibrational (or pulsational) stability (Eqs.
336 \begin_inset LatexCommand \cite{baker}
341 Using Baker's notation:
345 \begin_inset Formula \begin{eqnarray*}
346 M_{r} & & \textrm{mass internal to the radius }r\\
347 m & & \textrm{mass of the zone}\\
348 r_{0} & & \textrm{unperturbed zone radius}\\
349 \rho _{0} & & \textrm{unperturbed density in the zone}\\
350 T_{0} & & \textrm{unperturbed temperature in the zone}\\
351 L_{r0} & & \textrm{unperturbed luminosity}\\
352 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
360 and with the definitions of the
369 (see Fig.\SpecialChar ~
371 \begin_inset LatexCommand \ref{FigGam}
376 \begin_inset Formula \begin{equation}
377 \tau _{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\, ,
387 \begin_inset Formula \begin{equation}
388 \tau _{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\, ,
394 \begin_inset Formula \( K \)
398 \begin_inset Formula \( \sigma _{0} \)
401 have the following form:
402 \begin_inset Formula \begin{eqnarray}
403 \sigma _{0} & = & \frac{\pi }{\sqrt{8}}\frac{1}{\tau _{\mathrm{ff}}}\\
404 K & = & \frac{\sqrt{32}}{\pi }\frac{1}{\delta }\frac{\tau _{\mathrm{ff}}}{\tau _{\mathrm{co}}}\, ;
410 \begin_inset Formula \( E_{\mathrm{th}}\approx m(P_{0}/{\rho _{0}}) \)
414 \begin_inset Formula \begin{equation}
416 \delta =-\left( \frac{\partial \ln \rho }{\partial \ln T}\right) _{P}\\
423 is a thermodynamical quantity which is of order
424 \begin_inset Formula \( 1 \)
428 \begin_inset Formula \( 1 \)
431 for nonreacting mixtures of classical perfect gases.
432 The physical meaning of
433 \begin_inset Formula \( \sigma _{0} \)
437 \begin_inset Formula \( K \)
440 is clearly visible in the equations above.
442 \begin_inset Formula \( \sigma _{0} \)
445 represents a frequency of the order one per free-fall time.
447 \begin_inset Formula \( K \)
450 is proportional to the ratio of the free-fall time and the cooling time.
451 Substituting into Baker's criteria, using thermodynamic identities and
452 definitions of thermodynamic quantities,
453 \begin_inset Formula \[
454 \Gamma _{1}=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{S}\, ,\; \chi ^{}_{\rho }=\left( \frac{\partial \ln P}{\partial \ln \rho }\right) _{T}\, ,\; \kappa ^{}_{P}=\left( \frac{\partial \ln \kappa }{\partial \ln P}\right) _{T}\]
459 \begin_inset Formula \[
460 \nabla _{\mathrm{ad}}=\left( \frac{\partial \ln T}{\partial \ln P}\right) _{S}\, ,\; \chi ^{}_{T}=\left( \frac{\partial \ln P}{\partial \ln T}\right) _{\rho }\, ,\; \kappa ^{}_{T}=\left( \frac{\partial \ln \kappa }{\partial \ln T}\right) _{T}\]
464 one obtains, after some pages of algebra, the conditions for
474 \begin_inset Formula \begin{eqnarray}
475 \frac{\pi ^{2}}{8}\frac{1}{\tau _{\mathrm{ff}}^{2}}(3\Gamma _{1}-4) & > & 0\label{ZSDynSta} \\
476 \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} \\
477 \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}
482 For a physical discussion of the stability criteria see Baker (
483 \begin_inset LatexCommand \cite{baker}
488 \begin_inset LatexCommand \cite{cox}
495 We observe that these criteria for dynamical, secular and vibrational stability,
496 respectively, can be factorized into
499 a factor containing local timescales only,
502 a factor containing only constitutive relations and their derivatives.
506 The first factors, depending on only timescales, are positive by definition.
507 The signs of the left hand sides of the inequalities\SpecialChar ~
509 \begin_inset LatexCommand \ref{ZSDynSta}
514 \begin_inset LatexCommand \ref{ZSSecSta}
519 \begin_inset LatexCommand \ref{ZSVibSta}
523 ) therefore depend exclusively on the second factors containing the constitutive
525 Since they depend only on state variables, the stability criteria themselves
528 functions of the thermodynamic state in the local zone
531 The one-zone stability can therefore be determined from a simple equation
532 of state, given for example, as a function of density and temperature.
533 Once the microphysics, i.e.
539 the thermodynamics and opacities (see Table\SpecialChar ~
541 \begin_inset LatexCommand \ref{KapSou}
545 ), are specified (in practice by specifying a chemical composition) the
546 one-zone stability can be inferred if the thermodynamic state is specified.
547 The zone -- or in other words the layer -- will be stable or unstable in
548 whatever object it is imbedded as long as it satisfies the one-zone-model
550 Only the specific growth rates (depending upon the time scales) will be
551 different for layers in different objects.
558 \begin_inset LatexCommand \label{KapSou}
567 <lyxtabular version="2" rows="4" columns="2">
568 <features rotate="false" islongtable="false" endhead="0" endfirsthead="0" endfoot="0" endlastfoot="0">
569 <column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
570 <column alignment="left" valignment="top" leftline="false" rightline="false" width="" special="">
571 <row topline="true" bottomline="false" newpage="false">
572 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
580 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
586 \begin_inset Formula \( T/[\textrm{K}] \)
593 <row topline="true" bottomline="false" newpage="false">
594 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
599 Yorke 1979, Yorke 1980a
602 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
608 \begin_inset Formula \( \leq 1700^{\textrm{a}} \)
615 <row topline="false" bottomline="false" newpage="false">
616 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
624 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
630 \begin_inset Formula \( 1700\leq T\leq 5000 \)
637 <row topline="false" bottomline="true" newpage="false">
638 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
646 <cell multicolumn="0" alignment="center" valignment="top" topline="true" bottomline="false" leftline="true" rightline="false" rotate="false" usebox="none" width="" special="">
652 \begin_inset Formula \( 5000\leq \)
667 \begin_inset Formula \( ^{\textrm{a}} \)
672 We will now write down the sign (and therefore stability) determining parts
673 of the left-hand sides of the inequalities (
674 \begin_inset LatexCommand \ref{ZSDynSta}
679 \begin_inset LatexCommand \ref{ZSSecSta}
684 \begin_inset LatexCommand \ref{ZSVibSta}
690 stability equations of state
695 The sign determining part of inequality\SpecialChar ~
697 \begin_inset LatexCommand \ref{ZSDynSta}
702 \begin_inset Formula \( 3\Gamma _{1}-4 \)
705 and it reduces to the criterion for dynamical stability
706 \begin_inset Formula \begin{equation}
707 \Gamma _{1}>\frac{4}{3}\, \cdot
712 Stability of the thermodynamical equilibrium demands
713 \begin_inset Formula \begin{equation}
714 \chi ^{}_{\rho }>0,\; \; c_{v}>0\, ,
720 \begin_inset Formula \begin{equation}
726 holds for a wide range of physical situations.
728 \begin_inset Formula \begin{eqnarray}
729 \Gamma _{3}-1=\frac{P}{\rho T}\frac{\chi ^{}_{T}}{c_{v}} & > & 0\\
730 \Gamma _{1}=\chi _{\rho }^{}+\chi _{T}^{}(\Gamma _{3}-1) & > & 0\\
731 \nabla _{\mathrm{ad}}=\frac{\Gamma _{3}-1}{\Gamma _{1}} & > & 0
736 we find the sign determining terms in inequalities\SpecialChar ~
738 \begin_inset LatexCommand \ref{ZSSecSta}
743 \begin_inset LatexCommand \ref{ZSVibSta}
747 ) respectively and obtain the following form of the criteria for dynamical,
748 secular and vibrational
753 \begin_inset Formula \begin{eqnarray}
754 3\Gamma _{1}-4=:S_{\mathrm{dyn}}> & 0 & \label{DynSta} \\
755 \frac{1-3/4\chi ^{}_{\rho }}{\chi ^{}_{T}}(\kappa ^{}_{T}-4)+\kappa ^{}_{P}+1=:S_{\mathrm{sec}}> & 0 & \label{SecSta} \\
756 4\nabla _{\mathrm{ad}}-(\nabla _{\mathrm{ad}}\kappa ^{}_{T}+\kappa ^{}_{P})-\frac{4}{3\Gamma _{1}}=:S_{\mathrm{vib}}> & 0\, . & \label{VibSta}
761 The constitutive relations are to be evaluated for the unperturbed thermodynami
763 \begin_inset Formula \( (\rho _{0},T_{0}) \)
767 We see that the one-zone stability of the layer depends only on the constitutiv
769 \begin_inset Formula \( \Gamma _{1} \)
773 \begin_inset Formula \( \nabla _{\mathrm{ad}} \)
777 \begin_inset Formula \( \chi _{T}^{},\, \chi _{\rho }^{} \)
781 \begin_inset Formula \( \kappa _{P}^{},\, \kappa _{T}^{} \)
785 These depend only on the unperturbed thermodynamical state of the layer.
786 Therefore the above relations define the one-zone-stability equations of
788 \begin_inset Formula \( S_{\mathrm{dyn}},\, S_{\mathrm{sec}} \)
792 \begin_inset Formula \( S_{\mathrm{vib}} \)
796 See Fig.\SpecialChar ~
798 \begin_inset LatexCommand \ref{FigVibStab}
803 \begin_inset Formula \( S_{\mathrm{vib}} \)
807 Regions of secular instability are listed in Table\SpecialChar ~
814 Vibrational stability equation of state
815 \begin_inset Formula \( S_{\mathrm{vib}}(\lg e,\lg \rho ) \)
820 \begin_inset Formula \( >0 \)
823 means vibrational stability
827 \begin_inset LatexCommand \label{FigVibStab}
838 The conditions for the stability of static, radiative layers in gas spheres,
839 as described by Baker's (
840 \begin_inset LatexCommand \cite{baker}
844 ) standard one-zone model, can be expressed as stability equations of state.
845 These stability equations of state depend only on the local thermodynamic
850 If the constitutive relations -- equations of state and Rosseland mean opacities
851 -- are specified, the stability equations of state can be evaluated without
852 specifying properties of the layer.
856 For solar composition gas the
857 \begin_inset Formula \( \kappa \)
860 -mechanism is working in the regions of the ice and dust features in the
862 \begin_inset Formula \( \mathrm{H}_{2} \)
865 dissociation and the combined H, first He ionization zone, as indicated
866 by vibrational instability.
867 These regions of instability are much larger in extent and degree of instabilit
868 y than the second He ionization zone that drives the Cephe\i \"{\i}
871 \layout Acknowledgement
873 Part of this work was supported by the German
876 sche For\SpecialChar \-
877 schungs\SpecialChar \-
887 project number Ts\SpecialChar ~
891 \bibitem [1966]{baker}
894 1966, in Stellar Evolution, ed.
905 Cameron (Plenum, New York) 333
907 \bibitem [1988]{balluch}
916 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton)
919 \bibitem [1969]{cox69}
924 1969, Academia Nauk, Scientific Information 15, 1
926 \bibitem [1980]{mizuno}
933 \bibitem [1987]{tscharnuter}
939 \bibitem [1992]{terlevich}
944 31, Relationships between Active Galactic Nuclei and Starburst Galaxies,
950 \bibitem [1980a]{yorke80a}
956 \bibitem [1997]{zheng}
958 Zheng, W., Davidsen, A.