1 #LyX 2.1 created this file. For more info see http://www.lyx.org/
6 \use_default_options true
7 \maintain_unincluded_children false
9 \language_package default
14 \font_typewriter default
16 \font_default_family default
17 \use_non_tex_fonts false
23 \default_output_format default
25 \bibtex_command bibtex
26 \index_command default
27 \paperfontsize default
32 \use_package amsmath 0
33 \use_package amssymb 0
36 \use_package mathdots 1
37 \use_package mathtools 0
39 \use_package stackrel 0
40 \use_package stmaryrd 0
41 \use_package undertilde 0
43 \cite_engine_type default
47 \paperorientation portrait
57 \paragraph_separation indent
58 \paragraph_indentation default
59 \quotes_language english
62 \paperpagestyle default
63 \tracking_changes false
73 \begin_inset Note Note
76 \begin_layout Plain Layout
81 This is an example LyX file for articles to be submitted to the Journal
82 of Astronomy & Astrophysics (A&A).
83 How to install the A&A LaTeX class to your LaTeX system is explained in
88 \begin_layout Plain Layout
90 http://wiki.lyx.org/Layouts/Astronomy-Astrophysics
96 \begin_inset Newline newline
99 Depending on the submission state and the abstract layout, you need to use
100 different document class options that are listed in the aa manual.
103 \begin_inset Newline newline
111 If you use accented characters in your document, you must use the predefined
112 document class option
116 in the document settings.
125 Hydrodynamics of giant planet formation
128 \begin_layout Subtitle
131 \begin_inset Formula $\kappa$
140 \begin_inset Flex institutemark
143 \begin_layout Plain Layout
153 \begin_layout Plain Layout
164 \begin_inset Flex institutemark
167 \begin_layout Plain Layout
177 \begin_layout Plain Layout
190 \begin_layout Plain Layout
191 Just to show the usage of the elements in the author field
197 \begin_inset Note Note
200 \begin_layout Plain Layout
203 fnmsep is only needed for more than one consecutive notes/marks
211 \begin_layout Offprint
216 \begin_layout Address
217 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
219 \begin_inset Newline newline
223 \begin_inset Flex Email
226 \begin_layout Plain Layout
227 wuchterl@amok.ast.univie.ac.at
236 \begin_layout Plain Layout
245 University of Alexandria, Department of Geography, ...
246 \begin_inset Newline newline
250 \begin_inset Flex Email
253 \begin_layout Plain Layout
254 c.ptolemy@hipparch.uheaven.space
263 \begin_layout Plain Layout
264 The university of heaven temporarily does not accept e-mails
273 Received September 15, 1996; accepted March 16, 1997
276 \begin_layout Abstract (unstructured)
277 To investigate the physical nature of the `nuc\SpecialChar \-
278 leated instability' of proto
279 giant planets, the stability of layers in static, radiative gas spheres
280 is analysed on the basis of Baker's standard one-zone model.
281 It is shown that stability depends only upon the equations of state, the
282 opacities and the local thermodynamic state in the layer.
283 Stability and instability can therefore be expressed in the form of stability
284 equations of state which are universal for a given composition.
285 The stability equations of state are calculated for solar composition and
286 are displayed in the domain
287 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
291 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
295 These displays may be used to determine the one-zone stability of layers
296 in stellar or planetary structure models by directly reading off the value
297 of the stability equations for the thermodynamic state of these layers,
298 specified by state quantities as density
299 \begin_inset Formula $\rho$
303 \begin_inset Formula $T$
306 or specific internal energy
307 \begin_inset Formula $e$
311 Regions of instability in the
312 \begin_inset Formula $(\rho,e)$
315 -plane are described and related to the underlying microphysical processes.
316 Vibrational instability is found to be a common phenomenon at temperatures
317 lower than the second He ionisation zone.
319 \begin_inset Formula $\kappa$
322 -mechanism is widespread under `cool' conditions.
323 \begin_inset Note Note
326 \begin_layout Plain Layout
327 Citations are not allowed in A&A abstracts.
333 \begin_inset Note Note
336 \begin_layout Plain Layout
337 This is the unstructured abstract type, an example for the structured abstract
342 template file that comes with LyX.
350 \begin_layout Keywords
351 giant planet formation --
352 \begin_inset Formula $\kappa$
355 -mechanism -- stability of gas spheres
358 \begin_layout Section
362 \begin_layout Standard
365 nucleated instability
367 (also called core instability) hypothesis of giant planet formation, a
368 critical mass for static core envelope protoplanets has been found.
370 \begin_inset CommandInset citation
376 ) determined the critical mass of the core to be about
377 \begin_inset Formula $12\, M_{\oplus}$
381 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
384 is the Earth mass), which is independent of the outer boundary conditions
385 and therefore independent of the location in the solar nebula.
386 This critical value for the core mass corresponds closely to the cores
387 of today's giant planets.
390 \begin_layout Standard
391 Although no hydrodynamical study has been available many workers conjectured
392 that a collapse or rapid contraction will ensue after accumulating the
394 The main motivation for this article is to investigate the stability of
395 the static envelope at the critical mass.
396 With this aim the local, linear stability of static radiative gas spheres
397 is investigated on the basis of Baker's (
398 \begin_inset CommandInset citation
404 ) standard one-zone model.
407 \begin_layout Standard
408 Phenomena similar to the ones described above for giant planet formation
409 have been found in hydrodynamical models concerning star formation where
410 protostellar cores explode (Tscharnuter
411 \begin_inset CommandInset citation
418 \begin_inset CommandInset citation
424 ), whereas earlier studies found quasi-steady collapse flows.
425 The similarities in the (micro)physics, i.
426 \begin_inset space \thinspace{}
430 \begin_inset space \space{}
433 constitutive relations of protostellar cores and protogiant planets serve
434 as a further motivation for this study.
437 \begin_layout Section
438 Baker's standard one-zone model
441 \begin_layout Standard
442 \begin_inset Float figure
447 \begin_layout Plain Layout
448 \begin_inset Caption Standard
450 \begin_layout Plain Layout
451 \begin_inset CommandInset label
458 \begin_inset Formula $\Gamma_{1}$
463 \begin_inset Formula $\Gamma_{1}$
466 is plotted as a function of
467 \begin_inset Formula $\lg$
471 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
475 \begin_inset Formula $\lg$
479 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
492 In this section the one-zone model of Baker (
493 \begin_inset CommandInset citation
499 ), originally used to study the Cepheı̈d pulsation mechanism, will be briefly
501 The resulting stability criteria will be rewritten in terms of local state
502 variables, local timescales and constitutive relations.
505 \begin_layout Standard
507 \begin_inset CommandInset citation
513 ) investigates the stability of thin layers in self-gravitating, spherical
514 gas clouds with the following properties:
517 \begin_layout Itemize
518 hydrostatic equilibrium,
521 \begin_layout Itemize
525 \begin_layout Itemize
526 energy transport by grey radiation diffusion.
530 \begin_layout Standard
532 For the one-zone-model Baker obtains necessary conditions for dynamical,
533 secular and vibrational (or pulsational) stability (Eqs.
534 \begin_inset space \space{}
538 \begin_inset space \thinspace{}
542 \begin_inset space \thinspace{}
546 \begin_inset CommandInset citation
553 Using Baker's notation:
556 \begin_layout Standard
560 M_{r} & & \textrm{mass internal to the radius }r\\
561 m & & \textrm{mass of the zone}\\
562 r_{0} & & \textrm{unperturbed zone radius}\\
563 \rho_{0} & & \textrm{unperturbed density in the zone}\\
564 T_{0} & & \textrm{unperturbed temperature in the zone}\\
565 L_{r0} & & \textrm{unperturbed luminosity}\\
566 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
574 \begin_layout Standard
576 and with the definitions of the
585 \begin_inset CommandInset ref
587 reference "fig:FigGam"
594 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
606 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
612 \begin_inset Formula $K$
616 \begin_inset Formula $\sigma_{0}$
619 have the following form:
622 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
623 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
629 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
636 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
643 is a thermodynamical quantity which is of order
644 \begin_inset Formula $1$
648 \begin_inset Formula $1$
651 for nonreacting mixtures of classical perfect gases.
652 The physical meaning of
653 \begin_inset Formula $\sigma_{0}$
657 \begin_inset Formula $K$
660 is clearly visible in the equations above.
662 \begin_inset Formula $\sigma_{0}$
665 represents a frequency of the order one per free-fall time.
667 \begin_inset Formula $K$
670 is proportional to the ratio of the free-fall time and the cooling time.
671 Substituting into Baker's criteria, using thermodynamic identities and
672 definitions of thermodynamic quantities,
675 \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}
683 \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}
688 one obtains, after some pages of algebra, the conditions for
695 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
696 \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}\\
697 \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}
702 For a physical discussion of the stability criteria see Baker (
703 \begin_inset CommandInset citation
710 \begin_inset CommandInset citation
719 \begin_layout Standard
720 We observe that these criteria for dynamical, secular and vibrational stability,
721 respectively, can be factorized into
724 \begin_layout Enumerate
725 a factor containing local timescales only,
728 \begin_layout Enumerate
729 a factor containing only constitutive relations and their derivatives.
733 \begin_layout Standard
734 The first factors, depending on only timescales, are positive by definition.
735 The signs of the left hand sides of the inequalities
740 \begin_inset CommandInset ref
747 \begin_inset CommandInset ref
754 \begin_inset CommandInset ref
760 ) therefore depend exclusively on the second factors containing the constitutive
762 Since they depend only on state variables, the stability criteria themselves
765 functions of the thermodynamic state in the local zone
768 The one-zone stability can therefore be determined from a simple equation
769 of state, given for example, as a function of density and temperature.
770 Once the microphysics, i.
771 \begin_inset space \thinspace{}
775 \begin_inset space \space{}
778 the thermodynamics and opacities (see Table
783 \begin_inset CommandInset ref
785 reference "tab:KapSou"
789 ), are specified (in practice by specifying a chemical composition) the
790 one-zone stability can be inferred if the thermodynamic state is specified.
791 The zone -- or in other words the layer -- will be stable or unstable in
792 whatever object it is imbedded as long as it satisfies the one-zone-model
794 Only the specific growth rates (depending upon the time scales) will be
795 different for layers in different objects.
798 \begin_layout Standard
799 \begin_inset Float table
804 \begin_layout Plain Layout
805 \begin_inset Caption Standard
807 \begin_layout Plain Layout
808 \begin_inset CommandInset label
822 \begin_layout Plain Layout
825 <lyxtabular version="3" rows="4" columns="2">
826 <features rotate="0" tabularvalignment="middle">
827 <column alignment="left" valignment="top" width="0pt">
828 <column alignment="left" valignment="top" width="0pt">
830 <cell alignment="center" valignment="top" topline="true" usebox="none">
833 \begin_layout Plain Layout
839 <cell alignment="center" valignment="top" topline="true" usebox="none">
842 \begin_layout Plain Layout
843 \begin_inset Formula $T/[\textrm{K}]$
853 <cell alignment="center" valignment="top" topline="true" usebox="none">
856 \begin_layout Plain Layout
857 Yorke 1979, Yorke 1980a
862 <cell alignment="center" valignment="top" topline="true" usebox="none">
865 \begin_layout Plain Layout
866 \begin_inset Formula $\leq1700^{\textrm{a}}$
876 <cell alignment="center" valignment="top" usebox="none">
879 \begin_layout Plain Layout
885 <cell alignment="center" valignment="top" usebox="none">
888 \begin_layout Plain Layout
889 \begin_inset Formula $1700\leq T\leq5000$
899 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
902 \begin_layout Plain Layout
908 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
911 \begin_layout Plain Layout
912 \begin_inset Formula $5000\leq$
928 \begin_layout Plain Layout
929 \begin_inset Formula $^{\textrm{a}}$
937 We will now write down the sign (and therefore stability) determining parts
938 of the left-hand sides of the inequalities (
939 \begin_inset CommandInset ref
946 \begin_inset CommandInset ref
953 \begin_inset CommandInset ref
961 stability equations of state
966 \begin_layout Standard
967 The sign determining part of inequality
972 \begin_inset CommandInset ref
979 \begin_inset Formula $3\Gamma_{1}-4$
982 and it reduces to the criterion for dynamical stability
985 \Gamma_{1}>\frac{4}{3}\,\cdot
990 Stability of the thermodynamical equilibrium demands
993 \chi_{\rho}^{}>0,\;\; c_{v}>0\,,
1006 holds for a wide range of physical situations.
1008 \begin_inset Formula
1010 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1011 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1012 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1017 we find the sign determining terms in inequalities
1018 \begin_inset space ~
1022 \begin_inset CommandInset ref
1024 reference "ZSSecSta"
1029 \begin_inset CommandInset ref
1031 reference "ZSVibSta"
1035 ) respectively and obtain the following form of the criteria for dynamical,
1036 secular and vibrational
1041 \begin_inset Formula
1043 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1044 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1045 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1050 The constitutive relations are to be evaluated for the unperturbed thermodynami
1052 \begin_inset Formula $(\rho_{0},T_{0})$
1056 We see that the one-zone stability of the layer depends only on the constitutiv
1058 \begin_inset Formula $\Gamma_{1}$
1062 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1066 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1070 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1074 These depend only on the unperturbed thermodynamical state of the layer.
1075 Therefore the above relations define the one-zone-stability equations of
1077 \begin_inset Formula $S_{\mathrm{dyn}},\, S_{\mathrm{sec}}$
1081 \begin_inset Formula $S_{\mathrm{vib}}$
1086 \begin_inset space ~
1090 \begin_inset CommandInset ref
1092 reference "fig:VibStabEquation"
1097 \begin_inset Formula $S_{\mathrm{vib}}$
1101 Regions of secular instability are listed in Table
1102 \begin_inset space ~
1108 \begin_layout Standard
1109 \begin_inset Float figure
1114 \begin_layout Plain Layout
1115 \begin_inset Caption Standard
1117 \begin_layout Plain Layout
1118 \begin_inset CommandInset label
1120 name "fig:VibStabEquation"
1124 Vibrational stability equation of state
1125 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1130 \begin_inset Formula $>0$
1133 means vibrational stability
1146 \begin_layout Section
1150 \begin_layout Enumerate
1151 The conditions for the stability of static, radiative layers in gas spheres,
1152 as described by Baker's (
1153 \begin_inset CommandInset citation
1159 ) standard one-zone model, can be expressed as stability equations of state.
1160 These stability equations of state depend only on the local thermodynamic
1165 \begin_layout Enumerate
1166 If the constitutive relations -- equations of state and Rosseland mean opacities
1167 -- are specified, the stability equations of state can be evaluated without
1168 specifying properties of the layer.
1172 \begin_layout Enumerate
1173 For solar composition gas the
1174 \begin_inset Formula $\kappa$
1177 -mechanism is working in the regions of the ice and dust features in the
1179 \begin_inset Formula $\mathrm{H}_{2}$
1182 dissociation and the combined H, first He ionization zone, as indicated
1183 by vibrational instability.
1184 These regions of instability are much larger in extent and degree of instabilit
1185 y than the second He ionization zone that drives the Cepheı̈d pulsations.
1189 \begin_layout Acknowledgement
1190 Part of this work was supported by the German
1193 sche For\SpecialChar \-
1194 schungs\SpecialChar \-
1200 \begin_inset space ~
1206 \begin_layout Standard
1207 \begin_inset CommandInset bibtex
1209 btprint "btPrintAll"
1210 bibfiles "biblioExample"
1216 \begin_inset Note Note
1219 \begin_layout Plain Layout
1224 If you cannot see the bibliography in the output, assure that you have
1225 gievn the full path to the BibTeX style file
1229 that is part of the A&A LaTeX-package.