1 #LyX 2.4 created this file. For more info see https://www.lyx.org/
5 \save_transient_properties true
6 \origin /systemlyxdir/examples/Articles/
8 \use_default_options true
9 \maintain_unincluded_children no
11 \language_package default
14 \font_roman "default" "default"
15 \font_sans "default" "default"
16 \font_typewriter "default" "default"
17 \font_math "auto" "auto"
18 \font_default_family default
19 \use_non_tex_fonts false
22 \font_typewriter_osf false
24 \font_sf_scale 100 100
25 \font_tt_scale 100 100
27 \use_dash_ligatures false
29 \default_output_format default
31 \bibtex_command bibtex
32 \index_command default
33 \paperfontsize default
38 \use_package amsmath 1
39 \use_package amssymb 1
42 \use_package mathdots 1
43 \use_package mathtools 1
45 \use_package stackrel 1
46 \use_package stmaryrd 1
47 \use_package undertilde 1
49 \cite_engine_type authoryear
53 \paperorientation portrait
65 \paragraph_separation indent
66 \paragraph_indentation default
68 \math_numbering_side default
73 \paperpagestyle default
74 \tracking_changes false
75 \postpone_fragile_content false
85 \begin_inset Note Note
88 \begin_layout Plain Layout
93 This is an example \SpecialChar LyX
94 file for articles to be submitted to the Journal of
95 Astronomy & Astrophysics (A&A).
96 How to install the A&A \SpecialChar LaTeX
97 class to your \SpecialChar LaTeX
98 system is explained in
102 \begin_layout Plain Layout
104 https://wiki.lyx.org/Layouts/Astronomy-Astrophysics
110 \begin_inset Newline newline
113 Depending on the submission state and the abstract layout, you need to use
114 different document class options that are listed in the aa manual.
117 \begin_inset Newline newline
125 If you use accented characters in your document, you must use the predefined
126 document class option
130 in the document settings.
139 Hydrodynamics of giant planet formation
142 \begin_layout Subtitle
145 \begin_inset Formula $\kappa$
154 \begin_inset Flex institutemark
157 \begin_layout Plain Layout
167 \begin_layout Plain Layout
178 \begin_inset Flex institutemark
181 \begin_layout Plain Layout
191 \begin_layout Plain Layout
204 \begin_layout Plain Layout
205 Just to show the usage of the elements in the author field
211 \begin_inset Note Note
214 \begin_layout Plain Layout
217 fnmsep is only needed for more than one consecutive notes/marks
225 \begin_layout Offprint
230 \begin_layout Address
231 Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse
233 \begin_inset Newline newline
237 \begin_inset Flex Email
240 \begin_layout Plain Layout
241 wuchterl@amok.ast.univie.ac.at
250 \begin_layout Plain Layout
259 University of Alexandria, Department of Geography, ...
260 \begin_inset Newline newline
264 \begin_inset Flex Email
267 \begin_layout Plain Layout
268 c.ptolemy@hipparch.uheaven.space
277 \begin_layout Plain Layout
278 The university of heaven temporarily does not accept e-mails
287 Received September 15, 1996; accepted March 16, 1997
290 \begin_layout Abstract (unstructured)
291 To investigate the physical nature of the `nuc\SpecialChar softhyphen
292 leated instability' of proto
293 giant planets, the stability of layers in static, radiative gas spheres
294 is analysed on the basis of Baker's standard one-zone model.
295 It is shown that stability depends only upon the equations of state, the
296 opacities and the local thermodynamic state in the layer.
297 Stability and instability can therefore be expressed in the form of stability
298 equations of state which are universal for a given composition.
299 The stability equations of state are calculated for solar composition and
300 are displayed in the domain
301 \begin_inset Formula $-14\leq\lg\rho/[\mathrm{g}\,\mathrm{cm}^{-3}]\leq0$
305 \begin_inset Formula $8.8\leq\lg e/[\mathrm{erg}\,\mathrm{g}^{-1}]\leq17.7$
309 These displays may be used to determine the one-zone stability of layers
310 in stellar or planetary structure models by directly reading off the value
311 of the stability equations for the thermodynamic state of these layers,
312 specified by state quantities as density
313 \begin_inset Formula $\rho$
317 \begin_inset Formula $T$
320 or specific internal energy
321 \begin_inset Formula $e$
325 Regions of instability in the
326 \begin_inset Formula $(\rho,e)$
329 -plane are described and related to the underlying microphysical processes.
330 Vibrational instability is found to be a common phenomenon at temperatures
331 lower than the second He ionisation zone.
333 \begin_inset Formula $\kappa$
336 -mechanism is widespread under `cool' conditions.
337 \begin_inset Note Note
340 \begin_layout Plain Layout
341 Citations are not allowed in A&A abstracts.
347 \begin_inset Note Note
350 \begin_layout Plain Layout
351 This is the unstructured abstract type, an example for the structured abstract
356 template file that comes with \SpecialChar LyX
365 \begin_layout Keywords
366 giant planet formation –
367 \begin_inset Formula $\kappa$
370 -mechanism – stability of gas spheres
373 \begin_layout Section
377 \begin_layout Standard
380 nucleated instability
382 (also called core instability) hypothesis of giant planet formation, a
383 critical mass for static core envelope protoplanets has been found.
385 \begin_inset CommandInset citation
392 ) determined the critical mass of the core to be about
393 \begin_inset Formula $12\,M_{\oplus}$
397 \begin_inset Formula $M_{\oplus}=5.975\,10^{27}\,\mathrm{g}$
400 is the Earth mass), which is independent of the outer boundary conditions
401 and therefore independent of the location in the solar nebula.
402 This critical value for the core mass corresponds closely to the cores
403 of today's giant planets.
406 \begin_layout Standard
407 Although no hydrodynamical study has been available many workers conjectured
408 that a collapse or rapid contraction will ensue after accumulating the
410 The main motivation for this article is to investigate the stability of
411 the static envelope at the critical mass.
412 With this aim the local, linear stability of static radiative gas spheres
413 is investigated on the basis of Baker's (
414 \begin_inset CommandInset citation
421 ) standard one-zone model.
424 \begin_layout Standard
425 Phenomena similar to the ones described above for giant planet formation
426 have been found in hydrodynamical models concerning star formation where
427 protostellar cores explode (Tscharnuter
428 \begin_inset CommandInset citation
436 \begin_inset CommandInset citation
443 ), whereas earlier studies found quasi-steady collapse flows.
444 The similarities in the (micro)physics, i.
445 \begin_inset space \thinspace{}
449 \begin_inset space \space{}
452 constitutive relations of protostellar cores and protogiant planets serve
453 as a further motivation for this study.
456 \begin_layout Section
457 Baker's standard one-zone model
460 \begin_layout Standard
461 \begin_inset Float figure
466 \begin_layout Plain Layout
467 \begin_inset Caption Standard
469 \begin_layout Plain Layout
470 \begin_inset CommandInset label
477 \begin_inset Formula $\Gamma_{1}$
482 \begin_inset Formula $\Gamma_{1}$
485 is plotted as a function of
486 \begin_inset Formula $\lg$
490 \begin_inset Formula $[\mathrm{erg}\,\mathrm{g}^{-1}]$
494 \begin_inset Formula $\lg$
498 \begin_inset Formula $[\mathrm{g}\,\mathrm{cm}^{-3}]$
511 In this section the one-zone model of Baker (
512 \begin_inset CommandInset citation
519 ), originally used to study the Cepheïd pulsation mechanism, will be briefly
521 The resulting stability criteria will be rewritten in terms of local state
522 variables, local timescales and constitutive relations.
525 \begin_layout Standard
527 \begin_inset CommandInset citation
534 ) investigates the stability of thin layers in self-gravitating, spherical
535 gas clouds with the following properties:
538 \begin_layout Itemize
539 hydrostatic equilibrium,
542 \begin_layout Itemize
546 \begin_layout Itemize
547 energy transport by grey radiation diffusion.
551 \begin_layout Standard
553 For the one-zone-model Baker obtains necessary conditions for dynamical,
554 secular and vibrational (or pulsational) stability (Eqs.
555 \begin_inset space \space{}
559 \begin_inset space \thinspace{}
563 \begin_inset space \thinspace{}
567 \begin_inset CommandInset citation
575 Using Baker's notation:
578 \begin_layout Standard
582 M_{r} & & \textrm{mass internal to the radius }r\\
583 m & & \textrm{mass of the zone}\\
584 r_{0} & & \textrm{unperturbed zone radius}\\
585 \rho_{0} & & \textrm{unperturbed density in the zone}\\
586 T_{0} & & \textrm{unperturbed temperature in the zone}\\
587 L_{r0} & & \textrm{unperturbed luminosity}\\
588 E_{\textrm{th}} & & \textrm{thermal energy of the zone}
596 \begin_layout Standard
598 and with the definitions of the
607 \begin_inset CommandInset ref
609 reference "fig:FigGam"
616 \tau_{\mathrm{co}}=\frac{E_{\mathrm{th}}}{L_{r0}}\,,
628 \tau_{\mathrm{ff}}=\sqrt{\frac{3\pi}{32G}\frac{4\pi r_{0}^{3}}{3M_{\mathrm{r}}}}\,,
634 \begin_inset Formula $K$
638 \begin_inset Formula $\sigma_{0}$
641 have the following form:
644 \sigma_{0} & = & \frac{\pi}{\sqrt{8}}\frac{1}{\tau_{\mathrm{ff}}}\\
645 K & = & \frac{\sqrt{32}}{\pi}\frac{1}{\delta}\frac{\tau_{\mathrm{ff}}}{\tau_{\mathrm{co}}}\,;
651 \begin_inset Formula $E_{\mathrm{th}}\approx m(P_{0}/{\rho_{0}})$
658 \delta=-\left(\frac{\partial\ln\rho}{\partial\ln T}\right)_{P}\\
665 is a thermodynamical quantity which is of order
666 \begin_inset Formula $1$
670 \begin_inset Formula $1$
673 for nonreacting mixtures of classical perfect gases.
674 The physical meaning of
675 \begin_inset Formula $\sigma_{0}$
679 \begin_inset Formula $K$
682 is clearly visible in the equations above.
684 \begin_inset Formula $\sigma_{0}$
687 represents a frequency of the order one per free-fall time.
689 \begin_inset Formula $K$
692 is proportional to the ratio of the free-fall time and the cooling time.
693 Substituting into Baker's criteria, using thermodynamic identities and
694 definitions of thermodynamic quantities,
697 \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}
705 \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}
710 one obtains, after some pages of algebra, the conditions for
717 \frac{\pi^{2}}{8}\frac{1}{\tau_{\mathrm{ff}}^{2}}(3\Gamma_{1}-4) & > & 0\label{ZSDynSta}\\
718 \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}\\
719 \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}
724 For a physical discussion of the stability criteria see Baker (
725 \begin_inset CommandInset citation
733 \begin_inset CommandInset citation
743 \begin_layout Standard
744 We observe that these criteria for dynamical, secular and vibrational stability,
745 respectively, can be factorized into
748 \begin_layout Enumerate
749 a factor containing local timescales only,
752 \begin_layout Enumerate
753 a factor containing only constitutive relations and their derivatives.
757 \begin_layout Standard
758 The first factors, depending on only timescales, are positive by definition.
759 The signs of the left hand sides of the inequalities
764 \begin_inset CommandInset ref
771 \begin_inset CommandInset ref
778 \begin_inset CommandInset ref
784 ) therefore depend exclusively on the second factors containing the constitutive
786 Since they depend only on state variables, the stability criteria themselves
789 functions of the thermodynamic state in the local zone
792 The one-zone stability can therefore be determined from a simple equation
793 of state, given for example, as a function of density and temperature.
794 Once the microphysics, i.
795 \begin_inset space \thinspace{}
799 \begin_inset space \space{}
802 the thermodynamics and opacities (see Table
807 \begin_inset CommandInset ref
809 reference "tab:KapSou"
813 ), are specified (in practice by specifying a chemical composition) the
814 one-zone stability can be inferred if the thermodynamic state is specified.
815 The zone – or in other words the layer – will be stable or unstable in
816 whatever object it is imbedded as long as it satisfies the one-zone-model
818 Only the specific growth rates (depending upon the time scales) will be
819 different for layers in different objects.
822 \begin_layout Standard
823 \begin_inset Float table
828 \begin_layout Plain Layout
829 \begin_inset Caption Standard
831 \begin_layout Plain Layout
832 \begin_inset CommandInset label
846 \begin_layout Plain Layout
849 <lyxtabular version="3" rows="4" columns="2">
850 <features tabularvalignment="middle">
851 <column alignment="left" valignment="top" width="0pt">
852 <column alignment="left" valignment="top" width="0pt">
854 <cell alignment="center" valignment="top" topline="true" usebox="none">
857 \begin_layout Plain Layout
863 <cell alignment="center" valignment="top" topline="true" usebox="none">
866 \begin_layout Plain Layout
867 \begin_inset Formula $T/[\textrm{K}]$
877 <cell alignment="center" valignment="top" topline="true" usebox="none">
880 \begin_layout Plain Layout
881 Yorke 1979, Yorke 1980a
886 <cell alignment="center" valignment="top" topline="true" usebox="none">
889 \begin_layout Plain Layout
890 \begin_inset Formula $\leq1700^{\textrm{a}}$
900 <cell alignment="center" valignment="top" usebox="none">
903 \begin_layout Plain Layout
909 <cell alignment="center" valignment="top" usebox="none">
912 \begin_layout Plain Layout
913 \begin_inset Formula $1700\leq T\leq5000$
923 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
926 \begin_layout Plain Layout
932 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
935 \begin_layout Plain Layout
936 \begin_inset Formula $5000\leq$
952 \begin_layout Plain Layout
953 \begin_inset Formula $^{\textrm{a}}$
961 We will now write down the sign (and therefore stability) determining parts
962 of the left-hand sides of the inequalities (
963 \begin_inset CommandInset ref
970 \begin_inset CommandInset ref
977 \begin_inset CommandInset ref
985 stability equations of state
990 \begin_layout Standard
991 The sign determining part of inequality
996 \begin_inset CommandInset ref
1003 \begin_inset Formula $3\Gamma_{1}-4$
1006 and it reduces to the criterion for dynamical stability
1007 \begin_inset Formula
1009 \Gamma_{1}>\frac{4}{3}\,\cdot
1014 Stability of the thermodynamical equilibrium demands
1015 \begin_inset Formula
1017 \chi_{\rho}^{}>0,\;\;c_{v}>0\,,
1023 \begin_inset Formula
1030 holds for a wide range of physical situations.
1032 \begin_inset Formula
1034 \Gamma_{3}-1=\frac{P}{\rho T}\frac{\chi_{T}^{}}{c_{v}} & > & 0\\
1035 \Gamma_{1}=\chi_{\rho}^{}+\chi_{T}^{}(\Gamma_{3}-1) & > & 0\\
1036 \nabla_{\mathrm{ad}}=\frac{\Gamma_{3}-1}{\Gamma_{1}} & > & 0
1041 we find the sign determining terms in inequalities
1042 \begin_inset space ~
1046 \begin_inset CommandInset ref
1048 reference "ZSSecSta"
1053 \begin_inset CommandInset ref
1055 reference "ZSVibSta"
1059 ) respectively and obtain the following form of the criteria for dynamical,
1060 secular and vibrational
1065 \begin_inset Formula
1067 3\Gamma_{1}-4=:S_{\mathrm{dyn}}> & 0\label{DynSta}\\
1068 \frac{1-3/4\chi_{\rho}^{}}{\chi_{T}^{}}(\kappa_{T}^{}-4)+\kappa_{P}^{}+1=:S_{\mathrm{sec}}> & 0\label{SecSta}\\
1069 4\nabla_{\mathrm{ad}}-(\nabla_{\mathrm{ad}}\kappa_{T}^{}+\kappa_{P}^{})-\frac{4}{3\Gamma_{1}}=:S_{\mathrm{vib}}> & 0\,.\label{VibSta}
1074 The constitutive relations are to be evaluated for the unperturbed thermodynami
1076 \begin_inset Formula $(\rho_{0},T_{0})$
1080 We see that the one-zone stability of the layer depends only on the constitutiv
1082 \begin_inset Formula $\Gamma_{1}$
1086 \begin_inset Formula $\nabla_{\mathrm{ad}}$
1090 \begin_inset Formula $\chi_{T}^{},\,\chi_{\rho}^{}$
1094 \begin_inset Formula $\kappa_{P}^{},\,\kappa_{T}^{}$
1098 These depend only on the unperturbed thermodynamical state of the layer.
1099 Therefore the above relations define the one-zone-stability equations of
1101 \begin_inset Formula $S_{\mathrm{dyn}},\,S_{\mathrm{sec}}$
1105 \begin_inset Formula $S_{\mathrm{vib}}$
1110 \begin_inset space ~
1114 \begin_inset CommandInset ref
1116 reference "fig:VibStabEquation"
1121 \begin_inset Formula $S_{\mathrm{vib}}$
1125 Regions of secular instability are listed in Table
1126 \begin_inset space ~
1132 \begin_layout Standard
1133 \begin_inset Float figure
1138 \begin_layout Plain Layout
1139 \begin_inset Caption Standard
1141 \begin_layout Plain Layout
1142 \begin_inset CommandInset label
1144 name "fig:VibStabEquation"
1148 Vibrational stability equation of state
1149 \begin_inset Formula $S_{\mathrm{vib}}(\lg e,\lg\rho)$
1154 \begin_inset Formula $>0$
1157 means vibrational stability
1170 \begin_layout Section
1174 \begin_layout Enumerate
1175 The conditions for the stability of static, radiative layers in gas spheres,
1176 as described by Baker's (
1177 \begin_inset CommandInset citation
1184 ) standard one-zone model, can be expressed as stability equations of state.
1185 These stability equations of state depend only on the local thermodynamic
1190 \begin_layout Enumerate
1191 If the constitutive relations – equations of state and Rosseland mean opacities
1192 – are specified, the stability equations of state can be evaluated without
1193 specifying properties of the layer.
1197 \begin_layout Enumerate
1198 For solar composition gas the
1199 \begin_inset Formula $\kappa$
1202 -mechanism is working in the regions of the ice and dust features in the
1204 \begin_inset Formula $\mathrm{H}_{2}$
1207 dissociation and the combined H, first He ionization zone, as indicated
1208 by vibrational instability.
1209 These regions of instability are much larger in extent and degree of instabilit
1210 y than the second He ionization zone that drives the Cepheïd pulsations.
1214 \begin_layout Acknowledgement
1215 Part of this work was supported by the German
1217 Deut\SpecialChar softhyphen
1218 sche For\SpecialChar softhyphen
1219 schungs\SpecialChar softhyphen
1220 ge\SpecialChar softhyphen
1221 mein\SpecialChar softhyphen
1225 \begin_inset space ~
1231 \begin_layout Standard
1232 \begin_inset CommandInset bibtex
1234 btprint "btPrintAll"
1235 bibfiles "../biblioExample"
1241 \begin_inset Note Note
1244 \begin_layout Plain Layout
1249 If you cannot see the bibliography in the output, assure that you have
1250 given the full path to the Bib\SpecialChar TeX
1255 that is part of the A&A \SpecialChar LaTeX