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Thesaurus

06(03.11.1;16.06.1;19.06.1;19.37.1;19.53.1;19.63.1) Title

Hydrodynamics of giant planet formation Subtitle

I. Overviewing the _inset Formula κ inset

-mechanism Author

G. Wuchterl Address

Institute for Astronomy (IfA), University of Vienna, Tⁿrkenschanzstrasse 17, A-1180 Vienna Offprint

G. Wuchterl Email

wuchterl@amok.ast.univie.ac.at Date

Received September 15, 1996 / Accepted March 16, 1997 Abstract

To investigate the physical nature of the `nuc leated instability' of proto giant planets (Mizuno _inset LatexCommand [#mizuno##1###]

inset

), the stability of layers in static, radiative gas spheres is analysed on the basis of Baker's _inset LatexCommand [#baker##1###]

inset

standard one-zone model. It is shown that stability depends only upon the equations of state, the opacities and the local thermodynamic state in the layer. Stability and instability can therefore be expressed in the form of stability equations of state which are universal for a given composition. Abstract

The stability equations of state are calculated for solar composition and are displayed in the domain _inset Formula #math1#-14≤lgρ/[g cm^-3]≤ 0 inset

, _inset Formula #math2#8.8≤lg e/[erg g^-1]≤17.7 inset

. These displays may be used to determine the one-zone stability of layers in stellar or planetary structure models by directly reading off the value of the stability equations for the thermodynamic state of these layers, specified by state quantities as density _inset Formula ρ inset

, temperature _inset Formula T inset

or specific internal energy _inset Formula e inset

. Regions of instability in the _inset Formula #math3#(ρ e) inset

-plane are described and related to the underlying microphysical processes. Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone. The _inset Formula κ inset

-mechanism is widespread under `cool' conditions. Abstract

atex

keywords<#9#> efault giant planet formation -- _inset Formula κ inset

-mechanism -- stability of gas spheres atex <#9#> Section

Introduction Standard

In the on nucleated instability default (also called core instability) hypothesis of giant planet formation, a critical mass for static core envelope protoplanets has been found. Mizuno ( _inset LatexCommand [#mizuno##1###]

inset

) determined the critical mass of the core to be about _inset Formula #math4#12 M_⊕ inset

( _inset Formula #math5#M_⊕ = 5.975 10²⁷ g inset

is the Earth mass), which is independent of the outer boundary conditions and therefore independent of the location in the solar nebula. This critical value for the core mass corresponds closely to the cores of today's giant planets. Standard

Although no hydrodynamical study has been available many workers conjectured that a collapse or rapid contraction will ensue after accumulating the critical mass. The main motivation for this article is to investigate the stability of the static envelope at the critical mass. With this aim the local, linear stability of static radiative gas spheres is investigated on the basis of Baker's ( _inset LatexCommand [#baker##1###]

inset

) standard one-zone model. The nonlinear, hydrodynamic evolution of the protogiant planet beyond the critical mass, as calculated by Wuchterl ( _inset LatexCommand [#wuchterl##1###]

inset

), will be described in a forthcoming article. Standard

The fact that Wuchterl ( _inset LatexCommand [#wuchterl##1###]

inset

) found the excitation of hydrodynamical waves in his models raises considerable interest on the transition from static to dynamic evolutionary phases of the protogiant planet at the critical mass. The waves play a crucial role in the development of the so-called nucleated instability in the nucleated instability hypothesis. They lead to the formation of shock waves and massive outflow phenomena. The protoplanet evolves into a new quasi-equilibrium structure with a on pulsating default envelope, after the mass loss phase has declined. Standard

Phenomena similar to the ones described above for giant planet formation have been found in hydrodynamical models concerning star formation where protostellar cores explode (Tscharnuter _inset LatexCommand [#tscarnuter##1###]

inset

, Balluch _inset LatexCommand [#balluch##1###]

inset

), whereas earlier studies found quasi-steady collapse flows. The similarities in the (micro)physics, i.e., constitutive relations of protostel lar cores and protogiant planets serve as a further motivation for this study. Section

Baker's standard one-zone model Standard

_float wide-fig Standard

atex

rule<#20#>0.4pt<#20#><#21#>4cm<#21#>

parbox[b]<#22#>55mm<#22#><#272#> Caption

Adiabatic exponent _inset Formula Γ inset

.

_inset Formula #math6#Γ₁ inset

is plotted as a function of _inset Formula lg inset

internal energy _inset Formula #math7#[erg g^-1] inset

and _inset Formula lg inset

density _inset Formula #math8#[g cm^-3] inset

_inset LatexCommand

inset

atex <#272#> float In this section the one-zone model of Baker ( _inset LatexCommand [#baker##1###]

inset

), originally used to study the Cephe∩d pulsation mechanism, will be briefly reviewed. The resulting stability criteria will be rewritten in terms of local state variables, local timescales and constitutive relations. Standard

Baker ( _inset LatexCommand [#baker##1###]

inset

) investigates the stability of thin layers in self-gravitating, spherical gas clouds with the following properties: Itemize

hydrostatic equilibrium, Itemize

thermal equilibrium, Itemize

energy transport by grey radiation diffusion. Standard

For the one-zone-model Baker obtains necessary conditions for dynamical, secular and vibrational (or pulsational) stability [Eqs. _separator (34a, atex

, efault b, atex

, efault c) in Baker _inset LatexCommand [#baker##1###]

inset

]. Using Baker's notation: _inset Formula

inset

and with the definitions of the on local cooling time default (see Fig. _separator

_inset LatexCommand #FigGam#71>

inset

) Standard

_inset Formula

inset

and the on local free-fall time Standard

_inset Formula

inset

Baker's _inset Formula K inset

and _inset Formula #math12#σ₀ inset

have the following form: _inset Formula

inset

where _inset Formula #math14#E_th #tex2html_wrap_inline900#
m(P₀/ρ₀) inset

has been used and Standard

_inset Formula

inset

is a thermodynamical quantity which is of order _inset Formula 1 inset

and equal to _inset Formula 1 inset

for nonreacting mixtures of classical perfect gases. The physical meaning of _inset Formula #math16#σ₀ inset

and _inset Formula K inset

is clearly visible in the equations above.

_inset Formula #math17#σ₀ inset

represents a frequency of the order one per free-fall time.

_inset Formula K inset

is proportional to the ratio of the free-fall time and the cooling time. Substituting into Baker's criteria, using thermodynamic identities and definitions of thermodynamic quantities, _inset Formula

#math18#

inset

Standard

_inset Formula

#math19#

inset

one obtains, after some pages of algebra, the conditions for on stability default given below: Standard

_inset Formula

inset

For a physical discussion of the stability criteria see Baker ( _inset LatexCommand [#baker##1###]

inset

) or Cox ( _inset LatexCommand [#cox##1###]

inset

). Standard

We observe that these criteria for dynamical, secular and vibrational stability, respectively, can be factorized into Enumerate

a factor containing local timescales only, Enumerate

a factor containing only constitutive relations and their derivatives. Standard

The first factors, depending on only timescales, are positive by definition. The signs of the left hand sides of the inequalities _separator ( _inset LatexCommand #ZSDynSta#178>

inset

), ( _inset LatexCommand #ZSSecSta#179>

inset

) and ( _inset LatexCommand #ZSVibSta#180>

inset

) therefore depend exclusively on the second factors containing the constitutive relations. Since they depend only on state variables, the stability criteria themselves are on functions of the thermodynamic state in the local zone default . The one-zone stability can therefore be determined from a simple equation of state, given for example, as a function of density and temperature. Once the microphysics, i.e. the thermodynamics and opacities (see Table _separator

_inset LatexCommand #KapSou#181>

inset

), are specified (in practice by specifying a chemical composition) the one-zone stability can be inferred if the thermodynamic state is specified. The zone -- or in other words the layer -- will be stable or unstable in whatever object it is imbedded as long as it satisfies the one-zone-model assumptions. Only the specific growth rates (depending upon the time scales) will be different for layers in different objects. Standard

_float tab Caption

Opacity sources _inset LatexCommand

inset

Standard center multicol4 4 2 0 0 -1 -1 -1 -1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 2 0 0 2 0 0 0 2 1 0 0 0 0 0 2 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0 0 2 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0 0 8 1 0 0 0 0

Source
T/[K]
Yorke 1979, Yorke 1980a

_inset Formula #math21#≤1700^a inset

Krⁿgel 1971

_inset Formula #math22#1700≤T≤5000 inset

Cox ;SPMamp; Stewart 1969

_inset Formula 5000≤ inset

Standard

_inset Formula #math23#a inset

This is footnote a float _float wide-tab Caption

Regions of secular instability _inset LatexCommand

inset

Standard

atex

vspace<#186#>4cm<#186#> float We will now write down the sign (and therefore stability) determining parts of the left-hand sides of the inequalities ( _inset LatexCommand #ZSDynSta#187>

inset

), ( _inset LatexCommand #ZSSecSta#188>

inset

) and ( _inset LatexCommand #ZSVibSta#189>

inset

) and thereby obtain on stability equations of state default . Standard

The sign determining part of inequality _separator ( _inset LatexCommand #ZSDynSta#190>

inset

) is _inset Formula #math24#3Γ₁ - 4 inset

and it reduces to the criterion for dynamical stability Standard

_inset Formula

inset

Stability of the thermodynamical equilibrium demands _inset Formula

inset

and Standard

_inset Formula

inset

holds for a wide range of physical situations. With Standard

_inset Formula

inset

we find the sign determining terms in inequalities _separator ( _inset LatexCommand #ZSSecSta#221>

inset

) and ( _inset LatexCommand #ZSVibSta#222>

inset

) respectively and obtain the following form of the criteria for dynamical, secular and vibrational on stability default , respectively: Standard

_inset Formula

inset

The constitutive relations are to be evaluated for the unperturbed thermodynamic state (say _inset Formula #math30#(ρ₀, T₀) inset

) of the zone. We see that the one-zone stability of the layer depends only on the constitutiv e relations _inset Formula #math31#Γ₁ inset

, _inset Formula #math32#∇_ad inset

, _inset Formula #math33#χ_T, χ_ρ inset

, _inset Formula #math34#κ_P, κ_T inset

. These depend only on the unperturbed thermodynamical state of the layer. Therefore the above relations define the one-zone-stability equations of state _inset Formula #math35#S_dyn, S_sec inset

and _inset Formula #math36#S_vib inset

. See Fig. _separator

_inset LatexCommand #FigVibStab#253>

inset

for a picture of _inset Formula #math37#S_vib inset

. Regions of secular instability are listed in Table _separator

_inset LatexCommand #TabSecInst#255>

inset

. Standard

_float fig Standard

atex

vspace<#256#>5cm<#256#> Caption

Vibrational stability equation of state _inset Formula #math38#S_vib(lg e, lgρ) inset

.

_inset Formula ;SPMgt; 0 inset

means vibrational stability. _inset LatexCommand

inset

float Section

Conclusions Enumerate

The conditions for the stability of static, radiative layers in gas spheres, as described by Baker's ( _inset LatexCommand [#baker##1###]

inset

) standard one-zone model, can be expressed as stability equations of state. These stability equations of state depend only on the local thermodynamic state of the layer. Enumerate

If the constitutive relations -- equations of state and Rosseland mean opacities -- are specified, the stability equations of state can be evaluated without specifying properties of the layer. Enumerate

For solar composition gas the _inset Formula κ inset

-mechanism is working in the regions of the ice and dust features in the opacities, the _inset Formula #math39#H₂ inset

dissociation and the combined H, first He ionization zone, as indicated by vibrational instability. These regions of instability are much larger in extent and degree of instabilit y than the second He ionization zone that drives the Cephe∩d pulsations. Acknowledgement

Part of this work was supported by the German on Deut sche For schungs ge mein schaft, DFG default project number Ts _separator 17/2--1. Bibliography