A Systematic Study of the Final Masses of Gas Giant Planets

We construct an analytic model for the rate of gas accretion onto a planet embedded in a protoplanetary disk as a function of planetary mass, disk viscosity, disk scale height, and unperturbed surface density in order to study the long-term accretion and f

ASYSTEMATICSTUDYOFTHEFINALMASSESOFGAS

GIANTPLANETS

TakayukiTanigawaandMasahiroIkoma

arXiv:0705.4343v1 [astro-ph] 30 May 2007DepartmentofEarthandPlanetarySciences,TokyoInstituteofTechnology,Ookayama,Meguro-ku,Tokyo152-8551,Japantanigawa@geo.titech.ac.jpABSTRACTWeconstructananalyticmodelfortherateofgasaccretionontoaplanetembeddedinaprotoplanetarydiskasafunctionofplanetarymass,diskviscosity,diskscaleheight,andunperturbedsurfacedensityinordertostudythelong-termaccretionand nalmassesofgasgiantplanets.We rstderiveananalyticalformulaforsurfacedensitypro http://www.77cn.com.cningitintheempiricalformulalinkingsurfacedensitywithgasaccretionratethatisderivedbasedonhydrodynamicsimulationsofTanigawa&Watanabe(2002,ApJ586,506),wethensimulatethemassevolutionofgasgiantplanetsinviscously-evolvingdisks.We nallydeterminethe nalmassasafunctionofsemi-majoraxisoftheplanet.We ndthatthediskcanbepidedintothreeregionscharacterizedbydi erentprocessesbywhichthe nalmassisdetermined.Intheinnerregion,theplanetgrowsquicklyandformsadeepgaptosuppressthegrowthbyitselfbeforediskdissipation.The nalmassoftheplanetinthisregionisfoundtoincreasewiththesemi-majoraxisinasimilarwaytothemassgivenbytheviscousconditionforgapopening,buttheformerislargerbyafactorofapproximately10than

thelatter.Intheintermediateregion,viscousdi usionofthediskgaslimitsthe

gasaccretionbeforetheplanetformadeepgap.The nalmasscanbeuptothe

diskmass,whendiskviscousevolutionoccursfasterthandiskevaporation.In

theouterregion,planetscaptureonlytinyamountsofgaswithinthelifetimeof

thedisktoformNeptune-likeplanets.Wederiveanalyticformulaeforthe nal

massesinthedi erentregionsandthelocationsoftheboundaries,whichare

helpfultogainasystematicunderstandingofthemassesofgasgiantplanets.

Subjectheadings:methods:analytical—methods:numerical—solarsystem:

formation

We construct an analytic model for the rate of gas accretion onto a planet embedded in a protoplanetary disk as a function of planetary mass, disk viscosity, disk scale height, and unperturbed surface density in order to study the long-term accretion and f

1.Introduction

Afundamentalbutunresolvedissuewithplanetformationishowthemassofagiant

planetis xed.Inthesolarsystemtherearefourgiantplanets,whicharecharacterized

bytheirmassivehydrogen/heliumenvelopes.JupiterandSaturnconsistmostlyofhydro-

gen/helium,whileUranusandNeptunearemostlyoficebuthavesigni cantamountsofhy-

drogen/helium.Thegiantplanetsaredi erentinmass:Jupiter’smassis1×10 3M⊙(=MJ),

Saturn’smassis～0.3MJ,andUranus’andNeptune’smassesare～0.05MJ.Furthermore,

theextrasolarplanetsdetectedsofaralsorangeinmassfromaboutoneNeptunemassto

～10MJ1;thoseplanetsarebelievedtohavemassivehydrogen/heliumenvelopeslikethe

giantplanetsinthesolarsystem.

Therearetwocompetingideasfortheformationofgiantplanets,thecoreaccretion

model(e.g.,Mizuno1980;Bodenheimer&Pollack1986)andthediskinstabilitymodel(e.g.,

Cameron1978;Boss1989).Inthecoreaccretionmodel,arocky/icycore rstformsthrough

collisionalaggregationofplanetesimals,followedbyenvelopeformationduetosubstantial

accretionofgasfromthecircumstellar(protoplanetary)disk.Inthediskinstabilitymodel,

density uctuationofthediskgasgrowstoformagaseousplanet,followedbycoreformation

duetosedimentationofheavyelementsinitsinterior.Theadvantagesanddisadvantages

ofbothmodelsarediscussedinseveralliteratures(e.g.,Boss2002),whichisnotrepeated

here.Inthispaperweconsidergiantplanetformationinthecontextofthecoreaccretion

model.

Inthecoreaccretionmodel,theprocessoftheaccumulationoftheenvelopesispided

intotwophases,thesubcritical-accretionandsupercritical-accretionphases,intermsofthe

dominantenergysource.Thetransitionfromtheformertothelatteroccurswhenthemass

ofacorereachesacriticalvalue.Inthesubcritical-accretionphase,incomingplanetesimals

supplyenergytotheenvelopesothattheenvelopeisinthehydrostaticequilibrium.Thecore

accretionthuscontrolsthegasaccretioninthisphase.Thephase2foundbyPollacketal.

(1996)isapartofthesubcritical-accretionphase.Inthesupercritical-accretionphase,

theenergysuppliedbyplanetesimalsisinsu cienttokeepthehydrostaticstructureofthe

envelope,sothattheenvelopesubstantiallycontractsandreleasesitsgravitationalenergy,

resultinginrunawayaccretionofthediskgas.

Thesupercritical-accretionphasecanbefurtherpidedintotwosubphases.Inthe

formersubphase,thegasaccretioniscontrolledbycontractionoftheenvelopeandthusoc-

cursontheKelvin-Helmholtztimescale(Bodenheimer&Pollack1986;Pollacketal.1996;

We construct an analytic model for the rate of gas accretion onto a planet embedded in a protoplanetary disk as a function of planetary mass, disk viscosity, disk scale height, and unperturbed surface density in order to study the long-term accretion and f

Ikoma,Nakazawa,&Emori2000;Ikoma&Genda2006).However,becausethegasaccre-

tiondrivenbytheenvelopecontractionacceleratesrapidlywithtime,thesupplyofthedisk

gasinevitablybecomesunabletokeepupwiththedemandofthecontractingenvelope.Thus,

inthelattersubphase,thedisk-gassupplylimitsthegasaccretion(Tanigawa&Watanabe

2002,hereafterTW02).

Inthispaperwefocusonthegasaccretioninthelattersubphaseofthesupercritical-

accretionphase,i.e.,thephaseinwhichtheKelvin-Helmholtzcontractionoftheenvelopeis

underwaybutthegasaccretionislimitedbydisk-gassupply.Aftertheonsetoftherunaway

gasaccretion(i.e.,inthesupercritical-accretionphase),theplanetalwaysexperiencesthe

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