Diffusive Migration of Low-Mass Proto-planets in T

arXiv:astro-ph/0603235v228Apr2006DiffusiveMigrationofLow-MassProto-planetsinTurbulentDisksEricT.JohnsonandJeremyGoodmanDepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NJ08544andKristenMenouDepartmentofAstronomy,ColumbiaUniversity,550W.120thSt.,NewYork,NY10027ABSTRACTTorquefluctuationsduetomagnetorotationalturbulenceinproto-planetarydisksmaygreatlyinfluencethemigrationpatternsandsurvivalprobabilitiesofnascentplan-ets.Providedthattheturbulenceisastationarystochasticprocesswithfiniteamplitudeandcorrelationtime,theresultingdiffusivemigrationcanbedescribedwithaFokker-Planckequation,whichwereducetoanadvection-diffusionequation.Wecalibratethecoefficientswithexistingturbulent-disksimulationsandmean-migrationestimates,andsolvetheequationbothanalyticallyandnumerically.Diffusiontendstodominateoveradvectionforplanetsoflow-massandthoseintheouterregionsofproto-planetarydisks,whethertheyaredescribedbytheMinimumMassSolarNebula(MMSN)orbyT-Taurialphadisks.Diffusionsystematicallyreducesthelifetimeofmostplanets,yetitallowsadecliningfractionofthemtosurviveforextendedperiodsoftimeatlargeradii.Meanplanetlifetimescanevenbeformallyinfinite(e.g.inaninfinitesteadyMMSN),thoughmedianlifetimesarealwaysfinite.Survivingplanetsmaylingernearspecificradiiwherethecombinedeffectsofadvectionanddiffusionareminimized,oratlargeradii,dependingonmodelspecifics.Thestochasticnatureofmigrationinturbulentdiskschallengesdeterministicplanetformationscenariosandsuggestsinsteadthatawidevarietyofplanetaryoutcomesarepossiblefromsimilarinitialconditions.Thiswouldcontributetothediversityof(extrasolar)planetarysystems.Subjectheadings:accretion,accretiondisks—planetarysystems:formation—plane-tarysystems:proto-planetarydisks1.IntroductionAdecadeofextrasolarplanetdiscoverieshasshownthattheprocessofplanetformationismorecomplexthanoriginallyanticipated.Itleadstoaremarkablediversityofplanetaryconfigurations,rangingfrommigratinghotJupiterstoeccentricgiantplanets,aswellasourown“circular”Solar–2–System.Followingthesediscoveries,progresshasbeenmadeinunderstandingplanetformation,butthetheoryisstillincomplete(e.g.Marcyetal.2000;Wuchterletal.2000;Papaloizou&Terquem2006;Armitage&Rice2006).Theleadingscenariofortheformationofgiantplanetsisthecoreaccretionmechanism.Icyplanetesimalslocatedbeyondthesnowlineintheirhostdiskcolliderepeatedlytogrowacorewithamodestgaseousatmosphere.IfthiscoresucceedsinreachingacriticalmassofafewtensofEarthmasses,runawayaccretionofamassivegaseousenvelopeproceedsandleads,ultimately,totheformationofagaseousgiantplanet(Safronov1969;Mizuno1980;Pollacketal.1996).Someevidencesupportingthisscenariohasemergedinrecentyears,intheformofametallicitytrendforstarshostingplanets(e.g.Santosetal.2004;Gillilandetal.2000;Weldrakeetal.2005),thediscoveryofahighdensityhotJupiter(Satoetal.2005)andthatofasurprisinglylow-massmicro-lensingplanet(Beaulieuetal.2006).However,along-standingdifficultyforthecoreaccretionscenario,whichhasnotyetbeenfullyelucidated,isthefactthatthetimescalerequiredtobuild-upcriticalcoremassesandthuslargegaseousenvelopes(∼106-107yr)iscomparabletothelifetimesofproto-planetarydisks(e.g.Pollacketal.1996;Papaloizou&Terquem1999).Orbitalmigrationaddsalayerofcomplicationtotheoriesofplanetformation.Asaresultofgravitationalinteractionswiththeirgaseousdisk(Goldreich&Tremaine1980;Lin&Papaloizou1986),theorbitsofplanetsintheterrestrialmassrangearepredictedtodecayontimescales(∼105yr)shortcomparedtodisklifetimes(e.g.Korycansky&Pollack1993;Ward1997a,b).Migrationisslowerforplanetsofmuchsmallerormuchlargermasses:inthefirstcasebecausethetorquecausingmigrationisquadraticinplanetmass,andinthesecondcasebecausetheplanetopensagapandthenmigratesonthedisk’saccretiontimescale,whichcanbecomparabletoitslifetime.Whileitispossibleorprobablethatmanyterrestrialplanetsformbyagglomerationofsmallerbodiesafterthegasisgone,thisisnotanoptionforthesolidcoresofJovianplanetssince,inthecoreaccretionscenario,thecoresmustformbeforethegaseousenvelopescanbeaccreted.TheprevalenceofJovianplanetswithorbitalperiodsofonlyafewdaysdeepensthemysteryasitsuggeststhattheseplanetsdidmigratebutstoppedshortofmergingwiththeirstarsatorbitalradiiwhereeventheaccretiontimescalewouldseemtohavebeenveryshort(e.g.Linetal.1996).Analyticcalculationsandmosthydro-dynamicalsimulationsofmigrationusuallyassumeadiskthatislaminarapartfromthewavesandshocksexcitedbytheplanetitself(e.g.Korycansky&Pollack1993;Ward1997a;Kleyetal.2001;Nelson&Benz2003a,b;D’Angeloetal.2003;Lufkinetal.2004;Schaeferetal.2004).Buttheeffectiveviscosityofdisksprobablyinvolvesturbulence.Nelson&Papaloizou(2004),Laughlinetal.(2004),andNelson(2005,hereafterN2005)havefoundin3Dsimulationsofmagneto-rotationalturbulencethattheinstantaneoustorqueexertedonaplanetintheterrestrialmassrangeissubjecttofluctuationsmanytimesitsmeanvalue,apparentlycausedbyturbulentdensityfluctuationsintheplanet’svicinity.Infact,noobviousseculardecaymanifestsitselfintheorbitsofplanetswithmassMp.10M⊕,althoughbecausethesimulationsarelimitedto∼102planetaryorbits—compare&105forJupiterduringthelifetimeofthepr