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EPJWebofConferenceswillbesetbythepublisher DOI:willbesetbythepublisher (cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2013 3 1 0 2 n a Small flow rate can supply inwardly migrating shortest-period J planets 7 1 ] StuartF.Taylor1,2,a P E 1ParticipationWorldscope/GlobalTelescopeScience HongKong,SARChinaandSedona,Arizona,U.S.A. . h 2JobSeeking p - o Abstract. The number of exoplanets found with periods as short as one day and less r t was surprising given how fast these planets had been expected tomigrate intothe star s due to the tides raised on the starby planets at such close distances. It has been seen a [ asimprobablethatwewouldfindplanetsinsuchasmallfinalfractionoftheirlives[1]. Thefavoredsolution hasbeen thatthetidal dissipationismuch weakerthan expected, 1 whichwouldmeanthatthefinalinfallwouldbealargerfractionoftheplanets’life. We v findnoreason, however, toexcludetheexplanation thatasmallnumber ofplanetsare 9 continuouslysentmigratinginwardssuchthattheseplanetsindeedareinthelastfraction 2 oftheirlives.Followingtheobservationthatthedistributionofmediumplanetsdisfavors 2 tidal dissipation being significantly weaker than has been found from observations of 4 . binary stars [2], we now show that the numbers of planets in such a “flow” of excess 1 planets migrating inwards islow enough that even depletion of the three-day pileup is 0 a plausible source. Then the shortest period occurrence distribution would be shaped 3 byplanetscontinuouslybeingsentintothestar,whichmayexplainthedepletionofthe 1 pileupintheKeplerfieldrelativetothesolarneighborhood[3].BecauseKeplerobserves : v abovethegalacticplane,[3]suggestedtheKeplerfieldmayincludeanolderpopulation i X of stars. The tidaldissipation strength instarsdue togiant planets maybe not greatly weakerthanitisinbinarystars. r a 1 Introduction The discovery of too many planets found in the last fraction of their lifetimes is among the most unexplainedexoplanetdiscoveries[1]. Eithertidalmigrationisslowerthanexpected,ortheseplanets haverecentlybeensentinward.Thefavoredexplanationhasbeenthattidaldissipationwithinthestar fromtidesonthe starraisedbythese planetsis muchweakerthanthe tidaldissipationasmeasured frombinarystarstatistics[1],butweshowthemoreplausibleexplanationisthatmoreplanetsmigrate inwards than expected. It is thought that inwardly migrating planets circularize and nearly cease inwardmigration,avoidinginfallintothestar. Weproposethatplanetsinthepileupmaybecausedto migratemorerapidlythanexpectedthanifmostorbitsarecircular. Weshowthatthediscrepancybetweentheinferredtidalstrengthfoundby[2,4]isalsosupported bydirectlycalculatingtheinfalltimesfortheKeplerplanetcandidates(hereafter“planets”). ae-mail:[email protected] EPJWebofConferences n 0.01 Data Fit Data Fit oitcarF tenalP00.0.000011 FoInri tiQal* O ocfc Dauta F8i1.tr00 renc1e7.00 160.0 (a) For Q*I onitfi al8 1.00Occu17r.00renData c1Fi6t0.0e (b) 0.1 1.0 10.00.1 1.0 10.0 Period (days) Period (days) Figure1. Migratedevolutioncomparedtodataformassesandradiiforanexampleageof4.5Gyrforplanets withmassessummedovertherangesgiveninthetext,(a)for“large”planets,and(b)for“medium”planets. 2 Inward migration as sourceof shortestperiod planets Thestrengthofdissipationhadbeenexpectedtobestrongerthanatidaldissipationstrength“Q1 ”of ˚ 107,whichisexpressedasQ1 being“lessthan”107,wherethedissipationstrengthisproportionalto ˚ 1{Q1. However,carefulstudiesbasedontheevolutionoftheoccurrencedistributionhavepresented ˚ planetdistributionto be consistentwith Q1 weakerthan 107 (e.g., [5]), when assumingno inputof ˚ new planets migrating inwards. We find that the rate of planet migrating inwards need only be on theorderof10´12 orlessplanetsperyearinorderforthepresenceoftheplanetswiththeseshortest periodsto beconsistentwith a tidaldissipation Q1 strenghtof107 orstronger. Thisnumberis low ˚ enoughtobesuppliedbyadecayinthelongobservedthreedaypileupofgiantplanets,oritcouldbe duetoahypothesizedflowofplanetsfromfurtherout. Bothexplanationshavesupport: Thenumber ofplanetsinthepileupofplanetcandidates(hereafter“planets”)foundbyKeplerhasbeenreported tobe40%lowerthanfortheplanetsfoundinthesolarneighborhoodby[3],whosuggestthatperhaps theKeplerfield,beingabovethegalacticplane,hasanolderpopulationofstars. Thepileupmayhave decayed.Thehypothesisofaflowof“higheccentricitymigration”(HEM)ofgiantplanetssupplying thepileuphasbeensuggestedby[6],thoughsucheccentricplanetshavenotyetbeenfound[7]. Theshortestperioddistributionofplanetshasafalloffthatwithinlargeuncertaintyisconsistent withmigrationduetotidesinthestarcausedbytheplanet[2]. Tidalmigrationduetothese“stellar” tidesproduceanoccurrencedistributionwithafalloffthathasapowerindexof13/3basedonequa- tionsof[8]. Thetwo-powerlawfittotheKeplerplanetoccurrencedistributionby[3]givesafalloff intheshortestperiodregionthatforgiantandmediumradii planetsisreasonablycloseto13/3. We obtainthepowerindexofthefallofffromthefitby[3],bysumingtheirtwoindicesβ`γ,whichis thepowerindexoftheirfitfunctioninthelimitoftheperiodgoingtozero.Wecallthethreerangesin radiiusedby[3]large,mediumand(relatively)small. Weassignthefollowingradiiandmassranges touseincalculationsoftidalmigration,inearthradiiRC,andearthmasses, MC: “Large”planetsof radiifrom8to16RC andofmassfrom100to2000MC,“medium”planetsofradii4to8RC andof massfrom100to2000MC, and“small”planetsofradii2to4 RC andofmassfrom10to100MC. Thepowerindicesfoundby[3],forgiantplanets,4.5˘2.5,andformediumplanets,4.8˘1.3,are bothconsistentwiththevalueof13/3indicatingcurrentlyinwardlytidallymigratingplanets,butthe powerindexforsuperearthplanets,2.9˘0.4,istoolowtobefromongoingtidalmigration.Theslope ofthesuperearthplanetslikelyoriginatedwiththeirformation. WepresenttheevolutionofthefalloffasafunctionofQ1 forasummedrangeofstellaragesin ˚ figure1. Wecalculatetidalmigrationoperatingonaninitialoccurrencedistributionofasinglepower lawtoproduceadistributionwithafalloff,givingthetwo-powerlawdistributionthatresemblesthe actualdistribution.Theresult,asshowninfigure1issimilartothetwo-powerlawdistributionof[3]. HotPlanetsandCoolStars Figure2. Futurefallinratesforthethreerangesofplanetradii(panelsforlarge,medium,andsmallradii)each withplotsfortidaldissipationvaluesoflogpQ1 qvaluesof106.5 (dotted,top),107.0,107.5,108.0,and108.5. Rate ˚ offallincalculatedbasedonfit. 10 Figure3. RateoffallincalculatedbymodelingwhenKeplercandidateswouldfallin,withthesamerangeof valuesasinfigure2. Infigure1,wecomparefitsby[3]withfalloffscalculatedforarangeoftidaldistributions. We showtheoccurrencedistributionsof[3]forgiantandmediumradiiplanetsinfigure1,plottedagainst ourcalculationsofoccurrencedistributionssummedformassafterarepresentativemigrationtimeof 4.5Gyr,shownforseveralvaluesoftidaldissipationstrengthsQ1 . Thefalloffforgiantandmedium ˚ planetscouldbeinterpretedasgivingdifferentvaluesofQ1 fordifferentlysizedplanets,butitcould ˚ alsobeexplainedbymoregiantplanetsmigratinginwards.Thedifferenceappearstocorrespondwith the radii where the pile up of giant planets occurs, which suggests that the pile up is related to the higheroccurrenceoftheshortestperiodgiantplanets. Thisdifferenceappearsto bebestreconciled byalargerrate(flow)ofplanetsmigratingfromfurtherout. Thismayormaynotberelatedtoamore distant HEM flow of giant planets that [6] propose may have supplied at least a part of the pileup, becausedepletionofthepileupmaybesufficient. In the threepanelsof figure2 we showforthe three size ranges of planetsthe calculatedfuture infall rates plotted for stellar tidal dissipation values of Q1 from 106.5 to 108.5, using the fits of [3] ˚ as the initialoccurrencedistributions. We also show infallcalculateddirectlyusing data directlyin figure3,whereweseethesamediscrepancy,thoughtheresultisnoisy.ThecorrectvalueofQ1 isthe ˚ onethatgivesinfallratesthatdecreaseataratenofasterthanthesupplyofplanetsdecreasesasstars age. Theonlywaytomaintainthepresenceofmoregiantthanmediumplanetsintheshortestperiod rangeis if there isa largerinwardflow ofgiantplanetsthan formediumplanets. (Thiswouldonly start addingto the infall afterthe new planetsmigrate in). Arrivingplanetswouldproducea flatter curveforQ1 closerto107. ˚ Inward planet migration might cause pollution of the star if other planets orbits are disrupted. ThecorrelationbetweenstellarFe/H andeccentricitypresentedin figure3of [4]suggeststhathigh eccentricityplanetscould be associatedwith recentpollution. Forsystemsfoundby radialvelocity withperiodslessthan200days(exoplanets.org),wefindalessthan3%chancethatFe/Hvaluesfor starshostingplanetswithorbitsofeccentricityaboveandbelow0.35representthesamepopulation. EPJWebofConferences Figure4. Thedifference inplanet infall rateperyear asafunction of Q1 between theflowat 1and 10Gyr, ˚ shownforthreeradiiranges.Thisshowsthatthelikelyrateofinwardflowofgiantplanets(dotted)ishigherthan thatformedium(dashed)andsmaller(dot-dashed)planets. NolineisshownforQ1 valuesnotneedingaflow. ˚ Ittakesonlythisrateofinwardplanetflowtomaintainaconstantfutureinfallrate. 3 Small supply required – pileup depletion probably sufficient Wefindthattherateofplanetinfallrequiredtoprovidetheshortestperiodplanetpopulationissmall enoughto reasonablyexpect that depletion of the pileup or a migration of planets from furtherout couldsupplyenoughplanetstoexplainshortperiodplanetscontinuallyinfalling. Weshowtherates requiredinfigure4,wherewehavetakenthedifferenceintheratesofinfall(asinfigure2)at1and 10Gyr. Therateofthisflowonlyneedstobelessthan10´12 giantplanetsperyearperstarforthe fallinratestobeconsistent. The 40% smaller size in the pileup in the Kepler field, [3], compared to ground surveys that predominantlysample the solarneighborhood[9] couldindicatethatthe pileupdecreaseswith age. Thisdecreaseislikelylargeenoughto supplythesmallongoinginfallofplanetsmigratingintothe star. Itis importantto studywhetherplanetsin the pileupregionmightbe havingtheireccentricity pumpedup,ormightbedisruptedfromplanetsmigratingfromfurtherout,decreasingthenumbersof planetsinthepileup. Thefirstsignofstrongertidalmigrationcouldcomeinonlya fewyears,whenenoughtimehas elapsedtomeasuredecreasesintheshortestperiodplanets [1]. Animportantmeasureoftherateof planet infall will be observing transients from planet/star mergers, which will likely have sufficient luminositytobeobservablefromnearbygalaxiesbyupcomingorcurrenttransientsurveys[10,11]. References [1] Hamilton,D.P.,Nature460,1086,(2009) [2] Taylor,S.F.arXiv:astro-ph/1206.1343(2012a) [3] Howard,A.W.,Marcy,G.W.,&Bryson,S.T.etal.,ApJS201,15(2012) [4] Taylor,S.F.arXiv:astro-ph/1211.1984(2012b) [5] Penev,K.,Jackson,B.,Spada,F.&Thom,N.,Ap.J.751,96(2012) [6] Socrates,A.,Katz,B.,Dong,S.&Tremaine,S.,ApJ750,106(2012) [7] Dawson,R.I.,Murray-Clay,R.A.,&Johnson,J.A.,arXiv:astro-ph/1211.0554(2012) [8] Jackson,B.,Barnes,R.&Greenberg,R.,ApJ,698,1357(2009) [9] Marcy,G.,Butler,R.P.,Fischer,D.,etal.,PTPS158,24(2005) [10] Taylor,S.F.arXiv:1009.4221,(2010) [11] Metzger,B.D.,Giannios,D.&Spiegel,D.S.,MNRAS,425,27(2012)