On the Fate of Close-in Extrasolar Planets

arXiv:astro-ph/0209061v14Sep2002OntheFateofClose-inExtrasolarPlanetsIng-GueyJiang1,Wing-HuenIp1andLi-ChinYeh21InstituteofAstronomy,NationalCentralUniversity,Chung-Li,Taiwan2DepartmentofMathematics,NationalHsinchuTeachersCollege,Hsin-Chu,TaiwanReceived;accepted–2–ABSTRACTIthasbeenshownthatthereisapossiblemass-periodcorrelationforextrasolarplanetsfromthecurrentobservationaldataandthiscorrelationis,infact,relatedtotheabsenceofmassiveclose-inplanets,whicharestronglyinfluencedbythetidalinteractionwiththecentralstar.WeconfirmthatthemodelinP¨atzold&Rauer(2002)isagoodapproximationfortheexplanationoftheabsenceofmassiveclose-inplanets.Wethusfurtherdeterminetheminimumpossiblesemimajoraxisfortheseplanetstobedetectedduringtheirlifetimeandalsostudytheirmigrationtimescaleatdifferentsemimajoraxesbythecalculationsoftidalinteraction.Weconcludethatthemass-periodcorrelationatthetimewhentheseplanetswerejustformedwaslesstightthanitisnowobservediftheseorbitalmigrationsaretakenintoaccount.Subjectheadings:celestialmechanics–planetarysystems–solarsystem:formation–solarsystem:general–stellardynamics–3–1.IntroductionThenumberofdiscoveredextrasolarplanetsisincreasingquicklyduringrecentyears.AccordingtotheExtrasolarPlanetsCatalogmaintainedbyJeanSchneider(),inMay2002,thereareabout77extrasolarplanetsaround69mainsequencestars.Theseplanetswithmassrangefrom0.16to17Jupitermasses(MJ)havesemimajoraxesfrom0.04AUto4.5AUandalsoawiderangeofeccentricities.Interestingly,thereisaplanetmovingonanextremelyelongatedorbit(e=0.927)aroundthesolar-typestarHD80606(Naefetal.2001).Theseexcitingdiscoveriesprovidegreatopportunitiestounderstandtheformationandevolutionofplanetarysystems.Forexample,Jiang&Ip(2001)showedthattheinteractionwithdiscisimportanttoexplaintheoriginalorbitalelementsduringtheplanetaryformation.Yeh&Jiang(2001)analyticallyshowedthatthescatteredplanetsshouldingeneralmoveonaneccentricorbitandthustheorbitalcircularizationmustbeimportantforscatteredplanetsiftheyarenowmovingonnearlycircularorbits(SeeJiang&Yeh2002a,Jiang&Yeh2002bforthefollowingup).Inadditiontothedynamicalstudies,Tabachnik&Tremaine(2002)usedthemaximumlikelihoodmethodtoestimatethemassandperioddistributionsofextrasolarplanetsandfoundthereisamass-periodcorrelation,buttheyattributedtheirfindingtotheobservationalselectioneffect.However,Zucker&Mazeh(2002)claimedthatthismass-periodcorrelationcannotbecompletelyexplainedbytheobservationalselectioneffect.TheydidsomeMonteCarlosimulationsandshowtherealdependencybetweenthemassandperiodofextrasolarplanets.Thismass-periodcorrelationgivesthepaucityofmassiveclose-inplanets.Sincetheyaresupposedtobetheeasiesttodetect,Zucker&Mazeh(2002)saidthispaucitywas–4–unlikelytobetheresultofanyselectioneffect.P¨atzold&Rauer(2002)havereportedthepossibleexplanationabouttheabsenceofmassiveclose-inplanetsbytidalinteraction.Theydefined“criticalmass”tobethemaximummassthattheplanetcanhaveandsurviveunderthetidalinteractionfromthecentralstarforagivenparticularsemimajoraxis.Theydeterminedthecriticalmassasfunctionofsemimajoraxisforsomeassumedstellardissipationfactorsandtheagesoftheplanetarysystems.Theirresultsshowedthatmostplanetarysystemsarelocatedatthepermittedregionofthe“criticalmass-semimajoraxis”plot(theirFigure3)excepttheτBoosystem,whichneedsmorecarefultreatmentfortheassumedparametervalues.However,iftheseplanetscouldbeformedabitfartherfromthecentralstarinitially,theyshouldstillsurviveunderthetidalinteractionandthusmightbedetectedduringtheinwardmigration.Oneshouldkeepinmindthatthelocationwheretheplanetsaredetectedarenotwheretheyareformed.Theplanetsfromfartherplacecouldmigrateinwardtotheregionclosertothecentralstarandprobablyhavechancestobedetectedbyus.Tofurtherinvestigatethisproblem,wecarefullystudytheplanetarymigrationduetotidalinteraction.WetrytoincludetheeffectoforbitaleccentricityatthebeginningandweconfirmthatthatthemodelusedinP¨atzold&Rauer(2002)isagoodapproximation.WethususethesimilarmodelinP¨atzold&Rauer(2002)fortherestcalculations.WedescribeourbasicmodelsfortidalinteractioninSection2andtheresultswillbeinSection3.WeprovideconcludingremarksinSection4.2.TheModelsforTidalInteractionAtideisraisedonthecentralstarbytheclose-inplanetbecausetheforceexperiencedbythesideofthecentralstarfacingtheplanetisstrongerthanthatexperiencedbythe–5–farsideofthecentralstar.Weconsiderbelowthemodelsforplanetsonbothcircularandeccentricorbits.2.1.CircularOrbitsIftheclose-inplanetismovingonacircular,equatorialorbit,accordingtothetidalpotentialtheory,thisplanetwouldchangeitsorbitfollowingbelowformula:dadt=sign(Ω−n)3kQmM(Ra)5na,(1)whereaisthesemimajoraxis,tisthetime,Ωistherotatingangularspeedofthecentralstar,kisthestellarLovenumber,Qisthetidaldissipationfunction,mistheplanetarymass,Misthemassofthecentralstar,Risthecentralstar’sradiusandnistheorbitalmeanmotionwhichisdeterminedbyn=sG(M+m)a3.(2)Wesetk=0.2(Murray&Dermott1999)andtakeQ=3.0×105(theaveragevalueinP¨atzold&Rauer2002).Theaboveformulaprovidesagoodsimpletooltostudythetidalorbitaldecayforclose-inplanets.Ho