The mass-period distribution of close-in exoplanets

3.7 MB·English

by  P. Benitez-Llambay

Astronomy&Astrophysicsmanuscriptno.15774 (cid:13)c ESO2011
January19,2011
The mass-period distribution of close-in exoplanets
P.Ben´ıtez-Llambay1,F.Masset2,andC.Beauge´1
1 ObservatorioAstrono´mico,UniversidadNacionaldeCo´rdoba,Argentina
2 InstitutodeCienciasF´ısicas,UniversidadNacionalAuto´nomadeMe´xico,ApdoPostal48-3,62251Cuernavaca,Mor.,Me´xico⋆⋆
Preprintonlineversion:January19,2011
ABSTRACT
1 Context.ThelowerlimittothedistributionoforbitalperiodsPforthecurrentpopulationofclose-inexoplanetsshowsadistinctive
1 discontinuitylocatedatapproximatelyoneJovianmass.Mostsmallerplanetshaveorbitalperiodslongerthan P ∼ 2.5days,while
0 highermassesarefounddowntoP∼1day.
2 Aims.Weanalyzewhetherthisobservedmass-perioddistributioncouldbeexplainedintermsofthecombinedeffectsofstellartides
andtheinteractionsofplanetswithaninnercavityinthegaseousdisk.
n
Methods.Weperformedaseriesofhydrodynamicalsimulationsoftheevolutionofsingle-planetsystemsinagaseousdiskwithan
a
innercavitymimickingtheinnerboundaryofthedisk.Thesubsequenttidalevolutionisanalyzedassumingthatorbitaleccentricities
J
aresmallandstellartidesaredominant.
8
Results.We find that most of the close-in exoplanet population isconsistent with an inner edge of the protoplanetary disk being
1 locatedatapproximatelyP&2daysforsolar-typestars,inadditiontoorbitaldecayhavingbeencausedbystellartideswithaspecific
] ttyidpaelIpIamraimgreatetiroonn,ctrhoessoirndgerthoefgQa′∗p≃,a1n0d7fi.Tnahlelydhataaltiisngbraotatdhleyicnotenrsiiosrte2n/t1wmitehanp-lamnoettisomnorerseomnaansscievewtihthanthoendeisJkupeidtegrem.Samssaullnedreprlgaonientgs
P
do not open a gap in the disk and remain trapped in the cavity edge. CoRoT-7b appears detached from the remaining exoplanet
E
population,apparentlyrequiringadditionalevolutionaryeffectstoexplainitscurrentmassandsemimajoraxis.
.
h
p
-
o 1. Introduction
r
st Close-inplanets(semimajoraxisa < 0.1AU)constituteaspe- 8
a cial subset of the exoplanetary population. Since it is unclear
[
whetherin-situformationoccurs,thecurrentorbitalandphysical

1 characteristicsoftheseplanetsprovideimportantconstraintson 4
v theirpastevolutionandformationprocess.Severalmechanisms d]
5 havebeenproposedtoexplainthepile-upofhotplanetswitha d [
o
4 three day orbital period, including a truncation of the gaseous eri 2 GJ876d
P
5 diskbythestar(Linetal.1996,KuchnerandLecar2002),plan-
GJ1214b
3 etaryscatteringcombinedwithKozairesonanceandtidalcircu-
.
1 larization(Nagasawaetal. 2008),planetaryevaporation(Davis 1
0 and Wheatly 2009), and tidal interactions with the parent star CoRoT-7b
1 (Jacksonetal.2009).
1 In particular, Kuchner and Lecar (2002) suggested that a
0.5
: giant planet in circular orbit could halt its migration when its 0.01 0.1 1 10
v
mass [M ]
i orbital period was half that of the inner edge of the disk. In Jup
X
this configuration,allthe planet’scircular Lindbladresonances
Fig.1. Distribution of orbital periods and planetary masses for
r wouldlie in theinnercavity(IC)andnofurtherinterchangeof
close-in exoplanets with orbital period P < 12 days. Data
a
angularmomentumwouldtakeplace.Masset etal. (2006)per-
fromhttp://exoplanets.eu.Blackcirclesdenoteplanetswithboth
formedaseriesofhydrodynamicalsimulationstofollowtheevo-
Doppler and transit data, while gray circles mark bodies with-
lutionoflow-massplanetsindisksincludinganIC.Theyfound
outdetectedtransits.Emptysquarescorrespondtoplanetswith
thatallbodiesmigrateduntilreachingapointslightlyexteriorto
retrogradeorbitswithrespecttothestellarspin.
thecavityedge,wheretheywereeffectivelytrappedina stable
configurationin almostcircular orbits. Althoughthis resultap-
pearsdifferentfromthatpredictedbyKuchnerandLecar(2002),
eachis valid,as we shallsee, fora differentrangeofplanetary
masses. al.(2007)andDavisandWheatley(2009)extendedtheanalysis
The first reference to a possible correlation between mass toalargertransitingpopulation,findingasimilarresultalthough
and orbital periodfor close-in planetswas proposedby Mazeh with a much broader dispersion. They proposed that smaller
etal.(2005)foronlysixtransitingbodies.Theyfoundthatboth planetsclosertothestarmighthavebeenlostbecauseofevap-
parameters seemed to follow a linear law, with more massive oration,similartothatcurrentlyongoingatleastinHD209458b
bodiesbeing located at smaller semimajor axes. Southworthet (Vidal-Madjaretal.2003)andWASP-17(Andersonetal.2010).
2 P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets
Jacksonetal.(2009)alsoanalyzedthedistributionofclose-
in planets, this time focusing on the correlation between the
semimajor axis of the planet and the age of its star. They
foundthat the lower limit of the semimajor axis was lower for
youngerstars, which impliesthattidal effectscouldbe respon-
sible. Exoplanets with very short orbital periods in older stars
would have had enough time to be tidally disrupted, thus they
wouldonlybepresentlyobservableinrelativelyyoungsystems.
In this paper, we revisit the mass-period distribution, tak-
ingadvantageoftherecentincreaseintheexoplanetpopulation.
Figure1showstheorbitalperiodsP,asafunctionofthemassm,
forthe knownpopulationofexoplanetswith P ≤ 12days(137
planets).Blackcirclescorrespondtocasesforwhichbothtran-
sitsandDopplerdataareavailable;bodieswithoutdetectedtran-
sits are shown in gray. Exoplanets in apparent retrograde mo-
tion with respect to stellar rotation are identified by an empty
square. These are WASP-8b (Queloz et al. 2010), WASP-17b
(Anderson et al. 2010), WASP-33b (Collier Cameron 2010),
Hat-P-7b(Winnetal.2009),WASP-2b(Triaudetal.2010),and
WASP-15bandWASP-17b(Triaudetal.2010).Althoughitmay
bearguedthatthesebodiesarenotconsistentwithplanetarymi-
gration(Triaudetal.2010),theymayalsopointtowardsprimor-
dial spin-orbit misalignment and not be related to subsequent
orbitalevolutionoftheplanets(Laietal.2010). 1 x 10-4
The distribution exhibits a noticeable “step”, exoplanets Planet
9 x 10-5
largerthanoneJupitermass(M )appeartohavealowerinner
Jup
boundary(downto∼1day)whileform< MJupthedistribution 8 x 10-5
seems restricted to larger values of P. The only exceptionsare
threebodiesintheSuper-Earthrange,CoRoT-7b,GJ1214b,and 7 x 10-5
GJ876d,whichareallmarkedintheplot.Ofthese,thelattertwo
plalranmeatsssbeesl,ornegsptoecltoivwe-lmy)a,stshsutsarcso(nMst∗itu=te0.s1p7ecainadlcMas∗e=s.0C.o3R2osTo-- Σ(r) 6 x 10-5
5 x 10-5
7b, however, belongs to a solar-type star (Rouan et al. 2009).
Thisplanethasaveryshortorbitalperiod(∼0.85days)butalso
4 x 10-5
averylowmass(m∼0.015M ),anddoesnotseemtocomply
Jup
withtherestoftheexoplanetdistribution.Inparticular,Jackson 3 x 10-5
et al. (2009)pointedoutthatCoRoT-7b couldreachthe Roche
radiusontimescalesof107−109years,dependingonthevalue 2 x 10-5
ofthespecifictidalparameterQ′.
Regardlessoftheseisolated∗cases, thereseemstobea very 1 x 10-5
0.5 1.0 1.5 2.0 2.5 3.0 3.5
cleardiscontinuity(orbump)inthemass-perioddistribution,lo-
r
catedatapproximatelym= M .Moreover,forhighmassesthe
Jup
Fig.2.Snapshotofthesurfacedensityprofileofoneofoursim-
lowerlimitinorbitalperiodsappearsveryclosetoa2/1mean-
ulations. The inner cavity is centered around r = 1.8. In the
motionresonancewiththediskedgeforsmallplanetarybodies.
top frame, the low-density regions are shown in black, while
This appears consistent with a scenario in which the planetary
the high-density regions are shown in white. The planet (m =
trapsproposedbyMassetetal.(2006)woulddominatethelow-
0.1M )islocatedonthex-axis.
massregion,whilethemechanismofKuchnerandLecar(2002) Jup
wouldbemainlyresponsiblefortheupperendofthemassspec-
trum.
mechanismsisconsistentwiththeobserveddistributionofclose-
Themainobjectiveofthisstudyistotestwhetherthecom-
in planets. Section 3 is devoted to the subsequent evolution of
bined action of planetary traps in the gaseous disk plus subse-
exoplanetsunder the stellar tide and their effects on any initial
quent tidal interactions with the parent star could explain the
disk-drivendistribution in the mass-period diagram.In Section
observeddistributionofclose-inexoplanets.Sinceexoplanetsin
4, we analyze the case of the CoRoT-7 planetary system and
retrogradeorbitsshouldhaveexoticdisk-planetandtidalevolu-
presentpossibleexplanationsofthepresentlocationofCoRoT-
tions, thestudy ofthese exosystemsis beyondthe scopeof the
7b.Finally,conclusionsclosethepaperinSection5.
present work, and we focus mainly on bodies believed to have
orbitalmotioninthesame directionasthestellar spin.Evenin
thiscase,weassumeazeroinclinationwithrespecttothestellar 2. Hydrodynamicalsimulations
equator.
In Section 2, we present a series of hydrodynamicalsimu- 2.1.Initialconditions
lations adopting different planetary masses and analyzing the Our simulations were carried out using the FARGO1 hydro-
relative halting distance from the central star. Not only are we
code(Masset2000a,b)withacentralstarofonesolarmass.We
interested in seeing whether such a hybrid and mass-selective
processispossible,butalsowhethertheboundarybetweenboth 1 Seehttp://fargo.in2p3.fr
P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets 3
place for the planet, as long as its mass is sufficiently low to
2.4
avoiddisruptionofthedensity(andtorque)profile.
2.1 m=0.1 MJup
2.2.Testsimulationsofplanettrapping
IC
1.8 Afterthediskdensityprofilehadstabilized,weincludedaplanet
r ofmassminaninitiallycircularorbitatr = 2.5r0.Thesystem
m=10 MJup was allowed to evolve until the planet reached a stationary so-
1.5 lutionandtheorbitaldecayeffectivelystopped.Figure3shows
theresultsoftworuns,thefirstwithm=0.1M andthesecond
Jup
withm=10M .ThelocationofthecenteroftheICismarked
1.2 2/1 MMR Jup
bythetophorizontaldashedline,whilethelowercorrespondsto
theinterior2/1mean-motionresonance(MMR)withthecenter
0.9 oftheIC.
100 1000 10000 1e+05
time The smaller planet suffers an almost constantorbital decay
untilitishaltedatalocationslightlyoutsidethecavity,inaccor-
Fig.3. Evolution of semimajor axis of two one-planetsystems
dance with the findingsof Masset et al (2006).The more mas-
migratingandhaltingnearthediskinnercavityedge(IC).Black
sive planet, however, suffers a Type II migration that opens a
curvecorrespondstom=0.1M andgraytom=10M .The
Jup Jup gapin its co-orbitalregionand completelydisruptsthe density
IC is centeredat r = 1.8, while its interior2/1 mean-motion
IC profile of the disk in its vicinity. The cavity edge is therefore
resonance(MMR)islocatedatr ≃1.13.Timeisinorbitalperi-
incapable of generating a strong corotation torque and fails to
ods.
traptheplanet,whichcontinuesitsorbitaldecayinsidethecav-
ity.Themigrationisfinallystoppedveryclosetothelocationof
considertwo-dimensional,locallyisothermaldisks.Theunitof the2/1MMRwiththeICwherethedifferentialLindbladtorque
length, r , is arbitrary (our simulations are scale free and can reducestozero,aspredictedbyKuchnerandLecar(2002).
0
berescaledtodisksofarbitrarysizes),butitcanbeconsidered, Inallcases,theeccentricitiesweredamped,andthefinalor-
for our purpose, that r0 ∼ 0.01 AU. Our disks, prior to a 1D bitswereobservedtobequasi-circularwithe∼10−3.Thisfinal
relaxation dedicated to the creation of the IC, have a uniform eccentricityisdifferentfromtheresultspresentedbyRiceetal.
surfacedensityΣ = 10−4 M⊙.r0−2,whichtranslates,inourcase, (2008), where a significant eccentricity excitation is observed
toΣ∼900g.cm−2.Theasymptoticradiusofourplanetsisvirtu- forTypeIImigrationinsidetheinnerdiskedge.Fromwhatwe
allyindependentofthisvalue,whichessentiallyprescribeshow havebeenable to see, the discrepancyarisesfromthe different
longittakesforthemtoreachit.Inasimilarmanner,thekine- numericalsetup.Riceetal(2008)truncatethediskatthemesh’s
maticviscosityofourdisksisuniformoutsidethecavity,andis inner edge, and the planets that they consider essentially orbit
set to ν = 10−5r2Ω−1, where Ω is the orbital frequency at ra- in a vacuum.This isin contrastto oursituation in whichsome
0 0 0
diusr .Someinitialtestrunswereperformedwithothervalues residual material surrounds the orbit. This material, by the ac-
0
of Σ and ν with no significant change in the results, except for tion of co-orbitalLindbladresonances, is a powerfulsource of
themigrationtimescale.Thediskaspectratioissetuniformlyto eccentricitydamping.
H/r = 0.05,hencethe valueof the α parameterthatcharacter-
izes turbulence (Shakura & Sunyaev,1973) just outside the IC
(seebelow)isα∼3·10−3. 2.3.Scalingandcomparisonwithobservations
Ourdisksaredescribedonapolargridwith384zonesinaz-
Werepeatedthesimulationsforatotalof18runs,withplanetary
imuthand306zonesinradius.Theinnerradiusofthemeshisat
massesrangingfrom0.01to20Jupitermasses,allstartingfrom
0.2r ,whiletheouterradiusisat3.5r .TheICwasgeneratedus-
0 0 initial circular orbits at r = 2.5r . The orange curve in the top
inganadhocstepinkinematicviscosityaroundr=1.8r ,using 0
0 plot of Figure 4 presents the final orbital period as function of
thesamerecipeasdescribedinMassetetal.(2006),adoptinga
the planetary mass. For comparison,the observationaldata are
value of F = Σ /Σ = 10 for the ratio of the surface densi-
o i againdrawnincircles.
tiesoutsideandinsidethecavity.WeobtainarathersharpICof
width∆r ≃ 0.5r .Boundaryconditionswerechosentobenon- SincethelocationofourICwasinarbitraryunits,we have
0
reflectingfortheinneredgeandsuchthattherewasacontinuous a degree of freedom in the sense that we may shift the orange
outer source mass to maintain the surface density in the exter- curvearbitrarilyinthey-axistofittheobservedexoplanets.We
nal regions of the disk (Masset et al. 2006 used non-reflecting findthatthebestfitgivesaninnercavityedgelocatedatapprox-
boundaryconditionsforbothedges).Noplanetwasconsidered imatelya ∼ 0.03AU correspondingtoanorbitalperiodof∼ 2
atthispoint. days.Thisvalueiscomparablewiththeinferredinnergasradii
An initial one-dimensionalrun was performedto allow the ofTTauristar disksfromCO spectroscopy(Najitaetal. 2007,
cavity to form and reach a steady state configuration. The re- Carr 2007)which seems to be located around0.03−0.04AU.
sultingdensityprofileandtotaltorquewereanalogoustothose Although our best-fit distance is slightly lower, any difference
showninMassetetal.(2006),showingtheexistenceofastable couldbeduetostellarparametersandthemaximumCOveloc-
fixedpointinthetorqueatadistancer∗ slightlylargerthanthe ity(Carr2007).
nominaledge of the IC (Figure 2). A planet located outside r∗ ExoplanetswithmassesbelowoneJovianmassseemtohave
wouldfeelanegativetorqueandsufferanegativeorbitaldecay alower-limitorbitalperiodofabout2−3days,consistentwith
towardsthe star. Conversely,a bodyplaced ata radialdistance the position of the equilibrium point being slightly beyond the
slightlysmallerthanr∗ wouldfeelapositivetorqueandanout- IC(Massetetal.2006).Ontheotherhand,theorbitalperiodsof
wardradialmigration.Thus,thepointr = r∗ isastablenesting moremassiveplanetshavealowerlimit,thistimeinaccordance
4 P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets
(seeFerraz-Melloetal.2008,Jacksonetal.2009).Thisexpres-
8 sioncanbeeasilyintegratedtoyield
117 G 1/2 R5M
4 a(t)13/2 = ∗ p t+a13/2, (2)
d [d] " 4 M∗! Q′∗ # 0
erio 2 where a0 = a(t = 0) is the initial value of the semimajor axis.
p Aftersomesimplealgebraicmanipulations,wecanexpressthis
solutionas
1
a¯(t)13/2 =a¯13/2−βt, (3)
0
0.5
0.01 0.1 1 10 where
mass [M ]
Jup 117 G 1/2 M
0.1 β= R 5 p (4)
0.08 4 M⊙! ⊙ Q′∗
0.06 and
M 1/13 R 10/13
0.04 a¯ =a ∗ ⊙ . (5)
a M⊙! R∗!
Equation(3)isthenindependentofthestellarparameters,whose
0.02 valuesareincorporatedintothe“normalized”semimajoraxisa¯.
We note that the rate of orbital decay is given by β, which is
linearly proportional to the planetary mass and inversely pro-
portionalto the stellar dissipation parameter Q′. Is it of course
0.01 ∗
0.01 0.1 1 10 probablethatthetidalparameteritselfdependsonthestellarpa-
mass [M ] rameters(seeBarkerandOgilvie2009),buttherehassofarbeen
Jup
noclearindicationofhowitmayvarynorbywhatmagnitude.
Fig.4. Gray circles reproduce the mass-period distribution of
The lower plot in Figure 4 shows the distribution of a¯ as
known close-in exoplanets. The broad orange band represents
a function of m for the close-in exoplanet population. The or-
the finalsemimajorobtainedfromtheruns,scaled alongthe y-
ange curve marks the position of the simulated lower limit to
axisto fit the observationaldata. In the lowerframe,a¯ denotes
thesemimajoraxis.Asexpected,bothGJ876dandGJ1214bare
thenormalizedsemimajoraxisdefinedinEq.(5).
nowabovethelowerlimit,althoughCoRoT-7bstillremainsde-
tached.We notethatthebumpinthedistributionisstillclearly
visible.
We cannowuseequation(3)to checkwhetherthepresent-
withthelocationofthe2/1mean-motionresonancewiththeIC,
day distribution could originate solely from tidal evolution. To
aspredictedbyKuchner&Lecar(2002).
test this idea, we generated a synthetic population of 104 ficti-
tious exoplanetsdistributed randomlyin planetary mass across
theintervalm ∈ [0.01,12]M andwithrandomstellar agesT
Jup
3. Tidalevolutionofclose-inexoplanets betweenoneandeightGyr.Thenormalizedsemimajoraxisa¯ of
each planet was then evolved tidally for the age of the system
Althoughthe resultspresentedin the topgraphofFigure4 ap-
usingastellartidalparameterQ′ =106.
peartobeencouraging,therearetwoimportantapproximations ∗
Figure5showsresultsconsideringthreedifferentinitialdis-
thatmustbeexamined.First,theexoplanetsorbitstarswithdif-
tributions,chosentobeuniforminorbitalperiodPandlog(m).
ferent radii and masses, and we have assumed that the scaling
In all cases, the lowest mass was equal to m = 0.01M , and
in the y-axis is the same. Although we have not assumed any Jup
thelargesttotenJovianmasses.Sincethispreliminaryanalysis
originfortheinnercavity(MRI,stellar wind,etc.),itisalmost
is only intended to be illustrative, we assumed solar-type stars
certain that the location of the IC should be a function of the
forallbodies;consequently,thenormalizedsemimajoraxisa¯ is
stellar mass. If we assume a very simple model in which r
IC equaltothenominalsemimajoraxisa.
scales with R , then the position of both GJ1214band GJ876d
∗ Forthetopplots,we assumedfictitiousplanetswith orbital
wouldbedisplacedabovethebroadorangecurve,thuseliminat-
periodsthathavenolowerlimit.Theinitialdatasetisshownin
ing their incongruitywith respect to the simulations. However,
theleftplot,whiletheright-handplotpresentsthefinaldistribu-
CoRoT-7 has a stellar mass of 0.93M and its location in the
⊙ tionafterevolutionthroughstellartidesforT =8Gyr.Sincethe
diagramwouldstillbeconflicting.
decayrateβisproportionaltotheplanetarymass,moremassive
A second approximation is that we have neglected the
planetsfallmorerapidlytowardsthestar,leadingtoafinalpop-
later evolution of the planetary periods caused by stellar tides.
ulationthathasalackoflargebodieswithsmallorbitalperiods.
Assumingalmostcircularplanarorbitsfortheclose-inplanets,
This distribution shows little relation to the real distribution of
wecanneglecttheplanetarytidesandapproximatethedifferen-
planets.
tialequationfororbitaldecayas
Forthe middleplots,weconsideredalowerlimittotheor-
bitalperiodsequalto3days.Thisvalueissufficientlylargefor
da =− 9 G 1/2 R5∗Mp a−11/2 (1) terrestrial-type bodies to be virtually unaffected, although gi-
dt "2 M∗! Q′∗ # ant planets still suffer significant orbital decay. Depending on
P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets 5
original today 0.10
0.08
77
8 8 QQ** == 1100
00..0066
d] 4 4 0.04
d [ a
o 2 2
eri
p 0.02
1 1
0.5 0.5 0.01
0.01 0.1 1 10 0.01 0.1 1 10 0.01 0.1 1 10
mass [M ]
Jup
8 8
0.10
0.08
6
4 4 Q* = 10
d] 00..0066
d [
erio 2 2 0.04
p a
1 1
0.02
0.5 0.5
0.01 0.1 1 10 0.01 0.1 1 10
0.01
8 8 0.01 0.1 1 10
mass [M ]
Jup
4 4
d] 0.10
d [ 0.08 5
o 2 2 Q* = 10
eri 00..0066
p
1 1 0.04
a
0.5 0.5
0.01 0.1 1 10 0.01 0.1 1 10
mass [M ] mass [M ] 0.02
Jup Jup
Fig.5. Tidal evolution of fictitious planets (stellar tides alone)
with Q′ = 106.Initialdistributionsareshownontheleft,while 0.01
∗ 0.01 0.1 1 10
the evolved distributions are shown on the right. Stellar ages
mass [M ]
wherechosenuniformlybetweenoneandeightGyr. Jup
Fig.6. Distribution of close-in exoplanets in the a¯-m diagram.
Graycurvesshowtheendevolutionduetostellartides(atT =8
Gyr) of initial constant values of a¯. Each curve correspondsto
adifferenta¯ .Stellartidalparametersaregiveninthetopright-
theirinitialsemimajoraxisandstellarage,manymassiveplan- 0
handsideofeachgraph.Thebroaddashedcurveinthetoppanel,
ets reach the Roche radius and are engulfed by the star, but a
correspondingtoa¯ =0.035,givesthebestfittothelowerlimit
portionofthepopulationremains.Wenotethefinalstep-likedis- 0
ofthecurrentexoplanetpopulation.
tribution,whichisreminiscentoftherealpopulation.However,
thebumpisnowlocatedat∼0.3M ,thusatalowermassthan
Jup
fortherealexoplanets.
areshowninFigure6forthreevaluesofthestellartidalparam-
Finally,inthelowerplotsofFigure5weconsideredanini- eter.Eachgraycurveshowsthefunction
tial population with a step in the orbital period, similar to that
resultingfromthehydrodynamicalsimulations.Eventhoughthe 2/13
initialconditionsaredifferentfromthosedepictedinthemiddle a¯ = a¯13/2−βT with T =8Gyr (6)
0 !
plots, there is no significant difference in the final distribution.
Thisseemstoindicatethatanyevidenceofaninitialdisk-related fordifferentvaluesofa¯ .Forcomparison,eachplotalsorepro-
0
structurewouldbesmearedbythelatertidalevolution.Thus,a ducesthepresentdistributionofclose-inexoplanets.
bumpinthepresentrealpopulationisnotnecessarilyindicative The present-day lower limit to the normalized semimajor
of a dynamicalstructure priorto the dissipation of the gaseous axis shows a very good agreement assuming a¯ ≃ 0.035 and
0
disk. a tidal evolution with Q′ = 107, represented by a bold dashed
∗
Theseresultsindicatetheclearpossibilitythattheobserved curveinthetopgraph.Thisvalueofthenormalizedsemimajor
bumpin the exoplaneta¯-m distributioncould be mainly due to axiscorrespondstooriginalorbitalperiodsof P ∼ 2.5daysfor
stellartidaleffects.Totestthisproposalinmoredetail,wecal- solar-type stars. We note that the other plots, correspondingto
culatedhowalowerboundinthevaluesofa¯ wouldbemodified, smallervaluesof Q′ donotshowgoodcorrespondenceforany
∗
fordifferentvaluesofQ′,afteratimescaleofT =8Gyr.Results adoptedvalueofa¯ .Thisresultappearstoindicatethatsmaller
∗ 0
6 P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets
today original (Q* = 107) theSimbaddatabaseforstellarpropertiesandfromJacksonetal.
0.1 0.1 (2009).
0.08 0.08
Thepresent-daya¯-mdistributionforthisreducedpopulation
0.06 0.06
is shown as black circles in the left-hand plot of Figure 7. For
0.04 0.04 comparison,graycirclesshowthoseplanetsforwhichcomplete
a a stellar data is currently unavailable. The location of the lower
boundtothesemimajoraxisdeducedfromthehydrosimulations
0.02 0.02
is shown as a bold dashed curve.Unfortunately,manysystems
with giantplanetswith a¯ < 0.025do nothavecompletestellar
0.01 0.01 properties,andthebumpinthedistributionisnotclearlyvisible
0.01 0.1 1 10 0.01 0.1 1 10
mass [M ] mass [M ] forthesmallerpopulation.
Jup Jup
Theright-handsideplotofFigure7showsthe“original”lo-
Fig.7. Left: Current-day distribution of exoplanets in the a¯-m cationofthesmallerpopulationafterthebackwardsintegration
diagram.Blackcirclesshowplanetsforwhichvaluesof M∗,R∗ for each stellar age. For comparison, the inner-cavity-induced
andstellaragesT∗areavailable.Graycirclesshowthosebodies lower limit is again shown as a bold dashed curve, although
for which stellar data is missing. Right: Reconstructed origi- the scaling in the y-axis has been modified to fit the new val-
naldistributionassumingbackwardorbitalevolutionfromstel- ues of semimajor axis. Although the distribution still shows a
lartidesfortheageofeachparentstar.Inbothplots,thedashed goodagreementforplanetarymassesuptom ∼ 1M ,mostof
Jup
curvesshowthelowerlimittonormalizedsemimajoraxis,asa the higher masses have increased their semimajor axis beyond
functionoftheplanetarymass,obtainedfromthehydrodynami- a¯ ∼0.03,andlittleevidenceremainsofthebump.
calsimulationsandscaledalongthey-axistofittheexoplanets. Apossibleexplanationcouldbethatmostgiantplanetswith
smallvaluesofa¯ weresubsequentlylostduetotidaldisruption.
0.1 0.1 Tocheckthishypothesis,theleftplotofFigure8showsthere-
0.08 0.08 lation between the current values of the normalized semimajor
0.06 0.06 axis as a functionof the stellar age. Althoughplanets with rel-
0.04 0.04 ativelylargesemimajoraxesexistforallvaluesofT∗,itseems
a a veryclear thatgiantplanetswith a¯ < 0.025,which areprimar-
T = 8 Gyr
* ilyresponsibleforthebumpinthedistribution,belongtoyoung
0.02 0.02
systemswithamaximumstellarageofT ≃3Gyr.
T = 1 Gyr ∗
* From Eq. (3), it is possible to estimate, for each planetary
0.01 0.01 mass, the critical value of the normalized semimajor axis (i.e.
0 3 6 9 12 0.01 0.1 1 10
stellar age [Gyr] mass [MJup] a¯∗0) that falls towards the star of a given stellar age T∗. This is
givenapproximatelyby
Fig.8. Left: Current values of the normalized semimajor axis
a¯∗ =(βT )2/13, (7)
a¯ for close-in exoplanets as a function of the stellar age T∗. 0 ∗
Notice that bodieswith a¯ < 0.03 are limited to youngsystems whichissimply obtainedbysettingthe finalsemimajoraxisto
(T∗ <3Gyr),whilelargersemimajoraxesarefoundforallstellar beequaltozero.Althoughtheplanetisbelievedtodisruptupon
ages.Right:Originaldistributioninthem-a¯ diagram.Diagonal reaching the Roche radius and, thus, before impacting the star
curvesgive limits to the orbitaldecay leadingto disruptionfor itself,asshownbyJacksonetal.(2009)theorbitaldecayatsuch
twostellarages:thecontinuouslineshowresultsforT∗ =1Gyr, smallsemimajoraxesissoswiftthatthetimescalesforbothsce-
whilethedashedlinecorrespondstoT∗ =8Gyr. nariosarepracticallyequal.
The right-hand plot of Figure 8 once again reproduces the
“original”distributionofexoplanetsin the m-a¯ diagramshown
valuesofQ′,andconsequentlyfasterorbitaldecaysduetostel- beforeinFigure7.Thediagonallinesshowthevaluesofa¯∗,asa
∗ 0
lartides,areinconsistentwiththecurrentpopulationofclose-in functionoftheplanetarymass,fortwostellarages:T = 1Gyr
∗
exoplanets. (continuousline) and T = 8 Gyr (dashed line). It is clear that
∗
Althoughthetimescale(T =8Gyr)chosenforFigure6may evenifgiantplanetsweredepositedbyahypotheticaldiskinner
seemarbitrary,equations(4)and(5)showthatthemostrelevant cavity at small values of the semimajor axis (a¯ < 0.025), they
parameter for the orbital evolution is actually the ratio T/Q′. would be rapidly absorbed by the parent star because of tidal
∗
Sincetheuncertaintyinthetidalparameterismuchlargerthan effects, even for planetary systems as young as 1 Gyr. Thus, it
in theage ofthe system, itseemsjustified to use a single fixed is not unexpected that if any original bump in the distribution
value of T and to assume that no qualitative differenceswould werecreatedbydisk-planetinteractions,subsequenttidaleffects
beobservedforotherstellarages. wouldhaveeliminatedmosttraces.
As a final test, we can attempt to reconstruct the original
mass/semimajor-axis distribution of the exoplanet population,
4. CoRoT-7b
integratingeachcurrentvalueofa¯backwardsintimeusingequa-
tion(6),adoptinginthiscasethevalueofT equaltotheageof CoRoT-7 appears to be a special case. As seen from the lower
each parent star T . To perform this calculation, we require in plotofFigure4,itslowmassandshortorbitalperiodmeanthat
∗
additiontothestellarmass M andradiusR ,estimatesofeach itiswellseparatedfromthem-a¯ distributionobservedforother
∗ ∗
stellarageT .However,thisinformationisnotavailableforall close-inexoplanets.Werecallthatthisplanetorbitsasolar-type
∗
planetarysystems.Outofthe133originalplanetsfromourdata starofmassM ≃0.93M .
∗ ⊙
setin directorbits(i.e.eliminatingthebodiesbelievedto be in Jackson et al. (2009) proposed that these planetary bodies
retrogrademotion),wewereonlyabletoobtainafullsetofstel- shouldbe undergoingsignificanttidalevolutionandorbitalde-
larparametersfor94exoplanets.Stellardatawereobtainedfrom cay.Mostbodiesinthismassrangewouldthenbeabsorbedby
P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets 7
thestarontimescalesshorterthantheageofthestar.According for T Tauri stars as estimated from CO spectroscopy(Najita et
to this idea, CoRoT-7b owes its present existence solely to its al.2007,Carr2007).
starbeingrelativelyyoung(∼1.5Gyr). Planetsbelowacertaincriticalmassm ∼ M aretrapped
c Jup
However, numerical simulations of the tidal evolution of just outside the IC as foundby Masset et al. (2006).The loca-
the CoRoT-7 planetary system (Ferraz-Mello et al. 2010) in- tion of the stationary solution with respect to the IC is practi-
dicate that the current eccentricities should be extremely low cally mass-independent.In contrast, bodieswith m > m enter
c
(e ≪ 10−3), and that any primordial departure from circular a regime characterized by a Type II migration that causes sig-
motionwouldhavebeenrapidlydampedbeforeanysignificant nificant perturbations to the density profile of the disk; conse-
orbital decay occurred. Thus, tidal evolution would have been quentlytheICedgecannotgenerateasignificantpositivecoro-
given primarily by stellar tides alone. In this case, the original tationaltorqueanddoesnotstoptheorbitaldecay.Aspredicted
locationofCoRoT-7bintheprimordialm-a¯ distributionshould byKuchnerandLecar(2002),migrationonlybrakesinsidethe
begivenbytheright-handplotsofFigures7and8;onceagain inneredgeata2/1mean-motionresonancewiththecavityedge.
this planet appears to be detached from the remaining close-in
Forreasonablevaluesofthediskviscosity,weexpectagap
planetarypopulation.
opening to occur when the height of the disk is approximately
Since CoRoT7 harbors at least one additional planet, it is
equal to the Hill radius of the planet. For a scale height equal
possible that mutual dynamicalinteractions might also explain to H/r ≃ 0.05, this implies a minimum mass of m ≃ 0.4M ,
Jup
this planet’s proximity to the star. In multiple-planet systems,
a value similar to our critical mass m . Adopting a value of
c
scatteringisbelievedtohaveplayedanimportantroleinsculpt-
one Jovian mass for the critical mass leads to a slightly larger
ing the general exoplanet distribution, and the same phenom- valueofH/r ≃ 0.07.However,giventheuncertaintiesinvolved
ena may have occurred in this system. According to this idea,
in both the gap openingcriteria and its dependenceon the vis-
CoRoT7-b could have had a close encounter with CoRoT7-
cosity,the valuesmay be consideredto be virtuallyequivalent.
c (or with an additional ejected planet) and suffered a signif-
Thus, the location of the bump in the observed distribution of
icant reduction in its semimajor axis. Subsequent tidal inter-
close-in planets appears to be consistent with a mass thresh-
actions would have circularized its orbit to its present state.
old for gap opening (Crida et al. 2006 and referencestherein).
However,CoRoT7isnottheonlymultiple-planetsysteminthis
Although the present data are sparse and plagued by the ad-
region. Both the GI581 and HD40307 planetary systems have ditional effects described in this work, we may expect that the
two known super-Earths with short orbital periods that never-
aforementionedresultswillbeconfirmedbyfurtherdetections.
theless lie above the expected lower limit. Hence once again In contrast, when the statistics become sufficiently robust, the
CoRoT7seemstobedifferent.
location of this bump may be used to place constraints on the
A possible explanation may lie elsewhere. Valencia et al. physicalpropertiesof theinnerdisk, in termsofbothtempera-
(2010)andJacksonetal.(2010)proposedthatCoRoT-7bcould tureandeffectiveviscosity,asthesequantitiesfeatureinthegap
bethesolidcoreofaprimordialgiantplanetwhosegaseousen- openingcriterion.
velopewaslostduetoevaporation.Arecentre-evaluationofthe
SincetheedgeoftheICcreatedinoursimulationsisplaced
radialvelocitydatabyseveralauthorsindicatethatthemassof
atanarbitrarydistancefromthestar,wehaveacertaindegreeof
CoRoT-7b could be as high as 9M (Ferraz-Mello et al. 2010)
⊕ freedomwhen fitting the numericalmass-perioddistribution to
or as low as 2M (Pont et al. 2010). Although a high current
⊕ therealplanets(asseeninFigure4).Wehaveadoptedavertical
massappearsindicativeofarocky/ironcompositionandthatthe
displacementthatminimizesthenumberofexoplanetinsidethe
original mass was at most twice the present value (Valencia et
cavityedge,butthisisnottheonlyoption.Itmaybearguedthat
al.2010),alowermassisconsistentwithalightercomposition
itwouldbebetterto fitthesyntheticcurvewiththe locationof
andaenvelope-depletedgasgiant.
sub-giants(i.e.m ∼ 0.5M )inthem-Pdiagram.We notethat
IfCoRoT-7bwereindeedthesolidcoreofaprimordialgas Jup
these bodies exhibit a smaller dispersion in orbital period than
giant, then its location in the m-a¯ diagram (right-hand plot of
observed for any other mass range, and can be seen as a com-
Figure7)wouldbelocatedclosetothedashedcurveand,thus,
pactgroupinbothplotsofFigure4.However,thisdisplacement
consistentwiththerestoftheplanetarypopulation.Westillneed
would lead to our accepting a larger number of small planets
ofcoursetoexplainhowthisplanetsufferedanevaporationofits
insidethecavityedge,bodieswhosesubsequenttidalevolution
gaseousenvelopeontimescalesshorterthanoneGyr,andwhyit
shouldhavebeenverysmall.
appearstobetheonlyexampleofthiseffect.
Whateverthechoice,thequalitativeresultsarenotaffected.
Moreover, the ratio of the stopping values of P (for small and
large bodies) is scale-independent, and found to be slightly
5. Conclusions
largerthan2.Thisisbecausealthoughhighermassesarestopped
Wehaveattemptedtounderstandthedynamicaloriginandevo- ina2/1MMRwiththeIC,thesmallbodiesaretrappedoutside
lutionofthemass-perioddistributionofclose-inexoplanets.The thediskedge.Onceagain,thedistributionofrealplanetsseems
present-daypopulationshowsadistinctivediscontinuitylocated toyieldasimilarratio.
atapproximatelyoneJovianmass.Smallerplanetshaveorbital Thesubsequenttidalevolutionoftheclose-inplanetsiscon-
periodslongerthanP∼2.5days,whilehighermassesarefound sistentwithastellar tidalparameterof Q′ = 107,avaluesimi-
∗
tohaveperiodsasshortasP∼1day. lartothatpredictedbySchlaufmanetal.(2010)fromsynthetic
We have found that the combined effects of tidal evolution populationmodels. Smaller parameters, leading to higher rates
and disk-planet interactions with an inner cavity (IC) in the of orbital decay, do not lead to distributions similar to the ob-
gaseous disk can explain most of the observed characteristics. served population. This is also consistent with the analysis of
The current distribution appears to be compatible with an in- Ogilvie & Lin (2007). A consequence of the tidal evolution is
nerdisk edge locatedapproximatelyatdistancesof a¯ ≃ 0.035, theremovalofmostoftheoriginalgasgiantswithshortorbital
whichforsolar-typestarscorrespondsroughlytoorbitalperiods periodsandtheirsubstitutionbyexoplanetsthatwereoriginally
ofP≃2.5days.Thisvalueisconsistentwiththeinnergasradii fartheraway.Thus,weexpectthatmanyoftheprimordialplan-
8 P.Ben´ıtez-Llambay1etal.:Themass-perioddistributionofclose-inexoplanets
ets with P < 2 days and m > 1MJup might have been tidally Najita,J.R.,Carr,J.S.,Glassgold,A.E.,etal.2007,in:B.Reipurth,D.Jewittand
disruptedandabsorbedbytheirparentstars. K.Keil(eds.),ProtostarsandPlanetsV,Tucson,UniversityofArizona,p.
507
Although this scenario is consistent with the properties of
Ogilvie,G.I.,&Lin,D.N.C.2007,ApJ,661,1180
most of the exoplanetary population, it appears difficult to ex-
Queloz,D.,Bouchy,F.,Moutou,etal.2009,A&A,506,303
plain the present-day mass and orbit of CoRoT-7b. A possible Queloz,D.,Anderson,D.,CollierCameron,A.,etal.2010.A&A,inpress
explanationis to assume that the planetstarted its life as a gas Pont,F.,Aigrain,S.,&Zucker,S.2010,MNRAS,submitted
giantwhosegasenvelopewascompletelyevaporated(Valencia Rice,W.K.M.,Armitage,P.J.,&Hogg,D.F.2008,MNRAS,384,1242
Rouan,D.,LegerA.,&SchneiderJ.2009,In:CoRoTInternationalSymposium
etal.2010,Jacksonetal.2010).
I
Lastofall,inthepresentscenariowehaveneglectedtherole Schlaufman,K.C.,Lin,D.N.C.,&Ida,S.2010,ApJ,724,L53
of the orbitalinclinations;however,the same results should be Shakura,N.I.,&Sunyaev,R.A.1973,A&A,24,337
Shu,F.H.,Najita,J.R.,Shang,H.,etal.2000.In“ProtostarsandPlanetsIV,ed.
expected as long as the inclinations are not very large. Three-
V.Mannings,A.P.BossandS.S.Russel,Univ.ArizonaPress,Tucson,pp
dimensionalstudiesofdisk-planetinteractionsin thelinearap-
789
proximation(e.g.TanakaandWard2004),aswellasnumerical Southworth,J.,Wheatly,P.J.,&Sams,G.2007,MNRAS,379,L11
simulations (e.g. Cresswell et al. 2007), show little effect on a Starczewski,S.,Gawryszczak,A.J.,Wu¨nsch,R.,etal.2007,ActaAstronomica,
finiteinclinationonthemigrationtimescale.Similarresultsare 57,123
Tanaka,H.,&Ward,W.R.2004,ApJ,602,388
alsofoundfororbitalevolutionduetotidaleffects(e.g.Ferraz-
Triaud,A.H.M.J.,Collier Cameron,A.,Queloz, D.,etal.2010.A&A524,
Melloetal.2008,BarkerandOgilvie2009).Forplanetsinret- id.A25
rogradeorbitsresultsare,however,ismoredifficulttoevaluate. Valencia,D.,Ikoma,M.,Guillot,T.,etal.2010,A&A,516,A20
It is unclear whether gas disks in retrograde motion (with re- Vidal-Madjar A.,Lecavelier des Etangs A.,De´sertJ.-M.,et al.2003, Nature,
422,143
spectto thestellar spin)wouldhaveinnercavitiesorhowsuch
Winn,J.N.,Johnson,J.A.,Albrecht,S.,etal.2009,ApJL,703,L99
astructurewouldinteractwithplanetsinitsvicinity.Similarly,
the tidalevolutionof bodiesin retrogrademotionispoorlyun-
derstood, thus it is impossible at present to ascertain how the
resultsofthisworkcouldbeextendedtothesesystems.
Acknowledgments
This work has been partially supported by the Argentinian
Research Council -CONICET-. F.M. and C.B. would like to
acknowledge the invitation to participate in the XII Brazilian
ColloquiumofOrbitalDynamics,wheretheoriginalideaforthis
work was discussed. A substantial part of the work was devel-
opedduringtheprogram”DynamicsofDisksandPlanets”,held
fromAugust15toDecember12,2009attheNewton’sInstitute
ofMathematicalScienceattheUniversityofCambridge(UK).
C.B. would like to thank the organizersof the program and to
fruitful discussions with all the participating researchers. Most
ofthe numericalsimulationsperformedin thiswork havebeen
run on a 140 core cluster funded by the program Origine des
Plane`tesetdela Vie (OPV)of theFrenchInstitutNationaldes
Sciencesdel’Univers(INSU).
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