Extrasolar planet population synthesis IV. Correlations with disk metallicity, mass and lifetime

3.5 MB·English

by  C. Mordasini

Astronomy&Astrophysicsmanuscriptno.aamontecarloivarxiv (cid:13)c ESO2012
January6,2012
Extrasolar planet population synthesis
IV. Correlations with disk metallicity, mass and lifetime
C.Mordasini1,Y.Alibert2,3,W.Benz2,H.Klahr1,andT.Henning1
1 Max-Planck-Institutfu¨rAstronomie,Ko¨nigstuhl17,D-69117Heidelberg,Germany
2 PhysikalischesInstitut,UniversityofBern,Sidlerstrasse5,CH-3012Bern,Switzerland
3 InstitutUTINAM,CNRS-UMR6213,ObservatoiredeBesanc¸on,BP1615,25010Besanc¸onCedex,France
Received27.05.2011/Accepted27.12.2011
ABSTRACT
2
1 Context.Thisisthefourthpaperinaseriesshowingtheresultsofplanetpopulationsynthesiscalculations.InPaperI,wepresented
0 our methods. In Paper II, we compared statistically the synthetic and the observed planetary population. Paper III addressed the
2 influencesofthestellarmassonthepopulation.
Aims.Ourgoalinthisfourthpaperistosystematicallystudytheeffectsofimportantdiskproperties,namelydiskmetallicity,mass
n
andlifetimeonfundamentalpropertiesofplanetslikemassandsemimajoraxis.
a
Methods.Foralargenumberofprotoplanetarydiskswhichhavepropertiesfollowingdistributionsderivedfromobservations,we
J
calculateapopulationofplanetswithourformationmodel.Themodelisbasedontheclassicalcoreaccretionparadigmbutincludes
4
self-consistentlyplanetmigrationanddiskevolution.

] dRieffseureltnst.rWoleesfi:nFdoarhviegrhymlaregtaellnicuimtiebse,rgoifanctorprlealnaetitosnasr:eRmeograerdfirnegquthenetp.lFaonrethairgyhinMitial,mgiaasnstfpulnacnteiotsna,rmeemtaolrliecimtya,sMsivdiesk.Fanodrlτodnisgkτhave,
P disk disk
giantplanetsarebothmorefrequentandmassive.Atlowmetallicities,verymassivegiantplanetscannotform,butotherwisegiant
E
planetmassandmetallicityarenearlyuncorrelated.Incontrast,(maximal)planetmassesanddiskgasmassesarecorrelated.The
h. formationofgiantplanetsispossibleforinitialplanetesimalsurfacedensitiesΣS ofatleast6g/cm2 at5.2AU.Thesweetspotfor
p giantplanetformationisat∼ 5AU.In-andoutsidethisdistance,higherΣS arenecessary.Lowmetallicitiescanbecompensated
- byhigh Mdisk,andviceversa,butnotadinfinitum.Atlowmetallicities,giantplanetsonlyformoutsidetheiceline,whileathigh
o metallicities,giantplanetformationoccursthroughoutthedisk.Theextentofmigrationincreaseswith M andτ andusually
disk disk
r decreaseswithmetallicity.Noclearcorrelationofmetallicityandthesemimajoraxisdistributionofgiantplanetsexistsbecausein
t
s lowmetallicitydisks,planetsstartfurtherout,butmigratemore,whilethecontraryappliesforhighmetallicities.Thefinalsemimajor
a axisdistributioncontainsanimprintoftheiceline.Close-inlowmassplanetshavealowermeanmetallicitythanHotJupiters.The
[ frequencyofgiantplanetsvariesapproximatelyasM1.2 andτ2 .
disk disk
Conclusions.Thepropertiesofprotoplanetarydisks-theinitialandboundaryconditionsforplanetformation-aredecisiveforthe
1
propertiesofplanets,andleavemanyimprintsinthepopulation.
v
6 Keywords.Stars:planetarysystems–Stars:planetarysystems:formation–Stars:planetarysystems:protoplanetarydisks–Planets
3 andsatellites:formation–Solarsystem:formation
0
1
.
1 1. Introduction (2010),hereafterPaperIII,wediscussedtheinfluenceofthestel-
0 larmassonthesyntheticplanetarypopulation,studyingforex-
2 Thenumberofknownextrasolarplanetshasgrownlargeenough ampletheeffectontheplanetaryinitialmassfunction,thesemi-
1 tolookatthestatisticalpropertiesofthepopulationasawhole, major axis distribution or the “metallicity effect” (the increase
: rather than at the properties of single objects, and to compare
v ofthedetectionprobabilityofgiantplanetswithmetallicity).In
theactualpopulationwithasyntheticpopulationobtainedfrom
i thisPaperIV,wefocusbackontosolar-likestars.
X a theoretical planet formation model. In this way all discov-
Overthelastyears,observationalconsiderableprogresswas
r eredplanetsserve(providedthedetectionbiasisknown)tocon-
a strainthemodel,andtoimproveourunderstandingofplanetfor- achieved in the characterization of the end products of plane-
tary formation process, i.e. the planet themselves. Progress has
mation. We used our extended core accretion formation model
also been substantial in the characterization of the initial and
(Alibert et al. 2005a) to generate in a Monte Carlo way popu-
boundary conditions for this process, i.e. the properties of pro-
lations of synthetic extrasolar planets (Mordasini et al. 2009a,
toplanetary disks. This was made possible to a large measure
hereafter Paper I). Then, we compared the detectable synthetic
thankstonewobservationalfacilitieslikeSpitzer(e.g.Fangetal.
planetswithanobservationalcomparisonsampleofactualexo-
2009).Observationsofdisksaroundyoungstarshaveprovided
planets using statistical methods (Mordasini et al. 2009b, here-
uswithknowledgeofthedistributionsofdiskmasses(Beckwith
after Paper II). We found that we could reproduce in a statisti-
&Sargent1996;Andrewsetal.2009),disksizes(McCaughrean
cally significant way some of the most important properties of
& Odell 1996; Andrews et al. 2010) and lifetimes (Haisch et
the observed extrasolar giant planets. Finally, in Alibert et al.
al.2001;Fedeleetal.2010).Additionally,correlationsbetween
Send offprint requests to: Christoph MORDASINI, e-mail: disk properties and stellar mass were discovered (Kennedy &
[email protected] Kenyon 2009; Mamajek 2009), which have important implica-
2 C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV
tionsfortheformationofplanetsaroundstarsofdifferentmasses themareknown.Foractualexoplanetsthisisobviouslynotthe
(Currie2009,PaperIII).AswasshowninPaperI,thediversity case.Observationally,thehoststarsmetallicityonlycanbede-
of extrasolar planets is a direct consequence of the diversity of termined and assumed to be a proxy for the disks dust-to-gas
protoplanetary disks, which means that the properties of disks ratio f . Other fundamental parameters of the disks, such as
D/G
are critical in defining the outcome of planet formation taking theirinitialgasmass(whichtogetherwith f setstheabsolute
D/G
placeinsuchdisks. amount of solids), and their lifetime, cannot be observationally
In this article we focus on correlations between disk and determined. It is therefore no surprise that the observationally
planetary properties. We study systematically the influences of inferredcorrelationsinvolveessentiallyonlythemetallicity.
diskmetallicity[Fe/H],disk(gas)mass M andlifetimeτ
disk disk Arecentlydiscussedcorrelationthatcouldpartiallychange
onimportant,observablepropertiesofsyntheticplanetsbycom-
thisisthepossibleenhancedlithiumdepletionmeasuredinSun-
putingalargenumberofmodelsofplanetformation.
likestarshostingplanets(Israelianetal.2009,butseeBaumann
etal.(2010)foranopposedview).Thisisbecausesuchadeple-
1.1. Observedcorrelations tioncouldresultfromapositivecorrelationofdisklifetimeand
likelihood of forming giant planets (Bouvier 2008). The influ-
From an observational point of view, a number of correlations enceofthedisklifetimeτ ontheoccurrenceofgiantplanets,
disk
were inferred in the past years, with various degrees of signif- asobservedinourmodels,isdiscussedinSect.8.3.
icance (see Udry & Santos 2007 or Mayor et al. (2011) for an
In general however, only probability distributions for disk
overview).Weaddressinthispaperthefollowingcorrelations:
masses and lifetimes can be inferred from the observations of
(1) The clearest correlation is the link between the stellar
starformingregions.Ourstudyallowstoacertaindegreetonar-
metallicity (to first order an indicator of the initial dust-to-gas-
rowdowntheseprobabilitydistributionsforaspecificstar-planet
ratio in the disk, see Santos et al. 2003) and the likelihood of
system,asnotalltypesofplanetscanforminalltypesofdisks.
detectingagiantplanet.This“metallicityeffect”isobservation-
ally known for a long time and very well established for solar-
like stars by numerous studies (e.g. Gonzalez 1997; Fischer &
Valenti2005,Udry&Santos2007).The“metallicityeffect”was 1.3. Earlierworks
studiedasanobservationalconstraintinPaperIIandisherefur-
Severalstudieshavealreadyaddressedcorrelationsbetweendisk
theraddressedinSect.4.2,4.3and8.1.
and planetary properties: Ida & Lin (2004b) studied the influ-
(2) Interestingly, lower mass Neptunian and Super-Earth
encesofthemetallicityasmodeledbythedust-to-gasratio.They
planets seem in contrast not to be found preferentially around
find that the normalized mass and semimajor axis distributions
high [Fe/H] stars (Mayor & Udry 2008; Sousa et al. 2008;
ofthepotentiallyobservable(giant)syntheticplanetsarerather
Ghezzietal.2010).WestudythisinSect.4.1and7.
independent of [Fe/H], while the frequency of giant planets in-
(3) Already much less compelling is a possible absence of
creases with it, in agreement with observations. Kornet et al.
very massive planets (in a mass range of about 5 to 20 Jupiter
(2005)modeledtheevolutionofthesolidsfromdustsizetoplan-
masses, M ) orbiting within a few AUs low metallicity solar-
etesimalsand,usingthefinalplanetesimalsurfacedensity,esti-
likestarsasfoundinradialvelocity(RV)searches.Suchapossi-
(cid:88) mated the giant planet formation capability. Such an approach
bleabsencewasnotedbyUdryetal.(2002),Santosetal.(2003) alsoreproducesthe“metallicityeffect”.Dodson-Robinsonetal.
andFischer&Valenti(2005).ThisisdiscussedinSect.4.3.3.
(2006) derived a fit for the time until gas runaway accretion is
(4) In contrast, no secure correlations were found between
triggeredbyaplanetarycoreatafixeddistanceof5.2AU,asa
stellarmetallicityandplanetarysemimajoraxis,atleastamong
function of the planetesimal surface density. They used this fit
RV detections (Udry & Santos 2007; Valenti & Fischer 2008;
to estimate the frequency of giant planets as a function of sev-
Ammler-vonEiffetal.2009),eventhoughpossiblecorrelations
eral disk properties. We show our results regarding this subject
were discussed in the literature. Sozzetti (2004) for example
in Sect. 8, comparing with their results. Matsuo et al. (2007)
studiedwhetherstarswithPegasiplanetshaveparticularlyhigh
usedparameterizedformationmodelstodeterminethediskpa-
[Fe/H], even among planet hosts. Correlations between migra-
rameterswheregiantplanetformationispossibleeitherbycore
tion, semimajor axis and metallicity are addressed in Sect. 6.1
accretion or direct gravitational collapse. We present a similar
and6.3.
studyinSect.5.
(5)GiantplanetsaremorefrequentlyfoundorbitingAstars
than orbiting solar-like stars, and are also more massive (Lovis
& Mayor 2007; Bowler et al. 2010) on average. In Paper III it
1.4. Structureofthepaper
was shown that this correlation can be best reproduced if one
assumes a roughly linear scaling between disk (gas) mass and
Thepaperisorganizedasfollows:Section2describesthemeth-
stellarmass.Thisimplies,atleastpartially,acorrelationbetween
ods used to obtain the result concerning the synthetic popula-
diskmassandplanetmass(Sect.4.1.2and4.4).
tionpresentedinSect.3.Section4studiesvariousaspectsofthe
Inaddition,weaddressbelowseveralothercorrelationsthat
impact of disk properties on the mass of extrasolar planets, in-
appear in the models but that have not yet been reported in the
cluding the planetary initial mass function. Section 5 analyzes
observation, but which could become observable in the future
under which disk conditions giant planets can form. Section 6
withbetterinstrumentsandlarger,morecompletedatabases.
addresses the correlationsof disk properties with planetary mi-
gration and the final semimajor axis distribution. In Sect. 7 we
study the metallicity of close-in planets. Section 8 shows how
1.2. Specialroleof[Fe/H]
diskpropertiesdeterminethefractionofstarswithgiantplanets,
Populationsynthesisisapowerfultooltoinvestigatesuchcorre- whileSect.9assestheconsequencesifdiskmassesandlifetimes
lations, because all properties of the (numerical) parent disks, arecorrelated.Finally,inSect.10wesummarizetheresultsand
and those of the synthetic planet populations emerging from presentourconclusions.
C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV 3
2. Methods tential correlations arising during disk formation itself e.g. via
opacityeffects.ThefactthatwedrawΣ and M˙ independently
As our approach to planet formation and population synthesis 0 w
implies,onaverage,longerdisklifetimeτ formoremassive
wasdescribedindetailsinPaperI,welimithereourselvestoa disk
disks. The importance of this coupling is discussed in sect. 9.
shortoverview.
We do not consider here the influence of varying initial disk
radii(Kornetetal.2005).Thiswillbeconsideredinfuturework,
2.1. GeneralProcedure takingintoaccountrecentobservationalresults(Andrewsetal.
2009,2010).
To obtain a synthetic population of planets, we proceed in five
steps:(1)Theprobabilitydistributionsfortheinitialconditions
are derived from observations of protoplanetary disks. (2) A 2.3. PlanetFormationModel
largenumbersofsetsofinitialconditionsaredrawnfromthese
Weuseaslightlymodifiedversion(seeMordasinietal.2009a)
probabilitydistributionsinaMonteCarlofashion.(3)Thecorre-
oftheextendedcoreaccretionformationmodeldescribedinde-
spondingfinaloutcomesoftheplanetformationprocess(plane-
tailsinAlibertetal.(2005a).AsinPollacketal.(1996),wecom-
tarymassandposition)arecomputedusingourplanetformation
putetheevolutionoftheplanetarycoreandenvelopestructure,
model. This results in a population of synthetic planets, most
butincludediskevolutionusingtheαformalism,andplanetary
of them however undetectable with current observational tech-
migration(isothermaltypeIandtypeII).Wehavebeenableto
niques. (4) To obtain the subset of the potentially observable
show(Alibertetal.2005b)thatthemodelreproducesmanyob-
syntheticplanets,weapplyanappropriatedetectionbias.Since
servationalconstraintsimposedbyourowngiantplanets.
mostplanetshavebeendiscoveredbyradialvelocitytechniques,
weuseabiasbasedonthevelocityamplitudeofthehoststar.(5)
Correlations between the initial (disk) conditions and the plan- 2.3.1. Relevantmodelassumptions
etary properties are searched and compared to observations (if
existing). Forthedisk-planetcorrelationsdiscussedinthiswork,anumber
ofmodelassumptionswerefoundtobeparticularlyrelevantand
havedirectlyvisibleconsequencesinthecorrelations.
2.2. InitialConditions-Probabilitydistributions
The first one concerns the structure of the protoplanetary
We use four Monte Carlo variables to specify the initial con- disk, and specifically the location of the iceline. In the nomi-
ditions. (1) The dust-to-gas-ratio f which we link to [Fe/H] nal model, the position of the ice line is an increasing function
D/G
as [Fe/H]=log(f /f ), where f is the dust-to-gas ra- ofdiskmassduetoviscousdissipation(PaperIandIII;Minet
D/G D/G,(cid:12) D/G,(cid:12)
tio of the solar nebula for which we assume a value of 0.04. al.2011).Itisfoundthatthishasimportantconsequencesboth
The choice of this value was discussed in details in Paper I. on the mass of giant planets (Sect. 4.3.1) and their formation
Here we only briefly mention that the factor two to three by location(Sect.5.2.1).
which 0.04 is larger than the measured photospheric Z of the A second assumption which is directly relevant for the
sun (Lodders 2003) is a first order representation of the effects massesofgiantplanets(Sect.4.3.2,4.4)isthatweassumethat
ofdustevolutionanddrift.Thesemechanismsleadtoanincrease gap formation does no reduce the gas accretion rate of giant
ofthe“planetesimal” f comparedtotheoriginal“dust” f planets (Lubow et al. 1999). As explained in Paper I, the rea-
D/G D/G
intheinner,planetformingpartsofthediskbyasimilarfactor sonforthisisthatKley&Dirksen(2006)haveshownthatwhen
through the advection of material from the outer disk (Kornet agiantplanetbecomessufficientlymassive,thedisk-planetsys-
et al. 2004). The probability of occurrence of a given [Fe/H] temcanundergoaneccentricinstability.Theplanetthenleaves
is derived from the metallicity distribution of the FGK stars in the clean parts of the gap, resulting in a substantial increase of
the CORALIE planet search sample (Udry et al. 2000) which the gas accretion rate. This means that the gas accretion rate is
is representative for the [Fe/H] distribution of the target stars obtained for low mass planets (Mcore (cid:46) 10M⊕) by solving the
inthevariousmajorradialvelocitysearchcampaigns(PaperI). planetarystructureequationswhileformoremassiveplanetsin
Thus, we assume that the observed stellar metallicity is a good the disk limited, runaway gas accretion phase it is equal to the
indicator of the primordial disk metallicity (Santos et al. 2003, rate at which gas viscously flows towards the star in the disk
but see also Pasquini 2007). (2) The initial gas surface density M˙enve = M˙disk = 3πνΣ where ν is the disk viscosity, and Σ the
Σ at a = 5.2 AU which we link to the initial disk gas mass gassurfacedensity.
0 0 √ √
M = 4πΣ a3/2( a − a ).Theprobabilitydistribution Athirdsettingwhichisrelevantforthecorrelationofmigra-
disk 0 0 max min
of circumstellar disk masses is derived from the observations tion and metallicity (6.1.2) is that we assume that the planetes-
of the ρ Ophiuchi star formating region (Beckwith & Sargent imal accretion rates are independent of the migration rate. The
1996).(3)Therateofphoto-evaporationM˙w.Thedistributionof M˙core is calculated in the same way as in Pollack et al. (1996).
M˙ isconstrainedbytheobserveddiskagedistribution:Wehave Possibleshepherdingeffectsarenotincludedforthereasonsex-
w
adjustedourM˙ distributiontoobtaintogetherwithourvalueof plained in Paper I. This means that cores in low solid surface
w
the viscosity parameter α a distribution of disk lifetimes τ densitydisksstillcangrowrelativelymassivebymigration.
disk
thatisingoodagreementwiththeobserveddistribution(Haisch Fourth, we assume for type II migration that as soon as the
etal.2001).(4)Theinitialsemimajoraxisoftheplanetaryseed planet is more massive than the local disk mass (M > Σa2
planet
a .Thedistributionofthestartingpositionsofplanetaryem- where a is the semimajor axis), the migration rate is given as
start
bryosisnotconstrainedbyobservationsandonlytheoreticalar- −(3ν/a)×(Σa2/M )p. In the nominal model, we use p = 1
planet
gumentscanbeused.Theyindicate(e.g.Mordasinietal.2009a) (“fully suppressed” case, Armitage 2007). This is important to
that runaway bodies should emerge with a uniform distribution understandtheabsenceofastrongcorrelationof[Fe/H]andthe
inlog(a ),asalreadyadoptedbyIda&Lin(2004a). semimajoraxis(Sect.6.1.1,6.3.3)
start
Inaddition,weassumethattheseMonteCarlovariablesare Notethatwesimplifytheproblemsignificantlybyassuming
independent variables. Hence, we do not take into account po- thatonlyasingleplanetcanforminagivendisk.Thiswouldbe
4 C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV
correct in the limit that proto-planets can form within a single
system without influencing each other. This is clearly an ideal-
ization, but in the context of this study this is not necessarily
a disadvantage. It allowsone to see clearly disk-planet correla-
tions,whichotherwisemightgetpartiallyblurredduetotheran-
domcharacterofthe(gravitational)interactionsbetweenseveral
protoplanets.Thommesetal.(2008)discussextensivelysomeof
theeffectsinducedbytheconcurrentformationofseveralplan-
etsinonedisk.
Thepopulationdiscussedhereisessentiallyobtained,except
if otherwise mentioned, with identical parameters and Monte
CarlodistributionsasthenominalpopulationpresentedinPaper
IandII.Inparticular,thismeansthatthestellarmassisequalto
onesolarmass M ,thegasdiskviscosityparameterαis0.007,
(cid:12)
andthetypeImigrationefficiencyfactor f is0.001.
I
However,thereisonerelevantaspectinwhichtheprocedure
used here differs from the one used in paper I and II. We draw
the initial semimajor axis of the starting seed strictly uniform
in log(a ) and disregard the additional criteria mentioned in
start
PaperI.Thiseliminatessomecorrelationspresentalreadyinthe
initial conditions and thus makes it easier to identify the influ-
enceofdiskpropertiesontheplanetaryproperties.
Asapracticalunitforthediskgasmasseswedefine,inanal-
ogyto[Fe/H],arelativelogarithmicunit,denoted[M /M ] =
log(Σ /Σ ), where we assume an initial gas surfacDe deSnNsity Fig.1. Mass-distance diagram of the synthetic population ana-
at 5.20AU0,SΣN = 200 g/cm2. This is somewhat more than a lyzed in this work. The dashed curve shows the feeding limit
0,SN (cf.PaperI):planetsreachingthislimithavebeenarbitrarilyset
plain MMSN values - Hayashi’s (1981) value would be about
145 g/cm2 - but with this choice [M /M ] covers, similar as to0.1AU.
D SN
[Fe/H],anearlysymmetricrangearoundzerobetween-0.6and
+0.7forthevaluesofΣ consideredhere(50-1000g/cm2,cor-
0
responding to disk masses between about 0.004 and 0.09 M ).
(cid:12)
As the disk gas mass M and the initial gas surface density Σ M⊕ only. Further growth from these masses to the final masses
D 0
by giant impacts (e.g. Marcus et al. 2010) will then only oc-
aredirectlyproportionaltoeachother,thesetwotermsareoften
cur after the damping influence of the gas disk is gone (Ida &
usedinaninterchangeableway.
Lin2010),aphasewhichishowevernotincludedinourmodel
at the moment. This effect would populate the depleted region
from“below”.
3. Mass-DistanceDiagram
Second,thehighobservedfrequencycouldbeindicativeof
To provide an overview of the synthetic planet population we quiteefficienttypeImigration,atleastinsomepartsofthedisk.
analyzeinthisworkweplotinFig.1themass-distancediagram This would populate the depleted region from “outside”. Large
oftheentirepopulation.Inthefigure,anumberofstructuresal- quantitiesoflowmass,close-inplanetsarefoundinsomesyn-
ready discussed in Paper I can be identified again, like the low theticpopulationstoo,butonlyforhightypeIefficiencyfactors
mass “failed cores”, the approximately Neptunian mass “hori- (f ∼ 0.1 − 1) as shown in Fig. 11 of paper II. On the other
I
zontal branch” or the massive “outer group” planets outside a hand can the distribution of semimajor axes of giant planets at
fewAU.Wealsoseethattherearenoverymassiveplanetsclose- larger distances only be reproduced when the type I migration
in. This is a consequence of the fact that planets with a mass rate (as found from linear theories for isothermal disks) is re-
largerthanthelocaldiskmassmigrateatareducedratebecause ducedbyasignificantfactor(f (cid:46)0.01,PaperII;Schlaufmanet
I
theirinertiaistoolargeforthefiniteangularmomentumfluxin al.2009).HereweareusingthesamelowtypeIefficiencyfactor
thedisk.Theirabsenceatsmalladependsupontheassumedde- f =0.001whichleadinPaperIItothebestreproductionofthe
I
greeofreductionofthetypeIImigrationrateoncetheplanetis observed properties of giant planets. Unfortunately, this causes
massive(Syer&Clarke1995). inthesametimenecessarilyalsothedepletedregion.
Theanswertothisdilemmaisprobablyasignificantlymore
complex migration pattern than what can be mimicked with
3.1. Paucityoflowmass,close-inplanets
global(independentofplanetmassanddistance)efficiencyfac-
There is also a relative paucity of low mass (4 (cid:46) M (cid:46) 10M⊕) torslike fI forisothermaltypeIratesasdonehere.Recentstud-
close-in(a(cid:46)0.3AU)planets.Recentobservations(e.g.Howard iesoftypeImigrationdroppingtheofteninappropriateassump-
etal.2010;Mayoretal.2011)ratherindicateahighfrequency tion of isothermality (e.g. Paardekooper et al. 2010; Masset &
oflowmassSuperEarthplanetsclosetotheirparentstar. Casoli 2010) indeed find complex patterns with rapid in- and
Concerningthisissue,wefirstremindthatourmodelshows outward migration. First exploratory planet population synthe-
the state of (proto-)planets at the moment when the disk disap- ses using such updated type I migration models (Mordasini et
pears, and not the final one after billions of years of evolution. al.2011b)findthattypeImigrationisdirectedoutwardinsome
Atthisstage,thelowmassprotoplanetsatsmalldistanceshave partsofthedisk,andinwardinothers,whichleadstotheexis-
amassaboutequalthelocal(solid)isolationmass(typeImigra- tenceofconvergencezones(migrationtraps)whichcanresultin
tionisstronglyreducedinthissimulation),whichisoforder0.1 a pile-up of many low mass planets in certain parts of the disk
C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV 5
Fig.2.Planetaryinitialmassfunctionatthemomentwhenthegaseousdiskdisappears.Notethatthemodelsdoesn’tincludeany
growthprocesseslikegiantimpactsafterdiskdispersalwhichcansignificantlymodifythemassdistributionafterwards,especially
forlowmassplanetswith M (cid:46) 10M .Left:PIMFbinnedaccordingtometallicity.Center:PIMFbinnedaccordingtothegasdisk
⊕
mass.Left:PIMFbinnedaccordingtothedisklifetime.Themeaningofthedifferentlines,andthelimitingvalues,areindicatedin
theplot.
(cf.Lyraetal.2010;Sandoretal.2011).Thisimportantsubject 4.1.1. PIMFasfunctionofmetallicity
willbeaddressedindedicatedwork(Dittkristetal.inprep.).
A third mechanism occurs when several planets form and
The left panel shows that the metallicity just modifies the im-
migrate concurrently. Then, lower mass planets can be pushed
portance of the three basic families of planets visible in the
closetothestarafterbeingcapturedinmeanmotionresonances
PIMF(Jovianplanets,Neptunianplanets,proto-terrestrialplan-
of a more massive, more rapidly migrating outer planet. This
ets), without however changing the general shape of the PIMF
willbestudiedinoncomingsimulationsallowingtheformation
(in contrast to the other two disk properties). For giant planets
ofmanyplanetsperdisk(Alibertetal.inprep.)
this means that in a high metallicity environment, more giant
planets form, but their mass distribution is similar. The reason
forthisisrelatedtothefactthatacertain[Fe/H]actsasathresh-
4. Mass
oldforgiantplanetformation(thresholdsolidsurfacedensityto
4.1. Planetaryinitialmassfunction(PIMF) reachacriticalcoremass,seeSect.5),butisnotimportantinde-
terminingthefinaltotalmass,becauseintheend,gasmakesup
Acentraloutcomeofpopulationsynthesisistheplanetaryinitial
formostofthemassofagiantplanet,andnotsolids.Therefore,
massfunctionPIMF(PaperIIandIII).Figure2showsthePIMF
a high metallicity mainly allows a larger number of high mass
of all synthetic planets with a mass larger than 1 M , binned
⊕ planets,butnotofahighermass(exceptforverylargemasses,
intolow,mediumandhighmetallicity(leftpanel),diskgasmass
seesect.4.3).Thisincreaseinfrequencyisofcoursetheunder-
(central panel) and disk lifetime (right panel). The diving val- lyingreasonfortheobservedmetallicityeffect,i.e.theincrease
ues are indicated in the plot and are chosen for all three cases ofthedetectionrateofgiantplanetswith[Fe/H](seeSec.8.1).
in a way that the central bin contains about 70% of the plan-
ets,andtheothertwoabout15%each.Werecallthatourmod- Moving down in mass, we see that in the Neptunian mass
els start with an initial seed mass of 0.6 M , so that we cannot domain, the dependence of planet frequency on metallicity is
⊕
reliably make predictions about the exact form of the PIMF in weak. This means that no metallicity effect is predicted in this
the (cid:46) 10M domain as mentioned in Paper I and II. Note that domain. This is in good agreement with recent observations
⊕
each bin has been normalized individually, so that the absolute (Mayoretal.2011).Movingfurtherdowninmasstotheproto-
heightofdifferentbins(e.g.lowvs.mediummetallicity)cannot terrestrial domain, the lines are inverted relative to the giant
becomparedinabsoluteterms,buttakingintoaccountthatthe planet domain, which means that an inverse metallicity effect
initialdistributionof[Fe/H]followsthedistributionobservedin occurs(lowmassplanetsaremorefrequentatlow[Fe/H]com-
the solar neighborhood. However, one can directly see the rel- paredtohigh[Fe/H]).Concerningthislastpoint,wemusttake
ative importance of a given planetary type within a bin (e.g. at into account that in our model, only one embryo can form per
highmetallicitytherearemoreJovianthanNeptunianplanets,in disk.Thesituationthatinhigh[Fe/H]disks,bothagiantplanet
contrast to the low [Fe/H] case.). One recognizes again a num- anda“byproduct”terrestrialplanetformisthereforenotpossi-
ber of features discussed in Paper I, like the high mass tail in ble.Thiscouldartificiallystrengthenthe[Fe/H]-lowmassplanet
theSuper-Jupiterdomain,thegiant’splateauintheJovianmass anti-correlation,sothatmoresecurepredictionswillbepossible
regime, the minimum at about 30-40 M corresponding to the with models dropping the one-embryo-per-disk simplification
⊕
planetarydesert(Ida&Lin2004a),theNeptunianbumpandthe (Alibert et al. in prep.). Observationally, future high precision
strongrisetowardssmallmasses. observations(e.g.withESPRESSO)willtestthisprediction.
6 C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV
Fig.3.Planetarymassfunctionbinnedac-
cording to [Fe/H] as in Fig. 2, left panel,
butnowonlyforthesub-populationofpo-
tentially detectable synthetic planets with
an RV precision of 10 m/s (left panel)
and 1 m/s (right panel). Green dashed
linesagainrepresenthigh,bluesolidlines
mediumandreddottedlineslowmetallic-
ities,wherethethresholdmetallicitiesare
thesameasinFig.2.
Onefurthernotesthatforallthreebins,thehighestpeakin longer lifetimes, cores will be able to grow to the critical mass
themassfunctionisfoundfortheproto-terrestrialplanets,with andaccretegasinarunawayfashionevenforrelativelylowsolid
the clearest dominance at low metallicity. For high metallicity, surface density. Therefore, the disk lifetime acts as a threshold
the second highest peak occurs for the giant planets, whereas for giant planet formation, similar to the metallicity. The life-
for intermediate and even more clearly for low metallicity, the time of disks however also affects the total mass of the planets
Neptunianplanetsareresponsibleforthesecondhighestpeak. similartothediskmass,as,tofirstorder,theplanet’sfinalmass
will be equal to the accretion rate times the duration of the ac-
cretion phase. This twofold effect is clearly seen in the figure.
4.1.2. PIMFasfunctionofdiskgasmass
Giantplanetsformedinlong-liveddisksarebothnumerousand
ofahighermass.Comparedtometallicityanddiskmasses,the
Thecentralpanelshowsthatthediskmassdistributiondirectly
affects the shape of the PIMF and not only the height of the disk lifetimes has thus a more complex influence on the result-
ingplanetpopulation.Thedisklifetimebothscalesanddistorts
peaks of the distribution as metallicity does. For giant plan-
the shape of the PIMF. Similarly to the primordial disk mass,
ets,ahigh[M /M ]shiftstheformationofgiantplanetsfrom
D SN it is difficult to deduce the disk lifetime for any give observed
lowermasstohighermasses(comparetheblueandgreenline).
For lower disk masses, massive giant planets ((cid:38) 6M ) can- system.Wecomebacktothatinsect.8.
notformatall.Inthiscase,thereissimplynotenoughgaseous
(cid:88)
materialavailable.Thedistributionofintermediate-massplanets
(30M (cid:46) M (cid:46) 1M ), on the other hand, is little affected. We 4.2. Observabledistributionasfunctionof[Fe/H]
⊕
concludethatinourmodelthereisadirectcorrelationbetween
(cid:88) It is interesting to look whether some of the correlations be-
the (maximal) mass of giant planets and the disk gas masses.
tween disk properties and the underlying mass function (of all
Thisisessentiallyduetothefactthatgiantplanetsaccretemost
planets) discussed in the section above can already be seen in
of their mass in a regime where their accretion rate is propor-
the observational data we have today, which represent only a
tionaltothediskmass,seethediscussioninSection4.4.
smallfractionofallexistingplanets.Forthis,weplotinFig.3
In Paper III, we showed that, in our alpha-disk model, the themasshistogrambinnedaccordingto[Fe/H],butnowinclud-
disk mass has to be scaled roughly linearly with M in order
∗ ingonlyplanetsdetectablebya10-yeardurationradialvelocity
to reproduce the observed correlation between the star’s accre- (RV) survey with a precision of either 10 m/s (left panel) or 1
tion rate and its mass. As shown in this paper, this translated m/sprecision(rightpanel).Notethat,likeasinPaperIII,weuse
into the formation of planets of a higher mass orbiting stars of averysimplevelocityamplitudecut-offcriteriumtodetermine
a larger mass, in good agreement with observation (Lovis &
detectability.
Mayor2007).Eventhoughwedidnotvarythestellarmassinthe
present study, the fact that more massive planets form in disks
takenfromthehigh-massendofthedistributionstemsfromthe
4.2.1. 10m/sradialvelocityprecision
samereason.Unfortunately,inpracticewecannotinferthepri-
mordial disk mass for any actually detected exoplanet and so The figure shows that at a precision of 10 m/s, the mass distri-
thiscorrelationisdifficulttotestobservationalincontrasttothe
butionhasaverysimilarshapeforallthreemetallicitybins,as
metallicitycorrelation.Thepanelfurthershowsthattherelative
expected from the discussion above and the previous work by
importanceofNeptune-likeplanetsstronglydependsonthedisk
Ida&Lin(2004b).However,acloserlookindicatesthatthelow
gasmass.Neptune-likeplanetsformparticularlyeasilyindisks [Fe/H]binhasasomewhatnarrowerdistribution,andthatthere
withsmallprimordialdiskmasseswhiletheydonotseemtobe isacertainsystematicdifferenceattheuppermassend(seethe
abletoforminmassivedisks.Thisagainisaconsequenceofthe
nextsection).However,themediumandhighmetallicitybinsto
sameeffectdiscussedaboveandalreadypointedoutinPaperIII.
whichmostofthepresentdayknownplanetpopulationbelongs,
areverysimilar.Itisthereforenotsurprisingthat,giventherel-
ativelysmallnumberofobservedplanetscomparedtothelarge
4.1.3. PIMFasfunctionofdisklifetime
numberofsyntheticplanetsusedhere,nosignificantcorrelation
ThethirdpanelattherightofFig.2finallyshowsthedisklife- betweentheshapeofthemassdistributionofgiantplanetsand
time. The disk lifetime has a twofold influence. In disks with thehoststarmetallicityhasbeennoticedthusfar.
C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV 7
4.2.2. 1m/sradialvelocityprecision
Ataprecisionof1m/sonthecontrary,theinfluenceof[Fe/H]
on the PIMF becomes visible as measured by the relative fre-
quencyofNeptunianversusJovianplanets,atrendthathasbeen
observed(Udryetal.2006).Wefindthattheratioofthenumber
ofNeptune-masstoJupiter-massplanetsstronglycorrelateswith
themetallicity.Forthelowmetallicity,theNeptune-massplan-
etsarealmostasnumerousasJupiter-massplanetswhileforhigh
metallicitydiskstheyaresignificantlylessfrequent.Thiscorre-
sponds well with the observed result (Sousa et al. 2008). The
numerical values of the ratios of synthetic Jupiter-to-Neptune
numberofplanets(dividingmassdefinedat30M )obtainedfor
⊕
a 10 year search at 1 m/s is in our nominal simulation 4.7, 5.8
and 7.6 for the low, medium and high [Fe/H] bin respectively.
ThesevaluesaredifficultdirectlywithSousaetal.(2008)dueto
severalreasons(differing[Fe/H]bins,differingprimarymasses,
differingsurveyduration).However,theseauthorsfindvaluesfor
theJupiter-to-Neptuneratioof1.6-5for[Fe/H]<−0.15,and7.5-
30for[Fe/H]>0.15.Notethattheratiosevenat1m/sprecision
arehigherthanthoseinthefullunderlyingdistributionwithout
detectionbias.Thereasonisthatevenat1m/smanyNeptune-
likeplanetsremainundetected.Finally,wepointoutthatdueto
our assumption of a single planet-per-disk simplification, it is
verylikelythatourratioisanupperlimit.Ahintinthesamedi- Fig.4. The mass of synthetic planets (in Jupiter masses M ) as
J
rectioncanalsobeinferredwhencomparingoursyntheticresult a function of [Fe/H] for the nominal population (small dots).
withtheobservedoneshowninFig.10ofMayoretal.(2011). There is an absence of very massive planets around low metal-
Otherwise,observedandsyntheticresultaresimilar. licitystars.Intheplot,linesindicateapproximativelytheregion
From the two plots we also understand that the decrease of ofhighmassandlowmetallicitywherenosyntheticgiantplan-
theobservedmassdistributionat10m/stowardssmallmasses, etsarefoundinothernon-nominalpopulations:withaniceline
whilebeingclearlytheconsequenceoftheobservationaldetec- fixedto2.7AU(dottedline),orwiththeeffectofgapformation
tionbiasagainstlowmassplanets,isamplifiedbytheactualtrue onthegasaccretionratemodeledasinVeras&Armitage(2004)
decreaseoftheunderlyingmassdistribution.Thisdegreeofde- (solidline).ThedashedlineshowsthelimitgiveninFischer&
pletionofintermediatemassplanets(30−100M⊕)ispotentially Valenti(2005).
an important constraint on the gas accretion rate at the begin-
ning of runaway gas accretion. In Mordasini et al. (2011a), we
presentedadetaileddiscussionaboutvariousassumptionsabout 4.3.1. Effectcausedbytheicelineposition
gasaccretionthatdirectlydeterminethisportionofthetheoreti-
A less pronounced absence of massive planets at low metallic-
calPIMF.
ity can also be obtained if we fix the position of the ice line at
2.7AUindependentlyof[M /M ],asexpectedforanoptically
D SN
4.3. (Maximal)planetarymassesandmetallicity thindiskirradiatedfroma1M star(Ida&Lin2004a).Thisdif-
(cid:12)
ferenceisduetoaninterestingchainofcorrelations:
Toemphasizetheimportanceofthemetallicity(measuredinour
1)Werecallthatinthenominalcase,thepositionoftheiceline
model by the ratio of dust-to-gas in the disk) on the resulting
isanincreasingfunctionofdiskmass.
high-endofplanetarymasses,weshowinFigure4themassof
2)Atlowmetallicities,highdiskgasmassesareneededforgiant
synthetic planets (in units of Jupiter masses) as a function of
planet formation (Matsuo et al. 2007; Sect. 5.3). Thus, the ice
metallicity. While we have indicated in Sect. 4.1 that metallic-
linepositioninlowmetallicitydisksforminggiantplanetswill
itydoesnotchangesignificantlythedistributionofthemassfor
typicallybelocatedatlargedistances.
thebulkofthepopulation,weseeherethatthemetallicitydeter-
3)Hence,thetypicalstartingpositionofgiantplanets-to-bewill
minesthemaximummassaplanet1 cangrowtoinagivendisk, alsobelargeatlow[Fe/H](a (cid:38)a ,seeSect.5.2).
in particular for subsolar metallicities. This can be understood start ice
4)This distant starting position implies a slower growth of the
asfollows.Togrowtoaveryhighplanetarymass,acriticalcore
core (e.g. Paper I), disadvantageous for successful giant planet
mustformbeforeviscousevolutionandphotoevaporationofthe
formation.
diskhavehadtimetosignificantlydepletethediskmass.Sucha
Thus,apositivecorrelationofthediskgasmassandtheiceline
veryearlystartisnotpossibleinalowmetallicityenvironment positionrendersgiantplanetformationatlow[Fe/H]evenmore
inwhichgrowinglargecorestakeslonger. difficult,andmakesitimpossibletopopulatetheupperleftcor-
This second order correlation via the core formation
ner of Fig. 4 with the nominal model. It also means that there
timescalehasbeencheckedandconfirmedbyartificiallyreduc- is a difference between a metal poor-gas rich and a metal rich-
ingthestartingtimeoftheseedembryosbyafactoroftwo.In
gas poor environment, even if the solid surface density beyond
this test, the maximal mass of planets at [Fe/H]=-0.4 increases theicelineareinprinciplethesame.Alow[Fe/H]cannotinall
toabout18M comparedto∼7M inthenominalcase.
circumstancesbecompensatedahigh[M /M ].
D SN
1 Weherecal(cid:88)lforsimplicityallobje(cid:88)ctsthatforminthepopulation In the case with an ice line independent of [MD/MSN], this
synthesisplanets,eveniftheirmassislargerthanthedeuteriumburning complexchainofcorrelationsbetween[Fe/H],[MD/MSN],aice,
limitatabout13M . astart andtstart isbroken,andplanetswithabout10 M canstill
(cid:88) (cid:88)
8 C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV
format[Fe/H]=-0.5,andthatmaximalmassesbecomeindepen-
dentof[Fe/H]alreadyfor[Fe/H](cid:38)−0.3.Theupperlimitofpos-
siblemassesisindicatedforthiscasebythedottedlineinFigure
4. It is clear that this chain of correlations depends quite sensi-
tively on a number of specific model assumptions, making it a
lessrobustprediction.
4.3.2. Verymassiveplanets
Theabsenceofverymassivecompanionsformedbycoreaccre-
tion at very low metallicities is also seen in the models of Ida
&Lin(2004b)andMatsuoetal.(2007).However,comparedto
theseworks,planetsofsignificantlyhighermasscanforminour
simulations.Thisisadirectconsequenceofthefactthatwedo
not limit gas accretion due to gap formation, as mentioned in
Sect.2.3.1.Ifweinsteadlimitinanon-nominalpopulationthe
gas accretion rate by giant planets due to gap formation by us-
ingthefitofVeras&Armitage(2004),themaximalmassofall
planetsinthepopulationarereducedto∼7M .Theabsenceof
themostmassiveobjectsatlow[Fe/H]wouldremainsimilarbut
with weaker dependence of the maximal mas(cid:88)s on [Fe/H] when
scaled to smaller absolute masses. The approximative limiting
envelopeforsuchapopulationisalsoshowninFig.4asasolid
line. This envelope is now similar to the one of Matsuo et al.
Fig.5. Observed and synthetic planetary masses as a function
(2007).Wethusseethatwhilethetendencytowardsanabsence
of metallicity. Black dots are synthetic planets in the nominal
ofverymassiveplanetsatlow[Fe/H]isageneralpredictionof
model, while solid, dashed and dotted lines again indicate the
thecoreaccretiontheory,thequantitativeresultsdependonspe-
limitingenvelopeofnon-nominalmodels,asinFig.4.Redstars
cificmodelsettings.
areobservedRVcompanionsaroundstarswith0.8< M /M <
∗ (cid:12)
Whenstudyingthefigureoneshouldkeepinmindthatvery 1.2.GreensquaresareobservedRVplanetsaroundstarswitha
massive planets ((cid:38) 10M ) are in fact very rare outcomes in masslargerthan1.2M .Forbothcases,apointinthemiddleof
(cid:12)
our simulations, in agreement with the observed “brown dwarf thesymbolindicatesthatthestarisasubgiantorgiant.Bluetri-
(cid:88)
desert”asdiscussedinPaperII.Figure4mayprovidethesome- anglesshowplanetsaroundHR8799.Namesofrelevantobjects
whatmisleadingimpressionthatthesearecommonobjectswhile aregivenintheplot.
in fact these objects appear only because the underlying syn-
thetic population is extremely large (about 200 000 initial con-
ditions). Even if such a population vastly exceeds the actually tribute,stillsuffersfromlownumberstatistics(seealsoSozzetti
observedpopulation,thesehighnumbersofplanetsarerequired
&Desidera2010).
to investigate the different correlations some depending on two
In Fig. 5, red stars symbolize planets detected by the ra-
variables(likediskmassand[Fe/H]).
dialvelocitymethodaroundprimarieswithamassbetween0.8
and 1.2 M , which are the most relevant cases (in the model
(cid:12)
M = 1M ).Greensquaresarecompanionstostarsmoremas-
4.3.3. Comparisonwithobservation ∗ (cid:12)
sivethanthis,alsodetectedbyradialvelocity.Starswhichhave
The absence of very high mass planets (or of a very high to- evolvedoffthemainsequence(ifknown)aremarkedinthefig-
tal mass locked up in planets in multi-planet systems) at low urewithadot.Notethatstellarevolutioncouldbeofrelevance
metallicitieswasnotedalsoobservationallyinUdryetal.(2002), here, as it might invalidate our underlying assumption that the
Santos et al. (2003) and Fischer & Valenti (2005). The dashed photospheric composition measured today correlates with the
line in Fig. 4 is taken from the latter work. It is clear that such bulkcompositionofthediskmaterialatformation.Thisassump-
an absence could also simply be a small number effect: Giant tioncouldbeinvalidincaseofenhancedheavyelementsettling,
planets around low [Fe/H] hosts are rare, and high mass giant which for the sun seems to have already lead to a reduction of
planets((cid:38) 5M )arerareatallmetallicities,whichcouldcom- thepresentdayZascomparedtotheprimordialZ0bymorethan
bine into the observed paucity. It is interesting to compare our 10%(Lodders2003).
(cid:88)
results with the observational database especially since the lat- Looking at Fig. 5, we see that the bulk of all observation-
ter has been significantly extended since the above mentioned ally detected planets falls into regions of the plot where syn-
papershavebeenpublished. thetic planets are also found. Hence, from this observation we
Such a comparison is presented in Fig. 5 which shows the could conclude that core accretion can account for almost all
massesofactualandsyntheticplanetsasafunctionofmetallic- planets currently known. This conclusion was already reached
ity.ObservationaldatawastakenfromJ.SchneidersExtrasolar byMatsuoetal.(2007).Itshouldbenotedthatsurveysthatlook
Planet Encyclopedia. Note that this database does not include atverymetalpoorstarsdoexist(Santosetal.2009).
companions larger than about 20 M . However, such compan-
ions are anyway extremely rare, both in the model (only 0.1%
(cid:88) 4.3.4. Relevantindividualobjects
of the synthetic planets have a mass larger than 20 M ) but
alsoobservationally.Asaresult,ourobservationalknowledgeof However,eightRVplanetsorbitingfivestarsclearlylieoutside
(cid:88)
thismassrange,whereseveralformationmechanismcouldcon- theregionwheregiantplanetsformorbitinga1 M starinour
(cid:12)
C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV 9
nominal model. This group is characterized by a much higher probably has a mass larger than 1.2 M (1.5±0.3M ), so the
(cid:12) (cid:12)
fractionofgiantstarsandofstarsmoremassivethan1.2M than synthetic population does not apply directly, but it is neverthe-
∗
thefullsample.Italsohasahighmultiplicityforgiantplanets. lessinterestingtonotethattheseplanetscometolieinaregion
Assomeoftheobjectsarequitepeculiar,itisworthtodiscuss inthe[Fe/H]-massplanewhichisatorclosetothelimitwhere
individuallysomeoftheseobjects. giantplanetscanoriginatefromcoreaccretion.Thisisafinding
HD155358 has two planetary companions of rather low independentfromthefactthatthelargesemimajoraxesofthese
mass ((cid:46) 1M ) at 0.6 and 1.2 AU (Cochran et al. 2007). The planetsmakescoreaccretionastheformationmechanismdiffi-
starhasamassof0.87±0.07M anda[Fe/H]= −0.68±0.07. cult(Dodson-Robinsonetal.2009),ifnoadditionalmechanism
(cid:12)
(cid:88)
HD155358 is a particular star (Fuhrmann & Bernkopf 2008). causingoutwarddisplacementlikescatteringisacting.
It is a very old, thick disk subgiant with a chemical composi-
tion very far from scaled solar composition. It is significantly
enriched in alpha-chain nuclei. [Mg/H] is for example with - 4.3.5. Summaryconcerningthe[Fe/H]-M correlation
0.36 much higher than [Fe/H]. What matters for the ability to
Insummaryweseethatthereareonlyextremelyfewexamples
formasufficientlymassivecoreisthesurfacedensityofallcon- of bona fide, massive companions (5 (cid:46) M (cid:46) 20M ) orbiting
densible elements beyond the ice line which can attribute im-
solar-like, main sequence stars at small orbital distances which
portant quantities of matter, in particular the α elements O, Si do not fall into the [Fe/H]-M parameter space cove(cid:88)red by our
and Mg (Dodson-Robinson et al. 2006; Gonzalez 2009). Due
implementationofthecoreaccretionmodel.
to the enrichment in α elements, the disk around HD155358
The solid line in Fig. 5 shows also the limiting envelope of
wasdepletedinplanetesimalsmaybeonlybyafactor2belowa
syntheticplanetarymassesderivedinthecasewherethegasac-
[Fe/H]=0solarcompositiondisk,equivalenttoa[Fe/H]∼ −0.3
cretionrateduetogapformationislimitedbyusingthesimple
if it were to have a scaled solar composition. At such a value,
one parameter fit of Veras & Armitage (2004) to hydrodynam-
formingtheanywaylowmassgiantplanetsiscertainlypossible
ical simulations of Lubow et al. (1999). It is clear that such a
withthecoreaccretionmechanism.
simple fit can only be a rough approximation of the real effect
HD114762 with its Msini = 11M , a=0.36 AU compan-
and that e.g. disk viscosity also has an influence on the degree
ion (Latham et al. 1989) was discussed in this context already
(cid:88) ofquenchingofgasaccretion(Lissaueretal.2009).Butthesig-
byUdryetal.(2002).Ithasbeenoftenconsideredthatthesys-
nificant number of planets lying above the solid line indicates
temisseennearlypoleon,sothatthecompanionmightinfact
that mechanisms like the eccentric instability (Kley & Dirksen
bealateMdwarf(Cochranetal.1991;Hale1995).Analterna-
2006)thatallowgrowthbeyondthegapbarrierseemtoplayan
tivehypothesisisbasedonthefindingthatHD114762ischem-
importantroleinnature.
ically and evolutionary very similar to HD155358 (Fuhrmann
& Bernkopf 2008) which would mean that its effective surface Wefinallynotethatthemostmassivesyntheticplanetscould
also, at least partially, be an artifact of the one embryo per
densityofplanetesimalswasmuchhigherthanonewouldinfer
fromthelow[Fe/H]andscaledsolarcomposition. disk approximation. In a disk where several protoplanets con-
HD47536 is an old K1III, [Fe/H]=-0.68 giant which is or- currently grow, planets compete for gas and ejection can even
removesomeplanets(Thommesetal.2008).Inthissense,look-
bited by one (Setiawan et al. 2003), possibly two (Setiawan et
ing at the total mass in the system should be a more adequate
al.2008)quitemassivecompanions(5and7 M )insideafew
quantitythanindividualplanetmasses.
AU. While Setiawan et al. (2003) originally quoted a possible
(cid:88)
stellarmassofabout1to3 M ,daSilvaetal.(2006)morere-
(cid:12)
cently determined a mass of 0.94±0.08M . With this primary
∗ 4.4. Planetarymassesanddiskmass
mass, the object falls clearly out of the envelope of synthetic
planetsinourmodel.Inparticularifitstwo-planetconfiguration Figure6showsplanetarymassesasafunctionoftheinitialdisk
is confirmed, this is an interesting object to study with forma- gasmass,againinunitsofJupitermassestofocusonthemassive
tion models, as one then cannot invoke, due to orbital stability, planets.Herewefindadifferentbehaviorthanfor[Fe/H]:Asex-
a nearly pole on orientation as a possible explanation. Also for pectedfromthediscussionoftheinfluenceof[M /M ]onthe
D SN
thedirectcollapsemodelthiscaseisprobablynotobvioustoex- PIMF, we see that there is a positive correlation of disk mass
plain due to the small semimajor axis of the planets (about 1.5 andplanetarymassoverthefulldomainofdiskmassesconsid-
AUfortheinnerplanet),seee.g.Boley(2009). eredhere.Thisisillustratedbythedottedlinecorrespondingtoa
BD+202457isaK2IIgiant,withanestimatedmassof2.8± massthatisequalhalftheinitialdiskmass.Thisshowsthatthere
1.5M ,averylowmetallicityof[Fe/H]=−1.00±0.07withtwo is,forthedomainof[M /M ] (cid:38) −0.2anapproximativelylin-
(cid:12) D SN
verymassivecompanionsintightorbits(Niedzielskietal.2009). ear correlation between the maximal planet mass and the disk
Eveniftheverydifferentprimarymassmakesacomparisonwith mass. The reason for this is that giant planets accrete in our
theresultsheredifficult,thesecompanionsseemtobefarfrom model(seeMordasinietal.2009a)mostoftheirmassinthedisk
thepossibleparameterspaceforcoreaccretion. limited accretion regime where the planetary gas accretion rate
Theobjectaroundγ1 LeoA(HIP50583)wasfoundtohave isassumedtobethesameasinthedisk(M˙env = M˙disk =3πνΣ),
a projected mass of about 8.8 M (Han et al. 2010). The orbit which is itself proportional to the initial disk mass for a given
ofthecompanionwashoweverrecentlyastrometricallydetected distance and moment in time, at least if the disk evolution is
by Reffert & Quirrenbach (2011(cid:88)) in the Hipparcos data. Their closetoself-similaritysolutions(Hartmannetal.1998).
resultsindicateaclearlylargeractualmassofabout66M (with Thedifferentinfluenceof[Fe/H]anddiskgasmassesonthe
substantial incertitude). This moves the companion out of the massofgiantplanetscanbequantified:Forplanetswithamass
(cid:88)
relevantmassdomain. larger than 100 M , one finds that the median planetary mass
⊕
The plot also includes HR 8799 with a measured [Fe/H]=- growsforanincreaseof[Fe/H]from-0.45to0.45onlybyabout
0.47 (which might however not reflect the initial metallicity, afactor1.5(from512to747M ).Foranincreaseoverthesame
⊕
Maroisetal.2008)andplanetsdetectedbydirectimagingi.e.at [M /M ]domainincontrast,themediangiantplanetmassin-
D SN
largeorbitaldistances(Maroisetal.2008,2010).HR8799also creasesfrom182to814M i.e.byaboutafactor4.5.
⊕
10 C.Mordasinietal.:ExtrasolarplanetpopulationsynthesisIV
value of M˙ = 0.04M˙ is hit. The consequence is as shown
env disk
by the figure that the clear correlation of disk mass and planet
massisnowbroken,asexpectedfromthefunctionalformofthe
equation: One finds that the maximal mass now depends only
weakly on the disk mass, approximatively as (M /M )1/4 for
D SN
M (cid:38) M .Aweakcorrelationremainsbecauseofthefloorac-
D SN
cretion rate. At lower disk masses, the correlation is somewhat
stronger,forthesamereasonasforthenominalpopulation.
Thefactthatforthenominalpopulation,maximalmassesare
linearly proportional to disk masses is interesting as one there-
fore expects to see an imprint of the disk gas mass distribution
ontheplanetarymassdistribution.Thisisreminiscentofthesit-
uationforstars,wheretheprotostellarcoremassfunctionCMF
hasaremarkablysimilarfunctionalformasthestellarIMF(e.g.
McKee & Ostriker 2007). This question will be addressed fur-
therinforthcomingwork.
5. Diskconditionsleadingtogiantplanetformation
5.1. Minimalsolidsurfacedensity
Withthepopulationsynthesiscalculations,onecanstudyapos-
teriori which combinations of disk properties allow the forma-
tion of giant planets, and in particular which are the most ex-
Fig.6. The mass of synthetic planets (in Jupiter masses) as a tremeones.ComparablestudieshavebeendoneinKornetetal.
function of the initial disk mass for the nominal population (2006), Dodson-Robinson et al. (2006), Ikoma et al. (2000) or
(black dots). The dotted line indicates a mass half as large as Thommesetal.(2008).
the initial gas disk mass, showing the linear correlation of disk Figure 7 shows the relative planetesimal surface density
massandmaximalplanetmassformostofthedomain.Thesolid Σ /Σ at 5.2 AU, where Σ = f Σ is the planetesimal
S S,SN S D/G 0
lineshowstheupperlimitingenvelopeinanon-nominalpopula- surface density in a given disk and Σ = f Σ = 8
tionwherethelimitingeffectofgapformationonthegasaccre- g/cm2 is the planetesimal surface denSs,iStNy in oDu/rGs,(cid:12)ola0r,SNnebula
tionrateistakenintoaccount,usingthefitofVeras&Armitage reference disk, which lead to the formation of a giant planet
(2004). (M ≥ 300M ),asafunctionofthestartingpositionoftheseed
⊕
embryo. The core accretion rate M˙ ∝ Σ (e.g. Alibert et al.
core S
2005a), therefore this quantity is within the core accretion for-
For[M /M ] (cid:46) −0.2,themaximalmassesdecreasefaster mationparadigmacentralcontrolparameter.Bluefilledcircles
D SN
then linearly with decreasing [M /M ] and are thus smaller representssyntheticplanetsofthenominalpopulationinallfour
D SN
than indicated by the dotted line. This is because at such low panels.Avalueofe.g.ΣS/ΣS,SN = 3thuscorrespondstoadisk
disk masses, the low resulting solid surface Σ = f Σ den- with roughly three times more solids than the amount we as-
S D/G 0
sities becomes important in a second order effect, so that the sumeforthesolarnebula.Inalldisks,thesolidsurfacedensity
final planetary mass is no more just determined by the amount scalesasa−1.5 withajumpofafactor4attheiceline,therefore
ofgasthatcanbeaccreted,butalsobythetimeneededtoform this value corresponds for a starting position of the embryo of
asupercriticalcore.Thisisaneffectthatisanalogoustothede- e.g.astart = 10AU(whichisoutsidetheicelineforallΣ0 con-
pendenceofthemaximalmassseenatlow[Fe/H].Notethatthe sidered here) to a local planetesimal surface density at astart of
exactvalueofthehighestefficiencyofconvertingdiskgasinto about3×8×(10/5.2)−1.5 =9g/cm2.
planetaryenvelopematerial(hereabout0.5)dependsontheex-
act treatment of the back-reaction of the planet’s accretion on
5.1.1. Sweetspotforgiantplanetformation
the disk, but is found to lie in a domain of 0.3 to 0.5. The lin-
earcorrelationbetweendiskmassandmaximalplanetarymass The plot shows that the sweet spot for giant planet formation
is however not affected by different treatments. This indicates (i.e. the lowest solid surface densities which allow that) occurs
thatapositivecorrelationofdiskmassandgiantplanetmassis at about 5 AU, at a value of about 0.75, corresponding to 6
astablepredictionofthecoreaccretiontheory,whilethequanti- g/cm2. This value is similar to the result of Dodson-Robinson
tativedegreeismodeldependent.Forthelargemajorityofgiant et al. (2006) who find that a solid surface density of about 6.5
planets (M ≥ 1M ) the efficiency of converting disk gas into g/cm2 (also at 5.2 AU) is needed to bring an embryo to run-
planetary material is lower and to order of magnitude 0.1, but away gas accretion in 7 Myr, which corresponds to the longest
(cid:88)
with a very wide spread of possible values, depending on e.g. living disks in our population (Paper I). At a starting position
the photoevaporation rate, the metallicity or the disk gas mass of1AU(whichisinsidetheiceline),aΣ /Σ ofatleast10
S S,SN
itself. isneeded,correspondingtoalocalplanetesimalsurfacedensity
ThesolidlineinFigure6indicatesagaintheapproximative of about 0.25×10×8×(1/5.2)−1.5 ≈ 240 g/cm2, which is in
upper envelope obtained for the non-nominal population using good agreement with Kornet et al. (2006). We see that both at
gas accretion rates limited because of gap formation according smallerandatlargerdistances,moremassivedisksofplanetesi-
totheVeras&Armitage(2004)fit.Inthiscase,theplanetarygas malsareneeded.Thisisinagreementwiththeworksmentioned
accretion rate is still proportional to M˙ but decreases expo- earlier.Theabruptincreasebyaboutafactor4atabout3-4AU
disk
nentiallyasexp(−M /1.5M )withplanetmassuntilafloor simplycorrespondstotheincreasenecessarytocompensatethe
planet
(cid:88)