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The formation and retention of gas giant planets around stars with a range of metallicities

a r X i v :a s t r o -p h /0408019v 1 2 A u g 2004The formation and retention of gas giant planets around stars with a range of

metallicities

Shigeru Ida

Tokyo Institute of Technology,Ookayama,Meguro-ku,Tokyo 152-8551,Japan

ida@geo.titech.ac.jp and D.N.C.Lin UCO/Lick Observatory,University of California,Santa Cruz,CA 95064lin@https://www.wendangku.net/doc/374559834.html, ABSTRACT The apparent dependence of detection frequency of extrasolar planets on the metal-licity of their host stars is investigated with Monte Carlo simulations using a determin-istic core-accretion planet formation model.According to this model,gas giants formed and acquired their mass M p through planetesimal coagulation followed by the emer-gence of cores onto which gas is accreted.These protoplanets migrate and attain their asymptotic semi-major axis a through their tidal interaction with their nascent disk.Based on the observed properties of protostellar disks,we generate M p -a distribution.Our results reproduce the observed lack of planets with intermediate mass M p =10–100M ⊕and a 3AU and with large mass M p 103M ⊕and a 0.2AU.Based on the simulated M p -a distributions,we also evaluate the metallicity dependence of fraction of

stars harboring planets that are detectable with current radial velocity survey.If pro-tostellar disks attain the same fraction of heavy elements which are contained in their host stars,the detection probability around metal-rich stars would be greatly enhanced because protoplanetary cores formed in them can grow to several Earth masses prior to their depletion.These large masses are required for the cores to initiate rapid gas ac-cretion and to transform into giant planets.The theoretically extrapolated metallicity dependence is consistent with the observation.This correlation does not arise naturally in the gravitational-instability scenario.We also suggest other metallicity dependence of the planet distributions that can be tested by on-going observations.

Subject headings:planetary systems:formation –solar system:formation –stars:statics

1.Introduction

One of the most important observed characteristics of known extrasolar planets is that their detection frequencyηJ increases with the metallicity Z?of their host stars(Fischer&Valenti 2003;Santos et al.2004).Although Z?could be signi?cantly changed by the accretion of plan-ets/planetesimals onto the host stars after they have evolved onto the main sequence,e.g.,(Sandquist et al.1998;Murray&Chaboyer2002),the lack of any Z?dispersion among members of stellar clus-ters(Quillen2002;Wilden et al.2002)indicate that the impact of this process is very limited.The ηJ-Z?correlation may also be interpreted as evidence that the formation probability of gas giant planets is greatly enhanced in metal-rich protostellar disks,in accordance with the core-accretion scenario for formation of giant planets,e.g.,(Mizuno1980;Bodenheimer&Pollack1986;Pollack et al.1996;Ikoma et al.2000).

In the conventional core-accretion scenario,heavy elements(metals)in a protoplanetary disk condense into grains which form km-sized planetesimals.Through gravitational interaction and coagulation,planetesimals evolve into cores through runaway(Greenberg et al.1978;Wetherill& Stewart1989;Aarseth et al.1993;Kokubo&Ida1996)and oligarchic growth(Kokubo&Ida1998, 2002).The cores’eccentricity is e?ectively damped by their tidal interaction with the ambient disk gas(Artymowicz1993;Ward1993).After the cores have swept up all residual planetesimals within a“feeding-zone width”～10Hill’s radii(Kokubo&Ida1998,2002),their growth in the inner regions of the disks is stalled with an“isolation”mass M c,iso.If a core attains the critical mass, M c,acc～several M⊕,before the disk gas depletion,rapid gas accretion onto it is initiated and it transforms into a gas giant planet.Since M c,iso increases with the surface density(Σd)of dusts (composed of metals)in protoplanetary disks,formation of gas giants tends to be more proli?c in a metal-rich environment.

Here,using the deterministic planet formation model developed by Ida&Lin(2004)(hereafter referred to as Paper I),we calculate the metallicity dependence of fraction(ηJ)of stars harboring planets that are detectable with the current radial velocity surveys.In§2,we brie?y describe our model.In§3,we simulate the mass(M p)–semimajor axis(a)distributions of extrasolar planets and their metallicity dependence through a series of Monte Carlo simulations.The theoretically constructed distributions are consistent with observed https://www.wendangku.net/doc/374559834.html,ing the simulated distributions,we evaluateηJ.Our model explains the observed metallicity dependence ofηJ.§4is summary and discussions.

2.Model

We carry out Monte Carlo simulations to reproduce the observed M p-a distribution of ex-trasolar planets.We basically follow the methods of Paper I except slightly di?erent choice of parameters.Initial conditions of the Monte Carlo simulations are the a of planets and the dust and gas surface density(Σg andΣd)at a in their nascent disks.For a given set ofΣg(a)andΣd(a),

we numerically compute the entire formation and migration sequence of individual cores and gas giant planets.Here we brie?y summarize our prescription.For details,see Paper I.

Since their observational data over the relevant length scales(a few AU’s)for planet-forming regions is not available,we scale their quantitative values with global factors f d and f g,to those of the empirical minimum mass nebula(MMN)model for our Solar system as

Σd=10ηice f d(a/1AU)?3/2[g cm?2],

(1)

Σg=2.4×103f g(a/1AU)?3/2[g cm?2],

where the compositional scaling parameterηice=1inside the ice condensation radius[a ice= 2.7(M?/M⊙)2AU and4.2outside it(Hayashi1981).[Note thatηice outside a ice can be slightly smaller(～3.0)(Pollack et al.1994).]The mass of the host star M?is scaled with that of the Sun M⊙.The dependence on the power-law index forΣd andΣg are examined in a separate paper.To represent the decline in the observational signatures of protostellar disks on timescalesτdep～1–10 Myrs(Haisch et al.2001),we assume exponential decay ofΣg as

f g=f g,0exp(?t/τdep),(2) due to both viscous evolution and photoevaporation.We neglect gas replenishment onto the disk and any time variation in f d.We discuss the distributions of f d and f g,0below.

With prescription(1),the cores’mass at time t after the formation of the disk is deduced to be(Paper I)

M c(t)? τdep1022g ?2/5 a M⊙ 1/2 1?exp ?2t

1AU 3/4

&Ida1999;Rice&Armitage2003)up to M c,noiso,which is given with M c,noiso?πΣd a2(feeding zone width?a～a)by(Paper I)

f d a

M c,noiso?1.2η3/2

ice

M⊕ ?c yrs,(6) with b=10and c=3,which is a?tting formula for Pollack et al.(1996)’s results with negligible planetesimal accretion.τKH for M p～1–10M⊕can be longer if modest planetesimal accretion induced by gas accretion is assumed[e.g.,“phase2”by Pollack et al.(1996)].But,the planetesimal accretion may be signi?cantly reduced as discussed above.In this case,phase2cannot be sustained. On the other hand,if the core’s migration is very fast,the growth of a core does not decline until it acquires mass～M c,noiso.In this case,phase2does not exist either(Alibert et al.2004).So,in the nominal case,we adopt(b,c)=(10,3),but(b,c)=(9,3)and(11,3.5)are also examined(the latter corresponds to the gas accretion including phase2).

The gas accretion rate increases rapidly with M p.Such runaway gas accretion is terminated when the residual gas is depleted globally on timescaleτdep or it is severely depleted locally in the vicinity of their orbits due to gap opening,which is assumed to occur if their Hill radius exceeds1.5 times the disk scale height(Paper I).When M p becomes a signi?cant fraction of M J(Jupiter mass), planets’tidal torque on the disk becomes larger than the viscous torque of the disk gas,resulting in a partial gap opening,e.g.,(Lubow et al.1999).It reduces gas accretion rate.Bondi accretion limit also becomes important in the reduction.However,the elongated accretion timescale may be still shorter thanτKH for M p～several M⊕,so that the limited accretion at high M p may not a?ect

the mass distribution of gas giants signi?cantly.The critical mass M c,acc for actual formation of gas giants corresponds to the value of M p for whichτKH～τdep.Equation(6)shows M c,acc～several M⊕.From eqs.(4)and(3),we?nd that,for su?ciently large f d,it is possible for M c(τdep)>M c,acc and M c,iso>M c,acc so that gas giant can form readily.

After the gap opening,the planets undergo“type-II”migration,in contrast to the cores’“type-I”migration without a gap(Ward1997).In order to compute the protoplanets’migration,we adopt theα-prescription for e?ective viscosity(Shakura&Sunyaev1973)with a uniformα=10?4for all the disks such that their viscous di?usion time scale(τν)at a～10AU is comparable toτdep=1–10 Myrs.The planets migrate(Lin&Papaloizou1985,1993)with a viscously evolving disk on time scales(Paper I)

τmig=a

M J a

homogeneity among young stars in the Pleiades cluster(Wilden et al.2002)and mature binary systems(Desidera et al.2004).

3.Numerical Results

The simulated M p–a distributions with?Z?≡Z??Z⊙=0are shown in Figure1a.This is the nominal case where m=1022g,M c,iso andτKH=1010(M p/M⊕)?3yrs are adopted.For comparison,the observed distributions are also plotted in Figure1b.The observational limits in the current Doppler survey,M p 100(a/1AU)1/2M⊕and a 3AU(around F,G,K dwarfs within ～50pc)are marked by dotted lines.A de?cit of planets with an asymptotic mass M p=10–100M⊕and a 3AU(within a“planet desert”)is apparent in Fig.1a.This sparsely populated region is primarily caused by the runaway gas accretion onto cores with M p several M⊕(Paper I). This region divides the terrestrial(rocky)planet,gas and ice giant domains in the planets’mass-period distribution.Planetary migration generally sharpens the boundaries of the domains(Paper I;Udry et al.(2003)).Since migration is slower for larger M p(Eq.[7]),relatively massive planets are less likely to migrate to inner regions,resulting in a de?cit of planets with M p 103M⊕at a 0.2AU(Fig.1a;also see discussion by Udry et al.(2003)).If largerαis adopted,τνbecomes short compared withτdep,so that even relatively massive planets can migrate signi?cantly,resulting in less clear de?cit.We found that the two planet-depleted regions exist for other?Z?as well, because the“planet desert”and the other are determined by the core accretion model for gas giants andτdep/τν,respectively.The observed distributions may exhibit the above two de?cits and are consistent with our theoretical models withτdep～τνat～10AU.Adopting the same condition, the simulated a-distribution of the gas giant planets can match with the observation[Figure1c; also see Armitage et al.(2002)and Trilling et al.(2002)].

The0.04AU cut-o?in the planets’a distribution in our results may also be less abrupt and more consistent with the observed distributions if we adopt more relaxed inner boundary conditions. Many of giant planets may have migrated toward the vicinity of their host stars(Lin et al.1996). The theoretically determined ratio of the giant planets with a<0.06AU(hot Jupiters)to those between0.2–2AU is～1–10for stars with Z?～Z⊙,which is order of magnitude larger than its observed value(～0.2).This discrepancy suggests that more than90%of the hot Jupiters which migrated to the stellar vicinity are either consumed(Sandquist et al.1998)or tidally disrupted(Gu et al.2003)by their host stars.Provided these disruptive events occur before the host stars become main sequence stars with relatively shallow convection zones,their apparent chemical homogeneity may be preserved to match the observation of the Pleiades cluster(Wilden et al.2002).As the migrating protoplanets sweep planetesimals along their paths,this process may self-regulate the amount of residual heavy elements in the disk.

Figures1c and d show that the normalized slopes of the a and M p distributions are independent of?Z?,because they are determined byτdep/τνand the core accretion model for gas giants, respectively.In these?gures,we do not include planets stalled at0.04AU,since a large fraction of

them may plunge into their host stars.

Although the normalized distributions are independent of?Z?,we found that formation and retention rates of gas giant planets rapidly increase with?Z?.In order to account for the?Z?-dependence in the M p and a distribution,we?rst describe how planet formation depends on f d[for details,see Paper I and Kokubo&Ida(2002)].In a disk similar to the MMN(f d=f g,0=1),for aM c,acc,the timescale required for M c to reach M c,acc is?τdep.The cores evolve into ice giants,similar to Uranus and Neptune.In the disk similar to th MMN,gas giants can form only slightly outside the ice boundary,because the cores may acquire M c,acc there within a few Myr and enter a rapid gas accretion phase prior to the global disk depletion.Gas giants can form inside a ice in some massive disks(with f d 5).Rocky cores are formed with M c,iso M⊕in the inner regions and ice giants are formed with M c(τdep) M⊕in the outer regions of low-mass disks with f d 0.5,without formation of gas giants.

Around metal-de?cient stars,formation e?ciency of gas giants is low because disks with suf-?ciently large f d are only at the tail end of the distribution.Around the metal-rich stars,the e?ciency is high because both dM c/dt and M c,iso increase rapidly with f d(eqs.3and4)and M c,iso M c,acc even interior to a ice.These account for the?Z?-dependence.In order to directly compare with the observed frequency of extrasolar planets,we plot in Figures2the theoretical de-termination for fraction(ηJ)of stars which bear giant planets detectable with the current Doppler survey,as a function of their metallicity.Here,the distribution of f d is the same as that in Figures 1.For each f d,a is selected as log(a j+1/a j)=0.2(j=1,2,...).We do not include planets stalled at0.04AU in the evaluation ofηJ.

These results indicateηJ increases linearly with?Z?,which is consistent with observational results of Fischer&Valenti(2003)(open circles in Figures2).As shown in Figure2a,ηJ does not depend on the choice ofτKH[(b,c)=(9,3),(10,3)or(11,3.5);τKH=10b(M p/M⊕)?c]nor that of asymptotic core mass[min(M c,iso,M c,noiso)or M c,noiso].The reasons are as follows.As long as τKH has strong dependence on M p,even signi?cant change in b results in relatively small change of M c,acc,the core mass that enables rapid gas accretion prior to disk gas depletion.Since in our simulations,most gas giant planets are formed with such parameter ranges where f d and a are relatively large and M c,iso M c,noiso,the distributions hardly change even if M c,noiso is adopted in entire f d and a(although it a?ects the distributions of terrestrial planets[Paper I]).

Although the absolute values ofηJ are higher/lower for faster/slower core accretion,the depen-dence on?Z?is similar(Figure2b).If disk depletion is caused by viscous di?usion,the assumption thatτdep～τνat～10AU is reasonable.However,the observation of dense clusters suggest that gas depletion in protostellar disks is primarily driven by photo-evaporation(Johnstone et al.1998)

in these clusters,where a few massive stars contribute to nearly the entire UV ionizing?ux.In order to consider the possibility of rapid gas depletion in a dense cluster,we carried out the addi-tional simulations withτdep=105–106yrs.In this case,formation rates of giant planets are lower, resulting in smallerηJ(Figure2b).But,the dependence on?Z?does not change also in this case. This general agreement is a generic feature of the core accretion scenario and the assumption for

f d and f g,0.It may not arise naturally from the gravitational instability scenario(Boss1997).

4.Discussion

Through Monte Carlo simulation using the deterministic planet formation model developed in Paper I,we?nd that the formation probability of gas giant planets increases rapidly with the metallicity of their host stars.Our model quantitatively accounts for observational correlation between stellar metallicity and the fraction of stars harboring giant planets that are detectable with the current Doppler survey.Our model also explains the two observationally suggested planet-depleted regions in M p-a distributions of extrasolar planets:the de?cit in intermediate mass M p= 10–100M⊕and a 3AU,and that in large mass M p 103M⊕and a 0.2AU.

Our model can be tested by more detailed observation.We show that the normalized slope of the a and M p distributions of gas giants and the planet-depleted regions are independent of ?Z?,because these are regulated byτdep/τνand the core accretion model.The lower limit mass M p,lim of hot Jupiters depend on?Z?as M p,lim∝10?(3/2)?Z?(Paper I).The outer boundary radius of the distribution of gas giants a out∝10(10/27)?Z?(Paper I),although interactions among gas giants,which are not taken into account in the present paper,may make the boundary less clear(Marzari&Weidenschilling2002).These predictions of?Z?-dependence can be tested by statistics of extrasolar planets with increased detection number in a few years.

Our results provide supporting evidences for the core-accretion scenario and they suggest that the detection of gas giant planets may be used to infer the presence of rocky planets around the same host stars.”Habitable planets”which can retain liquid water on their surface(Kasting et al. 1993)may have0.1M⊕ M p 10M⊕and a～0.8–1.5×(M?/M⊙)2AU.The results in Figure1 indicate that the frequency of”habitable planets”is comparable to or more than that of detectable giant planets(at present,more than100planets have been detected around F,G,K dwarfs within 50pc),although some fraction of the habitable planets may be destabilized by migrating giant planets.

REFERENCES