A SYSTEMATIC STUDY OF THE FINAL MASSES OF GAS GIANT PLANETS

Transcription

1 The Astrohysical Journal, 667:557Y570, 2007 Setember 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. A SYSTEMATIC STUDY OF THE FINAL MASSES OF GAS GIANT PLANETS Takayuki Tanigawa and Masahiro Ikoma Deartment of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo , Jaan; tanigawa@geo.titech.ac.j Received 2007 February 16; acceted 2007 May 29 ABSTRACT We construct an analytic model for the gas accretion rate onto lanets embedded in rotolanetary disks as a function of lanetary mass, viscosity, scale height, and unerturbed surface density, and systematically study the long-term accretion and final masses of gas giant lanets. We first derive an analytical formula for the surface density rofile near the lanetary orbit from considerations of the balance of force and dynamical stability. Using it in the emirical formula of normalized gas accretion rate that is derived based on hydrodynamic simulations, we then simulate the mass evolution of gas giant lanets in viscously evolving disks. We finally determine the final mass as a function of semimajor axis of the lanet. We find that the disk can be divided into three regions characterized by different rocesses by which the final mass is determined. In the inner region, the lanet grows quickly and forms a dee ga to suress the growth by itself before disk dissiation. The final mass shows the same trend as the mass determined by the viscous condition for ga oening, but is about 10 times larger than that. In the intermediate region, the disk s viscous diffusion limits gas accretion onto lanets before dee ga formation. The final mass can be u to the disk mass, when the disk s viscous evolution occurs faster than disk evaoration. In the outer region, lanets cature only tiny amounts of gas within the disk lifetime to form Netune-like lanets. We also derive analytic formulae for the final masses in the different regions and the locations of the boundaries, which are helful to gain a systematic understanding of the masses of gas giant lanets. Subject headinggs: methods: analytical methods: numerical solar system: formation 1. INTRODUCTION A fundamental but unresolved issue with lanet formation is how the mass of a giant lanet is fixed. In the solar system there are four giant lanets, which are characterized by their massive hydrogen/ helium enveloes. Juiter and Saturn consist mostly of hydrogen/ helium, while Uranus and Netune consist mostly of ice but have significant amounts of hydrogen/ helium. The giant lanets are different in mass: Juiter s mass (M J )is1; 10 3 M, Saturn s mass is 0.3 M J, and Uranus s and Netune s masses are 0.05 M J. Furthermore, the extrasolar lanets detected so far also range in mass from about one Netune mass to 10 M J ; 1 those lanets are believed to have massive hydrogen/ helium enveloes like the giant lanets in the solar system. There are two cometing ideas on the formation of giant lanets, the core accretion model (e.g., Mizuno 1980; Bodenheimer & Pollack 1986) and the disk instability model (e.g., Cameron 1978; Boss 1989). In the core accretion model, a rocky/icy core first forms through collisional aggregation of lanetesimals, followed by enveloe formation due to substantial accretion of gas from the circumstellar ( rotolanetary) disk. In the disk instability model, density fluctuation of the disk gas grows to form a gaseous lanet, followed by core formation due to sedimentation of heavy elements in its interior. The advantages and disadvantages of both models are discussed in the literature (e.g., Boss 2002) and are not reeated here. In this aer we consider giant lanet formation in the context of the core accretion model. In the core accretion model, the rocess of the accumulation of the enveloes is divided into two hases, the subcritical accretion and suercritical accretion hases, in terms of the dominant energy source. The transition from the former to the latter occurs when the mass of a core reaches a critical value. In the 1 See htt://exolanet.eu. 557 subcritical accretion hase, incoming lanetesimals suly energy to the enveloe so that the enveloe is in hydrostatic equilibrium. The core accretion thus controls the gas accretion in this hase. Phase 2 found by Pollack et al. (1996) is a art of the subcritical accretion hase. In the suercritical accretion hase, the energy sulied by lanetesimals is insufficient to kee the hydrostatic structure of the enveloe, so the enveloe substantially contracts and releases its gravitational energy, resulting in runaway accretion of the disk gas. The suercritical accretion hase can be further divided into two subhases. In the former subhase, the gas accretion is controlled by contraction of the enveloe and thus occurs on the Kelvin-Helmholtz timescale (Bodenheimer & Pollack 1986; Pollack et al. 1996; Ikoma et al. 2000; Ikoma & Genda 2006). However, because the gas accretion driven by the enveloe contraction accelerates raidly with time, the suly of the disk gas inevitably becomes unable to kee u with the demand of the contracting enveloe. Thus, in the latter subhase, the disk gas suly limits the gas accretion (Tanigawa & Watanabe 2002, hereafter TW02). In this aer we focus on the gas accretion in the latter subhase of the suercritical accretion hase, i.e., the hase in which the Kelvin-Helmholtz contraction of the enveloe is under way but the gas accretion is limited by disk-gas suly. After the onset of the runaway gas accretion (i.e., in the suercritical accretion hase), the lanet always exeriences the limited gas suly (i.e., the latter subhase) by the end of its growth. Although there is an idea that the masses of Uranus and Netune were fixed in the subcritical accretion hase because of dissiation of the disk gas ( Pollack et al. 1996), we exlore another ossibility that the masses of those lanets were fixed in the suercritical accretion hase. One idea is that the mass of a giant lanet is comletely fixed when the lanet oens a ga in the disk. From a hysical oint of

2 558 TANIGAWA & IKOMA Vol. 667 view, the ga oens when the gravitational scattering by the lanet overwhelms both viscous diffusion (the viscous condition) and ush due to the ressure gradient (the thermal condition; e.g., Lin & Paaloizou 1993). In normal rotolanetary disks, the final mass of a giant lanet is determined by the thermal condition (e.g., Ida & Lin 2004), which is equivalent to the condition that the Hill radius is equal to the disk scale height at the lanet s location. The final mass is thus determined only by temerature of the disk gas for given stellar mass and lanet s location. In rotolanetary disks similar to the minimum-mass solar nebula (Hayashi 1981), the thermal condition yields a final mass of aroximately 1 M J at 5 AU but redicts that the masses of giant lanets increase with the semimajor axis, which is in clear contradiction to the current configuration of the solar system. Furthermore, because in reality the ga is not a vacuum but a low-density region, continuous gas accretion through the ga takes lace, as suggested by several studies (e.g., Artymowicz & Lubow 1996). The effects of the subsequent accretion on the final masses of giant lanets have been oorly understood. Another idea is that gas accretion is truncated by deletion of the disk gas before ga oening (e.g., viscous dissiation and hotoevaoration of the disk gas). This means that the final mass of a giant lanet is the total mass of the disk gas that the lanet has catured until the disk gas dissiates comletely. This may be resonsible for the current configuration of the solar system, since it takes more time for outer giant lanets to accrete rincially because of their long orbital eriods. However, there have been no quantitative studies of this ossibility. The urose of this aer is to gain a systematic understanding of how the final mass of a giant lanet is determined. Secifically, we clarify (1) how much the mass of an accreting giant lanet increases via slow, continuous accretion through the ga, (2) when and where disk dissiation dominates ga oening, and finally (3) how the final masses of giant lanets deend on several disk arameters. To do so, we simulate the long-term accretion of a giant lanet in a self-consistent fashion. The essential ingredients of the modeling are global viscous evolution of the disk and the flow attern of the accreting gas, as described below; the latter effect in articular was comletely neglected by revious studies. Furthermore, we derive analytical formulas for the final masses of giant lanets and the boundaries between several different regimes. During the viscous evolution of the disk, gas radially migrates from outer regions to the site of giant lanet formation. The mass flux limits gas accretion onto giant lanets in some situations. Because of the limited CPU seed of comuters, azimuthally averaged one-dimensional simulations of the disk evolution are necessary in order to follow the long-term accretion. Several revious studies simulated the radial mass flux in viscously evolving disks and then calculated gas accretion rate onto giant lanets. Lecar & Sasselov (2003) and Guillot & Hueso (2006) also simly assumed that gas accretion rate onto giant lanets is always roortional to the viscous mass flux. Veras & Armitage (2004) and Alibert et al. (2004) used an aroximate formula for gas accretion rate that fits the results of two-dimensional hydrodynamic simulations of disk gas with an embedded lanet done by Lubow et al. (1999) and D Angelo et al. (2002). However, the formula was derived based on a limited number of numerical simulations. Hence, whether the formula alies in a variety of situations is in question. To understand the detailed attern of accretion flow and to obtain the gas accretion rate onto the lanet, several authors (e.g., Kley 1999; Lubow et al. 1999; D Angelo et al. 2002, 2003; Bate et al. 2003) erformed two- or three-dimensional hydrodynamic simulations of interaction between a lanet and disk gas. Since they simulated global disks, the effect of ga formation on accretion rate was automatically included. However, because of the timeconsuming simulations, they did not simulate the growth of giant lanets on timescales of the viscous disk evolution. This study is clearly different from the revious studies in that this study includes the effects of the flow attern of the accreting gas in simulating long-term growth of lanets embedded in globally evolving disks. To do so, we use the emirical relation between gas accretion rate and unerturbed surface density that was derivedbytw02(seex 3 for the details). They carried out detailed two-dimensional hydrodynamic simulations of gas accretion flow onto a lanet and demonstrated the imortance of the local flow attern on the accretion rate. In this aer, we first derive an analytical formula for radial surface density distribution of rotolanetary disks with an embedded lanet in x 2. Next, in x 3 we describe how to include the effect of the flow attern of accreting gas, following TW02, to calculate gas accretion rate onto giant lanets. In x 4, we simulate the long-term accretion of giant lanets, based on the rescrition given in xx 2 and 3, and obtain the final masses of giant lanets for wide ranges of several disk arameters. In addition, we derive analytical aroximate formulas for the final masses of giant lanets and the boundaries between several regimes. Finally, we conclude this aer in x SURFACE DENSITY We consider a lanet embedded in a gas disk, both of which are assumed to rotate with the Kelerian velocity around a star. We derive an aroximate formula for azimuthally averaged surface density as a function of radial distance to the central star, r. We adot a local coordinate system in which all quantities excet the surface density are indeendent of r. We assume steady states and no mass flow toward the central star. We do not consider migration of the lanet. The validity of these assumtions is discussed in x 4.4. We emloy a local coordinate system corotating with the lanet on a circular Kelerian orbit around the central star (a local shearingsheet aroximation). The origin of the coordinate system is the lanet s osition. The x-axis is on the line from the star through the lanet; the direction of the y-axis is the same as that of the velocity vector of the lanet. The equation of motion of the disk gas in the y-direction in this coordinate system is v v yv @v þ v y ; where is the surface density, v x and v y are the velocities in the x- and y-directions, is the viscosity coefficient, and v y is the forceactingonthegaserunitmass.herewehaveomittedtheadvection term in the y-direction because of axisymmetry. We also eliminate all the terms on the left-hand side of this equation on the assumtions of steady states (@/@t ¼ 0) and no mass flow (v x ¼ 0). Hereafter we consider only x > 0 because of symmetry. To derive an analytical formula for (x), we assume that v y is equal to the Kelerian velocity, v y ¼ 3 2 x, where is the Kelerian angular velocity at the lanet s osition. This assumtion is shown to be reasonable below. Equation (1) is thus transformed into 3 þ v y ¼ 0: The gravity of the lanet erturbs the motion of the disk gas. We follow the imulse rescrition (e.g., Lin & Paaloizou 1979) ð1þ ð2þ

3 No. 1, 2007 FINAL MASSES OF GAS GIANT PLANETS 559 that aroximates scattering of the disk gas as small-angle scattering of a article by a oint-mass object. On the rescrition, v y is given by v y ¼ 4 M 2 r 5 9 M 2 x 4 ; ð3þ where r is the distance of the lanet from the central star and M and M are the masses of the lanet and the central star, resectively. This aroximation is inaroriate for x P minf2r H ; r ½(M /M )/hš 1 =2 g,wherer H r (M /3M ) 1 =3,namely, the Hill radius, because this region is what is called the horseshoe region, in which gas is not ushed away from the lanet. However, since the gas in the horseshoe region has no contribution to the accretion flow of interest (see x 3), we neglect such a region. Note that Crida et al. (2006) roosed a more elaborate formula for the torque (a y-direction force in our formulation) that includes the radial transfer of torque through waves, which they found in their two-dimensional hydrodynamic simulations. However, since the formula is unable to aly for wide ranges of the arameters, as they mentioned, we do not use their formula in this study. Substituting equation (3) into equation (2), we obtain ðþ¼ x 1 ex x l 3 vis ðþ; x where 1 is the surface density at infinity (i.e., far away from the lanet s orbital radius) and l is defined as 2 l ¼ 0:146! 3 1 M 2 r 2 5 M 1=3 r ð4þ ð5þ! 1=3 M 2= r r : ð6þ M This surface density rofile is basically the same as that derived by Lubow & D Angelo (2006) excet for the value of the coefficient. The dynamical stability of the surface density rofile given by equation (4) must be checked using the well-known Rayleigh criterion (e.g., Chandrasekhar 1961). For the surface density rofile to be stable, in the local coordinate 2 must be fulfilled. The velocity v y can be obtained by solving the equation of motion for the x-comonent, 3 2 x þ 2 v y c ¼ 0; where c is the sound seed. Substituting equation (4) into equation (8), we obtain ð7þ ð8þ " v y ¼ 3 2 x 1 h 2 # x 5 ; ð9þ l l Fig. 1. Surface density rofile for M /M ¼ 10 3, /(r 2 ) ¼ 10 5,and h/r ¼ 0:1. Solid line, vis given by eq. (4); dashed line, R given by eq. (13). The two curves connect smoothly with each other at x ¼ x m (i.e., x ¼ 2:08h in this case). The thick line is the actual surface density we use in this aer. where h is the disk scale height defined as h c/. From equations (7) and (9), we find that condition (7) is not fulfilled for x < x m, where x m 12 1=5 h 2=5 l ¼ 12 1=5 l 3=5 h ð10þ l h! h 2=5 1=5 M 2=5 ¼ 0:207 0:1r 10 5 r r : M ð11þ In the region x x m, the density gradient thus has to be small enough to satisfy the Rayleigh condition: the density rofile would be relaxed to be marginally stable for the Rayleigh condition. y /@x ¼ 2 inward from x m we obtain v y ¼ v y;m 2 ðx x m Þ ð12þ in the region x x m. Substituting this into equation (8), we finally obtain " (x) ¼ 1 ex 1 x 2 h 5 x 2 m þ 1 x 2 m x # 3 m R ; 4 h 32 h l ð13þ for x x m. In summary, the equilibrium rofile of surface density is given as (x) ¼ R(x) for x x m ; ð14þ vis (x) for x x m : An examle of the density rofile is shown in Figure 1. As mentioned above, we have assumed v y ¼ 3 2 x in deriving equation (4). However, based on equation (9), the velocity at x ¼ x m is v y;m ¼ 11 8 x m : ð15þ

4 560 TANIGAWA & IKOMA Vol. 667 This means that the maximum deviation from the Kelerian shear velocity, 3 2 x, is less than 10% of the absolute value. Hence, the assumtion of the Kelerian shear velocity is reasonable for x x m. Finally, we briefly describe the deendence of the surface density on the arameters. Figures 2a, 2b, and 2c show the surface density as a function of x/r for several different values of lanetary mass M, viscosity, and scale height h, resectively. Figures 2a and 2b show that the ga becomes deeer and wider with increasing lanetary mass and decreasing viscosity. The width and deth of the ga are determined by cometition between gravitational scattering by the lanet and viscous diffusion of the disk gas. The ga becomes wider and deeer when the lanet s gravity is stronger or viscous diffusion is less efficient. From a mathematical oint of view, this is because both vis and R deend on M and in the form of (M 2/)1 =3 through l. As for the deendence of on h (see Fig. 2c), all the curves are indeendent of h for x x m,where ¼ vis, which is indeendent of h (see eq. [4]). In other words, surface density in x x m is determined by the balance between viscous torque and gravitational torque from the lanet and has nothing to do with ressure. On the other hand, difference in surface density aears at x < x m,where is determined by the Rayleigh condition (see eq. [13]) and thus deends on all of h,, andm through x m. Symbols used in this aer are listed in Table GAS ACCRETION RATE ONTO PLANETS In this study, we describe the gas accretion rate onto the lanet, Ṁ. Through a series of local high-resolution hydrodynamic simulations of the gas accretion flow onto giant lanets in rotolanetary disks, TW02 obtained an emirical relation between accretion rate normalized by unerturbed surface density and h/r H, which is the only arameter of the local system (see eq. [18] of TW02). Furthermore, TW02 demonstrated that only the gas in the two bands at jxj 2r H accretes onto the lanet. Thus, by writing equation (18) of TW02 in the exlicit form, for 0:5 h/r H 1:8, we have Ṁ ¼ Ȧ acc; ð16þ where Ȧ 0:29 h 2 M 4=3 r 2 r M ; ð17þ which is the area in which the gas is to be accreted onto the lanet er unit time (hereafter the accretion area), and acc ¼ (2r H ). The reason why small h or large M yields high Ṁ is exlained as follows. Disk gas loses its energy by assing through siral shocks around the lanet and consequently accretes onto the lanet. Small h (i.e., small sound velocity) or large M (i.e., strong erturbation on gas motion) yields a large Mach number of the disk gas, which results in strong shocks. In addition, an increase in M exands the lanetary feeding zone, resulting in large Ṁ. Note that the osition of the accretion band, x acc, is located at 2r H in the highest mass case of TW02 (i.e., C iso ¼ 0:5 in their notation; see Fig. 7 of TW02, where x acc is shown to deend slightly on h/r H ). When C iso < 0:5, a ga exists around the lanetary orbit. In such situations, aroriate choice of x acc is crucial in determining Ṁ because a small difference in x acc makes a large difference in acc and thus Ṁ, while the choice of x acc has little Fig. 2. Surface density rofiles as a function of distance from the lanetary orbit. (a) Planetary mass deendence: M /M ¼ 10 4 (solid line), (longdashed line), 10 3 (short-dashed line), (dotted line), and 10 2 (dotdashed line). (b) Viscosity arameter deendence: /(r 2 ) ¼ 10 7 (solid line), 10 6 (long-dashed line), 10 5 (short-dashed line), and 10 4 (dotted line). (c) Scale height deendence: h/r ¼ 10 2 (solid line), (long-dashed line), 10 1 (short-dashed line), and (dotted line). The standard values of the arameters are M /M ¼ 10 3, /(r 2 ) ¼ 10 5,andh/r ¼ The lus signs indicate where x ¼ x m, and crosses in (a) and vertical lines in (b) and (c) indicate where x ¼ 2r H ; 1 /e is also shown as a horizontal line in each anel.

5 No. 1, 2007 FINAL MASSES OF GAS GIANT PLANETS 561 TABLE 1 List of Notations Variable Meaning Definition M... Planetary mass... M... Mass of the central star R... M disk... Disk mass at the initial condition 10 2r ss (r; 0) dr M Bnal... Final mass of a lanet Eq. (23) M Bnal;ga... Final mass of a lanet in the ga-limiting region Eq. (36) M Bnal;diA... Final mass of a lanet in the diffusion-limiting region Eqs. (38) and (39) M trans... Planetary mass when acc is (1/e) 1 in the case of 2r H x m Eq. ( B2) M ;init... Initial lanetary mass... M ;crit... Maximum lanetary mass that does not violate a geometrical limit for eq. (17) Eq. (48) M ;local... Final mass if Ṁ is assumed to be Ṁ ;local Eq. (35) M ;disk... Final mass if Ṁ is assumed to be Ṁ disk Eq. (35) M ;vis... Planetary mass when viscous condition for ga oening is fulfilled Eq. (44) Ṁ... Accretion rate onto a lanet Eqs. (16) or (32) Ṁ ;local... Accretion rate onto a lanet using eqs. (14), (16), and (26) See x Ṁ disk... Radial mass flux of the disk due to viscous evolution Eq. (30) 1... Unerturbed surface density at lanet s orbit Eqs. (4), (13), (21), and (26) acc... Surface density at the accretion band (x ¼ 2r H ) Eq. (16) vis... Surface density determined by the balance between viscous and gravitational torques Eq. (4) R... Surface density determined by marginally stable state of the Rayleigh condition Eq. (13) ss... Surface density of self-similar solution for viscous evolving disk Eq. (27) 1;init... Initial unerturbed surface density at lanet s orbit Eq. (21) r... Semimajor axis of a lanet... r b... Boundary osition between the ga-limiting region and the diffusion-limiting region See x r e... Boundary osition between the diffusion-limiting region and the no-growth region See x l... Position where vis is (1/e) 1 Eqs. (4) or (5) x m... Position where the Rayleigh condition is marginally fulfilled for vis Eq. (10) div... Time when the accretion rate is the maximum Eq. (19) de... Exonential deletion timescale of a disk Eq. (21) vis... Viscous evolution timescale of a disk Eq. (29) lifetime... Effective disk lifetime (i.e., shorter one of de and vis ) Eq. (34)... Normalized arameter (surface density times disk deletion time) Eq. (24) influence on Ṁ in low-mass cases. That is why we adot the value of x acc (=2r H ) for the highest mass case of TW02. Note also that, as shown by D Angelo et al. (2003), the gas accretion rate obtained by two-dimensional simulations is usually higher than that obtained by three-dimensional simulations. Thus, the accretion rate given by equation (16) could be overestimated. We discuss this issue in x EVOLUTION In this section we show the evolution of the growth rate and mass of an accreting lanet. To gain a roer understanding of the basic behavior of the lanetary accretion, we first exlore two simle cases with no disk dissiation (i.e., constant 1 ) and with exonentially decreasing 1 in xx 4.1 and 4.2, resectively. Then we investigate the accretion of a giant lanet embedded in a viscously evolving rotolanetary disk in x 4.3. The numerical rocedure is as follows. Excet in x 4.1, we first calculate the unerturbed surface density, 1,fromequation(21) in x 4.2 or from equation (26) in x 4.3. For given values of the arameters, h/r and /(r 2 ), we then calculate acc and A,using equation (4) if x acc x m or equation (13) if x acc x m. Finally, we integrate Ṁ with resect to time using acc and A (see eq. [16]) to obtain the time evolution of the lanetary mass Case without Disk Dissiation We first show the evolution of the lanetary mass without global disk dissiation. In this case the inut arameters are 1 r 2/M, h/r,and/(r 2 ) General Proerties Figure 3 shows the evolution of the lanetary mass and the gas accretion rate for several values of the three arameters. Without global disk deletion, the only way to suress gas accretion is by oening a dee ga around the lanetary orbit by the lanet itself. As seen in Figure 3, the evolution can be divided into two hases. The first hase is growth without a ga (rega hase), while the second hase is growth with a dee ga (ostga hase). Proerties of the evolution in both hases are exlained in analytical ways below. In the rega hase, Ṁ is almost roortional to M 4 =3 (see Figs. 3c, 3f, and3i). In this hase, the lanetary mass is insufficient to oen a ga, so acc 1. Then we can easily integrate equation (16) with equation (17): Ṁ ¼ S 1 r 2 h ; r 2 "! M 1=3 ;init S 1 h 2 # t 4 M 3 r ; M =r 2 1 ð18þ where S corresonds to 0.29 in equation (17) and M ;init is the initial mass of the lanet that corresonds to the mass beyond which the gas accretion is limited by disk-gas suly. This equation

6 562 TANIGAWA & IKOMA Vol. 667 Fig. 3. Evolution of lanetary mass and accretion rate without disk dissiation. Figures in the left column show accretion rate onto the lanet as a function of time, those in the middle column show lanetary mass as a function of time, and those in the right column show accretion rate vs. lanetary mass. Figures in the to row show the deendence on surface density [ 1 /(M r 2 ) ¼ 10 6 (solid line), 10 5,10 4,10 3,and10 2 (dot-dashed line)], those in the middle row show the deendence on scale height [h/r ¼ 10 2 (solid line), ,10 1, and (dotted line)], and those in the bottom row show the deendence on viscosity coefficient [/(r 2 ) ¼ 10 7 (solid line), 10 6,10 5, and 10 4 (dotted line)]. The standard values of the arameters are 1 /(M r 2) ¼ 10 4, /(r 2 ) ¼ 10 5,andh/r ¼ imlies that if acc is constant, the gas accretion rate diverges at a time div ¼ 3! M 1=3 1 ;init 1 h 2 S M M =r 2 1 r! ¼ 4:8 ; 10 4 M 1=3 1 ;init M 10 5 M =r 2 h 2 ; 10 1:5 1 r ; ð19þ div corresonds to the end of the rega hase. In the ostga hase, Ṁ decreases with M in an exonential fashion (see Figs. 3c,3f, and 3i). Such a stee decrease in Ṁ is due to a stee decrease in acc with resect to M via l, x m, and r H (see eqs. [4] and [13]). We can derive aroximate exressions for the accretion rate in this hase, which shows Ṁ is inversely roortional to time as shown in Figures 3aY3c (see Aendix A for the details) Deendence on the Parameters We now see the deendence of the evolution of lanetary mass on the unerturbed surface density, 1 (Figs. 3aY3c). The evolution timescale is inversely roortional to 1 (e.g., Fig. 3b). This is simly because acc is roortional to 1, whereas Ȧ is indeendent of 1. The eak value of Ṁ is thus accordingly roortional to 1 (e.g., Fig. 3c). The deendence on scale height, h, is shown in Figures 3dY3f. In the rega hase, Ṁ is simly /h 2 and the evolution timescale is thus /h 2. This is because the accretion area A / h 2 (see

7 No. 1, 2007 FINAL MASSES OF GAS GIANT PLANETS 563 Fig. 4. Evolution of lanetary mass and accretion rate with disk dissiation when h/r ¼ 0:1, /(r 2 ) ¼ The three anels have the same axes as Fig. 3. The thick lines show different initial surface densities [ init /(M r 2) ¼ 10 6:5 (solid line), (long-dashed line), and (short-dashed line)], and evolutions without disk dissiation are shown as thin lines. eq. [17]) in the rega hase where acc ¼ 1. Even in the ostga hase, the deendence of Ṁ on h is the same as that in the rega hase if h (i.e., x m )issmall(h/r P 0:03). As long as h is so small that x m 2r H, acc is determined by vis and is thus indeendent of h (see eqs. [4] and [14]). When h is so large that x m 2r H, acc ¼ R, which deends on h. Hence, acc increases with h for large h. The transition between the low-h and high-h cases takes lace when x m ¼ 2r H, namely, from equation (11),! M 1=6 1=2 h ¼ 0:037r 10 3 M 10 5 r 2 : ð20þ The deendence on viscosity,, isshowninfigures3gy3i. In the rega hase, the evolution of Ṁ is almost the same for different values of because acc is almost constant ( 1 )in the rega hase and A is indeendent of. Thus, the timing when Ṁ reaches at the eak value does not deend on either (see eq. [19]). In the ostga hase, Ṁ at a given t increases with. In Aendix B, we have derived aroximate solutions for Ṁ in the ostga hase. From equations (B2) and (B3), one finds that M / in high- cases (i.e., x acc > x m ). In low- cases (i.e., x acc < x m ), Ṁ also increases with, but the deendence is rather weak (see eqs. [B7] and [B8]). In addition, the accretion rate is inversely roortional to time (see eqs. [B3] or [B8]), so the maximum accretion rate can be roughly estimated by equations (B3) or (B8) at t ¼ div Case with Exonential Disk Dissiation Next we examine the evolution of the lanetary mass in a simle case where the disk surface density decreases in an exonential fashion with a time constant of de : 1 ¼ 1;init ex t de ; ð21þ where 1;init is the initial surface density at infinity. Figure 4 shows the evolution of M and Ṁ for de ¼ and three values of 1;init. In the high- 1;init case [ 1;init /(M r 2) ¼ 10 4:5 ], Ṁ increases with time to reach a eak at t 1:5 ; and then decreases with time, which is similar to the evolution without disk dissiation shown in x 4.1. This is because div < de in this case. In the low- 1;init case [ 1;init /(M r 2) ¼ 10 6:5 ], on the other hand, Ṁ decreases without exeriencing a significant increase that occurs in the high- 1;init case. This is because div > de in this case ( div / 1 1;init ; see eq. [19]). Because of such different evolution of Ṁ,the mass evolution also differs between high- and low- 1;init cases (see Fig. 4, middle): The lanet catures a significant amount of gas to be a Juiter-like lanet in the high- 1;init case, whereas the lanet catures only a small amount of gas to be a Netunelike lanet in the low- 1;init case. The boundary between the two regimes is determined by the condition de ¼ div,namely, from equation (19), 1;init M =r 2 ¼ 3 S M 1=3 ;init h M r Figure 5 shows the final mass defined by M Bnal Z 1 0! 2 1 de : ð22þ Ṁ dt 1 ð23þ as a function of a quantity defined by!! 1;init de M =r 2 1 : ð24þ Fig. 5. Final mass of lanets as a function of (defined at eq. [24]) in the case with h/r ¼ 0:1, /(r 2 ) ¼ The three lines are for different initial lanetary masses (M ;init /M ¼ 1 ; 10 5,3; 10 5,and1; 10 4 ).

8 564 TANIGAWA & IKOMA Vol. 667 Fig. 6. Contour lines of log (M Bnal /M )asafunctionof and /(r 2 ). Left, h ¼ 0:1r ; right, h ¼ 10 1:5 r. The initial mass is set to 10 5 M. ThedeendenceofM Bnal on de is the same as that on 1;init because the ratio of div (/ 1 1;init )to de determines the evolution. Figure 5 illustrates that when is small (P1), the final mass is almost the same as the initial mass. In this regime, the lanet catures only a tiny amount of gas and becomes a Netunelike lanet. When 10, on the other hand, the lanet catures enough gas to be a gas giant lanet like Juiter, and the final mass does not deend on the initial lanetary mass. This is because the lanet becomes large enough to form a ga and the evolution slows down significantly after ga formation; the condition for ga oening is determined by the lanetary mass. The transition occurs when div ¼ de, namely, from equation (19), M 1=3 ;init h 2 t 4: : ð25þ M 0:1r Figure 6 shows that the final mass as a function of and in the cases with h/r ¼ 0:1 and A ortion characterized by a stee gradient is illustrated to shift to the left with decreasing h/r ( t 5 for h/r ¼ 0:1, while t 0:1 for h/r ¼ 0:032; see eq. [25]). On the left side of the ortion (low- case), the final mass is almost equal to the initial mass and thus almost indeendent of viscosity. On the right side, the final mass increases with viscosity. This is because the growth is limited by oening a dee ga; the mass for ga oening deends on viscosity Case with Global Disk Evolution The Model We have so far neglected global evolution of the disk, although we examined a simle case with an exonential deletion of the disk gas in x 4.2. In reality, the unerturbed surface density, 1, changes because of global disk evolution. In this section, we examine the effects of global disk evolution on lanetary growth. Because a detailed descrition of disk evolution is beyond the scoe of this aer, we assume that the disk surface density changes with time in such a way that 1 ¼ ss (r; t) ex t ; ð26þ de instead of equation (21): ss reresents a change in the surface density due to viscous diffusion of the disk gas and e t = de is introduced to mimic hotoevaoration of the disk. For ss,we adot the self-similar solution with -rescrition for disk viscosity given by (Hartmann et al. 1998), ss (r; t) ¼ M disk 2R 2 out r R out 1 3=2 ss ex r=r out ; ð27þ ss where M disk is the initial disk mass and R out is the initial disk size; ss is defined as ss ¼ t vis þ1; with a tyical timescale of global viscous evolution ð28þ vis R out 2 3 out ¼ 5:3 ; h 2 1AU R out 0: :5 yr; AU 100 AU ð29þ where out is the viscous coefficient of the disk gas at r ¼ R out.in deriving equation (27), we have assumed a temerature distribution, T / r 1 =2, which results in / r. Using the equations above, we calculate the local gas accretion rate onto the lanet, denoted hereafter by Ṁ ;local, in the same way as we did in x 4.2. However, when Ṁ ;local is larger than the radial mass flux due to viscous diffusion in the disk, the latter limits the lanetary growth. The mass flux through a ring with radius r is given by Ṁ disk (r; t) ¼ 2r ss (r; t)v r ¼ M disk vis 1 2 r ss R out 3=2 ss ex r ss R out ; ð30þ

9 No. 1, 2007 FINAL MASSES OF GAS GIANT PLANETS 565 Fig. 7. Contour lines of log (M Bnal /M ) when global viscous evolution is considered in the case with de ¼ The horizontal axis is normalized surface density ; 10 6 [i.e., it corresonds to when 1;init ¼ ss (r ; 0)], and the corresonding disk mass assuming R out ¼ 10r is also shown on the to. The vertical axis is normalized viscosity, and the corresonding is also shown on the right. The left anel shows the case with h/r ¼ 0:1, and the right anel shows that with h/r ¼ 0:032. with the radial drift velocity of the diffusing gas in a Kelerian disk v r ¼ lnðþ þ 1 : ð31þ ln r 2 Thus, we calculate the gas accretion rate onto the lanet in such a way that ( Ṁ ;local if Ṁ ;local < Ṁ disk ; Ṁ ¼ ð32þ Ṁ disk if Ṁ ;local > Ṁ disk : There are several studies that modeled gas accretion rate onto gas giant lanets. Guillot & Hueso (2006) used a formula in which the gas accretion rate onto the lanet is 0.3 times the radial mass flux of the disk gas due to global viscous diffusion. However, their formula is an emirical one and alies in a limited situation. Lubow et al. (1999) carried out a series of hydrodynamic simulations to obtain gas accretion rate onto lanets. Their simulations demonstrated that the lanetary accretion rate can be larger than the diffusion flux because of gradients imosed by the ga. However, what they found is robably a transient henomenon that occurs on a timescale much shorter than the viscous timescale. Veras & Armitage (2004) used an aroximate formula based on hydrodynamic simulations done by Lubow et al. (1999) and D Angelo et al. (2002) Deendence on the Disk Parameters Figure 7 is similar to Figure 6 and shows the final mass of the lanet, M Bnal, as a function of the initial surface density, 1;init,and viscosity,, in the case where h/r ¼ 0:1 (left) and (right), R out ¼ 10r,and de ¼ Thevalueof de corresonds to ¼ 10 6 ½ 1;init /(M r 2 )Š, so the ranges of Figure 7 are the same as those of Figure 6 for the horizontal axis as well as the vertical axis. Since we have adoted the self-similar solution for the surface density evolution of the global disk, there aear two additional limits to the final mass. One arises from the aearance of the viscous timescale vis, in addition to the timescale of exonential decay de.when vis < de, the effective disk lifetime is vis.thus, in order for lanets to be massive, the growth timescale div should be shorter than vis, so the additional limit is given by vis ¼ div : r 2 7 ; 10 5 M 1=3 ;init h M 0:1r! 1 R out ; 10 4 M =r 2 ; ð33þ 10r which runs from the lower left to the uer right in Figure 7 because is linearly roortional to the surface density. As described above, the effective disk lifetime, lifetime, is the shorter of the two timescales, so we here define lifetime ¼ de if de < vis ; ð34þ vis if de > vis : The other limit is the total mass of the disk. This limit does not aear in Figure 6 (i.e., the case without global viscous evolution). This can be clearly seen in the case of h/r ¼ 0:032 on the right anel of Figure 7 where contour lines are vertical. Note that the stee boundary (being equivalent to ¼ t ) characterized by div ¼ de that is seen in Figure 6 is not found in Figure 7. This is simly because the two limits described above are more severe than the condition of div ¼ de in the cases shown in Figure 7. However, when de is smaller, the boundary determined by div ¼ de moves rightward, so the vertical stee boundary determined by div ¼ de emerges even in the case with global disk evolution described in this subsection Classification by Semimajor Axis In Figure 7, the deendence of the final mass was shown for the normalized disk arameters. However, one might want to

10 566 TANIGAWA & IKOMA Vol. 667 where M trans, defined by equation (B2), is a lanetary mass when acc ¼ 1 /e and 2r H x m. Excet for the weak deendence on the other arameters included in the log term, the final mass given by equation (36) is determined rincially by and increases with. This is because larger lanetary mass is required for the lanet to oen a ga in the case of higher viscosity. Since M trans / /(r 2 )and / r in our disk model, M trans / r 1 =2, which is consistent with the result shown in Figure 8. In the intermediate region (1 AU P r P 100 AU), Figure 8 illustrates M Bnal M ;disk : ð37þ Fig. 8. Final mass of lanets as a function of semimajor axis in AU. The solid line shows the standard case: ¼ 0:01, h/r ¼ 0:032 at 1 AU, de ¼ 10 6 yr, M disk ¼ 1:3 ; 10 2 M, and R out ¼ 100 AU. The dashed line shows M ;disk (i.e., mass when all viscous-accreting gas is assumed to accrete onto lanets), and the dotted line shows M ;ss (i.e., mass when the global viscous evolution with selfsimilar solution is not assumed). The initial mass of the lanets is set as 3:2 ; 10 5 M, which corresonds to 10 M in the solar mass system. know its deendence on semimajor axis. We thus show the final mass as a function of semimajor axis in Figure 8 for a tyical case with ¼ 0:01, h/r ¼ 0:032 at 1 AU, de ¼ 10 6 yr, R out ¼ 100 AU, M ;init ¼ 3:2 ; 10 5 M, which corresonds to 10 M in the solar mass system, and M disk ¼ 0:013M, which yields a surface density roughly 0.1 times that of the minimum-mass solar nebula at 1 AU. 2 In addition to the final mass given by equation (23), two kinds of mass defined by M ;local Z 1 0 Ṁ ;local dt and M ;disk Z 1 0 Ṁ disk dt ð35þ are shown to clarify the limiting rocess for the final mass. As shown in Figure 8, the final mass (solid line) is determined basically by the smaller of M ;local (dotted line)andm ;disk (dashed line), although the limiting rocess varies with time. Thus, one finds three characteristic regions that are divided at the intersection of the two curves. In the inner region (r P 1AU),M Bnal is almost equal to M ;local and is roortional to r 1 =2. The duration of the rega hase is shorter than the disk lifetime (i.e., div < lifetime )inthisregion.the gas accretion is thus suressed because of ga oening. This region is hereafter called the ga-limiting region. In this situation the final mass can be roughly estimated by integrating Ṁ from div to lifetime.since div < lifetime in the ga-limiting region for the values of the arameters used in drawing Figure 8, M Bnal is readily calculated as (see Aendix B) Z lifetime M trans M Bnal div t ¼ M trans log dt lifetime div M Bnal;ga ; ð36þ 2 This surface density is much lower than in the minimum-mass solar nebula model because the sloe of the surface density is shallower and the disk size is larger. In the region, viscous diffusion of the disk gas limits the gas accretion onto the lanet. This region is hereafter called the diffusion-limiting region. In this region, radial mass flux due to viscous diffusion is uniform where r TR out, so the final mass is usually insensitive to r. In the case with vis < de, which is the case of Figure 8, most of the disk gas is transferred toward the central star by viscous diffusion, and thus the final mass is roughly the disk mass ( 0:013M ). Hence, we have M Bnal;diA M disk : ð38þ On the other hand, when vis 3 de, evaoration dominates viscous diffusion in disk dissiation. In this situation, the final mass at rtr out is given by M Bnal;diA Z de Ṁ disk de vis dt M disk : ð39þ Note that, even when vis 3 de, the gas accretion rate onto the lanet is limited by viscous diffusion. In the outer region (r k 100 AU), the final mass suddenly decreases to M ;init. This is because the lanetary growth finishes in the rega hase because of the dissiation of the disk gas (i.e., div k lifetime ). We call this region the no-growth region. The ositions of the boundaries between the three regions are given as follows. At the boundary between the ga-limiting region and the diffusion-limiting region (its osition being denoted by r b ), M Bnal;ga ¼ M Bnal;diA.When vis < de, from equations (36) and (38), we obtain 2 h 4 1AU r b :5 AU M 2 disk logð vis = div Þ 2 ; 10 2 : ð40þ M 5 Here we have assumed the log factor in equation (36) is constant. When vis > de, from equations (36) and (39), we have 2 h 4 1AU M 2 disk r b :5 AU 10 2 M 2 2 de log de = div ; : ð41þ vis 5

11 No. 1, 2007 FINAL MASSES OF GAS GIANT PLANETS 567 of the mass given by the viscous condition for ga oening (Lin & Paaloizou 1979, 1993): M ;vis ¼ 40 r 2 ¼ 4 ; 10 4 M 10 2 ð44þ r 1=2 M : ð45þ 1AU The deendence also agrees with that of M trans, which is the mass when acc is (1/e) 1 in the case of 2r H x m (see eq. [B2]), and rewritten as M trans 9 ; 10 4 r 1= M : ð46þ 1AU Fig. 9. Final mass of lanets as a function of semimajor axis in AU for several disk models. The thick solid line shows the standard case as described in Fig. 8. The thick long-dashed line shows the case where M disk is 10 times smaller, the thick short-dashed line shows that where viscous is 10 times smaller, the thick dotted line shows that where de is 10 times shorter, the thick dotyshortdashed line shows that where the scale height is 3 times higher, and the thick dotylong-dashed line shows that where the gas accretion rate is reduced by a factor of 10. The thin solid line and thin dashed line show the masses determined by the viscous and thermal condition ( ¼ 0:01), resectively. The location of the boundary between the diffusion-limiting region and the no-growth region is basically determined by div lifetime and is denoted by r e.when vis < de, from div ¼ vis,weobtain M 1=3 ;init 1;1 AU r e M 10 5 M =AU 2 h 4 1AU 1 R out ; 10 1:5 AU 10 2 AU: 100 AU When vis > de, from div ¼ de, we obtain M 1=3 ;init 1;1 AU r e M 10 5 M =AU 2 h 2 1AU de ; 10 1:5 AU 10 6 AU: yr ð42þ ð43þ Figure 9 shows the final mass for several different values of the disk arameters. The discussion above suffices to understand this figure. Note that there is a hollow at 50 AU in the case where the value of is lower by a factor of 10 than its nominal value (thick short-dashed line). This is, however, artificial. In the self-similar solution we have adoted, the direction of the radial gas flow due to viscous diffusion changes at r ¼ (R out /2) ss,which means that the mass flux is zero at the oint, and the accretion rate onto the lanet is accordingly also zero (see eq. [32]). Since ss increases with time (see eq. [28]), the oint moves outward with time. When de > vis, the oint moves significantly with time and integration of mass flux at a articular oint eliminates the effect. In contrast, when de T vis, the oint hardly changes before the exonential deletion; the lanet around r ¼ R out /2 thus cannot grow significantly. Another imortant asect is on the final mass in the ga-limiting region. We can see that the r deendence of M Bnal agrees with that Both of the two masses indicate ga-forming masses, and they agree with each other within a factor of about 2. However, the final mass shown in our model is larger by a factor of 10 than M ;vis or M trans. This is because the two masses corresond to the masses at which a ga is about to form and does not mean the masses at which growth stos. A further increase in M after reaching M ;vis or M trans is reflected in the log term in equation (36) Remarks on the Uncertainties in the Model We have assumed that the distribution of the disk gas around the lanetary orbit (i.e., the surface density) is always in the equilibrium state that is determined by the balance between viscous stress and gravitational scattering by the lanet. However, for the equilibrium state to be achieved, nonequilibrium distributions must be relaxed by diffusion. The diffusion timescale, vis;local, is estimated by the ga width (2 ; 2r H ) divided by the viscosity as! vis;local ¼ 7:7 ; 10 3 M 2= M 10 5 r 2 1 : ð47þ This timescale should be comared with the tyical growth timescale (i.e., div ; see eq. [19]). The comarison indicates that vis;local < div in most of the cases resented in this aer. Hence, the assumtion is aroriate unless quite low viscosities ( yielding large vis;local ) or high surface densities (yielding small div ) are considered. The emirical formula for the accretion rate based on local two-dimensional isothermal hydrodynamic simulations ( TW02) can be different from that derived based on more realistic simulations including, for examle, three-dimensional accretion flow (D Angelo et al. 2003; Bate et al. 2003), a nonisothermal equation of state (TW02), and a magnetic field (Machida et al. 2006). The modification would change the form of div, which would yield quantitatively different results. However, even in that case, by following our rescrition given in this aer, one can easily calculate the mass evolution of lanets in a similar way. The formula given by TW02 can be considered as the highest limit; thus, for examle, r e is exected to shift inward if more realistic models for the accretion rate are used. We comare the accretion rate (eq. [16]) with those obtained by global simulations (Kley 1999; Lubow & D Angelo 2006). Although the value of our accretion rate is usually larger (by u to a factor of 10) than those given by the simulations, the deendence on viscosity and lanetary mass is consistent.

12 568 TANIGAWA & IKOMA Vol. 667 We have not taken into account a limit in terms of geometry of the accretion flow, namely, one within which equation (17) is alicable. We can make a simle estimate of the maximum accretion rate if we assume all the gas within 0 x 2 ffiffi 3 r H in y > 0and 2 ffiffiffi 3 r H x 0iny < 0 accrete to the lanet, where x ¼ 2 ffiffiffi 3 r H at y 3 r H is the oint where otential energy on the rotating frame is the same as that at the Lagrange oints L1 or L2. In this case, the maximum accretion rate (normalized by surface density) is given by 3 1 =3 6r 2 (M /M ) 2 =3, and thus A (eq. [17]) becomes larger than the maximum value when M > 5:6 ; 10 3 h 3 M M ;crit: ð48þ 0:032r Although lanetary mass can be larger than M ;crit deending on the arameters, such a massive lanet should already have a dee ga, so the accretion rate is reduced greatly when the lanetary mass reaches M ;crit. Consequently, the geometric effect has no significant influence on the final mass. Note that the discussion here is based on an aroximation that streamlines can be well described by article motions on the framework of the restricted three-body roblem. Since this aroximation is not valid when r H < h (Masset et al. 2006), it is not alicable for the cases of small-mass lanets. However, we consider the uer limit of alicable mass, where the lanet mass is, in most cases, large enough to satisfy r H > h, so the aroximation is valid for the situations that we consider here. In this aer, we have ut off the issue of lanetary migration. Since we consider a hase in which a ga exists around the lanet, we may have to consider the tye II migration (e.g., Ward 1997; Ward & Harn 2000), esecially in the ga-limiting region (e.g., r < 1 AU in Fig. 8). The dee ga created by the lanet blocks the accretion flow toward the central star. It follows that the lanet is ushed inward by the gas exterior to the lanet s orbit. We will include the effect of lanetary migration in our future work. 5. SUMMARY To gain a systematic understanding of the final masses of gas giant lanets, we have simulated the long-term accretion of gas giant lanets after the onset of the suercritical gas accretion in a variety of situations, deending on four disk arameters such as disk mass, viscosity, scale height, and semimajor axis. To do so, we have made a semianalytical model to simulate the mass evolution, which enabled us to study the final mass of gas giant lanets for extensive ranges of all the arameters. We have first made a one-dimensional analytical model of the equilibrium surface density rofile around a rotolanet, from consideration of the balance of torque and the dynamical stability (x 2). Combining the surface density rofile with an emirical formula for gas accretion rate that was obtained on the basis of hydrodynamic simulations by TW02, we have obtained a formula for gas accretion rate as a function of lanetary mass, viscosity, scale height, and unerturbed surface density (x 3). We have then integrated the gas accretion rate numerically with resect to time to simulate the long-term accretion of gas giant lanets (x 4). To understand the basic behavior of the lanetary accretion, we have exlored two simle cases with no disk dissiation (x 4.1) and with exonentially decreasing surface density (x 4.2). Finally, we have simulated the long-term accretion of gas giant lanets embedded in a viscously evolving and evaorating disk to obtain the final mass of gas giant lanets as a function of semimajor axis (x 4.3). We have consequently found the following three different regions deending on limiting rocesses on the final mass. In the inner region (r P r b ; see eqs. [40] and [41]), the lanet grows quickly to form a dee ga to suress the gas accretion from the disk by itself within the disk lifetime ( ga-limiting region). We have found that the final mass in this region is roughly 10 times larger than that determined by the viscous condition for ga oening (Lin & Paaloizou 1993). This is because the condition for ga oening only exresses the condition when a ga begins to form, and is by no means equivalent to the condition that the growth is terminated. In the intermediate region (r b P r P r e ; see eqs. [42] and [43]), radial transfer of the disk gas toward the lanetary orbit limits the gas accretion before the lanet oens a dee ga; the final mass is thus limited by viscous diffusion of the disk ( diffusion-limiting region). We have found that when the evaoration timescale de is shorter than the viscous-diffusion timescale vis, the relationshi between the final mass M Bnal and the disk mass M disk is given by M Bnal 1 2 ( de/ vis )M disk, whereas M Bnal M disk when de > vis. In the outer region (r k r e ), the lanet catures only a tiny amount of gas by the time the disk gas comletely dissiates ( nogrowth region). Saturn and ossibly Uranus/Netune are likely to have exerienced the situation. In this study, we have gained a clear understanding of the final masses of gas giant lanets, deriving analytical exressions for them in three characteristic regions (eqs. [36]Y[39]) and the locations of the boundaries between the three regions (eqs. [40]Y[43]). To understand the mass-eriod distribution of gas giant lanets in extrasolar systems found by radial velocimetry, we need to take several additional rocesses into consideration. Planets in the ga-limiting region would be esecially suscetible to the tye II migration because the gas exterior to the lanetary orbit blocked by the ga ushes the lanet inward. Inclusion of lanetary migration is our future work. In addition, inclusion of core accretion rocesses and the gas accretion rocess governed by the Kelvin-Helmholtz contraction of the enveloe is needed esecially to determine the initial mass and the origin time of our model. Although the final masses of gas giant lanets were focused on in this aer, the accretion rocess for reaching the final mass is also imortant to resolve issues relevant to lanet formation. Growing giant lanets dynamically affect other bodies in a lanetary system. Satellites are likely to form in subdisks around accreting gas giant lanets (e.g., Canu & Ward 2002, 2006). The longterm accretion of gas giant lanets may affect the internal structure and evolution of isolated young gas giants (Marley et al. 2007). This would be imortant for future direct detection of young gas giants. We are grateful to S. Ida for fruitful discussion and continuous encouragement. We also thank H. Tanaka for critical comments on our modeling. Valuable comments and suggestions from the anonymous referee were quite helful in imroving this aer. This work was suorted by Ministry of Education, Culture, Sorts, Science and Technology of Jaan (MEXT) Grant-in- Aid for Scientific Research on Priority Areas, Develoment of Extrasolar Planetary Science (MEXT ).