ANGULAR MOMENTUM ACCRETION ONTO A GAS GIANT PLANET · 2008-12-04 · x˜; 0; ð14Þ ˜ 0 ¼ 1 ffiffiffiffiffiffi 2 (specific) angular momentum p exp ˜z2 2 at the Jovian orbit when

ANGULAR MOMENTUM ACCRETION ONTO A GAS GIANT PLANET

Masahiro N. Machida,1Eiichiro Kokubo,

2Shu-ichiro Inutsuka,

1and Tomoaki Matsumoto

3

Received 2008 January 21; accepted 2008 May 28

ABSTRACT

We investigate the accretion of angular momentum onto a protoplanet system using three-dimensional hydro-dynamical simulations.We consider a local region around a protoplanet in a protoplanetary disk with sufficiently highspatial resolution. We describe the structure of the gas flow onto and around the protoplanet in detail. We find that thegas flows onto the protoplanet system in the vertical direction, crossing the shock front near the Hill radius of theprotoplanet, which is qualitatively different from the picture established by two-dimensional simulations. The spe-cific angular momentum of the gas accreted by the protoplanet system increases with the protoplanet’s mass. Ata Jovian orbit, when the protoplanet’s mass Mp is MpP 1MJup, where MJup is the Jovian mass, the specific angularmomentum increases as j / Mp. On the other hand, it increases as j / M2/3

p when the protoplanet’s mass isMpk1MJup.The stronger dependence of the specific angular momentum on the protoplanet’s mass for MpP 1MJup is due tothe thermal pressure of the gas. The estimated total angular momentum of a system of a gas giant planet and acircumplanetary disk is 2 orders of magnitude larger than those of the present gas giant planets in the solar system.A large fraction of the total angular momentum contributes to the formation of the circumplanetary disk. We alsodiscuss the formation of satellites from the circumplanetary disk.

Subject headinggs: accretion, accretion disks — hydrodynamics — planetary systems —planets and satellites: formation — solar system: formation

1. INTRODUCTION

Up to the present time, more than 200 extrasolar planets (or‘‘exoplanets’’) have been detected, mainly by measuring the ra-dial motion of their parent star along the line of sight. Almost allexoplanets observed by this method are giant planets, like Jupiterand Saturn in our solar system, because massive planets are pref-erentially observed. Although these planets are supposed to havebeen formed in the disk surrounding the central star (i.e., the cir-cumstellar disk or protoplanetary disk), their formation processhas not yet been fully understood. In the core accretion scenario(Hayashi et al. 1985), a solid core or protoplanet with ’10M�,where M� is the Earth mass, captures a massive gas envelopefrom the protoplanetary disk by self-gravity to become a gas giantplanet.

The evolution of gaseous protoplanets has been studied withthe approximation of spherical symmetry including radiative trans-fer (e.g., Mizuno 1980; Bodenheimer & Pollack 1986; Pollacket al. 1996; Ikoma et al. 2000). Ikoma et al. (2000) showed thatrapid gas accretion is triggered when the solid core mass exceeds’5Y20M�, and the protoplanet quickly increases its mass by gasaccretion. However, the angular momentum of the accreting gaswas ignored in these studies, because they assumed spherical sym-metry. Since the gas accretes onto the solid core with a certainamount of angular momentum, a circumplanetary disk formsaround the protoplanet, analogously to the formation of a proto-planetary disk around a protostar. The difference in the disk for-mation between the protostar and protoplanet is the region fromwhich the central object acquires the angular momentum. Theprotostar acquires the angular momentum from a parent cloud,

while the protoplanet acquires it from the shearing motion in theprotoplanetary (circumstellar) disk. In addition, the gravitationalsphere of the protostar spreads almost infinitely, while the gravi-tational sphere (i.e., the Hill sphere) of the protoplanet is limitedto the region around the protoplanet because the gravity of the cen-tral star exceeds that of the protoplanet outside the Hill sphere.Numerical simulations are useful for investigating the gas ac-

cretion onto a protoplanet and its circumplanetary disk (here-after, we just call them a ‘‘protoplanet system’’). Korycanskyet al. (1991) studied giant planet formation using a one-dimensionalquasi-spherical approximation with angular momentum transfer.They showed that as the protoplanet contracts, outer layers of theenvelope containing sufficiently high specific angular momentumremain in bound orbit and form a circumplanetary disk. However,due to their assumption of spherical symmetry, the accretion flowfrom the protoplanetary disk to the protoplanet could not be in-vestigated in their study. In order to study the accretion flowonto aprotoplanet and the acquisition process of the angular momentumin detail, a multidimensional simulation is necessary. Sekiya et al.(1987) investigated the gas flow around a protoplanet with rela-tively low resolution and found that the spin rotation vector of theprotoplanet becomes parallel to the orbital rotation vector. Re-cently, the flow pattern has been carefully investigated by manyauthors (Miyoshi et al. 1999; Lubow et al. 1999; Kley et al. 2001;D’Angelo et al. 2002, 2003). However, since the main purpose ofthese studies was to clarify the planet migration process on a largescale (i.e., outside the Hill radius), they did not investigate theregion near the protoplanet (i.e., inside the Hill radius) with suf-ficient resolution. Thus, in their simulations, we cannot study thegas stream inside the Hill radius.To investigate the accretion flow onto the protoplanet system,

we need to cover a large spatial scale, from the region far from theHill sphere to that in proximity to the protoplanet. For Jupiter,since the Hill radius is rH ¼ 744rp, where rp is, in this case, theJovian radius (in general, it is the radius of the protoplanet), wehave to resolve scales that differ by factors of at least�1000. Tocover a large dynamical range in scales, a few authors have used

1 Department of Physics, Graduate School of Science, Kyoto University,Sakyo-ku, Kyoto 606-8502, Japan; [email protected], [email protected].

2 Division of Theoretical Astronomy, National Astronomical Observatory ofJapan, Osawa, Mitaka, Tokyo 181-8588, Japan; [email protected].

3 Faculty of Humanity andEnvironment, HoseiUniversity, Fujimi, Chiyoda-ku,Tokyo 102-8160, Japan; [email protected].

1220

The Astrophysical Journal, 685:1220Y1236, 2008 October 1

# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.

the nested grid method. D’Angelo et al. (2002, 2003) investi-gated the relation between the spiral patterns within the Hillradius and the migration rate using a three-dimensional nestedgrid code. Although they resolved the region inside the Hill ra-dius, they did not investigate the structure in close proximity tothe protoplanet because they adopted a sink cell at 0:1rH, whichcorresponds to�70rp at the Jovian orbit (5.2 AU). Thus, we can-not observe a circumplanetary disk at rTrH in their calculation.Tanigawa & Watanabe (2002a) also investigated the gas flowaround a protoplanet using a two-dimensional nested grid code.They resolved the region from 12rH to 0:005rH. They found thata circumplanetary disk with 100M� is formed around the proto-planet. Tanigawa&Watanabe (2002b) andMachida et al. (2006a)investigated the evolution of the protoplanet system, using athree-dimensional nested grid code. They found that the gas flowpattern in three dimensions is qualitatively different from that intwo dimensions: the gas is flowing into the protoplanet systemonly in the vertical direction in the three-dimensional simulations(see also Kley et al. 2001).

In the present study, we calculated the evolution of the proto-planet system using a three-dimensional nested grid code. Wefound that after the flow around the protoplanet reaches a steadystate, the angular momentum accreting onto the protoplanet sys-tem is well converged, regardless of both the cell width andthe size of the sink cell region, while the mass accretion rate isnot well converged. Although we calculated the evolution ofthe protoplanet system with a spatial resolution that was muchhigher than those in previous studies, we will need a still higherspatial resolution in order to determine the mass accretion rateonto the protoplanet. Thus, in this paper, we focus on the gasflow onto and around the protoplanet system and the process ofthe accretion of the angular momentum, and we do not deal withthe mass accretion rate (we plan to investigate the mass accre-tion rate with higher spatial resolution, using a higher perfor-mance computer, in a subsequent paper). Note that the specificangular momentum accreting onto the protoplanet system with afixed protoplanet mass does not strongly depend on the massaccretion rate as described in the following sections.

The structure of the paper is as follows. The frameworks of ourmodels are given in x 2, and the numerical method is described inx 3. The numerical results are presented in x 4. Section 5 is devotedto discussions. We summarize our conclusions in x 6.

2. MODEL

2.1. Master Equations

We consider a local region around a protoplanet, using theshearing sheet model (e.g., Goldreich & Lynden-Bell 1965). Weassume that the temperature is constant and that the self-gravityof the disk is negligible. The orbit of the protoplanet is assumedto be circular in the equatorial plane of the circumstellar disk.

We set up local rotating Cartesian coordinates with the originat the protoplanet and the x-, y-, and z-axes in the radial, azi-muthal, and vertical directions of the disk, respectively.We solvethe equations of hydrodynamics without self-gravity:

@�

@tþ: = �vð Þ ¼ 0; ð1Þ

@v

@tþ v = :ð Þv ¼ � 1

�:P �:�eA � 2�pz < v; ð2Þ

where �, v, P, �eA, and�p are the gas density, velocity, gas pres-sure, effective potential, and Keplerian angular velocity of the

protoplanet, respectively, and z is a unit vector directed along thez-axis. In the above equations, the curvature terms are neglected.We adopt an isothermal equation of state,

P ¼ c2s �; ð3Þ

where cs is the sound speed. The Keplerian angular velocity ofthe protoplanet is given by

�p ¼GMc

a3p

!1=2

; ð4Þ

whereG,Mc, and ap are the gravitational constant, themass of thecentral star, and the orbital radius of the protoplanet, respectively.The effective potential �eA is given by

�eA ¼ ��2

p

23x2 � z2� �

� GMp

r; ð5Þ

whereMp and r are the mass of the protoplanet and the distancefrom the center of the protoplanet, respectively (for details, seeMiyoshi et al. 1999). The first term is composed of the gravita-tional potential of the central star and the centrifugal potential, andhigher orders in x, y, and z are neglected. The second term is thegravitational potential of the protoplanet. Using the Hill radius

rH ¼ Mp

3Mc

� �1=3

ap; ð6Þ

we can rewrite equation (5) as

�eA ¼ �2p � 3x2 � z2

2� 3r3H

r

� �: ð7Þ

2.2. Circumstellar Disk Model

Our initial settings are similar to those of Miyoshi et al. (1999)and Machida et al. (2006a). The gas flow has a constant shear inthe x-direction as

v0 ¼ 0; � 3

2�px; 0

� �: ð8Þ

For hydrostatic equilibrium, the density is given by

�0 ¼�0ffiffiffiffiffiffi2�

phexp � z2

2h2

� �; ð9Þ

where�0 (�R1�1 � dz) is the surface density of the unperturbeddisk.

The scale height h is related to the sound speed cs as h ¼ cs/�p.In the standard solar nebular model (Hayashi 1981; Hayashi

et al. 1985), the temperature T, sound speed cs, and gas density �0are given by

T ¼ 280L

L�

� �1=4ap

1 AU

� ��1=2; ð10Þ

where L and L� are the protostellar and solar luminosities;

cs ¼kT

�mH

� �1=2

¼ 1:9 ; 104T

10 K

� �1=22:34

�

� �1=2

cm s�1;

ð11Þ

ANGULAR MOMENTUM ACCRETION 1221

where � ¼ 2:34 is the mean molecular weight of the gas com-posed mainly of H2 and He; and

�0 ¼ 1:4 ; 10�9 ap

1 AU

� ��11=4g cm�3; ð12Þ

respectively. When the values of Mc ¼ 1 M� and L ¼ 1 L� areadopted, using equations (4), (10), and (11), we can describe thescale height h as

h ¼ 5:0 ; 1011ap

1 AU

� �5=4cm: ð13Þ

2.3. Scaling

Our basic equations can be normalized by unit time, ��1p , and

unit length, h. The density is also scalable in equations (1) and (2),since we neglect the self-gravity of the disk in equation (2). Wenormalize the density by �0/h. Hereafter the normalized quan-tities are expressed with a tilde on top: x ¼ x/h, � ¼ �/ �0/hð Þ,t ¼ t�p, etc. The nondimensional unperturbed velocity and den-sity are given by

v ¼ 0; � 3

2x; 0

� �; ð14Þ

�0 ¼1ffiffiffiffiffiffi2�

p exp � z2

2

� �: ð15Þ

Thus, the nondimensional equations corresponding to equa-tions (1), (2), (3), and (7) are

@�

@ tþ : = �vð Þ ¼ 0; ð16Þ

@v

@ tþ�v = :

�v ¼ � 1

�:P � :�eA � 2z < v; ð17Þ

P ¼ �; ð18Þ

�eA ¼ � 1

23x2 � z2� �

� 3r3Hr

: ð19Þ

Thus, the gas flow is characterized by only one parameter, thenondimensional Hill radius: rH ¼ rH/h. In this paper, we adoptvalues of rH ¼ 0:05Y4.21 (see Table 1). As functions of the or-bital radius and the mass of the central star, the parameter rH isrelated to the actual mass of the protoplanet in the unit of Jovianmass MJup as

Mp

MJup

¼ 0:12Mc

1 M�

� ��1=2ap

1 AU

� �3=4r3H: ð20Þ

For example, in ourmodelwith rH ¼ 1:0, ap ¼ 5:2AU, andMc ¼1 M�, the protoplanet’s mass is Mp ¼ 0:4MJup (model M04 inTable 1). Hereafter, we call model M04 the ‘‘fiducial model.’’For each model, the protoplanet’s mass for ap ¼ 5:2 AU andMc ¼ 1 M� is presented in Table 1. In our parameter range, at aJovian orbit (ap ¼ 5:2 AU), protoplanets have masses of 0.05Y30MJup. We will show our results, assuming the values of ap ¼5:2 AU and Mc ¼ 1 M�, in the following. We will discuss thedependence on the orbital radius ap in x 5.2.

In subsequent sections, we use nondimensional quantities (e.g.,�, x, y, and z) when we show the structure of the protoplanetsystem (see Figs. 1, 4, 5, 7, 8, 12, and 13). On the other hand, tocompare the physical quantities derived from numerical results

with those of the present Jupiter, we use dimensional quantitiesat the Jovian orbit when we show the time evolution or radialdistribution (see Figs. 2, 3, 6, 9, 10, and 11) of the mass M and(specific) angular momentum J ( j) of the protoplanet system, inwhich the conversion coefficient from physical quantities at theJovian orbit (5.2 AU) into those at any orbit (ap) is described ineach axis. In addition, for convenience, we redefine the time unitas tp � t/ 2�ð Þ ¼ 1/ 2��p

� �, which corresponds to the orbital pe-

riod of the protoplanet. When ap ¼ 5:2 AU is assumed, tp ¼ 1corresponds to 11.86 yr.

3. NUMERICAL METHOD

3.1. Nested Grid Method

To estimate the angular momentum acquired by a protoplanetsystem from the protoplanetary disk, we need to cover a large dy-namic range in the spatial scale. Using the nested grid method(for details, seeMachida et al. 2005, 2006b), we cover the regionnear the protoplanet by the grids with high spatial resolution andthe regions remote from the protoplanet by the grids with coarsespatial resolution. Each level of the rectangular grid has the samenumber of cells (=32 ; 128 ; 8), but the cell width�s lð Þ dependson the grid level l. The cell width is reduced by 1/2 when the gridlevel is increased (l ! l þ 1). In our fiducial model, we use eightgrid levels (lmax ¼ 8). The box size of the coarsest grid, l ¼ 1, ischosen to be Lx; Ly; Lz

� �¼ 30; 120; 7:5ð Þ, and that of the finest

grid, l ¼ 8, is Lx; Ly; Lz� �

¼ 0:234; 0:938; 0:059ð Þ. The cellwidth in the coarsest grid, l ¼ 1, is �s(1) ¼ 0:9375, and it de-creases with �s(l ) ¼ 0:9375/2l�1 as the grid level l increases.Thus, the finest grid has a cell width of �s 8ð Þ ’ 7 ; 10�3. Weassume the fixed boundary condition in the x- and z-directionsand the periodic boundary condition in the y-direction. Adopt-ing equation (15), we find that the density at the z-boundary(zb ¼ 7:5) is about 10�12 (=�zb /�0) times lower than that at theequatorial plane (z ¼ 0). Since this value (10�12) is too small, weput the uniform density of 1:5 ; 10�6 [=1/ 2�ð Þ1/2 expð� z2uni/2Þ]in the region of z > zuni ¼ 5. To check our results, we have cal-culated the disk evolution, changing zb and zuni, and confirmedthat our results do not depend on the values of zb and zuni whenthe z-boundary for the uniform density region is adopted to besufficiently far from the equatorial plane: zuni 3 3.

TABLE 1

Model Parameters

Model rH Mp(5.2 AU)a lmax

� s(lmax)

(10�3)

rsink(10�2)

M005...................... 0.5 0.05 8 7.3 . . .

M01........................ 0.63 0.1 8 7.3 . . .

M02........................ 0.8 0.2 8 7.3 . . .M04........................ 1.0 0.4 8 7.3 . . .

M06........................ 1.15 0.6 8 7.3 . . .

M1.......................... 1.36 1 8 7.3 . . .M3.......................... 1.95 3 8 7.3 . . .

M10........................ 2.91 10 8 7.3 . . .

M30........................ 4.21 30 8 7.3 . . .

M04L5.................... 1.0 0.4 5 59 . . .M04L6.................... 1.0 0.4 6 29 . . .

M04L7.................... 1.0 0.4 7 15 . . .

M04L9.................... 1.0 0.4 9 3.7 . . .

M04L10.................. 1.0 0.4 10 1.8 . . .M04S01.................. 1.0 0.4 8 7.3 1

M04S03.................. 1.0 0.4 8 7.3 3

a Units are Jupiter masses MJup.

MACHIDA ET AL.1222 Vol. 685

We do not explicitly adopt the smoothing length for the grav-itational potential of the protoplanet. In our numerical setting,the physical quantities are defined at the cell center, while theorigin (the protoplanet’s position) is defined at the cell boundary.Thus, since the region inside r < rs �

ffiffiffi3

p�s lmaxð Þ/2 has a uni-

form gravitational potential, we can regard rs as the efficientsmoothing length in our settings. In the fiducial model (lmax ¼ 8),the value of rs is 6 ; 10�3, which is much smaller than the Hillradius, and roughly 3 times as large as the Jovian radius.

In real units, using the standard solar nebular model, the scaleheight at the Jovian orbit (ap ¼ 5:2 AU) is h ¼ 0:27 AU. Thecomputational domain in the azimuthal direction corresponds to120 ; 0:27 AU ¼ 32:4 AU, which is equivalent to the circum-ference of the Jovian orbit around the Sun: 2�ap (=32.7 AU).Although we imposed a periodic boundary condition in the az-imuthal direction, this domain size is valid, except that we ignorethe curvature. Note that the computational domain is not neces-sarily the same as the real circumference of a planet as long as thedomain is sufficiently large. To verify our results, in somemodels,we calculate the evolution of the protoplanet system with differ-ent levels of the finest grid (or different maximum grid levels):lmax ¼ 5, 6, 7, 9, and 10. In these models, the cell width oflmax¼ 5 is�s(5)¼ 5:9 ;10�2 (2:34 ; 1011 cm�3 at ap ¼ 5:2AU),which corresponds to 33 times the Jovian radius, while thatof lmax ¼ 10 is �s(10) ¼ 1:83 ; 10�3 (7:32 ; 109 cm�3 at ap ¼5:2 AU), which corresponds to 1.02 times the Jovian radius. Themaximum grid level, lmax, and the cell width of the finest grid foreach model are summarized in Table 1.

3.2. Test Simulation

At first, we show the evolution of the protoplanet system cal-culated with lmax ¼ 8. The cell width of the maximum grid level(l ¼ 8) that covers the region in the proximity of the protoplanetis �s(8) ¼ 7:3 ; 10�3 (�s ¼ 2:92 ; 1010 cm at ap ¼ 5:2 AU).Figure 1 shows the evolution for model M04 (the fiducial model),in which a protoplanet with 0:4MJup is adopted at ap ¼ 5:2 AU.The top panels in Figure 1 (Figs. 1aY1e) show the time sequenceof the region far from the protoplanet (rk 15) at the l ¼ 1, 2,and 3 grid levels, in which the three different grid levels aresuperimposed, while the bottom panels (Figs. 1 fY1 j) show theregion near the protoplanet (rP 3:5) at grid levels of l ¼ 3, 4,and 5. Each bottom panel is a 4 times magnification of the cor-responding top panel. In this figure, the protoplanet is located atthe origin: x; y; zð Þ ¼ 0; 0; 0ð Þ. The elapsed time tp is denotedabove each of the top panels. The central density �c is also de-noted above each of the top panels.

Figures 1aY1c and 1 fY1h show that the density is enhanced ina narrow bandwith a spiral pattern that is distributed from the topleft region to the bottom right region. The density gaps that ap-pear on the right ( left) side of the spiral pattern in the region ofy > 0 ( y < 0) are also seen in these panels. In Figure 1c, the den-sity of the spiral pattern around the protoplanet is � ’ 2, whilethat of the gap is � ’ 0:2. Thus, there is a density contrast of�10 between the spiral patterns and the gaps. In addition, thecentral density increases up to � � 106 for tpk 1. Figures 1hY1 jshow a round shape near the protoplanet. Since the Hill radius isrH ¼ 1 in this model, the gravity of the protoplanet is predom-inant in the region of rT1. Thus, the central region of rT1has a round structure. Figure 1 shows that the density distribu-tions in panels cYe (or panels hYj ) are similar. In all models, thestructure around the protoplanet hardly changes for tpk 1, whichseems to indicate that a steady state has already been achieved fortpk1. Tanigawa &Watanabe (2002a) investigated the evolutionof the protoplanet system in the same condition as ours but in

two dimensions, and they showed that the gas stream around theprotoplanet is in a steady state after a short timescale of tp � 1.

3.3. Convergence Test

In x 3.2, we showed the evolution of the protoplanet system atthe maximum grid level lmax ¼ 8. In this subsection, to check theconvergence of our calculation, we compare the evolutions ofthe protoplanet system with different maximum grid levels (ordifferent cell widths of the finest grid). Since our purpose is toinvestigate the angular momentum acquired by the protoplanetsystem, we use the average specific angular momentum as amea-sure of the convergence. As a function of the distance from theprotoplanet r, we define the average specific angular momen-tum jr as

jr ¼Jr

Mr

; ð21Þ

where the mass

Mr ¼Z r

0

4�r2� dr ð22Þ

and angular momentum

Jr ¼Z r

0

4�r2�$v� dr ð23Þ

are integrated from the center, r ¼ 0, to a distance r, where $ andv� are the cylindrical radius from the protoplanet and the azi-muthal velocity, respectively. Here we adopt a value of r ¼0:5rH (i.e., j0:5rH). The dependence of jr on r is discussed in x 4.5.We often show the mass and (specific) angular momentum ofthe protoplanet system in real units at ap ¼ 5:2 AU in order tocompare numerical results with the present values of gaseousplanets. The dimensional valuesMr 5:2 AUð Þ, Jr 5:2 AUð Þ, andjr 5:2 AUð Þ at ap ¼ 5:2AUcan be converted intoMr ap

� �, Jr ap� �

,and jr ap

� �at any orbit as

Mr ap� �

¼ Mr 5:2 AUð Þ ap

5:2 AU; ð24Þ

Jr ap� �

¼ Jr 5:2 AUð Þ ap

5:2 AU

� �2; ð25Þ

jr ap� �

¼ jr 5:2 AUð Þ ap

5:2 AU: ð26Þ

In the following, we show the values at 5.2 AU.Whenwe refer todimensional physical quantities without any mention of the or-bit, they are the values at ap ¼ 5:2 AU.

The evolution of j0:5rH formodelsM04L5YM04L10 (seeTable 1)are shown in Figure 2. In these models, the protoplanet’s mass isfixed, and only the maximum grid level (or cell width of the finestgrid) is changed. Figure 2 shows that the average specific angularmomentum rapidly increases initially, and then it saturates at acertain value in eachmodel. Althoughwe adopted the samemassfor the protoplanet, the saturation levels of the average specificangular momenta are different. The average specific angular mo-mentum saturates at j0:5rH ’ 2 ; 1016 cm2 s�1 for model M04L5,whereas it is saturated at j0:5rH ’ 7Y8 ; 1016 cm2 s�1 for modelsM04L7, M04, M04L9, and M04L10. Thus, there are only smalldifferences for models with lmax > 7. For example, modelM04L7has a value of j0:5rH ¼ 6:9 ; 1016 cm2 s�1, while model M04L9has a value of j0:5rH ¼ 8:2 ; 1016 cm2 s�1 at tp ¼ 10. Therefore,

ANGULAR MOMENTUM ACCRETION 1223No. 2, 2008

Fig. 1.—Time sequence for model M04. The density (color scale) and velocity distributions (arrows) on the cross section in the z ¼ 0 plane are plotted. The bottompanels (l ¼ 3) are 4 times the spatial magnification of the top panels (l ¼ 1). Three levels of grids are shown in each top (l ¼ 1, 2, and 3) and bottom (l ¼ 3, 4, and 5) panel.The level of the outermost grid is denoted in the top left corner of each panel. The elapsed time tp and the central density �c on the midplane are denoted above each of thetop panels. The velocity scale in units of the sound speed is denoted below each panel.

the average specific angular momenta are sufficiently con-verged within a relative error of about 15% if lmax > 7 or�s <1:4 ; 10�2. In the following, we safely calculate the evolutionof the protoplanet system with lmax ¼ 8 as the maximum gridlevel.

3.4. Models with and without Sink Cells

When we adopt the maximum grid level of lmax ¼ 8, the cellwidth is �s(8) ¼ 7:3 ; 10�3. In real units, when the protoplanetis located at 5.2 AU, the cell width corresponds to �s ¼ 2:9 ;1010 cm, which is 4.1 times the Jovian radius. Interior to the gasgiant planet, there exists a solid core that is’10M� in mass and’109 cm in size (Mizuno 1980). The size of the solid core ismuchsmaller than the cell width adopted in our calculation. To inves-tigate the evolution of a protoplanet through gas accretion, in prin-ciple, we need to resolve a solid core with a sufficiently small cellsize (�sT109 cm). However, since our purpose is to investi-gate the angular momentum flowing into the protoplanet sys-tem, we do not always need to resolve a central solid core and aprotoplanet. As is discussed in D’Angelo et al. (2002), for a limitednumerical resolution, the gas around the central region makes anartificial pressure gradient force, which may affect the gas accre-tion onto the protoplanet system. To check this, we also calculatedthe evolution of the protoplanet system, adopting the sink cell insome models. We parameterized the size of the sink: rsink ¼ 0:01and 0.03 (models M04S01 and M04S03, respectively). Thesemodels are also summarized in Table 1.

In real units, the radius of the sink in model M04S01 is 4:0 ;1010 cm (’5.6 times the Jovian radius),while that inmodelM04S03is 1:2 ; 1011 cm (’17 times the Jovian radius). During the calcu-lation, we remove the gas from the region inside the sink radiusin each time step and integrate the removedmass and angular mo-mentum that are assumed as the mass and angular momentum ofthe protoplanet system.

After the steady states are achieved at tp ’ 20, we estimate theaverage specific angular momenta jr as a function of r. Figure 3shows the distribution of the average specific angular momentumjr against the distance from the protoplanet r in real units for eachmodel. The solid line represents jr without a sink cell, while theother lines represent those with sink cells. Inside the sink radius,the average specific angular momentum is assumed to be a con-stant value. In Figure 3, the vertical dashed line indicates the Hillradius.

In Figure 3, the solid line shows that jr increases from thecenter to a peak around the Hill radius and then drops sharply justoutside the Hill radius. The drop indicates that the angular mo-mentum becomes negative at r > rH. Thus, the rotational direc-tion turns around between the regions inside and outside the Hillradius. As is shown in Sekiya et al. (1987), Miyoshi et al. (1999),and Tanigawa & Watanabe (2002a), the protoplanet formed bythe gas accretion in the protoplanetary disk has a prograde spin,and thus it has a positive (specific) angular momentum. On theother hand, gas far outside of the Hill sphere seems to rotate ret-rogradely against the protoplanet because it rotates with a nearlyKeplerian velocity with respect to the central star [vy ¼ � 3/2ð Þ�px,as shown in eq. (8)]. As a result, gas inside the Hill radius has apositive angular momentum, while that outside the Hill radius hasa negative angular momentum.

Figure 3 shows that the values of jr in models with a sink celldepend on the radius of the sink in the region rTrH. However,they do not depend sensitively on the size of the sink in the regionof rk 0:5rH. For example, at r ¼ rH, the model without the sink(model M04) has a value of jrH ¼ 5:3 ; 1016 cm2 s�1, whilemodelM04S03 has a value of jrH ¼ 6:4 ; 1016 cm2 s�1. Thus, thedifference of the average specific angular momentum amongthese models at the Hill radius is’20%. This difference decreaseswith the sink radius. In this study, we focus on the angular mo-mentum of the protoplanet system, not that of the planet itself.As is shown in Figure 3, the angular momentum acquired by theprotoplanet system (planet + circumplanetary disk) extends upto the Hill radius, in which almost all the angular momentum isdistributed in the region of r3 rp or r3 rsink. In the following,we calculate the evolution of the protoplanet system without thesink cell.

4. RESULTS

4.1. Typical Gas Flow

Figure 4 shows the gas structure around the Hill sphere formodel M04 at tp ¼ 20. The left panel of Figure 4 shows thestructure on the cross section in the z ¼ 0 plane, in which redlines indicate the streamlines. In this panel, the gas enters thel ¼ 4 grid from the upper ( lower) y boundary for x > 0 (x < 0)

Fig. 2.—Evolution of average specific angular momenta derived in the regionof r < 0:5rH for models with different finest grid levels lmax.

Fig. 3.—Average specific angular momenta jr against the distance from theorigin for models with different sink radii rsink. The vertical dashed line representsthe Hill radius.

ANGULAR MOMENTUM ACCRETION 1225

and goes downward (upward) according to the Keplerian shearmotion. The shocks (crowded contours near the Hill radius) canbe seen in the upper right and lower left regions near the proto-planet. In this model, the shock front almost corresponds to theHill radius (Fig. 4, left). When the gas approaches the proto-planet, the streamlines are bent by the gravity of the protoplanet.According to Miyoshi et al. (1999), the gas flow is divided intothree regions: the pass-by region ( xj jk rH), the horseshoe region(xP rH and yj jk rH), and the atmospheric region (rP rH). Notethat although Miyoshi et al. (1999) classified the flow pattern intheir two-dimensional calculation, their classification is useful fora global flow pattern in three dimensions. In the pass-by region,the flow is first attracted toward the protoplanet and then causes ashock after passing by the protoplanet. At the shock front, thedensity reaches a local peak and the streamlines bend suddenly.On the other hand, the gas entering the horseshoe region turnsaround because of the Coriolis force and goes back. The outer-most streamlines in the horseshoe region (i.e., the streamlinespassing very close to the protoplanet) pass through the shockfront, whereas the gas on the streamlines far from the protoplanetdoes not experience the shock. In the atmospheric region, the gasis bound by the protoplanet and forms a circumplanetary diskthat revolves circularly around the protoplanet in the prograde(counterclockwise) direction.

Although the streamlines on the z ¼ 0 plane (Fig. 4, left) aresimilar to those in recent two-dimensional calculations (e.g., Lubowet al. 1999; Tanigawa & Watanabe 2002a), there are important

differences. In the two-dimensional calculations, a part of the gasnear the Hill sphere can accrete onto the protoplanet. Lubow et al.(1999) showed that only gas in a narrow band distributed from thelower left to the upper right region against the protoplanet fory < 0 spirals inward toward the protoplanet, passes through theshocked region, and finally accretes onto the protoplanet (for de-tails, see Figs. 4 and 8 of Lubow et al. 1999). On the other hand,in our three-dimensional calculation, gas only flows out from theHill sphere and thus does not accrete onto the protoplanet on themidplane. The left panel of Figure 4 shows that although gas flowsinto the Hill sphere, some of the gas flows out from the centralregion. The right panel of Figure 4 is a three-dimensional view atthe same epoch as the left panel. In this panel, only the stream-lines flowing into the high-density region of rTrH are drawnfor z � 0 and are inversely integrated from the high-density re-gion. This panel clearly shows gas flowing into the protoplanet inthe vertical direction.To investigate the gas flowing into the protoplanet system in

detail, in Figure 5 we plot the streamlines at the same epoch asFigure 4, but with different grid levels (l ¼ 3, 5, and 7). In thisfigure, each of the top panels shows a three-dimensional view,while each of the bottom panels shows the structure on the crosssection in the y ¼ 0 plane. Note that, in the bottom panels, thestreamlines are projected onto the y ¼ 0 plane. Figure 5a showsonly the streamlines in a narrow bundle flowing into the proto-planet system. This feature is similar to that shown in two-dimensional calculations (Lubowet al. 1999;Tanigawa&Watanabe

Fig. 4.—Structure around the Hill sphere for model M04 on the midplane (left) and in three dimensions, shown in bird’s-eye view (right). The gas streamlines (redlines), density structure (color), and velocity vectors (arrows) are plotted in both panels. The dashed circle in the left panel represents the Hill radius. The size of the domainis shown in each panel.


2002a). However, the streamlines in Figures 5a and 5b indicatethat gas rises upward near the shock front and then falls into thecentral region in the vertical direction. Gas flowing in the verticaldirection spirals into the inner region (Fig. 5c). In this process,vortices appear, as shown in Figure 5d. As shown in the left panelof Figure 4, and also in Figure 5d, gas is flowing out from the cen-tral region on the z ¼ 0 plane. Gas in the proximity of the pro-toplanet rotates circularly in the prograde direction, as shown inFigure 5e. Figure 5 f shows that some of the gas flowing into theupper boundary of the l ¼ 7 grid level contributes to the forma-tion of the disk around the protoplanet,while the remainder is bentand flows out from the central region.

Whenwe look down the protoplanetary disk from above alongthe z-axis, the streamlinesmay seem to be almost the same as thosein the two-dimensional calculations. However, gas also moves inthe vertical direction: streamlines go upward near the shock front(r � rH) and vertically fall into the central region at rTrH. Thisfeature of streamlines is also seen inmodelsM04S01 andM04S03,in which the sink cell is adopted. Thus, the different features ofthe streamlines in two- and three-dimensional calculations arecaused not by the effect of the pressure gradient force, but by thedimensionality (because the same feature appears inmodels bothwith and without the sink). This flow pattern is also seen in otherthree-dimensional calculations (Tanigawa & Watanabe 2002b;D’Angelo et al. 2003). In two-dimensional calculations, sincethe vertical motion is restricted, the flow pattern is different fromthat in three dimensions. This difference affects the accretion rateonto the protoplanet and themigration rate. D’Angelo et al. (2003)showed that themigration rate is different between two- and three-dimensional calculations. In the present study, however, since we

focus on the angular momentum of a protoplanet system, wedo not discuss these issues anymore.Wewill discuss the accretionand migration rates in subsequent papers.

Finally, we comment on the circumplanetary disk. In the bot-tom panels of Figure 5, the green surface (i.e., the isodensitysurface) indicates the high-density structure around the proto-planet. These panels show the disklike structure in the proximityof the protoplanet, and the disk becomes thinner as it approachesthe protoplanet (i.e., the origin). We discuss the circumplanetarydisk in x 5.3.

4.2. Dependence on Protoplanet Mass

We have shown the evolution of the protoplanet systemfor the model in which the mass of the protoplanet is 0:4MJup

in x 4.1. In this subsection, we investigate the evolution of theprotoplanet system with different protoplanet masses. The toppanels of Figure 6 show the accumulated masses (eq. [22])within r < 0:1 (M0:1; Fig. 6a) and r < 0:05 (M0:05; Fig. 6b)against the elapsed time for different models, while the middlepanels of Figure 6 show the corresponding angular momenta(eq. [23]) in the same regions (Fig. 6c for J0:1, and Fig. 6d forJ0:05). In Figure 6, both the masses and the angular momenta forall models increase rapidly for values of tp < 0:1. This is becausethe protoplanet with amass of 0.05Y0:6MJup suddenly appears inthe protoplanetary disk at tp ¼ 0. However, this rapid growthphase ( tp < 0:1) is not real, because the gas is considered to beginto accrete onto the protoplanet when the mass of the solid coreexceeds ’10 M� (Mizuno 1980; Bodenheimer & Pollack 1986;Ikoma et al. 2000). The growth rates of the mass and angularmomentum begin to decrease at tp � 0:1 in all models, and then

Fig. 5.—Structure around the protoplanet for model M04 with different grid levels: l ¼ 3 (left), 5 (middle), and 7 (right). The top panels present the structure in threedimensions, shown in bird’s-eye view, and the bottom panels show the structure on the cross section in the y ¼ 0 plane. The gas streamlines (red lines), density structure(color), and velocity vectors (arrows) are plotted in each panel. The size of the domain is shown in each panel.


both the masses and the angular momenta increase with an al-most constant rate until the end of the calculation (0:1P tpP 20).Tanigawa & Watanabe (2002a) calculated the mass accretionrate onto the protoplanet as

M ¼ 8:0 ; 10�3 M�Mp

10 M�

� �1:3

yr�1: ð27Þ

Thus, the growth time �grow ¼ M /M is �grow ¼ 430Y1000 yr(i.e., tp ¼ 36Y88). In our calculation, we continue to calculatethe evolution of the protoplanet system for t � 230 yr ( tp � 20)by fixing the protoplanet’s mass. We think that this treatment isnot problematic, since the growth timescale is longer than ourcalculation time. Note that since Tanigawa & Watanabe (2002a)calculated the evolution of the protoplanet system in two di-mensions, the growth rate might be different from that in three-dimensional calculations.

The bottom panels of Figure 6 show the evolution of the aver-age specific angularmomentum in the regions of r < 0:1 (Fig. 6e;j0:1) and r < 0:05 (Fig. 6 f; j0:05). The masses and angular mo-menta flowing into the protoplanet system increase with a constantrate for tp < 0:1, while the average specific angular momentaare saturated at certain values for tp > 0:1. This saturation meansthat the flow around the protoplanet is in a steady state. Figures 6e

and 6f also indicate that the average specific angular momen-tum brought into the protoplanet system increases with the massof the protoplanet. In Figures 6e and 6f, the average specific an-gular momenta in the region of r < 0:1 are larger than those inthe region of r < 0:05, which indicates that the protoplanet sys-tem has a larger average specific angular momentum farther awayfrom the protoplanet. We will investigate the angular momentumacquired in the protoplanet system in x 4.5.

4.3. Gas Structure around a Protoplanet

Figures 7 and 8 show the density distributions (top) and Jacobienergy contours (bottom) on the cross section in the z ¼ 0 plane(Fig. 7) and the y ¼ 0 plane (Fig. 8) around the Hill sphere afterthe steady state is achieved ( tp � 20) for modelsM02 (left), M04(middle), and M06 (right). The white dashed line in each panelindicates the Hill radius rH. In each of the top panels, the shockappears from the upper left to lower right near the Hill radius.These shocks are frequently seen in similar calculations (Sekiyaet al. 1987; Miyoshi et al. 1999; Lubow et al. 1999; Tanigawa &Watanabe 2002a; D’Angelo et al. 2002). In the top panels ofFigure 7, round structures are seen in proximity to the proto-planet (rTrH), while ellipsoidal structures are seen in the re-gions where r ’ rH. This is because gas distributed near theprotoplanet is more strongly bound by the protoplanet. The toppanels of Figure 8 show that contours of the central region sag in

Fig. 6.—Evolution of (a, b) the accumulated masses, (c, d) the angular momenta, and (e, f ) the average specific angular momenta in the regions of (a, c, e) r < 0:1 and(b, d, f ) r < 0:05 against the elapsed time for models M005, M01, M02, M04, and M06.

MACHIDA ET AL.1228

Fig. 7.—Density ( false color and contours) and velocity vectors (arrows), plotted on the cross section in the z ¼ 0 plane in the top panels for models M02 (left), M04(middle), andM06 (right). The Jacobi energy ( false color and contours) is plotted in the bottom panels for the samemodels as in the panels above. Three levels of the grid(l ¼ 4, 5, and 6) are superimposed in each panel. The dashed circle represents the Hill radius.

the center of a concave structure and thin disks are formed aroundthe protoplanet (rTrH). In addition, the butterfly-like structureis also seen inside the Hill radius in the top panels of Figure 8.These structures are considered to be formed by the rapid rotationof the central circumplanetary disk; similar structures are seen in arapidly rotating protostar (e.g., Fig. 1 of Saigo& Tomisaka 2006).We will discuss the disk structure in x 5.3.

In contrast to celestial mechanics, it is difficult to find fluid ele-ments bound by the protoplanet because thermal energy is im-portant in addition to the gravitational and kinetic energies. Todiscern the gas bound by the gravity of the protoplanet, we use theJacobi energy as an indicator (Canup & Esposito 1995; Kokuboet al. 2000). In our notation, the Jacobi energy is given by

EJ ¼1

2˙x2 þ ˙y2 þ ˙z2� �

� 3

2x2 þ 1

2z2 � 3r3H

r: ð28Þ

In equation (28), the first term is the kinetic energy, the second andthird terms are the tidal energy of the central star, and the fourthterm is the gravitational energy of the protoplanet. The Jacobi en-ergy is the conserved quantity in a rotating system. In the case of afluid, the Jacobi energy is not strictly appropriate, because thethermal energy is ignored. However, we can use this to make arough estimation of the gas bound by the protoplanet. We deter-mine the fluid elements bound by the protoplanet from the con-tours of the Jacobi energy in the bottom panels of Figures 7 and 8.

Fluid elementswith lower Jacobi energies are strongly bound bythe protoplanet, as described in equation (28). The bottom panelsof Figures 7 and 8 show that inside the region of rP0:5rH, eachcontour has a closed ellipse. Since fluid elements move on thisclosed orbit, it is considered that gas distributed in the region ofrP 0:5rH is bound by the protoplanet when the thermal effect canbe ignored. We discuss the thermal effect in x 4.5. On the otherhand, although fluid elements exist inside the Hill radius, outsideof 0:5rHP rP rH, they are not bound by the protoplanet, becausethe contours of the Jacobi energy straddle the Hill radius. For ex-ample, in the bottom left panel of Figure 7, the contour of EJ ¼�2:5 straddles the Hill radius, and therefore fluid elements with

this Jacobi energy freely move on this contour. Thus, althoughthese elements transiently stay inside the Hill radius, they alsoflowout from theHill sphere. In summary, Figures 7 and 8 suggestthat fluid elements inside rP0:5rH are bound by the protoplanet.

4.4. Angular Momentum of a Protoplanet System

The top panel of Figure 9 shows the accumulated massMr fordifferentmodels after the steady state has been achieved ( tp ’ 20).In this panel, a thin solid line indicates the initial distribution ofthe mass (or that of the protoplanetary disk), and the circles rep-resent the Hill radius rH for each model. The accumulated massin anymodel is larger than the initial value, because gas flows intothe Hill sphere. This panel indicates that the massive protoplanethas a massive envelope. Since these mass distributions are in asteady state at the fixed mass of the protoplanet, it can be con-sidered that different curves correspond to snapshots at differentevolutionary phases. Namely, a gas envelope increases its masswith time (or the protoplanet’s mass). Outside the Hill radius, theaccumulated mass in each model converges to the initial value,which indicates that the mass distribution for r3 rH does notchange from the initial state because the influence of the proto-planet is small.The bottom panel of Figure 9 shows the absolute value of the

angular momentum Jrj j. There are spikes in all of the models, atwhich points the sign of the angular momentum is reversed. As isshown in x 3.4, the angularmomentum has a positive sign aroundthe protoplanet (rTrH) and becomes negative outside the Hillradius (r3 rH). The sign of the angular momentum is reversedoutside the Hill radius for models M01, M02, M04, and M06,while it is reversed inside the Hill radius for model M005. Thisis because the protoplanet system for model M005 does not ac-quire a sufficient amount of mass and angular momentum, dueto the shallow gravitational potential and the relatively largethermal pressure (for details, see x 4.5). The top panel of Figure 9shows that the protoplanet system has an envelope mass of onlyM ’ 0:06M� for model M005, in which the protoplanet’s massis 0:05MJup. Except for in model M005, the angular momentagradually decrease after they reach their peak around r ’ rH and

Fig. 8.—Same as Fig. 7, but in the cross section in the y ¼ 0 plane.


then become negative at r ’ 1Y2rH. Thus, the angular momentabound by the protoplanet system are limited in the region ofr < 2rH at themaximum. The bottompanel of Figure 9 shows thatthe angular momentum keeps an almost constant value around theHill radius (0:5rHP rP1rH), until it is reversed. This means thatthe positive and negative angularmomenta aremixed in this region,as is shown in the left panel of Figure 4. As a result, wherever weestimate the angular momentum in the range of 0:5rHP rP1rH,we can obtain almost the same values for the angular momentum.

The distributions of the average specific angular momentumjr for different models are shown in Figure 10. The circles in thisfigure represent the Hill radii rH. The crosses indicate the Jacobiradii rJ, within which gas is considered to be bound by the gravityof the protoplanet.We determine the Jacobi radii from the contourof the Jacobi energies as in the bottom panels of Figures 7 and 8.The Jacobi radii in all models are distributed in the range ofrJ ’ 0:5Y1rH. Figure 10 indicates that a more massive proto-planet has an envelope with a larger amount of specific angularmomentum. Thus, the specific angular momentum accreting ontothe protoplanet system increases as the protoplanet’s mass in-creases. The rapid drops at large radii indicate the reverse of therotation axis, as shown in Figures 3 and 9. In Figure 10, in modelM01 (M ¼ 0:1MJup), the value of jr at r ¼ rJ is twice that atr ¼ rH, whereas in models M02, M04, M06, M1, and M3 (M >0:2MJup), there are little differences between the average specificangular momentum derived from the Hill radius jrH and that de-rived from Jacobi radius jrJ . Thus, we can safely estimate theaverage specific angular momentum using either the Hill radiusor the Jacobi radius for models with M > 0:2MJup.

4.5. Evolution of the Specific Angular Momentum

To properly calculate the angular momentum of the proto-planet system, we have to calculate the planetary growth fromthe solid core with ’10 M� up to the present mass. However, it

takes a huge computation time to calculate all evolution phases.Thus, we estimate the angular momentum of the protoplanet sys-tem according to the following procedure: (1) we calculate theaverage specific angular momenta of the gas flowing into theprotoplanet system at fixed masses of the protoplanet (i.e., underthe same parameter rH) in models with different masses of theprotoplanet, (2) derive the relation between the average specificangular momentum and the mass of the protoplanet and describeit as a function of the protoplanet’s mass, and (3) estimate the an-gular momentum of the protoplanet system, integrating the aver-age specific angular momentum by mass up to the present valueof the gas giant planets.

At first, we analytically estimate the specific angular momen-tum of the protoplanet system, and then we compare it withnumerical results. We assume that the gas that overcomes the Hillpotential flows into the Hill sphere (or a protoplanet system) atthe Keplerian velocity of the protoplanet at the Hill radius rH.When the protoplanet’s mass isMp, the Keplerian velocity at rHis given by

vK;rH ¼ffiffiffiffiffiffiffiffiffiffiGMp

rH

r¼

ffiffiffi3

p�prH / M 1=3: ð29Þ

Note that �p is constant when ap is fixed. The specific angularmomentum can be written as

j ¼ffiffiffi3

p�pr

2H: ð30Þ

Thus, the specific angular momentum is proportional to r2H. Inaddition, when the mass of the central star is fixed, the specificangular momentum is proportional to

j / M 2=3p : ð31Þ

Thus, the specific angularmomentum increases with the 2/3 powerof the protoplanet’s mass.

Fig. 9.—Distribution of the accumulated mass (top) and absolute value of theangular momentum (bottom) against the distance from the protoplanet for modelsM005, M01, M02, M04, and M06. The thin solid lines represent the initial dis-tribution of the mass (top) and the angular momentum (bottom). The circles rep-resent the Hill radii.

Fig. 10.—Average specific angular momentum jr against the distance fromthe protoplanet for models M01, M02, M04, M06, M1, and M3. The circles andcrosses represent the Hill and Jacobi radii, respectively.


Figure 11 shows the average specific angular momentum de-rived from all models against the protoplanet’s mass. The aver-age specific angular momenta are estimated in the regions of r <rH ( plus signs), r < rJ (circles), r < 0:5rH (triangles), and r <0:1rH (squares). Although we fixed the protoplanet’s mass ineach model, we can consider the horizontal axis in Figure 11as a time sequence of the protoplanet system. Figure 11 clearlyshows that more massive protoplanets can acquire an envelopewith a larger average specific angular momentum. Although theaverage specific angular momenta for r < 0:5rH differ from thosefor r < rH for models withMp < 0:2MJup, there is little differencefor models with Mp > 0:2MJup. In addition, when we adopt theaverage specific angular momenta in the region of r < 0:1rH,we underestimate them for any model, as is shown in Figure 11.Therefore, we can properly estimate the average specific angularmomentum of the protoplanet system in any region where r <0:5Y1rH for models with Mp > 0:2MJup.

In Figure 11, the red and blue lines represent fitting formulaeof the evolution of the average specific angular momentum as afunction of the protoplanet’s mass forMp < MJup ( lowmass, jlm;red line) and Mp > MJup (high mass, jhm; blue line). These aregiven by

jlm ¼ 1:4 ; 1017Mp

1MJup

cm2 s�1 for Mp < 1MJup; ð32Þ

jhm ¼ 1:6 ;1017Mp

1MJup

� �2=3

cm2 s�1 for Mp > 1MJup: ð33Þ

Figure 11 shows that the evolution of the average specific an-gular momentum for Mpk 1MJup is well described by j / M 2/3

p ,which corresponds to equation (31). On the other hand, the growthrate of the average specific angular momentum forMp < 1MJup islarger than j / M 2/3

p and can be fitted by j / Mp. We ignored thethermal effect whenwe derived equation (31).WhenMp is small,the gas flowing into the protoplanet system is affected rela-

tively strongly by the thermal pressure.We can estimate the massat which the gravity dominates the thermal pressure force from thebalance between the thermal pressure gradient and gravitationalforces. Near the Hill radius, the thermal pressure gradient force isstronger than the gravitational force when the Keplerian speed isslower than the sound speed (i.e., vK < cs). On the other hand,when vK > cs, the gas flow is controlled mainly by the gravity ofthe protoplanet, even near the Hill radius. Using equation (29),we find that vK ¼ cs is realizedwhen rH ¼ cs/(

ffiffiffi3

p�p), which cor-

responds to M ¼ 0:08MJup for the protoplanet’s mass at ap ¼5:2 AU. Thus, the gas flow is largely affected by the thermalpressure for MpT0:08MJup, while it is not as affected by thethermal pressure forMp 30:08MJup. In Figure 11, the evolutionof the angular momentum forMp > 1MJup corresponds well withthe analytical solution ( j / M 2/3

p ), which indicates that the ther-mal effect is negligible for Mpk1MJup. However, for Mp <1MJup, the average specific angular momentum is smaller thanjhm (blue line). Thus, when the protoplanet’s mass isMp < 1MJup,the thermal effect is not negligible for the acquisition process ofthe angular momentum because the gravitational potential is rela-tively shallow.Thus, the thermal pressure seems to remain importantfor M P 1MJup, although vK ¼ cs is realized at M ¼ 0:08MJup.To verify the relation of the thermal and gravitational effects,

in Figure 12we plot the ratio of the azimuthal to the Keplerian ve-locity (v�/vK) around the protoplanet on the midplane for modelsM02,M04, andM06. In this figure, closed contours inside theHillradius indicate that gas revolves around the protoplanet. For ex-ample, the closed contour of v�/vK ¼ 0:5 in Figure 12 means thatgas rotates along the contour with 50% of the Keplerian velocity.The black circles represent the contours of v� ¼ cs, inside whichgas revolves around the protoplanetwith supersonic velocity. Thus,the gravitational force of the protoplanet is dominant inside theblack circles, while the thermal pressure gradient force is domi-nant outside the black circles. Figure 12 shows that the radius ofthe v� ¼ cs contour increases with the protoplanet’s mass, indi-cating that the region that is dominated by the gravity of the

Fig. 11.—Specific angular momenta in the regions of r < rH ( plus signs), r < rJ (circles), r < 0:5rH (triangles), and r < 0:1rH (squares) against the protoplanet’smass. The blue and red lines represent the fitting formulae for Mp > 1MJup and Mp < 1MJup, respectively.


protoplanet extends with the protoplanet’s mass. As the proto-planet’s mass increases, the flow speed inside the Hill radiusapproaches the Keplerian velocity, and the region dominated bythe gravity of the protoplanet spreads outward. In this way, sincethe thermal effect decreases as the protoplanet’s mass increases, thegrowth rate of the average specific angular momentum approachesthe analytical solution (eq. [31]).

5. DISCUSSION

5.1. Effect of Thermal Pressure

In this paper, we investigated the evolution of a protoplanetsystem under the isothermal approximation, which might not beproblematic in the circumplanetary disk, but may not be valid invery close proximity to the protoplanet. To investigate the ther-mal effect around the protoplanet, we adopted a sink cell in someof our models. As is shown in x 3.4, the cell width in the fiducialmodel with lmax ¼ 8 is about 4 times the Jovian radius. Thus, gasinside r < 4rJup has the same thermal energy. If the protoplanethas almost the same size as a present gas giant planet, the thermalenergy around the protoplanet may be overestimated in modelswithout the sink cell, because an actual protoplanet is embeddedin a small part of the innermost cell. On the other hand, when weadopt the sink cell, the thermal energy around the protoplanet isunderestimated, because the thermal energy is artificially removedfrom the sink cell. However, as is shown in x 3.4, when the sinkradius is much smaller than the Hill radius [rsinkT 1/100ð ÞrH],there is little difference in the angular momentum acquired bythe protoplanet system between models with and without thesink cell. This is because a large part of the angular momentumof the protoplanetary system is distributed around the Hill radius

(0:5rHP rP rH), as is shown in x 4.4. The Jovian radius is muchsmaller than theHill radius (rH ¼ 744rJup). In addition, the angularmomentum flowing into the protoplanet system is determined bythe shearing motion in the region of r ’ rH (see x 4.5). There-fore, under the assumption that the protoplanet is smaller than theinnermost cell width, it is expected that the thermal effect fromthe protoplanet is sufficiently small for the acquisition process ofthe angular momentum to take place when we are using muchsmaller cells than the Hill radius.

Mizuno (1980), Bodenheimer & Pollack (1986), and Ikomaet al. (2000) suggested that gaseous protoplanets have enve-lopeswith high temperatures. For example,Mizuno (1980) showedthat, for the Jovian case, gas distributed in the range of rk 2 ;1011 cm behaves isothermally, while that distributed in the rangeof rP 2 ; 1011 cm behaves adiabatically in their spherically sym-metric calculations (see Fig. 4 of Mizuno 1980). Thus, gas inthe range of rP 2 ; 1011 cm (�30rJup) has a higher temperaturethan that of the ambient medium. However, it is expected that thegaseous envelope hardly affects the angular momentum of theprotoplanet system, because the size of the envelope (�30rJup)is much smaller than the Hill radius (rH ¼ 744rJup). A large partof the angular momentum is distributed in the range of 0:5rHPrP rH, as is shown in x 4.4. On the other hand, the size of thecircumplanetary disk may be affected by the thermal envelope.Thus, when we investigate the formation of the circumplanetarydisk, we have to include realistic thermal evolution around theprotoplanet. However, it is difficult to study the thermal evolu-tion around the protoplanet, because we need to solve the radi-ation hydrodynamics in three dimensions. In a subsequent paper,we will investigate the effect of the thermal envelope under sim-ple assumptions.

Fig. 12.—Ratio of the azimuthal to the Keplerian velocity, v�/vK (color and contours), around the protoplanet for models M02 (left), M04 (middle), and M06 (right).Four levels of the grid are superimposed in each panel. The outer and inner white dashed circles represent the Hill radius, r ¼ rH, and half of the Hill radius, r ¼ 0:5rH,respectively. The black curves represent the contours of v� ¼ cs, inside which gas rotates with a supersonic velocity v� > cs.


5.2. Angular Momentum for Jupiter and Saturn

In x 4.5, we fixed the physical quantities to be those at ap ¼5:2AU. However, in our calculation, since we use dimensionlessquantities, we can rescale these quantities to be those at any orbitap. Under the standard model (Hayashi et al. 1985), we can gen-eralize equation (32) as

jlm ¼ 7:8 ; 1015Mp

MJup

ap

1 AU

� �7=4cm2 s�1

¼ 2:5 ; 1013Mp

M�

ap

1 AU

� �7=4cm2 s�1: ð34Þ

We assume that the protoplanet system acquires the gas withthe average specific angularmomentumof equation (34)when theprotoplanet has a mass of Mp < 1MJup. Thus, in order to estimatethe angular momentum of the protoplanet system, we need tointegrate equation (34) by mass, up to the present values of gasgiant planets. For example, the angular momentum in our modelfor Jupiter is

JJup ¼Z MJup

jlm 5:2 AUð Þ dM ¼ 1:3 ; 1047 g cm2 s�1; ð35Þ

which is about 30 times larger than that of the present Jupiter(4:14 ; 1045 g cm2 s�1). On the other hand, the angular momen-tum in our model for Saturn is

JSat ¼Z MSat

jlm 9:6 AUð Þ dM ¼ 3:6 ; 1046 g cm2 s�1; ð36Þ

which is 50 times larger than that of the present Saturn (7:2 ;1044 g cm2 s�1), where MSat ¼ 95:16M� is the Saturnian mass.Note that the orbital angular momenta of the present Jovian andSaturnian satellites can be ignored in the protoplanet system be-cause they are considerably smaller than the Jovian and Saturnianspin angular momenta.

The above estimation corresponds to the angular momentumof a protoplanet-disk system, which includes the central planetand protoplanetary disk. We showed that the angular momentumof the protoplanet system is 30Y50 times larger than the pres-ent spin of the gaseous planets. Note that, in reality, we are com-paring the angular momentum of the protoplanet system withthat of the present gaseous planet system, because the orbitalangular momenta of the satellites in the present gas giant systemsare negligibly small compared with the spin angular momenta ofthe gas planets. The angular momenta flowing into the Hill sphereare distributed into the spin of the protoplanet and the orbitalmotion of the circumplanetary disk. It is expected that a large frac-tion of the total angular momentum contributes to the formationof the circumplanetary disk and the residual contributes to the spinof the planet. Although the fraction of the angular momentum ofthe circumplanetary disk to that of the spin of the protoplanet isnot correctly estimated in our calculation, some of the angularmomentum is certainly distributed into the disk. Thus, it is neces-sary to know the angular momentum transfer and dissipationmechanism for the circumplanetary disk in order to follow fur-ther evolution of the protoplanet system. Takata & Stevenson(1996) proposed the de-spin mechanism of a protoplanet by theplanetary dipole magnetic field, in which an initially rapidly ro-tating protoplanet can be spun down to the present value by themagnetic interaction between the protoplanet and the circum-planetary disk.

5.3. Disk Formation and Implications for Satellite Formation

When the circumplanetary disk is formed around the proto-planet, it is possible for satellites to form in the disk. While thereare many scenarios for satellite formation (e.g., Stevenson et al.1986), regular satellites around gas giant planets are supposed tobe formed in the gaseous disk as for the planet’s formation in theprotoplanetary disk (e.g., Korycansky et al. 1991; Canup&Ward2002). Observations showed that the regular satellites aroundJupiter and Saturn are distributed only in the close vicinity of theplanets (rP 50rp) and are on prograde orbits near the equatorialplane. Thus, it is expected that these regular satellites formed inthe circumplanetary disk.We discussed the angular momentum of the protoplanet system

in x 4.5. As the angular momentumflowing into theHill sphere isbrought into both the protoplanet and the circumplanetary disk,we cannot estimate the fraction of the angular momentum thatgoes to the circumplanetary disk. When we assume that the cen-trifugal force is balanced with the gravity of the protoplanet, wecan derive the centrifugal radius rcf as

rcf ¼j2

GMp

: ð37Þ

To quantify the disk size, we adopt the specific angular momen-tum in equation (37) as those in equation (34). The centrifugalradius for a proto-Jovian disk withMp ¼ 1MJup and ap ¼ 5:2 AUis

rcf ;Jup ¼ 1:5 ; 1011 cm; ð38Þ

which is 22 times as large as the Jovian radius (rJup ¼ 7:1 ;109 cm�3). The Galilean satellites, which are regular satellites,are distributed in the range of 6rJupP rP 27rJup, which is con-sistent with the centrifugal radius derived from our calcula-tion. Note that the disk size is expected to be larger than equa-tion (38) due to the thermal effect around the protoplanet. In thesame way, we estimate the centrifugal radius of a proto-Saturniandisk as

rcf ;Sat ¼ 4:1 ; 1011 cm; ð39Þ

which is 68 times as large as the Saturnian radius (rSat ¼ 6:0 ;109 cm). The Saturnian representative regular satellites (Mimas,Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus)are distributed in the range of 3rSatP rP 60rSat, which also cor-responds to our result for the centrifugal radius.Figure 13 shows the structure around the protoplanet at the

end of the calculation ( tp ’ 20) for model M1, in which theprotoplanet has a mass of 1MJup. Each panel to the right is shownat 4 times themagnification of each panel to the left. Figures 13a,13b, and 13c show the shock fronts outside the Hill radius and athick disk between the shock front and the Hill sphere. In prox-imity to the protoplanet, a concave structure is seen in Figures 13eand 13f. The contours in these figures rapidly drop around therotation axis (i.e., the z-axis). Thus, even inside the Hill sphere,except for the central region, the disk has a thick torus-like struc-ture. Figures 13g, 13h, and 13i show the structure in very closeproximity to the protoplanet. On the midplane, elliptical struc-tures are seen on a large scale (Fig. 13d), while an almost roundshape is seen in close proximity to the protoplanet (Fig. 13g)because the force acting on the gas in this region is dominated


by the gravity of the planet. The contours in Figures 13h and 13ishow a very thin disk in the region of rP0:1 (or rP50rJup).Thus, regular satellites might be formed in these circumplanetarydisks. However, in order to study the satellite formation in moredetail, we need more realistic calculations.

6. SUMMARY

To study the gas flow pattern and the accretion of angular mo-mentum onto a protoplanet system, we have calculated the evo-lution of the protoplanet system in the circumstellar disk. First,we have investigated the dependence of the angular momentumaccreting onto the protoplanet system on the spatial resolution,using different cell widths, and confirmed its convergence forcell widths much smaller than the Hill radius. Next, adopting asink cell whose size is comparable to or slightly larger than theradius of the present gaseous planets in the solar system, wehave checked that, in models both with and without the sink,the thermal effect around the protoplanet barely affects the gasflow pattern and the angular momentum of the protoplanet sys-tem. Thirdly, with a sufficiently high spatial resolution, we havecalculated the evolution of the protoplanet system with the proto-planet’s mass in the range of 0:05MJup � Mp � 30MJup, where

Mp andMJup are the protoplanet and Jovian masses, respectively.The following results are obtained:

1. The gas flow pattern in three dimensions is qualitativelydifferent from that in two dimensions: the gas is flowing ontothe protoplanet system mainly in the vertical direction in three-dimensional simulations.

2. The specific angular momentum increases as j / Mp whenthe protoplanet’s mass is MpP 1MJup, whereas it increases asj / M 2/3

p when Mpk 1MJup.3. The angular momentum of the protoplanet system is 30Y

50 times larger than the present spin of the gaseous planets in thesolar system.

4. A thin disk is formed only in the region of rP20Y60rp,where rp is the radius of the protoplanet. This location agrees withthe orbital radii of regular satellites around the present gaseousplanets in the solar system.

These conclusions imply that a protoplanet system can acquiresufficient angular momentum from the circumstellar disk, andde-spin mechanisms of the protoplanet system are necessaryin order to realize the present spin of the gaseous planets in thesolar system. We are planning higher resolution simulations so

Fig. 13.—Density distribution (color and contours) on the cross section in the z ¼ 0 plane (top), the y ¼ 0 plane (middle), and the x ¼ 0 plane (bottom), with differentoutermost grid levels of l ¼ 3 (left), 5 (middle), and 7 (right) for modelM1. Panels (g), (h), and (i) show a 16 times enlargement of panels (a), (b), and (c), respectively. Thedashed circles represent the Hill radii, r ¼ rH.


as to further investigate the evolution of the angular momentumof the gas giant planets in the near future.

We have greatly benefited from discussions with H. Tanaka,T. Tanigawa, and T. Muto. We also thank T. Hanawa for con-tributing to the nested grid code. Numerical calculations were

carried out with a Fujitsu VPP5000 at the Center for Compu-tational Astrophysics, the National Astronomical Observa-tory of Japan. S. I. is grateful for the hospitality of KITP andinteractions with the participants of the program ‘‘Star For-mation through Cosmic Time.’’ This work is supported byGrants-in-Aid from MEXT (15740118, 16077202, 16740115,and 18740104).

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ANGULAR MOMENTUM ACCRETION ONTO A GAS GIANT PLANET · 2008-12-04 · x˜; 0; ð14Þ ˜ 0 ¼ 1 ffiffiffiffiffiffi 2 (specific) angular momentum p exp ˜z2 2 at the Jovian orbit when