EVOLUTION OF THE SOLAR NEBULA. VII. FORMATION AND SURVIVAL OF PROTOPLANETSFORMED BY DISK INSTABILITY

Alan P. Boss

Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road, NW,

Washington, DC 20015-1305; boss@dtm.ciw.edu

Received 2005 February 18; accepted 2005 April 28

ABSTRACT

A major concern for the disk instability mechanism for giant planet formation is survival of the self-gravitatingclumps that form in a marginally gravitationally unstable disk. Previous grid-based calculations have found thatthese clumpsmay only survive for an orbital period or two, an outcome that has been attributed to insufficient spatialresolution of the clumps. Here we use the highest spatial resolution grid-based models to date (effectively over8 ; 106 grid points, with a locally refined radial grid and 1024 azimuthal grid points) to demonstrate that clumpformation and survival are enhanced as the numerical resolution is increased, even with a full treatment of diskthermodynamics and radiative transfer. The overall disk evolution appears to be converging toward a solution withrobust spiral arms and self-gravitating protoplanets. The survival of these protoplanets is then further exploredby introducing ‘‘virtual protoplanets,’’ point masses representing massive protoplanets that accrete gas from thedisk and interact with the disk as they orbit around the protostar. While growing cores and protoplanets formed bycore accretion are thought to be subject to significant orbital migration, the virtual protoplanet models show thatprotoplanets formed by disk instability are likely to avoid rapid inward orbital migration, at least initially, becausethe self-gravitating disk gas flows inward, past the protoplanets, while the protoplanets orbit relatively undisturbed.Subsequent orbital migration in this case may depend primarily on the outer disk lifetime and hence on the star-forming environment, i.e., whether it is Taurus-like with relatively long-lived outer disks or Orion-like withrelatively short-lived outer disks. The latter environment should lead to minimal inward orbital migration, asappears to be the case for our solar system, while the former may lead to sufficient inward migration to produce theobserved short-period gas giant planets.

Subject headings: instabilities — planetary systems — solar system: formation

1. INTRODUCTION

Two fundamentally different mechanisms have been ad-vanced for giant planet formation, core accretion (bottom-up)and disk instability (top-down). Core accretion, the generallyaccepted mechanism, relies on the collisional accumulationof rock and ice planetesimals to assemble �10 M� solid cores,which then slowly accrete massive gaseous envelopes fromthe disk gas (Mizuno 1980; Lissauer 1987; Pollack et al. 1996;Kornet et al. 2002; Inaba et al. 2003; Hubickyj et al. 2005). Theopposite extreme is disk instability, where gas giant protoplan-ets form rapidly through a gravitational instability of the gas-eous portion of the disk (Cameron 1978; Boss 1997, 1998, 2000,2001, 2002a, 2002b, 2003, 2004; Mayer et al. 2002, 2004) andthen more slowly contract to planetary densities. In this sce-nario solid cores form simultaneously with protoplanet forma-tion by the coagulation and sedimentation of dust grains in therelatively low density, newly formed clumps of disk gas anddust. Hybrid formation mechanisms can also be envisioned thatuse the local pressure maxima of spiral arms and the associatedheadwinds and tailwinds to drive meter-sized solids toward thecenters of the arms, where they are more likely to collide andgrow to planetesimal size (Haghighipour & Boss 2003a, 2003b;Rice et al. 2004; Durisen et al. 2005).

Recently Rafikov (2005) has concluded that disk instabilitycannot form gas giant protoplanets at distances of �1 AU fromsolar-type stars because the disk gas would be too hot at suchdistances. This point was first made by Boss (1997) and wasconsidered one of the attractions of the ‘‘best of both worlds’’scenario presented therein, where the terrestrial planets had to

form slowly by collisional accumulation, while the gas giantscould form rapidly by gravitational instability of the cooler gasbeyond a few AU. Rafikov (2005) also concluded, however, thatdisk instability cannot form protoplanets at �10 AU, a find-ing that is at odds with detailed models of disk instabilities in-cluding a full treatment of disk thermodynamics, i.e., radiativetransfer and large-scale convective motions (Boss 2002b, 2004).One major reason for this discrepancy is Rafikov’s (2005) assump-tion that the instability requires a Toomre Q ¼ 1, i.e., a highlygravitationally unstable disk, as was assumed in the outermostregions of the pioneering models (Boss 1997). However, ithas long since been recognized that such a highly unstable diskis unlikely to occur and that marginally unstable disks withminimum Q-values of 1.5 (Pickett et al. 2003; Boss 2000) oreven higher are more likely to exist. Disks with initial mini-mum values ofQ ¼ 1:5 can still form clumps by disk instability(Boss 2002b, 2004). Because Rafikov’s (2005) analysis de-pends on a factor involving Q6, the difference between his as-sumption ofQ ¼ 1 and the disk instability models withQ ¼ 1:5is a factor of 1:56 � 11, sufficiently large to account for much ofthe apparent discrepancy. Rafikov’s (2005) main point is that inorder for a disk instability to form protoplanets, the disk must beboth gravitationally unstable and able to cool efficiently. De-tailed models have shown that both criteria can indeed be met inrealistic, relatively low mass disks at distances of �8 AU andbeyond (Boss 2002b, 2004).

1.1. Orbital Migration

In addition to raising the question of the formation mechanismof gas giant planets, the new era of extrasolar planet discovery

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The Astrophysical Journal, 629:535–548, 2005 August 10

# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

has forced theorists to consider mechanisms for starting andstopping inward orbital migration of planets. The first Jupiter-mass extrasolar planet discovered (51 Pegasi b) orbits its hoststar in 4.23 days (Mayor & Queloz 1995). Because forming aJupiter-mass planet so close to its host star seems to be difficult,if not impossible (Bodenheimer et al. 2000b), it appears thatsome giant planets must form at larger distances and then mi-grate inward to their final orbital distances (Lin et al. 1996). Thepossibility of significant orbital migration complicates the taskof making firm observational predictions based on the differentformation mechanisms.

Recently Sozzetti (2004) has shown that the metallicitiesof the host stars appear to correlate with the orbital periods ofthe planets, with the more metal-rich stars having significantlymore short-period planets, while long-period planets exist forboth metal-rich and metal-poor stars. This suggests that the well-known metallicity correlation has more to do with inward or-bital migration than with the formation mechanism, as highermetallicity should lead to shortened timescales for type II in-ward migration (Livio & Pringle 2003; Boss 2005). Livio &Pringle (2003) concluded that migration is insensitive to themetallicity, even though their analysis of type II migration foundthat migration timescales were halved by increases in the met-allicity by factors of �10. If the migration timescale is com-parable to the inner disk lifetime, as seems to be the case, giventhe wide spread in semimajor axes of the known extrasolar gasgiants, then even factors of 2 in migration rates could very wellaccount for the enhancement of short-period planets around themost metal-rich stars, regardless of how the gas giant planetsformed. Higher metallicity implies a higher surface density ofsolids, which hastens giant planet formation by core accretion(Hubickyj et al. 2005), but higher metallicity also implies higherdust opacity, which slows the accretion of the gaseous atmo-sphere (Hubickyj et al. 2005), making the overall effect on coreaccretion unclear. The disk instability mechanism is somewhatinsensitive to metallicity or dust opacity (Boss 2002a). Whilesome of the metallicity correlation may be caused by ingestionby the star of planetary material (Santos et al. 2004), it thus ap-pears that at least some of the metallicity correlation is causedby inward planet migration.

Orbital migration is a serious threat to the formation and sur-vival of planets in the core accretion scenario. Growing coreswith masses in the range from 3 to 30M� are subject to inwardmovement by type I migration on timescales of�104 to�105 yr(Nelson & Papaloizou 2004), timescales that are orders of mag-nitude shorter than the time needed (�4 ; 106 yr) for a pro-toplanet to grow large enough to open a gap in the disk andtransition to type II migration ( Inaba et al. 2003; Hubickyj et al.2005). Moreover, type III migration (Masset & Papaloizou 2003)appears to be an equally dangerous threat, as gas flowing throughthe disk gap can force a planet to migrate on timescales that areeven shorter than for type I migration, although the migrationcan be either inward or outward in the case of type III migra-tion. Nelson & Papaloizou (2004) found that type I migrationcould be a random walk rather than a monotonic process in re-gions of magnetorotational instability (MRI; Balbus & Hawley1991), perhaps thereby lengthening the lifetime of �10 M�cores. However, given that MRI is confined primarily to theionized surfaces of protoplanetary disks (Gammie 1996), thismechanism is unlikely to slow the migration of cores growingin the disk midplane.

Type I migration presents no problem for disk instabilitybecause this mechanism sidesteps the phase of growing coreswith masses of �3 M� and proceeds directly to protoplanets

with a Jupiter mass or more (�1MJ). Type III migration doesnot present a problem either, as it does not occur for such mas-sive protoplanets (�1MJ), even in disks with masses of Md �0:1 M� (F. Masset 2004, private communication). Type II mi-gration is mitigated by the fact that with a disk mass of Md �0:1 M�, the planet-forming midplane of the disk is again neu-tral and magnetically dead, and hence immune to MRI. As aresult, it is unclear if protoplanets in the disk instability scenariowill be subject to classic type II migration, where the disk is as-sumed to have an effective viscosity derived from some process(such as MRI) throughout its extent. Instead, protoplanetsformed in a disk capable of a disk instability will be subject tothe forces associated with a dynamically evolving, marginallygravitationally unstable disk, at least initially. It is unclear whena well-defined disk gap will form in this scenario, or what theexpected long-term outcome should be, although eventuallytype II migration should be a possibility for long-lived disks(i.e., Taurus-like).Ida & Lin (2004) presented an analysis of the type II migra-

tion of protoplanets formed by core accretion. While they didnot include the effects of type I migration, which are potentiallyfatal for growing cores, their concentration on type II migrationmeans that their analysis should apply to protoplanets formedby disk instability, once the disks lose sufficient mass and theprotoplanetary orbits become stable enough for disk gaps toopen and for type II migration to dominate. Ida & Lin (2004)showed that type II migration leads to distributions of planetarysemimajor axes and masses that resemble those of the knownextrasolar giant planets (e.g., their Fig. 12), a conclusion thatargues in favor of protoplanet formation by disk instability.The overabundance of hot Jupiters in the models of Ida & Lin(2004) also suggests that some of the observed correlation ofstellar metallicity with short-period planets is caused by inges-tion of unfortunate giant planets by their host stars.

1.2. Clump Survival

One of the major concerns regarding the viability of thedisk instability mechanism has been the survival of the newlyformed clumps. Boss (2000) found that in disk instability mod-els with increasingly higher spatial resolution in the crucialazimuthal (�) direction, clump formation became increasinglyrobust, to the point that with N� ¼ 512, a clump could last forabout two orbital periods before being torn apart by Keplerianshear, tidal forces, and internal gas pressure. Boss (2000) attrib-uted the eventual loss of the self-gravitating clumps to insufficientspatial resolution to allow the clumps to contract and becomesufficiently dense as to be immune to the forces that seek to de-stroy them. However, the Boss (2000) models employed locallyisothermal thermodynamics, which errs on the side of clumpsurvival. Therefore, Boss (2001) repeated the calculations witha full treatment of thermodynamics and radiative transfer, find-ing that with N� ¼ 512, a well-defined clump lasted for a littleover one orbital period before colliding with the innermost gridboundary at a radius of 4 AU. Clearly the fate of clumps formedby disk instability requires further investigation.

1.3. Clumps and Virtual Protoplanets

In this paper we pursue the question of the survival of clumpsformed by disk instability with several different approaches.First, we present a sequence of disk instability models similar tothose of Boss (2001), but with a locally refined radial grid andwith unprecedented azimuthal resolution, N� ¼ 1024. We thenexplore the extent to which the Poisson solver for the gravi-tational potential of the disk controls the fate of clumps by

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approximating the gravitational field of dense clumps withsoftened gravitational potentials from point masses. Finally, weexplore the orbital migration of self-gravitating clumps bypresenting the results of several models of marginally unstabledisks where a self-gravitating clump of gas and dust is replacedwith a ‘‘virtual protoplanet’’ (VP). AVP is a point mass that ac-cretes mass and angular momentum from the disk, therebydetermining its orbital evolution, subject to the gravitationalforces of the protostar and the disk, while the disk itself reactsto the gravitational force of the VP. The models show that theVPs appear to orbit stably for indefinite periods of time, even asthe unstable disk gas flows inward past the VP to accrete ontothe protostar.

2. NUMERICAL METHODS

The disk instability calculations were performed with a fi-nite volume code that solves the three-dimensional equations ofhydrodynamics and radiative transfer, as well as the Poissonequation for the gravitational potential. The code is the same asthat used in previous studies of disk instability (Boss 2001,2002a, 2002b, 2003). The code has been shown to be second-order accurate in both space and time through convergencetesting (Boss & Myhill 1992).

The equations are solved on spherical coordinate grids withNr ¼ 101, N� ¼ 23 in �/2 � � � 0, and N� ¼ 256, 512, or1024. The radial grid is either uniformly spaced with �r ¼0:16 AU between 4 and 20 AU or nonuniformly spaced with�r ¼ 0:16 AU from 4 to �8 AU, �r ¼ 0:08 AU from �8to �12 AU, �r ¼ 0:16 AU from �12 to �13 AU, and �r ¼0:32 AU from �13 to 20 AU. This nonuniform radial grid isreferred to as the local mesh refinement (LMR) grid, as it con-centrates the radial grid points at the radial distances of 8–12 AU where clumps first form in previous disk instabilitymodels. The LMR preserves the second-order accuracy of thecode as the finite-difference replacements were chosen to besecond-order accurate for nonuniform radial grids. The � gridis compressed into the midplane to ensure adequate verticalresolution (�� ¼ 0N3 at the midplane). The � grid is uniformlyspaced, to prevent any bias in the azimuthal direction. Thecentral protostar wobbles in response to the growth of non-axisymmetry in the disk, thereby rigorously preserving the lo-cation of the center of mass of the star and disk system. Thenumber of terms in the spherical harmonic expansion for thegravitational potential of the disk is NYlm ¼ 32 or 48. The Jeanslength criterion (Boss 2002b) is used to ensure that the clumpsthat form are not numerical artifacts: even at the maximumclump densities, the numerical grid spacings in all three co-ordinate directions remain less than 1

4of the local Jeans length.

The boundary conditions are chosen at both 4 and 20 AU toabsorb radial velocity perturbations (Boss 1998). Mass andlinear or angular momentum entering the innermost shell ofcells at 4 AU are added to the central protostar and therebyremoved from the hydrodynamical grid. Nomatter is allowed toflow outward from the central cell back onto the main grid.Similarly, mass and momentum that reach the outermost shellof cells at 20 AU are effectively removed from the calculation:the mass piles up in this shell and is assigned zero radial ve-locity. This matter is not allowed to return to the main grid. Notethat both the inner and outer boundary conditions are designedto absorb incident mass and momentum, rather than to reflectmass and momentum back into the main grid.

As in Boss (2001, 2002a, 2002b, 2003), the models treatradiative transfer in the diffusion approximation, which shouldbe valid near the disk midplane and throughout most of the disk

because of the high vertical optical depth. The radiative flux termis set equal to zero in regions where the optical depth � dropsbelow 10, in order to ensure that the diffusion approximationdoes not affect the solution in regions where it is not valid. Asa result, it has not been found necessary to include a flux limiterin the models (Boss 2001); artificial viscosity is also not em-ployed. The energy equation is solved explicitly in conserva-tion law form, as are the four other hydrodynamic equations.Cooling of the disk midplane appears to occur largely as a re-sult of convective cells, stretching from the midplane to thedisk surface, and capable of cooling the disk on a timescale of�50 yr at 10 AU (Boss 2002b, 2004).

Further details about the code may be found in Boss (2002b).The code has been extensively tested on a wide variety of testcases (Boss & Myhill 1992), including the nonisothermal testcase for protostellar collapse (Myhill&Boss 1993). Bodenheimeret al. (2000a) described the results of isothermal collapse cal-culations performed with several different three-dimensionalcodes, illustrating the excellent agreement between the resultsobtained with the Boss &Myhill (1992) code and with an adap-tive mesh refinement (AMR) code. Protostellar collapse cal-culations are considerably more computationally challengingin many ways than the calculations of Keplerian disks consid-ered here.

3. INITIAL CONDITIONS

The physical model under consideration consists of a 1 M�central protostar surrounded by a protoplanetary disk with a massof 0.091 M� between 4 and 20 AU. Disks with similar massesappear to be necessary to form gas giant planets by core accretion.Kornet et al. (2002) assumed a disk mass of 0.164 M� in theircore accretion models, while Inaba et al. (2003) required a diskwith a mass of 0.08 M� in order to form a Jupiter-mass planet.

The initial protoplanetary disk structure is based on the fol-lowing approximate vertical density distribution (Boss 1993)for an adiabatic, self-gravitating disk of arbitrary thickness innear-Keplerian rotation about a point mass Ms:

� R; Zð Þ��1¼ �o Rð Þ��1� � � 1

�

� �

;2�G� Rð Þ

K

� �Z þ GMs

K

1

R� 1

R2 þ Z 2ð Þ1=2

" #( );

where R and Z are cylindrical coordinates, �o(R) is the midplanedensity, and �(R) is the surface density. The adiabatic pressure(used only for defining the initial model: the radiative transfersolution includes a full thermodynamical treatment) is defined byp ¼ K�� , where K is the adiabatic constant and � is the adiabaticexponent. The adiabatic constant is K ¼ 1:7 ; 1017 (cgs units)and � ¼ 5/3 for the initial model. The radial variation of themidplane density is a power law that ensures near-Keplerianrotation throughout the disk

�o(R) ¼ �o4

�R4

R

�3=2

;

where �o4 ¼ 1:0 ;10�10 g cm�3 and R4 ¼ 4 AU. A lower den-sity halo �h of infalling molecular cloud gas and dust surroundsthe disk, with

�h(r) ¼ �h4

�R4

r

�3=2;

EVOLUTION OF SOLAR NEBULA. VII. 537No. 1, 2005

where �h4 ¼ 1:0 ;10�14 g cm�3 and r is the spherical coordi-nate radius. The initial velocity field vanishes inside the disk,except for Keplerian rotation, while in the halo the initial ve-locity field is given (based on conservation of energy) by

vr ¼ ��GMs

r

�1=2cos �;

v� ¼�GMs

r

�1=2

sin �;

v� ¼�GMs

r

�1=2:

The translational (vr, v�) velocity field in the halo is simply ver-tical infall toward the disk midplane, and the azimuthal velocityis taken to be Keplerian.

The initial temperature profile is based on the two-dimensionalradiative hydrodynamics calculations of Boss (1996) and is thesame as used in the previous models (Boss 2001, 2002a, 2002b).Two outer disk temperatures are investigated, To ¼ 40 or 50 K.As a result of the initial temperature and density profiles, thedisks initially haveQ gravitational stability parameters as low asQmin ¼ 1:5 for To ¼ 50 K or Qmin ¼ 1:3 for To ¼ 40 K. In lowoptical depth regions such as the disk envelope, the temperatureis assumed to be 50 K, consistent with heating by radiation fromthe central protostar at distances of order 10 AU.

4. LMR AND HIGH-N� MODELS

We first examine models with a locally refined radial grid(LMR) and varied azimuthal grid resolution. Table 1 lists all themodels discussed in this paper.

4.1. Technique

Previous calculations of disk instabilities by Boss (1997, 1998,2000, 2001, 2002a, 2002b, 2003, 2004) all used uniformly spacedradial grids. Here we present the results of three LMR models(AR, ARH, and ARH2) where the radial grid spacing was re-duced by a factor of 2 in the critical range of radii from 8 to12 AU where the clumps tend to form first. Because of the LMR,the effective radial grid resolution in the most important regionof the disk is effectively doubled. Model A had a uniformlyspaced radial grid and N� ¼ 256, for an effective total of over1 ;106 grid points. With the LMR grid, model AR had effec-tively over 2 ;106 grid points. Because of the increased valuesof N� ¼ 512 and 1024, models ARH and ARH2 had effectivelyover 4 ;106 and 8 ; 106 grid points, respectively. These appearto be the highest effective spatial resolutions achieved to date ina finite-difference calculation of a disk instability. Boss (2000,2001) achieved an effective resolution of 2 ; 106 grid pointswith a uniform radial grid and N� ¼ 512.

Because of the extreme computational burden associatedwith models ARH and ARH2, these two models were startedafter 298 yr of previous evolution of model A by taking the diskstructure of model A and interpolating it onto a grid with eithera doubled or quadrupled value of N�.

4.2. Results

Figures 1 and 2 show the formation of spiral arms and clumpsin the midplane of model A after 325 yr of evolution from theinitial disk configuration. The most prominent clump, locatedat about eleven o’clock in Figures 1 and 2, stands out from the

disk by only about a factor of 4 in density. In contrast, for modelARH2, shown in Figures 3 and 4, after 324 yr, this same clumpat about eleven o’clock is much better defined and is at least20 times denser than its surroundings. Comparing these twosets of figures, we can see that in model ARH2 the spiral arms,clumps, and shock fronts, while similar to those of model A, arebetter defined and more numerous. Model ARH, with inter-mediate spatial resolution, is similarly intermediate in terms ofthe development of its nonaxisymmetric structure compared tomodels A and ARH2. It is clear that while the calculations ap-pear to be converging toward a robust structure composed ofspiral arms and clumps, even the highest spatial resolutionmodel ARH2 is not necessarily converged to the continuumlimit, at least not on the scale of the densest clumps.In order to separate out the improvement achieved by LMR

from that of increasingN�, Figures 5 and 6 present the midplanedensities for model AR after 338 yr. Comparing to Figures 1and 2 for model A without LMR (although at a slightly earliertime) shows that the addition of LMR has significantly improvedthe sharpness of the arms and clumps in the 8–12 AU band, aswell as inside 8 AU. In fact, the results of model AR are muchmore comparable to those of model ARH2 with quadrupled N�,as shown in Figures 7 and 8 after 339 yr. Nevertheless, in modelARH2 the spatial definition of the clumps and arms has con-tinued to improve over that seen in model AR. Evidently bothLMR and quadrupling N� lead to significant improvements inthe spatial resolution.Boss (2000, 2001) found that as N� ! 1, clump forma-

tion became more robust. The same result evidently applieshere with the LMR models and models with N� ¼ 512 and1024.Figures 9 and 10 compare the midplane temperature distri-

butions for models AR and ARH2 after 338 and 339 yr of evo-lution, respectively. Again, the strong similarities between thetwo models suggest that the calculations are converging towarda solution where the spiral arms are compressionally heated sig-nificantly over the surrounding disk, although again the higherspatial resolutionmodel ARH2 continues to exhibit a somewhatgreater richness of detail.Density and temperature profiles through the densest clumps

in models AR and ARH2 are shown in Figures 11 and 12 andFigures 13 and 14, respectively. The clump in model ARH2 isslightly better defined than in model AR, with both clumpsstanding out over their surroundings by factors of 10–100 indensity. These clumps both occur at the inner edge (8 AU) ofthe LMR grid, where the midplane temperature begins to rise

TABLE 1

Numerical Specifications for the Three-Dimensional

Disk Instability Models

Model

To(K) min(Qi) �r N� NYlm �c VP

A................... 50 1.5 UNI 256 32 . . . . . .AR................ 50 1.5 LMR 256 32 . . . . . .

ARH............. 50 1.5 LMR 512 48 . . . . . .

ARH2 ........... 50 1.5 LMR 1024 48 . . . . . .E ................... 40 1.3 UNI 256 32 . . . . . .

ER ................ 40 1.3 LMR 256 32 . . . . . .

EP................. 40 1.3 UNI 256 32 Yes . . .

EV ................ 40 1.3 UNI 256 32 . . . Yes

Note.—UNI denotes a uniformly spaced radial grid, and LMR indicateslocal mesh refinement of the radial grid.

BOSS538 Vol. 629

toward the central protostar (Figs. 12 and 14). Midplane tem-peratures in the clumps tend to rise by a factor of 2 or less overtheir surroundings. Midplane temperature increases appear tobe held to these modest increases by the efficiency of convec-tive cooling of the disk midplane coupled with the assumedrapid loss of heat from the surface of the disk (Boss 2004).

Figures 15–20 present the time evolution of the clumps andspiral arms in model ARH2 over a period of time from 313 to340 yr, a total of 27 yr of evolution, or about an orbital period at10 AU. During this evolution, a number of separate clumps canbe followed as they form and orbit around the central protostar,as described in the captions to the figures. Two of the clumpsmerge into one clump, and a few more appear during the evo-lution. Clumps 1, 2, and 3 can be traced back to their formationafter around 301 yr of evolution in model ARH2, i.e., for aprevious half-orbit and 12 yr of existence. Models ARH andARH2 have both been running for well over 1 yr on dedicatedAlpha chip workstations, and the models continue to run to test

Fig. 3.—Equatorial density contours for model ARH2 after 324 yr.

Fig. 4.—Same as Fig. 3, with the hatch marks removed.

Fig. 2.—Same as Fig. 1, with the hatch marks removed in order to allow thedensity contours in the high-density clumps to be resolved. Density contoursagain represent factor of 2 changes in density. The log of the maximumequatorial density (RHOMAX) is noted above the plot.

Fig. 1.—Equatorial density contours for model A after 325 yr of evolution. Inthis figure, as well as in the subsequent figures, a 0.091M� disk is in orbit around a1 M� protostar. In all the figures, the entire disk is shown, with an outer radius of20 AU and an inner radius of 4 AU through which mass accretes onto the centralprotostar. Hatched regions denote density clumps and spiral arms with densitieshigher than 10�10 g cm�3. Density contours represent factor of 2 changes in density.

EVOLUTION OF SOLAR NEBULA. VII. 539No. 1, 2005

the survivability of the clumps. The major clumps continue toexist at 342 yr in model ARH2.

The properties of the most well-defined clumps evident inFigure 20 can be estimated as follows, where clumps are de-fined to include all adjacent disk gas that is denser than 1/20 ofthe maximum clump density. For clump 4, the maximum den-sity is 2:0 ; 10�9 g cm�3 with an average temperature of 92 K.The clump’s mass is 1.6MJ (where MJ is the mass of Jupiter),considerably larger than the Jeans mass at that density andtemperature of 0.67MJ, demonstrating that the clump is gravi-tationally bound. The average spherical radius of the clump is0.71 AU, smaller than the tidal radius of 0.72 AU, showing that

the clump is marginally stable to tidal disruption. This criticaltidal radius effectively defines the boundary of the clump: if notfor tidal forces, the clumps would be even larger. The clump hasan orbit with a semimajor axis of 9.0 AU and an eccentricity of0.06. For clump 5, the maximum density is 5:5 ;10�9 g cm�3

with an average temperature of 94 K. The clump’s mass is 1.7MJ,compared to the Jeans mass at that density and temperature of0.57MJ. The average spherical radius of the clump is 0.64 AU,equal to the critical tidal radius of 0.64AU. The clump has an orbitwith a semimajor axis of 7.9 AU and an eccentricity of 0.04.As previously argued by Boss (2000, 2001), these self-

gravitating clumps should be able to contract sufficiently on a

Fig. 6.—Same as Fig. 5, with the hatch marks removed.

Fig. 7.—Equatorial density contours for model ARH2 after 339 yr.

Fig. 8.—Same as Fig. 7, with the hatch marks removed.

Fig. 5.—Equatorial density contours for model AR after 338 yr.

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timescale shorter than an orbital period to become compactenough to allow their continued survival as gas giant proto-planets. While the present models offer further support of thisassertion, it is likely that an AMR code is needed to further but-tress this claim. The smoothed particle hydrodynamics (SPH)calculations of Mayer et al. (2002, 2004) used a locally definedsmoothing length to demonstrate the long-term survival of theclumps formed in their high-resolution disk instability calcu-lations. The present calculations are in full agreement with the

results in favor of long-term clump survival previously foundby Mayer et al. (2002, 2004).

5. POISSON SOLVER VARIATIONS

We next explore the extent to which the Poisson solver for thegravitational potential of the disk controls the fate of clumps byapproximating a portion of the gravitational field of denseclumps with the gravitational potentials of point masses.

5.1. Technique

Boss (2000, 2001) found that NYlm was equally important asspatial resolution for robust clump formation. Models where

Fig. 10.—Equatorial temperature contours for model ARH2 after 339 yr.

Fig. 11.—Equatorial density profile for model AR after 338 yr. The profilepasses through the dense clump located at six o’clock in Figs. 5 and 6. Theprofile is along a fixed azimuthal angle with K ¼ 64.

Fig. 12.—Equatorial temperature profile formodel AR after 338 yr, for the sameazimuthal angle as in Fig. 11. Themidplane temperature is constrained to a value of50 K or higher and rises to much higher values (�600 K) in the inner disk.

Fig. 9.—Equatorial temperature contours for model AR after 338 yr ofevolution. Contours denote changes in the midplane temperature by factors of1.3 here and in Fig. 10. The temperature is constrained to be less than 1000 K,but temperatures only rise to �650 K in the innermost disk.

EVOLUTION OF SOLAR NEBULA. VII. 541No. 1, 2005

NYlmwas increased from 32 to 48 developed significantly strongerclumps. However, increasing NYlm much beyond 48 cannot betolerated unless the gravitational potential solver is allowed todominate the entire computational effort, as the computationaleffort roughly scales as N2

Ylm. Hence, a different approach hasbeen tried in order to investigate how improved representationof the gravitational potential can affect the outcome of a diskinstability. Given that clump formation and survival are largelya battle between self-gravity of the clump on the one hand andgas pressure, Keplerian shear, and tidal forces on the other hand,improving the representation of the self-gravity of a growingclump is expected to be important for clump survival.

The new technique is motivated by the realization that as aclump forms and itsmaximumdensity increases, the densitymax-imum will become increasingly poorly resolved at fixed spatialresolution and fixed NYlm. Thus, in order to attempt to provide amore accurate representation of the gravitational potential of anincreasingly denser clump, the following technique is employed.The portions of a clump with densities greater than some criticaldensity �crit are effectively sliced off of the clump and their massdetermined. The mass of each computational cell associated with

Fig. 14.—Equatorial temperature profile for model ARH2 after 339 yr, forthe same azimuthal angle as in Fig. 13.

Fig. 15.—Equatorial density contours for model ARH2 after 313 yr. Clump 1:five o’clock. Clump 2: eight o’clock. Clump 3: ten o’clock.

Fig. 16.—Equatorial density contours for model ARH2 after 319 yr. Clump 1:two o’clock. Clump 2: five o’clock. Clump 3: seven o’clock. Clump 4: threeo’clock. Clump 5: eleven o’clock. Clump 6: nine o’clock.

Fig. 13.—Equatorial density profile for model ARH2 after 339 yr. The pro-file passes through the dense clump located at seven o’clock in Figs. 7 and 8.The profile is along a fixed azimuthal angle with K ¼ 220.

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� > �crit is then assigned to a point mass located at the center ofthe cell fromwhich it was removed. For cells above themidplane,this point mass is assumed to reside in the midplane, as thesecalculations all assume symmetry above and below the equatorialplane. The gravitational potential of the entire disk is then cal-culated, but using �crit as the density for the cells with � > �crit.The gravitational potential of the point masses is then added backinto the gravitational potential of the disk, with the gravitational

potential of the point masses within each cell with � > �crit beingsoftened (i.e., a minimum is placed on the radial distance from thepoint mass) by a fraction of the radial grid spacing,�r/5, in orderto avoid singularities in the densest cells.

5.2. Results

Model EP was evolved with �crit ¼ 3 ;10�10 g cm�3, startingafter 322 yr of evolution of model E, when well-defined clumps

Fig. 18.—Equatorial density contours for model ARH2 after 331 yr. Clump 1:eight o’clock. Clump 2/3: twelve o’clock. Clump 4: ten o’clock. Clump 5: sixo’clock. Clump 6: four o’clock.

Fig. 19.—Equatorial density contours for model ARH2 after 335 yr. Clump 1:seven o’clock. Clump 2/3: ten o’clock. Clump 4: eight o’clock. Clump 5: fouro’clock. Clump 6: two o’clock.

Fig. 20.—Equatorial density contours for model ARH2 after 340 yr. Clump 1:five o’clock. Clump 2/3: seven o’clock. Clump 4: six o’clock. Clump 5: twoo’clock. Clump 6: twelve o’clock.

Fig. 17.—Equatorial density contours for model ARH2 after 325 yr. Clump 1:ten o’clock. Clump 2 (merged with 3): three o’clock. Clump 4: twelve o’clock.Clump 5: eight o’clock. Clump 6: five o’clock.

EVOLUTION OF SOLAR NEBULA. VII. 543No. 1, 2005

had formed. Figure 21 shows one of the clumps after 475 yr ofevolution, which stands out over the surrounding disk by morethan a factor of 1000 in density (Fig. 22). Given thatmodel EP hasa uniform radial grid and Nphi ¼ 256, the clump is remarkablydense and well defined. However, because of the relatively lowspatial resolution of model EP, the Jeans length constraints in theradial and azimuthal directions are violated at the location of thehigh-density clump, so this approach could only be justified incombination with sufficiently increased spatial resolution as tosatisfy all the Jeans constraints. It is interesting to note, however,that in spite of this violation inmodel EP, high-density clumps stilldisappear after an orbit or so; thus, clearly violating the Jeans con-straints does not lead to artificially long-lived clumps.

While this Poisson solver improvement clearly leads to betterdefined, denser clumps, in practice this approach is not espe-cially practical because of the slow execution of the code as-sociated with calculating the gravitational potential at each gridpoint at each time step of all of the point masses in the cells with� > �crit. However, the idea of introducing point masses tobetter represent the densest regions of the disk leads naturally tothe next topic, the introduction of VPs.

6. VIRTUALPROTOPLANETSANDORBITALMIGRATION

We now consider models where a dense clump is representedin part by a VP in the dynamically evolving disk. This approachis intended to learn more about the ultimate survival of clumpsformed in the middle of a disk that is itself in the process ofundergoing a gravitational instability. A similar approach wasused by Rice et al. (2003) in their SPH models of the orbitalevolution of clumps produced by fragmentation in protostellardisks, based on the technique developed by Bate et al. (1995).

6.1. Technique

The VP is assumed to accrete mass M at the rate given by theBondi-Hoyle-Lyttleton formula (Livio 1986;Ruffert&Arnett 1994),

M ¼ f 4��c GMð Þ2= V 2 þ c2c� �3=2

;

where f is a dimensionless coefficient, G is the gravitationalconstant, M is the VP mass, �c is the local density, cc is the lo-cal sound speed, and V is the speed of VP through the localgas. The coefficient f should be less than unity because of var-ious effects neglected in the analysis, such as the accretion ofrotating gas of nonuniform density and temperature, and shockfronts. Nelson & Benz (2003a) explored a range of values of f,from 1 to 10�4. The VP also accretes orbital angular momen-tum from the disk gas, by accreting an amount of momentumfrom the local hydrodynamical cell proportional to the massbeing accreted from that cell, i.e., by ‘‘consistent advection,’’in such a way as to guarantee the conservation of total orbitalangular momentum. The mass and angular momentum accretedby the VP are then removed from the cell in which it residesduring the time step under consideration.The gravitational force of the VP affects the disk’s evolu-

tion through having the gravitational potential of a point mass(�GM/R) added into the total gravitational potential experi-enced by the disk, where the radius R is the distance from theVP’s position to the cell’s center. This radius is constrained to beno smaller than �r/2, where �r is the local grid spacing in theradial coordinate of the hydrodynamical grid, thereby softeningthe gravitational potential in order to avoid singularities whenthe VP approaches the center of a grid cell. The VP evolves as aresult of the mass and angular momentum accreted, subject tothe gravitational potential of the protostar and disk, as well as tothe effects of centrifugal force. Similarly, the disk evolves dueto the gravitational effect of the VP, as well as that of itselfand the protostar. Gas drag is neglected, as is appropriate for amassive protoplanet. Tests with M ¼ 0 and the gravitationalpotential of only a central protostar show that VPs on initiallycircular or elliptical orbits with semimajor axes of 5 AU orbitstably for at least 500 yr, and the VP’s angular momentum isconserved to at least eight digits.Nelson & Benz (2003a) compared the results from their

detailed, two-dimensional (thin disk) hydrodynamical calcu-lations with the predictions of analytical theories of gravita-tional torques in thin disks and found that the gravitational

Fig. 21.—Equatorial density contours for model EP after 475 yr.

Fig. 22.—Equatorial density profile for model EP after 475 yr. The profilepasses through the dense clump located at one o’clock in Fig. 21. The profileis along a fixed azimuthal angle with K ¼ 178.

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torques only agreed to within factors of 3–6 or so, implying thatsome of the assumptions made in the analytical theories (e.g.,that interactions occur only at Lindblad or corotation reso-nances) may not be justifiable in real disks. Comparing theresults of hydrodynamical codes with the results from analyticaltheories that involve approximations themselves can be of lim-ited usefulness in this case. On the other hand, Bate et al. (2003)found that their three-dimensional hydro code was able to repro-duce the migration timescales for linear, type I drift calculatedby Tanaka et al. (2002) for planets less massive than 0.1MJ.While the present code has not been used to try to reproduce thepredictions of linear theories for disk migration problems, testcases for protostellar collapse problems (e.g., Boss & Myhill1992; Myhill & Boss 1993; Bodenheimer et al. 2000a), involv-ing dynamic collapse over many orders of magnitude increasein density and decrease in spatial scale, lend credence to the va-lidity of the models presented here.

6.2. Results

The VP models all started at the same time (after 322 yr) ofmodel E (the same as model ed fromBoss 2002b). The VPmod-els were continued with the same spatial resolution as model E,with the only change being the addition of the VP at the locationof the maximum density clump in the disk. Models were runwith f ¼ 0:1, 10�4, and 10�5, starting with VPmasses of 1MJ atan initial distance of �8 AU from the protostar (Fig. 23).

With f ¼ 0:1, the VP grew quickly in mass, gaining nearly2MJ in its first orbit, a mass accretion rate of�0.1MJ yr

�1. Sucha high rate of mass accretion is not physically reasonable, asnoted by Nelson & Benz (2003a), who argue that such ratesshould be less than�10�4MJ yr

�1. With f ¼ 0:1 the VP contin-ues to gain mass rapidly and its orbit spirals gradually inward,reaching �7.7MJ after 23 orbits. At this time f was set equal tozero, freezing the VP’s mass, and the orbit expanded a smallamount and became more eccentric, eventually hitting the innerboundary at 4 AU after about another 28 orbits. Throughout the

total evolution of this model, including both the f ¼ 0:1 and 0phases, disk mass flowed freely inward, past the orbiting VP,and was absorbed by the inner grid boundary, located at 4 AU.The total amount of mass accreted by this central region was�14MJ, over a time period of 559 yr, leading to a mass accretionrate into the central regions of �2 ;10�5 M� yr�1.

The remaining models explored perhaps more reasonablevalues for the VP mass accretion rate. With f ¼ 10�4, the VPstill gains mass at a high rate of �2:5 ; 10�3MJ yr

�1, rising to�1.9MJ within 400 yr (Fig. 24). The VPmoves outward towardthe end of this period but is still orbiting stably on a slightlyeccentric orbit after over 1000 yr of existence (Fig. 25). Withf ¼ 10�5, the VP grows to 1.05MJ after 350 yr (Fig. 26), a massaccretion rate of �1:4 ;10�4MJ yr

�1, roughly consistent withthe Nelson & Benz (2003a) constraint. (Note that Figs. 24, 25,and 26 had to be scanned in from paper copies, as a failure of theRAID hard drive system used for data archiving led to the lossof the data files for these models.) In this model, the VP’s orbitexpands several AU to a semimajor axis of about 10AU, becomesmildly eccentric, and continues to orbit for at least 1052 yr, whenthe calculation was terminated. Throughout both of these evo-lutions, disk mass continues to accrete onto the central regionsat a vigorous rate while the VP orbits. Bate et al. (2003) found asimilar outcome, namely, that planets with up to a few MJ un-derwent little migration, whereas planets with masses of�10MJ

underwent large-scale migration, as with the f ¼ 0:1 model.Nelson & Benz (2003b) found that a Jupiter-mass planet suf-

fered significant, monotonic, inward orbital migration within1000 yr in their models of gravitationally stable (Q > 5) disks.In the absence of disk gravity, the usual assumption (e.g., Kley2000), their inward migration rate increased to about 1 AU in150 yr. The VPs in the models with f ¼ 10�4 and 10�5 were notlost to inwardmigration on these timescales, as might have beenexpected. They also did not transition to type II migration, in

Fig. 23.—Equatorial density contours for the initial state of EV modelswith VPs. These models start from an evolved state of model E at a time of322 yr. The VP is introduced in the clump at eight o’clock at a distance of8 AU from the central protostar.

Fig. 24.—Orbit of a VP in the midplane of the version of model EV withf ¼ 10�4, showing the first 400 yr of evolution (about 16 orbits). Arrows denotethe location and velocity of the VP every 100 time steps. The VP grows in massfrom 1.00MJ to 1.9MJ during this time. Disk gas passes by the VP throughoutthis period and into the innermost regions, bypassing the VP, which continueson a relatively stable orbit. The VP orbit expands somewhat at the end of thisinterval of time.

EVOLUTION OF SOLAR NEBULA. VII. 545No. 1, 2005

that well-defined disk gaps did not form: the disks continuedto resemble the structures evident in Figures 23 and 25, withbanana-shaped clumps and holes. The key difference here seemsto be including the effects of a marginally gravitationally un-stable disk, which evidently does not behave in the same way asnon–self-gravitating or gravitationally stable disks. The pres-ence of self-gravitating spiral arms more massive than the VPclearly leads to gravitational interactions that differ from thosein disks without self-gravity.

The models shown in Figures 24 and 26 show that orbitalmigration of massive protoplanets need not be monotonic orrapid in a marginally gravitationally stable disk, at least overtime periods of�1000 yr. While limited in the time period cov-ered, these models suggest that, to first order, giant protoplanetsformed by disk instability will not suffer rapid inward orbitalmigration shortly after their formation. A marginally gravita-tionally unstable disk has a high enough surface density to pre-vent cosmic rays from reaching and ionizing the midplane, sothat magnetically driven torques should be limited to the disksurface, while the midplane in the planet-forming region willremain magnetically dead (Gammie 1996). In the absence of anyother source of significant turbulent viscosity, type II migrationcannot occur. Once the disk’s surface density decreases enoughfor cosmic rays to ionize the entire disk, magnetic torques maybe able to viscously evolve the disk and hence eventually todrive inward type II migration of the protoplanets.

The results imply that protoplanets formed in gravitationallyunstable disks do not immediately open disk gaps and hence donot quickly transition to type II migration. Previous studies ofangular momentum transport in self-gravitating disks (Gammie2001; Lodato & Rice 2004) had suggested that such disks trans-port angular momentum in a manner analogous to standardviscous accretion disk theory, which is based on a local analysisof the gas. The new results suggest that global effects can alsobe important (Mejıa et al. 2005), as one might expect for a long-

range force such as gravity. Furthermore, the models show theformation of multiple clumps that could become protoplanets.The formation of a disk gap requires a resonant interaction be-tween the protoplanet and the disk gas lasting for several orbitalperiods. However, the ongoing dynamical interactions betweenthe protoplanet(s) and the disk gas result in continual changes inthe protoplanet’s orbit (e.g., Fig. 24), so that there is not enoughtime for the disk gas to be cleared away while in a specific or-bital resonance with the protoplanet.

7. MIGRATION, SURVIVAL, METALLICITY,AND ENVIRONMENT

While type I, II, and III orbital migration presents consid-erable problems for the core accretion mechanism for giantplanet formation, the present models show that orbital migra-tion is much less of a problem for the disk instability mecha-nism. Type I and III orbital migration is sidestepped completelyby the rapid formation of Jupiter-mass clumps, while type II mi-gration requires an effective source of turbulent viscosity in themagnetically dead midplane where the giant planets form andorbit. The present VP models show that in the disk instabilityscenario, gravitational torques between the VPs and the diskand the associated vertically driven convective motions (Boss2004) do not behave like a classic viscous accretion disk, wherea well-defined gap is expected to open around a massive proto-planet, with the disk’s viscous evolution forcing inward type IImigration of the protoplanet. Instead, the variations of f in modelEV show that only minor inward and outward orbital migrationis to be expected for newly formed protoplanets in the disk in-stability scenario, even while the disk mass is flowing rapidly in-ward past the protoplanets’ orbits and onto the central protostar.Even a marginally gravitationally unstable disk will even-

tually become gravitationally stable, probably primarily by lossof mass through accretion onto the central protostar. At thatpoint, the protoplanets should be able to open disk gaps and besubject to type II migration, driven perhaps by MRI turbulence.

Fig. 25.—Equatorial density contours for model EV with f ¼ 10�4 after1092 yr of evolution (about 30 orbits). The dense clump of disk gas that forms anatmosphere around the VP can be at eight o’clock at a distance of 11 AU. Thecalculation was arbitrarily stopped at this point because of the dense clumps thatformed at the artificial outer disk boundary. The VP seems capable of continuingto orbit indefinitely.

Fig. 26.—Orbit of a VP in the midplane of the version of model EV withf ¼ 10�5, showing the first 350 yr of evolution (about 12 orbits), as in Fig. 24.The VP grows in mass from 1.00MJ to 1.05MJ during this time. Disk gas againbypasses the VP and accretes onto the central protostar. The VP continues on arelatively stable orbit for 1052 yr before the calculation was arbitrarily halted.

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Protoplanets formed by disk instability will then be subjectto inward orbital migration, possibly explaining the correlationof short-period gas giants with the metallicity of the host star(e.g., Sozzetti 2004). Type II migration is expected to becomemore rapid as the metallicity of the star (and presumably of thedisk) increases because internal disk torques depend on the diskviscosity �, and in standard viscous accretion disk theory (e.g.,Ruden & Pollack 1991) � ¼ csh, where is a free parameter,cs is the sound speed, and h is the disk thickness. As the disk met-allicity decreases, the disk opacity decreases, leading to lowerdisk temperatures, lower sound speeds, and a thinner disk. As aresult, � decreases with lowered metallicity, and the timescalesfor type II migration increase. Ruden & Pollack (1991) haveshown that viscous disk evolution times lengthen by a factor ofabout 20 when � decreases by a factor of 10. Hence, stars thatare metal-rich because of their primordial abundances (as op-posed to having ingested planetary material) should have moreshort-period gas giant planets, regardless of whether the giantsformed by core accretion or disk instability, as seems to be thecase (Sozzetti 2004).

This argument for producing short-period Jupiters assumesthat the disk lasts long enough for type II migration to operate.Protoplanetary disks in regions of low-mass star formation, suchas Taurus and Auriga, should last longer than disks formed inregions of high-mass star formation, such as Orion and Carina,because in the latter regions far-ultraviolet (FUV) and extreme-ultraviolet (EUV) radiation photoevaporates the gas from theouter disk gas on timescales of �0.1 Myr (Bally et al. 1998).The rapid loss of the outer disk gas would prevent the transfer ofangular momentum outward by gravitational torques from theinner disk that is needed for type II migration to produce short-period Jupiters, freezing the orbits of the giant planets at closeto their initial distances. This situation would imply a lack ofmajor orbital migration since formation for the giant planets, asseems to be required for our solar system. This argument is con-sistent with the suggestion that our solar system formed in aregion of high-mass star formation, in order to use the FUVandEUV radiation to remove the outer disk gas and then to strip thegaseous envelopes from the outermost gas giant protoplanets,turning them into ice giants, all in the context of the disk insta-bility scenario (Boss et al. 2002). In Taurus-like regions, wherethe outer disk gas remains for a few million years or so, all thegiant protoplanets formed by disk instability would become gasgiant planets. These planets would also be subject to significantinward type II migration, leading to short-period Jupiters accom-panied by gas giants with increasingly longer periods. In eitherthe core accretion or disk instability scenario, the mechanismfor stopping inward migration and producing hot Jupiters islikely to be depletion of gas in the innermost disk or migrationinto a magnetized region of the disk (Terquem 2003).

8. CONCLUSIONS

The present models have shown that the chances for forma-tion and survival of clumps formed by disk instability continueto improve as the spatial resolution of the three-dimensionalcalculation and representation of the self-gravity of the clumpsimproves. While even the present calculations do not appearto have reached the continuum limit, the degree of convergenceevident in the three-dimensional models implies that the ulti-mate outcome of a disk instability may be the formation of strongspiral arms and dense, self-gravitating clumps that could be-come gas giant protoplanets. Further progress in demonstratingthis claim may await the application of AMR codes to diskinstability problems.

Remarkably, VPs in gravitationally unstable disks are able toaccrete mass at high rates and still orbit more or less stably atdistances of 5–10 AU, even as the unstable disk gas flows pastthe protoplanets and accretes onto the central protostar at ratesof �2 ; 10�5 M� yr�1. While the three-dimensional modelsonly demonstrate this orbital stability for time periods of slightlyover 1000 yr, there is no reason to believe that the protoplan-ets could not orbit stably for considerably longer periods oftime, based on the present models. In particular, it is impor-tant to note that the Jupiter-mass protoplanets do not open well-defined, steady gaps in these gravitationally unstable disks, as aresult of the somewhat chaotic evolution of the disk gas. Or-bital migration in a gravitationally unstable disk might be termed‘‘type IV’’ migration, although the present models imply thattype IV migration is only significant for fairly massive (>5MJ)protoplanets.

Eventually type IV nonmigration for <5MJ protoplanetsmust be succeeded by type II migration, when the disk massdecreases enough for the disk to be gravitationally stable anda classic disk gap can open. Thereafter the protoplanets willmigrate along with the disk, assuming that it has some intrinsicsource of turbulent viscosity, such asMRI. There should then bea distinct difference in the final orbits of giant planets that de-pends on the outer disk lifetimes, i.e., on the star-forming en-vironment. Relatively long-lived outer disks in regions likeTaurus should allow type II migration to be effective for longerperiods of time than in the case of the shorter lived outer disksin regions like Orion. The latter environment should lead tominimal inward orbital migration, while the former may lead tosufficient inwardmigration to produce the observed hot and warmJupiters. Spectroscopic planet search surveys have found that�15% of nearby stars have short-period, hot or warm Jupiters,while at least �25% have no short-period Jupiters but showevidence for possible long-period, cold Jupiters (A. Hatzes 2004,private communication). These rough estimates are consistentwith the idea that the short-period Jupiters are the result ofinward type II orbital migration associated with the relativelylong lives of the outer disks in regions of low-mass star for-mation like Taurus because most stars are thought to form inregions like Orion, not Taurus (Lada & Lada 2003). If planetarysystems can form in regions of high-mass star formation whereouter disks are short lived, then most gas giant planets shouldexperience only minor degrees of inward orbital migration, asseems to be the case based on the spectroscopic planet searchesto date. In fact, there are good reasons for arguing that the solarsystem itself formed in a region like Orion, as ice giant planetformation in the disk instability scenario (Boss et al. 2002) re-quires the presence of a strong source of EUV/FUVradiation. Iftrue, this would imply that solar system analog planetary sys-tems may be considerably more commonplace than would bethe case if the solar system formed in a region like Taurus.

I thank Caroline Terquem and Pawel Artymowicz for dis-cussions about the effects of metallicity on type II migration, thereferee for a number of helpful and perceptive comments, andSandy Keiser for her continued assistance with recalcitrantcomputers. Supported in part by NASA Planetary Geology andGeophysics grant NAG5-10201 and by NASA AstrobiologyInstitute grant NCC2-1056. The calculations were performedon the Carnegie Alpha Cluster, the purchase of which was par-tially supported by NSF Major Research Instrumentation grantMRI-9976645.

EVOLUTION OF SOLAR NEBULA. VII. 547No. 1, 2005

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