Giant Planet Formation†

EXOPLANETS

Space Science Series University of Arizona Press arXiv:1006.5486v1 [astro-ph.EP] 29 Jun 2010 Giant Planet Formation†

Gennaro D’Angelo NASA Ames Research Center and University of California, Santa Cruz

Richard H. Durisen Indiana University

Jack J. Lissauer NASA Ames Research Center

Gas giant planets play a fundamental role in shaping the orbital architecture of planetary systems and in affecting the delivery of volatile materials to terrestrial planets in the habitable zones. Current theories of gas giant planet formation rely on either of two mechanisms: the core accretion model and the disk instability model. In this chapter, we describe the essential principles upon which these models are built and discuss the successes and limitations of each model in explaining observational data of giant planets orbiting the Sun and other stars.

1. INTRODUCTION tion is essential for the formulation of theories describing the origins and evolution of terrestrial planets capable of Jupiter and Saturn are composed predominantly of Hy- sustaining life in the form of complex organisms. drogen and Helium and are therefore referred to as gas gi- Observations of exoplanets have vastly expanded our ants,althoughmostoftheseelementsarenotingaseous database by increasing the number of known planets by form at the high pressures of these planets’ interiors. The over one and a half orders of magnitude. The distribution majority of extrasolar planets (or exoplanets)discoveredso of observed exoplanets is highly biased towards those ob- far have masses in excess of about one quarter of Jupiter’s jects that are most easily detectable using the Doppler ra- mass (M )andareknowntobeorsuspectedofbeinggas J dial velocity and transit photometry techniques (see chap- giants. Since Helium and molecular Hydrogen do not con- ters by Lovis and Fischer and by Winn), which have been dense under conditions typically found in star forming re- by far the most effective methods of discovering planetary- gions and in protoplanetary disks, giant planets must have mass object orbiting other stars. Although these extraso- accumulated them as gasses. Therefore, giant planets must lar planetary systems are generally different from the so- form prior to the dissipation of protoplanetary disks. Opti- lar system, it is not yet known whether our planetary sys- cally thick dust disks typically survive for only a few mil- tem is the norm, quite atypical, or somewhere in between. lion years and protoplanetary disks lose all of their gaseous Nonetheless, some unbiased statistical information can be contents by an age of 107 years (see the chapter by ! distilled from available exoplanet data (see the chapter by Roberge and Kamp), hence giant planets must form on this Cumming). timescale or less. The mass distribution function of young compact ob- Giant planets contain most of the mass and angular mo- jects in star forming regions (e.g., Zapatero Osorio et al., mentum of our planetary system, and thus they must have 2000)extendsdownthroughthebrowndwarfmassrange played a dominant role in inﬂuencing the orbital properties to below the deuterium burning limit (12–14 M ). This ob- of smaller planets. Gas giants may also affect the timing and J served continuity implies that most isolated brown dwarfs efﬁciency of the delivery of volatile materials (Chambers and isolated high planetary-mass objects form via the same and Wetherill, 2001), such as water and carbon compounds, collapse process as do stars. to the habitable zones of planetary systems, where liquid However, star-like direct quasispherical collapse is not water can exist on a (rocky) planet’s surface. In the solar considered a viable mechanism for the formation of Jupiter- system, Jupiter is also believed to have reduced the impact mass planets, because of both theoretical arguments and ob- rate of minor bodies (such as comets) on the Earth (Horner servational evidence. The brown dwarf desert,aprofound et al., 2010). Therefore, understanding giant planet forma- dip over the range from 15 MJ to 60 MJ in the mass distribution function of companions∼ orbiting∼ within several †In EXOPLANETS, edited by S. Seager, to be published in the Fallof 2010 in the Space Science Series of the University of Arizona Press AU of Sun-like stars, suggests that the vast majority of gas (Tucson, AZ). giants form via a mechanism different from that of stars. 2.1 From Dust to Planetesimals 2COREACCRETIONMODELS

Atheorybasedonauniﬁedformationscenarioforrocky tions in a protoplanetary disk around a solar-mass star), the planets, ice giants much as Uranus and Neptune, and gas gi- gas accretion rate decreases as the planet mass increases due ants is the core accretion model,inwhichtheinitialphases to tidal interactions between the planet and the disk. The gi- of a giant planet’s growth resemble those of a terrestrial ant planet continues to grow at an ever decreasing rate until planet’s. The only alternative formation scenario receiving there is no gas available within the planet’s gravitational signiﬁcant attention is the disk instability model,inwhicha reach. giant planet forms directly from the contraction of a clump of gas produced via a gravitational instability in the proto- 2.1. From Dust to Planetesimals planetary disk. The formation of a heavy element core is an essential In this chapter, we introduce the basic physical concepts part in any core accretion model. Hence, for completeness, of gas giant planet formation according to core accretion we present a summary of the basic elements involved in models (Section 2)anddiskinstabilitymodels(Section3). this process. An in-depth discussion of these concepts can In Section 4,wepresenttheoreticalargumentssuggesting be found in the chapter by Chambers. that most of the giant planets known to date formed via core The core formation starts at some distance from the star, accretion and pose some of the outstanding questions that most likely beyond the “snow line” where disk tempera- still need an answer. Future prospects that may settle some tures allow for condensation of water ice and solid material. of the lingering issues are discussed in Section 5. This process begins from (sub-)micron-sized dust particles, which may have originated from the interstellar medium 2. CORE ACCRETION MODELS and/or condensed within the disk’s gas. Such small solid The initial phases of a giant planet’s growth by core nu- particles are well-coupled to the gas, on account of their cleated accretion proceed through an accumulation process large surface area to mass ratio, and therefore move with it. of solid material, in the same fashion as terrestrial planets As particles collide and stick together, they can grow form. Dust and small solid grains, which are entrained in larger and start to de-couple from the gas and interact with the predominant gas component of a protoplanetary disk, it because of differential rotation between solids and gas. coagulate into larger particles. Centimeter-sized particles The relative velocity stems from the fact that the gas ro- tend to settle towards the disk midplane, aggregating and tates about the star slower than do solids due to the radial eventually forming kilometer-sized agglomerates, referred pressure gradient that partially counteracts the gravitational to as planetesimals. attraction of the gas towards the star. The resulting inter- Planetesimals grow larger via pairwise collisions, lead- action can be described in terms of friction via gas drag on ing to the formation of a planetary embryo.Anembryomay the particle, producing an acceleration proportional to and have the ability to grow at a rate that increases as its mass in the opposite direction of the relative velocity between the increases, eventually consuming nearly all planetesimals in particle and the gas. This friction generally removes orbital the neighborhood, while rapidly gaining mass and becom- angular momentum from solid particles, causing them to ing a planetary core or protoplanet.Onceaplanetarycore drift towards the star. grows large enough and the escape velocity from its surface Along with the radial drift, small solids also experience a exceeds the thermal speed of the surrounding gas, a tenuous friction in the direction perpendicular to the disk midplane. envelope of gas begins to accumulate around the solid core. Assuming that the vertical motion of the gas is negligible For most of the following planet’s growth history, ther- (compared to that of solids), the vertical motion of a solid mal pressure effects within the envelope regulate the ac- particle is obtained by applying the second law of dynamics cretion of gas. The ability of the envelope to radiate away d2z 1 dz the gravitational energy released by incoming planetesimals 2 2 + + z Ω =0, (1) and by contraction limits the amount of gas that can be ac- dt τf dt creted by the planet. A slow contraction phase ensues in in which z represents the distance over the disk midplane, which the accretion rates of both solid material and gas are τf is the friction timescale (also referred to as “stopping small. However, as the protoplanet grows and its total mass time”), and Ω is the particle’s angular velocity about the exceeds the value beyond which the pressure gradient (in star. The second term on the lefthand side of equation (1) the envelope) can no longer balance the gravitational force, is the opposite of the frictional force per unit mass exerted the envelope undergoes a phase of rapid contraction, which by the gas and the third term is the opposite of the vertical allows more gas to be accreted. The augmented mass of the gravitational acceleration imposed by the star. In the limit envelope triggers further contraction and gas can be thereby τf 0,dustisperfectlycoupledtogasanddz/dt is equal accreted at an ever increasing rate. to the→ vertical gas velocity, which is zero by assumption. During this epoch, known as runaway gas accretion Hence z is a constant and no sedimentation would occur. phase, the gas accretion rate is regulated by the ability of Notice that for the gaseous part of the disk, equation (1)is the surrounding disk to supply gas to the planet’s vicinity. replaced by an equation for hydrostatic equilibrium in the In this stage, once the planet’s mass exceeds a few tenths of vertical direction, with the vertical component of the gas Jupiter’s mass (for typical temperature and viscosity condi- pressure gradient balancing the vertical gravitational force

D’Angelo, Durisen, & Lissauer 2of28 Giant Planet Formation 2.1 From Dust to Planetesimals 2COREACCRETIONMODELS

(see equation (3)). In the limit τf ,equation(1)as- Centimeter-sized icy/rocky aggregates would require only sumes the form of a harmonic oscillator,→∞ hence particles os- on the order of thousands of orbits to approach the disk’s cillate about the disk midplane with a period equal to 2π/Ω. midplane. For solid particles whose size is shorter than the mean Athinlayerofsolidmaterialmaythusaccumulateatthe free path of gas molecules and whose velocity relative to disk’s midplane in a relatively short amount of time com- the gas is slower than the gas sound speed, c,thefriction pared to disk lifetimes. If this layer is sufficiently dense, time is (Epstein, 1924) clumps may form through gravitational instabilities within the layer (e.g., Goldreich and Ward, 1973). Such process ρ R τ = sp sp , (2) may produce kilometer-sized bodies, known as planetesi- f ρ c ! "! " mals. The assumption that the gas vertical velocity is small where ρ and R are the solid particle’s volume density sp sp compared to the particle vertical velocity is, however, only and radius, respectively, while ρ is the gas density. In- valid in absence of sustained turbulent motions. In gen- dicating with H the vertical scale-height of the gaseous eral, protoplanetary disks are expected to be somewhat tur- component of the disk and with a the orbital distance to bulent and moderate amounts of turbulence could affect the the star, hydrostatic equilibrium in the vertical directionre- settling timescales of small grains (R 1cm). Ignor- quires that sp ! ing the second term in equation (5), particle speeds are 2 z˙ zΩ τf .Assumingaturbulentkinematicviscosity 1 ∂p GM! z + =0, (3) of≈− the form ν = αHc (Shakura and Syunyaev, 1973)and ρ ∂z a2 a ! " # $ indicating with λturb the vertical mixing length (the typical in which p = p(a, z) is the gas pressure and M! is the size of eddies), vertical gas speeds due to turbulence can be mass of the star. Writing the pressure as p = c2ρ,ap- estimated as z˙turb ν/λturb or z˙turb αHc/λturb = 2 | | ∼ | | ∼ proximating c as being independent of height, and integrat- αH Ω/λturb,andthus ing over the disk thickness one finds that c2 H2Ω2, 3 ≈ z˙ z λturb Ωτf where Ω = GM!/a is the Keplerian angular veloc- . (6) z˙ ∼ H2 α ity of the gas. (A more accurate determination of the & turb & ! "! " % & & disk’s rotation rate can be derived from imposing hydro- & 2 & Since z λturb/H& & 1 (eddies cannot be larger than the static equilibrium in the radial direction, which results in ≤ 2 3 2 disk’s thickness), we have that z/˙ z˙turb < Ωτf /α.No- Ω (GM!/a )[1 (H/a) ]). Thus, equation (2)gives | | $ − tice that in order for turbulent motions to be subsonic (i.e., z˙ c), the turbulent kinematic viscosity ν a ρsp Rsp | turb| ≤ ≤ Ωτf = . (4) c λ cH and thus α 1.Underthedisk’scondi- H ρ a turb ≤ ≤ # $ ! "! " tions adopted above, the ratio of the particle’s settling ve- Assuming typical values at a 5AUfor ρ and H/a around locity to the gas vertical turbulent speed is then z/˙ z˙turb < 10 ≈ 3 3 | | asolar-massstarof10− gcm− and 0.05,respectively, 10− ρsp Rsp/α (ρsp and Rsp are in cgs units). 1 3 and expressing both the density and radius of the particle in Therefore, if the turbulence parameter α " 10− ,the 3 cgs units, Ωτ 10− ρ R .Thisestimateisapplicable settling time of centimeter-sized particles may be affected f ∼ sp sp up to values of Rsp less than tens of centimeters, i.e., on the since vertical turbulent mixing could influence their verti- order of the mean free path of gas molecules ( 1/ρ)under cal transport. Studies of grain settling in the presence of the adopted gas conditions. ∝ turbulent motions indicate that particles tend to concentrate Equation (1)hastheformadampedharmonicoscillator in stagnant regions of the flow and that concentrations are and can be integrated once initial conditions are provided size-dependent, which may lead to the accumulation of sub- for position and velocity. Setting z = z0 and z˙ = dz/dt = centimeter-sized particles (Cuzzi et al., 2001). Previous 0 at time zero and taking into account the inequality Ωτf growth of these agglomerates would still rely on sticking 1 found above, the solution can be approximated as ( collisions of smaller particles. The growth from centimeter-sized to kilometer-sized 2 t Ω τf 2 2 t/τf bodies is still a poorly understood process and an ac- z z0 e− Ω τf e− . (5) $ − tive field of both theoretical and experimental research. # $ The second term in the above solution is a fast transient Nonetheless, there is observational evidence that it does that rapidly decays to zero, hence we can further approx- occur in nature: dust particles are observed in debris disks 2 t Ω τf around other stars and small bodies of tens to hundreds of imate the solution as z z0e− .Therefore,inor- der for the altitude above≈ the midplane to reduce by more kilometers in size are observed in the Kuiper belt around the Sun. than 99% of its initial value, Ωt>5/(Ωτf ) 5 3 ∼ × 10 /(ρsp Rsp),inwhichagainρsp and Rsp are expressed 1Based on observed gas accretion rates and other properties ofyoungstellar in cgs units. Micron-sized particles would take millions of objects with disks, typical values of α are in the range from 10 4 to ∼ − orbital periods (τrot =2π/Ω)tosettle,whichsuggeststhat 0.1 (e.g., Hueso and Guillot, 2005; Isella et al., 2009). they first need to aggregate and grow into larger particles. ∼

D’Angelo, Durisen, & Lissauer 3of28 Giant Planet Formation 2.2 From Planetesimals to Planetary Cores 2COREACCRETIONMODELS

2.2. From Planetesimals to Planetary Cores atangentialapproach(whentheprojectilegrazestheem- bryo). Conservation of speciﬁc energy during the collision We shall now assume that the solid component of the 2 2 requires that vrel vta 2GMs/R and therefore the grav- protoplanetary disk has had time to agglomerate into plan- ≈ − 2 etesimals, rocky/icy bodies of a kilometer (or larger) in itational enhancement factor, i.e., the ratio (Reﬀ /R) ,can be cast in the form size. These objects are massive enough (! 1015–1016 g 12 ∼ 10− M )thattheymaygravitationallyinteractwiththeir v2 ⊕ F =1+ esc , (8) neighbors and perturb their velocities. As a result of these g v2 interactions, planetesimals become prone to collisions. rel

Although the outcome of a collision between two plan- in which vesc = 2GMs/R is the escape velocity from the etesimals depends upon their relative velocity, we assume surface of the target planetesimal. Notice that, if the plan- " that collisions among such objects result in mergers rather etesimal radius is not negligible compared to the embryo than fragmentation. Under this assumption, the growth rate radius, the radius R in equation (7)andinvesc should be of a planetesimal of mass Ms can be written as (Safronov, replaced by the sum of the two radii. 1969) If relative velocities are high and vrel vesc,then dMs 2 & = πR vrel ρs Fg , (7) Fg 1 (i.e., Reff R)andequation(7)yieldsdMs/dt dt 2 ≈ 2/3 ≈ ∝ R Ms .Ifrelativevelocitiesarelowandvrel vesc, where R is the planetesimal radius, vrel is the relative ve- ∝ 2 2 2 4 '4/3 then Fg vesc/vrel R and dMs/dt R Ms .If locity between the two impacting bodies, ρs is the volume the escape≈ velocity is∝ very much larger than∝ the∝ relative ve- density of the solid component of the disk. The product locity, then three-body effects (star, planetary embryo, and vrel ρs represents a mass flux, that is the amount of solid planetesimal), neglected in deriving equation (8), must be material sweeping across the target planetesimal per unit taken into account to compute F (Greenzweig and Lis- 2 g time and unit area. The quantity Fg =(Reff /R) is a grav- sauer, 1992). itational enhancement factor, which is the ratio of the ef- In the high relative velocity regime, the growth timescale 2 fective cross-section (πReff )oftheaccretingplanetesimal τ = M (dM /dt) 1 (i.e., the time it takes for the mass of 2 s s s − to its geometrical cross-section (πR ). This factor accounts the embryo to increase by a factor e 2.7), is proportional for the ability of the growing body to bend towards itself 1/3 ≈ to M and therefore the growth rate, 1/τ ,ofanembryo the trajectories of other, sufficiently close, planetesimals. s s reduces as it grows larger. This implies an orderly growth It is often useful to express the accretion rate dM /dt s of large planetesimals, which tend to attain similar masses. in terms of the surface density of the solid material Σ = s From equation (7)castintermsofsolids’surfacedensity, ρ dz H ρ ,withH being the vertical thickness of s s s s one obtains that the timescale τ necessary to build a large the planetesimal∼ disk. In order to do so, one can assume og embryo, or planetary core, of mass M can be estimated as that! the gravitational force exerted by the star is the major component of the force acting on planetesimals in the ver- 1/3 1 4π 2/3 ρ2 M tical direction of the disk (as in equations (1)and(3)) and Ωτ sp . (9) og ∼ π 3 Σ that relative velocities between planetesimals are isotropic. # $ % s & Thus, Hs is of order vrel/Ω,whereΩ is the Keplerian angu- 4 To assemble a body of mass M 1.6 10− M (about lar velocity of the growing planetesimal. Equation (7)can ∼ × ⊕ 2 as massive as Ceres, the largest object in the asteroid belt) then be cast in the form dMs/dt = πR ΩΣs Fg.Since 2 3/2 within a solids’ disk of density 10 g cm− ,orderlygrowth the angular velocity along a Keplerian orbit is Ω a− , 6 ∝ would require a few 10 orbital periods, which already rep- the accretion rate of planetesimals is slower farther from resents a fairly long timescale compared to lifetime of pro- the star (neglecting variations of Σ F with distance). This s g toplanetary disks around solar-mass stars (" 107 years). would imply that dMs/dt at the current location of Uranus In the low relative velocity regime, the growth rate of (19.2AU)wasabout7 times as small as it was at the current an embryo is 1/τ M 1/3,andthusthelargertheplan- location of Jupiter (5.2AU). s s etary embryo the faster∝ it grows, a process known as run- If we neglect the collective gravitational action of the away growth.Duringrunawaygrowth,thelargestembryo other planetesimals and that of the star during an encounter, grows faster than any other embryo within its accretion re- two interacting bodies can be described in the framework gion. Although a two-body approximation (equation (8)) of a two-body problem. Assuming that the target planetesi- yields an unlimited gravitational enhancement factor as the mal has already grown somewhat larger than the neighbor- ratio v /v 0,gravitationalscatteringduetothree- ing bodies, hence becoming a planetary embryo, the im- rel esc body effects set→ a limit to F ,whichcannotexceedvalues pacting body can be thought of as a projectile. We can g much beyond several thousands (Lissauer, 1993). therefore use the approximation that the embryo sits on During the assembly of a planetary core through the the center of mass of the two-body system. In the rest growth of an embryo, relative velocities among planetesi- frame of the embryo, conservation of the specific angular mals play a crucial role in determining the accretion rates, momentum (i.e., angular momentum per unit mass) reads as indicated by equations (7)and(8). The velocity distribu- R v Rv ,wherev is the relative velocity for eff rel # ta ta tion of a swarm of planetesimals is affected by a number of

D’Angelo, Durisen, & Lissauer 4of28 Giant Planet Formation 2.3 Isolation Mass of Planetary Cores 2COREACCRETIONMODELS

physical processes, such as elastic and inelastic collisions, extent of this region is several Hill radii, RH,whichcanbe gravitational scattering, and frictional drag by the gas, the understood by recalling the deﬁnition of RH.Thisisthedis- results of which can be highly stochastic. tance of the equilibrium point from mass Ms (the embryo), If the orbit radial excursion of planetesimals, relative to on the line connecting (and in between) masses M! and Ms, the orbit of an embryo, is on the order of the embryo’s Hill in a reference frame rotating at angular velocity Ω.The radius2 force balance for a planetesimal of mass much smaller than 1/3 2 2 Ms both M! and Ms requires that MsRH− M!(a RH)− + RH = a , (10) 2 − − 3M! Ω (a RH)=0.Hence,toleadingorderinRH/a, ! " −2 2 2 M R− M a (1 + 2R /a)+M a (1 R /a)=0, or smaller, then the accretion is said to be shear-dominated s H ! − H ! − H whose solution− is the righthand side of equation− (10). because growth is dictated by Keplerian shear in the disk, Once the feeding zone has been severely depleted, the rather than by planetesimals’ random velocities. If the em- planetary embryo becomes nearly isolated. Numerical N- bryo’s orbit is nearly circular, this situation requires that or- body simulations suggest that isolation of an embryo oc- bital eccentricities and inclinations of planetesimals should curs once a region of width ∆a bR ,whereb 4,on be R /a,orsmaller.Therelativevelocitybetweenan H H each side of the embryo’s orbit becomes∼ nearly emptied≈ of embryo∼ and a planetesimal traveling on a circular Keplerian planetesimals (Kokubo and Ida, 2000). The isolation mass, orbit with radii a and a + ∆a (∆a/a 1), respectively, is M ,ofaplanetaryembryocanthereforebecalculatedas v a Ω(a+∆a) Ω(a) aΩ(a)"[1 3∆a/(2a)] 1 , iso rel M 4π a ∆a Σ 4π abR Σ .NoticethathereΣ hence# v| 3Ω∆a/− 2.Ifweapproximate| ≈ | − ∆a as the− half-| iso s H s s rel refers∼ to the initial value∼ of the solids’ surface density. By width| of the| ≈ region within which the gravity ﬁeld of the using equation (10), the isolation mass can be written as embryo dominates over that of nearby embryos, the accre- 2/3 tion rate is dMs/dt Ms Ω∆a ρsFg.Wewillseein 2 3 ∝ (4π a b Σs) Section 2.3 that ∆a is generally proportional to the em- Miso . (11) ∼ # 3M! bryo’s Hill radius, R .Sincev ΩR and ρ H | rel| ∼ H s ∼ Σs/Hs Σs/RH,theplanetesimalaccretionratebecomes According to equation (11), a planetary core of mass ∼ 2/3 dM /dt Ms ΩΣ F . 11 M (if b 4)wouldbecomenearlyisolatedat s s g ∼ ⊕ ≈ This result∝ is formally the same as the accretion rate in 5.2AU from a solar-mass star if the local surface density 2 2 of solids were equal to 10 g cm 2.ThisvalueforΣ is a the orderly regime (in which vesc/vrel 1 and Fg 1), − s 2 2 " ≈ couple of times as large as that predicted for a minimum except that now vesc/vrel 6RH/R 1,andtherefore ∼ ' 3 Fg 1.Accordingly,thegrowthtimescale(inunitsof mass proto-solar nebula (MMSN). Additionally, the isola- 1/Ω')isgivenbytherighthandsideofequation(9)dition mass increases with distance from the star, although vided by the gravitational enhancement factor, Fg.The this increase can be somewhat compensated for by a re- growth rate of an embryo is much larger than it is during duced surface density of solids, which is expected to de- the orderly growth phase (since F 1), but it reduces crease as a increases. g ' as the embryo mass grows larger (see equation (9)). This An order-of-magnitude estimate of the timescale, τiso, phase of growth, often referred to as oligarchic,maylead required to reach isolation can be obtained by taking the to the formation of massive embryos at regular intervals ratio of equation (11)totheplanetesimalaccretionrate(ex- in semi-major axis. Notice that, neglecting variations of pressed in terms of Σs,seeSection2.2), which yields

ΣsFg with orbital distance, as in the orderly growth regime 3 2/3 1/2 3/2 iso M! a ρsp dMs/dt M! a− .Thisimpliesthattheaccretionrate Ωτ C , (12) ∝ iso ∼ F a2Σ M of planetesimals reduces as the stellar mass decreases or as g $ s ! ! " the distance from the star increases. 2/3 5/3 where iso =(2π) (2/√3) b/π 2.44√b.Notice that forC given values of F and Σ ,thistimescaleincreases# 2.3. Isolation Mass of Planetary Cores g %s with increasing distance from the star and decreases with The rapid (runaway/oligarchic) growth of a planetary 5/2 2/3 increasing stellar mass: τiso a M!− .Atthecurrent embryo continues until its neighborhood, or feeding zone, ∝ location of Jupiter around a solar-mass star, Ωτiso 3 has been substantially cleared of planetesimals (Lissauer, 8 2 ∼ × 10 /Fg,ifΣs is 10 g cm− .Assumingasituationinwhich 1987). The feeding zone represents the domain within the gravitational enhancement factor is of order 1000 over which the dominant embryo is able to signiﬁcantly deﬂect the entire accretion epoch, τiso is several tens of thousands the paths of other planetesimals towards itself. The radial of orbital periods or several 105 years.

2 The Hill radius, RH,representsthedistanceoftheLagrangepointL1 from 3This is deﬁned as the amount of heavy elements (heavier than Helium) the secondary in the circular restricted three-body problem(e.g.,Murray observed in the planets and minor bodies of the solar system (mostly con- and Dermott, 2000). It provides a rough measure of the distance from tained in giant planets) augmented by an amount of gas such to render the the secondary beyond which the gravitational attraction of the primary and proto-solar nebula composition equal to that of the young Sun. Such def- centrifugal effects prevail over the gravity of the secondary. Note however inition constrains the total mass, 0.02 M ,and,toalesserextent,the that this region, which identiﬁes the Roche lobe, is not a sphere and its surface density distribution of the∼ minimum! mass solar nebula (see, e.g., volume is about athird that of a sphere of radius RH (e.g., Eggleton, 1983). Davis, 2005).

D’Angelo, Durisen, & Lissauer 5of28 Giant Planet Formation 2.4 Growth of Thermally-Regulated Envelopes 2 CORE ACCRETION MODELS

In deriving equation (12)weassumedthatthesurface At distances of several AU from a solar-mass star, relatively density of solids, Σs,iscomparabletoitsinitialvalueand small bodies (Ms 0.01 M )canretainanatmosphere.In the geometrical cross-section of the planet only depends on these early phases,∼ the atmospheric⊕ gas is optically thin and the core radius, R.Yet,thesurfacedensityofsolidsdrops thermal energy released by impacting planetesimals can be if accreted planetesimals are not replaced by others arriving readily radiated away, allowing for contraction of the enve- from outside of the core’s feeding zone, which would op- lope and for additional gas to be accreted. erate towards increasing τiso.Moreover,duringitsgrowth Prior to achieving isolation from the planetesimal disk, toward isolation, a planetary core also accretes gas from the the accretion rate of solids is much larger than that of gas. disk, although at a much smaller rate than it accretes solids. The growing core mass forces the envelope to contract and Once the atmospheric envelope becomes massive enough more gas can be collected from the surrounding gaseous to dissipate the kinetic energy of incoming planetesimals disk. However, as the envelope grows more massive, it via drag friction, the geometrical cross-section of the pro- becomes increasingly optically thick to its own radiation, toplanet in equation (7)becomessubstantiallylargerthan which therefore cannot escape to outer space as easily as in 2 πR .Thiseffectwouldoperatetowardsreducingτiso. earlier phases. As a consequence, envelope’s temperature Aplanetarycoredoesnotnecessarilystopgrowingonce and density exceed those of the local disk’s gas. attaining the mass given by equation (11). Perturbations The pressure gradient that builds up in the envelope among planetesimals and other embryos can supply addi- opposes gravitational contraction, preventing accretion of tional solid material to the core’s feeding zone via scatter- large amounts of gas. On the one hand, ongoing accretion ing. Furthermore, a planet in excess of a Mars mass can of solid material and growth of the core help envelope con- exert gravitational torques on the surrounding gas. The disk traction, on the other the gravitational energy released in the responds by exerting the same amount of torques onto the envelope by gas compression and that supplied by accreted planet, which modify the planet’s orbital angular momen- planetesimals help maintain a relatively large pressure gra- tum, forcing it to radially migrate within the disk. As a dient. Therefore, contraction becomes self-regulated.Once result of this radial displacement, a planet may reach disk the planet achieves (near) isolation from planetesimals, the regions that still contain planetesimals. However, as the accretion rate of solids becomes small and accretion of gas planet’s mass grows, so does its ability to scatter planetesi- continues to the extent allowed by envelope compression. mals away from the orbital path or to trap them into mean- The envelope enters a stage of quasistatic contraction motion resonances (so that the ratio of a planetesimal’s or- that can be characterized by long evolution timescales. If bital period to the planet’s orbital period is a rational num- the accretion of solids and gas was negligibly small, the ber). timescale over which the envelope contracts would be re- The order-of-magnitude estimates given above for the lated to the ratio of the gravitational energy released by con- isolation mass and the timescale to reach it neglect many traction, E ,totheenvelope’sluminosity,L.Thisratio | grav| aspects of the physical processes involved in the growth of deﬁnes the Kelvin-Helmholtz timescale τKH = Egrav /L. aplanetarycore.Nonetheless,detailedcalculationsofgi- Indicating the protoplanet’s core and envelope masses| with| ant planet formation by core accretion and gas capture indi- Mc and Me,respectively, Egrav GMcMe/R¯, R¯ being cate that those estimates are valid under appropriate con- the average radius comprising| most| ∼ the protoplanet’s mass, ditions. Calculations that start from a planetary embryo and thus the contraction timescale is of M 0.1 M (about equal to the mass of Mars) or- s GM M biting a∼ solar-mass⊕ star at 5.2AU and undergoing rapid τ c e . (14) KH ∼ RL¯ growth within a planetesimal disk of initial surface den- 2 sity Σs 10 g cm− ,showthataplanetarycorebecomes Using values from Figures 1 and 2 around the middle of the ≈ 5 nearly isolated within less than half a million years when planetesimal isolation phase, τKH would be of order 10 Miso 11 M (Pollack et al., 1996), as also indicated by years. However, for the case illustrated in the Figures, most the solid≈ line in⊕ left panel of Figure 1 (see Figure’s caption of the luminosity produced during this slow growth phase for further details). is due to gravitational energy generated by accretion rather than by contraction of the envelope. 2.4. Growth of Thermally-Regulated Envelopes The length of this epoch depends on several factors, but Gas can accrete onto a planetary embryo when the ther- principally on the opacity of the envelope and on the solids’ mal energy is smaller than the gravitational energy bind- accretion rate. The more opaque the gas is to outgoing ra- ing the gas to the embryo. This condition is satisﬁed diation, the less able the envelope is to dissipate the energy when the escape velocity from the surface of the embryo, provided by gas compression and/or by continued planetes- vesc (see Section 2.2), exceeds the local thermal speed imal accretion. Moreover, accreted solids can be consumed of the disk’s gas ( 8/πc), which occurs when Ms ! by ablation during their atmospheric entry, releasing dust 2 4(H/a) M!R/(πa),or and increasing the envelope’s opacity. Therefore, a reduced ! accretion of solids shortens this epoch. As shown in the left 3 M 3 H panel of Figure 1,theslowcontractionphase,fromthetime M ! . (13) s ! a3ρ a the core mass reaches near-isolation from surrounding plan- " sp # $

D’Angelo, Durisen, & Lissauer 6of28 Giant Planet Formation 2.4 Growth of Thermally-Regulated Envelopes 2 CORE ACCRETION MODELS

Figure 1. Formation models of a giant planet by core nucleated accretion need to take into account many physical effects. These include: ı)Calculationoftheplanetesimalaccretionrateontotheprotoplanet; ıı)Calculationoftheinteractionviagasdragofimpacting planetesimals with the protoplanet’s gaseous envelope; ııı)Thermodynamicscalculationoftheenvelopestructure;ıv)Calculationof the gas accretion rate from the protoplanetary disk onto the protoplanet during the phase of slow envelope growth; v)Hydrodynamics calculation of constraints on envelope size and disk-limited gas accretion rates during the phase of runaway gas accretion. Some results from one such calculation are reported here for a model of Jupiter’s formation (Lissauer et al., 2009). This model assumes that the 2 planetesimal disk has Σs 10 g cm− and the grain opacity in the protoplanet’s envelope is 2% the value of the interstellar medium. ≈ The gaseous disk is assumed to dissipate within 3Myr.Theleftpanelshowsthemassofsolids(solidline),ofgas(dotted line), and the total mass of the planet (dot-dashed line) as functions of time. The right panel displays the radius of the planet (solid line) and that of the planet’s solid core (dashed line) from the same model.

once the isolation mass is achieved (Hubickyj et al., 2005). It thus appears that the duration of the slow contraction phase represents the main uncertainty in determining whether a giant planet can form before the gaseous component of the disk dissipates (i.e., within a few to several millions of years; see the chapter by Roberge and Kamp). However, it offers a natural explanation for the formation of planets such as Uranus and Neptune in our solar system, which do not posses very massive envelopes. These planets may have not been able, because of core and envelope conditions, or may have not had enough time, because of a dissipating disk, to acquire a large envelope. In addi- tion, static atmospheric models indicate that the removal of grains by growth and settling from the radiative zone of a planet’s envelope may signiﬁcantly reduce their contribu- tion to the opacity (Movshovitz and Podolak, 2008), hence Figure 2. Luminosity as a function of time from the model re- shortening the contraction timescale of the envelope. ported in Figure 1.Fromthetimeoftheﬁrstluminositypeak Quasi-static calculations of the envelope thermal struc- (t 0.35 Myr)tothemiddleoftheplanetesimalisolationphase ≈ ture suggest that the growth timescale of a protoplanet’s en- (t 1.6Myr), the luminosity of the protoplanet decreases by 1 ∼ velope τe = M(dMe/dt)− can be cast in the following more than a factor of 25.Thisismainlyduetothereducedaccre- parametric form (e.g., Ida and Lin, 2008) tion rate of solids. M ξ τ =¯τ ⊕ , (15) e e M etesimals to the time the rapid gas accretion phase begins, ! " lasts about 2Myr.Theassumptionmadeforthisparticular in which M = Mc + Me is the total planet’s mass and model is that the grain opacity in the protoplanet’s enve- Mc ! Miso.Thetimescaleτ¯e depends on several factors, lope is 2% that of the interstellar medium and that plan- but mostly on the opacity of the envelope. For the model 10 etesimals continue to accrete throughout the slow contrac- reported in Figure 1, τ¯e 10 years and ξ 3.No- tion phase. All else being equal, this time-span increases to tice, though, that many factors∼ affect the values∼ of param- 6Myrwhen the full interstellar opacity is adopted and de- eters ξ and, especially, of τ¯e.Ingeneral,applicabilityof creases to 1Myrif planetesimal accretion ceases entirely equation (15)requiressomepreviousknowledgeoftheen- ≈

D’Angelo, Durisen, & Lissauer 7of28 Giant Planet Formation 2.5 Critical Core Mass and Envelope Collapse 2 CORE ACCRETIONMODELS velope thermal conditions, for example through theoretical Another assumption we shall rely upon is that the gra- arguments or numerical modeling. When Me ! Mc,the dient = d ln T/dln p is nearly constant in the envelope. inner (densest) parts of the envelope, which contain most of The temperature∇ gradient can thus be cast in the form the mass, effectively contribute to the compression of the dT T dp outer parts. Yet, around this stage, the envelope may be- = . (18) dr p dr ∇ come gravitationally unstable and collapse (see Figure 1). ! "! " 2.5. Critical Core Mass and Envelope Collapse In order for energy to be transported via radiation alone, the gradient must be smaller than its adiabatic value, In order to gain some insight into the conditions lead- ∇ ad.Foraperfectmonoatomicgasandnegligibleradia- ing to envelope collapse, we can construct simplified ana- tion∇ pressure, =2/5.If ,theenvelopebe- lytical models that qualitatively describe the physical state ad ! ad comes convectively∇ unstable. By∇ substituting∇ equation (16) of the envelope. The first assumption we shall make is in place of the pressure gradient in equation (18)andusing that a slowly contracting envelope evolves through stages the equation of state for a perfect gas p/ρ = T/µ ( is of quasihydrostatic equilibria. Early studies of a static enve- the gas constant), we have that dT/dr µGMR /( Rr2), lope surrounding a planetary core (e.g., Perri and Cameron, ∼− ∇ R where M = Mc + Me,andhenceT µGM /( r).In 1974)suggestedthattheremaynotbeanystaticstructure the above relations, µ indicates the mean-molecular∼ ∇ R weight, beyond some value of the core mass. Past such a critical that is the average number of atomic mass units per gas par- core mass,theextendedouterportionsofthegaseousenve- ticle. Notice that the approximation m M made above lope become hydrodynamically unstable and collapse onto determines how deep in the envelope this∼ temperature strat- the core. Simplified models of hydrostatic envelopes can ification is applicable. shed some light over the concept of critical core mass. One Equation (17)canbewrittenas such model was used by Stevenson (1982)tointerpretre- sults of static envelope calculations performed by Mizuno dm 44π2 σ dT = rad r4 T 3 , (19) (1980). dr − 3¯κ L dr Let us assume that a gaseous envelope of mass Me sur- from which it follows that rounds a core of mass Mc and that the envelope is spheri- cally symmetric around the core. Hydrostatic equilibrium 4 dm 44π2 σ µGM 1 in an envelope shell of radius r and thickness dr implies = rad ∇ . (20) dr 3¯κ L r that there is a balance between gravitational force and pres- ! R " sure acting at the shell surface boundaries. Therefore, the Integrating both sides of this equation along r,fromthecore condition for hydrostatic equilibrium reads radius, Rc,totheenveloperadius,R,andneglectingopac- dp Gmρ ity variations4,oneobtains = 2 , (16) dr − r 4 (µ ) 4 where p and ρ are, respectively, the pressure and density at M Mc = 0 ∇ M , (21) − A κ¯ L radius r,andm is the mass within r. 2 4 An important assumption we shall make is that energy in which 0 =(π σrad/3)(4G/ ) ln (R/Rc). is transported through the envelope via radiative diffusion We seekA for the largest valueR of the core mass that can (i.e., this model applies to a radiative envelope) and that have a fully radiative envelope in hydrostatic equilibrium. the envelope is stable against convection. In general, this This can be obtained from equation (21), which gives an may be true only in the outer layers of the envelope (Bo- explicit function Mc = Mc(M),ifL is independent of both denheimer and Pollack, 1986). M and Mc.ThevalueofM for which Mc is maximum is The amount of radiation energy transported through an given by envelope shell per unit surface area and unit time, i.e., the 1/3 cr 1 κ¯ L radiation flux, is Frad = dErad/dr,wheretheradia- M = 4 . (22) −D 4 #4 0 (µ ) $ tion energy density Erad T and T is the temperature. A ∇ The radiation diffusion coefficient,∝ ,isproportionaltothe D Physically significant solutions of equation (21)canonly speed of light times the mean free path of photons, 1/(¯κρ), be obtained for M/Mcr < 41/3,andthereforethemass where κ¯ is a frequency-integrated opacity (or mass absorp- of strictly hydrostatic envelopes does not exceed 41/3 M cr. tion coefficient). Hence, for an envelope shell of mass The critical core mass for radiative envelopes is found by dm,theradiationfluxisalsoequalto(e.g.,Kippenhahn substituting equation (22)inequation(21), which gives 2 4 and Weigert, 1990) Frad = 16πr σrad/(3¯κ)(dT /dm), − 3 where σrad is the Stefan-Boltzmann constant. If the lumi- M cr = M cr. (23) nosity through the envelope, L,isnearlyconstantthenone c 4 2 can write Frad =4πr L,andthus 4Amoregeneralopacitylaw,givenbytheproductofsomepowerof p and dT 4 3¯κ L some power of T ,wouldresultinthesamerelationasequation(21), but = . (17) 2 4 with a modified form of coefficient 0 (Wuchterl et al., 2000). dm −64π σrad r A

D’Angelo, Durisen, & Lissauer 8of28 Giant Planet Formation 2.5 Critical Core Mass and Envelope Collapse 2 CORE ACCRETIONMODELS

According to these simple analytical arguments, the crit- The largest core mass that bears a fully convective enve- cr ical core mass, Mc ,isindependentofthelocaldensityand lope in hydrostatic equilibrium corresponds to a total mass temperature of the protoplanetary disk, it is weakly depen- 1/2 dent on envelope opacity and luminosity, but more strongly 3 3 cr 1 3Γ 1 TD dependent on µ and .Usingradiiandluminosityvalues M = − R . (28) 12π Γ µG ad ρD before collapse from∇ Figures 1 and 2, µ =2.25, =0.235 % ! "! ∇ " & ∇ 2 1 (Kippenhahn and Weigert, 1990), and κ¯ =0.02 cm g− , In order for equation (27)toadmitphysicallysignificant cr one obtains Mc 16 M and a total mass of 21 M cr √ ∼ ⊕ ∼ ⊕ solutions, M/M < 3.Accordingtothesesimplear- (recall that these values would apply to a strictly hydrostatic guments, strictly hydrostatic and convective envelopes are and fully radiative envelope). limited in mass to √3 M cr.Fromequations(27)and(28), As mentioned in Section 2.4,mostoftheprotoplanet’s one finds that the critical core mass of a convective envelope luminosity during the slow growth phase may be due to is gravitational energy released by accreted planetesimals. In 2 M cr = M cr. (29) these circumstances the luminosity can be approximated as c 3 L (GMc/Rc)(dMc/dt),inwhichdMc/dt is the plan- Unlike the solution for radiative envelopes, the critical ∼ 2/3 etesimal accretion rate, thus L Mc (dMc/dt).Sub- core mass for convective envelopes explicitly depends on ∝ stituting in equation (21), one obtains Mc as an implicit the disk density and temperature. Using values ρD 10 3 ∼ function of M.Bydifferentiationofthisfunction,onecan 10− gcm− and TD 100 K,whichwouldapplytoa cr cr ∼ readily show that again Mc =(3/4)M . protoplanetary disk at 5AU from a solar-mass star if The deepest and densest layers of an envelope are typ- there was no planet, equation∼ (28)givesM cr 170 M . ically convective (e.g., Bodenheimer and Pollack, 1986). However, once M achieves several tens of Earth∼ masses,⊕ Simple analytical models of hydrostatic envelopes can also both ρD and TD are affected by disk-planet interactions be constructed under the hypothesis that energy is trans- and hence depend on M.Inthesecases,becauseofthe ported only via convection, as that proposed by Wuchterl gravitational perturbation produced by the protoplanet, lo- (1993). By employing the adiabatic gradient ad,thetem- cal disk densities as well as temperatures can be smaller ∇ 2 perature gradient becomes dT/dr µGM ad/( r ). than the corresponding values in absence of the protoplanet ∼− ∇ R Instead of an energy equation, in a fully convective enve- (D’Angelo et al., 2003a), if effects of stellar irradiation can Γ lope one can use the polytropic law T/TD =(ρ/ρD) , be neglected. in which TD and ρD indicate values where the envelope The two analytic solutions for fully radiative and fully merges with the disk. In a perfect monoatomic gas with convective envelopes can be combined to obtain an estimate negligible radiation pressure, Γ =(5/3) ad.Bydifferen- of the critical core mass for composite convective-radiative ∇ tiation of the polytropic law, it follows that envelopes. Following the same line of argument, we con- (Γ 1) sider a two-layer model in which the “convective” solution dT T ρ − = Γ D , (24) applies to the inner convective layer that extends from the dρ ρ ρ ! D "! D " core radius, Rc,totheradiusRclb,whichplaystheroleof which combined with the temperature gradient above yields envelope radius (R)intheconvectivesolutiongivenabove. (Γ 1) Accordingly, temperature Tclb and density ρclb at the con- − TD ρ dρ µGM ad vective layer boundary, Rclb,replacethediskvaluesTD Γ = ∇2 . (25) ρD ρD dr − r and ρ ,respectively.Then,indicatingwithM the mass ! "! " ! R " D clb Equation (25)canbeintegratedinradiustoobtainthe within Rclb (core mass plus mass of the convective layer), cr cr envelope’s density stratification equation (29)givesacriticalvalueMclb =(3/2)Mc . The “radiative” solution applies to the outer radiative Γ ρ µ GM layer, whose boundaries are at the convective layer (outer) = ∇ad . (26) ρD − TD r radius Rclb,whichplaystheroleofcoreradius(Rc)in ! " ! R "! " the radiative solution above, and at the envelope radius The approximation m M,introducedabovetoderivethe R.Thus,themasswithintheconvectivelayerboundary, temperature gradient, again∼ determines the envelope depths M ,replacesthecoremass(M )intheradiativesolu- to which this density stratification applies. In equation (26) clb c tion above. Equation (23)givesthecriticalvalueofthe the constant of integration is set to zero by choosing as outer total mass M cr =(4/3)M cr .Therefore,thelargestplanet envelope radius R = µGM /( T ).Byintegrating clb ad D mass for which both the convective and radiative layers, and the quantity 4πr2ρdr from the∇ coreR radius, R ,toR we get c hence the entire envelope, can be in hydrostatic equilibrium Γ µ 3 is M M = ρ ∇ad M 3, (27) 4 − c B0 D 3Γ 1 T M cr = M cr =2M cr . (30) ! − "! D " 3 clb c 3 (3 1/Γ) where 0 =4π(G/ ) 1 (Rc/R) − ,whichcan According to equation (30), a composite convective- B R − 3 be approximated to 4π(G/ ) since Rc R and 3 radiative envelope collapses once the envelope mass is # $ 1/Γ > 1 when radiation pressureR can be ignored.% −

D’Angelo, Durisen, & Lissauer 9of28 Giant Planet Formation 2.6 Disk-Limited Gas Accretion Rates 2COREACCRETIONMODELS

equal to the core mass. This prediction of the simple an- (c = HΩ is the gas sound speed) and marks the distance alytic model is in agreement with the results from the de- beyond which the thermal energy of the gas is larger than tailed calculation shown in the left panel of Figure 1.Equa- the gravitational energy binding that gas to the planet. In cr tion (22)canstillbeusedtoestimateM ,wherenow the case that RB

GM 3 RB = (32) M! H H c2 Mtr = C . (37) √3 B a & C ! " 5 In this phase, we designate the accretion rate of gas simply as dM/dt since This simple gas accretion model can be compared total mass variations are overwhelmingly determined by accretion of gas, regardless of whether there is continued accretion of solids. against results from three-dimensional hydrodynamical simulations of a planet gravitationally interacting with a

D’Angelo, Durisen, & Lissauer 10 of 28 Giant Planet Formation 2.7 Gap Formation 2COREACCRETIONMODELS

Figure 3. Left: Mass growth rate, 1/τD =(dM/dt)/M ,versusmassofaprotoplanetaccretinggasatalimitingrateprovided by a protoplanetary disk. These results were obtained from a three-dimensional, high-resolution hydrodynamical modelofagrowing 2 protoplanet interacting with a disk of initial surface density at a =5.2AUof 100 g cm− ,relativethicknessH/a =0.05,andturbulence 3 parameter α =4 10− (D’Angelo and Lubow, 2008). The timescale τD is in units of orbital periods. The dashed line represents × the growth rate given by equation (36): the slanted portion corresponds to the rate in the Bondi-type gas accretion regime (1/τB with B 2.6)andthehorizontalportioncorrespondstotheHill-typegasaccretionrate(1/τH with H 0.9). At large planet masses, C " C " the growth rates drop due to the formation of the tidally produced gap in the local density distribution. Right:Averagesurfacedensity distribution of the disk’s gas, Σ,neartheplanetrelativetothelocalunperturbed(i.e.,without a planet) value, Σ0,asafunctionofthe planet’s mass. disk and growing in mass at a disk-limited gas accretion whose magnitude is of order rate. Figure 3 (left panel) shows numerical results (solid 3 2 line) and the growth rate given by equation (36)(dashed Σ 4 2 a M g a Ω , (38) line). Agreement is found for values of the constants T ≈ f ∆a M! ! " # $ B 2.6 and H 0.9.Notethatthemassscaling, Ci.e., the! slopes ofC the! two dashed line segments, is correctly which leads to exchange of orbital angular momentum be- predicted by the simple model and so is the transition mass tween the planet and the disk (see the chapter by Lubow 5 and Ida for a derivation of this torque). The factor f is typ- between the two regimes (Mtr 4 10− M!,or 14 M ≈ × ≈ ⊕ ically of order unity. It can be shown that ∆a is the larger if M! =1M ). The disk thickness in the example shown | | in Figure 3 is#H/a =0.05,whichcorresponds,fortypical between the Hill radius, RH,andthediskscale-height,H. The sign of is positive for material lying outside of the disk properties, to a local temperature T 100 K.Since Tg the gas sound speed c √T ,afactorof∼ 2 increase in planet’s orbit (∆a>0)andnegativeformaterialinside the local temperature would∝ reduce the planet’s growth rate of it (∆a<0). Hence, material orbiting outside of the by an order of magnitude during the Bondi-type accretion planet’s orbit gains angular momentum, moving towards regime. larger radii, whereas material inside of the planet’s orbit As the planet’s mass increases, the effective radius for loses angular momentum, moving towards smaller radii. gas capture may exceed the disk thickness. Moreover, den- This process tends to deplete the disk of gas along the sity perturbations due to tidal interactions of the protoplanet planet’s orbit, thus forming an annular gap in the local den- with the disk are no longer negligible. The gas density sity distribution (e.g., Lin and Papaloizou, 1986). However, along the planet’s orbit starts to be affected (see right panel in a viscous (Keplerian) disk, because of differential rota- of Figure 3), and the simple accretion model becomes in- tion gas is also subject to a viscous torque generated by applicable beyond some value of the protoplanet mass. In viscous friction between adjacent disk rings. The viscous the example shown in Figure 3,thisoccursformasses torque exerted by material inside of the orbital radius a on 4 material outside of a is (Lynden-Bell and Pringle, 1974) M ! 10− M!,whendisk-planetinteractionshavechanged the local surface density by more than 20%ofitsunper- 3πν a2 ΣΩ. (39) turbed value (i.e., when there is no planet in the disk), as Tv ≈ illustrated in the right panel. This torque tends to smooth out density gradients and redis- tribute material across the planet’s orbit, ﬁlling in the gap. 2.7. Gap Formation Aconditionforgapformationrequiresanetgainofor- Aprotoplanetembeddedinadiskexertsagravitational bital angular momentum for material outside the radius a, torque on the gas interior of a ∆a and exterior of a+ ∆a that is > 0,andanetlossoforbitalangularmo- −| | | | Tg − Tv mentum for material inside a, g + v < 0.IfH RH, the condition > translatesT intoT a condition≥ for the |Tg| Tv

D’Angelo, Durisen, & Lissauer 11 of 28 Giant Planet Formation 2.8 Final Masses of Giant Planets 2COREACCRETIONMODELS minimum planet’s mass necessary to open up a density gap in a disk M 2 H 5 3πfα , (40) M ! a ! ! " ! " where the turbulence parameter α is defined in Section 2.1. In case of very cold disks or very massive planets RH >H 2 and the torque inequality gives (M/M!) ! πfα (H/a) . For conditions adopted in the simulations reported in Fig- ure 3,substantialgasdepletionisexpectedforM ! 2 4 × 10− M! (or M ! 60 M for a disk around a solar-mass star). In accord with this⊕ estimate, the right panel of the Figure indicates that the average surface density along the planet’s orbit is reduced by 40%ofitsunperturbedvalue 4 when M ! 2 10− M!. The timescale× for gap formation can be estimated from Figure 4. Disk-limited gas accretion rates as a function of the the equation of motion of a thin viscous Keplerian disk, planet mass obtained from three-dimensional, high-resolution hy- which also evolves under the action of gravitational torques drodynamical calculations of a planet interacting with a proto- exerted by an embedded object (Lin and Papaloizou, 1986) planetary disk. Accretion rates are in units of the unperturbed surface density at the planet’s orbital radius, a,andtheplanet’s ∂Σ ∂ 1 ∂ orbital period, τrot.ThetopaxisusesM! =1M .Filledcir- πa = ( v g) , (41) ! ∂t ∂a aΩ ∂a T − T cles correspond to results for a disk with a turbulence parameter # $ 3 α =4 10− (see Section 2.1). Empty circles are for a disk with × 4 where radius a is now interpreted as a variable indicat- α =4 10− .Thedisk’saspectratioisH/a =0.05.Thecurves × ing the distance from the star. Equation (41)canbede- represent fits to the data. rived from imposing conservation of mass and angular momentum in an axisymmetric and flat disk. Ignoring vis- orbiting a solar-mass star at 5AU in a disk, whose cous torques ( v < g )andtakingtheapproximation ∼ ∂/∂a 1/∆aTin equation|T | (41), defining the gap formation gas surface density just outside the gap region is of order → 1 100 g cm 2,mayaccretegasatrates 10 3–10 2 M per timescale as τgap = Σ ∂Σ/∂t − and using equation (38), − − − ∼ 4 5 ⊕ it follows that | | year, thereby reaching a Jupiter’s mass in 10 –10 years.

M 2 ∆a 5 2.8. Final Masses of Giant Planets Ωτ πf ! | | , (42) gap ∼ M a It is not yet entirely clear what all the factors are that ! " ! " determine the final mass of a giant planet. Yet, tidal trunca- with ∆a equal to the larger of H and R .Thistimescale | | H tion of the disk by gravitational interactions with the planet is typically short, on the order of tens of orbital periods. is likely one of the main factors. As indicated in the right Once gap formation starts, accretion rates drop because panel of Figure 3,gapscanbecomequitedeeparounda of gas depletion around the planet. According to con- Jupiter-mass planet, even at moderate values of the turbu- dition (40), lower disk viscosity allows for gas depletion lence parameter α.However,gascanfilterthroughthetidal around smaller mass planets. By using equation (37), one barrier (i.e., the gap) and continue to accrete towards the finds that for planet. H α CH , (43) This can be seen in Figure 5,whichshowsthegapinthe " 9πf a CB ! " surface density distribution around a giant planet together gas depletion begins before the transition to a Hill-type gas with trajectories of material moving towards the planet accretion and the growth rate starts to decline for smaller along the inner and outer edges of the gap. The trajecto- mass planets. In a disk of thickness H/a 0.05,thismay ries are drawn in a frame co-moving with the planet. This 4 ≈ occur for α less than few times 10− . material becomes trapped in the inner parts of the planet’s Effects of gap formation can be seen in Figure 4.The Roche lobe and is eventually accreted. As mentioned in the Figure shows disk-limited gas accretion rates, derived from previous section, accretion through a gap can be quite ef- 2 hydrodynamical simulations (see D’Angelo et al., 2003b, ficient. Assuming a gas surface density of 100 g cm− ∼ for details), for protoplanets embedded in a moderately just outside the gap region at 5AU,thegrowthtimescale 3 ∼ 1 viscous (α =4 10− )andinalow-viscosity(α = (due to disk-limited gas accretion), τD = M(dM/dt)− , 4 × 3 4 10− )disk.Notice,though,thatifτ > τ ,i.e., of a Jupiter-mass planet in a disk with α 10− and × gap H ∼ f a2Σ/M (M/M )2 (a/H)4,Hill-typegasaccre- H/a 0.05 would be several thousands of orbital peri- CH ! ! ! ≈ tion may persist until RH H. ods (see Figure 4). Around a solar-mass star, the tidal bar- Disk-limited% & gas accretion∼ rates can be quite large, un- rier alone would constrain the planet’s final mass to about less the disk’s gas density is low. A Saturn-mass planet 6–7 MJ.

D’Angelo, Durisen, & Lissauer 12 of 28 Giant Planet Formation 2.9 Orbital Migration of Protoplanets 2COREACCRETIONMODELS

on a Kelvin-Helmholtz timescale (equation (14)). The luminosity is mostly due to gravitational energy released in the contraction.

2.9. Orbital Migration of Protoplanets Gravitational interactions between a protoplanetary disk and an embedded planet result in an exchange of orbital angular momentum (see the chapter by Lubow and Ida for a thorough discussion on this topic). Assuming that H ! RH and that condition (40)forgapformationisnotsatisfied, the disk’s surface density is largely unperturbed and equation (38)applies. If the planet exerts a torque g on the gas outside its orbit, then conservation of angularT momentum dictates that this gas exerts the same torque (with an opposite sign) on the planet. Therefore, the planet loses orbital angular momentum to material exterior of its orbit. By the same principle, the planet gains orbital angular momentum from material interior of its orbit. The torque expression in equation (38)issymmetricalwithrespecttotheplanet’sradial Figure 5. Surface density of a gaseous disk containing a pro- position, which would result in a zero net torque acting on 3 toplanet whose mass is M =10− M! (i.e., one Jupiter’s mass the planet. However, one can show that, because of global for a solar-mass star). The relative disk thickness is H/a =0.05 variations of the disk properties across the planet’s orbital 3 and the turbulence parameter is α =4 10− .Theplanetislo- radius a,thenettorqueexertedontheplanetbydiskmate- × cated at ( 1, 0),thestarat(0, 0),andthediskisrotatinginthe − rial is of order p (H/a) g (Goldreich and Tremaine, counterclockwise direction. The greyscale bar is in cgs units. The 1980; Ward, 1997|T |),∼ hence |T | plot shows the density gap along the planet’s orbit and the wave pattern generated by disk-planet interactions. The two black lines 2 2 4 2 a M represent trajectories of gas in the co-moving frame of the planet. p IΣa Ω . (44) T ≈ C H M! These gas parcels move along the inner and outer gap edges, be- ! " # $ come gravitationally bound to the planet, and are eventuallyac- Quantity depends on the radial gradients of surface creted. I density, temperature,C and pressure of the disk across the planet’s orbit. In typical circumstances, this quantity is neg- Alowerdisktemperature,i.e.,asmallerdiskthickness, ative and of order unity (Tanaka et al., 2002). can increase the planet’s tidal barrier and reduce gas ac- The orbital angular momentum of a planet on a circu- 2 cretion. However, it is unclear whether accretion can be lar orbit is Lp = Ma Ω.Iftheplanetisacteduponbya stopped around M 1 M for reasonable values of H/a torque ,conservationofangularmomentumimposesthat J Tp (Lubow and D’Angelo∼ , 2006). Lower disk viscosity also dL /dt =0,andthus p − Tp helps reduce disk-limited gas accretion rates, as indicated 1 dM 1 da in Figure 4.Assumingadiskaroundasolar-massstarwith + = Tp . (45) 4 2 α 10− ,forthesameexamplemadeabove,thetidalbar- M dt 2a dt Ma Ω ∼ rier would limit the planet’s final mass to 2 MJ . If the first term on the lefthand side of equation (45)isneg- The considerations above do not take∼ into account the ligible compared to the second, then da/dt =2 p/(MaΩ). fact the a protoplanetary disk evaporates due to irradia- This phenomenon is referred to as orbital migrationT .Since tion from the central star and, possibly, from other exter- the torque experienced by the planet is generally negative, nal sources (Hollenbach et al., 2000). Both observations the orbit shrinks. When the torque acting on a (non-gap and theory suggest that photoevaporation timescales are on opening) planet is of the type in equation (44), migration the order of a few million years. Therefore, disk consump- 1 is said of type I.Themigrationtimescale,a da/dt − ,in tion by photoevaporation may become a relevant process in these circumstances is | | the final stages of a giant planet’s growth. It is likely that acombinationofthesethreefactors,i.e.,areducingdisk M M H 2 Ωτ ! ! . (46) thickness H/a,adecayingturbulenceα,andgasdepletion I ∼ a2Σ M a by photoevaporation, all of which may take place as a proto- # $# $# $ planetary disk ages, plays a significant role in determining Orbital decay due to type I migration can be quite rapid. 5 final masses of gas giant planets. A 10 M (M =3 10− M!)planetarycoreinteracting ⊕ × 2 Once gas accretion stops, a giant planet evolves at a con- with a disk, whose surface density is Σ 100 g cm− and ∼ stant mass. The planet gradually cools down and contracts H/a 0.05,wouldmigrateonacharacteristictimescale ∼

D’Angelo, Durisen, & Lissauer 13 of 28 Giant Planet Formation 2.10 Examples of Giant Planet Evolution Tracks 2 CORE ACCRETION MODELS

4 of 4 10 orbital periods. For a planet orbiting at da/dt = 2 v/(MaΩ),andthemigrationtimescaleis 5AU∼ around× a solar-mass star this period would amount∼ − T 5 1 a 2 M to several times 10 years, a factor of a few shorter than the ΩτED . (48) thermally-regulated envelope phase (see Section 2.4)shown ∼ 6πα H a2Σ ! " # $ in the left panel of Figure 1.Suchrapidmigrationratesmay Inertia effects are expected to be important at small orbital be difficult to reconcile with formation via core accretion radii, when a disk is sufficiently depleted. Migration of gi- of giant planets that orbit at large distances from their stars, ant planets provides a natural explanation for the occurrence unless the time spent by a planet in the slow growth stage of Jupiter-mass planets orbiting within a few tenth of AU and the migration timescale τI are actually closer (see, e.g., from their parent stars, and whose existence would be oth- Alibert et al., 2005). erwise difficult to explain by means of in situ formation via Once a planet becomes a gas giant, a density gap forms core accretion or disk instability (as we shall see in Sec- along its orbit (because condition 40 is typically satisfied) tion 3). where the surface density is drastically depleted (see right panel of Figures 3 and 5). In the limit of a very clean gap, 2.10. Examples of Giant Planet Evolution Tracks there is a balance between viscous torques and gravitational An illustrative example of models combining planet’s torques at gap edges (interior and exterior of the planet’s or- growth and orbital migration is displayed in Figure 6.These bit) and the planet remains locked in the gap. This argument results are obtained from calculations that encapsulate in is valid as long as the disk inside the orbit of the planet is asimplifiedmannerallthebasicaspectsofacoreaccre- not significantly depleted. The planet drifts towards the star tion model and standard migration theory discussed thus carried by the viscous diffusion of the disk, if the local mass far. We consider here the case of a planetary embryo em- of the disk is comparable to the planet’s mass, or larger. bedded in an axisymmetric (around the star) disk, which This is referred to as type II migration. Viscous diffusion evolves under the action of viscous torques (equation (39)), through the disk occurs on a timescale a da/dt 1 a2/ν, − gravitational torques, and photoevaporation induced by the and therefore | | ∼ 1 a 2 central star. All three effects are accounted for in equa- Ωτ . (47) II ∼ α H tion (41)oncethelefthandsideofthatequationisreplaced with πa∂(Σ + Σ )/∂t,inwhich∂Σ /∂t is the mass loss In typical circumstances, there! " is residual gas in the pe pe flux due to photoevaporation. gap region and there may not be an exact balance between Tidal torques exerted by the planet on the disk are incor- viscous and gravitational torques at gap edges. Nonethe- porated in equation (41)throughasetoftorquedensitydis- less, results from multi-dimensional simulations indicate tributions (torque per unit disk mass as a function of radius), that migration occurs on a timescale of order given by equa- which cover the relevant range of planet masses and are ob- tion (47)inthepresenceofasufficientlydeepgap(e.g., tained from three-dimensional hydrodynamical simulations D’Angelo and Lubow, 2008). In a disk with α 10 3 − of disk-planet interactions (see D’Angelo and Lubow, 2008, and H/a 0.05,agap-openingplanetwouldundergoor-∼ for details). This procedure allows us to encompass the bital migration∼ on timescales of many tens of thousands of different regimes of orbital migration experienced by the orbital periods, or many 105 years at 5AUfrom a solar- planet as it grows and discussed in Section 2.9. mass star. ∼ We use a simplified form of the mass loss rate from According to equation (47), τ a3/2 (if H/a is ap- II the disk that is zero within the “gravitational” radius proximately constant) and thus the migration∝ timescale be- a 10 (M /M )AU,wherethethermalenergyof comes shorter as a giant planet approaches the star. How- g ! the gas≈ equals in magnitude" its gravitational energy, oth- ever, efficient exchange of orbital angular momentum be- ˙ g 5/2 tween a gas giant and a protoplanetary disk requires that erwise ∂Σpe/∂t = Σpe (ag/a) (see the chapter by the local disk mass is at least comparable with the planet’s Roberge and Kamp). The mass loss flux at radius ag is 2 ˙ g 13 41 1/2 3/2 mass, i.e., πa Σ M.Ifthisisnotthecase,planet’sin- Σpe =3.7 10− Φpe (M!/M )− in units of ! " 2 × 41 ertia acts to slow down migration. Strict type II migration M AU− per year. The constant Φpe is the rate of ioniz- " % & 41 1 of a 1 MJ planet would therefore necessitate a gas surface ing photons emitted by the star, Φpe,inunitsof10 s− . 2 density Σ at both gap edges greater than 100 g cm− at At time t =0,thediskcontains 0.022 M of gas 3 ∼2 ≈ " 5AUfrom the star and 2.8 10 gcm− at 1AU. within about 40 AU (Davis, 2005). The disk evolution ! × If the disk inside the planet’s orbit is significantly de- equation is solved numerically, adopting a turbulence pa- 3 pleted (due to the tidal barrier of a massive planet), conser- rameter α 4 10− and a disk scale-height H/a =0.05. vation of angular momentum at the gap edge exterior of the The evolution≈ begins× at 13 AU from a solar-mass star. planet’s orbit requires that dLp/dt + v =0(e.g., Syer and Aplanetembryo,ofinitialmass0.1 M ,accretessolids T ⊕ Clarke, 1995), where the first term on the lefthand side is at an oligarchic rate ( M 2/3,seeSection2.2)andbe- the rate of change of the planet’s orbital angular momentum comes isolated when M∝ 13 M .Thegasaccretionrate, ≈ ⊕ and the second term is the (positive) viscous torque exerted dM/dt,isgivenbythesmallerofM/τe (equation (15)) 41 on the disk, given by equation (39). Thus, for a nearly- and the upper fit to data in Figure 4.ThreevaluesofΦpe truncated disk at the gap edge exterior of the planet’s orbit are used: 1, 10,and50.Thickerandthinnerlinesrepresent

D’Angelo, Durisen, & Lissauer 14 of 28 Giant Planet Formation 3DISKINSTABILITYMODELS

Figure 6. Evolution tracks of giant planet formation via core accretion obtained from models that account for orbital migration bytidal interactions with the protoplanetary disk (see Section 2.10 for details). The total mass (left) and orbital radius (right) of the planet are shown as a function of time. Two sets of tracks represent casesinwhichtheplanetevolutionbeginsattimet =0(thicker lines) and at time t =1.5Myr(thinner lines). Within each set of tracks, three different line types represent different values of the applied rate of 41 1 42 1 42 1 ionizing photons emitted by the star: Φpe =10 s− (solid lines), 10 s− (long-dashed lines), and 5 10 s− (short-dashed lines). × cases in which the planet’s evolution starts at time t =0 3. DISK INSTABILITY MODELS and 1.5Myr,respectively.Theevolutiontracksillustrated Stars form through the gravitational collapse of inter- in Figure 6 highlight the importance of the disk’s initial con- stellar clouds, and Boss (1997)haschampionedtheidea ditions in which a giant planet’s core begins to evolve and (Kuiper, 1951; Cameron, 1978)thatgasgiantplanetsmight on the details of disk photoevaporation. It also supports the form from protoplanetary disks in a similar way. This is contention that early core accumulation may lead to sub- commonly referred to as the disk instability theory of gas gi- stantial amounts of radial migration, whereas late-stage for- ant planet formation, because the mechanism by which the mation scenarios may imply much less orbital decay. planets form is the onset of gas-phase gravitational instabil- 2.11. Summary of Core Accretion Models ities (hereafter referred to as GIs) leading to fragmentation of a protoplanetary disk into bound, self-gravitating clumps. The formation of a gas giant via core nucleated accre- An important distinction with core accretion is that, if the tion may take about one to a few Myr and is initiated by conditions prevail for disk instability to work, then planets asolidplanetarycoreofatleastafewEarthmasses.Core can form directly out of the gas phase within only a few formation requires that the heavy elements of the disk have to tens of disk orbit periods (τ =2π/Ω). The planets condensed out and coagulated into planetesimals. Although rot gain most of their complement of gas immediately and sub- the process of planetesimal formation is still under scrutiny, sequently sediment heavy element cores and/or sweep up small bodies of 10 to 100 km in size are observed planetesimals over a longer timescale. Disk instability is in the Kuiper belt∼ and Neptune-mass∼ planets are observed top down and initially rapid; core accretion is bottom up around the Sun and other stars. Core accretion models pro- and initially slow. The critical issue for disk instability is vide an explanation for the existence not only of the four when and where the conditions necessary for fragmentation giant planets in the solar system, but also for the majority might actually occur in protoplanetary disks. of giant planets that have been observed around other stars. We will begin by discussing how GIs operate and what Giant planets contain large amounts of Hydrogen and is required for the outcome to be fragmentation. To decide Helium, which are acquired in gaseous form from proto- whether real disks fragment, we must rely heavily on de- planetary disks. Therefore, gas dispersal timescales of disks tailed numerical simulations. set a strict upper limit on the time available to form these planets (a few to 10 Myr). The phase of slow envelope 3.1. Gravitational Instabilities: Linear Instability contraction that precedes∼ runaway gas accretion seems to Gravitational instabilities are caused by the self-gravity represent the greatest hurdle to overcome since it may last of the disk, but, unlike star formation, the onset of planet longer than disk’s gases, at least in some situations. Orbital formation via disk instability is much more strongly inﬂu- decay by tidal torques may require formation in evolved enced by rotation, and it happens in a different thermody- rather than young disks to avoid amounts of radial migration namic environment. that are too large. Additionally, since the oligarchic growth 1/2 To understand how GIs work, consider ﬁrst the basic timescale is a3/2M − ,coreformationatorbitaldis- ! equilibrium of a gas disk orbiting a central star for the case tances of many∝ tens of AU or around low-mass stars (e..g., where disk self-gravity cannot be ignored. For the vertical red dwarfs) may take too long to produce a gas giant. equilibrium in a thin disk where parameters vary smoothly

D’Angelo, Durisen, & Lissauer 15 of 28 Giant Planet Formation 3.1 Gravitational Instabilities: Linear Instability 3 DISKINSTABILITYMODELS

and slowly with orbital distance a,equation(3)becomes with the same period as τrot.Asviewedfromitsunper- turbed equilibrium position, an oscillating fluid element ap- 1 ∂p GM z + ! +2πGΣ(a, z)=0, (49) pears to execute a small elliptical “epicycle”. The epicycle ρ ∂z a2 a has the opposite or retrograde sense of rotation compared ! " # $ with the orbital motion of the disk. where Σ(a, z)= z ρdz is the surface density of gas at z " Equation (51)revealsagreatdealaboutthebehaviorof radial position a between− z and z,withz =0being the perturbations in self-gravitating disks. As long as the right- midplane. In what% follows,− we use Σ = Σ(a) to refer to hand side (RHS) of equation (51)ispositive,iωt 2πa/λ Σ(a, ).Theadditional(third)termonthelefthandside is a complex number and so the ripples are waves which± os- of equation∞ (49)characterizestheverticalcomponentofthe cillate and propagate radially. However, when the RHS is gravitational field due to the gas disk. The radial equilib- negative, there are stationary solutions that grow exponen- rium for a thin disk is then tially (the argument of the exponential is real). The disk is 1 ∂p GM then unstable and tends to break apart into dense concentric aΩ2 + ! gD =0, (50) ρ ∂a − a2 − a rings. ! " An examination of the individual terms on the RHS of D where ga is the additional radial component of the grav- equation (51)revealsthestabilizinganddestabilizingin- itational field due to disk self-gravity, which depends on fluences. The first and third terms are always positive and the detailed distribution of the disk surface density Σ(a). represent stabilization by rotation and gas pressure, respec- When H/a is small, disk self-gravity can be important in tively. As λ gets large, the first term dominates the RHS of equation (49)withoutnecessarilybeingamajorcontribu- equation (51); in other words, rotation stabilizes long wave- tion in equation (50). So, to derive simple relations in later lengths. Because the central gravitational force of the star sections, we will sometimes approximate Ω by using the appears in equation (51)throughκ,itisalsoastabilizing 3 Keplerian angular speed, GM!/a . influence. When λ is small, the third term dominates; in GIs arise through perturbation of the equilibrium quan- other words, gas pressure stabilizes small wavelengths. In & tified by equations (49)and(50). A classic linear sta- contrast to the first and third terms, the self-gravity of the bility analysis by Toomre (1981)foraxisymmetricradial disk represented by the second term on the RHS, which is ripples is very instructive. Equations (49)and(50)de- only important at intermediate wavelengths, is always neg- fine the zero-order equilibrium state of the disk, which ative and hence is a destabilizing influence. If the accel- we denote by a subscript zero, e.g., p0, ρ0,etc.Physi- eration imparted by disk gas, G Σ,islargeenoughrelative cal variables in the perturbed disk can then be represented to κ2λ and c2/λ,somemiddlerangeofwavelengthswill by these zero-order quantities plus small perturbations, i.e., be unstable. Because the destabilizing term is due to disk p = p0 + δp,with δp p0.Tosimplifytheanaly- self-gravity, we refer to such instabilities as gravitational | | # sis, the radial wavelength λ of the perturbations is taken instabilities. to be small compared with both a and the scale length of Setting ω =0in equation (51)andsolvingforλ,one radial changes in Σ,butlargecomparedwiththevertical can see by simple algebra that there is a range of unstable scale-height H.Thethinverticalstructureofthediskis wavelengths when assumed to remain in hydrostatic equilibrium according to cκ equation (49)evenwhenperturbed.So,aftertheaccelera- Q = < 1 . (52) tion term ∂v /∂t ∂(δv )/∂t is introduced into the left- πGΣ a $ a hand side of equation (50)inordertodescribethedynamics Notice that the numerator of the Toomre stability param- of the perturbations, the radial equation is integrated over z. eter Q is, in effect, a geometric mean of the stabilizing in- Equation (50)allowsthezero-ordertermstobecancelled, fluences of pressure and rotation, while the denominator and only terms to first order in the perturbed quantities are represents the destabilizing influence of disk self-gravity. retained in the linear analysis (for more details, see Binney Thus, fixing the mass of the star, a protoplanetary gas disk and Tremaine, 1987). becomes unstable to GIs if it is sufficiently cold (low c)or By looking for wave-like perturbations such that, e.g., massive (high Σ). The critical wavelength λcr that first be- δp exp (iωt 2πa/λ),onecanderivethedispersionre- comes unstable when Q dips below unity is lation∝ ± 2 2 2 2 2 2 4π GΣ 4π c 2π GΣ 2πH ω = κ + , (51) λ = , (53) − λ λ2 cr κ2 ≈ Q where λ is positive-definite and c2 is the sound speed squared. The quantity κ in equation (51)istheepicyclic fre- as can be derived by setting ω =0and Q =1in equa- quency at which a perturbed fluid element oscillates about tion (51). For the approximate relation on the right in equation (53), we use c HΩ,whichstillholdstowithina its equilibrium position (Binney and Tremaine, 1987). For ≈ astrictlyKepleriandisk,κ = Ω,because,whentheradial factor of order unity for a self-gravitating disk. equilibrium is dominated by the central force of the star, a As summarized in various reviews (e.g., Durisen et al., slightly perturbed fluid element follows an elliptical orbit 2007), multi-dimensional simulations of disks show that

D’Angelo, Durisen, & Lissauer 16 of 28 Giant Planet Formation 3.2 Occurrence of Gravitational Instabilities 3 DISK INSTABILITY MODELS

2 nonaxisymmetric perturbations, i.e., spiral waves, become The typical solids’ surface density Σs 10 g cm− dis- unstable for cussed in Section 2 for core accretion∼ models scales to cκ Q = gz . protoplanetary disks. Disks may well have masses this large As expected, GIs appear when the self-gravity of the gas or larger during the accretion phase, when gas is falling onto disk becomes sufficiently important, i.e., when the second 5 the young star/disk system at rates of 10− M per year or and third term on the lefthand side of equation (49)become higher (e.g., Vorobyov and Basu, 2006). Alternatively," a comparable. disk might have a localized enhancement of surface den- The discussion associated with equation (51)makesit sity due to accumulation of mass in or at the edges of a clear how self-gravity could break a disk into ringlets, but dead zone (Gammie, 1996), where radial transport of mass how do spiral waves grow? The likely mechanism is ex- becomes inefficient. This could lead to episodic eruptive plained in Toomre (1981)andBinney and Tremaine (1987). phenomena related to FU Orionis outbursts (e.g., Armitage Consider part of a leading spiral wave spanning its own et al., 2001; Boley and Durisen, 2008; Zhu et al., 2009). corotation radius (CR hereafter), the radial location in the Although we have not yet discussed the conditions for disk where the angular pattern speed of the wave equals Ω, fragmentation, it is instructive to estimate how much mass i.e., where one arm of the wave goes around the disk in ex- is involved in the GIs. Using equation (53)asanestimate actly τrot.Shearinthediskcausesthearcto“swing”from for the radial extent of the instability, we expect the mass in leading to trailing. Because κ Ω in a nearly Keplerian one arm of a k-armed unstable wave at 5.2AUto be about disk, the fluid elements of the disk≈ in the crest of the wave aλcrΣ 11 near the CR are executing epicyclic oscillations that keep Mfrag 2π MJ . (56) them in the crest. This allows self-gravity to amplify the ≈ k ∼ k density maximum at the crest of the wave. Numerical stud- Simulations show that k is typically four or five at the onset ies of protoplanetary disks in which spiral waves grow show of GIs (Mej´ıa et al., 2005; Boley and Durisen, 2008)when that the conditions are indeed right for the swing amplifi- MD 0.14 M!,andthereforethearmsofthespiralswould cation mechanism to operate (Pickett et al., 1998; Mayer indeed∼ have a gas inventory sufficient to produce a giant et al., 2004), and the spiral waves that grow always straddle planet. their CR (see. e.g., Mej´ıa et al., 2005). Even if a disk is stable (Q>Qcr)attheradiusof Jupiter’s orbit, it may be susceptible to GIs at larger radii. 3.2. Occurrence of Gravitational Instabilities 1/2 Suppose that the disk temperature T a− (i.e., H/a 1/4 ∝ 3/2 ∝ To get a feeling for when GIs are expected in protoplan- a ,a“flared”disk).Weknowthatκ Ω a− ,soif s ≈ ∝ etary disks, we can evaluate equation (54)forthetypical Σ a− , ∝ parameters used in earlier sections, namely H/a 0.05 1/2 ≈ cκ T Ω s 7/4 at a =5.2AUfor a disk orbiting a solar-mass star (corre- Q = a − . (57) sponding to a temperature T 75 K if c is the adiabatic πGΣ ∝ Σ ∝ ∼ sound speed). Let us assume that Qcr 1.7.Then, Thus, for any disk with s<7/4, Q becomes less than Q ≈ cr 3 2 if the disk extends to a sufficiently large radius. For a disk 3 10 gcm− a 2 s Q Q × . (55) with s<2, MD(a)= 0 2πa#Σda# a − ,andsothe ≈ cr Σ ∼ ! " disk mass MD(a) interior of a diverges as a increases. As a result, if a disk with s<7#/4 is extended enough to have GIs

D’Angelo, Durisen, & Lissauer 17 of 28 Giant Planet Formation 3.3 Nonlinear Growth of Gravitational Instabilities and Fragmentation 3 DISK INSTABILITY MODELS in its outer regions, it is likely to have a mass that is a signif- umn through the disk. The factor of two in equation (58) icant fraction of the star’s mass. Again, this is most likely to accounts for radiation from both the top and bottom of the occur during the early accretion phase of the star/disk sys- disk. This equation makes the approximation that all en- tem. If we put in parameters appropriate for this situation ergy loss from the disk is vertical, which need not be true in at a few hundred AU around a solar-type star, Mfrag can complicated dynamical situations. become tens of Jupiter’s masses or more (Stamatellos and When τcool is comparable to or less than τrot,thebalance Whitworth, 2009). of heating and cooling discussed above may not be possi- The above arguments are not very sensitive to the tem- ble. The gas could cool profoundly between the spiral arms 0 1 perature distribution T (a).IfweuseT (a) a or a− before it is reheated by the shock in the next arm. Another instead, then we get H/a a1/2 or H/a ∝equal to∝ a con- way tosay the same thingis that, becausethe growthtime of ∝ s 3/2 s 2 stant, respectively, and Q a − or Q a − ,respec- GIs is τ ,itistheshortesttimescaleonwhichGIscanre- ∝ ∝ rot tively. Parameter Q will still become less than Qcr at large plenish thermal energy in the gas. There will then be large enough radii for surface densities that do not fall off too compressions behind the shocks where the self-gravity of rapidly. During the early accretion phase, one expects that, the gas could win out entirely. The arms may then break in spatially extended disks, the gas temperature levels off up into dense clumps, which become bound in the sense to a constant value (T a0)atlargeradiiduetoenvelope that the magnitude of the clump self-gravitational energy irradiation. At small radii,∝ some disks may not be irradiated exceeds the clump’s own internal and rotational energies. by their central stars due to shadowing. Such disks would The precise value of τcool for which dense clumps form 1/2 have steeper temperature fall-offs than a− and may not must be determined numerically. The first systematic treat- be flared. For more detailed analytic arguments about grav- ment by Gammie (2001)forasimpleequationofstateina itational instabilities in disks during the accretion phase, see local patch of a razor thin disk showed that fragmentation Clarke (2009)andRafikov (2009). occurs for Ωτcool ! 3,orequivalentlyτcool/τrot ! 1/2. Global three-dimensional simulations by several groups us- 3.3. Nonlinear Growth of Gravitational Instabilities ing a variety of simulation techniques have verified that the and Fragmentation fragmentation criterion is

There are two possible nonlinear outcomes for GIs: τcool steady-state balance of heating and cooling or fragmenta-

D’Angelo, Durisen, & Lissauer 18 of 28 Giant Planet Formation 3.4 Realistic Radiative Cooling 3DISKINSTABILITYMODELS

4 prad = aradT /3,thestandardgrey atmosphere solution to equation (61)is 3 T 4 = T 4 (τ + q) , (62) 4 eff where q depends on boundary conditions and approximations but is roughly 2/3.Equation(62)strictlyappliesonly to a semi-infinite atmosphere with a constant upward flux. Adiskisfiniteandhasafluxwhichvarieswithτ (Hubeny, 1990). Nevertheless, within a factor of order unity, equation (62)isausefulestimaterelatingthemidplanetemper- ature Tmid = T (z =0),i.e.,T (τ) at τ = τ(0),tothe expected effective temperature Teff in radiative equilibrium given a midplane optical depth τ(0). When τ(0) q,thediskisopticallythick.Inequa- & tion (58), one can approximate the integral of Eint to be Figure 7. Midplane density contours after 374 years for an 3Σ Tmid/(2µ),where is the gas constant and µ is the isothermal simulation of a protoplanetary gas disk spanning 4 to mean-molecularR weight (seeR Section 2.5). Then, 20 AU around a solar-mass star with MD =0.09 M and an ! initial Qmin =1.3.Thedenseclumpneartwelveo’clockisgrav- 2 9 κ¯ Σ (3 4s)/2 itationally bound and has about 5 M .AdaptedfromBoss (2000). τ R a − , (63) J cool ∼ 32 σ µ T 3 ∝ " rad #" #" mid # where, for simplicity, the final dependence on a is based range between tenths to multiples of MJ for disks with 1/2 on assuming that κ¯ is constant and Tmid a− .(If MD 0.1 M! that are a few tens of AU in radius (see, 0 1 2s ∝ 3 2s ∼ T a or a− then τ a− or τ a − , e.g., Mayer et al., 2004). A discussion of fragmentation mid ∝ ∝ cool ∝ cool ∝ spectra for disks with more extended radii can be found in respectively). At 5.2AU for the usual parameters, equa- Stamatellos and Whitworth (2009). tions (60)and(63)give

3.4. Realistic Radiative Cooling 3 κ¯ Σ τ(0) 10 2 1 3 2 (64) ∼ 1cm g− 3 10 gcm− The pertinent question now is whether τcool for realistic " #" × # dust opacities is actually short enough to cause fragmenta- and tion. The vertical optical depth down into a disk is defined 2 by 5 κ¯ Σ τcool 10 yr 2 1 3 2 . (65) ∞ ∼ 1cm g− 3 10 gcm− τ(z)= κρ¯ dz" , (60) " #" × # z ! Note that, for H/a =0.05 at 5.2AU, Tmid 75 K if we where κ¯,withunitsofcross-sectionperunitmass,isthe ≈ use an adiabatic sound speed with Γ1 =5/3 and µ 2.4. mass absorption coefficient for radiation (opacity) averaged ≈ In an unstable disk at 5.2AU,whereτrot 12 years, equa- in an appropriate way over frequency. If κ¯ does not vary 4 ≈ tion (65)givesτcool/τrot 10 ,whichisordersofmagni- much with z,thenthemidplaneopticaldepthisτ(0) ≈ ≈ tude greater than ffrag for any reasonable value of Γ1. κ¯Σ/2.Thisisessentiallythenumberofphotonmeanfree We have just seen that, for our standard parameters, paths between the midplane and z = . ∞ equations (54), (55), (59), and (63)togetherimplythat,at If the radiation emitted by the gas and dust is thermal and 5.2AU,adiskwhichisdenseenoughtobegravitationally if a frequency-independent (grey) opacity characterized by unstable will cool too slowly to fragment. Rafikov (2005) κ¯ is a good approximation, then momentum conservation first pointed out this conundrum. Although it depends on for radiation gives (e.g., Gray, 1992) detailed assumptions about how disk properties vary with a, it can reasonably be argued that this problem extends out to dprad arad = Frad (61) about 40 AU or so, exactly the region where one would like dτ 4σrad " # to form the giant planets in our own solar system. More- for plane-parallel geometry, where prad is the radiation over, if we relax our choice of H/a =0.05,thenlowering pressure, Frad is the net upward flux of radiant energy, τcool at fixed κ¯ requires a lower Σ and/or a higher Tmid, 15 3 4 and arad 7.57 10− erg cm− K− is the radiation- both of which would tend to make the disk more stable. density constant$ (notice× that the ratio in parentheses in One can instead imagine lowering κ¯ itself because dust equation (61)isequaltotheinverseofthespeedoflight). grows and settles to the midplane, but there is a limit to the For no radiation shining down on the disk, for a con- efficacy of this ploy. The most efficient cooling by thermal 4 stant upward flux (Frad = σradTeff ), and for a radiation emission occurs for τ(0) 1.Forlowerτ(0),thedisk field that is well described by black-body properties, e.g., is a poor absorber and, hence,≈ by Kirchhoff’s law, a poor

D’Angelo, Durisen, & Lissauer 19 of 28 Giant Planet Formation 3.5 Computational Results for Disk Instability 3 DISK INSTABILITY MODELS

mentation should not occur inside 40 AU or so. Evi- dence supporting this conclusion comes from simulations using both Smoothed Particle Hydrodynamics (SPH) (Nel- son et al., 2000; Stamatellos and Whitworth, 2008; For- gan et al., 2009)andgrid-basedhydrodynamics(Cai et al., 2006; Boley et al., 2006, 2007; Boley and Durisen, 2008). These papers employ a variety of radiative algorithms. Fig- ure 8 illustrates a typical midplane density structure for a nonfragmenting simulation. These studies treat 0.5–1 M ! stars with a disk mass MD 0.1 M! and with outer disk radii of ten to tens of AU.Thedisksdonotcoolrapidly∼ enough to fragment. In the cases where τcool is explicitly calculated, nonfragmentation is consistent with the appli- cation of equation (59)fortherelevantequationofstate. Some efforts have been made by those who do not see fragmentation to evolve similar disks and to test their radiative schemes against analytic expectations, e.g., compare Boley Figure 8. Midplane density greyscale of a nonfragmenting and et al. (2007)andStamatellos and Whitworth (2008). radiatively cooled disk with M! =0.5 M , MD =0.07 M , ! ! Evidence for fragmentation comes from two groups, one and an initial Qmin =1.5 after about 2000 years. The box shown is 80 AU on a side. Adapted from Boley et al. (2006). using grid-based simulations (Boss, 2001, 2002a,b, 2007, 2008)andtheotherusingSPH(Mayer et al., 2007). The disk parameters are similar to those of the nonfragmenting thermal emitter. All else being equal, equation (64)shows simulations, but these researchers find that their disks frag- that a reduction of τ(0) to unity requires a reduction of κ¯ ment into dense, bound clumps, with masses of typically a 3 by a factor of 10 .Withthisreduction,τcool/τrot becomes few MJ (see equation (56)). These groups do not agree in about 10,whichisstilllargerthanthelargestexpectedvalue detail on the conditions necessary for fragmentation. Boss of ffrag. finds that fragmentation is robust, i.e., fairly independent These analytic arguments suggest that gas giant planet of many physical and numerical parameters, while Mayer formation by disk instability probably does not work in- et al. find that fragmentation requires somewhat higher disk side several tens of AU.Ontheotherhand,equations(57) masses and high mean molecular weight and that it is sen- and (63)furthersuggestthattheinstabilitycriterion(equa- sitive to some choices of numerical parameters. Cai et al. tion (54)) and the fragmentation criterion (equation (59)) (2010), using an improved radiative cooling scheme simi- may be satisfied simultaneously at large enough radii, lar to that of Boley et al. (2006)butwithhighernumeri- roughly 100 AU and beyond, when 0

D’Angelo, Durisen, & Lissauer 20 of 28 Giant Planet Formation 3.5 Computational Results for Disk Instability 3 DISK INSTABILITY MODELS times by orders of magnitude. Boley and Durisen (2006) with micron-sized particles but including aggregation and explain the upwellings associated with spiral shocks, which growth, give similar times. The sedimentation timescale, are also seen in nonfragmenting simulations, as hydraulic τsed,isshortcomparedwiththeKelvin-Helmholtzcontrac- jumps, a dynamic phenomenon very different from thermal tion timescale, τKH,oftheprotoplanet,whichisafewtimes convection. 105 years (Helled et al., 2006). Turbulence slows sedimen- The disagreement about cooling times leads to disagree- tation, but τsed for centimeter-sized particles is still shorter ment about how the disk metallicity affects fragmentation. than τKH.Therefore,gasgiantsformedbydiskinstability Boss (2002b)studiesvariationsinmetallicitybyvaryingκ¯ probably are able to produce heavy element cores out of up and down by factors of ten relative to solar composition their initial complements of heavy elements. A 1 MJ mass and finds that fragmentation is not sensitive to κ¯.Onthe of solar composition could form a 6 M core. other hand, Cai et al. (2006), using simulation techniques Jupiter and Saturn are strongly∼ enhanced⊕ in heavy ele- similar to those in Boley et al. (2006), find that GIs are sen- ments compared with solar composition, as are some tran- sitive to metallicity in simulations where κ¯ is varied only siting exoplanets (e.g., HD 149026 b). There are two ways from 1/4 to 2 times the solar opacity. No simulation in that gas giants formed by disk instability could end up Cai et al. produces fragmentation, but the GIs in the lower with complements of heavy elements which are fractionally metallicity disks are stronger, which is expected if τcool is larger than in their natal gas disk: accreting planetesimals mostly determined by radiative processes. from the surrounding disk after formation or forming in a Finally, Boss (2002a), Cai et al. (2008), and Stamatellos gas clump already enhanced in solids above the original gas and Whitworth (2008)doagreewithanalyticargumentsby disk composition. Matzner and Levin (2005)thatsufficientlystrongexternal Detailed computations of planetesimal accretion (Helled irradiation of a disk can weaken and suppress GIs. Strongly et al., 2006; Helled and Schubert, 2009)fora1 MJ proto- irradiated disks should not fragment. An easy way to es- planet formed by disk instability show that total masses of 4 timate the necessary flux of irradiation Firr = σradTirr to accreted planetesimals can vary anywhere from 1 to over stabilize a disk is to compute the irradiation temperature, 100 M ,dependingupontheinitiallocationoftheplanet ⊕ Tirr,suchthattheresultingsoundspeedc(Tirr) satisfies the and upon the assumed mass and radial density distribution inequality in equation (54). At 5.2AU, T 75 K suffices of the parental disk. Large accretion rates of heavy elements irr ∼ to suppress GIs altogether for MD 0.1 M .Nomatter are made possible by the appreciable starting radii of gas what else one might believe about disk≈ instability! in the re- giants in the disk instability model coupled with their mod- gion within tens of AU,itseemsprettyclearthatirradiation erately long contraction timescale. Such calculations are of of this order or higher prevents planets from forming by course sensitive to assumptions, and a great deal more work GIs. is needed, but it appears that heavy element enrichment of planets may not be a strong discriminator between disk in- 3.5.2. Enrichment of Heavy Elements stability and core accretion models. In the standard disk instability model, heavy element The other intriguing possibility is that, at the time of cores form after the bulk of the gas in the planet becomes fragmentation, spiral arms may already be enhanced in agravitationallyboundequilibriumobject.Althoughsuch solids relative to the ambient disk due to gas drag effects. protoplanets are likely to be rotating (see, e.g., Mayer et al., How might this occur? We have seen that solid particles 2004), let us approximate them here as hydrostatic equi- respond to pressure gradient forces by drifting relative to librium spheres, with the gas pressure gradient balancing the gas in a direction opposite to the gradient. For exam- gravity. As discussed in Section 2.1,solidparticlesre- ple, particles drift or sediment toward the pressure maxi- spond to gas pressure forces only indirectly through gas mum at the disk midplane (Section 2.1)oratthecenterof drag. Thus, in the absence of turbulence, small solid par- aprotoplanet(asdiscussedabove).Thesamethingapplies ticles slowly sink to the center of a gas giant protoplanet at to radial drift of solid particles in the protoplanetary disk: adriftspeedvd g τf (see Section 2.1), where g is the they drift radially in a direction opposite to the direction of magnitude of the≈ gravitational acceleration inside the pro- the radial pressure gradient (Weidenschilling, 1977), which toplanet and τf is the friction time. For a newly formed affect the gas orbital velocity, and concentrate at a radial 1 M protoplanet with a radius R 0.5AU,approxi- pressure maximum (Haghighipour and Boss, 2003). J ≈ mately half λcr in equation (53)forourstandardparame- The physics in this case is complicated by angular mo- 2 3 2 ters, g GMJ /R 2.3 10− cm s− ,andtheinternal mentum exchange. Equation (50)determinestheequilib- temperature≈ would be#T ×100 K.Usingequation(2), one rium rotation rate of the gas, Ω,whichincludestheeffectof gets a sedimentation time∼ the gas pressure term. Centrifugal balance against gravity in the absence of gas pressure requires a different angular 3 2 3 D R 3 1gcm− 1cm speed for solids, Ωs,suchthatΩs = GM!/a ga /a. τsed 6 10 yr , (66) − ≈ v ∼ × ρ R From equation (50)wehavethatif∂p/∂a<0,then d ! sp "! sp " Ω < Ωs,whileif∂p/∂a>0,thenΩ > Ωs.Asa in rough agreement with an estimate of Boss (1997). consequence, gas drag exerts a torque (per unit mass) on Detailed calculations by Helled et al. (2008), starting solid particles equal to (Ω Ω ) a2/τ .Then,byconser- − s f

D’Angelo, Durisen, & Lissauer 21 of 28 Giant Planet Formation 3.7 Summary of Disk Instability Models 3DISKINSTABILITYMODELS vation of angular momentum, the radial drift velocity is da/dt =(a/π)(Ω Ωs)(τrot/τf ) (see Section 2.9). In disk regions with a negative− radial pressure gradient, an orbiting particle feels a head wind and drifts inward due to loss of orbital angular momentum. In disk regions with a positive radial pressure gradient, a solid particle feels a tail wind and drifts outward due to gain of orbital angular momentum. If there is a pressure maximum at some value of a, solids drift toward the pressure maximum from both sides and become concentrated relative to the gas. The magnitude of the radial drift speed depends on the ratio τf /τrot.Forsmallparticles,τf /τrot 1 (see equation (4)), and particles are essentially entrained" with the gas. The head and tail winds are very small (Ωs Ω), and so the radial drift is slow. Drift speeds increase# with particle size in the small particle regime. Very large particles, for which τ /τ 1,orbitatangularspeedΩ = Ω. f rot $ s % They can experience considerable head and tail winds, but Figure 9. Logarithmic surface density (in cgs units) for a radia- radial drift speeds are again small because of the long fric- tively cooled simulation of a fragmenting protoplanetary gas disk 4 tion times. Drift speeds decrease with particle size in this around a solar-mass star that is accreting at a rate of 10− M " regime. Maximum radial drift speeds ( a Ω Ω )occur per year. The dense clumps have masses ranging from 4 MJ to ∼ | − s| for τf /τrot 1,correspondingtoparticlesofaboutame- 14 MJ .AdaptedfromBoley (2009) ter in radius∼ (Weidenschilling, 1977). Such particles can be swept into radial pressure maxima on timescales of only a Unless planets formed by core accretion can migrate or few hundred years (Haghighipour and Boss, 2003). be scattered to large distances, it seems reasonable to con- It is not obvious apriorithat this phenomenon general- jecture that planets found in large orbits are formed in situ izes to the dynamic environment of a GI-active disk with by disk instability. But, if nonfragmentation of GI-active spiral waves, but, in fact, solid particles of this optimal size disks in the inner tens of AU prevails as the consensus view, do drift into the pressure maxima of spiral arms on simi- we still have to rely on core accretion to explain the bulk larly short timescales (Rice et al., 2003). This concentra- of gas giant exoplanets discovered so far, whose orbits are tion of solids relative to gas in spiral arms can trigger GIs in within a few AU of their stars. This leads to the notion the solids themselves (Rice et al., 2006)andcouldevenin- (Boley, 2009)thatgasgiantplanetformationmayactually crease the tendency of the gaseous arms to fragment (Mayer be bimodal, with different mechanisms dominating in dif- et al., 2007). In the latter case, a protoplanet formed by disk ferent regions of protoplanetary disks. Even in this case, it instability could initially be enhanced in heavy elements rel- is possible that gas-phase GIs play a role in core accretion ative to the disk gas. by accelerating the formation of solid planetesimals through 3.6. Bimodal and Hybrid Planet Formation Models migration of solids into structures formed in the disk by GIs (e.g., Haghighipour and Boss, 2003; Rice et al., 2004, 2006; As argued analytically in Section 3.4 and illustrated in Durisen et al., 2005). Figure 9,evencodesthatdonotproducefragmentation in disks within tens of AU around solar-type stars con- 3.7. Summary of Disk Instability Models ﬁrm that fragmentation can and does occur in radially ex- Once the necessary conditions exist, namely Q 1.5– tended and massive disks, beyond about 100 AU (Boley, ! 1.7 and τ /τ

D’Angelo, Durisen, & Lissauer 22 of 28 Giant Planet Formation 4.1 Solar and Extrasolar Planets 4 HIGHLIGHTS AND OUTSTANDING QUESTIONS unlikely to fragment within a few tens of AU,becausethe metallicity and frequency of giant planet detections (e.g., conditions of low Q and a low τcool/τrot cannot be simulta- Udry and Santos, 2007): 25% of stars whose metallicity neously satisfied. The inclusion of stellar irradiation, which is twice that of the Sun host∼ a gas giant (within a few AU). tends to stabilize these regions by maintaining a high Q, This percentage reduces to only 5% for solar metallicity makes the problem worse. Disk instability is most likely to stars. Disk instability models do∼ not seem to predict this occur in the outer regions of disks relatively early in disk trend. In fact, there is some evidence that GIs are stronger evolution, when disks are massive and still accreting from in lower metallicity disks (Cai et al., 2006). Core accretion their protostellar cloud at a high rate. The ultimate fate and models do predict this trend (Kornet et al., 2005; Mordasini survivability of planets formed at such an early phase of et al., 2009), provided that the density of solids in a disk is disk evolution is unclear. The principle of parsimonious ex- proportional to the metallicity of the central star. planations makes appeal to more than one formation mech- Planet searches around M dwarf stars (M! " 0.5 M ) anism for gas giants unappealing. seem to suggest that the occurrence of giant planets is less" likely than it is around more massive stars (Johnson et al., 4. HIGHLIGHTS AND OUTSTANDING QUESTIONS 2007; Eggenberger and Udry, 2010). Although these sur- veys are still incomplete, there is mounting observational 4.1. Solar and Extrasolar Planets evidence that planets of mass M " 0.1 MJ orbiting M Any theoretical model of giant planet formation has to dwarfs may be common (Forveille et al., 2009). If con- provide a quantitative explanation for the existence of the firmed by more complete statistics, a correlation between four outer planets of the solar system. Although Jupiter’s stellar masses and the masses of hosted planets may pro- core mass remains uncertain, there is theoretical evidence vide valuable information on the process of planet forma- that Saturn, Uranus, and Neptune have cores of 10–20 M ⊕ tion. According to core accretion models, gas giants orbit- (Podolak et al., 1995; Saumon and Guillot, 2004). All four ing low-mass stars should be rare (Kennedy and Kenyon, planets have massive Hydrogen/Helium envelopes, which 2008), whereas Neptune-mass planets should be common represent most of the mass of Jupiter and Saturn and more (Laughlin et al., 2004). According to disk instability mod- than 10%ofthemassofUranusandNeptune(e.g.,Guillot, els, if conditions are appropriate for disk fragmentation, the 2005). occurrence rate of giant planets should not depend strongly Core accretion models offer a natural explanation for on the stellar mass (Boss, 2006). both the formation of heavy element cores and the accre- Interpretations of observational data based on either of tion of massive gaseous envelopes, such as those of Jupiter these mechanisms remain difficult and predictions are still and Saturn. In this scenario, Uranus and Neptune reached rather qualitative. There is a number of unresolved issues their current masses in an environment that may have been that prevent us from making more quantitative predictions. deprived of gases because their cores took too long to form, Such issues are generally related to poor constraints on the and never underwent a runaway gas accretion phase. It initial conditions used in the formation models. Some of seems unlikely, instead, that any of these planets formed these open questions are outlined below. via GIs in the early solar nebula, principally because theo- What factors determined the final masses of Jupiter and retical arguments show that fragmentation may only occur other gas giants? The mass of Saturn may have been influ- beyond many tens of AU from a solar-type star. enced by the presence of Jupiter and similar conditions may Any formation theory also needs to explain the wide have operated in some extrasolar planetary systems with range of physical and orbital properties of giant planets multiple gas giants. However, a more general answer to around other stars (see the chapter by Cumming). About this question will require a much deeper understanding of 6% of Sun-like stars have giant planets orbiting within the physical conditions that existed in the solar nebula and 4AU.Jupiter-massplanetsinorbitwithinseveralAU of other protoplanetary disks. Sun-like stars are more common than planets of several Are there peaks in the mass distribution function of mas- Jupiter’s masses, and substellar companions more massive sive planets? During the final phases of growth, giant plan- than 10 M are rare. Neptune-mass planets may be ∼ J ets are in a symbiotic relationship with their parent disks. rather common compared to planets a few times the mass of Therefore, we need better constraints on masses, tempera- Neptune. These observational results are broadly predicted tures, and lifetimes of protoplanetary disks to make predic- by core accretion models, provided that gas in protoplan- tions on the relative abundances of giant planet masses. etary disks survives for at least a few million years. Disk What is the occurrence rate of gas giants and Neptune- instability models, instead, would predict an abundance of mass planets as a function of the stellar mass? What is the massive objects (! 10 MJ )relativetoJupiter-massplanets, percentage of red dwarfs that host massive planets? Cur- since formation begins at very early times in massive disks. rently, a handful of cases are known, as the two Jupiter- GIs, though, provide an appealing prospect for the forma- mass planets orbiting Gliese 876 (http://exoplanet. tion of 10 M objects observed at distances 100 AU ∼ J ! eu/), but conclusions cannot be drawn from such small- from their stars, where core formation timescales are prob- number statistics. ably too long. Is there a numerous population of gas giants orbiting at Exoplanet data show a strong correlation between stellar large separations from their host stars, as GIs models seem

D’Angelo, Durisen, & Lissauer 23 of 28 Giant Planet Formation 5FUTUREPROSPECTS to predict? If so, how do their orbital properties and enve- 5. FUTURE PROSPECTS lope composition differ from those of other known gas gi- To achieve a more comprehensive understanding of the ants revolving on smaller orbits? What is the likelihood that formation of planetary systems, at least some of the ques- they formed closer to the star (perhaps via core accretion) tions highlighted in Section 4 need be addressed. We en- and were later scattered or migrated to larger distances? vision that answers to those questions can indeed be found 4.2. Impact of Orbital Migration in the near future, as further progress is made along three main avenues: data collection from the solar system; obser- Planetary cores of 10 M that undergo type I migra- vations of star-forming regions, protoplanetary disks, and tion may be subject to∼ rapid orbital⊕ decay. One of the main exoplanets; theoretical studies of star and planet formation. advantages of disk instability models is that they would by- Interplanetary spacecrafts (e.g., Voyager, Galileo,and pass this phase of planetary growth altogether and avoid this Cassini)haveprovidedawealthofinformationaboutthe potential threat. Planets formed via GIs, instead, may be origins and evolution of the outer solar system. The Juno subject to prolonged periods of type II migration. But, how mission may finally reveal the mass of Jupiter’s core by di- relevant is planetary migration during planet formation? rect measurements of the planet’s gravity field. Although it Although a complete answer is not presently known, seems implausible that such information alone will provide there is evidence that substantial migration may occur dur- definitive evidence for one or the other formation scenario, ing the formation process. A significant fraction of multi- it is essential for refining interior models of the gas giant planet extrasolar systems are close to or in a mean-motion nearest to us. resonance (Udry and Santos, 2007). Since it is unlikely To better constrain current estimates of sizes, masses, that such orbital configurations can be produced by in situ dust contents, and lifetimes of protoplanetary disks, it is im- formation models, these systems may lend support to con- portant to know the detailed physical conditions that exist at vergent migration hypotheses. Recent studies aimed at ex- various stages of a disk’s evolution, starting from the latter plaining the structure of the outer solar system (e.g., Mor- phases of stellar formation. The observation of star-forming bidelli et al., 2009)suggestthatJupiterandSaturnmayhave regions and disks surrounding young stars plays a pivotal undergone convergent migration, as a result of which Saturn role in establishing more comprehensive and accurate mod- became temporarily captured into a mean motion resonance els of disk evolution. These models can be then applied with Jupiter. in the context of increasingly sophisticated calculations of The presence of Neptune-mass planets and gas giants or- planet formation that include disk dispersal. biting stars within a few tenths of AU may also constitute Space observatories, like Hubble and Spitzer space evidence of orbital migration, although alternative interpre- telescopes, have provided valuable data that has signifi- tations are possible. Planet-planet scattering and Kozai cy- cantly improved our knowledge of star-forming regions and cles (see the chapter by Lubow and Ida)can,inprinciple, planet-forming environments. Over the next decade, a lot drive gas giants very close to their host stars. However, more is expected to be learned about protoplanetary disks while these last two mechanisms could produce a large mis- with both existing and new-generation instruments, such as alignment between the stellar spin axis and the planet’s orbit the Atacama Large Millimeter Array (Tarenghi, 2008), the axis, disk-induced orbital migration would preserve spin- Stratospheric Observatory for Infrared Astronomy airborne orbit alignment. Some of the transiting planets allow for observatory (Gehrz et al., 2009), and the James Webb Space the measurement of the sky-projected angle between stellar Telescope (http://www.jwst.nasa.gov/). These spins and orbit axes (see the chapter by Winn). Currently, facilities, among others, will improve in completeness the about two-thirds of these measurements are consistent with mass distribution function of Neptune-mass and larger plan- aclosespin-orbitalignment(Simpson et al., 2010), suggest- ets. They will likely improve our knowledge on the rela- ing that migration by tidal interaction with the parent disk tionships between stellar metallicity and dust content of drove these planets close to their stars. young disks. They should also allow direct imaging of The standard theory of orbital migration by disk-planet high planetary-mass objects and low-mass brown dwarfs, interactions seems able to reproduce the main features of orbiting at several tens of AU from stars, and provide pre- the orbital period distribution of exoplanets (e.g., Armitage, cious data on the abundances of heavy elements and on the 2007), which suggest that considerable migration may have chemical compositions of their envelopes. occurred. Yet, the question of whether or not substantial Ground-based interferometers (e.g., Very Large Tele- migration is the norm can only be assessed once observa- scope and Keck telescope) and astrometry space observations of long-period planets become available. A more com- tories, such as Gaia (http://sci.esa.int/GAIA) plete distribution function of orbital periods can also pro- and the Space Interferometry Mission (http://sim. vide quantitative indications on whether current estimates jpl.nasa.gov/), will extend the orbital period dis- of the migration rates of Neptune-mass planets are too rapid tribution function of detected exoplanets towards longer- and, if so, to what extent they are. period orbits. Correlations between planet detection frequency and stellar metallicity and between stellar mass and planet mass are also expected to become more accu-

D’Angelo, Durisen, & Lissauer 24 of 28 Giant Planet Formation REFERENCES

rate and statistically significant. If a significant fraction Bodenheimer, P., and Pollack, J. B. (1986) Calculations of the ac- of M dwarf stars possess gas giants, observational data cretion and evolution of giant planets: the effects of solid cores. should allow us to compare their orbital and physical prop- Icarus, 67,391–408. 8, 9 erties with those of planets orbiting more massive stars. If Boley, A. C. (2009) The two modes of gas giant planet formation. alargepopulationofhigh-massplanetsisfoundatlarge Astrophys. J., 695,L53–L57.22 separations, their atmospheric composition can be com- Boley, A. C., and Durisen, R. H. (2006) Hydraulic/Shock jumps in protoplanetary disks. Astrophys. J., 641,534–546.21 pared to those of transiting planets, whose number should Boley, A. C., and Durisen, R. H. (2008) Gravitational instabilities, also grow significantly thanks to space missions such as chondrule formation, and the FU Orionis phenomenon. Astro- COROT (http://smsc.cnes.fr/COROT/), Kepler phys. J., 685,1193–1209. 17, 20 (http://kepler.nasa.gov/), and TESS (http:// Boley, A. C., Durisen, R. H., Nordlund, A.,˚ and Lord, J. (2007) space.mit.edu/TESS). Within the next five to ten Three-dimensional radiative hydrodynamics for disk stability years, we should be able to test the bimodal distribution simulations: a proposed testing standard and new results. As- hypothesis for gas giant formation based on statistical anal- trophys. J., 665,1254–1267.20 ysis of orbital properties, mass distribution functions, and Boley, A. C., Mej´ıa, A. C., Durisen, R. H., Cai, K., Pickett, envelope compositions. M. K., and D’Alessio, P. (2006) The thermal regulation of The trend of exoplanet detections and characterization gravitational instabilities in protoplanetary disks. III.Simula- witnessed over the past decade will carry forward to the tions with radiative cooling and realistic opacities. Astrophys. J., 651,517–534. 20, 21 next decade. Theoretical studies need to tackle still un- Boss, A. P. (1997) Giant planet formation by gravitational insta- solved problems on the timescales and physical conditions bility. Science, 276,1836–1839. 15, 21 required for gas giant formation. Core accretion models Boss, A. P. (2000) Possible rapid gas giant planet formation in the must be extended to include proper feedback on the pro- solar nebula and other protoplanetary disks. Astrophys. J., 536, toplanetary disk and tidally-induced orbital migration in a L101–L104. 18, 19 self-consistent manner. Disk instability models need to be Boss, A. P. (2001) Gas giant protoplanet formation: disk instabil- refined in order to make quantitative predictions on surviv- ity models with thermodynamics and radiative transfer. Astro- ability, final orbital radii, and final masses of planetary- phys. J., 563,367–373.20 mass objects formed after disk fragmentation. As data of Boss, A. P. (2002a) Evolution of the solar nebula. V. Disk instabil- better and better quality become available, it may be chal- ities with varied thermodynamics. Astrophys. J., 576,462–472. lenging, from a theoretical standpoint, to combine all pieces 17, 20, 21 Boss, A. P. (2002b) Stellar metallicity and the formation of extra- in a coherent framework. Some of the pieces now regarded solar gas giant planets. Astrophys. J. Lett., 567,L149–L153. as important may be discarded and replaced by others. Yet, 20, 21 the prospects are in place to turn a blurred picture into one Boss, A. P. (2004) Convective cooling of protoplanetary disks and with much higher definition within the next ten years. rapid giant planet formation. Astrophys. J., 610,456–463. 20 Boss, A. P. (2006) Rapid formation of gas giant planets aroundM Acknowledgments. We thank Aaron C. Boley, Uma dwarf stars. Astrophys. J., 643,501–508.23 Gorti, and two anonymous reviewers for useful com- Boss, A. P. (2007) Testing disk instability models for giant planet ments. G.D. acknowledges support from NASA Origins formation. Astrophys. J. Lett., 661,L73–L76.20 of Solar Systems (OSS) Program grants NNX08AH82G Boss, A. P. (2008) Flux-limited diffusion approximation models and NNX07AI72G. J.J.L. was supported by NASA OSS of giant planet formation by disk instability. Astrophys. J., 677, grant 811073.02.07.02.4. R.H.D. was partially supported 607–615. 20 during manuscript preparation by NASA OSS grants Cai, K., Durisen, R. H., Boley, A. C., Pickett, M. K., and Mej´ıa, A. C. (2008) The thermal regulation of gravitational instabili- NNG05GN11G and NNX08AK36G. G.D. acknowledges ties in protoplanetary disks. IV. Simulations with envelopeir- computational resources provided by the NASA High-End radiation. Astrophys. J., 673,1138–1153. 21 Computing (HEC) Program through the NASA Advanced Cai, K., Durisen, R. H., Michael, S., Boley, A. C., Mej´ıa, A. C., Supercomputing (NAS) Division at Ames Research Center. Pickett, M. K., and D’Alessio, P. (2006) The effects of metallicity and grain Size on gravitational instabilities in protoplanetary disks. Astrophys. J., 636,L149–L152. 20, 21, 23 REFERENCES Cai, K., Pickett, M. K., Durisen, R. H., and Milne, A. M. (2010) Alibert, Y., Mordasini, C., Benz, W., and Winisdoerffer, C. (2005) Giant planet formation by disk instability: a comparison sim- Models of giant planet formation with migration and disc evo- ulation with an improved radiative scheme. Astrophys. J. Lett., lution. Astron. Astrophys., 434,343–353.14 in press. 20 Armitage, P. J. (2007) Massive planet migration: theoretical pre- Cameron, A. G. W. (1978) Physics of the primitive solar accretion dictions and comparison with observations. Astrophys. J., 665, disk. Moon and Planets, 18,5–40.15 1381–1390. 24 Chambers, J. E., and Wetherill, G. W. (2001) Planets in the aster- Armitage, P. J., Livio, M., and Pringle, J. E. (2001) Episodicac- oid belt. Meteoritics and Planetary Science, 36,381–399. 1 cretion in magnetically layered protoplanetary discs. Mon. Not. Clarke, C. J. (2009) Pseudo-viscous modelling of self-gravitating R. Astron. Soc., 324,705–711. 17 discs and the formation of low mass ratio binaries. Mon. Not. Binney, J., and Tremaine, S. (1987) Galactic dynamics. Princeton R. Astron. Soc., 396,1066–1074. 18, 20, 22 University Press, Princeton, New Jersey. 16, 17 Clarke, C. J., Harper-Clark, E., and Lodato, G. (2007) The re-

D’Angelo, Durisen, & Lissauer 25 of 28 Giant Planet Formation REFERENCES

sponse of self-gravitating protostellar discs to slow reduction Astrophys. J., 241,425–441. 13 in cooling time-scale: the fragmentation boundary revisited. Goldreich, P., and Ward, W. R. (1973) The formation of planetes- Mon. Not. R. Astron. Soc., 381,1543–1547. 18 imals. Astrophys. J., 183,1051–1062.3 Cuzzi, J. N., Hogan, R. C., Paque, J. M., and Dobrovolskis, A. R. Gray, D. F. (1992) The observation and analysis of stellar photo- (2001) Size-selective concentration of chondrules and other spheres. Cambridge University Press, Cambridge. 19 small particles in protoplanetary nebula turbulence. Astrophys. Greenzweig, Y., and Lissauer, J. J. (1992) Accretion rates ofpro- J., 546,496–508. 3 toplanets. II - Gaussian distributions of planetesimal velocities. D’Angelo, G., Henning, T., and Kley, W. (2003a) Thermohydro- Icarus, 100,440–463. 4 dynamics of circumstellar disks with high-mass planets. Astro- Haghighipour, N., and Boss, A. P. (2003) On pressure gradients phys. J., 599,548–576.9 and rapid migration of solids in a nonuniform solar nebula. As- D’Angelo, G., Kley, W., and Henning, T. (2003b) Orbital migra- trophys. J., 583,996–1003.21, 22 tion and mass accretion of protoplanets in three-dimensional Hayashi, C. (1981) Structure of the solar nebula, growth and de- global computations with nested grids. Astrophys. J., 586,540– cay of magnetic fields and effects of magnetic and turbulent 561. 12 viscosities on the nebula. Progress of Theoretical Physics Sup- D’Angelo, G., and Lubow, S. H. (2008) Evolution of migrating plement, 70,35–53.17 planets undergoing gas accretion. Astrophys. J., 685,560–583. Helled, R., Podolak, M., and Kovetz, A. (2006) Planetesimal cap- 11, 14 ture in the disk instability model. Icarus, 185,64–71.21 Davis, S. S. (2005) The surface density distribution in the solar Helled, R., Podolak, M., and Kovetz, A. (2008) Grain sedimenta- nebula. Astrophys. J. Lett., 627,L153–L155.5, 14, 17 tion in a giant gaseous protoplanet. Icarus, 195,863–870.21 Desch, S. J. (2007) Mass distribution and planet formation inthe Helled, R., and Schubert, G. (2009) Heavy-element enrichment solar nebula. Astrophys. J., 671,878–893. 17 of a Jupiter-mass protoplanet as a function of orbital location. Durisen, R. H., Boss, A. P., Mayer, L., Nelson, A. F., Quinn, T., Astrophys. J., 697,1256–1262. 21 and Rice, W. K. M., 2007. Gravitational instabilities in gaseous Hollenbach, D. J., Yorke, H. W., and Johnstone, D. (2000) Disk protoplanetary disks and implications for giant planet forma- dispersal around young stars. In: Mannings, V., Boss, A. P., tion. In: Reipurth, B., Jewitt, D., and Keil, K. (Eds.), Proto- and Russell, S. S. (Eds.), Protostars and Planets IV. University stars and Planets V. University of Arizona Press, Tucson. pp. of Arizona, Tucson. pp. 401–428. 13 607–622. 16 Horner, J., Jones, B. W., and Chambers, J. (2010) Jupiter – friend Durisen, R. H., Cai, K., Mej´ıa, A. C., and Pickett, M. K. (2005) A or foe? III: the Oort cloud comets. International Journal of hybrid scenario for gas giant planet formation in rings. Icarus, Astrobiology, 9,1–10.1 173,417–424. 22 Hubeny, I. (1990) Vertical structure of accretion disks - A simpli- Eggenberger, A., and Udry, S., 2010. Detection and characteri- fied analytical model. Astrophys. J., 351,632–641. 19 zation of extrasolar planets through doppler spectroscopy.In: Hubickyj, O., Bodenheimer, P., and Lissauer, J. J. (2005) Accre- T. Montmerle, D. Ehrenreich, & A.-M. Lagrange (Ed.), EAS tion of the gaseous envelope of jupiter around a 5–10 Earth- Publications Series. Vol. 41 of EAS Publications Series. pp. mass core. Icarus, 179,415–431. 7 27–75. 23 Hueso, R., and Guillot, T. (2005) Evolution of protoplanetary Eggleton, P. P. (1983) Approximations to the radii of Roche lobes. disks: constraints from DM Tauri and GM Aurigae. Astron. Astrophys. J., 268,368–369. 5 Astrophys., 442,703–725.3 Epstein, P. S. (1924) On the resistance experienced by spheres in Ida, S., and Lin, D. N. C. (2008) Toward a deterministic model of their motion through gases. Physical Review, 23,710–733. 3 planetary formation. IV. Effects of type I migration. Astrophys. Forgan, D., Rice, K., Stamatellos, D., and Whitworth, A. (2009) J., 673,487–501. 7 Introducing a hybrid radiative transfer method for smoothed Isella, A., Carpenter, J. M., and Sargent, A. I. (2009) Structure and particle hydrodynamics. Mon. Not. R. Astron. Soc., 394,882– evolution of pre-main-sequence circumstellar disks. Astrophys. 891. 20 J., 701,260–282. 3 Forveille, T., Bonfils, X., Delfosse, X., Gillon, M., Udry, S., Johnson, J. A., Butler, R. P., Marcy, G. W., Fischer, D. A., Vogt, Bouchy, F., Lovis, C., Mayor, M., Pepe, F., Perrier, C., Queloz, S. S., Wright, J. T., and Peek, K. M. G. (2007) A new planet D., Santos, N., and Bertaux, J. (2009) The HARPS search for around an M dwarf: revealing a correlation between exoplanets southern extra-solar planets. XIV. Gl 176b, a super-Earth rather and stellar mass. Astrophys. J., 670,833–840. 23 than a Neptune, and at a different period. Astron. Astrophys., Johnson, B. M., and Gammie, C. F. (2003) Nonlinear outcome of 493,645–650. 23 gravitational instability in disks with realistic cooling. Astro- Gammie, C. F. (1996) Layered accretion in T Tauri disks. Astro- phys. J., 597,131–141.18, 20 phys. J., 457,355–362.17 Kalas, P., Graham, J. R., Chiang, E., Fitzgerald, M. P., Clampin, Gammie, C. F. (2001) Nonlinear outcome of gravitational insta- M., Kite, E. S., Stapelfeldt, K., Marois, C., and Krist, J. (2008) bility in cooling, gaseous disks. Astrophys. J., 553,174–183. Optical images of an exosolar planet 25 light-years from Earth. 18 Science, 322,1345–1348.22 Gehrz, R. D., Becklin, E. E., de Pater, I., Lester, D. F., Roel- Kennedy, G. M., and Kenyon, S. J. (2008) Planet formation around lig, T. L., and Woodward, C. E. (2009) A new window on the stars of various masses: the snow line and the frequency of cosmos: the Stratospheric Observatory for Infrared Astronomy giant planets. Astrophys. J., 673,502–512. 23 (SOFIA). Advances in Space Research, 44,413–432. 24 Kippenhahn, R., and Weigert, A. (1990) Stellar structure andevo- Guillot, T. (2005) The Interiors of giant planers: models andout- lution. Springer-Verlag, Berlin. 8, 9 standing questions. Annual Review of Earth and Planetary Sci- Kokubo, E., and Ida, S. (2000) Formation of protoplanets from ences, 33,493–530. 23 planetesimals in the solar nebula. Icarus, 143,15–27.5 Goldreich, P., and Tremaine, S. (1980) Disk-satellite interactions. Kornet, K., Bodenheimer, P., Ró˙zyczka, M., and Stepinski,T.F.

D’Angelo, Durisen, & Lissauer 26 of 28 Giant Planet Formation REFERENCES

(2005) Formation of giant planets in disks with different metal- Nelson, A. F., Benz, W., Adams, F. C., and Arnett, D. (1998) Dy- licities. Astron. Astrophys., 430,1133–1138.23 namics of circumstellar disks. Astrophys. J., 502,342–371. 17 Kuiper, G. P. (1951) On the Origin of the Solar System. Proceed- Nelson, A. F., Benz, W., and Ruzmaikina, T. V. (2000) Dynamics ings of the National Academy of Science, 37,1–14.15 of circumstellar disks. II. Heating and cooling. Astrophys. J., Laughlin, G., Bodenheimer, P., and Adams, F. C. (2004) The core 529,357–390. 20 accretion model predicts few jovian-mass planets orbiting red Perri, F., and Cameron, A. G. W. (1974) Hydrodynamic instability dwarfs. Astrophys. J. Lett., 612,L73–L76.23 of the solar nebula in the presence of a planetary core. Icarus, Lin, D. N. C., and Papaloizou, J. (1986) On the tidal interaction 22,416–425. 8 between protoplanets and the protoplanetary disk. III - Orbital Pickett, B. K., Cassen, P., Durisen, R. H., and Link, R. (1998) migration of protoplanets. Astrophys. J., 309,846–857. 11, 12 The effects of thermal energetics on three-dimensional hydro- Lissauer, J. J. (1987) Timescales for planetary accretion and the dynamic instabilities in massive protostellar disks. Astrophys. structure of the protoplanetary disk. Icarus, 69,249–265. 5 J., 504,468–491. 17 Lissauer, J. J. (1993) Planet formation. Annu. Rev. Astron. Astro- Pickett, B. K., Cassen, P., Durisen, R. H., and Link, R. (2000) phys., 31,129–174. 4 The effects of thermal energetics on three-dimensional hydro- Lissauer, J. J., Hubickyj, O., D’Angelo, G., and Bodenheimer, P. dynamic instabilities in massive protostellar disks. II. High- (2009) Models of Jupiter’s growth incorporating thermal and resolution and adiabatic evolutions. Astrophys. J., 529,1034– hydrodynamic constraints. Icarus, 199,338–350. 7 1053. 18 Lodato, G., and Rice, W. K. M. (2004) Testing the locality of trans- Podolak, M., Weizman, A., and Marley, M. (1995) Comparative port in self-gravitating accretion discs. Mon. Not. R. Astron. models of Uranus and Neptune. Planet. Space Sci., 43,1517– Soc., 351,630–642. 18 1522. 23 Lubow, S. H., and D’Angelo, G. (2006) Gas flow across gaps in Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., protoplanetary disks. Astrophys. J., 641,526–533.13 Podolak, M., and Greenzweig, Y. (1996) Formation of the gi- Lubow, S. H., and Ida, S. (2010) Planet Migration. ArXiv e-prints, ant planets by concurrent accretion of solids and gas. Icarus, 1004.4137. 11, 13, 24 124,62–85.6 Lynden-Bell, D., and Pringle, J. E. (1974) The evolution of vis- Rafikov, R. R. (2005) Can giant planets form by direct gravita- cous discs and the origin of the nebular variables. Mon. Not. R. tional Instability? Astrophys. J., 621,L69–L72.19 Astron. Soc., 168,603–637. 11 Rafikov, R. R. (2007) Convective cooling and fragmentation of Marois, C., Macintosh, B., Barman, T., Zuckerman, B., Song, I., gravitationally unstable disks. Astrophys. J., 662,642–650. 20 Patience, J., Lafrenière, D., and Doyon, R. (2008) Direct imag- Rafikov, R. R. (2009) Properties of gravitoturbulent accretion ing of multiple planets orbiting the star HR 8799. Science, 322, disks. Astrophys. J., 704,281–291. 18, 20, 22 1348–1352. 22 Rice, W. K. M., Armitage, P. J., Bate, M. R., and Bonnell, I. A. Matzner, C. D., and Levin, Y.(2005) Protostellar disks: formation, (2003) The effect of cooling on the global stability of self- fragmentation, and the brown dwarf desert. Astrophys. J., 628, gravitating protoplanetary discs. Mon. Not. R. Astron. Soc., 817–831. 21 339,1025–1030. 18, 22 Mayer, L., Lufkin, G., Quinn, T., and Wadsley, J. (2007) Fragmen- Rice, W. K. M., Lodato, G., and Armitage, P. J. (2005) Inves- tation of gravitationally unstable gaseous protoplanetarydisks tigating fragmentation conditions in self-gravitating accretion with radiative transfer. Astrophys. J. Lett., 661,L77–L80. 20, discs. Mon. Not. R. Astron. Soc., 364,L56–L60.18 22 Rice, W. K. M., Lodato, G., Pringle, J. E., Armitage, P. J., and Mayer, L., Quinn, T., Wadsley, J., and Stadel, J. (2002) Forma- Bonnell, I. A. (2004) Accelerated planetesimal growth in self- tion of giant planets by fragmentation of protoplanetary disks. gravitating protoplanetary discs. Mon. Not. R. Astron. Soc., Science, 298,1756–1759.18 355,543–552. 22 Mayer, L., Quinn, T., Wadsley, J., and Stadel, J. (2004) The evolu- Rice, W. K. M., Lodato, G., Pringle, J. E., Armitage, P. J., and tion of gravitationally unstable protoplanetary disks: fragmen- Bonnell, I. A. (2006) Planetesimal formation via fragmentation tation and possible giant planet formation. Astrophys. J., 609, in self-gravitating protoplanetary discs. Mon. Not. R. Astron. 1045–1064. 17, 19, 21 Soc., 372,L9–L13.22 Mej´ıa, A. C., Durisen, R. H., Pickett, M. K., and Cai, K. (2005) Safronov, V. S., 1969. Evolution of the protoplanetary cloudand The thermal regulation of gravitational instabilities in proto- formation of the Earth and planets. Moscow, Nauka Press. En- planetary disks. II. extended simulations with varied cooling glish translation: NASA TTF-677, 1972. 4 rates. Astrophys. J., 619,1098–1113. 17, 18 Saumon, D., and Guillot, T. (2004) Shock Compression of Deu- Mizuno, H. (1980) Formation of the giant planets. Progress of terium and the Interiors of Jupiter and Saturn. Astrophys. J., Theoretical Physics, 64,544–557. 8 609,1170–1180. 23 Morbidelli, A., Brasser, R., Tsiganis, K., Gomes, R., and Levison, Shakura, N. I., and Syunyaev, R. A. (1973) Black holes in bi- H. F. (2009) Constructing the secular architecture of the solar nary systems. Observational appearance. Astron. Astrophys., system. I. The giant planets. Astron. Astrophys., 507,1041– 24,337–355. 3 1052. 24 Simpson, E. K., Pollacco, D., Hebrard, G., Gibson, N. P., Barros, Mordasini, C., Alibert, Y., Benz, W., and Naef, D. (2009) Extraso- S. C. C., Bouchy, F., Collier Cameron, A., Boisse, I., Watson, lar planet population synthesis. II. Statistical comparison with C. A., and Keenan, F. P. (2010) The spin-orbit alignment of observations. Astron. Astrophys., 501,1161–1184. 23 the transiting exoplanet WASP-3b from Rossiter-McLaughlin Movshovitz, N., and Podolak, M. (2008) The opacity of grains in observations. Mon. Not. R. Astron. Soc.,inpress.24 protoplanetary atmospheres. Icarus, 194,368–378.7 Stamatellos, D., and Whitworth, A. P. (2008) Can giant planets Murray, C. D., and Dermott, S. F. (2000) Solar system dynamics. form by gravitational fragmentation of discs? Astron. Astro- Cambridge University Press, Cambridge. 5 phys., 480,879–887. 20, 21

D’Angelo, Durisen, & Lissauer 27 of 28 Giant Planet Formation REFERENCES

Stamatellos, D., and Whitworth, A. P. (2009) The properties of brown dwarfs and low-mass Hydrogen-burning stars formed by disc fragmentation. Mon. Not. R. Astron. Soc., 392,413– 427. 18, 19, 22 Stevenson, D. J. (1982) Formation of the giant planets. Planetary and Space Science, 30,755–764. 8 Syer, D., and Clarke, C. J. (1995) Satellites in discs: regulating the accretion luminosity. Mon. Not. R. Astron. Soc., 277,758–766. 14 Tanaka, H., Takeuchi, T., and Ward, W. R. (2002) Three- dimensional interaction between a planet and an isothermal gaseous disk. I. Corotation and lindblad torques and planet migration. Astrophys. J., 565,1257–1274. 13 Tarenghi, M. (2008) The Atacama Large Millimeter/Submilli- meter Array: overview & status. Astrophys. Space Sci., 313, 1–7. 24 Tomley, L., Cassen, P., and Steiman-Cameron, T. (1991) On the evolution of gravitationally unstable protostellar disks. Astro- phys. J., 382,530–543.18 Toomre, A. (1981) What amplifies the spirals. In: Fall, S. M., and Lynden-Bell, D. (Eds.), Structure and Evolution of Nor- mal Galaxies. pp. 111–136. 16, 17 Udry, S., and Santos, N. C. (2007) Statistical properties of exoplanets. Annu. Rev. Astron. Astrophys., 45,397–439. 23, 24 Vorobyov, E.I.,andBasu, S.(2006) Theburstmodeof protost ellar accretion. Astrophys. J., 650,956–969. 17, 22 Ward, W. R. (1997) Protoplanet migration by nebula tides. Icarus, 126,261–281. 13 Weidenschilling, S. J. (1977) Aerodynamics of solid bodies in the solar nebula. Mon. Not. R. Astron. Soc., 180,57–70.21, 22 Winn, J. N. (2010) Transits and Occultations. ArXiv e-prints, 1001.2010. 1, 24 Wuchterl, G. (1993) The critical mass for protoplanets revisited - Massive envelopes through convection. Icarus, 106,323–334. 9 Wuchterl, G., Guillot, T., and Lissauer, J. J. (2000) Giant planet formation. In: Mannings, V., Boss, A. P., and Russell, S. S. (Eds.), Protostars and Planets IV. University of Arizona, Tuc- son. pp. 1081–1109. 8 Zapatero Osorio, M. R., Béjar, V. J. S., Mart´ın, E. L., Rebolo, R., Barrado y Navascués, D., Bailer-Jones, C. A. L., and Mundt, R. (2000) Discovery of young, isolated planetary mass objects in the σ Orionis star cluster. Science, 290,103–107.1 Zhu, Z., Hartmann, L., Gammie, C., and McKinney, J. C. (2009) Two-dimensional simulations of FU Orionis disk outbursts. As- trophys. J., 701,620–634.17

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D’Angelo, Durisen, & Lissauer 28 of 28 Giant Planet Formation