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Earth, Moon and Planets manuscript No.(will be inserted by the editor)

Numerical simulations of disc-planet interactions

Richard P. Nelson · Sijme-Jan Paardekooper

Received: date / Accepted: date

Abstract The gravitational interaction between a protoplan-etary disc and planetary sized bodies that form within itleads to the exchange of angular momentum, resulting inmigration of the planets and possible gap formation in thedisc for more massive planets. In this article, we review thebasic theory of disc-planet interactions, and discuss the re-sults of recent numerical simulations of planets embeddedin protoplanetary discs. We consider the migration of lowmass planets and recent developments in our understandingof so-called type I migration when a fuller treatment of thedisc thermodynamics is included. We discuss the runawaymigration of intermediate mass planets (so-called type IIImigration), and the migration of giant planets (type II mi-gration) and the associated gap formation in this disc. Theavailability of high performance computing facilities hasen-abled global simulations of magnetised, turbulent discs tobe computed, and we discuss recent results for both low andhigh mass planets embedded in such discs.

Keywords planet formation· accretion discs· numericalsimulations

1 Introduction

Planets are believed to form in the circumstellar discs of gasand dust that are observed around young stars (1). Withinthese discs, sub-micron sized interstellar dust grains col-lide and stick, slowly growing to form bodies comparablein mass to the Earth (M⊕) via a multi-stage process which

R.P. NelsonAstronomy Unit, Queen Mary University of London, Mile End Rd,London, E1 4NS, U.K.E-mail: R.P.Nelson@qmul.ac.uk

S.-J. PaardekooperDAMTP, Wilberforce Road, Cambridge CB3 0WA, U.K.E-mail: S.Paardekooper@damtp.cam.ac.uk

involves bodies of increasing size colliding and sticking.These planetary bodies which form with masses up to≃ 10M⊕ can accrete gas envelopes, leading to the formation ofa gas giant planet (2), a process which, models predict, re-quires a few million years. The precursor≃ 10 M⊕ bodieswhich accrete gas are usual referred to as planetary cores,and their formation is most efficient at locations in the discwhere the density of solids is highest. Thus gas giant plan-ets are thought to form in the cool disc regions beyond the‘snowline’, where volatiles can condense into ice grains. Inmost disc models, and apparently in our own early SolarSystem, the snowline is located at a radius∼ 3 AU fromthe central star. The discovery of the first extrasolar planetin 1995 (3), a gas giant planet orbiting at 0.05 AstronomicalUnits (AU) from the star 51 Pegasi therefore came as a bigsurprise, and posed a problem for planet formation theory.

The solution to this problem is thought to be that planetssuch as 51 Peg b probably did form several AU away fromthe star, but migrated inward afterwards, while still embed-ded in the circumstellar disc. The large number of extraso-lar planets now known (more than 350 at the time of writ-ing), with broad variation in their masses and orbital config-urations, indicates that migration is a common phenomenonduring planet formation.

Prior to the discovery of extrasolar planets, it was al-ready known that embedded bodies experience a force fromthe surrounding gas disc (4; 5; 6; 7) leading to orbital migra-tion. This can be appreciated in a qualitative way from Fig.1, where we show the surface density perturbation in a discby a low-mass planet. The planet launches two tidal wavesthat are sheared into spiral waves by differential rotationinthe disc. The gravitational pull from these waves leads toorbital migration of the planet.

Disc-planet interaction has become the focus of intenseresearch since 1995, and three types of migration can be dis-tinguished: Type I migration for low-mass planets (compara-

2

ble to the Earth), which do not strongly perturb the surround-ing disc, Type II migration for high-mass planets (compara-ble to Jupiter), which open deep gaps around their orbit, andType III migration for intermediate-mass planets (compara-ble to Saturn), embedded in massive discs (8).

A significant fraction of the research in this area hasutilised numerical simulations, and a number of insights havebeen gained from the results of such simulations which havesubsequently been intepreted within the fraemwork of an an-alytic model. The increasing availability of parallel comput-ing facilities, combined with the development of simulationcodes, has allowed increasingly complex simulations to beperformed, including physics such as MHD turbulence andradiation transport. In this article we will review the basictheory of disc-planet interactions, highlight some recentde-velopments, and summarise the state of the field as it cur-rently stands.

2 Type I migration

Type I migration is thought to apply to low-mass planets(approximately 1-20 M⊕). It is important to understand thisregime, because not only does it apply to terrestrial planets,also the solid cores of gas giant planets have been subjectto Type I migration before they obtained their gaseous enve-lope. This stage in giant planet formation requires more than106 years (2). In this section, we will review the basic the-ory of Type I migration, focusing on laminar (non-turbulent)discs.

Type I migration has long been regarded as the simplestmigration mechanism to understand. Since low-mass plan-ets induce only small perturbations in the disc, a linear anal-ysis should provide accurate estimates of the migration rate(9). These can later be verified by numerical hydrodynami-cal simulations (e.g. 10).

2.1 Linear analysis

It is convenient to express the perturbing potential due to theplanet acting on the discΦp as a Fourier series

Φp(r,ϕ , t) =− GMp

|r− rp|=

∞

∑m=0

Φm(r)cos(mϕ −mΩpt), (1)

whereϕ is the azimuthal angle andΩp is the orbital fre-quency of the planet, and to evaluate the response of thedisc due to each component separately. For simplicity, wewill consider only two-dimensional, isothermal discs in thissection. The vertically integrated pressure is then given byP = c2Σ , wherec is the sound speed andΣ is the surfacedensity of the disc defined byΣ =

∫ ∞−∞ ρ dz. In vertical hy-

drostatic equilibrium, the pressure scale height of the discequalsH = c/Ω , whereΩ is the angular velocity of the

disc. A typical disc hash ≡ H/r ≈ 0.05. The unperturbedsurface density follows a power law in radius,Σ ∝ r−α .

When working with discs, angular momentum is a keyquantity, dominating the dynamics. The angular momentumof the planet isJp = Mpr2

pΩp ∝ √rp for a circular orbit.

Therefore, changes in angular momentum are directly re-lated to a change in orbital radius, which makes the torque,the change in angular momentum, an important quantity tomeasure.

Writing the perturbed surface density as

Σ ′(r,φ , t) = Σ ′m(r)exp(imϕ − imΩpt), (2)

one can numerically solve the linearised fluid equations toobtain the disc response. The torque on the planet,

Γ =

∫

DiscΣr×∇ΦpdS, (3)

is directly related to the imaginary part of the surface densityperturbation:

dΓm

dr=−πmΦmI m(Σ ′

m), (4)

and the torqueΓm can be found by integrating over the wholedisc. The total torque is equal to the sum over allΓm.

Neglecting pressure effects, the disc can exchange an-gular momentum with the planet at a Lindblad resonance,which is located at a position in the disc wherem(Ω −Ωp)=

±κ , or at a corotation resonance, whereΩ = Ωp. Here,κ isthe epicyclic frequency, equal toΩ in a Keplerian disc. Ata Lindblad resonance, a pressure wave is excited that propa-gates away from the planet, which cause an oscillatory discresponse in the solutions to linearised equations describedabove. The waves transport angular momentum, resulting ina torque on the planet. The outer wave removes angular mo-mentum from the planet, while the inner wave gives angularmomentum to the planet. The torque therefore depends onthe difference in the strengths of the two waves.

There are several effects that conspire to make the outerwave stronger (11), the most important one being that outerLindblad resonances are located closer to the planet than in-ner Lindblad resonances. Therefore, for all reasonable sur-face density profiles the total Lindblad torque on the planetis negative, resulting in inward migration. The most recentlinear calculations for a 2D disc (9) resulted in a Lindbladtorque

ΓL,lin =−(3.2+1.468α)q2

h2 Σpr4pΩ2

p , (5)

whereq denotes the mass of the planet in units of the centralmass,rp denotes the orbital radius of the planet andΣp is thesurface density at the location of the planet. Note thatα isusually thought to be positive, resulting in a negative torqueon the planet.

3

∆Σ

0.00 0.05 0.10 0.15 0.20

-1.0 -0.5 0.0 0.5 1.0 x

-1.0

-0.5

0.0

0.5

1.0

y

Fig. 1 Surface density perturbation for a 4 M⊕ planet (located at(x,y) = (−1,0)) embedded in a disc withh= 0.05, showing the promi-nent spiral wakes associated with Lindblad torques.

In figure 1 we show the surface density perturbation dueto a low-mass planet from a numerical hydrodynamical cal-culation. Clearly visible are the two spiral waves launchedby the planet that result in the above Lindblad torque.

As mentioned above, angular momentum exchange alsotakes place at a corotation resonance. Since the location ofthis resonance is very close to the orbit of the planet, theresulting corotation torque is potentially very strong. Sincethe relative velocity of the gas and the planet is subsonicnear the corotation resonance, the planet is unable to excitewaves at this location (12). Semi-analytical calculations(9)result in a linear corotation torque

Γc,lin = 1.36

(

32−α

)

q2

h2 Σpr4pΩ2

p , (6)

and therefore a total linear torque

Γlin = ΓL,lin +Γc,lin =−(1.16+2.828α)q2

h2 Σpr4pΩ2

p . (7)

Again, sinceα is usually thought to be positive, we find thatlow-mass planets will migrate inward. In equation 6, thequantity in brackets is the power law index of the specificvorticity, or vortensity(∇ × v)/Σ , wherev is the velocityin the disc. The corresponding migration rate is (assuming acircular orbit):

rp =2Γlin

rpΩpMp=−(1.16+2.828α)

qh2

Σpr2p

M∗rpΩp. (8)

A major problem with the resulting inward migrationrates is that they are much too high: Type I migration wouldquickly take all low-mass planets, including the cores of gasgiant planets, very close to or even into the central star (11).

Models of planet formation including Type I migration (13)have great difficulty explaining the observed distributionofplanets.

Several mechanisms have been proposed to slow downType I migration. Within linear theory, it is first of all im-portant to note that the two-dimensional approximation isnot valid once the gravitational sphere of influence of theplanet (the Hill sphere) is smaller than the pressure scaleheight of the disc. It is easy to see that for a small enoughplanet, the upper layers of the disc do not feel the gravita-tional pull of the planet, while in the above two-dimensionaltheory they fully contribute to the torque. A torque formulathat accounts for this effect reads (9)

Γ3D,lin =−(1.364+0.541α)q2

h2 Σpr4pΩ2

p . (9)

Note that the difference between the 2D and the 3D resultstrongly depends on the surface density profile. In favourablecases, migration speeds can be reduced by a factor of a few.In the 2D numerical simulations presented below, a gravita-tional softening parameterb is used to account for 3D effectsin an approximate way:

Φp =− GMp√

|r − rp|2+ r2pb2

. (10)

It is expected that forb ∼ h, 3D effects on the Lindbladtorque can be captured approximately. The same may not betrue for the non-linear torque discussed below. In (14), equa-tion 9 has been successfully compared to 3D isothermal nu-merical simulations. Note, however, that their adopted den-sity profile allows for weak corotation torques only.

The above analysis was simplified by neglecting mag-netic fields, self-gravity and detailed thermodynamics. In-cluding magnetic resonances in the linear analysis can re-duce the torque significantly if a toroidal magnetic field ispresent (15). On the other hand, self-gravity tends to accel-erate Type I migration due to a shift in resonances (16; 17).More recently, releasing the assumption of an isothermaldisc has been the subject of intensive study, since simula-tions presented by (18) which included radiation transportin the disc suggest that type I migration may be stopped orreversed in a disc which cools inefficiently. A linear analysis(19) suggested that a linear effect due to the presence of a ra-dial entropy gradient may reduce or even reverse migration.However, it was subsequently shown (20) that this effect isin fact small and can not account for the results obtained in(18).

2.2 Beyond the linear model

Although intuitively, one would expect that low-mass plan-ets induce linear perturbations only, recent work has shownthat this is not the case. In a sense, introducing a planet in

4

0.96 0.98 1.00 1.02 1.04r

3.05

3.10

3.15

3.20

φ

Fig. 2 Schematic view of the streamline pattern close to the planet,which is located at(r,ϕ) = (1,π) and is denoted by the black circle.Material inside the black streamline (the separatrix) executes horseshoeturns on both sides of the planet.

a disc amounts to a singular perturbation at corotation, sug-gesting that the corotation torque can differ from its linearvalue.

The corotation torque is due to material that, on average,corotates with the planet. Two ways of obtaining the corota-tion torque can be found in the literature: one can perform alinear analysis of perturbed circular orbits around the coro-tation resonance (4; 9) (see above), or, alternatively, onecanlook at the expected pattern of the streamlines close to theplanet (see figure 2) and analyse the torque due to materialexecuting horseshoe turns (21). We will denote the formerresult thelinear corotation torque, and the latter thehorse-shoe drag. It is important to stress that there isnohorseshoeregion in linear theory, since these bends are a genuinelynon-linear phenomenon.

2.2.1 Isothermal results

Considering again two-dimensional, isothermal (or, more gen-eral, barotropic) flow, in which case specific vorticity is con-served along streamlines, it is easy to see that when execut-ing the turn, material should change its density. Since it isforced to a different orbit by the planet, the vorticity of thematerial changes, and conservation of vortensity then dic-tates that there should be an associated change in density.This change in density is of opposite sign on the differentsides of the planet, hence the planet feels a torque. Thistorque depends sensitively on the width of the horseshoe re-gionxs (21):

Γc,hs=34

(

32−α

)

x4sΣpr4

pΩ2p . (11)

Note thatxs is in units ofrp. Just as in the linear case, thehorseshoe drag is proportional to the radial gradient in spe-cific vorticity. In the special case ofα = 3/2, the density

0 5 10 15 20 25 30t (orbits)

-1000

-500

0

Γ/q2

α=5/2α=3/2α=1/2α=-1/2

Fig. 3 Torques on aq= 1.26·10−5 planet (4 M⊕ around a Solar massstar) embedded in an inviscid disc withh= 0.05. All torques have beennormalised toΣp = rp = Ωp = 1. Results for different density profilesare shown, with dotted horizontal lines indicating resultsobtained fromlinear theory. A gravitational softening ofb/h= 0.6 was used.

change needed to compensate for the change in vorticityexactly amounts to the background profile, resulting in notorque on the planet.

Until recently, it has never been clarified how equations6 and 11 are related. There is no reason why the value ofxs,unspecified in the original theory (21), should adjust itselfto match the linear torque. Detailed modeling of the horse-shoe region (22) shows thatxs ∝

√

q/h, so that the horse-shoe drag scales in exactly the same way as the linear coro-tation torque. The only way they differ is in magnitude, andin general the horseshoe drag is much stronger (23).

It is intuitively clear that these two distinct torques cannot exist at the same time: material either follows a per-turbed circular orbit or executes a horseshoe turn. At suf-ficiently early times after inserting the planet in the disc,therefore, before any turns can have occurred, one wouldexpect linear theory to be valid. Once material starts to exe-cute horseshoe turns, the linear corotation torque will getre-placed by the non-linear horseshoe drag, which, as remarkedabove, will in general be much stronger.

This can be verified using numerical hydrodynamic sim-ulations. Resolution is crucial, since for low-mass planetsxs ≪ h≪ 1, and in order to obtain reliable estimates of thetorque, the horseshoe bends need to be resolved. Simulationresults for different values ofα are displayed in figure 3,showing the total torque on the planet as a function of time.Overplotted are the results from solving the linear equations(dotted lines). It is clear that only in the case whereα = 3/2the non-linear hydrodynamic calculations match linear the-ory. In this special case, the corotation torque vanishes. Forall other values ofα, a departure from linear theory can beseen, pointing to a stronger corotation torque than expectedfrom linear theory. However, the total torque is consistent

5

with the linear corotation torque replaced by the strongerhorseshoe drag, showing that indeed non-linear effects playan important role in the corotation region.

Another important aspect is the time scale to set up thetorque. Since the linear corotation torque has no definitescale, it takes approximately one orbit to develop. The horse-shoe drag, on the other hand, does have an intrinsic scale,namelyxs. An associated time scale is the libration time,which is basically the time it takes for a fluid element, ini-tially at a distancexs from the orbit of the planet, to completeone whole horseshoe orbit:

τlib =8π3xs

Ω−1p . (12)

It takes a fraction of the libration time for the horseshoe dragto develop, which is therefore different for different planetmasses sincexs ∝ √

q. The transition between the linear re-sult (obtained after approximately 1-2 orbits), and the moreslowly developing non-linear horseshoe drag can be clearlyseen in figure 3.

2.2.2 Extension to non-isothermal discs

These simple models of Type I migration lack a crucial in-gredient: a proper treatment of the disc thermodynamics.Most previous simulations were indeed done in the locallyisothermal limit, where the temperature of the disc is a pre-scribed and fixed function of radius. Such a model wouldapply if any excess energy could be radiated away very ef-ficiently. However, in discs that can not cool efficiently (i.e.are optically thick), the presence of a radial entropy gradientleads to strong coorbital torques that can reverse the direc-tion of migration (24). This can again be understood in termsof the horseshoe drag. When the gas is not able to cool (theadiabatic limit), entropy is conserved along streamlines.Ina similar way as the vortensity gradient led to the barotropichorseshoe drag, a radial entropy gradient leads to a stronghorseshoe drag related to density changes as the fluid tries toconserve its entropy when making the horseshoe turn (20).When the entropy decreases outward, this horseshoe draggives a positive contribution to the torque.

This is illustrated in figure 4, where we compare adia-batic and isothermal simulations withα = 3/2 for differenttemperature profiles (parametrised byβ =−d logT/d logr).An adiabatic exponentγ = 5/3 was chosen. Due to the isen-tropic nature of linear waves, the density of the wakes islowered by a factorγ compared to isothermal simulations,which makes the Lindblad torque a factorγ smaller. Notethat the density profile is such that the barotropic corotationtorque vanishes. The entropy follows a power law, initially,with index−ξ = −β +(γ −1)α. Therefore, the isothermaldisc (β = 0) with no vortensity gradient (α = 3/2) does havean entropy gradient (ξ = −1), giving rise to a strong horse-shoe drag. This is clear from the grey solid line, which shows

0 5 10 15 20 25 30t (orbits)

-1400

-1200

-1000

-800

-600

-400

-200

0

Γ/q2

α=3/2, β=0α=3/2, β=2

γ=1γ=5/3

Fig. 4 Torques on aq= 1.26·10−5 planet (4 M⊕ around a Solar massstar) embedded in an inviscid disc withh = 0.05. All torques havebeen normalised toΣp = rp = Ωp = 1. The black line indicates theisothermal result, grey curves results from adiabatic calculations fortwo different temperature profiles. A gravitational softening of b/h =0.4 was used.

0 2 4 6 8 10 12 14t (orbits)

-1000

-500

0

500

1000

Γ/q2

isothermalρp=10-11

ρp=10-12

ρp=10-13

Fig. 5 Torques on aq= 1.26·10−5 planet (4 M⊕ around a Solar massstar) embedded in a 3D inviscid disc withh = 0.05. All torques havebeen normalised toΣp = rp = Ωp = 1. Results for different midplanedensities are shown, and the differences between the curvesare a re-sult of the corresponding change in opacity, which affects the coolingproperties of the disc.

a large swing between 2 and 10 orbits as a result. When wemodify the temperature structure to reverse the entropy gra-dient, a corresponding opposite swing is observed (dashedgrey line in figure 4). Even without an additional contribu-tion from a vortensity gradient, the torque is reduced by afactor of 3 compared to the linear value.

Finally, we release the two-dimensional approximation,and enable the disc to cool through radiation. The resulting3D radiation-hydrodynamical models (18) are the most re-alistic simulations of low-mass planet migration so far. Infigure 5, we show the results forα = 1/2 andβ = 1. Dif-ferent midplane densitiesρp were used, whereρp = 10−11

6

g cm−3 approximately corresponds to the minimum massSolar nebula at 5 AU. Note that the torques are still nor-malised toΣp = 1. Therefore, different densities mainly in-dicate different opacities.

The high density case is almost equivalent to an adia-batic simulation. This is easy to understand, since the discis very optically thick in this case and basically can not coolthrough radiation. This again leads to a positive torque, andoutward migration. Lowering the density leads to more effi-cient cooling, and one would expect to return to the isother-mal result when the opacity is low enough. This is indeedthe case: upon decreasing the density by a factor of 10 thetorque is already negative, and with another factor of 10we can almost reproduce the isothermal result. Therefore,the entropy-related horseshoe drag is important in the denseinner regions of protoplanetary discs. This mechanism canthen act as a safety net for low-mass planets, since they stoptheir inward journey as soon as they reach the dense innerregions of the disc.

2.2.3 Saturation

In absence of any diffusive process in the disc, the horse-shoe region is a closed system, and it can therefore only givea finite amount of angular momentum to the planet. Phasemixing leads to a flattening of the vortensity/entropy pro-files, which results in a vanishing torque after the gas hasmade several horseshoe turns. In other words, the corota-tion torque saturates (25). The horseshoe region needs to befed fresh material in order to sustain the torque. This can bedone through viscous diffusion in the isothermal case (26),and it has been observed that by including a small kinematicviscosity and heat diffusion the torque can be sustained (20).This has also been shown with radiative cooling instead ofheat diffusion in (27).

2.3 Outstanding issues

The non-linear torque strongly depends on the width of thehorseshoe regionxs. It is therefore of critical importanceto have a good physical understanding of what determinesthis width when three-dimensional effects are taken into ac-count.

It is not clear yet how the entropy and vortensity relatedtorques work together. Vortensity is not conserved in generalalong streamlines in 3D non-barotropic discs, but neverthe-less there appears to be a contribution from the radial gradi-ent in vortensity to the horseshoe drag in this case. A properunderstanding of these processes will lead to an improvedmigration law for low-mass planets.

2.4 Concluding remarks

Type I planetary migration is not as well-understood as pre-viously thought. Even in the simple isothermal case, strongdeviations from linear theory occur whenever the disc has aradial gradient in specific vorticity. In other words, the coro-tation torque is always non-linear. Therefore, one should becareful in applying the widely used linear torque formulapresented in (9). Even more so for non-isothermal discs,where a radial entropy gradient can contribute to the non-linear torque to the effect of reversing the direction of migra-tion. In order for the torque to be sustained, some diffusionof energy and momentum is needed.

3 Type III migration

The most recently discovered migration mechanism, first re-ferred to as runaway migration (8), relies on strong corota-tion torques induced by the radial movement of the planetitself as it migrates. Later, it has become known as Type IIImigration (e.g. 28).

3.1 Theory

We will return to the simple case of a laminar, isothermaldisc. Consider the horseshoe region as depicted in the leftpanel of figure 6, and for now assume that it has constantspecific vorticity, either through saturation or due to the ini-tial density profile. The corotation torque therefore vanishes.If we now let the planet migrate radially, the horseshoe re-gion as viewed in a frame comoving with the planet changesits shape (middle panel of figure 6). If the planet is movinginward, as is the case in figure 6, some fresh material fromthe inner disc will enter the horseshoe region, make the turnand move into the outer disc, exerting a torque on the planetin the process. It is easy to see that this torque is proportionalto the migration speed, and acts to accelerate the migration.There is therefore a positive feedback loop, and the possibil-ity of a runaway process.

The mass flow relative to a planet migrating at a rate ˙rp

(we assume a circular orbit, so that the semimajor axisa=

rp) is given by 2πΣsrp, with Σs the surface density at the up-stream separatrix. Since we assume the ordinary, vortensity-related corotation torque to be fully saturated, the only horse-shoe drag on the planet comes from this fresh material exe-cuting a horseshoe turn, giving rise to a torque (29):

Γc,3 = 2πΣsrpΩpr3pxs. (13)

The system of interest that is migrating is the planet itself,mass of gas bound within its Roche lobe, and the horseshoeregion. The drift rate of this system is given by

(Mp+MHS+MR)Ωprprp = 4πr2pxsΣsΩprprp+2ΓL, (14)

7

0.96 0.98 1.00 1.02r

0

1

2

3

4

5

6

φ

0.96 0.98 1.00 1.02r

0

1

2

3

4

5

6

φ

0.96 0.98 1.00 1.02r

0

1

2

3

4

5

6

φ0.96 0.98 1.00 1.02

r

0

1

2

3

4

5

6

φ0.96 0.98 1.00 1.02

r

0

1

2

3

4

5

6

φ

0.96 0.98 1.00 1.02r

0

1

2

3

4

5

6

φ

Fig. 6 Streamlines near the corotation region for different migration speeds, in a frame moving with the planet. Left panel: nomigration, middlepanel: slow inward migration, right panel: fast inward migration.

0.0 0.5 1.0 1.5 2.0 2.5 3.0M∆

-3

-2

-1

0

1

2

3

Z

Fig. 7 Non-dimensional migration rate as a function of the mass deficitparameterM∆ .

which can be rewritten as

rp =4πr2

pxsΣs−MHS

Mprp+

2ΓL

rpΩp, (15)

where we have redefinedMp as the mass of the planet plusthe mass within its Roche lobe. Here,ΓL is the Lindbladtorque, which may deviate from the linear Lindblad torquedue to shock formation of the appearance of a gap aroundthe orbit of the planetl. The numerator of the first term onthe right hand side is called thecoorbital mass deficit(8).Basically, this is the difference between the mass the horse-shoe region would have in a disc withΣ = Σs and its ac-tual mass. It is clear that when this coorbital mass deficitbecomes comparable toMp, very high migration rates arepossible. However, in this limit the above analysis becomesinvalid. A more careful treatment (29; 30) results in:

Z− 23

M∆ sign(Z)(1− (1−|Z|<1)3/2) =

2ΓL

MpΩprprp,f, (16)

10-6 10-5 10-4 10-3 10-2

q

0.0001

0.0010

0.0100

0.1000

µ dGravitational stability limit

Type I Type II

Type III

100

10

1

Q

Fig. 8 The three migration regimes in disc-planet mass parameterspace, forh= 0.05.

whereM∆ denotes the coorbital mass deficit divided by theplanet mass,

rp,f =3x2

s

8πrpΩp (17)

is the drift rate required for the planet to migrate a distancexs in one libration time, andZ= rp/rp,f . A quantity with sub-script< 1 indicates that the minimum of that quantity and 1is to be taken. Equation 16 gives rise to interesting behavior.Assuming Lindblad torques to be small, solutions to equa-tion 16 are plotted in figure 7. ForM∆ < 1, only one solution,Z = 0, is possible. There is a bifurcation atM∆ = 1, beyondwhich there are three possible solutions: steady inward oroutward Type III migration, plus an unstable solutionZ = 0.IncorporatingΓL in this picture removes this symmetry be-tween inward and outward migration (30), favouring migra-tion towards the central star, although outward migration isstill possible (31).

8

Fig. 9 This figure shows a snapshot of a planet with inital mass of 1 Jupiter mass embedded in a protoplanetary disc. Note the formation of nonlinear spiral waves and the accompanying gap.

3.2 Occurence

There are different ways in which a coorbital mass deficitcan be obtained. The first possibility is when the planet opensup a (partial) gap (8). Clearly, in this case there is a dif-ference between the actual mass of the horseshoe regionand the mass of the horseshoe region withΣ = Σs, hencethere is a coorbital mass deficit. For a high enough discmass, this deficit can become comparable to the planet mass.More massive planets require a more massive disc to en-ter the Type III migration regime. On the other hand, low-mass planets do not open up a gap and therefore their coor-bital mass deficit is zero. It turns out that in a typical disc,Saturn-mass planets are most susceptible to Type III mi-gration (8). This is depicted in figure 8, where we showthe regime of Type III migration for different disc masses.Here,µd = πr2

pΣp/M∗ is a dimensionless disc mass param-eter, andQ is the corresponding Toomre stability parameteragainst fragmentation. For the Minimum Mass Solar Nebula(MMSN) andh= 0.05,Q> 10 inside 10 AU, making TypeIII migration unlikely. However, the MMSN is effectivelya lower limit on the true disc mass profile, and for massivediscs Type III migration may be an important effect.

Another way for a coorbital mass deficit to arise is whenthe background disc has a sharp edge. This was studied forJupiter-mass planets (q = 0.001) in (28; 31). Such a coor-bital mass deficit would also affect low-mass planets, butthis has not been studied so far.

Type III migration is a numerically challenging problem,since it strongly depends on material close to the planet. Forexample, in the above analysis we redefined the planet massto include all material inside the Roche lobe. If the planet ob-tains a significant envelope, torques from within the Roche

lobe can be large (32) and oppose Type III migration. How-ever, since self-gravity is neglected, this material has effec-tively no inertia, making this calculation inconsistent (29).

To make further progress and deal with such a massiveenvelope, it is necessary to incorporate effects of self-gravity,at least in an approximate way (33). Moreover, to avoid anartificially large mass build-up inside the Roche lobe due tothe isothermal approximation, a temperature profile for theenvelope needs to be supplied. Only then can simulations ofType III migration, including torques from inside the Rochelobe, be shown to be converged (33).

Type III migration can be very fast. The typical timescale isrp/rp,f = 2/3x2

s orbits. Forxs = h, appropriate forSaturn-mass planets (22), the migration timescale is approx-imately 200 orbits, or 2360 year for a planet starting at 5.2AU. However, it is not clear at this point how long the coor-bital mass deficit can be retained. For example, unless thesurface density profile decreases steeply with radius, the coor-bital mass deficit will naturally decrease as the planet movesinward because of the changing factorr2

pΣs in equation 15.Outward Type III migration does not suffer from this effect,but simulations show that in this case the planet acquires avery massive envelope that pulls it into the Type II regime(31). Therefore, it seems that Type III migration is of limitedimportance for high-mass planets, changing the semimajoraxis by a factor of a few (28; 31).

4 Type II migration in viscous, laminar discs

When a planet grows in mass, the spiral density waves ex-cited by the planet in the disc can no longer be treated aslinear perturbations in the manner described in section 2.The planetary wake becomes a shock-wave in the vicinity

9

Fig. 10 The left panel shows the orbital radius of giant planet as it migrates in the disc model described in the text. The right panel shows themass of the planet as it concurrently accretes gas from the disc.

of the Lindblad resonances where it is launched. Shock dis-sipation, as well as the action of viscosity, leads to the depo-sition of angular momentum associated with the spiral wavelocally in the disc. Material is thus pushed away from the lo-cation of the planet on either side of its orbital location, andan annular gap begins to form, centred on the planet semi-major axis. The equilibrium structure and width of the gapis determined by a balance between the gap-closing viscousand pressure forces and gap-opening gravitational torques(e.g. 34).

In order for a gap to form and be maintained, two ba-sic conditions must be satisfied. The first is that the discresponse,via the launching of spiral wakes, should be nonlinear, such that gravitational forces due to the planet over-whelm pressure forces in the neighbourhood of the planet.This condition for non linearity is equivalent to the planetHill sphere radius exceeding the disc vertical scale height,and may be expressed mathematically as:

a

(

Mp

3M∗

)1/3

> H. (18)

wherea is the planet semimajor axis. The second conditionis that angular momentum transport by viscous stresses inthe disc be smaller than planetary tidal forces. This can beexpressed approximately as

Mp

M∗>

40a2Ων

. (19)

whereν is the kinematic viscosity. For more discussion aboutthese aspects of gap opening, see the following papers (35;36; 37; 34).

For standard disc parameters,H/r = 0.05, andν/(a2Ω)=

10−5, equation (18) givesMp/M∗> 3.75×10−4,while equa-tion (19) givesMp/M∗ > 4×10−4. The estimates are in rea-sonable agreement with the results of numerical simulationsof gap opening by (36) and (38). Accordingly, for typicalprotoplanetary disk parameters, we can expect that a planet

with a mass exceeding that of Saturn will begin to open anoticeable gap.

Because the disc response to a massive embedded pro-toplanet is non linear, the study of the orbital evolution andgas accretion by such a planet has been the subject of exten-sive studyvia numerical simulations. Both 2D (e.g 36; 38;37; 39) and 3D simulations (40; 41) have been performed,and overall the results found in these studies are in goodagreement. Differences obviously arise in the details of thegas flow near the planet when comparing 2D and 3D simu-lations, but the global properties of the models are otherwisesimilar as the gas is pushed away from the planet vicinity bythe process of gap opening.

Two snapshots from a 2D simulation of a disc withH/r =0.05 andν/(a2Ω) = 10−5 are shown in figure 9, wherethe initial mass of the planetMp = 1 MJup (where MJup isJupiter’s mass). The formation of a deep annular gap in thevicinity of the planet is apparent. The parameters used in thissimulation are such that both the condition for non linearityand the condition for tides to overwhelm viscous stresses aresatisfied, as described above.

The evolution of the planet semimajor axis is shown inthe left panel of figure 10. As the gap forms and the planetmigrates inward, the migration rate drops significantly be-low that which is predicted for type I migration. Once aclean gap is formed, the planet orbital evolution is controlledby the rate at which the outer gap edge viscously evolves,such that the migration time,τmig, can be approximated by

τmig ≃2r2

p

3ν. (20)

For typical disc parameters the time to migrate from a dis-tance ofrp = 5 AU to the central star isτmig ≃ 2×105 years.

As the planet mass increases, or the disc mass decreases,the inertia of the planet can become important and begin toslow the migration. This occurs when the local disc mass

10

contained within the orbital radius of the planet falls beneaththe planet mass (πr2

pΣ(rp) < Mp). In this case the migra-tion rate is controlled by the rate at which mass can build-up at the edge of the gap such that the disc mass becomescomparable to the planet mass locally (42). The slowingof migration is apparent in the left panel of figure 10, anda simple linear extrapolation of the migration history indi-cates that the time required for this planet to migrate intothe central star is≃ 3× 104 orbits (or∼ 3×105 years fora planet initially located at 5 AU. The right hand panel offigure 10 shows the evolution of the planet mass as it ac-cretes gas from the disc. The mass accretion rate is slowedby the formation of the gap, but nonetheless gas is still ableto diffuse through the gap onto the planet allowing it togrow. Linear extrapolation of the planet mass shown in fig-ure 10 indicates that the planet will have grown to a massof Mp ≃ 3.5MJupby the time it reaches the central star, suchthat migration and concurrent gas accretion occur during thegap opening phase of giant planet formation.

The simulations for gap forming planets presented inthis section indicate that planets grow substantially throughcontinued gas accretion as they migrate. This raises an inter-esting question about the type of correlation which expectedbetween planetary semimajor axes and masses. Taken at facevalue, a model which predicts that planets grow as they mi-grate inward will predict an inverse correlation between plan-etary mass and semimajor axes. The strength of this cor-relation will be weakened by variations in protoplanetarydisc masses, planet formation times, disc lifetimes etc, butif these parameters are peaked around certain typical valuesin nature, then the correlation should be apparent in the ex-trasolar planet data. The absence of such a correlation mayin part be explained by post-formation interaction within aplanetary system, leading to significant gravitational scat-tering. The possibility that significant migration may occurprior to the last stages of giant planet formation may alsobe important. Nonetheless, indirect evidence for the past oc-curence of Type II migration in some extrasolar planetarysystemsis provided by multi-planet systems which displaythe 2:1 mean motion resonance (or other resonances of lowdegree). Such a configuration is most naturally explained bytwo giant planets undergoing convergent migration at ratespredicted by Type II migration theory, as illustrated by thenumerical simulations presented in (e.g. 43; 44).

4.1 Outstanding issues

It has turned out to be difficult to improve on the non-self-gravitating, isothermal results in which gas inside the Rochelobe is considered to be part of the planet (8). Approximatecorrections for self-gravity and non-isothermality are neces-sary to make progress on high-mass planets (33). The firstfully self-gravitating calculations indicate that this can have

important effects (45). More work is necessary in this area,also to completely release the isothermal assumption to ob-tain a realistic envelope structure for the planet.

5 Turbulent protoplanetary discs

The majority of calculations examining the interaction be-tween protoplanets and protoplanetary discs have assumedthat the disc is laminar. The mass accretion rates inferredfrom observations of young stars, however, require an anoma-lous source of viscosity to operate in these discs. The mostlikely source of angular momentum transport is magnetohy-drodynamic turbulence generated by the magnetorotationalinstability (MRI) (46) Numerical simulations have been per-formed using the local shearing box approximation (see ref-erence 47, for a review) and using global geometry (e.g. 48).In each of these set-ups, runs have been conducted using nonstratified models in which the vertical component of grav-ity is ignored, and in more recent work the vertical compo-nent of gravity has been included in shearing box (49) andglobal models (50, and references therein). These model in-dicate that the nonlinear outcome of the MRI is vigorousturbulence, and dynamo action. Recent work has indicatedthat the picture is more complicated than previously thought,with there being a strong dependence of the saturated stateof the turbulence on the value of the magnetic Prandtl num-ber (51; 52).

Fig. 11 This figure shows a snapshot of a 10 M⊕ protoplanets embed-ded in the turbulent protoplanetary disc model described inthe text.

Prior to the realisation that the strength of the MHD tur-bulence depends on the Prandtl number, it was recongnisedthat non ideal MHD effects might be important in proto-planetary discs. The ionisation fraction in cool, dense pro-

11

toplanetary discs is expected to be small in the planet form-ing region between 0.5 – 10 AU. Only the surface layersof the disc, which are exposed to external sources of ioni-sation such as X–rays from the central star or cosmic rays,are likely to be sufficiently ionised to sustain MHD turbu-lence (e.g. 53; 54; 55; 56). Predictions about the structureand depth of the interior non turbulent “dead zone” dependon complex chemical reaction networks and the degree ofdust grain depletion, issues which themselves may be in-fluenced by the process of planet formationvia dust graingrowth.

Although there are a number of uncertaintities remainingabout the nature and strength of disc turbulence, it is clearthat an accurate understanding of planet formation and disc-planet interaction requires the inclusion of turbulence inthemodels. Work conducted over the last four or five years hasexamined the interaction between planets of various massesand turbulent protoplanetary discs, where these studies haveusually simulated explicitly MHD turbulence arising fromthe MRI. We now review the results of these studies, andpresent some more recent models.

Fig. 13 This shows the variation of semimajor axis with time (mea-sured in planet orbits atr = 3). The green line shows the planet evolu-tion in a turbulent disc, the red line shows the evolution in an equivalentlaminar disc.

5.1 Low mass protoplanets in turbulent discs

The interaction between low mass, non gap forming proto-planets and turbulent discs has been examined by (10; 57;58; 59) These calculations show that interaction betweenembedded planets and density fluctuations generated by MHDturbulence can significantly modify type I migration, at leastover time scales equal to the duration of simulations that arecurrently feasible (t ∼ 150 planet orbits). The influence ofthe turbulence is to induce a process of ‘stochastic migra-tion’ rather than the monotonic inward drift expected for

planets in laminar discs. The simulations presented in thepapers cited above considered the interaction of planets withnon stratified (no vertical component of gravity) disc mod-els (for reasons of computational expense), but the work of(50) indicates that the amplitude of midplane density fluctu-ations in stratified models is weaker than in their non strat-ified counterparts because of magnetic buoyancy effects re-moving net magnetic flux from the simulation domain. It istherefore important to consider the evolution of planets instratified models.

One such model is presented in figure 11 which showsa snapshot of the midplane density in a turbulent disc withan embedded 10 Earth mass planet. The disc model in thiscase has aspect ratioH/r = 0.1, and was initiated with a ra-tio of magnetic to thermal gas pressurePg/Pm = 25 though-out the computational domain. As such the model is verysimilar to those presented in (50). The disc extends froman inner radius of 1.6 AU to 16.6 AU, and has a verticaldomain covering a total of 9 scale heights. The resolutioncorresponds to (Nr , Nφ , Nθ ) = (464, 280, 1200). Figure 11shows the turbulent density fluctuations are of higher am-plitude than the spiral wakes generated by the planet, as theplanet induced wakes are not visible against the perturba-tions induced by the MHD turbulence (see also 10; 58). In-deed, density fluctuations generated by turbulence in simu-lations are typicallyδΣ/Σ ≃ 0.15 – 0.3, with peak fluctua-tions beingO(1). Thus, on a length scale equivalent to thedisc thicknessH, density fluctuations can contain more thanan Earth mass in a disc model a few times more massive thana minimum mass nebula.

The left panel of figure 12 shows the variation in thetorque per unit mass experienced by the planet shown in fig-ure 11. The green line shows the torque experienced by aplanet in the turbulent disc model, and the red line showsthe torque experienced by the same planet in an equivalentlaminar disc model (the torque value here is≃ 3× 10−6).It is clear that the planet in the turbulent disc experiencesstrongly varying stochastic torques whose magnitudes aresignificantly larger than the expected type I torques. Theright hand panel shows the running time average of the torqueexperienced by the planet. The light blue, green and darkblue lines correspond to the torque from the outer disc, theinner disc, and the total torque, respectively. The red linesshow the equivalent quatities from the laminar disc model.It is clear that while the various torque contributions are ofsimilar magnitudes when comparing the laminar and turbu-lent models, the net average torque in the turbulent modelis positive, indicating that the planet has migrated outwardslightly over the duration of this simulation.

This expectation is confirmed by figure 13, which showsplanet semimajor axes for both turbulent (green line) andlaminar (red line) models. It is clear that for the durationof this run the planet in the turbulent disc has undergone

12

Fig. 12 The green line in left panel shows the torque experienced by the planet embedded in the turbulent disc. The red line shows the torqueexperienced by a planet embedded in an equivalent laminar disc. The right panel shows the time averaged values of the torques. From top tobottom, red lines are the torques from the laminar disc calculation corresponding to the disc interior to the planet, thetotal torque, and the discexterior to the planet. Similarly the turquoise, purpole and green lines are from the turbulent disc simulation.

Fig. 14 The left panel shows the distribution of torques experienced by the planet embedded in the turbulent disc. The right panel shows theautocorrelation function of the torque plotted in the left panel of Fig. 13.

stochastic migration, resulting in it having moved outwardslightly. A key question is whether the stochastic torques cancontinue to overcome type I migration over time scales upto disc life times. A definitive answer will require very longglobal simulations, but simple estimates can be made basedon the simulated torques and their statistical properties.

If we assume that the planet experiences type I torquesfrom the turbulent disc, on top of which are linearly super-posed Gaussian distributed fluctuations with a characteristiccorrelation timeτc, then we can write an expression for thetime averaged torque experienced by the protoplanet,T, inthe form

T =< T >+σT√ttot

(21)

whereσT is the standard deviation of the torque amplitude,< T > is the underlying type I torque, andttot is the to-tal time elapsed, measured in units of the torque correlationtime τc. In other words we treat the calculation as a simplesignal-to-noise problem. Convergence toward the underly-ing type I value is expected to begin once the two terms on

the right hand side become equal, as the fluctuating com-ponent should begin to average out. The torque distributionfunction for this simulation is shown in the left panel of fig-ure 14, and the torque autocorrelation function is shown inthe right hand panel. Expressed in code units, the rms torquevalue is∼ 4×10−5 and the correlation time isτmathrmc≃ 1planet orbital period. The type I torque is< T >≃ 3.5×10−6, so the time over which type I torques should be appar-ent is 130 orbits, slightly more than the simulation run time.This prediction suggests that we should expect to see theplanet begin to migrate inward in figure 13, but this clearlyis not the case.

The lack of inward migration in figure 13 may just bea consequence of not having run the simulation for longenough, such that the fluctuating torques have not had timeto begin to average out. The similarity between the run timeof the simulation and the estimate of the time when type Ieffects should become apparent suggests that this may bethe case. If we look at the right panel of figure 12, how-

13

ever, we can see that the averaged torque due to the disc thatlies interior to the planet has converged to a value whichis very close to that obtained in a laminar disc model. Theouter disc torque, however, remains somewhat lower in am-plitude, such that the net torque experienced by the planet ispositive over the simulation run time. It is well known thatthe stochastic nature of the disc turbulence, and the associ-ated stresses it generates, can lead to structuring of the discradial density profile due to local variations in the effectiveviscous stress. This may provide a means of modifying thebalance between inner and outer disc torques such that con-vergence to the expected type I value does not happen inpractice for an individual planet in a disc, although it shouldhappen on average for an ensemble of planets. One poten-tial way in which the underlying type I torques may not berecovered is if local variations in the viscous stress lead toundulations in the radial density profile. In this situationthecorotation torque may become important in regions wherethe disc surface density increases locally with radius. Theeffect of this is illustrated by equation (6), which shows thatif the surface density profile becomes positive then the coro-tation torque experiences a boost becauseα becomes neg-ative. This phenomenon has been dubbed a ‘planet trap’ by(60). Over longer viscous time scales such undulations in thedensity distribution will be smoothed out, but it is possiblethat planets may be able to migrate from one such densityfeature to another such that the next migration rate is sub-stantially reduced.

Significantly longer simulations are required in order toexamine the long term evolution of low mass planets em-bedded in stratified, global, turbulent disc models and to de-termine the efficacy of disc turbulence in modifying type Imigration. Furthermore, in order to build up a statisticallysignificant picture of the distribution of outcomes, a numberof such simulations will be required. In addition, the issueofhow dead zones affect the stochastic torques also needs to beexamined. It is worth noting that the simulation presented inthis section required approximately 55 days of wall-clockrun time on 256 processing cores. Clearly a full explorationof this problem remains a very challenging computationaltask.

5.2 High mass protoplanets in turbulent discs

The interaction of high mass, gap forming planets with tur-bulent protoplanetary discs has been considered in a numberof papers (61; 62; 57; 10). In all these works, the disc modelconsidered was non stratified such that the vertical compo-nent of gravity was neglected. It is well-known, however,that the flow of material in the vicinity of a giant planet de-velops a well-defined three dimensional structure, such thatfull 3D simulations are required to examine the details of thedisc-planet interaction.

Fig. 15 This figure shows a snapshot of the disc midplane density fora protoplanet with initial mass of 1/3 MJup embedded in a turbulentdisc. This snapshot corresponds to a time when the planet mass hasreached≃ 4 MJup by accreting gas from the disc.

Figure 15 shows a snapshot of the midplane density fora turbulent disc with an embedded 4 MJup protoplanet. Thedisc model in this case is the same as that described in sec-tion 5.1 with H/R= 0.1. The planet had an initial massMp = 1/3 MJup and was able to accrete gas from the disc.The gas accretion algorithm consists of the standard proce-dure of removing a fraction of the gas contained within onequarter of the planet Hill sphere at each time step. Whenmagnetic fields are present in the simulation this procedurecan lead to the formation of a magnetically dominant enve-lope forming around the planet. This problem was circum-vented by allowing the gas within the planet Hill sphere tohave a finite resistivity such that the field diffuses from thisregion as it builds up.

An initial planet mass ofMp = 1/3 MJup is below the gapopening mass according to the criterion expresed by equa-tion (18). As the planet accretes mass, however, gas is re-moved from the vicinity of the planet, and the growth ofthe planet mass allows tides to begin to open a gap oncethe Hill sphere radius begins to exceed the local disc scaleheight. This is in agreement with (57), who considered thetransition from fully embedded to gap forming planets us-ing local shearing box and global simulations of turbulentdiscs. These simulations showed that gap formation beginswhen the disc reponse to the planet gravity starts to becomenon linear. The combined Maxwell and Ryenolds stressesin simulations with zero–net magnetic flux (as consideredhere) typically give rises to an effective viscous stress pa-rameterαvisc ≃ 5×10−3, such that the viscous criterion forgap formation is satisfied when the criterion for non lineardisc response is satisfied. Hereαvisc is defined to beαvisc =

〈T〉/〈P〉, whereT is the total (Maxwell plus Reynolds) stress,

14

Fig. 16 The left panel shows the gas density at the disc midplane, andthe right panel shows contours of the quantityB2 at the disc midplane.

P is the thermal pressure, and angled brackets represent ver-tical and azimuthal averages.

Global simulations allow the net torque on the planet dueto the disc to be calculated and hence the migration time tobe estimated. Simulations presented in (61; 10) for massiveplanets indicate migration times of∼ 105 yr, in line withexpections for type II migration. The left panel of figure 16shows the torques per unit mass experienced by the planetdue to the disc as a function of time. Although gap forma-tion has not reached a state of completion (this is found torequire approximately 400 planet orbits in laminar disc runssuch as those presented in section 4), such that the torqueshave not reached their final values, this figure can be usedto estimate a lower limit on the migration time. We expressthis asτmig = j/(d j/dt), wherej is the specific angular mo-mentum of the planet. The specific torque att = 100 orbitsis d j/dt = 2× 10−5, (expressed in code units). The planetspecific angular momentumj =

√3 such that the migration

time corresponds toτmig = 3×104 years. Completion of gapformation will reduce the torques experienced by the planetconsiderably as the gas is pushed away from its vicinity bytides, causing the final migration time to significantly ex-ceed this value. But, as noted in section 5.1, the run timesassociated with these full 3D stratified disc models only al-low full 3D magnetised and stratified disc simulations to runfor ≃ 100 orbits at the present time.

A number of interesting features arise in simulations ofgap forming planets embedded in turbulent discs, which dif-ferentiate them from runs performed using laminar but vis-cous disc models. Comparing figure 15 with figure 9 showsthat the spiral arms in the turbulent disc appear less well-defined and more diffused than in the laminar models. Thispoint is reinforced by the left panel of figure 17, which con-tains a snapshot of the planet-disc system in ther–φ plane.The magnetic field topology is significantly modified in thevicinity of the protoplanet, as illustrated by the right panelin figure 17 which displays a snapshot of the quantityB2 inthe disc in the vicinity of the planet. It can be seen that the

magnetic field is compressed and ordered in the postshockregion associated with the spiral wakes, increasing the mag-netic stresses there. Accretion of gas into the protoplanetHill sphere causes advection of field into the circumplan-etary disc that forms there, creating a strong magnetic fieldin the circumplanetary disc, and this field links between theprotoplanetary disc and the circumplanetary disc, producingan effect of magnetic braking of the circumplanetary mate-rial. Indeed, a direct comparison between a simulation of a3 MJup protoplanet in a viscous laminar disc and an equiva-lent turbulent disc simulation presented by (10) suggests thatmass accretion onto the planet in a magnetised disc may beenhanced by this effect. Simulations are underway at presentto explore this effect within the 3D stratified model pre-sented in this section. It is worth noting, however, that theseglobal simulations are of modest resolution, especially whenconsidering complex physical processes occuring within theplanet Hill sphere, and more high resolution work needs tobe done to examine these issues in greater detail.

6 Summary

We have reviewed recent progress in the field of disc-planetinteractions in the context of orbital migration. This progresshas come about to a large degree from large scale two andthree dimensional simulations that have utilised the most upto date supercomputer resources. These have allowed veryhigh resolution simulations to be performed which exam-ine the detailed behaviour of the disc in the corotation re-gion, leading to a fuller understanding of the potential in-fluence of the corotation torque in modifying the migrationof low mass and fully embedded planets. Simulations havealso enabled the study of migration in non isothermal discsin which radiation transport is included, showing the impor-tance of relaxing the widely used assumption that the discgas can be treated as being locally isothermal. The inclu-sion of MHD effects in full 3D global simulations is now

15

Fig. 17 The left panel shows the torque experienced by the giant as a function of time. The right panel shows the evolution of the giant planetmass as it accretes gas from the disc.

possible, allowing the influence of magnetohydrodynamicturbulence on planet formation and migration to be studied.

Key questions for the future relate to how these variousphysical effects combine to change our current understand-ing of planetary migration. For example, the long-term ac-tion of corotation torques requires viscous diffusion to op-erate in the disc in order to prevent saturation. An importantquestion is how these corotation torques operate in discswhere the effective viscous stresses are provided by mag-netohydrodynamic turbulence ? Can the corotation torquesremain unsaturated within a dead zone ? These and other im-portant questions are the subject of on-going research. Theincreasing power of available computing facilities will con-tinue to facilitate the development of increasingly realisticdisc models, which will in turn lead to an increase in ourunderstanding of planetary formation and migration.

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1.00 1.05 1.10

-1.0 -0.5 0.0 0.5 1.0 x

-1.0

-0.5

0.0

0.5

1.0

y

0.96 0.98 1.00 1.02 1.04r

3.05

3.10

3.15

3.20

φ

0 5 10 15 20 25 30t (orbits)

-1000

-500

0

Γ/q2

α=5/2α=3/2α=1/2α=-1/2

0 10 20 30 40t (orbits)

-400

-200

0

200

400

Γ/q2

isothermal

ξ=0

ξ=-1.2

0 2 4 6 8 10 12 14t (orbits)

-1000

-500

0

500

1000

Γ/q2

isothermalρp=10-11

ρp=10-12

ρp=10-13