planet migration

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on the orbital evolution of low mass solids embedded in a

gaseous disk that orbits a star (Weidenschilling, 1977; Cuzzi

& Weidenschilling, 2006). The gas is subject to a radial

pressure force, in addition to the gravitational force of the

star. For a disk with smooth structural variations in radius,

this force induces slight departures from Keplerian speeds

of order (H/r)2r, where H is the disk thickness and isthe angular speed of the disk at radius r from the star. For athin disk (H r) whose pressure declines in radius, as istypically expected, the pressure force acts radially outward.

The gas rotation rate required to achieve centrifugal bal-

ance is then slightly below the Keplerian rate. Sufficientlyhigh mass solids are largely dynamically decoupled from

the gas and orbit at nearly the Keplerian rate, but are sub-

ject to drag forces from the more slowly rotating gas. The

drag leads to their orbit decay. Both the drag force and the

inertia increase with the size of the solid, R. The drag forceon the object increases with its area ( R2), while its iner-tia increases with its volume ( R3). In the high mass solidregime, the dominance of inertia over drag causes the or-

bital decay rate to decrease with object size as 1/R or withobject mass as 1/M1/3.

As we will see in the next section, the orbital migration

rate due to gravitational interactions between an object and

a disk increases with the mass of the object. There is thena cross-over mass above which the orbital changes due to

disk gravitational forces dominates over those due to drag

forces. For typical parameters, this value is much less than

an Earth mass, 1M. Consequently, for the purposes ofplanet migration, we will ignore the effects of aerodynamic

gas drag.

2.2 Ballistic Particle Model

As an initial description of gravitational disk-planet in-

teractions, we consider the disk to consist of noninteracting

particles, each having mass M much less than the mass of

the planet Mp. The description is along the lines of Lin& Papaloizou, 1979. The particles are considered to en-

counter and pass by the planet that is on a fixed circular

orbit of radius rp about the star of mass Ms. As a resultof the interaction, the planet and disk exchange energy and

angular momentum. This situation is a special case of the

famous three-body problem in celestial mechanics.

The tidal or Hill radius of the planet where planetary

gravitational forces dominate over stellar and centrifugal

forces is given by

RH = rpMp

3Ms1/3

, (1)

where radius RH is measured from the center of the planet.Consider a particle that approaches the planet on a circular

orbit about the star with orbital radius radius r sufficientlydifferent from rp to allow it to freely pass by the planet witha small deflection. This condition requires that the closest

approach between the particle and planet |r rp| to be

somewhat greater than a few times RH. The solid line inFig. 1 shows the path of a particle deflected by a planet

whose mass is 106Ms, Hill radius RH 0.007rp, andorbital separation r rp 3.5RH. The planet and particleorbit counter-clockwise in the inertial frame with angular

speeds p and (r), respectively. In the frame of the planet,the particle moves downward in the figure, since its angular

speed is slower than that of the planet ((r) < p for r >rp).

To estimate the angular momentum change of a parti-

cle like that in Fig. 1, we consider a Cartesian coordinate

system centered on the planet. The x axis lies along a linebetween the star and planet and points away from the star.The dashed line in Fig. 1 traces the path that the patricle

would take in the absence of the planet, while the solid line

shows the path in the presence of the planet. In both cases,

the particle path is generally along the negative y direction.The two paths are nearly identical prior to the encounter

with the planet (for y > 0). In that case, the particle veloc-ity in the frame that corotates with the planet is then given

by

u r((r) p)ey x rp ddr

ey 32

px ey. (2)

Upon interaction with the planet, the particle is deflectedslightly toward it. The particle of mass M experiencesa force Fx = GMMpx/(x2 + y(t)2)3/2. This forcedominantly acts over a time t when |y(t)| |x| and isFx GMMp/x2. From equation (2), it follows that theencounter time t |x/u| 1/p, of order the orbitalperiod of the planet, independent of x. As a result of theencounter, the particle acquires an x velocity

ux FxtM

GMpx2p

. (3)

The particle is then deflected by an angle

uxu

MpMs

rpx

3

RHx

3(4)

after the encounter (see Fig. 1).

To determine the change in angular momentum of the

particle, we need to determine its change in velocity along

the direction, that is the same as the y direction near theplanet. To determine this velocity change, uy , we ignorethe effects of the star during the encounterand apply conser-

vation of kinetic energy between the start and end of the en-

counter in the frame of the planet. The velocity magnitude uis then the same before and after the encounter, although the

direction changes by angle .1 Since the particle in Fig. 1

1We are applying the so-called impulse approximation. The approxima-

tion involves the assumption that the duration of the interaction is much

shorter than the orbital period and is only marginally satisfied here. Con-

sequently, the expressions for deflection angle and the torque T cannot

be determined with high accuracy in this approximation. They contain the

proper dependences on various physical quantities. But the approximation

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moves in the negative y direction, its pre-encounter y ve-locity is u and its post-encounter y velocity is u cos .We then have that the change of the velocity of the particle

along the y direction is

uy = u cos + u, (5)where u px is the velocity before the encounter (seeequation (2)). We assume that the perturbation is weak,

0. Thenet angular momentum change is given by equation (7) and

is quadratic in the planet mass. The quadratic dependence

does not lead to the correct dimensionless numerical coefficients of pro-

portionality for these quantities. An exact treatment of the particle torque

in the limit of weak perturbations exerted by a small mass planet is given

in Goldreich & Tremaine, 1982.

on Mp is a consequence of the deviations in torque betweenthe perturbed and the unperturbed paths, since the angu-

lar momentum change along the unperturbed path is zero.

These deviations involve the product of the linear depen-

dence of the force on planet mass with the linear depen-

dence of the path deflection on planet mass (equation (4))

and so are quadratic in Mp.As indicated by equation (7), the particle in Fig. 1 gains

angular momentum. The reason is that the path deflection

occurs mainly after the particle passes the planet, that is for

y < 0. The planet then pulls the particle toward positive

y, causing it to gain angular momentum. Just the oppositewould happen for a particle with r < rp, for x < 0 in equa-tion (7). The particle would approach the planet in the pos-

itive y direction, be deflected towards the planet for y > 0,and be pulled by the planet in the negative y direction, caus-ing it to lose angular momentum. This process behaves like

friction. The particle gains (loses) angular momentum if it

moves slower (faster) than the planet. The angular momen-

tum then flows outward as a result of the interactions. That

is, for d/dr < 0 as in the Keplerian case, a particle whoseorbit lies interior to the planet gives angular momentum to

the planet, since the planet has a lower angular speed than

the particle. The planet in turn gives angular momentum to

a particle whose orbit lies exterior to it.Fig. 4 shows the results of numerical tests of equation

(7). It verifies the dependence ofJ on x/rp and Mp/Ms.Departures of the expected dependences (solid lines) occur

when x 3RH. At somewhat smaller values of x, theparticle orbits do not pass smoothly by the planet.

We now apply equation (7) to determine the torque on

the planet for a set of particles that form a continuous disk

with surface density that we take to be constant in theregion near the planet. The particle disk provides a flux of

mass past the planet between x and x + dx

dM

uy dx

p xdx. (8)

We evaluate the torque Tout on the planet due to diskmaterial that extends outside the orbit of the planet from

r = rp + r to or x from r to , where r > 0.We use the fact that the torque the planet exerts on the disk

is equal and opposite to the torque the disk exerts on the

planet. We then have

Tout r

J

M

dM

dxdx, (9)

Tout = CTr4p2p

MpMs

2 rpr

3, (10)

where CT is a dimensionless positive constant of orderunity. The torque on the planet due to the disk interior to

the orbit of the planet from x = to x = r withr < 0 evaluates to Tin = Tout or

Tin = CT2pr

4p

MpMs

2rp

|r|3

. (11)

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The equations of motion for particles subject only to

gravitational forces are time-reversible. If we time-reverse

the particle-planet encounter in Fig. 1, we see that the ec-

centric orbit particle would approach the planet (both on

clockwise orbits) and then lose angular momentum (apply

t t in Fig. 2). What determines whether a disk particlegains or loses angular momentum? We have assumed that

the particles always approach the perturber on circular or-

bits. The particles in the disk re-encounter the planet on the

synodic period, of order rp/x planetary orbital periods. Butwe saw in Fig. 2 that the particles acquire eccentricity after

the encounter. For this model to be physically consistent,we require that this eccentricity damp on the synodic pe-

riod. The eccentricity damping produces an arrow of time

for the angular momentum exchange process that favors cir-

cular orbits ahead of the encounter as shown in Fig. 1, re-

sulting in a gain of angular momentum for particles with

r > rp.Equations (10) and (11) have important consequences.

The torques on the planet arising from the inner and outer

disks are quite powerful and oppose each other. This does

not mean that the net torque on the planet is zero because

we have assumed perfect symmetry across r = rp. Thesymmetry is broken by higher order considerations, such

as the radial density gradient. For this reason, migrationtorques are often referred to as differential torques.

Since the torques are singular in r, they are domi-nated by material that comes close to the planet. Conse-

quently, the asymmetries occur through differences in phys-

ical quantities at radial distances r from the planet. Con-sider for example the effect of the density variation in radius

that we have ignored up to this point. If we expand the disk

density in a Taylor expansion about the orbit of the planet,

we have that

out in 2r ddr

, (12)

where in and out are the surface densities at r = rp

r and r = rp + r, respectively. Consequently, for|d/dr| p/rp, we expect that the sum of the inner andouter torques to be smaller than their individual values by

an amount of order r/rp. Similar considerations apply tovariations in other quantities. That is, we have that the abso-

lute value of the net torque T on the planet is approximatelygiven by

|T| = |Tin + Tout|, (13) |Tin| |r|

rp, (14)

2pr

4p Mp

Ms2

rp|r|2

. (15)

Equation (15) must break down for small r, in orderto yield a finite result. For values of |r| 3RH, theparticles become trapped in closed orbits in the so-called

coorbital region (see Fig. 9). We exclude this region from

current considerations, since the torque derivation we con-

sidered here does not apply in this region. In particular, the

assumption that particles pass by the planet with little de-

flection is invalid in this region. Using equation (15) with

|r| RH, we obtain a torque

|T| 2pr4p

MpMs

4/3. (16)

Another limit on r comes about due to gas pressure.We have not yet described gas pressure effects, but will do

so in Sections 2.4 and 2.5. One effect of gas pressure is to

cause the disk to have a nonzero thickness H. The gas den-sity is then smeared over distance H out of the orbit plane.

Near the planet, the gas gravitational effects are smoothedover distance H. Distance r is then in effect limited to H.The torque is then estimated as

|T| 2pr4p

MpMs

2 rpH

2. (17)

Which form of the torque applies (equation (16) or (17))

to a particular system depends on the importance of gas

pressure. We expect equation (16) to be applicable in the

case that H < RH and equation (17) to be applicable oth-erwise. For typically expected conditions in gaseous proto-

stellar disks, it turns out that equation (17) is the relevant

one for planets undergoing (nongap) Type I migration. Themigration rate T /Jp with planet angular momentum Jp isthen linear in planet mass, since T is quadratic while Jpis linear in planet mass. Therefore, the Type I migration

rate increases with planet mass, as asserted in Section 2.1.

This somewhat surprising result that more massive planets

migrate faster is in turn a consequence of quadratic varia-

tion ofJ with planet mass in equation (7). This quadraticdependence occurs because the possible linear dependence

ofJ on planet mass vanishes due to the antisymmetry ofthe torque as a function of time along the unperturbed par-

ticle path, as discussed in the second paragraph following

equation (7).

Based on equation (17) with typical parameters forthe minimum mass solar nebula at rp = 5AU ( =150 g/cm2, = 1.8 108 s1, and H = 0.05rp), weestimate the planetary migration timescale Jp/T for a plan-etary core of mass Mp = 10M embedded in a minimummass solar nebula at 5AU as 4 105y. This timescale isshort compared to the disk lifetime, estimated as several

106y or the Jupiter formation timescale of 106y in thecore accretion model. The relative shortness of the mi-

gration timescale is a major issue for understanding planet

formation. Since the migration is generally found to be in-

ward, as we will see later, the timescale disparity suggests

that a planetary core will fall into the central star before it

develops into a gas giant planet. Research on planet migra-

tion has concentrated on including additional effects such

as gas pressure and improving the migration rates by means

of both analytic theory and multi-dimensional simulations.

A more detailed analysis reveals that the torque in equa-

tion (17) does provide a reasonable estimate for migration

rates in gaseous disks in the so-called Type I regime in

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which a planet does not open a gap in the disk. The model

provides considerable physical insight, but is crude. In fact,

it does not specify whether the migration is inward or out-

ward (i.e., whether T is negative or positive). The densityasymmetry about r = rp (see equation (12)) typically in-volves a higher density at smaller r, as in the case of theminimum mass solar nebula. This variation suggests that

torques from the inner disk dominate, implying outward mi-

gration, as was thought to be the case in early studies. But

this conclusion is incorrect. As we will see in Section 2.4,

inward migration is typically favored. We have not included

the effects of gas pressure in a proper way. A disk with gaspressure propagates density waves launched by the planet.

The analysis has only considered effects of material that

passes by the planet. In addition, there are effects from ma-

terial that lies closer to the orbit of the planet. It is trapped

in librating orbits of the coorbital region (see Fig. 9). This

region can also provide torques. We have also assumed that

the disk density is undisturbed by presence of the planet.

Feedback effects of the disk disturbances and gaps in the

disk can have an important influence on migration. Finally,

there are other physical effects such as disk turbulence that

should be considered.

2.3 Waves in 2D Gas Disks

In this section and the next section, we improve the par-

ticle disk description of the migration torque by consider-

ing a fluid disk, along the lines of Goldreich & Tremaine,

1979, 1980 and Meyer-Vernet & Sicardy, 1987 (see also

Shu, 1991). A fluid disk introduces some additional physi-

cal effects. Unlike free particle orbits, disk fluid streamlines

cannot cross each other. Tidal disturbances in a fluid disk

generally result in the propagation of waves, often referred

to as density waves.

The disk is taken to be thin so that its thickness

H =

c

r, (18)where c is the gas sound speed. We model the disk astwo-dimensional and utilize a cylindrical coordinate system

(r, ) centered on the star. The disk has a density distribu-tion and velocity field (u, v). The equations for massconservation, radial, and azimuthal force balance in the in-

ertial frame are

t+

(ru)

rr+

(v)

r= 0, (19)

u

t+ u

u

r+

v

r

u

v

2

r= 1

p

r+ fvr

r, (20)

and

v

t+ u

v

r+

v

r

v

+

uv

r= 1

r

p

+ fv

r, (21)

where is the gravitational potential due to the star andplanet, p is the gas pressure, and fv is the viscous force perunit disk mass to model the effects of turbulence. For the

purposes of describing waves in this section, we will ignore

viscous forces, as well as gas self-gravity.

We now consider the linear departures from the unper-

turbed state consisting of an axisymmetric disk that orbits

the central star. The planet is taken to be on a circular orbit

about the star. We express these perturbations as complex

quantities for which it is implicit that the real part should be

taken. We consider a Fourier decomposition of the gravita-

tional potential in angle and time given by

(r,,t) = m

m(r)exp[im( pt)], (22)

where m is a nonnegative integer, m is real, and p isthe orbital frequency of the planet. Physical quantities are

expressed as

(r,,t) =m

m(r)exp[i(m( pt)], (23)

u(r,,t) =m

um(r)exp[im( pt)], (24)

v(r,,t) =m

vm(r)exp[im( pt)], (25)

p(r,,t) = m

pm(r)exp[im(

pt)]. (26)

We assume here that in lowest order the axisymmetric den-

sity component 0(r) is equal to the unperturbed disk den-sity. That is, the presence of the planet does not substan-

tially 0. We return to this point in Section 2.7. We alsoassume that in the lowest order the axisymmetric azimuthal

velocity v0 is equal to the Keplerian speed K(r)r. Pres-sure pm is assumed to involve an imposed axisymmetriclocally isothermal (density independent) temperature varia-

tion. We then have that

pm(r) = c2(r)m(r). (27)

Such a temperature variation arises in the case of an opti-

cally thin disk that is heated by a central star. This tempera-

ture distribution corresponds to a gas sound speed described

as c(r).For each m > 1, the linearized forms of equations (19) -

(21) are

im + d(rum0)rdr

+imvm0

r= 0, (28)

ium 2vm = 10

dpmdr

dmdr

, (29)

and

ivm + 2Bum = impmr0

immr

, (30)

where = m(p ) is the Doppler shifted frequency,B(r) = +1/2r(d/dr). For a Keplerian disk, B = /4.

We seek solutions to equations (28) - (30) by applying

the WKB or tight-winding approximation. The approxima-

tion is that perturbed quantities vary much more rapidly in

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the radial direction than in the azimuthal direction. We re-

gard azimuthal wavenumber m to be of order unity. Theapproximation is that |dym/dr| m |ym|/r, where ym ism, um, vm, or pm. From eqs (29) and (30) it follows that|um| |vm|. Under these approximations, equations (28) -(30) simplify in lowest order to

im + 0 dumdr

= 0, (31)

i(2 2)

um = c2

0

dmdr

dmdr

+2mm

r, (32)

where equation (32) resulted from combining equations

(29) and (30).

If we ignore the driving terms involving m in the aboveequations, we determine the behavior of free waves having

frequency mp. We express

ym(r) = ym exp[i

rkr(r

)dr], (33)

where ym is um or m, kr(r) is the radial wavenumber,and ym on the right-hand side is taken to be constant. Weassume |kr|r m (tight-winding approximation) and ap-ply equation (33) in equations (31) and (32) with m = 0.

We then obtain the following dispersion relation

2 2 = k2rc2, (34)

where is the epicyclic oscillation frequency defined by2 = (1/r3)(dr42/dr) = 4B. For a Kepleriandisk, = . The dispersion relation determines radialwavenumber kr(r) as a function of frequency (r). Thedispersion relation describes sound waves that are modified

by the presence of rotation and shear.

Dispersion relation (34) shows that the waves are prop-

agating (kr is real) for || > and evanescent otherwise.Equation (34) tells us that kr

/c

H1 for m of order

unity.2 Therefore, the WKB approximation, krrp m,then requires that H rp, as stated in equation (18). Thatis, the disk is thin and the gas is cold. For a given m, eachradius where

kr(rm) = 0 (35)

is a radial wave turning point. The wave turning points

coincide the locations of resonances, called Lindblad res-

onances. The turning points (or resonances) occur for

= , (36)

where we follow the notation throughout that the upper

sign denotes the condition for the inner Lindblad resonance

(ILR) and the lower sign for the outer Lindblad resonance

2Since the radial wavelength is comparable to the disk thickness, 3D effects

may be important. If the waves have a locally isothermal equation of state,

the 2D assumption is valid. But if not, e.g., they are adiabatic or if the disk

temperature varies with height from the orbit plane, then the waves have a

3D character (Lubow & Pringle, 1993; Lubow & Ogilvie, 1998). In such

cases, the wave propagation and damping properties are quite different.

(OLR). The ILR occurs for r < rp and the OLR for r > rp.For a Keplerian disk, the turning point locations rm are de-termined by

K(rm) =mp

m 1 (37)or

rm = rp

m 1

m

2/3. (38)

Fig. 6 plots rm as a function of m with lighter dots. Forlarge m, the resonances approach the orbital radius of theplanet,

|rm

rp

| 2rp/(3m).

The sign of kr determines the direction of wave propa-gation. The radial group velocity of these waves is given

by

cg =d

dkr, (39)

=krc

2

. (40)

Since is negative for an ILR and positive for an OLR,it then follows that for kr > 0, the waves propagate awayfrom the orbit of the planet (cg < 0 at an ILR and cg > 0 atan OLR). Such waves also have the property that they are

trailing. That is, their wave fronts bend towards negative with increasing r (see Fig. 5). Along a path of constantwave phase krr + m = 0, so d/dr = krr/m < 0for kr > 0. Leading waves have the opposite properties:they have kr < 0 and propagate towards the orbit of theplanet.

Up to this point we have disregarded the effects of the

driving by the potential m. The key physical point is thatthe propagating waves generally do not interact with the po-

tential because their wavelengths are short compared to the

radial variations of the potential. To see this, consider the

tidal torque per unit disk radius on the disk

dTddr

(r) = r

2

0

(r, )

d, (41)

= irm

m m(r) m(r), (42)

=m

dTmdr

(r), (43)

where we applied equations (22) and (23) in going from

equation (41) to (42) and used the fact that m is real. It isimplicit that the real part of equation (42) should be taken.

Each term in the sum involves the product mm. Butin the wave propagating region, m(r) is rapidly varying in

radius for a cold disk, since |kr|r 1, while m(r) is non-oscillatory and varies more slowly in r. Consequently, overa finite radial interval r 2/|kr| in the wave propagat-ing region, the net torque

r+rr (dTm/dr

) dr is in somesense small. However, near the Lindblad resonance, where

kr is small (see eq (35)), the radial oscillations are not rapidand this integral develops a more substantial contribution.

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We then expect that torques develop near r = rm that weevaluate below.

We determine the solution for equations (28) - (30) in

the vicinity of the resonance located at radius rm, whereequation (35) is satisfied, by applying a Taylor expansion

about r = rm2 2 Dmx, (44)

where x = (r rm)/rm and Dm = rd(2 2)/drevaluated at r = rm. For a Keplerian disk, using = and equation (37), we have Dm = 3m(rm)p =

3m22p/(m

1).

We reconsider equations (31) and (32) to find a solutionfor driven waves that this valid near a Lindblad resonance.

We eliminate m in favor ofum, again apply the approxi-mation |dm/dr| m|m|/r together with equation (44)to obtain that

c2

r2m

d2umdx2

umDmx = i(rm)m(rm)rm

, (45)

where

m =2m

m r dm

dr. (46)

The left-hand side of equation (45) can readily be seen to be

equivalent to the dispersion relation equation (34) by recog-nizing d/dx ikrrm and Dmx 2 2. The right-hand side introduces the forcing. In addition, from equation

(29) and (30), it follows from solving for um and vm nearthe resonance that

vm = 2iBum

(47)

= ium2

. (48)

We seek solutions with the property that the waves prop-

agate away from and not towards a resonance. That is, we

impose a causality condition that waves are emitted from a

resonance. These waves behave as trailing waves far fromthe resonance, as we saw in equation (40). That is, we re-

quire

kr(r) = i d log um(r)dr

> 0 (49)

in the wave propagating region away from the resonance.

The solution with these properties is

um = Cm[Ai x

w

i Gi

x

w

], (50)

with

w3 = c

2

r2m|Dm| (51)

and

Cm = mrmDmw , (52)

where again the upper (lower) signs refer to inner (outer)

Lindblad resonances. Airy function Ai(x) satisfies Ai(x)

xAi(x) = 0 and Airy function Gi(x) satisfies Gi(x) xGi(x) = 1/. These functions are described inAbramow-icz & Stegun, 1972.

2.3 Lindblad Torques

To determine the torque on the disk for a given m, weapply the equation of mass conservation (28) and eliminate

vm by equation (48), to express m near the resonance interms ofum as

m(r) = i

d(rum0)

rdr mum0

2r

. (53)

We then obtain from equations (41) - (43) that

Tm =

dTm

dr(r)dr, (54)

= im

rm(r)m(r)dr, (55)

= m

0m

umdr, (56)

where the integrals extend over all r. Equation (56) is ob-tained from equation (55) by applying the expression for

m given by equation (53), integrating by parts, and drop-ping the surface term. We used the fact that 2 = 2 nearthe resonance and applied m given by equation (46). If weassume 0, m, , and are nearly constant in the regionnear the resonance, we obtain from equations (50) and (56)

that

Tm = 2m2mDmw

Ai(x/w)dx, (57)

Tm = 2m2m

Dm . (58)

Fig. 7 shows that the torque contributions to the integral in

equation (57) come from a radial region of order wrp rp.There is no dissipation in this model. One might ex-

pect the system to be time-reversible, so that the net torque

on the planet is zero. But, we have imposed an arrow of

time by the causality condition (49) that waves are only

emitted at the resonance, implying that only trailing waves

are present. In a dissipationless system of finite size, the

trailing waves would reflect at the disk center and outer

edge as leading waves that propagate back to the resonance

(Fig. 5). These waves would cancel the torque from the

trailing waves, resulting in a zero net torque. Consequently,

the model implicitly assumes some dissipation somewhere

in the disk prevents leading waves from returning to the res-

onance. Dissipation is needed for gas to produce a torque,

as it was needed in the case of ballistic particles in Section

2.2.

For a given m value there are ILR and OLR contributionsfrom the gas interior and exterior to the orbit of the planet,

respectively. We denote these contributions with subscripts

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in and out. The torque on the planet T is minus the sum ofthe torques on the disk

T = Td = m

(Tm,in + Tm,out). (59)

Tm,in is Tm evaluated at the ILR and similarly Tm,out is Tmevaluated at the OLR.

Since Dm > 0 at the ILRs, the torques given by equation(58) on the planet due to the inner disk, Tm,in, are posi-tive, i.e, outward. The direction changes to inward for the

outer disk containing OLRs, since

Dm < 0. This behavior

is the same as we found for particle disks. Notice that theresonant torque T has a similar form to the particle torquesin equations (10) and (11). Both depend on the square of

the planet mass Mp (equation (58) involves 2m that de-

pends on M2p ). For a Keplerian disk, it can be shown thatthe correspondence between the particle torque and the gas

torque on the planet is nearly exact. Notice that gas torque

is independent of the gas sound speed in the Keplerian case,

since Tm in equation (58) has that property. However, theresonance description for the gas disk torques is more pre-

cise than the particle description and allows us to include

the nonKeplerian effects due to gas pressure. In this way,

the sign and magnitude of the net torque can be determined.

For large m, the resonances lie close to the planet (lightdots in Figs. 6 and 8) and the torque T becomes singular.The singularity is mainly due to the 2m factor in equation(58) (m/Dm is nearly independent ofm). This singularityis related to the ones we encountered for Tout and Tin inequations (10) and (11), since m 1/r. Since the physi-cal torque cannot be infinite, something must be wrong with

our description. One issue is that we have approximated

the disk as being two-dimensional. A 3D disk would have

thickness H = c/. Within a radial distance r = r rpof order H, we expect this description to break down, sincethe gas will be spread away from the planet in the direction

perpendicular to the orbit plane. There is, however, anothereffect that yields a finite torque in the large m limit for a2D disk. The dispersion relation (34) can be shown to have

corrections due to azimuthal wave propagation. This ex-

tended WKB approximation involves the replacement of

k2r k2r + k2 , where k2 = m2c2/r2 (Artymowicz, 1993a).We then have that

2 2 = k2rc2 +m2c2

r2. (60)

Recall that the site of wave excitation occurs where the

wavelength is relatively large, equation (35). For large m,the wave turning point (where kr(rm) = 0) in equation (60)

becomes = mc/rp, since || . For a Kepleriandisk, we have

rm = rm rp crp

H, (61)

for large m. So unlike the case of the standard WKB ap-proximation (equations (34) and (36)), the resonances do

not get arbitrarily close to the planet with increasing m, asindicated in Fig. 6. As a consequence, m does not divergeas m goes to infinity, as seen in Fig. 8. Instead, it decaysexponentially for m rp/c and the total torque T is fi-nite. 3D effects due to the disk thickness produce a similar

drop-off. This means that m is effectively limited to

m mcr =prp

c=

r

H. (62)

This critical value mcr is referred to as the torque cutoff.The m limit implies that the contributing resonances come

from a region no closer to the planet than |r| H, as weapplied in our torque estimate equation (17).

2.4 Differential Lindblad Torques

In this section, we describe the net torque on the planet

that results from the effects of the opposing torque contri-

butions from the ILRs and OLRs, along the lines of Ward,

1997. A more rigorous and complete way to determine the

torque involves considering the equations of motion in 3D

and improving the validity of the equations near r = rp. Inthis section, we will describe the main qualitative features

of the torque that remain valid when more accurate treat-

ments are applied.We now describe the main reason why planets migrate

inward. At the same distance d > 0 inside and outside theorbit of the planet, quantity 2m(rp + d) outside the planetorbit is slightly larger than 2m(rp d) inside. This asym-metry is due to the circular geometry of the orbit. Quantity

m, defined in equation (46), contains a linear combina-tion of radial and azimuthal derivatives of the gravitational

potential m that are multiplied by radius r. It is the mul-tiplication by r that causes 2m(rp + d) >

2m(rp d).

In addition, the resonance radius rm at the OLR is slightlycloser to the planet orbital radius rp than is rm at the ILR.This effect further strengthens torques at OLRs. It can also

be seen that the effects of factor D1m favor the Tm,out overTm,in. As a result, inward planet migration is the suggestedoutcome.

These geometric effects are called curvature effects.

They are of course not the only influences on the torque. In

the remainder of this section, we will more fully describe

various effects on migration.

We consider a simplified disk model in which c(r) istaken to be constant in radius and the unperturbed density

distribution is taken to be a power law in radius with con-

stant defined by

0(r)

r. (63)

We consider some contributions to the gas disk rotation rate

2 = 2K + 2pr +

2pg, (64)

which are respectively due to the gravity of the star (Ke-

plerian rotation), the axisymmetric component of the disk

pressure force, and the axisymmetric component of planet

8


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gravity. The effects of disk self-gravity are ignored. The

planet and disk are subject to nearly the same axisymmetric

gravitational forces at the location of the planet. But, the

disk feels the gravitational force of the planet that is not felt

by the planet (no self-force). The disk feels pressure, while

the planet does not. The angular frequency of the planet is

the Keplerian rate

p = K(rp). (65)

The torque sum T in equation (59) is dominated by con-tributions having m

mcr = prp/c. We consider an

expansion of resonance radius rm and orbital frequencysquared 2(rm) in powers of = 1/mcr 1 of the form

rm = rp + rm,1 + 2rm,2 + . . . , (66)

2K(rm) = 2K +

d2Kdr

rm,1 + . . . , (67)

2pg(rm) = 2 q3rp

2rm,12p + . . . , (68)

2pr(rm) = 2 2p + . . . , (69)(r) = K + 1 + . . . , (70)

where all functions ofr on the right-hand side of the aboveare evaluated at the planets orbital radius rp. The scaled

planet to star mass ratio q3 = Mp/(3Ms) is assumed to beof order unity.

Substituting expansions (66) - (70) into equation (60),

we obtain

rm,1rp

= 23

1 + (m/mcr)2, (71)

rm,2rp

= 19

m2

9m2cr

3 , (72)

where quantity describes radius changes that are equaland opposite about r = rp.

We see that the lowest order radius shifts rm,1 are equal

and opposite. As we described above, the curvature asym-metry favors inward migration for resonances located at

equal distances inside and outside radius rp. The torquecontributions due to these resonance position and angular

velocity shifts are obtained by applying equations (58) and

(59).3 We first consider the differential torque contribution

that does not involve the density factor in equation (58).We denote this differential torque as Tm,cpg, since it in-volves curvature, pressure, and planet gravity. After con-

siderable algebra, it follows that Tm,cpg due to the ILR andOLR for some 1 m mcr is given by

Tm,cpg =

mcrr

4p

2pMp

Ms2

f1

|rm,1|rp

+ f2

|rm,1|rp

rm,2

rp

(73)

3The derivation of equation (58) assumed that 2 = 2 at the Lindlabd

resonances. There are corrections to this condition and therefore the torque

in the extended WKB approximation (equation (60)). For simplicity, we

ignore such effects here.

and

rm,2 = 19

m2

9m2cr

3, (74)

where f1 and f2 are dimensionless, order unity, positivefunctions of|rm,1|/rp that contains a curvature term. Quan-tity rm,2 is the mean shift of resonance locations for theILR and OLR. The first term, -1/9, is a curvature term,

while the next two terms involve pressure. Equation (73)

shows that the first order antisymmetric shift rm,1 providesa net torque at the same level as the second order symmet-

ric (mean) shift, rm,2. Since typically > 0, we have

that rm,2 < 0 and both terms on the right-hand side ofequation (73) contribute to inward migration. The effect of

a negative rm,2 can be understood as moving the OLRcloser to the planet and the ILR further away. These shifts

in turn cause the OLR torque to dominate and provide a

negative torque contribution (see equation (10)). The ef-

fects ofpg are contained in the term. But since doesnot contribute to the mean shift rm,2, the planet gravitydoes not influence migration, provided that Mp 3Ms.

We now consider the contribution to the differential

torque due to the density factor that appears in equation

(58). For a given m value, the net torque depends on thedensity change between the ILR and OLR that is related to

. This torque can be seen to be

Tm, = C

mcr

|rm,1|rp

|Tm,in|, (75)

where constant C is positive. For > 0 as is typically thecase, torque Tm, is positive. It can be shown that torqueTm, is be comparable to and opposite to Tm,cpg, but some-what weaker. As a result, the differential (net) Lindblad

torque is typically negative.

Notice that by summing equations (73) plus (75) over m,we have that the differential torque

T m

Tm,cpg (76)

mcrTmcr,cpg (77)

m2crr4p2p

MpMs

2(78)

rp

H

2r4p

2p

MpMs

2, (79)

which agrees with equation (17). A more careful evaluation

of the sum can be performed to provide the detailed depen-

dence of T on . We will reconsider this issue in Section2.6.

2.5 Coorbital Torques

Thus far we have considered torques that arise from gas

that passes the planet in the azimuthal direction (the y di-rection in Fig. 1). These torques can be described by Lind-

blad resonances, as we saw in Section 2.4. Gas that resides

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closer to the planet, |r rp| 3RH, in the so-called coor-bital region, does not pass by the planet. Instead, it follows

librating orbits in the corotating frame of the planet as seen

in Fig. 9. We consider orbits that nearly fully circulate, the

so-called horseshoe orbits. They approach the planet at both

< p and > p, as seen in the figure. As the gas ap-proaches the planet, it gets pulled by the planet from r > rpto r < rp or vice versa. This change in angular momen-tum of the gas in turn causes a torque to be exerted on the

planet. Away from the region where the orbit transitions

between different radii, the gas follows an approximately

circular Keplerian path.For a disk with no dissipation and a planet on a fixed cir-

cular orbit, the net torque on the planet is zero. The reason

is that the gas follows periodic orbits in the frame of the

planet, so there is no change in angular momentum of the

gas over a complete period of its motion. Whatever angular

momentum is gained by a gas element as it changes from

the inner radius ri to outer radius ro is lost when it laterencounters the planet and shifts from ro to ri. So althoughgas on horseshoe orbits can come close to the planet, the

symmetry limits the torque that it exerts on the planet.

To analyze this situation more carefully, consider the la-

beled regions in Fig. 9. We describe the disk as consisting

of a gas fluid, but will ignore the effects of gas pressure,and follow the approach of Ward, 1991. Gas loses angular

momentum in going from position o+ to i+. This angularmomentum is continuously gained by the planet. Similarly,

the planet continuously loses angular momentum from the

gas that passes from i to o. The gas spends a relativelyshort time in transition between ri and ro compared to thetime it spends between encounters that is of order the syn-

odic timescale rp/((ro rp)p). Between encounters,the gas may experience the effects of turbulent viscosity. It

acts to establish a characteristic density profile 0(r) thatwould be present in the absence of the planet. As the gas

re-encounters the planet, it does so with the turbulence-

enforced background density. Such effects give rise to anet torque on the planet.

The net torque on the planet due to a streamtube that

extends on both the < p and > p sides of the planetis given by

Tco = (M+ M) dr2

drw, (80)

where w = ro ri is the radius change along the stream-tube, M is the flux of mass passing the planet for < pand analogously for M+. Both mass fluxes are defined tobe positive and are approximately given by

M M rp((ri) p)r, (81)

where r is the radial width of the streamtube away fromthe planet, on the circular portion of the orbit. The ratio

of the mass fluxes depends on the density just ahead ofthe encounter, with density (ri) at position i and density

(ro) at position o+. The ratio is given by

M+

M (ro) B(ri)

(ri) B(ro), (82)

where the ratio of B values comes into play because of achange in the area of fluid elements between ri and ro. Wethen obtain an expression for the torque on the planet due

to the streamtube as

Tco M

M+

M 1

prpw, (83)

2prpw2

B

B

r, (84)

2pw3d log (/B)

d log rr, (85)

where = oi and B = BoBi. Integrating overall streamtubes to width w, we estimate the coorbital torqueas

Tco 2pw4d log(B/)

d log r, (86)

where we now interpret w as the width of the coorbital re-gion that is a few times RH.

Quantity B/ is sometimes called the vortensity, sinceB is related to the vorticity or curl ofre . The coorbitaltorque then depends on the gradient of the vortensity. A

time-reversible system (no dissipation), with arbitary ini-

tial conditions would evolve towards a state in which the

vortensity is constant in the coorbital region. This process

occurs through phase mixing on the libration timescale

rp/(pw) (see Fig. 10). The coorbital torque drops tozero or is said to be saturated. As discussed above, main-

taining a density gradient and a nonzero torque could be

accomplished by the effects of turbulent viscosity.

We compare the coorbital torque with the net Lind-

blad torque for a particle (pressureless) disk as we dis-

cussed in Section 2.2. Taking w RH in equation(86), we see that the unsaturated coorbital torque (taking

|d log(B/)/d log r| 1 and nonzero) is comparable tothe differential Lindblad torque in equation (16). It can be

shown that this is also true for a disk where pressure effects

are important, where H > RH. For such a disk, the coor-bital torque is generally of the same order as the differential

Lindblad torque, equation (17). The direction of the coor-

bital torque contribution depends on the sign of the vorten-

sity gradient. For the minimum mass solar nebula model,

r3/2 and B r3/2. Consequently, the coorbitaltorque is zero in that case. For smaller values of < 3/2,the torque is positive. For the turbulent viscosity to prevent

torque saturation, the viscous timescale across the coorbital

region should be shorter than the libration timescale. This

constraint can be translated into a condition on the minu-

mum magnitude of the turbulent viscosity required for a

nonzero corotation torque (see equation (89)).

With pressure effects, the gas communicates distur-

bances with neighboring gas. But the gas is incapable of

10


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launching a propagating wave because the region near rpis evanescent, as seen in Fig. 5. Instead, the acoustic dis-

turbances remain trapped within a radial distance H of theplanets orbit. The trapping of disturbances near the coro-

tation radius rp limits the amount of angular momentumthat a planet can gain. But in the case of a viscous disk,

mass and angular momentum in the coorbital region can be

transferred to the remainder of the disk by viscous torques.

The corotation torque can then act without saturation.

Although the coorbital torque is of the same order as the

differential Lindblad torque, more detailed calculations as

described in the next section show it typicallydoes not dom-inate over Lindblad torques.

2.6 Type I Migration Torques

Thus far, our estimates of migration rates have assumed

that the axisymmetric gas surface density is not perturbed

by the presence of the planet. This regime is sometimes

called the Type I case of migration.

More detailed 3D linear analytic calculations of the Type

I migration rates have been carried out by Tanaka et al,

2002. They assumed that the gas sound speed is constant

in radius. For the case of saturated (zero) coorbital coro-

tation torques, where only differential Lindblad torques areinvolved, the migration rate is given by

T = (2.34 0.10 ) p 2p r4p

MpMs

2 rpH

2, (87)

where is given by equation (63). The torque on the planetresulting from the action of both Lindblad and (unsaturated)

coorbital corotation torques is given by

T = (1.36 + 0.54 ) p 2p r4p

MpMs

2 rpH

2. (88)

These migration rates are consistent with the estimate in

equation (79). Notice that the Lindblad-only torque, equa-tion (87), contains a small coefficient of . The reason isthat there is a near cancellation of the inward migration ef-

fects of the pressure in equation (73) (in rm,2) with theoutward migration effects of the density in equation (75).

The differential Lindblad torque is then mainly due to the

curvature effects. In going from equation (87) to equation

(88), the effects of the coorbital torque reduce the inward

migration rate for < 1.5 and nearly vanish for = 1.5.4

This behavior is consistent with equation (86) for which

the coorbital torque is positive for < 1.5 and is zero for = 1.5.

Based on scaling arguments, the condition on the turbu-

lent viscosity for the coorbital torque to be effective (unsat-

urated), as discussed in Section 2.5, is given by

MpMs

3/2 rH

7/2. (89)

4The small nonzero coorbital torque at = 1.5 is due to 3D effects.

where the dimensionless disk viscosity parameter in thestandard disk model (Ward, 1992). For disk parameters = 0.004 and H/r = 0.05, this constraint implies that forplanets of order 10 M or greater, the corotation torquesshould be saturated (small) and equation (87) should be ap-

plied.

Nonlinear 3D hydrodynamical calculations have been

carried out to test the migration rates, under similar disk

conditions used to derive the analytic model. Figs. 11 and

12 show that the migration rates agree well with the expec-

tations of the theory.

We examine the comparison between simulations andtheory in more detail by comparing torque distributions in

the disk as a function of disk radius. We define the distribu-

tion of torque on the planet per unit disk mass as a function

of radius as dT/dM(r) = 1/(2r(r)) dT/dr(r). Fig. 13plots (Ms/Mp)

2dT/dM as a function of radial distancefrom the planet based on 3D simulations. The distributions

show that the torque from the region interior (exterior) to

the planet provides and positive (negative) torque, as pre-

dicted in equations (11) and (10) for a particle disk and as

shown in Section 2.3 for a gas disk. Also, the integrated

total torque is negative, as expected. The theory predicts

that the torque density peak and trough occur at distance

from the planet r rp = r H, where the torquecut-off takes effect as indicated in equation (61). For the

case plotted in Fig. 13 that adopts H = 0.05r, the predictedlocations agrees well with the locations of the peaks and

troughs in the figure. The theory in Section 2.4 and 2.5 also

predicts that for a fixed gas sound speed, the shape of the

scaled torque density distribution (Ms/Mp)2dT/dM(r) isindependent of planet mass. The reason is that the width of

each contributing resonance is independent of planet mass,

but depends on the sound speed (see equation (51)). In ad-

dition, the set of of contributing resonances (range ofm val-ues) is also independent of planet mass (see equation (62)).

Furthermore, the overall torque scales with the square of the

planet mass. As seen in Fig. 13, these expectations are wellmet for the two cases plotted.

2.7 Disk Response

The various torque expressions (e.g., equations (17),

(73), and (88)), contain a density term. This density is the

lowest order density at the radius of the planet. Up to this

point, we have assumed that this density is the unperturbed

disk density, the disk density that occurs in the absence of a

planet. If the planet mass is sufficiently small or the level of

turbulent viscosity is sufficiently large, then the presence of

the planet does not substantially modify the density distri-

bution. But these conditions may be violated and the den-

sity can be affected.

In particular, some our approximations, such as in going

from equation (56) to equation (58), were reliant on set-

ting the lowest order density distribution 0 to the unper-turbed density distribution. Modifications to 0 caused bythe planet need not be small and can produce a significant

11


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change in the migration torque.

To analyze the disk response, we consider equations (19)

and (21) and include the effects of shear viscosity with kine-

matic viscosity = cH by taking

fv = Gr

, (90)

G = 2r3 ddr

, (91)

and we will ignore fvr . We write the azimuthal velocity as

v = (r)r + v

, (92)

where velocity v denotes the nonaxisymmetric departuresfrom circular rotation that are assumed small compared to

the axisymmetric circular velocity r.Multiplying equation (21) by r, applying equation

(19), and integrating in , we obtain to high accuracy thetorque equation

Mdr2

dr=

Tdr

Fwr

Gr

, (93)

with

M = r20

ud, (94)

Tdr

= r20

d, (95)

Fw = r20

20

uvd, (96)

where M is the mass flux through the disk, dTd/dr is thetorque per unit radius on the disk, and Fw is the flux of an-gular momentum carried by waves. By integrating equation

(19) in we have that

0t = 1r

Mr . (97)

We briefly consider some specific cases of interest be-

low.

a) Unperturbed viscous disk

Consider the case of a steady state viscous disk in which

no planet is present, Fw = Td = 0. For a steady state,equation (97) requires that M is constant. From equation(93), we have that Mr2 = G. For a Keplerian disk, wethen recover the standard result that M = 3(r)0(r).

b) Disk with planet and little dissipation

Consider the case that there is no dissipation in some re-

gion of space where waves are generated (i.e., near the res-

onance). In that region M = 0, since there is reversibility.By equation (93) with G = 0 we have that Td = Fw. Thatis, the disk torque exerted near the planet is transferred by

the wave flux Fw to some other region of the disk. Wher-ever the wave damps, there is an imbalance between Td andFw, resulting in a nonzero mass flux M that will generallyresult in a density change (equation (97)). This situation is

reminiscent of the case of ocean waves. They are gener-

ated by wind far from land, but undergo final decay when

they break at the shore. Notice that the planet obtains its

torque from gas near the Lindblad resonance, independent

of where the waves damp and the disk gets torqued. Re-

call that it is necessary that the waves damp somewhere for

there to be a nonzero Lindblad migration torque.

c) Disk with strong dissipation and tidal forces

In this case, the waves damp immediately by the strong

dissipation and therefore Fw = 0. We have seen that disk-planet torques from an outer disk cause an inward torque

on the planet and an outward torque on the disk. Similarly,the interactions cause an inward torque on the inner disk.

As a result of these torques, material is pushed away from

the orbit of the planet. In the case that the planet produces

strong tidal force, we may expect that a gap is created near

the orbit of the planet. Consequently, we expect that near

the planet M = 0 and Td = G in equation (93).

2.8 Type II migration

Type II migration occurs for case c above when the re-

gion near the planet is strongly depleted of gas and a gap

forms (Lin & Papaloizou, 1986). Gap formation occurs for

Td = G in equation (93). Tidal forces on the disk interioror exterior to the planet are estimated by equation (10) with

r H, as we argued at the end of Section 2.3. For agiven level of disk turbulent viscosity, the gap opening con-

dition Td = G becomes a constraint on the planet mass.The condition is estimated as

MpMs

40

r2p

1/2Hrp

3/2. (98)

For disk parameters = 0.004 and H/r = 0.05, the pre-dicted gap opening at the orbit of Jupiter occurs for planets

having a mass Mp 0.2MJ. This prediction is in good

agreement with the results of 3D numerical simulations (seeFig. 14).

In addition to the above viscous condition, an auxiliary

condition for gap opening has been suggested based on the

stability of a gap. This condition is to preclude gaps for

which steep density gradients would cause an instability

that prevents gap opening. This condition, called the ther-

mal condition, is given by rH H (Lin & Papaloizou,1986). The critical mass for gap opening by this condition

is given by

MpMs

3

H

rp

3. (99)

For H/r = 0.05, this condition requires a larger planetmass for gap opening than equation (98) for 0.01.Even if both the thermal and viscous conditions are satis-

fied, a substantial gas flow, M 3(r)0(r), may oc-cur in the presence of a gap in certain circumstances (Arty-

mowicz & Lubow 1996).5

5Near the end of Section 2.7 we asserted that M = 0, since the density

12


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The migration rate of a planet embedded in a gap is quite

different from the Type I (nongap) case that we have already

considered. A planet that opens a gap in a massive disk, a

disk whose mass is much greater than the planets mass,

would be expected to move inward, pushed along with the

disk accretion inflow. The planet simply communicates the

viscous torques across the gap by means of tidal torques

that balance them. The Type II migration timescale is then

of order the disk viscous timescale

tvis r2p

r2p

cH

rpH

2 1

p, (100)

which is 105 years for = 0.004, H = 0.05rp, andp = 2/12 y

1. Therefore, the migration timescale can

be much longer than the Type I migration timescale for the

higher mass planet that open gaps, as is found in simulations

(Mp > 0.1MJ in Figs. 11 and 12). However, this timescaleis still shorter than the observationally inferred global disk

depletion timescales 106 107 years. The actual mi-gration rate may be somewhat smaller, in order to explain

the abundant population of observed extrasolar giant plan-

ets beyond 1AU (Ida & Lin 2008). Such retardation is also

required to retain terrestrial planets in habitable zones that

are located inside the likely formation regions of gas giants

that may be beyond the snow lines.In practice, the conditions for pure Type II migration are

unlikely to be satisfied. The disk mass may notbe very large

compared to the planet mass and the disk gap may not be

fully clear of material. However, simulations have shown

that to within factors of a few, the migration timescale is of

order the viscous timescale of the disk over a wide range

of parameters, provided the tidal clearing is substantial and

the disk mass is at least comparable to the planet mass (e.g.,

Fig. 12). This appears to be true even if there is a significant

mass flow in the gap. The planet in Fig. 12 has mass 1MJat a time 1000 orbits and is migrating at the disk viscous

rate. At this time, the planet is accreting mass through a

deep gap at a substantial rate, comparable to the accretion

rate of the disk in the absence of a planet.

Another way of understanding the Type II torques is to

recognize that the distribution of tidal torques per unit disk

mass, as seen in Fig. 13 still applies, even if the disk has

a deep gap that is not completely clear of material. The

disk density throughthe gap region adjusts so that the planet

migration rate is compatible with the evolution of the disk-

planet system.

3. OUTSTANDING QUESTIONS

3.1 Limiting Type I Migration

As seen in Fig. 11, the timescales for Type I migration

are short compared to disk lifetimes. They become even

is low in a tidally produced gap. But, the mass flux need not be small,

even if the density is small, provided that the radial velocity u increases

sufficiently in the gap, as occurs certain situations.

shorter for a disk having a mass greater than the minimum

mass solar nebula. For retention of habitable planets and

giant planet cores, the short timescales for Type I migration

is a serious problem. To be consistent with the ubiquity of

extrasolar gas giants and formation of Jupiter and Saturn,

some studies suggest that the Type I migration rates must

be reduced by more than a factor of 10 (Alibert et al. 2005;

Ida & Lin 2008). One major question is whether there are

processes that could slow the migration. Several ideas have

been proposed. The major uncertainty with them concerns

our knowledge of the true state of the disk.

Many young stars, such as the T Tauri stars, are sur-rounded by gaseous disks and have observational signatures

of gas accretion. How the accretion operates on a global

scale is not known. There are observational signatures of

accretion onto T Tauri stars. Some form of turbulence is

likely needed to produce the observationally inferred accre-

tion rates. The nature of disk turbulent viscosity has an in-

fluence on the global disk structure and the disk response

to the presence of a planet. Disk turbulence within planet-

forming regions of the disks in early stages of evolution is

likely to be dominated by gravitational instability and later

by magnetic instability. In the latter case, a major unknown

is the level of disk ionization. Unless the gas is sufficiently

ionized, magnetic instability will not occur, e.g., Salmeron& Wardle, 2008.

We briefly discuss below a few of the several suggested

mechanisms for slowing Type I migration.

Low viscosity regions

Disk viscosity suppresses the formation of the weak disk

perturbations that are produced by a small mass planet,

as suggested by equation (98). The perturbations are

smoothed by viscous diffusion. But if the disk turbulent

viscosity is sufficiently low, the disk density distribution

can be affected by the presence of a low mass planet. We

have seen that the ILR and OLR torques push material

away from the orbit of the planet. If the planet is on a fixed

orbit, the density perturbation is almost symmetric aboutr = rp. This near symmetry is broken by the migrationof the planet. In the comoving frame of the planet, there

is a radial steady-state gas flow past the planet. For an

inwardly migrating planet, this flow causes a feedback ef-

fect that enhances the gas density, interior to the orbit, and

lowers the density exterior to the orbit (Ward, 1997). The

feedback then enhances the positive torques that arise in

the inner disk and slows inward migration. This feedback

grows with planet mass. Above some critical planet mass

Mcr the steady-state radial flow of gas past the planet isnot possible. As a result, the planet migration stops, and

gap formation begins. Planet migration can be halted for

a certain critical planet mass Mcr that depends on the gassound speed, the turbulent viscosity, and the rate at which

wave damping occurs. Shocks can provide wave damp-

ing, although the damping is not instantaneous. The waves

launched by a low mass planet propagate some distance

as they steepen and ultimately dissipate. The values of

Mcr caused by shocks in 2D disks are typically 10M

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(Rafikov, 2002; Li et al, 2008). Suppose a planet core could

form in less than the migration timescale prior to reach-

ing this mass, typically less than a few times 105y. In thiscase, the overall migration timescale may be comparable

to Type II migration timescale that is also long due to the

small values of (see equation (100)). Type I migrationmight then nearly stall before the planet migrates through

the disk. Analytic calculations and simulations suggest that

the feedback is strong, typically for values up to of a fewtimes 104. For 0.001, Type I migration proceeds withlittle reduction (see Fig. 15).

Disk Property JumpsWe have seen in Section 2 that the torque on the planet

depends on differences in disk properties across rp. Weexpect the density and temperature to generally smoothly

decrease in radius with the outcome being inward migra-

tion (see equation (88)). However, this need not always be

the case. It is possible that sudden changes in disk prop-

erties with radius could occur as a consequence of sud-

den changes in disk opacity, turbulent viscosity, or gravity

variations (due to the planet itself). In such regions, it is

possible that inner disk contributions could be enhanced or

even be greater than the outer disk contributions. A planet

could then experience slowed migration or even be trapped

in such a region with no further inward migration (Menou& Goodman, 2004; Masset et al, 2006).

Turbulent Fluctuations

We have modeled the effects of disk turbulence by means

of a turbulent viscosity in equation (91). But, in addition

there are time-dependent small-scale density fluctuations

that give rise to a fluctuating random torque on the planet.

Unlike the Type I torque that acts continuously in the same

direction, the fluctuating torque undergoes changes in direc-

tion on timescales characteristic of the turbulence that are

short compared with the migration timescale. The fluctuat-

ing torque causes the planet to undergo a random walk. For

the random walk to compete with Type I migration, the am-

plitude of the fluctuating torque must be much larger thanthe Type I torque. The change in angular momentum of the

planet due to a random torque TR is given by

N TR,where N is the number of fluctuations felt by the planetand is the characteristic timescale for the torque fluctu-ation. The angular momentum change caused by Type I

torques Tin over time t is simply tTin. Since N = t/,it follows that for the random migration to dominate, we

require TR >

N Tin. If we take to be the orbital pe-riod of the planet and t to be the Type I migration timescale 105y, the condition becomes TR > 300Tin. Torque TRdepends linearly on the planet mass, while torque Tin in-creases quadratically with the planet mass and the migra-

tion time t decreases with the inverse of the planet mass.The random torque is then more important for lower mass

planets. The nature of the random torque depends on the

properties of the disk turbulence, in particular its power

spectrum, that are generally not well understood. If there

is power in the turbulence spectrum at low frequencies,

then the fluctuating torques are more effective at counter-

acting the Type I torques. The reason is that the effective

N value is smaller. Some simulations and analytic mod-els suggest that turbulent fluctuations arising from a mag-

netic instability (the magneto-rotational instability Balbus

& Hawley, 1991) are important for migration of lower mass

planets (Nelson, 2005; Johnson et al., 2006; see Fig. 18).

However, if turbulent fluctutations are important for mi-

gration, then the eccentricities of planetesimals are pumped

up so highly that collisions between them may result in de-

struction rather than accretion (Ida et al., 2008). Therefore,

although the turbulent fluctuations may inhibit the infall of

planetary cores into the central star by migration, they tendto inhibit the build up of the cores necessary for giant planet

formation in the core accretion model.

3.2 Other Forms of Migration

Kozai Migration

A planet that orbits a star in a binary star system can peri-

odically undergo a temporary large increase in its orbital ec-

centricity through the a process known as the Kozai effect.

Similar Kozai cycles occur in multi-giant planet systems.

The basic idea behind Kozai migration is that the increased

eccentricity brings the planet closer to the star where it loses

orbital energy through tidal dissipation. In the process, theplanets semi-major axis is reduced and inward migration

occurs (Wu & Murray, 2003). We describe this in more de-

tail below.

Consider a planet in a low eccentricity orbit that is

well interior to the binary orbit and is initially highly in-

clined with respect to it. The orbital plane of the planet

can be shown to undergo tilt oscillations on timescale of

P2b /Pp, where Pb is the binary orbital period, Pp is theplanets orbital period, and by assumption Pb Pp. Un-der such conditions, it can be shown that the component

of the planets angular momentum perpendicular to the bi-

nary orbit plane (the z-component) is approximately con-

served, Jz = Mp

GMsap(1 e2p)cos I, where ap and epare respectively the semi-major axis and eccentricity of the

planets orbit and I is the inclination of the orbit with re-spect to the plane of the binary. The conservation ofJz iseasily seen in the case that the binary orbit is circular and the

companion star is of low mass compared to the mass of the

star about which the planet orbits. On such long timescales

Pb, the companion star can be considered to be a contin-uous ring that provides a static potential. In that case, the

azimuthal symmetry of the binary potential guarantees that

Jz is conserved. By assumption, we have cos I 1 andep 1 in the initial state of the system. As the planetsorbital plane evolves and passes into alignment with the bi-nary orbital planet, cos I 1, conservation of Jz requiresep 1. In other words, Jz is initially small because of thehigh inclination of the orbit. When the inclination drops,

the orbit must become more eccentric (radial), in order to

maintain the same small Jz value. The process then period-ically trades high inclination for high eccentricity.

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During the times of increased eccentricity, the planet

may undergo a close encounter with the central star at peri-

astron distance ap(1 ep). During the encounter, the tidaldissipation involving the star and planet results in an energy

loss in the orbit of the planet and therefore a decrease in ap.This process then results in inward planet migration. The

energy loss may occur over several oscillations of the orbit

plane.

Another requirement for the Kozai process to operate is

that the system must be fairly clean of other bodies. The

presence of the other object could induce a precession that

washes out the Kozai effect. Given the special requirementsneeded for this process to operate, it is not considered to

be the most common form of migration. However, there is

good evidence that it does operate in some systems. The

Kozai effect can also occur in two-planet systems, where

the outer planet plays the role of the binary companion. The

process can be robust due to the proximity of outer planet

(Nagasawa et al. 2008).

Runaway Coorbital Migration

In Section 2.5 we saw that the coorbital torque will be

saturated (reduced to zero) unless some irreversibility is in-

troduced, such as turbulent viscosity. However, planet mi-

gration itself introduces irreversibility and could therefore

act to prevent torque saturation. The coorbital torque modelpresented in Section 2.5 ignored the effects of planet mi-

gration on the horseshoe orbits near the planet (Fig. 9). The

coorbital torque for a migrating planet could then depend

on the rate of migration. Under some conditions, the coor-

bital torque could in turn cause faster migration and in turn

a stronger torque, resulting in an instability and a fast mode

of migration (Masset & Papaloizou, 2003). The resulting

migration is sometimes referred to as Type III migration. To

see how this might operate in more detail, we consider the

evolution of gas trapped in the coorbital region (Artymow-

icz, 2004; Ogilvie & Lubow, 2006). For a sufficiently fast

migrating planet, the topology of the streamlines changes

with open streamlines flowing past the planet and closedstreamlines containing trapped gas (see Fig. 16). The lead-

ing side of the planet contains trapped gas acquired at larger

radii, while the gas on the trailing side is ambient material

at the local disk density. The density contrast between ma-

terial on the trailing and leading sides of the planet gives

rise to a potentially strong torque. A major question centers

around the conditions required for this form of migration to

be effective. If the planet mass is very small, the process

appears ineffective. Once a more massive planet forms a

gap, there is an insufficient amount of gas in the coorbital

region to cause a substantial torque. Some simulations sug-

gest that this type of migration requires a somewhat massive

planet that is not allowed grow in mass to be immersed in

a disk and allowed to migrate before gap opening is com-

plete, artificially bypassing the gap opening that would oc-

cur naturally for a growing planet (e.g, Zhang et al, 2008;

DAngelo & Lubow 2008). These simulations suggest that

this form of migration may not typically arise, due the the

special conditions required.

Migration Driven By Nonisothermal Effects in the Coor-

bital Region

Many studies of disk planet interactions simplify the

disk temperature structure to be locally isothermal. The

locally isothermal assumption, frequently applied in nu-

merical simulations and as we applied in equation (27),

means that the temperature structure is prescribed and is

independent of disk density. The behavior in the isother-

mal limit tends to suggest that coorbital torques do not typ-

ically dominate migration (e.g., equation (88)). The non-

isothermal regime has been recently explored in simula-

tions by Paardekooper & Mellema, 2006 who find that out-ward migration due to coorbital torques may occur in cer-

tain regimes. The conditions required for this possible ef-

fect is an active area of investigation.

Migration in a Planetesimal Disk

After the gaseous disk is cleared from the vicinity of the

star, after about 107y, there remains a disk of solid materialin the form of low mass planetesimals. This disk is of much

lower mass than the original gaseous disk. But the disk is

believed to have caused some migration in the early solar

system with important consequences ( Hahn & Malhotra,

1999; Tsiganis et al, 2005).

There is strong evidence that Neptune migrated outward

due to the presence of Kuiper belt objects that are reso-nantly trapped exterior, but not interior, to its orbit. The

detailed dynamics of a planetesimal disk are somewhat dif-

ferent from the case of a gaseous disk, as considered in

Section 2. The planetesimals behave as a nearly collision-

less system of particles. Jupiter is much more massive than

the other planets and can easily absorb angular momen-

tum changes in Neptune. As Neptune scatters planetesi-

mals inward and outward, it undergoes angular momentum

changes. It is the presence of Jupiter that breaks the sym-

metry in Neptunes angular momentum changes. Once an

inward scattered planetesimal reaches the orbit of Jupiter, it

gets flung out with considerable energy and does not inter-

act again with Neptune. As a result of the loss of inwardscattered particles, Neptune gains angular momentum and

migrates outward, while Jupiter loses angular momentum

and migrates slightly inward.

A similar process occurs in gaseous decretion disks of

binary star systems (Pringle, 1991). The circumbinary disk

gains angular momentum at the expense of the binary. The

binary orbit contracts as the disk outwardly expands. A gap-

opening planet embedded in a circumbinary disk (or under

some conditions, a disk that surrounds a star and massive

inner planet) would undergo a form of Type II migration

that could carry the planet outward (Martin et al, 2007). In

the solar system case, the Sun-Jupiter system plays the role

of the binary. Viscous torques are the agent for transfer-

ring the angular momentum from the binary outward in the

gaseous circumbinary disk, while particle torques play the

somewhat analogous role in the planetesimal disk.

In a planetesimal disk, another process can operate to

cause migration. This process is similar to the runaway

coorbital migration (Type III migration) described above,

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but applied to a collisionless system of particles (Ida et al.

2000). Interactions between the planestimals and the planet

in the planets coorbital zone can give rise to a migration

instability.

Migration of Eccentric Orbit Planets

The analysis in Section 2 assumed that planets reside on

circular orbits. This assumption is not unreasonable, since

there are strong damping effects on eccentricity for a planet

that does not open a gap in the disk (Artymowicz, 1993b).

Some eccentricity may be continuously produced by turbu-

lent fluctuations in the gas, as described above, or by inter-

actions with other planets. In general, eccentricity dampingis faster than Type I migration. For planets that open a gap,

it is possible that they reside on eccentric orbits in the pres-

ence of the gaseous disk. In fact, one model for the observed

orbital eccentricities of extra-solar planets attributes the ex-

citation of eccentricities to disk-planet interactions (Goldre-

ich & Sari, 2003; Ogilvie & Lubow, 2003).

A planet on a sufficiently eccentric orbit embedded in a

circular disk will orbit more slowly at apoastron than the ex-

terior gas with which it tidally interacts. Similarly, a planet

can orbit more rapidly than the tidally interacting gas at pe-

riastron. These angular velocity differences can change the

nature of the friction between the planet and the disk dis-

cussed in Section 2.2. For example, at apoastron the moreslowly orbiting planet could gain angular momentum from

the more rapidly rotating gas. This situation is then just the

opposite of the case in Section 2.2. Furthermore, since the

planet spends more time at apoastron than periastron, the

effects at apoastron could dominate over effects at perias-

tron. It is then possible that outward migration could occur

for eccentric orbit planets undergoing Type I migration, as-

suming such planets could maintain their eccentricities (Pa-

paloizou, 2002).

In the case of a planet that opens a gap, simulations

suggest that outward torques dominate as the planet gains

eccentricity from disk-planet interactions (DAngelo et al,

2006). If giant planet orbits evolve this way, then their or-bital distribution might favor their presence at larger radii,

beyond the snowline where they may form (see Chapter 19).

The situation is complicated by the fact that the gaseous

disk generally gains eccentricity from the planet by a tidal

instability (Lubow, 1991). For the outward torque to be ef-

fective, there needs to be a sufficient difference in the mag-

nitude and/ororientation between the planet and disk eccen-

tricities, so that the planet moves slower than nearby disk

gas at apoastron.

Multiplanet Migration

Thus far, we have only considered single planet systems.

Of the more than 200 planetary systems detected to date by

Doppler techniques, over 20 reside in multi-planet system

(Butler et al, 2006). About 5 systems have been found to

have orbits that lie in mutual resonance, typically the 2:1

resonance. The resonant configurations are likely to be the

result of convergent migration, migration in which the sep-

aration of the orbital radii decreases in time. This process

occurs as the outer planet migrates inward faster than the

inner planet. Planets can become locked into resonant con-

figurations and migrate together, maintaining the planetary

orbital frequency ratio of the resonance. The locking can

be thought of as a result of trapping the planets within a

well of finite depth. Just which resonance the planets be-

come locked into depends on their eccentricities and the

relative rate of migration that would occur if they migrated

independently. As planets that are initially well-separated

come closer together, they lock into the first resonance that

provides a deep enough potential to trap them against the

effects of their convergence. We discuss below the conse-

quences of resonant migration.To maintain a circular orbit, a migrating planet must ex-

perience energy and angular momentum changes that sat-

isfy dE/dt = p(t)dJ/dt, where p is the angular speedof the planet. As the planets migrate together, their mutual

interactions cause deviations from this relation. As a result,

their energies and angular momenta evolve in a way that is

incompatible with maintaining a circular orbit. Orbital ec-

centricities, as well as mutual inclinations, can develop (Lee

& Peale, 2002; Yu & Tremaine, 2001; Thommes & Lissauer,

2003).

To see how this process operates in more detail, consider

the case that the mass of the inner planet Mi is much less

than the mass of the outer planet planet Mo. We assumethe planets undergo convergent migration. As the planets

migrate together locked in a resonance, the inner planet un-

dergoes energy and angular momentum changes as result

of its interaction with the outer planet. The much more

massive outer planet is unperturbed by the small mass in-

ner planet. We ignore the effects of the disk interactions

on the inner planet compared with the effects of the outer

planet. We assume that the outer planet migrates inward

due to its interaction with the disk and maintains a circu-

lar orbit. We consider a cylindrical coordinate system as in

Section 2. The energy of the inner planet is given by

Ei = Miv2i

2+ Mis(ri) + Mio(ri, ro(t), o(t) i),

(101)

where s is the potential due to the star. Potentialo(ri, ro(t), o(t) i) is due to the outer planet. It con-tains an explicit time dependence due to the position of the

outer planet at (ro(t), 0(t)). Taking the time derivative ofthe energy of the inner planet, we obtain

dEidt

= Mivi

dvidt

+ i(s + o)

+ Miot

, (102)

where the gradient i is taken in the inner planet coordi-nates (r

i,

i). The first term on the right-hand side of equa-

tion (102) vanishes as a consequence of the equation of mo-

tion of the planet. We then have that

dEidt

= Miot

, (103)

= Midrodt

oro

Mio oi

, (104)

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where o = do/dt and we have used the fact that o de-pends on o(t) i. We recognize that Mioi is thetorque on the inner planet that is equal to dJi/dt. The ratioof the first term to the second term in equation (104) is then

easily shown to be Mo/Ms 1. We then have to highaccuracy that

dEidt

= o(t)dJidt

. (105)

The inner planet is assumed to initially be on an ap-

proximately circular orbit, but locked in resonance, at

t = 0. Using the standard Keplerian relations that

Ei = GMsMi/(2ai), o = GMs/a3o, and Ji =Mi

GMsai(1 e2i ), with semi-major axis ai, it is straight-forward to show from equation (105) that

1 ei(t)2 = + (1 )

ai(0)

ai(t), (106)

where = i/o > 1 is the ratio of orbital frequenciesthat is a constant for resonantly locked planets. The result

implies that eccentricity formally goes to unity where

ai(t)

ai(0)=

1

2

. (107)

For planets locked in a 2:1 resonance, we have that = 2and the eccentricity goes to unity for ai(t) = ai(0)/4. Theanalysis has assumed the orbits remain coplanar. Numerical

simulations have shown that before reaching a radial orbit,

the inner planet becomes strongly inclined relative to the

outer one, if it starts with a small nonzero initial inclination.

If the inner planet manages to miss striking the star as it

passes ei = 1, further inward migration can cause the innerplanets orbit to flip over and change the sense of its orbital

motion to be counterrotating. Qualitatively similar effects

occur for planetary systems with nonextreme mass ratios.

Eccentricity is generated in both planets (see Fig. 17). Ex-

citation of inclination requires that Mo > Mi/2.Planetary system GJ876 is a well-studied case in which

the planets are in a 2:1 resonance. If the systems measured

eccentricities are due to resonant migration, then accord-

ing to theory (Lee & Peale, 2002), the system migrated in-

ward by less than 10%. Such a small amount of migration is

hard to understand. It is possible that eccentricity damping

through disk-planet interactions could have limited the ec-

centricities to the observed levels as the planets underwent

further migration. But the required damping rate is quite

high. This level of damping is more than an order of mag-

nitude higher than is found in hydrodynamic simulations of

this system (Kley et al, 2005). Once the disk dissipates, mi-

gration ceases and eccentricity growth by this mechanism is

terminated. The disk might have dissipated after the plan-

ets achieved convergent migration and underwent a small

amount of further migration, but the timing seems some-

what unlikely.

3.3 Validity of Numerical Simulations

Numerical simulations provide an important tool for an-

alyzing planet migration. They can provide important in-

sights in cases where nonlinear and time-dependent effects

are difficult to analyze by analytic methods. Some power-

ful grid-based hydrodynamics codes (such as the Zeus code

Stone & Norman, 1992) have been adapted to the study of

disk-planet interactions. In addition, particle codes based

on the Smoothed Particle Hydrodynamics (SPH) (Mon-

aghan, 1992) have sometimes been employed. A systematic

comparison between many of the codes has been carried out

by de Val-Borro et al, 2006.

We discuss a few basic points. For planets that opena gap, grid-based codes offer an advantage over particle-

based codes. The reason is that the resolution of grid based

codes is determined by the grid spacing, while the resolu-

tion of particle-based codes is determined by the particle

density. If a planet opens an imperfect gap, the particle den-

sity and resolution near the planet is low. Poor resolution

near the planet can give rise to artificial torques. Higher res-

olution occurs where the particle density is higher, but this

occurs in regions that interact less strongly with the planet.

There have been variable resolution techniques developed

for grid-based codes in which the highest resoluti

planet migration

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