Planet Driven Disk Evolution

Roman Rafikov

IAS

Outline

• Introduction

- Planet-disk interaction

- Basics of the density wave theory

• Density waves as drivers of disk evolution

• Nonlinear evolution of density waves

- Nonlinear wave propagation

- Wave dissipation

- Effective viscosity

• Gap formation by shocked density waves

• Applications

Planet-disk interaction

Lubow et al. (2002b)

• Tidal interaction of planet with the disk leads to the formation of spiral density perturbation which, because of the differential rotation, leads (trails) planet in the inner (outer) disk.

• As a result, planet is pulled forward (backward) and its angular momentum increases (decreases).

• Inner (outer) disk loses (gains) angular momentum.

Thus, planet repels disk. This might lead to a gap formation.

Slight imbalance between the torques exerted on the inner/outer disk leads to planet migration. Its direction is usually inward.

Takeuchi et al. (1996)

Gravitational interaction between the planet on circular orbit and disk leads to the density wave generation at Lindblad resonances

2222 )( pm

Density waves carry energy and angular momentum.

is the natural frequency of a fluid element (epicyclic frequency)

(disk thickness) contribute most of the angular momentum

ham /Harmonics with

/sch

Density waves

222222 ||2)( kckGm sp

Dispersion relation

1

G

cQ s

Toomre’s Q

How density waves affect the disk?If waves experience damping of any kind their energy and angular momentum get transferred to the disk fluid disk evolves.

dwvis FFrrrrt

11

Dissipation of density waves has analogous effect on the disk:

• they change its angular momentum

• they lead to energy dissipation in the disk

This can be quantified by introducing effective viscosity chosen e.g. to match the energy dissipation in a viscous disk

eff

In thin a accretion disk theory (Shakura & Sunyaev 1973) viscosity drives disk evolution; its effect is quantified by - parameter:

hcs 2~ scE

Linear wave damping mechanisms

Thus, linear damping cannot effectively dissipate density waves

• Dissipation due to the disk viscosity (Takeuchi et al. 1996)

Damping length But viscosity in protoplanetary disks is low,

• Radiative losses from the disk surface (Cassen & Woolum 1996)

Optical depth of protoplanetary disks is too high to allow effective leakage of radiation: and (but see Morohoshi & Tanaka 2003)

• Radiative losses within the disk (Goodman & RR 2001)

Short wavelength of the perturbation makes horizontal radiation transfer more efficient:

hhld 3/1310

hthl coold )(~

hthl coold 3/1)(~

1 coolt

Nonlinear damping

Nonlinear evolution leading to shock formation can dissipate density waves (Larson 1989).

Wave evolution can be split into two distinct regimes (Goodman & RR’01):

• Linear generation - at multiple Lindblad resonances, occurs mostly at from the planet, nonlinear effects are weak, can be neglected

• Nonlinear propagation - takes place beyond - tidal forcing is no longer important, can be neglected Linear

Nonlinear

Lubow et al. (2002)

Nonlinear damping

Goodman & RR 2001

Linear excitation Nonlinear evolution

Lubow et al 2002

Determine rate of wave dissipation obtain damping prescription

Separation is valid for 4/33

1 112

AU

aM

G

cM s

1MM p

Nonlinear damping

Basics of nonlinear evolution

Propagation speed is different for different parts of the wave profile. This leads to the profile steepening and breaking

shock forms

0t 12 tt 01 tt

At large k density waves behave as sound waves: 222 kcs

Nonlinear damping

Nonlinear damping

5/21/ MMhr psh

Validity regime

1MM p

As a result

Wave profile breaking and shock formation due to nonlinear steepening is facilitated by

• Increasing radial wavenumber k with the distance from the planet

• Increasing the amplitude of the wake caused by the conservation of the angular momentum flux

pp rrrk )(

2/12 ,)(/)( ppJ rrrF

JF

Applications

Shock damping is not exponential – waves travel large distance in the disk.

Dissipation of a system of density waves driven by a planetary system is equivalent to an effective viscosity which drives global disk evolution (typical timescale ).yr76 1010

RR’02a

Accretion rateRR’02a

Protoplanetary disk dispersal by planetary torques

Gap opening

• In an inviscid disk a nonmigrating planet of infinitesimal mass would open a gap.

• While planetary torques repel disk away, planet might migrate out of the forming gap. This is important for small planets:

12 , pmigpopen MtMt

• Feedback from the surface density inhomogeneities slows the migration ( ) facilitating gap opening (Ward & Hourigan’89).

Repulsive flux Repulsive fluxMigration flux Migration flux

• Viscosity opposes gap formation by filling the gap

Gap formation

One has to take into account 3 effects simultaneously (RR’02b):

• Realistic density wave damping

• Planetary migration

• Feedback from disk inhomogeneities

Important timescales:

• - time to cross gap due to migration

• - time to fill the gap due to viscosity in the disk

• - time to clear the gap by planetary torques

migt

t

opent

• planetary torques win over migration

• viscosity wins over planetary torques

1/ openmig tt

1/ ttopen

Gap formation

Gap formationGap opening planetary mass (RR’02b)

0

6103

5103

4103

0

6103

5103

4103

Typical gap forming planet has mass ( for )410

4/3

1 1)42()3.015.0(

AU

aMMM p

This translates into at 5 AU and at 30 AUM10 M40

Conclusions

• Planet-disk interaction leads to angular momentum exchange and disk evolution

• Nonlinear effects play important role in the propagation of the density waves

• They lead to shock formation and its subsequent dissipation

• Corresponding transfer of the angular momentum to disk material leads to

- disk evolution ( )

- gap formation

• Self-consistent study including nonlinear dissipation leads to a new estimate of a gap-forming mass . gapM

yreff7643 1010~,1010