# Planets in Magnetised, Turbulent Discs

### Introduction

This is an ongoing project which examines how disc-planet interactions change when the protostellar discs in which planets form are turbulent, with particular emphasis on the flow morphology and migration rates. Previous studies of disc-planet interactions have considered laminar, viscous disc models, in which the anomalous disc viscosity is modeled using the alpha' prescription. The most probable source for this anomalous viscosity is magnetohydrodynamic turbulence generated by the magnetorotational instability or MRI (Balbus & Hawley 1991), so it is crucial to include the effects of turbulence when considering disc-planet interactions. The results presented here were obtained by performing three dimensional MHD simulations of turbulent protostellar discs with embedded planets of different masses, and as such represent the first step toward modeling planet formation in realistic disc models.

The project consists of two distinct themes:

#### Low Mass Protoplanets

Simulations of 3, 10, and 30 Earth mass protoplanets embedded in turbulent disc models with aspect ratio H/R=0.07 have been performed. Low mass planets such as these are expected to modestly perturb the disc without gap formation. A long standing, unresolved problem in planet formation is that low mass protoplanets undergo rapid inward migration due to interaction with the disc in which they form, on a time scale that is significantly shorter than the formation time of giant planets (e.g. Ward 1997; Pollack et. al. 1996). One important aim of this work is to examine the migration rate of low mass protoplanets in turbulent discs to see if this problem remains. The main results of this work are:

• The turbulent density fluctuations are of higher amplitude than the planetary spiral wakes for planets with masses less than around 30 Earth masses
• The dominance of these turbulent density fluctuations cause the planet to undergo stochastic migration, essentially following a random walk
• For planets with mass less than around 30 Earth masses, neither the rate or direction of migration are well defined over simulation run times of 20 - 30 planet orbits
• Although the long term orbital evolution of low mass planets in turbulent discs is yet to be computed, stochastic migration may provide a resolution of the problem of rapid migration for planetary cores

These points are illustrated by the images and movies presented below.

Images and movies

(click to enlarge)

Commentary

The image on the left shows a snapshot of a 30 Earth mass protoplanet embedded in a turbulent disc. The planet is located at (x,y)=(-3,0). An mpeg movie showing details of the evolution in the vicinity of the planet is available below.

30 Earth masses  The image on the left shows a close-up of a 30 Earth mass protoplanet embedded in a turbulent disc. The usual spiral wakes generated by the protoplanet are apparent, as are the wakes generated by the turbulence. For comparison purposes, an image of a 30 Earth mass planet embedded in a laminar disc is provided below.

The image on the left shows a snapshot of a 30 Earth mass protoplanet embedded in a viscous, laminar disc model.

Torque on 30 Earth mass planet in turbulent disc  This image shows the torque per unit mass exerted on the 30 Earth mass planet by the turbulent disc as a function of time. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line the total torque. The erratic behaviour of the torque contrasts sharply with a laminar disc model (see below), suggesting the planet will experience a random walk' through the disc.

This image shows the torque per unit mass exerted on the 30 Earth mass protoplanet by a laminar disc as a function of time. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line shows the total torque. A well defined torque and associated migration rate is established within a couple of orbits.

Torque on 30 Earth mass planet in turbulent disc

This image shows the running mean of the torque exerted on the 30 Earth mass protoplanet by the turbulent disc as a function of time. The blue line is the inner disc torque, the green line is the outer disc torque, and the red line is the total torque. The white line is the torque due to the laminar disc. The running mean suggests inward migration, but the rate of migration has not converged.

The image on the left shows a snapshot of a 10 Earth mass protoplanet embedded in a turbulent disc with aspect ratio H/R=0.07. The planet is located at (x,y)=(-3,0). An mpeg movie showing details of the evolution in the vicinity of the planet is available below.

The image on the left shows a close-up of a 10 Earth mass protoplanet embedded in a turbulent disc. The spiral wakes generated by the protoplanet are barely apparent, as the turbulent wakes have larger amplitude. For comparison purposes, an image of a 10 Earth mass planet embedded in a laminar disc is provided below.

The image on the left shows a snapshot of a 10 Earth mass protoplanet embedded in a viscous, laminar disc model.

This image shows the torque per unit mass exerted on the 10 Earth mass planet by the turbulent disc. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line the total torque. The rapid variation of the torque suggests the planet will migrate as a random walk.

This image shows the torque per unit mass exerted on the 10 Earth mass protoplanet by a laminar disc as a function of time. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line shows the total torque. A well defined torque and associated migration rate is establishedwithin a couple of orbits.

This image shows the running mean of the torque per unit mass exerted on the 10 Earth mass protoplanet by the turbulent disc. The blue line is the inner disc torque, the green line is the inner disc torque, and the red line is the total torque. The white line is the torque due to the laminar disc. The running mean varies in magnitude and sign over the run-time, such that neither the rate or direction of migration are well defined.

The image on the left shows a snapshot of a 3 Earth mass protoplanet embedded in a turbulent disc. The planet is located at (x,y)=(-3,0). An mpeg movie showing details of the evolution in the vicinity of the planet is available below.

The image on the left shows a close-up of a 3 Earth mass protoplanet embedded in a turbulent disc. The spiral wakes generated by the protoplanet are undetectable as the turbulent wakes have significantly larger amplitude. For comparison purposes, an image of a 3 Earth mass planet embedded in a laminar disc is provided below.

3 Earth mass laminar disc  The image on the left shows a snapshot of a 3 Earth mass protoplanet embedded in a viscous, laminar disc model.

This image shows the torque per unit mass exerted on the 3 Earth mass planet by the turbulent disc. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line the total torque. The erratic behaviour of the torque suggests the planet will experience a random walk' through the disc.

This image shows the torque per unit mass exerted on the 3 Earth mass protoplanet by a laminar disc as a function of time. The blue line shows the inner disc torque, the green line shows the outer disc torque, the red line shows the total torque. A well defined torque and associated migration rate is established within a couple of orbits.

This image shows the running mean of the torque per unit mass exerted on the 3 Earth mass protoplanet by the turbulent disc. The blue line is the inner disc torque, the green line is the outer disc torque, and the red line is the total torque. The white line is the torque due to the laminar disc. By the end of the simulation the running mean suggests outward migration, but neither the direction or rate of migration have converged to a well defined value.

The image on the left shows a snapshot from a simulation with three 30 Earth mass protoplanets embedded in a turbulent disc. In this simulation the planets do not interact with each other gravitationally, but undergo orbital evolution due to interaction with the disc. Three planets were evolved in order to sample the range of outcomes due to stochastic forcing by the turbulence.

The image on the left shows a snapshot from a simulation with six 10 Earth mass protoplanets embedded in a turbulent disc. In this simulation the planets do not interact with each other gravitationally, but undergo orbital evolution due to interaction with the disc. Six planets were evolved in order to sample the range of outcomes due to stochastic forcing by the turbulence. The resulting semimajor axis evolution is shown in the image below.

Semimajor axes of the 10 Earth mass planets in turbulent disc

This image shows the evolution of the semimajor axes of the six 10 Earth mass protoplanets embedded in the turbulent disc. The effects of stochastic migration are clear. The white dotted lines show the evolution of planets embedded in an equivalent laminar disc.

#### High Mass Protoplanets

Simulations of 5 and 3 Jupiter mass protoplanets embedded in turbulent disc models have been performed. Massive planets such as these are expected to form gaps and to migrate inward on the viscous time scale of the disc, and these expectations are bourne out by the simulations. However some interesting and important differences arise when the disc sustains MHD turbulence when compared with viscous, laminar models. These include:

• The disc shows a more time dependent behaviour, with the spiral waves induced by the planet having a more diffused appearance
• The gap which is formed is wider in turbulent models
• The planet can compress and order the magnetic field in its vicinity, thereby increasing the magnetic stress in the shock region associated with the spiral wakes
• The magnetic field from the protostellar disc is advected into the planet Hill sphere as gas accretes onto the protplanet. Magnetic linkage between the circumplanetary disc and the surrounding protostellar disc appears to cause magnetic braking of the circumplanetary disc, and modification of the accretion rate onto the planet.

These points are illustrated by images and mpeg movies presented below.

Images and Movies

(click to enlarge)

Commentary

The image on the left shows a snapshot of a 5 Jupiter mass planet embedded in a turbulent disc with aspect ratio H/R=0.1. For comparison purposes, an mpeg movie of a gap forming planet embedded in a laminar disc is provided below.

The image on the left shows a snapshot of a 1 Jupiter mass planet embedded in a viscous, laminar disc model.

The image on the left shows a close up image of the density in the vicinity of the planet in the turbulent disc. An equivalent plot showing the distortion of the magnetic field due to the planet is shown below.

Magnetic field vectors

The image on the left shows the magnetic field vectors in the vicinity of the planet. The ordering and compression of the field in the region near the planetary wakes is apparent, leading to an enhancement of the magnetic stress (and effective alpha' value) there.

Magnetic field vectors

The image on the left shows another close up of magnetic field vectors in the vicinity of the planet. The advection of field from the protoplanetary disc into the Hill sphere of the planet can be observed, as well as the linking of field lines from the circumplanetary disc to the surrounding protostellar disc. This magnetic linkage appears to cause magnetic braking of the circumplanetary disc, as discussed below.

The image on the left shows contours of the magnetic energy, and illustrates how the planet affects the strength and topolgy of the field in its vicinity. The field strength is increased in the spiral shocks associated with the planet, but decreases on average in the gap region.

The image on the left shows the velocity field in the vicinity of the planet. The existence of a circumplanetary disc is apparent, as well as the horsehoe orbits in the corotation region.

This image shows the azimuthally averaged midplane density for two simulations. The red line corresponds to a viscous, laminar disc run with a 3 Jupiter mass planet. The green dotted line corresponds to an equivalent turbulent disc run. Gas entering the Hill sphere forms a rotationally supported circumplanetary disc. The higher density of gas sitting on the planet in the turbulent run is evidence for magnetic braking of the circumplanetary disc. Note also the wider gap that forms in the turbulent disc simulation.

#### Resulting Publications

The interaction of planets with a disc with MHD turbulence IV: Migration rates of embedded protoplanets
Authors: Richard P. Nelson & John C.B. Papaloizou
Status: To appear in MNRAS 2004

The interaction of planets with a disc with MHD turbulence III: Flow morphology and conditions for gap formation in local and global simulations
Authors: John C.B. Papaloizou, Richard P. Nelson & Mark D. Snellgrove
Status: To appear in MNRAS 2004

The interaction of a giant planet with a disc with MHD turbulence - II. The interaction of the planet with the disc
Authors: R.P. Nelson & J.C.B. Papaloizou
Status: Appeared in MNRAS, 2003: volume 339, page 993

The interaction of a giant planet with a disc with MHD turbulence - I. The initial turbulent disc models
Authors: J.C.B. Papaloizou & R.P. Nelson
Status: Appeared in MNRAS, 2003: volume 339, page 983