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:
Papers arising from this work may be downloaded
here
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.
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Images and Movies
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Commentary
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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.
Click on the image to enlarge.
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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.
Click on the image to download an mpeg movie of this simulation.
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The image on the left shows a snapshot of a 30 Earth mass protoplanet
embedded in a viscous, laminar disc model.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to download an mpeg movie of this simulation.
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The image on the left shows a snapshot of a 10 Earth mass protoplanet
embedded in a viscous, laminar disc model.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to download an mpeg movie of this simulation.
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The image on the left shows a snapshot of a 3 Earth mass protoplanet
embedded in a viscous, laminar disc model.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to download an avi movie of this simulation.
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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.
Click on the image to download an avi movie of this simulation.
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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.
Click on the image to enlarge.
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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.
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Images and Movies
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Commentary
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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.
Click on the image to download an mpeg movie of this simulation.
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The image on the left shows a snapshot of a 1 Jupiter mass planet
embedded in a viscous, laminar disc model.
Click on the image to download an mpeg movie of this simulation.
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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.
Click on image to enlarge
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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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.
Click on the image to enlarge.
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