理論天文学懇談会 ポスター発表 2006年12月25日〜27日

The Effect of Poloidal Magnetic
Field on Planetary Migration
Takayuki Muto
in collaboration withShu-Ichiro Inutsuka
Department of Physics, Kyoto University
Planetary migration is the radial motion of protoplanets due to the gravitational interaction between
protoplanets and circumstellar gas. Recent work has shown that when a protoplanet of an Earth mass
is formed at 1AU, it falls towards the central star before the dispersal of the protoplanetary disk.
In order for protoplanets to survive, some mechanism should act to stop the infall. Recently, it has
been shown that toroidal magnetic field may stop the inward planetary migration under a certain
condition, so magnetic field should be important. In this poster, the effect of weak poloidal magnetic
field is investigated.

(Type I) Planetary Migration

planetAngular momentum transfer
From Dr F Masset’s web page
http://www.maths.qmul.ac.uk/~masset/moviesmpegs.html• Protoplanets generatedensity waves on
protoplanetary disk
• Waves transport
angular momenta• Outer wave wins
• The planets fall
towards the centralstar due to thebackreaction of thewaves

Timescale of Planetary Migration
Tanaka, Takeuchi and Ward ApJ 565 1257 (2002):surface density:central star mass

:thermal velocity (isothermal):planet mass

Minimum Mass Solar Nebula:For

Protoplanets Fall into Central Star
BEFORE Gas Dispersal

Stopping Inward Planetary
Migration by …
Terquem MNRAS 341 1157 (2003)
Toroidal magnetic field can strengthen the wave
inside the planet orbit.
１．Magnetic Resonance appears
２．When magnetic field decreases slower thanΩB

planets migrate outward

Magnetic Field may be Important when
Considering Planetary Migration

Ongoing Research
Muto and Inutsuka in prep

• Terquem considered toroidal field• Do poloidal fields do any good?
• Polodal fields usuallyoriginate outside the diskΩ

– Always exist
B• Need considering diskthickness• Any change in wave
propagation on the disk?
• Relation with MRI?

Basic Equations (1)Ideal MHD＋One Planet’s Gravitational Force
Background:No planet in backgroundNo structure in z direction, but disk exists only at -H<z<H
H: Scale height of the disk
Neglect disk self-gravity
Fourier expand the perturbation: linearisation

Basic Equations (2)Consider stationary perturbation：
gamma: small viscosity is taken into account if necessary

Isothermal perturbation:
Local approximation:
Lagrangian displacement：withPerturbation Equations
EoC:
EoM (r component):

EoM (phi component):

EoM (z component):

withTorque Exerted on the Disk
Once density perturbation is known,
torque exerted on the disk is calculated:with

Torque on the planet is back reaction

Equation Governing the Density
Perturbation

Basically, this equation is of the form:

Wave Propagation on DiskWKB Approximation:
but

Lindblad resonance:orAlfven resonance:

Sound resonance:

Slow resonance:
Evanescent

For example:

Propagation

Planetrslow

SRD-ARD+

c=0, weak B Case
For simplicity, we consider:
Rem: disk is MRI unstableandPropagation

Evanescent

Propagation

Planetr
AR

D+

Solve the wave equation withappropriate boundary conditions in
the vicinity of resonances

Boundary Conditions
Propagation

Evanescent

Propagation

PlanetrAR

consistency ofoutgoing boundary
and the symmetryof the equation

D+

Waves should not
grow exponentially

Outgoing wave

Rem: this poster focuses on Lindblad resonanceLindblad Torque FormulaThis formula generalises Goldreich and Tremaine formula

Goldreich and Tremaine ApJ 233 857 (1979)‘… the torque exerted at an isolated resonance isindependent of particular physics at work …’Meyer-Vernet and Sicardy Icarus 69 157 (1987)

Discussion: How Torque Affected?
Lindblad resonance is further apart from planet
TorqueTorque
m=100
m=50
m=10
Increasing v_A*k_z
100m01
V_A*k_z / Omega

Torque is suppressed by about a factor of 5Migration timescale is 5 times longer

Summary• We considered the effect of poloidal
magnetic field on planetary migration.
• Under WKB approximation, we derived ananalytic torque formula at Lindblad
resonance when sound speed is negligible.
• Lindblad torque is suppressed in thepresence of magnetic field.
• More detailed analysis is under way.