Type I Migration with
Stochastic Torques
Fred C. Adams & Anthony M. Bloch
University of Michigan
Dynamics of Discs and Planets
Cambridge, England, 2009
OUTLINE
 The Type I Migration Problem
 Solution via Turbulent Torques
 Fokker-Planck Formulation
 Effects of Outer Disk Edge
 Effects of Initial Planetary Locations
 Long Term Evolution (eigenfunctions)
 Time Dependent Torques
Previous Work
 Nelson & Papaloizou 2004: numerical
 Laughlin, Adams, Steinacker 2004:
basic numerical + back-of-envelope
 Nelson 2005+: longer term numerical
 Johnson, Goodman, Menou 2006:
Fokker-Planck treatment
This work: Effects of outer disk edge,
long time evolution, time dependent
forcing terms, predict survival rates…
Type I Planetary Migration
Planet embedded in
gaseous disk creates
spiral wakes. Leading
wake pushes the planet
outwards to larger semimajor axis, while trailing
wake pulls back on the
planet and makes the
orbit decay. The planet
migrates inward or
outward depending on
distribution of mass
within the disk.
(Ward 1997)
Net Type I Migration Torque
mP 
2  r 
2
T1  f1  r r  
H 
M 
2
2
(Ward 1997; 3D by Tanaka et al. 2002)
Type I Migration Problem
t1  J 0 /3T1(J 0)
where
J 0  mP GM r0
2
mP 
2  r 
2
T1  f1  r r  
H 
M 
2
and
For typical parameters, the Type I migration
time scale is about 0.03 Myr (0.75 Myr) for
planetary cores starting at radius 1 AU (5 AU).
We need some mechanism to save the cores…
MRI-induced turbulence enforces order-unity surface
density fluctuations in the disk. These surface density
perturbations provide continuous source of stochastic
gravitational torques.
Turbulent Torques
We can use results
of MHD simulation
to set amplitude
for fluctuations of
angular momentum
acting on planets
(LSA04, NP04, &
Nelson 2005)
Turbulence -> stochastic torques -> random walk
-> outward movement -> some cores saved
Working Analytic Model for
Characterizing MHD Turbulence
MHD instabilities lead to surface density variations in
the disk. The gravitational forces from these surface
density perturbations produce torques on any nearby
planets. To study how this process works, we can
characterize the MHD turbulent fluctuations using the
following basic of heuristic potential functions:
k 
(rrc )2 /
Ae
1/ 2
r
2
t
cos[m    c t ]sin[  ]
t
(LSA2004)
Estimate for Amplitude
due to Turbulent Fluctuations
Td  2 G  r mP ,
(J)  4Porb fT Td
fT  0.05  fraction of physical scale
2
2


J 
16 r

3
 10 
  fT
2 
  
 J k
M
1000gcm 
see also Laughlin et al. (2004), Nelson (2005), Johnson et al. (2006)
Power-Law Disks
Surface Density (r)  r
Temperature T(r)  r
p
q
Keplerian Rotation (r)  r
Scale Height H(r)  r(1q )/ 2
3 / 2
FOKKER-PLANCK EQUATION
P
  1 

b
   a P   2 x P 
t
x x  x
2

  f1


m   r 2 Gr 
1
P


10
Myr
   
M  H  GM r 
1AU

2  

3 r
2

  f  f T 2      1 Myr 1

 M  

1AU
a  2( p  q) and b  7  4 p
FOKKER-PLANCK EQUATION
P
  1 

   2 P   2 x P 
t
x x  x
2

  f1


m   r 2 Gr 
1
P


10
Myr
   
M  H  GM r 
1AU

2  

3 r
2

  f  f T 2      1 Myr 1

 M  

1AU
p  3/2, q  1/2, a  2, and b  1
DIMENSIONLESS PARAMETER
8f H f 
3 pq
Qmig 
r
mP
f1
2
T
2
(depends on radius, time)
Distributions vs Time
time = 0 - 5 Myr
Survival Probability vs Time
(fixed diffusion constant)
  0, 1, 3, 5, 10, 20

Survival Probability vs Time
(fixed Type I migration)
  0.1, 0.3, 0.5, 1, 3, 10

DIFFUSION COMPROMISE
 If diffusion constant is too small, then
planetary cores are accreted, and the
Type I migration problem is not solved
 If diffusion constant is too large, then
the random walk leads to large radial
excursions, and cores are also accreted
 Solution required an intermediate value
of the diffusion constant
Optimization of Diffusion Constant
time=1,3,5,10,20 Myr
Survival Probability vs
Starting Radial Location
time=1,3,5,10 Myr
Distributions in Long Time Limit
time=10 - 50 Myr
Only the lowest order eigenfunction
survives in the asymptotic (large t) limit
Lowest Order Eigenfunctions
 /   0.01, 0.1, 1

Surviving planets live in the outer disk…
Time Varying Torques
(r,t)  0 (r) s(t)
s(t)  exp[t /t 0 ]

2


P
 1

2
b
  s(t)
x P
 a P   s (t)
2 
t
x x 
x
Survival Probability with
Time Varying Surface Density
t0 / Myr 1, 3, 10, 30, 

Survival Probability with
Time Varying Mass and Torques
  mP exp[mP /mC ]
and
mP  m1(t / Myr) 3
SUMMARY
 Stochastic migration saves planetary cores
 Survival probability ‘predicted’ 10 percent
 Outer boundary condition important -disk edge acts to reduce survival fraction
 Starting condition important -- balance
between diffusion and Type I torques
 Optimization of diffusion constant
 Long time limit -- lowest eigenfunction
 Time dependence of torques and masses
UNRESOLVED ISSUES
 Dead zones (turn off MRI, turbulence)
 Disk structure (planet traps)
 Outer boundary condition
 Inner boundary condition (X-point)
 Fluctuation distrib. (tails & black swans)
 Competition with other mechanisms
(see previous talks…)
Reference
F. C. Adams and A. M. Bloch (2008): General
Analysis of Type I Planetary Migration with
Stochastic Perturbations, ApJ, 701, 1381
[email protected]