1. Type I Migration with Stochastic Torques Fred C. Adams & Anthony M. Bloch University of Michigan Fred C. Adams & Anthony M. Bloch University of Michigan Dynamics of Discs and Planets Dynamics of Discs and Planets Cambridge, England, 2009 Cambridge, England, 2009

2. OUTLINEOUTLINE  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  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

3. 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  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…

4. Type I Planetary Migration Planet embedded in gaseous disk creates spiral wakes. Leading wake pushes the planet outwards to larger semi- major 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)

5. Net Type I Migration Torque (Ward 1997; 3D by Tanaka et al. 2002)

6. Type I Migration Problem 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…

7. MRI-induced turbulence enforces order-unity surface density fluctuations in the disk. These surface density perturbations provide continuous source of stochastic gravitational torques. Turbulence -> stochastic torques -> random walk -> outward movement -> some cores saved We can use results of MHD simulation to set amplitude for fluctuations of angular momentum acting on planets (LSA04, NP04, & Nelson 2005) Turbulent Torques Turbulent Torques

8. 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: (LSA2004)

9. Estimate for Amplitude due to Turbulent Fluctuations see also Laughlin et al. (2004), Nelson (2005), Johnson et al. (2006)

10. Power-Law Disks

11. FOKKER-PLANCK EQUATION

13. DIMENSIONLESS PARAMETER (depends on radius, time)

14. Distributions vs Time time = 0 - 5 Myr

15. Survival Probability vs Time (fixed diffusion constant)

16. Survival Probability vs Time (fixed Type I migration)

17. 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  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

18. Optimization of Diffusion Constant time=1,3,5,10,20 Myr

19. Survival Probability vs Starting Radial Location time=1,3,5,10 Myr

20. Distributions in Long Time Limit Only the lowest order eigenfunction survives in the asymptotic (large t) limit time=10 - 50 Myr

21. Lowest Order Eigenfunctions Surviving planets live in the outer disk…

22. Time Varying Torques

23. Survival Probability with Time Varying Surface Density

24. Survival Probability with Time Varying Mass and Torques

25. SUMMARYSUMMARY  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  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

26. 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…)  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…)

27. ReferenceReference F. C. Adams and A. M. Bloch (2008): General Analysis of Type I Planetary Migration with Stochastic Perturbations, ApJ, 701, 1381 fca@umich.edu