1 / 28

Type I Migration with Stochastic Torques

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

teddy
Download Presentation

Type I Migration with Stochastic Torques

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. 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

  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 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. 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

  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 Amplitudedue to Turbulent Fluctuations see also Laughlin et al. (2004), Nelson (2005), Johnson et al. (2006)

  10. Power-Law Disks

  11. FOKKER-PLANCK EQUATION

  12. 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

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

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

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

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

  22. Time Varying Torques

  23. Survival Probability withTime Varying Surface Density

  24. Survival Probability withTime Varying Mass and Torques

  25. 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

  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…)

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

More Related