1. Chaotic Case Studies: Sensitive dependence on initial conditions in star/planet formation Fred C. Adams Physics Department University of Michigan With: E. David, M. Fatuzzo, G. Laughlin, E. Quintana

2. Outline Turbulent fluctuations in the formation of cloud cores through ambipolar diffusion Terrestrial planet formation in binaries Giant planet migration through disk torques and planet-planet scattering events Solar system scattering in the solar birth aggregate Long term stability of Earth-like planets in binary star systems

3. None of these problems actually has an answer!

4. None of these problems actually has an answer! Instead, each of these problems has a full DISTRIBUTION OF POSSIBLE OUTCOMES

5. Star Formation takes place in molecular cloud cores that form via ambipolar diffusion

6. Problem: Observations of the number of starless cores and molecular clouds cores with stars indicate that the theoretically predicted ambipolar diffusion rates are too slow

7. Ambipolar diffusion in a gaseous slab (Shu 1983, 1992)

8. where Nonlinear Diffusion Equation let Fatuzzo and Adams, 2002, ApJ, 570, 210

9. Diffusion time vs fluctuation amplitude Amplitude expected from observed line-width

10. Distribution of Ambipolar Diffusion Times Shu 1983 solution 10,000 simulations

11. Simulations of Planet Formation Mercury code (J. Chambers 1999) Late stages of planetessimal accumulation: from 100s of bodies to a few large bodies Initially circular orbits w/ a = 0.36 - 2.05 AU Evolution time of 100 Myr (PhD Thesis for E. Quintana)

12. Planet position as a function of binary periastron Notice the wide range of values

13. Distribution for the number of terrestrial planets (30 realizations)

14. WORK IN PROGRESS STAY TUNED…

15. Observed planets: California and Carnegie Planet Search

17. MIGRATION Disk Torques Planet Scattering MIGRATION with multiple planets and disk torques Good for a Poor for e Good for e Poor for a the disk is present other planets are present

18. 10 Planet Scattering Simulations Place 10 planets on initially circular orbits within the radial range a = 5 - 30 AU (one solar mass central star) Use 4 different planetary mass functions: (1) equal mass planets with (2) equal mass planets with (3) planet masses with random distribution (4) planet mass with log-random distribution Mercury code: Chambers 1999, MNRAS, 304, 793

19. Half-life for 10 planet solar systems

20. Ejection time is not a well-defined quantity but the half-life is well determined

21. Distributions of final-state orbital elements (Log-random planet mass distribution)

22. Distributions of orbital elements for the innermost planet (at end of simulation)

23. Final orbital elements for migrating planets with planet-planet scattering only

24. Final orbital elements for migrating planets with planet-planet scattering only Giant planet desert

25. Maximally Efficient Scattering Disk Starting energy: Scattering changes energy:

26. Surface density distribution ( in discrete form) in the limit:

27. 2 Planet simulations with disk torques and scattering B-S code Starting periods out of resonance: Random planet masses: Viscous evolution time = 0.3 Myr Allow planetary collisions Relativistic corrections Tidal interactions with the central star Random disk lifetime = 0 - 1 Myr Eccentricity damping time = 1 - 3 Myr

28. Final Orbital Elements for Migrating Planets with disk torques and planet scattering a (AU) eccentricity Adams and Laughlin 2003, Icarus, 163, 290

29. Solar Birth Aggregate Supernova enrichment requires large N Well ordered solar system requires small N

30. Probability of Supernovae

31. Probability of Scattering Event Scattering rate: Survival probability: Known results provide n, v, t as function of N (e.g. BT87) need to calculate the interaction cross sections

32. Basic Cluster Considerations Velocity: Density and time:

34. Monte Carlo Experiments Jupiter only, v = 1 km/s, N=40,000 realizations 4 giant planets, v = 1 km/s, N=50,000 realizations KB Objects, v = 1 km/s, N=30,000 realizations Earth only, v = 40 km/s, N=100,000 realizations 4 giant planets, v = 40 km/s, N=100,000 realizations

35. Red Dwarf saves Earth sun red dwarf earth

36. Red dwarf captures the Earth Sun exits with one red dwarf as a binary companion Earth exits with the other red dwarf Sun and Earth encounter binary pair of red dwarfs 9000 year interaction

37. Eccentricity e Semi-major axis a Jupiter SaturnUranus Neptune Scattering results for our solar system

38. Cross Section for Solar System Disruption

39. Stellar number N Probability P(N) Expected Size of the Stellar Birth Aggregate survival supernova Adams and Laughlin, 2001, Icarus, 150, 151

40. Cumulative Distribution: Fraction of stars that form in stellar aggregates with N < N as function of N Charlie Phil Median point: N=300

41. Constraints on the Solar Birth Aggregate (1 out of 120)

42. Probability as function of system size N Adams and Myers 2001, ApJ, 553, 744

43. The Case for Extra-solar Terrestrial Planets Our solar system has terrestrial planets and no instances of astronomical creation are unique Our solar system has made large numbers of moons, asteroids, and other rocky bodies Terrestrial planets discovered in orbit around the Pulsar PSR 1257+ 12 (Wolszczan & Frail 1992)

44. Earth-like planet in a binary system Variables:

45. Distribution of Ejection Times

47. Planet survival time for companion mass

48. Ejection Time versus Binary Periastron

50. Conservative Estimate: Need binary periastron p > 7 AU for the long term stability of and Earth-like planet What fraction of the observed binary population has periastron p > 7 AU? David et al. 2003, PASP, 115, 825

51. Observed Distributions e.g., Duqunenoy & Mayor 1991, A&A, 248, 485

52. Fraction of binaries with periastron less than given value p

53. Conclusions Turbulent fluctuations increase the rate of ambipolar diffusion and replace a single time scale with a distribution of time scales Binary companions limit the region of terrestrial planet formation, lead to distributions of a & N Planet scattering -> e (not a) Disk torques -> a (not e) Planetary migration with both disk torques and planet scattering can explain observed a-e plane Determination of scattering cross sections For external enrichment scenario, Sun forms in stellar aggregate with N = 2000 50 percent of binary systems allow for Earth-like planet to remain stable for the age of solar system

54. Bibliography M. Fatuzzo and F. C. Adams, 2002, ApJ, 570, 210 E. Quintana and F. C. Adams, 2003, in preparation F. C. Adams and G. Laughlin, 2003, Icarus, 163, 290 F. C. Adams and G. Laughlin, 2001, Icarus, 150, 151 E. David, F. C. Adams, et al. 2003, PASP, 115, 825 F. C. Adams and P. C. Myers, 2001, ApJ, 553, 744

56. Chaotic Case Studies Sensitive dependence on initial conditions in star and planet formation Fred C. Adams Physics Department University of Michigan