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

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

None of these problems actually has an answer!

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

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

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

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

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

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

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

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 = AU Evolution time of 100 Myr (PhD Thesis for E. Quintana)

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

Distribution for the number of terrestrial planets (30 realizations)

WORK IN PROGRESS STAY TUNED…

Observed planets: California and Carnegie Planet Search

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

10 Planet Scattering Simulations Place 10 planets on initially circular orbits within the radial range a = 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

Half-life for 10 planet solar systems

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

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

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

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

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

Maximally Efficient Scattering Disk Starting energy: Scattering changes energy:

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

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 = Myr Eccentricity damping time = Myr

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

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

Probability of Supernovae

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

Basic Cluster Considerations Velocity: Density and time:

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

Red Dwarf saves Earth sun red dwarf earth

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

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

Cross Section for Solar System Disruption

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

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

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

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

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 (Wolszczan & Frail 1992)

Earth-like planet in a binary system Variables:

Distribution of Ejection Times

Planet survival time for companion mass

Ejection Time versus Binary Periastron

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

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

Fraction of binaries with periastron less than given value p

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 = percent of binary systems allow for Earth-like planet to remain stable for the age of solar system

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

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