Chaotic Case Studies:Sensitive dependence on initial

conditions in star/planet formation

Fred C. AdamsPhysics 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

€

ˆ z

€

B

Ambipolar diffusion in a gaseous slab

€

g

€

g

(Shu 1983, 1992)

€

∂∂t

B(1+ ξ )

ρ (1+ η )

⎡

⎣ ⎢

⎤

⎦ ⎥=

€

1

1+ η

∂

∂σD

∂

∂σ

⎧ ⎨ ⎩

B(1+ ξ )[ ]

€

D =B2(1+ ξ )2

4πγCρ1/ 2(1+ η )3 / 2

€

}

where

Nonlinear Diffusion Equation

€

B → B(1+ ξ ),ρ → ρ(1+ η )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 1983solution

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 = 0.36 - 2.05 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

Observed planets: California and Carnegie Planet Search

MIGRATION

Disk Torques Planet Scattering

MIGRATION with multiple planets and disk torques

Good for aPoor for e

Good for ePoor 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 = 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

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mP =1mJ

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mP = 2mJ

Mercury code: Chambers 1999, MNRAS, 304, 793

Half-life for 10 planet solar systems

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 planetdesert

Maximally Efficient Scattering Disk

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E0 = −GM∗mP

2a0

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ΔE = −GM∗μ

2r

Starting energy:

Scattering changes energy:

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a1 = a0(1+ μ /mP )−1, an = a0(1+ μ /mP )−n

fn = a0 /an = (1+MS /mP

n)n

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n →∞lim fn = exp[MS /mP ]

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MMES = mP ln[a0 /a f ]

Surface density distribution

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σ(r) =μ

2πrk=1

n

∑ δ(r − ak )

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n → ∞, μ → 0, nμ → const

σ MES (r) →mP

2πr2

(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 = 0 - 1 Myr• Eccentricity damping time = 1 - 3 Myr

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P1 =1900d,P2 = P1 × 3.736...

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mP = 0 − 5mJ

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rP = 2rJ

Final Orbital Elements for Migrating Planets with disk torques and planet scattering

a (AU)

ecc

en

tric

ity

Adams and Laughlin 2003, Icarus, 163, 290

Solar Birth Aggregate

Supernova enrichment requires large N

€

M∗ > 25Mo

Well ordered solar system requires small N

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FSN = 0.000485

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ε(Neptune) < 0.1

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ΔΘ j < 3.5o

Probability of Supernovae

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PSN (N) =1− fnotN =1− (1− FSN )N

Probability of Scattering Event

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Γ=n σvScattering rate:

Survival probability:

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Psurvive ∝ exp − Γdt∫[ ]

Known results provide n, v, t as function of N (e.g. BT87) need to calculate the interaction cross sections

€

σ

Basic Cluster Considerations

Velocity:

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v ≈1 km /s

Density and time:

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n ≈ N /4R3

tR ≈ (R /v) N /(10lnN)

(nvt)0 ≈ N 2 /40R2 lnN ∝N μ

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 dwarfSun and Earth encounter

binary pair of red dwarfs

9000 year interaction

Ecc

en

t ric

ity

e

Semi-major axis aJupiter

Saturn UranusNeptune

Scattering results for our solar system

Cross Section for Solar System Disruption

€

σ ≈(400AU)2

Stellar number N

Pro

babili

ty P

(N)

Expected Size of the Stellar Birth Aggregate

survivalsupernova

Adams and Laughlin, 2001, Icarus, 150, 151

Cumulative Distribution: Fraction of stars that form in stellar aggregates with N < N as function of N

CharliePhil

Median point: N=300

Constraints on the Solar Birth Aggregate

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N ≈ 2000 ±1100

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P ≈ 0.0085 (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 1257+ 12 (Wolszczan & Frail 1992)

Earth-like planet in a binary system

€

(MC ,ab,εb ,ibE )Variables:

Distribution of Ejection Times €

Distribution of Ejection Times

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Δτ ej /τ ej < 0.01⇔ N > 2625

Planet survival time for companion mass

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M∗ = 0.1Mo

Ejection Time versus Binary Periastron

€ €

Ejection Time versus Binary Periastron

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τ SS

€

(4.6Gyr)

Conservative Estimate:

Need binary periastron p > 7 AUfor the long term stability of and Earth-like planet

What fraction of the observed binarypopulation has periastron p > 7 AU?

David et al. 2003, PASP, 115, 825

Observed Distributions

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dPa = f (lna)d ln a =1

2πσexp −

(x − x0)2

2σ 2

⎡

⎣ ⎢

⎤

⎦ ⎥

€

dPε = 2εdε, dPμ = w(μ)dμ

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

Fraction of binaries with periastron less than given value p

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Ffit ( p) = 0.711exp −(0.101ξ + 0.0287ξ 2)[ ]

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ξ =ln 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 = 2000 • 50 percent of binary systems allow for Earth-like

planet to remain stable for the age of solar system

€

σ

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 StudiesSensitive dependence on initial

conditions in star and planet formation

Fred C. Adams

Physics Department

University of Michigan