chaotic case studies: sensitive dependence on initial conditions in star/planet formation fred c....
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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
€
mP =1mJ
€
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
€
E0 = −GM∗mP
2a0
€
ΔE = −GM∗μ
2r
Starting energy:
Scattering changes energy:
€
a1 = a0(1+ μ /mP )−1, an = a0(1+ μ /mP )−n
fn = a0 /an = (1+MS /mP
n)n
€
n →∞lim fn = exp[MS /mP ]
€
MMES = mP ln[a0 /a f ]
Surface density distribution
€
σ(r) =μ
2πrk=1
n
∑ δ(r − ak )
€
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
€
P1 =1900d,P2 = P1 × 3.736...
€
mP = 0 − 5mJ
€
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
€
FSN = 0.000485
€
ε(Neptune) < 0.1
€
ΔΘ j < 3.5o
Probability of Supernovae
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PSN (N) =1− fnotN =1− (1− FSN )N
Probability of Scattering Event
€
Γ=n σvScattering rate:
Survival probability:
€
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:
€
v ≈1 km /s
Density and time:
€
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
€
N ≈ 2000 ±1100
€
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
€
Δτ ej /τ ej < 0.01⇔ N > 2625
Planet survival time for companion mass
€
M∗ = 0.1Mo
Ejection Time versus Binary Periastron
€ €
Ejection Time versus Binary Periastron
€
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
€
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
€
Ffit ( p) = 0.711exp −(0.101ξ + 0.0287ξ 2)[ ]
€
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
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