L8: Pre-planetesimal growth Ormel (2016) [Star & Planet Formation || Lecture 8: pre-planetesimal...
Transcript of L8: Pre-planetesimal growth Ormel (2016) [Star & Planet Formation || Lecture 8: pre-planetesimal...
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L8: Pre-planetesimal growth
Chris Ormel (2016) [Star & Planet Formation || Lecture 8: pre-planetesimal growth] 2/25
Pre-planetesimal growth
● Collision physics– elastic, surface energy, sticking
● Coagulation equation– Smoluchowski equation, discrete vs continous, size
distribution function, analytical kernels, kernels
● Synthesis– fractal growth, compaction, fragmentation, “the
great divide”, many movies!
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Elasticity & surface tension
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– contact physics, Hertz force, elastic energysurface energy, contact area
– collision time, coagulation equation, size distributions
Blackboard
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JKRS potential
Q: Why negative δ and negative Ueq?
contact radius
disp
lace
men
t
energy minimum
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Collision kernelTime for particle i to collide with particles j:
Collision rate between i and j:
Kij = σcol Δv = collision kernel (= reaction rate).
Smoluchowski's equation describes evolution of distribution function
t col=(n j σ col Δ v)−1
dnij
dt=K ij ni n j
mass
num
ber
1 2 3 4 5
∂ni
∂ t=
12∑j+k=i
K jk n j nk−ni∑all j
K ij n j
Smoluchowski equation – discrete form:
Smoluchowski equation – continuous form:
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Collision kernel Kij
Collision kernel: Kij = σij * Δvij; orK(m1,m2) = ...
Three “analytical” kernels:
– constant K = 1– linear, K = (m1+m2)– product, K = m1*m2
one can solve analytically for the size distribution function f(m,t)
mass distribution for theconstant kernel →
Smoluchowski equation – continuous form:
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Collision kernel Kij
Collision kernel: Kij = σij * Δvij; orK(m1,m2) = ...
Three “analytical” kernels:
– constant K = 1– linear, K = (m1+m2)– product, K = m1*m2
one can solve analytically for the size distribution function f(m,t)
mass distribution for theconstant kernel →
Smoluchowski equation – continuous form:
Chris Ormel (2016) [Star & Planet Formation || Lecture 8: pre-planetesimal growth] 9/25
Collision kernel Kij
Collision kernel: Kij = σij * Δvij; orK(m1,m2) = ...
Three “analytical” kernels:
– constant K = 1– linear, K = (m1+m2)– product, K = m1*m2
one can solve analytically for the size distribution function f(m,t)
mass distribution for theconstant kernel →
Smoluchowski equation – continuous form:
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Linear kernel
Linear kernel out-paces the constant kernel!
Growth is exponential(m ~ exp[t]); mass-doubling time stays constant
Both distributions are self-similar (same shape)
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Product kernel
After t = 1:mass is no longer conserved; the small particles seems to get “eaten” by something.
Q: What happened?
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Kernel summary
Constant Linear Product
Analytical Kij = cnst Kij ∝ m1+m2 Kij ∝ m1 m2
Similar to:(if σgeo ∝ m2/3)
Brownian Motion(Δv ∝ m–1/2)
Turbulence(Δv ∝ m1/3)
Gravitational focusing(σ ∝ m4/3)
Size distribution Narrow Broad Discontinuous(runaway bodies)
However:– We have assumed perfect sticking!– Constant internal density (ρ•) is not
guaranteed! ρ• will evolve with time!
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Dust aggregates (fractals)
Seizinger et al. (2013)
Q: which aggregate is the largest?– by size– by mass– by aerodynamical size (tstop)
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Fractal dimension S
ize
Min et al. (2006)
Fractal lawmass ~ (size)Df
Df = fractal dimens.(1 < Df < 3)
Consequence:Particles stay aerodynamically small, since:
tstop ~ mass/Area
at least as long as Δv is small (hit-and-stick growth)
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Compaction, Fragmentation, Bouncing...
Paszun & Dominik (2009)
Hit-and-stick (low velocity) Restructuring/compaction(modest velocity) → fractal dimension increases
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Bouncing
Lab experiments (microgravity)(TU-Braunschweig)
Weidling et al. (2012)
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Fragmentation
Lab experiments (TU-Braunschweig)
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The great divide
● Fractal growth proceeds until planetesimals[e.g. Wada et al. 2008, Okuzumi et al. 2012, Kataoka et al. 2013]
collisional compaction inefficient; Δv lowaggregates keep a fractal structure; growth outpaces drift; narrow size distribution (only growth); planetesimals form quickly (and fluffy)
● Fragmentation barrier[e.g. Guettler et al. 2010, Zsom et al. 2010, Birnstiel et al. 2010]
collisional compaction efficient; Δv increases; fragmentation barrier; growth terminates; broad size distribution (b/c fragmentation); planetesimals form by gravitational instability (→Lecture 9)
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Summary: Birnstiel et al. (2010)
● Model include:– Coagulation +
fragmentation (size distribution)
– Radial drift (material transport)
● 3 Cases– Growth only– Growth & drift– Growth, drift, &
fragmentation orbital radius [AU]
grai
n si
ze [c
m]
Vertically-integrated distribution function
∫m2 f (m ,r , z)dz
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Growth only
Birnstiel et al. (2012)
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Growth + Drift
Birnstiel et al. (2012)
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Growth, drift, & fragmentation
Birnstiel et al. (2012)
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Use K(m',m'') = K(m'',m')somewhere
What is key property of Dirac-δ function?
What [physical principle] is expressed...
Exercise 1.5
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Exercise 1.6
See fig 1.5
Dimensional argument
z
σ = E* ε = E* ΔL/L
Equivalent to (dimensionless units)
U c=δac−23
δ ac3+
15
ac5