planetesimal* formaon* - university of...
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Planetesimal Forma.on
gas drag se1ling of dust turbulent diffusion damping and excita.on mechanisms for planetesimals embedded in disks minimum mass solar nebula par.cle growth core accre.on
Radial dri= of par.cles is unstable to streaming instability Johansen & Youdin (2007); Youdin & Johansen (2007)
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Gas drag
Gas drag force
where s is radius of body, v is velocity difference • Stokes regime when Reynold’s number is less than 10
• High Reynolds number regime CD ~0.5 for a sphere • If body is smaller than the mean free path in the gas Epstein regime (note mean free path could be meter sized in a low density disk)
FD = CD⇥gas�s2v2/2
CD � 24Re�1
FD =4�
3⇥gass
2csvEssen.ally ballis.c except the cross sec.on can be integrated over angle
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Drag Coefficient
cri.cal drop moves to the le= in main stream turbulence or if the surface is rough
Note: we do not used turbulent viscosity to calculate drag coefficients
Re =sv
�
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Stopping .mescale
• Stopping .mescale, ts, is that for the par.cle to be coupled to gas mo.ons
• Smaller par.cles have short stopping .mescales
• Useful to consider a dimensionless number tsΩ which is approximately the Stokes number
F = Mdv/dt ts =Mv
FD
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Se1ling .mescale for dust par.cles
• Use gravita.onal force in ver.cal direc.on, equate to drag force for a terminal velocity
• Timescale to fall to midplane
• Par.cles would se1le unless something is stopping them
• Turbulent diffusion via coupling to gas
Fz = Mdvz
dt�M
vz
tsFz = �M�2z
tsettle = z/z = z/vz = ��1(�ts)�1
vz = (ts�)z�
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Turbulent diffusion
• Diffusion coefficient for gas • For a dust par.cle • Schmidt number Sc • Stokes, St, number is ra.o of stopping .me to eddy turn over
.me • Eddy sizes and veloci.es
– Eddy turnover .mes are of order t~Ω-‐1
• In Epstein regime
• When well coupled to gas, the Diffusion coefficient is the same as for the gas
• When less well coupled, the diffusion coefficient is smaller
D0 = �cshD = �csh/Sc
Sc = (1 + St)
St � 34�
M
s2
�⇥gcs
Following Dullemond & Dominik 04
l �⇥
�h dv �⇥
�cs
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Mean height for different sized par.cles Diffusion vs se1ling
• Diffusion processes act like a random walk
• In the absence of se1ling • Diffusion .mescale
• To find mean z equate td to tse1le
• This gives
and so a predic.on for the height distribu.on as a func.on of par.cle size
z =�
Dt
z =�
< z2 >
td � z/ ˙z ˙zz =D
2td = 2z2/D
tsettle = ��1(�ts)�1
z2 =D
2�(�ts)�1
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Equilibrium heights
Dullemond & Dominik 04
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Effect of sedimenta.on on SED
Dullemond & Dominik 04
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Minimum Mass Solar Nebula
• Many papers work with the MMSN, but what is it? Commonly used for references: Gas Dust
• Solids (ices) 3-‐4 .mes dust density • The above is 1.4 .mes minimum to make giant planets
with current spacing – Hyashi, C. 1981, Prog. Theor. Physics Supp. 70, 35
• However could be modified to take into account closer spacing as proposed by Nice model and reversal of Uranus + Neptune (e.g. recent paper by Steve Desch)
�g = 2400� r
1AU
⇥�3/2g cm�2
�d = 10� r
1AU
⇥�3/2g cm�2
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Larger par.cles (~km and larger)
• Drag forces: – Gas drag, collisions, – excita.on of spiral density waves, (Tanaka & Ward) – dynamical fric.on – All damp eccentrici.es and inclina.ons
• Excita.on sources: – Gravita.onal s.rring – Density fluctua.ons in disk caused by turbulence (recently Ogihara et al. 07)
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Damping via waves • In addi.on to migra.on both eccentricity and inclina.on on averaged damped for a planet embedded in a disk. Tanaka & Ward 2004
• Damping .mescale is short for earth mass objects but very long for km sized bodies
• Balance between wave damping and gravita.onal s.rring considered by Papaloizou & Larwood 2000
e/e = �0.78 t�1wave
i/i = �0.54 t�1wave
twave = ⇥�1p
�Mp
M⇥
⇥�1⇤
�g(rp)r2p
M⇥
⌅�1 �cs
VK
⇥4
Note more recent studies get much higher rates of eccentricity damping!
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Excita.on via turbulence Stochas.c Migra.on
• Johnson et al. 2006, Ogihara et al. 07, Laughlin 04, Nelson et al. 2005
• Diffusion coefficient set by torque fluctua.ons divided by a .mescale for these fluctua.ons
• Gravita.onal force due to a density enhancement scales with
• Torque fluctua.ons • parameter γ depends on density fluctua.ons δΣ/Σ • γ~α though could depend on the nature of incompressible
turbulence • See recent papers by Hanno Rein, Ketchum et al. 2011
tfluc � ��1
Mp2�G�g
(⇥�)2 � (�Mp2⇤rG⇥g)2
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Eccentricity diffusion because of turbulence
• We expect eccentricity evolu.on • with
• or in the absence of damping
• constant taken from es.mate by Ida et al. 08 and based on numerical work by Ogihara et al.07
• Independent of mass of par.cle • Ida et al 08 balanced this against gas drag to es.mate when
planetesimals would be below destruc.vity threshold
< e2 >= Dt
D � (⇥�)2
L2⇤ � �2
�⇥gr2
M�
⇥2
⇤
e � 100�
��gr2
M�
⇥(⇥t)1/2
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In-‐spiral via headwinds
• For m sized par.cles headwinds can be large – possibly stopped by
clumping instabili.es (Johanse & Youdin) or spiral structure (Rice)
• For planetary embryos type I migra.on is problem – possibly reduced by
turbulent sca1ering and planetesimal growth (e.g., Johnson et al. 06)
Heading is important for m sized bodies, above from Weidenschilling 1977
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Density peaks Pressure gradient trapping
• Pressure gradient caused by a density peak gives sub Keplerian veloci.es on outer side (leading to a headwind) and super Keplerian veloci.es on inner side (pushing par.cles outwards).
• Par.cles are pushed toward the density peak from both sides.
figures by Anders Johansen
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Forma.on of gravita.onal bound clusters of boulders
points of high pressure are stable and collect par.cles
Johansen, Oishi, Mac Low, Klahr, Henning, & Youdin (2007)
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The restructuring/compac4on growth regime (s1 ≈ s2 ≈ 1 mm…1 cm; v ≈ 10-‐2…10-‐1 m/s)
Collisions result in s4cking
Impact energy exceeds energy to overcome rolling fric4on (Dominik and Tielens 1997; Wada et al. 2007)
Dust aggregates become non-‐fractal (?) but are s4ll highly porous
Low impact energy: hit-‐and-‐s.ck collisions
Intermediate impact energy: compac.on Blum & Wurm 2000 Paszun & Dominik, pers. comm.
From a talk by Blum!
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contour plot by Weidenschilling & Cuzzi 1993
1 AU
collisions bounce
collisions erode
from a talk by Blum
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Diameter
Dia
met
er
1 µm
100 m
100 µm
1 cm
1 m
1 µm 100 m 100 µm 1 cm 1 m
Non-fractal Aggregate Growth (Hit-and-Stick) Erosion
Non
-fra
ctal
Agg
rega
te
Stic
king
+ C
ompa
ctio
n
Cra
teri
ng/F
ragm
en-
tati
on/A
ccre
tion
Cratering/ Fragmentation
Restructuring/ Compaction
Eros
ion
N
on-f
ract
al A
ggre
gate
Gro
wth
(H
it-a
nd-S
tick
)
Non-fractal Aggregate Sticking + Compaction
Cratering/Fragmen-tation/Accretion
Cra
teri
ng/
Fr
agm
enta
tion
Mass loss
Mass conservation
Mass gain
* *
* *
* for compact targets only
Blum & Wurm 2008
1 AU
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Clumps
• In order to be self-‐gravita.ng clump must be inside its own Roche radius
• Concentra.ons above 1000 or so at AU for minimum solar mass nebula required in order for them to be self-‐gravita.ng
�s > M⇥/r3 � 6� 10�7g cm�3
⇤M⇥M⇤
⌅ � r
1AU
⇥�3
s < rH s < r
��ss3
M�
⇥1/3
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Clump Weber number
• Cuzzi et al. 08 suggested that a clump with gravita.onal Weber number We< 1 would not be shredded by ram pressure associated turbulence
• We = ra.o of ram pressure to self-‐gravita.onal accelera.on at surface of clump
• Analogy to surface tension maintained stability for falling droplets
• Introduces a size-‐scale into the problem for a given Concentra.on C and velocity difference c. Cuzzi et al. used a headwind velocity for c, but one could also consider a turbulent velocity
We =�gc2
G�2ds
2= C�2
⇤c
vK
⌅2 �r
s
⇥2⇤
M⇥/r3
�g
⌅
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Growth rates of planetesimals by collisions
• With gravita.onal focusing
• Density of planetsimal disk ρ, • Dispersion of planetesimals σ • Σ=ρh, σ=hΩ • Ignoring gravita.onal focusing • With solu.on
dM
dt= �s2⇥⇤
�1 +
v2esc
⇤2
⇥
dM
dt⇥�1 � �(M/�d)2/3
s � t
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Growth rate including focusing
• If focusing is large
• with solu.on quickly reaches infinity • As largest bodies are ones where gravita.onal focusing is important, largest bodies tend to get most of the mass
Fg =�
1 +v2
esc
�2
⇥� M
s
dM
dt��1 �M4/3
s(t) ⇥ 1C � �t
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Isola.on mass
• Body can keep growing un.l it has swept out an annulus of width rH (hill radius)
• Isola.on mass of order
• Of order 10 earth masses in Jovian region for solids le= in a minimum mass solar nebula
M = 2�a�rH rH = a(M/3M�)1/3
Miso � a3�3/2M�1/2⇥
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Self-‐similar coagula.on
• Coefficients dependent on s.cking probability as a func.on of mass ra.o
• Simple cases leading to a power-‐ law form for the mass distribu.on but with cutoff on lower mass end and increasingly dominated by larger bodies
• dN(M)/dt = – rate smaller bodies combine to make mass M
– subtracted by rate M mass bodies combine to make larger mass bodies
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Core accre.on (Earth mass cores)
• Planetesimals raining down on a core • Energy gained leads to a hydrosta.c envelope • Energy loss via radia.on through opaque envelope
• Maximum limit to core mass that is dependent on accre.on rate sezng atmosphere opacity
• Possibly a1rac.ve way to account for different core masses in Jovian planets
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Giant planet forma.on Core accre.on vs Gravita.onal Instability
• Two compe.ng models for giant planet forma.on championed by
– Pollack (core accre.on) – Alan Boss (gravita.onal instability of en.re disk)
• Gravita.onal instability: Clumps will not form in a disk via gravita.onal instability if the cooling .me is longer than the rota.on period (Gammie 2001)
Where U is the thermal energy per unit area
– Applied by Rafikov to argue that fragmenta.on in a gaseous circumstellar disk is impossible. Applied by Murray-‐Clay and collaborators to suggest that gravita.onal instability is likely in dense outer disks (HR8799A)
• Gas accre.on on to a core. Either accre.on limited by gap opening or accre.on con.nues but inefficiently a=er gap opening
tcool =U
�T 4� ��1
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Connec.on to observa.ons
• Chondrules • Composi.on of different solar system bodies
• Disk deple.on life.mes
• Disk velocity dispersion as seen from edge on disks
• Disk structure, composi.on
• Binary sta.s.cs
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Reading • Weidenschilling, S. J. 1977, Areodynamics of Solid bodies in the solar nebula,
MNRAS, 180, 57 • Dullemond, C.P. & Dominik, C. 2004, The effect of dust se1ling on the appearance
of protoplanetary disks, A&A, 421, 1075 • Tanaka, H. & Ward, W. R. 2004, Three-‐dimensional Interac.on Between A Planet
And An Isothermal Gaseous Disk. II. Eccentricity waves and bending waves, ApJ, 602, 388
• Johnson, E. T., Goodman, J, & Menou, K. 2006, Diffusive Migra.on Of Low-‐mass Protoplanets In Turbulent Disks, ApJ, 647, 1413
• Johansen, A., Oishi, J. S., Mac-‐Low, M. M., Klahr, H., Henning, T. & Youdin, A. 2007, Rapid Planetesimal forma.on in Turbulent circumstellar disks, Nature, 448, 1022
• Cuzzi, J. N., Hogan, R. C., & Shariff, K. 2008, Toward Planetesimals: Dense Chondrule Clumps In The Protoplanetary Nebula, ApJ, 687, 143
• Blum, J. & Wurm, G. 2008, ARA&A, 46, 21, The Growth Mechanisms of Macroscopic Bodies in Protoplanetary Disks
• Armitage, P. 2007 review • Ketchum et al. 2011 • Rein, H. 2012