Evolution of rocky and icy planetary bodies in the …fnimmo/website/CPS_2012_short.pdfWhy think...
Transcript of Evolution of rocky and icy planetary bodies in the …fnimmo/website/CPS_2012_short.pdfWhy think...
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Evolution of rocky and icy planetary bodies in the Solar System
Francis Nimmo
(U. C. Santa Cruz)
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Approach & Resources
http://www.es.ucsc.edu/~fnimmo/eart290q_09
http://www.es.ucsc.edu/~fnimmo/eart290q_11
http://www.es.ucsc.edu/~fnimmo/eart290c Planet formation & accretion:
Geophysics & heat transfer:
Satellites & tides:
• Pedagogical, not research • Generic, not specific, processes • Order of magnitude arguments (+equations)
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Mercury Venus
Earth
Moon
Mars
Ganymede
Io
Small N, lots of information (e.g. chemistry)
Large N, little information (especially for solid bodies!)
Same processes operating
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Why think about solid bodies? • Habitability (sigh) • Their surfaces, interiors,
orbits and chemistry give clues to their history (and that of the solar system)
• Gas giants (mostly) don’t do this
Present-day state
= Initial Conditions
Subsequent Evolution
+
Enceladus
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Outline
I. Accretion (long)
II. Heat Sources (short)
III. Heat Transfer (short)
IV. Tides (long)
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Part I - Accretion
The young Sun gas/dust nebula
solid planetesimals
• Why is accretion important?: – Universal process – It sets the initial conditions from which the bodies evolve – It can yield diverse outcomes
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Accretion - Topics
• 0. Introduction • 1. Bulk composition • 2. Spin/orbit state • 3. Later events • 4. Energy delivery (see below)
Resources: J. Chambers, in Exoplanets, Sara Seager (ed.), U.Az. Press, 2010 P.J. Armitage, Astrophysics of Planet Formation, C.U.P., 2010 A. Morbidelli, in Solar & Planetary Systems, French & Kalas, eds., Springer, 2012
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Sequence of events • 1. Nebular disk
formation • 2. Initial coagulation
(~10km, ~104 yrs) • 3. Runaway growth (to
Moon size, ~105 yrs) • 4. Oligarchic growth (to
Mars size, ~106 yrs), migration (?), gas loss
• 5. Late-stage collisions (~107-8 yrs)
?
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Accretion
Planet density ρ
Planetesimal swarm, surface density σ
R
Mean motion n
( ) 2 1 RndtdM
πσθ+=
2
2
rel
esc
vv
=θ
3/23/2 ~ Mn
dtdM
ρσ
3/423/1
~ MvGn
dtdM
relρσ
:1>>θ
:1<<θ
Runaway
Oligarchic
( ) 3/1/~ starH MMar
Accretion slows down once all material in a planet’s Hill Sphere rH has been accreted:
Semi-major axis a
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Warning! • Our Solar System does not resemble many
other planetary systems – high eccentricities & inclinations, “hot Jupiters” etc.
• Intuition developed by studying accretion of our (dynamically “cold”) Solar System may not apply to other planetary systems
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Late-stage Accretion
Raymond et al. 2006
Colours denote water abundance
• Volatiles arrive late – change in oxidation state? • Chem. evidence (Schonbachler et al. 2010, Rubie et al. 2011)
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Ensemble Outcomes
• Stochastic process – diversity
• Some radial mixing
• Close-by gas giants can have a significant effect (e.g. asteroid belt)
• Last, largest impacts dominate
O’Brien et al. (2006)
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Individual Growth History
Bodies experience collisions with comparable-mass objects
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Planet vs. Satellite Accretion
Decreasing tidal strain (H/Rs)
Vorb/Vesc decreases
• Satellites experience larger tides • Impact velocities are comparable to planets • Satellites require larger total surface densities
(but not all material present at same time)
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Consequences of Accretion • Large amounts of energy delivered for bodies greater
than ~ Mars-size (see below) • Initially homogeneous body differentiates into core plus
mantle • Magma oceans develop, leading to further
differentiation (e.g. lunar plagioclase crust)
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Core formation chronometry Kleine et al. (2002) Hf-W isotopic system
Rock+ Metal
Rock Metal
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1. Bulk Composition • Accretion is not 100% efficient! • Examples: Earth/Moon, Mercury, asteroids . . . • Chemical evidence? (non-chondritic Earth)
Asphaug et al. (2006)
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Diverse Outcomes
Asphaug (2009)
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Not all impacts add mass
ξ
Dwyer et al. (2010)
Accretion Efficiency
• Can generate oddball outcomes (e.g. Mercury)
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Debris Disks • Are some debris disks the
result of recent impacts?
Jackson & Wyatt (2012)
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2. Spin/Orbit Agnor et al. (1999)
• Spin rate close to break-up (on average)
• Spin orientation close to random (Uranus?)
• Are these results due to simplified models?
• Tides can modify subsequently (see below)
• Eccentricities/inclinations can be perturbed (but larger angular momentum budget)
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3. Waning Accretion • Earth suffered declining impact flux:
• Moon-forming impact (~10% ME, ~4.4 Ga) • “Late veneer” (~1% ME, 4.4-3.9 Ga) • “Late Heavy Bombardment” (0.001% ME, 3.9 Ga)
Bottke et al. 2010
• The LHB may have represented a “spike” due to reorganization of gas giant orbits (“Nice model”)
• Consequences for volatiles unclear – addition or blowoff?
Moon
Earth Mars
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Nice Model
Initial edge of planetesimal swarm
30 AU
Ejected planetesimals (Oort cloud)
48 AU 18 AU
J S U N
“Hot” population
Early in solar system
2:1 Neptune resonance
J S U N Neptune stops at original edge
Planetesimals transiently pushed out by Neptune 2:1 resonance
“Hot” population
“Cold” population
See Gomes et al., Nature 2005
3:2 Neptune resonance (Pluto)
Present day
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Part I (Accretion) - Summary • Late-stage planetary growth involves collisions
between like-size objects • Collisions are stochastic events - diversity • Accretion is not 100% efficient (though it is
usually modelled as such)
• Geochemical constraints on growth process exist • The last, large impact determines the initial
boundary conditions
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Part II – Heat Sources • Why are heat sources important?:
– High temperatures cause observable effects (differentiation, melting, dynamos etc.)
– Initial heating can influence long-term evolution – Long-term (Gyr) evolution controlled by balance between
heat sources and heat loss
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Topics
• 1. Insolation • 2. Radioactive decay • 3. Gravitational energy (impacts) • 4. Tides (see below) • 5. Induction heating (not covered)
Resources: Rubie et al.,Treatise Geophys., 2007
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1. Insolation • Determines surface temperature • Greenhouse effect, runaways (Venus) • Lava-ocean planets & “Eyeball Earths”
Pierrehumbert (2011) and Leger et al. (2011)
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2. Radioactive decay
• 26Al decay (t1/2=0.7 Myr) is extremely energetic • Planet growth time relative to 26Al decay time matters • 26Al was definitely present when some asteroids (and
perhaps Mars) formed and melted • K,U,Th provide main long-lived (Gyr) energy source
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3. Accretion • “Onion-shell model”, assuming no radiative losses
early later RGME
2
53
=Δ
3/23/13/2
K 40,000
≈≈Δ
Ep MM
CGMT ρ
• CAVEATS!: • For slow accretion and small impactors , radiation may be important • Impacts are large, discrete and stochastic, not continuous and small • Spatial heterogeneity may be important
None the less, Earth-mass bodies almost certainly started life molten
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Impacts • “Small” impacts (very roughly
< 1% of target mass) cause local heating
• “Large” impacts have global effects
• Size spectrum of impactors is very important
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“Small” Impacts
• Temperatures highest near surface • Melting only at shallow depths for Mars-sized object • Accretion entirely from small bodies makes cold bodies
1000 Radial distance, km
2000 3000 4000 0
500
1000
1500
2000
2500
Te
mpe
ratu
re, K
3000 Mantle melting
Core melting
Temperature
Method of Squyres et al. (1988)
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“Big” Impacts • Accretion involves
collisions between comparable-mass objects
• Assume all energy deposited into interior
M m=γM
Rubie et al. 2007
3/2
1.0K 6,000
≈ΔEMMT γ
• This is averaged temperature increase • Heating will in reality be (initially) spatially variable
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Magma “sea” readjustment
Tonks and Melosh 1992
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“Big” Impacts
• Whether melting occurs depends on both M and γ – impactor size spectrum is important
• 0.1 ME body suffering a single giant impact (γ=0.1) will be hot
M m =γM
Melting occurs Rubie et al. 2007
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Combined Effects
• Relative importance of impacts and radioactivity depends on body mass, impactor size and timescale
• Melting unavoidable for Earth-sized objects
Melting occurs
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Evolution with time
dtdMMGE ~ 3/23/1ρ
Accretion dominates for ~100 Myr Continuum approximation to discrete, stochastic, spatially variable process!
Io Earth
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N-body simulations
• Gravity is the dominant heat source
10,000 K
1000 K
100 K
Rubie et al. Treatise Geophys. 2007 Simulations courtesy Craig Agnor
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Differentiation and Core Formation
• Differentiation occurs when temperatures get high enough for melting to occur
• Differentiation releases further potential energy • Cores of Earth-size bodies start life hot (assuming rapid
transport of core material) • Hot cores are good for driving planetary dynamos
(Earth, Mars, Mercury?) • Differentiation leaves isotopic signatures (Hf/W) • Similar arguments apply to rock/ice mixtures
Rock+Metal
Rock
Metal
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Incomplete differentiation (?) • Titan (likely*) and Callisto (possibly) have not
completely differentiated – requires low T • This implies they were put together slowly, out of
small objects – constraint on accretion process
Barr and Canup (2010)
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Part II (Heat Sources) - Summary • Insolation only sets boundary conditions • Grav. energy depends on size-spectrum of bodies • Global melting is inevitable for Earth-sized
objects (magma oceans) • Radioactivity most important for small objects • For Earth-mass bodies, two epochs:
– Early (~100 Myr): accretion dominates – Later: long-lived radionuclides
• Melting leads to differentiation (core formation)
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Part III – Heat transfer • Why is heat transfer important?:
– It controls the duration and magnitude of a body’s geological activity (outgassing, dynamo etc.)
– It can (potentially) be remotely measured
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Topics
• 1. Magma oceans • 2. Solid state convection • 3. Advection (melt)
Resources: Rubie et al., Treatise Geophys., 2007
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Magma ocean evolution • Large bodies started life pervasively molten
• Magma ocean lifetimes highly uncertain: • Convective/radiative: few kyr (e.g. Solomatov 2000) • Conductive: tens of Myr (flotation crust, small bodies only) • Thick steam atmosphere: ~100 Myr (Zahnle et al. 2007)
• Magma oceans can produce unstable density structures (subsequent overturn)
• Is lifetime long or short compared to interval between “big” impacts?
• For how long would the IR emission be visible?
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Early thick atmosphere?
Zahnle et al. 2007
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Mantle convection 3/4
1-
3/1212-3/4
3/12
K 01.0s Pa 10 mWm 4~~
−−
γη
γκηαρ
bbsl
GRkF
3/13/1212-3/4
3/12
km 6000s Pa 10 mWm 150~~
Δ
RTGRkFbb
pt ηκηαρ
Io
Solomatov (1995)
Earth
Mantle viscosity temperature-dependent η=η0exp(-γT)
Stagnant lid
Isoviscous
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Stagnant lid vs. plate tectonics
• Yield strength (compared to convective stress) • Earth vs. Venus – water is important! • What does “yield strength” really mean?
Tackley (2002)
Low yield strength
High yield strength
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Icy satellite plate tectonics?
Sullivan et al., (1998)
20km 30 km
South Pole
O’Neill & Nimmo (2010)
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Dynamos • Dynamos (usually) depend on
how rapidly heat is being extracted by the mantle
• Whether or not plate tectonics operates can control dynamo activity (e.g. Earth vs. Venus)
• Early dynamos (Moon, Mars) are affected by initial hot core
• So initial conditions (accretion) may control dynamo operation
• Mechanically-stirred dynamos? (Dwyer et al. 2011)
Critical heat flux
Stagnant Lid
Plate tectonics
Hot core
Nimmo & Stevenson (2000)
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Melting
Solid lid
Partially- molten mantle
Solid mantle
~500
km
~50k
m
• Advection can be an efficient heat transfer mechanism
• E.g. Io 2 Wm-2 (!) • Near-surface melt transfer is
macroscopic (e.g. dikes) • Mantle melt transfer is
microscopic (porous flow) • Dihedral angle matters!
θ<60o
Rapid flow θ>60o
No flow
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Melting
Io
Earth
φρη
ρφ H
melt
nmelt LgcdF Δ2~
permeability driving/resistive stresses
advected heat
After Moore, Icarus, 2001
Stagnant lid
Isoviscous M
eltin
g st
arts
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Melting/Density • Deep mantle melting behaviour controls whether magma
ocean solidifies from top or bottom – important! • Melt-solid density contrast controls whether magma can
move upwards or not – affects e.g. CMB heat flux • E.g. “Deep magma ocean” on Earth
Labrosse et al. (2007)
solid
melt
melt
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Part III (Heat Transfer) - Summary • Molten or partially-molten mantles cool rapidly • Solid-state mantles cool slowly • Mantles spend a long time close to the melting
point 100 Myr 1 Gyr 10 Gyr 1 Myr 10 Myr
26Al Accretion
K,U,Th Sour
ces
Magma oceans
Mantle convection Melt advection Si
nks
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Part IV – Tides • Planetary tides are important for two reasons:
– We can use observations of tidal effects to constrain the internal structures of planetary bodies
– Tides play an important role in the orbital (and thermal) evolution of some bodies
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Decreasing tidal strain (H/Rs) satellites
exoplanets
satellites
exoplanets
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Topics
• 0. Introduction • 1. k2 and Q • 2. Despinning • 3. Tidal heating • 4. Inclination and obliquity
Resources: Murray & Dermott, Solar System Dynamics, CUP, 1999
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Basics
22 1 enah −= Angular momentum per unit mass. Compare with na2 for a circular orbit
Orbital angular momentum is conserved unless an external torque is acting upon the body
a ae
focus
r
b2=a2(1-e2) b
e is the eccentricity, a is the semi-major axis h is the angular momentum
m
An elliptical orbit has a smaller angular momentum than a circular orbit with the same value of a
GMan =32
aGMmE2
−=M
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Tides
a Rs M
H
m
3
2s
sRMH h R
m a =
Decreasing tidal strain (H/Rs)
H strongly influences tidal torques and tidal dissipation
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Rigidity • Reduces the tidal amplitude • Gravity competes with rigidity µ:
3/2
GPa 100 7.0~~
=MM
gRE
s
µρµ
µ
• E.g. Love number h2 for a uniform body:
)~1(1
25
2192 µ+
=h
• Rigidity dominant for small bodies, moderate for Earth-mass bodies, small(?) for larger bodies
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Tidal torques on the primary Torque spins down primary and moves secondary outwards (Reversed if within synchronous distance – exoplanets!)
Synchronous distance
Tidal bulge
Primary
Secondary
The Moon has moved outwards from ~5 RE to 60 RE over 4.5 billion years. The current measured recession rate (4 cm/yr) tells us how large the torques are, and thus how dissipative the Earth is, at present.
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Ariel’s orbit expands faster than Miranda’s because Ariel is so much more massive
Orbital evolution
• (Murray and Dermott 1999) • Passage through resonance may have led to transient eccentricities
and heating • Note that diverging paths do not allow capture into resonance
(though they allow passage through it), while converging paths do.
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Tidal torques on the secondary Synchronous distance
Tidal bulge
Tide raised by primary on secondary is large, so torque is large Primary
Secondary
• Synchronization is rapid for close-in objects (see later) • Rotation period may not exactly equal orbit period (see later) • Even synchronous objects generally experience tides . . .
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Diurnal Tides
• From a fixed point on the satellite, the resulting tidal pattern can be represented as a static tide (permanent) plus a much smaller component that oscillates (the diurnal tide)
Fixed point on satellite’s surface
Empty focus Planet
Tidal bulge
a
N.B. it’s often helpful to think about tides from the satellite’s viewpoint
eHHd 3=
Orbit with eccentricity e
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1. k2 and Q • Torques and dissipation both depend on k2/Q
Gribb and Cooper 1998
Qk2 Phase
lag
Tidal bulge amplitude. Depends on rigidity and
density.
Tidal bulge phase. Depends on viscosity & rigidity.
• Q is ~ number of cycles for energy to dissipate • Large Q means small phase lag/torque (!) • Q depends strongly on mechanical properties
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Observational constraints • Earth as a whole has a Q of 12 (oceans) • The solid Earth is not very dissipative (Q~300) • Mars is dissipative (Q~80) • So is the Moon (Q~30 at tidal periods)
• Io and Enceladus are generating observable heat, so we can infer k2/Q directly
• Gas giants (Saturn, Jupiter) have astrometrically-determined Q~104-105 (Lainey et al. 2009)
• Q is frequency-dependent!
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Observations of Q
Increasing dissipation
Maxwell model
Frequency-dependence (α~0.3)
Frequency-dependence (α~0.2)
0 -1 -2 -3 -4 -5 -6 -7 -8 -9
Apples vs. oranges?
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An observational constraint! Batygin et al. 2009 HATP-13
• Both k2 (and Q) have been inferred
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Tidal torques
Tns!k2s
Qs
Rs
a
"
#$
%
&'
6
GM2
Rs
eTTnssynch3≈
3
2
=aR
mMRhH s
s
Torque on non-synchronous satellite by primary:
Size of (static) tidal bulge:
Torque on synchronous satellite by primary:
Torque on primary by satellite can be calculated using symmetry
Rs
a M m
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2. Despinning to synchronous • Fast for close-in non-synchronous objects • Subsequent evolution (synchronous) is slower
−
QkRa
Ra
GkQ p
pp
s
/10
10/
kyr 30~~2
32/92/9
2/32/12 ρ
ρτ
Io
Mercury
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Non-synchronous rotation • Torque on a synchronous satellite is given by:
• This torque should increase the satellite’s rotation rate slightly above synchronous (Greenberg & Weidenschilling 1984) . . .
• . . . As long as there are no permanent mass asymmetries • Potentially very important for eccentric close-in exoplanets
Torque opposes spin Smaller
Torque increases spin Larger
Eccentric orbit
Periapse Apoapse planet
satellite
eTTnssynch3≈
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3. Tidal Heating • Diurnal tides – deformation -> heating • Heat output allows k2/Q determination (~0.01)
23/5
2/15
3/5
2/52/32~ em
RaG
QkE
ps
p
−
ρ
ρ
k2/Q=10-3
Synchronous rotator. Valid for small e.
Io
Enceladus Io
Enceladus
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Eccentricity Damping
3/2
32
32/133/22/13
6/132/1
6/10
2 10/
/10
10/
Myr 30~~
− mMQk
RamM
Ra
GkQ p
pp
s
ρρ
τ
k2/Q=10-3
• Energy from orbit, e should damp to zero • e-damping time long compared to despin time
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Resonances
• These ultimately involve transfer of (rotational) angular
momentum from the primary to the secondaries • In steady-state (de/dt=0), the dissipation rate in the
secondaries depends only on k2/Q of the primary
~ nk2p
Qp
Rp
a
!
"##
$
%&&
6
Gm2
Rp
~ n Tns
J I E G
One of the conjunctions occurring due to the Laplace resonance. Note that there is never a triple conjunction.
• Eccentricities will damp, unless they are being excited • Mean-motion resonances can excite eccentricities:
A possible observational constraint . . .
!Esteady
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Feedbacks and coupling Ojakangas & Stevenson 1986
• Dissipation in primary increases eccentricity
• Dissipation in satellite decreases eccentricity
• Heat transfer, dissipation and e-damping are coupled, because Q is strongly temperature-dependent
• Complex (periodic?) behaviour can result
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4. Inclination and obliquity
Primary
Inclination i
Obliquity θ
• Inclination damping is slow
• Many satellites occupy a Cassini state, in which the obliquity is controlled by the inclination
dtda
adtdii 4
11= E
GMma
dtda
22=
• τdespin << τecc << τinc
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Cassini state
2
23
2
rate precessionspin rate precessionorbit
kcJ
Mm
RaR pp≈=ε
θ and i are not independent, related via ε
θ∼-εi θ∼i
Winn & Holman 2005 Things we would like to know
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Obliquity tides & heating • Bulge moves “up-and-down”, rather than “side-to-side” • Otherwise heating effect same as eccentricity tides:
23/5
2/15
3/5
2/52/32~ θ
ρ
ρm
RaG
QkE
ps
p
−
• Crucial distinction: obliquity damps much more slowly than eccentricity (because controlled by inclination)
• So obliquity tides can be a good long-term source of heat for bodies in Cassini states (within limits – see Fabrycky et al. 2007)
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Summary • Tides depend strongly on a/Rp – important for
our satellites and many exoplanets • Tidal processes happen at different rates:
τdespin << τecc << τinc
• Tidal heating important in our solar system and likely elsewhere (resonances, inclinations)
• Orbital observations can constrain k2 etc. • Coupling between thermal and orbital evolution –
complicated problem . . . • . . . But may allow us to use orbital observations to
constrain interior state, or vice versa
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What have we learnt? • Late-stage impacts: generate initial diversity;
dominate the thermal budget for ~100 Myr • Tides and radioactivity: longer-term energy sources • Planets start hot; stay “slightly molten” for Gyrs • Orbit-interior coupling: challenge and opportunity
100 Myr 1 Gyr 10 Gyr 1 Myr 10 Myr
26Al Accretion
K,U,Th
Sour
ces
Magma oceans
Mantle convection Melt advection Si
nks
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Lessons • Observations of exoplanets will be limited, but:
– Young solid planets are good targets (luminous) – Bulk densities may be diagnostic of impact history – Tidal/interior coupling (e.g. HATP-13) – Likewise atmosphere/interior coupling
• Solid bodies are complex systems which defy simple predictions (Earth vs. Venus, Mimas vs. Enceladus)
• Chemistry helps! (in this Solar System) • Our Solar System is likely not typical - biases
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References (1) • Agnor et al. Icarus 142, 219-237, 1999 • Asphaug et al., Nature 439, 155-160, 2006 • Asphaug, Ann. Rev. Earth Planet. Sci. 37, 413-448, 2009 • Barr and Canup, Nature Geosci. 3, 164-167, 2010. • Batygin et al., Ap. J. L. 704, 49-53, 2009. • Bottke et al., Science 330, 1527-1530, 2010. • Dwyer et al., LPSC 41, 2271, 2010. • Dwyer et al., Nature 479, 212-214, 2011. • Fabrycky et al., Ap. J. 665, 754-766, 2007. • Gomes et al., Nature 443, 466-469, 2005. • Greenberg and Weidenschilling, Icarus 58, 186-196, 1984. • Gribb and Cooper, JGR 103, 27267-27279, 1998. • Jackson and Wyatt, MNRAS 2012 (submitted) • Kleine et al., Nature 418, 952-955, 2002. • Kokubo and Ida, Icarus 131, 171-178, 1998. • Labrosse et al., Nature 450, 866-869, 2007. • Lainey et al., Nature 459, 957-959, 2009. • Leger et al., Icarus 213, 1-11, 2011.
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References (2) • Moore, Icarus 154, 548-550, 2001. • Nimmo and Stevenson, JGR 105, 11969-11979, 2000. • O'Brien et al., Icarus 184, 39-58, 2006. • Ojakangas and Stevenson, Icarus 66, 341-358, 1986. • O’Neill & Nimmo, Nature Geosci 3, 88-91, 2010. • Pierrehumbert, Ap. J. L. 726, L8, 2011. • Raymond et al., Icarus 183, 265-282, 2006. • Rubie et al., Treatise Geophys. 9, 51-90, 2007. • Rubie et al., EPSL 301, 41-42, 2011. • Schonbachler et al., Science 328, 884-887, 2010. • Solomatov, Phys. Fluids 7, 266-274, 1995. • Solomatov, in Origin of the Earth and Moon, Canup and Righter, eds., U. Az. Press,
323-338, 2000. • Squyres et al., JGR 93, 8779-8794, 1988. • Sullivan et al., Nature 391, 371-373, 1998. • Tackley, Science 288, 2002-2007, 2000. • Tonks and Melosh, JGR 98, 5319-5333, 1993. • Winn and Holman, Ap. J. 628, L159-162, 2005. • Zahnle et al., Space Sci. Rev. 129, 35-78, 2007.
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Eccentricity damping
• Damping releases a lot of energy:
!Eecc
Egrav
~ 0.3M /m
104
"
#$
%
&'
2/3
10
a / Rp
"
#$$
%
&''!e
0.1
"
#$
%
&'
2
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Planetary Growth & Accretion • Early growth – from dust/gas to ~ 1 km (e.g.
Weidenschilling 1997) – Occurs over ~104 yrs at 1 AU • Runaway growth (e.g.Wetherill & Stewart 1989) – dM/dt ~ M4/3 – Terminates when v~vesc, ~105 yrs at 1AU • Oligarchic growth (e.g. Kokubo & Ida 1998) – dM/dt ~ M2/3
– Terminates at ~0.1 ME , ~106 yrs at 1 AU • Late-stage accretion (e.g. Agnor et al. 1999) – Stochastic, large impacts, 107-108 yrs