EART 160: Planetary Science

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EART 160 Planetary Science
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Last Time

• Celestial Mechanics
• Keplers Laws
• Newtons Laws of Motion
• Law of Universal Gravitation

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Today

• Celestial Mechanics
• Explain Kepler with Newton
• Conservation
• Solar System Origin
• Formation of the Solar System
• Nebular Theory
• Distribution of solar system materials
• Planet formation, composition, structure

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Explanation of Keplers Laws

• Kepler observed orbital periods and distances,
  but didnt know what caused it.
• Third Law only works for the Sun, using Earth as
  a reference.
• Newton finds force of gravity is what moves
  planets toward the sun.
• Can extend Keplers Third Law for any object.
• Lets do that now!

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Keplers Third Law

• Compare orbital velocity to period
• Ill show this for a circular orbit
• Works for elliptical orbit as well, but the
  derivation is unpleasant and not very
  informative.
• Should recover Keplers version if we stick in
  the Suns Mass, keep times in years, and
  distances in AU.

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Circular Velocity

• Gravity imposes a centripetal acceleration to an
  orbiting object.

v
a
r
This is why planets dont fall into the Sun. And
why its so hard to get to Mercury!
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Conservation Laws

• Momentum
• If the vector sum of the external forces on a
  system is zero, the total momentum of the system
  is constant.
• Momenta of individual objects can change.
• Angular Momentum
• When the net external torque on a system is zero,
  the total angular momentum of the system is
  constant.
• Angular Momenta of individual objects can change.
• Energy
• Cannot be created or destroyed
• Can be converted from one form to another(e.g.
  from potential to kinetic)

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Escape Velocity

• How fast does an object have to go to escape the
  gravitational pull of a planet?
• Conservation of Energy
• Balance the Potential Energy due to gravity
  against the Kinetic Energy due to motion
• Collapse of solar nebula ? lots of potential
  energy lost. Where does it go?

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Keplers Second Law
v
v- v sin j
j
r
dq
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Law of Areas

• Conservation of Angular Momentum
• Object moves fast near periapse (short lever
  arm), slow near apoapse (long lever arm.
• Energy shifts from kinetic to potential and back.

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Earth-Moon System

• The Moon is moving away from the Earth!
• The day is getting longer!
• Earths spin angular momentum turns into Moons
  orbital angular momentum.
• This will continue until the spins and orbits
  match (syncrhonous rotation)
• Common for nearly all satellites

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Keplers First Law

• Derivation is unpleasant
• Requires Differential Equations
• Pure mathematics, no science involved
• Shall we skip it?
• Bound orbits are ellipses (or circles)
• Not enough KE to escape, keep orbiting
• Negative total energy! KE lt -U ? KE U lt 0
• Unbound orbits are hyperbolae (or parabolae)
• One pass and gone for good (e.g. many comets)
• Positive total energy. KE U gt 0.

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Collisions

• Conservation of Momentum
• Inelastic collison Kinetic energy not conserved
• But total energy is! Some goes into heating or
  deformation
• Objects may stick together (completely inelastic)
• Elastic collision Kinetic energy is conserved

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Inelastic Collisions
Impacts

• Dust Grains colliding during solar system
  formation

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Elastic Collisions
Collision with no impactJust Gravity
Without this, solar system explortation would be
slow and expensive.Saved 19 years off Voyager
2s trip to Neptune!
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Two-body problem

• All this is derived for two bodies, as if nothing
  else exists in the universe.
• Good approximation if one body is very large.
• Third body causes perturbations
• Three-body problem is analytically unsolvable in
  general.
• Good treatment of restricted three-body problem
  in Murray and Dermott (1999) Solar System
  Dynamics.

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Solar System Formation

• Why do we care?
• Current state of the solar system controlled by
  the initial conditions
• Recall Stevenson 2000
• Composition, Distribution, Rotation
• To understand planets, we need to know how they
  got here.

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A successful theory must explain

• All planets orbits in a single plane.
• Suns rotation in same plane.
• Prograde orbits of all planets
• Planetary orbits nearly circular
• Angular momentum distribution
• Some meteorites contain unique inclusions
• Correlation of planetary composition with solar
  distance.

• Meteorites different from terrestrial and lunar
  rocks
• Spacing of the planets
• Giant impacts on all planetary bodies
• Prograde rotation, low obliquity of most planets
• Similar rotation periods for many planets
• Spherical distribution of comets
• Satellite sysems of giant planets

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Prelude to the Solar System

• Big Bang (14 Ga)
• Creation of Matter (75 H, 25 He)
• Star Formation
• Nucleosynthesis (All elements up to Fe)
• Supernovae
• All other elements formed
• Material ejected into interstellar space
• Nebula Dense cloud of gas and dust
• Thousands of M?

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The Solar Nebula

• Gravitational collapse of part of cloud
• Virial Theorem
• Central part heats up
• Why?
• Conservation of Energy!
• Protostar (becomes star if M gt 0.08 M?)
• Rotation rate increases
• Why?
• Conservation of Angular Momentum!
• Flattens into protoplanetary disk (proplyd)

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Jeans Collapse

• A perturbation will cause the density to increase
  locally
• Runaway Process
• Increased density ? increased gravity ? more
  material gets sucked in

Gravitational potential energy
M,r
Thermal energy
R
Equating these two and using MrR3 we get
Does this make sense?
Mmass rdensity Rradius kBoltzmanns
constant Ttemperature (K) Nno. of atoms
matomic weight MHmass of H atom
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Proplyds in the Orion Nebula
Disks radiate in the infrared All very young few
My
Beta Pictoris 50 ly
Bipolar Outflow
HH-30 in Taurus
HST Images Courtesy NASA/ESA/STSci
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Minimum Mass Solar Nebula
Density drops off with distance. COINCIDENCE?!?!?
!

• We can use the present-day observed planetary
  masses and compositions to reconstruct how much
  mass was there initially

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Timeline of Planetary Growth

• 1. Nebular disk formation
• 2. Initial coagulation (10km, 104 yrs)
• 3. Runaway growth (to Moon size, 105 yrs)
• 4. Oligarchic growth, gas loss(to Mars size,
  106 yrs)
• 5. Late-stage collisions (107-8 yrs)

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Collisional Accretion (104 y)
Inelastic Collisions between dust grains
Dust grains also accrete onto chondrules
solidified molten fragments
Forms Planetesimals R lt few km
Vertical Motions canceled out Disk orientation
controlled by angular momentum Disks gravity
also draws material toward midplane
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Runaway Growth (105 y)

• Slow-moving planetesimals accrete
• Protoplanets grow to size of moon (3500 km)

Fg GMm / R2
vorbital lt vesc
vorbital gt vesc
The rich get richer! -- Bender
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Oligarchic Growth (105 y)

• Cosmic Feudal System
• Only a few dozen big guys left (oligarchs)
• And a lot of very small stuff (serfs?)
• Oligarchs sweep up everything in their feeding
  zones
• Gas drag slows large objects down, circularizes
  orbits
• Brightening sun clears away nebular gas.

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Composition

• Solar Nebula
• 98.4 gas (H, He)
• 1.1 ices (e.g. H2O, NH3, CH4)
• 0.4 rock (e.g. MgSiO4)
• 0.1 metal (mostly Fe, Ni)
• How do we know this?
• Look at the Sun!
• Absorpiton lines indicate elements
• Discovery of He

Volatile
Refractory
Image courtesy N.A.Sharp, NOAO/NSO/Kitt Peak
FTS/AURA/NSF
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Condensation in the Nebula
Metals and Rocks Ices 1600 K 180 K
The Frost Line
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Terrestrial v. Jovian

• Only refractories in inner SS
• Planets can only grow to Earth-size
• Too small to hold onto gas
• Ices also available beyond frost line
• Much more material
• Ice-rock planets up to 20 M? possible
• Big enough to accrete H, He ? can get huge, 300
  M?
• How big do we need to get?
• Why no giant planets farther out than Neptune?

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Final Compositions
Io
Ganymede

• Terrestrial Planets
• Iron Core (Red), Silicate Mantle (Grey)
• Mercury has v. thin mantle. Why?
• Very few volatiles, thin atmospheres?
• Jovian Planets
• Rock (Grey) and Ice (Blue Cores)
• Gas envelope (Red, Yellow)
• Jupiter and Saturn mostly H, He
• Uranus, Neptune mostly ice

Guillot, Physics Today, (2004).
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Satellites

• Satellites formed from mini-accretion disks about
  giant planets
• Explains why they all orbit the same way and in
  the same plane
• Irregular satellites (including Marss moons)
  captured later (high e, i)
• What about our own freakishly large Moon?

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Problems with this

• Why exactly four terrestrial planets?
• Numerical models cant do this.
• What is up with the Moon?
• Gas Loss Timing
• As star heats up, gas in disk is blown away
• Gas causes planets to spiral in
• Gas must stick around long enough to form giant
  planets
• Why are Uranus and Neptune so shrimpy?
• Why are extrasolar planets so close in?
• Alan Boss
• Rapid giant planet formation by disk instability
  (100s of years)
• But computer models dont go all the way to end
  state
• Migration (next time)

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Next Time

• Late-stage Accretion
• Formation of the Moon
• The Late Heavy Bombardment
• Planetary Migration
• You should now have everything you need to
  complete the homework

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Late-stage accretion (107-108 y)

• Oligarchic growth results in dozens of
  planetesimals
• Perturb each other until orbits cross
• Giant Impacts
• Retrograde rotation of Venus
• Obliquity of Uranus
• Formation of the Earths Moon

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Jupiter The Cosmic Bully

• Its huge! Perturbs anything nearby
• Disrupted accretion at 2-3 AU
• No planet here where we expected one.
• Location of the asteroid belt
• Ejected icy planetesimals
• Gravitational slingshot effect
• Scattered in all directions ? The Oort Cloud

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From Albarede, Geochemistry An introduction
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Observations (2)

• We can use the present-day observed planetary
  masses and compositions to reconstruct how much
  mass was there initially the minimum mass solar
  nebula

• This gives us a constraint on the initial nebula
  conditions e.g. how rapidly did its density fall
  off with distance?
• The picture gets more complicated if the planets
  have moved . . .
• The observed change in planetary compositions
  with distance gives us another clue silicates
  and iron close to the Sun, volatile elements more
  common further out