EART 160: Planetary Science

Last Time

• Celestial Mechanics– Kepler’s Laws– Newton’s Laws of Motion– Law of Universal Gravitation

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

Explanation of Kepler’s Laws

• Kepler observed orbital periods and distances, but didn’t 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 Kepler’s Third Law for any object.– Let’s do that now!

Kepler’s Third Law

• Compare orbital velocity to period

• I’ll show this for a circular orbit

• Works for elliptical orbit as well, but the derivation is unpleasant and not very informative.

• Should recover Kepler’s version if we stick in the Sun’s Mass, keep times in years, and distances in AU.

Circular Velocity

• Gravity imposes a centripetal acceleration to an orbiting object.

r

va

2

a

r

v

This is why planets don’t fall into the Sun.

And why it’s so hard to get to Mercury!

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)

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?

Kepler’s Second Law

d

r

v┴ = v sin v

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.

Earth-Moon System

Earth

Moon

r

•The Moon is moving away from the Earth!

•The day is getting longer!

•Earth’s spin angular momentum turns into Moon’s orbital angular momentum.

•This will continue until the spins and orbits match (syncrhonous rotation)

•Common for nearly all satellites

Kepler’s 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 < -U KE + U < 0

• Unbound orbits are hyperbolae (or parabolae)– One pass and gone for good (e.g. many comets)– Positive total energy. KE +U > 0.

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

Inelastic Collisions

Dust Grains colliding during solar system formation

Impacts

Elastic Collisions

“Collision” with no impactJust Gravity

Without this, solar system explortationwould be slow and expensive.

Saved 19 years off Voyager 2’s tripto Neptune!

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.

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.

A successful theory must explain:

• All planets’ orbits in a single plane.

• Sun’s 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

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

The Solar Nebula

• Gravitational collapse of part of cloud– Virial Theorem

• Central part heats up– Why? – Conservation of Energy!– Protostar (becomes star if M > 0.08 M)

• Rotation rate increases– Why? – Conservation of Angular Momentum!

• Flattens into protoplanetary disk (proplyd)

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 R

GM 2

~

R

M,Thermal energy

HM

MkTkTN

~~

Equating these two and using M~R3 we get:

2~

RG

kTcrit

Does this make sense?

M=mass; =density; R=radius;k=Boltzmann’s constant; T=temperature (K)N=no. of atoms; =atomic weight; MH=mass of H atom

Proplyds in the Orion Nebula

HST Images Courtesy NASA/ESA/STSci

Beta Pictoris – 50 ly

HH-30 in Taurus

Bipolar Outflow

Disks radiate in the infraredAll very young; few My

Minimum Mass Solar Nebula

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

Density drops off with distance.

COINCIDENCE?!?!?!

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)

Collisional Accretion (104 y)

Inelastic Collisions between dust grains

Vertical Motions canceled outDisk orientation controlled by angular momentumDisk’s gravity also draws material toward midplane

Dust grains also accrete onto chondrules: solidified molten fragments

Forms PlanetesimalsR < few km

Runaway Growth (105 y)• Slow-moving planetesimals accrete• Protoplanets grow to size of moon (3500 km)

Fg = GMm / R2

vorbital < vesc

vorbital > vesc “The rich get richer!”-- Bender

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.

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

Condensation in the Nebula

Polar jets

Stellar magnetic field (sweeps innermost disk clear, reduces stellar spin rate)

Disk cools by radiation

Dust grains Infallingmaterial

Nebula disk(dust/gas)

Hot, high

Cold, low

Metals and Rocks Ices1600 K 180 K

The Frost Line

Gas giants Ice giants Terrestrial planets

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?

Final Compositions

•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).

Io

Ganymede

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 Mars’s moons) captured later (high e, i)

• What about our own freakishly large Moon?

Problems with this

• Why exactly four terrestrial planets?– Numerical models can’t 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 don’t go all the way to end state

• Migration (next time)

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

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 Earth’s Moon

Jupiter: The Cosmic Bully

• It’s 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

From Albarede, Geochemistry: An introduction

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