EART 160: Planetary Science

29 February 2008

HW 5 Common Mishaps

• Sum of logs is not log of sums!

• Hydrostatic Equilibrium

• Mass of a column

• Scale Height, H– If we took the mass of the

entire column and compressed it to the surface density, 0, the column would have height H

BABA

BABA

ln*ln)ln(

lnln)ln(

dzM

dVM

c

gdzdP

ghP

If constant, h well defined

In general, (z increases up)

Histogram

0

1

2

3

4

5

60 3 6 9

12 15 18 21 24 27 30

Mor

e

Bin

Frequency

Mean: 22Max: 27Min: 8St. Dev.:

Grades

• All HWs and midterm grades on WebCT.– “Final” Grade computed from 80% HW avg,

20% midterm

• Few are not doing so well, – Please talk to me!– Everybody can still pass

• Paper Discussions– 70% for summary of your paper– 30% for general discussion

• Participate more in last few to get full credit for this!

Last Time

• Tidal Interactions– Spin-orbit exchange– Orbital Resonances– Tidal Heating

• Paper Discussions– Namouni and Porco (2002)– Porco et al. (2006)

Today

• Tidal Interactions– Roche Limit– Rings– Orbital Niftiness

• Comets– Introduction

• Paper Discussion– Levison et al., 2002 -- KBO’s

Comet Hyakutake, 1996, NASA photo

Roche Limit• If a satellite gets too close to a planet, it will be

pulled apart by tidal forces (e.g. comet SL-9)• The distance from the planet that this happens is

called the Roche limit• It determines where planetary rings are found

• The satellite experiences a mean gravitational acceleration of GMp/a2

• But the point closest to the planet experiences a bigger acceleration, because it’s closer by a distance Rs (i.e. tides)

• The net acceleration of this point is

• If the (strengthless) satellite is not to break apart, this acceleration has to be balanced by the gravitational attraction of the satellite itself:

3

2

a

RGM sp

2s

s

R

GM

• This expression is usually rewritten in terms of the densities of the two bodies, and has a numerical constant in it first determined by Roche:

a

RpRs

Mp

Ms s

p

3/1

456.2

s

p

pR

a

Rings

• When a body passes the Roche limit it breaks apart – forms a ring system

• Rings are collections of tiny particles and “moonlets” – particles have enough strength

• What’s the cutoff between moon and ring particle?

Zoom Out

“One ring to rule them all, and in the darkness bind them”-- J. R. R. Tolkien

Rings are just collections of icy particles and dust

Each ring particle acts like a moon. Each particle obeys Kepler’s Laws

The Fellowship of the Ring

Jupiter’s Rings

Moons control thickness of rings

Saturn’s Rings Close-up

Esposito: Good For Astronomy But Can’t

Draw. Saturn’s Main Rings.

Thinner than a sheet of paper

Why are Saturn’s rings so nifty?

•Rings are probably only a few million years old!

•Tidal disruption? Not common enough.•Micrometeorites grind down small moons

•Continuously make particles•Particles keep getting ground smaller

•Small particles spiral into planet

•Saturn must have lost a moon recently•Enceladus is the source of the E-ring

What’s with the gaps?

• Resonances

• Mimas clears gaps– Cassini Division (2:1)– Encke Gap (5:3)– Same as Jupiter clearing out Asteroid Belt

Shepherd Moons

Prometheus and Pandora

F Ring

Roche limits

Roche limits

Jupiter Saturn

How do we get satellites inside the Roche limit?

Roche limits

Roche limits

Uranus Neptune

Summary• Tides arise because gravitational attraction varies

from one side of a body to the other• Tides generate torques • Dissipation in the primary normally causes the

primary to spin down, and the satellite to move out• In the absence of external torques, orbital angular

momentum is conserved (e.g. Earth-Moon system)• Rate at which energy is dissipated controls the

satellite recession rate• Tidal torques result in synchronous satellite orbits• Tides can rip a body apart if it gets too close to the

primary (Roche limit)• Diurnal tides (for eccentric orbit) can lead to

heating and volcanism (Io, Enceladus)

Orbital Stability

• Lagrange Points– Equilibrium points in two-

body rotating system– L1-3 unstable

• Horseshoe orbits– Follows equipotential– Two moons can exchange

ang. mom., swap to other orbit

– E.g. Janus & Epimetheus

“Home, home on Lagrange …”

The Moon• Phase-locked to the Earth (its rotation rate was

slowed by torques from tides raised by the Earth)

Image taken by Galileo (the spacecraft, not the man)

• Has moonquakes which repeat once every month in the same – presumably triggered by tidal stresses

Lunar Recession• The Apollo astronauts left laser reflectors on the

surface (as well as seismometers)• So we can measure the rate at which the Moon is

receding due to tidal torques: ~4 cm per year• As a result, the Earth is spinning down, by about 2s

per 100,000 years (conservation of angular momentum)

McDonald Observatory, TexasApollo 14 laser reflectometer

The Problem• The Moon must only have

formed 1-2 Gyr ago!• Major embarrassment for

geophysicists• Also used as an argument

by Creationists

time~1.5 Gyr ago

distance

Present day

Known recessionrate (gives us Q)

What is the solution?

Constant Q

4.5 Gyr ago

Higher Q in past

• The Earth’s Q must have been higher (i.e. less dissipation) in the past

• What controls dissipation in the Earth?

• Bulk of the dissipation occurs in the oceans

• What controls dissipation in the oceans?

• Bathtub effect – sloshing gets amplified if the driving frequency equals the resonant frequency of the basin.

• What controls the resonant frequency?

The Solution (cont’d)

Plate Tectonics!• Resonant frequency of an ocean basin is

controlled by its length• So as continental drift occurs, the length of the

ocean basins changes and so does the amount of dissipation

• There will also be an effect from sea-level changes – much of the dissipation occurs on shallow continental shelves

• So in the past, when the continental configuration was different, oceanic dissipation was smaller and the Moon retreated more slowly

So the evolution of the Moon’s orbit is controlled by plate tectonics!

Comets / TNOs• General Structure: What is a comet?• Dynamics:

– Where are the comets?– Kuiper Belt and Oort Cloud– What happens to a comet when it gets near the Sun?– Orbits tell us about how the early solar system was

assembled

• What are comets made of?– Pristine (or nearly pristine) samples of high volatile

components of original nebula– Important source of volatiles and organic matter to inner

solar system (astrobiology, atmospheres)

Comets

• Coma – Greek for “hair”• Mostly ice with some rock mixed in

• “"Ah, yes. Comets, the icebergs of the sky. By jackknifing from one to the next at breakneck speed, we might just get some kind of gravity boost, or something."

-- Zapp Brannigan

• How do we know what they’re made of?– Spectroscopy!

• Proof that the Heavens changed• Ill omen in ancient times

Classic Anatomy of Comet

• Nucleus: 10 km

• Coma: cloud of gas surrounding nucleus, 104 km

• Tails: 107 km long– Plasma Tail points

away from Sun– Dust Tail trails a bit

behind direction of motion

Why do the tails point in different directions?

Question

• What is Sun’s gravity at edge of Oort cloud?

• What is Sun’s gravity at Earth’s orbit?

Next time

• More on Comets– Composition– Homemade comet– Dynamics

• Trans-Neptunian Objects– Populations