Francis Nimmo

1
ESS 298 OUTER SOLAR SYSTEM

• Francis Nimmo

Io against Jupiter, Hubble image, July 1997
2
Course Outline

• Week 1 Introduction, solar system formation,
  exploration highlights, orbital dynamics
• Weeks 2-3 Galilean satellites
• Week 4 Titan and the other Saturnian
  satellites, Cassini results
• Week 5 Gas giants and ice giants structure,
  atmospheres, rings, extra-solar planets
• Weeks 6-7 Computer project
• Weeks 8-9 Student presentations
• Weeks 10 The Outer Limits Pluto/Charon,
  Kuiper Belt, Oort Cloud, future missions

This schedule can be modified if someone is
interested in a particular topic
3
Logistics

• Set texts see website for suggestions
• http//www2.ess.ucla.edu/nimmo/ess298
• Office hours make appointments by email
  nimmo_at_ess.ucla.edu or drop by (4642 Geology)
• Auditing?
• Student presentations 30 min. talk on
  controversial research topic. Sign-up sheets week
  4.
• Grading based on performance in student
  presentation (70) and computer project writeup
  (30). P/NP or letter grade.
• Location/Timing Tues/Thurs 1115-1230 in room
  4677 Geology
• Questions? - Yes please!

4
This Week

• Where and what is the outer solar system?
• What is it made of ?
• How did it form?
• How do we know? (spacecraft missions and
  ground-based observations)
• Highlights
• Orbital dynamics
• Keplers laws
• Moment of inertia and internal structure
• Tidal deformation

5
Where is it?

• Everything beyond the asteroid belt ( 3AU)
• 1 AUEarth-Sun distance 150 million km
• Jupiter, Saturn, Uranus, Neptune, Pluto, plus
  satellites
• Kuiper Belt
• Oort Cloud

Inner solar system
1.5 AU
5 AU
30 AU
Outer solar system
6
Where is it? (contd)
Distances on this figure are in AU. Areas of the
planets are scaled by their masses. Percentages
are the total mass of the solar system (excluding
the Sun) contained by each planet. Note that
Jupiter completely dominates. We conventionally
divide the outer solar system bodies into gas
giants, ice giants, and small bodies. This is a
compositional distinction. How do we know the
compositions?
7
Basic Parameters
a (AU) Period (yr) e Rotation (hr) R (km) M (1024 kg) r Ts (K) m x10-4
Earth 1.0 1.0 .017 23.9 6371 6.0 5.52 290 0.61
Jupiter 5.20 11.9 .048 9.93 71492 1899 1.33 165 4.3
Saturn 9.57 29.4 .053 10.7 60268 568 0.69 134 0.21
Uranus 19.2 84.1 .043 R17.2 25559 87 1.32 76 0.23
Neptune 30.1 164 .010 16.11 24764 102 1.64 72 0.13
Pluto 39.5 249 .25? R6.38d 1152 0.01 1.9 40 ?
Data from Lodders and Fegley, 1998. a is
semi-major axis, e is eccentricity, R is radius,
M is mass, r is relative density, Ts is
temperature at 1 bar surface, m is magnetic
dipole moment in Tesla x R3.
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Compositions (1)

• Well discuss in more detail later, but briefly
• (Surface) compositions based mainly on
  spectroscopy
• Interior composition relies on a combination of
  models and inferences of density structure from
  observations
• We expect the basic starting materials to be
  similar to the composition of the original solar
  nebula (how do we know this?)
• Surface atmospheres dominated by H2 or He

Solar Jupiter Saturn Uranus Neptune
H2 83.3 86.2 96.3 82.5 80
He 16.7 13.6 3.3 15.2 (2.3 CH4) 19 (1 CH4)
(Lodders and Fegley 1998)
9
Compositions (2)

• Jupiter and Saturn consist mainly of He/H with a
  rock-ice core of 10 Earth masses
• Uranus and Neptune are primarily ices covered
  with a thick He/H atmosphere
• Pluto is probably an ice-rock mixture

90 H/He
75 H/He
10 H/He
10 H/He
Figure from Guillot, Physics Today, (2004). Sizes
are to scale. Yellow is molecular hydrogen, red
is metallic hydrogen, ices are blue, rock is
grey. Note that ices are not just water ice, but
also frozen methane, ammonia etc.
10
Temperatures

• Obviously, the stability of planetary
  constituents (and thus planetary composition)
  depends on the temperature as the planets formed.
  Well discuss this in a second.
• The present-day surface temperature may be
  calculated as follows

• Here F is the solar constant (1367 Wm-2), Ab is
  the Bond Albedo (how much energy is reflected), a
  is the distance to the Sun in AU, e is the
  emissivity (typically 0.9) and s is the
  Stefan-Boltzmann constant (5.67x10-8 in SI
  units).
• Where does this equation come from?

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Temperatures (contd)

• Temperatures drop rapidly with distance
• Volatiles present will be determined by local
  temperatures
• Volatiles available to condense during initial
  formation of planets will be controlled in a
  similar fashion (although the details will differ)

Neptune
Jupiter
Saturn
Uranus
Plot of temperature as a function of distance,
using the equation on the previous page with
Ab0.1 to 0.4
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Solar System Formation - Overview

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

13
Observations (1)

• Early stages of solar system formation can be
  imaged directly dust disks have large surface
  area, radiate effectively in the infra-red
• Unfortunately, once planets form, the IR signal
  disappears, so until very recently we couldnt
  detect planets (see later)
• Timescale of clearing of nebula (1-10 Myr) is
  known because young stellar ages are easy to
  determine from mass/luminosity relationship.

This is a Hubble image of a young solar system.
You can see the vertical green plasma jet which
is guided by the stars magnetic field. The white
zones are gas and dust, being illuminated from
inside by the young star. The dark central zone
is where the dust is so optically thick that the
light is not being transmitted.
Thick disk
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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 change in planetary compositions with
  distance gives us another clue silicates and
  iron close to the Sun, volatile elements more
  common further out

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Cartoon of Nebular Processes

• Scale height increases radially (why?)
• Temperatures decrease radially consequence of
  lower irradiation, and lower surface density and
  optical depth leading to more efficient cooling

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Temperature and Condensation
Nebular conditions can be used to predict what
components of the solar nebula will be present as
gases or solids
Mid-plane
Photosphere
Earth
Saturn
Condensation behaviour of most abundant elements
of solar nebula e.g. C is stable as CO above
1000K, CH4 above 60K, and then condenses to
CH4.6H2O. From Lissauer and DePater, Planetary
Sciences
Temperature profiles in a young (T Tauri) stellar
nebula, DAlessio et al., A.J. 1998
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Accretion timescales (1)

• Consider a protoplanet moving through a
  planetesimal swarm. We have
  where v is the relative velocity and f is
  a factor which arises because the gravitational
  cross-sectional area exceeds the real c.s.a.

f is the Safronov number
Planet density r
vorb
Where does this come from?
R
fR
where ve is the escape velocity, G is the
gravitational constant, r is the planet density.
So
Planetesimal Swarm, density rs
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Accretion timescales (2)

• Two end-members
• 8GrR2 ltlt v2 so dM/dt R2 which means all
  bodies increase in radius at same rate orderly
  growth
• 8GrR2 gtgt v2 so dM/dt R4 which means largest
  bodies grow fastest runaway growth
• So beyond some critical size (Moon-size), the
  largest bodies will grow fastest and accrete the
  bulk of the mass
• If we assume that the relative velocity v is
  comparable to the orbital velocity vorb, we can
  show (how?) that

f
Here f is the Safronov factor as before, n is the
orbital mean motion (2p/period), ss is the
surface density of the planetesimal swarm and r
is the planet density
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Accretion Timescales (3)

• Rate of growth decreases as surface density ss
  and orbital mean motion n decrease. Both these
  parameters decrease with distance from the Sun
  (as a-1.5 and a-1 to -2, respectively)
• So rate of growth is a strong function (a-3) of
  distance

a, AU ss,g cm-2 n, s-1 t, Myr
1 10 2x10-7 5
5 1 2x10-8 500
25 0.1 2x10-9 50,000
Approximate timescales t to form an Earth-like
planet. Here we are using f10, r5.5 g/cc. In
practice, f will increase as R increases.
Note that forming Neptune is problematic!
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Runaway Growth

• Recall that for large bodies, dM/dtR4 so that
  the largest bodies grow at the expense of the
  others
• But the bodies do not grow indefinitely because
  of the competing gravitational attraction of the
  Sun
• The Hill Sphere defines the region in which the
  planets gravitational attraction overwhelms that
  of the Sun the distance from which planetesimals
  can be accreted to a single body is a few times
  this distance rH, where

Where does this come from?
Here M and Ms are the planet and solar mass
(2x1030 kg), and a is semi-major axis. Jupiters
Hill Sphere is 0.5 AU
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Late-Stage Accretion

• Once each planet has swept up debris out to a few
  Hill radii, accretion slows down drastically
• Size of planets at this point is determined by
  Hill radius and local nebular surface density,
  Mars-size at 1 AU
• Collisions now only occur because of mutual
  perturbations between planets, timescale 107-8
  yrs
• This stage can be simulated numerically

Agnor et al. Icarus 1999
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Complications

• 1) Timing of gas loss
• Presence of gas tends to cause planets to spiral
  inwards, hence timing of gas loss is important
• Since outer planets can accrete gas if they get
  large enough, the relative timescale of planetary
  growth and gas loss is also important
• 2) Jupiter formation
• Jupiter is so massive that it significantly
  perturbs the nearby area e.g. it scattered so
  much material from the asteroid belt that a
  planet never formed there
• Jupiter scattering is the major source of the
  most distant bodies in the solar system (Oort
  cloud)
• It must have formed early, while the nebular gas
  was still present. How?

23
Giant planets?

• Why did the gas giants grow so large, especially
  in the outer solar system where accretion
  timescales are slow?
• 1) original gaseous nebula develops
  gravitational instabilities and forms giant
  planets directly
• 2) solid cores develop rapidly enough that they
  reach the critical size (10-20 Me) to accrete
  local nebular gas (runaway)
• Hypothesis 1) cant explain why the gas/ice
  giants are so different to the original nebular
  composition, and require an enormous initial
  nebula mass (1 solar mass)
• Hypothesis 2) is reasonable, and can explain why
  Uranus and Neptune are smaller with less H/He
  they must have been forming as the nebula gas was
  dissipating (10 Myr)
• In this scenario, the initial planet radius was
  rH, but the gas envelope subsequently contracted
  (causing heating)

24
Summary

• The Outer Solar System is Big and Cold
• Cold - because disk density lower, radiative
  cooling more efficient. Means that volatiles can
  be accreted . . .
• Big planets are large because of runaway effect
  of accreting volatiles (while nebular gas is
  present)
• Big lengthscales separating planets set by Hill
  Sphere, which increases with planet mass and
  distance from the Sun

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Spacecraft Exploration

• Three major problems (how do we solve them?)
• Power
• Communications
• Transit time
• Pioneers 10 11 were the first outer solar
  system probes, with fly-bys of Jupiter (1974) and
  Saturn (1979)

Saturn with Rhea in the foreground
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Voyagers 1 and 2

• A brilliantly successful series of fly-bys
  spanning more than a decade
• Close-up views of all four giant planets and
  their moons
• Both are still operating, and collecting data on
  solar/galactic particles and magnetic fields

Voyagers 1 and 2 are currently at 90 and 75 AU,
and receding at 3.5 and 3.1 AU/yr Pioneers 10
and 11 at 87 and 67 AU and receding at 2.6 and
2.5 AU/yr
The Death Star (Mimas)
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Galileo
antenna

• More modern (launched 1989) but the high-gain
  antenna failed (!) leaving it crippled
• Venus-Earth-Earth gravity assist
• En route, it observed the SL9 comet impact into
  Jupiter
• Arrived at Jupiter in 1995 and deployed probe
  into Jupiters atmosphere
• Very complex series of fly-bys of all major
  Galilean satellites
• Deliberately crashed into Jupiter Sept 2003
  (why?)
• Well discuss results in a later lecture

28
Cassini

• Cassini is the last of the Cadillacs, a large
  (6 ton why? ), very expensive and very
  sophisticated spacecraft.
• Launched in 1997, it did gravity assists at
  Venus, Earth and Jupiter, and has now arrived in
  the Saturn system.
• It carries a small European probe called Huygens,
  which will be dropped into the atmosphere of
  Titan, the largest moon
• Cassini will do flybys of most of Saturns moons
  (particularly Titan), as well as investigating
  Saturns atmosphere and magnetosphere
• Well discuss the new results later in the course

False-colour Cassini image of Titans surface
greens are ice, yellows are hydrocarbons, white
is methane clouds
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Outer Solar System Highlights
(NB these reflect my biases!)

• 1) The most volcanically active place in the
  solar system

• 3) An ocean 3 times larger than Earths

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Highlights (contd)

• 4) Active nitrogen geysers
• 5) Hot Jupiters

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Next time . . .

• Orbital mechanics

32
Orbital Mechanics

• Why do we care?
• Fundamental properties of solar system objects
• Examples synchronous rotation, tidal heating,
  orbital decay, eccentricity damping etc. etc.
• What are we going to study?
• Keplers laws / Newtonian analysis
• Angular momentum and spin dynamics
• Tidal torques and tidal dissipation
• These will come back to haunt us later in the
  course
• Good textbook Murray and Dermott, Solar System
  Dynamics, C.U.P., 1999

33
Keplers laws (1619)

• These were derived by observation (mainly thanks
  to Tycho Brahe pre-telescope)
• 1) Planets move in ellipses with the Sun at one
  focus
• 2) A radius vector from the Sun sweeps out equal
  areas in equal time
• 3) (Period)2 is proportional to (semi-major axis
  a)3

ae
a
b
apocentre
pericentre
focus
e is eccentricity
34
Newton (1687)

• Explained Keplers observations by assuming an
  inverse square law for gravitation

Here F is the force acting in a straight line
joining masses m1 and m2 separated by a distance
r G is a constant (6.67x10-11 m3kg-1s-2)

• A circular orbit provides a simple example and is
  useful for back-of-the-envelope calculations

Period T
Centripetal acceleration rw2 Gravitational
acceleration GM/r2 So GMr3w2 (this is a
useful formula to be able to derive) So (period)2
is proportional to r3 (Kepler)
Centripetal acceleration
M
r
Angular frequency w2 p/T
35
Angular Momentum (1)

• The angular momentum vector of an orbit is
  defined by
• This vector is directed perpendicular to the
  orbit plane. By use of vector triangles (see
  handout), we have
• So we can combine these equations to obtain the
  constant magnitude of the angular momentum per
  unit mass
• This equation gives us Keplers second law
  directly. Why?
• C.f. angular momentum per unit mass for a
  circular orbit r2w
• The angular momentum will be useful later on
  when we calculate orbital timescales and also
  exchange of angular momentum between spin and
  orbit

36
Elliptical Orbits Two-Body Problem

• Newtons law gives us

r
m1
r
m2
where mG(m1m2) and is the unit vector (The
m1m2 arises because both objects move)
See Murray and Dermott p.23
The tricky part is obtaining a useful expression
for d 2r/dt2 (otherwise written as ) . By
starting with rr and differentiating twice,
you eventually arrive at (see the handout for
details)
Comparing terms in , we get something which
turns out to describe any possible orbit
37
Elliptical Orbits

• Does this make sense? Think about an object
  moving in either a straight line or a circle
• The above equation can be satisfied by any conic
  section (i.e. a circle, ellipse, parabola or
  hyberbola)
• The general equation for a conic section is

qfconst.
e is the eccentricity
ae
a
r
f
For ellipses, we can rewrite this equation in a
more convenient form using
focus
b
b2a2(1-e2)
38
Timescale

• The area swept out over the course of one orbit
  is
• where T is the period
• Lets define the mean motion (angular velocity)
  n2p/T
• We will also use
  (see previous slide)
• Putting all that together, we end up with two
  useful results

Where did that come from?
We can also derive expressions to calculate the
position and velocity of the orbit as a function
of time
39
Energy

• To avoid yet more algebra, well do this one for
  circular coordinates. The results are the same
  for ellipses.
• Gravitational energy per unit mass
• Eg-GM/r why the
  minus sign?
• Kinetic energy per unit mass
• Evv2/2r2w2/2GM/2r
• Total sum EgEv-GM/2r (for elliptical orbits,
  -m/2a)
• Energy gets exchanged between k.e. and g.e.
  during the orbit as the satellite speeds up and
  slows down
• But the total energy is constant, and independent
  of eccentricity
• Energy of rotation (spin) of a planet is
• ErCW2/2 C is moment of inertia, W angular
  frequency
• Energy can be exchanged between orbit and spin,
  like momentum

40
Summary

• Mean motion of planet is independent of e,
  depends on m (G(m1m2)) and a
• Angular momentum per unit mass of orbit is
  constant, depends on both e and a
• Energy per unit mass of orbit is constant,
  depends only on a

41
Tides (1)

• Body as a whole is attracted with an acceleration
  Gm/a2
• But a point on the far side experiences an
  acceleration Gm/(aR)2

a
R
m

• The net acceleration is 2GmR/a3 for Rltlta
• On the near-side, the acceleration is positive,
  on the far side, its negative
• For a deformable body, the result is a
  symmetrical tidal bulge

42
Tides (2)
P
planet
satellite

• Tidal potential at P
• Cosine rule
• (R/a)ltlt1, so expand square root

(recall acceleration - )
43
Tides (3)

• We can rewrite the tide-raising part of the
  potential as
• Where P2(cos j) is a Legendre polynomial, g is
  the surface gravity of the planet, and H is the
  equilibrium tide
• Does this make sense? (e.g. the Moon at 60RE,
  M/m81)
• For a uniform fluid planet with no elastic
  strength, the amplitude of the tidal bulge is
  (5/2)H
• An ice shell decoupled from the interior by an
  ocean will have a tidal bulge similar to that of
  the ocean
• For a rigid body, the tide may be reduced due to
  the elasticity of the planet (see next slide)

This is the tide raised on the Earth by the Moon
44
Effect of Rigidity

• We can write a dimensionless number which
  tells us how important rigidity m is compared
  with gravity

(g is acceleration, r is density)

• For Earth, m1011 Pa, so 3 (gravity and
  rigidity are comparable)
• For a small icy satellite, m1010 Pa, so 102
  (rigidity dominates)
• We can describe the response of the tidal bulge
  and tidal potential of an elastic body by the
  Love numbers h2 and k2, respectively
• For a uniform solid body we have

• E.g. the tidal bulge amplitude is given by h2 H
  (see previous slide)
• The quantity k2 is important in determining the
  magnitude of the tidal torque (see later)

45
Effects of Tides
In the presence of friction in the primary, the
tidal bulge will be carried ahead of the
satellite (if its beyond the synchronous
distance) This results in a torque on the
satellite by the bulge, and vice versa. The
torque on the bulge causes the planets rotation
to slow down The equal and opposite torque on the
satellite causes its orbital speed to increase,
and so the satellite moves outwards The effects
are reversed if the satellite is within the
synchronous distance (rare why?) Here we are
neglecting friction in the satellite, which can
change things see later.

• 1) Tidal torques

The same argument also applies to the satellite.
From the satellites point of view, the planet is
in orbit and generates a tide which will act to
slow the satellites rotation. Because the tide
raised by the planet on the satellite is large,
so is the torque. This is why most satellites
rotate synchronously with respect to the planet
they are orbiting.
46
Tidal Torques

• Examples of tidal torques in action
• Almost all satellites are in synchronous rotation
• Phobos is spiralling in towards Mars (why?)
• So is Triton (towards Neptune) (why?)
• Pluto and Charon are doubly synchronous (why?)
• Mercury is in a 32 spinorbit resonance (not
  known until radar observations became available)
• The Moon is currently receding from the Earth (at
  about 3.5 cm/yr), and the Earths rotation is
  slowing down (in 150 million years, 1 day will
  equal 25 hours). What evidence do we have? How
  could we interpret this in terms of angular
  momentum conservation? Why did the recession rate
  cause problems?

47
Diurnal Tides (1)

• Consider a satellite which is in a synchronous,
  eccentric orbit
• Both the size and the orientation of the tidal
  bulge will change over the course of each orbit

• 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)

N.B. its often helpful to think about tides from
the satellites viewpoint
48
Diurnal tides (2)

• The amplitude of the diurnal tide is 3e times the
  static tide (does this make sense?)
• Why are diurnal tides important?
• Stress the changing shape of the bulge at any
  point on the satellite generates time-varying
  stresses
• Heat time-varying stresses generate heat
  (assuming some kind of dissipative process, like
  viscosity or friction). NB the heating rate goes
  as e2 why?
• Dissipation has important consequences for the
  internal state of the satellite, and the orbital
  evolution of the system (the energy has to come
  from somewhere)
• We will see that diurnal tides dominate the
  behaviour of some of the Galilean satellites

49
Angular Momentum Conservation

• Angular momentum per unit mass
• where the second term uses
• Say we have a primary with zero dissipation
  (this is not the case for the Earth-Moon system)
  and a satellite in an eccentric orbit.
• The satellite will still experience dissipation
  (because e is non-zero) where does the energy
  come from?
• So a must decrease, but the primary is not
  exerting a torque to conserve angular momentum,
  e must decrease also- circularization
• For small e, a small change in a requires a big
  change in e
• Orbital energy is not conserved dissipation in
  satellite
• NB If dissipation in the primary dominates, the
  primary exerts a torque, resulting in angular
  momentum transfer from the primarys rotation to
  the satellites orbit the satellite (generally)
  moves out (as is the case with the Moon).

50
How fast does it happen?

• The speed of orbital evolution is governed by the
  rate at which energy gets dissipated (in primary
  or satellite)
• Since we dont understand dissipation very well,
  we define a parameter Q which conceals our
  ignorance
• Where DE is the energy dissipated over one cycle
  and E is the peak energy stored during the cycle.
  Note that low Q means high dissipation!

• It can be shown that Q is related to the phase
  lag arising in the tidal torque problem we
  studied earlier

e
51
How fast does it happen(2)?

• The rate of outwards motion of a satellite is
  governed by the dissipation factor in the primary
  (Qp)

Here mp and ms are the planet and satellite
masses, a is the semi-major axis, Rp is the
planet radius and k2 is the Love number. Note
that the mean motion n depends on a.

• Does this equation make sense? Recall
• Why is it useful? Mainly because it allows us to
  calculate Qp. E.g. since we can observe the rate
  of lunar recession now, we can calculate Qp. This
  is particularly useful for places like Jupiter.
• We can derive a similar equation for the time for
  circularization to occur. This depends on Qs
  (dissipation in the satellite).

52
Tidal Effects - Summary

• Tidal despinning of satellite generally rapid,
  results in synchronous rotation. This happens
  first.
• If dissipation in the synchronous satellite is
  negligible (e0 or QsgtgtQp) then
• If the satellite is outside the synchronous
  point, its orbit expands outwards (why?) and the
  planet spins down (e.g. the Moon)
• If the satellite is inside the synchronous point,
  its orbit contracts and the planet spins up (e.g.
  Phobos)
• If dissipation in the primary is negligible
  compared to the satellite (QpgtgtQs), then the
  satellites eccentricity decreases to zero and
  the orbit contracts a bit (why?) (e.g. Titan?)

53
Modelling tidal effects

• We are interested in the general case of a
  satellite orbiting a planet, with Qp Qs, and we
  can neglect the rotation of the satellite
• Angular momentum conservation
• 
  (1)
• Dissipation
• 
  (2)

Dissipation in primary and satellite
Rotational energy
Grav. energy

• Three variables (Wp,a,e), two coupled equations
• Rate of change of individual energy and angular
  momentum terms depend on tidal torques
• Solve numerically for initial conditions and Qp,Qs

54
Example results
1.
2.

• 1. Primary dissipation dominates satellite
  moves outwards and planet spins down

• 2. Satellite dissipation dominates orbit
  rapidly circularizes
• 2. Orbit also contracts, but amount is small
  because e is small

55
Summary

• Tidal bulges arise because bodies are not point
  masses, but have a radius and hence a gradient in
  acceleration
• A tidal bulge which varies in size or position
  will generate heat, depending on the value of Q
• If the tidal bulge lags (dissipation - finite Q),
  it will generate torques on the tide-raising body
• Torques due to a tide raised by the satellite on
  the primary will (generally) drive the satellite
  outwards
• Torques due to a tide raised by the primary on
  the satellite will tend to circularize the
  satellites orbit
• The relative importance of these two effects is
  governed by the relative values of Q

56
(No Transcript)