Title: Francis Nimmo
1ES 290Q OUTER SOLAR SYSTEM
Io against Jupiter, Hubble image, July 1997
2Course 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 - Week 6 Computer project
- Weeks 7-8 Student presentations
- Week 9 The Outer Limits Pluto/Charon, Kuiper
Belt, Oort Cloud, future missions - Week 10 Computer project writeup LPSC
This schedule can be modified if someone is
interested in a particular topic
3Logistics
- Set texts see website for suggestions
- http//es.ucsc.edu/fnimmo/eart290q
- Office hours make appointments by email
fnimmo_at_es.ucsc.edu or drop by (A219) - 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 Mon/Weds 900-1045 in room
D250 - Questions? - Yes please!
4This 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
5Where 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
6Where 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?
7Basic Parameters
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.
8Compositions (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
(Lodders and Fegley 1998)
9Compositions (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.
10Temperatures
- 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?
11Temperatures (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
12Solar 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)
13Observations (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
14Observations (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
15Cartoon 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
16Temperature 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
17Accretion 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
18Accretion timescales (2)
- Two end-members
- 8GrR2 bodies increase in radius at same rate orderly
growth - 8GrR2 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
19Accretion 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
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!
20Runaway 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
21Late-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
22Complications
- 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 only 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?
23Giant 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)
24Summary
- 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
25Spacecraft 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
26Voyagers 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)
27Galileo
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
28Cassini
- 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 carried a small European probe called Huygens,
which was dropped into the atmosphere of Titan,
the largest moon, and produced images of the
surface - Cassini is doing 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
29Outer 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
30Highlights (contd)
- 4) River channels and ice cobbles
- 5) Hot Jupiters
31Next time . . .
32Orbital 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
33Keplers 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
empty focus
e is eccentricity a is semi-major axis
34Newton (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
35Angular 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? What does constant angular
momentum mean physically? - 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
36Elliptical Orbits Two-Body Problem
r
m1
r
where mG(m1m2) and is the unit vector (The
m1m2 arises because both objects move)
m2
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
37Elliptical 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
e is the eccentricity, a is the semi-major axis h
is the angular momentum
qfconst.
ae
r
a
f
For ellipses, we can rewrite this equation in a
more convenient form (see MD p. 26) using
focus
b
b2a2(1-e2)
38Timescale
- 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
39Energy
- 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
40Summary
- 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
41Tides (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 R
- 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
42Tides (2)
P
planet
satellite
- Tidal potential at P
- Cosine rule
- (R/a)
(recall acceleration - )
43Tides (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
44Effect of Rigidity
- We can write a dimensionless number which
tells us how important rigidity m is compared
with gravity
Note that this m is different from previous
definition!
(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)
45Effects 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.
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.
46Tidal 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?
47Diurnal 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
48Diurnal 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
49Angular 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).
50How 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
51How 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).
52Tidal 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 QsQp) 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 (QpQs), then the
satellites eccentricity decreases to zero and
the orbit contracts a bit (why?) (e.g. Titan?)
53Summary
- 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
54(No Transcript)
55Modelling 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
56Example 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