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Planetary system dynamics
Mathematics tripos part III / part III Astrophysics
Lecturer: Dr Mark Wyatt
Schedule: Lent 2015 – Mon Wed Fri 10am MR12, 24 lectures, start Fri 16 Jan, end Wed 11 Mar
Problems: My office is Hoyle 38 at the Institute of Astronomy, or email [email protected]
Examples sheets: 4 examples sheets, handed out around Mon 19 Jan, 2 Feb, 16 Feb, 2 Mar
Examples classes: 3-5pm in HCR (IoA) on Tue 3 Feb, 17 Feb, 3 Mar (*Ryle Meeting Room), 28 Apr
Course content
1.  Two body problem
2.  Small body dynamics
3.  Three body problem
4.  Close approaches
5.  Collisions
6.  Disturbing function
7.  Secular perturbations
8.  Resonant perturbations
Main textbook
Other useful textbooks
Planetary system dynamics
Course content
0.
Planetary system architecture:
overview of Solar System and extrasolar systems, detectability,
planet formation
1. 
Two-body problem:
equation of motion, orbital elements, barycentric motion, Kepler's equation,
perturbed orbits
2. 
Small body forces:
3. 
Three-body problem:
4. 
Close approaches:
5. 
Collisions:
6. 
Disturbing function:
7. 
Secular perturbations:
8. 
Resonant perturbations:
stellar radiation, optical properties, radiation pressure, Poynting-Robertson drag,
planetocentric orbits, stellar wind drag, Yarkovsky forces, gas drag, motion in protoplanetary disc, minimum mass
solar nebula, settling, radial drift
restricted equations of motion, Jacobi integral, Lagrange equilibrium points,
stability, tadpole and horseshoe orbits
hyperbolic orbits, gravity assist, patched conics, escape velocity, gravitational
focussing, dynamical friction, Tisserand parameter, cometary dynamics, Galactic tide
accretion, coagulation equation, runaway and oligarchic growth, isolation mass, viscous stirring,
collisional damping, fragmentation and collisional cascade, size distributions, collision rates, steady state, long
term evolution, effect of radiation forces
elliptic expansions, expansion using Legendre polynomials and Laplace coefficients,
Lagrange's planetary equations, classification of arguments
Laplace coefficients, Laplace-Lagrange theory, test particles, secular resonances,
Kozai cycles, hierarchical systems
geometry of resonance, physics of resonance, pendulum model, libration
width, resonant encounters and trapping, evolution in resonance, asymmetric libration, resonance overlap
1
Components of the Solar System
Material gravitationally bound to the Sun (out to ~100,000 AU, ~0.5 pc)
•  The Sun
•  Mass/luminosity/evolution
•  Planets and their moons and ring systems
•  Terrestrial planets: Mercury, Venus, Earth, Mars
•  Jovian planets: Jupiter, Saturn, Uranus, Neptune
•  Dwarf planets: Pluto (Ceres, Eris)
•  Minor planets
•  Asteroids: Asteroid Belt, Trojans, Near Earth Asteroids
•  Comets: Kuiper Belt, Oort Cloud
•  Dust
•  Zodiacal Cloud
The planets – overview/mass
Mass
Sun
Distance
300000Mearth
0.0046AU
Mercury
0.06 Mearth
0.39 AU
Venus
0.82 Mearth
0.72 AU
Earth
1.0 Mearth
1.0 AU
Mars
0.11 Mearth
1.5 AU
Jupiter
318 Mearth
5.2 AU
Saturn
98 Mearth
9.5 AU
Uranus
15 Mearth
19.2 AU
Neptune
17 Mearth
30.1 AU
0.002 Mearth
39.5 AU
Pluto
Terrestrial
planets
Jovian
planets
Dwarf planet
1 Mearth = 6 x 1024 kg = 3x10-6 Msun , 1 AU = 1.5 x 1011 m
2
The planets - orbits
Aphelion
ae
Perihelion
Orbits defined by:
•  Semimajor axis, a (tper=a1.5)
2a
•  Eccentricity, e
•  Inclination, I (relative to the ecliptic, the plane of Earth’s orbit)
a, AU
e
I, deg
Mercury
0.39
0.206
7.0
Venus
0.72
0.007
3.4
Earth
1.0
0.017
0.0
Mars
1.5
0.093
1.9
Jupiter
5.2
0.048
1.3
Saturn
9.5
0.054
2.5
Uranus
19.2
0.047
0.8
Neptune
30.1
0.009
1.8
Pluto
39.5
0.249
17.1
•  Evenly spaced, orbiting in
same direction in same plane
(Sun’s rotation axis inclined by
7.3o) with nearly circular orbits
•  La Grande Inequalite (JS near
5:2 resonance) and NP in 3:2
resonance
•  System is stable for >4.5Gyr,
though Mercury’s orbit evolves
chaotically on such timescales
Other examples of resonances
Mean motion resonances:
Jupiter’s satellites in 4:2:1
resonance causes strong tides
and vulcanism on Io and liquid
water under surface of Europa
Spin-orbit resonances:
The Moon’s rotation period =
orbital period, synchronous
rotation, means Moon keeps
same face to us (caused by
tidal evolution)
3
Secular interactions between planets
Secular interactions
between the planets cause
the obliquity and
eccentricity of Earth’s orbit
to vary on 100,000 yr
timescales
This changes the insolation
of upper atmosphere
And is reflected in global
temperature changes
measured in ice cores
Minor planets in the inner solar sytem
•  The Asteroid Belt is the
20,000-strong belt of rocky
asteroids orbiting 2-3.5 AU
from the Sun (green)
Jupiter
•  Some asteroids in the Earth
region (Near Earth Asteroids in
red) that originate in AB until
orbits become chaotic
•  Another family of asteroids
are the Jupiter Trojans at ±
60o from Jupiter at L4 and L5
points (blue, other planets also
have Trojans)
4
Minor planets: dynamical structures
Kirkwood gaps in the distribution of
asteroids at mean motion resonances
with Jupiter; Yarkovsky forces move
~100m sized asteroids into these
unstable regions where they may be
perturbed into Earth-crossing orbits
Orbital distribution of
KBOs: resonant (e.g.,
Pluto in 3:2 with
Neptune), classical
(low e,I, outer edge
47AU), scattered disc
(high e, but perihelia
near Neptune),
detached (e.g., Sedna
with perihelion at 44AU)
Minor planets: mutual collisions
Asteroid orbits are
clustered into Hirayama
(1918) families created in
the break-up Gyr-ago of
large asteroids
Nesvorny (2003)
found evidence of
families created
when medium-sized
asteroids collided
just ~1Myr ago
In last few years
there is evidence of
dust created in
collision in asteroid
belt
5
Minor planets: size distribution
Asteroid belt’s size distribution is that of a
collisional cascade that extends from
1000km objects down to micron-sized
dust, and is reason many are rubble piles
Asteroids also collide with planets
and moons, and crater counts give
size distribution and imply more
massive population in past (e.g.,
most Moon craters from Late
Heavy Bombardment epoch 3.8Ga)
Dust: Zodiacal cloud
PR drag moves dust from AB toward the
Sun; sunlight scattered by this cloud is
visible as the zodiacal light, and its
thermal emission is the brighest thing in
the IR sky; some dust is accreted by Earth
Zodiacal cloud structure is
affected by planets; e.g.,
brighter behind Earth because of
a coorbiting clumpy ring of
resonantly trapped particles
Sun
Earth
6
Kuiper Belt: origin of comets
•  Belt of comets orbiting the
Sun >30AU; discovered
1992, now ~1000 known
•  Scattered by giant planets
until they reach inner SS, or
Jupiter ejects them, or collide
with a planet (origin of H2O?)
•  Few km nucleus of frozen
gases and embedded dust
released when heated at
perihelion of eccentric orbit
•  Long period comets
originate in Oort Cloud
1000-100,000AU; perturbed
by Galactic tides
Circumplanetary material: captured
minor planets
Most of the giant planets’ satellites are irregulars: small
(2-200km) and on eccentric (~0.4) inclined (~400)
more often retrograde orbits filling a large fraction of
Hill sphere; origin in capture from passing asteroids/
comets
Mars has two 6-10km
satellites: Phobos (will
spiral into Mars in few
Myr) and Deimos; thought
to be captured asteroids,
but origin of equatorial
orbits I<10 is mystery.
7
Giant impacts
Moon (0.012 Mearth, 3.3g/cm3)
formed from Earth crust
stripped in collision 50Myr
after Earth formed
Giant
impacts
also
explain:
Pluto satellite Charon is half its diameter
(in mutually synchronous rotation, keeping
same face to each other), and two
60-165km satellites in 6:4:1 orbital period
ratio, all thought to have collisional origin
Uranus’ tilted spin axis, Mercury’s high density, Mars’ hemispheric dichotomy
Kuiper Belt - evolution
Missing mass problem:
total mass now is
0.05-0.3Mearth but 100x
that required to form Pluto
and KBO binaries
Currently favoured model
starts with a more
massive Kuiper belt
outside a more compact
planetary system
Planetary system
becomes unstable after
800Myr scattering Uranus
and Neptune into Kuiper
belt causing depletion and
Late Heavy Bombardment
8
How to detect extrasolar planets?
Effect on motion of parent star
•  Astrometric wobble
•  Timing shifts
•  Radial velocity method
2-body motion:
both bodies orbit
centre of mass
Mpl
Effect on flux from parent star
•  Planetary transits
•  Gravitational microlensing (rather flux from
another star)
Direct detection
•  Direct imaging
M*
Other techniques
•  Disc structures
Methods using motion of parent star
Astrometric wobble = in plane of sky
Angular scale is
2x10-3(apl/d*)(Mpl/MJ)(Msun/M*) arcsec
so Jupiter around 1Msun at 10 pc is 1mas
[hard]
Timing shifts = out of sky plane
ms radio pulsar timing variation is
Δt = 3apl(Mpl/Mearth)(Msun/Mstar) ms
so Earth around 1Msun gives a 3 ms shift
Radial velocity = out of sky plane
Stellar radial velocity semi-amplitude is
30apl-0.5(Mplsini/MJ)(M*/Msun)-0.5 m/s
so Jupiter around 1Msun gives 13 m/s
9
Pulsar Planets
First extrasolar planets
detected around 6.2 ms
pulsar PSRB1257+12
(Wolszczan & Frail 1992)
The planets are small,
coplanar, low eccentricity
(Konacki & Wolszczan 2003)
A
B
C
a, AU M, Mearth
I
e
0.19
0.02
-
0
4.3
530
0.019
3.9
470
0.025
0.36
0.47
Gravitational interactions
between B and C near 3:2
resonance (Malhotra 1992)
detected so orbital planes
and masses derived
Possible fourth planet or
asteroid belt beyond C
(Wolszczan et al. 2000)
Radial Velocity Planets
First extrasolar planet around main sequence star 51 Peg (G2 at 15pc) used radial
velocity method to detect >0.45Mjupiter planet at 0.05AU near circular orbit (Mayor &
Queloz 1995; Marcy & Butler 1997) = HOT JUPITER
Now 591 planets discovered using this method
(see http://exoplanet.eu or http://exoplanets.org)
and >5% of stars have planets
10
Planet discovery space
To interpret observed stats need to understand detection bias: e.g.,
instrument sensitivity 30apl-0.5(Mpl/MJ)(M*/Msun)-0.5 m/s and survey duration
1% stars have
HJs: tides
circularise
orbits, mass
loss, formed
further out
then migrated
or scattered
in?
Long period
Jupiters: new!
Eccentric Jupiters
around ~5%
stars: origin of
eccentricity?
Super-Earths common (30-50%?): cores
of evaporated Jupiters or massive Earths?
Planet eccentricity distribution
Giant planets at few AU have eccentric orbits: mean 0.32, up to 0.92
(compared with <0.05 for the Solar System)
Theories for origin of high
eccentricities range from:
•  planet-planet scattering
•  planet-disk interactions,
•  scattering by passing
stars
•  perturbations of
companion stars
11
Multiple bodies: dynamical interactions
There are 176 systems with multiple planets, and 57 planets in multiple
stellar systems
e.g., GJ876 planets in 2:1 mean motion
eccentricity and secular resonances: pericentres
oscillate 340 about ϖb=ϖc and line of apsides
precesses at -410/yr (Laughlin et al. 2005; Beauge et al.
2006); also 3 body resonance (Rivera et al. 2010)?
e.g. γ Cephei has 1.7MJ
at 2.1AU and 0.4Msun at
28AU with e=0.4
strongly perturbing
planet
Transit detection method
If orientation just right, star gets fainter
when the planet passes in front of it
e.g., HD209458b discovered by rv;
transit lasts 3hrs every 3.5days
confirming the planet and giving its
mass, size, density
Space mission Kepler already detected 1Mearth planets (3538 planet candidates,
238 of which confirmed, see keplerscience.arc.nasa.gov)
12
Transit Timing Variations (TTV)
Transits are precise clock meaning perturbations detectable from other planets
or satellites (e.g., Nesvorny & Beauge 2010; Veras et al. 2011), and TTVs used to confirm
planet and constrain planet masses; e.g., Kepler 9 has two transiting planets
close to 2:1 resonance (also imply additional planet; Holman et al. 2010)
Direct imaging of outer planetary systems
Four planets imaged around 60Myr A star HR8799 with
masses 5-13Mjup at 14-68AU (Marois et al. 2010)
Are outer planetary systems common, how do they relate to inner planetary
systems, resonance required for stability (Fabrycky & Murray-Clay 2010), formation?
13
Planet interacting with debris disk
<2Mjup planet imaged at inner edge of debris disk around 200Myr A5V
star Fomalhaut (Kalas et al. 2008, 2013)
But emission spectrum is
circumplanetary dust not planet,
so rings or irregular satellite
swarm (Kennedy & Wyatt 2011)?
Also, planet’s orbit is eccentric,
crossing the disk, which would
rapidly destroy it unless planet is
young or low mass
0Myr
5Myr
Planets inevitably affect disk structure
50
Disk: 20-60AU
AU
Planet: 1Mjup, 5AU,
e=0.1, I=5o
spiral
stirring
0
Time: 100Myr of
secular perturbations
So disk structures tell
us about planets
e.g., β Pic’s warped
edge-on disk was used
to predict a planet that
was later imaged
-50
offset
warp
-50
0
AU
2003
50
100
2010
14
Clumps: planet migration or collision
Mid-IR image
shows clump at
52AU (Telesco et al.
2005)
52AU
Origin could be:
•  Outward migration of
planet which trapped
planetesimals into
resonances
•  Collisional destruction
of Mars-sized
protoplanet
Hot dust puzzle
Star
Eta Corvi’s disk: a 150AU Kuiper
belt, and dust at 1.5AU
Evident in both images and
spectrum, as Tdust=278.L*1/4.r-1/2
Hot
dust at
1.5AU
Cold dust
at 150AU
Collisions would have depleted
any asteroid belt over 1Gyr age
of star, so where does hot dust
come from?
Recent collision, or comets
scattered in during an epoch
similar to the Late Heavy
Bombardment?
15