Physics in the solar system
• Orbital resonances
• Tidal forces
• Radiation pressure
Review
Kepler’s Laws
1.
A planet orbits the Sun in an ellipse, with the Sun at one focus
(supporting the Copernican heliocentric model and disproving
Brahe’s hypothesis)
2. A line connecting a planet to the Sun sweeps out equal areas in
equal time intervals
4 2 a 3
2
3. The period P, semimajor axis a, are related by:
P 
G ( M  m)
Escape velocity
vesc
2GM

r
Vis-viva equation
1 1 
v (r )  2GM  

 r 2a 
2
Recall
How do we know when a planet has completed one sidereal period
(i.e. is in the same position relative to the background stars?
It is easy to observe the synodic period: this is the time between
successive oppostions (when the Earth, Sun and planet are
aligned).
The angular velocity of a circular
orbit is 360/P. The synodic rate
is the rate of the planet relative
to the Earth. So:
360 360
360


PSyn PEarth P planet
Kirkwood gaps
• The distribution of asteroid periods (or semi-major axes) in the
main asteroid belt is not smooth, but shows gaps and peaks
Resonances
•
•
If the orbit of a small body around
a larger one is a small-integer
fraction of the larger body’s period,
the two bodies are commensurable.
Some resonances (3:2 resonance of
Jupiter) actually have a stabilising
effect.
Examples of resonant orbits:
1. An asteroid in a 1:2 resonance with
Jupiter completes two revolutions,
while Jupiter completes one
2. Mercury completes 1.5 rotations in
one trip around the Sun. Tidal
forces are strongest at perihelion,
and this controls the rotation of
Mercury.
Kirkwood gaps
• Gaps in the distribution of asteroids correspond to resonances
with Jupiter
Lagrangian points
 An analytic solution to the 3-body problem is possible for a specific
case: with two co-orbiting bodies with nearly circular orbits and a
third body with nearly the same revolution period P as the other
two.
 There are five points at which the third body can be placed and it
will remain fixed relative to the other two bodies. Only L4 and L5
are stable.
Trojan asteroids
• Two groups of asteroids, occupying the L4 and L5 points of Jupiter.
• Perturbations from other planets are significant, so the Trojans
drift well away from the Lagrangian points
Horseshoe orbits
• Two small moons of Saturn, Janus and
Epimetheus, only separated by about 50 km.
• As inner (faster moving) moon catches up
with slower moon, it is given a gravitational
kick into a higher orbit.
• It then moves more slowly and lags behind
the other moon.
Tides
• Tides are due to differential gravitational forces on a body.
 Consider the Earth and Moon: the gravitational force on the Moon
due to Earth is stronger on the near side than on the far side.
 This net difference in force will cause the body to stretch along the
line between the bodies.
Tidal Forces
What force is exerted on body M1, by the tidal bulges raised on
body M2?
r12
M1
M2
Tidal Friction
• Tides result in a net force which slows Earth’s rotation and speeds the
Moon’s orbital velocity.
• As a result the day is getting
longer by ~1 second/century and
the distance between the Earth
and Moon is increasing. There is
evidence for this in the fossil
record on Earth
Tidal Friction
In the past, when the moon was 0.25 as far from Earth as it is now:
a) How much more massive was Earth’s average tidal bulge?
b) How much stronger was the net accelerating effect of this bulge
on the moon?
Synchronous Rotation
• Why does the Moon always show the same face to Earth?
 Moon’s rotation period and orbital period are the same
 Due to drag force caused by tidal bulge
• This effect causes Pluto and Charon to always show the same face to one
another (as Earth and Moon will do eventually)
• Similarly, Mercury rotates exactly 3 times for every two orbits of the
Sun
 Ensures Mercury’s tidal bulge always aligns with Sun at perihelion.
Roche limit
• Tidal force gets
very large as the
distance between
objects decreases.
• At a critical
distance, the tidal
forces will exceed
the gravitational
force holding the
satellite together,
and it will be torn
apart.
Roche limit
Calculate the Roche limit for two equal-mass particles, just
touching and with their centres separated by a distance dr. If
these particles are a distance r away from a much larger mass M,
at what distance (the Roche limit) will tidal forces overwhelm the
gravitational force holding them together?
r
m
m
M
dr
Tidal heating
• Calculations of tidal heating led to
prediction of volcanoes on Io, in 1979
• Tidal bulge is raised on Io, due to
Jupiter. However other large moons
perturb Io’s orbit, which cause it to
vary its distance to Jupiter
• This causes the tidal bulge to
rise and fall, generating internal
heat
 200 times more heat per gram of
mass than radioactive heat that
drives Earth’s geology
 Energy obtained at the expense
of orbital energy: Io is very
gradually approaching Jupiter
Break
Rings
The “gosssamer” ring
of Jupiter is very
faint
• related to
degradation of small,
inner satellite
Almathea
Many gaps – large and
small – in Saturn’s ring
structure.
•The rings of Uranus are thin, narrow, and dark
compared to other planetary ring systems.
•The ring particles reflect as little light as
charcoal, although they are really made of ice
chucks darkened by rock.
Due to resonances
Rings of
Neptune show
thin ringlets,
and ring arcs
Rings
• Ring features (gaps,
edges) due primarily
to resonant
perturbations
• In densest regions,
ring particles
collide with one
another every few
hours.
• Extend out to
Roche limit: these
are swarms of
debris which cannot
coalesce to form a
moon
• In Jupiter and
Saturn systems,
small moonlets are
associated with the
outer ring edges,
near the Roche
limit.
Thinness of rings
• Saturn’s rings are very thin: only a few tens of metres thick (270,000 km in diameter)
• Why?
Shepherding satellites
• Narrow rings can be maintained by gravitational action of small
moons in or between rings.
Two shepherding moons straddling
the brightest ring around Uranus
Two small moons on either side of
Saturn’s narrow F ring
Shepherding moons
• Craters on Pandora appear to be covered over by some sort of material,
providing a more smooth appearance than sponge-like Hyperion, another
small moon of Saturn. Curious grooves and ridges also appear to cross the
surface of the small moon.
• Pandora is partly interesting because, along with its companion moon
Prometheus, it helps shepherd the particles of Saturn's F ring into a
distinct ring.
Gap Moons
• Gap moons have the opposite effect: clearing a gap in the ring
structure
Rings of Uranus
• several distinct rings,
mostly narrow
• dark, sooty particles
• some banded structure
• only tens of metres thick
• mass ~1/4000 Saturn’s
system
Radiation pressure
• Photons carry momentum: can push an object away from the Sun
• If object has a large surface area but small mass, radiation
pressure can overcome gravity
• Imagine a typical stone (r~3000 kg/m3) with radius a, a distance r
from the Sun. Under what conditions will radiation pressure
balance the gravitational attraction to the Sun?
f 2
F  a Q
c
r
Radiation pressure
Force due to radiation is
maximal for particles about
0.1-1 mm in size, but greater
for darker particles.
Solar Wind
• Small particles in comet tails are being
accelerated away from the Sun faster than can
be accounted for by radiation pressure.
• This suggests the interplanetary gas itself is
moving away from the Sun, carrying cometary
material with it.
• The solar wind
interacts with
Earth’s magnetic
field to create the
aurora
Heliopause
• The solar wind creates a bubble in interstellar space called the
heliosphere. Solar matter is dominant inside the heliosphere, while matter
from other stars dominates outside
• Voyager 1 crossed the termination shock (94 AU) on Dec. 16, 2004
Poynting-Robertson Effect
• Small dust particles moving through the stream of photons from
the Sun are impacted more frequently on their leading side
• Net process is complex: but leads to a net loss of energy so
particle spirals into the Sun.
Yarkovsky Effect
• Sunlight warms one side of a larger body.
• The warm side rotates away from the Sun and radiates thermal
energy as photons which provide a “thrust”
• This can move particles either in or out
Clearing of small bodies
Particle size (m)
10 - 9 10 - 8 10 - 7 10 - 6 10 - 5 10 - 4 10 - 3 10 - 2 10 - 1 10 0 10 1 10 2 10 3
Solar wind
Radiation
pressure
PoyntingRobertson
Yarkovsky
Negligible
Thus small bodies are generally cleared from the solar system,
apart from resonant orbits and ring systems.
Next Lecture
Light and Matter
• Blackbody radiation
• Spectral line formation
• States of matter