Dynamics
Solar System Explorers 05
How does the Sun affect objects in the Solar System?
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Basic Newton
F gravity on Earth = —
G mEarth m2
_______________
r2
m2 (kg)
r (m)
Fgravity
Sun
1.99e30 1.50e11
3.52e22
Venus
4.87e24 4.14e10
1.13e18
Moon
7.35e22 3.84e08
1.98e20
Jupiter
1.90e27 6.29e11
1.91e18
Winner!
178 X
Moon
What about the Moon?
mEarth = 5.97e24 kg
Earth-Moon has FEarth ~ 1.98e20, Sun-Moon has FSun ~ 4.34e20
Dynamics: Kepler I
Kepler I: planetary orbits are ellipses with the Sun at a focus
a (1 - e2)
rSun =
______________
1 + e cos f
e, eccentricity = (1 - b2minor/a2major)1/2
f (or θ, or ν), true anomaly = angle between perihelion and current position
Dynamics: Kepler I
Kepler I: planetary orbits are ellipses with the Sun at a focus
a (1 - e2)
rSun =
______________
1 + e cos f
e, eccentricity = (1 - b2minor/a2major)1/2
f (or θ, or ν), true anomaly = angle between perihelion and current position
Newton I : both bodies move along elliptical paths, with one focus of
each ellipse located at the center of mass
m1r1 + m2r2
rCM =
_________________
M
M = m1 + m2
Application: discovery of extrasolar planets
Dynamics: Kepler II
Kepler II: a line between a planet and the Sun sweeps out equal areas
in equal times
dA/dt = constant
Newton II : a line connecting two bodies (or connecting one body to
the center of mass position) sweeps out equal areas in equal times
dL/dt = 0
(conservation of angular momentum)
Application: spectroscopic binary orbits; prediction of planet locations
Dynamics: Kepler III
Kepler III: planetary orbital periods and distances from the Sun are
directly (and simply) related as long as you assume SS units
P2 (yr) = a3 (AU)
Newton III: it also works outside of the Solar System
4π2a3
P2 =
__________________
G (m1 + m2)
a3
or
Mtotal =
_______
P2
solar masses, AU, yrs
Application: stellar and planetary masses
need fractional mass, f, for individual masses
double dirty little secret of exoplanet masses …
Orbital Elements
a
e
i
P
T
Ω
ω
semimajor axis
eccentricity
inclination (~0 in SS, edge on = 90 outside)
orbital period
epoch of periastron
longitude of ascending node
argument of periastron
size
shape
tilt angle
time
a date
spin angle
tωist angle
Spin: Longitude of Ascending Node
Tωist: Longitude of Periastron
Orbital Elements
a
e
i
P
T
Ω
ω
equinox
f
semimajor axis
size
eccentricity
shape
inclination
tilt
orbital period
time
epoch of periastron
a date
longitude of ascending node
flip angle
longitude of periastron
twist angle
equinox of date
sets direction of equinox
fractional mass
a number
Two observations will not yield an orbit. Why?
Each point has (position X, position Y, time). There are 7 classical
unknowns, so you need a third point to give you 9 pieces of data to
solve equations.
1245 AC
PushingGJ
Towards
Exoplanets
12
New Orbits in Solar System
located
44.7 AU
Psun ~ 300 yrs
HST WFPC2 images
V = 23.1
Porb
a
mtot
at least 77 multiple TNOs known
www2.lowell.edu/users/grundy/tnbs/status.html
590 ± 40 days
22400 ± 900 km
0.02% Pluto
Reality Check: 3-body Systems
theory: about 7:1 ratio in semimajor
axis is critical point
two well-defined sets of triples:
Fekel’s spectroscopic triples
SETI sample projected separations
our Solar System is different … why?
Counter-Intuitive Dynamics
Lagrangian Points: where objects feel no net force in rotating frame;
gravitational force of two masses cancels centrifugal force because of
rotation
5 per two body system
Trojan asteroids at Jupiter (>5000), Mars (6+), Neptune (7+)
small moons at Sat/Tethys (Telesto+Calypso) and Sat/Dione (Helene+Polydeuces)
Earth orbiting spacecraft
WMAP
SOHO
Gaia
JWST
15
Counter-Intuitive Dynamics
Tadpole orbits: librating positions around L4 and L5 (note corotating
frame!)
Trojan asteroids at Jupiter, Mars, and Neptune
Counter-Intuitive Dynamics
Horseshoe orbits: orbit swapping due to particles passing in orbits, or
in resonance with larger bodies (note corotating frame!)
Janus and Epimetheus (Saturn) swap orbits every 4 years
Cruithne and Asteroid 2002 AA29 around Earth
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Counter-Intuitive Dynamics
Horseshoe orbits: Cruithne --- each loop takes 1 yr
http://www.astro.uwo.ca/%7Ewiegert/3753/3753.html
18
Counter-Intuitive Dynamics
Horseshoe orbits: Asteroid 2002 AA29 --- each vertical loop takes 1 yr
http://www.astro.uwo.ca/%7Ewiegert/AA29/AA29.html
“at least three others”
http://www.astro.uwo.ca/%7Ewiegert/3753/3753.htm
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Counter-Intuitive Dynamics
Chaotic motion: trajectories that begin arbitrarily close together will
diverge exponentially with time (note that 4.6 Gyr is often not sufficient
“time”)
Mars’ axis tilt
Hyperion rotation in Saturn-Titan tug-of-war
Resonances: orbital periods with ratios A : B (both integers)
Io : Europa : Ganymede (1 : 2.008 : 4.044) … oblate? tides?
Neptune : Plutinos (3:2)
Asteroids : Jupiter (lots) --- pumped up e leads to Kirkwood gaps
Saturn ring particles : Saturn moons (Mimas, Atlas, …)
Saturn’s Rings
300,000 km wide X 10 m (!) thick
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particles forced into plane by orbits
albedo 0.8 … shiny snowballs of H2O
Saturn’s Rings
total mass only that of small moon
young … likely formed by Roche limit crossing
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…………………………
Saturn’s Ring Structure
6 major regions … 2 divisions … 1000s of ringlets
ABC rings seen from ground … major
DEF rings seen from Voyager/Cassini … minor
Cassini Division
A ring
Encke Gap
F ring (braided)
E ring
spokes
Mimas 2:1 resonance
Atlas on edge (in 3:2 with Mimas)
Pan within A ring
shepherds Prometheus + Pandora
Enceladus volcanism (outside Roche)
collisions in rings
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The Saturn System
Saturn’s Rings Details
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Jupiter’s Rings
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Jupiter’s Rings Details
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Jupiter’s Rings Details
29
Tides
tides are a differential gravitational force ~ 1 / r3
a. cause bulk motions of fluid components
ocean tides, moonquakes, Io volcanoes
2 reasons --- bulge amplitude changes, position of bulge changes
b. cause torques (~1 / r6) that lead to rotational changes
Earth day lengthening, Pluto-Charon locked,
Mercury 3:2 spin:orbit resonance
eccentric orbit…Mercury elongation…resonance
c. cause shape changes if tidal force > tensile strength/self-gravity
evidence for Moon’s shape --- closer in the past
d. may create rings
Tidal Forces
2 x G m2
F tidal ≈
______________
r3
r is distance between two bodies
x is distance along axis separating two bodies (surface = body’s radius)
tides on Earth
m2 (kg)
r (km)
Ftidal/2RG
Sun
1.99e30
1.50e08
5.90e05
Venus
4.87e24
4.14e07
6.86e01
Moon
7.35e22
3.84e05
1.30e06
Jupiter
1.90e27
6.29e08
7.63e00
Winner! 2.2 X Sun
Earth-Moon tidal force is 81 X Moon-Earth tidal force
Jupiter-Io tidal force is 20,000 X Moon-Earth tidal force
Tides in Earth-Moon System
angular momentum is conserved, but …
can be swapped between rotation and revolution via tidal torques
on Earth:
twice per 25 hours due to Moon
twice per 24 hours due to Sun (1/2 strength)
on Moon:
mostly fixed because of synchronous rotation
but not entirely because of eccentricity … nutation
bulge torque:
Earth rotates faster than Moon orbits
Earth not perfectly elastic, so bulge not on Earth-Moon line
Moon pulls back on bulge --- Earth slows down
Earth bulges pull on Moon --- torque acc. Moon outward ~ 1/r6
death spirals:
moons moving retrograde, or faster than planet rotates
Moon’s Shape
bulge frozen at 2/3 current Moon distance
Roche Limit
Roche limit: point beyond which an object is ripped apart by tidal forces
aRoche = 2.456 Rp (ρp/ρs)1/3
moons inside Roche limit !?!?!
Jupiter has 3 +
Saturn has ~ 0
Uranus has 8 +
Neptune has 4 +
(1) assumes fully deformable (fluid) moon
(2) assumes no “tensile strength”
… resulting orbital systems have …
… large moons … small moons … ring particles
Roche Limits and Ring Systems
35
Neptune’s Rings
Adams
Lassell
LeVerrier
Galle
Billions of Years from Now…
Solar System Explorers 04
How does the Sun affect objects in the Solar System?
1.
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3.
4.
5.
6.
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10.
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13.
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