Dynamics. Solar System Explorers 05 How does the Sun affect objects in the Solar System? 1. 2. 3. 4....
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Transcript of Dynamics. Solar System Explorers 05 How does the Sun affect objects in the Solar System? 1. 2. 3. 4....
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Dynamics
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Solar System Explorers 05How does the Sun affect objects in the Solar System?
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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Basic Newton G mEarth m2
F gravity on Earth = — _______________
r2
m2 (kg) r (m) Fgravity
Sun 1.99e30 1.50e11 3.52e22 Winner! 178 X Moon
Venus 4.87e24 4.14e10 1.13e18
Moon 7.35e22 3.84e08 1.98e20
Jupiter 1.90e27 6.29e11 1.91e18
What about the Moon? mEarth = 5.97e24 kgEarth-Moon has FEarth ~ 1.98e20, Sun-Moon has FSun ~ 4.34e20
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Dynamics: Kepler IKepler I: planetary orbits are ellipses with the Sun at a focus
a (1 - e2) rSun = ______________
1 + e cos f
e, eccentricity = (1 - b2minor/a2
major)1/2
f (or θ, or ν), true anomaly = angle between perihelion and current position
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Dynamics: Kepler IKepler I: planetary orbits are ellipses with the Sun at a focus
a (1 - e2) rSun = ______________
1 + e cos f
e, eccentricity = (1 - b2minor/a2
major)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
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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
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Dynamics: Kepler IIIKepler 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 a3
P2 = __________________ or Mtotal = _______
G (m1 + m2) P2
solar masses, AU, yrs
Application: stellar and planetary massesneed fractional mass, f, for individual masses
double dirty little secret of exoplanet masses …
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Orbital Elementsa semimajor axis sizee eccentricity shapei inclination (~0 in SS, edge on = 90 outside) tilt angleP orbital period timeT epoch of periastron a dateΩ longitude of ascending node spin angleω argument of periastron tωist angle
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Spin: Longitude of Ascending Node
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Tωist: Longitude of Periastron
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Orbital Elementsa semimajor axis sizee eccentricity shapei inclination tiltP orbital period timeT epoch of periastron a dateΩ longitude of ascending node flip angleω longitude of periastron twist angleequinox equinox of date sets direction of equinoxf 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.
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GJ 1245 ACPushing Towards Exoplanets
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New Orbits in Solar System
located 44.7 AUPsun ~ 300 yrs
HST WFPC2 images V = 23.1
Porb 590 ± 40 daysa 22400 ± 900 km
mtot 0.02% Pluto
at least 77 multiple TNOs knownwww2.lowell.edu/users/grundy/tnbs/status.html
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Reality Check: 3-body Systemstheory: 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?
Reality Check: 3-body Systems
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Counter-Intuitive DynamicsLagrangian 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 systemTrojan asteroids at Jupiter (>5000), Mars (6+), Neptune (7+)small moons at Sat/Tethys (Telesto+Calypso) and Sat/Dione (Helene+Polydeuces)
Earth orbiting spacecraft
SOHO
WMAP
GaiaJWST
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Counter-Intuitive DynamicsTadpole orbits: librating positions around L4 and L5 (note corotating frame!)
Trojan asteroids at Jupiter, Mars, and Neptune
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Counter-Intuitive DynamicsHorseshoe 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 yearsCruithne and Asteroid 2002 AA29 around Earth
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Counter-Intuitive DynamicsHorseshoe orbits: Cruithne --- each loop takes 1 yr
http://www.astro.uwo.ca/%7Ewiegert/3753/3753.html
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Counter-Intuitive DynamicsHorseshoe 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 DynamicsChaotic motion: trajectories that begin arbitrarily close together will diverge exponentially with time (note that 4.6 Gyr is often not sufficient “time”)
Mars’ axis tiltHyperion 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 gapsSaturn ring particles : Saturn moons (Mimas, Atlas, …)
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300,000 km wide X 10 m (!) thick
Saturn’s Rings
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Saturn’s Ringsparticles forced into plane by orbits
albedo 0.8 … shiny snowballs of H2O
total mass only that of small moon
young … likely formed by Roche limit crossing
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6 major regions … 2 divisions … 1000s of ringletsABC rings seen from ground … majorDEF rings seen from Voyager/Cassini … minor
Cassini Division Mimas 2:1 resonanceA ring Atlas on edge (in 3:2 with Mimas)Encke Gap Pan within A ringF ring (braided) shepherds Prometheus + PandoraE ring Enceladus volcanism (outside
Roche)spokes collisions in rings
Saturn’s Ring Structure
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The Saturn System
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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
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Tidestides 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 changesEarth 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-gravityevidence for Moon’s shape --- closer in the past
d. may create rings
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Tidal Forces 2 x G m2
F tidal ≈ ______________
r3
r is distance between two bodiesx is distance along axis separating two bodies (surface = body’s radius)
tides on Earth m2 (kg) r (km) Ftidal/2RG
Sun 1.99e30 1.50e08 5.90e05
Venus 4.87e24 4.14e07 6.86e01
Moon 7.35e22 3.84e05 1.30e06 Winner! 2.2 X Sun
Jupiter 1.90e27 6.29e08 7.63e00
Earth-Moon tidal force is 81 X Moon-Earth tidal forceJupiter-Io tidal force is 20,000 X Moon-Earth tidal force
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Tides in Earth-Moon Systemangular momentum is conserved, but …
can be swapped between rotation and revolution via tidal torques
on Earth: twice per 25 hours due to Moontwice per 24 hours due to Sun (1/2 strength)
on Moon: mostly fixed because of synchronous rotationbut not entirely because of eccentricity … nutation
bulge torque: Earth rotates faster than Moon orbitsEarth not perfectly elastic, so bulge not on Earth-Moon lineMoon pulls back on bulge --- Earth slows downEarth bulges pull on Moon --- torque acc. Moon outward ~ 1/r6
death spirals: moons moving retrograde, or faster than planet rotates
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Moon’s Shape
bulge frozen at 2/3 current Moon distance
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Roche LimitRoche 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 ~ 0Uranus 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
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Roche Limits and Ring Systems
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Neptune’s Rings
Adams
Lassell
LeVerrierGalle
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Billions of Years from Now…
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Solar System Explorers 04How does the Sun affect objects in the Solar System?
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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