Dynamics

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.

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

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

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

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 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 …

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

Spin: Longitude of Ascending Node

Tωist: Longitude of Periastron

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

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

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

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

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

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

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

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

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

Moon’s Shape

bulge frozen at 2/3 current Moon distance

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

Neptune’s Rings

Adams

Lassell

LeVerrierGalle

Billions of Years from Now…

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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