Dynamics

Solar System Explorers 05 How does the Sun affect objects in the Solar System?

Basic Newton G m Earth m 2 F gravity on Earth = — _______________ r 2 m 2 (kg)r (m)F gravity Sun1.99e301.50e113.52e22Winner!178 X Moon Venus4.87e244.14e101.13e18 Moon7.35e223.84e081.98e20 Jupiter1.90e276.29e111.91e18 What about the Moon? m Earth = 5.97e24 kg Earth-Moon has F Earth ~ 1.98e20, Sun-Moon has F Sun ~ 4.34e20

Dynamics: Kepler I Kepler I: planetary orbits are ellipses with the Sun at a focus a (1 - e 2 ) r Sun = ______________ 1 + e cos f e, eccentricity = (1 - b 2 minor /a 2 major ) 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 - e 2 ) r Sun = ______________ 1 + e cos f e, eccentricity = (1 - b 2 minor /a 2 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 m 1 r 1 + m 2 r 2 r CM = _________________ M M = m 1 + m 2 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 P 2 (yr) = a 3 (AU) Newton III: it also works outside of the Solar System 4π 2 a 3 a 3 P 2 = __________________ or M total = _______ G (m 1 + m 2 ) P 2 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 asemimajor axissize eeccentricityshape iinclination (~0 in SS, edge on = 90 outside) tilt angle Porbital periodtime Tepoch of periastrona date Ωlongitude of ascending nodespin angle ω argument of periastrontωist angle

Spin: Longitude of Ascending Node

Tωist: Longitude of Periastron

Orbital Elements asemimajor axissize eeccentricityshape iinclinationtilt Porbital periodtime Tepoch of periastrona date Ωlongitude of ascending nodeflip angle ω longitude of periastrontwist angle equinoxequinox of datesets direction of equinox ffractional massa 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.

12 GJ 1245 AC Pushing Towards Exoplanets

New Orbits in Solar System located 44.7 AU P sun ~ 300 yrs HST WFPC2 images V = 23.1 P orb 590 ± 40 days a ± 900 km m tot 0.02% Pluto at least 77 multiple TNOs known www2.lowell.edu/users/grundy/tnbs/status.html

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? Reality Check: 3-body Systems

15 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 SOHO WMAP Gaia JWST

Counter-Intuitive Dynamics Tadpole orbits: librating positions around L4 and L5 (note corotating frame!) Trojan asteroids at Jupiter, Mars, and Neptune

17 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

18 Counter-Intuitive Dynamics Horseshoe orbits: Cruithne --- each loop takes 1 yr

19 Counter-Intuitive Dynamics Horseshoe orbits: Asteroid 2002 AA each vertical loop takes 1 yr “at least three others”

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 : : 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, …)

21 300,000 km wide X 10 m (!) thick Saturn’s Rings

22 Saturn’s Rings particles forced into plane by orbits albedo 0.8 … shiny snowballs of H 2 O total mass only that of small moon young … likely formed by Roche limit crossing

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24 6 major regions … 2 divisions … 1000s of ringlets ABC rings seen from ground … major DEF rings seen from Voyager/Cassini … minor Cassini DivisionMimas 2:1 resonance A ringAtlas on edge (in 3:2 with Mimas) Encke GapPan within A ring F ring (braided)shepherds Prometheus + Pandora E ringEnceladus volcanism (outside Roche) spokescollisions in rings Saturn’s Ring Structure

The Saturn System

26 Saturn’s Rings Details

27 Jupiter’s Rings

28 Jupiter’s Rings Details

29 Jupiter’s Rings Details

Tides tides are a differential gravitational force ~ 1 / r 3 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 / r 6 ) 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 m 2 F tidal ≈ ______________ r 3 r is distance between two bodies x is distance along axis separating two bodies (surface = body’s radius) tides on Earthm 2 (kg)r (km)F tidal/ 2R  G Sun1.99e301.50e085.90e05 Venus4.87e244.14e076.86e01 Moon7.35e223.84e051.30e06Winner!2.2 X Sun Jupiter1.90e276.29e087.63e00 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/r 6 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 a Roche = R p (ρ p /ρ s ) 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

35 Roche Limits and Ring Systems

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?

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