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*Planetary Dynamics Dr Sarah Maddison Centre for Astrophysics & Supercomputing Swinburne University*

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Planetary Dynamics Dr Sarah Maddison Centre for Astrophysics & Supercomputing Swinburne University

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Planetary DynamicsDr Sarah MaddisonCentre for Astrophysics & SupercomputingSwinburne University

AstroFest 2007

ABSTRACT:Observations of the Solar System over the past 50 - particularly minor bodies - has demonstrated a lot of dynamic structure (e.g. satellite resonances, planetary rings, synchronous locking, Kirkwood gaps) and thus it is important for us to understand planetary dynamics, which is driven by gravity.

AstroFest 2007

OUTLINE: This lecture will cover the gravitational theory behind planetary dynamics, including: Keplers laws and Newtons laws, resonances, tides, and orbits and orbital elements.

To understand simulations of planetary dynamics, well also cover: the N-body problem.

AstroFest 2007

Laws of Motion..

AstroFest 2007

Keplers Laws

Kepler (1609, 1619) presented three empirical laws of planetary motion from obs made by Tycho Brahe:(1) Planets move in an ellipse with the Sun at one focus(2) The radial vector from the Sun to a planet sweeps out equal area in equal time (3) The orbital period square is proportional to the semi-major axis cubed (T2 a3)

But empirical laws with no physical understanding of why planets obey them

AstroFest 2007

Newtons Laws

Newtons (1687) three laws of motion:(1) Bodies remain at rest or in uniform motion in a straight line unless acted on by a force(2) Force equals the rate of change of momentum (F = dp/dt = ma) (3) Every action has an equal and opposite reactions (F12= -F21)Plus his universal law of gravitation: F = Gm1m2 / d2Probably first derived by Robert Hooke, but Newton used it to explain Keplers laws.

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Newtons laws revolutionized science and dynamical astronomy in particular.E.g. extending Newtons law of gravitational to N > 2 showed that the mutual planetary interactions resulted in ellipses not fixed in space orbital precession Planetary orbits rotate in space over ~105 years

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But its an approximation (though a pretty good one!) Mercury should precess at a rate of 531/century, but 43/century greater. Precession of Mercurys perihelion explained using Einsteins theory of General Relativity.

AstroFest 2007

Resonances..

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Resonances Lots of discoveries of minor bodies in the last 50 years: ~100 new satellites over 10,000 catalogised asteroids over 500 reliable comet orbits over 1000 KBOs dust bands in the asteroid belt planetary rings of all giants with unique characteristics All follow Newtons laws and experience subtle gravitational effects of resonances

AstroFest 2007

Resonances result from a simple numerical relationship between periods: rotational + orbital periods spin-orbit coupling orbital periods of N bodies orbit-orbit coupling plus more complex resonances

Dissipative forces drive evolutionary processes in the Solar System connected with the origins of some of these resonances.

AstroFest 2007

Examples of Solar System resonances:(1) spin-orbit coupling of the Moon: Trot = Torb 1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth

AstroFest 2007

Examples of Solar System resonances:(1) spin-orbit coupling of the Moon: Trot = Torb 1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth(2) spin-orbit coupling of Mercury: 3Trot = 2Torb 3:2 spin-orbit couplingtwo Mercury years = three sidereal days on Mercury

AstroFest 2007

Examples of Solar System resonances:(1) spin-orbit coupling of the Moon: Trot = Torb 1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth(2) spin-orbit coupling of Mercury: 3Trot = 2Torb 3:2 spin-orbit coupling(3) orbit-orbit resonances of planets: - Jupiter + Saturn in 5:2 near resonance, perturbs both planets orbital elements on ~900 year timescale - Neptune + Pluto in 3:2 orbit-orbit resonance, maximises separation at conjunction and avoids close approaches - other planets involved in long term secular resonances associated with the precession of their orbits

AstroFest 2007

Examples of Solar System resonances cont..(4) Galileans satellites spin-spin resonances : - Io + Europa 2:1 resonance - Europa + Ganymede 2:1 resonance 1 2 3 4 5 6 7 8 9

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- average orbital angular velocity or mean motion defined as n = 360/T (degrees per day) - mean motions of the Galileans: nI = 203.448 o/d, nE = 101.374 o/d, nG = 50.317 o/d so nI/nE=2.0079 and nE/nG=2.01469 and hence nI - 3nE + 2nG = 0 (to within obs errors of 10-9 o/d) This is the Laplace relation, prevents triple conjunctions - 2:1 Io:Europa resonances results in active volcanism on Io Examples of Solar System resonances cont..

AstroFest 2007

Examples of Solar System resonances cont..(5) Saturns satellites have widest variety of resonances : - Mimas + Tethys 4:2 resonance (nM/nT=2.003139) - Enceladus + Dione 2:1 resonance (nE/nD=1.997) - Titan + Hyperion 4:3 resonance (nT/nH=1.3343) - Dione & Tethys 1:1 resonance with small bodies on their orbits - Janus + Epimetheus on 1:1 horseshoe orbits (swap orbits every 3.5 years) http://ssdbook.maths.qmw.ac.uk/animations/Coorbital.mov - 2:1 resonant perturbation of Mimas causes gap in rings (Cassini division) - structure of F ring due to Pandora + Prometheus http://photojournal.jpl.nasa.gov/animation/PIA07712 - spikes in Encke gap due to PanCassini divisionEncke gap

AstroFest 2007

Examples of Solar System resonances cont..(6) Uranuss satellites also in resonance: - Rosalind + Cordelia in close 5:3 resonance - Cordelia + Ophelia bound to narrow ring by 24:25 and 14:13 resonances with the inner and outer ring edge - resonances not due to the major satellites, though high inc of Miranda suggests resonances of the past, may have produced resurfacing eventsAriel9 rings of Uranus

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Examples of Solar System resonances cont..(7) Pluto: - Pluto + Charon in synchronous spin state - totally tidally despun(both keep same face towards each other, fixed above same spot)Pluto & CharonAve separation ~17 RPluto

AstroFest 2007

Examples of Solar System resonances cont..(7) Pluto: - Pluto + Charon in synchronous spin state - totally tidally despun(both keep same face towards each other, fixed above same spot)(8) Kuiper Belt: - predicted by Edgeworth (1951) and Kuiper (1951) and observed in 1992 (Jewitt & Luu) - three main classes: Classical, Resonant and Scattered - Third of all KBOs in 3:2 resonance with Neptune, i.e. PlutinosPluto & CharonAve separation ~17 RPluto

AstroFest 2007

Examples of Solar System resonances cont..(9) Asteroid Belt: - Resonant structure found by Kirkwood (1867), noticed gaps at important Jupiter resonances: 4:1, 3:1, 5:2, 7:3, 2:1 but also concentrations at 3:2 and 1:1Resonances not totally cleared, some asteroids captured by Jupiter

AstroFest 2007

Tides..

AstroFest 2007

Tidal forces Small bodies orbit massive object due to gravity, but are also subject to tidal forces that may tear the satellite apart. The satellite feels a stronger gravitational force on its near side to its far side tidal forces are differential. Oscillations can develop and deform or disrupt the satellite.gravity at near surface is stronger than at far surfaceas satellite approaches massive object, tidal forces get stronger and satellite is distorted

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The Roche limit Neglecting internal satellite forces, disruption occurs when differential tidal force exceeds the satellites self-gravitation: Maximum orbital radius for which tidal disruption occurs is the Roche limit. Substituting average densities the equation becomes:where Ms and Mm are the masses of the satellite and central body; r is their separation; and Rs is the radius of the satellite.

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The Hill radius For an N-body system a satellite can feel tidal forces from several massive bodies, e.g. the Moon feels a tidal force from the Earth and from the more distant (but more massive) Sun.

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The Hill radius 2 The Hill radius is the radius of a sphere around a planet within which the planetary tidal forces on a small body are larger than the tidal forces of the Sun.As a rough guide, the Hill radius is: 0.35 AU for Jupiter, 0.44 AU for Saturn, 0.47 AU for Uranus, and 0.78 AU for Neptune. For one test particle and two massive bodies (e.g. the Sun and a planet), the Hill radius, RH, is:

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Orbits..

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Eccentricity of the ellipse defined by:The Geometry of Ellipsesr2r1Equation of the ellipse:In Cartesian coordinates:Let:Thus :Simple algebra shows that the following relations hold:

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Specifying a point on the ellipseCartesian coordinates with the origin at the centre of the ellipse, we have:From the equation of the ellipse, and by substituting the equations that define x, y, b and e, it is possible to show that:

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Orbital elements Orbits are uniquely specified in space by six orbital elements. semi-major axis a eccentricity e = c/aThe inclination, i, describes tilt of orbital plane with respect to reference planeThe size and shape of an orbit determined by the semi-major axis, a, and eccentricity, e

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PThe argument of pericentre*, , and longitude of the ascending node, , determine the orientation of the orbit and where the line of nodes crosses the reference plane.* Pericentre = periastron, perihelion, periapse depending on system in question - point of closest approach to the focus

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The true anomaly, f, tells where orbiting body is at a particular instant in time and is measured from pericentre to orbiting body.

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a, the semi-major axis of the ellipse; e, the eccentricity of the ellipse; i, the inclination of the orbital plane; , the argument of pericentre; , the longitude of the ascending node; and (say) time T when planet is at perihelion

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Cartesian vs Keplerian orbital elements The Cartesian orbital elements are: position (x, y, z), and velocity (vx, vy, vz). Cartesian & Keplerian are equally precise ways of describing an orbit. Relatively simple equations exist for transforming between the two coordinate systems. (x,y,z)(vx,vy,vz)(0,0)

Cartesian

Keplerian

x

a

y

e

z

i

vx

(

vy

(

vz

f

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Orbital Energy

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Energy and Orbit Types The shape of an orbit depends if body is bound or unbound, which depends on system total energy of the system. Total energy is the sum of the kinetic energy, KE, and the gravitational potential energy, U:where:and

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If E < 0, orbiting body m2 does not have sufficient velocity to escape from the gravitational field of m1 the orbit is bound. If E > 0, orbiting body m2 has sufficient velocity to escape the orbit is unbound

Thus total system energy is:

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Different types of orbits:

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N-body Problem

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N-body Problem Analytic solution exists for the 2-body problem. But no solution for the 3-body problem and stable orbits difficult to obtain. Can simplify to a restricted 3-body problem (two bodies in circular orbit about COM and third body with m3