Planetary Dynamics Dr Sarah Maddison Centre for Astrophysics

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

OUTLINE: This lecture will cover the gravitational theory behind planetary dynamics, including: • Kepler’s laws and Newton’s laws, • resonances, • tides, and • orbits and orbital elements. To understand simulations of planetary dynamics, we’ll also cover: • the N-body problem. Astro. Fest 2007

Laws of Motion…. . Astro. Fest 2007

Kepler’s 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 (T 2 a 3) But empirical laws with no physical understanding of why planets obey them… Astro. Fest 2007

Newton’s Laws Newton’s (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 (F 12= -F 21) Plus his universal law of gravitation: F = Gm 1 m 2 / d 2 Probably first derived by Robert Hooke, but Newton used it to explain Kepler’s laws. Astro. Fest 2007

Newton’s laws revolutionized science and dynamical astronomy in particular. E. g. extending Newton’s 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 Astro. Fest 2007

But it’s an approximation (though a pretty good one!) Mercury should precess at a rate of 531”/century, but 43”/century greater. Precession of Mercury’s perihelion explained using Einstein’s theory of General Relativity. Astro. Fest 2007

Resonances…. . Astro. Fest 2007

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 Newton’s laws and experience subtle gravitational effects of resonances Astro. Fest 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. Astro. Fest 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 G H 3 rd quarter F Sun’s rays A New moon Full moon 1 st quarter B E D C Dark side of the Moon A B C Near side of the Moon (the face that we see!) D E F G H Phases as seen from Earth Astro. Fest 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: 3 Trot = 2 Torb 3: 2 spin-orbit coupling two Mercury years = three sidereal days on Mercury Astro. Fest 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: 3 Trot = 2 Torb 3: 2 spin-orbit coupling (3) orbit-orbit resonances of planets: - Jupiter + Saturn in 5: 2 near resonance, perturbs both planet’s 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 Astro. Fest 2007

Examples of Solar System resonances cont. . (4) Galileans satellite’s spin-spin resonances : - Io + Europa 2: 1 resonance - Europa + Ganymede 2: 1 resonance 1 2 3 4 5 6 7 8 9 Io passes Europa every 2 nd orbit and Ganymede every 4 th orbit Astro. Fest 2007

Examples of Solar System resonances cont. . - average orbital angular velocity or mean motion defined as n = 360/T (degrees per day) - mean motions of the Galileans: n. I = 203. 448 o/d, n. E = 101. 374 o/d, n. G = 50. 317 o/d so n. I/n. E=2. 0079 and n. E/n. G=2. 01469 and hence n. I - 3 n. E + 2 n. G = 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 Astro. Fest 2007

Examples of Solar System resonances cont. . (5) Saturn’s satellites have widest variety of resonances : - Mimas + Tethys 4: 2 resonance (n. M/n. T=2. 003139) - Enceladus + Dione 2: 1 resonance (n. E/n. D=1. 997) - Titan + Hyperion 4: 3 resonance (n. T/n. H=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/PIA 07712 Cassini division - spikes in Encke gap due to Pan Encke gap Astro. Fest 2007

Examples of Solar System resonances cont. . (6) Uranus’s 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 9 rings of Uranus - resonances not due to the major satellites, though high inc of Miranda suggests resonances of the past, may have produced resurfacing events Ariel Astro. Fest 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) Ave separation ~17 RPluto & Charon Astro. Fest 2007

Examples of Solar System resonances cont. . (7) Pluto: Ave separation ~17 R - Pluto + Charon in synchronous spin state “totally tidally despun”(both keep same face towards each other, fixed above same spot) Pluto & Charon (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. Plutinos Pluto Astro. Fest 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: 1 Resonances not totally cleared, some asteroids captured by Jupiter Astro. Fest 2007

Tides…. . Astro. Fest 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. gravity at near surface is stronger than at far surface • Oscillations can develop and deform or disrupt the satellite. as satellite approaches massive object, tidal forces get stronger and satellite is distorted Astro. Fest 2007

The Roche limit • Maximum orbital radius for which tidal disruption occurs is the Roche limit. • Neglecting internal satellite forces, disruption occurs when differential tidal force exceeds the satellite’s self-gravitation: 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. • Substituting average densities the equation becomes: where Rm is the radius of the central body. Astro. Fest 2007

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. Forces on the near side of the Moon from the Sun and Earth Forces on the far side of the Moon from the Sun and Earth Astro. Fest 2007

• 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. • For one test particle and two massive bodies (e. g. the Sun and a planet), the Hill radius, RH, is: 2 is the reduced mass of the second body given by 2 = M 2/(M 2+M 1) 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. Astro. Fest 2007

Orbits…. . Astro. Fest 2007

b The Geometry of Ellipses a ae Equation of the ellipse: r 2 r 1 In Cartesian coordinates: Let: Thus : Eccentricity of the ellipse defined by: Simple algebra shows that the following relations hold: Astro. Fest 2007

(x, y) y r Specifying a point on the ellipse Cartesian coordinates with the origin at the centre of the ellipse, we have: (0, 0) c f F 1 2 a x From the equation of the ellipse, and by substituting the equations that define x, y, b and e, it is possible to show that: Astro. Fest 2007

Orbital elements Orbits are uniquely specified in space by six orbital elements. The size and shape of an orbit determined by the semi-major axis, a, and eccentricity, e • semi-major axis a • eccentricity e = c/a c a ne la it p orb i The inclination, i, describes tilt of orbital plane with respect to reference plane p c i t p i ecl Astro. Fest 2007

The 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. descending node c a ne la it p P 0 o in orb i lane p c i t p i ecl ascending Pisces node * Pericentre = periastron, perihelion, periapse depending on system in question - point of closest approach to the focus Astro. Fest 2007

The true anomaly, f, tells where orbiting body is at a particular instant in time and is measured from pericentre to orbiting body. node c a P orb i true anomaly 0 o in Pisces ane l p it lane p c i t p i ecl node Astro. Fest 2007

• 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 node c a ne la it p P orb i 0 o in Pisces lane p c i t p i ecl node Astro. Fest 2007

Cartesian vs Keplerian orbital elements • The Cartesian orbital elements are: – position (x, y, z), and – velocity (vx, vy, vz). (x, y, z) (0, 0) (vx, vy, vz) • Cartesian & Keplerian are equally precise ways of describing an orbit. • Relatively simple equations exist for transforming between the two coordinate systems. Astro. Fest 2007

Orbital Energy… Astro. Fest 2007

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 Astro. Fest 2007

Thus total system energy is: • If E < 0, orbiting body m 2 does not have sufficient velocity to escape from the gravitational field of m 1 the orbit is bound. • If E > 0, orbiting body m 2 has sufficient velocity to escape the orbit is unbound Astro. Fest 2007

Different types of orbits: Ellipses and circles 0 e < 1 Bound Total energy is negative Parabola e=1 Unbound Total energy is zero Hyperbola e>1 Unbound Total energy is positive Astro. Fest 2007

N-body Problem… Astro. Fest 2007

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 m 3 << m 1, m 2) • Numerical simulations needed to studying systems of 3 or more objects N-body problem. Astro. Fest 2007

Setting up a Numerical Experiment The basic steps involved in using a computer to find a numerical “solution” to an N-body problem are: Physical phenomena Mathematical model the mathematical model must choose thethe physical system is that be converted a continuous you approximated wish from to investigate by a ore. g. differential equation into an mathematical the motion model, of N planets which algebraic approximation around uses a some star, simplifying where Nwhich ≥ 3 computerstocan solve. the assumptions describe Both time must be workings ofand the space physical system discretised, which can produce numerical errors Discretise the model Astro. Fest 2007

The next steps are: Numerical algorithm Computer program finally get to run your Choice of discretisation is often writingyou the computer code is computer you related to experiments, theofalgorithm chosen where most the hard but work need to know what you’re to solve the discrete system. lies. The code needs to be well testing you’re to Need tofor, be what able to solvetrying thethe engineered to capitalise on find, andproblem how to do analysis on discrete rapidly available computing power, and the data that your otherwise having aexperiment computer is it should be be easy to use and produces. pictures are of no help at Pretty all! modify. course vital at this stage! Computer experiment Astro. Fest 2007

The Mathematical Model Two main parts of codes for solving N-body problems: • the force calculation and • the time evolution. Both can be described by a mathematical model - a set of mathematical equations which tell of the future state of the system, given a set of initial conditions. The relevant equations for a dynamical N-body code are just: • Newton’s law of gravitation for the forces; and • the equation of motion for the time evolution. Astro. Fest 2007

Gravitational Forces Newton’s universal law of gravitation between two bodies is: m 1 r F 1 F 2 m 2 Astro. Fest 2007

What about an N=5 system? The force exerted on body 1 by the other 4 bodies would be given by: m 4 m 3 F 13 m 2 F 14 F 12 m 1 F 15 m 5 the sum of the individual forces acting on it: F 1 = F 12 + F 13 + F 14 + F 15. Astro. Fest 2007

Also need to calculate the force on particle 2 due to the other 4 particles: and the force on particle 3 due to all the other particles: and the force on particle 4 due to all the other particles: m 4 m 3 m 2 m 1 m 5 and the force on particle 5 due to all the other particles: A computer would be helpful : -) Astro. Fest 2007

The force equation becomes: For each N particle i we need to sum over all the other N-1 particles. The mathematical model for gravitational force is quite easy to discretise for N particles. (Note that this is an Nx(N-1) or O(N 2) calculation). However. . . Astro. Fest 2007

(1) Force is actually a vector quantity, so it has a magnitude and a direction. (2) As the particles get closer together, the forces get larger. As particle i approaches j the denominator rij of the force equation approaches zero so the force become infinite. Need to soften the gravity Equation becomes: The softening parameter must be carefully chosen - if too large it affects the physics (like an outward force) - if too small the forces become large (and time must slow down) Astro. Fest 2007

The Equations of Motion The time evolution of the system is governed by the equations of motion: We can easily discretise by writing the differential as a finite difference: where i = initial f = final The symbol represents a small but finite change. Astro. Fest 2007

Newton’s second law relates force to acceleration via the equation: Substituting F by Fgrav from Newton’s law of gravitation gives: Need to solve for the position and velocity of the system at the next timestep. Hence: Astro. Fest 2007

Taking a small timestep t between the old and new states of the system, the final velocity and position are given by: Once we’ve solved for the gravitational force, F, at the initial state of the system, we can work out the position and velocity for each body in the system at the next timestep. In practice there are many more sophisticated ways to discretise the equations of motion that produce more accurate time stepping, but the essential principles have been described here. Astro. Fest 2007

Accuracy and Stability • Two more things that we need to be careful about: our choice of t and N. • Timestep t controls the stability. If t = constant, we get large errors when two particles get close. Need a numerical scheme with a variable timestepping which automatically decreases t if particles are too close and increases t as particles move apart. • The particle number N gives the resolution. Ideally we want N to be as large as possible, but this means more calculations. Supercomputers can help us here. Astro. Fest 2007

The N-body Algorithm We’re now armed with our mathematical model for the gravitational force and equations of motion; we have a discrete algorithm for the mathematical model, and we’re ready to write our computer code to run our computer experiments. Our computer algorithm will look like: Set initial conditions Choose N and t, set initial particle mi, ri, vi, Fi Solve equations of motion ai = vi/ ti vi = ri/ ti Calculate forces Fi = j Gmimj/rij 2 Update time counter tnew = told + t Output data rnew, vnew, Fnew, tnew Astro. Fest 2007