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

2. 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.

3. AstroFest 2007 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.

4. AstroFest 2007 Laws of Motion…..

5. AstroFest 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…

6. AstroFest 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.

7. AstroFest 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 ~10 5 years

8. AstroFest 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.

9. AstroFest 2007 Resonances…..

10. AstroFest 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

11. 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.

12. AstroFest 2007 Examples of Solar System resonances: (1) spin-orbit coupling of the Moon: T rot = T orb  1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth A B C D E F G H A B C D E F G H Phases as seen from Earth Sun’s rays New moon Full moon 3rd quarter 1st quarter Sun’s rays Dark side of the Moon Near side of the Moon (the face that we see!)

13. AstroFest 2007 Examples of Solar System resonances: (1) spin-orbit coupling of the Moon: T rot = T orb  1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth (2) spin-orbit coupling of Mercury: 3T rot = 2T orb  3:2 spin-orbit coupling two Mercury years = three sidereal days on Mercury

14. AstroFest 2007 Examples of Solar System resonances: (1) spin-orbit coupling of the Moon: T rot = T orb  1:1 or synchronous spin-orbit coupling same face of the Moon always faces Earth (2) spin-orbit coupling of Mercury: 3T rot = 2T orb  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

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

16. AstroFest 2007 - 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 - 3n E + 2n 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 Examples of Solar System resonances cont..

17. AstroFest 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 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 http://photojournal.jpl.nasa.gov/animation/PIA07712 - spikes in Encke gap due to Pan Cassini division Encke gap

18. AstroFest 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 - resonances not due to the major satellites, though high inc of Miranda suggests resonances of the past, may have produced resurfacing events Ariel 9 rings of Uranus

19. 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) Pluto & Charon Ave separation ~17 R Pluto

20. 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. Plutinos Pluto & Charon Ave separation ~17 R Pluto

21. 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 1:1 Resonances not totally cleared, some asteroids captured by Jupiter

22. AstroFest 2007 Tides…..

23. AstroFest 2007 Small bodies orbit massive object due to gravity, but are also subject to tidal forces that may tear the satellite apart. Tidal forces 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 surface as satellite approaches massive object, tidal forces get stronger and satellite is distorted

24. AstroFest 2007 Neglecting internal satellite forces, disruption occurs when differential tidal force exceeds the satellite’s self-gravitation: The Roche limit Maximum orbital radius for which tidal disruption occurs is the Roche limit. Substituting average densities the equation becomes: where M s and M m are the masses of the satellite and central body; r is their separation; and R s is the radius of the satellite. where R m is the radius of the central body.

25. AstroFest 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

26. AstroFest 2007 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, R H, is:  2 is the reduced mass of the second body given by  2 = M 2 /(M 2 +M 1 )

27. AstroFest 2007 Orbits…..

28. AstroFest 2007 The Geometry of Ellipses r2r2 r1r1 aae b Equation of the ellipse: In Cartesian coordinates: Let: Thus : Eccentricity of the ellipse defined by: Simple algebra shows that the following relations hold:

29. AstroFest 2007 Specifying a point on the ellipse Cartesian coordinates with the origin at the centre of the ellipse, we have: F1F1 2a (0,0) y x  f r c (x,y) From the equation of the ellipse, and by substituting the equations that define x, y, b and e, it is possible to show that:

30. AstroFest 2007 Orbital elements Orbits are uniquely specified in space by six orbital elements. ecliptic plane orbit plane i a c semi-major axis a eccentricity e = c/a The inclination, i, describes tilt of orbital plane with respect to reference plane The size and shape of an orbit determined by the semi-major axis, a, and eccentricity, e

31. AstroFest 2007 ecliptic plane orbit plane P  i a c  0 o in Pisces ascending node descending node 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. * Pericentre = periastron, perihelion, periapse depending on system in question - point of closest approach to the focus

32. AstroFest 2007 ecliptic plane orbit plane P  i a c  0 o in Pisces node The true anomaly, f, tells where orbiting body is at a particular instant in time and is measured from pericentre to orbiting body. true anomaly

33. AstroFest 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 ecliptic plane orbit plane P  i a c  0 o in Pisces node

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

35. AstroFest 2007 Orbital Energy…

36. AstroFest 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

37. AstroFest 2007 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 Thus total system energy is:

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

39. AstroFest 2007 N-body Problem…

40. AstroFest 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.

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

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

43. AstroFest 2007 The Mathematical Model Two main parts of codes for solving N-body problems: The relevant equations for a dynamical N-body code are just: 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. Newton’s law of gravitation for the forces; and the equation of motion for the time evolution. the force calculation and the time evolution.

44. AstroFest 2007 Newton’s universal law of gravitation between two bodies is: Gravitational Forces r m1m1 m2m2 F2F2 F1F1 F1F1

45. AstroFest 2007 What about an N=5 system? m1m1 m2m2 m3m3 m4m4 m5m5 F 12 F 13 F 15 F 14 The force exerted on body 1 by the other 4 bodies would be given by: the sum of the individual forces acting on it: F 1 = F 12 + F 13 + F 14 + F 15.

46. AstroFest 2007 Also need to calculate the force on particle 2 due to the other 4 particles: m1m1 m3m3 m4m4 m5m5 m2m2 and the force on particle 3 due to all the other particles: and the force on particle 4 due to all the other particles: and the force on particle 5 due to all the other particles: A computer would be helpful :-)

47. AstroFest 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...

48. AstroFest 2007 (1) Force is actually a vector quantity, so it has a magnitude and a direction.  Need to soften the gravity Equation becomes: (2) As the particles get closer together, the forces get larger. As particle i approaches j the denominator r ij of the force equation approaches zero so the force become infinite. 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)

49. AstroFest 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.

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

51. AstroFest 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.

52. AstroFest 2007 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. Accuracy and Stability

53. AstroFest 2007 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: The N-body Algorithm Set initial conditions Solve equations of motion Calculate forces Update time counter Output data Choose N and  t, set initial particle m i, r i, v i, F i a i =  v i /  t i v i =  r i /  t i F i =  j Gm i m j /r ij 2 t new = t old +  t r new, v new, F new, t new