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MTH724U/ASTM001 SOLAR SYSTEM Nick Cooper/Carl Murray Lecture 9: Chaos and Long- Term Evolution

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Introduction The equations we have used so far describe systems that are usually conservative (i.e. energy is a constant) and deterministic. We expect that the future and past states of the systems are determined by the equations and the starting conditions alone. Therefore we expect that if we are given the state (positions and velocities) of a system at one time then we can confidently predict the future or past state at another time. What we have learned in recent years is that systems as simple as the three-body problem can give rise to chaotic motion whereby the motion, although deterministic, is nevertheless unpredictable over sufficiently long time scales.

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What is Chaos? A system is chaotic if it shows a sensitive dependence on initial conditions, whereby a small change in starting values can produce a radically different subsequent state (the “butterfly hypothesis”).

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Examples of Chaotic Orbits (1) The orbit of (2060) Chiron (a=13.75AU, e=0.385) is chaotic. It has a 1 in 8 chance that a close approach to Saturn will cause it to leave the Solar System. It has a 7 in 8 chance that a close approach to Saturn will cause its orbit to evolve into the inner solar system. The orbit of Comet Shoemaker-Levy 9 is/was chaotic. It had a close approach to Jupiter in 1992 leading to its tidal disruption and eventual impact with Jupiter in July 1994. It was probably captured by Jupiter in 1929 (+/-9y) and prior to that it probably had the orbit of a short-period Jupiter- family comet. Interplanetary spacecraft move on chaotic trajectories. The gravity assist technique makes use of the sensitive dependence on initial conditions to make large changes.

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Examples of Chaotic Orbits (2) But chaotic orbits can be more subtle. Consider the following numerical integration: The starting longitudes differed by 0.000001 degrees resulting in a clear divergence in the behaviour of the eccentricity after about 160 Jupiter periods.

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Regular and Chaotic Orbits (1) Consider motion in the circular restricted three-body problem (CRTBP): In the CRTBP the Jacobi constant is a constant or integral of the motion:

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Regular and Chaotic Orbits (2) Therefore, if we know, for example, the x, y and xdot values we can always find the ydot value (up to a sign change) by using the Jacobi constant. Suppose we then decide to look at the x and xdot values only when y=0 and ydot positive. This will give a sequence of points in the (x,xdot) space. This gives a Poincaré surface of section.

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Regular and Chaotic Orbits (3) Let us look at some results of numerical integrations showing regular and chaotic orbits: First the evolution of e and a for a regular orbit:

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Regular and Chaotic Orbits (4) Now the same data plotted as a surface of section: The points on the “islands” are produced in sequence, one on the first, one on the second, one on the third. Islands like these are usually associated with resonance, in this case the 7:4 resonance with Jupiter.

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Regular and Chaotic Orbits (5) Now plots of e and a variation for a chaotic orbit: The time variation still shows some obvious periods but the behaviour is fundamentally different from the regular case.

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Regular and Chaotic Orbits (6) Now the same data plotted as a surface of section: Note that the previous three islands are still there, outlined by the “fuzzy” chaotic region. Note also the tendency for points to “stick” around the resonant islands.

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The Lyapounov Exponent (1) We can quantify the degree of divergence of nearby trajectories by calculating the maximum Lyapounov characteristic exponent. Exponential divergence of nearby trajectories is one of the properties of chaotic motion. For example:

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The Lyapounov Exponent (2) The maximum LCE is found numerically by calculating: It is usually better to “renormalise” the separation vector at fixed intervals so that the local rate of divergence is measured.

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The Lyapounov Exponent (3) If the orbit is chaotic then the separation will vary with time by: The maximum LCE is then: If renormalisation is used then:

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The Lyapounov Exponent (4) If the orbit is chaotic then we expect a log vs. log t plot to tend to a finite value. If the orbit is regular the plot should have a slope of -1.

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Chaos in the CRTBP (1) The amount of the chaos in the CRTBP depends on the value of the Jacobi constant. This is a sequence of surfaces of section for the region interior to Jupiter’s orbit in the CRTBP:

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Chaos in the CRTBP (2) This is a sequence of surfaces of section for the region exterior to Jupiter’s orbit in the CRTBP:

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Algebraic Mappings One of the biggest advances in studies of the dynamical evolution of the solar system was the use of algebraic mappings to study orbits over long time periods. The mappings most used in solar system work can be classified into three major types: 1.Resonance maps 2.Encounter maps 3.N-Body maps We will look at teach of these in turn, but first it helps if we look at an even simpler map — the standard map.

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The Standard Map (1) The standard map is derived from the Hamiltonian of a simple pendulum where the point of suspension is oscillating. In suitable coordinates the Hamiltonian is To this we add an infinite number of short period terms: If we write

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The Standard Map (2) Then the Hamiltonian can be written as: where we now have a periodic Dirac delta function. The equations of motion are now: which we can integrate to give:

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The Standard Map (3) Hence, the relationships between the old (unprimed) and new (primed) values of the action and angle variables are: This provides a mapping taking us from one point in the phase space to the next. This mapping, despite its simplicity, has incredibly intricate structure, illustrating resonance, chaos, etc. k 0 is a perturbation parameter.

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The Standard Map (4) k 0 = 0.8k 0 = 1.2

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Resonance Maps (1) Jack Wisdom was the first person to apply Chirikov’s technique in the derivation of the standard map to devise an algebraic mapping to study asteroid motion. He chose to study the 3:1 resonance, partly because it was easy to isolate the relevant terms without having to worry about nearby resonances. Wisdom’s idea was to replace the resonant terms by an infinite number of short-period terms which, because they now looked like Dirac delta functions, could be applied as impulses using the analytical solution to the secular part in between impulses.

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Resonance Maps (2) The resulting map relied on (i) using an analytical secular solution and (ii) approximating the (albeit artificial) resonant terms by impulses. The map was approximately 1000 times faster than solving the full equations of motion. Murray & Fox showed that the map was as good as the averaged second-order part of the disturbing function from which it was derived and that it could be used as a good indicator of regular and chaotic regions in the solar system. Note that Wisdom’s map was based on a second-order expansion of the disturbing function and so we would not expect it to be a good approximation to the real motion when the eccentricity was large.

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Encounter Maps Encounter maps make use of a constant of the motion, such as the Jacobi constant, to predict the outcome of a conjunction between two objects (usually a test particle and a planet). Encounter maps are useful for approximating the localised behaviour of the particle’s motion as it encounters a planet. It can be used in association with other maps (particularly the N-body map) to provide an accurate map for close approaches.

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N-Body Maps (1) In 1991 Wisdom & Holman developed an algebraic mapping for the N-Body problem that was to revolutionize studies of the long-term dynamical evolution of the solar system. The map relied on delta functions but not a truncated expansion of the disturbing function. The key to the map is the expression of the Hamiltonian in a Jacobian centre of mass coordinate system. This allowed the Hamiltonian to be split neatly between a keplerian part and an interaction part: Furthermore,

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N-Body Maps (2) Relationship between the standard Cartesian frame and the Jacobian centre of mass frame: The Wisdom-Holman map was subsequently adapted by Duncan and Levison to produce an off-the-shelf integrator for solar system orbits that is widely used today.

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Separatrices and Resonance Overlap (1) In the previous lecture we saw that the precise semi-major axis corresponding to a particular resonant argument generally depends not only on the mean motions of the bodies in question, but also on their apsidal and nodal rates. This led to the concept of resonance splitting. Also, the separation between adjacent resonances becomes smaller as the perturber is approached. Hence, depending on their relative widths, there may be overlap between adjacent resonances.

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Separatrices and Resonance Overlap (2) In phase-space, the separatrix represents the boundary between libration and circulation. Chaotic motion tends to be located in the area of phase-space around the separatrix. Why ? On the separatrix, the period of the unperturbed motion tends to infinity (giving numerous resonances with other frequencies). As resonances begin to overlap, chaotic regions associated with adjacent separatrices begin to merge. Chaotic motion is therefore closely associated with resonance overlap.

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Prometheus and Pandora (1) Saturn’s F ring

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Prometheus and Pandora (2) Prometheus was found to be lagging behind its expected position in 1995 by ~ 19 degrees, while Pandora was leading by about the same amount (French et al. 2003). However, the orbits were modelled using precessing ellipses (which ignore possible interactions between the satellites).

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Prometheus and Pandora (3) Goldreich & Rappaport (2003) : Numerical simulations using the full equations of motion showed that the observed discrepancies can be accounted for by chaotic gravitational interactions between the 2 satellites themselves. The interactions have a 24.8 day period (at conjunctions) with the strongest interactions every 6.2 years, when the pericentres are anti-aligned. Lyapunov exponent ~ Chaos is the result of overlap between four 121:118 Pandora resonances.

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Prometheus and Pandora (4) Goldreich & Rappaport (2003) : Numerical simulations using the full equations of motion showed that the observed discrepancies can be accounted for by chaotic gravitational interactions between the 2 satellites themselves.

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The Kirkwood Gaps (1)

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The Kirkwood Gaps (2) At first glance the orbits appear to be randomly distributed. In fact, the belt has a clear structure, determined by resonances.

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The Kirkwood Gaps (3) The gaps were first noted by Daniel Kirkwood in 1867, based on a sample of <100 asteroids.

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The Kirkwood Gaps (4) In 1979 Froeschlé & Scholl classified the theories for the origin of the gaps into four main groups: 1.Statistical 2.Collisional 3.Cosmogonic 4.Gravitational Any theory for the gaps has to explain not only the absence of asteroids at certain resonances but also the presence of asteroids at other locations. Consider the outer region of the belt. A plot of e against a for the asteroids can be superimposed on a plot of maximum libration widths for the main resonances:

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The Kirkwood Gaps (5) This suggests that resonance overlap is responsible for clearing asteroids in the outer part of the belt. What about the main belt?

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The Kirkwood Gaps (6) Dermott & Murray suggested that the gravitational hypothesis was the correct one based on their study of maximum libration widths and the close connection with the observed distribution of asteroids:

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The Kirkwood Gaps (7) However, it was Wisdom’s study of the 3:1 resonance, using his map, that provided the first clues that chaos was involved.

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The Kirkwood Gaps (8) Wisdom went on to show that asteroids close to the 3:1 resonance could become Earth-crossing thereby providing an explanation for the delivery of meteorites from the asteroid belt within their cosmic ray exposure ages.

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The Kirkwood Gaps (9) Wisdom also provided an analytical explanation for the observed motion of asteroids at the resonance. The 3:1 resonance mechanism may not be as simple as first thought because secular resonance also plays a role.

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The Kirkwood Gaps (10) Gladman et al. (1997) studied the long-term evolution of asteroids. For example, consider this bizarre evolution:

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The Kirkwood Gaps (11) In the space of a decade the problem went from one where no mechanism for removal was known to one where the mechanisms were too efficient.

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Solar System Stability (1) The classical Laplace-Lagrange secular theory appeared to prove that the Solar System was stable. However, the theory was derived using several assumptions which are not strictly valid. An analytical approach to the problem has always proved very difficult. The advent of high speed digital computers has allowed people to investigate the orbits of the planets numerically. Laskar & Gastineau (2009) modelled the evolution of the Solar System over 5 Gyr, incorporating the effects of general relativity (important for Mercury) and including effect of the Moon on the Earth’s orbit.

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Solar System Stability (2) Laskar & Gastineau integrated 2501 sets of orbits with initial conditions consistent with current knowledge of the positions of the planets. Their results confirmed earlier work which showed that the orbits are chaotic. However they also found: In 1 % of solutions, Mercury’s eccentricity increases by a large enough factor to allow collisions with Venus or the Sun. In one possible solution, a subsequent sudden decrease in Mercury’s eccentricity destabilises the entire inner Solar System ~ 3.34 Gyr from now, resulting in possible collisions of Mercury, Mars or Venus with the Earth.

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Solar System Stability (3) Laskar & Gastineau (2009)

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Solar System Stability (4) Laskar & Gastineau (2009)

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