1. The long-term evolution of planetary orbits TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

2. The problem: A point mass is surrounded by N > 1 much smaller masses on nearly circular, nearly coplanar orbits. Is the configuration stable over very long times (up to 10 12 orbits)? Why is this interesting? one of the oldest and most influential problems in theoretical physics (perturbation theory, Hamiltonian mechanics, nonlinear dynamics, resonances, chaos theory, Laplacian determinism, design of storage rings, etc.) occupied Newton, Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Arnold, Moser, etc. what is the fate of the Earth? where do asteroids and comets come from? why are there so few planets? calibration of geological timescale over the last 50 Myr can we explain the properties of extrasolar planetary systems?

3. The problem: A point mass is surrounded by N > 1 much smaller masses on nearly circular, nearly coplanar orbits. Is the configuration stable over very long times (up to 10 12 orbits)? many famous mathematicians and physicists have attempted to find analytic solutions (e.g. KAM theorem), with limited success only feasible approach is numerical solution of equations of motion by computer, but: –needs up to  10 14 timesteps so lots of CPU –inherently serial, so little gain from massively parallel computers –needs sophisticated algorithms to avoid buildup of errors symplectic integration algorithms mixed-variable methods warmup symplectic correctors optimal floating-point arithmetic

4. The solar system Known small corrections include: satellites ( < 10 -7 ) general relativity (fractional effect < 10 -8 ) solar quadrupole moment (< 10 -10 even for Mercury) Galactic tidal forces (fractional effect < 10 -13 ) Unknown small corrections include: asteroids (< 10 -9 ) and Kuiper belt (< 10 -6 even for outermost planets) mass loss from Sun through radiation and solar wind, and drag of solar wind on planetary magnetospheres (< 10 -14 ) passing stars (closest passage about 500 AU) Masses m j known to better than 10 -9 M  Initial conditions known to fractional accuracy better than 10 -7 d 2 x i d t 2 = ¡ G P N j = 1 m j ( x i ¡ x j ) j x i ¡ x j j 3 + sma ll correc t i ons 1 AU = 1 astronomical unit = Earth-Sun distance Neptune orbits at 30 AU solar mass

5. To a very good approximation, the solar system is an isolated, conservative dynamical system described by a known set of equations, with known initial conditions All we have to do is integrate the equations of motion for ~10 10 orbits (4.5  10 9 yr backwards to formation, or 7  10 9 yr forwards to red-giant stage when Mercury and Venus are swallowed up) Goal is quantitative accuracy (   1 radian) over 10 8 yr and qualitative accuracy over 10 10 yr The solar system

6. an integration of the solar system (Sun + 9 planets) for § 4.5 £ 10 9 yr = 4.5 Gyr figures show innermost four planets Ito & Tanikawa (2002) 0 - 55 Myr -55 – 0 Myr -4.5 Gyr +4.5 Gyr

7. Ito & Tanikawa (2002)

8. Pluto’s peculiar orbit Pluto has: the highest eccentricity of any planet (e = 0.250 ) the highest inclination of any planet ( i = 17 o ) perihelion distance q = a(1 – e) = 29.6 AU that is smaller than Neptune’s semimajor axis ( a = 30.1 AU ) How do they avoid colliding?

9. Pluto’s peculiar orbit Orbital period of Pluto = 247.7 y Orbital period of Neptune = 164.8 y 247.7/164.8 = 1.50 = 3/2 Resonance ensures that when Pluto is at perihelion it is approximately 90 o away from Neptune Cohen & Hubbard (1965)

10. Pluto’s peculiar orbit early in the history of the solar system there was debris (planetesimals) left over between the planets ejection of this debris by Neptune caused its orbit to migrate outwards if Pluto were initially in a low-eccentricity, low-inclination orbit outside Neptune it is inevitably captured into 3:2 resonance with Neptune once Pluto is captured its eccentricity and inclination grow as Neptune continues to migrate outwards other objects may be captured in the resonance as well Malhotra (1993) period ratio inclination of Pluto eccentricity of Pluto semi-major axis of Pluto resonant angle 17 o

11. Kuiper belt objects Plutinos (3:2) Centaurs comets

12. Two kinds of dynamical system Regular highly predictable, “well- behaved” small differences in position and velocity grow linearly:  x,  v  t e.g., simple pendulum, all problems in mechanics textbooks, baseball, golf, planetary orbits on short timescales Chaotic difficult to predict, “erratic” small differences grow exponentially at large times:  x,  v  exp(t/t L ) where t L is Liapunov time appears regular on timescales short compared to Liapunov time  linear growth on short times, exponential growth on long times e.g., roulette, dice, pinball, billiards, weather

13. The orbit of every planet in the solar system is chaotic (Laskar 1988, Sussman & Wisdom 1988, 1992) separation of adjacent orbits grows  exp(t / t L ) where Liapunov time t L is 5-20 Myr  factor of at least 10 100 over lifetime of solar system 300 million years Jupiter factor of 1000 saturated ln(distance)

14. Integrators: double-precision (p=53 bits) 2 nd order mixed-variable symplectic (Wisdom-Holman) method with h=4 days and h=8 days double-precision (p=53 bits) 14 th order multistep method with h=4 days extended-precision (p=80 bits) 27 th order Taylor series method with h=220 days saturated Hayes (astro-ph/0702179) Liapunov time t L =12 Myr 200 Myr

15. Chaos in the solar system the orbits of all the planets are chaotic with e-folding times for growth of small changes (Liapunov times) of 5-20 Myr (i.e. 200-1000 e-folds in lifetime of solar system positions (orbital phases) of planets are not predictable on timescales longer than 100 Myr – future of solar system over longer times can only be predicted probabilistically the solar system is a poor example of a deterministic universe dominant chaotic motion for outer planets is in phase, not shape (eccentricity and inclination) or size (semi-major axis) dominant chaotic motion for inner planets is in eccentricity and inclination, not semi-major axis

16. Laskar (1994) birth death

17. Laskar (2008) 1000 integrations of solar system for 5 Gyr using simplified but accurate equations of motion all four inner planets exhibit random walk in eccentricity (more properly, in e cos ! and e sin ! ) probability Mercury will reach e = 0.6 in 5 Gyr is about 2% 1 curve per 0.25 Gyr

18. Causes of chaos orbits with 3 degrees of freedom have three actions J i, three conjugate angles µ i, and three fundamental frequencies  i. in spherical potentials,  1 =0. In Kepler potentials  1 =  2 =0 all resonances involve perturbations to Hamiltonian of the form © (J i ) cos (m 1 µ 1 +m 2 µ 2 +m 3 µ 3 - ! t) two types of resonance: –secular resonances have m 3 =0. If ©» O( ® ) ¿ 1 the characteristic oscillation frequency in the resonance is O( ® 1 / 2 ) but width of resonance is O(1) –mean-motion resonances have m 3  0. Width of resonance is O( ® 1 / 2 ) but resonances with different m 1,m 2 and same m 3 overlap (“fine-structure splitting). resonance overlap leads to chaos chaos arises from secular resonances in inner solar system and mean- motion resonances in outer solar system chaos in outer solar system arises from a 3-body resonance with critical argument  = 3  (longitude of Jupiter) - 5  (longitude of Saturn) - 7  (longitude of Uranus) (Murray & Holman 1999) small changes in initial conditions can eliminate or enhance chaos cannot predict lifetimes analytically

19. Laskar (2008) change PPN parameter 2 ° - ¯ from +1 to -2 precession of Mercury / 2 ° - ¯ so changes from +43 arcsec/100 yr to -86 arcsec/100 yr Mercury collides with Venus in < 4 Myr or < 0.1% of solar system age eccentricity of Mercury time (Myr - not Gyr!)

20. Holman (1997) age of solar system J S U N

21. we can integrate the solar system for its lifetime the solar system is not boring on long timescales planet orbits are chaotic with e-folding times of 5-20 Myr the orbits of the planets are not predictable over timescales > 100 Myr the shapes (eccentricities, inclinations) of the inner planetary orbits execute a random walk the phases of the outer planetary orbits execute a random walk “Is the solar system stable?” can only be answered probabilistically it is unlikely (2- ¾ ) that any planets will be ejected or collide before the Sun dies most of the solar system is “full”, and it is likely that planets have been lost from the solar system in the past Summary

22. (261 planets)

23. Given the star mass M (known from spectral type), radial-velocity observations yield: orbital period P semi-major axis a combination of planet mass m & inclination I, m sin I orbit eccentricity e current accuracy ¢ v » 3 m/s in best observations Jupiter  v ~ 13 m/s, P = 11.9 yr – just detectable Earth  v ~ 0.1 m/s, P = 1 yr – not detectable yet I ' 0 I ' 90 ±

24. 51 Peg: m sin I = 0.46 M J P = 4.23 d a = 0.05 AU e = 0.01 70 Vir: m sin I = 7.44 M J P = 116.7 d a = 0.48 AU e = 0.40 1 M J = 1 Jupiter mass = 0.001 M ¯ = 318 M ©

25. What have we learned? 277 extrasolar planets known (March 2008) 261 from radial-velocity surveys 36 from transits 5 from timing 6 by microlensing 5 by imaging ( see http://exoplanet.eu)

26. What have we learned? planets are remarkably common (~ 5% even with current technology) probability of finding a planet / Z a, a ' 1-2, where Z = metallicity = fraction of elements heavier than He (Fischer & Valenti 2005) log 10 Z/Z ¯

27. What have we learned? M V E Mars Jupiter giant planets like Jupiter and Saturn are found at very small orbital radii record holder is Gliese 876d: M = 0.018 M Jupiter, P = 1.94 days, a = 0.0208 AU maximum radius set by duration of surveys (must observe 1-2 periods at least for good orbit)

28. What have we learned? wide range of masses up to 15 Jupiter masses down to 0.01 Jupiter masses = 3 Earth masses lower cutoff is determined entirely by observational selection upper cutoff is real maximum planet mass incomplete Earth mass Jupiter mass

29. What have we learned? eccentricities are much larger than in the solar system biggest eccentricity is e = 0.93 tidal circularization

30. What have we learned? current surveys could (almost) have detected Jupiter Jupiter’s eccentricity is anomalous therefore the solar system is anomalous

31. Theory of planet formation: Theory failed to predict: –high frequency of planets –existence of planets much more massive than Jupiter –sharp upper limit of around 15 M Jupiter –giant planets at very small semi-major axes –high eccentricities

32. Theory of planet formation: Theory failed to predict: –high frequency of planets –existence of planets much more massive than Jupiter –sharp upper limit of around 15 M Jupiter –giant planets at very small semi-major axes –high eccentricities Theory reliably predicts: –planets should not exist

33. Planet formation can be divided into two phases: Phase 1 protoplanetary gas disk  dust disk  planetesimals  planets solid bodies grow in mass by 45 orders of magnitude through at least 6 different processes lasts 0.01% of lifetime (1 Myr) involves very complicated physics (gas, dust, turbulence, etc.) Phase 2 subsequent dynamical evolution of planets due to gravity lasts 99.99% of lifetime (10 Gyr) involves very simple physics (only gravity)

34. Modeling phase 2 (M. Juric, Ph.D. thesis) distribute N = 3-50 planets randomly between a = 0.1 AU and 100 AU, uniform in log(a) choose masses randomly between 0.1 and 10 Jupiter masses, uniform in log(m) choose small eccentricities and inclinations with specified  e 2 ,  i 2  include physical collisions repeat 200-1000 times for each parameter set N,  e 2 ,  i 2  follow for 100 Myr to 1 Gyr

35. Characteristic radius of influence of a planet of mass m at semi-major axis a orbiting a star M is r Hill =a(m/M) 1/3 (Hill, Roche, or tidal radius) Crudely, planetary systems can be divided into two kinds: inactive: –large separations or low masses ( ¢ r >> r Hill ) –eccentricities and inclinations remain small –preserve state they had at end of phase 1 active: –small separations or large masses ( ¢ r < a few times r Hill ) –multiple ejections, collisions, etc. –eccentricities and inclinations grow Modeling phase 2 - results

36. most active systems end up with an average of only 2-3 planets, i.e., 1 planet per decade inactive partially active active

37. all active systems converge to a common spacing (median  r in units of Hill radii) solar system is not active inactive partially active active solar system (all) solar system (giants) extrasolar planets median ¢ r/r Hill 10 8 yr

38. a wide variety of active systems converge to a common eccentricity distribution initial eccentricity distributions

39. a wide variety of active systems converge to a common eccentricity distribution, which agrees with the observations initial eccentricity distributions

40. late dynamical evolution (Phase II), long after planet formation is complete, may establish some of the properties of extrasolar planetary systems, such as the eccentricity distribution Questions: why are there planets at all? why are giant planets (M > 0.1 M Jupiter = 30 M Earth ) so common? how common are smaller planets (< 30 M Earth )? how do giant planets form at semi-major axes 200 times smaller than any solar-system giant? why are there no planets more massive than » 15 M J ? what are the relative roles of Phase I and II evolution in determining the properties of extrasolar planetary systems? why is the solar system unusual? (anthropic principle?) Summary:

41. Kozai oscillations Kozai (1962); Lidov (1962) arise most simply in restricted three-body problem (two massive bodies on a Kepler orbit + a test particle $ binary star + planet orbiting one member of binary) in Kepler potential © =-GM/r, eccentric orbits have a fixed orientation generic axisymmetric potential Kepler potential

43. Kozai oscillations in Kepler potential © =-GM/r, eccentric orbits have a fixed orientation because of conservation of the Laplace-Runge-Lenz or eccentricity vector vector e points towards pericenter and |e| equals the eccentricity e ´ v £ ( r £ v ) GM ¡ ^ r e e

44. Kozai oscillations now subject the orbit to a weak, time-independent external force because the orbit orientation is fixed even weak external forces can act for a long time in a fixed direction relative to the orbit and therefore change the angular momentum or eccentricity if F external / ² then timescale for evolution / 1/ ² but nature of evolution is independent of ² so long as all other external forces are negligible, i.e. ¿ ² F external e

45. Kozai oscillations Consider a planet orbiting one member of a binary star system: can average over both planetary and binary star orbits both energy (a) and z-component of angular momentum (GMa) 1/2 (1-e 2 ) 1/2 cos I of planet are conserved thus the orbit-averaged problem has only one degree of freedom (e.g., eccentricity e and angle in orbit plane from the equator to the pericenter,  ), so © = © (e, ! ) motion is along level surfaces of © (e, ! )

46. Kozai oscillations initially circular orbits remain circular if and only if the initial inclination is < 39 o = cos -1 (3/5) 1/2 for larger initial inclinations the phase plane contains a separatrix ! can either librate around ¼ /2 or circulate orbits undergo coupled oscillations in eccentricity and inclination (Kozai oscillations) the separatrix includes circular orbits, so circular orbits cannot remain circular and are excited to high eccentricity (surprise number 1) are chaotic (surprise number 2) circular radial Holman, Touma & Tremaine (1997) circular orbits have inclination 60 ±

47. Kozai oscillations initially circular orbits remain circular if and only if the initial inclination is < 39 o = cos -1 (3/5) 1/2 for larger initial inclinations the phase plane contains a separatrix ! can either librate around ¼ /2 or circulate orbits undergo coupled oscillations in eccentricity and inclination (Kozai oscillations) the separatrix includes circular orbits, so circular orbits cannot remain circular and are excited to high eccentricity (surprise number 1) are chaotic (surprise number 2) as the initial inclination approaches 90 ±, the maximum eccentricity achieved in a Kozai oscillation approaches unity ! collisions (surprise number 3) mass and separation of companion affect period of Kozai oscillations, but not the amplitude (surprise number 4) circular radial

48. eccentricity oscillations of a planet in a binary star system a planet = 2.5 AU companion has inclination 75 ±, semi- major axis 750 AU, mass 0.08 M ¯ (solid) or 0.9 M ¯ (dotted) (Takeda & Rasio 2005) M 3 =0.9M ¯ M 3 =0.08 M ¯

49. Kozai oscillations initially circular orbits remain circular if and only if the initial inclination is < 39 o = cos -1 (3/5) 1/2 for larger initial inclinations the phase plane contains a separatrix ! can either librate around ¼ /2 or circulate orbits undergo coupled oscillations in eccentricity and inclination (Kozai oscillations) the separatrix includes circular orbits, so circular orbits cannot remain circular and are excited to high eccentricity (surprise number 1) are chaotic (surprise number 2) as the initial inclination approaches 90 ±, the maximum eccentricity achieved in a Kozai oscillation approaches unity ! collisions (surprise number 3) mass and separation of companion affect period of Kozai oscillations, but not the amplitude (surprise number 4) small additional effects such as general relativity can strongly affect the oscillations (surprise number 5) circular radial

50. Kozai oscillations may excite eccentricities of planets in some binary star systems, but probably not all planet eccentricities: not all have stellar companion stars (so far as we know) suppressed by additional planets suppressed by general relativity (!) Desidera & Barbieri (2007) filled circle  binary star dot size represents planet mass

51. Kozai oscillations – applications the Moon would crash into the Earth if its inclination were 90 ±, not near zero distant satellites of the giant planets have inclinations near 0 or 180 ± but not near 90 ± may be responsible for the high eccentricities of stars near the Galactic center long-period comets and asteroids survival of binary black holes in galactic nuclei homework: why do Earth satellites stay up?