1. AST3020. Lecture 07 Migration type I, II, III Talk by Sherry on migration Ups And and the need for disk-planet interaction during the formation of multiplanetary systems Torques and migration type I Numerical calculations of gap opening and type II situation, as well as fast migration Test problem and difference between codes Last Mohican scenario and the viability of Earths

2. Orbital radii + masses of the extrasolar planets (picture from 2003) These planets were found via Doppler spectroscopy of the host’s starlight. Precision of measurement: ~3 m/s Hot jupiters Radial migration

3. Marcy and Butler (2003)

4. ~2003 2005

5. m sin i vs. a Zones of avoidance? multiple single Blurry knowledge of exoplanets in 2006

6. m sin i vs. a Zones of avoidance? Migration? Distance mass Pile-up

7. Eccentricity of exoplanets vs. a and m sini m, a, e somewhat correlated: a e ? m a e ? m a e ? m

8. Eccentricity of exoplanets vs. a and m sini m, a, e somewhat correlated: a e ? m a e ? m a e ? m

9. Upsilon Andromedae And the question of planet-planet vs. disk planet interaction

10. The case of Upsilon And examined: Stable or unstable? Resonant? How, why?...

11. Upsilon Andromedae’s two outer giant planets have STRONG interactions Inner solar system (same scale)

13. . Definition of logitude of pericenter (periapsis) a.k.a. misalignment angle

14. In the secular pertubation theory, semi-major axes (energies) are constant (as a result of averaging over time). Eccentricities and orbit misalignment vary, such as to conserve the angular momentum and energy of the system. We will show sets of thin theoretical curves for (e2, dw). [There are corresponding (e3, dw) curves, as well.] Thick lines are numerically computed full N-body trajectories. Classical celestial mechanics

15. eccentricity Orbit alignment angle 0.8 Gyr integration of 2 planetary orbits with 7th-8th order Runge-Kutta method Initial conditions not those observed!

16. Upsilon And: The case of very good alignment of periapses: orbital elements practically unchanged for 2.18 Gyr unchanged

17. N-body (planet-planet) or disk-planet interaction? Conclusions from modeling Ups And 1. Secular perturbation theory and numerical calculations spanning 2 Gyr in agreement. 2. The apsidal “resonance” (co-evolution) is expected and observed to be strong, and stabilizes the system of two nearby, massive planets 3. There are no mean motion resonances 4. The present state lasted since formation period 5. Eccentricities in inverse relation to masses, contrary to normal N-body trend tendency for equipartition. Alternative: a lost most massive planet - very unlikely 6. Origin still studied, Lin et al. Developed first models involving time-dependent axisymmetric disk potential

18. Diversity of exoplanetary systems likely a result of: cores? disk-planet interactiona m e (only medium) yes planet-planet interactiona m? e yes star-planet interactiona m e? yes disk breakup (fragmentation into GGP) a m e? Metallicity no X X X X XX X X

19. resonances and waves in disks, orbital evolution migration type I - embedded planets Disk-planet interaction

22. ... SPH (Smoothed Particle Hydrodynamics) Jupiter in a solar nebula (z/r=0.02) launches waves at LRs. The two views are (left) Cartesian, and (right) polar coordinates.

23. Inner and Outer Lindblad resonances in an SPH disk with a jupiter

26. Laboratory of disk-satellite interaction

27. A gap-opening body in a disk: Saturn rings, Keeler gap region (width =35 km) This new 7-km satellite of Saturn was announced 11 May 2005. To Saturn

28. Migration Type I : embedded in fluid Migration Type II : more in the open (gap)

29. Illustration of nominal positions of Lindblad resonances (obtained by WKB approximation. The nominal positions coincide with the mean motion resonances of the type m:(m+-1) in celestial mechanics, which doesn’t include pressure.) Nominal radii converge toward the planet’s semi-major axis at high azimuthal numbers m, causing problems with torque calculation (infinities!). On the other hand, the pressure-shifted positions are the effective LR positions, shown by the green arrows. They yield finite total LR torque.

30. Wave excitation at Lindblad resonances (roughly speaking, places in disk in mean motion resonance, or commensurability of periods, with the perturbing planet) is the basis of the calculation of torques (and energy transfer) between the perturber and the disk. Finding precise locations of LRs is thus a prerequisite for computing the orbital evolution of a satellite or planet interacting with a disk. LR locations can be found by setting radial wave number k_r = 0 in dispersion relation of small-amplitude, m-armed, waves in a disk. [Wave vector has radial component k_r and azimuthal component k_theta = m/r] This location corresponds to a boundary between the wavy and the evanescent regions of a disk. Radial wavelength, 2*pi/k_r, becomes formally infinite at LR.

31. One-sided and differential torques, type I migration

32. Migration Type I, II Time-scale (years) Underlying fig. from: “Protostars and Planets IV (2000)”;

33. gap opening: thermal criterion viscous criterion migration type II - non-embedded planets Disk-planet interaction

34. The diffusion equation for disk surface density at work: additional torque to to planet added. Type II migration inside the gap. Speed = viscous speed (timescale = t_dyn * Re)

35. This case illustrates the fact that outer parts of a disk spread OUT, carrying the planet with it. In any case, migration type II is very slow, since the viscous time scale is ~1 Myr or a significant fraction thereof.

36. Eccentricity evolution Disk-planet interaction

38. --> m(z/r) Eccentricity pumping Eccentricity in type-I situation is always strongly damped.

40. Migration Type I : embedded in fluid Migration Type III partially open (gap) Migration Type II : in the open (gap)

41. Migration Type I, II, and III Time-scale (years) Underlying fig. from: “Protostars and Planets IV (2000)”; cf. “Protostars and Planets V (2006)” & this talk for type III data type III ?

42. Disk-planet interaction: Numerics

43. ANTARES/FIREANT Stockholm Observatory 20 cpu (Athlons) mini-supercomputer (upgraded in 2004 with 18 Opteron 248 CPUs inside SunFire V20z workstations)

44. AMRA

46. MNRAS (2006)

47. Code comparison project: EU RTN, Stockholm

48. AMRA FARGO FLASH-AGFLASH-AP Comparison of Jupiter in an inviscid disk after t=100P FLASH-AP

49. RH2D NIRVANA-GD PARA-SPH Jupiter in an inviscid disk t=100P RODEO

50. Surface density comparison

52. jupiter vortex L4L4 Surface density

53. Vortensity = specific vorticity = vorticity / Sigma De Val Borro et al (2006, MNRAS)

54. Disk-planet interaction: how do supergiant planets (~10 M_Jup) form?

55. Mass flows through the gap opened by a jupiter-class exoplanet ==> Superplanets can form Gas flows through & despite the gap. This result explains the possibility of “superplanets” with mass ~10 M J Migration explains hot jupiters.

56. Binary star on circular orbit accreting from a circumbinary disk through a gap. Surface density Log(surface density) An example of modern Godunov (Riemann solver) code: PPM VH1-PA. Mass flows through a wide and deep gap!

57. Shepherding by Prometheus and Pandora Pan opens Encke gap in A-ring of Saturn

58. Prometheus (Cassini view)

59. 1. Early dispersal of the primordial nebula ==> no material, no mobility 2. Late formation (including Last Mohican scenario)

60. What the permeability of gaps tells us about our own Jupiter: - Jupiter was potentially able to grow to 5-10 m J, if left accreting from a standard solar nebula for ~1 Myr - the most likely reason why it didn’t: the nebula was already disappearing and not enough mass was available. What the permeability of gaps tell us about exoplanets: - some, but not too many, grew in disks to become superplanets - most didn’t, and we can’t invoke the perfect timing argument. One way to uderstand the ubiquitous small exo-giants is that we see the LAST MOHICANS (survivors from an early epoch of planet migration and demise inside the suns).

61. Disk-planet interaction: Direction & rate of fast migration

62. Migration Type I : embedded in fluid Migration Type II : more in the open (gap) (1980s & 90s) (1980s) Migration Type III partially open (gap) (2003)

63. AMR PPM (Flash) simulation of a Jupiter in a standard solar nebula. 5 levels/subgrids. (Peplinski and Artymowicz 2004)

64. Variable-resolution PPM (Piecewise Parabolic Method) [Artymowicz 1999] Jupiter-mass planet, fixed orbit a=1, e=0. White oval = Roche lobe, radius r_L= 0.07 Corotational region out to x_CR = 0.17 from the planet disk gap (CR region)

65. Outward migration type III of a Jupiter Inviscid disk with an inner clearing & peak density of 3 x MMSN Variable-resolution, adaptive grid (following the planet). Lagrangian PPM. Horizontal axis shows radius in the range (0.5-5) a Full range of azimuths on the vertical axis. Time in units of initial orbital period.

66. Simulation of a Jupiter-class planet in a constant surface density disk with soundspeed = 0.05 times Keplerian speed. PPM = Piecewise Parabolic Method Artymowicz (2000), resolution 400 x 400 Although this is clearly a type-II situation (gap opens), the migration rate is NOT that of the standard type-II, which is the viscous accretion speed of the nebula.

67. Consider a one-sided disk (inner disk only). The rapid inward migration is OPPOSITE to the expectation based on shepherding (Lindblad resonances). Like in the well-known problem of “sinking satellites” (small satellite galaxies merging with the target disk galaxies), Corotational torques cause rapid inward sinking. (Gas is trasferred from orbits inside the perturber to the outside. To conserve angular momentum, satellite moves in.)

68. Now consider the opposite case of an inner hole in the disk. Unlike in the shepherding case, the planet rapidly migrates outwards. Here, the situation is an inward-outward reflection of the sinking satellite problem. Disk gas traveling on hairpin (half-horeseshoe) orbits fills the inner void and moves the planet out rapidly (type III outward migration). Lindblad resonances produce spiral waves and try to move the planet in, but lose with CR torques.

69. Saturn-mass protoplanet in a solar nebula disk (1.5 times the Minimum Nebula, PPM, Artymowicz 2003) Type III outward migration Condition for FAST migration: disk mass in CR region ~ planet mass. Notice a carrot-shaped bubble of “vacuum” behind the planet. Consisting of material trapped in librating orbits, it produces CR torques smaller than the matrial in front of the planet. The net CR torque powers fast migration. radius 123 Azimuthal angle (0-360 deg)

70. AMR PPM (FLASH). Jupiter simulation by Peplinski and Artymowicz (in prep.). Red color marks the fluid initially surrounding the planet’s orbit.

71. Variable-resolution PPM (Piecewise Parabolic Method) 1. Gas surface density, accentuating LR-born waves (surf) 2. Vortensity, showing gas flow (rip-tide) 0.1 Jupiter mass planet in a z/r=0.05 gas nebula Horizontal tick mark = 0.1 a Corotational region out to x CR = 0.08 a away from the planet 0.8 1 1.2 1.4 radius azimuth

72. What is more important: Lindblad Resonances (waves) or C orotation?

73. Impulse approximation

77. No migration of planetOutward migration Flow fields obtained from simplification of Hill’s equations of motion. (Guiding center trajectories.) One result: x CR ~ 2.5 r L

78. Animation: Eduardo Delgado Guiding center trajectories in the Hill problem Unit of length = Hill sphere Unit of da/dt = Hill sphere radius per dynamical time

79. NO MIGRATION: In this frame, comoving with the planet, gas has no systematic radial velocity V = 0, r = a = semi-major axis of orbit. Symmetric horseshoe orbits, torque ~ 0 r a 0 disk Librating Corotational (CR) region Librating Hill sphere (Roche lobe) region x CR protoplanet x CR = half-width of CR region, separatrix distance

80. SLOW MIGRATION: In this frame, comoving with the planet, gas has a systematic radial velocity V = - da/dt = -(planet migr.speed) asymmetric horseshoe orbits, torque ~ da/dt FAST MIGRATION: CR flow on one side of the planet, disk flow on the other Surface densities in the CR region and the disk are, in general, different. Tadpole orbits, maximum torque r r a 0 0 a

82. M

83. Migration type III, neglecting LRs & viscous disk flow independent of planet mass, e.g., in MMSN at a= 5 AU, the type-III time-scale = 48 P orb

84. Peplinski and Artymowicz (MNRAS, 2006, in prep.) AMR code FLASH adaptive multigrid, PPM, Cartesian coordinates local resolution up to 0.0003 a = 0.0015 AU = 225000 km = 3 Jupiter radii NUMERICAL CONVERGENCE when gas given higher temperature near the planet - results not sensitive to gravitational softening length - or resolution

85. time Radius (a) Disk gap Smooth initial disk 4 jupiter masses 1 jupiter mass 0 50 100 P 1 2 As theorized - no significant dependence on mass:

86. time Radius (a) Disk gap Smooth initial disk 4 jupiter masses 1 jupiter mass 0 50 100 P 1 2 As theorized - no significant dependence on mass:

88. Outward migr. Inward migr. ALL TORQUES RESTORED (LRs, viscous)

89. Mass deficit Migration rate Global migration reverses at the outer boundary

91. How can there be ANY SURVIVORS of the rapid type-III migration?! Migration type III Structure in the disk: gradients of density, edges, gaps, dead zones Migration stops, planet grows/survives

92. Unsolved problem of the Last Mohican scenario of planet survival in the solar system: Can the terrestial zone survive a passage of a giant planet? §N-body simulations, N~1000 (Edgar & Artymowicz 2004) §A quiet disk of sub-Earth mass bodies reacts to the rapid passage of a much larger protoplanet (migration speed = input parameter). §Results show increase of velocity dispersion/inclinations and limited reshuffling of material in the terrestrial zone. §Migration type III too fast to trap bodies in mean- motion resonances and push them toward the star  Evidence of the passage can be obliterated by gas drag on the time scale passage of a pre- jupiter planet(s) not exluded dynamically.

93. Edges or gradients in disks: Magnetic cavities around the star Dead zones

96. Summary of type-III migration §New type, sometimes extremely rapid (timescale > LRs §Direction depends on prior history, not just on disk properties. §Supersedes a much slower, standard type-II migration in disks more massive than planets §Conjecture: modifies or replaces type-I migration §Very sensitive to disk density (or vortensity) gradients. §Migration stops on disk features (rings, edges and/or substantial density gradients.) Such edges seem natural (dead zone boundaries, magnetospheric inner disk cavities, formation-caused radial disk structure) §Offers possibility of survival of giant planets at intermediate distances (0.1 - 1 AU), §...and of terrestrial planets during the passage of a giant planet on its way to the star. §If type I superseded by type III then these conclusions apply to cores as well, not only giant protoplanets.

97. Migration: type 0 type I type II & IIb type III N-body Timescale of migration: from ~1e2 yr to disk lifetime (up to 1e7 yr) > 1e4 yr > 1e5 yr > 1e2 - 1e3 yr > 1e5 yr (?) Interaction: Gas drag + Radiation press. Resonant excitation of waves (LR) Tidal excitation of waves (LR) Corotational flows (CR) Gravity