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AST3020. Lecture 07 Migration type I, II, III Talk by Sherry on migration

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## AST3020. Lecture 07 Migration type I, II, III Talk by Sherry on migration

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**AST3020.**Lecture 07 Migration type I, II, III Talk by Sherry on migration Ups And and the need for disk-planet interaction during the formationof 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**Orbital radii + masses of the extrasolar planets (picture**from 2003) Radial migration Hot jupiters These planets were found via Doppler spectroscopy of the host’s starlight. Precision of measurement: ~3 m/s**2005**~2003**m sin i vs. a**Blurry knowledge of exoplanets in 2006 Zones of avoidance? multiple single**m sin i vs. a**Zones of avoidance? Migration? mass Pile-up Distance**Eccentricity of exoplanets vs. a and m sini**a e ? m a e ? m m, a, esomewhat correlated: a e ? m**Eccentricity of exoplanets vs. a and m sini**a e ? m a e ? m m, a, esomewhat correlated: a e ? m**Upsilon Andromedae**And the question of planet-planet vs. disk planet interaction**The case of Upsilon And examined:**Stable or unstable? Resonant? How, why?...**Upsilon Andromedae’s two outer giant planets**have STRONG interactions Inner solar system (same scale)**Definition of logitude of pericenter**(periapsis) a.k.a. misalignment angle .**Classical celestial mechanics**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.**0.8 Gyr integration of 2 planetary orbits**with 7th-8th order Runge-Kutta method Initial conditions not those observed! eccentricity Orbit alignment angle**Upsilon And: The case of very good alignment of periapses:**orbital elements practically unchanged for 2.18 Gyr unchanged unchanged**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**Diversity of exoplanetary systems likely a result of:**cores? disk-planet interaction a m e (only medium) yes planet-planet interaction a m? e yes star-planet interaction a m e? yes disk breakup (fragmentation into GGP) a m e? Metallicity no X X X X X X X X**resonances and waves in disks, orbital evolutionmigration**type I - embedded planets Disk-planet interaction**.**. . 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.**Inner and Outer Lindblad resonances in an SPH disk with a**jupiter**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**Migration Type I :**embedded in fluid Migration Type II : more in the open (gap)**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.**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.**Migration Type I, II**Underlying fig. from: “Protostars and Planets IV (2000)”; Time-scale (years)**gap opening: thermal criterionviscous criterionmigration**type II - non-embedded planets Disk-planet interaction**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)**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.**Eccentricity evolution**Disk-planet interaction**Eccentricity in type-I**situation is always strongly damped. Eccentricity pumping --> m(z/r)**Migration Type I :**embedded in fluid Migration Type II : in the open (gap) Migration Type III partially open (gap)**Migration Type I, II, and III**Underlying fig. from: “Protostars and Planets IV (2000)”; cf. “Protostars and Planets V (2006)” & this talk for type III data ? type III Time-scale (years)**ANTARES/FIREANT**Stockholm Observatory 20 cpu (Athlons) mini-supercomputer (upgraded in 2004 with 18 Opteron 248 CPUs inside SunFire V20z workstations)**FARGO**AMRA Comparison of Jupiter in an inviscid disk after t=100P FLASH-AG FLASH-AP FLASH-AP**RH2D**NIRVANA-GD Jupiter in an inviscid disk t=100P RODEO PARA-SPH