PLANETARY MIGRATION A. CRIDA
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1 PLANETARY MIGRATION A. CRIDA (the migration of Jupiter carrying its satellites. Alegory.) 1
2 INTRODUCTION A planet orbiting in a protoplanetary disk launches a one-armed, spiral wake. p star planet Orange = gas Clear/white = over-density, Dark = underdensity. This is a pressuresupported wave, caused by the planet's gravity perturbing the trajectories of the gas particles. The wave corotates with the planet at the same angular velocity p = (GM*/ap3)1/2.
3 INTRODUCTION Wake = over-density. This mass in turn attracts the planet gravitationnally. These forces exerted on the planet have a momentum with respect to the central star. It leads to a torque, and thus a change in the planet's orbital angular momentum Lp=Mp(GM*ap)1/2, thus a change in ap : Orange = gas Clear/white = over-density, Dark = underdensity. planetary migration. star planet
4 TORQUES In the outer disk, the keplerian rotation is slower, so the wake trails behind the planet. Thus, it exerts on the planet a negative torque. It slows the planet down, and pushes it towards the central star.
5 TORQUES In the inner disk, the keplerian rotation is faster, so the wake leads the planet. Thus, it exerts on the Sens de planet a positive rotation torque. It accelerates the planet, and pulls it Dans le disque externe, outwards. l onde spirale est en retard sur la planète.
6 TORQUES In general, the (negative) outer torque is larger (in magnitude) than the (positive) inner torque (Ward, 1997). The planet loses orbital angular momentum, and its orbital radius ap decreases. The inner and outer torques are called one sided Lindblad torques. The total is called the differential Lindblad torque and is proportional to : 0 = q2 ap4 p2 (H/r)-2 where q=mp/m*, is the surface density of the gas, and H/r is the aspect ratio of the disk. Note: in the linear regime, the amplitude of the wake is proportionnal to the planetary mass, and the force is proportionnal to the product of the planetary mass and that of the wake. Therefore, 0 is proportionnal to q2.
7 TYPE I MIGRATION Migration in the linear regime is often called type I migration. It concerns small mass proto-planets (typically a few Earth masses). The migration speed is proportionnal to the planet's mass. (proof : dlp/dt = (Mp/2)(GM*/ap)1/2 dap/dt = dl so dap/dt is proportionnal to Mp. ) Ex : Mp=1MEarth, =1700 g/cm2, H/r=0.05 => tmigr = years. Mp=5MEarth, =5100 g/cm2, H/r=0.1 => tmigr = years.
8 GIANT PLANETS In the case of giant planets, the linear regime in not valid anymore. The wake shocks, and deposits angular momentum. Thus, giant planets perturb the density profile of the disk. The outer disk is taking angular momentum from the planet, and is accelerated by the planet. Therefore, it shifts outwards. The inner disk is giving angular momentum to the planet, and it is slowed down by the planet, because the wake is leading in front of the planet. Therefore, the inner disk loses orbital angular momentum, and shifts towards the star.
9 GAP OPENING The outer wake has a larger angular velocity than the local gas ( p > gas). It accelerates the gas, gives angular momentum to the gas. In contrast, the inner wake carries a negative angular momentum flux, given to the gas. So, the gas moves inwards. In the end, the planet tends to open a gap around its orbit, and to split the disk into an inner disk and an outer disk, separated by an empty region.
10 GAP OPENING The amount of gas remaining in the gap depends on the competition between the torques from the planet, wihch repel the gap, and the effects of the viscosity and pressure in the gap, which tend to make the profile smooth. The more viscous or thick the disk is, the highest the planetary mass must be. Therefore, there is an opening criterion (Crida et al. 2006) : 3H / [4rp(Mp/3)1/3] + 50ν / [rp2 pmp] < 1 Application : In a standard disk, H = ~ 0.05 rp, = ~10-5 rp2 p, and Mp>MSaturn is enough.
11 GAP OPENING In ~100 orbits, a giant planet of a Jupiter mass repels the disk around its orbit, and opens a gap. Small planet. Giant planet.
12 TYPE II MIGRATION The planet is repelled outwards by the inner disk inwards by the outer disk. It is locked in the middle of the gap, and can not migrate withe respect to the gas of the disk anymore. Inner disk Outer disk star But the disk falls onto the star (accretion), driving the planet inwards. 12
13 TYPE II MIGRATION The planet follows the viscous accretion of the disk towards the star
14 TYPE II MIGRATION Many exoplanets are giant planets, close to their host star, where they couldn't form : the hot Jupiters. This is a strong indicatino of type II migration...
15 TYPE II MIGRATION
16 MIGRATION in RESONANCE If two planets in the disk + convergent migration then resonance capture is possible.
17 MIGRATION in RESONANCE Two planets in their own gaps migrate in parallel. Inner disk Outer disk star Two planets in a same gap approach each other résonance. Inner disk Outer disk star Once locked in resonance, the pair of planet migrates in this configuration. The outer disk pushes the outer planet inwards ; in turn, the outer planet pushes the inner planet inward, due to the resonance. 17
18 MIGRATION in RESONANCE PROBLEM : As the outer planet pushes the inner one, the eccentricity of the latter increases dramatically, up to unrealistic values. Model : Model : (da2/dt)/a2 = -1 / τ a (da2/dt)/a2 = -1 / τ 0 a < (de2/dt)/e2 = K x (da2/ dt)/a2, K = constant ~ However, if an (unrealistically strong) eccentricity damping is applied to the outer planet, the eccentricities settle, though to too high values. 18
19 MIGRATION in RESONANCE We can compute realistic hydrodynamic simulations of the GJ876 system. Initial conditions : a1 = 0.21, a2 = 0.36, e1=e2=0. Disk parameters : α = 10-2 ; H/r = 7% ; Rinf= Result : resonance locking at t=100 years in 2:1 ; eccentricity growth until a limit value. At t = 270, the orbital elements are close to the observed ones. If the disk force on the inner planet is removed, the eccentricity rises too much. 19
20 MIGRATION in RESONANCE Conclusion : GJ876 s 2 more massive planets tend to lock in 2:1 MMR in the gas disk. The observed eccentricities are the natural outcome of the planetary evolution in resonance in the gas disk. If the disk disappears when the planets are at their present semi-majoraxes, it leads to the observed configuration. This is explained by the influence of the inner disk. 20
21 MIGRATION in RESONANCE Application to HD45634 Rein, Papaloizou, Kley (2009) : This system is in 3:2 MMR. However, capture in 2:1 is more likely, and is stable. Solution : the outer planet was in type III migration when it encountered the 2:1. No capture, because too quick migration. Then, capture in 3:2. Migration theories constain observations and help understand the systems formation. 21
22 How to prevent our Jupiter from becoming «hot»?
23 Masset and Snellgrove, :2 res
24 Once locked in resonance, the evolution of Jupiter and Saturn depends on disk parameters (Morbidelli and Crida, 2007, Icarus) Release time H/r = 0.05 : ~stationary solution.
25
26
27 CONCLUSION Jupiter can migrate outwards, if it is in resonance with Saturn. Then, the ice giants (Uranus and Neptune) can be trapped in mean motion resonance as well... 27
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