Planet migration: different kinds Type I migration (small mass planets) Type II migration (high mass planets) Type III migration (rare type II variant)
Two main ways to calculate torque: 1.Follow gas packets in time, and see how they exchange angular momentum with the planet. –Impulse approximation 2.Analyse how azimuthal asymmetries in the steady-state gas distribution in the disk Σ(r, ϕ ) gravitationally pull on the planet. Note: With 2-D/3-D time-dependent hydrodynamic simulations you essentially do both, because you simulate the entire thing in full glory.
Planet-induced spiral waves in the protoplanetary disk
Spiral wave: Pitch angle Δv(a) β Δv perp (a) To ensure that the spiral wave is stationary in the reference frame corotating with the planet, the component of the orbital velocity Δv(a) perpendicular to the spiral wave (i.e. Δv perp (a)) must be precisely equal to the sound speed (assuming the wave is not a shock). spiral wave gas orbital velocity vector toward sun
Spiral wave: Launching point Δv(a) β Δv perp (a) This angle becomes ≈1 (i.e. very large) when spiral wave gas orbital velocity vector With we can write So we have: So with the inner/outer wave is launched at: launching point
Spiral wave: Gravitational „drag“ D‘Angelo, Henning & Kley (2002) The gravitational force acting on the planet by the material in the spiral arms adds and subtracts angular momentum to/from the planet. In general the inward force is a tiny bit stronger, and so the planet migrates inward.
Spiral wave: Gravitational „drag“ D‘Angelo, Henning & Kley (2002) The other way of looking at this is that gas parcels are slung by the planet and spend some time „behind“ the planet. The torque acting on the planet by these waves is called the Lindblad torque
Time scale of type I migration Time scale of inward type I migration (1 solar mass star): Review Thommes & Duncan in “The Formation of Planets” 2005 3-D estimates: 10 5...10 6 (Tanaka et al. 2002)
In „horseshoe orbits“ the gas parcels „librate“ back and forth. At turning point A the gas parcels give angular momentum to the planet (pushing the planet outward). At turning point B the gas parcels retrieve angular momentum from the planet (pushing the planet inward). Normally both forces cancel because each parcel passes as many times point A as point B. A B Horseshoe drag
Horseshoe drag: close-up Close-up view of the planet fly-bys that add and remove angular momentum to/from the planet pushing the planet outward / inward. Image: D‘Angelo, Henning & Kley (2002)
„Unsaturated“ horseshoe drag r s (entropy) Suppose, as is to be expected, that the specific entropy s of the gas in the disk increases with radial distance from the star. At the turning points (the fly-by points) gas parcels change radius, but (if they do not cool/heat quickly) retain their entropy. horseshoe region The inward moving gas parcel finds itself with „too much“ entropy while the outward moving gas parcel has „too little“ entropy compared to the local „standard“.
„Unsaturated“ horseshoe drag Image: D‘Angelo, Henning & Kley (2002) The fact that the fly-by gas parcels keep their entropy, but have to adjust their density to keep in local pressure balance, they will create an imbalance in the two torques. NOTE: After many libration periods this would „saturate“. IF gas radiatively cools/ heats during libration, it can remain unsaturated. Radiation-hydro problem! Excess entropy: under- density. Deficit entropy: over- density.
Hill sphere: sphere of gravitational influence of planet: If Hill radius larger than h of disk: disk can be regarded as thin compared to potential. This happens for massive enough planets. Planet may then open a gap. But this depends also on other things, e.g. viscosity. P. Ciecielag Gap opening in a disk
by Frederic Masset http://www.maths.qmul.ac.uk/~masset/moviesmpegs.html Role of disk viscosity: Planet pushes gas away, out of the co-orbital region. Viscosity tries to move gas back in to the co-orbital region. Low viscosity larger gap, extending beyond the co-orbital region less gas near the planet less torque.
Behavior of Type II migration Case of M planet <>M disk : –Disk cannot push planet. Planet migration is very slow. –Gap can be very deep, completely halting inward gas flow through the gap: inner disk „choked“ and vanishes on the viscous time scale. Large inner hole forms.
Type III migration Masset & Papaoloizou Type III migration takes place when the planet migration time across the co-orbital region is shorter than the libration time.
Type III migration Masset & Papaoloizou Type III migration takes place when the planet migration time across the co-orbital region is shorter than the libration time. By the time a parcel has librated to the other fly-by point, it might find itself no longer inside the co-orbital region. A strong asymmetric horseshoe drag follows.
Type III migration by Frederic Masset Note: this movie has opposite rotation as discussion above. http://www.maths.qmul.ac.uk/~masset/moviesmpegs.html