The role of magnetic fields for planetary formation

Transcription

1 fields for esimal Leiden Observatory, Leiden University IAU Symposium 259: Cosmic Magnetic Fields: from s, to Stars and Galaxies Puerto Santiago, Tenerife, November 2008

2 Star and The four stages of low mass star : a b c d esimal!=0 yr 1 pc!"10 # 4 yr 10 4AU!"10 # 5 yr 10 3AU!"10 # 6 yr 10 2 AU (Lada & Wilking 1984; Adams, Lada, & Shu 1987) 1 Molecular cloud core collapses under its own gravity 2 Star forms in the center, fed by continued collapse 3 Angular momentum conservation and cooling leads to of circumstellar disc of a few hundred AUs in radius 4 Envelope emptied, Keplerian accretion disc feeds star

3 esimal Infrared excess above black body of star reveals the presence of cool dust orbiting the star Discs seen by Hubble as dark spots on bright background vary in mass from 0.001M to 1M (Beckwith et al. 1990) Sizes typically of a few hundred astronomical units Life-times of approximately 5 million years (Bouwman et al. 2006)

4 Meteorites and fields esimal Murchison meteorite Natural Remanent Magnetism found in most meteorites Levy & Sonett (1978) Meteorites have remnant magnetisation of G Frozen in during cooling past Curie temperature Parent bodies of undifferentiated meteorites very unlikely to have field Magnetic field internal to disc, dragged in from envelope or originating in the young sun?

5 Accretion esimal How do proto accrete? Keplerian shear flows are (linearly) hydrodynamically stable (Ji et al. 2006) Gravitational instabilities can transport angular momentum, but need gas column densities more than 20 times higher than in the solar nebula Stratorotational (Shalybkov & Rüdiger 2005) Baroclinic (Klahr & Bodenheimer 2003) Streaming and Kelvin Helmholtz instabilities (Weidenschilling 1980; Youdin & Goodman 2005; et al. 2006; Youdin & 2007; & Youdin 2007)

6 Balbus & Hawley (1991): A weak vertical field renders Keplerian shear flows linearly unstable to the magnetorotational esimal Figure from Balbus & Hawley (1998) MRI transports angular momentum through Maxwell and Reynolds stresses if degree of ionisation high enough Based on earlier ideas by Velikhov and by Chandrasekhar

7 Non-linear evolution of MRI esimal Illustration of non-linear evolution of MRI from Sano et al. (2004): Initial patch of vertical field Slow magnetosonic wave is linearly unstable Makes entire box turbulent Maxwell and Reynolds stress gives α-viscosity of around α = No significant dependence on boundary conditions

8 Accretion disc dynamo? esimal z/h Lesur & Longaretti (2007) and Fromang et al. (2008) found that MRI generally only self-sustained at high Prandtl numbers Pm = ν/η Ongoing efforts to find limiting Prandtl number in ultra-high resolution simulations BUT: Pm dependence only shown for non-stratified simulations so far show Stratified models e.g. by Brandenburg et al. 1995, Arlt & Rüdiger 2001, Dziourkevitch et al Parker and stratification? (Tout & Pringle 1994, & Levin 2008) t=0.1 T orb ln ρ z/h t=20.0 T orb ln ρ y/h y/h 5.0

9 s form in proto from dust grains that collide and stick together (planetesimal hypothesis of Safronov, 1969). From dust to planetesimals µm m: Contact forces in collisions cause sticking m km:??? From planetesimals to protoplanets km 1,000 km: Gravity From protoplanets to planets Terrestrial planets: Protoplanets collide Gas planets: Solid core attracts gaseous envelope esimal Image references: (1) en et al. (1998); (2) William K. Hartmann

10 esimals esimal Kilometer-sized objects massive enough to attract each other by gravity (two-body encounters) Building blocks of planets Formation: µm cm: Dust grains collide and stick (Blum & Wurm 2000) cm km: Sticking or gravitational (Safronov 1969, Goldreich & Ward 1973, Weidenschilling & Cuzzi 1993) Dynamics of turbulent gas important for modelling dust grains and boulders William K. Hartmann

11 Dust dynamics esimal Gas accelerates dust through drag force: w t =... 1 τ f (w u) Dust velocity Particle radius a versus friction time τ f (at r = 5 AU): a /m Epstein regime (a<λ) Ωτ f Gas velocity Important non-dimensional parameter: Ω K τ f (Stokes number).

12 Diffusion-sedimentation equilibrium esimal Diffusion-sedimentation equilibrium: H dust δ t = H gas Ω K τ f H dust = scale height of dust-to-gas ratio H gas = scale height of gas δ t = turbulent diffusion coefficient, like α-value Ω K τ f = Stokes number, proportional to radius of solid particles ( & Klahr 2005)

13 Diffusion coefficient esimal Definition of Schmidt number: Sc = ν t /D t = α t /δ t From the scale-height of the dust one can calculate the diffusion coefficient: δ t = δ t (H dust ) Sc 10 1 Sc x Sc z Sc x (L y =4) Sc z (L y =4) α & Klahr (2005): Sc z 1.5, Sc x 1 (Turner et al. 2006: Sc z 1; Fromang & Papaloizou 2006: Sc z 3) Carballido, Stone, & Pringle (2005): Sc x 10, Klahr, & Mee (2006): The ratio between diffusion and viscosity depends on the strength of an imposed field

14 the Schmidt number esimal Safronov (1969): Dust grains coagulate and gradually decouple from the gas Sediment to form a thin mid-plane layer in the disc esimals form by continued coagulation or self-gravity (or combination) in dense mid-plane layer HOWEVER: MRI-driven very efficient at diffusing dust

15 the Schmidt number esimal Safronov (1969): Dust grains coagulate and gradually decouple from the gas Sediment to form a thin mid-plane layer in the disc esimals form by continued coagulation or self-gravity (or combination) in dense mid-plane layer HOWEVER: MRI-driven very efficient at diffusing dust Need to look at how larger particles react to

16 Boulders in, Klahr, & Henning (2006): 2,000,000 boulders moving in magnetorotational esimal

17 Gas density bumps esimal Strong correlation between high gas density and high particle density (, Klahr, & Henning 2006) Solid particles are caught in gas overdensities (Whipple 1972, Klahr & Lin 2001, Haghighipour & Boss 2003) Gravoturbulent of planetesimals

18 Gas density bumps esimal Strong correlation between high gas density and high particle density (, Klahr, & Henning 2006) Solid particles are caught in gas overdensities (Whipple 1972, Klahr & Lin 2001, Haghighipour & Boss 2003) Gravoturbulent of planetesimals

19 Pressure gradient trapping Outer edge: Gas sub-keplerian. Particles forced by gas drag to move inwards. Inner edge: Gas super-keplerian. Particles forced by gas drag to move outwards. esimal

20 esimal movie Time is in Keplerian orbits (1 orbit 10 years) esimal Keplerian flow et al (Nature, 448, 1022) Keplerian flow

21 esimal movie Time is in Keplerian orbits (1 orbit 10 years) esimal Keplerian flow et al (Nature, 448, 1022) Keplerian flow

22 Global models Fromang & Nelson (2005): Dust concentrates in long-lived vortex Dust density (5 cm and 25 cm): Gas density and vorticity (ω z ): esimal

23 Global models Lyra,, Klahr, & Piskunov (2008): Global disc with boulders on Cartesian grid (disk-in-a-box) esimal Gas density ( ) Particle density (10 6 particles)

24 Increasing box size esimal Stratified shearing box simulations with increasing box size t/t orb t/t orb x/h Σ g/<σg> t/t orb x/h x/h Density amplitude ˆρ(k x ) k 2 x Life-time of high pressure bumps increases with box size

25 Zonal flow esimal t/t orb ρ/<ρ> x/h u y/c s x/h t/t orb d/dt[k x ^uy 2 (kx,0,0)] k x Large scale variation in Maxwell stress launches zonal flows Pressure bumps form as zonal flows are slightly compressive Balance between turbulent diffusion and compression gives ˆρ k 2 x, Klahr, & Youdin (in preparation): in proto Adv Cor Lor

26 Inverse cascade esimal Plots show power contribution of different terms in the induction equation: Magnetic energy cascades to largest scales in the box Happens through the advection term Excites large scale variation in Maxwell stress Very little large scale activity in the vertical field component ^ 2 db x /dt ^ 2 db y /dt ^ 2 db z /dt Adv Adv (K) Com Adv Adv (K) Com Adv Adv (K) Com Str Res Str Res Str Str (K) Res 1 10 k

27 Dead zones esimal Transition from active accretion to dead zones triggers Rossby wave in pile up of gas Rossby vortices trap particles Formation of Mars or Earth size planets by self-gravity Lyra,, Klahr, & Piskunov (2009)

28 Dead zones esimal Transition from active accretion to dead zones triggers Rossby wave in pile up of gas Rossby vortices trap particles Formation of Mars or Earth size planets by self-gravity Lyra,, Klahr, & Piskunov (2009)

29 esimal Magnetic fields play an important part in the of planets The solar nebula, and probably proto in general, contain dynamically important fields are excited by the radial variation in the Maxwell stress of magnetorotational Convergence zones concentrate solids and allow the of 1000 km sized planet embryos by gravity