Non-ideal MHD, Stability, and Dissipation in Protoplanetary Disks

Transcription

1 August 4, 2014 Non-ideal MHD, Stability, and Dissipation in Protoplanetary Disks NBI in Copenhagen The Formation and Evolution of Protoplanetary Disks: The Critical Effects of Non-Ideal MHD Shu-ichiro Inutsuka (Nagoya Univ.) M. Machida (Kyushu Univ), T. Matsumoto (Hosei Univ.), S. Takahashi, Y. Tsukamoto, K. Iwasaki (Nagoya Univ.)

2 Phases of Star Formation Collapse Phase Molecular Cloud Cores ~10 4 AU ~ yr ~ yr Accretion Phase First Core ~10AU ~ yr Protoplanetary Disk Protostar ~ yr TTauri, MS Planet Formation 2/36

3 Phases of Star Formation Collapse Phase Molecular Cloud Cores ~ yr ~10 4 AU ~ yr Accretion Phase First Core ~10AU ~ yr Protoplanetary Disk t = t * Protostar Planet Formation ~ yr TTauri, MS 3/36

4 Outline of Part 1 Basic Problems of Star Formation 1st Collapse 1st Core 2nd Collapse 2nd Core=Protostar Outflows vs. Jets Properties & Driving Mechanism Formation of Magnetized PPD and Fragmentation Dead Zone and Envelope Dispersal Focus on Formation of Single Stars

5 Basic Problems in Star Formation 1. Angular Momentum Problem: Protostar: h * = W * R 2 * ~ (10 11 cm) 2 /(10 5 s) ~ cm 2 /s Molecular Cloud: h core = dv core R core ~ 0.1km/s cm ~ cm 2 /s h * ~ 10-4 h core 2. Magnetic Flux Problem When? Protostar: F * ~ B * R *2 ~ kg(10 11 cm) 2 Molecular Cloud: F core ~ B core R core2 ~ 10mG(10 17 cm) 2 F * ~ 10-4 F core

6 Self-Gravitational Collapse Homologous Collapse P r g, C 2 S r g-1 r 1/R 3, M = const. F P (1/r)dP/dR C S2 /R F G GM/R 2 1/R 2 F P / F G R -(3g-4) Run-Away Collapse Larson 1969 r 1/R 2 g crit = 4/3 if g <4/3 unstable (g 1 in Molecular Clouds) t ff ~ (Gr) -0.5 Run-Away

7 Log(Surface Density) Rotating Run-Away Collapse Isothermal Run-Away Collapse W(t) (Gr) 1/2 F C r W(t) 2 r r 1/r 2 F G p Gr r z/r c r r 1/r 2 F C / F G = const. < Self-Similar Collapse Saigo, Matsumoto & Hanawa 2000 No Run-Away Collapse if g > 1. log r Narita, Miyama, Hayashi 1984 See also Norman, Wilson, Burton 1980 Tomisaka, Basu, Matsumoto, etc. g crit =1... isothermal

8 Convergence to Self-Similar Solution? Isothermal Rotating Run-Away Collapse Self-Similar Collapse Convergence of Time-Evolution without Dissipation? Same Protostars with Same Disks? Unique Initial Condition of Planet Formation? Answer: Isothermal (Barotropic) Hydrodynamics is Hamiltonian. Liouville s Theorem L = (1/2)rv 2 re Conserved Phase Space Density = No Convergence Convergence in Region of Vanishing Volume & Mass

9 Evolution from Molecular Cloud Core to Protostar Initial State Isothermal Collapse First Core Formation Adiabatic Collapse n c =10 4 cm -3 n c =10 9 cm -3 n c =10 11 cm -3 n c =10 13 cm -3 n c =10 15 cm -3 n c =10 17 cm -3 n c =10 21 cm -3 n c =10 22 cm -3 Outflow Driving Magnetic Dissipation Second Collapse Protostar & Jet

10 Temperature Evolution at Center g eff =1.1 < g crit =4/3 g eff = 7/5 Second Core Second Collapse dense core: n=10 5 /cc Effective Ratio of Specific Heats First Collapse g eff = 5/3 First Core (Larson 1969) Dissociation of H 2 E bind = 4.48 ev Masunaga, Miyama, & SI 1998, ApJ 495, 369; Masunaga & SI 2000, ApJ 531, 350

11 See also Commerçon & Hennebelle Temperature Evolution at Center Tomida et al. 2013, ApJ 763, 1

12 Effect of Non-Ideal MHD Weakly Ionized Gas Low density Ambipolar Diffusion Intermediate Hall Current Effect High density Ohmic Dissipation e.g., Nakano, Mouchouvias, Wardle, Tassis, Galli, R e,m Ionized 1st Core Second Core Temperature Dead Zone h Number Density, n[/cc]

13 Evolution of Ionization Degree Because of uncertainty of dust grain properties, we have parameterized resistivity. Okuzumi 2012 Machida, SI, & Matsumoto (2007) ApJ 670, 1198

14 Stage 1: Outflow driven from the first core The evolution of the Outflow around the first core This animation start after the first core is formed at n~10 10 cm -3 Grid level L =12 (Side on view) Model for (a, w)=1, 0.3 Grid level L =12 (Top on view) L = 1 (~10 4 AU) L = 12 (360AU) 360 AU Same as in Tomisaka 2002

15 Stage 3: Jet driven from the protostar The evolution of the Jet around the protostar This animation start before the protostar is formed at n~10 19 cm -3 Grid level L =21 (Side on view) Model for (a, w)=1, AU

16 Difference in Driving Mechanism Magnetocentrifugally driven Wind Magnetic Pressure driven Wind Strong B Outflow Jet Weak B Wide Opening Angle outflow around first core B r B z B f Narrow Opening Angle jet around protostar B z << B f only at launching region, not in distant region Machida, SI, & Matsumoto (2008) ApJ 676, 1088

17 Summary of Machida et al. Stiffening of EoS 2 different flows (outflow/jet) Outflow driven by the first core has wide opening angle and slow speed. Jet driven by the protostar has well-collimated structure and high speed. Velocities = Escape speeds from the first core & protostar. Machida, SI, Matsumoto (2008) ApJ 676, 1088 Observational Proof Velusamy, T., et al ApJ 668, L159, Velusamy, T. et al ApJ 741, 60

18 How About SPH? Godunov SPH: SI (2002) JCP 179, 238 Godunov SPH for Ideal MHD: Iwasaki & SI (2011) MN 418, 1668 Godunov SPH for Resistive MHD: Tsukamoto, Iwasaki & SI (arxiv: ) See also Bate & Price Iwasaki 2013

19 Magnetic flux Magnetic Flux Loss Magnetic flux largely removed from First Core when n =10 12 ~ cm -3 B = kg or less Ideal MHD Resistive MHD Machida, SI, & Matsumoto (2007) ApJ 670, 1198

20 Sufficient Flux Loss? Machida, SI, & Matsumoto (2008) B protostar ~ kgauss in standard model of h resistivity my guess: Turbulent Diffusion in Convection on Hayashi-Track Decrease of B in T-Tauri Phase(kGG?) For Turbulent Diffusion, see Lazarian & Vishniac (1999)

21 Formation & Gravitational Evolution of Disks

22 Formation of a Protostar Outflow jet 360 AU first core protostar v~5 km/s v~50 km/s Machida et al. ( ), Banerjee & Pudritz (2006), Hennebelle et al. (2008), Duffin & Pudritz (2011), Commerçon et al. (2011), Tomida et al. (2011) Outflows & Jets are Natural By-Products!

23 Formation of Planetary Mass Companions in Protoplanetary Disk t c ~10 5 yr Protoplanetary Disk Protostar ~0.1 M sun Protoplanet M~8 M Jup 300 AU R sep ~10-20 AU Machida, SI, Matsumoto (2009)

24 Resistive MHD Calc. from Mole. Cloud Core SI, Machida, & Matsumoto (2010) ApJ 718, L58

25 End of Formation Phase Spiral Structure with/without a Planet Machida, SI & Matsumoto (2011)

26 End of Formation Phase Machida, SI & Matsumoto (2011) PASJ 63, 555 Disk Growth Correlated with Depletion of Envelope Mass

27 Formation of Protoplanetary Disk L M disk = M * =0 M disk < M env M disk > M env L ~ l J R disk 10 2 AU 10 1 AU Formation of First Core t = t * dead zone Inutsuka (2012) PTEP 2012, 01A307 t

28 Formation of Protoplanetary Disk t = t * SI, Machida, & Matsumoto (2010) ApJ 718, L58

29 Evolution of Stellar Mass & Disk Mass Local Criterion for Gravitational Instability: Q C s /(pgs) SI, Machida, & Matsumoto (2010) ApJ 718, L58

30 Can planets cool and collapse? if cooling make g eff < 4/3 Gravitational Collapse Energy that should be radiated Resultant Luminosity Luminosity of Planet at T=T p Cooling with T=10 3 K Collapse by 10-5 in 10 2 yr Luminosity ~ 0.2L SI, Machida, & Matsumoto (2010) ApJ 718, L58

31 2D Modeling by Vorobyov & Basu 2006 Solution to Luminosity Problem? 2D Simulations of Infinitesimally Thin Disk

32 Time- Variability Vorobyov & Basu 2006 Observational Clue: Variability (FU Ori?) in deeply embedded protostar! Machida, SI & Matsumoto (2011) ApJ 729, 42

33 When? Machida, SI & Matsumoto (2011) ApJ 729, 42 Highly Eccentric Orbit! Outburst due to Migration Theory with e 0

34 Orbits of Planets & Their Fates Multiple Episodes of Planet Formation and Orbital Decay Machida, SI & Matsumoto (2011) ApJ 729, 42

35 Final Outcome? Machida, SI & Matsumoto (2011) ApJ 729, 42

36 Ang. Mom. Problem Summary of Part 1 Outflows from First Core & Jets from Protostar Ang. Mom. & Mag. Flux Problem Mag Flux Problem Disk Emerges in Dead Zone and Both Expand Together! Disk Grows Rapidly after Dispersal of Envelope! The First Core becomes Protoplanetary Disk! M disk > M * in disk formation phase Successive Formation of Planetary-Mass Objects Episodic Accretion and Planet Falling Outburst of Protostellar Luminosity Summarized in Inutsuka (2012) PTEP 2012, 01A307

37 August 4, 2014 Non-ideal MHD, Stability, and Dissipation in Protoplanetary in Copenhagen The Formation and Evolution of Protoplanetary Disks: The Critical Effects of Non-Ideal MHD B Part 2 Shu-ichiro Inutsuka (Nagoya Univ) Takeru Suzuki (Nagoya Univ) Satoshi Okuzumi (Tokyo Tech) Takayoshi Sano (Osaka Univ) Special Focus on Net B z Case with Ohmic Dissipation B A A

38 Evolution of Circumstellar Discs Early Phase Rapid Gas Accretion due to Gravitational Torque of m=2 Spiral Modes Typically a > 0.1 Later Phase Slow Accretion due to Magnetorotational Instability (MRI) Balbus & Hawley 1991 (Velikhov 1959, Chandrasekhar 1961) Remarkable Difference from Protostellar Collapse Phase Rotationally Supported & t evo > 10 3 t dyn! Need for Good Codes and Theories

39 Basic Energetics 1 Lynden-Bell & Pringle 1974 specific angular momentum: h = r v f specific energy: e = v f2 / 2 + y = (h/r) 2 / 2+ y(r) total energy: E = m 1 e 1 + m 2 e 2 total angular momemtum: H = m 1 h 1 + m 2 h 2 m 1 m 2 h W R Transfer angular momentum (dh) between 1 & 2 : dh = m 1 dh 1 + m 2 dh 2 = 0 de = m 1 ( e/ h) dh 1 + m 2 ( e/ h) dh 2 = m 1 dh 1 (W 1 - W 2 ) < 0 for dh 1 < 0 i.e., Outward transfer of angular momentum should be unstable.

40 Basic Energetics 2 Next, transfer mass & angular mom. dm = dm 1 + dm 2 = 0 dh = d(m 1 h 1 ) + d(m 2 h 2 ) = 0 de = d(m 1 e 1 ) + d(m 2 e 2 ) = dm 1 { (e 1 h 1 W 1 ) (e 2 h 2 W 2 ) } + d(m 1 h 1 ) ( W 1 - W 2 ) where d (e h W) /dr = d(- v f2 /2 + y )/dr = - r v f dw /dr > 0 Thus, (e 1 h 1 W 1 ) (e 2 h 2 W 2 ) < 0 m 1 m 2 de < 0 for dm 1 > 0 and d( m 1 h 1 ) < 0 Mass Inward Lynden-Bell & Pringle 1974 Ang Mom Outward Important only in rotationally supported case! How to transfer of Angular Momentum and Mass? h W R

41 Basic Eq. where, No Cooling!

42 Central Star Weak Magnetic Field Lines Magneto-Rotational Instability with Vertical Mag Flux Local Linear Analysis with Bousinesq approx. d e i( k z + w t ), k = 2p/l z l z Angular Momentum Transfer Dispersion Relation in Ideal MHD (h=0) Case w 4 - w 2 [ 2 +2(k v A ) 2 ] + (k v A ) 2 [ (k v A ) 2 +R dw 2 /dr ] = 0 Balbus & Hawley 1991

43 Simple Explanation for Instability Equivalent Model with a Spring! Connect two bodies with a spring K s =(k v A ) 2. dw x - 2W y = -xr dr y + 2W x = - K y 4 w 2 -w s 2 - K x dw ( 2 + 2K ) + K K + R = 0 If K s =(k v A ) 2, this is equiv. to w 4 - w 2 [ 2 +2(k v A ) 2 ] + (k v A ) 2 [ (k v A ) 2 +R dw 2 /dr ] = 0 s s s s dr 2 Balbus & Hawley 1998, Rev. Mod. Phys. 70, 1

44 Basics of MRI In Ideal MHD Linear Growth Rate: w max (3/4) W Kepler Exponential Growth from Small Field Kinetic Dynamo for Rot. Supported System l max 2p v a /W kepler Inverse Cascade Lucky for Simulations k x =0 axisymmetric (m=0) mode growth rate wavenumber R m v A (v A /W) /h larger h Sano & Miyama 1999, ApJ 515, 776

45 Non-Linear Stage of MRI Hawley & Balbus (1991) Hawley, Gammie & Balbus (1995, 1996) Matsumoto & Tajima (1995) Brandenburg et al. (1995). Balbus & Hawley (1998) Rev. Mod. Phys. 70, 1 Sorry for not citing your contribution Analytical Efforts: Pessah+(2007), Pessah (2010), Vishniac (2009) Okuzumi+(2014), Takeuchi+(2014)

46 Global Disk Simulation by T. Suzuki Wait for other talks on global simulations! Global Calculation by T. K. Suzuki (2010) Focus on local simulations! Periodic Boundary Condition in Azimuthal Direction (Df = 2p/16)

47 ex.) Sano+2004:2nd-order Godunov Method + MoC-CT

48 2D Axisymmetric Calculation Magnetic Raynolds Number: R M < 1 Uniformly Random Turbulent State h-dependent Saturation Level (Sano et al. 1998) b 0 = 3200, R m = 0.5

49 2D Axisymmetric Calculation Magnetic Raynolds Number: R M > 1 simple growth of the most unstable mode Channel Flow indefinite growth of B b 0 = 3200, R m = 1.5

50 R M > 1 2D Axisymmetric Calculation simple growth of the most unstable mode Channel Flow indefinite growth of B

51 3D Calculations Re M > 1 Exponential Growth of Most Unstable Mode channel flow dissipation due to reconnection f R B 2 Sano, SI, Turner, & Stone 2004, ApJ 605, 321

52 Nonlinear Time Evolution When Re M > 1, Spicky Feature in Time Evolution of Energy = Recurrence of Exponential Growth and Magnetic Reconnection Gas Pressure Gas Pressure Magnetic Pressure Magnetic Pressure time

53 Energy Budget Red Line: de/dt Green Line: Maxwell Stress + Reynolds Stress time t / t rot

54 Fluctuation vs Dissipation 1 B G r p 2 2 v u y dv Thermal Energy Hawley et al dg 1 P 3 BB r y r d dt = W - 2 r 2 面 4p 2 x y v v u S da Lx vx vy da yz BB R f dg M WR f rvrdvf -. 4p dt If saturated, Poynting Flux Stress Tensor, Wxy BB R f = = 0, then, = = rvrdvf -, t t dt t 2 4p 2 2 v B dg ru 3W where denotes time-average, and denotes time- and spatial- average. Note that v = d v = B = B = 0. R f R f Sano & SI (2001) ApJ 561, L179 Saturation Value of B 2 Dissipation Rate 0.03W B 2 SI & Sano (2005) ApJL 628, L155

55 Powerful Quasi-Steady Disk Wind after 210 rotations Powerful MHD Wind from Disk just like Solar Wind Suzuki & SI (2009) ApJ 691, L49

56 Powerful Quasi-Steady Disk Wind Distance from Midplane, Z Suzuki & SI (2009) ApJ 691, L49 Suzuki, Muto, & SI (2010) ApJ 718, 1289

57 Inner Hole Creation and Dispersal Distance from Central Star Dispersal Timescale ~ a few Myr for typical disk models in Ideal MHD Suzuki, Muto, & SI (2009) ApJ 718, 1289

58 Transport of Vertical Flux? Vertical magnetic flux should leak outward! Evolution of Vertical Flux by T. Suzuki Preliminary! Radius

59 Std Model of Protoplanetary Disks r g e ( r) fs 2 S = S 1-2 r T( r) = 280 K 1AU C ( r) 10 1AU cm r m cm 1AU 2.34 s r M * m 3 fs 1AU M O 2.34 r ( r, z) g/cm r bc B( r) 1.9 fs G, 1AU 100 b c 2C va S 2 2

60 Ionization Degree in PP Disks neutral gas + ionized gas + dust grains z CR = s -1 cosmic ray ionization resistivity Classical Models: Sano et al. 2000, ApJ 543, 486 Glassgold et al. 2000, PPIV Fromang et al. 2002, MN 329, 18 Salmeron & Wardle 2003, MN 345, 992, etc. Sano, et al. 2000, ApJ 543, 486 Dead Zone The site of planet formation seems to be "Dead Zone. But this depends on dust grain properties. Highly Uncertain

61 Dependence on Density and Ionization Source Resistivity: h = c 2 /(4p c ) = (T) 1/2 / x e Magnetic Reynolds Number: Re, M = v A2 /h W C s H/bh = (10/b)(x e /10-13 ) (H/0.1AU) b = (2C s2 )/v a 2 Required Electron Ionization: x e Relative Abundance electron log(n H /z -17 )

62 Ionization Degree Required for MRI resistivity: h = c 2 /(4p c ) = (T) 1/2 / x e Re, M = v A2 /hw = (10/b) (x e /10-13 ) (H/0.1AU) > 1 Required Electron Ionization: x e Scaling Relation in High Density Regime with Dust grains For x e 10-13, 10 7 < n H /z -17 < x e (z/n H ) i.e., (x e / ) ( ) -1 (z/ s -1 ) (10 15 cm -3 /n H ) requires an ionization process 30,000 times higher than the standard cosmic rays rate z CR =10-17 /s. For n H =10 15 cm -3, n H z = 300 cm -3 s -1 Is it possible?

63 Feedback from MRI Turbulence MRI MHD Turbulence Re M > 1 Ionization Energy Budget for Sustaining Ionization The ratio f ionize of energy required for the ionization to the energy available in the MRI driven turbulence, f ionize = e ionize z n H / [de/dt] = 0.03 (n H /10 15 cm -3 ) 2 (2pyr -1 /W) (6G/B) 2 (e / 13.6eV) is sufficiently small. Thermalization Radiation SI & Sano (2005) ApJL 628, L155 Muranushi, Okuzumi & SI (2013)

64 Thermal Ionization? Saturation State of MRI driven Turbulence = High b plasma Magnetic Energy < Thermal Energy of Gas Magnetic dissipation does not result in thermal ionization. NB. Magnetic Reconnection with Re M > 1? What is the mechanism of saturation? thermal ionization of Alkali (see, eg., Umebayashi 1983) not promising

65 Microphysics (1) Electric Currents 4p j = c B j c B/(4p L) j =S i e q i n i v i e q i n i v i f i j We can estimate the average velocities of charged species. v v e cb B 0.03AU 10 cm 10 cm = 42 km/s 4πLene 6G L nh xe fm+ cb B 0.03AU 10 cm 10 cm = 42 fm 4πLenM+ 6G L nh xm+ M+ + km/s The electron bulk velocity is surprizingly large! Inutsuka & Sano (2005) ApJL 628, L155

66 electron distribution function f (p) in weakly ionized plasma very small change of e at each collision with H 2 because m e /M < «1 Fokker-Planck approx. for e or p. Electrons have large velocity dispersion: e 19 ev Inutsuka & Sano (2005) ApJL 628 This is expected to provide sufficient ionization. Microphysics (2) Druyvesteyn , 0.43., 6 ) (, 3 exp ) ( ), ) cos( ( ) (, )],, ( ),, ( [ 1 v z v z v v M m m eel m eel m eel f m p f T A p f p f p f f Ee d p t f p t f N p f f T B p p f p f e t f M M M T T M e e e e e z = = = = - = + = = = - e e g g e g e E p E Let

67 Simulation of MRI with Discharge New New New New Muranushi, Okuzumi & SI (2012) ApJ 760, 56 Wait for Talk by Okuzumi!

68 Turbulent Mixing of Ionization In region with no dust grains, recombination happens in gas phase: dx e /dt = z S j b j x e n j = z b' n x e 2 where b'= S j b j (x j /x e ) Time evolution of ionization degree can be solved as x e (t) -1 = x e, ( b'/ s -1 ) (n H /10 15 cm -3 )(t/1yr). i.e., within an eddy turn-over time the ionization degree keeps the level of x e > Thus, the ionized region penetrates into the neutral region by the turbulent motions and homogenize the ionization. the most of the region in the protoplanetary disk will be sufficiently ionized. SI & Sano (2005) ApJL 628, L155 See also Ilgner & Nelson (2006), Turner+

69 Summary Results of 3D Resistive MHD Calculation When Magnetic Reynolds Number (Re m ) > 1 Exponential Growth from very small B Growth Rate = (4/3)Ω independent on B Field Strength cf. Kinematic Dynamo λ maximum growth becomes larger as B becomes greater. Inverse Cascade of Energy Saturated States Energy Equipartition Classified by Re m Re m < 1 quasi-steady saturation similar to 2D results Re m > 1 recurrence of Channel Flow & Reconnection Fluctuation-Dissipation Relation Energy Dissipation Rate ρv x δv y - B x B y /4π Mass Accretion Rate

70 To Do List Saturation Level a with Net Vertical Flux Vertical & Radial Stratification Global Simulation Disk Winds Driven even in Local Simulation Suzuki, Magneto-Centrifugally Driven only in Global Simulation Non-Ideal MHD Effects in Disks Non-Linear Ohms Law & Impact Ionization Okuzumi Hall Term Wardle, Ambipolar Diffusion Gressel, Bai, Magnetic Flux Loss from Disks! Transport Mass inward, Angular Momentum Outward, Flux Outward! Who?