The inner -magnetized- regions of accretion discs

1. Jonathan Ferreira - LAOG (France) Collaborators: N. Bessolaz, C. Zanni, P. Garcia, C. Combet, S. Cabrit, C. Dougados, F. Casse The inner -magnetized- regions of accretion discs

2. Outline I- Large scale magnetic fields in circumstellar discs II- The magneto-rotational instability (MRI), dead zone and disc ionization issues III- Jet Emitting Discs (JEDs) vs Standard Accretion Discs (SADs) IV- Star-disc interactions and the stellar spin-down issue V- Concluding remarks

3. I- Magnetic fields around YSOs CTTs are oriented randomly with respect to local Bz(Ménard & Duchêne 2004) BUT - sources with only discs: parallel - sources with strong outflows: perpendicular (Strom et al 86) • Suggests strong impact of large scale Bz in jet formation. But B required in discs for « viscosity » via MRI, (Balbus & Hawley 91, Lesur & Longaretti 2006)

4. Heiles et al. 93 Basu & Mouschovias 94 How strong is this field? • Infall stage: a decoupling matter/field must be at work. • ideal MHD (B  n) gives far too strong fields • ambipolar diffusion (also Banerjee & Pudritz 06) : • Crucial issue of matter ionization (unsolved yet) Presence of Bz will be an assumption and amount of flux F = pBzr2 a free parameter (IAU Grenoble 2007: now also a motto from F. Shu) We expect however F  M (Virial theorem and Jeans mass).

5. Effects of Bz on a SAD (1) • Can it drive a jet ? No • Magnetic field bending is not enough to launch jets Blandford & Payne (82) jets require Br+/Bz > 1 This is possible only with a magnetic Reynolds number But in a SAD, turbulent « viscous » torque gives and in turbulent media nv~nm … Lubow et al. 94a Heyvaerts, Priest & Bardou 96 Ogilvie & Livio 98

6. Effects of Bz on a SAD (2) • Does it affect accretion ? Yes • => no large scale torque (magnetic braking) if Bz small • << 1 • if disc magnetization << 1 • Thus, « magnetic braking » is negligible in weakly magnetized accretion discs. • => BUT, B triggers the magneto-rotational instability (Balbus & Hawley 91) giving rise to the turbulent radial transport of angular momentum

7. Magnetic field advection in SADs A SAD can transport Bz such as to increase the disc magnetization towards the center : Diffusion equation leads to Since where one gets with e ~ 1 for typical values of d Note: the only real issue is ionization and B/plasma coupling Ferreira et al 06a, A&A

8. A picture for the innermost disc regions Jet Emitting Disc with ~ equipartition large scale Bz field Ferreira & Pelletier 95 => Existence of high Bz recently confirmed by spectro-polarimetric observations around FU Or (~ kG @ 0.05 AU) Donati et al 05, Nature

9. II- Magneto-Rotational Instability, dead zone and disc ionization Chandrasekhar 53 Velikhov 59 Balbus & Hawley 91 Balbus 2003, ARA&A Flows in differential rotation: B= 0 - small perturbation: ang momentum Wr2= Cst => centrifugal force pulls back to equilibrium (Rayleigh) B≠ 0 - perturbation: magnetic torque Ff= JrBz < 0 => W < WK(r) : destabilizing effect which amplifies the deviation (with Bz or Bf) BUT magnetic tension is stabilizing Fr= JfBz > 0 ∂t ur = (W2 - W2K)r + Fr/r => B must be small B radius

10. MRI: dispersion relation Balbus & Hawley 91 Axisymmetric modes, incompressible flow, isothermal disc, homogeneous Bz, no radial modes Instability ifw2<0 The most unstable mode has a dynamical growth rate In an accretion disc l < lmax= h => VA < Wh = Cs Thus, an equipartition field (m ~ 1) quenches the MRI.

11. MRI: turbulent transport In a SAD, radial angular momentum transport is due to a turbulent « viscosity » n, giving rise to an accretion rate Mass conservation Angular momentum conservation Where the « viscous » tensor is (Shakura & Sunyaev 73) Energy equation: Q- = sT4 = Qe with Alpha prescription: n = av Cs h = r < ur uf> - < Br Bf>/m0 av= sRf/P0 Numerical simulations av ~ m1/2 (Lesur & Longaretti 07)

12. MRI: the role of Ohmic resistivity Gammie 96 • Previous arguments/calculations neglected the Ohmic resistivity h (electron-ions collisions): • - any wavelength l will be dissipated on a time scale l2/h • the growth rate of a magnetic perturbation is l/VA • MRI is quenched whenever l2/h < l/VA In a disc, relevant wavelengths are l < lmax= h • MRI quenched at all scales when Rm = h VA/h < 1 Critical threshold is hard to define precisely (eg Fleming et al 2000) < Bz > = 0 Rmcrit = h Cs/h ≈ 104 < Bz > ≠ 0 Rmcrit = h Cs/h ≈ 102 To conclude: MRI sets in only if Rm = h VA/h between 1 and 100 avbetween 10-3 and 1 ? (av ~ m1/2)

13. Rm Thedead zone Gammie 96 MRI quenched at all scales when Rm = h VA/h < 1 • Estimates: • r VA2 ≈ avP= av r Cs2 => VA ≈ av1/2 Cs • Ionization fraction x = ne/nH • Collisional ionization (ions K, Na) requires T> 1000 K => r < 0.1 au (SAD) • At large distances, ionization due to Cosmic Rays, but absorbed when S > So = 100 g cm-2 (Umebayashi & Nakano 81) • A layered accretion disc: active surface + inner dead zone? Hayashi 81 => Around 1 au, Rm < 1 for x < 10-13 How small is x ?

14. An unsteady accretion ? Gammie 96 • Only active upper zones with Sa = So will undergo MRI: • The disc accretion rate varies like Sa na Sa =103 g cm-2is a constant: Macc varies like na # T r3/2 Temperature T(r) depends on opacity regimes (eg. Bell & Lin 94) Thus Macc(r) is a function of the radius. • Mass accumulation in the dead zone frontier? • Time dependent disc accretion rate (eg. self-gravity sets-in, or heating and ionization) ? We are facing a difficulty: need for a self-consistent model taking into account both macro (fluid laminar equations) and microphysics (turbulence + ionization states)…

15. Models of disc ionization • Glassgold et al 97: only X-rays, disc = Minimum Solar Nebula • S(r) = 1700 R-3/2 g cm-2 • T(r) = 280 R-1/2 K => dead zone between 1- 10-30+au • Fromang et al 02: only X-rays, disc = av standard (shallower gradients) • v = 0.001 => all the disc is « dead » • v = 0.01 Ma= 10-8 Msun/yr => dead zone between 0.2-100 au • v = 0.1 Ma< 10-7 Msun/yr => no dead zone Matsumura & Pudritz 03: X-rays, CR and radioactivity, disc = Chiang & Goldreich 97 Passive disc with Ma=0 (more consistent) => Ionization CR > X-rays (unless kTX ≈ 5-10 keV) => dead zone between 0.2- 3 au => Results highly depend only chosen value So ≈ 103 g cm-2 => Interesting proposition: low massive planets formed in dead zone? But is there really a « dead zone » ?

16. Fleming & Stone 03 Magnetic energy Ohmic resistivity h(z) Maxwell stress The dead zone is not quite « dead » Turbulent mixing and mass exchange between the 2 regions… Reynolds stress Kinetic energy

17. III- Jet Emitting Discs (JEDs) Why do jets need to be magnetized anyway ? See Ferreira, Dougados, Cabrit 2006, A&A, 453, 785 1000 AU B: only necessary in the acceleration zone not anymore in the asymptotic regime

18. Constraints from T Tauri Jets Images: degree of collimation, evolution (HH objects) Spectroscopy: jet kinematics (line profiles, PV diagrams) and physical conditions such as density and temperature (line ratios). => Strong contrainsts on all MHD models

19. The 3 basic steady-state jet models Blandford & Payne 82 Ferreira & Pelletier 93,95,97 Wardle & Königl 93 Casse & Ferreira 00, 04 Shu et al 94, 95 Fendt 9, 00 Shang et al 98, 02 Weber & Davis 68 Hartmann & McGregor 80 DeCampli 82 Sauty & Tsinganos 94, 02 • share the same physics: rotating body + large scale Bz • Governed by the same set of MHD equations • Apart the « extended disc wind » model, mass flux is imposed => Can observations discriminate between these models ?

20. First: are jets indeed rotating? DG Tau Observations HST/STIS Bacciotti et al 00 The collimation degree increases with the jet speed (higher closer to the axis) Unresolved acceleration scale <20 au => MHD: launching radius at 2-3 au (Anderson et al 03, Pesenti et al 04)

21. Putting all constraints together Ferreira, dougados, Cabrit 2006 • Observations: • jet radius r • velocity Vphi • velocity Vp • Models: • anchoring radius ro • magnetic lever arm => l ~ 10 needed • Jet velocity gradients incompatible with current X-wind models • Observed mass fluxes incompatible with stellar winds only. • IF velocity shifts are rotation: launching radius from 0.2 to 3 AU

22. Accretion-Ejection Systems Axisymmetric jets are nested magnetic surfaces of constant magnetic flux: a(r,z) = Cst Blandford & Payne 82, Ferreira & Pelletier 95 • Single fluid MHD description • Non-relativistic equations (no light-cylinder) • Steady-state • Usually: polytropic energy equation • Ideal MHD (no viscosity, no diffusivity) • A complex interplay between disc and jets is necessary to assess the mass loading => Magnetized Accretion-Ejection Structure (MAES)

23. Governing MHD equations • Mass • Momentum • Energy • Perfect gas • Ohm’s law • Induction

24. Obtaining Self-Similar solutions • Separation of variables: all power-law indices are constrained • using requires The magnetic field distribution is intimately linked with the disc ejection efficiency (only radially self-similar class of solution compatible with a Keplerian disc) Propagation of a solution (Runge-Kutta for stiff equations) from disc midplane, needs a matrix inversion with several singular points: MHD critical points (ideal MHD jet) • Slow => m= B2/P • Alfvén => x where Mdot rx • Fast => g Note: all parameters are constants

25. A typical super-FM solution x= 0.03 e= h/r= 0.03 am= 1 kBP= 0.12 lBP= 23 Ferreira & Casse 04, ApJL

26. Self-similar MAES x= 0.005 x= 0.05 Ferreira 97 • Narrow parameter space: equipartition field • All solutions recollimate (Blandford & Payne 82,Pelletier & Pudritz 92) • A gradient in x could forbid it (Ferreira 97) • All solutions (here sub-fast) terminate with a shock 0.1 < m < 1 am~1

27. Numerical studies of accretion-ejection systems In most numerical simulations the disc is a boundary condition and mass loss imposed (eg. Ouyed & Pudritz 97,99,03, Pudritz et al 06, Ustyugova 99, Krasnopolsky et al 99, 03, Anderson et al 05…) Casse & Keppens 02, 04 Zanni et al 04, 07 => Main results from self-similar calculations are confirmed by MHD simulations where the disc is also computed.

28. JEDs vs SADs A JED is weakly dissipative: lack of disc emission (Ferreira & Pelletier 93) A JED is accreting faster : time scales and variability…? A JED is thinner and less dense than a SAD with same Mdot: ~ h/r << 1 with ms = ur/Cs ~ 1 in a JED, ms= av h/r <<1 in a SAD

29. JEDs vs SADs (3) A JED is thinner and less dense than a SAD with same Mdot: • The SAD/JED transition provides a trap stopping migration of protoplanetary cores (Masset et al 06) • Different radial structure than a SAD (planet formation?) Optically thick region Optically thin region Combet & Ferreira, in prep

30. JEDs vs SADs • (3) A JED is thinner and less dense than a SAD with same Mdot: • Different properties with respect to X-ray screening and ionization (recombination time scales) => dead zone ? Dead zone (Gammie 96) if Rm= hVA/h < 1 => translates into an ionization rate (only coll. recomb) Ferreira et al, in prep kTX= 3.9 keV, LX=3 1030erg/s Mdot=10-7 Msun/yr

31. The need for a warm « corona » • Thin discs => enthalpy negligible in jets: « cold jet » models • In that case, l~ 50-100 • Mass fluxes too low < 10% observed • Velocities too high (Garcia et al 01a,b) Higher mass fluxes with l~ 10 can only be done if some energy is deposited at the disc surface (Casse & Ferreira 00b). Warm jets from thin discs can arise if Stellar UV/X illumination(Garcia et al, in prep) Local dissipationof accretion energy(coronal heating, Galeev 79, Heyvaerts & Priest 89)

32. IV- The star-disc magnetic interaction See review in PPV Alencar et al • (1) Evidences of a magnetospheric interaction(Edwards et al. 94, 98, Calvet 04, Muzerolle et al 01, Bouvier et al. 99, Günther et al 99, Johns-Krull et al 99,01, Feigelson & Montmerle 99, COUP survey) • (2) T Tauri stars are slow rotators despite contraction + accretion(Bertout et al 89, Bouvier et al 97, Rebull et al 02) • Accretion must proceed AND help to remove stellar angular momentum • => What kind of magnetic configuration?

33. Rotational evolution of PMS stars • Velocity dispersion requires disc interaction (« disc locking » paradigm) Bouvier et al. 97 • T Tauri rotate at ≈ 10 % break-up (Bertout 89)

34. The Gosh & Lamb configuration The disc-locking paradigm Accretion must still proceed Gosh & Lamb 79 Cameron & Campbell 93, Li 96 Matt & Pudritz 04 • Rt > Rco: star is spun down => accretion is prevented • Rt< Rco: star is spun up => accretion is allowed But 2 major difficulties: • Efficiency of disc viscosity?? • Magnetic shear and reconnection => Loss of causal connection Aly 86 Lovelace et al 95, 99, Bardou & Heyvaerts 96 Uzdensky et al 02, Matt & Pudritz 05

35. Two (over-)simplified configurations Camenzind 90, Shu et al 94a, 95 (Matt & Pudritz 05) • Assumes Rt= Rco • « X-wind » is actually a disc-wind from a small disc region: no stellar spin-down per se (Uchida & Low 81, Hirose 97) Ferreira, Pelletier, Appl 00 - Assumes Rt= RX = Rco - « Reconnection X-winds » at the magnetic neutral line: very efficient stellar spin down (but unsteady events)

36. Heavy numerical simulations Hayashi et al 96, Miller & Stone 97, Hirose et al 97, Goodson et al 97,99, Kueker et al 03, Romanova et al 02,03,04,05, Long et al 05, 06 von Rekowski & Brandenburg 04,05 Zanni et al, in prep • Funnel flows are a natural feature in a non force-free magnetosphere, BUT lead to stellar spin up. • Main difficulties (and differences between works): • Treatment of disc physics (mass, nv and nm) • Boundary conditions at the star (!!) • Numerical resolution

37. The disc truncation radius Rt But why should Rt= Rco? • In fact, 2 conditions define Rt < Rco: • Accretion is halted • Disc material is loaded onto stellar field lines • Pmag ~ Pth • Bphi ~ Bz (Bessolaz et al, to be subm) => Accretion funnels with B ~ 140 G only at Rt= 2 R* (and no X-wind)

38. Torques on the accreting protostar • Gosh & Lamb configuration is NOT established • - After some time, most field lines are disconnected from the star (nm≈ Csh) • - Final stage = 2 electric circuits = 2 torques (+ no X-wind !!) • Positive torque due to accretion (dominant) • Negative torque due to open field lines (ready for wind…) Zanni et al, in prep

39. Reconnection X-winds: Interplay of dynamo + fossil fields • Protostellar core at break-up (Class 0) • Bipolar fossil field • t=0: dynamo produces a dipole field • Magnetic neutral line at RX= R* • Contraction of the protostar, spin-down by the ReX-wind + accretion funnels • rx≈ rco increases (magnetosphere expands) • Accretion rate onto the star is regulated • Stellar open field increases (Class I, II?)

40. Reconnection X-winds Stellar field n = 4 Ferreira, Pelletier, Appl 00

41. Momentum flux Vs Bolometric luminosity Class 0 103 yr 104 yr Class I 105 yr Ferreira et al. 00 Bontemps et al. 95

42. Saturation mechanism for B*? • ReX-wind power diminishes in time withW* • The disc magnetic flux F must be limited otherwise B* too large, W* too low • One would expect a correlation between W*and F • Does the weak dispersion in stellar periods reflect a weak dispersion in disc magnetic flux ? (remember F  M) n = 4

43. V- Concluding remarks (1/3) • Effects of large scale Bz are most probably very important in the innermost disc zones • MRI (accretion) • jet launching (« disc winds ») • star-disc interaction and possibility to drive ReX-winds • planet migration halting • Models have reached some maturity but • numerical experiments needed to assess time-dependent behavior and complex fields (eg Gregory et al, 06, 07) • important issues such as ionization, thermodynamics left over • challenge to find out observational constraints

44. V- Concluding remarks (2/3) • The « stellar angular momentum problem » requires a wind as a sink: • Accretion-powered stellard winds ? • Reconnection X-winds ? Matt & Pudritz 05 Ferreira, Pelletier & Appl 00 • Both scenarii suffer from lack of full dynamical calculations : • Enhanced stellar winds require RA/R* > 14 ! • ReX-winds rely on a magnetic neutral line… • => High quality MHD experiments + HRA observations are needed

45. V- Concluding remarks (3/3) Time ? Bz: an unavoidable ingredient ? • => A possible evolutionary sequence for the disc magnetic field • Classes 0 and I: • Mainly disc winds + ReX-winds (stellar spin-down) • Classes II (and III): • Mainly « enhanced » (accretion-powered) stellar winds