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The inner -magnetized- regions of accretion discs

This research explores the role of magnetic fields in the inner regions of accretion discs, including their effects on jet formation, accretion, and the magneto-rotational instability. The study also examines the presence of strong magnetic fields and the issue of disc ionization. The findings have implications for understanding the dynamics of young stellar objects and circumstellar discs.

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The inner -magnetized- regions of accretion discs

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  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

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