1. Accretion-ejection and magnetic star-disk interaction: a numerical perspective Claudio Zanni Laboratoire d’Astrophysique de Grenoble 5 th JETSET School January 8 th – 12 th 2008 Galway - Ireland

2. Outline Observational evidences supporting these scenarios: - accretion-ejection (disk-winds) (45 min) - magnetically controlled accretion (45 min) What analytical models can do? - pros: exact solutions, analysis of the parameter space - cons: stationarity, self-similarity What numerical simulations can do? - pros: time-dependent, no self-similarity, 3D - cons: can you trust them?

3. Ejection: jets from YSO They are directly observed ! - dynamics (speed, rotation) - thermodynamics (temperature, chemistry) … but not close enough to the central source to give direct informations on their origin

4. Proposed scenarios Succesful models require large-scale magnetic fields with plasma flowing along the magnetic surfaces: - extended disk wind : B z distributed on a large radial extension - X-wind : B z exists only in a tiny region around the magnetopause - stellar wind : opened magnetic field anchored on the star Extended disk windX-windStellar wind

5. Why extended disk winds are important ? Ferreira, Dougados, Cabrit (2006) For a given footpoint r 0 relation between toroidal and poloidal speed: Extended disc winds, X-winds, and stellar winds occupy distinct regions in the plane Only extended disk winds give results consistent with observations

6. Ejection: how it works? The magneto-centrifugal mechanism Magnetic field lines frozen in a disk rotating at Keplerian rate  k “Bead on the wire” accelerated with constant  k if fieldline is open  > 60 o Angular momentum extraction  accretion At Alfven surface matter inertia bends the lines and field gets wound up Toroidal magnetic field controls collimation (magnetic “hoop stress”) and pushes the outflow

7. Framework: MHD Conservation of: Mass Momentum Energy Induction equation: with Solenoidality of the field:

8. Analytical solutions (1) Invariants: Assumptions: - stationarity - axisymmetry - self-similarity (Radially self-similar solution) Mass loading Specific angular momentum (lever arm) Field angular velocity Entropy Energy (Bernoulli equation)

9. Analytical solutions (2) Blandford & Payne (1982) - Trans-Alfvenic solution - B z / r -5/4 An entire class of radially self-similar MHD solutions can be constructed (Vlahakis et al. 1998, see Rammos’ poster) Examples (Contopoulos & Lovelace 1994): B z / r -1.2 B z / r -1.1 B z / r -0.98

10. Numerical solutions Why time-dependent simulations? - Test the analytical models - Go beyond self-similarity - time-dependent  variability - 3D models – stability - combine different components (stellar wind)

11. Ejection: initial conditions Keplerian rotation + injection boundary Boundary conditions (rotation/injection) + non-rotating magnetized corona (Ouyed & Pudritz 1997) A) Initial analytical solution (one ore more superposed) + boundary conditions (Gracia et al. 2006, Matsakos et al. 2008, and see Stute’s poster) B)

12. Ejection: boundary conditions Injection boundary: number of incoming characteristics = number of fixed variables. Other variables must be free to evolve. Outer boundaries: even if the flow is super-fastmagnetosonic, pay attention at the direction of the Mach cones (  below or beyond the separatrix) Example: “outflow” condition on B  at r out Artificial collimating effect Ustyugova et al. (1999)

13. Testing stationary models (1) Acceleration mechanism IS magneto-centrifugal: dominant forces are centrifugal (C) and Lorentz (M) Ustyugova et al. (1999) Axisymmetric MHD invariants are almost constant Ustyugova et al. (1999)

14. Testing stationary models (2) Matsakos et al. (2008) Wave-structure and characteristic surfaces of analytical solutions are recovered MHD invariants Matsakos et al. (2008)

15. Non-stationarity / variability 1) 2) When the outflow is too mass-loaded, the flow “lags behind” the Keplerian rotation and falls towards the center (Anderson et al. 2004) “Overdetermined” boundary conditions force the propagation of MHD shocks along the jet (Ouyed & Pudritz 1997b)

16. 3D simulations Some technical issues: How to put a circle inside a square: smoothly reduce the rotation to zero between r 0 and r max Ensure r v = 0 and r B = 0 in the injection boundary Ouyed, Clarke & Pudritz (2003)

17. 3D simulations – stability (1) Ouyed, Clarke & Pudritz (2003) “Corescrew” or wobbling solutions are found which are not destroyed by the non-axisymmetric (m=1) modes A self-regulatory mechanism is found which maintains the flow sub-Alfvenic and therefore more stable (Ray 1981, Hardee & Rosen 1999)

18. 3D simulations – stability (2) Anderson et al. (2006) Asymmetric outflow stabilized by a (light) fast- moving outflow near the axis with a poloidally dominated magnetic field.

19. … what about accretion? Additional elements must be taken into account … - Accretion (mass conservation) - Disk vertical equilibrium (mass loading) - Field diffusion

20. … what about accretion? Mass conservation:  : ejection efficiency Disk vertical equilibrium: Only thermal pressure can uplift matter at the disk surface Magnetic field diffusion: Diffusion must counteract advection of the footpoints of the fieldlines

21. Analytical self-similar solutions Ferreira (1997) Radially self-similar solution now depends on the disk parameters: magnetization disk thickness magnetic diffusion Important results: - jet parameter space strongly reduced - field must be around equipartition (  » 1) and  m » 1 (or strongly anisotropic)

22. What simulations can do? Casse & Keppens (2004) Zanni et al. (2007) … And give a look to Tzeferacos’ poster

23. Initial-boundary conditions Self-similar Keplerian disk in equilibrium with gravity, pressure gradients and Lorentz forces. Disk parameters: Resolution: FLASH – AMR / 7 levels of refinement / 512x1536 eq. resolution

24. Resistivity parameter  m = 1 Smooth, trans-Alfvenic, trans-fastmagnetosonic outflow is accelerated

25. … as seen in 3D

26. Mass loading - acceleration Thermal pressure gradients supports the disk against gravity and magnetic pinch Pressure provides the mass loading and then Lorentz forces accelerate the outflow Lorentz toroidal force changes sign at the disk surface Magnetic field extracts angolar momentum from the disk and transfer to the outflow P M G

27. Current circuits - collimation Lorentz force (JxB) perpendicular to electric current circuits (rB  = const) Outflow collimated only towards the axis. Outer part still uncollimated Zanni et al. (2007)Ferreira (1997)

28. Axisymmetric MHD invariants Flow perpendicular to the fieldlines in the disk and parallel in the jet (resistive – ideal MHD transition) Inner fieldlines more stationary Radial dependency of and k Weber & Davis (1967) r 0 = 2 r 0 = 4 r 0 = 8 r 0 = 2 r 0 = 4 r 0 = 8

29. Resistivity parameter  m = 0.1 Footpoints of the fieldline advected towards the central object Differential rotation along the fieldlines triggers a “magnetic tower”

30. … as seen in 3D

31. Parameter study - diagnostics Increasing  m Ejection efficiencies consistent with observations (Cabrit 2002) Terminal speeds around 1-2 times the escape velocity ! Simulated spatial scale too small to check rotation ! But » 9 in the outer fieldlines of the outflow (see Ferreira et al. 2006)

32. Is everything ok? Despite having the same disk parameters (  » 0.6,  m » 1,  » 0.1), analytical and numerical solutions have different jet parameters Analytical: - k » 2£ 10 -2 - » 35 -  » 0.01 Numerical: - k » 0.1 - 0.3 - » 4 - 9 -  » 0.09 Analytical solution less mass loaded and faster ( )

33. A physical reason No analytical trans-Alfvenic solutions found when the electric current enters the surface of the disk (mass loading too high) Inner boundary forces the current to enter at the surface of the disk in its inner radii. The mass outflow is strongly enhanced in this region Zanni et al. (2007) Casse & Keppens (2004)

34. A numerical reason With a resolution 4 times lower it is possible to find stationary solutions even with  m » 0.1 Radial numerical diffusion of B z Density jump at the disk surface under-resolved in current simulations Numerical solutions closer to “warm” analytical models. Dissipation at the disk surface Casse & Ferreira (2000)

35. Perspectives Parameter space analysis - Magnetization (see Tzeferacos’ poster) - Transition between jet emitting and non-emitting disks (standard accretion disk) - The missing link between the small and the large scale - Interaction with an inner component (Meliani et al. 2006) Go to 3D …