Accretion-ejection and magnetic star-disk interaction: a numerical perspective

1
Accretion-ejection and magnetic star-disk
interaction a numerical perspective

• Claudio Zanni
• Laboratoire dAstrophysique de Grenoble

5th JETSET School January 8th 12th
2008 Galway - Ireland
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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?

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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

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Proposed scenarios
Extended disk wind
X-wind
Stellar wind

• Succesful models require large-scale magnetic
  fields with
• plasma flowing along the magnetic surfaces
• - extended disk wind Bz distributed on a
  large radial extension
• - X-wind Bz exists only in a tiny region
  around the magnetopause
• - stellar wind opened magnetic field anchored
  on the star

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Why extended disk winds are important ?
Ferreira, Dougados, Cabrit (2006)
For a given footpoint r0 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
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Ejection how it works?The magneto-centrifugal
mechanism

• 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

• Magnetic field lines frozen in a disk
• rotating at Keplerian rate ?k
• Bead on the wire accelerated with
• constant ?k if fieldline is open
• ? gt 60o
• Angular momentum extraction
• ? accretion

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Framework MHD

• Conservation of

• Mass

• Momentum

• Energy

• Induction equation

with

• Solenoidality of the field

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Analytical solutions (1)

• Invariants

• Specific angular momentum
• (lever arm)

• Mass loading

• Field angular velocity

• Entropy

• Energy (Bernoulli equation)

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Analytical solutions (2)

• An entire class of radially self-similar
• MHD solutions can be constructed
• (Vlahakis et al. 1998, see Rammos poster)

• Examples (Contopoulos Lovelace 1994)

Bz / r -1.2
Bz / r -0.98
Bz / r -1.1

• Blandford Payne (1982)
• - Trans-Alfvenic solution
• - Bz / r -5/4

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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)

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Ejection initial conditions
Keplerian rotation injection boundary
Initial analytical solution (one ore more
superposed) boundary conditions (Gracia et al.
2006, Matsakos et al. 2008, and see Stutes
poster)
B)
A)
Boundary conditions (rotation/injection)
non-rotating magnetized corona (Ouyed Pudritz
1997)
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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 rout
Artificial collimating effect
Ustyugova et al. (1999)
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Testing stationary models (1)

• Axisymmetric MHD invariants
• are almost constant

Ustyugova et al. (1999)
Ustyugova et al. (1999)

• Acceleration mechanism IS
• magneto-centrifugal dominant
• forces are centrifugal (C) and
• Lorentz (M)

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Testing stationary models (2)

• MHD invariants

Matsakos et al. (2008)
Matsakos et al. (2008)

• Wave-structure and
• characteristic surfaces of
• analytical solutions are
• recovered

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Non-stationarity / variability

• When the outflow is too
• mass-loaded, the flow
• lags behind the
• Keplerian rotation and falls
• towards the center
• (Anderson et al. 2004)

1)

• Overdetermined
• boundary conditions
• force the propagation of
• MHD shocks along the jet
• (Ouyed Pudritz 1997b)

2)
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3D simulations

• Some technical issues

• How to put a circle inside a
• square smoothly reduce the
• rotation to zero between r0 and rmax

• Ensure r v 0 and r B 0 in
• the injection boundary

Ouyed, Clarke Pudritz (2003)
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3D simulations stability (1)

• Corescrew or wobbling solutions are
• found which are not destroyed by the
• non-axisymmetric (m1) modes

• A self-regulatory mechanism is found
• which maintains the flow sub-Alfvenic
• and therefore more stable (Ray 1981,
• Hardee Rosen 1999)

Ouyed, Clarke Pudritz (2003)
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3D simulations stability (2)

• Asymmetric outflow stabilized by
• a (light) fast- moving outflow
• near the axis with a poloidally
• dominated magnetic field.

Anderson et al. (2006)
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what about accretion?

• Additional elements must be taken into account
  
• - Accretion (mass conservation)
• - Disk vertical equilibrium (mass loading)
• - Field diffusion

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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
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Analytical self-similar solutions
Radially self-similar solution now depends on the
disk parameters
magnetization
disk thickness
Ferreira (1997)
magnetic diffusion

• Important results
• - jet parameter space strongly reduced
• - field must be around equipartition (? 1)
  and ?m 1 (or strongly anisotropic)

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What simulations can do?
Zanni et al. (2007)
Casse Keppens (2004)
And give a look to Tzeferacos poster
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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
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Resistivity parameter ?m 1
Smooth, trans-Alfvenic, trans-fastmagnetosonic
outflow is accelerated
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as seen in 3D
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Mass loading - acceleration
P
G
M

• Lorentz toroidal force changes
• sign at the disk surface
• Magnetic field extracts angolar
• momentum from the disk and
• transfer to the outflow

• Thermal pressure gradients supports
• the disk against gravity and magnetic
• pinch
• Pressure provides the mass loading
• and then Lorentz forces accelerate
• the outflow

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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)
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Axisymmetric MHD invariants
r0 2
r0 8
r0 4
r0 4
r0 2
r0 8
Flow perpendicular to the fieldlines in the disk
and parallel in the jet (resistive ideal MHD
transition)
Weber Davis (1967)
Inner fieldlines more stationary
Radial dependency of ? and k
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Resistivity parameter ?m 0.1
Footpoints of the fieldline advected towards the
central object Differential rotation along the
fieldlines triggers a magnetic tower
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as seen in 3D
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Parameter study - diagnostics
Increasing ?m
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)
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Is everything ok?
Despite having the same disk parameters (? 0.6,
?m 1, ? 0.1), analytical and numerical
solutions have different jet parameters
Numerical - k 0.1 - 0.3 - ? 4 - 9 - ?
0.09
Analytical - k 2 10-2 - ? 35 - ? 0.01
Analytical solution less mass loaded and faster (
)
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A physical reason
Zanni et al. (2007)
Casse Keppens (2004)

• 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

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A numerical reason
Casse Ferreira (2000)

• Density jump at the disk surface
• under-resolved in current
• simulations
• Numerical solutions closer to
• warm analytical models.
• Dissipation at the disk surface

• With a resolution 4 times lower it is
• possible to find stationary solutions
• even with ?m 0.1
• Radial numerical diffusion of Bz

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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