Magnetic star-disk
interaction
Claudio Zanni
Laboratoire d’Astrophysique de Grenoble
5th JETSET School
January 8th – 12th 2008
Galway - Ireland
Observational evidences
• CTTS have dynamically important surface magnetic fields: B* » 1-3 kG
(Valenti & Johns-Krull 2004)
• Redshifted absorption features in inverse P-Cygni profiles of H- lines
reveal accreting material at free-fall speed (> 100 km s-1) (Edwards et al.
1994)
• Hot spots can be inferred from photometric and colour variability (Bouvier et
al. 1995)
• Rotational modulation of light curves suggests star rotation periods around 310d (Bouvier et al. 1993)
origin of stellar spin-down?
A simple model …
• Accretion disk is truncated at a few
stellar radii by the interaction with the
(dipolar) stellar magnetosphere.
• The flow is channelled into funnel
flows terminating with an accretion
shock on the star surface
… with some limitations
• Spectropolarimetric observations of
CTTs suggest that the stellar field is
more complex than dipolar.
• Ex. V2129 Oph (Donati et al. 2007)
- octupole 1.2 kG
- dipole 0.35 kG
• Photometric and spectroscopic
variations of AA Tau determined by
periodic occultations of a disk
warped by the interaction with an
inclined dipolar magnetosphere:
intrinsically 3D problem
The ingredients of the problem
• “Outer” accretion disk
(torque: viscous, disk wind)
• Stellar magnetosphere
connecting the rotating star and
the disk
• Accretion columns
• Outflows: disk winds, “reconnection driven” outflows, stellar winds
• System characterized by three radii:
- Truncation radius Rt
- Corotation radius Rco
- Outer radius Rout
What analytical models can do?
• No analytical model which takes into account all the elements of the
scenario (accretion disk, accretion columns, stellar magnetosphere,
outflows) is currently (and probably it will never be) available.
• Parts of the problem can be solved separately:
- Localization of the truncation radius Rt
- Structure of the accretion columns
- Structure of the magnetically torqued accretion disk
- Angular momentum exchange between the star and the disk
…. with some approximation
The truncation radius
(N.B. located below corotation radius)
• Alfvén radius (Elsner & Lamb 1977):
(ram pressure of a spherical envelope accreting at free-fall speed = magnetic pressure of a dipole)
• The constant k:
(Ghosh & Lamb 1979,
Konigl 1991, Long 2005)
(Arons 1993, Wang 1996,
Ostriker & Shu 1995)
(Bessolaz et al. 2008)
Can a “weak” (» 100G) dipolar
component truncate a disk
accreting at 10-8 Msun yr-1?
Probably not…
Accretion columns
• Trans-sonic solutions can be calculated (ex. Koldoba et al. 2002):
• Passage of the sonic point and therefore
accretion is controlled by thermal pressure
at the base of the accretion column
• Thermal energy greater than what is
available in a thin accretion disk
• Limitations: sub-Alfvenic flow, force-free dipolar fieldlines
Torqued disk structure
• It is possible to calculate the effects of the magnetospheric torques on
the structure of the disk (ex. Kluzniak & Rappaport 2007):
• The magnetospheric torque brakes
down the disk rotation inside the
corotation radius and forces the disk to
co-rotate with the star
• Limitations: vertically averaged disk model, a-priori hypothesis on B
Putting the pieces together:
numerical simulations
• Many numerical simulations do not
have strong enough magnetic fields to
truncate the disk and produce accretion
columns (Hayashi et al. 1996, Miller &
Stone 1997, Kuker et al. 2003)
Kuker et al. (2003)
• First accretion columns simulated in
2002 (Romanova et al. 2002) assuming
a magnetic field in equipartition with the
disk energy ( » 1 kG)
Romanova et al. (2002)
Typical initial conditions
• Dipolar field aligned with the rotation
axis of the star
• Resistive (viscous) Keplerian accretion disk
Resistivity (viscosity):
• Field in equipartition with the thermal
pressure of the disk at the initial
truncation radius Rin
dominant magnetic torque
• “star” (M* = 0.5Msun, R* = 2Rsun) modeled as
perfect conductor rotating with a 4.5 days
period (* = 0.1k, Rco = 4.6 R*)
• MHD fluid equations solved with the PLUTO code (Godunov + CT method)
Movies…
As seen in 3D…
In 2D…
Disk truncation
• Disk truncated in equipartition
Rin
conditions
Pth
• Pram =  vr2 < Pth
Pmag
Pram
• Magnetosphere represents a
“magnetic wall” for such an
accretion flow
• Confirms analytical results contained in Bessolaz et al. (2008)
Funnel flow dynamics
rPth
v/r
Fkin
Fmag
FLorentz
ggrav
• Thermal pressure gradient uplifts matter at Rin into the funnel flow and slows down
matter fall
pressure comes from the compression against the magnetic wall
• Centrifugal barrier always negligible
matter is braked along funnel flow
• Transport of angular momentum dominated by advection (Fkin = rVVp) at the
base of the funnel and by magnetic torque (Fmag = rBBp) at the star surface
Star-disk torques: general ideas
• Accretion torque (
) can
only spin up the star rotation (which is
still contracting anyway)
• How it is possible to extract this excess
angular momentum?
• Extended magnetosphere, connected beyond Rco (Ghosh & Lamb 1978): does
not work due to limited size of magnetosphere (Matt & Pudritz 2005)
• X-wind extracting the disk angular momentum BEFORE it falls onto the star
surface (Shu et al 1994) … is a wind like that possible?
• Stellar wind: accretion powered stellar wind (Matt & Pudritz 2005),
reconnection X-Wind (Ferreira et al. 2000)
Interaction regimes
“Accretor”
“Propeller”
(Matt & Pudritz 2004)
• Compact magnetosphere (Rin < Rout < Rco)
no braking torques are present
except for outflows
• Extended magnetosphere (Rin < Rco < Rout)
disk can extract angular momentum
(“disk locked” state)
• Propeller (Rin > Rco)
disk truncated beyond corotation, no
accretion columns, only spin-down
torques
State 1: “compact magnetosphere”
( = 0.1 B* » 800 G)
• All fieldlines beyond corotation magnetic surface (yellow line) are opened
• The opened stellar and disk fieldlines are separated by a strong current sheet
along which numerical reconnection phenomena can occur as well as episodic
mass outflows
• CME-like ejection site close to the base of the accretion column. No X-winds.
State 1: “compact magnetosphere”
( = 0.1 B* » 800 G)
• After initial strong transients the accretion rate (and “hot spot luminosity”) shows
an almost stationary behavior
• Variability may occur on the longer accretion time-scale
• The magnetic torque measured on the closed fieldlines really small (weak
accretion rate)
• The star is always braked along the opened field lines
State 2: “extended magnetosphere”
( = 1 B* » 800 G)
• Magnetosphere stays connected up to a radius » 2.5 (Rco » 1.6)
• The current sheet is located further from the star and the episodic outflows are
weaker
• The disk viscosity is efficient enough in the connected region in order to remove
radially both the disk and the stellar angular momentum as to provide mass to the
accretion columns.
State 2: “extended magnetosphere”
( = 1 B* » 800 G)
• Accretion rate (and “hot spot luminosity”) regularly oscillates with a 1.5-2 P*
period (mismatch between magnetospheric and viscous torque)
• Even if part of the disk magnetically connected to the star beyond Rco the disklocked torque always spins up the star
• The star is always braked along the opened field lines: “zero-torque” state?
Magnetic braking: stellar wind
• Mwind » 8 £ 10-11 Msun yr-1
• Strongly magnetized ( » 10-3)
• Lever arm RA/R0 » 15
State 3a: “propeller”
( = 1 B* » 1.6 kG)
• Propeller regime…
• The trucation criterium ( » 1) valid also beyond corotation…
• Can be this state maintained for long timescales?
State 3a: “propeller”
( = 1 B* » 1.6 kG)
• The star is braked both along the closed and opened field lines
• The “accretor” solutions are in an almost “zero-torque” condition
• The “propeller” solution always spins-down the star
Beyond dipoles and
axisymmetry: 3D simulations
• Technical issue: curvilinear geometries (cylindrical, spherical) introduce
singularities; cartesian geometry cannot describe correctly the surface of
the star and the disk (putting a sphere in a cube)
Koldoba et al. (2003)
• Optimal solution: cubed sphere
• Problems: non-orthogonal metric, interpolation between 6 sectors
Romanova et al. (2003, 2004)
• Inclined dipole varying the angle  between the
rotation axis and the magnetic moment 
Romanova et al. (2003)
 = 15o
 = 60o
• Two streams funnel flow
• FF located »
downstream
(FF rotates faster than the star)
30o
• Warped accretion flow
perpendicular to 
• Funnel flow more complicated
• Direct accretion on the poles
• Disk depleted of material
Romanova et al. (2003)
• Higher accretion rate for higher 
• Torque on the star always positive
• Higher initially for higher  but then
less matter is accreted in outer part
of the disk
Romanova et al. (2004)
•  = 15o
• Kinetic energy flux on the star
converted in radiation
• One peak of intensity during
one period for i < 60o
• Two peaks for i > 60o
Long et al. 2007
• Accretion on inclined quadrupolar+dipolar field