The Accretion of Poloidal Flux by
Accretion Disks
Princeton 2005
What is an Accretion Disk?
Original theoretical description by Shakura and Sunyaev

Flattened, rotationally supported, gas accreting onto a
central object over the course of many rotation periods.

A good fluid, i.e. a high collision rate and a short mean
free path.

Negligible gravity due to disk. (ignoring galaxies, outer
parts of AGN disks)

Not necessarily ionized, but here we will discuss only
good conductors. (ignoring intermediate radii in
protostellar disks)
Fundamental Questions About Disks
1.
Are accretion disks turbulent? Yes - the
Magnetorotational Instability (MRI)
2.
How is angular momentum transported outward?
Internal stresses? Stress transmitted through an external
field?
3.
Does the environment play a role? Is accreted magnetic
flux important?
How is magnetic flux accreted?

The usual assumption is that entrained flux moves with the
accreting plasma. This process is balanced by the outward
turbulent diffusion of bent magnetic field lines. This gives
(van Ballegooijen)
 
 cs H r :  Br B  DT H :  cs  Br B 


z
z
In other words, the field lines bend very slightly as they
pass through a narrow disk, which limits the accretion of
external flux to a negligible level.
The use of turbulent diffusion is rather controversial.
Magnetic Helicity and Turbulent Diffusion
The electromotive force in a turbulent medium, has a
piece of the form:
r
DT J,
DT : v2
 eddy
3
When the turbulence is anisotropic due to shearing,
the diffusion coefficients differ and attach to the
appropriate gradient. When the back reaction is
large, the diffusion is suppressed by the factor
. The radialI diffusion
of entrained flux is
b̂b̂
suppressed by the ratio of the perpendicular
B 2
components to the total field.
2
B
Outward diffusion = inward drag when….
Balancing inward drag and outward diffusion, as before, we
get a critical poloidal field strength
1/2
1/2
H

&
Bcrit : B
: M r
r
This is just big enough that the radial and vertical field
components will be similar (Lubow, Papaloizou and
Pringle 1994) and the stress (and angular momentum
transport) through the magnetosphere will be comparable
to the internal transport. When the poloidal field is weaker
than this, it will be dragged inward. Due to this, and the
scaling of disk properties, the poloidal field will typically
affect the innermost part of the accretion disk, or none of
it. When it does, it can drive a Blandford-Payne jet.
  

When does the environment supply enough
magnetic flux?
Since the accreted poloidal field has nowhere to go, whether
or not one gets an inner annulus dominated by a large scale
poloidal field will depend on the environment. In general,
it’s not obvious what this means for systems that accrete
from the ISM. We can look at binary accreting systems,
and assume that the accretion disk can accrete some
fraction of the donor star’s poloidal field, determined by
the size of the disk. This will be important if
Bp

rdisk

2
rinner
 Bcrit
 
M r
Plugging in Numbers….
Obviously, for otherwise identical accreting systems, this is
more likely to give us a strong poloidal field when the
accreting object is a compact object, the smaller the better.
A white dwarf near the Chandrasekhar limit, accreting at
10^18 gm/sec, with an externally imposed field of about a
gauss, and an inner disk radius of ~3x10^10 cm, will fail to
acquire a strong poloidal field by a bit more than an order
of magnitude. In the same kind of system a black hole will
easily acquire a strong poloidal field.
Jets?
Conclusions (&speculations)

The suppression of turbulent diffusion by
magnetic helicity conservation implies that
available flux is strongly concentrated at small
radii
 The accretion of a large scale field from the
environment can produce an organizing field for
a jet. This may explain the systematic difference
between accreting white dwarfs and black holes.
 Since the critical magnetic field strength
increases with the accretion rate, systems with
fluctuating mass accretion rates are more likely
to have magnetically dominated inner regions
during low states, and show a transient burst
during the transition to the high state.