Transcript 20040907103511001-148699

Magneto-rotational
instability in the solar
core and Ap star
envelopes
Rainer Arlt
Astrophysikalisches Institut Potsdam
and Günther Rüdiger, Rainer Hollerbach
Solar rotational evolution

Wind model by Stępień (1988)
The solar tachocline and the core

Thompson et al. 2003 from various sources
Stellar radiative envelopes
Star of spectral type A and B
 Small convective core
 Extensive radiative zone
 10% of these stars
have magnetic fields
 These 10% are slow
rotators

Differential-rotation decay
Rotation of solar core is
slow and uniform
 Rotation period has increased by factor
of 10 during life
 Viscosity too small to reduce rotation
homogeneously throughout the Sun
 Magnetic Ap stars rotate much slower
than „normal“ A stars
 Did MRI reduce the internal rotation of
Sun and Ap stars?

Magneto-rotational instability
Angular velocity decreasing with
axis distance
 Magnetic field or arbitrary geometry

 Instability
of 
with growth rate of the order
Lower limit for MRI
k of most unstable mode
depends on B
Diffusive
decay rate
increases with
k
 MRI suppressed below
certain B
in Gauss
Upper limit for MRI
Wavelength
of most unstable MRI
mode exceeds
object size
in kG
Numerical simulations

Spherical spectral code (Hollerbach 2000)

  u  0, and   B  0
Initial conditions
Vertical cut
through radiative
zone
 Left:
Magnetic field
 Right:
Angular velocity

Differential-rotation
decay – close-up
Rm = 104
Pm = 1
Ra = 0
Differential-rotation
decay – close-up
Resolution at high Reynolds number
t = 4 rotations
t = 8 rotations
t = 1 rotation
Resolution at high Reynolds number
t = 4 rotations
t = 8 rotations
t = 1 rotation
Differential-rotation decay
Steepness of rotation profile versus time
 Initially

0
1 (2s )

2q
Rayleigh-stable
Differential-rotation decay time
versus Reynolds number
Effect of negative buoyancy
Rm = 2·104
Pm = 1
Ra = -108
Effect of negative buoyancy
Differential-rotation decay
Extrapolation to stellar parameters
 Decay time of 10-100 million years
 Short compared with the age of the Sun
(5 billion years)  MRI may have
provided the enormous angularmomentum transport for slow-down
 Considerable fraction of Ap star ages
(life-time < 109 yr)  MRI may still be
operating in them.

The End