impact of magnetic fields on stellar structure and...
TRANSCRIPT
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Impact of magnetic fields on
stellar structure and evolution
Jean-Paul Zahn
Ecole de Physique Stellaire
Stellar Magnetism
La Rochelle 24-28 September 2007
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Impact on
stellar structure
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main forces that rule hydrostatic equilibrium in stars:
pressure and gravity
sound speed in solar interior !
!
cs
2 "P
#"GM
R
!
cs" 510
7cm /s
magnetic pressure is of same order as gas pressure
when Alfvén velocity equals sound speed :
!
!
B " 2 108G = 210
4T
Are magnetic fields strong enough
to play a role in the structure of stars ?
!
P
HP
= "GM
r2
magnetic force Lorentz, or rather Laplace ?
!
cA
=B
4"#$ c
s
!
r j "
r B
c
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Laplace or Lorentz ?
Pierre-Simon Laplace
1749 - 1827
force exerted by a magnetic field on
an element of electric courant
force exerted by an magnetic field on
a moving charged particle
!
r F = q
r V "
r B
Hendrik Lorentz
1853 - 1928
!
dr F = I d
r l "
r B
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Models built without magnetic field
compared with helioseismic data
Can a field of MegaGauss strength exist in the Sun?
new composition
(Asplund et al.)
old composition
(Grevesse et al.)
Bahcall & coll. 2005
due to a field of ! 3 MG ?
observed - model sound speed
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A closer look at helioseismology:
frequencies are split by mag. field
Can a field of MegaGauss strength exist in the Sun?
Anita, Chitre & Thompson 2000
! upper limit ! 300 kG
Splitting coefficients expected from a 4 MGtoroidal field located in the tachocline ("=0.04 R
!)
same, averaged over
30 neighbouring modes
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gas pressure:
Magnetic pressure dominates gas presure
in the surface layers
magnetic pressure:
!
Pm
=B2
8"
!
Pg =R"T
µ
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Magnetic fields interfer
with thermal convection
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Strong magnetic fields block convective heat transport
in sunspots
SST
La Palma
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Strong magnetic fields suppress
the convective instability
Linear instability in a unstably stratified magnetized medium
!
" #exp s t + i
r k $
r x [ ]perturb by
dispersion relation, neglecting thermal and Ohmic diffusion :
Chandrasekhar; Weiss 1960’s
! most unstable for horizontal wave-vector,
may be stabilized by sufficiently strong horizontal field
~ 107G below the solar CZ
~ 103G at surface
• displaces somewhat the boundary of CZ;
effect on Li burning during PMS ?
!
s2 =
kh
k
"
# $
%
& '
2
g
HP
( )(ad[ ] )r k *
r V A( )
2
• explains why inhomogeneities in surface composition of Ap stars
are not smooted out by convection
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Magnetic fields couple
stars to their environment
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Magnetized star coupled to accretion disk
! explains the relatively slow rotation
of TT stars
Fendt 1994
Bouvier et al. 1997
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Magnetized winds ! strong angular momentum loss
Schatzman 1962Fessenkov 1949 Schatzman 1954
If Sun loses matter at equator : but Sun loses matter at distance D
(Alfvén radius) :
!
d
dtI" = R
2"d
dtM
!
d
dtk2MR
2" = R
2"d
dtM
!
(R2") f
(R2")i
=Mf
Mi
#
$ %
&
' (
p
!
Mf
Mi
"
# $
%
& ' = 0.99
!
(R2") f
(R2")i
= 0.85
!
p = k"2"1=16
!
d
dtI" = D
2"d
dtM
!
d
dtk2MR
2" = D
2"d
dtM
!
(R2") f
(R2")i
=Mf
Mi
#
$ %
&
' (
p
!
Mf
Mi
"
# $
%
& ' = 0.99
!
D /R = 5
!
p = (D /R)2k"2"1= 425
!
(R2") f
(R2")i
= 0.014
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Disc-coupling and mass loss by magnetized wind
determine the rotation of stars
Monitor Project
(Irwin et al. 2006, 2007) ! the young Sun was a fast rotator
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Rotational mixing in radiation zones
Meridional circulation
Classical picture: circulation is due to thermal imbalancecaused by perturbing force (centrifugal, magn. field, etc.)
Eddington (1925), Vogt (1925), Sweet (1950), etc
Eddington-Sweet time
Revised picture: after a transient phase of about tES,
circulation is driven by the loss (or gain) of angular momentum
and structural changes due to evolution Busse (1981), JPZ (1992), Maeder & JPZ (1998)
• no AM loss: no need to transport AM ! weak circulation
• AM loss by wind: need to transport AM to surface ! strong circulation
!
tES
= tKH
GM
"2R3
with tKH
=GM
2
RL
shear-induced turbulence and internal gravity waves
contributes to AM transport
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Rotational mixing in magnetized radiation zones
Transport of angular momentum
turbulent diffusionadvection thru MC Laplace torque
!
+ rsin"r e #$r L
Even a weak field can inhibit the transport of AM
: characteristic time for AM loss
!
B2
> 4"# R2$
tAML
tAML
!
For " =1g /cm3
R = 7 1010cm R# =10
7cm /s tAML =10
9yr
!
" Bcrit# 20G
But the exact figure depends sensitively on the topology of magnetic field
internal gravity waves
!
"d
dtr2sin
2#$( ) = %& ' " r2sin
2# $r
U ( ) +sin
2#
r2(
r")
vr4(
r$( ) % & ' " r
2F
IGW( )
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Rotational mixing in magnetized radiation zones
Evolution of an axisymmetric field
poloidal (meridian) field
!
r B
P=" #
r A ,
r A = A
r e $
toroidal (azimuthal) field
!
r B
T= B
T
r e "
induction equations
!
"t A +1
s
r U # $ sA( ) =% $2
A &A
s2
'
( )
*
+ , s = rsin-
!
"tBT+ s
r U # $
BT
s
%
& '
(
) * = +B
T$ #
r U + s
r B
P# $,+ - $2
BT+
BT
s2
%
& '
(
) *
advection diffusion
diffusionadvection stretching
2D equations are projected on spherical harmonics
to be implemented in stellar evolution codes (thesis S. Mathis)
#-effect
suppressed when # cst on field lines of BP (Ferraro law)
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microscopic diffusion
distribution of chemical elements
meridional circulation turbulent transport
magnetic field
rotation
convection
internal gravity waves
penetration, overshoot
standard model
rotational mixing type I
rotational mixing type II
Rotational mixing in radiation zones
(in tachocline)
Mathis & Zahn 2005
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The solar tachocline problem (Spiegel & Z 1992)
Brun 2006
Hydrostatic and geostrophic equilibrium
conservation of angular momentum
conservation of thermal energy
Boussinesq approximation
solutions are separable :
Differential rotation #($)
applied at top of RZ
In thin layer approximation, for
In present Sun, differential rotation would have spread down to r = 0.3 R
!
Another physical process must confine the tacholine
Anisotropic turbulence ? Spiegel & Z 1992
Fossil magnetic field ? Gough & McIntyre 1998
! not observed - why is the tachocline so thin ?
Sunat 3.1 Gyr
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Can the tachocline be confined by a fossil field ?
Numerical simulations by Sacha Brun
diff. rotation imposed at top of RZ
initial dipolar penetrates in CZ
ASH code
tuned for RZ
optimized for massively
parallel machines
193x128x256
% Ferraro# ~ cst on field lines of Bpol
no
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Can the tachocline be confined by a fossil field ?
initial dipolar field burried in RZ
% Ferraro
No : such a field eventually connects with the CZ
and imprints its differential rotation on the RZBrun & Z 2006
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Magnetic fields generate
instabilities
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MHD instabilities
Theoretical results, mostly by Tayler & collaborators
• A purely poloidal field is unstable
to non-axisymmetric perturbations
(Markey & Tayler 1973)
• A purely toroidal field is unstable
to non-axisymmetric perturbations
(Tayler 1973; Wright 1973; Goossens et al. 1981)
• Rotation stabilizes somewhat a purely toroidal field,
but it cannot suppress entirely the instability
(Pitts & Tayler 1973)
• Stable fields are probably
a mix of poloidal and toroidal fields
of comparable strength
Results obtained in the ideal case (no thermal and Ohmic diffusions)
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MHD instabilities
Linear analysis, adding diffusion (Acheson 1978; Spruit 1999, 2002)
Radiation zone, stable stratification
buoyancy frequency :
Purely toroidal field
Alfvén frequency :
!
N2 = Nt
2 + Nµ
2 =g
HP
"ad #"( ) +g
HP
d lnµ
d lnP
$
% &
'
( )
!
"A
2=
B#2
4$% s2s = rsin&
Diffusivities - thermal: Ohmic:
!
" #107 cm2/s
!
" #103 cm2/s
Instability for
Perturbation, near axis:
Spruit’s conjectures :
• instability saturates when turbulent ! ensures marginal stability
• turbulence operates a dynamo in radiation zone
!
" #expi l s+ m$ + n z %& t( )
!
Im(" ) = 0
!
"A
4 > C#$ l2$
%N
t
2 + Nµ
2&
' ( )
* + C = O(1)
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Tayler instabilities in the solar radiation zone(magnetic tachocline simulation)
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Tayler instabilities in the solar radiation zone(magnetic tachocline simulation, cont.)
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Tayler instabilities in the solar radiation zone(magnetic tachocline simulation)
instability of poloidal field
instability of toroidal field
poloidal field decays steadilyBrun & JPZ 2006
JPZ, Brun & Mathis 2007
Poloidal field
is not
regenerated
! no dynamo
Decay of
poloidal field
not enhanced
by instability
! no eddy diff.
! no mixing
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The dynamo loop
Z, Brun & Mathis 2007
! It cannot work as explained by Spruit and Braithwaite
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Why does Braithwaite find a dynamo ?
How Braithwaite’s 2006 simulation (in black) differs from ours (in red)
boundary conditions :
field normal to surface
potential field
128x256x19264x64x63
Euler, numerical dissipation : Rm=?
DNS, enhanced diffusion : Rm=105
was the simulation pursued long enough ?
geometry
resolution
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Magnetic fields may inhibit
instabilities
Another example: thermohaline instability in RG
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Thermohaline mixing in red giant stars
extra mixingfirst dredge-up due to inversion of µ-gradient
produced by 3He(3He,2p)4He
Eggleton, Dearborn & Lattanzio et al. 2006
1/µ
mass
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In fact, Eggleton et al. observed convective instability, which occurs when
Ledoux criterion
!
" >"ad
+d lnµ
d lnP
In reality, as the µ-gradient builds up,
the first instability to arise
as soon as
!
d lnµ
d lnP< 0
is the thermohaline instability.
Charbonnel & Z 2007
who use Ulrich’s 1972 prescription
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Such extra-mixing destroys 3He ;
which explains its Galactic abundance However, observations show
that a small fraction of stars
(~4%) avoid this extra-mixing(Charbonnel & do Nascimento 1998)
Moreover,
2 PNe have been observed
with high 3He abundance ~10-3
(NGC 3242, J320)(Balser et al. 2006)
Our explanation:
the thermohaline instability is
suppressed by magn. field ~105G
in those RGB stars that are
the descendants of Ap stars
(Charbonnel & Z, submitted to A&A)
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Conclusions
Magnetic fields play little rôle in the structure of stars
but they have an impact on their evolution
• by suppressing instabilities
• by interfering with mixing processes operating in RZ:
rotational mixing, thermohaline mixing
• possibly by triggering MHD instabilities
• by determining their rotation state
Obviously, the effect depends on field strength! observational constraints are highly needed