Magnetic field and stellar structure
Jiří Krtička
Masaryk University
Magnetic field and stellar structure
Magnetic field and global stellar structure
Magnetic field significantly influences stellar structure if the magnetic field energy is comparable with gravitational energy, that is,
4 oß2 GM2 -ttR3— -
3      4tt R
In solar units,
B « 108 G
This limit is never reached in any star. In nondegenerate stars, the magnetic field is always significantly lower than the above limit. The most strongly magnetized non-degenerate star ever discovered is Babcock star HD 215441, which has M « 2/W0, R « 2/?0I and B « 34 kG, i.e., several orders of magnitude below the limit. Neither this is fulfilled in white dwarfs, where the magnetic field is up to 108G, but R ^ 10~2Rq. The limit is not reached even in magnetars with B upto 1015G, but very small radii R ^ 10~sRq. Magnetic field field does not influence the global structure of stars, but may influence local processes like convection, angular momentum transport, and magnetosphere.
Magnetic field and convection
The interplay between magnetic field and convection establishes a very complex problem, which shall be treated usign MHD simulations in its general form, coining the term magnetoconvection. At least, an analogue of Schwartzchield stability condition can be formulated as
din 7     k-\ 6,
< -+
V
d In p        >c + 8tv>cp
where 6V is the vertical component of the magnetic field. This means that the magnetic field may stabilize atmosphere against the convection if
3 = — < 1
M g2   ^ 11
8tt
2
Magnetic field and convection
The interplay between magnetic field and convection establishes a very complex problem, which shall be treated usign MHD simulations in its general form, coining the term magnetoconvection. At least, an analogue of Schwartzchield stability condition can be formulated as
d In t k
< —
+
a
V
d In p        >c + 8tv>cp
where 6V is the vertical component of the magnetic field. This means that the magnetic field may stabilize atmosphere against the convection if
s = -p-<1
8tt
The magnetic field stabilizes the solar atmosphere against convection in solar spots. As a result, the heat transport becomes inefficient and the spot becomes cool.
Magnetic buyonancy
Let us assume the horizontal | z
magnetic flux tubewith internal and
external pressure p\ and pe, ^-1-^
respectively. The hydrostatic -1-g
equilibrium requires that
B2
Pi + — = Pe-47T
This implies that the density inside the flux tube is lower than outside leading to magnetic buyonancy (Parker & Jensen 1955).
The magnetic flux tubes rise and appear on the stellar surface in the form of Greek letter ft. This explains the appearance of stellar spots with oposite polarities.
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Magnetic field decomposition
Magnetic field decomposition
In many cases, the stars retain some kind of symmetry, for example an axial symmetry around the rotational axis. In such case it becomes convenient to decompose the magnetic field into poloidal and toroidal components,
B = Bp + Bt.
Denoting t unit vector in the azimuthal direction, the components fulfill
Bp • t = 0,       Bt = B^t.
Introducing scalar P such that the potential of Bp is A = At = —Pt/R, where R is radius in cylindrical coordinates (distance from z axis)
(P \ 1 Bp = rot4 = -rot f —t j = x t
Here we used rot (^C) = ^rot C +      x A and rot (t/R) = 0.
Magnetic field decomposition: the current
As a result of axial symmetry, from the Ampere's law (e.g., writing in components)
;'t = ^rotep>
This means that poloidal magnetic field creates toroidal currents and vice versa. The Lorentz force density is
1111
-j x B = — j x Bt + -jt x ep + — j x ep .
c c K c c K
S-v-'      S-v-'
poloidal toroidal
From symmetry, the Gauss's law for magnetism in the differential form is
div B = div ep + div Bt = div Bp = 0.
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Ferraro isorotation law
Induction equation in the rotating star
Let us study rotation of spherically symmetric star with rotational velocity
vt = RQt,
where R is the radius in cylindrical coordinates, ft is angular frequency, and t is unit toroidal vector. Separating the magnetic field into poloidal and toroidal components, B = Bp + Bt, the induction equation is
dB
— = rot (vt x B) = rot [RQt x (Bp + Bt)] = rot (RQt x Bp).
Evaluating this in the cylindrical component, the tp component of induction equation is
^ = /?(Bp-V)fi.
If angular velocity changes along Bp, then the toroidal field is generated from the original poloidal field.
Equation of motion in rotating star
The equation of motion in the direction of ip is
dv.
dt
pR
00. ~dt
1 ,.
{Jt +iP) x (Bp + Bt)] t = - (jp x Bp) t
Using the previously derived relations and A • (6 x C) = C • (A x 6)
<9ft     /? c
St        C 47T
(v(/?B„) x |)
x 6,
t = —Bp-v(/?ev)
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Torsional waves
Combining the induction and momentum equations and neglecting the changes of 6P we arrive at the wave equation of torsional waves
pR'
02Q dt2
47T
Bp-V[/?2(Bp-V)ft
02B,
dt2
K(Bp-V)
_4vr/o/?2
ep • V{RBV)
Taking into account small variations of Bp and R
d2Q
B2 d2Q
dt2     Airp ds2
v
A
d2Q
ds2
where s the element length along 6P. The torsional waves propagate with Alfven speed relatively quicky through the star (within 102 — 104yr)
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Ferraro isorotation law
The torsional waves dampen down during the stellar lifetime, dB^/dt = 0, what implies (Bp • V) ft = 0. For Bp =     VP x t we have (VP x t) • Vft = 0. Again using the identity A • (B x C) = C • (A x 6) we have t • (VP x Vft) = 0. As a result of symmetry around the z axis, the term in the bracket should have a component just in the direction of t. This implies VP x Vft = 0, or
ft = ft(P).
This is Ferraro isorotation law, which says that the angular rotation frequency is constant along the magnetic field line. In most cases, this means solid body rotation for magnetic stars.
From the equation of motion in the stationary state dQ/dt = 0 and therefore 6P • V(P6^) = 0. Again using 6P = —-^VP x t we have RB^ = f(P). Therefore the current
,v=iV(ffEv)x(=_i_f<VPX(=jiBp
flows along the magnetic field and does not produce any force.
Solar dynamo
Dynamo: creating magnetic field by motion
Many stars show strong magnetic fields. One of the possibilities how to explain such fields is by a stellar dynamo, that is, by creating magnetic fields by flow motion. As we shall see, the dynamos work in cool stars in their convective envelopes and in convective cores of hot stars.
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Biermann battery
Let us explore the possibility that the polarization electric field is the source of the magnetic field. The polarization electric field is
E = ~2eg-
From hydrostatic equilibrium equation g = Vp/p we have E = —ATijVp/ (2ep). For n\ = A7e we have p = 2pe and therefore
1
eh--Vpe = o.
nee
The Faraday's law of induction —1/cdB/dt = rot E provides a possibility to create magnetic field if rot (Vpe/ne) ^ 0.
In a spherically symmetric nonrotating star rotE = 0 and therefore there is no induced magnetic field. However, in a rotating star where ft = ft(z) from the hydrostatic equilibrium equation rot E 7^ 0 providing a posibility to induce toroidal magnetic field. This is called Biermann battery. However, the analysis shows that the field is very weak, but it may serve as a seed field for the dynamos.
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Cowling antidynamo theorem: the essence
a
Let us assume axial symmetry. The toroidal magnetic field 6P creates from the Ampere's law toroidal currents jt. This results in a flow velocity jt x 6P in the direction to the neutral point. To prevent this, we would need a flow source in the neutral point 0, which is impossible due to the conservation of mass.
This means that any axisymmetric magnetic field cannot be sustained against the Ohmic decay via axisymmetric flow of matter.
This is Cowling antidynamo theorem. As a result, we need nonaxisymmtric flow to obtain stable dynamo.
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Cowling antidynamo theorem: detailed analysis
Let us search for a stable dynamo assuming axial symmetry. The toroidal
component of the Ohm's law is
■
- = Et + -vp x ep.
a c
Assuming the vector potential in the form of A = —Pt/R, we have the poloidal magnetic field
Bp = --VPx t.
Decaying axially symmetric magnetic field creates toroidal electric field et = — 1/cdA/dt (as a result of axial symmetry, there is not contribution from V(f. In a stationary state, et = 0 and the toroidal current is
jf = —rotS = —rot rot A = —rot (4 VP xt| =
47T
^ div VP + VP div (± J + (VP • V) f ^
=0, axisym.
-1/RdP/dRt
--
—0. axisym,
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Cowling antidynamo theorem: detailed analysis
The right hand side of the Ohm's law is
-cvp x Bp = --Lvp x (VP xt) = ~ K • 0 VP +     („p . VP) • t.
=0
Combining the last two equations we have the Ohm's law as
„p.VP=^(divVP i«py
p        47rcr V P <9P/
In the neutral point 0 we have VP = 0 (dP/dR = 0), but div VP 7^ 0. This means that the above condition is fullfilled either just for a —>* 00 (for the frozen field) or in the vicinity of 0 we shall have vp —>► 00, which does not make a sense. Consequently, any axisymmetric magnetic field cannot be sustained against the Ohmic decay via axisymmetric flow of matter.
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Towards the working dynamo: model due to Babcock & Parker
The Cowling antidynamo theorem shows that to sustain a working dynamo, one needs to employ the toroidal field component. The dynamo model is based on oscillarory process:
• The process starts with a dipolar field. As a result of differential rotation, the magnetic field is wind up. Therefore, the differential rotation creates toroidal field from initial poloidal field.
• In a second step, the poloidal field should be recreated from initial toroidal one. This happends due to the magnetic buyonancy and convection, due to which the magnetic flux tubes move upward. The convective bubles expand, what creates the the Coriolis force, which winds us the toroidal field creating the poloidal field.
These steps lead to reversal of the magnetic poles. Therefore the real period of the dynamo is twice that given by these processes.
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Parker's dynamo: poloidal component of induction equation
We shall assume general magnetic field that contains both poloidal and toroidal components induced by stellar rotation
ep = rot (At),       Bt = B^t,       v=vp + QRt.
The general induction equation including the resistivity A dB
—— = rot (v x B) - A rot rot B = rot(v x B) - AAB.
The poloidal component has the form of
d (rot (At)) , , ,A.
v ^—— = rot (v x B) - A rot rot rot (At),
what gives for A after integration with Bp = rot (At) = — t x V/4
9^Qt^ = - v x (t x VA) - A rot rot (At). With rot rot (At) = V div (At) - A (At) = - A/l - /I//?2 we have
=0 (sym.)
dA ( 1
_ + „p.W\ = Af A--^ | A 16
Parker's dynamo: toroidal component of induction equation
The toroidal component of the induction equation is
dB<
dt
= rot (ft/? t xBp + i/pX (B^t)) -A(B<pt).
The first right hand term is after some manipulation
rot (ft/?t x ep) = rot (t x Bp) ft/? - (t x Bp) x V (ft/?) =
t div Bp +tBp • V (/?ft) - V (/?ft) • t Bp.
-'
=o =o After evaluating in components after some math the second term is rot (vp x tB^) = rot (vp x t) 6^ - (vp x t) x V6^ = -t/?div (i/p//?) Bcp-tvp-        = -t/?div(i/pe^//?).
As a result, the toroidal component of the induction equation is
d^l + tfdiv (±*vP) = RBP ■ Vft + a (a - ±\ e„.
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Dynamo equations
Let us summarize that we obtained the following system of equations:
A
^ + tfdiv (±13^ = RBP ■ Vft + A (a - ij) Bv
for toroidal and poloidal components of the magnetic field. The term RBP • Vft describes the generation of toroidal magnetic field from the poloidal field due to differential rotation (winding up of the field lines). This is the first step of the dynamo.
However, due to topological difference between these field components, there is no such term in the equation for the poloidal component. Therefore, there has to be a term propotional to      to close the dynamo loop. In such case the poloidal equation reads
— + vp-VA = aBa+\(A- — )A.
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The magic with a
The possibility how to create poloidal field from the toroidal field is the followind one. As a result of the convective motion, hot buble moves upward. The buble expands as it moves up. This leas the circular movement of the gas in the bubble due to to the action of the Coriolis force. The magnetic field follows the matter in the plume, creating the poloidal field from the toroidal one.
This does not violate the Cowling antidynamo theorem, because the motion due to the convection is not axisymmetric.
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Convenient model of dynamo
Let us for simplicity study the dynamo in cartesian coordinates with axis y in toroidal direction and axis z corresponding to the rotation. We shall impose the axial symmetry d/dz = 0 and the vector of the differential rotation velocity v = (0, v(z), 0). In such a case
dA OA
dz1   y' dx
fulfills divB = 0 constraint. The induction equations then read
d By fdA
~df ~   ~dx+ ^ ^
dt
where v' = dv/dz.
dA
= aBy + r)AA.
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Oscillatory dynamo
We will search the solution in the form of exp (/7cx + pt) with constants k and p. This yields the system of algebraic equations
pBy = ikvA - r]k2Byi
pA = aBy - r}k2A.
From the second equation we have By = A/a (p + 77/c2), which inserting into the first equation yields (p + 77/c2)2 = iav'/c, or
p + 77/c2 = (1 ± i)^\Ď\r]k2,      with D
av
2t?2/c3
and ± depending on the sign of D. The solution has the form of
A ~ e
Dl-1 ŕ
Therefore, for \D\ > 1 we obtain exponentially growing wave (with
amplitude constrained due to damping) with period 2tt/ yr]k2^/\D
D > 0 (y' > 0) it disseminates towards the poles, while for D < 0 (y' < 0) it disseminates towards the equator, explaining the cyclical movement of solar spots.
For
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Magneto-rotational instability
Magneto-rotational instability: principle
The magneto-rotational instability is a possible source of anomalous viscosity in accretion disks. Let us study the stability of the material in the disk imersed in magnetic field.
The magnetic field follows the density perturbation. For a strong field (/3 < 1): magnetic field returns the blob to its original position. On the other hand, for weak field (/3 > 1) the centrifugal force wins and the material is further accelerated leading to instability. Therefore, the magnetorotational isntability is a weak field instability.
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Magneto-rotational instability: perturbations
We will study the magneto-rotational instability (MRI) in so-called Boussinesq approximation assuming constant density and introducing just density variations in the buyonancy term. We will study the perturbations in the disk with radially variable angular velocity ft(P). We will assume that the stationary magnetic field is homogeneous and has nonzero component just in the direction perpendicular to the disk. Thus, the velocity in cylindrical coordinates is
v = (5vr, QR + 5v<p,5vz)
and the magnetic field
B = (5BR,5B^,BZ + 5BZ). We will search for the axisymmetric perturbations in the form of waves,
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MRI: continuity and induction equations
Withinh our assumptions, the continuity equation div v = 0 is
I<r5vr + kzSvz = 0.
The induction equation dB/dt = rot(v x 6) gives with for the R, z, and ip components (neglecting second order terms)
— iuj5Br = ikzBz5vR,
— iujSBz = —H<rBz5vr,
-iuSB^ = ikzBz5v^ + QRikzSBz + (nR)'6BR + QRikR5BR,
where we used the continuity equation in the z-component. The condition /?div56 = 0 gives ikRRSBR + 5Br + ikzRSBz = 0, which cancels several terms in the (^-component equation giving
-iuoSB^ = ikzBz5v^ + Q'RSBr.
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MRI: equation of motion
The perturbations in the equation of motion
_ + ,,.v, + -v(p+-J-*- — e.ve = o
give for the /?, z, and components
iu5vR - 205vv + ^5p-%% +^BZ5BZ - ^BZ5BR = 0.
p pz on    4ivp 4ivp
vl/r -^
^ Boussinesq
iu5vz + '— Sp    ^77- = 0, p        /r az
^2 r //C^
where the epicyclic frequency is
2 _ 1 d (OR2)2 K ~ /?3 d/?
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En route we find...
The strategy is to evaluate all perturbations in term of Svr and insert the terms into the /?-component of the momentum equation,
iujSvr - 2ildVp h--dp--^ — + -—BZSBZ - -—BZSBR = 0.
p pz on    4ivp 4ivp
From the induction equation we have
e n            Bz                             kf( Bz 5Br =-OVr,       ()Bz =-OVr
uj uj
and from the (^-component of equation of motion (inserting induction equation)
/                            n2         Q! R --(oj2 - k2zvlz) 5vv + — 5vR +-kzSBR = 0
uj z\l. uj
or ^
-- {uj2 - k2zvlz) Sv^ + £-5vR + ^-k2BzSvR = 0.
uj *        ' ■     2ft uj2 z
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Employing the Boussinesq approximation
Assuming the entropy conservation for gas with k = 5/3
Dln(pp-5/3) dln(pp-5/3)
Dt
dt
+ i/-Vln (pp~5/3) =0.
which within the Boussinesq approximation gives
. 5 5p        d\n(pp-s/3) d\n(pp-s/3)
ICJ---h ovz----h ovR-—-        = 0,
3 p oz oR
This gives from the z-component of the equation of motion
Sp uj
= —5v,
Sp dp
P
kzp2 dz
uj 3 1 dp
kz 5 kzp dz
d\n(pp 5/3) dIn (pp 5/3
Svz-^-J-+5vR —
dz
dR
Dispersion relation
Denoting Co2 = uj2 — k2v\z and inserting into the /?-component of the momentum equation we finally derive the dispersion relation
~4 ,
uj +
kl + kl
3 (kf>dp dp
bp\kzdz dRj\k
kRd\n (pp"5/3)   din (pp"5/3)
dz
OR
-k
uj -
4ft'
kl
k\ + k\
<z = o.
From the relation p = p(p) follows
dp d In (pp"5/3)     öp d In (pp-5/3)
9/?
dz
dz
9/?
Finally, we introduce pieces of Brunt-Vaisala frequency
N
r
3 dpdln(pp~5/3) 5p"dR OR
N:
3 dp din (pp~5/3) 5p dz dz
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Dispersion relation
We arrive at the dispersion relation
k2 + k2
k2 +
kR
NZ-NR
Co2 -AQ2k2zvlz = 0.
The discriminant of the dispersion relation is always positive, therefore always oj2 and also oj2 are real. Therefore, it is sufficient to investigate the instability around the root oj2 = 0, where the imaginary part of u appears. In this case the dispersion relation becomes in terms of kp>
kR{k2zvlz + Ni) - 2kRkzNRNz + k:
dft:
+ N<R + k2zv2z) =0.
din R
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Magneto-rotational instability
This equation is never fullfilled (and therefore uj2 never passes through 0) if the discriminant is negative, that is (in terms of k2)
For arbitrary kz, the condition of stability is fulfilled only if
dft2
dR
> 0.
However, this never happens in real disks where dfi/d/? < 0. Therefore, the instability, which is called the magneto-rotational instability, always appears in real disks (Fricke 1969, Balbus & Hawley 1991). The instability leads to turbulence, which is considered to be the main source of anomalous viscosity in disks.
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Magneto-rotational instability: conditions
Neglecting the Brunt-Vaisala frequencies, the instability appears (the previous condition is not fulfilled) for small wavenumbers
1/2
^ ^z,max
din/?
VAz
For larger wavenumbers the magnetic stresses are so large that they return the blob to its original position.
Denoting a typical vertical disk thickness as 23/2/-/ with H = aR/v^, where     is the Keplerian rotation velocity and a is the sound speed, only the modes with wavelength lower than 23^2H can exist in the disk, giving the condition for the lowest wavelength leading to instability as 2vr//cZ5max < 23/2/-/. Inserting the Alfven speed with the midplane density P0i the conditions for the development of instability gives
— ■ \ 1/2
2 avKM
7t VrR2
3 /2
where the disk mass- loss rate is M = (2tt)0/ vRp0RH. 31
Suggested reading
L. Mestel: Stellar magnetism
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