1992ApJ...395..250G
THE
AsTROPHYSICAL
JouRNAL, 395:250--258, 1992 August 10
©
1992.
The
American
Astronomical Society.
AH
rights
reserved.
Printed
in
U.S.A.
MAGNETIC
FIELD
DECAY
IN
ISOLATED
NEUTRON
STARS
PETER
GOLDREICH
AND
ANDREAS REISENEGGER
California
Institute
of
Technology,
Pasadena,
CA
91125
Received
1991
September 23; accepted 1992 February
14
ABSTRACT
We investigate three mechanisms
that
promote the loss
of
magnetic flux from
an
isolated neutron star.
Ohmic decay produces a diffusion
of
the magnetic field with respect to the charged particles.
It
proceeds
at
a
rate
that
is inversely proportional to the electric conductivity
and
independent of the magnetic field strength.
Ohmic decay occurs in
both
the fluid core and solid crust
of
a neutron star,
but
it
is
too
slow
to
directly affect
magnetic fields
of
stellar scale.
Ambipolar diffusion involves a drift of the magnetic field and charged particles relative to the neutrons. The
drift speed is proportional to the second power
of
the magnetic field strength if the protons form a normal
fluid. Variants
of
ambipolar diffusion include
both
the buoyant rise
and
the dragging by superfluid neutron
vortices of magnetic flux tubes. Ambipolar diffusion operates in the outer
part
of the fluid core where the
charged particle composition
is
homogeneous, protons
and
electrons being the only species. The charged par-
ticle flux associated with ambipolar diffusion decomposes into a solenoidal
and
an irrotational component.
Both components are opposed by frictional drag. The irrotational component perturbs the chemical equi-
librium between neutrons, protons, and electrons, thus generating pressure gradients
that
effectively choke it.
The solenoidal component
is
capable
of
transporting magnetic flux from the outer core
to
the crust
on
a short
time scale. Magnetic flux
that
threads the inner core, where the charged particle composition
is
inhomoge-
neous, would be permanently trapped unless particle interactions could rapidly smooth departures from
chemical equilibrium.
Magnetic fields undergo a
Hall drift related to the Hall component
of
the electric field. The drift speed is
proportional to the magnetic field strength. Hall drift occurs throughout a neutron star. Unlike ohmic decay
and ambipolar diffusion which are dissipative, Hall drift conserves magnetic energy. Thus, it cannot by itself
be responsible for magnetic field decay. However, it can enhance the rate of ohmic dissipation. In the solid
crust, only the electrons are mobile and the tangent
of
the Hall angle
is
large. There, the evolution
of
the
magnetic field resembles
that
of
vorticity in
an
incompressible fluid
at
large Reynolds number. This leads us to
speculate
that
the magnetic field undergoes a turbulent cascade terminated by ohmic dissipation
at
small
scales. The small-scale components
of
the magnetic field are also transported by Hall drift waves from the
inner crust where ohmic dissipation is slow to the outer crust where it
is
rapid. The diffusion
of
magnetic flux
through the crust takes - 5
x
10
8
/B
12
yr, where B
12
is the crustal magnetic field strength measured in units
of 10
12
G.
Subject headings: stars: magnetic - stars: neutron
1.
INTRODUCTION
Young neutron stars are seen as ordinary radio pulsars
and
X-ray pulsars. Their surface magnetic field strengths are
deduced to be
of
order
10
12
-10
13
G. Older neutron stars are
observed as recycled pulsars
and
low mass X-ray binaries.
Their surface fields are weaker,
;5
10
10
G. The association
of
weaker fields with older objects suggests
that
the magnetic
fields of neutron stars are subject to decay. Since the neutron
stars found in recycled pulsars
and
low-mass X-ray binaries
have accreted substantial amounts
of
matter, it
is
difficult to
resolve whether the decay results from age
or
accretion
(Bisnovatyi-Kogan & Kornberg 1975). Evidence favoring age
comes from some statistical studies
of
ordinary, single, radio
pulsars which conclude
that
the magnetic fields of these objects
decay
on
time scales
of
order
10
7
yr (Lyne, Manchester, &
Taylor 1985;
Narayan
& Ostriker
1990).
However, other
studies reach the opposite conclusion (Bhattacharya et al.
1992).
The detection in y-ray burst spectra
of
what appear to be
cyclotron lines formed in
10
12
-10
13
G fields (Murakami et al.
1988)
would provide evidence in favor of accretion should the
bursts emanate from old neutron stars (Shibazaki et al.
1989).
250
The purpose
of
this paper
is
to identify decay mechanisms
for the magnetic field
of
an
isolated neutron star
and
to esti-
mate their time scales. We
do
not
address questions related to
the origin of the field. We merely assume
that
the initial field
threads the interior
of
the
star
and
inquire as to how it would
evolve.
To
do so,
we
solve the equations
of
motion for charged
particles in the presence of a magnetic field
and
a fixed back-
ground
of
neutrons while allowing for the creation
and
destruction
of
particles by weak interactions. Strictly speaking,
these equations apply to normal neutrons
and
protons.
However,
we
extend
our
interpretations of their solutions to
cover cases
of
neutron superfluidity
and
proton
superconduc-
tivity.
The organization
of
the paper
is
set out below. We present
continuity equations
and
equations
of
motion for the protons
and electrons in
§
2.
These equations are manipulated to prove
that, in the presence
of
a magnetic force, the charged particles
cannot be simultaneously in magnetostatic equilibrium and in
chemical equilibrium with the neutrons. In
§
3,
the equations
are solved and two mechanisms for the decay
of
the magnetic
energy are identified,
Ohmic dissipation and ambipolar diffusion.
Speculations concerning turbulent field evolution by Hall drift
©American Astronomical Society • Provided by the NASA Astrophysics Data System
1992ApJ...395..250G
MAGNETIC
FIELD
DECAY
IN
ISOLATED
NEUTRON
STARS
251
are offered in §
4.
Finally, § 5 contains a discussion
of
the
application
of
our
results to real neutron stars.
Each of the three mechanisms
we
investigate, ohmic decay,
am bipolar diffusion,
and
Hall drift,
has
already received atten-
tion in relation
to
neutron
star
magnetic fields. Baym, Pethick,
& Pines (1969b) were the first to properly calculate the ohmic
decay time in the fluid core under the assumption
that
the
neutrons
and
protons were normal (not superfluid
and
superconducting). Ewart, Guyer, & Greenstein (1975)
and
Sang &
Chanmugam
(1987) estimated the ohmic decay
of
fields
supported by currents in the solid crust. The ambipolar diffu-
sion time scale for normal neutrons
and
protons was evaluated
by Haensel, Urpin, & Yakovlev (1990), although these authors
mistakenly attributed it
to
enhanced ohmic decay (Pethick
1991).
Harrison (1991) properly appreciated the connection
between am bipolar diffusion
and
the buoyant rise in flux tubes.
Hall drift was
part
of
the picture
of
the thermoelectric gener-
ation
of
magnetic fields detailed by Blandford, Applegate, &
Hernquist (1983). Jones (1988) proposed
that
Hall drift could
transport magnetic flux across neutron star crusts. Relations
between
our
results and those obtained in earlier papers are
mentioned
in§
5.
2. EQUATIONS
OF
MOTION
FOR
THE
CHARGED
PARTICLES
We model the interior
of
a neutron star as a lightly ionized
plasma consisting
of
neutrons, protons, and electrons labeled
by the indices
n,
p,
e.
The equation
of
state for each particle
species is taken to be
that
of
an
ideal, completely degenerate,
gas. Modifications associated with the presence
of
other parti-
cle species
and
the strong interactions are discussed in
§§
3.5
and
5.2.
We neglect thermal contributions to the Brunt-
ViiisiiHi
frequency
on
the grounds
that
the thermal conductivity
of
neutron star interiors is so high that they are unimportant for
the slow motions of interest here.
We specify the local state
of
each species by its internal
chemical potential,
J-1;,
which
is
equal to the Fermi energy
including rest mass. The protons
and
electrons are described as
two separate fluids coupled by electromagnetic forces.
Drag
forces due
to
elastic binary collisions impede the relative
motions of the different particle species. Weak interactions
tend to erase pertubations away from chemical equilibrium
among the neutrons, protons, and electrons.
The neutrons are assumed to form a fixed background in
diffusive equilibrium. This assumption, while
not
entirely rea-
listic, simplifies the algebra
and
does
not
lead us astray. Its
justification
is
that
the combined fluid
of
neutrons, protons
and electrons
IS stably stratified (Reisenegger & Goldreich
1992).
The stratification
is
associated with the chemical com-
position gradient; the equilibrium ratio
of
the number densities
of
charged particles
to
neutrons increases with depth. The ratio
of
the magnetic field stress to the pressure
of
the charged par-
ticles is small. Thus, the magnetic field cannot force significant
displacements
of
the combined fluid,
at
least
not
ones in which
the composition is frozen. We show
in§
5
that
the interactions
which smooth perturbations of chemical equilibrium are so
slow
that
these are the only displacements
of
practical interest.
The density profile
of
the neutrons, as determined by
f.-In
+
mnl/1
= constant ,
(1)
gives rise to a Newtonian gravitational potential,
1/1;
contribu-
tions to
1/1
by protons and electrons are neglected, as are cor-
rections due to general relativity.
The charged particles satisfy the equations
of
motion:
~
(
~
)
m*
- +
m*(v
· V)v = -
V"
- e E + - x B
e
ot e e e
re
c
m:
ve
m:(v. -
vp)
(3)
'ten
rep
Here,
m:
=
JJ..fc
2
is
the effective inertia of the electrons, E
and
B are the electric and magnetic fields, V; is the mean velocity
of
the particles
of
species i,
and
T:;i
is
the relaxation time for
collisions of particles
of
species i against particles
of
species
j.
The average velocity
of
the neutrons is assumed to vanish,
vn
=
0.
Conservation
of
momentum implies
that
mP/7:
pe
=
m:fT:w
We ignore relativistic corrections to
both
the inertia
of
the neutrons
and
protons and to the gravitational forces acting
upon them.
To
be consistent,
we
also
drop
the gravitational
force acting
on
the electrons
and
take the neutron and
proton
masses to be equal. Without the essential additions
of
the
forces due to pressure and gravity,
our
equations
of
motion
would yield
an
electrical conductivity tensor similar to
that
applied by Haensel, Urpin, & Yakovlev (1990).
The processes under consideration involve small velocities
that
change over time scales much longer than any of the
relaxation times. Thus,
we
neglect the acceleration terms
on
the left-hand sides of equations
(2)
and
(3).
Then, combining
equations
(1),
(2),
and
(3),
we
arrive
at
fB
(A
)
mpvp
m:v.
_
(m"
m:)
--V
!J.f.-1
=
+--=
---"-+-
v,
nc
T:pn
T:en
T:pn
T:en
(4)
where
L\p
=
f.J.p
+
f.-le
-
f.-In
is
the departure from chemical equi-
librium,
nc
~
nP
~
n. is the number density
of
charged par-
ticles,JB
is
the magnetic force density,
I"
_j
X B
JB-
'
c
(5)
with the electric current,j, given by
j = enc(vp- v.) .
(6)
Each of the terms in equation
(4)
admits a simple interpreta-
tion. Clearly,
fBinc
is
the magnetic force per proton-electron
pair.
From
the thermodynamic identity (opfop)r = 1/n, it
follows that -
V(L\p)
is
the net of the forces due
to
particle
pressure plus gravity acting
on
a proton-electron pair. Equa-
tion
(4)
shows
that
magnetostatic equilibrium requiresfB/nc to
be the gradient
of
a potential. Only in this special circumstance
can the gradient
of
the perturbed chemical potential balance
the magnetic force density.
If
magnetostatic equilibrium does
not
apply, the forces drive the charged particles through the
fixed background
of
neutrons
at
the ambipolar diffusion
veloc-
ity,
v,
defined by the second equality in equation
(4).
Weak interactions tend to erase chemical potential differ-
ences between the charged particles and neutrons. The differ-
ence between the rates, per unit volume,
at
which the reactions
p +
e-
-+
n + v e and n
-+
p +
e-
+ v e occur
is
L\r
=
r(p
+ e -
-+
n + v .) -
r(
n-+
p + e - + v
.)
=
A.L\p
,
(7)
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1992ApJ...395..250G
252
GOLDREICH
&
REISENEGGER
Vol. 395
where the coefficient
A is a temperature-dependent proportion-
ality constant
in
the limit Ajj
~
kB
T.
The protons and electrons each satisfy a continuity equation
an.
-'
+ V •
(n;v;)
=
-AAJ1.
at
(8)
Approximate charge neutrality implies
nP
~
ne
=
ne
from
which it follows
that
one
+ V •
(n
w)
=
-AI!"
at
e
r'
(9)
where
w =
vP
+
Ve
= v _
(mp/'r:pn
- m:lr:e•) _j_ . (
1
0)
2
mp/r:pn
+
me/r:en
2nee
Since the Eulerian variations
of
ne
are of order
ncB
2
/Pe
~
1,
where
Pe
is
the electron pressure, equation
(9)
simplifies to
(11)
3. OHMIC DISSIPATION
AND
AMBIPOLAR DIFFUSION
In
this section
we
study the dissipation
of
magnetic energy in
a fluid mixture
of
neutrons, protons, and electrons
that
is
close
to
both
magnetostatic and chemical equilibrium.
To
avoid the
proliferation
of
inessential terms,
we
neglect gravity and treat
m:, r
P•'
r:e••
and
A as constants throughout most of the section.
Moreover,
we
assume
that
the magnetic field is spatially
bounded
and
that
the fluid medium is
of
infinite extent.
In
the
final subsection,
§
3.5,
we
consider extensions and refinements
of
our
results
to
inhomogeneous, gravitating media.
3.1. Magnetic Field Evolution
The evolution
of
the magnetic field is related to the electric
field,
E,
by Faraday's induction law,
aB
at=
-cV
x
E.
(12)
The electric field, obtained from a suitable combination
of
equations
(2)
and
(3)
without the inertial terms, reads:
E = j_ _
'!_
x B + (mp/r:P•- m:/r:••) j X B
u
0
c
mp/r:pn
+
m:fr:en
neec
(r
p./mp)V
Jlp-
(r:eJm:)v
Jle
+
e(r:p./mp
+ r:eJm:) '
(13)
where
2 ( 1 1
)-l
Uo
=
nee
--*
+ *
'T:ep/me
!p./mp +
'T:enlme
(14)
is the electrical conductivity
in
the absence
of
a magnetic field.
Substituting equation
(13)
into equation
(12),
we
obtain the
governing equation for the magnetic field,
- =
-cV
X
.1_
+ V X
(v
X
B)
aB
(
·)
at
Uo
(15)
where j
is
related
to
B by Ampere's law,
.
cV
x B
]=~·
(16)
The terms on the right-hand side
of
equation
(15)
describe, in
order, the effects
of
ohmic decay, am bipolar diffusion,
and
Hall
drift. Since
j
and
v are linear
and
quadratic functionals
of
B,
these terms scale as B, B
3
,
and B
2
,
respectively.
3.2. Dissipation
of
Magnetic Energy
The total magnetic energy is given by
EB=__!_fd
3
xiBI
2
•
(17)
8n
We write its time derivative, with the aid of equation
(12)
and
after
an
integration by parts,
in
the form
dEB
1 f
3
•
-;u=-
4
n d
xrE.
(18)
Neither the Hall term
nor
the potential term in the electric field
contribute
to
dEB/dt.
The former is orthogonal to j and the
latter is eliminated by the use
of
Ampere's law in the derivation
of
equation
(18).
Thus,
dEB
(dEB) (dEB)
dt
=
dt
ohmic
+
dt
ambip
• (
19
)
The contribution from ohmic dissipation reads
(
dEB)
= _
__!_
f d
3
x I
jl
2
•
dt
ohmic
4n
(J
o
(20)
The
am
bipolar term
is
given by
(
dEB)
1 f
3
-
=--
dxv·fB
dt
ambip
4n
=-
fd
3
xne
(~+
m:)lvl
2
rpn
ren
- f d
3
xne
v ·
V{Ajj)
,
(21)
where
we
arrive
at
the second expression by using equation
(4)
to eliminate
fB
in favor of v
and
Ajj. Another integration by
parts, together with equation
(11),
yields
(
ddEB)
. = -
fd
3
x[ne(~
+
m:)
I v 1
2
+
A(A}J)
2
].
(22)
t
amb•p
r
pn
!en
The first piece in the integrand arises from energy lost to fric-
tional drag. The second piece accounts for the energy carried
away by the neutrinos and anti-neutrinos
that
are emitted
during the inverse
and
direct beta decays
that
smooth depar-
tures from chemical equilibrium.
As
is evident from equations
(20)
and
(22),
ohmic dissipation
and
ambipolar diffusion always act to decrease the magnetic
energy.
3.3. Ambipolar Drift Velocity
To
relate the chemical potential imbalance, l!jj,
and
the drift
velocity,
v,
to the magnetic force,JB,
we
start from equations
(4)
and
(11).
It
is
convenient to resolve v and
fB
into solenoidal
(divergence-free) and irrotational (curl-free) components,
v•
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1992
MAGNETIC
FIELD
DECAY
IN
ISOLATED
NEUTRON
STARS
253
and f'B, and
vir
and
/~.
1
Because
V(~tt)
is
irrotational, the
solenoidal and irrotational components of equation
(4)
can be
written as
s f'B
V=(l
*I'
ncmplrpn+m.
r •• )
(23)
ir
f~
-
nc
V(~tt)
v -
(24)
-
nc(mp/rpn
+
m:/r
••
) .
Note
that
v"
is
directly proportional to the local value of f'B
with a coefficient that
is
inversely proportional to the frictional
coupling between the charged particles and neutrons. Because
vir
perturbs the chemical equilibrium between the neutrons and
charged particles, its response to
f~
is more complicated. The
details are worked
out
below.
Since the fractional variations
of
nc
are of order B
2
/p.
~
1,
equation
(11)
simplifies further to
(25)
Taking the divergence of equation
(24)
and using equation
(25)
to eliminate V ·
vir,
we
obtain
(26)
where the length scale a satisfies
[
A
(m
m*)]-
1
/2
a=
-
___!!+_e
nc
rpn
r
••
(27)
The solution of equation
(26)
is
conveniently expressed in
terms of the Green's function
as
G(x _ x') = _ exp
(-
I x -
x'
1/a)
4nlx-
x'l
(28)
~tt(x)
=
J..
f d
3
x'G(x-
x')V' ·
f~(x')
.
(29)
nc
Next,
we
relate
vir
to
f~
by substituting equation
(29)
into
equation
(24)
and performing an integration by parts:
vir(x)
=
An~
2
~~(x)-
f d
3
x'G(x-
x')V'[V'
·f~(x')]
J.
(30)
Let us denote by L the characteristic length scale over which
f~
varies. The response of
vir
to
f~
depends upon the relative
sizes of
L and
a.
For
L/a
~
1 the second term in equation
(30)
is
smaller than
the first by a factor of order
(a/L)
2
~
1,
and
.
Aa2
.
~~
v"
~
-
2
/';
=
(31)
nc
nc(mp/rpn
+
m:fr
••
).
In
this limit chemical equilibrium is achieved so rapidly that
only the frictional drag exerted by the neutrons on the charged
particles
is
available to balance the magnetic force.
In
the opposite limit,
Lfa
~
1,
the relation between
vir
and
f~
is
nonlocal, and therefore more complicated.
It
is
best
revealed in Fourier space, since the Fourier components of the
irrotational parts of vector fields are parallel to
k.
Taking the
1
This decomposition is unique since the fields are spatially bounded.
Fourier transforms
of
equations
(25)
and
(26)
yields
k .
Vir(k)
=
Aa2
k ·Jir(k)
,....,
AI} k •fir(k)
(32)
n;(l
+
k2a2)
B
,....,
n; B '
for L =
k-
1
~a.
For
L/a
~
1,/~
is
balanced by the pressure
gradient, leaving only f'B to be balanced by frictional drag.
3.4. Decay Time Scales
Here,
we
collect formulae giving the characteristic time
scales over which ohmic decay and ambipolar diffusion dissi-
pate magnetic energy. We reserve until§ 5 the numerical evalu-
ation
of
these time scales under different hypotheses
concerning the state of matter in neutron star interiors.
The time scale for ohmic decay, which follows immediately
from equations
(15)
and
(16),
has the familiar form
,....,
4nu
0
13
tohmic
2 ·
c
(33)
Ohmic decay involves a diffusion of the magnetic field lines
with respect to the charged particles. Note that
tohmic
is
pro-
portional to
J3
and independent of the field strength.
There are two time scales for am bipolar diffusion, one for the
solenoidal component of the charged particle flux and the
other for the irrotational component. Following equations
(23)
and
(32),
we
find
s .
,....,
~
,....,
4nnc
J3
(!:!!:£
m:)
tambtp
s
B2
+ '
v
rpn
r
••
(34)
(35)
Am
bipolar diffusion involves the motion of the magnetic field
lines together with the charged particles relative to the neu-
trons. Note that both expressions for
tambip
are inversely pro-
portional to
B
2
• Also, for
L/a
~
1,
fambip
~
(L/a)
2
t:mbip·
We show
in§
5.2
that
t~mbip
is
larger than the Hubble time.
However, if it were not,
we
would be compelled to consider
displacements of the combined fluid of neutrons and charged
particles. This is because magnetic forces would drive a sole-
noidal flux of baryons (neutrons plus protons) if particle inter-
actions could maintain chemical equilibrium. This solenoidal
motion of the combined fluid would
not
suffer the frictional
retardation that the solenoidal component of the charged par-
ticle fluid does.
It
would only have the milder effects of vis-
cosity to contend with.
3.5. Extensions and Refinements
It
is
easy to extend most of the results obtained in this
section so that they apply
to
inhomogeneous media in gravita-
tional fields.
The expressions for the dissipation of magnetic energy by
ohmic decay and ambipolar diffusion given by equations
(20)
and
(22)
are unchanged in an inhomogeneous medium.
However, the derivation of
(dEBfdt)ambip
is complicated by
the
spatial variations
ofm:,
rP"' r
••
, and
A.
We leave the proofs to
the reader.
The flow of charged particles in a homogeneous medium
tends to upset chemical equilibrium ifV •
(nc
v)
#
0.
This gener-
alizes in an inhomogeneous medium to V ·
(nc
w)
# 0
(see
eq.
[9]).
It
is useful
to
resolve the charged particle flux
nc
w into its
solenoidal and irrotational components.
If
beta reactions do
©
American
Astronomical Society •
Provided
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1992ApJ...395..250G
254
GOLDREICH
&
REISENEGGER
Vol. 395
not
erase perturbations from chemical equilibrium, the irrota-
tional component
is
choked by pressure gradients. We note
that
w differs from the ambipolar diffusion velocity v by a term
proportional
to
the current density
j.
Since V · j = 0 as a conse-
quence
of
charge neutrality,
V
·(new)=
V •
(nev)
_j_
· V
(mrfrpn-
m:/ren)
.
(36)
2e
mp/r:pn
+
mefr:en
The difference between V ·(new) and V •
(ne
v)
vanishes
in
either the limit
mP
r:en
~
m:
r:pn
or
the limit
m:
r:pn
~
mP
r:en·
The
first limit would be relevant if the protons were normal since
that
would imply
reJrpn
~
1 because neutron-proton scat-
terings are mediated by the strong force, whereas neutron-
electron scatterings are due to electromagnetic interactions
involving the neutron's magnetic moment. The consequences
of
proton
superconductivity are less clear. However,
we
shall
assume
that
V
·(new)~
V ·
(nev)
wherever ambipolar diffusion
might be
important
inside neutron stars. Thus,
we
write
(37)
from here on.
Am bipolar diffusion
in
a homogeneous medium is driven by
unbalanced magnetic stresses.
In
an
inhomogeneous medium
subject to a gravitational field buoyancy forces also play a role
(Parker
1979).
To
estimate the buoyancy forces, consider a
thin, circular, magnetic flux tube
of
outer radius r
that
sur-
rounds the center
of
a spherical star. The pressure
of
the
charged particles,
Pe•
mostly due
to
electrons, is lower inside
the tube
than
outside by
<>pe
~
- B
2
/(8n). The density deficit
inside the tube is
()pjp
~
-3B
2
/(32npJ. Thus, the buoyancy
force density
is
given by
3B
2
p 3B
2
A
/buoy
~
-
32npe
g
~
32nH r '
(38)
where r
is
the radial unit vector,
and
H is the pressure scale
height
of
the charged particle fluid.
It
is easy
to
show that the
magnitude
of
fbuey
exceeds
that
of
the inward directed force
density due
to
magnetic tension provided H < 3r/4. The buoy-
ancy force density is
to
be
compared
to
B
2
/(8nL), the charac-
teristic magnitude
of
the force density associated with a
magnetic field
of
scale
L.
Since L
;S
H in the fluid core
of
a
neutron star, the addition
of
buoyancy forces does
not
alter the
time scales for ambipolar diffusion given by equations
(34)
and
(35).
Our
treatment
of
ambipolar diffusion
is
predicated on the
assumption
that
the charged particle ftuid
is
homoge:neous;
more specifically,
that
it
is
composed
of
equal number densities
of
protons
and
electrons. This crucial assumption insures
that
the charged particle fluid is neutrally stratified. The solenoidal
component
of
the charged particle flux does
not
perturb the
density and pressure
of
a homogeneous fluid. However, it
is
likely
that
additional species
of
charged particles
appear
in the
equilibrium composition
at
pressures below the central pres-
sure
of
a neutron star. We refer to this region, where there
is
a
gradient in the charged particle composition, as the inner core.
Unfortunately, the size
and
composition
of
the inner core are
uncertain. However, it
is
clear
that
the charged particle fluid
in
the inner core is stably stratified. This has serious implications
for
am
bipolar diffusion. Displacements of the charged particle
fluid
at
frozen composition would raise the potential energy.
Unless particle interactions could rapidly erase perturbations
from chemical equilibrium, ambipolar diffusion could
not
occur
in
the inner core.
4.
HALL
DRIFT
AND
MAGNETIC
TURBULENCE
In
this section
we
examine the third term in equation
(15),
the one
that
describes advection of the field by Hall drift. This
term does
not
change the total magnetic energy. However, it
cannot be ignored
in
neutron star interiors because, in places,
its magnitude exceeds
that
of
the terms which account for
ambipolar diffusion
and
ohmic decay. We begin by describing
Hall drift waves. Then,
we
go
on
to consider the possibility
that
the magnetic field in the crust evolves through a turbulent
cascade.
We simplify the induction equation
(15)
by taking the limit
7:pn--+
0 and
'ren--+
oo.
With the protons immobilized, the elec-
trons carry all the current
and
ambipolar diffusion
is
elimi-
nated. The medium resembles a metallic solid. Then, the
reduced version
of
equation
(15)
reads
oB
c c
2
- = -
--
V X [(V X
B)
X B] + - V
2
B . (39)
ot
4nnee 4nuo
Application
of
dimensional analysis to equation (39) yields a
relation between the linear size,
L,
and
characteristic evolution
time scale,
tHan•
of
field structures:
4nne
ei3
tHan=
cB
(40)
Jones (1988) proposed
that
Hall drift could transport mag-
netic field from the inner crust where ohmic decay is slow to
the outer crust where it proceeds rapidly. Here
we
show
that
there is a class
of
Hall drift waves
that
carry magnetic energy
and
whose dispersion relation is closely related
to
equation
(40).
To
obtain the dispersion relation for linear waves in a
uniform magnetic field
B
0
,
we substitute the elementary dis-
turbance
B
1
= 1
1
exp i(k · x - wt) into equation (39). After a
little algebra,
we
obtain
ckik
• B
0
l
w=
4nne
e
(41)
where k = I k
1-
Fork
• B
0
~
0,
the corresponding group veloc-
ity
is
v = +
_ck--=[6_
0
"-+____,_(k_·
JJ~
0
~)k~A]
gp
-
4nne
e '
(42)
where k = k/k.
There is reason to
doubt
whether these waves could trans-
port
magnetic energy from the inner to the outer crust.
In
particular, they might be reflected as they propagate upward
toward lower density.
To
expose the problem,
we
interpret
equation
(41)
as a WK.BJ dispersion relation. Consider a plane-
parallel model for the crust with
ne
decreasing monotonically
in the z-direction.
The
validity
of
the WKBJ approximation
requires
kz
H
~
1,
where H
is
the local scale height. Let us
assume
that
a wave packet which satisfies this inequality
is
launched upward from the lower crust.
For
the moment,
we
focus
on
the special case with B
0
constant
and
aligned along
the x-axis. As the wave packet propagates toward lower
density,
k must decrease in direct proportion to
ne,
since w
remains constant. Because
of
the symmetry
of
the problem, the
decrease of
k comes entirely
at
the expense
of
kz.
Since H also
decreases with height, the inequality
kzH
~
1 must eventually
be violated.
It
is plausible
that
the wave packet would be reflec-
ted downward
at
about
the level where
kz
H -
1.
Although the
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Astronomical Society •
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