358

Astrophysics, Vol. 45, No. 3, 2002

0571-7256/02/4503-0358$27.00 ©2002 Plenum Publishing Corporation

STRUCTURE OF THE MAGNETIC FIELD OF A NEUTRON STAR

D. M. Sedrakian and K. M. Shahabasyan UDC: 524.354.6

The magnetic field distribution in the superfluid, spherical, hadronic core of a rotating neutron star, whichconsists of vortex and vortex-free zones, is investigated. Due to the effect of “entrainment” of superconduct-ing protons by rotating superfluid neutrons, a nonuniform magnetic field, the average value of which isconstant, is formed in the vortex zone of the neutron star, directed parallel to the star’s axis of rotation. It isshown that at the stellar surface, near the equatorial plane, there is a vortex-free zone of macroscopic size inwhich there is no magnetic field. The magnetic field near the boundaries of the vortex-free zone falls offexponentially with depth into the interior of this zone. This result essentially alters earlier concepts about themagnetic field distribution in the superfluid hadronic core of a neutron star. Outside the hadronic core themagnetic field has a dipole character with a magnetic moment on the order of 1030 g×cm3.

Key words: stars: neutron – stars: magnetic field.

1. Introduction

In model neutron stars with a hard equation of state, when the central mass density is higher than the nuclear

density, ρc = 2⋅1014 g/cm3, the core of the neutron star consists mainly of superfluid neutrons, superconducting protons,

and normal electrons [1]. The radius of this core is on the order of 10 km. The star’s envelope consists of fully ionized

plasma, in which the atomic number decreases with decreasing mass density. The average size of the envelope is about

a kilometer. In [2] we showed that the “entrainment” of the superconducting proton condensate by the superfluid neutron

condensate in the core of the neutron star results in the generation of neutron vortices with a certain magnetic induction

flux, contained in the ordinary proton vortices.

Whereas the neutron vortices appear due to the rotation of the neutron star, the proton vortices, as shown in [3,

4], may appear due to the magnetic fields produced by currents of “entrainment” of neutron vortices. The average field

generated in a neutron vortex does not depend on the star’s angular velocity and is fully determined by the microscopic

parameters of the superconducting proton fluid and the coefficient of entrainment k of the proton fluid by neutrons,

. 4

2

20

λξ

πλ

Φ=

kB (1)

Translated from Astrofizika, Vol. 435-442, July-September, 2002. Original article submitted March 29, 2002.

Yerevan State University, Armenia, e-mail: dsedrak@www.physdep.r.am.

359

This is actually an average field of the network of proton vortices which appears inside a neutron vortex. If we

assume, following [5], that the core of a neutron star contains a network of “mixed” neutron vortices, then to find the

magnetic field inside a neutron star we must determine the region of the star in which we can introduce the concept of

a continuous density of neutron vortices. For this the distance between neutron vortices must be far less than the

characteristic dimensions of their arrangement.

There are three characteristic distances in a rotating star: the star’s size R, the size ∆R of the vortex-free zone, and

the size b of a neutron vortex. The latter two are determined by the star’s angular velocity Ωr

and the average magnetic

induction of the core of the neutron star. As shown in [6], the entire region of the core is divided into two parts: the

inner region, where for b << R there is a well-developed structure of neutron vortices with a density nn >> 1, and an outer

region with a size ∆R >> b, called the vortex-free zone, with a density nn = 0. We should therefore expect that the average

magnetic induction in the vortex zone should be constant and equal to B , while in the vortex-free zone the magnetic

field should be absent owing to blocking of the propagation of the average magnetic induction into this zone by Meissner

currents of superfluid protons. To confirm this result, we must find the magnetic field distribution within the core of a

neutron star in both the vortex and the vortex-free zones.

The purpose of the present work is to solve the Ginzburg–Landau equation written for a proton superfluid and

determine the magnetic field distribution in the aforementioned zones with allowance for the actual shape of the core

of a neutron star, i.e., its sphericity.

2. Ginzburg–Landau Equation for the Core of a Neutron Star

The Ginzburg–Landau equation for superfluid protons in the core of a neutron star has the form [7]

( ), 41rotrot **2

Ψ∇Ψ−Ψ∇Ψπ=λ

+mi

ceAA h

rr

(2)

where e and m are the charge and mass, respectively, of a Cooper pair of protons while

21

2

2i

04

and

π=λΨ=Ψ φ

sne

mce (3)

are the order parameter and the London depth of penetration of the magnetic field into the superconducting proton

condensate. The density of the proton condensate is ns = |Ψ

0|2 while f is the phase of the order parameter. Substituting

(3) into (2), we obtain

,1rotrot2

fAArrr

=λ

+ (4)

where

.12 2

φ∇λ

=ecf hr

(5)

The circulation of the vector fr

along the contour L, which contains only one proton vortex, is

, 12 2

02 ∫∫ λ

Φ=φ∇

λ=

LL

ldecldf

rh

rr

(6)

where echπ=Φ 0 is the quantum flux of the magnetic field of one proton vortex. Using (6), from Eq. (4) we obtain

the London equation with the right side for the magnetic field induction,

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, rotrot 02

zpenBBrrr

Φ=+λ (7)

where np is the density of proton vortices and zer

is the unit vector of the z axis. The distribution of this density in a

neutron star is nonuniform. The proton vortices are concentrated mainly near the center of a neutron vortex at a distance

r ≤ 0.1b. However, the average density of proton vortices and hence the average magnetic field within one neutron vortex

do not depend on the size of the vortex and are determined by Eq. (1). Since the size of the core of a neutron star is

far greater than the distances between neutron and proton vortices, a good approximation for finding the magnetic field

distribution will be the assumption that the average density pn of proton vortices is independent of the coordinate, i.e.,

const=pn . This means that we have

. 0rotrotandrot 0 =Φ= fenf zp

rrr(8)

We assume that the neutron star is spherical and we direct the star’s axis of rotation along the z axis. We seek

the solution of the system of equations

,11rotrot22

fAArrr

λ=

λ+ (9)

.0rotrot =fr

(10)

in the spherical coordinates r, ϑ, ϕ. We introduce the auxiliary vector Mr

in accordance with the formula

.rotrot MArr

= (11)

With allowance for (10), Eq. (9) then takes the form

.01rotrot2

=λ

+ MMrr

(12)

Instead of solving the system of Eqs. (9) and (10), we can now seek the solution of the system of Eqs. (11) and (12). Since

the magnetic fields being sought do not depend on the ϕ coordinate, the only nonzero component of the vector Mr

will

be Mϕ, which can be sought in the form

( ) ( ) ,sin , ϑ=ϑ ϕϕ rMrM (13)

where the function Mϕ(r) satisfies the equation

( )( ) ( ) .0121222

2=

λ+− ϕϕ rM

rrrM

dr

dr (14)

The general solution of this equation, satisfying the condition Mϕ(r) → 0 as r → 0, is written in the form

( ) . chsh 2

λλ−

λ=ϕ

rrr

r

CrM (15)

Let us turn to the solution of Eq. (11). It is easy to see that the solution (13) for Mϕ(r, ϑ) with allowance for (15) is a

general solution of Eq. (11) if we take Mϕ(r, ϑ) = Aϕ(r, ϑ). On the other hand, a particular solution of Eq. (11) will be

( ) . sin, 0 ϑ=ϑϕ rcrA (16)

Consequently, the general solution of the inhomogeneous equation (11) can be written in the form

( ) ( )( ) . sin, 0 ϑ+=ϑ ϕϕ rcrMrA (17)

If we introduce the average magnetic field B = Φ0n

p produced by proton vortices, then in accordance with (7) and (16),

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the constant c0 will be B/2. The final expression for the vector potential A

r in the vortex zone of a neutron star will

therefore be

( ) ( ) . sin2

, ϑ

+=ϑ ϕϕ rBrMrA (18)

The constant C appearing in Eq. (15) for Mϕ(r) must be determined from the conditions of continuity of the magnetic

field components at the boundary of the core of a neutron star.

3. Magnetic Field in the Vortex Zone of a Neutron Star

Let us proceed to calculate the magnetic fields in two zones of a neutron star. As shown in [6], The vortex zone

occupies almost the entire volume of the hadronic core of a neutron star.

The vortex-free zone lies near the equatorial plane and enters the core of the neutron star at a distance ∆R from

the boundary of that core. The depth ∆R of the vortex-free zone is far less than the radius a of the core of the neutron

star. For the standard model of a neutron star with a hard equation of state with a ≈ 10 km, ∆R is only a few meters.

In finding the magnetic fields in the vortex zone, therefore, we can neglect the size of the vortex-free zone.

So to find the fields in the vortex zone, we assume the volume of that zone to be the entire volume of the core

of the neutron star. Then the constants appearing in the solution given above should be determined from the condition

of continuity of the magnetic field components at the surface of the core, i.e., at r = a.

To write these conditions, we first find the magnetic field in the vortex zone. The magnetic field components

are expressed in terms of the vector potential as follows:

( ), sinsin1 ϑ

ϑ∂∂

ϑ= ϕA

rBr (19)

( ). 1 rArr

B ϕϑ ∂∂−= (20)

Substituting Aϕ(r, ϑ) from Eq. (18) into (19) and (20), we obtain

( )

( )( ) .sin1

, cos2

ϑ

+∂∂−=

ϑ

+=

ϕϑ

ϕ

BrrMrr

B

Br

rMB

i

ir

(21)

Substituting the expression for Mϕ(r) from (15) into (21), we finally have

, sin chsh sh

, cos chsh 2

2

2

2

3

ϑ

+

λλ

−λ

λ−λλ

−=

ϑ

+

λλ−

λ=

ϑ Brrr

r

r

r

CB

Brrr

r

CB

i

ir

(22)

The magnetic field outside the core of a neutron star is a dipole field and its components have the form

, sin

, cos 2

3

3

ϑΜ=

ϑΜ=

ϑr

B

rB

e

er

(23)

362

where M is the total magnetic moment of the neutron star. The condition of continuity of the magnetic field components

at r = a determines the constants C and M in Eqs. (22) and (23) to be

( )

. 3cth312

, sh23

23

2

λ−λ

λ+=

λλ−=

aa

aBaM

aaC

(24)

Substituting (24) into (22), we obtain the following final expressions for the magnetic field components in the vortex

zone:

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )[ ]( ) ,sin

sh

chshsh

231

, cos sh

chsh 31

2

2

ϑ

λλλ−λλ−λ

−−=

ϑ

λλλ−λ

λ

−=

ϑ a

rrrrr

raBB

a

rrr

rraBB

i

ir

(25)

The field in outer space is described by Eqs. (23), while the star’s total magnetic moment M is given by Eq. (24).

For realistic models of neutron stars, we have a ≈ 10 km and λ ≈ 10–10 cm, so that l/a is far smaller than unity.

As seen from the solutions obtained, in the approximation of λ/a → 0 the magnetic field within the vortex zone is

constant, directed along the z axis of rotation of the star, and equal to B. This is consistent with the fact that the star’s

total magnetic moment in this approximation [see Eq. (24)] is M = Ba3/2 = (3B/8π)V, where V is the volume of the neutron

star’s vortex zone and 3B/8π is the magnetic polarization vector.

4. Magnetic Field in the Vortex-Free Zone

As noted above, the vortex-free zone lies near the equatorial plane and, in a fairly good approximation, consists

of the space between two cylinders, where the radius of the outer cylinder is a and the length of a cylinder is Ral ∆= ,

iRaR −=∆ .

Here Ri is the radius of the radius of the inner cylinder [6]. The angle at which the vortex-free zone is seen from

the center of the star is on the order of

.10-3≈∆=≈ϑ∆ aRal (26)

In the region of the vortex-free zone we therefore have ϑ = π/2. In accordance with the solutions (22) and (23),

the nonzero component of the magnetic field will be Bϑ, which at the inner boundary r = Ri of the vortex-free zone is

BBi −=ϑ , while at the external boundary r = a of that zone it is 2BBe =ϑ . The magnetic field in the vortex-free zone

is therefore perpendicular to the star’s equatorial plane and has different values and opposite directions at the boundaries

of that zone. Those values of the magnetic field must be boundary conditions in finding the magnetic field distribution

in the vortex-free zone. It follows from the foregoing that the magnetic field distribution in this zone has cylindrical

symmetry. Then the general solution of Eq. (2) without the right side is

( ) , 1211

λ

+

λ

=ϕrKArIArA (27)

where I1 and K

1 are first-order Bessel functions of an imaginary argument, which are two independent solutions of the

363

equation

. 0111222

2

=

λ+−+ ϕ

ϕϕA

rdr

dA

rdr

Ad(28)

The nonzero magnetic field component Bz is defined as

( )( ). 1 rrAdrd

rBz ϕ= (29)

Substituting (27) into (29), we obtain

, 0201

λ

+

λ

= rKArIABz (30)

where I0 and K

0 are zeroth-order Bessel functions. The constants A

1 and A

2 are determined from the conditions of

continuity of the magnetic field component at the inner and external boundaries of the vortex-free zone:

. 2

,

0201

0201

BaKAaIA

BR

KAR

IA ii

=

λ

+

λ

−=

λ

+

λ

(31)

Expressing A1 and A2 in terms of B and substiting into the solution (30), we finally obtain the following expression

for the magnetic field in the vortex-free zone:

, 22 0

20

1

λ∆

+

λ∆

−= rKBDrI

BDBz (32)

where the coefficients D1, D

2, and ∆ are expressed in terms of Bessel functions as follows:

,

, 2

, 2

0000

002

001

λ

λ−

λ

λ

=∆

λ

+

λ

=

λ

+

λ

=

ii

i

i

RKaIaK

RI

RIaID

RKaKD

(33)

If we allow for the fact that a, Ri, and ∆R are far larger than λ, then the magnetic field has a simpler form in this

approximation:

. exp2

exp

λ−−+

λ

−−−= raBRr

BB iz (34)

As seen from this solution, with depth into the vortex-free zone, from both the inner and the external boundary,

the magnetic field decreases exponentially and approaches zero. The field differs from zero only at a distance on the

order of λ from the boundaries of the vortex-free zone, while λ, as noted above, is negligible compared to the thickness

∆R of that zone. The magnetic field is therefore absent from most of the vortex-free zone.

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5. Conclusion

The magnetic field thus has a nonuniform distribution within the hadronic core of a neutron star. Whereas the

average magnetic field in the vortex zone is uniform, it is absent from the vortex-free zone. The field outside the hadronic

core has a dipole character with a moment on the order of 1030 G×cm3. At the boundaries of the vortex-free zone the

magnetic field decreases exponentially with depth into that zone. The characteristic depth of penetration of the field into

the vortex-free zone is on the order of the London depth of penetration λ, which is many orders of magnitude less than

the size of the vortex-free zone.

The existence of a vortex-free zone with a zero magnetic field radically alters the former concepts about the

magnetic field distribution in the superfluid hadronic core of a neutron star. In particular, the presence of a vortex-free

zone with macroscopic dimensions will play an essential role in calculations of the intensity of generation of magnetic

energy in the annihilation of proton vortices near the external boundary of that zone, responsible for slowing of the

neutron star’s rotation.

The authors thank the FAR/ANSEF fund for financial support with ANSEF grant No. PS51-01. This work was

completed while one of us, D. M. Sedrakian, was at the Paris Institute of Astrophysics in the “Jumelage France–Armenia”

program of the CNRS. D. M. Sedrakian thanks the International Science and Technical Center for support by grant No.

A- 353.

REFERENCES

1. D. M. Sedrakian and K. M. Shahabasyan, Usp. Fiz. Nauk, 161, No. 7, 3 (1991).2. D. M. Sedrakian and K. M. Shahabasyan, Astrofizika, 16, 727 (1980).

3. D. M. Sedrakian, Astrofizika, 19, 135 (1982).4. D. M. Sedrakian, K. M. Shahabasyan, and A. G. Movsessian, Astrofizika, 19, 303 (1983).5. D. M. Sedrakian, K. M. Shahabasyan, and A. G. Movsessian, Astrofizika, 21, 547 (1984).6. D. M. Sedrakian, Astrofizika, 43, 377 (2000).7. D. M. Sedrakian and K. M. Shahabasyan, Dokl. Akad. Nauk Arm. SSR, 70, No. 1, 28 (1980).