507

MAGNETIC FIELD ENERGY AS A SOURCE OF THE RADIO LUMINOSITYOF QUASARS

D. M. Sedrakian, M. V. Hayrapetyan, and N. S. Ayvazyan

Energy release in the superconducting core of a neutron star as neutron vortices move toward the

boundary of the star’s core and crust is examined. It is shown that the rate of energy release is on the

order of 1026-1030 erg/s, or sufficient to provide the radio luminosity of known pulsars. The energy

release rates calculated under the assumption of asymmetric energy release are compared with

observational data on the radio luminosity of 575 pulsars.

Keywords: pulsars: magnetic field: energy

1. Introduction

Pulsars were discovered more than 40 years ago as a source of periodic radio pulses and have become a cosmic

laboratory for the study of phenomena under extreme conditions of high density and superhigh magnetic fields.

Observations of more than 1700 pulsars have yielded a large amount of data that still remain to be explained. The

model of a magnetized neutron star with superfluid properties in its core has made it possible to explain the major

observed changes in the angular velocity of pulsars: secular spin-down, glitches, and relaxation of the angular

velocity. The mechanism of radio emission from pulsars remains as a major unsolved problem in the physics of

neutron stars. It is known that the overwhelming majority of detected isolated pulsars are observed in the radio

frequency range. The radio luminosity of pulsars lies within a range of 1026-1030 erg/s, which represents 10-4-10-6 of

the total rotational energy loss for a pulsar. Assuming that the source of the radio emission is the rotational energy

of a pulsar, then any theoretical model aiming to explain the features of the radio emission from pulsars must elaborate

the physical mechanisms by which the kinetic energy of the pulsar’s rotation is converted into the energy of

Astrophysics, Vol. 54, No. 4, December , 2011

0571-7256/11/5404-0507 ©2011 Springer Science+Business Media, Inc.

Original article submitted June 13, 2011. Translated from Astrofizika, Vol. 54, No. 4, pp. 571-580 (November2011).

Erevan State University, Armenia; e-mail: dsedrak@ysu.am

508

electromagnetic radiation.

While magnetic dipole radiation by a magnetized neutron star is the generally accepted mechanism for the

slowing down of pulsars, there is still no generally accepted theory of the radio emission from pulsars that is consistent

with the complete set of observational data [1-3]. Theoretical models of radio emission from pulsars are based on

the existence of a dense electron-positron plasma that is formed near the magnetic poles and flows out along open

magnetic field lines [2-6]. It is assumed that a coherent mechanism of radio emission develops as a result of various

kinds of instability that develop in the electron-positron plasma. However, the basic assumptions upon which the

existing theories of the emission from pulsars are based [the existence of a strong electric field near the pulsar’s

surface, the production of ultrarelativistic electrons, the presence of internal (at the pulsar’s surface) and external (near

the light cylinder) gaps, the location of the radiating region, etc.] are completely unjustified [1,2]. Besides these

uncertainties, the assumption in the theory of pulsar radiation that the source of the radio emission is in the

magnetosphere is questionable. Corotational motion of the magnetosphere plasma with the star has still not been

confirmed, although this is necessary for the very strict temporal sequence of the observed radio pulses.

All of these difficulties in understanding the physical phenomena taking place in the magnetospheres of

neutron stars support the idea that the source of the radio emission from pulsars is not in a plasma corotating with

a star, but on the star’s surface. The possible conversion of the rotational energy of interior superfluid regions into

the energy of radio emission was first pointed out in Ref. 7. It is known that as a neutron star rotates, a system of

vortices parallel to the axis of rotation develops in its inner superfluid regions. Owing to entrainment of superconducting

protons by the superfluid neutrons, a cluster of proton vortices develops in the central region of each neutron vortex,

and the magnetic field generated by the entrainment currents penetrates to the core of the neutron star through it [8].

As the star slows down, the neutron vortices move to the boundary of the core and crust of the neutron star. As they

do this, their length contracts, so that part of the magnetic energy included in the cluster is released at the boundary

of the “npe” phase. Calculations [7] show that the rate of energy release owing to slowing down of the magnetized

neutron vortices is on the order of 1028-1031 erg/s, with the energy mainly being released near the equatorial plane.

As pointed out above, the radio luminosity of pulsars is on the order of 1026-1030 erg/s, so that, if there is a mechanism

for conversion of this energy release into the radiant energy of pulsars, it becomes possible to explain their radio

emission.

An analysis of observational data on 291 pulsars [9] has shown that there is a gap in the interval

5727log4427 .L. ≤≤ in the distribution of the number of pulsars with respect to their radio luminosity. Based on

these data, it was proposed that there are two subpopulations of radio pulsars. The first included pulsars for which

the logarithm of their radio luminosity obeys 4427log .L ≤ , and the second, pulsars for which 5727log .L ≥ . It was

possible to calculate [10] the radii of the cores of the neutron stars by comparing the expressions for the energy release

rate I from Ref. 7 with the observational data. This calculation was in good agreement with the theory of the structure

of neutron stars, according to which the cores of neutron stars can have radii within the range mk12mk6 ≤≤ r .

Since 1986, when Ref. 10 was published, the number of detected pulsars has tripled. Thus, it has become

necessary to carry out a new statistical analysis of data on the radio luminosity of pulsars and to compare these data

with the theory of Ref. 10. The purpose of this article is to establish whether the appearance of two subpopulations

of pulsars depends on the number of observed pulsars, and also to prove that the source of the observed radio

509

luminosities of pulsars might be an energy release associated with the motion of vortex clusters in the core of a

neutron star.

2. Energy release rate in the core of a neutron star

The density of neutron vortices in the core of a rotating neutron star is proportional to the star’s angular

rotation velocity, i.e.,

( ) ( ),

2

κΩ

=t

tn s (1)

where nmh 2=κ , and mn is the mass of a neutron. As the star slows down, the angular velocity of the superfluid

component also decreases, so that the neutron vortices in the core develop a radial motion at a velocity

, 2

rs

sr Ω

Ω−=

�

� (2)

where r is the coordinate of a vortex in the equatorial plane. As the vortices move toward the boundary between

the core and crust of the neutron star, the vortex length decreases. The magnetic energy of a cluster of proton vortices

surrounding a neutron vortex is proportional to the length of the vortex and is given by

, 8

21

2

�rB

W ππ

= (3)

where B is the average magnetic induction in the cluster, r1 is the cluster radius, and � is the vortex length. Equation

(3) shows that, as the vortex length � decreases, part of the magnetic energy should be released at the boundary

between the star’s core and crust. The rate of energy release is given by [7,10]

, 43

323

2

20

ΩΩ

⎟⎠⎞⎜

⎝⎛

λξ

⎟⎠⎞⎜

⎝⎛

πλΦπ=

�k

Rk

I (4)

where 70 102 −⋅=Φ G.cm2 is the quantum of magnetic flux; λ is the penetration depth of the magnetic field; ξ is

the coherence length of the protons; 50.k ≈ is the entrainment coefficient for protons by neutrons; and R is the

radius of the neutron star’s core. Here it is also assumed that there must exist a mechanism by which the magnetic

energy of the contracting vortices is converted into the energy of radio emission from pulsars.

In order to apply Eq. (4) to the problem of pulsar radio luminosity, it is also necessary to provide a reason

for the asymmetry of the energy release with respect to the angle ϕ . Owing to the rotation of the star, only in that

510

case will the radio emission have a pulsed character for an earthbound observer. It is known that, besides the

generated dipole magnetic field whose axis is parallel to the axis of rotation of the star, pulsars also have a residual

magnetic field whose axis is at an angle with respect to the axis of rotation and can, in particular, be perpendicular

to the axis of rotation. It has been shown [11] that the existence of two perpendicular vortex systems can lead to

an asymmetry in the energy release near the boundary between the core and crust of a neutron star. In particular,

this asymmetry can show up as an enhancement in the magnetic field in the direction perpendicular to the two axes

of the dipole moments. If we include the fact that there is a vortex-free zone [12] bounded by a vortex with a length

on the order of 0.01R, then a circular “spot” with a diameter equal to the length of the last vortex will appear

perpendicular to this direction. In this case the characteristic “spot” size will be related to the circumference of the

equatorial cross section of the star in the same way that the temporal width w of the average radio pulse is related

to the rotation period p. Thus, Eq. (4) for the energy release rate can be rewritten in the form [10]

, 107616

34

4

3613

τ⎟⎠⎞⎜

⎝⎛

λξ

λ⋅= −

p

wR.I (5)

where R = 106 R

6 cm is the radius of the core and 6

6102 τ=ΩΩ=τ � years is the lifetime of the pulsar. A formula

that is more convenient for comparison with observations can be obtained from Eq. (5):

, logloglog xAI += (6)

where

34

4133

61

10761, ⎟⎠⎞⎜

⎝⎛

λξ

λ⋅== −.KKRA (7)

and

. 6τ

=p

wx (8)

Equation (5) shows that the energy release rate depends strongly on the properties of the superconducting proton

condensate in the star’s core, i.e., on the penetration depth λ and the coherence length ξ. It also depends on the radius

R of the star’s core. Changes in the values of λ, ξ, and R will correspond to different equations of state and different

neutron star models. Thus, by choosing the central density and the equation of state it is possible to obtain energy

release rates I within the range 1026-1030 erg/s and comparable to the observed radio luminosity of pulsars.

511

3. Comparison with observations

As pointed out above, early studies of the radio luminosity of pulsars revealed the existence of a gap in the

distribution of the number of pulsars with respect to radio luminosity. Pulsars were arbitrarily divided into two

subgroups, those with low 4427log .I < and those with high 5727log .I > radio luminosities. It was shown [10]

that a model of neutron stars with an average radius on the order of 5≈R km, i.e., a model with low central density

and mass, is applicable to the pulsars with lower radio luminosities, while a model of neutron stars with an average

radius 9≈R km, i.e., a model with a high central density and mass, is applicable to the second subgroup. An analysis

of the observational data showed, however, that there were a large number of the then known 291 pulsars that did

not conform to acceptable models of neutron stars based on the internal structure of these objects. Thus, high pulsar

radio luminosities could have been explained by assuming the existence of neutron stars with radii on the order of

20 km or more, but this was inconsistent with model calculations of neutron stars, even with extremely rigid equations

of state.

Because of the discovery of a large number of pulsars over the intervening years and improved understanding

of the properties of the neutron-proton superconducting condensate in the core of neutron stars, there is a need for

a new comparison of the theory of energy production in the cores of neutron stars developed in Ref. 10 with

observational data.

First of all, it should be noted that when the nearly 1700 pulsars are examined [13,14], the gap in the

distribution of the number of pulsars with respect to their radio luminosity vanishes. (See Fig. 1.) This figure shows

that there are quite a few previously unknown pulsars with radio luminosities in the range 5727log4427 .I. ≤≤ . It

can also be seen that it is no longer possible to divide the pulsars into subgroups with high and low radio luminosities.

This result confirms the fact that explaining the radio emission of pulsars based on the above mechanism (see section

2) does not require the use of models of neutron stars with extremely small (R < 5 km) and large (R > 20 km) radii.

Fig. 1. Distribution of the number of pulsars with respect totheir radio luminosity, L [13,14].

logL

N

��

��

�� �� �� � �

��

�

��

512

We have made a comparison of the theory with data on the radio emission from 575 pulsars [13,14] using

models of neutron stars with core radii of 8, 10, and 12 km, which conform to the generally accepted standard models

of neutron stars. In order to obtain high energy release rates I according to Eq. (5), we have also varied the field

penetration depth λ, on which the intensity I has a strong dependence. We considered values of λ between

5.10-12 and 1011 cm. As for the coherence length ξ, we assumed that it varies in a way such that the ratio 10.=λξ .

Figures 2 and 3 are log-log plots in which the points correspond to pulsars for which data are available on their radio

luminosity and the parameter x. Also shown there are plots of logI as a function of logx for three values of the core

radius R of the neutron star, R = 9, 10, and 12 km. For the plots in Fig. 2 we took λ = 10-11 cm and for those in

Fig. 3, λ = 5.10-12 cm. A comparison of Figs. 2 and 3 shows that a low value of the penetration depth λ is optimal

for the theory, since then most of the pulsars lie below the logI vs. logx curves. For almost 90% of the pulsars, the

energy release rate associated with the motion of magnetized vortices in the core of the neutron star is sufficient to

explain these observations of the radio luminosity of pulsars, and it is also possible to take into account the partial

conversion of the released energy into heat. In Fig. 3, a small fraction of the pulsars (about 10%), for which

2log −<x , lies above these curves. This group of pulsars includes pulsars with both low and high radio luminosities.

These pulsars can also be incorporated into the above energy release model if we note that, in writing down Eq. (5),

we have reduced the total energy release rate by multiplying Eq. (4) by the factor w/p, which is on the order of

0.03-0.05.

As noted above, in the calculations [11] of energy release rate, the existence of a proton vortex system parallel

to the residual magnetic field of the neutron star was also taken into account because of the motion of the vortices.

These are pinned to vortices parallel to the axis of rotation because of the gain in the condensation energy of the

vortex core. As the vortices that we examined earlier move with the slowing down of the star, the vortices that are

pinned to them move outward together with them and undergo contraction. Here the additional rate of energy release

within an element of angle ϕθdd is given by [11]

Fig. 2. The rate of energy release from pulsars as a function of the

parameter x for 1110−=λ m.

logx

logI

��

��

�� �� � �� �� � �

R6 = 0.8

R6 = 1.0

R6 = 1.2��

��

��

��

�

�

�

513

, sinln4

cos2

3 332

202

0

0äîï ϕθθ

ΩΩ

⎟⎟⎠

⎞⎜⎜⎝

⎛ξλ

⎟⎠⎞⎜

⎝⎛

πλΦ

ϕΦ

= ddRB

dI�

(9)

where B0 is the component of the magnetic field perpendicular to the axis of rotation of the neutron star. Because

of the factor ϕ2cos in Eq. (9), the energy release rate is highly asymmetric and is mostly localized in the directions

0→ϕ , 2π→θ and π→ϕ , 2π→θ . Integrating Eq. (9), taking 130 10≈B G, and using Eq. (5), we find that

including the proton vortices parallel to the residual magnetic field can increase the total energy release rate by a

Fig. 3. The rate of energy release from pulsars as a function of the

parameter x for 12105 −⋅=λ m.

logx

logI

��

��

�� �� � �� �� � �

R6 = 0.8

R6 = 1.0

R6 = 1.2

��

��

��

��

�

�

�

Fig. 4. The rate of energy release from pulsars as a function of the parameter x for12105 −⋅=λ m. The topmost line includes the effect of the residual magnetic

field of a neutron star.

logx

logI

��

��

�� �� � �� �� � �

R6 = 0.8

R6 = 1.0

R6 = 1.2

R6 = 1.2

+asymmetry��

��

��

��

�

�

�

514

factor of 1.5 over Eq. (5). Figure 4 shows some plots of )(loglog xI for 12105 −⋅=λ ; there the topmost line includes

the additional energy release discussed in Ref. 9. In that case, the observational data for even more pulsars can be

explained by the theory of Refs. 7, 10, and 11.

In sum, the above analysis of radio luminosity data from 575 pulsars demonstrates the effectiveness of the

proposed mechanism of energy release at the boundary between the core and crust of neutron stars. The magnetic

energy contained in the proton vortices is sufficient, in terms of the generally accepted models of neutron stars, to

produce the radio luminosity observed in most of the known pulsars. A complete treatment of the problem of radio

emission from pulsars must also include a mechanism for the conversion of vortex magnetic energy into the

electromagnetic emission from neutron stars, along with a mechanism for the formation of “spots” on the stars’ surfaces

which lead to the observed pulsed radio signals from pulsars. These questions and some related problems will be

discussed in future papers.

We (M.V.H. and N.S.A.) thank the State Committee for Science of Armenia (grant 11-1s107) and the Volkswagen

Stiftung (grant Az: 85182) for support of this work.

REFERENCES

1. D. B. Melrose, J. Astrophys. Astron. 16, 137 (1995).

2. V. S. Beskin, Usp. Fiz. Nauk 169, 1169 (1999).

3. D. Melrose, in: F. Camilo and B. M. Gaensler, eds., Young Neutron Stars and Their Environments, IAU Symposium

218, (2004), astro-ph/0308471.

4. P. Goldreich and W. H. Julian, Astrophys. J. 157, 869 (1969).

5. P. A. Sturrok, Astrophys. J. 164, 529 (1971).

6. M. A. Ruderman and P. G. Sutherland, Astrophys. J. 196, 51 (1975).

7. D. M. Sedrakian, Astrofizika 25, 323 (1986).

8. A. D. Sedrakian and D. M. Sedrakian, Astrophys. J. 447, 305 (1995).

9. S. Pineault, Astrophys. J. 301, 145 (1986).

10. D. M. Sedrakian, Astrofizika 30, 547 (1989).

11. D. M. Sedrakian and A. D. Sedrakian, Zh. Eksp. Teor. Fiz. 100, 353 (1991).

12. D. M. Sedrakian, Astrofizika 43, 377 (2000).

13. R. N. Manchester, G. B. Hobbs, and M. Hobbs, Astrophys. J. 129, 1993 (2005).

14. ATNF Pulsar Catalogue, http://www.atnf.csiro.au/research/pulsar/psrcat/