DOI 10.1007/s10511-018-9520-2 Astrophysics, Vol. 61, No. 1, March , 2018

MAGNETIC FIELD GENERATION IN HYBRID STARS

D. M. Sedrakian,1 M. V. Hayrapetyan,1 and D. S. Baghdasaryan2

The mechanism for magnetic field generation in hybrid neutron stars (containing “npe,” hadron, “2SC” and “CFL” quark phases) is discussed. It is assumed that the rotational vortices in “npe” and “CFL” phases with a quantum of circulation h/2m also continue in the “2SC” phase. Since the superconducting components in the “npe” and “2SC” phases are charged, entrainment currents develop around the vortices and generate a magnetic field. The average magnetic field in the quark phase is on the order of 5 ˜ 1015 G and exceeds the field in the “npe” phase by 2-3 orders of magnitude. The magnetic field

penetrates into the “CFL” phase by means of magnetic vortices with a flux 2) 0 and it can partially destroy the proton superconductivity in the “npe” phase. On the star’s surface, the magnetic field reaches 5 ˜ 1014 G, a level comparable to the magnetic field of magnetars. Magnetars may, therefore, contain quark matter. Keywords: magnetars: hybrid stars: quarks: superconductivity

1. Introduction

Over the half century of observations of pulsars and neutron stars (since 1967) a lot of observational data has been collected and various explanatory theories have been proposed. Nevertheless, many problems from the begin-

(1) Erevan State University, Armenia; e-mail: [email protected], [email protected] (2) V. Ambartsumyan Byurakan Astrophysical Observatory, Armenia; e-mail: [email protected]

Original article submitted October 6, 2017. Translated from Astrofizika, Vol. 61, No. 1, pp. 131-140 (February 2018). 0571-7256/18/6101-0113 ©2018 Springer Science+Business Media, LLC

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nings of neutron star physics remain unsolved. One of these concerns the magnetic fields of neutron stars. Estimates of the magnetic fields on the surfaces of neutron stars based on a magnetic dipole mechanism for slowing down of pulsars cover a rather wide range from 108 G (millisecond pulsars) to 1014-1015 G (magnetars). The overwhelming majority of pulsars (radio pulsars) have fields on the order of 1012 G [1]. The values of the magnetic fields of radio pulsars can be explained by freezing of the magnetic flux of a star’s progenitor during its collapse and the fields of millisecond pulsars can be explained by the evolution of the initial magnetic field during the lifetime of these objects (on the order of 107 years). The magnetic fields of order 1014-1015 of magnetars are difficult to explain in these terms, because the number of progenitor stars with a relatively high magnetic field is too low to provide for the rate at which magnetars are born [2]. A mechanism has been proposed [3,4] for generation of the magnetic field of pulsars that takes the superfluidity and superconductivity of the hadron phase of neutron stars into account. According to this mechanism, because of nuclear interactions some of the protons are entrained in the superfluid neutron motion. The resulting driven currents produce magnetic fields of up to 1012 G (see section 2). Thus, the ultrahigh magnetic fields of magnetars cannot be explained in terms of the standard model for neutron stars with superfluid hadronic matter. There is yet another problem associated with the presence of quark matter in the interiors of compact objects. It turns out that hybrid stars can also contain quark matter; in them the quark phase is surrounded by an “npe” hadron phase and, depending on the density of the material, there are two quark phases, “2SC” and “CFL.” The “2SC” phase consists of an equal amount of paired u, d, and s quarks and electrons compensated by the excess positive charge of the quarks [5]. The “CFL” phase consists of equal amounts of paired u, d, and s quarks but contains no electrons [6-8]. There have been many studies of the configurations of stable hybrid stars, i.e., neutron stars with a quark core [9-12]. According to Ref. 12, depending on the parameters of the phase transition between the hadron and quark phases, as well as between the “2SC” and “CFL” phases, a triplet of stars with a given mass can exist: a neutron star with a hadron phase, a hybrid star with a “2SC” phase, and a hybrid star with “2SC” and “CFL” phases. Thus, theoretical calculations confirm the possibility of quark matter in the interiors of compact stars, but it is necessary to study those properties of hybrid stars which might be observable. It is known that superfluidity and superconductivity of the “npe” phase can generate magnetic fields of order 12

10 G in neutron stars. In this paper we examine the possibility of magnetic field generation in a superfluid hybrid star. The purpose is to show that entrainment currents can also appear in the “2SC” phase of a hybrid star with subsequent generation of superstrong magnetic fields. The mechanism for magnetic field generation in the “npe” phase is briefly discussed in section 2, this mechanism is applied to the “2SC” phase in section 3, and energy release in the quark core of hybrid stars is examined in section 4. It is shown that on the surface of a hybrid star the magnetic 14 field reaches 5 ˜ 10 G, which is comparable to the magnetic fields of magnetars. Thus, magnetars may be compact

stars that contain quark matter. The features of the vortex structure of hybrid stars are studied, and it is shown that the slowing down of stars is accompanied by energy release at a rate on the order of the total rotational energy loss of pulsars.

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2. Magnetic field generation in the “npe” phase

It has been noted that in the superfluid core of a rotating neutron star a network of vortex filaments develops parallel to the star’s axis of rotation. The velocity of the neutron fluid in the vicinity of a filament in the rotating coordinate system is given by

v

Here F

F . r

(1)

h 2 m and m is the neutron mass. The density of the neutron vortices depends on the angular rotation

velocity and is given by

n

2: , F

(2)

h . 2m:

(3)

and the radius of the vortex is

b

Because of strong interactions between the neutrons and protons, the superconducting protons are entrained in the superfluid motion of the neutrons around a vortex. The entrained protons produce an electric current r jp

r e k Up v , m

(4)

where U p is the proton density, k is the entrainment coefficient, with k | 0.5 for the “npe” phase [13]. The entrainment currents generate a magnetic field that is determined by the Maxwell equation r rot H

4S r jp . c

(5)

It is known that protons form a type II superconductor. In places where the magnetic field exceeds the critical field H c1 for a proton superconductor, magnetic vortices develop; the magnetic field penetrates into the proton supercon-

ductor through the normal cores of the vortices. Thus, a cluster of magnetic proton vortices is formed around each of the neutron vortices. The radius of the magnetic vortex cluster around the neutron vortices is found by solving Eq. (5) [4]:

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H r

k )0 2 SO2p

ln

b , r

(6)

where r is the distance from the core of a neutron vortex and O p is the penetration depth of the magnetic field for the proton superconductor:

O2p

where ) 0

Sh c e

2 ˜ 10 7

˜

2

m2c2 4S e 2 U p

,

(7)

is the magnetic field quantum. In deriving Eq. (6) it has been assumed that the

magnetic field is zero at a distance b from the vortex core, i.e., in the core of neighboring vortices. For the density np of the proton vortices, we have [4]

np

H r  H c1 , )0

(8)

where

H c1

)0 6 SO2p

ln

Op

(9)

[p

is the value of the first critical field for a spherical proton superconductor [14] and [ p is the coherence length of the protons. Equation (8) shows that the density of the proton vortices is high in the core of a neutron vortex, i.e., for r

[ p , and it gradually falls to zero as the distance increases to H r1 H c1 corresponding to the condition r1.

Equations (6) and (9) yield the radius r1 of the vortex cluster:

r1 b

§Op ¨ ¨ [p ©

· ¸ ¸ ¹

1 3 k

.

(10)

We now calculate the average magnetic induction of a neutron vortex. It equals the total magnetic flux penetrating the vortex (or vortex cluster) divided by the area of the vortex surface. Assuming cylindrical symmetry, for the average induction of a neutron vortex we obtain

B

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1 S b2

r1

³ H r  H 0

c1

2S rdr

k )0 § [ p ¨ 4SO2p ¨© O p

2

·3 k ¸ . ¸ ¹

(11)

It follows from Eq. (11) that the average magnetic field of a neutron vortex is independent of the radius of the vortex itself, but depends on the microscopic parameters of the proton superconductor. Substituting the characteristic values [ p

10 12  2 ˜ 10 12 cm and O p

10 11 cm [4] in Eq. (11), for the average magnetic field we obtain

B ~ 1011  1012 G. Note also that the radius of the cluster is r1 | 0.1 b according to Eq. (10), and the average induction

of a vortex cluster is b r1 2 | 100 times this value of B , i.e., BC ~ 1013  1014 G. Therefore, the magnetic field generated in the “npe” phase by the entrainment currents is concentrated in the magnetic clusters, but we can assume that in the “npe” phase there is a uniform field on the average with an induction on the order of B ~ 1011  1012 G. These values correspond to the magnetic fields of most neutron stars and radio pulsars.

3. Magnetic field generation in the quark phase

We now consider hybrid stars containing quark “CFL” and “2SC” phases. The electrically neutral “CFL” phase with superconducting properties should be analogous to a neutron fluid. In particular, the rotational vortices referred to as semisuperconducting M1 vortices [15-17], also appear in the “CFL” phase. The velocity field surrounding these vortices is likewise determined by Eq. (1). M1 vortices differ from neutron vortices in that they carry a magnetic flux 2) 0 because of rotational electromagnetism. In order for the spin of a star to be independent of the coordinate Z, we assume that rotational vortices with the velocity distribution (1) also appear in the “2SC” phase, which lies between the “npe” and “CFL” phases. Thus, each neutron vortex will have a “continuation” in the quark phase. Then, by analogy with the mechanism for magnetic field generation in the “npe” phase, there should be a mechanism for magnetic field generation in the “2SC” phase. In fact, the superfluid motion of the quark condensate relative to the electrons produces currents. As opposed to the “npe” phase, where currents are produced only by entrainment of protons, in the “2SC” phase the entire quark fluid participates in the motion around the vortex and contributes to the entrainment currents. In this case, the entrainment current can be rewritten for the “2SC” phase by replacing e by e/3 and m by 2mq in Eq. (4) and setting k = 1: r jq

r e Uq v , 6 mq

(12)

where U q is the density of quark matter in the “2SC” phase and mq is the mass of a quark. Then, on solving the r equation analogous to Eq. (5) with jq from Eq. (12), we obtain the magnetic field distribution around the quark vortex:

H r

)0 SO2q

b ln , r

(13)

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where O2q

2 mq 2 c 2

4Se 3 2 U q

is the penetration depth of the magnetic field for a quark superconductor. To find the radius

of a vortex cluster in the “2SC” phase, we note that the first critical field H c1 for a spherical “2SC” phase is given by [18]

H c1

)q

ln

6 SO2q

Oq

,

[q

(14)

where ) q is the magnetic flux through the normal core of a quark vortex. For the radius of the vortex cluster, from the condition H r1 H c1 and Eqs. (13) and (14) we obtain

b r1

§ Oq ¨ ¨ © [q

· ¸ ¸ ¹

) q 6) 0

.

(15)

A solution of the Ginzburg-Landau equations in the form of magnetic vortices with flux ) q

6) 0 has been obtained

in studies of quark superconductors [18-20]. In this case, Eq. (15) gives

Oq

b r1

[q

.

(16)

To determine the average induction in the “2SC” phase we use the definition of B according to Eq. (11) and substitute it in Eqs. (13) and (14). Then, given Eq. (16), we obtain the following expression for B :

B

) 0 § r1 · ¨ ¸ 2SO2q © b ¹

2

) 0 §¨ [ q 2SO2q ¨© O q

2

· ¸ . ¸ ¹

(17)

The radius r1 of a vortex cluster and the average inductance B in the “2SC” phase can be estimated using the following values of [ q and O q for quark matter: [ q ~ 1 fm and O q ~ 5  10 fm [15]. Then Eqs. (16) and (17) yield r1 | 0.1 b  0.2 b and B ~ 5 ˜ 10 14  5 ˜ 1015 G. For the average inductance of the cluster we obtain an estimate of BC

) 0 2SO2q | 3 ˜ 10 16  1.5 ˜ 10 17 G. Thus, the average induction of the field generated in the “2SC” phase is three

orders of magnitude higher than that obtained by the same mechanism in the “npe” phase. We now examine the behavior of a magnetic field generated in the “2SC” phase in the “CFL” and “npe” phases. The superfluid and superconducting state of the “CFL” phase consists of equal amounts of paired “u,” “d,” and “s” quarks of all three colors [6-8] with complete absence of electrons, so that there are no entrainment currents. Thus, a magnetic field is not generated in the “CFL” phase. A magnetic field can penetrate into the “CFL” phase through the normal cores of rotational M1 vortices with a magnetic flux 2) 0 , but there are not enough of these 118

vortices to ensure that the magnetic field will penetrate through them. In the “CFL” phase an external magnetic field can produce new nonabelian magnetic vortices with flux 2) 0 , which are a continuation of the magnetic vortices in the “2SC” phase. Here a single magnetic vortex in the “2SC” phase with flux 6) 0 is divided into three magnetic vortices with flux 2) 0 . Thus, over the entire quark core the magnetic field will be almost uniform with an average inductance B ~ 5 ˜ 10 14  5 ˜ 1015 G. As for the “npe” phase, when the magnetic field penetrates it, the cluster magnetic field BC can partially or entirely destroy the proton superconductivity, since these values of BC are higher than the second critical field for a proton superconductor, H c 2 | 4 ˜ 1015  1016 G [4]. When the protons undergo a transition to the normal state, the field outside the quark core will have a dipole character. We now estimate the magnetic field at the surface of a hybrid star. To do this we use a model of a star with a radius R | 11 km in which the quark core has a radius Rq | 6.5 km [12]. The magnetic moment in the quark core will be B Rq3

M

2

,

and the field at the surface is estimated to be

Bext |

B 2

§ Rq ¨ ¨ R ©

3

· ¸ | 5 ˜ 1013  5 ˜ 1014 G ¸ ¹

It is remarkable that these estimates correspond to the magnetic fields of magnetars, so it can be stated that magnetars may contain a quark core with a sufficiently large volume.

4. Energy release in the quark core of a hybrid star

The magnetic field concentrated near the rotational vortices can become a source of radiated energy from hybrid stars. In fact, Eq. (2) implies that the densities of the vortices should decrease as a star slows down, i.e., the vortices should move away from the star’s axis of rotation. Because of the magnetization of neutron stars, however, the width of the vortex-free zone at the boundary of the “npe” phase increases from a size on the order of the intervortex separation b~10-3 cm to macroscopic sizes on the order of 5 m [21]. In this case, the last neutron vortex that reaches the boundary of the vortex-free zone and is annihilated will have a length on the order of 100 m. Then the cluster of magnetic vortices produced by the entrainment currents also collapses and the magnetic energy of the collapsing cluster can serve as a radio emission source [22,23]. The rate of energy release can be as high as 10251030 erg/s, which is on the order of the radio luminosity of pulsars. Now we consider the quark phase of a hybrid star and calculate the width of the vortex-free zone near the “2SC” phase. It has been shown [21] that the width of the vortex-free zone depends on the ratio of the densities of magnetic energy and rotational kinetic energy, and its radius is given by

119

§ B 2 / 8S R  1 |¨ R1 Rq ¨ U q : 2 Rq2 ©

Rq

· ¸ ¸ ¹

1/ 2

G1 / 2 ,

(18)

where R1 is the radius of the vortex zone. We use a model of a hybrid star with a quark core radius Rq | 6.5 km and an average magnetic field B ~ 1015 G in the quark phase and note also that objects with magnetic fields higher than 1014 G on their surface will have an angular rotation velocity on the order of : ~ 5 s-1. Substituting these values in Eq. (18) gives G ~ 1 . The solution of Eq. (18) shows that the average radius of the vortex zone is R1 | 5 km and the width of the vortex-free zone is of order ' R | 1.5 km. In this case, the length of the last vortex cluster is

l





2 2 Rq  ' R ' R | 8 km. On reaching the boundary of the vortex-free zone and collapsing, all the magnetic

energy contained in the vortex clusters is radiated over the surface of a cylinder of radius R1 and length l . The rate of energy release can be estimated using the formula [22]

I

Substituting R16 = 0.5 and l 6

& B2 : R1 ˜ 2 S R1 l 8S :

2.5 ˜ 10 35 B152

0.8 in Eq. (19), we obtain I

& : :

2 l6 . R16 12

(19)

5 ˜ 10 34 erg/s for an average magnetic field in the quark

core of B ~ 1015 G. We note that most of the rotational kinetic energy in magnetars is lost through x-ray emission at a rate on the order of 1034-1035 erg/s. Thus, the presence of quark matter might also be confirmed by observations of the emission from pulsars.

5. Conclusion

A hybrid star model for magnetars can explain the ultra-high magnetic fields of order 1014-1015 G in these objects. This means that magnetars may contain superfluid and superconducting quark matter, and within a substantial part of a star’s volume. A high magnetic field in the quark core also leads to energy release rates on the order of 1034-1035 erg/s at its surface. The observed rates of loss of rotational energy from magnetars lie in this range and this may be yet another manifestation of quark matter in the interiors of these objects.

REFERENCES 1. R. N. Manchester, G. B. Hobbs, A. Teoh, et al., Astron. J. 129, 1993 (2005), astroph/0412641, http://www.atnf.csiro.au/ research/pulsar/psrcat. 2. S. Mereghetti, J. A. Pones, and A. Melatos, Magnetars: Properties, Origin and Evolution, in: V. S. Belkin, et al., eds., The Strongest Magnetic fields in the Universe, Springer (2016), astro-ph.HE/1503.06313.

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3. D. M. Sedrakian and K. M. Shakhabasyan, Usp. Fiz. Nauk 161(7), 3 (1991). 4. A. D. Sedrakian and D. M. Sedrakian, Astrophys. J. 447, 305 (1995). 5. D. Bailin and A. Love, Phys. Rep. 107, 325 (1984). 6. M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys B 537, 443 (1999). 7. T. Schäfer and F. Wilczek, Phys. Rev. Lett. 82, 3956 (1999). 8. T. Schäfer, Nucl. Phys. B 575, 269 (2000). 9. N. Ippolito, M. Ruggieri, D. Rischke, et al., Phys. Rev. D 77, 023004 (2008). 10. B. Knippel and A. Sedrakian, Phys. Rev. D 79, 083007 (2009). 11. N. S. Ayvazyan, G. Colucci, D. Rischke, et al., Astron. Astrophys. 559, A118 (2013). 12. M. G. Alford and A. Sedrakian, astro-ph. HE/1706. 01592. 13. G. A. Vardanyan and D. M. Sedrakian, Zh. Eksp. Teor. Fiz., 81, 919 (1981). 14. D. M. Sedrakian, K. M. Shakhabasyan, and A. G. Movsisyan, Astrophysics, 20, 656 (1985). 15. K. Iida and G. Baym, Phys. Rev. D 66, 014015 (2002), hep-ph/0204124. 16. K. Iida, Phys. Rev. D 71, 054011 (2005). 17. A. P. Balachandran, S. Digal, and T. Matsuura, Phys. Rev. D 73, 074009 (2006). 18. D. Blaschke, D. M. Sedrakian, and K. M. Shahabasian, Astron. Astrophys. 350, L47 (1999). 19. D. Blaschke and D. M. Sedrakian, nucl-th/0006038. 20. M. Alford and A. Sedrakian, J. Phys. G 37, 075202 (2010). 21. D. M. Sedrakian, Astrophysics, 43, 275 (2000). 22. D. M. Sedrakian and M. V. Hayrapetyan, Astrophysics, 55, 377 (2012). 23. D. M. Sedrakian and M. V. Hayrapetyan, Astrophysics, 58, 131 (2015).

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