Landau level-superfluid modified factor and effective X/\gamma-ray coefficient of a magnetar

Landau level-superfluid modified factor and effective X/-ray coefficient of a magnetar

Z. F. Gao11affiliationmark: 22affiliationmark: 33affiliationmark: Q. H. Wang 44affiliationmark: N. Wang 11affiliationmark: 22affiliationmark: C. K. Chou 55affiliationmark: W. S. Huo 66affiliationmark: 77affiliationmark:
Abstract

As soon as the energy of electrons near the Fermi surface are higher than , the threshold energy of inverse decay, the electron capture process will dominate. The resulting high-energy neutrons will destroy anisotropic neutron superfluid Cooper pairs. By colliding with the neutrons produced in the process , the kinetic energy of the outgoing neutrons will be transformed into thermal energy. The transformed thermal energy would transported from the star interior to the star surface by conduction, then would be transformed into radiation energy as soft X-rays and gamma-rays. After a highly efficient modulation within the pulsar magnetosphere, the surface thermal emission (mainly soft X/-ray emission) has been shaped into a spectrum with the observed characteristics of magnetars. By introducing two important parameters: Landau level-superfluid modified factor and effective X/-ray coefficient, we numerically simulate the process of magnetar cooling and magnetic field decay, and then compute magnetars’ soft X/-ray luminosities . Further, we obtain aschematic diagrams of as a function of magnetic field strength . The observations are compared with the calculations.

slugcomment: Not to appear in Nonlearned J., 45.\@footnotetext

Xinjiang Astronomical Observatory, CAS, 40-5 South Beijing Road, Urumqi Xinjiang, 830011, China zhifugao@uao.ac.cn \@footnotetextKey Laboratory of Radio Astronomy, Chinese Academy of Sciences£¬Nanjing, 210008, China \@footnotetextGraduate University of the Chinese Academy of Scienes, 19A Yuquan Road, Beijing, 100049, China \@footnotetextDepartment of Astronomy, Nanjing University, Nanjing, 2100093, China \@footnotetextXinjiang Astronomical Observatory, CAS, 40-5 South Beijing Road, Urumqi Xinjiang, 830011, China \@footnotetextKey Laboratory of Radio Astronomy, Chinese Academy of Sciences£¬Nanjing, 210008, China \@footnotetextNational Astronomical Observatories, Chinese Academy of Sciences, Beijing, 10012, China \@footnotetextSchool of Physics, Xinjiang University, Urumqi Xinjiang, 830011, China \@footnotetext Xinjiang University-National Astronomical Observatories Joint Center for Astrophysics,Urumqi Xinjiang, 830011, China


Keywords Magnetar. Landau levels. Electron capture. Neutron star. Fermi energy

1 Introduction

Magnetars are ultra-magnetized neutron stars (NSs) with magnetic fields largely in excess of the quantum critical field = 4.414 G, at which the energy between Landau levels of electrons equals the rest-mass energy of a electron (Duncan & Thompson, 1992; Thompson & Duncan, 1995, 1996). Unlike ordinary radio pulsars, powered by their rotational energy loss, or shining in X-rays thanks to the accretion of matter from their companion stars, magnetars¡¯ persistent X-ray luminosities, are instead believed to be powered by the decay of their exceptionally strong magnetic fields (Colpi et al., 2000; Thompson & Duncan, 1995, 1996; Woods & Thompson, 2004; Mereghetti, 2008).

The majority of magnetars are classified into two NS populations historically that were independently discovered through different manifestations of their high-energy emission (Colpi et al., 2000; Kouveliotou et al., 1998; Woods & Thompson, 2004): the soft gamma-ray repeaters (SGRs), which give sporadic bursts of hard X-rays/soft-rays as well as rare, very luminous ( erg ) ¡®giant flares¡¯, and the anomalous X-ray pulsars (AXPs), so named due to their high X-ray luminosities ( erg ) and unusually fast spin-down rates, with no evidence of variation due to binary motion, which are distinct from both accreting X-ray binaries and isolated radio pulsars. Both AXPs and SGRs have common properties: stochastic outbursts (lasting from days to years) during which they emit very short X/-ray bursts; rotational periods in a narrow range s; compared to other isolated neutron stars, large period derivatives of ; rather soft X-ray spectra ( keV) that can be fitted by the sum of a blackbody model with a temperature 0.5 keV and a power-law tail with photon index (Mereghetti et al., 2002) and, in some cases, associated with supernova remnants (SNRs)(Duncan & Thompson, 1992; Mereghetti, 2008). In a few AXPs, good fits are obtained equivalently with two blackbodies (Halpern et al., 2005) or other combinations of two spectral components.

With the exception of SGR 0526-66, SGRs tend to have harder spectra below 10 keV than AXPs, and also suffer of a larger interstellar absorption, which makes the detection of blackbody-like components more difficult. For SGR 1806-20 and SGR 1900+14, most of their soft X-ray spectra have been well fit with power-laws of photon index 2. Nevertheless, when good quality spectra with sufficient statistics are obtainable, blackbody-like components with 0.5 keV can be detected also in these sources(Mereghetti et al., 2005, 2006a, 2006b). These data demonstrate that emissions from magnetars in the soft X-ray band are predominantly of thermal origin, but the emerging spectrum is far more intricate than that with a simple Planckian. This is not surprising if we take into account the presence of a strongly magnetized atmosphere and/or the effects of resonant cyclotron scattering within the magnetosphere of a magnetar (Tong et al., 2009).

Observations from the Rossi X-Ray Timing Explorer (RXTE) and the International Gamma-Ray Astrophysics Laboratory (INTEGRAL)have revealed that magnetars are luminous, persistent sources of 100 keV X-rays (Beloborodov & Thompson, 2007). This high-energy component, as distinct from the soft X-ray component, has a harder spectrum, and peaks above 100 keV. The luminosity in this band could be comparable or even exceed the thermal X-ray luminosity from the star surface. These hard X-rays could be emitted only in the exterior of a magnetar, which illustrates the existence of an active plasma corona (Beloborodov & Thompson, 2007). However, what we care about is the mechanism for the magnetar soft X-ray/ -ray emission in this article.

To explain the soft X-ray/-ray emission of magnetars, various ideas have been put forward (Heyl & Hernquist, 1997; Heyl & Kulkarni, 1998; Pons et al., 2006; Rheinhardt & Geppert, 2003; Thompson & Duncan, 1996; Thompson et al., 2000, 2002). There are also promising physical models that explain the absence of cyclotron features and the hard X-ray tails (Lyutikov & Gavrill, 2006; Nobili et al., 2008). The thermal emission model has been investigated extensively in the last decade, and significant progress has been achieved by many authors. However, some of the most fundamental questions are still unanswered. For example, the question of where the heat comes from is becoming more and more significant.

In the ¡®fall-back disk¡¯ model, different mechanisms for the disk formation and the relationship between the disk and the observed magnetar soft X-ray/-ray luminosity have been taken into consideration (Alpar, 2001; Chatterjee et al., 2000; Ghosh et al., 1997; Marsden et al., 2001; Tong et al., 2010; van Paradijs et al., 1995). The enhanced X-ray emission and the expected optical/IR flux from magnetars can be interpreted as due to the evolution of disks after they have been pushed back by the bursts. The transient behavior of XTEJ1810-197 has been instead explained in terms of a fall-back disk subject to viscous instability (Ertan & Erkut, 2008). However, the primary flaw of this model lies in the fact that it is unable to give a clear accounting for the bursts and flares (Ertan et al., 2006). As a result, other possible mechanisms for the magnetar soft X-ray/-ray emission have to be added.

The origin of magnetars is another interesting and important issue. A currently popular hypothesis is that magnetars are formed from rotating proto-neutron stars, and rapid differential rotation and convection would result in an efficient dynamo (Duncan & Thompson, 1992, 1996; Thompson & Duncan, 1993; Vink & Kuiper, 2006). In the context of the dynamo model, a mechanism (called a twisting magnetosphere) has been proposed to describe the radiative properties of magnetars (Thompson et al., 2000). According to the ¡®twisting magnetosphere¡¯ model, the energy caught in the twisting magnetic field gradually dissipates into X-rays. For a magnetar, the persistent soft X-ray emission can be induced by the twisting of the external magnetic field, while the persistent hard X-ray emission originates in a transition layer between the corona and the atmosphere (Beloborodov & Thompson, 2007; Thompson et al., 2000). The gradual dissipation of the magnetospheric currents can produce the persistent soft -ray emission (Thompson & Beloborodov, 2005). In some sources (primarily AXPs), it is very difficult to evaluate the bolometric losses by a steep power-law component in the X-ray spectrum, which is not observed below 0.5 keV because of absorption by the star¡¯s inner matter (Thompson et al., 2002). Unfortunately, we have not so far found any evidence supporting the idea that magnetars are formed from rotating proto-neutron stars. There is as yet no mechanism to explain such a high efficiency of energy transformation in the model. Moreover, when investigating solar flare using this model, apart from the difficulty in explaining such a high energy transformation efficiency, there are also many other problems to be settled at present. According to the above analysis, the dynamo is still just a hypothesis (Gao et al., 2011b; Hurley et al., 2005; Mereghetti, 2008; Vink & Kuiper, 2006).

In this paper, we focus on the interior of a magnetar, where the reaction of electron capture (EC) proceeds. As we know, EC, also called ‘the inverse -decay’, is a key physical process for nucleosynthesis and neutrino production in supernova (SN), especially for core-collapsed SN (including type SNII, SNIb and SNIc)(Langanke & Martinez-Pindo, 2000; Luo et al., 2006; Peng, 2001). It not only carries away energy and entropy in the form of neutrinos, but also reduces the number of electrons in the interior of a SN. The ways of calculating parameters relating to EC are dependent on models. For example, specific but representative parameters encountered during the initial stages of core formation in a SN are temperatures K( 2.4 MeV), densities g cm, electron Fermi energies 35 MeV, and the Fermi energies of neutrinos produced in the process of EC, 25 MeV (Lamb & Pethick et al., 1976; Mazurek, 1976). Unlike any other way of dealing with EC, by introducing related parameters and comparing our results with the observations, we numerically simulate the whole EC process accompanied by decay of the magnetic field and fall of the internal temperature. According to our model, superhigh magnetic fields of magnetars could be from the induced magnetic fields at a moderate lower temperature due to the existence of anisotropic neutron superfluid, and the maximum of induced magnetic field is estimated to be (3.0 4.0) G (Peng & Tong, 2007, 2009). In the interior of a magnetar(mainly in the outer core), superhigh magnetic fields give rise to the increase of the electron Fermi energy (Gao et al., 2011a), which will induce EC (if the energy of a electron is higher than the value of , the threshold energy of inverse decay). The resulting high-energy neutrons will destroy anisotropic neutron Cooper pairs, then the anisotropic superfluid and the superhigh magnetic field induced by the Cooper pairs will disappear. By colliding with the neutrons produced in the process , the kinetic energy of the outgoing neutrons will be transformed into thermal energy. The transformed thermal energy would be transported from the star interior to the star surface by conduction, then would be transformed into radiation energy as soft X-rays and gamma-rays. For further details, see Sec. 3.

In order to be feasible for our model, two factors must be taken into account. The first is that too much energy is lost in the process of thermal energy transportation, due to the inner matter’ absorption and the emission of neutrinos escaping from the interior of a magnetar. The second is that most of thermal energy transported to the star surface is carried away by the surface neutrino flux, only a small fraction can be converted into radiation energy as soft X/-rays. Taking into account of the above-mentioned two factors, we introduce the energy conversion coefficient and the thermal energy transportation coefficient in the latter calculations.

The remainder of this paper is organized as follows: in Sect.2we calculate the electron Fermi energy , the average kinetic energy of the outgoing EC neutrons , and the average kinetic energy of the outgoing EC neutrinos . In Sect. 3, we describe our interpretation of soft X/-ray emission of magnetars, compute the luminosity and obtain a diagram of as a function of . A brief conclusion is given in Sect.4, and a special solution of the electron Fermi energy is derived briefly in Appendix.

2 The calculations of , and

2.1 Electron Fermi energy and energy state densities of particles

In the presence of an extremely strong magnetic field (), the Landau column becomes a very long and very narrow cylinder along the magnetic field, the electron Fermi energy is determined by

(1)

where , and are three non-dimensional variables, which are defined as , and , respectively; is the modification factor; 6.02 is the Avogadro constant; , here , , and are the electron fraction, the proton fraction, the proton number and nucleon number of a given nucleus, respectively (Gao et al., 2011a). From Eq.(1), we obtain the diagrams of vs. , shown as in Fig. 1.

Fig. 1 : The relation of and . The range of is (1.6 3.0)G, and =2.8 g cm. Dot-dashed line, dotted line and dashed line for =0.05, =0.08 and =0.11, respectively.

From Fig. 1, it’s obvious that increases with the increase in when and are given. The high could be generated by the release of magnetic field energy. According to Eq.(11.5.2) of Page 316 in Shapiro & Teukolsky (1983), in empty space for each species we would have

(2)

where denotes the normalization volume, is the particle number of species , is the energy state density for species (the particle number per unit energy), and is the degeneracy for species ( = 2 for fermions). In the vicinity of a Fermi surface, (). Since protons and neutrons are degenerate and nonrelativistic, we get the approximate relations and , then we obtain and , where and are the neutron Fermi kinetic energy and the proton Fermi kinetic energy, respectively (Shapiro & Teukolsky, 1983). Since and , the energy state densities of neutrons and protons can be approximately written as

(3)

and

(4)

respectively. If we define the electron Fermi momentum by

(5)

then we find that the electron number in a unit volume is

(6)

The electron energy density is determined by

(7)

Since the neutrino is massless, by energy conservation in the process of EC, we obtain the expression of the neutrino energy state density

(8)

where is the neutrino momentum.

2.2 The calculations of , and

We focus on non-relativistic, degenerate nuclear matter and ultra- relativistic, degenerate electrons under -equilibrium implying the following relationship among chemical potentials (called the Fermi energies )of the particles: , where the neutrino chemical potential is ignored. In the case of , the following expressions are hold approximately: MeV, MeV, = 1.29 MeV (c.f. Chapter of (Shapirol & Teukolsky, 1983), where =2.8 g cm is the standard nuclear dencity. In this work, for convenience, we set , yielding the threshold energy of electron capture reaction =1.29 MeV+(60-1.9)MeV =59.39 MeV, then the range of is (59.39 MeV ). For an outgoing neutrons the value of is not less than that of , otherwise the process of will not occur, so the range of is assumed to be (). By employing energy conservation via , the Fermi energies of neutrinos resulting in the process of EC, , can be calculated by

(9)

The electron capture rate is defined as the number of electrons captured by one proton per second, and can be computed by using the standard charged-current -decay theory. The expression for reads:

(10)

where =1.4358 erg cm, is the universal Fermi coupling constant in the Weinberg-Salam-Glashow theory; = 0.9737 is the vector coupling constant; is the ratio of the coupling constant of the axial vector weak interaction constant to that of the vector weak interaction, = 1.253 experimentally; the quantity is the neutrino ‘blocking factor’giving the fraction of unoccupied phase space for neutrinos. For both electrons and neutrinos we shall assume Fermi-Dirac equilibrium distributions,

(11)

To find the total electron capture rate per proton, we integrate over all initial electron states and over

(12)

where Eq.(7) for the electron energy state density is used. In the interior of a NS, for neutrinos (antineutrinos), = 1; for electrons, when , = 1, and when , = 0. The average kinetic energy of the outgoing neutrinos can be calculated by

(13)

where . The average kinetic energy of the outgoing neutrinos can be estimated by

(14)

In this paper, for the purpose of convenient calculation, we set , and and =43.44 MeV, further details are presented in Appendix. The range of is assumed to be (1.5423 3.0) G, where 1.5423 G is the minimum of denoted as . When drops below , the direct Urca process is quenched everywhere in the magnetar interior. If we want to get the value of in any superhigh magnetic field, the quantities , , and must firstly be computed. The calculation results are shown in Table 1.

B aafootnotemark: bbfootnotemark: ccfootnotemark:
(G) (MeV) (MeV) (MeV) (MeV)
1.6 59.94 0.55 0.41 60.14
2.0 63.38 3.99 3.01 60.98
2.5 67.01 7.62 5.78 61.84
3.0 70.14 10.75 8.19 62.56
4.0 75.37 15.98 12.48 63.50
5.0 79.69 20.30 15.63 64.67
7.0 86.69 27.30 21.15 66.15
9.0 92.31 32.92 25.62 67.30
1.0 94.77 35.38 27.58 67.80
1.5 104.88 45.49 35.70 69.79
2.0 112.70 53.31 42.01 71.30
2.5 119.17 59.78 47.25 72.53
2.8 122.59 63.20 50.03 73.17
3.0 124.72 65.33 51.76 73.57
\appgdef\tblnote@list\pt@width

Note:  The signs and denote: the values of
and are calculated by using the relations of
43.44 MeV , and MeV, respectively. We set and , further details are presented in Appendix.

Table 1 : The calculated values of and .

3 The calculation of magnetar soft X/-ray luminosity

This section is composed of three subsections. For each subsection we present different methods and considerations.

3.1 Physics on of a magnetar

In this part, we briefly present a possible explanation for the soft X/-ray luminosity of a magnetar.

As mentioned above, once the energy of electrons near the Fermi surface are higher than the Fermi energy of neutrons (60 MeV, c.f. Shapiro & Teukolsky 1983) the process of EC will dominate. Owing to superhigh density of the star internal matter, the outgoing neutrons can’t escape from the star. In the interior of a NS, decay and inverse decay always occurs simultaneously, as required by charge neutrality (Gamov & Schoenberg, 1941; Pethick, 1992). In the ’recycled’ process: inverse decay decay inverse decay, the kinetic energies of electrons resulting in the process of decay are still high (higher than the neutron Fermi kinetic energy), most of the electron energy loss is carried away by neutrinos (antineutrinos) produced in this ‘recycle’ process, only a small fraction of this energy loss can effectively contribute to heat the star internal matter. If one outgoing neutron collide with one Cooper pair, the Cooper pair with low energy gap will be destroyed quickly. The outgoing neutron will react with the neutrons produced in the process , the kinetic energy of the outgoing neutrons will be transformed into thermal energy. When accumulating to some extent, the transformed thermal energy would transport from the star interior to the star surface by conduction, then would be converted into radiation energy as soft X-rays and -rays, keV. However, most of thermal energy transported to the star surface is carried away by the surface neutrino flux, only a small fraction can be converted into radiation energy as soft X/-rays. After a highly efficient modulation within the pulsar magnetosphere, the surface thermal emission(mainly soft X/-rays) has been shaped into a spectrum with the observed characteristics of magnetars. It is worth noting that because of the absorption of the star matter and the emission of neutrinos escaping from the interior of a magnetar, the overwhelming majority of the thermal energy will be lost in the process of energy transportation. This lost thermal energy may maintain a relative thermodynamic equilibrium in the interior of a magnetar. The energies of neutrinos escaping from the star interior could be high as MeV due to neutrinos’ coherent scattering caused by electrons(protons) and neutrons (Shapiro & Teukolsky, 1983), and the heat carried away by neutrinos(antineutrinos) could be slightly larger than that absorbed, despite of the compactness of the star matter. Therefore, the whole electron capture reaction process (here we focus on the direct Urca process) can be seen as a long-term process of magnetic field decay, accompanied by magnetar’s inner cooling. In a magnetar is ultimately determined by magnetic field strength , and is a weak function of the internal temperature . It should be noted that the surface temperature is controlled by crustal physics, and is independent of the evolution of the core, while the internal temperature is only equivalent to background temperature and decreases with decreasing .

3.2 The Calculation of

Actually, only the neutrons lying in the vicinity of the Fermi surface are capable of escaping from the Fermi sea. In other words, for degenerate neutrons, only a fraction can effectively contribute to the electron capture rate . The rate of the total thermal energy released in the EC process is calculated by

(15)

where denotes the volume of anisotropic neutron superfluid (; is the normalized volume; , is the fraction of phase space occupied at energy (Fermi-Dirac distribution), factors of reduce the reaction rate, and are called ‘blocking factor’, each factor of must be multiplied by , in the interior of a NS, for neutrinos (antineutrinos), = 1; for electrons, when , = 1, when , = 0; for neutrons, when , = 0, when , = 1; for protons: when , = 1, when , = 0, so can be ignored in the latter calculations; the 4 powers of originate as follows: The reaction in equilibrium gives , where is the concentration of the th species of particle (Shapiro & Teukolsky, 1983); in addition to this, for each degenerate species, only a fraction can effectively contribute , both and are proportional to . In the interior of a NS, the neutrons are ’locked’ in a superfluid state, the rates for all the reactions including -decay and inverse -decay are cut down by a factor , where 0.048 MeV, is the superfluid energy gap (Elgary et al., 1996). For convenience, we use the symbol , called ‘Landau level-superfluid modified factor’, to represent . The whole EC process (or the process of decay of magnetic fields) can be seen as a long-term process of the inner temperature’s fall. Due to the obvious effect of restraining direct Urca reactions by neutron superfluid, the process of magnetar cooling and magnetic field decay proceeds very slowly, so the value of ‘Landau level-superfluid modified factor’ can be treated as a constant in the latter calculations. Since magnetars are different, their initial reaction conditions (such as , , etc) are also different. However, for simplicity, we assume the initial magnetic field strength to be G for all magnetars in this work. The energy gap maximum of is 0.048 MeV (Elgary et al., 1996), the critical temperature of the neutron Cooper pairs can be evaluated as follows: 2.78 K, so the maximum of the initial internal temperature (not including the inner core) can not exceed (Peng & Luo, 2006). Keeping as a constant, we numerically simulate the process of magnetar cooling and magnetic field decay. The details are shown in Table 2.

11footnotemark: 22footnotemark: 33footnotemark: 44footnotemark: 55footnotemark:
(G) (K) (K) (K) (K) (K)
3.0 2.70 2.65 2.60 2.55 2.50
2.8 2.68 2.63 2.58 2.53 2.48
2.5 2.64 2.59 2.54 2.50 2.45
2.0 2.57 2.52 2.48 2.43 2.38
1.5 2.48 2.43 2.39 2.34 2.30
1.0 2.34 2.30 2.26 2.22 2.18
9.0 2.30 2.26 2.22 2.18 2.14
7.0 2.22 2.18 2.14 2.10 2.06
5.0 2.09 2.06 2.02 1.99 1.95
4.0 2.00 1.97 1.94 1.90 1.87
3.0 1.87 1.84 1.81 1.77 1.74
2.5 1.77 1.74 1.71 1.69 1.66
2.0 1.61 1.58 1.56 1.54 1.51
1.6 1.23 1.23 1.21 1.19 1.18
\appgdef\tblnote@list\pt@width

Note:  The signs denote that the values of are 4.031
, 3.598, 3.202 , 2.841 and 2.512
corresponding to column 2, column 3, column 4, column 5
and column 6, respectively.

Table 2 : Numerical simulating magnetar cooling and magnetic field decay.

From the simulations above, we infer that, the magnetic field strength is a weak function of the internal temperature , which is only equivalent to background temperature and decreases with decreasing . From Table 2, the mean value of is 3.237 . According to our model, the observed soft X/-ray output of a magnetar is dominated by the transport of the magnetic field energy through the core. In order to obtain , we must introduce another important parameter , called ‘ effective X/-ray coefficient’ of a magnetar. The main reasons for introducing are presented as follows:

  1. Firstly, the thermal energy transported to the surface of a magnetar could not be converted into the electromagnetic radiation energy entirely (to see Section 1), therefore, we introduce an energy conversion efficiency, , which is defined as the ratio of the amount of the soft X/-ray radiation energy converted to the amount of the thermal energy transported to the star surface by heat conduction.

  2. Second, in the process of heat conduction, the lost thermal energy could be either absorbed by the star inner matter, or carried away by neutrinos. Thus, we introduce the thermal energy transfer coefficient , defined as the ratio of the amount of net thermal energy transported to the star surface to the amount of the total thermal energy converted by the magnetic field energy.

  3. Finally, we define the effective soft X/-ray coefficient of a magnetr , . Due to special circumstances inside NSs (high temperatures, high-density matter and superhigh magnetic fields etc), the calculations of and have not yet appeared so far in physics community, so it is very difficult to gain the values of and directly. However, the mean value of of magnetars can be estimated roughly by comparing the calculations with the observations in our model. Furthermore, we make an assumption that in a magnetar.

By introducing parameters and , magnetar soft X/-ray luminosity can be computed by

(16)

Inserting and = into Eq.(16) gives

(17)

Eliminating -functions and simplifying Eq.(17) gives

(18)

Inserting Eq.(3) and Eq.(8) into Eq.(18) gives a general formula for ,

(19)

where the relation 1 MeV=1.6 erg is used. Inserting all the values of the following constants:=1.4358 erg cm, =0.9737, =1.253, =1.05 erg s, =6.63 erg s, =9.109 g, =1.67 g, =0.511 MeV and =3 cm s into Eq.(19) gives the values of in different superhigh magnetic fields. Now, the calculations are partly listed as follows (to see Table 3).

B
(G) (MeV) (MeV) (erg s)
2.0 63.38 60.98 3.924
4.0 75.37 63.50 5.051
5.0 79.69 64.67 1.973
7.0 86.69 66.15 1.010
9.0 92.31 67.30 2.888
1.0 94.77 67.80 4.350
1.5 104.88 69.79 1.849
2.0 112.70 71.30 4.701
2.5 119.17 72.53 9.308
3.0 124.72 73.57 1.589
\appgdef\tblnote@list\pt@width

Note:  We assume that = 3.237,
and =0.12 when calculating .

Table 3 : The relation of and .

If we want to determine the value of , we must combine our calculations with the observed persistent parameters of magnetars. The details are to see in 3.3 and 3.4.

3.3 Observations of magnetars

Up to now, nine SGRs (seven conformed) and twelve AXPs (nine conformed) at hand, a statistical investigation of their persistent parameters is possible. Observationally, all known magnetars are X-ray pulsars with luminosities of erg s, usually much higher than the rate at which the star loses its rotational energy through spin- down (Rea et al., 2010). In Table 4, the persistent parameters of sixteen conformed magnetars are listed in the light of observations performed in the last two decades.

Name bbfootnotemark: ccfootnotemark: eefootnotemark:
(s) (s  s) ( K) ( G) (erg s) (erg s)
SGR0526-66 8.0544 3.8 NO 5.6 1.4 2.9
SGR1806-20 7.6022 75 6.96 24 5.0 6.7
SGR1900+14 5.1998 9.2 5.45 7.0 (0.831.3) 2.6
SGR1627-41 2.5946 1.9 NO 2.2 2.5 4.3
SGR0501+4516 5.7621 0.58 8.1 1.9 NO 1.2
SGR0418+5729 9.0784 No NO
0.0006 0.075 3.2
SGR1833+0832 7.5654 0.439 No 1.8 NO 4.0
CXOUJ0100 8.0203 1.88 4.41 3.9 7.8 1.4
1E2259+586 6.9789 0.048 4.77 0.59 1.8 5.6
4U0142+61 8.6883 0.196 4.58 1.3 5.3 1.2
1E1841-045 11.7751 4.155 5.10 7.1 2.2 9.9
1RXSJ1708 10.9990 1.945 5.29 4.7 1.9 5.7
CXOJ1647 10.6107 0.24 7.31 1.6 2.6 7.8
1E1547.0-5408 2.0698 2.318 4.99 2.2 5.8 1.0
XTEJ1810-197 5.5404 0.777 1.67 2.1 1.9 1.8
1E1048.1-5937 6.4521 2.70 7.23 4.2 5.4 3.9
\appgdef\tblnote@list\pt@width

Note:  All data are from the McGill AXP/SGR online catalog of 10 April 2011(http://www.physics. mcgill. ca/pulsar/magnetar/ main.html) except for of SGR 1806 -20. The sign denotes: from Thompson & Duncan (1996). The sign denotes: the data of column 3 are gained from the original data by using the approximate relation 1 keV 1.16 K (, and are the energy of a photon and the Boltzmann constant, respectively). The sign denotes: The surface dipolar magnetic field of a pulsar can be estimated using its spin period, , and spin-down rate, , by G. The signs and denote: dim AXP and transient AXP, respectively. The sign denotes: A pulsar slow down with time as its rotational energy is lost via magnetic dipolar radiation, and the loss rate of a pulsar’s rotational energy is noted as .

Table 4 : The observational parameters of magnetars confirmed. .

From Table 4, three magnetars SGR0501+4516, SGR0418+5729 and SGR1833+0832 with no persistent soft X/-ray fluxes observed will not be considered in the latter calculations. Although the lack of optical identifications restrict accurate distance estimates for some magnetars, it is clear from their collective properties (such as high X-ray absorption and distribution in the Galactic plane, etc) that these sources have characteristic distances of at least a few kpc. Such values, supported in some cases by the distance estimates of the associated SNRs, imply typical in the range erg s, clearly larger than the rotational energy loss inferred from their period and values. Moreover, according to the magnetar modelDuncan & Thompson (1992, 1996); Thompson & Duncan (1996), the persistent soft X/-ray luminosity of a canonic magnetar shouldn’t be less than its rotational energy loss rate . In order to reduce the error in the calculation of the average value of , all the transient magnetars listed in Table 4 (including SGR1627-41, CXOJ1647, 1E11547.0-5408 and XTEJ 1810-197) are no longer to be considered in the latter calculations. Combining Eq.(20) with Eq.(4) gives the values of of magnetars. The calculations are shown as below (to see Table 5).

Name
(G) (MeV) (MeV)
SGR0526-66 5.6 81.98 63.93 5.32
SGR1806-20 2.4 117.96 72.30 6.07
SGR1900+14 7.0 86.69 66.15 1.28
CXOUJ0100 3.9 74.89 63.63 1.67
1E1841-045 7.1 86.99 66.21 2.05
1RXSJ1708 4.7 78.47 64.41 1.34
1E1048.1-5937 4.2 76.29 63.93 7.39
\appgdef\tblnote@list\pt@width

Note:  We assume that = 3.237, and
= 0.12 when calculating .

Table 5 : The values of of magnetars .

Clearly from Table 6, the values of of most magnetars are about . Since the value of is mainly determined by the magnetic field strength of a magnetar, the mean value of of magnetars can be roughly estimated by

(20)

Employing Eq.(20) gives the mean value of of magnetars 3.803 . Theoretically, using allows us calculate the value of in any ultrastrong magnetic field. The details are to see in 3.4.

3.4 Comparing calculations with observations

Inserting into Eq.(20) gives the value of in any strong magnetic field. Now, the calculations are partly listed as follows.

B
(G) (MeV) (MeV) (MeV) (erg s)
2.0 63.38 3.01 60.98 1.492
5.0 79.69 15.63 64.67 7.503
1.0 94.77 27.58 67.80 1.654
1.5 104.88 35.70 69.79 7.032
2.0 112.70 42.01 71.30 1.788
2.8 122.59 50.03 73.17 4.948
\appgdef\tblnote@list\pt@width

Note:  We assume that = 3.237, =3.083,
and =0.12 when calculating .

Table 6 : The calculated values of for magnetars .

Furthermore, employing the mean value of , we also gain the schematic diagrams of soft X/-ray luminosity as a function of magnetic field strength. The results of fitting agree well with the observational results in three models, which can be shown as in Figure 2.

Fig. 2 : The diagram of soft X/-ray luminosity as a function of magnetic field strength when = 0.12 and . Circles and squares are for AXPs and SGRS, respectively. The range of is assumed to be (1.60 3.0 G) considering that, when , the direct Urca process ceases, while the modified Urca process still occurs, from which weaker X-ray and weaker neutrino flux are produced.

Clearly from Figure 2, the magnetar soft X/-ray luminosity increases with increasing magnetic field obviously, and the steepness of every fitting curve corresponding to weaker magnetic fields ( G) is larger than that corresponding to higher magnetic fields G), because Eq.(1) is approximately hold only when . For SGR 1806-20 with the lowest value of , whose soft X/ -ray luminosity is cited from (Thompson & Duncan, 1996), its observed value of may be biased by the intense low-frequency absorption, corresponding to an electron column density of 6 cm (Murakami et al., 1994; Sonobe et al., 1994), as a result, a significant blackbody component contained in the X-ray bolometric flux is undetected, the luminosity in relativistic particles needed to power the plerion is 5 erg s (Thompson & Duncan, 1996). With respect to 1E 1048-59, which is discovered as a 6.4 s dim isolated pulsar near the Carina Nebula (Steward et al., 1986), substantial data were subsequently obtained, showing unambiguous evidence for a large flux increase coupled to a decrease in the pulsed fraction (Mereghetti et al., 2004), as well exemplified by the fact that its decreased from (12) erg s (Mereghetti et al., 2004) to 5.4 erg s between September 2004 and April 2011. Therefore, for SGR 1806-20 and 1E1048-59, their values of are lower than the mean value of of magnetars, which can be shown in Figure 2. For transient magnetars SGR1627-41, CXOJ1647, 1E11547 .0-5408 and XTEJ1810-197, their observed soft X/-ray luminosities are high than those calculated in theory (far from the fitted curve), the possible explanations are presented as follows:

  1. Firstly, with respect to SGR1627-41, its abnormal behavior suggests a connection between the bursting activity and the luminosity of transient magnetars: in 1998 more than 100 bursts in about 6 weeks were observed with different satellites (Woods & Thompson, 2004), however, no other bursts have been reported since then. Its soft X-ray counterpart was identified with BeppoSAX in 1998 at a level of erg s. Observations carried out in the following 13 years showed a monotonic decrease in its soft X/-ray luminosity, down to the current level of erg s (to see in Table 4).

  2. Second, the first transient AXP XTE J1810-197, discovered in 2003 (Ibrahim et al., 2004), displaced a persistent flux enhancement by a factor of 100 with respect to quiescent luminosity level increased from 7 erg s to 5 erg s between June 1992 and September 2004 (Bernardini et al., 2009; Gotthelf et al., 2004), however, the latest observations showed an obvious decline in (in 10 April 2011 1.9 erg s, to see in Table 6). Several other transient magnetars (SGR1627-41,CXOJ1647 and 1E1547.0-5408) have been discovered after XTEJ1810-197. When shining, they have spectral and timing properties analogous with those of the persistent sources. During their ‘quiescent’ phases they universally possess luminosity of erg s and soft thermal spectra, that make them similar to CCOs (Mereghetti, 2010). The long-term variations in of transient magnetars could be associated with the bursting activities. The source high state coincided with a period of strong bursting activity, while in the following years, during which no bursts were emitted, its luminosity decreased (Mereghetti, 2008).

  3. . Finally, the magnetic field strengthes of these four transient magnetars are in the range of G, and their values of are calculated to be erg s according to our model. It is not strange that the observed value of of a transient magnetar is higher than that calculated if we take into account of the long-term effect of the bursting activity on the soft X/-ray luminosity of a transient magnetar.

What must be emphasised here is that, for AXPs 4U 0142+61 and 1E 2259+586 their mechanisms for persistent soft X/-ray may be related with the accretion, and will be beyond of the scope of our model. The possible explanations are also presented as follows:

  1. Firstly, for AXPs 4U 0142+61 and 1E 2259+586 their magnetic field strengthes are lower than the critical magnetic field . According to our model, once , the direct Urca process ceases, while the modified Urca process still occurs, from which weaker X-ray and weaker neutrino flux are produced.

  2. Second, the observed properties of 1E 2259+586 seem consistent with the suggestion that it is an isolated pulsar undergoing a combination of spherical and disk accretion (White & Marshall, 1984). This magnetar could be powered by accretion from the remnant of Thorne -ytow object(T) (van Paradijs et al., 1995).

  3. Finally, as concerns AXP 4U 0142+61, which was previously considered to be a possible black hole candidate on the basis of its ultra-soft spectrum (White & Marshall, 1984), the simplest explanation for its involves a low-mass X-ray binary with a very faint companion, similar to 4U 1627-67(Israel et al., 1994).

Furthermore, timing observations show that the period derivative of SGR 0418+5729, 6.0 s s, which implies that the corresponding limit on the surface dipolar magnetic field of SGR 0418+5729 is 7.5 G (Rea et al., 2010). If the observations are reliable, then the value of of SGR 0418+5729 calculated in our model will be far less than erg s, which implies that our model is not in contradiction with the observation of SGR 0418+ 5729 (the real value of of SGR 0418+5729 is too low to be observed so far). SGR 0418+5729 may present a new challenge to the currently existing magnetar models. However, in accordance with the traditional view on the electron Fermi energy, the electron capture rate will decrease with increasing in ultrastrong magnetic fields. If the electron captures induced by field-decay are an important mechanism powering magnetar’s soft X-ray emission (Cooper & Kaplan, 2010), then will also decrease with increasing , which is contrary to the observed data in Table 4 and the fitting result of Figure 2.

4 Conclusions

In this paper, by introducing two important parameters: Landau level-superfluid modified factor and effective X/-ray coefficient, we numerically simulate the process of magnetar cooling and magnetic field decay, and then compute of magnetars. We also present a necessary discussion after comparing the observations with the calculations. From the analysis and the calculations above, the main conclusions are as follows:

1. In the interior of a magnetar, superhigh magnetic fields give rise to the increase of the electron Fermi energy, which will induce electron capture reaction.

2. The resulting high-energy neutrons will destroy anisotropic neutron Cooper pairs, then the anisotropic superfluid and the superhigh magnetic field induced by the Cooper pairs will disappear.

3.By colliding, the kinetic energy of the outgoing neutrons will be transformed into thermal energy. This transformed thermal energy would be transported from the star interior to the star surface by conduction, then would be converted into radiation energy as soft X-rays and -rays.

4.The largest advantage of our models is not only to explain but also to calculate magnetar soft X/-ray luminosity ; further, employing the mean value of , we obtain the schematic diagram of as a function of . The result of fitting agrees well with the observation result.

Finally, we are hopeful that our assumptions and numerical simulations can be combined with observations in the future, to provide a deeper understanding of the nature of soft X/ -ray of a magnetar.


Acknowledgements We are very grateful to Prof. Qiu-He Peng and Prof. Zi-Gao Dai for their help in improving our presentation. This work is supported by National Basic Research Program of China (973 Program 2009CB824800), Knowledge Innovation Program of The Chinese Academy Sciences KJCX -YW-T09, Xinjiang Natural Science Foundation No.2009211B35, the Key Directional Project of CAS and NSFC under projects 10173020,10673021, 10773005, 10778631 and 10903019.

References

  • Alpar (2001) Alpar, M. A., 2001, Astrophys. J., 554, 1245
  • Baiko & Yakovlev (1999) Baiko D. A., Yakovlev D. G., 1999, Astron. Astrophys., 342, 192-200
  • Bernardini et al. (2009) Bernardini, F., Israel, G. L., Dall’Osso, S., 2009, Astron. Astrophys., 498, 195
  • Beloborodov & Thompson (2007) Beloborodov, A.M., Thompson, C., 2007, 657, 967
  • Canuto & Chiu (1971) Canuto V., Chiu H. Y., 1971, Space Sci. Rev., 12, 3c
  • Canuto & Ventura (1977) Canuto V., Ventura J., 1977, Fund. Cosmic Phys., 2, 203
  • Chakrabarty et al. (1997) Chakrabarty S., Bandyopadhyay D., Pal S., 1997, Phys. Rev. Lett., 78, 75
  • Chatterjee et al. (2000) Chatterjee, P., Hernquist, L., Narayan, R., 2000, Astrophys. J., 534, 373
  • Colpi et al. (2000) Colpi, M., Geppert, U., Page, D., 2000, Astrophys. J. Lett., 529, L29
  • Cooper & Kaplan (2010) Cooper, R. L., Kaplan, D. L., 2010, Astrophys. J. Lett.,708, L80
  • Duncan & Thompson (1992) Duncan R. C., Thompson C., 1992, Astrophys. J., 392, L9
  • Duncan & Thompson (1996) Duncan R. C., Thompson C., In: Rothschild R.E., Lingenfelter R.E. (eds.) High-Velocity Neutron Stars and Gamma-Ray Bursts. AIP Conference Proc., vol. 366, p. 111. AIP Press, New York (1996)
  • Elgary et al. (1996) Elgary, ., et al., 1996, Phys. Rev. Lett., 77, 1482
  • Ertan et al. (2006)