`ISRN Astronomy and AstrophysicsVolume 2011 (2011), Article ID 843825, 5 pageshttp://dx.doi.org/10.5402/2011/843825`
Research Article

1State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China
2Laboratory of Informational Technologies, Joint Institute for Nuclear Research, Dubna 141980, Russia

Received 5 November 2011; Accepted 20 December 2011

Copyright © 2011 S. Bastrukov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In juxtaposition with the standard model of rotation-powered pulsar, the model of vibration-powered magnetar undergoing quake-induced torsional Alfvén vibrations in its own ultrastrong magnetic field experiencing decay is considered. The presented line of argument suggests that the gradual decrease of frequencies (lengthening of periods) of long-periodic-pulsed radiation detected from a set of X-ray sources can be attributed to magnetic-field-decay-induced energy conversion from seismic vibrations to magnetodipole radiation of quaking magnetar.

1. Introduction

There is a common recognition today that the standard (lighthouse) model of inclined rotator, lying at the base of current understanding of radio pulsars, faces serious difficulties in explaining the long-periodic ( s) -pulsed radiation of soft gamma repeaters (SGRs) and anomalous -ray pulsars (AXPs). The persistent -ray luminosity of AXP/SGR sources ( erg s−1) is appreciably (10–100 times) larger than expected from a neutron star deriving radiation power from the energy of rotation with frequency of detected pulses. Such an understanding has come soon after the detection on March 5, 1979 of the first 0.2-second long gamma burst [1], which followed by a 200-second emission that showed a clear 8-second pulsation period [2], and association of this event to a supernova remnant known as N49 in the Large Magellanic Cloud [3]. This object is very young (only a few thousand years old), but the period of pulsating emission is typical of a much older neutron star. In works [4, 5] it has been proposed that discovered object, today designated SGR 0526-66, is a vibrating neutron star, that is, the detected for the first time long-periodic pulses owe their origin to the neutron star vibrations, rather than rotation as is the case with radio pulsars. During the following decades, the study of these objects has been guided by the idea [6, 7] that electromagnetic activity of magnetars, both AXP’s and SGR’s, is primarily determined by decay of ultrastrong magnetic field ( G) and that a highly intensive gamma bursts are manifestation of magnetar quakes [810].

In this paper we investigate in some detail the model of vibration-powered magnetar which is in line with the current treatment of quasiperiodic oscillations of outburst luminosity of soft gamma repeaters as being produced by Lorentz-force-driven torsional seismic vibrations triggered by quake. As an extension of this point of view, in this paper we focus on impact of the magnetic field decay on Alfvén vibrations and magnetodipole radiation generated by such vibrations. Before doing so, it seems appropriate to recall a seminal paper of Woltjer [11] who was first to observe that magnetic-flux-conserving core-collapse supernova can produce a neutron star with the above magnetic field intensity of typical magnetar. Based on this observation, Hoyle et al. [12] proposed that a strongly magnetized neutron star can generate magnetodipole radiation powered by energy of hydromagnetic, Alfvén, vibrations stored in the star after its birth in supernova event (see, also, [13]). Some peculiarities of this mechanism of vibration-powered radiation have been scrutinized in our recent work [14], devoted to radiative activity of pulsating magnetic white dwarfs, in which it was found that the necessary condition for the energy conversion from Alfvén vibrations into electromagnetic radiation is the decay of magnetic field. As was stressed, the magnetic field decay is one of the most conspicuous features distinguishing magnetars from normal rotation-powered pulsars. It seems not implausible, therefore, to expect that at least some of currently monitoring AXP/SGR-like sources are magnatars deriving power of pulsating magnetodipole radiation from energy of Alfvénic vibrations of highly conducting matter in the ultrastrong magnetic field experiencing decay.

In approaching Alfvén vibrations of neutron star in its own time-evolving magnetic field, we rely on the results of recent investigations [1518] of both even parity poloidal and odd parity toroidal (according to Chandrasekhar terminology [19]) node-free Alfvén vibrations of magnetars in constant-in-time magnetic field. The extensive review of earlier investigations of standing-wave regime of Alfvénic stellar vibrations can be found in [20]. The spectral formula for discrete frequencies of both poloidal and toroidal -modes in a neutron star with mass , radius , and magnetic field of typical magnetar,  G, reads where numerical factor is unique to each specific shape of magnetic field frozen in the neutron star of one and the same mass and radius .

2. Alfvén Vibrations of Magnetar in Time-Evolving Magnetic Field

In above cited work it was shown that Lorentz-force-driven shear node-free vibrations of magnetar in its own magnetic field field can be properly described in terms of material displacements obeying equation of magneto-solid-mechanics The field is identical to that for torsion node-free vibrations restored by Hooke’s force of elastic stresses [18, 21] with , where is the nodeless function of distance from the star center and is Legendre polynomial of degree specifying the overtone of toroidal mode. In (4), the amplitude is the basic dynamical variable describing time evolution of vibrations, which is different for each specific overtone; in what follows, we confine our analysis to solely one quadrupole overtone. The central to the subject of our study is the following representation of the time-evolving internal magnetic field: where is the time-dependent intensity and is dimensionless vector-function of the field distribution over the star volume. Scalar product of (1) with the separable form of material displacements followed by integration over the star volume leads to equation for amplitude having the form of equation of oscillator with time-dependent spring constant The total vibration energy and frequency are given by It follows that This shows that the variation in time of magnetic field intensity in quaking magnetar causes variation in the vibration energy. In Section 3, we focus on conversion of energy of Lorentz-force-driven seismic vibrations of magnetar into the energy of magnetodipole radiation.

3. Vibration-Powered Radiation of Quaking Magnetar

The point of departure in the study of vibration energy-powered magnetodipole emission of the star (whose radiation power, , is given by Larmor’s formula) is We consider a model of a quaking neutron star whose torsional magnetomechanical oscillations are accompanied by fluctuations of total magnetic moment preserving its initial (in seismically quiescent state) direction: . The total magnetic dipole moment should execute oscillations with frequency equal to that for magnetomechanical vibrations of stellar matter, which are described by equation for . This means that and must obey equations of similar form, namely, It is easy to see that (11) can be reconciled if Given this, we arrive at the following law of magnetic field decay: The last equation shows that the lifetime of quake-induced vibrations in question substantially depends upon the intensity of initial (before quake) magnetic field : the larger the the shorter the . For neutron stars with one and the same mass and radius km, and magnetic field of typical pulsar  G, we obtain years, whereas for magnetar with  G, years.

The equation for vibration amplitude with help of substitution is transformed to permitting general solution [22]. The solution of this equation, obeying two conditions and , can be represented in the form where and are Bessel functions [23] and Here by , the average energy stored in torsional Alfvén vibrations of magnetar is understood. If all the detected energy of -ray outburst goes in the quake-induced vibrations, , then the initial amplitude is determined unambiguously. The impact of magnetic field decay on frequency and amplitude of torsional Alfvén vibrations in quadrupole overtone is illustrated in Figure 1, where we plot with pointed out parameters and . The magnetic-field-decay-induced lengthening of period of pulsating radiation (equal to period of vibrations) is described by On comparing given by (14) and (19), one finds that interrelation between the equilibrium value of the total magnetic moment of a neutron star of mass and radius km vibrating in quadrupole overtone of toroidal -mode is given by For a sake of comparison, in the considered model of vibration-powered radiation, the equation of magnetic field evolution is obtained in a similar fashion as that for the angular velocity in the standard model of rotation-powered neutron star which rests on which lead to where is angle of inclination of to . The time evolution of , and expression for are too described by (19). It is these equations which lead to widely used exact analytic estimate of magnetic field on the neutron star pole: . For a neutron star of mass and radius km, one has  G. Thus, the substantial physical difference between vibration- and rotation-powered neutron star models is that in the former model the elongation of pulse period is attributed to magnetic field decay, whereas in the latter one the period lengthening is ascribed to the slow-down of rotation [24, 25].

Figure 1: (Color online) The figure illustrates the effect of magnetic field decay on the vibration frequency and amplitude of quadrupole toroidal -mode presented as functions of .

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 10935001 and 10973002), the National Basic Research Program of China (Grant no. 2009CB824800), and the John Templeton Foundation.

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