Nonradial oscillations of the magnetized rotating stars with purely toroidal magnetic fields
Abstract
We calculate nonaxisymmetric oscillations of uniformly rotating polytropes magnetized with a purely toroidal magnetic field, taking account of the effects of the deformation due to the magnetic field. As for rotation, we consider only the effects of Coriolis force on the oscillation modes, ignoring those of the centrifugal force, that is, of the rotational deformation of the star. Since separation of variables is not possible for the oscillation of rotating magnetized stars, we employ finite series expansions for the perturbations using spherical harmonic functions. We calculate magnetically modified normal modes such as , , , , and inertial modes. In the lowest order, the frequency shifts produced by the magnetic field scale with the square of the characteristic Alfvén frequency. As a measure of the effects of the magnetic field, we calculate the proportionality constant for the frequency shifts for various oscillation modes. We find that the effects of the deformation are significant for high frequency modes such as  and modes but unimportant for low frequency modes such as , , and inertial modes.
keywords:
– stars: magnetic fields – stars: neutron – stars: oscillations.1 Introduction
Since the first discovery report of quasiperiodic oscillations (QPOs) in the tail of the giant X/ray flare from the soft ray repeater (SGR) 180620 (Israel et al. 2005), extensive theoretical studies have been carried out to identify physical mechanisms responsible for the QPOs. SGRs belong to what we call magnetars, neutron stars possessing an extremely strong magnetic field as strong as G at the surface. Giant flares have so far been observed only from three SGRs, that is, SGR 052666, 1900+44, and 180620, and just once for each of the SGRs, indicating that giant flares from magnetars are quite rare events. For magnetars, see reviews, for example, by Woods & Thompson (2006) and Mereghetti (2008). Analyzing archival data of another magnetar candidate SGR 1900+14, Strohmayer & Watts (2005) have succeeded in identifying QPOs in the Xray giant flare that was observed in 1998. QPOs frequencies now identified in the giant flares from the two magnetar candidates are 18, 30, 92.5, 150, 626 Hz for SGR 180620 (Israel et al. 2005; Strohmayer & Watts 2006; Watts & Strohmayer 2006), and 28, 53.5, 84, 155 Hz for SGR 1900+14 (Strohmayer & Watts 2005). Employing Bayesian statistics, Hambaryan et al. (2011) have reanalyzed the data for SGR 180620 to identify QPO frequencies at 16.9, 21.4, 36.8, 59.0, 61.3, and 116.3 Hz. For the giant flare in 1979 from SGR 052666, Watts (2011) mentioned in her review paper a report of a QPO at Hz, but she also suggested the difficulty in the frequency analysis in the impulsive phase of the burst. Here, it is worth mentioning a promising recent attempt to find QPOs in short recurrent bursts in SGRs. Huppenkothen et al. (2014) have succeeded in identifying candidate signals at 260, 93, and 127 Hz from J15505418, where they used Bayesian statistics for the analysis.
The QPOs are now regarded as a manifestation of global oscillations of the underlying neutron stars, and it is expected that they can be used for seismological studies of the magnetars. Seismological studies of magnetars may have started with a paper by Duncan (1998), who suggested that frequent starquakes in SGRs could excite crustal toroidal modes and the burst emission modulated at the mode frequencies would be detected. The detection of QPOs in the giant flare from SGR 180620 in 2004 (Israel et al. 2005) has given a huge trigger leading to subsequent intensive theoretical studies of QPOs in magnetars. In the early studies of QPOs in magnetars (e.g., Piro 2005; Lee 2007), the oscillations were assumed to be practically confined in the solid crust, as Duncan (1998) anticipated, and the effects of a magnetic field in the fluid core were ignored. Since magnetars are believed to possess an extremely strong magnetic field threading both the solid crust and the fluid core, to determine the oscillation frequency spectra of the stars, we need to correctly take account of the effects of the strong magnetic field in both regions on the oscillations. Applying a toy model, Glampedakis, Samuelsson, and Andersson (2006) discussed toroidal oscillations as global discrete modes residing in the fluid core and in the solid crust both threaded by a strong magnetic field, and showed that the modes that are most likely to be excited in magnetars are such that the crust and the core oscillate in concert. Also using a toy model, Levin (2006, 2007) put forward a different idea that Alfvén modes in the fluid core may lead to the formation of continuum frequency spectra and that toroidal crust modes will be rapidly damped as a result of resonant absorption in the core, and even suggested that there exist no discrete normal modes for the strongly magnetized neutron stars. However, assuming a pure poloidal field threading both the solid crust and the fluid core, Lee (2008) and Asai & Lee (2014) carried out normal mode calculations of axisymmetric toroidal modes and found discrete toroidal modes. Later on, van Hoven & Levin (2011, 2012), employing spectral method of calculation, have succeeded in suggesting the existence of discrete modes in the gaps between continuum frequency bands. In general relativistic frame work, Sotani et al. (2007) computed normal modes of magnetized neutron stars in the weak magnetic field limit.
Oscillations in magnetized stars are governed by a set of linearized partial differential equations. Normal mode analysis of the oscillations of magnetized stars are not necessarily easy to conduct, partly because separation of variables between the radial and the angular coordinates is in general impossible to represent the perturbations, and partly because the possible existence of continuum bands in the frequency spectra make it difficult to properly calculate oscillation modes, particularly belonging to the continua (see, e.g., Goedbloed & Poedts 2004). In normal mode calculations, we usually employ series expansions of a finite length to represent the perturbations so that the set of linearized partial differential equations reduces to a set of linear ordinary differential equations for the expansion coefficients. This process can be very cumbersome when we try to carry out modal analyses for various configurations of magnetic fields. This may be one of the reasons why MHD simulations have been used to investigate the small amplitude oscillations of magnetized neutron stars by many authors including, e.g., Sotani, Kokkotas & Stergioulas (2008), CerdáDurán et al. (2009), Colaiuda & Kokkotas (2011, 2012), Gabler et al. (2011, 2012, 2013a,b), Lander et al. (2010), Passamonti & Lander (2013, 2014). In the analyses with MHD simulations, QPOs are believed to be associated with continuum spectra and should be properly distinguished from discrete normal modes by closely watching motions and phases of various points in the interior.
Configurations of magnetic fields in magnetars are highly uncertain (e.g., Thompson & Duncan 1993, 1996; Thompson, Lyuitikov & Kulkarni 2002). As shown by the corecollapse supernova MHD simulations (e.g., Kotake, Sato & Takahashi 2006), toroidal fields can be easily produced and amplified when winding of the initial seed poloidal fields is effective in the differentially rotating core even if there is no initial toroidal fields. For modal analyses, it is desirable to examine various magnetic field configurations. Most of the modal analyses so far carried out have been for a purely poloidal magnetic field (Lee 2007, 2008; Sotani et al 2007, 2008; CerdáDurán et al. 2009; Colaiuda & Kokkotas 2011, 2012; Gabler et al. 2011, 2012; Passamonti & Lander 2013, 2014; Asai & Lee 2014). Recently, however, some authors, using MHD simulations, started investigating small amplitude oscillations for a purely toroidal magnetic field (Lander et al. 2010; Passamonti & Lander 2013), and even for mixed poloidal and toroidal field configurations (Gabler et al. 2013). Using MHD simulations, for example, Lander et al (2010), calculated rotational modes (modes and inertial modes) of magnetized stars and showed that modes at rapid rotation tend to magnetic modes in the slow rotation limit.
In this paper we carry out normal mode analysis of polytropic models with a purely toroidal magnetic field for various nonaxisymmetri oscillation modes, including , , modes, and rotational modes such as modes and inertial modes, where we include the effects of equilibrium deformation due to the magnetic field on the oscillations. To calculate rotational modes, we only take account of Coriolis force and include no effects of the centrifugal force. The oscillation equations for magnetically deformed rotating stars are derived by following the formulation similar to that by Saio (1981) (see also Lee 1993; Yoshida & Lee 2000a). The numerical method to compute normal modes of magnetized rotating stars is the same as that in Lee (2005) (see also Lee 2007), who employed series expansions of a finite length in terms of spherical harmonic functions for the perturbations. This paper is organized as follows. §2 describes the method used to construct a magnetically deformed equilibrium stellar model, and perturbation equations for nonaxisymmetric oscillation modes in magnetized rotating stars are derived in §3. Numeical results are summarized in §4 and we conclude in §5. The details of the oscillation equations solved in this paper and suitable boundary conditions imposed at the stellar center and surface are given in Appendix A.
2 Equilibrium model
We consider the oscillations of uniformly rotating polytropes with purely toroidal magnetic fields. Equilibrium structures of stars having purely toroidal magnetic fields have so far been studied with nonperturbative approaches within the framework of Newtonian mechanics (Miketinac 1973) and of general relativity (Kiuchi & Yoshida 2008; Frieben & Rezzolla 2012). In this study, we employ a perturbative approach to construct stars deformed by a purely toroidal magnetic field. Following Miketinac (1973), a purely toroidal magnetic field imposed on the stars in equilibrium is assumed to be given by
(1) 
where is a constant, is the parameter used for the strength of the magnetic field in the interior, is the density at the stellar center, and is the radius of the star. Here, we use spherical polar coordinates . The magnitude of the magnetic field is given by , where and . The fluid velocity in equilibrium is assumed to be given by
(2) 
where denotes the angular velocity of the uniformly rotating star. In this study, the deformation of the star is assumed to be solely caused by the magnetic fields and the effects of the centrifugal force are ignored. By the assumptions (1) and (2), the induction and continuity equations are automatically satisfied and need not be considered any further. For the toroidal field (1), we can write the Lorentz force per unit mass as a potential force, that is,
(3) 
The structure of a star in equilibrium is then determined by the hydrostatic equation, the Poisson equation, and the equation of state:
(4) 
(5) 
(6) 
where and are the polytropic index and the structure constant given by the mass and the radius of the star, is the gravitational constant, is the gravitational potential, and is the effective potential defined by
(7) 
where is the characteristic Alfvén frequency of the star and is a constant. Here, denotes the Legendre polynominal of order .
Since the potential is the quantity of order of with being the mass of the star, the ratio of the second term on the right hand side of equation (7) to may be given by where . For a neutron star model of the mass and radius , for example, we have for the field strength G, suggesting that the effects of the magnetic field on the equilibrium structure is not significant so long as G. In this paper, as mentioned before, we assume that the magnetic field is sufficiently weak so that the deformation of the equilibrium structure due to the magnetic field can be treated as a small perturbation to the nonmagnetic stars when . Under this assumption, we can regard appearing in the terms proportional to in equation (7) as the density in the nonmagnetic star. Thus, satisfies
(8) 
Since can be regarded as a function of from equations (4) and (6), if we expand as
(9) 
we may expand as
(10) 
Here, and are the gravitational potential and the density of the nonmagnetized star, and they satisfy , , and .
Substituting equations (9) and (10) into (8), we find
(11) 
from which we obtain the following linear ordinary differential equations for and :
(12) 
(13) 
where
(14) 
and
(15) 
In order to numerically integrate the differential equations (12) and (13) from the stellar center, we need to impose the regularity condition at the center, which may be obtained by substituting the expansion around the center
(16) 
into (12) and (13) for and , respectively. Since , , and as , where , , and are constants, values of the exponent are given by for the regular solution at the center and the expansion coefficients and remain undetermined. Assuming that the density at the center is independent of , we have for . The expansion coefficient for must be specified by applying the surface boundary condition:
(17) 
See Appendix B for the derivation of the boundary condition.
3 Perturbation equations
For the modal analysis of magnetically deformed stars, we introduce the parameter that labels equipotential surfaces of . The parameter is defined such that , that is,
(18) 
which may define the equipotential surface as given by a function . Assuming the deviation of the equipotential surface from the spherical surface is small, we define the function as
(19) 
and we assume that is the quantity of order of . By substituting equation (19) into (18), we obtain the explicit expression for the function up to the order of :
(20) 
where
(21) 
where , and denotes the mass inside the constant surface and .
Hereafter, we employ the parameter instead of the polar radial coordinate as the radial coordinate. In this coordinate system , the line element is given by
(22) 
Note that in this coordinate system, the pressure, the density and the effective potential of a magnetized star depend only on the radial coordinate , although the orthogonality of the basis vectors is lost.
The governing equations of nonradial oscillations of a magnetized and uniformly rotating star are obtained by linearizing the basic equations. As for rotation effects on the oscillations, as mentioned in the previous section, we consider only the effects of the Coriolis force and ignore those of the centrifugal force, where we assume the rotation axis is parallel to the magnetic axis. Since the equilibrium state is assumed to be stationary and axisymmetric, the perturbation quantities are proportional to , where is the frequency observed in an inertial frame and is the azimuthal wave number. Then, the linearized basic equations which govern the adiabatic, nonradial oscillations of a magnetized and uniformly rotating star are written in the coordinate system , to second order in , as (Saio 1981; Lee 1993; Yoshida & Lee 2000a)
(23) 
(24) 
(25) 
(26) 
where denotes the oscillation frequency observed in the corotating frame of the star, is the displacement vector, the prime () indicates the Eulerian perturbation, and are the LeviCivita permutation symbols, is the determinant of the metric , and
(27) 
and , , and are the basis vectors in the , , and directions, respectively. Here, the vector in equation (23) comes from the Coriolis force and is given by (see, e.g., Lee 1993; Yoshida & Lee 2000a)
(28) 
and in equation (25) denotes the Schwarzschild discriminant defined as
(29) 
where . In this paper, for simplicity, we employ the Cowling approximation, neglecting .
Because of the Lorentz and Coriolis terms in the equation of motion (23), separation of variables for the perturbations is impossible between the radial coordinate and the angular coordinates . We therefore expand the perturbations in terms of the spherical harmonic functions with different ’s for a given azimuthal index . The displacement vector is then given by (see e.g., Lee 2005, 2007)
(30) 
(31) 
(32) 
and the vector is given by
(33) 
(34) 
(35) 
where and for even modes, and and for odd modes, respectively, and .
The Euler perturbations of the pressure and density are given by
(36) 
In this paper, we usually use to obtain solutions with sufficiently highangular resolution. Substituting the expansions (30) (36) into the linearized basic equations (23)(26), we obtain a finite set of coupled linear ordinary differential equations for the expansion coefficients and , which we call the oscillation equations to be solved in the interior of magnetized and uniformly rotating stars. The set of oscillation equations obtained for the magnetized rotating star is given in Appendix A.
4 Numerical results
In this study, the three polytropes with indices , , and are used for modal analyses of the magnetized stars. The polytropes with and and with are regarded as simplified models of the neutron star and the normal star, respectively. For these polytopes, distributions of the density and the imposed magnetic fields are shown in Fig 1. In this figure, the equidensity and equimagnetic field strength contours on the meridional cross sections are given. We see that the density and magnetic field distribution of the models with a larger polytropic index tend to be concentrated in the central region of the star. Typical values of the mass and radius for the neutron star and the normal star are assumed to be and , respectively. Thus, we have for the neutron star with the field strength G, and for the normal star with the field strength G. For the polytrope with , the functions and for the magnetic deformation are plotted as functions of in Fig. 2.
4.1 , , and modes
We first calculate , low radial order , and modes of the polytropes taking account of the effects of the toroidal magnetic field. In these calculations, no effects of rotation are considered. To study oscillation modes for the neutron star and normal star models, the adiabatic indices for perturbations are assumed to be
(37) 
with being a constant, for which In this subsection, we use for the polytropes with the indices and 1.5, but for the polytrope with , we assume , and hence . Since all the magnetic terms in the oscillation equations are proportional to , an oscillation frequency of the modes may be written by (see Appendices A & C and Unno et al. 1989)
(38) 
where is the oscillation frequency of the nonmagnetized star, and is a proportionality coefficient and can be obtained by calculating the oscillation frequency of the mode for two different values of , that is, and , for example. Here, and are the quantities normalized by the Kepler frequency of the star . This coefficient for a mode may also be calculated by using the eigenfunctions of the nonmagnetized star by treating as a small parameter. We have used the symbol to denote the coefficient computed by using the eigenfunctions for the nonmagnetized star. The derivation and the explicit expression for are given in Appendix C.
mode  

0.00570  1.578()  1.490()  1.491()  
0.00770  2.051()  2.095()  2.095()  
0.01203  3.003()  3.063()  3.064()  
3.26931  6.162()  6.164()  3.553()  
5.09325  1.113()  1.114()  5.650()  
6.85013  1.550()  1.554()  7.586()  
0.00884  3.518()  3.488()  3.487()  
0.01152  2.427()  2.402()  2.402()  
0.01678  1.133()  1.130()  1.130()  
1.65562  4.107()  4.109()  3.567()  
3.79225  7.099()  7.103()  4.387()  
5.67886  1.173()  1.175()  6.713()  
7.48089  1.605()  1.608()  8.874()  
0.01136  7.583()  7.498()  7.498()  
0.01443  5.352()  5.319()  5.319()  
0.02008  2.732()  2.732()  2.731()  
1.97094  6.696()  6.696()  5.545()  
4.22956  8.280()  8.282()  5.196()  
6.18783  1.272()  1.272()  7.593()  
8.04091  1.701()  1.703()  9.900() 

We use the notation of .
mode  

0.00788  8.899()  8.120()  8.123()  
0.01057  1.179()  1.193()  1.193()  
0.01626  1.797()  1.856()  1.856()  
3.08199  5.763()  5.761()  9.406()  
4.64233  9.672()  9.668()  1.600()  
6.15692  1.329()  1.328()  2.190()  
0.01214  2.316()  2.286()  2.286()  
0.01564  1.628()  1.607()  1.607()  
0.02217  8.198()  8.153()  8.152()  
1.84930  4.027()  4.027()  1.621()  
3.55537  7.093()  7.093()  1.205()  
5.14850  1.073()  1.073()  1.808()  
6.69114  1.429()  1.428()  2.454()  
0.01547  4.950()  4.872()  4.872()  
0.01935  3.546()  3.511()  3.511()  
0.02596  1.941()  1.938()  1.938()  
2.15084  5.640()  5.640()  2.301()  
3.93952  8.393()  8.391()  1.504()  
5.58066  1.192()  1.191()  2.007()  
7.15896  1.544()  1.542()  2.653() 

We use the notation of .
mode  

0.88994  1.088()  7.875()  7.747()  
1.16154  1.270()  1.079()  9.976()  
1.68082  9.069()  7.333()  1.306()  
3.81006  2.906()  2.905()  2.689()  
5.01208  4.335()  4.331()  2.803()  
6.25522  5.541()  5.538()  3.836()  
1.35792  1.948()  1.930()  1.802()  
1.70580  1.502()  1.494()  1.332()  
2.29614  1.103()  1.099()  8.008()  
3.06379  2.193()  2.194()  1.814()  
4.14666  3.521()  3.521()  1.064()  
5.39097  4.792()  4.788()  7.335()  
6.65382  5.990()  5.978()  6.608()  
1.70370  4.161()  4.122()  3.942()  
2.07374  3.245()  3.227()  3.007()  
2.64527  2.338()  2.334()  1.920()  
3.12498  2.567()  2.566()  3.173()  
4.37567  3.947()  3.946()  1.118()  
5.68481  5.193()  5.189()  8.675()  
6.98164  6.392()  6.383()  8.076() 

We use the notation of .
In Tables 1–3, we tabulate the coefficients and as well as the frequency for the modes and low radial order  and modes of , 2, and 3 for the polytropes with the indices , , and . We observe that the two coefficients and are in good agreement with each other, except for a few very low frequency modes. In these tables, we also tabulate , which is the same as the coefficient but calculated ignoring all the equilibrium deformation effects due to magnetic stress. We see that the frequencies of the  and modes are strongly affected by the equilibrium deformation, but the deformation effects are not very important for the modes. This property of the frequency responses to the deformation due to magnetic field is quite similar to that found for the rotational deformation (Saio 1981). We note that the frequency we obtain for mode is consistent with that by Lander et al. (2010) for 0.1 (because they consider the effects of the second order of ).
From Tables 1 and 2, we see modal properties of , low radial order , and modes for the neutron star models. Because of the small value of , the frequencies of the modes are quite low, which may be consistent with almost isentropic stratifications expected in the deep interior of cold neutron stars. We find the ratio for the modes is much larger than the ratio for the  and modes, suggesting the low frequency modes are more susceptible to the magnetic field, reflecting their very low frequencies of order of . The ratio for the modes increases with , while the ratio for  and modes only weakly depends on . From Table 3, we see modal properties of , low radial order , and modes for the normal star model. The ratio have almost the same order of magnitudes for the , , and modes, except for the modes, for which the ratio is much smaller. It is also interesting to note that the magnitudes of the ratio for the modes is of order of 0.1 (except for the modes), the value of which is much smaller than those for the modes of the polytropes of and 1.5 with .
The coefficient in the tables 1 to 3 is computed by using two data points with different values of . For example, computed with 4 data points for modes of and for are and , and the coefficients for modes of and are and . We therefore think that the coefficients in the tables have at least three significant digits.
In Figure 3, we plot the expansion coefficients , , and of the , , and modes of for the polytrope with G. The first expansion coefficients associated with the harmonic degree and are dominant over the coefficients with and for , and the difference in the dominant expansion coefficients of the modes between the magnetized and nonmagnetized models is almost indiscernible. Because of the surface boundary condition (54) and an algebraic relation (44) in Appendix A, when and we have at the surface and hence at the surface can be very large for modes having very low frequencies for the normalization . In Figure 4, we plot magnetic field perturbations , , and of the , , and modes of for the polytrope with G, where , , and .
For slowly rotating stars, we may write the inertial frame oscillation frequency of a mode as
(39) 
where represents the response of the mode frequency to the slow rotation. Since for G and , the rotational effects may dominate the magnetic ones for .
4.2 rotational modes
We consider the effects of the magnetic field on rotational modes such as inertial modes and modes, for which the Coriolis force is the restoring force and the oscillation frequency is proportional to the rotation frequency . As shown by Yoshida & Lee (2000b), the stratification of the stellar interior strongly affects modal properties of the rotational mode. Since we are concerned with purely magnetic effects on the rotational mode, in this subsection, we focus on nonstratified stars or isentropic stars, in which the adiabatic index for perturbations is given by . To represent the effects of the magnetic field on the rotational modes, it is convenient to use the frequency ratio , where denotes the frequency observed in the corotating frame of the star, and for small values of we may write
(40) 
where the coefficient may depend on the rotation rate and the coefficient is a constant in the limit of (see Appendix C). Inertial modes and modes of uniformly rotating isentropic polytropes were studied, for example, by Lockitch & Friedman (1999) and Yoshida & Lee (2000a). The modes of and are nonaxisymmetric and retrograde modes and the frequency ratio tends to as . The ratio for an inertial mode also tends to a definite value as , depending on , , and the polytropic index (see, e.g., Yoshida & Lee 2000a), and we may use as a labeling of the inertial modes for a given .
Since stars with strong magnetic fields are frequently very slow rotators, we consider rotational modes in slowly rotating stars. In Table 4, the coefficients and as well as are tabulated for the modes and inertial modes for of isentropic (i.e., ) polytropes with three different indices . We use the symbol to denote the coefficient computed by using the eigenfunctions of nonmagnetized slowly rotating stars. The coefficient of the fitting formula , where and , can be calculated by a leastsquare method. In the table, means the modes, and correspond to inertial modes (see Yoshida & Lee 2000a), and the even and odd numbers of indicate even and odd parity, respectively. Since using , the frequencies of the rotational modes can be written by , the intercept of the fitting formula can also be calculated by a leastsquare method. We find that the coefficients and are in good agreement with each other. It is important to note that the effects of the magnetic deformation on the rotational modes are quite small, which is similar to the case of low frequency modes. We note that the frequency we obtain for mode is consistent with that by Lander et al. (2010).
0.66666  8.324()  8.326()  
0.55660  9.105()  9.171()  
1.10002  2.398()  2.401()  
1.02590  3.304()  3.316( 