Axisymmetric oscillations of magnetic neutron stars

We calculate axisymmetric oscillations of rotating neutron stars composed of the surface fluid ocean, solid crust, and fluid core, taking account of a dipole magnetic field as strong as $B_S\sim 10^{15}$G at the surface. The adiabatic oscillation equations for the solid crust threaded by a dipole magnetic field are derived in Newtonian dynamics, on the assumption that the axis of rotation is aligned with the magnetic axis so that perturbations on the equilibrium can be represented by series expansions in terms of spherical harmonic functions $Y_l^m(\theta,\phi)$ with different degrees $l$ for a given azimuthal wave number $m$ around the the magnetic axis. Although the three component models can support a rich variety of oscillation modes, axisymmetric ($m=0$) toroidal $_{l}t_n$ and spheroidal $_ls_n$ shear waves propagating in the solid crust are our main concerns, where $l$ and $n$ denote the harmonic degree and the radial order of the modes, respectively. In the absence of rotation, axisymmetric spheroidal and toroidal modes are completely decoupled, and we consider the effects of rotation on the oscillation modes only in the limit of slow rotation. We find that the oscillation frequencies of the fundamental toroidal torsional modes $_{l}t_n$ in the crust are hardly affected by the magnetic field as strong as $B_S\sim 10^{15}$G at the surface. As the radial order $n$ of the shear modes in the crust becomes higher, however, both spheroidal and toroidal modes become susceptible to the magnetic field and their frequencies in general get higher with increasing $B_S$. We also find that the surface $g$ modes and the crust/ocean interfacial modes are suppressed by a strong magnetic field, and that there appear magnetic modes in the presence of a strong magnetic field.


INTRODUCTION
Recent discoveries of quasi-periodic oscillations (QPOs) in the giant flares of Soft Gamma-Ray Repeaters SGR 1806-20 (Israel et al 2005) and SGR 1900+14 (Strohmayer & Watts 2005) and the confirmation of the discoveries  have made promising asteroseismology for magnetors, neutron stars with an extremely strong magnetic field (see, e.g., Woods & Thompson 2004 for a review on SGRs). Israel et al (2005) discovered QPOs of frequencies ∼ 18, ∼ 30 and ∼90Hz in the tail of the SGR 1806-20 hyperflare observed December 2004, and suggested that the 30Hz and 90Hz QPOs could be caused by seismic vibrations of the neutron star crust (see, e.g., Duncan 1998). Later on, in the hyperflare of SGR 1900+14, Strohmayer & Watts (2005) found QPOs of frequencies 28, 53.5, 84, and 155 Hz, and claimed that the QPOs could be identified as low l fundamental toroidal torsional modes in the solid crust of the neutron star. Although the interpretation in terms of the crustal torsional modes is promising, the mode identification cannot always be definite. In fact, if different classes of oscillation modes can generate similar periodicities and information other than the periods are not available for the QPOs, it is usually difficult to assign a class of oscillation modes to the periods observed in preference to other classes of modes. This is particularly true for high frequency QPOs (e.g., 625Hz QPO,  1835Hz QPO and less significant QPOs at 720 and 2384 Hz in SGR 1806-20, , for which there exist various classes of modes that can generate the periodicities observed. In this case, the pattern of observed frequencies, that is, the observed frequency spectrum could be a key for mode identification and hence understanding the underlying neutron stars themselves. McDermott et al (1988) have carried out detailed modal analyses for neutron star models with a solid crust, but without the effects of magnetic field and rotation on the oscillation modes. Since the detected QPOs are thought to come from magnetors inferred to possess a strong magnetic field, it is useful to calculate oscillations of rotating neutron stars that have a solid crust and a strong magnetic field.
An extensive theoretical modal analysis of magnetic neutron stars having a solid crust was first carried out by Carroll et al (1986), who however employed a cylindrical geometry for the analysis of the neutron star models, and assumed a uniform magnetic field whose axis is parallel to the axis of the cylinder. The maximum strength of the magnetic field they examined is BS ∼ 10 12 G, which may be too weak for mangetors for which BS > ∼ 10 14 G is inferred. Carroll et al (1986) have suggested the existence of Alfvén modes and also found the transformation of the g modes in the surface ocean into magnetic modes as the field becomes stronger. More recently, Piro (2005) solved a simplified set of oscillation equations for toroidal torsional modes employing improved shear modulus (Strohmayer et al 1991) and up to date microphysics for equations of state to construct back ground neutron star models. Since the fluid core was ignored in the calculations by Carroll et al (1986) and Piro (2005), no reliable modal analyses were possible for spheroidal modes that can have substantial amplitudes in the core. In this paper, we calculate various oscillation modes of neutron star models that have a solid crust and are threaded by a dipole magnetic field. Since a non-radial mode of a spherical neutron star thereaded by a dipole magnetic field cannot be represented by a single spherical harmonic function, we employ series expansions to represent the perturbations accompanied by a mode. In §2, we give a brief description of method of solution we empoy, numerical results are given in §3, and §4 is for conclusions. The oscillation equations are given in Appendix as well as boundary and jump conditions used in this paper.

Perturbation Equations
To derive oscillation equations for a magnetized and rotating neutron star with a solid crust, we follow Carroll et al (1986), Lee & Strohmayer (1996), and Lee (2004). We consider no general relativistic effects on oscillations, that is, the oscillation equations are derived in Newtonian dynamics. We employ spherical polar coordinates (r, θ, φ), whose origin is at the center of the star and the axis of rotation is the axis of θ = 0. We assume a dipole magnetic field given by where µm is the magnetic dipole moment. For simplicity, we also assume that the magnetic axis coincides with the rotation axis. Since the dipole field is a force-free field such that (∇ × B) × B = 0, the field does not influence the equilibrium structure of the star. Assuming the axis of rotation is also the magnetic axis, the temporal and angular dependence of perturbations can be given by a single factor e i(mφ+ωt) , where m is the azimuthal wavenumber around the rotation axis and ω ≡ σ + mΩ is the oscillation frequency observed in the corotating frame of the star where σ is the oscillations frequency in an inertial frame, and Ω is the angular frequency of rotation. The linearized basic equations applied in the solid crustal region of the star are then given by where ρ is the mass density, p is the pressure, c is the velocity of light, ξ is the displacement vector, B ′ and E ′ are the Euler perturbations of magnetic and electric fields, respectively, and the other physical quantities with a prime ( ′ ) denote their Euler perturbations, and A is the Schwartzshild discriminant defined by and Note that we have employed the Cowling approximation neglecting the Eulerian perturbation of the gravitational potential, and that no effects of rotational deformation are included. In equation (2), σ ′ denotes the Euler perturbation of the stress tensor and is obtained from the Lagrangian perturbation defined in Cartesian coordinates by with uij being the strain tensor defined by where δij denotes Kronecker delta, µ is the shear modulus, and u = 3 l=1 u ll . The last term on the right-hand side of equation (2) represents the contribution from the displacement current, where infinite conductivity has been assumed. The linearized basic equations for a fluid region may be obtained by simply replacing the terms ∇ · σ and ∇ · σ ′ by −∇p and −∇p ′ , respectively.
Since the angular dependence of perturbations on a rotating and magnetized star cannot be represented by a single spherical harmonic function, we expand the perturbed quantities in terms of spherical harmonic functions Y m l with different ls for a given m, on the assumption that the axis of rotation coincides with that of the magnetic field. The displacement vector ξ and the perturbed magnetic field B ′ are then approximately represented by finite series expansions of length jmax as and and the pressure perturbation, p ′ , for example, is given by where B0(r) = µm/r 3 , and lj = |m| + 2(j − 1) and l ′ j = lj + 1 for even modes, and lj = |m| + 2j − 1 and l ′ j = lj − 1 for odd modes, respectively, and j = 1, 2, 3, · · · , jmax. In this paper, we have used jmax = 12. Substituting the expansions (11)∼(13) into the linearized basic equations (2)∼(6), and making use of equations (9) and (10), we obtain a finite set of coupled linear ordinary differential equations for the expansion coefficients such as S l j (r) and b S l ′ j (r), which we call the oscillation equations solved in the solid crust. The oscillation equations to be solved in magnetic fluid regions can be derived in the same manner (e.g., Lee 2004). The sets of oscillation equations thus obtained for the solid and fluid regions are given in Appendix A for the case of axisymmetric modes with m = 0. The boundary conditions at the center and the surface of the star and the jump conditions imposed at the interfaces between fluid and crustal regions are discussed in Appendix B. It is important to note that in the case of Ω = 0 the oscillation equations for axisymmetric modes with m = 0 are decoupled into those for spheroidal modes and toroidal modes, respectively, and that axisymmetric spheroidal and toroidal modes are coupled only through the effects of rotation. For non-axisymmetric modes with m = 0, however there occurs no decoupling between spheroidal modes and toroidal modes even for Ω = 0.

Non-Magnetic Core
In a fluid region where the magnetic pressure pB ≡ B 2 0 (r)/8π is much smaller than the gas pressure p, that is, where the Alfvén velocity vA = 2pB/ρ is much smaller than the sound velocity ∼ p/ρ or rω, the magnetic perturbations b H and ib T suffer very rapid spatial oscillations except for extremely low frequency oscillations (e.g., Lee 2004, see also Biront et al 1982, Roberts & Soward 1983, Cambell & Papaloizou 1986). This rapid spatial oscillation happens in the fluid core for both spheroidal and toroidal modes. For axisymmetric toroidal modes, for example, the dimensionless wavenumber kr in the radial direction may be given by and the wavenumber can be very large in the magnetic fluid region immediately below the crust where pB/p ≪ 1 even for BS = B0(R) ∼ 10 15 G at the surface. Since this kind of rapid spatial oscillations usually do not occur in the solid crust, there exists an abrupt change in the property of the magnetic eigenfunctions across the boundary between the solid crust and the fluid core. Numerically, this means that we need an extremely large number of mesh points to correctly calculate very short magnetic perturbations in the fluid core, which is not feasible. Since the rapid spatial oscillations in the magnetic perturbations come from the fact that the magnetic pressure significantly weaker compared to the gas pressure in the fluid core, it is legitimate to assume that the fluid core is non-magnetic so that the oscillation equations for non-magnetic fluid stars (e.g., Lee & Saio 1990) could be used. Assuming non-magnetic core, we neglect a possible dissipation of the oscillation energy due to Joule heating in the core.

NUMERICAL RESULTS
In this paper we mainly discuss the case of Ω = 0, in which axisymmetric (m = 0) toroidal and spheroidal modes are decoupled.
We briefly discuss the effects of rotation on the axisymmetric modes, assuming slow rotation so that Ω/ GM/R 3 ≪ 1, in which case toroidal and spheroidal modes are only weakly coupled.

Equilibrium Models
The neutron star models we use consist of the surface fluid ocean, the solid crust, and the fluid core and are the same as those employed by McDermott et al (1988) for modal calculations without rotation and magnetic fields. The models are obtained from the fully general relativistic evolutionary cooling calculations of neutron stars by Richardson et al (1980). The outer crust extends down to the neutron drip point at ρ = 4.3 × 10 11 g cm −3 and is assumed to consist of bare Fe nuclei embedded in a uniform, neutralizing, degenerate electron gas. The strength of the Coulomb interaction between ions is characterized by the dimensionless parameter Γ, the ratio of the Coulomb energy to the termal energy kBT with kB being Boltzmann constant. The matter of the outer crust is assumed to undergo a first order fluid/solid phase transition at Γ = 155, and there exists the fluid ocean above the crystallization boundary at Γ = 155. In addition to the thermodynamic contributions from the nuclear and electronic kinetic energies, the equation of state used in the outer crust includes contributions from photons, the Coulomb interactions between nuclei and electrons, and nuclear vibrations and rotations. The inner crust extends from the neutron drip point at 4.3 × 10 11 g cm −3 to the base of the crust at 2.4 × 10 14 g cm −3 and it is assumed to consist of nuclei with Z ∼ 40, degenerate electrons, and degenerate, nonrelativistic neutrons, where the actual composition of the nuclei in the inner crust is taken from Negele & Vautherin (1973). For the inner crust, the zero temperature equation of state is that by Negele & Vautherin (1973), and the leading order thermal corrections for the nuclei, electrons, and free neutrons are also included. At densities greater than 2.4 × 10 14 g cm −3 , the lattice is assumed to dissolve, and the core of the neutron star is taken to consist of a mixture of free and highly degenerate neutrons, protons, and electrons. The equation of state in the fluid core is that by Baym, Bethe, & Pethick (1971), and are added the leading thermal corrections for the three species. The outer most fluid envelope models are from the calculations by Gudmundsson et al (1983), and are fitted to the evolutionaly core models by Richrdson et al (1980) at mass density ρ ≈ 10 10 g cm −3 . The more details of the models for oscillation calculations can be found in McDermott, van Horn, & Hansen (1988). The shear modulus µ of the solid lattice is that given by Pandharipande, Pines, & Smith (1976): where nN is the number density of the nuclei. An improved calculation for the shear modulus was given by Strohmayer et al (1991). This imporved shear modulus was used by several authors to calculate crustal oscillations of neutron stars, for example, by Duncun (1998) and Piro (2005). In this paper, we did not attempt to use imporved models for modal calculations since our main concerns are to investigate the modal properties of magnetic neutron stars and not to try a detailed comparison between theoretical predictions and observational results for the QPOs from the SGRs. For mode calculations in this paper, we mainly use neutron star models named NS05T7 and NS13T8, and the mass and radius of the former are M = 0.503M⊙ and R = 9.839km, respectively, and those of the latter are M = 1.326M⊙ and R = 7.853km, respectively. Both of the models have a solid crust, and the thickness of the crust is ∆r/R ∼ 0.24 for NS05T7 and ∆r/R ∼ 0.055 for NS13T8. The thickness of the surface ocean, on the other hand, is ∆r/R ∼ 3.8 × 10 −5 for NS05T7 and ∆r/R ∼ 2.3 × 10 −3 for NS13T8, respectively. The details of the physical parameters of these models are given in McDermott et al (1988).

Toroidal Modes
In Figure 1, the normalized frequenciesω ≡ ω/ GM/R 3 of the axisymmetric (m = 0) toroidal torsional modes l ′ tn are plotted versus BS for the models NS05T7 (top panel) and NS13T8 (bottom panel), where BS ≡ µm/R 3 is the strength of the dipole magnetic field at the surface. The frequenciesω are local ones and for convenience the redshifted frequencies defined by ν∞ ≡ ω(1 − 2GM/Rc 2 ) 1/2 /2π are also displayed on the right axis of the panels. We find that the frequenciesω of the fundamental toroidal modes are only weakly affected by the dipole magnetic field of strength as large as BS ∼ 10 15 G at the surface, which is consistent with the results by Piro (2005). Note that the frequencies of the fundamental modes l ′ t0 show a slight decrease with increasing BS for low values of l ′ . The overtone torsional modes l ′ tn having n ≥ 1 become more susceptible to the magnetic field as the radial order n gets higher and the wavelengths in the radial direction become shorter so that several wavelengths of the modes are spaned by the strong magnetic region in the outer most envelope (or crust). We find as a general trend the frequency of the overtone modes increases with increasing BS. Besides the magnetic effect just mentioned, there exists a different kind of magnetic effects closely related to the fact that the separation of variables using a single spherical harmonic function is not possible for the oscillations of stars with a dipole magnetic field. In this case we may consider crustal toroidal modes with different l ′ s but with the same parity are coupled in the presence of the magnetic field. Since the toroidal modes with a given radial order n ≥ 1 are nearly degenerate in frequency for different values of l ′ , the coupling effect brings about interference between them, which becomes significant as the field strength increases. As discussed by McDermott et al (1988) for non-magnetized neutron stars, the frequencies of the fundamental toroidal modes with different l ′ s approximately scale as 1 which is also confirmed even in the presence of a strong magnetic field BS ∼ 10 15 G. The eigenfunctions iT l ′ and ib T l of the axisymmetric fundamental 2t0 mode of the model NS05T7 are depicted in Figure  2 for BS = 10 12 G in panels (a) and (b) and for BS = 10 15 G in panels (c) and (d), where the solid and dashed lines stand for l ′ = 2 and 4 for iT l ′ and for l = 1 and 3 for ib T l , and the amplitude normalization is given by iT l ′ =2 = 1 at the surface of the star. Different from the case with no magnetic field, the toroidal components iT l ′ of the displacement vector is continuous at the crust/ocean interface (see Carroll et al 1986). However, since we have assumed the fluid core is non-magnetic, no toroidal components iT l ′ and ib T l exist in the core for Ω = 0. Both for BS = 10 12 G and BS = 10 15 G, iT l ′ =2 dominates iT l ′ =4 in the inner crust where pB ≪ p. For BS = 10 12 G, the eigenfunctions iT l ′ and ib T l are approximately constant as functions of r/R, but for BS = 10 15 G they are influenced by the strong magnetic field in the outer crust where the magnetic pressure pB dominates the gas pressure p.
Reflecting the difference in the thickness of the solid crust between the two neutron star models, the normalized frequency spectra of the toroidal torsional modes show substantial differences. For example, the fundamental l ′ t0 modes of the model NS13T8 have normalized frequenciesω lower than those of the model NS05T7, while the overtones of the former have higherω than the latter. This property remains the same even in the presence of a strong magnetic field. For the periods of the torsional modes of non-magnetic neutron stars, McDermott et al (1988) gave extrapolation formulae, in which the local periods are proportional to the radius of the star for the fundamental modes and to the crust thickness for the overtone modes for a given spherical harmonic degree l ′ . The normalized frequenciesω in the figure are consistent with what the extrapolation formulae predict.
On a closer look at the behavior of the low frequency torsional modes of the model NS13T8 in Figure 1, one may find that the torsional modes suffer mode crossings with magnetic modes, for which the oscillation frequency increases rapidly with increasing BS and the oscillation energy is dominated by the magnetic perturbations. In fact, if we calculate the energies defined by we find that EB(R) is much larger than EK (R) for the magnetic modes. Note that ∇ · ξ = 0 for toroidal magnetic modes. For more detail, see below §3.3.

Spheroidal Modes
Figure 3 plots the oscillation frequenciesω of the low radial order spheroidal shear modes l=2 sn and the core/crust interfacial modes l i2 for the models NS05T7 (top panel) and NS13T8 (bottom panel). For the model NS05T7, also plotted are magnetic modes, labeled m k , that are found in the presence of a strong magnetic field. Note that in this paper, we use the symbols l i1 and l i2 to denote the crust/ocean and core/crust interfacial modes, respectively, where the amplitudes of the former are strongly localized in a narrow region around the crust/ocean interface and the amplitudes of the latter are largest at the core/crust interface. The oscillation frequenciesω of the spheroidal shear modes increase with increasing BS and this frequency increase becomes more rapid for higher radial order n modes. As in the case of toroidal torsional modes, for a given radial order n, the spheroidal shear modes l sn with different degrees l are nearly degenerate in frequency, and mode couplings between the 1 As pointed out by one of the referees, the frequency scaling formula given by equation (16) is correct only for l ′ ≫ 1, and a much better scaling formula should beω l ′ ≈ (l ′ − 1)(l ′ + 2)/4ω l ′ =2 as one can easily verify by solving the simple case of a uniform density, uniform shear modulus oscillating crust.
shear modes become significant for strong BS. Figure 3 also shows that the frequency of the core/crust interfacial modes l i2 is hardly modified by the magnetic field as strong as BS = 10 15 G, except that the interfacial modes of the model NS05T7 expreience avoided crossings with magnetic modes, whose oscillation frequencies rapidly increase with increasing BS.
As in the case of the toroidal torsional modes, the normalized frequency spectra of the spheroidal modes of the two models appear differently, reflecting the difference in the thickness of the crust. The different appearance for the shear modes of the two models can be understood by using an approximation formula for the period, which is proportional to the crust thickness (e.g., McDermott et al 1988).
The eigenfunctions H l and b H l ′ of a 2s1 mode of the model NS05T7 are plotted in Figure 4 for BS = 10 12 G in panels (a) and (b) and for BS = 10 15 G in panels (c) and (d), where the amplitude normalization is given by S l 1 (R) = 1 at the surface. Note that because of the jump conditions applied at the interfaces between the solid crust and the fluid zones (see Appendix B), the horizontal component of the displacement vector is continuous at the outer interface, but it is discontinuous at the inner interface, at which almost free slippery jump condtions are employed. For BS = 10 12 G, the eigenfunctions b H l ′ exhibit zigzag behavior in the outer crust reflecting that of Γ1, which is caused by discontinuous change of equilibrium composition with increasing ρ. For BS = 10 12 G, the expansion coefficients H l 1 and b H l ′ 1 are dominant over others, but for BS = 10 15 G the first few components of H l and b H l ′ are comparable to each other, and the eigenfunctions in the outer crust are significantly modified by the strong field comaperd with those for the case of BS = 10 12 G. As shown by Figure 4, the amplitude of b H l ′ for BS = 10 12 G is much larger than that for BS = 10 15 G for the same amplitude normalization S l 1 (R) = 1. However, if we compare the quantities EK(r) and EB(r), we find EK(R) is much larger than EB(R) for BS = 10 12 G, but EK (R) and EB(R) are comparable to each other for BS = 10 15 G so that the magnetic perturbations is important to determine the modal properties of the shear modes in the presence of a strong magnetic field.
In Figure 5, the frequencyω of the fundamental 2f mode of the model NS05T7 is plotted versus BS. The 2f mode heavily suffers mode crossings with high radial order shear modes l sn as BS increases, and no pure 2f mode could be identified in the presence of a strong magnetic field. This is also the case for high frequency p modes for a strongly magnetized star. Note that, even at BS ∼ 10 12 G, the fundamental mode 2f withω = 1.8862 is affected by the shear mode 2sn=10 that has the frequencȳ ω = 1.8726 at BS = 0.
The frequencyω of the g1 mode in the fluid ocean and that of the core/ocean interfacial mode l=2 i1 are plotted versus BS for the model NS05T7 in Figure 6. The frequencies begin to decrease rapidly to zero beyond BS ∼ 10 5 G, suggesting the strong magnetic field suppresses these modes. This suppression may come from the dominance of the magnetic force over the buoyant force in the fluid ocean at large BS. We obtain almost the same suppression of the surface g modes and the crust/ocean interfacial modes for the model MS13T8, but they can survive much stronger magnetic field BS ∼ 10 10 G. This may be because the fluid ocean of the hotter model NS13T8 is much thicker and has a more developed radiative region, compared to the model NS05T7, giving much stronger buoyant force for the modes. It is interesting to note that the result for the g and l i1 modes in the fluid ocean in this paper is different from that obtained by Carroll et al (1986), who suggested that the frequency of g1 proportionally increases with increasing BS to be a magnetically dominating mode, which they called g/m modes. The reason for the discrepancy is probably the difference in the geometry of the models and the magnetic field configulation between the two calculations.

Magnetic Modes
For both toroidal and spheroidal modes, we find oscillation modes that can be regarded as magnetic modes, for which the frequencyω rapidly increases with increasing BS, and the magnetic energy EB(R) is dominant over the kinetic one EK(R), although its appearance looks different depending on the model structure, particularly, on the crust thickness. It is also important to note that there seem to exist critical strengths of the magnetic field beyong which magnetic modes are allowed to exist. We find that the low frequency torsional modes of the model NS13T8 are often affected by mode crossing with the magnetic modes, but that those of the model NS05T7 rarely suffer from the mode crossings within the frequency and magnetic field strength ranges we investigated in this paper. In the same parameter ranges as those for the toroidal torsional modes, we find magnetic modes interacting with the core/crust interfacial modes l i2 for the model NS05T7, but no examples of the mode crossings of this kind are found for the model NS13T8.
As an example of the mode crossings between the torsional modes and magnetic modes found for the model NS13T8, we plot, in Figure 7,ω of the 2t0 (left panel) and 2t1 (right panel) modes versus log Bs. The panels are magnifications of the corresponding parts from Figure 1, and clearly indicate mode interactions with magnetic modes as BS increases. It is interesting to note that between BS = 10 12 G and ∼ 10 13 G, the 2t1 mode also interact with sequences of modes whose frequencies decrease as BS increases. Figure 8 shows an example of the eigenfunctions of a toroidal magnetic mode havinḡ ω = 0.01557 at BS = 10 14.48 G for NS13T8, where the amplitude normalization is given by iT l ′ =2 = 1 at the surface. For this mode, the expansion coefficients iT l ′ =2 and ib T l=1 are dominant, and the magnetic perturbation ib T l shows almost discontinuous change at the crust/ocean interface, and has almost negligible amplitudes in the fluid ocean. Figure 9 is an example of the eigenfunctions of a spheroidal magnetic mode ofω = 0.07965 at Bs = 10 14.45 G for the model NS05T7, where the amplitude normalization is given by S l=0 (R) = 1. For this magnetic mode the first components Hl =2 and b H l ′ =1 of the series expansion are not necessarily dominant over other components associated with higher ls. Note that the relative thickness of the surface ocean of the model NS13T8 is much larger than that of NS05T7.

Effects of Slow Rotation
For slow rotation, we may approximate the oscillation frequency of axisymmetric (m = 0) modes as whereω0 denotes the oscillation frequency for the non-magnetic and non-rotating case. Note that the linear term in Ω is proportional to m and does not appear for axisymmetric modes, and that no effects of rotational deformation of the equilibrium structure on the oscillation are included in our calculation and hence the second order effects exclusively come from the Coriolis force, which is dominant for low frequency modes withω < ∼ 1 (e.g., Lee 1993). We tabulate in Table 1 the coefficients C2 for several toroidal and spheroidal modes including torsional l ′ tn and shear l sn modes for the models NS05T7 and NS13T8. Note that we have used the labels l i1 and l i2 to denote respectively the crust/ocean and core/crust interfacial modes for both the models. For the model NS05T7, the frequency of the 2i1 mode for BS = 0 is lower than that of the 2i2, but for the model NS13T8 the frequency of the 2i1 mode is higher than that of the 2i2 mode. From the table, we find that the second order response of the fundamental toroidal modes to rotation is negative and small, but that of the l ′ tn=1 is positive and rather large in the sense that the frequency correction C2Ω 2 can be comparable toω0 itself forΩ ∼ 0.1. This is particularly the case for the toroidal modes of the model NS13T8. In the same sense, the response of the low radial order spheroidal shear modes l sn with low l to rotation can be substantial as well, particularly for the model NS13T8. The effects of slow rotation on axisymmetric f and p1 modes, however, are not significant, the result of which is consistent with the calculation by Saio (1981) for a polytropic star with the index N = 3 and Γ1 = 5/3, who have however taken into account the effects of rotational deformation.

CONCLUSIONS
We have calculated axisymmetric (m = 0) oscillation modes of neutron star models that have a solid crust and are threaded by a dipole magnetic field. We find that the frequencies of the fundamental toroidal torsional modes are not affected significantly by the magnetic field as strong as BS ∼ 10 15 G at the surface, and that high radial order torsional and shear modes are susceptible to a magnetic field even if the strength at the surface is much less than BS ∼ 10 15 G. Because both spheroidal shear and toroidal torsional modes are almost degenerate in oscillation frequency for a given radial order n, the high radial order modes with different ls for a given n are easily coupled in the presence of a strong magnetic field. Since the f modes (and p modes) are embedded in the sea of high radial order shear modes with various ls that are sensitive to the magnetic field, the f modes suffer mode crossings with the shear modes as BS varies, and their identity may become ambiguous in the presence of a strong magnetic field.
We find that the g modes in the fluid ocean and the crust/ocean interfacial modes are suppressed in a strong magnetic field, the result of which contradicts that obtained by Carroll et al (1986), who showed that the ocean g modes will survive to be a magnetic mode, labeled m/g k in their paper, as the magnetic field becomes strong. The reason for the contradiction may be attributable to the difference in the geometry of the oscillating stars and of the magnetic field, that is, Carroll et al (1986) calculated oscillation modes of cylindrical stars threaded by a uniform magnetic field. The strength of the magnetic field necessary to completely suppress the ocean g modes and the crust/ocean interfacial modes depends on neutron star models, and if a neutron star has a hot buoyant radiative region in the ocean, the modes can survive a magnetic field of BS ∼ 10 10 G or stronger, although their frequency spectrum could be largely modified by the field.
As a model for burst oscillations observed in many low mass X-ray binary (LMXB) systems (e.g., Strohmayer et al 1997, Strohmayer & Bildsten 2004, van der Klis 2004, Heyl (2004) and Lee (2004) proposed that the oscillations are produced by low frequency buoyant r modes propagating in the surface fluid ocean of accreting neutron stars in the systems (see also Lee & Strohmayer 2005, Heyl 2005). Since no effects of magnetic field are properly taken into account in their analyses, it is needed to reexamine the r mode model for burst oscillations, whether the buoyant r modes can survive a strong magnetic field of the neutron stars and how their frequency spectrum is modified by the field.
We have also examined the effects of slow rotation on the axisymmetric oscillation modes, although no effects of rotational deformation are included. Since the first order effects of rotation do not appear for axisymmetric modes with m = 0, it is the second order effects of rotation that appear first and are due to the Coriolis force when no rotational deformation is considered. The second order effects can be important for torsional modes and shear modes with radial order n ≥ 1 in the sense that the second order corrections to the frequency due to rotation can be comparable to the frequency itself forΩ ∼ 0.1. But, for SGR 1806-20, for example, the rotation period is estimated to be 7.56s (e.g., Israel et al 2005), implying that the underlying object is a very slow rotator in the neutron star standard so thatΩ ≪ 1 and C2Ω 2 should be negligible compared toω0. The first order correction due to the Coriolis force for non-axisymmetric modes can be found in Lee & Strohmayer (1996).
We find magnetic modes for a strong magnetic field. Here, magnetic modes are regarded as a oscillation mode that exists only in the presence of a strong magnetic field, and whose frequency rapidly increases with increasing BS. We also note that the magnetic modes have their oscillation energy dominantly possessed by the magnetic perturbations. The modal properties of the magnetic modes found by Carroll et al (1986), labeled m/g k and a k , are not the same as the properties of the magnetic modes found in this paper. Note that although the magnetic modes computed by Carroll et al (1986) reside in the fluid ocean, the magnetic modes found in this paper have amplitudes both in the fluid ocean and in the solid crust, and a large fraction of the oscillation energy resides in the crust, as indicated by Figures 8 and 9. We note that a simple magnetohydrodynamical system can support Alfvén waves, as well as fast and slow magneto-acoustic waves in fluids (e.g., Sturrock 1994), and we may expect almost the same modal structure even for a magnetic solid, in which torsional waves propagate (e.g., Carroll et al 1986). At this moment, however, we have no clear classification scheme for the magnetic modes, which makes it difficult to obtain a good understanding of the modes. Further studies are definitely necessary for magnatic modes of neutron stars with a solid crust, since the magnetic modes could be important observationally for magnetor as an agent triggering instability for flares.
Because of the assumption of non-magnetic core employed in this paper ( §2.2), we completely ignore the possible existence of magnetosonic modes in the core and the possible coupling between core magnetic modes and crustal shear modes, for example. This assumption would be a serious flaw when we are interested in core (or more global) magnetic modes themselves. In fact, the crustal magnetic modes found in the present paper could be flawed in the sense that the assumption of non-magnetic core excludes from the beginning the possible existence of more global magnetic modes extending form the core to the crust. For non-magnetic modes such as crustal shear modes, however, so long as the field strength is less than ∼ 10 15 G at the surface so that the magnetic pressure is much smaller than the gas pressure (and/or the shear modulus) in the inner crust (see, e.g., Fig. 3 of Carroll et al (1986) or Fig. 1 of Piro (2005)), reflection of crustal shear waves at the bottom of the crust is effective to establish crustal shear modes well trapped in the crust, and the effects of the coupling with core magnetic modes would be minor, for example, on the frequency spectrum of the crustal shear modes. For the field strength much larger than 10 15 G, the confinement of crustal shear mode amplitudes into the crust would be imperfect and global treatment properly including the core will be necessary. As suggested by one of the referees, a differentially rotating magnetor progenitor would produce a toroidal magnetic field, for which the complexities possibly caused by mechanical coupling between the crust and the core could be avoided. This case may be important and worth careful examination. Recent discussions on the crust-core coupling of magnetic modes, leading to more global magnetic modes of neutron stars, are found in Levin (2006), Glampedakis, Samuelsson, & Andersson (2006), and Sotani, Kokkotas, & Stergioulas (2006).
The assumption of non-magnetic core also leads to neglect of another possible role played by perturbed magnetic field in the core. Several authors (e.g., Biront et al 1982, Roberts & Soward 1983, Cambell & Papaloizou 1986) have discussed for magnetic normal stars that in a deep fluid region where the magnetic pressure is much smaller than the gas pressure, magnetic perturbations become extremely short, decoupled from mechanical perturbations, which leads to a dissipation of the oscillation energy. In this paper, however, because of the numerical difficulty in properly treating the abrupt change of wave properties between the solid crust and the fluid core, we have employed a simplifying assumption that the fluid core is non-magnetic. Although the assumption of non-magnetic core would be reasonably justfied since in the fluid core the magnetic pressure pB is much smaller than the gas pressure p, appropriate estimations of the dissipative effects of extremely short magnetic perturbations in the fluid core are necessary to see whether the dissipation is significant enough to damp the excitation of the torsional oscillations. We have almost the same difficulty in treating the outer boundary conditions, for which we have assumed for simplicity no emission of electromagnetic waves from the surface even if we include the displacement current term. Although Carroll et al (1986) (see also McDermott et al 1988) suggested that the damping effects due to emission of electromagnetic waves from the surface are negligible except for magnetic modes, labeled m/g k and a k , fully consistent calculations including the damping effects are obviously necessary.
In the giant flare of SGR 1806-20 observed December 2004, there exist reports on detection of QPOs at 18, 30, and 92.5 Hz (Israel et al 2005), at 18, 92.5, and 626.5 Hz , and at ∼90, ∼150, 625, and 1835 Hz, and at ∼720 and 2384 Hz but with less significances . In the giant flare of SGR 1900+14 observed in August 1998, Strohmayer & Watts (2005 have also found detection of QPOs at 53.5, 84, and 155.5 Hz, and at 28 Hz with lower significance. Identifying the QPOs of low frequencies ν∞ < ∼ 100Hz with the fundamental toroidal torsionl modes with various ls is rather secure (McDermott et al 1988, Duncan 1998, Piro 2005, but identification of the middle to high frequency QPOs is not straightforward. In fact, QPOs with frequencies ∼ 100 to ∼ 1000 Hz can be generated by high l ′ l ′ t0 modes, or overtone modes l ′ t n≥1 with low l ′ , or low l spheroidal shear modes l s n≥1 , or core/crust interfacial modes l s2, depending on neutron star models, and there exist no unique solution to identification if the frequency is the only available information. For much higher frequency QPOs like that at 2384Hz, low l fundamental modes l f must be added to the list of candidates for the QPOs, although much more energy will be required to excite the modes to observable amplitudes than that for toroidal crustal modes. In this paper, we have been only concerned with axisymmetric modes with m = 0. If we extend our calculation to non-axisymmetric modes with m = 0, mode coupling effetcs will be much more significant in a strong magnetic field because both toroidal torsional and spheroidal shear modes are nearly degenerate in frequency for a given radial order n, which makes numerical analysis difficult and tedious. However, together with employing neutron star models constructed with up to date equations of state and shear modulus, the extension to non-axisymmetric modes will be inevitable in order to make possible serious comparions between theoretical predictions and observations. In this Appendix, we give the oscillation equations for axisymmetric (m = 0) modes separately for even modes and for odd modes. In the followings, for a given matrix F = (Fi,j ), we employ the symbols [F ], {F }, and F to indicate the matrices defined as for i, j = 1, 2, 3, · · ·.

A1 Even Modes
For the solid crust, we employ the dependent variables defined as where lj = 2(j − 1),lj = 2j, l ′ j = 1 + 2(j − 1) for j = 1, 2, 3, · · · , and the oscillation equations are given by where 1 denotes the unit matrix, and and and M and R are the mass and radius of the star, and G is the gravitational constant. Note that the terms proportional to (vA/c) 2 come from the displacement current. For fluid regions the dependent variables we use are defined as and the oscillation equations are given by and the oscillation equations are then given by the conditions lead to and Note that the conditions (B5) are obtained by integrating the right hand side of equation (5) along a closed, infinitisimally small, curve that crosses the interface. The conditions (B4), (B6), and (B7) guarantee the continuity of the radial component of B ′ at the interfaces because of equation (B3). Finally, we require the continuity of the three components of traction given by and where δτ S ij and δτ B ij are the ij components of the perturbed traction associated with bulk modulus and magnetic field, respectively, and they are given by    Figure 6.ω versus log(Bs) for the axisymmetric (m = 0) l=2 g n=1 mode and the crust/ocean interfacial mode l=2 i 1 whose amplitudes peak at the crust/ocean interface for the neutron star model NS05T7. 48 G for the model NS13T8, where the solid lines stand for iT l ′ =2 and ib T l=1 , and the dashed lines for iT l ′ =4 and ib T l=3 , respectively. The amplitude normalization is given by iT l ′ =2 = 1 at the surface.