Axisymmetric toroidal modes of magnetized neutron stars
We calculate axisymmetric toroidal modes of magnetized neutron stars with a solid crust. We assume the interior of the star is threaded by a poloidal magnetic field that is continuous at the surface with the outside dipole field whose strength at the magnetic pole is G. Since separation of variables is not possible for oscillations of magnetized stars, we employ finite series expansions of the perturbations using spherical harmonic functions to represent the angular dependence of the oscillation modes. For G, we find distinct mode sequences, in each of which the oscillation frequency of the toroidal mode slowly increases as the number of radial nodes of the eigenfunction increases. The frequency spectrum of the toroidal modes for G is largely different from that of the crustal toroidal modes of the non-magnetized model, although the frequency ranges are overlapped each other. This suggests that an interpretation of the observed QPOs based on the magnetic toroidal modes may be possible if the field strength of the star is as strong as G.
keywords:stars: neutron – stars: oscillations – stars : magnetic fields
Recent discovery of quasi-periodic oscillations (QPOs) of magnetar candidates is one of the observational manifestations of global oscillations of neutron stars. Israel et al (2005) detected QPOs of frequencies , and 92.5Hz in the tail of the SGR 1806-20 hyperflare observed December 2004, and suggested that the 30Hz and 92.5Hz 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 detected August 1998, Strohmayer & Watts (2005) found QPOs of frequencies 28, 53.5, 84, and 155 Hz, and claimed that the QPOs could be identified with the low fundamental toroidal torsional modes of the solid crust of the neutron star. These recent discoveries of QPOs in the giant flares of Soft Gamma-Ray Repeaters SGR 1806-20 (Israel et al 2005, Watts & Strohmayer 2006, Strohmayer & Watts 2006) and SGR 1900+14 (Strohmayer & Watts 2005) have made promising asteroseismology for magnetars, neutron stars with an extremely strong magnetic field (see, e.g., Woods & Thompson 2006 for a review of SGRs).
It is currently common to identify these QPOs with seismic vibrations caused by crustal toroidal modes of the neutron stars, since the frequency range of the modes overlap that of the observed QPOs and from the energetics point of view the crustal toroidal modes would be most easily excited to observable amplitudes by spending a least amount of available energies, for example, those released in magnetic field restructuring (e.g., Duncan 1998). Although the interpretation based on crustal torsional modes looks promising, we need detailed theoretical analyses of oscillations of magnetized neutron stars so that we could get information of physical conditions of the stars through the confrontation between theoretical modelings and observations. This is particularly true for high frequency QPOs (e.g., 625Hz QPO, Watts & Strohmayer 2006; 1835Hz QPO and less significant QPOs at 720 and 2384 Hz in SGR 1806-20, Strohmayer & Watts 2006), since there exist classes of modes other than the crustal toroidal modes that can generate the frequencies observed.
The presence of a magnetic field makes it possible for toroidal modes to exist in a fluid star even without rotation as does the presence of the shear modulus in the solid crust. In this paper we are interested in axisymmetric toroidal modes since axisymmetirc toroidal and spheroidal modes are decoupled for a poloidal field when the star is non-rotating. For non-axisymmetric modes, the toroidal and spheroidal components are coupled even without rotation and hence the modal analyses of magnetized stars would be much more complicated.
Theoretical calculations of toroidal modes of strongly magnetized neutron stars have been carried out by several authors, including Piro (2005), Glampedakis et al (2006), Sotani et al (2006, 2007), and Lee (2007). The analyses by Piro (2005), Glampedakis et al (2006), and Lee (2007) assume Newtonian gravity, while those by Sotani et al (2006, 2007) use general relativistic formulation. Although the studies by Piro (2006) and Lee (2007) ignore the effects of magnetic fields in the fluid core, those by Glampedakis et al (2006) and Sotani et al (2006, 2007) consider magnetic waves propagating in the fluid core, assuming the core is threaded by a magnetic field of substantial strength. Besides the differences mentioned above, most of the authors except Lee (2007) represent the angular dependence of the oscillations by a single spherical harmonic function . Since the shear modulus in the crust dominates the magnetic pressure in most parts of the crustal regions for a dipole field of strength G, this treatment may be justified so long as the crustal toroidal modes are well decoupled from the fluid core. But, if the torsional waves in the crust are strongly coupled with magnetic waves in the core, the treatment may not be justified because the angular dependence of the magnetic waves in the fluid core cannot be correctly represented by a single spherical harmonics. For example, the analysis by Reese, Fincon, & Rieutord (2004) of toroidal modes in a fluid shell have employed finite series expansions of long length for the perturbations.
In this paper, using the method of series expansions of perturbations we calculate toroidal modes of a strongly magnetized neutron star having a fluid core and a solid crust, where the entire interior is assumed to be threaded by a poloidal magnetic field. We employ two different sets of oscillation equations, one for fluid regions and the other for the solid crust, and solutions in the solid and fluid regions are matched at the interfaces between them to obtain an entire solution of a mode. The method of calculation we employ is presented in §2, and the numerical results are given in §3, and conclusions are in §4.
2 Method of Solution
2.1 Magnetic Fields in the Interior
We employ an inner magnetic field of Ferraro (1954) type. For a magnetized star in hydrostatic equilibrium, Ferraro (1954) assumed an interior poloidal field given by
where is a scalar function, and looked for a particular solution to the partial differential equation:
where is a constant to be determined. Assuming the scalar function is given by
and imposing the condition that the interior field is continuous at the stellar surface with the outside dipole field, Ferraro (1954) obtained
where denotes the strength of the magnetic dipole field at the pole, and is the radius of the star. For modal analysis shown below, we use this equilibrium magnetic field configuration with being a parameter.
2.2 Oscillation Equations
We assume the temporal and azimuthal angular dependence of perturbations is given by a single factor for oscillations of magnetized, non-rotating stars, where is the oscillation frequency in the inertial frame, and denotes the azimuthal wave number. The linearized basic equations governing axisymmetric () toroidal modes propagating in the solid crust of the magnetized star may be given by
In equation (5), denotes the Euler perturbation of the stress tensor and is obtained from the Lagrangian perturbation defined in Cartesian coordinates by
with being the strain tensor defined by
where denotes the Kronecker delta, is the shear modulus, and ( see, e.g., McDermott et al 1988, and Lee & Strohmayer 1996). The linearized equations for toroidal modes in the fluid regions are obtained by simply dropping the term in equation (5).
Since separation of variables is not possible for oscillations of magnetized stars, we employ series expansions of a finite length for the displacement vector and the perturbed magnetic field using spherical harmonic functions for a given . We have for axisymmetric toroidal modes
where and for even modes, and and for odd modes, respectively, and , and is a normalizing constant, which is set equal to . Note that, for axisymmetric toroidal modes with , we have to redefine and as and for even modes, and for odd modes for . Most of the numerical results shown below are obtained for . In this convention, the angular pattern of (of ) at the stellar surface is symmetric (anti-symmetric) about the equator for even modes and it is anti-symmetric (symmetric) for odd modes.
If we use as dependent variables the vectors and defined by
the oscillation equations for fluid regions are given by
and those for the solid crust are given by
where , and
and is the gravitational constant, and are the gravitational mass and the radius of the star, respectively, and
and . The definition of the matrices , , , , , and is found, for example, in Lee (1993), and the matrices , , and are defined by
for even modes, and
for odd modes, where for a matrix .
For the inner boundary condition we require that the functions and are regular at the stellar center. The boundary condition at the stellar surface is given by . The jump conditions at the solid-fluid interfaces is the continuity of the function and of the component of the traction, the latter of whcih leads to
2.3 Neutron Star Models
We integrate the Tolman-Oppenheimer-Volkoff equation (see, e.g., Shapiro & Teukolsky 1983) to obtain zero temperature () neutron star models, where no effects of the magnetic field on the equilibrium structure are included. The equation of state used for the inner crust and the fluid core is that given by Douchin & Haensel (2001). The equation of state in the outer crust is that given by Baym, Pethick, & Sutherland (1971), and for the fluid ocean we simply use the equation of state for a mixture of a completely degenerate electron gas and a non-degenerate gas of Fe nuclei. The boundary between the fluid ocean and the outer crust is set rather arbitrarily at the density g cm.
For the solid crust, we employ the average shear modulus (Strohmyer et al 1991), which in the limit of is given by
where is the number density of the nuclei and denotes the separation between the nuclei defined by
and is the Boltzmann constant.
For modal analyses, we calculate neutron star models of mass and . For the former model, the radius and the central pressure and density are , dyne/cm, and g/cm, respectively. The ratio of the thickness of the crust, , to the radius is , and the surface ocean above the crust is very thin. For the latter model, the parameters are , dyne/cm, g/cm, and , respectively.
Figure 1 plots the frequency of the crustal toroidal modes of the two models for low values of harmonic degree , where and , and no effects of magnetic field are included. For each value of , the lowest frequency mode is the fundamental mode that has no radial nodes of the eigenfunction. There exists no fundamental mode for . The figures show that the fundamental mode frequency rapidly increases with increasing , but the frequencies of the overtones remain almost constant for varying . As first discussed by Hansen & Cioffi (1980), the frequency of the fundamental toroidal mode in the solid crust is rather insensitive to the neutron star mass, and that of the overtones is inversely proportional to the thickness of the crust, which is consistent with Figure 1.
3 Numerical Results
With our numerical method employed to calculate toroidal modes of a magnetized star, we find numerous solutions to the oscillation equations for a given . Most of the solutions thus obtained, however, are dependent on , the length of the expansions, and we have picked up only the solutions that are independent of .
In Figure 2 the oscillation frequencies of the toroidal modes of the model are plotted versus the number of radial nodes of the expansion coefficient , where the left panel is for G and the right panel for G, and the filled circles and squares indicate odd and even modes, respectively. The figures clearly show the existence of distinct mode sequences, in each of which the frequency of the mode remains rather constant and increases only slowly as the number of nodes increases.
To compare the two cases of G and G, we plot in Figure 3 the frequency ratio versus , where the filled squares and circles are for even and odd modes, respectively. Except for the lowest frequency sequence of the even modes, the ratio is approximately equal to , and it decreases as the radial order increases, indicating that the frequency is approximately proportional to the field strength and is largely determined by the strength of the magnetic field. This is not the case for the lowest frequency sequence, for which the ratio rapidly increases from to as the radial order increases, demonstrating that the response of the oscillation frequency to the field strength is not the same as that for the overtone sequences. We note that the lowest frequency mode found in the figure is a much more slowly increasing function of than the modes in the overtone sequences.
It is convenient to write the mode frequency as
where the non-negative integer denotes the order of the mode sequences of a given parity (even or odd), and the integer is the number of nodes of the expansion coefficient , and the integer is set equal to for the even mode sequences and to for the odd mode sequences . Since there exists no fundamental modes that have no radial nodes of the expansion coefficients, we require even for , for which we have to replace with in equation (24). The quantities , which is set equal to the lowest frequency obtained, , and are assumed independent of and , and the quantity is assumed to be substantially small compared with and . Note also that we find no mode sequence having for odd modes, and hence we have to start with for them. Except for the case that involves the lowest frequency sequence of even modes, does not strongly depend on and , and slowly decreases as increases. It is interesting to note that the quantity is approximately proportional to the field strength , but the quantity is rather insensitive to . For example, for the model of , we have for G and for G, and for both cases.
Figure 4 is the same as Figure 2 but for the model of , the crust of which is thicker than that of the model. Although the frequency spectra look almost the same between the two models, for a given set of and , the frequency found for the model is higher than that for the model. This is because the fluid core radius (the Alfvén velocity ) of the model is smaller (larger) than that of the model, and hence the traveling time of magnetic perturbations in the core for the former is shorter than for the latter.
As examples of the eigenfunctions of the toroidal modes, we plot the expansion coefficients and of three even modes as functions of the fractional radius in Figures 5 to 7, where the solid, long-dahsed, short-dashed, and dotted lines are the expansion coefficients with to 4, respectively. For the lowest frequency sequence even modes plotted in Figures 5 and 6, the amplitudes are much larger than those of , indicating that the kinetic energy is dominating the magnetic energy. The magnetic perturbations have amplitudes also in the solid crust. This is not the case for the modes in the overtone mode sequences having as exemplified by Figure 7, where the amplitudes of the magnetic perturbations are much larger than and are strongly confined in the fluid core, having almost no amplitudes in the crust. Also note that the expansion coefficients with different values of seem to coalesce to a single curve in the outer core of the star.
Figures 8 to 10 give plots of the expansion coefficients and for three odd toroidal modes associated with , (1,5), and (5,5). The magnetic perturbations of the modes have much larger amplitudes than and they are strongly confined in the fluid core. The functions have comparable amplitudes both in the fluid core and in the solid crust. As in the case of even modes, the expansion coefficients of the mode (5,5) tend to coalesce to a single curve in the outer core region.
We need to examine how the existence of a solid crust affects the frequency spectrum of the toroidal modes. Figure 11 shows the toroidal mode frequencies of the model for G, where the modes are calculated by treating the entire interior as a fluid. Without the crust, the lowest frequency sequence is composed of odd modes, and the even mode sequence corresponding to , which appears as the lowest sequence when the solid crust is included, is missing. Therefore, for the fluid star we have to set equal to for odd mode sequences with and to for even mode sequences with in equation (24). In the lowest frequency odd mode sequence, there exists the fundamental mode having no radial nodes of the expansion coefficients, corresponding to . Since the oscillation equations (12) and (13) for fluid regions contain the oscillation frequency only in the form of the ratio , the frequency of a given is exactly proportional to the field strength . This suggests that the quantities , , , and defined in equation (24) are also proportional to . Note that if we write for the lowest frequency mode for the case with a solid crust, where the exponent may be weakly dependent on , we have for G, the feature of which is different from the case without the crust. Figure 11 indicates that the frequency separation between the lowest and second lowest sequences is approximately for the case without the crust, but this separation is given by for the case with the crust. Note that the frequency separation between the second and third lowest sequences, for example, is given by for both cases.
As suggested by Figures 2, 4, and 11, it is not always possible to numerically find independent modes associated with large values of and . Possible numerical reasons for the difficulty may be that the modes we want are immersed in a dense spectrum of dependent solutions, particularly for weak , and that a large and high numerical precision are required to correctly calculate the expansion coefficients in the vicinity of the stellar center for modes associated with large . From the theoretical point of view, however, it is not necessarily clear that the toroidal modes of the kind we find in this paper exist for arbitrary sets of integers and .
In this paper, we have calculated toroidal modes of magnetized stars with a solid crust, where the entire interior of the star is assumed to be threaded by a poloidal magnetic field that is continuous at the stellar surface to the outside dipole field. We find distinct mode sequences of the toroidal modes, in each of which the mode frequency remains rather constant and only slowly increases as the radial order of the modes increases. In the presence of a solid crust, the frequency separation between the lowest and second lowest frequency mode sequences is approximately given by , but that between the second and third lowest frequency mode sequences by , where the frequency separation is roughly proportional to the field strength . This frequency pattern of the low frequency sequences is different from that found for the model without the crust, for which the frequency separation between the sequential sequences of low frequency modes is given by . We also find that for the equation of state we use, is larger for smaller .
The eigenfunction of the modes belonging to the lowest frequency sequence have much larger amplitudes than and can penetrate into the solid crust. On the other hand, of the modes belonging to the higher frequency sequences is much smaller than , which is well confined into the fluid core and does not have any substantial amplitudes in the solid crust.
The frequency ranges of the toroidal modes we find for the magnetized neutron star with G overlap the QPO frequencies found for the magnetar candidates, SGR 1806-20 and SGR 1900+14. This suggests that we may interpret the observed QPOs based on the magnetic toroidal modes, and that detailed comparisons between observed frequency spectra and theoretical calculations make it relevant to infer physical parameters of the magnetar candidates, such as the equation of state and the strength of the magnetic field. But, we think it worth pointing out that except for the modes belonging to the lowest frequency sequence, the magnetic perturbations, which have much larger amplitude than , are well confined in the fluid core and do not have substantial amplitudes in the solid crust, which make it difficult for the modes to be directly observable. If the magnetar candidates do no have a crust, the problem of observability could be avoided. In this case, however, we have to use more realistic surface boundary conditions than used in the present calculations. Note that, although the existence of a solid crust affects the frequency pattern of the low frequency mode sequences, the frequency range of the magnetic modes itself is not very much dependent on the presence or absence of a solid crust unless the crust is extremely thick.
We have tried to find toroidal modes well confined in the solid crust or core toroidal modes that are in resonance with the crustal toroidal modes, but failed. We also find it extremely difficult to identify distinct toroidal mode sequences for magnetic fields weaker than G when a solid crust is included in the models. It is therefore not clear whether the toroidal mode sequences of the kind we find in the present paper can survive also for weakly magnetized neutron stars with a solid crust. If the field strength is much weaker than G, the magnetic fields may have only minor effects on the crustal toroidal modes (e.g., Lee 2007), and it will be justified to use frequency spectra of the crust modes theoretically obtained in the weak field limit to interpret observed QPOs. As briefly noted in the last section, it is not theoretically clear how the frequency spectra of the toroidal modes of magnetized stars should look like, and how the toroidal modes behave in the limit of . We may even speculate that the frequency spectra we obtained reflect the existence of continuous spectra (see, e.g., Goedbloed & Poedts 2004, see also Levin 2007), but the detailed numerical analysis of continuous frequency spectra is beyond the scope of this paper. It will be worthwhile to examine the effects of an interior toroidal field on magnetic modes. It is also needed to extend the present analysis to a general relativistic formulation (e.g., Sotani et al 2006, 2007).
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