A Oscillation equations, jump conditions, and boundary conditions

Non-axisymmetric low frequency oscillations of rotating and magnetized neutron stars

Abstract

We investigate non-axisymmetric low frequency modes of a rotating and magnetized neutron star, assuming that the star is threaded by a dipole magnetic field whose strength at the stellar surface, , is less than G, and whose magnetic axis is aligned with the rotation axis. For modal analysis, we use a neutron star model composed of a fluid ocean, a solid crust, and a fluid core, where we treat the core as being non-magnetic assuming that the magnetic pressure is much smaller than the gas pressure in the core. For non-axisymmetric modes, spheroidal modes and toroidal modes are coupled in the presence of a magnetic field even for a non-rotating star. Here, we are interested in low frequency modes of a rotating and magnetized neutron star whose oscillation frequencies are similar to those of toroidal crust modes of low spherical harmonic degree and low radial order. For a magnetic field of G, we find Alfvén waves in the ocean have similar frequencies to the toroidal crust modes, and we find no -modes confined in the ocean for this strength of the field. We calculate the toroidal crustal modes, the interfacial modes peaking at the crust/core interface, and the core inertial modes and -modes, and all these modes are found to be insensitive to the magnetic field of strength G. We find the displacement vector of the core -modes have large amplitudes around the rotation axis at the stellar surface even in the presence of a surface magnetic field G, where and are the spherical harmonic degree and the azimuthal wave number of the -modes, respectively. We suggest that millisecond X-ray variations of accretion powered X-ray millisecond pulsars can be used as a probe into the core -modes destabilized by gravitational wave radiation. If the -mode is excited, we will have the pulsation of the frequency with being the spin frequency of the star.

keywords:
stars: neutron – stars: oscillations – stars : magnetic fields

1 Introduction

Oscillation of strongly magnetized neutron stars has attracted an intense attention in recent years, particularly motivated by the discovery of quasi-periodic oscillations (QPOs) of magnetar candidates (e.g., Woods & Thompson 2006), which are believed to be one of the observational manifestations of global oscillations of neutron stars that have a strong global magnetic field of order of at the stellar surface (e.g., Duncun 1998, Israel et al 2005; Strohmayer & Watts 2005, 2006; Watts & Strohmayer 2006). Thus, recent theoretical studies of the oscillations of magnetized neutron stars have been mainly concerned with the stars having extremely strong surface magnetic fields G, and these studies have suggested that the QPOs observed in the magnetar candidates are attributable to the toroidal crust modes and Alfvén modes of the neutron stars (e.g., Piro 2005; Glampedakis, Samuelsson & Andersson 2006; Lee 2007, 2008; Sotani, Kokkotas & Stergioulas 2008; Cerdá-Durán, Stergioulas & Font 2009; Colaiuda, Beyer & Kokkotas 2009; Bastrukov et al 2009; Sotani & Kokkotas 2009). For toroidal modes of strongly magnetized neutron stars, however, it is intriguing, from the theoretical point of view, that Lee (2008) obtained discrete frequency spectra of the magnetic modes, but Sotani, Kokkotas & Stergioulas (2008), Cerdá-Durán, Stergioulas & Font (2009), and Colaiuda, Beyer & Kokkotas (2009), using a different numerical method from that used by Lee (2008), suggested the existence of continuum frequency spectra of the modes, as originally discussed by Levin (2006, 2007).

The burst oscillation observed in low mass X-ray binaries (LMXBs) can be another example in which a magnetic field plays an important role in the oscillations of neutron stars, although the strength of the field at the surface of the neutron stars in LMXBs is thought to be less than G, much weaker than that for the magnetar candidates. For the burst oscillation, the hot spot model (e.g., Strohmayer et al 1997; Cumming & Bildsten 2001; Cumming et al 2002) and the Rossby wave model (Heyl 2004; Lee 2004) have been proposed, but it seems none of the models is accepted as the one that fully explains the observational properties of the burst oscillation. In the Rossby wave model, Heyl (2004, 2005) , Lee (2004), and Lee & Strohmayer (2005) have examined the possibility that the burst oscillation is produced by low frequency Rossby waves, called -modes in the astrophysical literature, propagating in the surface fluid region (ocean), disregarding the effects of a magnetic field on the low frequency waves. We note that Bildsten & Cutler (1995) discussed the modal properties of modes propagating in the fluid ocean above the solid crust of mass accreting neutron stars as a possible mechanism responsible for the Hz QPOs observed in LMXBs. In their paper, on the assumption that the magnetic pressure, , is much smaller than the gas pressure, , the critical strength of a magnetic field, , below which the magnetic field has no significant effects on the -modes, was estimated to be G for the accreted envelopes composed of carbon (see also Piro & Bildsten 2005). Since their argument is based on a local analysis and on the assumption , we think it useful to carry out a global modal analysis of low frequency waves propagating in the magnetized fluid ocean of a neutron star.

Accretion powered millisecond X-ray pulsars in LMXBs show small amplitude, almost sinusoidal X-ray time variations, the dominant period of which is thought to be the spin period of the underlying neutron stars. Lamb et al (2009) argued that the millisecond X-ray variations are produced by an X-ray emitting hot spot located at a magnetic pole rotating with the star, and that so long as the symmetry center of the hot spot is only slightly off the rotation axis the X-ray variations produced have small amplitudes and become almost sinusoidal. They also suggested that if the hot spot is located close to the rotation axis, a slight drift of the hot spot away from the rotation axis leads to appreciable changes in the amplitudes and phases of the X-ray variations. Lamb et al (2009) pointed out that a temporal change in mass accretion rates and hence the radius of the magnetosphere, for example, can cause such a drift of the hot spot. We think it is also interesting to consider the possibility that global oscillations of the neutron stars work as a mechanism that perturbs the hot spot periodically.

In this paper, to calculate global oscillations of a rotating and magnetized neutron star, we use the method of series expansion, in which the angular dependence of the perturbations is represented by series expansion in terms of spherical harmonic functions with different spherical harmonic degrees for a given azimuthal wave number (e.g., Lee & Strohmayer 1996, Lee 2007). We calculate low , low frequency modes of a neutron star composed of a fluid ocean, a solid crust, and a fluid core, where the star is assumed to be threaded by a dipole magnetic field, the strength of which at the surface is smaller than G. The method of calculation we employ is presented in §2, and the numerical results are described in §3, and we discuss a local analysis for low frequency modes in the magnetized fluid ocean in §4, and we give discussion and conclusion in §5.

2 Method of calculation

We consider small amplitude oscillations of rotating and magnetized neutron stars in Newtonian dynamics, and no general relativistic effects on the oscillations are considered. We employ spherical polar coordinates , whose origin is at the center of the star and the axis of rotation is given by . We assume a dipole magnetic field given by

(1)

where 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 , the field does not influence the equilibrium structure of the star. When we assume the axis of rotation is also the magnetic axis, the temporal and azimuthal angular dependence of perturbation can be represented by a single factor , where is the azimuthal wavenumber around the rotation axis and is the oscillation frequency observed in the corotating frame of the star, where is the oscillation frequency in an inertial frame and is the angular frequency of rotation. The linearized basic equations used in a solid region of the star are given by

(2)
(3)
(4)
(5)

where is the mass density, is the pressure, is the displacement vector, and , , and are the Euler perturbations of the density, the pressure and the magnetic field, respectively, and is the Schwartzshild discriminant defined by

(6)

and

(7)

In equation (2), denotes the Eulerian perturbation of the stress tensor, which is derived by using the Lagrangian perturbation of the stress tensor defined, in Cartesian coordinates, by

(8)

where is the strain tensor given by

(9)

and is the shear modulus and , and is the Kronecker delta. 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.

We can obtain the equation of motion for a fluid region by replacing the terms and in equation (2) by and , respectively, and we do not need to consider equations (8) and (9) for fluid regions.

Since the angular dependence of perturbations in 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 with different s for a given , assuming that the axis of rotation coincides with that of the magnetic field. The displacement vector and the perturbed magnetic field are then represented by a finite series expansion of length as

(10)

and

(11)

and the pressure perturbation, , for example, is given by

(12)

where

(13)

and , and and for even modes, and and for odd modes, respectively, and . Substituting the expansions (10), (11) and (12) into the linearized basic equations (2)(5), and (8), we obtain a finite set of coupled linear ordinary differential equations for the expansion coefficients , , and so on, which we call the oscillation equation. For non-axisymmetric modes with , we cannot expect decoupling between spheroidal (polar) modes and toroidal (axial) modes even for , although for axisymmetric modes with , spheroidal modes and toroidal modes are decoupled when (e.g., Lee 2007).

In this paper, we treat for simplicity the fluid core being non-magnetic, which may be justified if we assume the magnetic pressure is much smaller than the gas pressure in the core. The oscillation equations used in the magnetized regions are given in the Appendix, in which the jump conditions imposed at the solid/fluid interfaces and the boundary conditions applied at the stellar center and the surface are also given. The oscillation equation used in the non-magnetic fluid core is found, for example, in the Appendix of Lee & Saio (1990).

3 numerical results

For the neutron star model used for modal analysis, we employ a cooling evolution model called NS05T7, which was calculated by Richardson et al (1982). The mass of the model is and the central temperature is K, and the model is composed of a fluid ocean, a solid crust and a fluid core, and because of density stratification -modes propagate in the fluid regions. The more detailed properties of the model, such as the equations of state used, are described, for example, in McDermott, Van Horn & Hansen (1988), who carried out modal analysis of the model assuming no rotation and no magnetic fields. Since this is a low mass model, the solid crust is rather thick, and the low radial order crustal modes of low spherical harmonic degree are well separated from the and modes of the model, which makes the modal analysis much simpler than the case in which the crustal modes have frequencies similar to those of the and modes as expected for more massive neutron stars with accreted envelopes.

Figure 1: Low frequency modes of the model NS05T7 are plotted as functions of for the case of G for even modes in the left panel and odd modes in the right panel, where and the positive and negative sides of the horizontal axis are for prograde and retorgrade modes, respectively.

Let us begin with the case of a weak magnetic field of strength G. In the presence of a magnetic field, the existence of Alfvén waves, whose oscillation frequency depends on the strength of the magnetic field and on the direction of wave propagation relative to the magnetic field, inevitably makes frequency spectra complex. For the case of G, the frequency spectra of low frequency modes are too complicated to calculate completely, as suggested by Figure 1, which plots the oscillation frequency of low frequency modes as functions of for even modes (left panel) and odd modes (right panel), respectively, where we have used . We notice that there appear in Figure 1 several kinds of modes, that is, the inertial modes and -modes in the fluid core, the toroidal crust modes, the interfacial modes whose amplitudes peak at the interface between the core and the solid crust, and the modes confined in the fluid ocean. The inertial modes and -modes are rationally induced modes, whose oscillation frequencies are approximately proportional to and are found on almost straight lines tending to the origin in the figure, where the -modes appear on the side of retrograde modes. The loci of the oscillation frequencies as functions of for the core inertial modes, -modes, the toroidal crust modes, and the interfacial modes are more clearly seen for the case of a stronger magnetic field of G as shown by Figure 2 below. We note that the toroidal crust modes are insensitive to the magnetic field of strength (e.g., Lee 2007), and that the inertial modes and -modes in the fluid core and the interfacial mode at the core/crust interface are not strongly affected by the magnetic field even if their eigenfunctions extend to the surface through the magnetized solid crust and fluid ocean.

However, the modes that are confined in the surface ocean are strongly influenced by a magnetic field as weak as G and show extremely complicated frequency spectra. Although there usually appear gravito-inertial modes and -modes as non-axisymmetric low frequency modes confined in the ocean for a non-magnetized neutron star, we find no ocean -modes in the presence of a magnetic field, even if it is as weak as G. We also find that Alfvén modes come into the frequency spectra of the low frequency modes, modifying the spectra of the gravito-inertial modes. We notice that there exist two branches of modes in Figure 1. In one branch of modes, the oscillation frequency increases as increases, while the oscillation frequency decreases with increasing in the other, although there occurs frequent avoided crossings between the two branches of modes as varies. Based on the local analysis presented in §4, we think the former can be regarded as (gravito-)inertial modes and the latter as Alfvén modes. However, we have to note that the eigenfrequencies of the modes in the two branches do not necessarily reach good convergence even if is increased to . In this sense we are not sure that the modes confined in the surface ocean we calculate are discrete modes with real frequencies.

Figure 2: Low frequency modes of the model NS05T7 plotted versus for the case of G for even modes in the left panel and odd modes in the right panel, where and the negative and positive sides of the horizontal axis are for retrograde and prograde modes, respectively. The symbols , , and denote the toroidal crust modes, the interfacial mode whose amplitudes peak at the core/crust interface, and the core -mode, respectively, where and denote the spherical harmonic degrees and the subscript is the radial order, which is usually equal to the number of radial nodes of the dominant component of the eigenfunctions.

In Figure 2, we plot low frequency modes against for the case of G for even modes (left panel) and odd modes (right panel), where we have used . In the figure, the symbols , , and respectively denote the toroidal crust modes, the interfacial mode whose amplitudes peak at the core/crust interface, and the core -modes whose frequency tends to as , where and denote the spherical harmonic degrees and the subscript indicates the radial order, usually corresponding to the number of nodes of the dominant eigenfunction in the propagation region. Note that we do not attach any labels to inertial modes in the fluid core whose frequencies are approximately proportional to and are found on almost straight lines tending to the origin of the figure. Because the fluid core is almost isentropic such that the Brunt-Väisälä frequency is extremely small, the odd -mode labeled is the only -mode we can find for a given value of (e.g., Yoshida & Lee 2000a). We note that the eigenfrequencies of the modes plotted in Figure 2 obtain good convergence when is increased to . We have calculated low frequency modes for G in the same frequency range as that in Figure 2, and obtained the result quite similar to that for the case of G.

Because of the effects of rotation, the frequencies of the toroidal crust modes and the interfacial mode vary as changes. If we expand the oscillation frequency of a mode as , where is the frequency of the mode for , the coefficient describes the first order response of the oscillation frequency to small , and a list of the coefficients of spheroidal modes of the model NS05T7 is tabulated in Lee & Strohmayer (1996), where no magnetic effects are considered. For the toroidal crustal modes, on the other hand, Lee & Strohmayer (1996) showed for non-magnetized stars. Note that for rotationally induced modes such as inertial modes and -modes, we have . Figure 2 indicates that the frequency behavior of the crust modes for slow rotation is consistent to that expected from the coefficients, even in the presence of a magnetic field of strength G, although it is also clear that as increases from , the deviation of from the expansion quickly becomes significant. The deviation of the frequency from the expansion may be partly caused by avoided crossings between two different modes. An example found in Figure 2 is the avoided crossing between and , which is a crossing between a toroidal mode and a spheroidal mode.

In Figure 3, we plot the eigenfunctions , , and of several low frequency modes for the case of G and , where , and the low frequency modes we plot are the even toroidal crust mode , the even interfacial mode , the odd -mode , and the odd toroidal crustal mode , and except for the -mode, which is a retrograde mode with , all the other modes are prograde modes with . Note that the toroidal crustal modes have appreciable amplitudes in the fluid core due to the effects of rotation. We find the eigenfunctions of the modes in the presence of the dipole magnetic field of G are quite similar to those found in the absence of the magnetic field (see, e.g., Lee & Strohmayer 1996, Yoshida & Lee 2001), suggesting that these modes are not strongly affected by magnetic fields of that strength. Note that in the vicinity of the stellar center, since the eigenfunctions and are approximately proportional to (e.g., Unno et al 1989), the functions and behave as for even modes of and as for odd modes of , as indicated by the panels (a) and (b).

Figure 3: Eigenfunctions (panel a), (panel b), and (panel c) plotted against for the low frequency modes for the case of G and , where we use . In each panel, the solid line is for the even toroidal crustal mode , the dashed line for the even interfacial mode , the dash-dotted line for the odd -mode , and the dotted line for the odd toroidal crustal mode . Note that the amplitude normalization is given by at the stellar surface. Except for the -mode, which is a retrograde mode with , all the other modes are prograde modes with .

From the observational point of view, it is useful to know the dependence of the displacement vector of the modes at the surface of the star. As noted in the previous section, for rotating and magnetized stars, the eigenfunction of an oscillation mode cannot be represented by a single spherical harmonic function and hence their surface pattern can be largely different from that for non-magnetic and non-rotating stars. To see the angular dependence of the displacement vector at the surface, we introduce the functions defined by

(14)
(15)
(16)

In Figure 4, we plot the functions of the low frequency modes presented in Figure 3 for the case of G and , where in each of the panels the solid, dashed, and dotted lines in each panel respectively denote the functions , , and , which are normalized by their maximum amplitudes. Since the modes have low frequencies, the maximum amplitudes of are much smaller than those of and , that is, the horizontal and/or toroidal components of the displacement vector are dominant over the radial component. It is to be noted that although the amplitudes of the ocean -modes tend to be confined to the equatorial regions (see, e.g., Lee 2004), this is not the case for the -modes in the fluid core, which have large amplitudes in the regions around the the poles.

Figure 4: Functions , , and versus for the low frequency modes for the case of G and for , where the functions are evaluated at the stellar surface, and the solid, dashed, and dotted lines denote the functions , , and , respectively. Panels (a), (b), (c), and (d) are for the even toroidal crust mode , the even interfacial mode , the odd core mode, and the odd toroidal crust mode , respectively, where only the -mode is a retrograde mode with , and the others are prograde modes with . The functions are normalized by their maximum amplitudes. The functions and of even (odd) modes are symmetric (antisymmetric) with respect to the axis of , while the function of even (odd) modes is antisymmetric (symmetric).

In Figure 5, we plot the frequencies of low frequency modes against for the case of G for even modes (left panel) and odd modes (right panel), where we have used . For the case of , the toroidal crustal mode appears as an even mode, while and as an odd mode. We note that no interfacial modes appear in the frequency range shown in the figure. Figure 6 shows the functions versus for the core -mode at for G (left panel) and G (right panel), where the functions, evaluated at the stellar surface, are normalized by their maximum amplitudes. This figure shows that the surface pattern generated by the core -mode is not affected by the field as strong as G, which is also the case for the core -mode.

Figure 5: Same as Figure 2 but for .
Figure 6: Functions , , and at the surface versus for the mode for the cases of G in the left panel and G in the right panel, where and are assumed. The solid, dashed, and dotted lines respectively denote the functions , , and , which are normalized by their maximum amplitudes.

4 Local Analysis

To gain an understanding of the low frequency modes confined in the shallow fluid ocean calculated for the case of G, we apply a local analysis to low frequency modes in a fluid region of a rotating and magnetized neutron star. Note that a local analysis of waves in rotating stars with no magnetic fields is found in, for example, Unno et al (1989). In this section, we assume that the and dependence of the perturbed quantities is given by the function , where is the wavenumber vector. Using the equation of motion for a fluid region and equations (3) to (5), we can derive a set of equations used for the local analysis:

(17)
(18)
(19)
(20)

which are combined to give

(21)

where is the adiabatic sound speed and is the Alfvén velocity and .

To make the local analysis tractable, we employ a local Cartesian coordinate system whose -axis is along the radial direction, and we assume that neglecting the local horizontal component of the rotation vector. Equation (21) can be rewritten into a form with being a matrix, and the condition leads to the dispersion relation:

(22)

where

(23)
(24)
(25)
(26)

where , , , , , , and is the Brunt-Väisälä frequency, and

(27)

Note that the term proportional to breaks the symmetry given by .

If we assume and , the dispersion relation reduces to

(28)

the non-trivial solution of which is

(29)

On the other hand, if we assume and , the dispersion relation reduces to

(30)

If we can further assume , the solutions of the dispersion relation are separated into

(31)

corresponding to the Alfvén waves, and to

(32)

For the case of and , it is difficult to analytically solve the dispersion relation (22) in general . Here, we numerically solve equation (22), which can be rewritten, by normalizing various quantities, into

(33)

where

(34)
(35)
(36)
(37)

where

(38)

To solve the dispersion relation for a given neutron star model with and and for a given magnetic field , we need to supply with appropriate values the following parameters,

(39)

although we note a relation given by

(40)

For a given value of , we have

(41)

and the parameters and must satisfy an inequality , that is,

(42)

which can be rewritten as

(43)

where

(44)

In the following discussions, instead of and , it will be convenient to use the parameters and defined by

(45)

Using these parameters we have and , and , , , , , , , and are the parameters we need to specify.

For the fluid ocean of the model NS05T7, typical values of the parameters and are found to be

(46)

and a typical value of the parameter , depending on , is for G and for G. Since for low frequency modes, we need and . In the following discussions, for simplicity, we assume and , and . Examples of numerical solutions of the dispersion relation (22) are given in Figure 7, where versus is plotted for (dotted line), 0.5 (solid line), and 0.9 (dashed line) in the left panel, and versus for (dotted line), (solid line), and (dashed line) in the right panel. It is interesting to note that the asymmetry due to the term proportional to is too weak to become noticeable in the figure. As shown by the left panel, for a given value of , the solution have two branches in this frequency region, and the upper and lower branches respectively correspond to gravito-inertial waves, for which when becomes large, and Alfvén waves, for which . The minimum frequency in the inertial mode branch and the maximum frequency in the Alfvén mode branch, which occur at , increase as increases. These two branches of modes are reminiscent of the low frequency waves in the ocean plotted in Figure 1. We think that the two different mode branches associated with a angle between and appear as a pair of mode branches in Figure 1 and that the minimum and maximum frequencies at in the pair depend on this angle, which may vary from one pair to another. As shown by the right panel of Figure 7, for a given value of , the frequency of the Alfvén modes increases as increases, and the inertial branch tends to the relation given by in the limit of .

Figure 7: Left panel: Low frequency solutions of the dispersion relation (22) plotted as functions of for (dotted line), 0.5 (solid line) and 0.9 (dashed line), where we have assumed , , , , , , and .; Right panel: Low frequency solutions of the dispersion relation (22) plotted as functions of for (dashed line), (solid line), and (dotted line), where we have assumed , , , , , , and .

5 discussion and conclusion

Lamb et al (2009) proposed that the small amplitude, almost sinusoidal millisecond X-ray pulsation observed in accretion powered millisecond X-ray pulsars may be well explained by the hot spot model, in which the hot spots are assumed to be located at the magnetic poles, which are nearly aligned with the rotation axis. As discussed by Lamb et al (2009), even a small drift of the hot spot could produce appreciable changes in pulsation amplitudes of the X-ray pulsation. If this proposition is correct, it is interesting to pursue a possibility of using the millisecond X-ray pulsation to probe the core -modes excited by gravitational wave radiation (Andersson 1998; Friedman & Morsink 1998). In fact, if the core -modes with are excited by the emission of gravitational wave, since the -mode, which is the most strongly destabilized mode among the -modes (e.g., Lockitch & Friedman 1999; Yoshida & Lee 2000a), produces the surface displacement vector whose horizontal and toroidal components at the surface have large amplitudes around the rotation axis as shown by Figure 6, the hot spot could suffer periodic disturbance from the -mode. Note that the -mode induced temperature perturbation, the surface pattern of which may be proportional to , might generate X-ray variations, the amplitudes of which should be very small. If we write the oscillation frequency of -modes as

(47)

the coefficient for the modes is given by

(48)

and the coefficient may depend on the equation of state and the deviation from the isentropic stratification in the core (e.g., Yoshida & Lee 2000a,b), where . Since the neutron star core is nearly isentropic such that , we only have to consider the -modes, for which we have , and we obtain the frequency for in the corotating frame of the star and the frequency in an inertial frame. It may be interesting to point out that if the -mode of is also excited by some mechanism, this -mode can produce long period variations in an inertial frame since the inertial frame frequency for this mode. Although no detection of periodicities whose frequency is approximately equal to