The Magnetosphere of Oscillating Neutron Stars in General Relativity

# The Magnetosphere of Oscillating Neutron Stars in General Relativity

## Abstract

Just as a rotating magnetised neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron-star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al. (2000), we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.

###### keywords:
stars: magnetic field – stars: neutron – stars: oscillations – pulsars: general
12

## 1 Introduction

Study of the internal structure of neutron stars (NSs) is of fundamental importance for subatomic physics since these objects provide a laboratory for studying the properties of high-density matter under very extreme conditions. In particular, there is the intriguing possibility of using NS oscillation modes as a probe for constraining models of the equation of state of matter at supranuclear densities. It was suggested long ago that if a NS is oscillating, then traces of this might be revealed in the radiation which it emits (Pacini & Ruderman, 1974; Tsygan, 1975; Boriakoff, 1976; Bisnovatyi-Kogan, 1995; Ding & Cheng, 1997; Duncan, 1998). Recently, a lot of interest has been focussed on oscillations of magnetized NSs because of the discovery of gamma-ray flare activity in Soft Gamma-Ray Repeaters (SGRs) which are thought to be the very highly magnetised NSs known as magnetars (for recent review on the SGRs see Woods & Thompson, 2006; Watts & Strohmayer, 2007). The giant flares in these objects are thought to be powered by global reconfigurations of the magnetic field and it has been suggested that the giant flares might trigger starquakes and excite global seismic pulsations of the magnetar crust (Thompson & Duncan, 1995, 2001; Schwartz et al., 2005; Duncan, 1998). Indeed, analyses of the observations of giant flares have revealed that the decaying part of the spectrum exhibits a number of quasi-periodic oscillations (QPOs) with frequencies in the range from a few tens of Hz up to a few hundred Hz (Israel et al., 2005; Strohmayer & Watts, 2006; Watts & Strohmayer, 2006) and there has been a considerable amount of theoretical effort attempting to identify these with crustal oscillation modes (Glampedakis et al., 2006; Samuelsson & Andersson, 2007; Levin, 2007; Sotani et al., 2007a, b). While there is substantial evidence that the observed SGR QPOs are caused by neutron star pulsations, there is a great deal of uncertainty about how stellar surface motion gets translated into the observed features of the X-ray radiation (Strohmayer, 2008; Strohmayer & Watts, 2006; Timokhin et al., 2007). To make progress with this, it is necessary to develop a better understanding of the processes occurring in the magnetospheres of oscillating neutron stars.

Standard pulsars typically have magnetic fields of around G while magnetars may have fields of up to G near to the surface. Rotation of a magnetized star generates an electric field:

 Erot∼ΩRcB , (1)

where is the magnetic field strength, is the speed of light and is the angular velocity of the star with radius . Depending on the rotation velocity and the magnetic field strength, the electric field may be as strong as V cm and it has a longitudinal component (parallel to ) which can be able to pull charged particles away from the stellar surface, if the work function is sufficiently small, and accelerate them up to ultra-relativistic velocities. This result led Goldreich & Julian (1969) to suggest that a rotating NS with a sufficiently strong magnetic field should be surrounded by a magnetosphere filled with charge-separated plasma which screens the accelerating electric field and thus hinders further outflow of charged particles from the stellar surface. Even if the binding energy of the charged particles is sufficiently high to prevent them being pulled out by the electric field, the NS should nevertheless be surrounded by charged particles produced by plasma generation processes (Sturrock, 1971; Ruderman & Sutherland, 1975), which again screen the longitudinal component of the electric field. These considerations led to the development of a model for pulsar magnetospheres which is frequently called the “standard model” (an in depth discussion and review of this can be found in, e.g., Michel, 1991; Beskin et al., 1993; Beskin, 2005).

Timokhin, Bisnovatyi-Kogan Spruit (2000) (referred to as TBS from here on) showed that an oscillating magnetized NS should also have a magnetosphere filled with charge-separated plasma, even if it is not rotating, since the vacuum electric field induced by the oscillations would have a large radial component which can be of the same order as rotationally-induced electric fields. One can show this quantitatively by means of the following simple arguments. To order of magnitude, the radial component of the vacuum electric field generated by the stellar oscillations is given by

 Eosc∼ωξcB , (2)

where is the oscillation frequency and is the displacement amplitude. Using this together with Eq.(1), it follows immediately that the electric field produced by oscillations will be stronger than the rotationally induced one for sufficiently slowly-rotating neutron stars, having

 Ω≲ωξR . (3)

For stellar oscillations with and kHz, the threshold is Hz. Within this context, TBS developed a formalism extending the basic aspects of the standard pulsar model to the situation for a non-rotating magnetized NS undergoing arbitrary oscillations. This formalism was based on the assumption of low current densities in the magnetosphere, signifying that the influence of currents outside the NS on electromagnetic processes occurring in the magnetosphere is negligibly small compared to that of currents in the stellar interior. This assumption leads to a great simplification of the Maxwell equations, which then can be solved analytically. As an application of the formalism, TBS considered toroidal oscillations of a NS with a dipole magnetic field, and obtained analytic expressions for the electromagnetic field and charge density in the magnetosphere. (Toroidal oscillations are thought to be particularly relevant for magnetar QPO phenomena.) They found that the low current density approximation (LCDA) is valid for at least half of all toroidal oscillation modes and analyzed the energy losses due to plasma outflow caused by these modes for cases where the size of the polar cap (the region on the stellar surface that is crossed by open magnetic field lines) is small, finding that the energy losses are strongly affected by the magnetospheric plasma. For oscillation amplitudes larger than a certain critical value, they found that energy losses due to plasma outflow were larger than those due to the emission of the electromagnetic waves (assuming in that case that the star was surrounded by vacuum). Recently, Timokhin (2007) considered spheroidal oscillations of a NS with a dipole magnetic field, using the TBS formalism, and found that the LCDA again holds for at least half of these modes. Discussion in Timokhin (2007) also provided some useful insights into the role of rotation for the magnetospheric structure of oscillating NSs.

The TBS model was a very important contribution and, to the best of our knowledge, remains the only model for the magnetosphere of oscillating NSs available in the literature. However, it should be pointed out that it does not include several ingredients that a fully consistent and realistic model ought to include. Most importantly, it does not treat the magnetospheric currents in a fully consistent way: although it gives a consistent solution for around half of the oscillation modes, the remaining solutions turn out to be unphysical and, as TBS pointed out, this is a symptom of the LCDA failing there. Also, rotation and the effects of general relativity can be very relevant; in particular, several authors have stressed that using a Newtonian approach may not give very good results for the structure of NS magnetospheres (see, e.g., Beskin, 1990; Muslimov & Tsygan, 1992; Mofiz & Ahmedov, 2000; Morozova et al., 2008). However, a more realistic model would naturally be more complicated than the TBS one whose relative simplicity can be seen as a positive advantage when using it as the basis for further applications.

The aim of the present paper is to give a general relativistic reworking of the TBS model so as to investigate the effects of the changes with respect to the Newtonian treatment. We derive the general relativistic Maxwell equations for arbitrary small-amplitude oscillations of a non-rotating spherical NS with a generic magnetic field configuration and show that they can be solved analytically within the LCDA as in Newtonian theory. We then apply this solution to the case of toroidal oscillations of a NS with a dipole magnetic field and find that the LCDA is again valid for at least half of all toroidal oscillation modes, as in Newtonian theory. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from these oscillations for all of the modes for which the size of the polar cap is small and discuss the influence of GR effects on the energy losses.

The paper is organized as follows. In Section 2 we introduce some definitions and derive the quasi-stationary Maxwell equations in Schwarzschild spacetime as well as the boundary conditions for the electromagnetic fields at the stellar surface. In Section 3 we sketch our method for analytically solving the Maxwell equations for arbitrary NS oscillations with a generic magnetic field configuration. In Section 4 we apply our formalism to the case of purely toroidal oscillations of a NS with a dipole magnetic field and also discuss the validity of the LCDA and the role of GR effects. In Section 5 we calculate the energy losses due to plasma outflow caused by the toroidal oscillations. Some detailed technical calculations related to the discussion in the main part of the paper are presented in Appendices A-C.

We use units for which , a space-like signature and a spherical coordinate system . Greek indices are taken to run from 0 to 3 while Latin indices run from 1 to 3 and we adopt the standard convention for summation over repeated indices. We indicate four-vectors with bold symbols (e.g. ) and three-vectors with an arrow (e.g. ).

## 2 General Formalism

### 2.1 Quasi-stationary Maxwell equations in Schwarzschild spacetime

The study of electromagnetic processes related to stellar oscillations in the vicinity of NSs should, in principle, use the coupled system of Einstein-Maxwell equations. However, such an approach would be overly complicated for our study here, as it is for many other astrophysical problems. Here we simplify the problem by neglecting the contributions of the electromagnetic fields, the NS rotation and the NS oscillations to the spacetime metric and the structure of the NS3, noting that this is expected to be a good approximation for small-amplitude oscillations. Indeed, for a star with average mass-energy density , mass and radius , the maximum fractional change in the spacetime metric produced by the magnetic field is typically of the same order as the ratio between the energy density in the surface magnetic field and average mass-energy density of the NS, i.e.,

 B28π¯ρc2≃10−7(B1015 G)2(1.4 M⊙M)(R10 km)3 . (4)

The corresponding fractional change in the metric due to rotation is of order

 0.1(ΩΩK)2=10−7(Ω1 Hz)2(1 kHzΩK)2 (5)

where is the Keplerian angular velocity at the surface of the NS. Moreover, in the case of magnetars, which we consider in our study, the oscillations are thought to be triggered by the global reconfiguration of the magnetic field. Due to this reason, the corrections due to the oscillations should not exceed the contribution due to the magnetic field itself given by estimate (4). Therefore, we can safely work in the background spacetime of a static spherical star, whose line element in a spherical coordinate system is given by

 ds2=g00(r)dt2+g11(r)dr2+r2dθ2+r2sin2θdϕ2 , (6)

while the geometry of the spacetime external to the star (i.e. for ) is given by the Schwarzschild solution:

 ds2=−N2dt2+N−2dr2+r2dθ2+r2sin2θdϕ2 , (7)

where and is the total mass of the star. For the part of the spacetime inside the star, we represent the metric in terms of functions and as

 g00=−e2Φ(r) ,g11=e2Λ(r)=(1−2m(r)r)−1 , (8)

where is the volume integral of the total energy density over the spatial coordinates. The form of these functions is given by solution of the standard TOV equations for spherical relativistic stars (see, e.g., Shapiro & Teukolsky, 1983) and they are matched continuously to the external Schwarzschild spacetime through the relations

 g00(r=R)=N2R ,g11(r=R)=N−2R , (9)

where . Within the external part of the spacetime, we select a family of static observers with four-velocity components given by

 (uα)obs≡N−1(1,0,0,0) . (10)

and associated orthonormal frames having tetrad four vectors and 1-forms , which will become useful when determining the “physical” components of the electromagnetic fields. The components of the vectors are given by equations (6)-(9) of Rezzolla & Ahmedov (2004) (hereafter Paper I).

The general relativistic Maxwell equations have the following form (Landau & Lifshitz, 1987)

 3F[αβ,γ]=Fαβ,γ+Fγα,β+Fβγ,α=0 , (11)
 Fαβ    ;β=4πJα , (12)

where is the electromagnetic field tensor and is the electric-charge 4-current. We consider the region close to the star (the near zone), at distances from the NS much smaller than the wavelength . In the near zone the electromagnetic fields are quasi-stationary, therefore we neglect the displacement current term in the Maxwell equations. Once expressed in terms of the physical components of the electric and magnetic fields, equations (11) and (12) become (see Section 2 of Paper I for details of the derivation)

 sinθ∂r(r2B^r)+N−1r∂θ(sinθB^θ)+N−1r∂ϕB^ϕ=0 , (13)
 (rsinθ)∂B^r∂t = N[∂ϕE^θ−∂θ(sinθE^ϕ)] , (14) (N−1rsinθ)∂B^θ∂t = −∂ϕE^r+sinθ∂r(rNE^ϕ) , (15) (N−1r)∂B^ϕ∂t = −∂r(rNE^θ)+∂θE^r , (16)
 Nsinθ∂r(r2E^r)+r∂θ(sinθE^θ)+r∂ϕE^ϕ=4πρer2sinθ , (17)
 [∂θ(sinθB^ϕ)−∂ϕB^θ] = 4πrsinθJ^r , (18) ∂ϕB^r−sinθ∂r(rNB^ϕ) = 4πrsinθJ^θ , (19) ∂r(NrB^θ)−∂θB^r = 4πrJ^ϕ , (20)

where is the proper charge density. We further assume that the force-free condition,

 →ESC⋅→B=0 , (21)

is fulfilled everywhere in the magnetosphere, implying that the magnetosphere of the NS is populated with charged particles that cancel the longitudinal component of the electric field. The charge density responsible for the electric field (cf. equation 17) is the characteristic charge density of the force-free magnetosphere; this is appropriate for describing the charge density in the inner parts of the NS magnetosphere. We will refer to as the space-charge (SC) electric field, while to as the SC charge density.

Finally, we introduce the perturbation of the NS crust in terms of its four-velocity, with the components being given by

 Missing or unrecognized delimiter for \bigg (22)

where is the relative oscillation three-velocity of the conducting stellar surface with respect to the unperturbed state of the star.

### 2.2 Boundary conditions at the surface of star

We now begin our study of the internal electromagnetic field induced by the stellar oscillations. We assume here that the material in the crust can be treated as a perfect conductor and the induced electric field then depends on the magnetic field and the pulsational velocity field according to the following relations (see Paper I for details of the derivation):

 E^rin=−e−Φ[δv^θB^ϕ−δv^ϕB^θ] , (23) E^θin=−e−Φ[δv^ϕB^r−δv^rB^ϕ] , (24) E^ϕin=−e−Φ[δv^rB^θ−δv^θB^r] . (25)

Boundary conditions for the magnetic field at the stellar surface can be obtained from the requirement of continuity for the radial component, while leaving the tangential components free to be discontinuous because of surface currents:

 B^rex|r=R=B^rin|r=R , (26) B^θex|r=R=B^θin|r=R+4πi^ϕ , (27) B^ϕex|r=R=B^ϕin|r=R−4πi^θ , (28)

where is the surface current density. Boundary conditions for the electric field at the stellar surface are obtained from requirement of continuity of the tangential components, leaving to have a discontinuity proportional to the surface charge density :

 E^rex|r=R = E^rin|r=R+4πΣs=−N−1R[δv^θB^ϕ−δv^ϕB^θ]|r=R+4πΣs , (29) E^θex|r=R = E^θin|r=R=−N−1R[δv^ϕB^r−δv^rB^ϕ]|r=R , (30) E^ϕex|r=R = E^ϕin|r=R=−N−1R[δv^rB^θ−δv^θB^r]|r=R , (31)

where is the surface charge density.

### 2.3 The low current density approximation

The low current density approximation was introduced by TBS, and in the present section we present a brief introduction to it for completeness. Close to the NS surface, the current flows along the magnetic field lines, and so in the inner parts of the magnetosphere it can be expressed as

 →J=α(r,θ,ϕ)⋅→B , (32)

where is a scalar function. The system of equations (13)–(20), (21) and (32) forms a complete set but is overly complicated for solving in the general case. However, within the LCDA these equations can, as we show below, be solved analytically for arbitrary oscillations of a NS with a generic magnetic field configuration.

The LCDA scheme is based on the assumption that the perturbation of the magnetic field induced by currents flowing in the NS interior is much larger than that due to currents in the magnetosphere, which are neglected to first order in the oscillation parameter :

 4πc→J≪∇×→B , (33)

and

 ∇×→B(1)=0 , (34)

where is the first order term of the expansion in . This also implies that the current density satisfies the condition

 J≪1r(B(0) ξR)c≈ρSC(R) c(cωr) , (35)

where is the SC density near to the surface of the star. Here we have used the relation , where is the velocity amplitude of the oscillation and is its frequency.

In regions of complete charge separation, the maximum current density is given by . Since the absolute value of decreases with increasing and because in the near zone, condition (35) is satisfied in the magnetosphere if there is complete charge separation there. Since the current in the magnetosphere flows along magnetic field lines, its magnitude does not change and so condition (35) is also satisfied along magnetic field lines in non-charge-separated regions as long as they have crossed regions with complete charge separation.

In the following, we solve the Maxwell equations assuming that condition (35) is satisfied throughout the whole near zone. As discussed above, a regular solution of the system of equations (13)-(20), (21) and (32) should exist for arbitrary oscillations and arbitrary configurations of the NS magnetic field and so, as shown by TBS, if a solution has an unphysical behaviour, this would imply that the LCDA fails for this oscillation and that the accelerating electric field cannot be screened only by a stationary configuration of the charged-separated plasma. In some regions of the magnetosphere, the current density could be as high as

 J≃ρSCc(cωr) . (36)

For a more detailed discussion of the LCDA and its validity, we refer the reader to Sections 2.3 and 3.2.1 of TBS.

## 3 The LCDA solution

### 3.1 The electromagnetic field in the magnetosphere

We now begin our solution of the Maxwell equations, assuming that the LCDA condition (35) is satisfied everywhere in the magnetosphere. Within the LCDA, equations (18)-(20) for the magnetic field in the magnetosphere take the form

 ∂θ(sinθB^ϕ)−∂ϕB^θ = 0 , (37) ∂ϕB^r−sinθ∂r(rNB^ϕ) = 0 , (38) ∂r(NrB^θ)−∂θB^r = 0 . (39)

As demonstrated in Paper I, the components of the magnetic field and can be expressed in terms of a scalar function in the following way:

 B^r = −1r2sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS] , (40) B^θ = Nr∂θ∂rS , (41) B^ϕ = Nrsinθ∂ϕ∂rS . (42)

Substituting these expressions into the Maxwell equations (14)–(16), we obtain a system of equations for the electric field components which has the following general solution

 E^rSC=−∂r(ΨSC) , (43) E^θSC=−1Nrsinθ∂t∂ϕS−1Nr∂θ(ΨSC) , (44) E^ϕSC=1Nr∂t∂θS−1Nrsinθ∂ϕ(ΨSC) , (45)

where is an arbitrary scalar function. The terms proportional to the gradient of are responsible for the contribution of the charged particles in the magnetosphere. The vacuum part of the electric field is given by the derivatives of the scalar function . Substituting (43)-(45) into equation (17), we get an expression for the SC charge density in terms of :

 ρSC=−14πr2[N∂r(r2∂rΨSC)+1N△ΩΨSC], (46)

where is the angular part of the Laplacian:

 △Ω=1sinθ∂θ(sinθ∂θ)+1sin2θ∂ϕϕ. (47)

### 3.2 The equation for ΨSC

Substituting expressions (40)–(42) and (43)–(45) for the components of the electric and magnetic fields into the force-free condition (21), we get the following equation for

 1sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]∂r(ΨSC) − 1sinθ[∂ϕ∂tS∂θ∂rS−∂θ∂tS∂ϕ∂rS] (48) − ∂θ∂rS∂θ(ΨSC)−1sin2θ∂ϕ∂rS∂ϕ(ΨSC)=0 .

If the amplitude of the NS oscillations is suitably small (), the function can be series expanded in terms of the dimensionless perturbation parameter and can be approximated by the sum of the two lowest order terms

 S(t,r,θ,ϕ)=S0(r,θ,ϕ)+δS(t,r,θ,ϕ) . (49)

Here the first term corresponds to the unperturbed static magnetic field of the NS, while is the first order correction to it. At this level of approximation, equation (48) for takes the form

 1sin2θ[sinθ∂θ(sinθ∂θS0)+∂ϕ∂ϕS0]∂r(ΨSC) − 1sinθ[∂ϕ∂t(δS)∂θ∂rS0−∂θ∂t(δS)∂ϕ∂rS0] (50) − ∂θ∂rS0∂θ(ΨSC)−1sin2θ∂ϕ∂rS0∂ϕ(ΨSC)=0 .

Next we expand in terms of the spherical harmonics:

 S=∞∑ℓ=0ℓ∑m=−ℓSℓm(t,r)Yℓm(θ,ϕ) . (51)

where the functions are given in terms of Legendre functions of the second kind by (Rezzolla et al., 2001)

 Sℓm(t,r)=−r2M2ddr[r(1−2Mr)ddrQℓ(1−rM)]sℓm(t) . (52)

Note that all of the time dependence in (52) is contained in the integration constants which, as we will see later, are determined by the boundary conditions at the surface of the star. We now series expand the coefficients and in terms of

 Sℓm(t,r)=S0ℓm(r)+δSℓm(t,r),sℓm(t)=s0ℓm+δsℓm(t) , (53)

where all of the time dependence is now confined within the coefficients and , while the coefficients and are responsible for the unperturbed static magnetic field of the star. Using these results, we can also express and in terms of a series in in the following way

 S0 = ∞∑ℓ=0ℓ∑m=−ℓS0ℓm(r)Yℓm(θ,ϕ), (54) δS = ∞∑ℓ=0ℓ∑m=−ℓδSℓm(t,r)Yℓm(θ,ϕ) . (55)

The variables and in the functions and can be separated using relation (52):

 S0ℓm(r) = Missing or unrecognized delimiter for \right (56) Sℓm(t,r) = Missing or unrecognized delimiter for \right (57)

### 3.3 The boundary condition for ΨSC

We now derive a boundary condition for at the stellar surface using the behaviour of the electric and magnetic fields in that region. Following TBS, we assume that near to the stellar surface the interior magnetic field has the same behaviour as the exterior one:

 B^r=−C1r2sin2θ[sinθ∂θ(sinθ∂θS)+∂ϕϕS] , (58) B^θ=C1e−Λr∂θ∂rS , (59) B^ϕ=C1e−Λrsinθ∂ϕ∂rS . (60)

Using the continuity condition for the normal component of the magnetic field at the stellar surface (Pons & Geppert, 2007) together with the condition , one finds that the integration constant is equal to one. The interior electric field components can then be obtained by substituting (58) – (60) (with ) into (23) – (25):

 E^rin = −e−(Φ+Λ)rsinθ{δv^θ∂ϕ∂rS−sinθδv^ϕ∂θ∂rS} , (61) E^θin = e−(Φ+Λ)rsinθ{δv^r∂ϕ∂rS+δv^ϕeΛrsinθ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]} , (62) E^ϕin = −e−(Φ+Λ)r{δv^r∂θ∂rS+δv^θeΛrsin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]} . (63)

The continuity condition for the component of the electric field across the stellar surface (30) gives a boundary condition for :

 ΨSC,θ|r=R=−{δv^ϕRsin2θ[sinθ∂θ(sinθ∂θS)+∂ϕ∂ϕS]+Nδv^rsinθ∂ϕ∂rS+1sinθ∂t∂ϕS}|r=R , (64)

while the continuity condition for (31) gives a boundary condition for :

 Missing or unrecognized delimiter for \right (65)

Integration of equation (64) or equation (65) over or respectively, gives a boundary condition for . We will use the result of integrating equation (64) over . Assuming that the perturbation depends on time as , we obtain the following condition, correct to first order in ,

 ΨSC|r=R=−∫{δv^ϕRsin2θ[sinθ∂θ(sinθ∂θS0)+∂ϕ∂ϕS0]+Nδv^rsinθ∂ϕ∂rS0+1sinθ∂t∂ϕ(δS)}dθ|r=R+eiωtF(ϕ) , (66)

where is a function only of which we will determine below.

The components of the stellar-oscillation velocity field are continuously differentiable functions of and . The boundary conditions for the electric field (30)-(31) imply that the tangential components of the electric field must be finite. The vacuum terms on the right-hand side of (44)-(45) and the terms on both sides of equation (45) are also finite. Consequently, the term

 −∂ϕ(ΨSC)sinθ|r=R (67)

should also be finite. Hence we obtain that and so the function in the expression for boundary condition (66) must satisfy the condition , where is a constant. Using gauge invariance, we choose

 ΨSC|θ=0;r=R=0, (68)

and from this and equation (66), we obtain our expression for the boundary condition for at the stellar surface:

 ΨSC|r=R=−∫θ0{δv^ϕRsin2θ[sinθ∂θ(sinθ∂θS0)+∂ϕ∂ϕS0]+Nδv^rsinθ∂ϕ∂rS0+1sinθ∂t∂ϕ(δS)}dθ|r=R . (69)

## 4 Toroidal oscillations of a NS with a dipole magnetic field

As an important application of this formalism, we now consider small-amplitude toroidal oscillations of a NS with a dipole magnetic field. For toroidal oscillations in the mode, a generic conducting fluid element is displaced from its initial location to a perturbed location with the velocity field (Unno et al., 1989),

 δv^r=0 ,δv^θ=dξθdt=e−iωtη(r)1sinθ∂ϕYℓ′m′(θ,ϕ) ,δv^ϕ=dξϕdt=−e−iωtη(r)∂θYℓ′m′(θ,ϕ) , (70)

where is the oscillation frequency and is the transverse velocity amplitude. Note that in the above expressions (70), the oscillation mode axis is directed along the -axis. We use a prime to denote the spherical harmonic indices in the case of the oscillation modes.

### 4.1 The unperturbed exterior dipole magnetic field

If the static unperturbed magnetic field of the NS is of a dipole type, then the coefficients involved in specifying it have the following form (see eq. 117 of Paper I)

 s010=−√3π2μcosχ ,s011=√3π2μsinχ , (71)

where is the magnetic dipole moment of the star, as measured by a distant observer, and is the inclination angle between the dipole moment and -axis. Substituting expressions (71) into (56) and then the latter into (54), we get

 S0=−3μr28M3[lnN2+2Mr(1+Mr)](cosθcosχ+eiϕsinθsinχ) (72)

The corresponding magnetic field components have the form

 B^r0=−3μ4M3[lnN2+2Mr(1+Mr)](cosχcosθ+sinχsinθeiϕ) , (73) B^θ0=3μN4M2r[rMlnN2+1N2+1](cosχsinθ−sinχcosθeiϕ) , (74) B^ϕ0=3μN4M2r[rMlnN2+1N2+1](−isinχeiϕ) . (75)

At the stellar surface, these expressions for the unperturbed magnetic field components become

 B^rR=fRB0(cosχcosθ+sinχsinθeiϕ) ,B^θR=hRB0(cosχsinθ−sinχcosθeiϕ) ,B^ϕR=−ihRB0(sinχeiϕ) , (76)

where is defined as . In Newtonian theory would be the value of the magnetic strength at the magnetic pole but this becomes modified in GR. The GR modifications are contained within the parameters

 hR=3R2NR8M2[RMlnN2R+1N2R+1] ,fR=−3R38M3[lnN2R+2MR(1+MR)] . (77)

For a given , the magnetic field near to the surface of the NS is stronger in GR than in Newtonian theory, as already noted by Ginzburg & Ozernoy (1964).

### 4.2 The equation for ΨSC

Substituting from (72) into equation (50), we obtain a partial differential equation containing two unknown functions and for arbitrary oscillations of a NS with a dipole magnetic field

 −2r2q1(r)(cosθcosχ+eiϕsinθsinχ)∂r(ΨSC)+∂r[r2q1(r)](sinθcosχ−eiϕcosθsinχ)∂θ(ΨSC) (78) −∂r[r2q1(r)]eiϕsinχsinθ∂ϕ(ΨSC)+∂r[r2q1(r)]sinθ[(sinθcosχ−eiϕcosθsinχ)∂ϕ∂t(δS)+ieiϕsinχsinθ∂θ∂t(δS)]=0 ,

where we have introduced a new function for simplicity of notation [see Eq. (119) for the definition of ].

From (69), the boundary condition for at the stellar surface is

 ΨSC|r=R=∫θ0{B0RfRδv^ϕ∂θ(cosθcosχ+eiϕsinθsinχ)−1sinθ∂t∂ϕ(δS)}dθ|r=R . (79)

Using the expressions for the velocity field of the toroidal oscillations (70) and for the boundary conditions for the partial derivatives of the SC potential (64)-(65), we find that is given by (see Appendix A for details of the derivation)

 ∂tδS(r,t) = ∞∑ℓ=0ℓ∑m=−ℓB0RfR~ηRℓ(ℓ+1)r2qℓ(r)R2qℓ(R) (80) × ∫4π[∂θYℓm(sinθcosχ−eiϕcosθsinχ)+ieiϕ∂ϕYℓmsinθsinχ]Y∗ℓ′m′(θ,ϕ)sinθdΩ .

From here on, for simplicity, we will consider only the case with . Although our solution depends on the angle between the magnetic field axis and the oscillation mode axis, focusing on the case does not actually imply a loss of generality because any mode with its axis not aligned with a given direction can be represented as a sum of modes with axes along this direction. We have developed a MATHEMATICA code for analytically solving equation (50) and hence obtaining analytic expressions for the electric and magnetic fields and for the SC density.

The solution of equation (78) for the case is given in Appendix B, where we show that the general solution has the following form

 ΨSC=−12m′2ℓ′(ℓ′+1)B0RfR~ηR∫rR∂r′[r′2