3.4 Nonaxisymmetric perturbations

3.4.1 Nonrotating limit

Thorne, Campolattaro, and Price, in a series of papers [308245303], initiated the computation of nonradial modes by formulating the problem in the Regge–Wheeler (RW) gauge [250] and numerically computing nonradial modes for a number of neutron star models. A variational method for obtaining eigenfrequencies and eigenfunctions has been constructed by Detweiler and Ipser [85]. Lindblom and Detweiler [201] explicitly reduced the system of equations to four first order ordinary differential equations and obtained more accurate eigenfrequencies and damping times for a larger set of neutron star models. They later realized that their system of equations is sometimes singular inside the star and obtained an improved set of equations, which is free of this singularity [86].

Chandrasekhar and Ferrari [61Jump To The Next Citation Point] expressed the nonradial pulsation problem in terms of a fifth order system in a diagonal gauge, which is formally independent of fluid variables. Thus, they reformulate the problem in a way analogous to the scattering of gravitational waves off a black hole. Ipser and Price [163] show that in the RW gauge, nonradial pulsations can be described by a system of two second order differential equations, which can also be independent of fluid variables. In addition, they find that the diagonal gauge of Chandrasekhar and Ferrari has a remaining gauge freedom which, when removed, also leads to a fourth order system of equations [244].

In order to locate purely outgoing wave modes, one has to be able to distinguish the outgoing wave part from the ingoing wave part at infinity. This is typically achieved using analytic approximations of the solution at infinity.

W -modes pose a more challenging numerical problem because they are strongly damped and the techniques used for f - and p-modes fail to distinguish the outgoing wave part. The first accurate numerical solutions were obtained by Kokkotas and Schutz [181], followed by Leins, Nollert, and Soffel [196]. Andersson, Kokkotas, and Schutz [15] successfully combine a redefinition of variables with a complex-coordinate integration method, obtaining highly accurate complex frequencies for w-modes. In this method, the ingoing and outgoing solutions are separated by numerically calculating their analytic continuations to a place in the complex-coordinate plane, where they have comparable amplitudes. Since this approach is purely numerical, it could prove to be suitable for the computation of quasi-normal modes in rotating stars, where analytic solutions at infinity are not available.

The non-availability of asymptotic solutions at infinity in the case of rotating stars is one of the major difficulties for computing outgoing modes in rapidly rotating relativistic stars. A method that may help to overcome this problem, at least to an acceptable approximation, has been found by Lindblom, Mendell, and Ipser [205].

The authors obtain approximate near-zone boundary conditions for the outgoing modes that replace the outgoing wave condition at infinity and that enable one to compute the eigenfrequencies with very satisfactory accuracy. First, the pulsation equations of polar modes in the Regge–Wheeler gauge are reformulated as a set of two second order radial equations for two potentials – one corresponding to fluid perturbations and the other to the perturbations of the spacetime. The equation for the spacetime perturbation reduces to a scalar wave equation at infinity and to Laplace’s equation for zero-frequency solutions. From these, an approximate boundary condition for outgoing modes is constructed and imposed in the near zone of the star (in fact, on its surface) instead of at infinity. For polytropic models, the near-zone boundary condition yields f -mode eigenfrequencies with real parts accurate to 0.01 – 0.1% and imaginary parts with accuracy at the 10 – 20% level, for the most relativistic stars. If the near zone boundary condition can be applied to the oscillations of rapidly rotating stars, the resulting frequencies and damping times should have comparable accuracy.

3.4.2 Slow rotation approximation

The slow rotation approximation is useful for obtaining a first estimate of the effect of rotation on the pulsations of relativistic stars. To lowest order in rotation, a polar l-mode of an initially nonrotating star couples to an axial l ± 1 mode in the presence of rotation. Conversely, an axial l-mode couples to a polar l ± 1 mode as was first discussed by Chandrasekhar and Ferrari [61Jump To The Next Citation Point].

The equations of nonaxisymmetric perturbations in the slow rotation limit are derived in a diagonal gauge by Chandrasekhar and Ferrari [61], and in the Regge–Wheeler gauge by Kojima [172174], where the complex frequencies σ = σR + iσI for the l = m modes of various polytropes are computed. For counterrotating modes, both σR and σI decrease, tending to zero, as the rotation rate increases (when σ passes through zero, the star becomes unstable to the CFS instability). Extrapolating σR and σI to higher rotation rates, Kojima finds a large discrepancy between the points where σR and σI go through zero. This shows that the slow rotation formalism cannot accurately determine the onset of the CFS instability of polar modes in rapidly rotating neutron stars.

In [173], it is shown that, for slowly rotating stars, the coupling between polar and axial modes affects the frequency of f - and p-modes only to second order in rotation, so that, in the slow rotation approximation, to 𝒪 (Ω ), the coupling can be neglected when computing frequencies.

The linear perturbation equations in the slow rotation approximation have recently been derived in a new gauge by Ruoff, Stavridis, and Kokkotas [256]. Using the ADM formalism, a first order hyperbolic evolution system is obtained, which is suitable for numerical integration without further manipulations (as was required in the Regge–Wheeler gauge). In this gauge (which is related to a gauge introduced for nonrotating stars in [27]), the symmetry between the polar and axial equations becomes directly apparent.

The case of relativistic inertial modes is different, as these modes have both axial and polar parts at order 𝒪 (Ω), and the presence of continuous bands in the spectrum (at this order in the rotation rate) has led to a series of detailed investigations of the properties of these modes (see [178Jump To The Next Citation Point] for a review). In a recent paper, Ruoff, Stavridis, and Kokkotas [257Jump To The Next Citation Point] finally show that the inclusion of both polar and axial parts in the computation of relativistic r-modes, at order 𝒪(Ω ), allows for discrete modes to be computed, in agreement with post-Newtonian [213] and nonlinear, rapid-rotation [293Jump To The Next Citation Point] calculations.

3.4.3 Post-Newtonian approximation

A step toward the solution of the perturbation equations in full general relativity has been taken by Cutler and Lindblom [7779Jump To The Next Citation Point198Jump To The Next Citation Point], who obtain frequencies for the l = m f -modes in rotating stars in the first post-Newtonian (1-PN) approximation. The perturbation equations are derived in the post-Newtonian formalism (see [36]), i.e., the equations are separated into equations of consistent order in 1∕c.

Cutler and Lindblom show that in this scheme, the perturbation of the 1-PN correction of the four-velocity of the fluid can be obtained analytically in terms of other variables; this is similar to the perturbation in the three-velocity in the Newtonian Ipser–Managan scheme. The perturbation in the 1-PN corrections are obtained by solving an eigenvalue problem, which consists of three second order equations, with the 1-PN correction to the eigenfrequency of a mode Δ ω as the eigenvalue.

Cutler and Lindblom obtain a formula that yields Δ ω if one knows the 1-PN stationary solution and the solution to the Newtonian perturbation equations. Thus, the frequency of a mode in the 1-PN approximation can be obtained without actually solving the 1-PN perturbation equations numerically. The 1-PN code was checked in the nonrotating limit and it was found to reproduce the exact general relativistic frequencies for stars with M ∕R = 0.2, obeying an N = 1 polytropic EOS, with an accuracy of 3 – 8%.

Along a sequence of rotating stars, the frequency of a mode is commonly described by the ratio of the frequency of the mode in the comoving frame to the frequency of the mode in the nonrotating limit. For an N = 1 polytrope and for M ∕R = 0.2, this frequency ratio is reduced by as much as 12% in the 1-PN approximation compared to its Newtonian counterpart (for the fundamental l = m modes) which is representative of the effect that general relativity has on the frequency of quasi-normal modes in rotating stars.

3.4.4 Cowling approximation

In several situations, the frequency of pulsations in relativistic stars can be estimated even if one completely neglects the perturbation in the gravitational field, i.e., if one sets δgab = 0 in the perturbation equations [224]. In this approximation, the pulsations are described only by the perturbation in the fluid variables, and the scheme works quite well for f, p, and r-modes [208]. A different version of the Cowling approximation, in which δgtr is kept nonzero in the perturbation equations, has been suggested to be more suitable for g-modes [99], since these modes could have large fluid velocities, even though the variation in the gravitational field is weak.

Yoshida and Kojima [331] examine the accuracy of the relativistic Cowling approximation in slowly rotating stars. The first order correction to the frequency of a mode depends only on the eigenfrequency and eigenfunctions of the mode in the absence of rotation and on the angular velocity of the star. The eigenfrequencies of f, p1, and p2 modes for slowly rotating stars with M ∕R between 0.05 and 0.2 are computed (assuming polytropic EOSs with N = 1 and N = 1.5) and compared to their counterparts in the slow rotation approximation.

For the l = 2 f -mode, the relative error in the eigenfrequency because of the Cowling approximation is 30% for weakly relativistic stars (M ∕R = 0.05) and about 15% for stars with M ∕R = 0.2; the error decreases for higher l-modes. For the p1 and p2 modes the relative error is similar in magnitude but it is smaller for less relativistic stars. Also, for p-modes, the Cowling approximation becomes more accurate for increasing radial mode number.

As an application, Yoshida and Eriguchi [328Jump To The Next Citation Point329Jump To The Next Citation Point] use the Cowling approximation to estimate the onset of the f -mode CFS instability in rapidly rotating relativistic stars and to compute frequencies of f -modes for several realistic equations of state (see Figure 6View Image).

View Image

Figure 6: Eigenfrequencies (in the Cowling approximation) of f -modes along a M = 1.8M ⊙ sequence of models, constructed with the WFF3-NV EOS. The vertical line corresponds to the frequency of rotation of the model at the mass-shedding limit of the sequence. (Figure 1 of Yoshida and Eriguchi [329]; used with permission.)

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