{'fraction_non_alphanumeric': 0.06297315898141775, 'fraction_numerical': 0.02860461114934618, 'mean_word_length': 4.0966787241039135, 'pattern_counts': {'":': 0, '<': 38, '<?xml version=': 0, '>': 50, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Resonant Tidal Excitations of Rotating Neutron Stars in Coalescing Binaries

In a coalescing neutron star-neutron star (NS-NS) or neutron star-black hole (NS-BH) binary, oscillation modes of the NS can be resonantly excited by the companion during the final minutes of the inspiral. The resonant energy transfer between the orbit and NS speeds up or slows down the inspiral and induces a phase change in the emitted gravitational waves from the binary. A tidal resonance, (jk,m), occurs when the mode frequency equals m times the orbital frequency. For f-mode resonance to occur before coalescence, the NS must have rapid rotation, with spin frequency nu_s>710Hz for (22,2)-resonance and nu_s>570Hz for (33,3)-resonance (M=1.4Mo and R=10km; however, for R=15km, these spin frequencies become 330Hz and 250Hz). Because of strong tidal coupling, f-mode resonances induce a large change in the number of orbital cycles, N_orb, with maximum N_orb~10-1000 for (22,2)-resonance and N_orb~1 for (33,3)- resonance. Such resonant effects, if present, must be included in constructing waveform templates used in searching for gravitational wave signals. Higher order f-mode resonances can occur at slower rotation rates, but N_orb<0.1. For the dominant g-mode (22,2)-resonance, even modest rotation (nu_s<100Hz) can enhance the resonant effect on the orbit by shifting resonance to a smaller orbital frequency. However, because of the weak coupling, N_orb lies in the range 10^-3-10^-2 (depending on the NS EOS) and is probably negligible for the purpose of detecting gravitational waves. R-mode resonances require misaligned spin-orbit inclinations, and the dominant resonances correspond to (22,3) and (22,1). Since tidal coupling depends strongly on rotation rate, N_orb<10^-2(R/10km)^(10)(M/1.4Mo)^(-20/3) is negligible for canonical NS parameters but can be appreciable if the NS radius is larger.


INTRODUCTION
The new generation of gravitational wave detectors such as the Laser Interferometer Gravitational Wave Observatory (LIGO; Abramovici et al. 1992) and its French-Italian counterpart VIRGO (Bradaschia et al. 1990) are designed to observe gravitational radiation in the frequency range from ∼ 10-1000 Hz. Coalescing neutron star-neutron star (NS-NS) and neutron star-black hole (NS-BH) binaries are promising sources for such radiation, with the last few minc 0000 RAS utes of the binary inspiral producing gravitational waves which sweep upward through this frequency range (e.g., Thorne 1987, Cutler et al. 1993, Thorne 1998. Due to the low signal-to-noise ratios, a good understanding of the mechanisms that generate and affect the gravitational waveform is required to create accurate theoretical templates that can be used to extract the signals from the noise. The binary inspiral and the resulting waveform can be described, to leading order, by Newtonian dynamics of two point masses, together with the lowest order dissipative effect corresponding to the emission of gravitational radiation via the quadrupole formula. To more accurately construct the theoretical templates, general relativistic effects must be taken into account by using post-Newtonian expansions (e.g., Blanchet et al. 1995) or techniques of numerical relativity (e.g., Teukolsky 1998;Brady et al. 1998). Other corrections come from hydrodynamical effects due to the finite size of neutron stars (Kochanek 1992;Bildsten & Cutler 1992;Lai et al. 1994). In particular, tidal interactions can have significant effects on the final stage of binary inspiral by destabilizing the orbit and accelerating the coalescence at small orbital radii (e.g., Lai et al. 1994;Lai & Shapiro 1995;Rasio & Shapiro 1995;Lai & Wiseman 1996;Baumgarte et al. 1997;New & Tohline 1997;Uryu & Eriguchi 1998; see also Shibata et al. 1993;Davies et al. 1994;Zhuge et al. 1996;Ruffert et al. 1997 for numerical simulations on binary coalescence).
The early stage of the inspiral, with the gravitational wave frequencies in the range of 10 Hz to a few hundred Hz, is particularly important for detecting the wave signal and for extracting binary parameters from the waveform using the matched filtering technique (e.g., Cutler et al. 1993;Cutler & Flanagan 1994). It is usually thought that tidal effects are completely negligible for this "low-frequency" regime. This is indeed the case for the "quasi-equilibrium" tides as the tidal interaction potential scales as D −6 (where D is the orbital separation; see Lai et al. 1994 for analytic expressions of the orbital phase error induced by static tides). The situation is more complicated for the resonant tides: As two compact objects inspiral, the orbit can momentarily come in resonance with the normal oscillation modes of the NS. By drawing energy from the orbital motion and resonantly exciting the modes, the rate of inspiral is modified, giving rise to a phase error in the gravitational waveform. This situation has been studied (Reisenegger & Goldreich 1994;Lai 1994) in the case of nonrotating neutron stars where the only modes that can be resonantly excited are g-modes (with typical mode frequencies < ∼ 100 Hz). It was found that the effect is negligible because the coupling between the gmode and the tidal potential is weak. However, the NS can be rotating, and consequently, these results need to be reexamined. As we show in this paper, even modest rotation of the NS can enhance the resonant g-mode excitation by shifting the resonance to a lower frequency (where the orbital decay is slower and resonance is stronger). In addition, f-modes, which possess strong tidal coupling, can have their normally high frequencies in the nonrotating case "dragged" or reduced by retrograde rotation to sufficiently low frequencies in the inertial frame, such that resonant excitations can occur well before binary merger. The NS spin also gives rise to a new class of modes which are absent in the nonrotating case: R-modes, which can have low frequencies and may also undergo resonant excitations.
The purpose of this paper is to study the resonant excitations of various modes of spinning NSs in inspiraling binary systems and to examine their effects on the orbital decay rate (particularly the phase error in the gravitational waveform). Section 2 develops the general formalism of resonant tides in coalescing NS binaries. In section 3 we apply the analysis to f-modes, g-modes, and r-modes when the spin axis is aligned or anti-aligned with the orbital angular momentum axis. Section 4 describes the modification to the results of the aligned (anti-aligned) case when the spin and orbital axes are misaligned by an inclination angle β. In section 5, we summarize our results and discuss their implications for gravitational wave detection.

Basic equations
Consider a NS with mass M , radius R, and spin Ωs in orbit with a companion M ′ , which can be another NS or a BH. We treat M ′ as a point mass. The orbital frequency vector is Ω orb . We define a coordinate system (x, y, z) centered on M with the z-axis along Ω orb , such that Ω orb = Ω orbẑ (with Ω orb ≥ 0). Define another coordinate system (x ′ , y ′ , z ′ ) with the z ′ -axis along Ωs, such that Ωs = Ωsẑ ′ (with Ωs ≥ 0) and the y ′ -axis in the orbital plane; these are the body axes. Let the angle between the z-axis and z ′ -axis be β (the spin-orbit inclination angle) and the angle between the y-axis and y ′axis be α. Thus the (x ′ , y ′ , z ′ ) frame is related to the (x, y, z) frame by Euler angle (α, β, γ = 0). The gravitational potential produced by M ′ can be expanded in terms of spherical harmonics: where r = (r, θ, φ) is the position vector of a fluid element in star M (r, θ, φ are the spherical coordinates with respect to the z-axis), D(t) = [D(t), π/2, Φ(t)] is the position vector of the point mass, M ′ , and W lm is defined in Press & Teukolsky (1977) as Here the symbol (−) p is zero if p is not an integer. For example, W2±2 = (3π/10) 1/2 and W3±3 = (5π/28) 1/2 . In equation (2.1), the l = 0 and l = 1 terms can be dropped since they are not relevant for tidal deformation. The tidal potential is specified by the index {lm} (relative to the orbital angular momentum axisẑ) and is nonzero only when (l+m) is even, reflecting the symmetry of the tidal potential. It is more convenient to define oscillation modes relative to the spin axisẑ ′ , and we will need to express the tidal potential in terms of Y lm ′ (θ ′ , φ ′ ), the spherical har- where the Wigner D-function is given by (e.g., Wybourne 1974) Thus {lm ′ } is the tidal potential index relative to the spin axis.
The linear perturbation of the tidal potential on M is specified by the Lagrangian displacement, ξ(r, t), of a fluid element from its unperturbed position. In the inertial frame, the equation of motion takes the form where v = Ωs × r is the unperturbed fluid velocity and C is a self-adjoint operator (a function of the pressure and gravity perturbations) acting on ξ (Friedman & Schutz 1978a). Our analysis is accomplished through an eigenmode decomposition of the Lagrangian displacement: ξ is taken to be the linear superposition of eigenmodes, ξα(r, t), namely We shall specify the normal mode by the index α = {njk} with respect to the body axes: n is the number of radial nodes, j gives the number of nodes in the θ ′ -eigenfunction, and k is the azimuthal index (about the z ′ -axis). In the following we shall suppress the radial index n so that α = {jk}. In the absence of external perturbation, the eigenmode behaves as where ωα is the α-mode frequency in the inertial frame. The mode eigenfunction satisfies the equation and we shall adopt the normalization Substituting (2.6) into (2.5), using (2.7) and (2.9), and noting that (2.8) gives a relation for C · ξ, the dynamical equation for the mode amplitude is where (this is called Bα in Lai 1997) and Q α,lm ′ is the tidal cou-pling coefficient or overlap integral defined by (2.12) Here δρα = −∇ · (ρ ξα) is the Eulerian density perturbation. Clearly, Q α,lm ′ is nonzero only when m ′ = k, thus (2.13)

Resonant Mode Energy
The right-hand-side of equation ( For Q jk,lk to be nonzero, the projected tidal potential [∝ Y lk (θ ′ , φ ′ )] must have the same θ ′ -symmetry, with respect to the equator, as the mode. This gives the requirement that l + j = even (for spherical modes) or l + j = odd (for r-modes) (see §3).
We consider now a specific resonance. Hereafter we shall adopt the convention ⋆ k > 0 and m > 0. Thus a resonance is designated by the indices (α, m) = (jk, m), which indicates that the mode α = {jk} is excited by tidal potential of azimuthal order m (with angular dependence e ±imΦ ). In general, there can be many l's that satisfies the conditions (2.14)-(2.16) † , but clearly the smallest l which satisfies these conditions will provide the dominant contribution. When appropriate, we shall also label a resonance using the notation (jk, lm), with l specifying the angular order of the dominant tidal field that contributes the resonance (jk, m).
In the special case when the spin and orbit are aligned or anti-aligned, one must have m = k, i.e., each mode can experience resonance only once. But in the general case of arbitrary spin-orbit inclination angle, β, for a given mode {jk}, there may be infinite number of m which satisfies the conditions (2.14)-(2.16); in other words, the mode can be resonantly excited many times at different orbital frequencies. Thus, strictly speaking, the energy transfer from the orbit to the mode at a given resonance (jk, m) depends on the amplitude of the mode induced by previous resonances (jk, m + 2), (jk, m + 4), · · · ‡ . However, as we shall see below, in almost all cases of interest, the energy transfer at the ⋆ Clearly, in equations (2.7) and (2.10), k and m can be positive or negative. † The exception is when the star is nonrotating, in which case m = k and l = j. ‡ Consider a spheroidal (f or g) mode, α = {jk}. For the (jk, m)resonance to be in effect, it is necessary for l + m = even [see eq. (2.15)] and l + j = even (otherwise Q jk,lk = 0). Now in order (jk, m)-resonance, ∆E jk,m , is much greater than the energy transfer at the earlier resonances, i.e., ∆E jk,m ≫ ∆E jk,m+2 . Therefore, throughout this paper, we shall treat different resonances of the same mode as being independent of each other. This corresponds to assuming that before each resonance, the mode amplitude is essentially zero.
To calculate the energy transfer to a given mode at a particular m-resonance, ∆E jk,m , we only need to keep the resonant driving terms in (2.10), namelÿ (2.17) Since aα ∝ e iωαt in the absence of external forcing [see eq. (2.7)], it is natural to define the mode amplitude cα via aα = cαe iωαt , (2.18) and we expect cα to be independent of time after the resonance § . Equation (2.17) then becomes Only one of the terms on the RHS of (2.19) can induce the resonance, which occurs when the argument of the exponential of either the first term or the second term goes through zero. The term that induces resonance depends on the sign of ωα.

General Properties of the Mode Frequencies
Before proceeding with our calculation, let us briefly discuss various possibilities for the mode frequency. In general, for a given mode index α = {jk} (recall our sign convention k > 0, and note that we have suppressed the radial mode index n), there are two possible solutions of opposite signs for ω (r) α , the mode frequency in the frame corotating with the star, which correspond to two different directions of propagation for the wave pattern. Recall from (2.7) that our convention for free oscillations is e iωαt+ikφ ′ relative to the spin axis, where Ωs = Ωsẑ ′ and Ωs ≥ 0. The mode frequency in the inertial frame, ωα, is related to ω (2.20) for the (jk, m + 1) to be in effect, it is necessary for l + m + 1 = even, which is incompatible with l + j = even. § Before and after a resonance, there is a small, non-resonant contribution to aα which is proportional to e ±imΦ(t) . This contribution can be neglected compared to the resonance-induced amplitude.
The first solution, with ω (r) α > 0, corresponds to the mode which propagates retrograde with respect to the spin (in the rotating frame). Rotation drags the mode frequency in the inertial frame to small values; thus the resonance is shifted to smaller orbital frequency. With increasing Ωs, the mode frequency ωα may become negative, corresponding to wave propagation in the same direction as the spin (in the inertial frame). This change in sign of ωα signals the onset of the so-called Chandrasekhar-Friedman-Schutz (CFS) instability (Chandrasekhar 1970;Friedman & Schutz 1978b) and can occur for both f-modes (e.g., Lindblom 1995) and g-modes (Lai 1998). At this point the mode energy in the inertial frame changes from positive (stable mode) to negative (unstable mode) ¶ . Note that when the resonant excitation of a positive-energy mode occurs, the orbital energy is transferred to the mode and the orbit decays faster; when a negative-energy mode is excited, the orbit actually gains energy and the orbital decay is slowed.
The second solution, with ω (r) α < 0, propagates prograde with respect to the spin (in the rotating frame). Rotation causes ωα to become even more negative. This mode always has positive energy and is stable. The resonance is shifted to even higher orbital frequency. Thus for f-modes, the ω (r) α < 0 modes do not experience any resonance. For g-modes, |ωα| can still lie in the appropriate range for resonance.
The r-modes are a special case for which ω (r) α > 0 but ωα < 0, meaning that in the rotating frame, the r-mode wave pattern always propagates opposite to the spin, while in the inertial frame it propagates in the same direction as the spin. This implies that the r-modes are always CFS unstable in the absence of fluid viscosity (see §3.3 below).

Mode with ωα > 0 (Spin-Retrograde Mode)
For a mode with ωα > 0, which corresponds to wave propagation opposite to the rotation, resonance occurs at mΦ = mΩ orb = ωα, where Ω orb is given by neglecting tidal and post-Newtonian corrections. In this case, only the second term on the RHS of (2.19) can induce resonance. Thus we havë Integrating over time around the resonance and noting that cα is almost constant after the resonance (and cα = 0 before the resonance since the mode amplitude induced by previous resonances is negligible; see the discussion at the beginning of §2), we find Resonance occurs when the argument of the exponential is equal to zero. Doing a Taylor expansion of the argument of the exponential around the resonance epoch and using mΦ = ωα, we have (2.24) D varies slowly with time, so that (2.25) Therefore the mode amplitude after the (α, m)-resonance is where Dα is the orbital radius where resonance occurs anḋ Ω orb is evaluated at Dα. The energy of the mode in the inertial frame is given by Friedman & Schutz (1978a) as Using eqs. (2.6)-(2.9), this gives the mode energy associated with the (α, m)-resonance as ∆Eα,m = 2 1 2 (2.28) The factor of two in the first line of (2.28) arises because, for each k > 0 and ωα > 0 resonant mode, there is also the identical k < 0 and ωα < 0 resonant mode (recall our sign convention m > 0 and k > 0). Hereafter we shall define Here we have defined Since the effect of the tidal interaction on the orbit is small, we use the quadrupole formula for the gravitational radiation of two point masses to obtain the orbital decay rate (e.g., Shapiro & Teukolsky 1983) where q = M ′ /M is the mass ratio. Equation (2.29) can be expressed in terms of index (α, m) = (jk, m) and quantities associated with the resonant mode, giving ∆Eα,m = 5π 96 In this case, the mode energy is always positive (see §2.2.1).

Mode with ωα < 0 (Spin-Prograde Mode)
For a mode with ωα < 0, corresponding to wave propagation in the same direction as the spin, the resonance occurs at mΩ orb = −ωα, and only the first term on the RHS of eq. (2.19) can induce resonance. Carrying out a similar calculation as in the ωα > 0 case, we find ∆Eα,m = − 5π 96  km . This difference reflects the "preferred" spin-orbit orientation for resonant excitation: In order to have resonance, the wave pattern of the mode in the inertial frame must propagate in the same direction as the orbital motion of the external mass, M ′ . For ωα > 0, the wave propagates opposite to the spin, thus the resonance will be strongest for retrograde spin-orbit inclination (β > 90 • ); For ωα < 0, the wave propagates in the same sense as the spin, thus we expect the resonance to be strongest for prograde spin-orbit inclination (β < 90 • ).

Change in Orbital Phase
We now calculate the orbital phase error due to the resonant energy transfer between the binary orbit and the NS oscillation mode. The number of orbital cycles, N orb , is obtained from where Etot = E orb + Estar is the total energy of the system, including both the orbit and the stellar mode. Comparing to the fiducial situation in which the mode energy is neglected and dN (0) orb = (Ω orb /2π)(dE orb /Ėtot), we find that the change in the number of orbital cycles due to the (α, m)-resonance is (Lai 1994 where tD ≡ |E orb /Ėtot| ≃ D/|Ḋ| is the orbital decay time, t orb = 2π/Ω orb is the orbital period, and |E orb | = GM M ′ /(2D) is the orbital energy. The subscript α implies that tD, t orb , and |E orb | are evaluated at the resonance radius Dα. Parameterizing by the resonance index (α, m) = (jk, m) and substituting in the expressions for ∆Eα,m, we obtain, for the mode with ωα > 0, If we keep only the dominant l-term and label the resonance by (α, lm), we arrive at Note that for ωα > 0, the mode energy is always positive, and thus (∆N orb ) α,lm is negative, which implies that the resonance causes the orbit to decay faster. Similarly, for the mode with ωα < 0, we find (∆N orb ) α,lm = + 25 6144 Note that when ωα < 0, the mode energy can be either positive or negative (∆Eα ∝ ωαεα), and thus (∆N orb ) α,lm can have both signs. This implies that the resonance can either slow down or speed up the inspiral.
The dominant quadrupolar gravitational wave emitted by the binary has frequency νgw = 2ν orb = Ω orb /π. Thus at the (α, m)-resonance, the gravitational wave frequency is where να = ωα/(2π) is the mode frequency in the inertial frame. In the following ( §3), we shall express our results in terms of νgw instead of |να|. The phase error in the gravitational wave due to the resonance is (∆Φgw)α,m = 4π(∆N orb )α,m. We expect that, for (∆Φgw)α,m ∼ 1, the phase error will not affect gravitational wave detection but could affect detailed analyses of the signals, such as parameter extraction (E. Flanagan 1998; S. Hughes 1999, private communication).
The orbital change, or phase error, does not occur instantaneously at D = Dα. The duration of the resonance is given by ( 2.44) This yields the number of orbital cycles during which resonance occurs, where M1.4 is the NS mass in units of 1.4M⊙ and recalling that νgw = (2/m)να. The resonance can be considered instantaneous if (δN orb )α,m is much less than dN orb /d ln νgw, the characteristic number of orbital cycles the binary spends near orbital frequency ν orb = νgw/2, as given by We now apply the general equations derived in §2 to calculate the effects of resonance for different types of modes in coalescing binary neutron stars. In this section we consider the cases when Ωs and Ω orb are either aligned (β = 0 • ) or anti-aligned (β = 180 • ). Various mode properties (ωα, Q α,lk , and εα) and our procedure to calculate them are summarized. The case of general β will be discussed in §4.

F-modes
For nonrotating NSs, the f-modes have frequencies of order (GM/R 3 ) 1/2 , which are too large for any resonance to occur during binary inspiral. Rotation can reduce the mode frequencies, making resonant excitations possible. In this paper, we shall consider only the f-modes with j = k. Since ωα = ω (r) α − kΩs, these j = k modes are more strongly affected by rotation. Only the modes with ω (r) α > 0 need to be considered, since rotation actually increases |ωα| for the ω (r) α < 0 modes, making their resonances impossible (see §2.2.1).
To calculate the f-mode properties relevant for tidal resonance, we shall model the NS as an incompressible Maclaurin spheroid, whose equatorial radius and polar radius are a1 and a3, respectively. The spin frequency Ωs is related to the eccentricity e = (1 − a 2 3 /a 2 1 ) 1/2 of the spheroid via the expression × 1 − e 2 1/2 3 − 2e 2 sin −1 e − 3e 1 − e 2 , (3.1) where ρ = 3M/(4πR 3 ) is the (uniform) density of the spheroid, and R = (a 2 1 a3) 1/3 is the mean radius. The frequency of the j = k mode in the rotating frame has been obtained by Bryan (1889) and Comins (1979): where the ± sign refers to modes which propagate opposite to Ωs and in the same direction as Ωs, respectively, and For j = k = 2, a more compact expression for the mode frequency is given by Chandrasekhar (1969) as where B11 is the index symbol (a dimensionless function of e) as defined in Chandrasekhar (1969). Also note that for Ωs = 0, the mode frequency becomes This is essentially the Kelvin mode. As discussed above, we shall only consider the mode with + sign in (3.2). Figure 1 shows the normalized spin frequency, at which resonance occurs [see eq. (2.14)] for several different f-modes (j = k = m = 2, 3, 4, 5). Higher order modes (k > 5) are not considered since, as shown below, their contribution to the orbital change is negligible. The spin frequency at which the star becomes secularly unstable (in the absence of viscosity) is where ωα becomes negative; for larger k-values, secular instability occurs at lower Ωs. Resonance can also occur at these rotation frequencies. For completeness, we shall include these unstable modes, even though they would require very high spin rates. The two vertical dashed lines in Fig. 1  .
To calculate the tidal coupling coefficient Q α,lk , we need to determine the Lagrangian displacement, ξα. We begin with the Euler equation in the corotating frame for the inviscid fluid: ∂v ∂t Φ is the self-gravitational potential. Linearizing and assuming small perturbations v, Ψ ∝ e iω (r) α t , eq. (3.9) reduces to Combining this with the continuity equation for an incompressible fluid, ∇ · v = 0, we find A specific set of solutions to this equation, corresponding to the j = k mode, is α ) can be obtained by substituting (3.13) into (3.11), giving (3.14) Adopting the normalization condition (2.9), we can write the displacement vector ξ as ξ(x, y, z) = (2k + 1)!! (2k)!! 1 4πρka 2k 1 a3 in Cartesian, cylindrical, and spherical coordinates, respectively. In (3.17), the radial and horizontal wavefunctions are given by [neglecting the factor (−1) k ] ξr = k ρa 2k 1 a3 and the spherical harmonic function Y kk is given by Applying appropriate boundary conditions (Comins 1979) for the eigenfunction then yields the expression for the mode frequency ω (r) α as given by (3.2). Using the spherical representation of ξ for the f-modes, the tidal coupling coefficient (for α = {kk} and l = m ′ = k), defined by (2.12), can be written as (3.20) Note that for a given θ, the radial variable ranges from 0 to rmax(θ), with where µ ′ = cosθ and ζ = 1 − e 2 1/2 /e. Equation (3.20) can then be evaluated explicitly, giving where we have normalizedεα and Q22,22 to their respective nonrotating values (rotation can change these by a factor of ∼ 20 and ∼ 1.5, respectively, for the range ofΩs < ∼ 0.57 considered in this paper). The upper sign (negative ∆N orb ) in (3.25) applies when ωα > 0 and β = 180 • , while the lower sign (positive ∆N orb ) applies when ωα < 0 and β = 0 • . Note that (∆N orb ) kk,kk diverges for k = 2 as νgw decreases. The reason for this is that at larger orbital radii (small resonant frequencies), the orbit decays more slowly, and the binary spends a longer time in resonance, giving rise to larger ∆N orb . For higher k-resonances, the same effect is also operative, but the tidal potential is weaker so that the net effect is that (∆N orb ) kk,kk ∝ ν (4k−11)/3 gw decreases with decreasing νgw.
It is evident from Fig. 2 that the (jk, m) = (22, 2) and (33, 3)-resonances contribute significantly to the orbital phase evolution. But, to attain such resonances during the binary inspiral, the NS of canonical mass (1.4M⊙) and radius (10 km) must be rotating very rapidly, i.e., νs > ∼ 710 Hz for the (22, 2)-resonance, νs > ∼ 570 Hz for the (33, 3)-resonance, νs > ∼ 460 Hz for the (44, 4)-resonance, and νs > ∼ 380 Hz for the (55, 5)-resonance (see Fig. 1). Such rapid rotation is physically possible but may be astrophysically unlikely. Observations of NS binary systems and the current theories regarding the formation of these systems indicate that such rapid rotation of the NS is not possible (see §5). If such resonances indeed occur, then their effect on the orbital phase of the binary inspiral must be taken into account when constructing theoretical wave templates in the search for gravitational waves from these systems.
It can also be seen from eqs. (3.1) and (3.2) that the NS spin frequency and mode frequencies depend on the radius of the NS and thus the NS equation of state (EOS). Current equations of state allow NSs with R = 15 km (e.g., Shapiro & Teukolsky 1983). This implies that the spin and mode frequencies are lowered, and resonance occurs at these lower frequencies as illustrated by Figure 3. Figure 4 shows the corresponding (∆N orb ) kk,kk as a function of the νgw for the R = 15 km NS. The lower limit to the spin frequency required for resonance is set by the associated gravitational wave frequency given by eq. (3.8). This limit is indicated by the dotted lines in Fig. 3 and Fig. 4 and yields νs > ∼ 330 Hz for the (22, 2)-resonance, νs > ∼ 260 Hz for the (33, 3)-resonance, νs > ∼ 200 Hz for the (44, 4)-resonance, and νs > ∼ 150 Hz for the (55, 5)-resonance.
It is necessary to check that the higher-l contribution to the (kk, m = k)-resonance is negligible. Except for the Ωs = 0 case, the l = k + 2, k + 4, · · · tides can also contribute to (∆N orb ) kk,k . The mode frequency is the same, but the new tidal coupling coefficient must be calculated. We shall consider just the (22,42)-resonance and the (33,53)-resonance here since the higher order terms are much smaller. Using the mode eigenfunction (3.17), we find Q22,42 = 45 112 and Q33,53 = 35 96 (3.27) Both expressions can be evaluated numerically. In Fig. 2 the results for the orbital change (∆N orb )22,42 is plotted, indicating that it is indeed much smaller than (∆N orb )22,22 . The orbital change (∆N orb )33,53 is not shown since it is below the limit of interest for this paper.

G-modes
Gravity modes in neutron stars arise from the composition (proton to neutron ratio) gradient in the stellar core (Reisenegger & Goldreich 1992;Lai 1994), density discontinuities in the crust (Finn 1987), as well as thermal buoyancy associated with finite temperatures, either due to internal heat (McDermott et al. 1988) or due to accretion (Bildsten & Cutler 1995). In this paper, we shall focus on the core Note that to calculate the total orbital change due to the (jk, m) = (22, 2)-resonance, one needs to use eq. (2.40) which includes contributions from all possible l-terms. But the smallness of (∆N orb ) 22,42 implies (∆N orb ) 22,2 ≃ (∆N orb ) 22,22 to a good approximation. The quantity (∆N orb ) 22,42 will also be used in §4 as a fiducial number for the (22, 44)-resonance. g-modes since the gradient of proton to neutron ratio in the NS interior provides the largest buoyancy. For a nonrotating NS, the typical frequencies of low order g-modes are around 100 Hz and are smaller for higher order modes. Thus we expect that even modest stellar rotation rate (∼ 100 Hz) can change the mode properties dramatically.
The core g-modes of rotating NS have been calculated by Lai (1998) using the traditional approximation. Two NS models are considered based on different EOS. Model UU gives a nonrotating NS radius R = 13.47 km for a M = 1.4M⊙, and its lowest order (n = 1) j = 2 g-mode has frequency να = 148 Hz and Q22,22 ≃ 0.62 × 10 −3 . Model AU gives a nonrotating NS radius R = 12.37 km for a M = 1.4M⊙, and its lowest order (n = 1) j = 2 g-mode has frequency να = 72 Hz and Q22,22 ≃ 0.11 × 10 −3 . Figure 5 shows the spin frequency, νs, as a function of the observed gravitational wave frequency, νgw = (2/m)|να| = |να|, for the m = 2 resonance of the {jk} = {22} g-modes (n = 1). The ω (r) α > 0 mode propagates in opposite direction as the spin [see the discussion following eq. (2.20)]. As νs increases, the mode frequency, να, in the inertial frame decreases; να can even become negative when νs exceeds a certain critical value (95 Hz for model UU and 47 Hz for model AU). This corresponds to the onset of secular instability for the g-mode (Lai 1998). We include these unstable modes because, as shown by Lai (1998), viscosity tends to stabilize these modes for typical parameters of the NS. Figure 5 also shows the results for the spin-prograde mode, which has ω (r) α < 0 [see the discussion following eq. (2.20)]. These are calculated in a similar manner as in Lai (1998). These modes always have negative ωα, but they have positive energy and thus are secularly stable. Rotation clearly increases the gravitational wave frequency at which resonance occurs for these modes. Figure 6 shows the orbital change, (∆N orb )22,22, due to the g-mode resonance as a function of the observed gravitational wave frequency, νgw. The mode parameters Q22,22 and εα have been obtained using the method of Lai (1998 for model AU. We have normalizedεα and Q22,22 to their respective nonrotating values (rotation can change these by a factor of ∼ 2 for the range ofΩs < ∼ 0.3 considered in this paper). The upper sign in (3.28) and (3.29) applies when ωα > 0 and β = 180 • , while the lower sign applies when ωα < 0 and β = 0 • . However, when ω (r) α < 0 and ωα < 0, εα < 0; therefore ∆N orb < 0 (a decrease in the number of orbits). Fig. 6 correspond to the case when the NS is nonrotating. We see that retrograde rotation (β = 180 • ) can increase |(∆N orb )22,22| for the spin-retrograde mode by shifting the resonance to smaller orbital frequency, where the orbital decay is slower. For the spin-prograde modes, rotation decreases |(∆N orb )22,22| from the nonrotating value. It is clear from Fig. 6 that the orbital change is small, especially in comparison to the orbital change generated by f-modes. However, the g-mode resonances can occur more easily than the f-modes since the g-mode resonance does not require the rapid rotation of the NS.

The large dots in
The above results apply for particular neutron star models. As in eq. (3.25), we have the scaling: (∆N orb )22,22 ∝ M −4.5 1.4 R 3.5 10 ; thus the orbital change is larger for low-mass NSs which typically have larger R/M .

R-modes
For nonrotating stars, r-modes are "degenerate" toroidal modes with zero frequency and zero density perturbation. Rotation breaks the degeneracy via the Coriolis force, giving rise to finite-frequency r-modes. For the analysis of the r-modes in this section and in §4.3, we shall consider the slowly-rotating [Ωs ≡ Ωs/(GM/R 3 ) 1/2 ≪ 1] incompressible NS models. As shown by Provost et al. (1981), only the j = k r-modes exist for barotropic stars.
Following Saio (1982) and Kokkotas & Stergioulas (1998), it is convenient to introduce a new radial coordinate a which is related to the radial distance r through r = a(1 + ǫ), where ǫ(a, θ) ∝Ω 2 s represents the centrifugal distortion of the star. In the new coordinates, all unperturbed quantities are functions of a only. The displacement eigenfunction of the α = {kk} r-mode can be written as where the first term represents the toroidal (axial) component and the second term represents the spheroidal (polar) component. The r-mode is predominantly toroidal, and the radial eigenfunctions scale with the rotation rate as K kk ∼Ω 0 s , while S k+1k , H k+1k ∼Ω 2 s . For a uniform density star, the mode frequency in the rotating frame is given, to orderΩ 3 s , by (Saio 1982;Kokkotas & Stergioulas 1998) In the inertial frame, this giveŝ Note that the r-mode always has positive ω (r) α and negative ωα, i.e, the wave pattern propagates opposite to spin in the rotating frame while it becomes spin-prograde in the inertial frame, indicating that the r-mode is secularly unstable, although viscosity tends to stabilize the mode for smallΩs (Andersson 1998;Andersson et al. 1998;Friedman & Morsink 1998;Lindblom et al. 1998).
The eigenfunction (3.30) implies the density perturbation for the r-mode δρ kk ∝ Y k+1k . Thus the tidal coupling coefficient, Q kk,lk , is nonzero only for l = k + 1. To evaluate Q kk,lk , we write the pressure perturbation of the r-mode as where g is the gravitational acceleration. The dimensionless function ζ k+1k (a) is given by (Kokkotas & Stergioulas 1998) (3.34) and the corresponding radial eigenfunction is (3.35) The Eulerian density perturbation is given by But the boundary condition at the stellar surface gives S k+1k (R) = ζ k+1k (R), which is obtained by requiring the Lagrangian pressure perturbation to vanish at the surface. We then find (3.37) Note that this value corresponds to the normalization as defined by (3.34). To obtain the proper normalization [see eq. (2.9)], we use eq. (3.30) and find, to leading order inΩs, Thus the renormalized Q kk,k+1k , expressed in units where Thus tidal coupling will be weak for slow rotation. With the ξα given by (3.30), we can similarly calculate the function Tα and εα [see eqs. (2.11) and (2.30)], yieldinĝ The next order correction toTα is of orderΩ 5 s and can be neglected. Now consider resonant excitation of the α = {kk} rmode. For aligned spin-orbit we require m = k, while to have non-vanishing Q kk,lk we require l = k + 1. Since W k+1k = 0, we conclude that for β = 0 or 180 • , there is no effective tidal excitation of the r-mode (at least for the barotropic stellar model considered here). Resonant excitations of r-modes can only occur for misaligned spin-orbit configurations, which we consider in §4.3, using the properties of the r-mode derived above.

NON-ALIGNED SPIN-ORBIT
In this section we consider tidal resonances for general spinorbit inclination angle, β.
The properties of the f-modes have already been calculated in §3.1. For ωα > 0 (the spin-retrograde mode), the relevant Wigner D-functions are From eq. (2.41), we find that the orbital change due to the (kk, lm)-resonance at inclination angle β can be written as  Fig. 7 that even when the rotation of the NS is mostly prograde with respect to the orbital motion (β < 90 • ), there still exist modes that can be resonantly excited and thus contribute to orbital energy dissipation. Also of note is the large orbital change caused by the (55,51)-resonance relative to the (55,55) and (55,53)resonances. It is thus possible for this resonance to produce an orbital change which is comparable to those produced by the (44,4m)-resonances.
For spin-prograde modes (ωα < 0), the above results still apply except that one should replace (∆N orb ) kk,lk (180 • ) ⋆⋆ This is because the eigenfunction of the α = {kk} mode is even (odd) with respect to the stellar equator for even (odd) k; the tidal potential must have the same symmetry in order to produce nonzero coupling. by (∆N orb ) kk,lk (0 • ) and, in the expression for f kk,lm (β), replace β by (180 • − β).

G-modes
The g-mode resonance considered in §3.2 corresponds to the (α, lm) = (22, 22). The function f kk,lm (β) derived in §3.1 equally applies to the g-mode resonance and is plotted in Fig. 7. Although other resonances of g-modes are also induced, their effects are below the limits of interest.

R-modes
Recall that only the j = k r-modes exist for barotropic stars, and Q kk,lk = 0 only if l = k + 1, in which case Q kk,k+1k is given by ( (4.8) For the (22,33)-resonance, we find, to leading order in Ωs, that the gravitational wave frequency is given by (4.11) The orbital change increases with increasing spin frequency. But note that our analysis of r-mode is based on the assumption of slow rotation, thus ( R 10 10 1 q(1 + q) 5/3 . (4.14) Thus we conclude that for canonical NS parameters, the (22,33) and (22,31)-resonances (and indeed all r-mode resonances) produce orbital changes that are below the limits of interest. However, if we consider a larger NS radius, such as R = 15 km for a M = 1.4M⊙ NS, which is allowed for some nuclear equations of state (e.g, Shapiro & Teukolsky 1983), then the orbital change for the (22, 33)-resonance may become significant (∆N orb ∼ 0.6).

SUMMARY AND DISCUSSION
In this paper, we have presented a systematic study on the resonant mode excitations of rotating neutron stars in coalescing binaries and their effects on the gravitational waveform. The essential effect of rotation is that it can modify the mode frequency in the inertial frame, thereby changing the strength of the g-mode resonance, which already exists for nonrotating neutron stars, and also making a variety of new resonances involving f-modes and r-modes possible.
We find that for the f-mode resonance to occur during the last few minutes of the binary inspiral, with the emitted gravitational wave frequency 10 Hz < νgw < 1000 Hz (the sensitivity band of the interferometric gravitational wave detector), the neutron star must have very rapid rotation (see Fig. 1), e.g., assuming canonical neutron star mass and radius, M = 1.4M⊙ and R = 10 km, respectively, νs > ∼ 710 Hz for the (jk, m) = (22, 2)-resonance (see the beginning of §2.2 for notation) and νs > ∼ 570 Hz for the (33, 3)-resonance. Observations (e.g., PSR 1913+16 has spin period 59 ms, and PSR 1534+12 has 38 ms) and our current understanding of the formation of neutron star binaries seem to indicate that neutron stars in compact binaries (with another neutron star or a black hole as a companion) can never achieve such a rapid rotation, although such a rotation rate is physically allowed. For a M = 1.4M⊙ and R = 15 km neutron star, the critical frequencies are lowered, e.g., νs > ∼ 330 Hz for the (22, 2)-resonance and νs > ∼ 260 Hz for the (33, 3)-resonance (see Fig. 3). If the neutron star in a coalescing binary indeed has the required rapid rotation, then the induced orbital change due to the f-mode resonance is significant (see Fig. 2 and Fig. 4) and must be included in the templates of waveforms used for searching gravitational wave signals.
The resonance of g-modes is strongly affected by even a modest rotation (see Fig. 5). Although there is large uncertainties associated with the property of the g-mode (since it depends on the symmetry energy of nuclear matter), we show that for a canonical 1.4M⊙ neutron star, the orbital change ∆N orb due to the g-mode resonance is below the 10 −2 level (see Fig. 6). Such a small phase error in the waveform is unimportant for the search of gravitational wave signals from the binary (E. Flanagan 1998, private communication). However, we note that if we consider very low-mass (M < ∼ 0.5M⊙) neutron stars, which have large R/M ratio, the orbital change due to g-mode resonance may well be significant. Of course, although such low-mass neutron stars are physically allowed, there is no astrophysical evidence for their existence.
Resonant excitations of r-modes occur for misaligned spin-orbit configurations. Since the r-mode always has negative energy, the resonance actually transfers energy to the orbit, slowing down the inspiral. Since the tidal coupling of the r-mode depends strongly on the stellar rotation rate, we find that for canonical neutron star mass and radius (1.4M⊙ and 10 km, respectively), the orbital change ∆N orb is negligible. However, if we consider a larger stellar radius, such as 15 km for a 1.4M⊙ neutron star (as is allowed by some nuclear equations of state), then ∆N orb may become important, especially in the high frequency band (νgw ∼ 1000 Hz) [see eqs. (4.10) and (4.11)].
Finally, we note that our calculations of the properties of the f-modes and r-modes are based on incompressible neutron star models. Our results for the r-modes should break down for very high rotation rates. Since neutron stars are not exactly barotropic, other r-modes, in addition to the j = k mode considered in this paper, can also exist. It is desirable to study the resonance effects using more realistic neutron star models.  10 , for the f-mode (jk, lm)-resonance. The labels in the figure give the values of (jk, lm). The solid lines are for the ω (r) α > 0 and ωα > 0 (spin-retrograde) stable modes with (∆N orb ) jk,lm (180 • ) < 0. The dot-dashed lines are for the ω (r) α > 0 and ωα < 0 (spin-prograde) unstable modes with (∆N orb ) jk,lm (0 • ) > 0. Note that both the (22,22) and (22,42) curves belong to the same (22,2)-resonance. The (22,42) curve serves as a baseline for the (22,44)-resonance (see §4 and Fig. 7). Also recall from Fig. 1 that the unstable modes require extremely high spin rates. The large dots with labels indicate the spin rate at that gravitational wave frequency. The vertical dashed lines indicate the frequency range of gravitational wave detectors, νgw = 10-1000 Hz. Figure 3. Spin frequency, νs, of the R = 15 km NS as a function of the gravitational wave frequency, νgw, for the f-mode (jk, m)resonance (taking j = k = 2-5), where νgw is related to the f-mode frequency by νgw = (2/m)|να|. The labels in the figure give the values of (jk, m). The solid lines are for the ω (r) α > 0 and ωα > 0 (spin-retrograde) stable modes. The dot-dashed lines are for the ω (r) α > 0 and ωα < 0 (spin-prograde) unstable modes. Note that the (22,4)-resonance is possible only for misaligned spin-orbit configurations (β = 0 • , 180 • ). It is also clear the unstable modes occur at extremely high spin rates. The vertical dotted line at νgw = 640 Hz denotes the maximum gravitational wave frequency allowed by eq. (3.8), which sets a lower limit to the spin frequency. The vertical dashed lines indicate the frequency range of gravitational wave detectors, νgw = 10-1000 Hz. . Orbital cycle change |(∆N orb ) jk,lm |, with M = 1.4M ⊙ , R = 15 km, q = 1, as a function of the gravitational wave frequency, νgw = (2/m)|να|, for the f-mode (jk, lm)-resonance. The labels in the figure give the values of (jk, lm). The solid lines are for the ω (r) α > 0 and ωα > 0 (spin-retrograde) stable modes with (∆N orb ) jk,lm (180 • ) < 0. The dot-dashed lines are for the ω (r) α > 0 and ωα < 0 (spin-prograde) unstable modes with (∆N orb ) jk,lm (0 • ) > 0. Note that both the (22,22) and (22,42) curves belong to the same (22,2)-resonance. The (22,42) curve serves as a baseline for the (22,44)-resonance (see §4 and Fig. 7). Also recall from Fig. 2 that the unstable modes require extremely high spin rates. The vertical dotted line at νgw = 640 Hz denotes the maximum gravitational wave frequency allowed by eq. (3.8). The large dots with labels indicate the spin rate at that gravitational wave frequency. The vertical dashed lines indicate the frequency range of gravitational wave detectors, νgw = 10-1000 Hz. Figure 5. Spin frequency, νs, of the NS as a function of the observed gravitational wave frequency, νgw, for the (jk, m) = (22, 2)resonance, where νgw is related to the g-mode frequency, να, by νgw = (2/m)|να| = |να|. UU and AU are two models of the NS EOS (see text for the NS parameters) given in Lai (1998). The solid lines are for the ω (r) α > 0 and ωα > 0 (spin-retrograde) stable modes. The dot-dashed lines are for the ω (r) α > 0 and ωα < 0 (spin-prograde) unstable modes. The dotted lines are for the ω (r) α < 0 and ωα < 0 (spin-prograde) stable modes. The vertical dashed line indicates the lower limit of the frequency range of gravitational wave detectors, νgw = 10-1000 Hz. Figure 6. Orbital cycle change |(∆N orb ) 22,22 |, with q = 1, as a function of the observed gravitational wave frequency, νgw = (2/m)|να| = |να|, for the g-mode (jk, lm) = (22, 22)-resonance. UU and AU are two models of the NS EOS (see text for the NS parameters) given in Lai (1998). The solid lines are for the ω  . f kk,lm (β) as a function of spin-orbit inclination angle, β. f kk,lm (β) is related to the induced orbital cycle change, (∆N orb ) kk,lm (β), by (∆N orb ) kk,lm (β) = (∆N orb ) kk,lk (180 • ) × f kk,lm (β). The labels in the figure give the values of (jk, lm).