Tidal Excitation of Oscillation Modes in Compact White Dwarf Binaries: I. Linear Theory
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 25 October 2018 (MN L A TEX style file v2.2)
Tidal Excitations of Oscillation Modes in Compact WhiteDwarf Binaries: I. Linear Theory
Jim Fuller (cid:63) and Dong Lai
Center for Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
25 October 2018
ABSTRACT
We study the tidal excitation of gravity modes (g-modes) in compact white dwarf bi-nary systems with periods ranging from minutes to hours. As the orbit of the systemdecays via gravitational radiation, the orbital frequency increases and sweeps througha series of resonances with the g-modes of the white dwarf. At each resonance, thetidal force excites the g-mode to a relatively large amplitude, transferring the orbitalenergy to the stellar oscillation. We calculate the eigenfrequencies of g-modes and theircoupling coefficients with the tidal field for realistic non-rotating white dwarf models.Using these mode properties, we numerically compute the excited mode amplitude inthe linear approximation as the orbit passes though the resonance, including the back-reaction of the mode on the orbit. We also derive analytical estimates for the modeamplitude and the duration of the resonance, which accurately reproduce our numer-ical results for most binary parameters. We find that the g-modes can be excited to adimensionless (mass-weighted) amplitude up to 0.1, with the mode energy approach-ing 10 − of the gravitational binding energy of the star. Therefore the low-frequency( < ∼ − Hz) gravitational waveforms produced by the binaries, detectable by LISA,are strongly affected by the tidal resonances. Our results also suggest that thousandsof years prior to the binary merger, the white dwarf may be heated up significantlyby tidal interactions. However, more study is needed since the physical amplitudes ofthe excited oscillation modes become highly nonlinear in the outer layer of the star,which can reduce the mode amplitude attained by tidal excitation.
Key words: white dwarfs – hydrodynamics – waves – binaries
It is well known that non-radial gravity modes (g-modes) areresponsible for the luminosity variations observed in someisolated white dwarfs (called ZZ Ceti stars) in the instabil-ity strip. These g-modes are thought to be excited by a con-vective driving mechanism operating in the shallow surfaceconvection zone of the star (see Brickhill 1983; Goldreich &Wu 1999; Wu & Goldreich 1999).In this paper we study the tidal excitation of g-modes incompact binary systems containing a white dwarf (WD) andanother compact object (white dwarf, neutron star or blackhole). The Galaxy is populated with ∼ WD-WD bina-ries and several 10 of double WD-NS binaries [Nelemans etal. (2001); see also Nelemans (2009) and references therein].A sizeable fraction of these binaries are compact enough sothat the binary orbit will decay within a Hubble time toinitiate mass transfer or a binary merger. Depending on thedetails of the mass transfer process (including the response (cid:63) Email: [email protected]; [email protected] of the WD to mass transfer), these ultra-compact binaries(with orbital period less than an hour) may survive masstransfer for a long time or merge shortly after mass trans-fer begins. A number of ultra-compact interacting WD-WDbinary systems have already been observed [including RXJ0806.3+1527 (period 5.4 min) and V407 Vul (period 9.5min); see Strohmayer 2005 and Ramsay et al. 2005]. Re-cent surveys (e.g., SDSS) have also begun to uncover non-interacting compact WD binaries (e.g., Badenes et al. 2009;Mullally et al. 2009; Kilic et al. 2009; Marsh et al. 2010;Kulkarni & van Kerkwijk 2010; Steinfadt et al. 2010). De-pending on the total mass, the systems may evolve into TypeIa supernovae (for high mass), or become AM CVn binariesor R CrB stars (for low mass). Many of these WD binariesare detectable in gravitational waves by the
Laser Interfer-ometer Space Antenna (LISA) (Nelemans 2009).In this paper we consider resonant tidal interaction inWD binaries that are not undergoing mass transfer. Thismeans that the binary separation D is greater than D min , theorbital radius at which dynamical merger or mass transfer c (cid:13) a r X i v : . [ a s t r o - ph . H E ] S e p J. Fuller and D. Lai occurs, i.e.,
D > ∼ D min (cid:39) . (cid:18) M t M (cid:19) / R, (1)where M is the WD mass, M t = M + M (cid:48) is the total massand R is the WD radius. This corresponds to orbital periodsof P > ∼ P min = 68 . (cid:18) R km (cid:19) / (cid:18) MM (cid:12) (cid:19) − / s . (2)Since WD g-mode periods are of order one minute or longer,they can be excited by the binary companion prior tomass transfer. In particular, as the binary orbit decaysdue to gravitational radiation, the orbital frequency sweepsthrough a series of g-mode frequencies, transferring orbitalenergy to the modes. Although the overlap integral of theg-mode eigenfunctions with the tidal potential is generallyquite small, a binary system that spends a long time at res-onance can still excite g-modes to large amplitudes.Previous studies of tidal interaction in WD binarieshave focused on quasi-static tides (e.g. Iben, Tutukov &Fedorova 1998; Willems, Deloye & Kalogera 2009), whichessentially correpond to non-resonant f-modes of the star.Such static tides become important only as the binary ap-proaches the tidal limit [equation (1)]. Racine, Phinney &Arras (2007) recently studied non-dissipative tidal synchro-nization due to Rossby waves in accreting ultra-compactWD binaries. Rathore, Blandford & Broderick (2005) stud-ied resonant mode excitations of WD modes in eccentricbinaries. They focused on f-modes, for which the resonanceoccurs when harmonics of the orbital frequency matches themode frequency. As mentioned above, for circular orbits,such resonance with the f-mode does not occur prior to masstransfer or tidal disruption. Their published analysis also didnot include back reaction of the excited mode on the binaryorbit.The problem of resonant mode excitations in compactbinaries has been studied before in the context of coalescingneutron star binaries: Reisenegger & Goldreich (1994), Lai(1994) and Shibata (1994) focused on the excitations of g-modes of non-rotating neutron stars; Ho & Lai (1999) andLai & Wu (2006) studied the effects of NS rotation – includ-ing r-modes and other inertial modes; Flanagan & Racine(2006) examined gravitomagnetic excitation of r-modes. Inthe case of neutron star binaries, the orbital decay rate (fororbital frequencies larger than 5 Hz) is large and the modeamplitude is rather small, so the back reaction of the excitedmode on the orbit can be safely neglected (see section 5 ofthe present paper). By contrast, in the case of WD binaries,the orbital decay is much slower and the excited mode canreach a much larger amplitude. It thus becomes essential totake the back reaction into account.In this paper, we consider WD binaries in circular or-bits, consistent with the observed population of compactWD binaries (e.g. Kulkarni & van Kerkwijk 2010). Such cir-cular orbits are a direct consequence of the circularization bygravitational radiation and/or the common envelope phaseleading to their formation. A key assumption of this pa-per is that we assume the WD is not synchronized withthe binary orbit. While it is true that the tidal circular-ization time scale is much longer than the synchronizationtime, the observed circular orbit of the WD binaries does not imply synchronization. While there have been numer-ous studies of tidal dissipation in normal stars and giantplanets (e.g., Zahn 1970,1989; Goldreich & Nicholson 1977;Goodman & Oh 1997; Goodman & Dickson 1998; Ogilvie &Lin 2004,2007; Wu 2005; Goodman & Lackner 2009), therehas been no satisfactory study on tidal dissipation in WDs.Even for normal stars, the problem is not solved (especiallyfor solar-type stars; see Goodman & Dickson 1999; see Zahn2008 for review). In fact it is likely that the excitations ofg-modes and other low-frequency modes play a role in thesynchronization process. The orbital decay time scale near g-mode resonances is relatively short (of order 10 years for or-bital periods of interest, i.e., minutes), so it is not clear thattidal synchronization can compete with the orbital decayrate. Given this uncertainty, we will consider non-rotatingWDs (or slowly-rotating WDs, so that the g-mode propertiesare not significantly modified by rotation) as a first step, andleaving the study of the rotational effects to a future paper.This remainder of the paper is organized as follows. Insection 2 we present the equations governing the evolution ofthe orbit and the g-modes. Section 3 examines the propertiesof WD g-modes and their coupling with the tidal gravita-tional field of the companion. In section 4, we numericallystudy the evolution of the g-modes through resonances, andin section 5 we present analytical estimates of the resonantg-mode excitation. We study the effect of mode damping onthe tidal excitation in section 6 and discuss the uncertaintiesand implications of our results in section 7. We consider a WD of mass M and radius R in orbit witha companion of mass M (cid:48) (another WD, or NS or BH). TheWD is non-spinning. The gravitational potential producedby M (cid:48) can be written as U ( r , t ) = − GM (cid:48) | r − D ( t ) | = − GM (cid:48) (cid:88) lm W lm r l D l +1 e − im Φ( t ) Y lm ( θ, φ ) , (3)where r = ( r, θ, φ ) is the position vector (in spherical coordi-nates) of a fluid element in star M , D ( t ) = ( D ( t ) , π/ , Φ( t ))is the position vector of M (cid:48) relative to M ( D is the binaryseparation, Φ is the orbital phase or the true anomaly) andthe coefficient W lm is given by W lm = ( − ) ( l + m ) / (cid:20) π l + 1 ( l + m )!( l − m )! (cid:21) / × (cid:20) l (cid:18) l + m (cid:19) ! (cid:18) l − m (cid:19) ! (cid:21) − , (4)(Here the symbol ( − ) p is zero if p is not an integer.) Thedominant l = 2 tidal potential has W ± = (3 π/ / , W = ( π/ / , W ± = 0, and so only the m = ± M isspecified by the Lagrangian displacement ξ ( r , t ), which sat- c (cid:13)000
D > ∼ D min (cid:39) . (cid:18) M t M (cid:19) / R, (1)where M is the WD mass, M t = M + M (cid:48) is the total massand R is the WD radius. This corresponds to orbital periodsof P > ∼ P min = 68 . (cid:18) R km (cid:19) / (cid:18) MM (cid:12) (cid:19) − / s . (2)Since WD g-mode periods are of order one minute or longer,they can be excited by the binary companion prior tomass transfer. In particular, as the binary orbit decaysdue to gravitational radiation, the orbital frequency sweepsthrough a series of g-mode frequencies, transferring orbitalenergy to the modes. Although the overlap integral of theg-mode eigenfunctions with the tidal potential is generallyquite small, a binary system that spends a long time at res-onance can still excite g-modes to large amplitudes.Previous studies of tidal interaction in WD binarieshave focused on quasi-static tides (e.g. Iben, Tutukov &Fedorova 1998; Willems, Deloye & Kalogera 2009), whichessentially correpond to non-resonant f-modes of the star.Such static tides become important only as the binary ap-proaches the tidal limit [equation (1)]. Racine, Phinney &Arras (2007) recently studied non-dissipative tidal synchro-nization due to Rossby waves in accreting ultra-compactWD binaries. Rathore, Blandford & Broderick (2005) stud-ied resonant mode excitations of WD modes in eccentricbinaries. They focused on f-modes, for which the resonanceoccurs when harmonics of the orbital frequency matches themode frequency. As mentioned above, for circular orbits,such resonance with the f-mode does not occur prior to masstransfer or tidal disruption. Their published analysis also didnot include back reaction of the excited mode on the binaryorbit.The problem of resonant mode excitations in compactbinaries has been studied before in the context of coalescingneutron star binaries: Reisenegger & Goldreich (1994), Lai(1994) and Shibata (1994) focused on the excitations of g-modes of non-rotating neutron stars; Ho & Lai (1999) andLai & Wu (2006) studied the effects of NS rotation – includ-ing r-modes and other inertial modes; Flanagan & Racine(2006) examined gravitomagnetic excitation of r-modes. Inthe case of neutron star binaries, the orbital decay rate (fororbital frequencies larger than 5 Hz) is large and the modeamplitude is rather small, so the back reaction of the excitedmode on the orbit can be safely neglected (see section 5 ofthe present paper). By contrast, in the case of WD binaries,the orbital decay is much slower and the excited mode canreach a much larger amplitude. It thus becomes essential totake the back reaction into account.In this paper, we consider WD binaries in circular or-bits, consistent with the observed population of compactWD binaries (e.g. Kulkarni & van Kerkwijk 2010). Such cir-cular orbits are a direct consequence of the circularization bygravitational radiation and/or the common envelope phaseleading to their formation. A key assumption of this pa-per is that we assume the WD is not synchronized withthe binary orbit. While it is true that the tidal circular-ization time scale is much longer than the synchronizationtime, the observed circular orbit of the WD binaries does not imply synchronization. While there have been numer-ous studies of tidal dissipation in normal stars and giantplanets (e.g., Zahn 1970,1989; Goldreich & Nicholson 1977;Goodman & Oh 1997; Goodman & Dickson 1998; Ogilvie &Lin 2004,2007; Wu 2005; Goodman & Lackner 2009), therehas been no satisfactory study on tidal dissipation in WDs.Even for normal stars, the problem is not solved (especiallyfor solar-type stars; see Goodman & Dickson 1999; see Zahn2008 for review). In fact it is likely that the excitations ofg-modes and other low-frequency modes play a role in thesynchronization process. The orbital decay time scale near g-mode resonances is relatively short (of order 10 years for or-bital periods of interest, i.e., minutes), so it is not clear thattidal synchronization can compete with the orbital decayrate. Given this uncertainty, we will consider non-rotatingWDs (or slowly-rotating WDs, so that the g-mode propertiesare not significantly modified by rotation) as a first step, andleaving the study of the rotational effects to a future paper.This remainder of the paper is organized as follows. Insection 2 we present the equations governing the evolution ofthe orbit and the g-modes. Section 3 examines the propertiesof WD g-modes and their coupling with the tidal gravita-tional field of the companion. In section 4, we numericallystudy the evolution of the g-modes through resonances, andin section 5 we present analytical estimates of the resonantg-mode excitation. We study the effect of mode damping onthe tidal excitation in section 6 and discuss the uncertaintiesand implications of our results in section 7. We consider a WD of mass M and radius R in orbit witha companion of mass M (cid:48) (another WD, or NS or BH). TheWD is non-spinning. The gravitational potential producedby M (cid:48) can be written as U ( r , t ) = − GM (cid:48) | r − D ( t ) | = − GM (cid:48) (cid:88) lm W lm r l D l +1 e − im Φ( t ) Y lm ( θ, φ ) , (3)where r = ( r, θ, φ ) is the position vector (in spherical coordi-nates) of a fluid element in star M , D ( t ) = ( D ( t ) , π/ , Φ( t ))is the position vector of M (cid:48) relative to M ( D is the binaryseparation, Φ is the orbital phase or the true anomaly) andthe coefficient W lm is given by W lm = ( − ) ( l + m ) / (cid:20) π l + 1 ( l + m )!( l − m )! (cid:21) / × (cid:20) l (cid:18) l + m (cid:19) ! (cid:18) l − m (cid:19) ! (cid:21) − , (4)(Here the symbol ( − ) p is zero if p is not an integer.) Thedominant l = 2 tidal potential has W ± = (3 π/ / , W = ( π/ / , W ± = 0, and so only the m = ± M isspecified by the Lagrangian displacement ξ ( r , t ), which sat- c (cid:13)000 , 000–000 esonant Oscillation Modes in White Dwarf Binaries isfies the equation of motion ∂ ξ ∂t + L · ξ = −∇ U, (5)where L is an operator that specifies the internal restoringforces of the star. The normal oscillation modes of the starsatisfy L · ξ α = ω α ξ α , where α = { n, l, m } is the usual modeindex and ω α is the mode frequency. We write ξ ( r , t ) as thesum of the normal modes: ξ ( r , t ) = (cid:88) α a α ( t ) ξ α ( r ) . (6)The (complex) mode amplitude a α ( t ) satisfies the equation¨ a α + ω α a α = GM (cid:48) W lm Q α D l +1 e − im Φ( t ) , (7)where Q α is the tidal coupling coefficient (also used by Press& Teukolski 1977), defined by Q α = (cid:104) ξ α |∇ ( r l Y lm ) (cid:105) = (cid:90) d x ρ ξ ∗ α · ∇ ( r l Y lm )= (cid:90) d x δρ ∗ α r l Y lm . (8)Here δρ α = −∇ · ( ρ ξ α ) is the Eulerian density perturbation.In deriving (7) we have used the normalization (cid:104) ξ α | ξ α (cid:105) = (cid:90) d x ρ ξ ∗ α · ξ α = 1 . (9)Resonant excitation of a mode α occurs when ω α = m Ω,where Ω is the orbital frequency.In the absence of tidal interaction/resonance, the WDbinary orbit decays due to gravitational radiation, with timescale given by (Peters 1964) t D = D | ˙ D | = 5 c G D MM (cid:48) M t = 3 . × (cid:18) M (cid:12) MM (cid:48) (cid:19)(cid:18) M t M (cid:12) (cid:19) / (cid:18) Ω0 . − (cid:19) − / s , (10)where M t = M + M (cid:48) is the total binary mass. When astrong tidal resonance occurs, the orbital decay rate can bemodified, and we need to follow the evolution of the orbitand the mode amplitudes simultaneously. The gravitationalinteraction energy between M (cid:48) and the modes in star M is W = (cid:90) d x U ( r , t ) (cid:88) α a ∗ α ( t ) δρ ∗ α ( r )= − (cid:88) α M (cid:48) MR D W lm Q α e − im Φ a ∗ α ( t ) , (11)where we have restricted to the l = 2 terms and set G = 1.The orbital evolution equations, including the effects of themodes, are then given by¨ D − D ˙Φ = − M t D − (cid:88) α M t D W lm Q α e im Φ a α − M t D (cid:16) A / + B / ˙ D (cid:17) , (12)¨Φ + 2 ˙ D ˙Φ D = (cid:88) α im M t D W lm Q α e im Φ a α − M t D B / ˙Φ . (13) The last terms on the right-hand side of equations (12) and(13) are the leading-order gravitational radiation reactionforces, with (see Lai & Wiseman 1996 and references therein) A / = − µ D ˙ D (cid:18) v + 2 M t D −
25 ˙ D (cid:19) , (14) B / = 8 µ D (cid:18) v − M t D −
15 ˙ D (cid:19) , (15)where µ = MM (cid:48) /M t and v = ˙ D + ( D ˙Φ) . In equations(12)-(15) we have set G = c = 1. We have dropped the otherpost-Newtonian terms since they have negligible effects ontidal excitations. The mode amplitude equation is given byequation (7), or,¨ b α − im Ω˙ b α + ( ω α − m Ω − im ˙Ω) b α = M (cid:48) W lm Q α D l +1 , (16)where b α = a α e im Φ . (17) The non-radial adiabatic modes of a WD can be found bysolving the standard stellar oscillation equations, as givenin, e.g., Unno et al. (1989). The g-mode propagation zonein the star is determined by ω α < N amd ω α < L l , where L l = (cid:112) l ( l + 1) a s /r is the Lamb frequency ( a s is the soundspeed), and N is the Br¨unt-V¨ais¨al¨a frequency, as given by N = g (cid:20) dρdP − (cid:18) ∂ρ∂P (cid:19) s (cid:21) , (18)where g the gravitational acceleration, and the subscript “ s ”means that the adiabatic derivative is taken. Alternatively, N can be obtained from (Brassard et al. 1991) N = ρg χ T P χ ρ (cid:16) ∇ s − ∇ + B (cid:17) , (19)where χ T = (cid:16) ∂ ln P ∂ ln T (cid:17) ρ, { X i } , χ ρ = (cid:16) ∂ ln P ∂ ln ρ (cid:17) T, { X i } , ∇ = d ln Td ln P , ∇ s = (cid:16) ∂ ln T∂ ln P (cid:17) s, { X i } . (20)The Ledoux term B accounts for the buoyancy arising fromcomposition gradient: B = − χ Y χ T d ln Yd ln P , (21)where χ Y = (cid:18) ∂ ln P∂ ln Y (cid:19) ρ,T , (22)and Y is the mass fraction of helium. This equation is validfor a compositional transition zone containing helium andone other element, as is the case for typical compositionallystratified DA WD models.Figure 1 shows the profiles of the Br¨unt Vais¨al¨a andLamb frequencies for one of the WD models adopted inthis paper. These models were provided by G. Fontaine (seeBrassard 1991). Since the pressure in the WD core is al-most completely determined by electron degeneracy pres-sure, N ∝ χ T is very small except in the non-degenerate c (cid:13) , 000–000 J. Fuller and D. Lai -8 -6 -4 -2 D en s i t y ( g c m - ) -6 -4 -2 F r equen cy S qua r ed ( s - ) Figure 1.
The square of the Br¨unt V¨ais¨al¨a (solid line) and Lamb(dotted line) frequencies and the density (thick solid line) as afunction of normalized radius in a DA WD model, with M =0 . M (cid:12) , R = 8 . × km, T eff = 10800 K. The spikes in theBr¨unt V¨ais¨al¨a frequency are caused by the composition changesfrom carbon to helium, and from helium to hydrogen, respectively. Table 1.
The eigenfrequency ¯ ω α , tidal overlap parameter ¯ Q α ,and numerical f-mode overlap c for the first six l = 2 g-modes ofa white dwarf model. The white dwarf model has T eff = 10800 K , M = 0 . M (cid:12) , and R = 8 . × km. Note that ¯ ω α and ¯ Q α are in dimensionless units such that G = M = R = 1, and( GM/R ) / / (2 π ) = 0 .
053 Hz.n ¯ ω α | ¯ Q α | c outer layers. As a result, g-modes are confined to the outerlayers of the star below the convection zone. Lower-ordermodes have higher eigenfrequencies, so they are confined toregions where N is especially large, i.e., just below the con-vection zone. Higher order modes have lower eigenfrequen-cies and can thus penetrate into deeper layers of the starwhere the value of N is smaller. Cooler WDs have deeperconvection zones that cause the modes to be confined todeeper layers where N is smaller. Consequently, the eigen-frequencies and associated values of Q α tend to be smallerin cooler WDs due to the decreased value of N in the regionof mode propagation.The other feature of WDs that strongly effects their g-modes is their compositionally stratified layers. The sharpcomposition gradients that occur at the carbon-helium tran-sition and the helium-hydrogen transition create large val-ues of the Ledoux term B [equation (21)], resulting in sharppeaks in N as seen in Figure 1. These peaks have a largeeffect on the WD g-modes, leading to phenomena such as Table 2.
Same as table 1, for a WD model of identical mass andcomposition but with T eff = 5080 K .n ¯ ω α | ¯ Q α | c mode-trapping (e.g., Brassard 1991) and irregular periodspectra. Thus, the eigenfrequencies and eigenfunctions ofWD g-modes are very sensitive to WD models.Tables 1-2 give the l = 2 f-mode and g-mode frequen-cies and their tidal coupling coefficients for two WD models.While the full oscillation equations need to be solved to ac-curately determine the f-modes, the Cowling approximation(in which the perturbation in the gravitational potential isneglected) gives accurate results for g-modes. Since high-order g-modes have rather small | Q α | , the mode eigenfunc-tion must be solved accurately to obtain reliable Q α . Toensure that this is achieved in our numerical integration, weuse the orthogonality of the eigenfunctions to check the ac-curacy of the value of Q α (see Reisengger 1994 for a studyon the general property of Q α ). Since the numerical deter-mination of an eigenfunction is not perfect, it will containtraces of the other eigenfunctions, i.e.,( ξ α ) num = c α ξ α + c ξ + c ξ + · · · , (23)with c α (cid:39) | c β | (cid:28) β (cid:54) = α . This means that thenumerical tidal overlap integral is( Q α ) num = (cid:104)∇ ( r l Y lm ) | ( ξ α ) num (cid:105) = c α Q α + c Q + c Q + . . . (24)Since | Q | (for the f-mode) is of order unity, while | Q α | (cid:28) Q α ) num accurately represents theactual Q α , we require | c | (cid:39) |(cid:104) ξ | ξ α (cid:105) num | (cid:28) | Q α | . (25)The results shown in tables 1-2 reveal that | c | is alwaysmore than an order of magnitude less than ¯ Q α , so the abovecondition is satisfied for the modes computed in this paper.We note from tables 1-2 that while in general higher-order g-modes tend to have smaller | Q α | , the dependence of | Q α | on the mode index n is not exactly monotonic. Thisis the result of the mode trapping phenomenon associatedwith composition discontinuities in the WD. To see this, wenote that a mode with amplitude ξ α has energy given by E α = ω α (cid:82) d x ρ | ξ α | , thus we can define the mode energyweight function dE α d ln P = ω α ρ r (cid:2) ξ r + l ( l + 1) ξ ⊥ (cid:3) H p , (26)where H p = dr/d ln P = P/ ( ρg ) is the pressure scale height,and we have used ξ α = [ ξ r ( r ) e r + rξ ⊥ ( r ) ∇ ] Y lm (27) c (cid:13)000
Same as table 1, for a WD model of identical mass andcomposition but with T eff = 5080 K .n ¯ ω α | ¯ Q α | c mode-trapping (e.g., Brassard 1991) and irregular periodspectra. Thus, the eigenfrequencies and eigenfunctions ofWD g-modes are very sensitive to WD models.Tables 1-2 give the l = 2 f-mode and g-mode frequen-cies and their tidal coupling coefficients for two WD models.While the full oscillation equations need to be solved to ac-curately determine the f-modes, the Cowling approximation(in which the perturbation in the gravitational potential isneglected) gives accurate results for g-modes. Since high-order g-modes have rather small | Q α | , the mode eigenfunc-tion must be solved accurately to obtain reliable Q α . Toensure that this is achieved in our numerical integration, weuse the orthogonality of the eigenfunctions to check the ac-curacy of the value of Q α (see Reisengger 1994 for a studyon the general property of Q α ). Since the numerical deter-mination of an eigenfunction is not perfect, it will containtraces of the other eigenfunctions, i.e.,( ξ α ) num = c α ξ α + c ξ + c ξ + · · · , (23)with c α (cid:39) | c β | (cid:28) β (cid:54) = α . This means that thenumerical tidal overlap integral is( Q α ) num = (cid:104)∇ ( r l Y lm ) | ( ξ α ) num (cid:105) = c α Q α + c Q + c Q + . . . (24)Since | Q | (for the f-mode) is of order unity, while | Q α | (cid:28) Q α ) num accurately represents theactual Q α , we require | c | (cid:39) |(cid:104) ξ | ξ α (cid:105) num | (cid:28) | Q α | . (25)The results shown in tables 1-2 reveal that | c | is alwaysmore than an order of magnitude less than ¯ Q α , so the abovecondition is satisfied for the modes computed in this paper.We note from tables 1-2 that while in general higher-order g-modes tend to have smaller | Q α | , the dependence of | Q α | on the mode index n is not exactly monotonic. Thisis the result of the mode trapping phenomenon associatedwith composition discontinuities in the WD. To see this, wenote that a mode with amplitude ξ α has energy given by E α = ω α (cid:82) d x ρ | ξ α | , thus we can define the mode energyweight function dE α d ln P = ω α ρ r (cid:2) ξ r + l ( l + 1) ξ ⊥ (cid:3) H p , (26)where H p = dr/d ln P = P/ ( ρg ) is the pressure scale height,and we have used ξ α = [ ξ r ( r ) e r + rξ ⊥ ( r ) ∇ ] Y lm (27) c (cid:13)000 , 000–000 esonant Oscillation Modes in White Dwarf Binaries Figure 2.
The mode energy weight functions for the n = 1(thickest line), n = 2 (thick line), and n = 5 (top line) modes(all for l = 2) for a WD with T eff = 10800K, M = 0 . M (cid:12) , R =8 . × km, displayed as a function of log P so that the structureof the outer layers of the WD is more evident. The y-axis fora given mode is intended only to show the relative value of theweight function. The squares of the Br¨unt V¨ais¨al¨a (thin solid line)and Lamb (dotted line) frequencies are displayed to demonstratehow their values constrain the region of mode propagation. ( e r is the unit vector in the r -direction). Figures 2 and 3 dis-play the weight functions for several g-modes of WD models.We can see that the weight functions for all the low-ordermodes are largest in the region below the convective zonenear the spikes in N produced by the composition gradi-ents. For the modes shown in Figure 2, the smooth fall-offof the weight function just below the convective zone indi-cates that these modes are confined by the falling value ofthe Lamb frequency in this region. The weight functions ofhigher-order modes and the modes in WDs with deeper con-vective zones may drop sharply at the convective boundary,indicating that these modes are trapped by the convectivezone rather than the decreasing Lamb frequency.The weight functions also reveal the phenomenonknown as mode trapping caused by the composition gra-dients. Mode trapping is especially evident for the n = 2mode, as it is confined to the helium layer between thespikes in N . It is clear that the mode is reflected by thecarbon-helium boundary at larger depths and by the helium-hydrogen boundary at shallower depths. See Brassard (1991)for a more detailed description of the effects of mode trap-ping.The weight function is essentially the energy of a modeas a function of radius, so it tells us where orbital energy isdeposited when a mode is excited. Since the weight functionis largest in the hydrogen and helium layers just below theconvection zone, most of the mode energy exists in this re-gion of the WD. Thus, if the mode is damped, most of themode energy will be damped out in this region. Figure 3.
Same as Figure 2, except for a WD model with T eff = 5080K. Note that the convection zone extends deeper inthis model, pushing the modes to larger depths. Having obtained the mode frequency and the tidal couplingcoefficient, we can determine the combined evolution of theresonant mode and the binary orbit using equations (7),(12), and (13). These are integrated from well before reso-nance until well after the resonance is complete. The initialmode amplitude b α and its derivative ˙ b α (prior to a reso-nance) are obtained by dropping the ¨ b α , ˙ b α and ˙Ω terms inequation (16), giving b α (cid:39) M (cid:48) W lm Q α D l +1 ( ω α − m Ω ) , (28)˙ b α (cid:39) (cid:20) − ( l + 1) ˙ DD + 2 m Ω ˙Ω ω α − m Ω (cid:21) b α , (29)with ˙Ω (cid:39) − (3 ˙ D/ D )Ω. These expressions are valid for for( ω α /m Ω) − (cid:29) ˙Ω / ( m Ω ) (cid:39) / (2 m Ω t D ) (see section 5).The evolution equations (7), (12), and (13) form a verystiff set of differential equations. The reason for this is thatthe problem involves two vastly different time scales: the or-bital decay time scale which is on the order of thousandsof years, and the orbital time scale (or the resonant modeoscillation period) which is on the order of minutes. Con-sequently, a typical Runge-Kutta scheme would require theintegration of millions of orbits, demanding a high degree ofaccuracy for each orbit. To avoid this problem, we employthe Rossenberg stiff equation technique (Press et al. 2007).The integrator requires a Jacobian matrix of second deriva-tives, meaning that we need to supply the 8 × n = 3 g-mode. Be-fore the resonance, the mode must oscillate with the samefrequency as the binary companion, so the amplitude | b α | is smooth. After the resonance, the mode oscillates at itseigenfrequency (which is now different from the forcing fre-quency). The amplitude of the mode continues to fluctuateafter resonance because it is still being forced by the bi- c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 4.
Mode amplitude | b α | and orbital distance D as afunction of time during resonance. The amplitude and distanceoscillate after resonance, causing the curves to appear as filledshapes due to the short period of the oscillations with respectto the total integration time. These oscillations occur due to thecontinued interaction between the excited mode and the orbit.Note the sharp drop in orbital distance at resonance, which iscaused by the transfer of orbital energy into the mode. The modeparameters are given in table I ( n = 3). The companion mass is M (cid:48) = M = 0 . M (cid:12) . Figure 5.
The average post-resonance amplitudes for the firstfive modes given in Table I. The open triangles mark the resultsobtained from numerical integration, and the asterisks indicatethe results predicted by the analytical estimate described in sec-tion 5. The analytical estimates are usually accurate within afactor of 10%. The n = 2 and n = 5 modes are trapped modes. nary companion, although the amplitude of these fluctua-tions diminishes over time as the orbit moves further fromresonance.Figure 5 displays the average post-resonance amplitudesfor the first five modes given in Table I. Note that no modeexceeds a maximum amplitude of | b α | = 0 .
1, and we expectthat our linear approximation is a reasonable first approachto the problem before non-linear effects can be included. Ingeneral, lower-order modes reach larger amplitudes (due totheir larger coupling coefficients), but higher-order modes
Figure 6.
The average post-resonance amplitude | b α | for the n = 4 mode given in Table I as a function of the mass of thebinary companion. The post-resonance amplitude increases withthe mass of the binary companion as predicted by the analyticalestimate except for very high companion masses. with an abnormally high value of Q α (trapped modes) mayreach large amplitudes as well. Here we provide an analytic estimate of the mode amplitudeattained during a resonance as well as the temporal dura-tion of the resonance (i.e., the characteristic time duringwhich the resonant mode receives most of its energy fromthe orbit).For a WD oscillation mode with frequency ω α , the res-onant orbital radius is D α = (cid:18) m M t ω α (cid:19) / . (30)Prior to the resonance, as the orbital radius D decreases,the mode amplitude grows gradually according to equation(28). At the same time, the orbit also loses its energy togravitational waves (GWs) at the rate˙ E GW = − MM (cid:48) ) M t c D = − MM (cid:48) / Dt D , (31)where t D is given by equation (10). We can define the be-ginning of resonance as the point where the orbital en-ergy is transferred to the mode faster than it is radiatedaway by GWs. That is, the resonance begins at the radius D = D α + ( > D α ), as determined by˙ E α = | ˙ E GW | , (32)where E α is the energy contained in the mode. Near the reso-nance, the mode oscillates at the frequency close to ω α , so wecan write the mode energy as E α (cid:39) ω α b α (including boththe m = 2 and m = − ω α (cid:29) | ˙ b α /b α | .Thus we have ˙ E α (cid:39) ω α b α ˙ b α . Using equations (28) and(29) for b α and ˙ b α , we find ω α − ( m Ω) = (cid:20) ω α M (cid:48) ( W lm Q α ) MD (cid:21) / at D = D α + (33) c (cid:13)000
The average post-resonance amplitude | b α | for the n = 4 mode given in Table I as a function of the mass of thebinary companion. The post-resonance amplitude increases withthe mass of the binary companion as predicted by the analyticalestimate except for very high companion masses. with an abnormally high value of Q α (trapped modes) mayreach large amplitudes as well. Here we provide an analytic estimate of the mode amplitudeattained during a resonance as well as the temporal dura-tion of the resonance (i.e., the characteristic time duringwhich the resonant mode receives most of its energy fromthe orbit).For a WD oscillation mode with frequency ω α , the res-onant orbital radius is D α = (cid:18) m M t ω α (cid:19) / . (30)Prior to the resonance, as the orbital radius D decreases,the mode amplitude grows gradually according to equation(28). At the same time, the orbit also loses its energy togravitational waves (GWs) at the rate˙ E GW = − MM (cid:48) ) M t c D = − MM (cid:48) / Dt D , (31)where t D is given by equation (10). We can define the be-ginning of resonance as the point where the orbital en-ergy is transferred to the mode faster than it is radiatedaway by GWs. That is, the resonance begins at the radius D = D α + ( > D α ), as determined by˙ E α = | ˙ E GW | , (32)where E α is the energy contained in the mode. Near the reso-nance, the mode oscillates at the frequency close to ω α , so wecan write the mode energy as E α (cid:39) ω α b α (including boththe m = 2 and m = − ω α (cid:29) | ˙ b α /b α | .Thus we have ˙ E α (cid:39) ω α b α ˙ b α . Using equations (28) and(29) for b α and ˙ b α , we find ω α − ( m Ω) = (cid:20) ω α M (cid:48) ( W lm Q α ) MD (cid:21) / at D = D α + (33) c (cid:13)000 , 000–000 esonant Oscillation Modes in White Dwarf Binaries or D α + = D α (cid:34) . (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / ¯ ω / α ¯ Q / α (cid:35) , (34)where we have used l = m = 2 and ¯ ω α , ¯ Q α are in dimen-sionless units where G = M = R = 1. The mode energy at D = D α + is E α ( D α + ) =0 . (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / × ¯ ω / α ( W lm ¯ Q α ) / M R . (35)Since for
D < D α + , the orbital energy will be depositedinto the mode much faster than it is being radiated away,we approximate that all the orbital energy between D α + and D α is transferred to the mode. Thus the mode energyincreases by the amount∆ E α = 2 × (cid:18) MM (cid:48) D α − MM (cid:48) D α + (cid:19) , (36)where we have multiplied by a factor of two to account forthe fact that energy is also deposited at a nearly equal ratebefore resonance as it is after resonance. Using equation(34), we find∆ E α =0 . (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / × ¯ ω / α ( W lm ¯ Q α ) / M R . (37)This is exactly four times of equation (35). Thus the maxi-mum mode energy after resonance is E α, max = E α ( D α + ) +∆ E α , or E α, max (cid:39) . × − (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / × (cid:16) ¯ ω α . (cid:17) / (cid:18) ¯ Q α − (cid:19) / M R . (38)The corresponding maximum mode amplitude is b α, max (cid:39) . × − (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / × (cid:16) ¯ ω α . (cid:17) − / (cid:18) ¯ Q α − (cid:19) / . (39)Figures 5 and 6 compare our numerical results with theanalytical expressions (38)-(39). We find good agreement forall the WD resonant modes considered. Figure 5 verifies thedependence of b α, max on the mode frequency and the valueof ¯ Q α , while figure 6 verifies the dependence of b α, max onthe mass of the binary companion (except for the highestmass cases discussed below). Therefore equations (38) and(39) provide fairly accurate estimates of the mode amplitudeand energy without performing numerical integrations.For the very high companion masses ( M (cid:48) > ∼ M )shown in Figure 6, our analytical formula significantly over-estimates the post-resonance amplitude. The reason for thisis that the gravitational decay time scale is shorter if thecompanion is more massive. If the companion is massiveenough, the orbit will decay through resonance due to grav-itational radiation before the orbital energy of equation (37)can be deposited in the mode (see below). Consequently, the amplitude to which a mode is excited decreases if the massof the companion becomes very high. Therefore our analyt-ical formula overestimates the post-resonance amplitude forextremely massive companions. For any reasonable WD orNS masses, our analytical estimate is accurate, but for asuper-massive black hole the estimate may become inaccu-rate.It is interesting that the above analytical results for theresonant mode energy is independent of the gravitationalwave damping time scale t D , in contrast to the NS/NS orNS/BH binary cases. In fact, for the above derivation to bevalid, the following four conditions must be satisfied at D α + :(i) ω α (cid:29) (cid:12)(cid:12)(cid:12) ˙ b α /b α (cid:12)(cid:12)(cid:12) , (40)(ii) ω α − ( m Ω) (cid:29) m ˙Ω , (41)(iii) ω α − ( m Ω) (cid:29) m Ω (cid:12)(cid:12)(cid:12) ˙ b α /b α (cid:12)(cid:12)(cid:12) , (42)(iv) ω α − ( m Ω) (cid:29) (cid:12)(cid:12)(cid:12) ¨ b α /b α (cid:12)(cid:12)(cid:12) . (43)Conditions (i) and (ii) both lead to ω α t D (cid:29) (cid:20) ω α ω α − ( m Ω) (cid:21) ; (44)condition (iii) gives ω α t D (cid:29) (cid:20) ω α ω α − ( m Ω) (cid:21) ; (45)and condition (iv) yields ω α t D (cid:29) √ (cid:20) ω α ω α − ( m Ω) (cid:21) / . (46)In equations (44)-(46), the right-hand sides should be eval-uated at D α + . Clearly, condition (iii) is most constraining.With ω α − ( m Ω) ω α = 3 (cid:18) D α + − D α D α (cid:19) = 0 . (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / (cid:16) ¯ ω α . (cid:17) / (cid:18) ¯ Q α − (cid:19) / , (47)we see that condition (iii) is satisfied if t D (cid:29) . × (cid:18) R GM (cid:19) / (cid:18) M (cid:48) M (cid:19) − / (cid:18) M t M (cid:19) / × (cid:16) ¯ ω α . (cid:17) − / (cid:18) ¯ Q α − (cid:19) − / . (48)Since t D is on the order of a thousand years or more fororbital frequencies comparable to WD g-modes, condition(iii) is always satisfied for WD/WD or WD/NS binaries. Onthe other hand, the conditions (i)-(iv) are not all satisfiedfor NS/NS or NS/BH binaries.For very massive companions, the inequality of equation(48) may not hold. Using equation (10) for t D , equation (48)implies (for M t /M (cid:39) M (cid:48) /M ) M (cid:48) M (cid:28) . × (cid:16) ¯ ω α . (cid:17) − / (cid:18) ¯ Q α − (cid:19) / × (cid:18) MM (cid:12) (cid:19) − / (cid:18) R km (cid:19) / . (49)The above inequality implies our estimates are valid for any c (cid:13) , 000–000 J. Fuller and D. Lai feasible companion except a super-massive black hole. Wecan also use this inequality to examine the inaccuracy of ourestimate in the highest mass cases of Figure 6. Figure 6 wasgenerated using the n = 3 mode parameters listed in Table1 for a WD of M = 0 . M (cid:12) and R = 8 . × km. Pluggingin these parameters, equation (49) requires M (cid:48) M (cid:28) . × (50)for our analytical estimates in Figure 6 to be accurate. Thisexplains why the analytical estimates of Figure 6 are ac-curate when M (cid:48) < ∼ M but diverge from the numericalresults when M (cid:48) > ∼ M .Given the maximum mode amplitude reached during aresonance, we can now estimate the temporal duration ofthe resonance. Letting a α = c α e − iω α t , the mode amplitudeevolution equation (7) becomes¨ c α − iω α ˙ c α = M (cid:48) W lm Q α D e iω α t − im Φ . (51)Assuming that during the resonance, ω α − m Ω (cid:39)
0, theright-hand-side of equation (51) can be taken as a constant,we then have ˙ c α (cid:39) iM (cid:48) W lm Q α ω α D , (52)i.e., the mode amplitude grows linearly in time. Thus, theduration of the resonance is of order t res = (cid:12)(cid:12)(cid:12)(cid:12) b α, max ˙ c α (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) . (cid:18) M (cid:48) M (cid:19) − / (cid:18) M t M (cid:19) / ¯ ω − / α ¯ Q − / α (cid:18) R M (cid:19) / = 3 . × (cid:18) M (cid:48) M (cid:19) − / (cid:18) M t M (cid:19) / × (cid:16) ¯ ω α . (cid:17) − / (cid:18) ¯ Q α − (cid:19) − / (cid:18) R M (cid:19) / . (53)Since the dynamical time ( R /GM ) / for typical WDs is onthe order of one second, the resonance duration is typicallyan hour or longer. Note that the above estimate is formallyvalid only when [ ω − m Ω( D α + )] t res (cid:28)
1, so that we canset ω α − m Ω ≈ ω α − m Ω( D α + )] t res ≈ E gw (cid:39) ( MM (cid:48) / D α )( t res /t D ), is much less than E α, max , justifyingour derivation of E α, max given by equation (38). Indeed, theabove condition simplifies to equation 49, since in both casesit is the energy carried away by gravitational waves that islimiting the mode growth.We can use the same method to solve for the size ofthe fluctuations in mode amplitude after resonance. Due tothe symmetry of the harmonic oscillator, the fluctuation inmode amplitude about the mean value after the resonance is Obviously, our estimate would not apply for a WD in a highlyeccentric orbit around an intermediate mass black hole, whichmay form in dense clusters as described by Ivanov & Papaloizou(2007).
Figure 7.
The amplitude of a mode as a function of time nearits resonance for different values of the damping coefficient γ α .The curves have γ α = 0 (black), γ α = 10 − (dark blue), γ α =10 − (yellow-green), γ α = 10 − (light blue), γ α = 10 − (red), γ α = 10 − (green), and γ α = 10 − (purple). For this mode, wehave set ω α = 0 . − and Q α = 1 × − so that t res ≈ s.Note that the damping term does not greatly affect the maximummode amplitude except when γ α > ∼ − , or when γ α ω α t res > ∼ γ α evolve on a muchshorter time scale because their orbits decay quickly due to theconversion of orbital energy into heat via mode damping. identical to the zeroth-order estimate of the mode amplitudebefore resonance [see eq. (28)], i.e.,∆ a ≈ M (cid:48) W lm Q α D l +1 ( m Ω − ω α ) . (54)These fluctuations occur with frequency m Ω − ω α , since thisis the difference in frequency between the eigenfrequencyat which the WD is oscillating and the orbital forcing fre-quency. So, as the orbital frequency continues to increase af-ter the resonance, the amplitude of the fluctuations becomessmaller while the frequency of the amplitude oscillations be-comes higher. The results in the previous two sections neglect mode energydamping in the WD. Since the duration of the resonance ismuch longer than the mode period [see equation (53)], inter-nal mode damping could affect the energy transfer duringresonance if the damping rate is sufficiently large. To ad-dress this issue, we incorporate a phenomenological damp-ing rate − γ α ω α ˙ a α to the mode equation (7) to study howmode damping affects energy transfer during a resonance.Figure 7 shows the excitation of a mode through resonancefor different values of γ α . We see that, as expected, whenthe internal damping time is larger than the resonance du-ration (equation 53), the maximum mode energy achievedin a resonance is unaffected.G-modes in white dwarfs are damped primarily by ra-diative diffusion. For sufficiently large mode amplitudes,non-linear damping is also important (e.g., Kumar & Good-man 1996; see section 7 for more discussion on this issue). c (cid:13)000
The amplitude of a mode as a function of time nearits resonance for different values of the damping coefficient γ α .The curves have γ α = 0 (black), γ α = 10 − (dark blue), γ α =10 − (yellow-green), γ α = 10 − (light blue), γ α = 10 − (red), γ α = 10 − (green), and γ α = 10 − (purple). For this mode, wehave set ω α = 0 . − and Q α = 1 × − so that t res ≈ s.Note that the damping term does not greatly affect the maximummode amplitude except when γ α > ∼ − , or when γ α ω α t res > ∼ γ α evolve on a muchshorter time scale because their orbits decay quickly due to theconversion of orbital energy into heat via mode damping. identical to the zeroth-order estimate of the mode amplitudebefore resonance [see eq. (28)], i.e.,∆ a ≈ M (cid:48) W lm Q α D l +1 ( m Ω − ω α ) . (54)These fluctuations occur with frequency m Ω − ω α , since thisis the difference in frequency between the eigenfrequencyat which the WD is oscillating and the orbital forcing fre-quency. So, as the orbital frequency continues to increase af-ter the resonance, the amplitude of the fluctuations becomessmaller while the frequency of the amplitude oscillations be-comes higher. The results in the previous two sections neglect mode energydamping in the WD. Since the duration of the resonance ismuch longer than the mode period [see equation (53)], inter-nal mode damping could affect the energy transfer duringresonance if the damping rate is sufficiently large. To ad-dress this issue, we incorporate a phenomenological damp-ing rate − γ α ω α ˙ a α to the mode equation (7) to study howmode damping affects energy transfer during a resonance.Figure 7 shows the excitation of a mode through resonancefor different values of γ α . We see that, as expected, whenthe internal damping time is larger than the resonance du-ration (equation 53), the maximum mode energy achievedin a resonance is unaffected.G-modes in white dwarfs are damped primarily by ra-diative diffusion. For sufficiently large mode amplitudes,non-linear damping is also important (e.g., Kumar & Good-man 1996; see section 7 for more discussion on this issue). c (cid:13)000 , 000–000 esonant Oscillation Modes in White Dwarf Binaries Wu (1998) presents estimates for the non-adiabatic radia-tive damping rates of WD g-modes in terms of ω i = γω r .Extrapolating Wu’s values to l = 2 modes for a white dwarfof temperature T = 10800 K, we find γ ∼ − for modesnear n = 1 and γ ∼ − for high-order modes with n > ∼ We have shown that during the orbital decay of compactwhite dwarf binaries (WD/WD, WD/NS or WD/BH), a se-ries of g-modes can be tidally excited to large amplitudes(up to 0 . < ∼ − Hz) gravitational waveforms emitted by the binary,detectable by LISA, will deviate significantly from the point-mass binary prediction. This is in contrast to the case ofneutron star binaries (NS/NS or NS/BH) studied previously(Reisenegger & Goldreich 1994; Lai 1994; Shibata 1994; Ho& Lai 1999; Lai & Wu 2006; Flanagan & Racine 2006), wherethe resonant mode amplitude is normally too small to affectthe binary orbital decay rate and the gravitational wave-forms to be detected by ground-based gravitational wavedetectors such as LIGO and VIRGO.In the case of WD binaries studied in this paper, thenumber of of orbits skipped as a result of a resonant modeexcitation is ∆ N orb = t D P orb E α, max E orb , (55)where t D is the gravitational wave decay time scale givenin equation (10), P orb is the orbital period, and E orb is theorbital energy at resonance. Using equation (38) for E α, max ,we find∆ N orb =3 . × (cid:18) M (cid:12) M M (cid:48) M t (cid:19) / (cid:18) R km (cid:19) / (cid:18) ¯ Q α − (cid:19) / (cid:18) Ω0 . − (cid:19) − / . (56)The number of skipped orbital cycles should be comparedto the number of orbits in a decay time, expressed by dN orb d ln Ω = 13 π Ω t D = 3 . × (cid:18) M (cid:12) MM (cid:48) (cid:19)(cid:18) M t M (cid:12) (cid:19) / (cid:18) Ω0 . − (cid:19) − / . (57)The huge number of skipped orbital cycles implies that sucha resonant interaction would be important, but because the number of skipped orbital cycles is much smaller than thenumber of orbital cycles in a decay time, resonances will notdominate the decay process.A second possible consequence of resonant mode exci-tations is that the large mode energy may lead to significantheating of the white dwarf prior to the binary merger. In-deed, equation (38) shows that for typical binary parame-ters, the mode energy can be a significant fraction ( ∼ − –10 − ) of the gravitational binding energy of the star, andcomparable to the thermal energy. Indeed, the thermal en-ergy of the WD is of order E th ≈ MkT c Am p , where T c is the coretemperature of the WD and A is the mean atomic weight.The ratio of post-resonance mode energy to thermal energyis then E α, max E th ≈ . (cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / (cid:16) ¯ ω α . (cid:17) / × (cid:18) ¯ Q α − (cid:19) / (cid:18) KT c (cid:19) (cid:18) MM (cid:12) (cid:19) (cid:18) km R (cid:19) . (58)This implies that the white dwarf may become bright thou-sands of years before binary merger.A third consequence that may result from a resonance issignificant spin-up of the WD. If we assume that all the an-gular momentum transferred to the WD during a resonanceeventually manifests as rigid body rotation of the WD, thechange in spin frequency of the WD is∆Ω s = E α, max I Ω , (59)where Ω s is the spin frequency of the WD and I is its mo-ment of inertia. Plugging in our expression for E α, max , wefind ∆Ω s =0 . (cid:18) . κ (cid:19)(cid:18) M (cid:48) M (cid:19) / (cid:18) M t M (cid:19) − / × (cid:16) ¯ ω α . (cid:17) − / (cid:18) ¯ Q α − (cid:19) / Ω , (60)where κ = I/ ( MR ) ≈ .
2. We can thus see that a givenresonance may deposit enough angular momentum to com-pletely spin up the WD (or significantly alter its spin) bythe time the mode damps out. This implies that mode res-onances are potentially very important in the spin synchro-nization process.However, before these implications can be taken seri-ously, one should be aware of the limitations of the presentstudy. One issue is the assumption that the white dwarfis non-rotating (and not synchronized), already commentedon in section 1. More importantly, we have assumed thatthe white dwarf oscillations can be calculated in the linearregime. While the mass-averaged dimensionless amplitude | a α | = | b α | of the excited g-mode is less than 0 . ξ = a α ξ α , with ξ α thenormalized eigenfunction (see section 3) and a α computedfrom equation (39) with M (cid:48) = M . In general, the linear ap-proximation is valid only if | ξ | (cid:28) | k r | − , where k r is the c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 8.
The horizontal (solid line) and radial (dashed line)displacements of the n = 1 (top panel) and n = 5 (bottom panel)modes as a function of radius. The physical displacements arecalculated using the analytical estimates for the post-resonanceamplitudes given in equation 39 using M = M (cid:48) , giving | a | =0 . | a | = 0 . /k r (dotted line). WKB wave number, given by k r (cid:39) l ( l + 1)( N − ω ) ω r . (61)Clearly, the three modes depicted in Fig. 8 strongly violatethe linear approximation beyond the radius r ≈ . R , nearthe jump in N associated with the carbon-helium bound-ary. Therefore, the results presented in this paper shouldbe treated with caution as nonlinear effects will likely limitmode growth. Rather than increasing to the large displace-ments shown in Figure 8, the white dwarf oscillations willundergo non-linear processes such as mode coupling thatwill transfer energy to high-order modes. These high-ordermodes have much shorter wavelengths and thus damp onvery short time scales. As nonlinearity is most important inthe outer layers of the white dwarf, we expect that the ex-cited oscillation will dissipate its energy preferentially in thisouter layer and will not reflect back into the stellar interior.We plan to address these issues in our next paper (Fuller &Lai 2010, in preparation). ACKNOWLEDGMENTS
We thank Gilles Fontaine (University of Montreal) for pro-viding the white dwarf models used in this paper and forvaluable advice on these models. DL thanks Lars Bildsten,Gordon Ogilvie and Yanqin Wu for useful discussions, andacknowledges the hospitality of the Kavli Institute for The-oretical Physics at UCSB (funded by the NSF throughGrant PHY05-51164) where part of the work was carried out. This work has been supported in part by NASA GrantNNX07AG81G and NSF grants AST 0707628.
REFERENCES
Badenes, C., et al. 2009, ApJ, 707, 971Brassard, P., et al. 1991, ApJ, 367, 601Brassard, P., et al. 1992, ApJ Supplement Series, 80, 369Brickhill, A.J., 1983, MNRAS, 204, 537Dobbs-Dixon, I., Lin, D.N.C., Mardling, R. 2004, ApJ, 610, 464Flanagan, E., Racine, E. 2007, Phys. Rev. D75, 044001Goldreich, P., Nicholson, P., 1977, Icarus, 30, 301Goldreich, P., Wu, Y. 1999, ApJ, 511, 904Goldman, J., Lackner, C. 2009, ApJ, 296, 2054Goodman, J., Dickson, E.S., 1998, ApJ, 507, 938Goodman, J., Oh, S.P. 1997, ApJ, 486, 403Ho, W.C.G., Lai, D. 1999, MNRAS, 308, 153Ivanov, P.B., Papaloizou J.C.B. 2007, A&A, 476, 121Kilic, M., et al. 2009, ApJ, 695, L92Kulkarni, S.R., van Kerkwijk, M.H., ApJ, 719, 1123Kumar, P., Goodman, J. 1996, ApJ, 466, 946Lai, D. 1994, MNRAS, 270, 611Lai, D., Wiseman, A.G. 1996, Phys. Rev. D54, 3958Lai, D., Wu, Y. 2006, Phys. Rev. D74, 024007Marsh, T., et al. 2010, arXiv:1002.4677Mullally, F., et al. 2009, ApJ, 707, L51Nelemans, G. 2009, Class. Quantum Grv., 26, 094030Nelemans, G., et al. 2001, AA, 365, 491Ogilvie, G.I., Lin, D.N.C. 2004, ApJ, 610, 477Ogilvie, G.I., Lin, D.N.C. 2007, ApJ, 661, 1180Peters, P.C. 1964, Phys. Rev., 136, B1224Press, W.H., Teukolsky, T.A. 1977, ApJ, 213, 183Press, W.H., et al. 1998, Numerical Recipes (Cambridge Univ.Press)Ramsay, G. et al. 2005, MNRAS, 357, 49Reisenegger, A. 1994, ApJ, 432, 296Reisenegger, A., Goldreich, P. 1994, ApJ, 426, 688Shibata, M. 1994, Prog. Theo. Phys., 91, 871Steinfadt, J., et al. 2010, ApJ, 716, L146Strohmayer, T.E. 2005, ApJ, 627, 920Unno, W., et al. 1989, Nonradial Oscillations of Stars (Universityof Tokyo Press)Wu, Y. 1998, PhD Thesis, CalTechWu, Y. 2005, ApJ, 635, 688Wu, Y., Goldreich, P. 1999, ApJ, 519, 783Zahn, J.P. 1970, AA, 4, 452Zahn, J.P. 1989, AA, 220, 112Zahn, J.-P., 2008, EAS, 29, 67c (cid:13)000