Dynamical Tides in Compact White Dwarf Binaries: Tidal Synchronization and Dissipation
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 October 2018 (MN L A TEX style file v2.2)
Dynamical Tides in Compact White Dwarf Binaries: TidalSynchronization and Dissipation
Jim Fuller (cid:63) and Dong Lai
Center for Space Research, Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
11 October 2018
ABSTRACT
In compact white dwarf (WD) binary systems (with periods ranging from minutesto hours), dynamical tides involving the excitation and dissipation of gravity wavesplay a dominant role in determining the physical conditions (such as rotation rate andtemperature) of the WDs prior to mass transfer or binary merger. We calculate theamplitude of the tidally excited gravity waves as a function of the tidal forcing fre-quency ω = 2(Ω − Ω s ) (where Ω is the orbital frequency and Ω s is the spin frequency)for several realistic carbon-oxygen WD models, under the assumption that the outgo-ing propagating waves are efficiently dissipated in outer layer of the star by nonlineareffects or radiative damping. Unlike main-sequence stars with distinct radiative andconvection zones, the mechanism of wave excitation in WDs is more complex due tothe sharp features associated with composition changes inside the WD. In our WDmodels, the gravity waves are launched just below the helium-carbon boundary andpropagate outwards. We find that the tidal torque on the WD and the related tidalenergy transfer rate, ˙ E tide , depend on ω in an erratic way, with ˙ E tide varying by ordersof magnitude over small frequency ranges. On average, ˙ E tide scales approximately asΩ ω for a large range of tidal frequencies.We also study the effects of dynamical tides on the long-term evolution of WDbinaries prior to mass transfer or merger. Above a critical orbital frequency Ω c , corre-sponding to an orbital period of order one hours(depending on WD models), dynam-ical tides efficiently drive Ω s toward Ω, although a small, almost constant degree ofasynchronization (Ω − Ω s ∼ constant) is maintained even at the smallest binary peri-ods. While the orbital decay is always dominated by gravitational radiation, the tidalenergy transfer can induce significant phase error in the low-frequency gravitationalwaveforms, detectable by the planned LISA project. Tidal dissipation may also leadto significant heating of the WD envelope and brightening of the system long beforebinary merger. Key words: white dwarfs – hydrodynamics – waves – binaries
Compact white dwarf (WD) binary systems (with orbitalperiods in the range of minutes to hours) harbor many inter-esting and unanswered astrophysical questions. An increas-ing number of such systems are being discovered by recentsurveys (e.g. Mullally et al. 2009; Kulkarni & van Kerkwijk2010; Steinfadt et al. 2010; Kilic et al. 2011; Brown et al.2011; see Marsh 2011 for a review). The orbits of these sys-tems decay via the emission of gravitational waves, whichcould be detected by the planned
Laser Interferometer SpaceAntenna (LISA) (Nelemans 2009). Depending on the WDmasses and the physics of the merger process, these merging (cid:63)
Email: [email protected]; [email protected]
WD systems may produce single helium-rich sdO stars, giantstars (R CrB stars), stable mass transfer AM CVn binaries,or possibly underluminous supernovae. Most importantly,compact WD binaries in which the total mass is near theChandrasekhar limit are thought to be the probable progen-itors of type Ia supernovae upon a stellar merger at the endof the orbital decay process (Webbink 1984; Iben & Tutukov1984). Recent studies have provided support for this “doubledegenerate” scenario (e.g., Gilfanov & Bogdan 2010; Di Ste-fano 2010; Maoz et al. 2010) and even sub-ChandrasekharWD mergers may lead to type Ia supernovae (van Kerkwijket al. 2010).Prior to merger, tidal interactions may affect the prop-erties of the binary WDs and their evolutions, including thephase evolution of the gravitational waves. Previous stud- c (cid:13) a r X i v : . [ a s t r o - ph . S R ] S e p J. Fuller and D. Lai ies have focused on equilibrium tides (e.g., Iben et al. 1998;Willems et al. 2010), corresponding to quasi-static deforma-tion of the star. Such equilibrium tides are unlikely to playa role in the tidal synchronization/dissipation process. Ibenet al. (1998) estimated the effect of tidal heating in the WDbased on the assumption that the (spherically averaged) lo-cal heating rate is equal to the rate of rotational energydeposition required to maintain synchronization. They sug-gested that the binary WDs may brighten by several mag-nitudes before merger.In fact, in a compact WD binary, as the orbital decayrate due to gravitational wave radiation increases rapidlywith decreasing orbital period, it is not clear if tidal effectsare sufficiently strong to drive the binary system towardsynchronous rotation. The critical orbital period for syn-chronization is unknown. For this reason, the majority ofrecent WD merger simulations (e.g., Segretain et al. 1997;Loren-Aguilar et al. 2009; Pakmor et al. 2010,2011) haveassumed the merging WDs to be non-synchronized prior tomerger. However, whether the WDs are spin-synchronizedmay affect the merger product and the possible supernovasignature: for example, the strong velocity shear betweenthe stars upon contact would be significantly reduced forthe merger of a synchronized binary. The degree of synchro-nization also determines the tidal luminosity of the binaryprior to merger. Indeed, it is possible that tidal dissipationcontributes significantly to the brightness of some of the re-cently observed WD binaries (e.g., Brown et al. 2011).In a recent paper (Fuller & Lai 2011, hereafter PaperI), we used linear theory to calculate the the excitation ofdiscrete gravity modes in a WD due to the tidal gravita-tional field of a compact companion star (a WD, neutronstar or black hole). The existence of discrete modes requiresthat gravity waves be reflected near the surface of the WD.In this case, tidal energy and angular momentum transfersbetween the WD and the binary orbit occur only during a se-ries of resonances, when the g-mode frequency σ α equals 2Ω(where Ω is the orbital frequency). Our calculations showedthat while the dimensionless (mass-weighted) amplitude ofthe resonantly excited g-mode is not extremely non-linear(it approaches ≈ . The dynamical tide of the WD (mass M ) is driven by theexternal gravitational potential of the the companion (mass M (cid:48) ). The leading order (quadrupole) potential is U ex ( r , t ) = U ( r ) (cid:104) Y ( θ, φ ) e − iωt + Y ∗ ( θ, φ ) e iωt (cid:105) (1)with U ( r ) = − GM (cid:48) W a r . (2)Here a is the orbital separation, ω = 2Ω is the tidal fre-quency for a non-spinning WD (we will account for the spineffect in Section 8), Ω is the orbital frequency, and W = (cid:112) π/
10. The actual fluid perturbations in the WD can bewritten as ξ ac ( r , t ) = ξ ( r , t )+ ξ ∗ ( r , t ) for the Lagrangian dis-placement and δP ac ( r , t ) = δP ( r , t ) + δP ∗ ( r , t ) for the Eu-lerian pressure perturbation, and similarly for other quanti-ties. In the following, we shall consider perturbations ( ξ , δP ,etc.) driven by the potential U ( r , t ) = U ( r ) Y ( θ, φ ) e − iωt .We shall adopt the Cowling approximation (so that the grav-itational potential perturbation δ Φ associated with the den-sity perturbation is neglected, i.e., δ Φ = 0), which is valid for c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries gravity waves in the star. We will consider adiabatic oscil-lations, for which the Lagrangian perturbations in pressureand density are related by ∆ P = a s ∆ ρ , where a s is theadiabatic sound speed. This is a good approximation in thebulk of the star where the thermal time is much longer thanthe wave period.Letting δP ( r , t ) = δP ( r ) Y ( θ, φ ) e − iωt , (3)and ξ ( r , t ) = (cid:2) ξ r ( r )ˆ r + ξ ⊥ ( r ) r ∇ ⊥ (cid:3) Y ( θ, φ ) e − iωt , (4)the fluid perturbation equations reduce to1 r (cid:0) r ξ r (cid:1) (cid:48) − ga s ξ r + 1 ρa s (cid:18) − L l ω (cid:19) δP − l ( l + 1) Uω r = 0 , (5)and 1 ρ δP (cid:48) + gρa s δP + (cid:0) N − ω (cid:1) ξ r + U (cid:48) = 0 , (6)where the (cid:48) denotes d/dr . In equations (5) and (6), L l and N are the Lamb and Br¨unt-Vais¨al¨a frequencies, respectively,given by [note we will continue to use the notations L l , l ( l +1), and m , although we focus on l = m = 2 in this paper] L l = l ( l + 1) a s r (7)and N = g (cid:18) dρdP − a s (cid:19) . (8)The other perturbation variables are related to δP and ξ r by ξ ⊥ = 1 rω (cid:18) δPρ + U (cid:19) , (9) δρ = 1 a s δP + ρN g ξ r . (10)Defining Z = χ − / r ξ r , where χ = r ρa s (cid:18) L l ω − (cid:19) , (11)equations (5) and (6) can be combined to yield Z (cid:48)(cid:48) + k ( r ) Z = V ( r ) . (12)Here, k ( r ) = χρ ( N − ω ) r + 12 (cid:18) χ (cid:48) χ (cid:19) (cid:48) − (cid:18) χ (cid:48) χ (cid:19) + ga s (cid:20) − ( g/a s ) (cid:48) g/a s + χ (cid:48) χ − ga s (cid:21) (13)and V ( r ) = χ − / (cid:20) l ( l + 1) ω (cid:18) − χ (cid:48) χ + ga s (cid:19) + 2 ra s (cid:21) U. (14)In the WKB limit | k | (cid:29) /H and | k | (cid:29) /r , where H = | P/P (cid:48) | (cid:39) a s /g is the pressure scale height, equation(13) simplifies to k ( r ) = 1 a s ω ( L l − ω )( N − ω ) . (15) This is the standard WKB dispersion relation for non-radialstellar oscillations (e.g., Unno et al. 1989). For ω (cid:28) L l and ω (cid:28) N , the wave equation (12) reduces to Z (cid:48)(cid:48) + l ( l + 1)( N − ω ) r ω Z (cid:39) − χ − / l ( l + 1) N ω Ug . (16)Then, as long as | Z (cid:48)(cid:48) /Z | (cid:29) | χ (cid:48)(cid:48) /χ | (which we expect to betrue because Z (cid:48)(cid:48) ≈ − k Z and χ (cid:48)(cid:48) ∼ χ/H ), equation (16)is identical to the oscillation equations used by Zahn (1975)and Goodman & Dickson (1998). Equations (5) and (6) or equation (12) can be solved withthe appropriate boundary conditions at r = r out near thestellar surface and at r = r in → Z ( r ) = c + Z + ( r ) + c − Z − ( r ) + Z eq ( r ) , (17)where c + , c − are constants. Z + ( r ) and Z − ( r ) are two inde-pendent solutions of the homogeneous equation Z (cid:48)(cid:48) + k Z =0, and Z eq ( r ) represents a particular solution of equation(12). We choose the outer boundary r out to be in the wavezone ( k > k ( r ) varies slowly such that | k (cid:48) /k | (cid:28) k >
0, then the two independent WKB solutions to the homoge-neous equation are Z ± ( r ) = 1 √ k exp (cid:18) ± i (cid:90) rr o kdr (cid:19) , (18)where r o is an interior point ( r o < r out ). For ω (cid:28) L l , theWKB wave dispersion relation [equation (15)] reduces to ω = N k ⊥ / ( k + k ⊥ ), where k ⊥ = l ( l +1) /r , which impliesthat the radial component of the group velocity is − ωk/ ( k + k ⊥ ). Thus, with ω > k > Z − ∝ e − i (cid:82) rro kdr representsan outgoing wave, while Z + ∝ e i (cid:82) rro kdr represents an ingoingwave. An approximate particular solution of equation (12)is Z eq ( r ) (cid:39) Vk − k (cid:18) Vk (cid:19) (cid:48)(cid:48) , (19)where the second term is smaller than the first by a factor of( kH ) or ( kr ) . This represents the “non-wave” equilibriumsolution. Throughout this paper, we adopt the radiative condi-tion at the outer boundary ( r = r out ), i.e., we require thatonly an outgoing wave exists: Z ( r ) (cid:39) Z eq ( r ) + c − √ k exp (cid:18) − i (cid:90) rr o kdr (cid:19) . (20)This implicitly assumes that waves propagating toward theWD surface are completely damped by radiative diffusion(Zahn 1975) or by non-linear processes. We will check this Note that the equilibrium tide usually refers to the f-mode re-sponse of the star to the tidal force. Here, we use the term “equi-librium” to refer to the “non-wave” solution.c (cid:13) , 000–000
J. Fuller and D. Lai assumption a posteriori from our numerical results (see Sec-tion 6.5). Thus, near the outer boundary, the radial displace-ment ξ r behaves as (for ω (cid:28) L l ) ξ r ( r ) = χ / r Z ( r )= ξ eq r ( r ) + c − (cid:112) ρr ( N − ω ) exp (cid:18) − i (cid:90) rr o kdr (cid:19) . (21)Here ξ eq r represents the equilibrium tide ξ eq r (cid:39) (cid:18) − Ug (cid:19) N N − ω (cid:20) − grl ( l + 1) a s ω N (cid:21) , (22)where we have retained only the first term of equation (19).For ω (cid:28) N , this further simplifies to ξ eq r (cid:39) − U/g (Zahn1975). The constant c − specifies the amplitude of the out-going wave which is eventually dissipated in the stellar en-velope; this is the constant we wish to determine from nu-merical calculations.In practice, to implement equation (21) at the outerboundary, we require a very accurate calculation of the non-wave solution ξ eq r . This can become problematic when theconditions | k | (cid:29) /H and ω (cid:28) L l are not well satisfied.Since the transverse displacement ξ ⊥ for gravity waves ismuch larger than the radial displacement in the wave zone,it is more convenient to use ξ ⊥ in our outer boundary con-dition. We define Z ( r ) = (cid:18) ρD (cid:19) / r ω ξ ⊥ ( r ) , (23)with D ≡ N − ω . Equations (5) and (6) can be rearrangedto yield Z (cid:48)(cid:48) + k ( r ) Z = V ( r ) , (24)where k ( r ) = − (cid:20)(cid:18) ln ρr D (cid:19) (cid:48) (cid:21) − (cid:18) ln ρr D (cid:19) (cid:48)(cid:48) − (cid:18) N g (cid:19) (cid:48) − N g (cid:18) ln r D (cid:19) (cid:48) + ω a s + l ( l + 1) Dr ω (25)and V ( r ) = − (cid:18) ρr D (cid:19) / × (cid:40) N g U (cid:20) ln (cid:18) r N Dg U (cid:19)(cid:21) (cid:48) − ω a s U (cid:41) . (26)For k (cid:29) /H and ω (cid:28) L l , the functions k ( r ) and V ( r )simplify to k ( r ) (cid:39) l ( l + 1)( N − ω ) ω r (27)and V ( r ) (cid:39) − (cid:18) ρr D (cid:19) / Dr (cid:18) Ur g N D (cid:19) (cid:48) . (28)Again, adopting the radiative boundary condition at r = r out , we have Z ( r ) (cid:39) V k − k (cid:18) V k (cid:19) (cid:48)(cid:48) + c − k exp (cid:18) − i (cid:90) rr o k dr (cid:19) , (29) where c − is a constant. Thus, the transverse displacement ξ ⊥ ( r ) behaves as ξ ⊥ ( r ) = (cid:18) Dρ (cid:19) / r ω Z ( r )= ξ eq ⊥ + c − (cid:18) k ρr (cid:19) / exp (cid:18) − i (cid:90) rr o k dr (cid:19) , (30)where several constants have been absorbed into c − . Theequilibrium tidal transverse displacement ξ eq ⊥ ( r ) is given by ξ eq ⊥ ( r ) (cid:39) − l ( l + 1) r (cid:18) Ur g N D (cid:19) (cid:48) . (31)For ω (cid:28) N , this reduces to (for l = 2) ξ eq ⊥ ( r ) (cid:39) − r (cid:18) Ur g (cid:19) (cid:48) , (32)in agreement with Goldreich & Nicholson (1989). Thus, weimplement the radiative boundary condition at r = r out as (cid:18) ξ ⊥ − ξ eq ⊥ (cid:19) (cid:48) = (cid:34) − (cid:16) ρr /k (cid:17) (cid:48) (cid:16) ρr /k (cid:17) − ik (cid:35)(cid:0) ξ ⊥ − ξ eq ⊥ (cid:1) , (33)with ξ ⊥ computed from ξ r and δP using equation (9).The inner boundary condition can be found by requiringthe radial displacement to be finite at the center of the star.This requires ξ r = lω r (cid:18) δPρ + U (cid:19) (cid:0) Near r = 0 (cid:1) . (34) As the wave propagates through the star, it carries an angu-lar momentum flux to the outer layers. At any radius withinthe star, the z component of the time-averaged angular mo-mentum flux is˙ J z ( r ) = (cid:28) (cid:73) d Ω r ρ (cid:0) δv r + δv ∗ r (cid:1)(cid:0) δv φ + δv ∗ φ (cid:1) r sin θ (cid:29) , (35)where (cid:10) ... (cid:11) implies time averaging. With δv r = − iωξ r ( r ) Y lm e − iωt (36)and δv φ = − iωξ ⊥ ( r ) r ∇ φ Y lm e − iωt = mωξ ⊥ ( r )sin θ Y lm e − iωt , (37)we find˙ J z ( r ) = 2 (cid:73) d Ω r ρω Re (cid:104) iξ ∗ r ( r ) Y ∗ lm mξ ⊥ ( r ) Y lm (cid:105) = 2 mω ρr Re (cid:16) iξ ∗ r ξ ⊥ (cid:17) . (38)In the wave zone, the fluid displacement consists ofan equilibrium (“non-wave”) component and a dynamical(wave) component, ξ = ξ eq + ξ dyn . Since the equilibriumtide component is purely real (assuming negligible dissipa-tion of the equilibrium tide), Re (cid:16) iξ eq ∗ r ξ eq ⊥ (cid:17) = 0, and theequilibrium tide does not contribute to angular momentumtransfer. The cross terms Re (cid:16) iξ dyn ∗ r ξ eq ⊥ (cid:17) , and Re (cid:16) iξ eq ∗ r ξ dyn ⊥ (cid:17) are opposed by a nearly equal and opposite Reynold’s stress c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries term (see Goldreich & Nicholson 1989) and do not con-tribute significantly to angular momentum transfer. Thus,the Re (cid:16) iξ dyn ∗ r ξ dyn ⊥ (cid:17) term dominates angular momentumtransfer. Equation (38) then becomes˙ J z ( r ) =2 mω ρr Re (cid:104) iξ dyn ∗ r ξ dyn ⊥ (cid:105) . (39)In the outer layers of the WD where ξ dyn is a pureoutgoing wave ( ∝ e − ikr ), equation (5) can be rearranged toobtain the relationship between ξ dyn ⊥ and ξ dyn r in the WKBapproximation ( k (cid:29) /H ) with ω (cid:28) L l : ξ dyn ⊥ (cid:39) − i krl ( l + 1) ξ dyn r . (40)Then the angular momentum flux is˙ J z (cid:39) ml ( l + 1) ω ρr k | ξ dyn ⊥ | (cid:39) (cid:112) l ( l + 1) ω ρr N | ξ dyn ⊥ | . (41)where we have used the dispersion relation (equation 27)with ω (cid:28) N and set m = 2. This expression agrees withthat found in Goldreich & Nicholson (1989). From the scal-ing of ξ dyn ⊥ provided in equation (33), we see that the angularmomentum flux is constant (independent of radius) in theouter layers of the star. Since the wave pattern frequency (inthe inertial frame) is Ω (the orbital frequency), the energyflux carried by the wave is given by ˙ E = Ω ˙ J z .Once we have solved our differential equations (5 and6) with the appropriate boundary conditions, we can useequation (39) to determine where angular momentum andenergy are added to the wave, i.e., where the wave is gener-ated. In the WD interior, the waves travel both inwards andoutwards and thus carry no net angular momentum, so thevalue of ˙ J z oscillates around zero. However, near the outerboundary, the value of ˙ J z is constant and positive becausethere only exists an outgoing wave. The region where thevalue of ˙ J z rises to its constant value is the region of waveexcitation, because it is in this region where energy and an-gular momentum are added to the waves (see Section 6.3).The energy and angular momentum carried by the out-going wave is deposited in the outer envelope of the star.Thus, the constant values of ˙ J z and ˙ E near the outer bound-ary represent the net angular momentum and energy trans-fer rates from the orbit to the WD. Since ξ dyn ⊥ ∝ M (cid:48) /a , theangular momentum and energy transfer rates can be writtenin the form ˙ J z = T F ( ω ) , ˙ E = T Ω F ( ω ) , (42)where T ≡ G (cid:18) M (cid:48) a (cid:19) R , (43)and F is a dimensionless function of the tidal frequency ω and the internal structure of the star. For WDs with rotationrate Ω s , the tidal frequency is ω = 2(Ω − Ω s ). It can be shown that in the WKB limit the Reynold’s stressassociated with the dynamical response is negligible.
Figure 1 depicts three WD models provided by G. Fontaine(see Brassard et al. 1991). These WD models are taken froman evolutionary sequence of a M = 0 . M (cid:12) WD, at effectivetemperatures of T = 10800K, T = 6000K, and T = 3300K.The WD has a radius R (cid:39) . × cm and a carbon-oxygen core surrounded by a 10 − M helium layer, whichin turn is surrounded by a 10 − M layer of hydrogen. Themodels shown have been slightly altered in order to ensurethermodynamic consistency (see Section 6.1).The Br¨unt-V¨ais¨al¨a frequency can be expressed as N = g ρP χ T χ P (cid:0) ∇ ad − ∇ + B (cid:1) , (44)where the symbols have their usual thermodynamic defi-nitions, and the Ledoux term B accounts for compositiongradients (see Brassard et al. 1991). In the core of the WD,the value of N is very small due to the high degeneracypressure, which causes χ T in equation (44) to be small. Thesharp spikes in N are created by the carbon-helium andhelium-hydrogen transitions, and are characteristic featuresof WD models. These sharp features in realistic WDs makeit difficult to construct toy WD models or to understandhow gravity waves propagate through the WD. From Figure1, it is evident that cooler WDs have smaller values of N throughout their interiors. However, the spikes in N havelittle dependence on WD temperature because they are pro-duced by composition gradients rather than thermal gradi-ents, thus these features are unlikely to be strongly affectedby tidal heating. To calculate the amplitude of the gravity waves excited ina WD by its companion, we integrate the inhomogeneousequations (5) and (6) with the appropriate boundary condi-tions given by equations (33) and (34). We use the relaxationmethod discussed by Press et al. (2007). The integration re-quires a grid of points containing stellar properties ( ρ , N , a s , g ) as a function of radius, and solves the equations on agrid of (possibly identical) relaxation points.When creating the grid of data points representing thestellar structure, one must be very careful in ensuring thatthe stellar properties are consistent with one another. Inparticular, the Br¨unt V¨ais¨al¨a frequency is given by N = − g (cid:18) ρ (cid:48) ρ + ga s (cid:19) . (45)If the value of N in our stellar grid is not exactly equalto the right hand side of the above equation as calculatedfrom the values of ρ , g , and a s , the stellar properties willnot be self-consistent. Such inconsistency may arise fromthe inaccuracy of the stellar grids, or from the interpolationof the stellar grids (even if the original grids are exactly self-consistent). We have found that even a small inconsistencycan lead to large error in the computed wave amplitude. Thereason for this can be understood by examining equation c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 1.
The square of the Br¨unt-V¨ais¨al¨a (thin solid lines) and Lamb (dashed lines) frequencies (for l = 2), in units of GM/R ,and the density (thick solid line) as a function of normalized radius in three WD models. The models are taken from an evolutionarysequence of a DA WD with M = 0 . M (cid:12) , R = 8 . × km, and effective temperatures of T = 10800K (top), T = 6000K (middle),and T = 3300K (bottom). The spikes in the Br¨unt V¨ais¨al¨a frequency are caused by the composition changes from carbon to helium, andfrom helium to hydrogen, respectively. Note the formation of a convective zone just below the carbon-helium transition zone as the WDcools. c (cid:13)000
The square of the Br¨unt-V¨ais¨al¨a (thin solid lines) and Lamb (dashed lines) frequencies (for l = 2), in units of GM/R ,and the density (thick solid line) as a function of normalized radius in three WD models. The models are taken from an evolutionarysequence of a DA WD with M = 0 . M (cid:12) , R = 8 . × km, and effective temperatures of T = 10800K (top), T = 6000K (middle),and T = 3300K (bottom). The spikes in the Br¨unt V¨ais¨al¨a frequency are caused by the composition changes from carbon to helium, andfrom helium to hydrogen, respectively. Note the formation of a convective zone just below the carbon-helium transition zone as the WDcools. c (cid:13)000 , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries ( ?? ). When tracing back to equations (5) and (6), one cansee that the N term on the right-hand side is actually thesum of two terms. That is, the value of N on the right-handside of equation (16) is calculated via equation (45) from ourgrid of ρ , a s , and g values, while the N term on the lefthand side of the equation is taken directly from our grid of N values. If these two values of N differ (by the amount δN ), then a “false”excitation term will be introduced onthe right-hand side of equation (16), given by V f (cid:39) − l ( l + 1) δN r ω Ug . (46)This false term can vary rapidly with radius depending onthe error in the stellar grid. In Section 7, we discuss howsharp changes in the excitation term can be responsible forthe excitation of the dynamical component of the tidal re-sponse. Thus, the false excitation term introduced by evensmall numerical inconsistencies can cause large errors in cal-culations of the dynamical tide.To test our methods, we calculated the tidal responsefor a simple massive star model. The results are discussedin Appendix A, and are consistent with previous studies ofgravity waves in massive stars (e.g., Zahn 1975,1977 andGoldreich & Nicholson 1989).
To understand wave excitation in WDs, we first examine atoy model constructed to mimic the structure of a WD. Ex-amining the T eff = 10800K model, we see that it contains asharp rise in N at the carbon-helium boundary, precededby a small dip in N near the top of the carbon layer. Conse-quently, we have created a toy model with a similar dip andrise in N in the outer part of the star. To create this model,we first construct a smooth density profile (identical to thatof an n = 2 polytrope, along with a smooth N profile thatmimics the dip-rise features associated with the C-He tran-sition in real WDs. Next, we compute a thermodynamicallyconsistent sound speed profile using the equation a s = (cid:18) dP/dρ − N g (cid:19) − . (47)Since the density profile is that of a polytrope, the dP/dρ term can be calculated analytically.We solve the forced oscillation equations as a functionof the tidal frequency ω . Figure 3 shows the energy flux andwave amplitude as a function of radius for a given value of ω . The small oscillations in energy flux are due to imperfectnumerical calculation of the dynamical component of thewave and do not actually contribute to energy or angularmomentum transfer. We see that waves are excited near thedip of N (before N rises to a maximum). This is similarto the location of wave excitation in real WD models (seeSection 6.3). The dip in N causes the wave to have a largerwavelength in this region, and so it couples to the companionstar’s gravitational potential best in this region of the star.Note that although N is smaller near the center of the star,no significant wave is excited there since U ∝ r is negligible.Figure 4 shows a plot of F ( ω ). For this model, F ( ω )is not a smooth, monotonic function of ω as it is for themassive star model studied in Appendix A. Instead, thereare many jagged peaks and troughs, causing the value of F ( ω ) to vary by two or three orders of magnitude over verysmall frequency ranges. These features are also present in thereal WD models, and will be discussed further in Section 7.Our numerical results indicate that the peaks of F ( ω ) can befitted by F ( ω ) ∝ ω , significantly different from the massivestar model. We now present our numerical results for tidal excitations inrealistic WD models. Using the outgoing wave outer bound-ary condition, we solved the oscillation equations (5) and (6)for the three WD models described in Section 6.2 (see Figure1). Figures 5 and 6 show plots of the outgoing energy flux asa function of radius for the model with T eff = 10800K andtidal frequencies of ω = 2Ω = 10 − and 1 . × − , in unitsof G = M = R = 1, respectively. The energy flux jumps toits final value near the carbon-helium transition zone. Onceagain, the oscillations in energy flux are due to imperfectnumerical calculation of the dynamical component of thewave and do not actually contribute to energy or angularmomentum transfer. In Figures 5 and 6, we have smoothedthe value of the energy flux to minimize the amplitude of theunphysical oscillations. Note that although Figure 6 corre-sponds to a larger tidal frequency, the outgoing energy fluxis about 100 times less than in Figure 5. Thus, as in our toyWD model (see Section 6.2), the tidal energy flux is not amonotonic function of tidal frequency as it is for early-typestars (see Section A).We have calculated the dimensionless tidal torque F ( ω ) = ˙ J z /T o [see equation (42)] as a function of ω forthe three WD models depicted in Figure 1. The results areshown in Figures 7, 8, and 9. In general, F ( ω ) exhibitsa strong and complicated dependence on ω , such that asmall change in ω leads to a very large change in F ( ω ) (seealso Figures 5-6). This dependence is largely due to “reso-nances”between the radial wavelength of the gravity wavesand the radius of the carbon core, as discussed in Section 7.We also find that the local maxima of F ( ω ) can be approx-imately fitted by the scaling F ( ω ) ∝ ω , similar to the towWD model discussed in Section 6.2. In an attempt to understand the erratic dependence of thetidal energy transfer rate ˙ E on the tidal frequency ω , herewe explore the possible relationship between ˙ E and the tidaloverlap integral. The energy transfer rate to the star due totidal interactions can be written as˙ E = − (cid:90) d x ρ ∂ ξ ( r , t ) ∂t · ∇ U (cid:63) ( r , t ) . (48)With ξ ( r , t ) = ξ ( r ) e − iωt and U ( r , t ) = U ( r ) Y e − iωt [seeequations (1)-(2)], we have˙ E = 2 ω GM (cid:48) W a Im (cid:20) (cid:90) d x ρ ξ ( r ) · ∇ ( r Y (cid:63) ) (cid:21) . (49)We decompose the tidal response ξ ( r , t ) into the superposi-tion of stellar oscillation modes (with each mode labeled by c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 2.
The square of the Br¨unt Vais¨al¨a (thin solid line) and Lamb (dashed line) frequencies (for l = 2), in units of GM/R , as afunction of normalized radius in a toy WD model. Also plotted is the stellar density (thick solid line). The stellar properties are onlyplotted out to r = 0 . R , the location where the outer boundary condition is imposed in the tidal excitation calculation. the index α ): ξ ( r , t ) = (cid:88) α a α ( t ) ξ α ( r ) , (50)where the mode eigenfunction ξ α is normalized via (cid:82) d x ρ | ξ α | = 1. Then the mode amplitude a α ( t ) satisfiesthe equation¨ a α + ω α a α + γ α ˙ a α = GM (cid:48) W Q α a e − iωt , (51)where ω α is the mode frequency, γ α is the mode (amplitude)damping rate, and Q α is the tidal overlap integral with mode α : Q α = (cid:90) d x ρ ξ (cid:63)α ( r ) · ∇ ( r Y ) . (52)The steady-state solution of equation (51) is a α ( t ) = GM (cid:48) W Q α a ( ω α − ω − iγ α ω ) e − iωt . (53)Thus the tidal energy transfer rate to mode α is˙ E α = 2 ω (cid:18) GM (cid:48) W | Q α | a (cid:19) γ α ω ( ω α − ω ) + ( γ α ω ) . (54)In paper I, we have computed ω α and Q α for adia-batic g-modes of several WD models used in this paper.The eigenfunctions of these modes satisfy the “reflective”boundary condition (i.e., the Lagrangian pressue perturba-tion ∆ P vanishes) at the WD surface. Our result showed that although the mode frequency ω α decreases as the ra-dial mode number n increases (for a given l = 2), the overlapintegral | Q α | is a non-monotonic function of n (or ω α ) due tovarious features (associated with carbon-helium and helium-hydrogen transitions) in the N profile of the WD models.On the other hand, our calculation of the tidal response ξ ( r , t ) presented in this paper adopts the radiative outerboundary condition; this implies significant wave dampingat the outer layer of the star. Because of the difference inthe outer boundary conditions, the mode frequency ω α (ascomputed using the ∆ P = 0 boundary condition) does nothave special significance. Nevertheless, we may expect thatwhen ω = ω α , the tidal energy transfer is dominated by asingle mode ( α ) and ˙ E is correlated to | Q α | .In Figures 7, 8, and 9 we show | Q α | as a function of ω α for a number of low-order g-modes. It is clear that thepeaks and troughs of F ( ω ) calculated with an outgoing waveouter boundary condition are associated with the peaks andtroughs in the value of | Q α | . Thus, the peaks in the value of F ( ω ) are not due to resonances with g-modes, but approxi-mately correspond to the tidal frequencies near the “intrin-sic frequencies” of the g-modes with large values of | Q α | .Note this correspondence between | Q α | and the local peaksof F ( ω ) is not precise (as they are calculated using differ-ent boundary conditions), as is clear from the T eff = 3300Kmodel (Figure 9). Another way to understand the erraticdependence of F ( ω ) on ω lies in the quasi-resonance cavityof the carbon core of the WD (see Section 7). c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure 3.
Dynamical tide in a toy WD model (based on the model depicted in Figure 2) driven by a companion of mass M (cid:48) = M ,with the tidal frequency ω = 2 . × − . Top: The energy flux ˙ E = Ω ˙ J z (dark solid line) as a function of radius, with ˙ J z calculated fromequation (39). All values are plotted in units of G = M = R = 1. Bottom: The real part of ξ dyn ⊥ (dark solid line) and imaginary part of ξ dyn ⊥ (dashed line) as a function of stellar radius. The value of N has been plotted (light solid green line) in both panels. In this model,the energy flux rises to its final value near the dip in N , showing that the wave is excited at this location. Our calculations in this paper adopt the outgoing waveboundary condition near the stellar surface. This implicitlyassumes that gravity waves are absorbed in the outer layer ofthe WD due to nonlinear effects and/or radiative damping.To analyze the validity of this assumption, we plot the mag-nitude of the displacement, | ξ dyn | , as a function of radius inFigure 10. We have shown the results for tidal frequenciesof ω = 2Ω = 0 .
028 and 0 . F ( ω ) shown in Figure 7) for our WD modelwith T eff = 10800K. We have also plotted the local radialwavelength k − r because we expect nonlinear wave breakingto occur when | ξ dyn | (cid:38) k − r .It is evident from Figure 10 that at relatively high tidalfrequencies, the gravity waves become nonlinear in the outerlayer of the star, justifying our outgoing wave boundary con-dition. In some cases, the waves formally reach nonlinear am-plitudes ( k − r | ξ dyn | >
1) in the helium-hydrogen transitionregion (demarcated by the dip in k − r at r (cid:39) . s = 0. If the WD has a non-negligible spin (Ω s ), a giventidal frequency ω = 2(Ω − Ω s ) would correspond to a higherorbital frequency Ω, further increasing the wave amplitudescompared to those shown in Figure 10. Furthermore, lowerfrequency waves may damp efficiently via radiative diffusionnear the stellar surface. We therefore expect our outgoingwave outer boundary condition to be a good approximationfor the frequencies considered in this paper for our warmestWD model.Our cooler WD models with T eff = 6000K and T eff =3300K do not formally reach the same nonlinear ampli-tudes as our warmest model. The cooler models have smallerBrunt-Vaisala frequencies, particularly in their outer lay-ers, as can be seen in Figure 1. Consequently, the gravity c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 4.
The dimensionless tidal torque F ( ω ) = ˙ J z /T o [see equation (42)] carried by the outgoing gravity wave as a function of thetidal frequency ω (solid line), for the toy WD model depicted in Figure 2. The frequency is in units of G = M = R = 1. The straightlight solid (green) line is calculated from equation (68) and is roughly proportional to ω . The dashed (red) line is our semi-analyticalapproximation, with α = 1 / β = 1 /
5, and δ = 0 (see Section 7). waves have smaller displacements (recall the WKB scaling ξ dyn ⊥ ∝ N / for a constant ˙ J z ) and larger wavelengths (re-call k r ∝ N ). Therefore, gravity waves are less likely todamp due to nonlinear effects in our cooler models, and ouroutgoing wave outer boundary condition may not be justi-fied at all frequencies considered. More detailed analyses ofthe nonlinear effects in dynamical tides are necessary (e.g.,Barker & Ogilvie 2010, Weinberg et al. 2011). To understand our numerical result for the tidal energytransfer rate ˙ E (Section 6.3), particularly its dependenceon the tidal frequency ω , here we consider a simple stellarmodel that, we believe, captures the essential physics of tidalexcitation of gravity waves in binary WDs. In this model,the star consists of two regions (see Figure 11): the outer re-gion with r > r a (region a) and the inner region with r < r b (region b). In each region, the stellar profiles are smooth,but N jumps from N b at r = r b to N a (with N a (cid:29) N b )at r = r a . The tidal frequency ω satisfies ω (cid:28) N b . As wewill see, although waves can propagate in both regions, thesharp jump in N makes the inner region behave like a res-onance cavity–this is ultimately responsible for the erraticdependence of F ( ω ) on the tidal frequency ω .We start from the wave equation (12) for Z ( r ) = χ − / r ξ r : Z (cid:48)(cid:48) + k ( r ) Z = V ( r ) , (55)with k ( r ) = l ( l + 1) N r ω + ∆ k ( r )= l ( l + 1) N r ω (cid:26) O (cid:20) r H ω l ( l + 1) N (cid:21)(cid:27) (56) V ( r ) = − χ − / l ( l + 1) N ω Ug (cid:20) − rH ω l ( l + 1) N (cid:21) , (57)where H = a s /g ( < ∼ r ) is the pressure scale height. The aboveexpressions are valid in both regions of the star, and we haveassumed ω (cid:28) L l and ω (cid:28) N [more general expressionsare given by equations (13) and (14)]. Since the stellar pro-files are smooth in each of the two regions, the non-wave(“equilibrium”) solution is given by Z eq ( r ) (cid:39) Vk − k (cid:18) Vk (cid:19) (cid:48)(cid:48) = Z + ∆ Z, (58)where Z = − χ − / r Ug , (59)∆ Z = Z βk H , (60) c (cid:13)000
5, and δ = 0 (see Section 7). waves have smaller displacements (recall the WKB scaling ξ dyn ⊥ ∝ N / for a constant ˙ J z ) and larger wavelengths (re-call k r ∝ N ). Therefore, gravity waves are less likely todamp due to nonlinear effects in our cooler models, and ouroutgoing wave outer boundary condition may not be justi-fied at all frequencies considered. More detailed analyses ofthe nonlinear effects in dynamical tides are necessary (e.g.,Barker & Ogilvie 2010, Weinberg et al. 2011). To understand our numerical result for the tidal energytransfer rate ˙ E (Section 6.3), particularly its dependenceon the tidal frequency ω , here we consider a simple stellarmodel that, we believe, captures the essential physics of tidalexcitation of gravity waves in binary WDs. In this model,the star consists of two regions (see Figure 11): the outer re-gion with r > r a (region a) and the inner region with r < r b (region b). In each region, the stellar profiles are smooth,but N jumps from N b at r = r b to N a (with N a (cid:29) N b )at r = r a . The tidal frequency ω satisfies ω (cid:28) N b . As wewill see, although waves can propagate in both regions, thesharp jump in N makes the inner region behave like a res-onance cavity–this is ultimately responsible for the erraticdependence of F ( ω ) on the tidal frequency ω .We start from the wave equation (12) for Z ( r ) = χ − / r ξ r : Z (cid:48)(cid:48) + k ( r ) Z = V ( r ) , (55)with k ( r ) = l ( l + 1) N r ω + ∆ k ( r )= l ( l + 1) N r ω (cid:26) O (cid:20) r H ω l ( l + 1) N (cid:21)(cid:27) (56) V ( r ) = − χ − / l ( l + 1) N ω Ug (cid:20) − rH ω l ( l + 1) N (cid:21) , (57)where H = a s /g ( < ∼ r ) is the pressure scale height. The aboveexpressions are valid in both regions of the star, and we haveassumed ω (cid:28) L l and ω (cid:28) N [more general expressionsare given by equations (13) and (14)]. Since the stellar pro-files are smooth in each of the two regions, the non-wave(“equilibrium”) solution is given by Z eq ( r ) (cid:39) Vk − k (cid:18) Vk (cid:19) (cid:48)(cid:48) = Z + ∆ Z, (58)where Z = − χ − / r Ug , (59)∆ Z = Z βk H , (60) c (cid:13)000 , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure 5.
Dynamical tide in a realistic WD model (with M = 0 . M (cid:12) , R = 8 . × km, and T eff = 10800K) driven by a companionof mass M (cid:48) = M , with the tidal frequency ω = 2Ω = 10 − . Top: The energy flux ˙ E = Ω ˙ J z (thick solid line) as a function of radius,calculated from equation (39). All values are plotted in units of G = M = R = 1. Middle: The real part of ξ dyn ⊥ (solid line) and imaginarypart of ξ dyn ⊥ (dashed line) as a function of stellar radius. Bottom: The same as the middle panel, but zoomed in on the outer layer ofthe WD. The value of N has been plotted as dashed (green) lines in each panel. The energy flux rises to near its final value around thecarbon-helium transition region, showing that the wave is excited at this location. and β is a constant (with β ∼ Note that the abovesolution for Z eq breaks down around r = r in (where ω = From equation (13), we find that ∆ k in equation (56) is givenby ∆ k = ( H − ρ ) (cid:48) / − (2 H ρ ) − + H − p [ − (ln H p ) (cid:48) + H − ρ − H − p ],where H p = H and H ρ = − ρ (cid:48) /ρ . In the isothermal region, H ρ = H , and we have ∆ k = − (2 H ρ ) − . In the region satisfying P ∝ ρ / , we have H ρ = (5 / H , and ∆ k (cid:39) / (12 H ρ ). Thus theparameter β in equation (59) ranges from | β | (cid:46) . . N ). At distances sufficiently far away from r in , we have k (cid:29) /H .The general solution to equation (12) consists of thenon-wave part Z eq and the wave part Z dyn . In region b thereexist both ingoing and outgoing waves. Thus Z ( r ) = Z eq ( r ) + A + exp (cid:18) i (cid:90) rr in k dr (cid:19) + A − exp (cid:18) − i (cid:90) rr in k dr (cid:19) (61) c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 6.
Same as Figure 5, but for the tidal frequency ω = 1 . × − . for r in < r < r b , where A + and A − are slow-varying func-tions of r . In region a, we require there be no ingoing wave.Thus for r > r a , Z ( r ) = Z eq ( r ) + A exp (cid:18) − i (cid:90) rr k dr (cid:19) (62)where r > r a is a constant, and A varies slowly with r .Note that Z eq is discontinuous between the two regions.At the inner boundary r = r in , gravity waves are per-fectly reflected. Thus we have A − = − e iδ A + , where δ isa constant phase that depends on the details of the dis-turbance around and inside r in . To determine A and A + we must match the solutions in the two regions. Althoughin reality r a is somewhat larger than r b , we shall makethe approximation r a (cid:39) r b , and label the physical quan- tities on each side with the subscript “a” or “b”. Notethat Z eq a − Z eq b (cid:39) − ∆ Z b since k a (cid:29) k b , and ( dZ eq /dr ) a − ( dZ eq /dr ) b (cid:39) − ( α/H )∆ Z b where α is a constant ( | α | ∼ Z and dZ/dr across r = r b (cid:39) r a , we obtain theexpression for the wave amplitude at r = r a : A exp (cid:18) − i (cid:90) r a r k dr (cid:19) = ∆ Z b (cid:20) − ( α/k b H ) tan ϕ i ( k a /k b ) tan ϕ (cid:21) , (63)which entails | A | = | ∆ Z b | | − ( α/k b H ) tan ϕ | [1 + ( k a /k b ) tan ϕ ] / , (64)where ϕ = (cid:90) r b r k dr − δ . (65) c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure 7.
The dimensionless tidal torque F ( ω ) = ˙ J z /T o [see equation (42)] carried by outgoing gravity waves as a function of thetidal frequency ω for our WD model with T eff = 10800K. The two boxed points correspond to ω = 10 − and 1 . × − , as depicted inFigures 5 and 6. The dashed (red) line is our semi-analytical approximation [see equation (67)], with α = 1 / β = 1 /
5, and δ = 0. Thesmooth solid line corresponds to the maximum values of F ( ω ) in our semi-analytical equation, and is calculated from equation (69). Thedot-dashed (blue) line corresponds to F ( ω ) = 20ˆ ω (see Section 8). The diamonds connected by the dotted line are the tidal overlapintegrals Q α associated with nearby gravity modes, and the n = 4 mode is the highest frequency mode shown. The frequency and Q α are plotted in units of G = M = R = 1. Clearly, | A | reaches the maximum | ∆ Z b | at ϕ = 0, and | A | (cid:39)| ∆ Z b ( α/k a H ) | at ϕ = π/ ξ dyn ⊥ (cid:39) − ikrl ( l + 1) ξ dyn r = − ikχ / l ( l + 1) r A exp (cid:18) − i (cid:90) rr k dr (cid:19) . (66)The tidal energy transfer rate ˙ E is equal to the energy fluxcarried by the wave. Using equation (41), we have˙ E = Ω ˙ J z = 4Ω k a | A | , (67)where k a = (cid:112) l ( l + 1) N a / ( r a ω ) and | A | are evaluated at r = r a . Using | A | max = | ∆ Z b | , we obtain the maximum tidalenergy transfer rate as a function of the tidal frequency ω and the orbital frequency Ω:˙ E max (cid:39) πβ ρ a r a N a N b [ l ( l + 1)] / g a (cid:18) r a H a (cid:19) (cid:18) M (cid:48) M t (cid:19) Ω ω . (68)The corresponding dimensionless tidal torque [see equation(42)] is F max ( ω ) = 6 πβ Gρ a r a N a N b [ l ( l + 1)] / g a R (cid:18) r a H a (cid:19) ω . (69) This scaling [ F ( ω ) ∝ ω ] agrees with our numerical resultsfor the toy WD models (Section 6.2) and realistic WD mod-els (Section 6.3).Realistic WD models are obviously more complicatedthan the analytical model considered in this section (seeFigure 11). To evaluate the tidal energy transfer rate ˙ E usingequation (67) [with | A | given by equation (64)] and ˙ E max using equation (68) for our WD models, we choose r b at thelocation where d ln N /dr is largest in the helium-carbontransition region. We then set the location of r a to be onehalf of a wavelength above r b , i.e., by finding the location r a such that the equation π = (cid:82) r a r b kdr is satisfied, where k isgiven by equation (27). For the three models considered inSection 6, we find that r a thus calculated typically lies nearthe peak in N associated with the carbon-helium transitionregion.In Figures 4, 7, 8, and 9, we compare the analyticalresults based on equations (67) and (68) to our numericalcalculations. We see that the erratic dependence of F ( ω )on the tidal frequency ω can be qualitatively reproduced byour analytical expression (67), and the maximum F max isalso well approximated by equation (69). Our analytical es-timate works best for the WD model with T eff = 10800K,but it does a poor job of approximating the value of F ( ω )for the WD model with T eff = 3300K. We attribute this dis- c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 8.
Same as Figure 7, except for the T eff = 6000K WD model. In this plot, the dot-dashed (blue) line corresponds to F ( ω ) = 200ˆ ω .The n = 15 mode is the highest frequency g-mode shown. agreement to the lower value of N in the cool WD modelbecause our assumption that N (cid:29) ω is not satisfied. In-stead, we find that gravity waves are excited near the spikein N associated with the helium-hydrogen transition regionin the cool WD model.For each model shown in Figures 7-9, our model alsobreaks down at the highest and lowest frequencies shown.These discrepancies are likely related to errors in our nu-merical methods. At the highest frequencies shown. the ap-proximation k r (cid:29) /H begins to break down, causing errorin our outer boundary condition. At the lowest frequenciesshown, extremely fine grid resolution is needed to resolvethe dynamical component of the tidal response, and so slightthermodynamic inconsistencies may introduce significant er-rors (see Section 6.1). The tidally-excited gravity waves and their dissipationscause energy and angular momentum transfer from the orbitto the star, leading to spin-up of the WD over time. In thissection, we study the spin-orbit evolution of WD binariesunder the combined effects of tidal dissipation and gravita-tional radiation. In general, the tidal torque on the primarystar M from the companion M (cid:48) and the tidal energy transferrate can be written as [see equations (42) and (43)] T tide = T F ( ω ) , ˙ E tide = T Ω F ( ω ) , (70) with T = G ( M (cid:48) /a ) R . In previous sections, we have com-puted F ( ω ) for various non-rotating (Ω s = 0) WD models(and other stellar models). To study the spin-orbit evolu-tion, here we assume that for spinning WDs, the function F ( ω ) is the same as in the non-rotating case. This is an ap-proximation because a finite Ω s can modify gravity wavesin the star through the Coriolis force (gravity waves becomethe so-called Hough waves) and introduce inertial waves,which may play a role in the dynamical tides. In otherwords, the function F generally depends on not only ω butalso Ω s . However, we expect that when the tidal frequency ω = 2(Ω − Ω s ) is larger than Ω s , i.e., when Ω > ∼ s /
2, theeffect of rotation on the gravity waves is small or modest.Also, we assume that the WD exhibits solid-body rotation,which would occur if different layers of the WD are stronglycoupled (e.g., due to viscous or magnetic stresses). Before proceeding, we note that in the weak frictiontheory of equilibrium tides (e.g., Darwin 1879; Goldreich &Soter 1966; Alexander 1973; Hut 1981), the tidal torque isrelated to the tidal lag angle δ lag or the tidal lag time ∆ t lag4 In a medium containing a magnetic field, we expect differen-tial rotation to be smoothed out by magnetic stresses on timescales comparable to the Alfven wave crossing time. The Alfvenwave crossing time is t A = R √ πρ/B (cid:39) B = 10 gauss and a density of ρ = 10 g/cm . Sincethe Alfven wave crossing time is always much smaller than theinspiral time for WDs, we expect solid body rotation to be a goodapproximation. c (cid:13)000
2, theeffect of rotation on the gravity waves is small or modest.Also, we assume that the WD exhibits solid-body rotation,which would occur if different layers of the WD are stronglycoupled (e.g., due to viscous or magnetic stresses). Before proceeding, we note that in the weak frictiontheory of equilibrium tides (e.g., Darwin 1879; Goldreich &Soter 1966; Alexander 1973; Hut 1981), the tidal torque isrelated to the tidal lag angle δ lag or the tidal lag time ∆ t lag4 In a medium containing a magnetic field, we expect differen-tial rotation to be smoothed out by magnetic stresses on timescales comparable to the Alfven wave crossing time. The Alfvenwave crossing time is t A = R √ πρ/B (cid:39) B = 10 gauss and a density of ρ = 10 g/cm . Sincethe Alfven wave crossing time is always much smaller than theinspiral time for WDs, we expect solid body rotation to be a goodapproximation. c (cid:13)000 , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure 9.
Same as Figure 7, except for the T eff = 3300K WD model. In this plot, the dot-dashed (blue) line corresponds to F ( ω ) =4 × ˆ ω . The n = 9 mode is the highest frequency g-mode shown. by T tide = 3 k T δ lag , with δ lag = (Ω − Ω s )∆ t lag , (71)where k is the Love number. Often, a dimensionless tidalquality factor Q tide is introduced (e.g. Goldreich & Soter1966) such that ∆ t lag = 1 / ( | ω | Q tide ) (valid only for ω (cid:54) = 0).Thus, if we use the weak-friction theory to parametrize ourdynamical tide, F ( ω ) would correspond to F ( ω ) = 3 k δ lag = 3 k (Ω − Ω s )∆ t lag = 3 k Q tide sgn(Ω − Ω s ) . (72)Obviously, the effective Q tide would depend strongly on ω asopposed to being a constant (assuming constant lag angle)or being proportional to 1 / | ω | (assuming constant lag time,appropriate for a viscous fluid).With equation (70) and the assumption in F ( ω ), theWD spin evolves according to the equation˙Ω s = T F ( ω ) I , (73)where I is the moment of inertia of the WD ( I (cid:39) . MR for our M = 0 . M (cid:12) WD models). The orbital energy E orb = − GMM (cid:48) / (2 a ) satisfies the equation˙ E orb = − ˙ E tide − ˙ E GW , (74)where ˙ E GW ( >
0) is the energy loss rate due to gravitationalradiation. The evolution equation for the orbital angular frequency Ω = ( GM t /a ) / is then˙Ω = 3 T F ( ω ) µa + 3Ω2 t GW , (75)where µ = MM (cid:48) /M t is the reduced mass of the binary, and t GW is the orbital decay time scale ( | a/ ˙ a | ) due to gravita-tional radiation: t GW = 5 c G a 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 , (76) Using our results for the function F ( ω ) obtained in previoussections, we integrate equations (73) and (75) numerically toobtain the evolution of the WD spin. Since at large a (smallΩ) the orbital decay time ∼ t GW ∝ Ω − / is is much shorterthan the time scale for spin evolution, t spin = Ω s / ˙Ω s ∝ / (Ω F ), we start our integration with Ω s (cid:28) Ω at a smallorbital frequency (an orbital period of several hours).The results for our three WD models are shown in Fig-ures 12 and 13. Note we only include the effects of tidesin the primary star ( M ), and treat the companion ( M (cid:48) ) asa point mass. All three models have the same WD masses( M = M (cid:48) = 0 . M (cid:12) ), but different temperatures. Also notethat the minimum binary separation (before mass transferor tidal disruption occurs) is a min (cid:39) . M t /M ) / R , corre- c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 10.
The magnitude of the gravity wave displacement vector | ξ dyn | (solid line) as a function of radius for a tidal frequency of ω = 0 .
028 (top) and ω = 0 . k − r (red dashed line). The wave displacement,wavelength, and frequency are in units where G = M = R = 1. sponding to the minimum orbital period P min (cid:39) (1 . M − / R / , (77)where M ≡ M/ (1 M (cid:12) ) and R = R/ (10 km). We seethat for all models, appreciable spin-orbit synchronization isachieved before mass transfer or tidal disruption. However,depending in the WD temperature, the rates of spin-orbitsynchronization are different.The basic feature of the synchronization process can beobtained using an approximate expression for the dimension-less function F ( ω ). We fit the local maxima of our numericalresults depicted in Figures 7-9 by the function F ( ω ) = fω = ˆ f ˆ ω , (78) where ˆ ω = ω/ ( GM/R ) / , and ˆ f (cid:39) , , × forthe T eff =10800K, 6000K, and 3300K models, respectively.Suppose Ω s (cid:28) Ω at large orbital separation. We can de-fine the critical orbital frequency , Ω c , at which spinup orsynchronization becomes efficient, by equating ˙Ω and ˙Ω s (with Ω s (cid:28) Ω). Note that since the orbital decay rate dueto tidal energy transfer [the first term in equation (75)]is much smaller than the spinup rate ˙Ω s , the orbital de-cay is always dominated by the gravitational radiation, i.e.,˙Ω (cid:39) / (2 t GW ). With T o = ¯ T o Ω and t GW = ¯ t GW Ω − / , c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries r b r a r N Region b Region a N b2 N a2 Figure 11.
A diagram showing a simplified model of a white dwarf used in our analytical estimate for gravity wave excitation. The arrowsindicate that region b contains both an inward and outward propagating wave, while region a contains only an outward propagatingwave. we findΩ c (cid:39) (cid:18) I f ¯ T o ¯ t GW (cid:19) / = (cid:34) κ f M / t M (cid:48) M / (cid:18) GMRc (cid:19) / (cid:35) / (cid:18) GMR (cid:19) / = (3 . × − s − ) (cid:32) κ . M / t M / ˆ fM (cid:48) R / (cid:33) / , (79)where κ = 0 . κ . = I/ ( MR ), M (cid:48) = M (cid:48) / (1 M (cid:12) ), and M t = M t / (1 M (cid:12) ). For Ω < ∼ Ω c , tidal synchronization is in-efficient. For Ω > ∼ Ω c , the spinup rate ˙Ω s becomes larger than˙Ω and the system will become increasingly synchronized. Infact, when Ω > ∼ Ω c , an approximate analytic expression forthe spin evolution can be obtained by assuming a posteriori ( ˙Ω s − ˙Ω) (cid:28) ˙Ω. With ˙Ω (cid:39) / (2 t GW ) (cid:39) ˙Ω s , we findΩ s (cid:39) Ω − Ω / c Ω − / (cid:0) for Ω > ∼ Ω c (cid:1) . (80)This expression provides an accurate representation of thenumerical solutions.Note that we can derive a similar equation as (80) formore general tidal torques. For example, assume˙Ω s = A Ω (Ω − Ω s ) n , (81)where n and A are constants. With ˙Ω = B Ω / (where B is a constant) and assuming ˙Ω s (cid:39) ˙Ω, we findΩ s (cid:39) Ω − Ω c (cid:18) Ω c Ω (cid:19) / (3 n ) , (82)for Ω > ∼ Ω c , where Ω c = (cid:18) BA (cid:19) / (3 n +1) . (83)Note that our equation (80) corresponds to n = 5, whichimplies Ω − Ω s (cid:39) Ω c for Ω > ∼ Ω c . By contrast, in the equilib-rium tide model (with constant lag time), n = 1, so (Ω − Ω s )changes moderately as the orbit decays. Figure 14 shows the tidal energy transfer rate (from theorbit to the WD) ˙ E tide = T Ω F ( ω ). For Ω < ∼ Ω c , ω (cid:39) s (cid:28) Ω), we see that ˙ E tide depends on Ω in arather erratic manner. However, when Ω > ∼ Ω c , efficient tidalsynchronization ensures ˙Ω (cid:39) ˙Ω s , or 3Ω / t GW (cid:39) T F ( ω ) /I ,and thus ˙ E tide simplifies to˙ E tide (cid:39) I Ω t GW (cid:0) for Ω > ∼ Ω c (cid:1) . (84)Since ˙ E tide / ˙ E GW (cid:39) I/ ( µa ) (cid:28)
1, the orbital decay is dom-inated by gravitational radiation. Nevertheless, the orbital c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 12.
Evolution of the spin frequency Ω s in units of the orbital frequency Ω as a function of the orbital period. The solid black,red, and blue lines correspond to our WD models with T eff = 10800K, T = 6000K, and T = 3300K, respectively. The black, red, andblue dashed lines correspond to evolutions using F = 20ˆ ω , F = 200ˆ ω , F = 4 × ˆ ω , respectively (these functions F ( ω ) approximatethe like-colored WD models, see Figures 7-9). The vertical dotted line denotes the critical orbital period, 2 π/ Ω c [see equation (79)],corresponding to the black dashed line. In these evolutions, M (cid:48) = M and the WDs initially have Ω s = 0. phase evolution is affected by the tidal energy transfer, andsuch a phase shift can be measurable for short period bina-ries such as the recently discovered 12 minute system SDSSJ0651 (Brown et al. 2011; see Section 9). Also, low-frequency(10 − − − Hz) gravitational waveforms emitted by the bi-nary, detectable by LISA, will deviate significantly from thepoint-mass binary prediction. This is in contrast to the caseof neutron star binaries (NS/NS or NS/BH) studied pre-viously (Reisenegger & Goldreich 1994; Lai 1994; Shibata1994; Ho & Lai 1999; Lai & Wu 2006; Flanagan & Racine2007), where the resonant mode amplitude is normally toosmall to affect the gravitational waveforms to be detectedby ground-based gravitational wave detectors such as LIGOand VIRGO, tidal effects only become important near theNS binary merger (e.g., Lai et al. 1994; Hinderer et al. 2010).The orbital cycle of a WD binary evolves according to dN orb = Ω2 π dE orb ˙ E orb . (85)Including tidal effects in ˙ E orb , we find dN orb d ln Ω = (cid:18) dN orb d ln Ω (cid:19) (cid:18) E tide ˙ E GW (cid:19) − , (86)where (cid:18) dN orb d ln Ω (cid:19) = Ω t GW π = 5 c πG / µM / t ( πf GW ) / = 2 . × (cid:18) M (cid:12) MM (cid:48) (cid:19)(cid:18) M t M (cid:12) (cid:19) / (cid:18) f GW .
01 Hz (cid:19) − / (87) is the usual result when the tidal effect is neglected ( f GW =Ω /π is the gravitational wave frequency). Thus, even though˙ E tide / ˙ E GW (cid:28)
1, the number of “missing cycles” due to thetidal effect, (cid:18) d ∆ N orb d ln Ω (cid:19) tide (cid:39) − (cid:18) dN orb d ln Ω (cid:19) ˙ E tide ˙ E GW , (88)can be significant. Since E tide ∝ I , proper modelling anddetection of the missing cycles would provide a measurementof the moment of inertia of the WD. The tidal energy transfer ˙ E tide does not correspond to theenergy dissipated as heat in the WD, because some of theenergy must be used to spin up the WD. Assuming rigid-body rotation, the tidal heating rate is˙ E heat = ˙ E tide (cid:18) − Ω s Ω (cid:19) . (89)Figure 14 shows ˙ E heat for our three binary WD models. Atlarge binary seperations (Ω < ∼ Ω c ) when Ω s (cid:28) Ω, virtu-ally all of the tidal energy transfer to the WD is dissipatedas heat. At smaller serparations, we have shown that theWD will retain a small degree of asynchronization. Insert- c (cid:13)000
1, the number of “missing cycles” due to thetidal effect, (cid:18) d ∆ N orb d ln Ω (cid:19) tide (cid:39) − (cid:18) dN orb d ln Ω (cid:19) ˙ E tide ˙ E GW , (88)can be significant. Since E tide ∝ I , proper modelling anddetection of the missing cycles would provide a measurementof the moment of inertia of the WD. The tidal energy transfer ˙ E tide does not correspond to theenergy dissipated as heat in the WD, because some of theenergy must be used to spin up the WD. Assuming rigid-body rotation, the tidal heating rate is˙ E heat = ˙ E tide (cid:18) − Ω s Ω (cid:19) . (89)Figure 14 shows ˙ E heat for our three binary WD models. Atlarge binary seperations (Ω < ∼ Ω c ) when Ω s (cid:28) Ω, virtu-ally all of the tidal energy transfer to the WD is dissipatedas heat. At smaller serparations, we have shown that theWD will retain a small degree of asynchronization. Insert- c (cid:13)000 , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure 13.
The spin frequency Ω s / (2 π ) in units of Hz as a function of orbital period. The solid black, red, and blue lines correspondto our WD models with T eff = 10800K, T = 6000K, and T = 3300K, respectively. The dashed line shows the orbital frequency, Ω / (2 π ).The dotted vertical black, red, and blue lines are the values of 2 π/ Ω c for F = 20ˆ ω , F = 200ˆ ω , F = 4 × ˆ ω , respectively. In theseevolutions, M (cid:48) = M and the WDs initially have Ω s = Ω / ing equation (80) into equation (89), we find˙ E heat (cid:39) ˙ E tide (cid:18) Ω c Ω (cid:19) / (cid:39) I Ω t GW (cid:18) Ω c Ω (cid:19) / (for Ω > ∼ Ω c ) . (90)Thus, as the orbital frequency increases, a smaller fractionof the tidal energy is dissipated as heat. Using equation (79)for Ω c , we have˙ E heat (cid:39) (6 . × erg s − ) κ / . ˆ f − / M / × ( M (cid:48) ) / R − / (cid:18) Ω0 . − (cid:19) / . (91)Note that ˙ E heat is relatively insensitive to ˆ f , so its pre-cise value is not important. Thus, tidal heating of the WDcan become significant well before merger. For example, forour T eff = 10800K WD model (with M = M (cid:48) = 0 . M (cid:12) , R = 8970 km and ˆ f ∼ E heat ∼ . × erg/s atthe orbital period P = 10 min, much larger than the “intrin-sic” luminosity of the WD, 4 πR σ SB T = 3 . × erg/s.Note that ˙ E heat is mainly deposited in the WD envelope, soan appreciable fraction of ˙ E tide may be radiated, and theWD can become very bright prior to merger. The 12 minutebinary SDSS J0651 (Brown et al. 2011) may be an exampleof such tidally heated WDs (see Section 9). We have studied the tidal excitation of gravity waves in bi-nary white dwarfs (WDs) and computed the energy and an-gular momentum transfer rates as a function of the orbitalfrequency for several WD models. Such dynamical tides playthe dominant role in spinning up the WD as the binarydecays due to gravitational radiation. Our calculations arebased on the outgoing wave boundary condition, which im-plicitly assumes that the tidally excited gravity waves aredamped by nonlinear effects or radiative diffusion as theypropagate towards the WD surface. Unlike dynamical tidesin early-type main-sequence stars, where gravity waves areexcited at the boundary between the convective core andradiative envelope, the excitation of gravity waves in WDsis more complicated due to the various sharp features as-sociated with composition changes in the WD model. Wefind that the tidal energy transfer rate (from the orbit tothe WD) ˙ E tide is a complex function of the tidal frequency ω = 2(Ω − Ω s ) (where Ω and Ω s are the orbital frequencyand spin frequency, respectively; see Figures 7-9), and the lo-cal maxima of ˙ E tide scale approximately as Ω ω . For mosttidal frequencies considered, the gravity waves are excitednear the boundary between the carbon-oxygen core andthe helium layer (with the associated dip and sharp rise inthe Brunt-V¨ais¨al¨a frequency profile). We have constructed asemi-analytic model that captures the basic physics of grav-ity wave excitation and reveals that the complex behavior of˙ E tide as a function of the tidal frequency arises from the par- c (cid:13) , 000–000 J. Fuller and D. Lai
Figure 14.
The tidal energy dissipation rate ˙ E tide (solid lines) and the tidal heating rate ˙ E heat (dashed lines) as a function of orbitalperiod. The black, red, and blue lines correspond to our WD models with T eff = 10800K, T = 6000K, and T = 3300K, respectively.Note that at small orbital periods, the ˙ E tide curves overlap for different WD models. The dotted line is the energy dissipation rate dueto gravitational waves, ˙ E GW . In these evolutions, M (cid:48) = M and the WDs initially have Ω s = 0. tial trapping of gravity waves in the quasi-resonance cavityprovided by the carbon-oxygen core.We have also calculated the spin and orbital evolutionof the WD binary system including the effects of both gravi-tational radiation and tidal dissipation. We find that above acritical orbital frequency Ω c [see equation (79)], correspond-ing to an orbital period of about an hour for our WD mod-els, the dynamical tide BEGINS to drive the WD spin Ω s towards synchronous rotation, although a small degree ofasynchronization is maintained even at small orbital periods:Ω − Ω s (cid:39) Ω c (Ω c / Ω) / [see equation (80)]. Thus, numericalsimulations of WD binary mergers should use synchronizedconfigurations as their initial condition – these may affectthe property of the merger product and possible supernovasignatures.We also show that, although gravitational radiation al-ways dominates over tides in the decay of the binary orbit,tidal effects can nevertheless affect the orbital decay andintroduce significant phase error to the low-frequency gravi-tational waveforms. Future detection of gravitational wavesfrom WD binaries by LISA may need to take these tidaleffects into account and may lead to measurements of theWDs’ moments of inertia. Finally, we have calculated thetidal heating rate of the WD as a function of the orbital pe-riod. For Ω > ∼ Ω c , since the tidal dissipation rate is largelycontrolled by the orbital decay rate due to gravitational ra-diation, it is a smooth function of orbital period [see equa-tion (90)]. We show that well before mass transfer or binarymerger occurs, tidal dissipation in the WD envelope can bemuch larger than the intrinsic luminosity of the star. Thus, the WD envelope may be heated up significantly, leading tobrightening of the WD binary well before merger. We planto study this issue in detail in a future paper.The recently discovered 12 minute WD binary SDSSJ0651 (Brown et al. 2011) can provide useful constraints forour theory. Applying equation (79) to this system, we findthat the orbital period (12.75 minutes) is sufficiently shortthat both WDs are nearly (but not completely) synchronizedwith the orbit. Because of the orbital decay ˙ P , the eclipsetiming changes according to the relation∆ t = ˙ P t / (2 P ) , (92)where t is the observing time. Gravitational radiation givesrise to ∆ t GW = 5 .
6s ( t/ . Using equation (84) to evalu-ate the orbital decay rate ˙ P tide due to tidal energy transfer,we find ∆ t tide (cid:39) .
28s ( t/ (see also Benacquista 2011).Thus, the orbital decay due to tidal effects should be mea-sured in the near future. Also, our calculated heating rate,equation (91), indicates the SDSS J0651 WDs have sufferedsignificant tidal heating, although to predict the luminos-ity change due to tidal heating requires careful study of thethermal structure of the WDs and knowledge of the locationof tidal heating. We note that Piro (2011) also consideredsome aspects of tidal effects in SDSS J0651, but his resultswere based on parameterized equilibrium tide theory.This paper, together with paper I, represents only thefirst study of the physics of dynamical tides in compact WDbinaries, and more works are needed. We have adopted sev-eral approximations that may limit the applicability of ourresults. First, we have not included the effects of rotation c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries (e.g., the Coriolis force) in our wave equations. In additionto modifying the properties of gravity waves (they becomegeneralized Hough waves), rotation also introduces inertialwaves that can be excited once the WD spin frequency be-comes comparable to the tidal frequency – this may lead tomore efficient tidal energy transfer and synchronization. Forexample, if we parameterize the spinup rate due to variousmechanisms (including inertial waves) by equation (87), thecritical orbital frequency for the onset of synchronization(Ω c ) is given by equation (83). For a stronger tidal torque(larger A ), Ω c is smaller. However, the tidal heating rate atΩ > ∼ Ω c becomes [cf. equation (90)]˙ E heat (cid:39) I Ω t GW (cid:18) Ω c Ω (cid:19) (3 n +1) / (3 n ) . (93)Thus, for stronger tidal torques, at a given orbital frequency(Ω > ∼ Ω c ), the tidal heating rate is reduced because the WDis closer to synchronization.Second, we have assumed that the WD rotates as a rigidbody. As the tidally-excited gravity waves deposit angularmomentum in the outer layer of the WD, differential rota-tion will develop if the different regions of the WD are notwell coupled. Thus it may be that the outer layer becomessynchronized with the companion while the core rotates ata sub-synchronous rate, analogous to tidal synchronizationin early-type main-sequence stars (Goldreich & Nicholson1989). Third, we have implicitly assumed that the outgoinggravity waves are efficiently damped near the WD surface.This may not apply for all WD models or all orbital frequen-cies. If partial wave reflection occurs, tidal dissipation willbe reduced compared to the results presented in this paperexcept when the tidal frequency matches the intrinsic fre-quency of a g-mode (cf. Paper I). More detailed studies onnonlinear wave damping (e.g., Barker & Ogilvie 2010; Wein-berg et al. 2011) and radiative damping would be desirable.Finally, we have only studied carbon-oxygen WDs inthis paper. Our calculations have shown that the strengthof dynamical tides depends sensitively on the detailed in-ternal structure of the WD. Recent observations (see ref-erences in Section 1) have revealed many compact WD bi-naries that contain at least one low-mass helium-core WD.The temperatures of these helium-core WDs tend to be high( T eff > ∼ K). These observations warrant investigation oftidal effects in hot, helium-core WDs, which have signifi-cantly different internal structures from the cool, carbon-oxygen WDs considered in this paper.
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 Bildstenand Gordon Ogilvie for useful discussions, and acknowl-edges the hospitality (Spring 2010) of the Kavli Institute forTheoretical Physics at UCSB (funded by the NSF throughGrant PHY05-51164) where part of the work was carriedout. This work has been supported in part by NSF grantAST-1008245.
REFERENCES
Alexander, M.E. 1973, Astrophys. Space Sci., 23, 459Barker, A., Ogilvie, G. 2010, MNRAS, 404, 1849Barker, A., Ogilvie, G. 2011, arXiv:1102.0861Benacquista, M. 2011, ApJ, 740, L54Brassard, P., Fontaine, G., Wesemael, F., Kawaler, S.D., TassoulM. 1991, ApJ, 367, 601Brown, R.B., Kilic, M., Hermes, J.J., Allende Prieto, C., Kenyon,S.J., Winget, D.E. 2011, arXiv:1107.2389v1Darwin, G.H. 1879, Phil. Trans. Toy. Soc., 170, 1Di Stefano, R. 2010, ApJ, 719, 474Flanagan, E., Racine, E. 2007, Phys. Rev. D75, 044001Fuller, J., Lai, D. 2011, MNRAS, 412, 1331Gilfanov, M., Bogdan, A., 2010, Nature, 463, 924Goldreich, P., Nicholson, P., 1989, ApJ, 342, 1079Golreich, P., & Soter, S. 1966, Icarus, 5, 375Goodman, J., Dickson, E.S., 1998, ApJ, 507, 938Hinderer, T., Lackey, B.D., Lang, R.N., Read, J.S. 2010, Phys.Rev. D81, 123016Ho, W.C.G., Lai, D. 1999, MNRAS, 308, 153Hut, P. 1981, A&A, 99, 126Iben, I., Tutukov, A. 1984, ApJS, 54, 335Iben, I., Tutukov, A., Fedorova, A. 1998, ApJ, 503, 344Kilic, M., Brown, W.R., Kenyon, S.J., Allende Prieto, C., An-drews, J., Kleinman, S.J., Winget, K.I., Winget, D.E., Her-mes, J.J. 2011, arXiv:1103.2354Kulkarni, S.R., van Kerkwijk, M.H., 2010, ApJ, 719, 1123Lai, D. 1994, MNRAS, 270, 611Lai, D., Rasio, F.A., Shapiro, S.L. 1994, ApJ, 420, 811Lai, D., Wu, Y. 2006, Phys. Rev. D74, 024007Loren-Aguilar, P., Isern, J., Garcia-Berro, E. 2009, AA, 500, 1193Maoz, D., Sharon, K., Gal-Yam, A. 2010, ApJ, 722, 1979Marsh, T. 2011, arXiv:1101.4970Mullally, F., Badenes, C., Thompson, S.E., Lupton, R. 2009, ApJ,707, L51Nelemans, G. 2009, Class. Quantum Grv., 26, 094030Ogilvie, G.I., Lin, D.N.C. 2007, ApJ, 661, 1180Pakmor, R., Kromer, M., Ropke, F.K., Sim, S.A., Ruiter, A.J.,Hillebrandt, W. 2010, Nature, 463, 61Pakmor, R., Hachinger, S., Ropke, F.K., Hillebrandt, W., 2011,AA, 528, A117Piro, T. 2011, ApJ, 740, L53Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.1998, Numerical Recipes (Cambridge Univ. Press)Reisenegger, A., Goldreich, P. 1994, ApJ, 426, 688Segretain, L., Chabrier, G., Mochkovitch, R. 1997, ApJ, 481, 355Shibata, M. 1994, Prog. Theo. Phys., 91, 871Steinfadt, J., Kaplan, D.L., Shporer, A., Bildsten, L., Howell, S.B.2010, ApJ, 716, L146Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H. 1989,Nonradial Oscillations of Stars (University of Tokyo Press)Van Kerkwijk, M.H., Chang, P., Justham, S. 2010, ApJL, 722,157Webbink, R.F. 1984, ApJ, 277, 355Weinberg, N., Arras, P., Quataert, E., Burkart, J. 2011,arXiv:1107.0946v1Willems, B., Deloye, C.J., Kalogera, V. 2010, ApJ, 713, 239Zahn, J.P. 1975, AA, 41, 329Zahn, J.P. 1977, AA, 57, 383
APPENDIX A: CALCULATION WITHMASSIVE STAR MODEL
To test the accuracy of our numerical calculations and espe-cially the importance of self-consistency in real stellar mod-els, we compute the tidal response of several toy models. The c (cid:13) , 000–000 J. Fuller and D. Lai
Figure A1.
The square of the Br¨unt-Vais¨al¨a (thin solid line) and Lamb (dashed line) frequencies (for l = 2), in units of GM/R , as afunction of the normalized radius in a simple massive star model. Also plotted is the stellar density profile (thick solid line) in units of M/R . The model has an inner convection zone extending to r = 0 . R . The stellar properties are only plotted out to r = 0 . R , wherean outgoing wave boundary condition is adopted in our calculation of the tidal excitation. first toy model we employ is shown in Figure A1 and is meantto mimic a massive early-type star. The model contains aninner convection zone surrounded by a thick radiative enve-lope. The convection zone extends to r = 0 . R , beyondwhich the value of N rises linearly to N ≈ GM/R .Dynamical tides in such massive stars have been studiedby Zahn (1975, 1977) and Goldreich & Nicholson (1989),who showed that the dominant effect arises from the grav-ity waves launched at the core-envelope boundary, whichthen propagate outwards and eventually dissipate near thestellar surface. Zahn (1975) derived an analytic solution forthe wave amplitude and the corresponding tidal torque. Al-though our model does not contain some of the details ex-hibited by realistic massive star models, it does capture themost important features. We can compare our result withZahn’s to calibrate our numerical method and to assess thedegree of self-consistency required to produce reliable resultsfor the tidal torque.Figure A2 shows an example of our numerical resultsfor the dynamical tides generated in a massive star by acompanion, for a given tidal frequency ω = 2Ω = 2 . × − (in units where G = M = R ). We see that gravity wavesare excited at the base of the radiative zone where N be-gins to rise above zero. A net energy flux ˙ E = Ω ˙ J z =Ω( GM (cid:48) R /a ) F ( ω ) flows outwards toward the stellar sur-face. Figure A3 shows our numerical result of the dimension-less function F ( ω ) ≡ ˙ J z /T o [see equation (42)], evaluated atthe outer boundary, as a function of the tidal frequency ω . The result can be fitted by F ( ω ) ∝ ω / , in agreement withthe scaling found by Zahn (1975).The power-law scaling of the energy flux can be derivedusing the method of Goldreich & Nicholson (1989). Assume | ξ dyn ⊥ | ≈ ξ eq ⊥ at r = r c + , which is located one wavelengthabove the convective boundary ( r = r c ). From the dispersionrelation (27), we find that the Br¨unt-Vais¨al¨a frequency at r c + is given by (for l = 2) N ( r c + ) ≈ (cid:18) dN dr r c (cid:19) / ω / . (A1)Using ξ eq ⊥ (cid:39) − (cid:2) / (6 r ) (cid:3)(cid:0) Ur /g (cid:1) (cid:48) (cid:39) − U/ (2 g ), we evaluateequation (41) to find˙ E ≈ π √ (cid:18) M (cid:48) M t (cid:19) ρr Ω ω / g ( dN /d ln r ) / . (A2)where M t = M + M (cid:48) , and all the quantities ( ρ , r , g , and dN /dr are evaluated at r = r c + (cid:39) r c ). The scaling of thisestimate nearly agrees Goldreich & Nicholson (1989), whoobtained ˙ E r ∝ Ω ω / , where ˙ E r is the energy flux carriedby outgoing gravity waves in the rotating frame of the star,not the total energy transfer rate from the orbit. These twoenergy transfer rates are related by ˙ E = Ω ˙ J z = 2Ω ˙ E r /ω .Goldreich & Nicholson (1989) estimates dN /dr ≈ g/H ≈ g/r , and with g (cid:39) πG ¯ ρr/ ρ is the mean density interior c (cid:13) , 000–000 idal Synchronization and Dissipation in White Dwarf Binaries Figure A2.
Dynamical tide in a massive star (based on the toy model depicted in Figure A1) driven by a companion of mass M (cid:48) = M ,with the tidal frequency ω = 2 . × − . Top: The energy flux (dark solid line) ˙ E = Ω ˙ J z as a function of radius, with ˙ J z calculated fromequation (39). All values are plotted in units of G = M = R = 1. Bottom: The real part of ξ dyn ⊥ (dark solid line) and imaginary partof ξ dyn ⊥ (dashed line) as a function of stellar radius. The value of N has been plotted in green (light solid line) in both panels. In thismodel, the energy flux rises to its final value just outside of the convective zone, showing that the wave is excited at this location. to r c ), equation (A2) becomes˙ E ≈ . (cid:18) M (cid:48) M t (cid:19) ρr Ω ω / ( G ¯ ρ ) / . (A3)The value of F ( ω ) based on equation (A2) is plotted in Fig-ure A3. Compared to our numerical results, we see thatequation (A2) overestimates F ( ω ) by an order of magni-tude [by contrast, equation (A3) would overestimate F ( ω )by more, since for our toy stellar model dN /d ln r (cid:29) g/r ].From our numerical results, we find that the dynamical partof the tide only reaches an amplitude of ξ dyn ⊥ ≈ ξ eq ⊥ /
4. If wehad used this wave amplitude in our estimate, equation (A2)would be a factor of 16 smaller and would provide an accu-rate approximation to F ( ω ) at all frequencies considered. c (cid:13) , 000–000 J. Fuller and D. Lai
Figure A3.
The dimensionless tidal torque F ( ω ) = ˙ J z /T o [see equation (42)] carried by the outgoing gravity wave as a function ofthe tidal frequency ω (solid line). The analytical estimate from equation (A2) is also plotted (dashed line). The frequency is in unitsof G = M = R = 1. The small wiggles at high frequencies are likely due to the slight inaccuracy of our implementation of the outerboundary condition due to the neglected terms which become non-negligible at higher tidal frequencies.c (cid:13)000