TIDAL NOVAE IN COMPACT BINARY WHITE DWARFS

Compact white dwarf (WD) binaries (with orbital periods in the range from minutes to hours) are important for several areas of astrophysics. The orbits of these systems decay via the emission of gravitational waves, constituting the largest signals for the next generation of space-based gravitational wave interferometers. Systems of sufficiently short orbital period will merge within a Hubble time, the result of which may produce a variety of exotic objects, such as helium-rich sd0 stars, R CrB stars, and AM CVn binaries. Most importantly, when the total binary mass is near the Chandrasekhar limit, the merged WDs may collapse into a neutron star or explode as a Type Ia supernova (SN Ia; e.g., Webbink 1984; Iben & Tutukov 1984). Recent studies have provided support for such "double degenerate" progenitors of SNe Ia. (e.g., Gilfanov & Bogdan 2010; Di Stefano 2010; Maoz et al. 2010; Li et al. 2011; Bloom et al. 2012; Schaefer & Pagnotta 2012).

The outcome of a WD binary merger depends on the masses of the WDs and their pre-merger conditions (e.g., Segretain et al. 1997; Yoon et al. 2007; Loren-Aguilar et al. 2009; van Kerkwijk et al. 2010; Dan et al. 2012; Raskin et al. 2012). Most previous studies of pre-merger binary WDs have focused on equilibrium tides and considered tidal dissipation in a parameterized way (e.g., Mochkovitch & Livio 1989; Iben et al. 1998; Willems et al. 2010; Piro 2011). None of these studies have sought to predict the magnitude and location of tidal heating due to dynamical tides, which dominate the tidal responses of the binary WDs.

In two recent papers (Fuller & Lai 2011, 2012, hereafter Paper I and Paper II, respectively), we presented the first ab initio calculations of dynamical tides in realistic WD models. In Paper I, we considered resonant excitations of WD g-modes during binary decay and showed that the modes reach nonlinear amplitudes near the surface of the star. This implies that, rather than exciting discrete g-modes, the binary companion will excite a continuous train of gravity waves, which propagate outward and dissipate in the outer envelope of the WD. We studied such continuous tidally excited waves in Paper II. For a canonical carbon/oxygen WD (consisting of CO core with an He–H envelope), we showed that the outgoing waves are primarily launched at the CO/He transition region, and propagate toward the WD surface, where they are likely dissipated through a combination of non-linear processes and radiative damping. We computed the energy and angular momentum flux carried by the waves in order to predict the orbital and spin evolution of WDs in compact binaries. We found that such dynamical tides cause the binary WDs to be nearly synchronized prior to merger. Furthermore, the tidal heating rate can be quite large at short orbital periods (exceeding tens of solar luminosities just before merger, depending on the system parameters), potentially leading to significant observable signatures.

In this Letter, we show that tidal heating may trigger a thermonuclear runaway hydrogen fusion event in a CO WD. The observational consequence of such an event would likely be an outburst that resembles a classical nova. We call this new phenomenon a "tidal nova" (TN). Unlike all other types of novae or supernovae, a TN does not rely on mass accretion or collapse. We present a simple two-zone model for the angular momentum evolution of a differentially rotating WD, which we use to calculate the radial tidal heating profile within the WD. We then evolve the WD model including tidal heating to calculate changes in its temperature, luminosity, and internal structure. For a wide range of physically plausible parameters, we demonstrate that tidal heating induces a thermonuclear runaway event. Finally, we discuss the observational signatures of such an event, and compare our predictions to observations of short-period WD binaries.

Using the method described in Paper II, we calculate the amplitude of tidally excited gravity waves inside a WD. We consider a circular orbit with angular frequency Ω. The WD spins at an angular frequency Ω_s, and the spin is aligned with the orbit. In the corotating frame, the frequency of the dominant l = m = 2 tidal potential is ω = 2(Ω − Ω_s). For a WD of mass M and radius R (and given internal structure) with a companion of mass M', the energy and angular momentum fluxes carried by the gravity waves can be written as

$$\begin{eqnarray} &&\dot{J}_z(\Omega,\omega) = T_0(\Omega) F(\omega), \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} && \dot{E}(\Omega,\omega) = \Omega T_0 F(\omega), \end{eqnarray} \tag{ 2 }$$

where

$$\begin{equation} T_0(\Omega) = \frac{G M^{\prime 2}}{a} \bigg (\frac{R}{a}\bigg)^5, \end{equation} \tag{ 3 }$$

with $\Omega =\sqrt{GM_t/a^3}$ the orbital angular frequency (M_t = M + M' is the total mass and a is the orbital semimajor axis).

The dimensionless function F(ω) (similar to the tidal lag angle in the language of equilibrium tide theory) determines the magnitude of wave excitation and is strongly dependent on the internal structure of the WD and the tidal frequency ω. In Paper II, we have calculated F(ω) for 0.6 M_☉ CO WD models of various surface temperatures and slow rotation. We found that F(ω) is an erratic function of ω because of the "quasi-resonance cavity" formed by the CO core inside the He/H shell. However, because of the strong dependence of F(ω) on ω (the envelope of F(ω) approximately scales as ω⁵), at sufficiently short orbital periods, tidal spin-up combined with orbital decay via gravitational radiation ensure that ω ≃ const. The orbital period at which this transition occurs is P_c ≃ 40 minutes, depending on the WD masses and temperatures (see Equation (79) of Paper II). At periods P ≲ P_c, the tidal energy transfer rate is

$$\begin{equation} \dot{E} \simeq \frac{3I\Omega ^2}{2t_{\rm GW}}, \end{equation} \tag{ 4 }$$

where I is the moment of inertia of the WD, and $t_{\rm GW}=|a/\dot{a}|$ is the binary inspiral time due to gravitational radiation,

$$\begin{equation} \hspace{-6pt} t_{\rm GW}= 4.2\times 10^{5}\,{\rm yr} \bigg (\frac{M_{\odot }^2}{MM^{\prime }}\bigg)\bigg (\frac{M_t}{2\ M_{\odot }}\bigg)^{\!\!1/3}\!\! \bigg (\!\frac{P}{10\,{\rm minutes}} \bigg)^{\!\! 8/3}. \end{equation} \tag{ 5 }$$

When the outgoing gravity waves damp in the WD envelope and locally deposit their angular momentum, some of the wave energy is converted into rotational kinetic energy, while the rest is converted to heat. The heating rate is

$$\begin{equation} \dot{E}_{\rm heat} = \dot{E} \bigg (1 - \frac{\Omega _{s}}{\Omega }\bigg). \end{equation} \tag{ 6 }$$

If the WD maintains some differential rotation, Ω_s in the above equation should be the rotation rate of the layer in which the waves damp, and heat will also be generated through viscous angular momentum transport.

Our calculations indicate that the gravity waves reach nonlinear amplitudes and break in the outer layers of the WD. The location of wave breaking depends on various parameters (e.g., orbital and tidal frequencies), but is always at r ≳ 0.92 R and the exterior mass ΔM ≲ 10⁻⁴M (Paper II). Since a small fraction of the stellar mass absorbs the entire angular momentum flux, the outer layer spins up rapidly. If it spins up faster than angular momentum can be transported to the core, the outer layer will rotate synchronously with the orbit. Outgoing waves approaching the synchronized envelope will be absorbed near corotation and deposit their angular momentum, causing the synchronized envelope to move to larger depths (see Goldreich & Nicholson 1989).

We consider a simple two-zone model for the spin evolution of the WD. In this model, the envelope of the star rotates synchronously with the orbit (Ω_env = Ω), while the core rotates sub-synchronously (Ω_core < Ω). The envelope and core are coupled, with angular momentum being transferred to the core according to a parameterized coupling time, t_coup. The angular momentum of the core–envelope system evolves according to

$$\begin{eqnarray} && \frac{d}{dt}\left(I_{\rm env}\Omega _{\rm env}\right) = \dot{J}_z(\Omega,\omega _{\rm core}) - \frac{I_{\rm env}}{t_{\rm coup}} (\Omega _{\rm env} - \Omega _{\rm core}), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} && \frac{d}{dt}\left(I_{\rm core}\Omega _{\rm core}\right) = \frac{I_{\rm env}}{t_{\rm coup}} (\Omega _{\rm env} - \Omega _{\rm core}), \end{eqnarray} \tag{ 8 }$$

where I_env = I − I_core is the moment of inertia of the envelope. Here, $\dot{J}_z$ is the angular momentum flux which can be calculated from Equation (1). We have assumed that the gravity waves are excited in the core and absorbed in the envelope.¹ Consequently, the angular momentum source term $\dot{J}_z$ is only present in the envelope evolution equation, although it is dependent on the tidal frequency in the core, ω_core = 2(Ω − Ω_core). Using Ω_env = Ω, Equations (7) and (8) can be integrated to find I_env and Ω_core as a function of time or orbital period. The mass ΔM_env of the envelope corresponds to I_env ≃ (2/3)ΔM_envR².

The thickness (or ΔM_env) of the envelope is dependent on the parameter t_coup. In stably stratified stars like WDs, angular momentum can be transported by magnetic fields. In the presence of a poloidal field B connecting the core and envelope, t_coup can be estimated from the Alfvén wave crossing time

$$\begin{equation} t_A = \frac{R\sqrt{4\pi \rho }}{B} \simeq 10^2 \ {\rm yr} \bigg (\frac{10^3 {\rm G}}{B}\bigg) \end{equation} \tag{ 9 }$$

for our CO WD model. For WDs without an intrinsic magnetic field, angular momentum may be transported via the Tayler–Spruit dynamo (Spruit 2002). To estimate t_coup, we calculate the effective viscosity for angular momentum transport via the Tayler–Spruit dynamo, ν_TS, as outlined in Spruit (2002).² We find t_TS ≡ ∫^R₀(r/ν_TS)dr ≈ 10³ yr (P/45 minutes)^3/2. Thus, we expect the coupling time to lie in the range t_coup ≲ 10³ yr for the short orbital periods of interest.

Figure 1 plots the value of ΔM_env as a function of orbital period for our 0.6 M_☉ WD model with a 0.3 M_☉ companion, using values of t_coup ranging from 1 yr to 10³ yr. We begin our calculation at P_orb > 1 hr and use I_{env, 0} = 0 and Ω_{core, 0} = 0, as appropriate at long orbital periods where tidal effects are negligible. We see that for the range of t_coup considered, ΔM_env remains small (≲ 10⁻² M_☉) at all orbital periods of interest. Thus, the synchronized envelope most likely does not extend down to the C/He transition layer where the gravity waves are excited, justifying our assumption that $\dot{J}_z$ is a function of Ω_core. However, the envelope does extend to very large optical depths, suggesting that binary WDs may be observed to be synchronized at large orbital periods even if their cores are not synchronized. Note that since I_env ≪ I, the core of the star contains most of the angular momentum, and its spin evolves in the same manner as discussed in Paper II.

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In the two-zone model discussed in Section 3, the total tidal heating rate $\dot{E}_{\rm heat}$ may be calculated from Equation (6) with Ω_s = Ω_core, and the tidal heat is deposited entirely at the base of the synchronized envelope where ΔM = ΔM_env. In a real WD, the heat deposition will occur over a range of depths that depends on the details of wave breaking and viscous angular momentum transport. For simplicity, here we choose to deposit the tidal heat uniformly per unit mass in the synchronized envelope. The heating rate per unit mass, $\dot{\varepsilon }_{\rm heat}$, is then

$$\begin{eqnarray} &\dot{\varepsilon }_{\rm heat} = 0 \quad {\rm for} \quad \Delta M > \Delta M_{\rm env} \\ \nonumber &\dot{\varepsilon }_{\rm heat} = \frac{\dot{E}_{\rm heat}}{\Delta M_{\rm env}} \quad {\rm for} \quad \Delta M < \Delta M_{\rm env}. \end{eqnarray} \tag{ 10 }$$

Although the radial dependence of this heating function is unlikely to be realistic, we find that the results below are not strongly dependent on the form of $\dot{\varepsilon }_{\rm heat}$.

To understand the effect of tidal heating on the WD properties, we evolve WD models using the extra heating term calculated via Equation (10). We use the one-dimensional stellar evolution code MESA (Paxton et al. 2011) to evolve our WD models, starting from an initial orbital period of 1 hr. We present results for a 0.6 M_☉ CO WD model with a ∼10⁻⁴ M_☉ hydrogen shell and a 0.3 M_☉ companion.

Figure 2 displays the surface temperature as a function of orbital period for our tidally heated WD. For comparison, we also show the temperature of a non-tidally heated WD and the "tidal heating temperature," defined as

$$\begin{equation} T_{\rm eff,heat}= \bigg (\frac{\dot{E}_{\rm heat}}{4\pi R^2 \sigma }\bigg)^{1/4}. \end{equation} \tag{ 11 }$$

At long orbital periods (P ≳ 45 minutes), the tidal heating has little effect on the surface temperature of the WD. At shorter periods (P ≲ 30 minutes), the temperature becomes substantially larger due to tidal heating. Several of the curves end abruptly due to the ignition of a thermonuclear runaway event, at which point we terminate our evolution calculations.

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For small values of t_coup, the tidal heat is deposited at shallow depths and quickly diffuses to the surface such that the luminosity of the WD is $L \simeq L_0 + \dot{E}_{\rm heat}$, where L₀ is the luminosity of a non-tidally heated WD. However, for larger values of t_coup, most of the tidal heat is deposited deeper in the WD where it cannot quickly diffuse outward. This leads to lower surface temperatures, although the internal temperature may increase substantially.

Figure 3 shows the interior temperature profile of our WD at three different orbital periods, using t_coup = 10³ yr. At long orbital periods, the temperature profile is similar to that of a non-tidally heated WD. As the orbital period decreases, the interior heats up, with the local temperature maximum at ΔM ∼ ΔM_env. If the base of the hydrogen layer reaches a temperature of ∼10⁷ K, hydrogen burning will be ignited.

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In the depicted model, the layer just above the He/H transition (at ΔM ≈ 10⁻⁴ M_☉) is composed of largely degenerate hydrogen gas. The ignition of fusion in this layer can thus spark a thermonuclear runaway. In general, our calculations show that these tidal novae occur only in initially cool WDs (T_eff ≲ 1.2 × 10⁴ K in the absence of tidal heating). They do not occur in hotter WDs because the hydrogen is not degenerate and can burn stably. Also, TNe require that the waves deposit some of the heat near the base of the hydrogen layer, i.e., 10⁻⁵ M_☉ ≲ ΔM_env ≲ 10⁻³ M_☉. In our two-zone model, such heating occurs for coupling times 10 yr ≲ t_coup ≲ 10⁴ yr. Overall, we find that the TNe occur at orbital periods 5 minutes ≲ P_orb ≲ 20 minutes, depending on the location of heat deposition, initial temperature of the WD, and companion mass.

We have shown that under rather general conditions (see the last paragraph of Section 4), tidal dissipation in compact WD binaries can lead to nova outbursts prior to binary merger or mass transfer. While we do not attempt to predict the detailed observational signal of a TN, we speculate that it may be very similar to a classic nova. However, in contrast to classical novae in CVs, a TN would occur in a compact system with no evidence for mass transfer. Our results indicate that a TN would precede the beginning of mass transfer or merger by about t_GW/4 ∼ 10⁵–10⁶ yr (see Equation (5)), provided the conditions outlined in the previous paragraph are satisfied.

In most classical novae, the initial outburst is followed by a period of stable hydrogen burning near the Eddington luminosity, in which the hydrogen shell of the WD inflates to a radius of order R_☉. However, the ultracompact nature of the WD system involved in a TN (where a ∼ R_☉/4) may preclude such a phase because the stably burning hydrogen shell would inflate beyond the WD's Roche lobe. This shell may then accrete onto the companion star or be ejected from the system. Therefore, we expect most of the hydrogen to be burned or ejected during in a TN. In the absence of mass transfer to supply fresh hydrogen, recurrent novae would be unlikely. Thus, the occurrence rate of these TNe may be comparable to that of WD mergers involving a CO WD.

Our theory can be constrained by comparing the prediction of our tidal heating calculations to observed compact WD binaries. The 12.75 minute system SDSS JJ065133+284423 provides the best opportunity (Brown et al. 2011). This system is composed of a primary with T_eff = 16400 K and mass 0.25 M_☉, and a secondary with T_eff ≈ 9000 K and mass 0.55 M_☉. Comparison with Figure 2 indicates that the luminosity of the secondary is likely dominated by tidal heating. Our result for a CO WD with an initial temperature of 5000 K and a value of t_coup = 10³ yr is most consistent with the observed temperature of the secondary. These results indicate that a TN may occur in this system in the future.

In principle, tidal heating may change the structure of the WD enough to alter the dynamics of gravity wave propagation. However, we find that this is not the case (i.e., no interior convection zone forms), with the exception of a thermonuclear runaway event. Our simple two-zone model for the WD obviously needs improvement, and we have neglected the effects of mixing induced by the breaking gravity waves and viscous angular momentum transport. If the mixing is strong enough to smooth out the WD composition gradients, the dynamics of gravity wave excitation and tidal heat deposition may be altered. Furthermore, if the surface hydrogen mixes into the WD interior where it burns, the surface hydrogen layer will be gradually depleted and a TN will not occur. Observations of the ejecta of classical novae indicate substantial enrichment with core elements, although the mixing mechanism is not well understood (Truran 2002). These and other aspects of TNe in compact WD binaries warrant further study.

Future observations may be able to test whether TNe occur and in turn provide information about the tidal processes at work in WD binaries. The observation of a nova-like event in a system with no evidence for mass transfer would be strong evidence for the existence of TNe and for the tidal heating mechanism studied in this paper. Measurements of hydrogen surface abundances in compact WD systems could also constrain our theory. The observation of a WD with a thick hydrogen envelope in a very tight (P ≲ 5 minutes) detached binary would indicate that TNe do not usually occur. If WDs in tight binaries are observed to have little to no hydrogen on their surface, this may indicate that TNe have stripped the surface hydrogen, or that the hydrogen has been destroyed due to efficient mixing processes. Observations of compact binary WDs detected in future surveys may provide opportunities to test these theories.

We thank Bill Paxton, Lars Bildsten, and Eliot Quataert for useful discussions. J.F. acknowledges the hospitality (during fall 2011) of the Kavli Institute for Theoretical Physics at UCSB (funded by the NSF through Grant 11-Astro11F-0016) where part of the work was carried out. This work has been supported in part by NSF grant AST-1008245, NASA grants NNX12AF85G and NNX10AP19G.