TIDAL INTERACTIONS IN MERGING WHITE DWARF BINARIES

Compact white dwarf (WD) binaries are important as progenitors for a variety of interesting astrophysical systems and/or events. Although the merger is poorly understood, it may result in a helium-rich sdB star (Saio & Jeffery 2000, 2002; Heber 2009), R CrB star (Webbink 1984), survive as mass-transferring AM CVn binaries (Postnov & Yungelson 2006) which may produce a.Ia supernova (Bildsten et al. 2007), or even result in a class of Type Ia supernovae (Iben & Tutukov 1984). Prior to Roche-lobe overflow, these are among the strongest gravitational wave sources in our Galaxy (Nelemans 2009). Tidal interactions are important at these short distances, which may alter the inspiral and determine the fate upon merger. Unfortunately, most tight binaries are also accreting (as studied in Marsh et al. 2004; Deloye et al. 2005), which complicates our ability to isolate the effects of tides.

The detached WD binary SDSS J065133.33+284423.3 (hereafter J0651; Brown et al. 2011) has an orbital period P = 765 s, the smallest of any such binary yet discovered. Furthermore, the system's light curve shows ellipsoidal variations, Doppler boosting, and primary and secondary eclipses, which allow the properties of this system to be tightly constrained. Since there is no accretion, J0651 is ideal for studying the role of tides.

In the following study I explore the effects of tidal interactions in J0651, focusing on tidal heating and eclipse measurements. In Section 2, I summarize the tidal and orbital evolution equations and estimate the asynchronicity and tidal heating expected. In Section 3, I follow the orbital evolution with time. I summarize how tides cause deviations in the period derivative and eclipse timing from what is expected if the evolution was driven merely by gravitational waves. This is in stark contrast to the Hulse–Taylor pulsar, which has a period derivative exactly equal to that predicted from Einstein's theory (Hulse & Taylor 1975; Weisberg et al. 2010). In Section 4, I summarize the results and discuss future work that can be done on this problem.

I begin by summarizing the equations used to study the time evolution of the binary. Following the convention in Brown et al. (2011), I refer to the less massive He WD as the primary, with M₁ = 0.25 M_☉ and R₁ = 0.035 R_☉, and the more massive C/O WD as the secondary, with M₂ = 0.55 M_☉ and R₂ = 0.013 R_☉.

The tidal force raises a tide on the primary, H₁, inducing a quadrupole moment

$$\begin{eqnarray} &&U_{\rm grav, 1}\sim \left(\frac{H_1}{R_1}\right)M_1R_1^2\sim M_2R_1^2\left(\frac{R_1}{a}\right)^3, \end{eqnarray} \tag{ 1 }$$

where a is the orbital separation.¹ The component of the quadrupole that contributes to secular changes in the orbit is (Goldreich & Soter 1966)

$$\begin{eqnarray} &&U_{\rm grav, 1}^{\rm sec} \sim U_{\rm grav, 1}\frac{\sigma _1}{n}\sin \epsilon _1 \sim \frac{M_2 R_1^2}{Q_1} \frac{\sigma _1}{n} \left(\frac{R_1}{a}\right)^3, \end{eqnarray} \tag{ 2 }$$

where ₁ ∼ 1/Q₁ is the angle of lag due to internal friction, σ₁ = 2(n − Ω₁) is the tidal forcing frequency in the rotating frame of the primary (the factor of two is due to the m = 2 perturbation represented by the tidal deformation), and n = 2π/P is the orbital frequency.

The tidal Q that is appropriate for WD binaries is not currently known. In other astrophysical systems, Q can vary quite dramatically from ∼10 for rocky bodies like the Earth (Goldreich & Soter 1966) to ∼10⁵–10⁶ for gaseous planets such as Jupiter or extrasolar Jupiters (Peale & Greenberg 1980; Ogilvie & Lin 2004; Wu 2005). For WD binaries, Campbell (1984) estimated synchronization times of ∼10⁸ yr, which correspond to Q ∼ 10¹². Willems et al. (2010) found Q ∼ 10¹⁵, but their analysis did not include dynamical tide effects, so this is merely an upper limit. Fuller & Lai (2011) looked at dynamical tides and then, appealing to Wu (1998) for damping times, estimate Q ∼ 10⁴–10¹¹ (depending the radial wavelength of the dominant excited mode). Other studies, such as Racine et al. (2007), are more focused on when accretion is occurring. Due to these uncertainties, I take Q to be a free parameter for the present study.

This secular quadrupole moment implies a torque

$$\begin{eqnarray} &&N_1 \sim U_{\rm grav, 1}^{\rm sec} n^2 \sim \frac{M_2 R_1^2}{Q_1} \left(\frac{R_1}{a}\right)^3 n\sigma _1. \end{eqnarray} \tag{ 3 }$$

Assuming solid-body rotation, the spin evolves as

$$\begin{eqnarray} &&\frac{d}{dt}\left(I_1\Omega _1 \right)\sim N_1 \sim \frac{M_2 R_1^2}{Q_1} \left(\frac{R_1}{a}\right)^3 n\sigma _1, \end{eqnarray} \tag{ 4 }$$

where I₁ = kM₁R²₁ and I set k = 0.2 (Marsh et al. 2004, but this may be modified for the hot He WD).

The angular momentum of the binary evolves as

$$\begin{eqnarray} &&\dot{J}_{\rm orb} = \dot{J}_{\rm gw} - N_{1} - N_2, \end{eqnarray} \tag{ 5 }$$

where the angular momentum loss to gravitational waves is (Landau & Lifshitz 1975)

$$\begin{eqnarray} &&\dot{J}_{\rm gw} = -\frac{32}{5}\frac{G^3}{c^5}\frac{M_1M_2M}{a^4}J_{\rm orb}, \end{eqnarray} \tag{ 6 }$$

where J_orb = (Ga/M)^1/2M₁M₂ and M = M₁ + M₂.

The evolution depends on two key timescales. The first is the gravitational wave timescale,

$$\begin{eqnarray} \tau _{\rm gw} &=& P\left|\frac{dP}{dt}\right|^{-1}= \frac{5}{96} \frac{c^5}{G^{5/3}} \frac{M^{1/3}}{M_1M_2}\left(\frac{P}{2\pi }\right)^{8/3} \nonumber \\\ms &=& 5.8\times 10^6 M_{0.25}^{-1}M_{0.55}^{-1}M^{1/3}_{0.8} P_3^{8/3}\ {\rm yr}, \end{eqnarray} \tag{ 7 }$$

where M_0.25 = M₁/0.25 M_☉, M_0.55 = M₂/0.55 M_☉, M_0.8 = M/0.8 M_☉, and P₃ = P/10³ s. The other is the torquing timescale found from Equation (4),

$$\begin{eqnarray} \tau _1 &=& \sigma _1 \left(\frac{d\Omega _1}{dt}\right)^{-1}= kQ_1\left(\frac{M_1}{M_2} \right)\left(\frac{GM}{R_1^3}\right)\left(\frac{P}{2\pi } \right)^3 \nonumber \\ \ms &=& 1.6\times 10^6k_{0.2} Q_{10}M_{0.25}^{-1}M_{0.55}M_{0.8}R_{9.3}^{-3}P_3^3\ {\rm yr}, \end{eqnarray} \tag{ 8 }$$

where R_9.3 = R₁/2 × 10⁹ cm, k_0.2 = k/0.2, and Q₁₀ = Q₁/10¹⁰. In previous calculations, such as Marsh et al. (2004), the strength of the tidal torquing is discussed in terms of a tidal synchronization timescale, τ_s. This is merely a factor of two smaller than the τ₁ defined here. This demonstrates that if Q₁ is assumed constant, the synchronization timescale decreases rapidly as the binary shrinks.

Given these timescales, I estimate the asynchronicity of the primary's spin. The time derivative of the tidal forcing frequency is

$$\begin{eqnarray} &&\frac{d\sigma _1}{dt} = 2\left(\frac{dn}{dt} - \frac{d\Omega _1}{dt} \right)= 2 \left(\frac{n}{\tau _{\rm gw}} - \frac{\sigma _1}{\tau _1} \right). \end{eqnarray} \tag{ 9 }$$

The competing effects of the gravitational wave emission promoting asynchronicity and the torques tidally locking the stars drive the system toward an equilibrium where dσ₁/dt ≈ 0. The tidal forcing frequency is then

$$\begin{eqnarray} \frac{\sigma _1}{n} & \approx &\frac{\tau _1}{\tau _{\rm gw}} = \frac{96}{5}\frac{G^{8/3}}{c^5}\frac{M_1^2M^{2/3}}{R_1^3}\left(\frac{P}{2\pi }\right)^{1/3}kQ_1 \nonumber \\ \ms &=&0.27 k_{0.2} Q_{10}M_{0.25}^2M_{0.8}^{2/3}R_{9.3}^{-3} P_3^{1/3}. \end{eqnarray} \tag{ 10 }$$

The general trend is that as the He WD gets closer to the C/O WD, the tidal torques become stronger (since τ₁∝P³ and τ_gw∝P^8/3), which in turn makes σ₁/n smaller. On the other hand, for a larger Q₁ there is less of a lever arm for torquing the He WD, and its spin becomes more asynchronous.

A tidal torque acting on an asynchronously spinning WD implies that work is being done. The rate of energy input is

$$\begin{eqnarray} &&\hspace{-1pc}\dot{E}_1 \approx \sigma _1 N_1 \approx \frac{M_2 R_1^2}{Q_1}\left(\frac{R_1}{a} \right)^3n \sigma _1^2 \nonumber \\ \ms &&= 2.3\times 10^{31} k_{0.2}^2 Q_{10}M_{0.25}^4M_{0.55} M_{0.8}^{1/3}R_{9.3}^{-1}P_3^{-13/3}{\rm erg\ s^{-1}}.\nonumber\\ \end{eqnarray} \tag{ 11 }$$

Whether or not this $\dot{E}_1$ is observed depends on how shallow it is deposited. One expects that the luminosity of the He WD is roughly given by $\dot{E}_1$ when the thermal timescale down to where the tide is damped is less than τ_gw. Iben et al. (1998) consider this issue in more detail by distributing the tidal heating evenly throughout the WD, and then following the WD's cooling. Brown et al. (2011) estimate the effective temperatures for the primary and secondary WDs to be 16, 400 K and 9000 K, respectively, providing luminosities of L₁ = 3.1 × 10³² erg s⁻¹ and L₂ = 3.8 × 10³⁰ erg s⁻¹. Using Equation (11), Q₁ ≈ 5 × 10¹⁰ is needed for $\dot{E}_1\approx L_1$. Equation (11) appears to imply that the heating increases indefinitely with larger Q₁, but the equilibrium condition only applies for τ₁ ≲ τ_gw. Using a maximum asynchronicity frequency of σ₁ = 2n, I find

$$\begin{eqnarray} &&\dot{E}_{\rm 1,max} \approx 5.9\times 10^{32} Q_{10}^{-1}M_{0.25} M_{0.8}^{-1}R_{9.3}^{5}P_3^{-5}\ {\rm erg\ s^{-1}}, \nonumber \\ \end{eqnarray} \tag{ 12 }$$

for the maximum heating rate.

The above estimates focus on the asynchronicity of the He WD, but the same arguments can be extended to the C/O WD. Using Equation (10),

$$\begin{eqnarray} &&\frac{\sigma _2}{\sigma _1} = \frac{Q_2}{Q_1}\left(\frac{M_2}{M_1}\right)^2\left(\frac{R_1}{R_2}\right)^3 \approx 95\frac{Q_2}{Q_1}, \end{eqnarray} \tag{ 13 }$$

where the term on the far right assumes the masses and radii appropriate for J0651. This result reflects the fact that the tidal torque is smaller on the C/O WD, so the asynchronicity is larger for the same Q. The ratio of the heating rates is

$$\begin{eqnarray} &&\frac{\dot{E}_2}{\dot{E}_1} = \frac{Q_2}{Q_1}\left(\frac{M_2}{M_1}\right)^3\left(\frac{R_1}{R_2}\right)\approx 29\frac{Q_2}{Q_1}. \end{eqnarray} \tag{ 14 }$$

If the observed luminosity of the C/O WD is also due to tidal heating, then Q₂ ≈ (Q₁/29)(L₂/L₁) ≈ 2 × 10⁷. The structures of the He and C/O WDs are quite different, with the former having a shallower convective region. In future work it would be interesting to theoretically explore whether such differences lead to Q₂ ≪ Q₁.

Numerical simulations of merging WDs often assume that the WDs are not tidally locked at merger (for example, Segretain et al. 1997, Guerrero et al. 2004). From our derivation we can see that this is not always the case, and in general it depends on the value of Q. For example, σ₁/n ≲ 0.1 merely require Q₁ ≲ 5 × 10⁹. If Q₂ = 2 × 10⁷ right up until merger, then σ₂/n ≈ 0.02 and the C/O WD must be tidally locked unless Q₂ increases by a couple of orders of magnitude as the binary nears merger.

In exploring the survival of binary WDs in forming AM CVn systems, Marsh et al. (2004) conclude that the synchronization time for the secondary must be less than 1000 yr. For the model presented here, that would imply Q₂ ≲ 10⁸ when accretion first begins. The current L₂ appears to already place a more stringent limit on Q₂ than this, although this conclusion again depends on how Q₂ changes as the orbit shrinks.

In Figure 1, I show numerical integrations of Equations (4) and (5), using the mass and radii appropriate for J0651. The tidal Q parameters are set to be constant at Q₁ = 7 × 10¹⁰ and Q₂ = 2 × 10⁷, so as to give heating rates at P = 765 s that are the same as the present luminosities of each star. The WDs are assumed to be non-spinning initially at a large orbital period, but are found to be quickly spun up by tides until dσ/dt ≈ 0 is reached, consistent with the assumptions for my analytic estimates. The integration ends when the Roche-lobe around the He WD becomes equal to its radius, which occurs at P ≈ 420 s (ignoring potential changes to R₁ due to tidal heating). The He WD is spinning significantly more slowly than the orbital period because of its large Q₁. The vertical dotted line denotes the current location of J0651 at 800, 000 yr before merger. If the values of Q remain constant, the luminosity of the He WD increases by a factor of ∼15 before tidal disruption.

Zoom In Zoom Out Reset image size

In the top panel of Figure 2, I calculate the rotational velocity of the primary V₁ = Ω₁R₁, as a function of orbital period. When the values of Q are chosen to match the current luminosities, V₁ ≈ 120 km s⁻¹. Another case where Q₁ = Q₂ = 10⁷ is also plotted, which represents what happens when the WDs are nearly tidally locked, resulting in V₁ ≈ 200 km s⁻¹. Since the binary is eclipsing, the difference between these cases may be measurable via the Rossiter–McLaughlin effect (Groot 2011).

Zoom In Zoom Out Reset image size

An important consequence of the tidal interactions is that the orbital period derivative deviates from what is expected if the system is purely driven by gravitational wave losses. Using Equation (7), gravitational waves alone give

$$\begin{eqnarray} &&|\dot{P}_{\rm gw}| = 1.7\times 10^{-4}M_{0.55}M_{0.25}M_{0.8}^{-1/3}P_3^{-5/3}\ {\rm s\ yr^{-1}}, \nonumber \\ \end{eqnarray} \tag{ 15 }$$

so for J0651 one would expect a period decrease of $|\dot{P}_{\rm gw}|=2.7\times 10^{-4}\ {\rm s\ yr^{-1}}$. When tidal effects are included, the period decreases faster because angular momentum is loss by the orbit to spin up the WDs. I assess this effect in the numerical models using the relation

$$\begin{eqnarray} &&|\dot{P}| = \frac{6\pi }{G^{2/3}}\left(\frac{P}{2\pi }\right)^{2/3}\frac{M^{1/3}}{M_1M_2}|\dot{J}_{\rm orb}|, \end{eqnarray} \tag{ 16 }$$

where $\dot{J}_{\rm orb}$ is found from Equation (5).

In the middle panel of Figure 2, I compare the period derivative for purely gravitational wave losses (dot-dashed line) with two different degrees of tidally locking (solid line and dashed line). This confirms that the period decreases more rapidly when tidal effects are included, and this deviation increases as the binary inspirals to smaller orbital periods.

In the bottom panel of Figure 2, I plot the fractional change in the period derivative, comparing the tidal cases with purely gravitational wave losses. For the values of Q that best fit the current luminosities, the period derivative is ≈3.0% larger, whereas if the WDs are nearly tidally locked, the period derivative changes by ≈5.6%. Looking for these deviations in future observations would provide an important measurement of the presence and level of tidal locking. An observed $|\dot{P}|\gtrsim |\dot{P}_{\rm gw}|$ implies strong tidal locking, but if $|\dot{P}|\approx |\dot{P}_{\rm gw}|$, then the asynchronicity and tidal heating must be large.

A non-zero $\dot{P}$ can also be measured by its affect on the eclipse timing. After a time t, this changes by

$$\begin{eqnarray} &&{\rm Change\ in\ eclipse\ time} = \frac{t^2}{2}\frac{ \dot{P}}{P}. \end{eqnarray} \tag{ 17 }$$

For purely gravitational wave emission, the change in the eclipse time is 5.5 s after one year. When tidal interactions are included, this change is 3%–6% larger, which corresponds to eclipses 5.6–5.8 s sooner. Associated phase shifts would be important in studies of this binary by the proposed Laser Interferometer Space Antenna (LISA) mission (Webbink & Han 1998; Nelemans et al. 2004).

I considered the effect of tides in the detached binary J0651. Assuming that the current luminosity of each WD reflects its tidal heating, I found Q₁ = 7 × 10¹⁰ and Q₂ = 2 × 10⁷ for the He and C/O WDs, respectively. The values of Q cannot be greater than this, otherwise the tidal heating would be inconsistent with the current luminosities. The degree of tidal locking can be measured in future observations, either via the Rossiter–McLaughlin effect or eclipse timing. If only gravitational waves are acting, then after one year the eclipses occur 5.5 s earlier. Tidal locking causes this difference to instead be as much as 5.8 s. Since the time of eclipses scales ∝t² (see Equation (17)), longer baseline observations will be important for making an accurate measurement.

Before deviations of $\dot{P}$ from merely gravitational wave losses can be inferred, the masses of the WDs must be measured sufficiently accurately. Since $\dot{P}_{\rm gw}\propto M_1M_2/M^{1/3}$ (Equation (15)), an uncharacteristically large $|\dot{P}|$ may instead imply larger masses. Mass measurements within a couple of percent would be ideal, which in turn means that the inclination must be highly constrained. At the same time, the $\dot{P}$ cannot be strictly due to gravitational waves, otherwise the tidal heating would be too large in comparison to the observed luminosities. Once $\dot{P}$ is measured in future observations, simultaneous modeling of the masses and luminosities should be done to constrain what values of Q, and thus degree of tidal locking, are allowed.

A critical question about J0651 is its age and how it relates to the larger population of WD binaries. The current merger time for J0651 is 800, 000 yr, while the age of the He WD inferred from its current effective temperature, presuming no tidal heating, is ≈10⁸ yr (Panei et al. 2007). On the face of it, this would imply that there should be ≈100 binaries with periods of P(τ_gw = 10⁸ yr) ≈ 50 minutes for every system like J0651. Although we are limited by small number statistics, such a large number of binaries have not been found (Kilic et al. 2011). This may point to an incompleteness of the current surveys. On the other hand, the progenitors to J0651 may be wider binaries that are older, cooler, and harder to detect. As the binary contracts, the tidal heating then makes the He WD bright and easier to observe. A more self-consistent model of the tidal heating plus stellar cooling, including changes to the He WD radius, is needed to assess what luminosity is expected as a function of time. Also, depending on when the heating takes place it can either be radiated readily or have time to heat the core and alter the stellar structure. I plan to explore these details better in a subsequent study.

Although there has been some work on estimating the Q of WD binaries (see Fuller & Lai 2011 and references therein), more can be done, especially in the context of He WDs. The presence of a thin layer of hydrogen (∼10⁻³ M_☉) on the He WD will alter the eigenfunctions of the excited modes, determining which specific modes are driven as well as affecting their thermal damping timescale, which in turn determines Q. In addition, the light curves presented by Brown et al. (2011) for J0651 show considerable dispersion (see the upper panel of their Figure 4) that could potentially point to stellar oscillations. The hot temperature of the He WD makes it well outside of the traditional instability strip associated with WDs, which requires T_eff  ≲  11, 000 K (Arras et al. 2006; Steinfadt et al. 2010). But strong tidal interaction may be another method of exciting observable oscillations, which should also be investigated.

Note. During the refereeing process of this manuscript, Benacquista (2011) also submitted a paper looking at the tidal interactions in J0651. In the limit when the WDs are tidally locked, he similarly finds that a 0.3 s shift in the eclipse time is expected after one year.

I thank Phil Arras, Lars Bildsten, Peter Goldreich, Paul Groot, Mukremin Kilic, Samaya Nissanke, Eran Ofek, Christian Ott, Sterl Phinney, and Marten van Kerkwijk for helpful feedback. This work was supported through NASA ATP grant NNX07AH06G, NSF grant AST-0855535, and by the Sherman Fairchild Foundation.