Tidal heating in detached double white dwarf binaries

TL;DR

This work develops a self-consistent tidal-heating model for detached short-period double white dwarf binaries with helium-core ELM WDs, using Hut's equilibrium tide formalism together with a temperature-dependent mass–radius relation to predict how tidal dissipation heats the donor and inflates its radius. By coupling orbital decay from gravitational waves with tidal torques and the donor's thermal response, the authors show that mass-transfer onset occurs at lower Roche frequencies ($f_{\rm RL}\sim$1–3 mHz) than cold-degenerate predictions, and surface temperatures can rise by tens of percent up to Roche contact. Applying the model to known Galactic DWDBs (e.g., J2243, J0538, J2029) demonstrates substantial temperature increases (up to ~50%) and radius growth (several to ~20%), with Roche contact occurring within a few 10^2 kyr; a focused study of J1539 explores the past evolution under different formation scenarios, highlighting the potential constraints on binary ages and prior temperatures. The findings imply that finite-temperature effects materially affect transient progenitor demographics (AM CVn, R CrB, Type .Ia SNe) and shift the gravitational-wave foreground expectations for LISA-like detectors, underscoring the need to incorporate thermal structure in population syntheses and GW analyses.

Abstract

Short--period ($P<$1 hr orbits) detached double white dwarf binary (DWDB) components identified with transient surveys (e.g. SDSS, ZTF) have hot surface temperatures ($>$10,000 K) and observed radii a factor two larger than completely degenerate white dwarfs. We formulate tidal heating in helium composition extremely low mass white dwarf (ELM WD) components of detached DWDBs which reach mass transfer within a Hubble time. We combine a mass radius relation which varies with surface temperature and the equilibrium tidal friction model of Hut 1981, where the additional orbital energy loss from tidal friction is accounted for by increases in the primary surface temperature, and hence increasing radius. Applying this heating model to the current sample of binaries with ZTF, we predict temperature increases from the present day of up to $\sim$40\% before the onset of mass transfer. We find that helium white dwarfs are generically hot and large at the onset of mass transfer, even for the oldest DWDBs whose components can cool to be degenerate by the present day. In the population of Galactic DWDBs, we find that the onset of mass transfer should occur at orbital periods as long as 1000s (17 minutes), or binary gravitational wave frequency of 2 mHz. This is over three times longer than periods expected for degenerate WD (5 minutes). Since mass transferring DWDBs are progenitors for a variety of transients and stellar populations e.g. RCrB stars, AM CVn binaries, so-called Type .Ia supernova, the finite temperature of donor white dwarfs should be taken into account.

Figures (5)

• Figure 1: The masses and radii of eclipsing ELM WD binary components in the Galactic detached sample (Table \ref{['tab:WD1']}) are shown by the stars, coloured according to their surface temperature. The majority of these ($m \lesssim 0.3 M_\odot$) are helium composition ELM WD. The secondary of J1539 which is likely CO composition is excluded, so there are a total of 14 stars in the plotting window. Comparing these stars to the completely degenerate (cold) relation (solid black line), the observed radii are typically larger by a factor between 2 to 3 for masses between 0.15--0.25$M_\odot$. Our empirical formulation of a temperature dependent mass radius relation are shown by the pastel contours, increasing in increments of 4,000 K. This relation is based on the helium composition models which include a hydrogen envelope of mass $3 \times 10^{-4}M_\odot$ in Panei2000, and is given by Equation (\ref{['eq:RMTPanei']}).
• Figure 2: Reverse cumulative distribution function for DWDBs as a function of $f$, using the estimate for ELM DWDB production rates $\mathcal{R}_\mathrm{ELM}$ via RCrB stars and also detached ELM DWDBs (orange) and AM CVn binaries (blue). The number expected by current surveys, $N_\mathrm{ZTF}$, which accounts for ELM selection effects via Equation (\ref{['eq:NZTF']}) is given by the black vertical axis on the left hand side. On the right hand side axis in grey is the intrinsic population in the Galaxy, $N_\mathrm{MW}$ (Equation (\ref{['eq:NMW']})). These two axes differ by the product of the eclipsing probability and observable volume ${V}_\mathrm{obs}$. Integrating from 1.3mHz to 3.8 mHz, the estimated $N_\mathrm{ZTF}$ are 3 (blue) and 59 (orange) respectively. The number in the observed sample (Table \ref{['tab:binaries']}) is 9. For the dashed portions of the lines, we suggest that both $N_\mathrm{ZTF}$ and $N_\mathrm{MW}$ may no longer be proportional to $f^{-8/3}$ for $f>2$ mHz as discussed in Section \ref{['sec:population_discussion']}.
• Figure 3: The ratio of the cooling timescale $\tau_\mathrm{cool}$ (via Equation (\ref{['eq:Lcool2']})) to tidal heating timescale $\tau_\mathrm{TF}$ (Equation (\ref{['eq:tauTF']})) given by $\tau_\mathrm{cool}/\tau_\mathrm{TH}$ is shown (log scale) against the surface temperature (linear scale). At $\tau_\mathrm{cool}/\tau_\mathrm{TH}=1$ (Log$_{10}(\tau_\mathrm{cool}/\tau_\mathrm{TH} ) = 0$), we divide the horizontal axis into cooling and tidal heating dominated temperature evolution regimes. When $\tau_\mathrm{cool}/\tau_\mathrm{TH}$ is less than 1 (grey shaded area), the current temperature evolution into the future is governed by white dwarf cooling. For each primary component in the binary population, its ratio from Table \ref{['tab:timescales']} and temperatures (Table \ref{['tab:WD1']}) are shown by the star shaped markers. J1539's primary (not in Table \ref{['tab:WD1']}) is included as the blue arrow symbol with its temperature upper bound. Components are shaded according to their time until Roche contact $\tau_{\mathrm{RL}}$, driven primarily by the emission of gravitational waves, using a yellow to black colour scheme.
• Figure 4: Evolution of temperature $T_\mathrm{eff}$ in K (left vertical axis) with gravitational wave frequency $f$ in mHz from the present day properties according to our tidal heating model for the three binaries J2243 (yellow), J2029 (indigo) and J0538 (pink). All three binaries have the same larger primary component of $m_1=0.32M_\odot$, so that we can plot their luminosity on one single right hand vertical axis in solar luminosity $L_\odot$. Each of the companion masses $m_2$ are listed in the legend. We plot each track until Roche lobe overflow (infinity markers). This is when $f= f_\mathrm{RL}$ according to Equation (\ref{['eq:fRL']}), given its temperature dependent radius from Equation (\ref{['eq:RMTPanei']}). In this temperature dependent model, the $f_\mathrm{RL}$ are 6.2, 7.3 and 7.7 mHz respectively. However, the cold mass radius relation predicts $f_\mathrm{RL}=16-17$ mHz, over two times larger.
• Figure 5: Temperature evolution of the primary ELM WD component in six J1539--like binaries. These models are evolved from temperatures of 4,000 K, 5,000 K, 6,000 K, 7,000 K, 8,000 K and 9,000 K at $f=1.5$ mHz, corresponding to the secondary CO WD's formation and hence DWDB formation. All tracks (solid lines) terminate at Roche contact when $R_1/a = R_\mathrm{RL}/a =$ 0.29 (grey dashed line) after around $\sim 5$ Myr. The present day gravitational wave frequency, $f=4.8$ mHz, and temperature upper bound of J1539's primary ELM WD component, $T<$10,000 K, are shown by the pink triangle. Our tidal heating model allows the primary of a J1539 to have a temperature at the CO component formation at $f=1.5$ mHz between 6,000-8,000 K (pink lines), and corresponding present day temperature between 8,000--10,000 K. We ultimately exclude the three black and maroon curves (see text for details).