Revised Stellar Parameters for V471 Tau, A Post-common Envelope Binary in the Hyades

V471 Tau was observed for roughly 80 days by the K2 Spacecraft during Campaign 4 for Guest Observer programs 4027 ⁶ and 4043. ⁷ The observations were acquired in short-cadence mode, with integration times of 58.89 s. Data from NASA's K2 Mission contain instrumental systematic effects due to pointing drift in the space telescope (Howell et al. 2014). To remove the instrumental systematic effects, we used the EPIC Variability Extraction and Removal for Exoplanet Science Targets (everest) software package (Luger et al. 2016, 2018). In addition to the expected instrumental systematic effects, the raw K2 light curve also contains anomalously high- and low-flux measurements occurring at regular intervals. We identified the period and ephemeris of the anomalous measurements using a Lomb–Scargle periodogram (P = 1.96066 days). We used the masking feature in the everest pipeline to exclude the anomalous data points and white-dwarf occultation events from the everest algorithm, as sharp features can adversely affect the quality of the systematic corrections. Manual inspection of the resulting light curve shows the periodic white-dwarf occultation events, flaring, and quasiperiodic variations consistent, which are consistent expectations from synchronous rotation and evolving magnetic spots on the main-sequence star.

For the radial-velocity measurements of the main-sequence star, we used 202 archival ground-based radial-velocity measurements and uncertainties from the literature (Bois et al. 1988). The measurements span the entire orbit of the main-sequence star and were reported in heliocentric Julian date. Radial-velocity measurements of the white dwarf are sparser owing to the high contrast between the the white dwarf and the main-sequence star at visible wavelengths. We used eight published measurements from O'Brien et al. (2001) taken with HST-GHRS after the COSTAR upgrade to the telescope. The measurements used the Stark-broadened wings of the white dwarf's Lyα absorption line to determine the white dwarf's radial velocity. We note that the zero point for the published white dwarf radial-velocity measurements is uncertain, and we chose to leave that as a free parameter in our fit to the data.

V471 Tau was also observed by the Space Telescope Imaging Spectrograph (STIS) on the HST, with publicly available reduced and calibrated data (Ayres 2010). Fourteen of the observations were taken using the E140M grating, which includes the white-dwarf Lyα absorption. The observations were originally collected to measure emission and absorption features in the system by Sion et al. (2012). They found that different species of absorption lines show different radial-velocity semiamplitudes, indicating nonisotropic emission. We therefore follow the approach of O'Brien et al. (2001) and use the Stark-broadened wings of the Lyα absorption line to represent the radial velocity of the white dwarf, excluding regions with other absorption or emission lines. We note that the STIS spectra are Doppler corrected for spacecraft motions onboard.

We originally fit model white-dwarf spectra to the data to determine radial velocities; however, the resulting radial velocities contained an obvious multiplicative bias depending on the choice of model temperature, owing to the temperature-dependent slope of the continuum. Instead, we constructed a spectral template from the data itself. We downloaded and coadded the spectra, then fit a piece-wise second-degree polynomial template to each side of the Lyα absorption line, ignoring regions with chromospheric emission, instellar absorption, and geocoronal emission (see Figure 1). We then fit the piece-wise polynomial template to each individual spectrum, allowing only radial velocity of the template and overall normalization to vary with each fit. Since the template was constructed from the data itself, it cannot serve as an absolute radial-velocity calibration, meaning we cannot measure the systemic radial velocity or the gravitational redshift of the white-dwarf component, both of which affect the radial-velocity zero point. But, the template can serve to measure relative changes in radial velocity. The resulting radial velocities clearly follow the expected relative radial-velocity variation due to the white-dwarf orbit.

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Following our approach with the GRS radial-velocity measurements, we treat the zero-point radial velocity of the STIS measurements as another free parameter in the full eclipsing binary fit. To estimate the uncertainty in the STIS radial-velocity measurements, we fit a simple sinusoid to the measurements versus orbital phase, fixing the sinusoidal period to that of the system and allowing the amplitude and offset to vary. We calculated the standard deviation between the measurements and the fitted sinusoid as 11.2 km s⁻¹. We adopted that standard deviation as the individual measurement uncertainties in each redial-velocity observation. Table 1 details the 14 observations and resulting radial-velocity measurements.

We note that since the template was constructed by coadding spectra, there will be some broadening or smoothing of the template due to the radial-velocity differences in those spectra. To test this effect, we recalculated the template, this time shifting the individual spectra to remove the orbital radial-velocity differences before coadding them. This approach had a negligible impact on the resulting radial velocities and estimated uncertainties, each changing by less than one tenth of the estimated measurement uncertainty.

The eb software package produces a model light curve for a given set of orbital parameters, and includes light travel time and limb-darkening effects. We used eb to model the eclipse events, with some custom modifications. The eb package does not include gravitational microlensing effects during eclipses, which can be significant for the transit of compact objects. During the eclipse of the K dwarf by the white dwarf (hereafter, the transit), gravitational microlensing will "fill in" the transit depth by the differential amount $2{R}_{\mathrm{Ein}}^{2}/{R}_{\mathrm{MS}}^{2}$ (Agol 2003), where R_Ein is the Einstein radius of the white dwarf during transit and R_MS is the radius of K dwarf. To modify the eb model to incorporate gravitational microlensing, we isolated the region of the model that contained the transit event, subtracted the flux contributed by the white dwarf, renormalized the light curve to a maximum of unity, then scaled the model transit to account for gravitational microlensing. If the Einstein radius is sufficiently large, the filling factor results in an inverted transit (e.g., Kruse & Agol 2014); in the case of V471 Tau, gravitational microlensing results in a shallower transit, similar to what was seen in the post-CE binary KOI-256 (Muirhead et al. 2013). For a set of orbital parameters, we computed the model light curve across the full orbital phase with a sampling of roughly 1 s. We convolved the model light curve with a normalized top hat function of width equal to integration time of the K2 short-cadence data (58.89 s), in order to capture the effect of finite integration time on the light curve. We then interpolated the model light curve onto the K2 time stamps using cubic spline interpolation.

The celerite software enables the treatment of stellar variability as correlated noise added to a "mean model," which in this case is the light-curve model produced by eb for a given set of orbital parameters. In this approach, the stellar variability is treated as correlated noise added to the purely geometric eclipsing binary model. However, since the eclipse events result in fractional changes in the overall system flux, a correct treatment would involve multiplying the stellar variability model by the the eclipse model, rather than adding them. We therefore chose to fit the log of the eclipse model, with added correlated noise, to the log of the K2 data, propagating the corresponding measurement errors appropriately. This results in the correct treatment of fractional changes in the system flux.

In a Gaussian-processes approach, correlated noise is described by a user-selected kernel function containing free parameters. The kernel function relates the degree of correlation between data points, which are then entered into the data covariance matrix for determining the likelihood probability density for a given choice of free parameters (both kernel parameters and mean-model parameters). Following the approaches described in Foreman-Mackey et al. (2017), for the kernel, we chose a damped harmonic oscillator with two free parameters: an amplitude α and an damping constant β. We fixed the periodic component of the damped harmonic oscillator to one half of the orbital period of the system P, which is the typical periodicity of the rotational variations as evidenced by visual inspection:

$$\begin{eqnarray}&&\kappa (\tau )=\alpha \,{e}^{-\beta \tau }\left(1+\cos \displaystyle \frac{\pi \tau }{P}\right)\end{eqnarray} \tag{ 1 }$$

where κ is the degree of correlation between K2 data points separated by τ in time.

To model the radial-velocity observations, we used a sinusoidal model for each component of V471 Tau corresponding to the orbit expected from the fitted orbital parameters. As noted in the literature, the radial-velocity zero point for the literature white-dwarf measurements is uncertain (Vaccaro et al. 2015). For the two sets of HST radial velocities of the white-dwarf component—the literature measurements from O'Brien et al. (2001) using GHRS and our measurements from archival STIS observations—we included a separate additive offset for each to account for the systematic offsets in the radial-velocity zero points.

For a given set of orbital parameters, we computed the corresponding sinusoidal radial-velocity variation for each component with a sampling of 1 s. We convolved the resulting model radial-velocity curve with a tophat of width 1800 s, to account for the typical integration times of the radial-velocity observations and resulting smoothing of the purely geometric radial velocities. In reality, the integration times for the radial-velocity observations varied; however, the choice of modeled exposure time had a negligible affect on the resulting best fit parameters, so we assume an 1800 s exposure for all radial-velocity observations.

For a given set of model orbital parameters and measured radial velocities and uncertainties, the log of the likelihood probability density was computed and added to the log of the likelihood probability density for the light-curve model.

To determine the best-fitting kernel parameters and orbital parameters, we sampled a Bayesian posterior probability function assuming Gaussian errors for all the measurements and assuming uniform prior functions on all free parameters with reasonable upper and lower bounds. For the independent measurement uncertainties on each data point, were used the uncertainties returned from the everest pipeline, the reported uncertainties for the radial-velocity measurements in the literature, and our estimated uncertainties from the STIS measurements. We assumed an eccentricity of zero, fitting each component's radial-velocity measurements with a sinusoidal function offset by π, and we assumed no third light contamination and no reflected light in the eclipse model. We assumed quadratic limb darkening in both stars, using predicted values for main sequence and white dwarfs with the expected surface gravity and surface temperature in the K2 bandpass (Claret 2018; Claret et al. 2020). We did not include ellipsoidal variations in the mean model, letting the Gaussian processes approach capture all stellar variability effects. The fitted Roche-lobe filling factor for the main-sequence star is 52% (here defined as the radius of the star divided by the distance to the L1 Lagrange point), implying that nonspherical effects should have a negligible impact on the mean model. All in all, we fit for the following parameters: kernel amplitude, kernel damping constant, orbital period, occultation ephemeris, sum of the radii in units of semimajor axis, radius ratio, surface brightness ratio in the K2 band, orbital inclination, sum of the masses, mass ratio, systemic radial velocity of the system, and the radial-velocity zero points for the two sets of white-dwarf RV measurements.

To find the mostly likely orbital parameters and their uncertainties, we sampled the Bayesian posterior using the emcee software. We used 100 chains, each starting with random parameters spread across the ranges of their respective uniform prior functions. We ran the sampling algorithm for 100,000 steps (a "burn-in") to find the highest likelihood parameters. We then divided the data by the best-fitting eclipse model and smoothed the resulting light curve with a median filter of with 61 data points (corresponding to roughly 1 hr of observations). Data points that differed from the smoothed light curve by more than three sigma were removed from the light curve, and ran another 100,000-step burn-in. This step removed the flare events in the light curve. Following the second burn-in, we removed chains with maximum posterior values that were significantly lower than the highest posterior value across all chains (5 out of 100 chains), and ran a 200,000-step "production" to sample the peak of the posterior probability density function as a function of free parameters.

Figure 2 shows the phase-folded K2 light curve and best-fitting model: that is, the sample of free parameters with the highest posterior probability. We recover the transit of the white dwarf in front of the main-sequence star, with a pronounced effect from gravitational microlensing. Figure 3 shows the radial-velocity measurements and the best-fitting model. Figures 4 and 5 show the sampling distributions for the fitted and derived parameters, respectively. Table 2 lists the fitted parameters corresponding to the sample with the highest posterior probability density, the median of the MCMC steps, and the standard deviations of the parameter samples, which serve as an estimate for the marginalized uncertainty in each parameter. Table 2 also lists several several system parameters derived from the fitted parameters, namely the individual stellar masses and radii.

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