A DEEPLY ECLIPSING DETACHED DOUBLE HELIUM WHITE DWARF BINARY

Three primary eclipses and two secondary eclipses of CSS 41177 were observed using the LT and RISE (Steele et al. 2004) between 2011 February and April. RISE is a high-speed frame transfer CCD camera with a single wideband V+R filter (Steele et al. 2008). All observations were made in 2 × 2 binning mode, giving a scale of 1.1 arcsec pixel⁻¹, and with exposure times of 12–13 s.

The raw data are automatically run through a pipeline that debiases, removes a scaled dark frame, and flat-fields the data. The source flux was determined with aperture photometry using a variable aperture, whereby the radius of the aperture is scaled according to the FWHM, using the ULTRACAM pipeline (Dhillon et al. 2007). Variations in observing conditions were accounted for by determining the flux relative to the nearby star G 54-11 (V = 14.6).

CSS 41177 was observed using GMOS on the 8 m Gemini Observatory North telescope (Hook et al. 2004) on 2011 March 15. We used the R831 grating with a 0.5 arcsec slit and 4 × 2 binning (spatial by spectral) to give a resolution of R ∼ 4400 centered at Hα. We used an exposure time of 10 minutes, recording 17 spectra covering an entire orbit.

The average bias and flat-field correction were carried out using the figaro package from starlink and nightly bias and tungsten frames. We optimally extracted each spectrum (Marsh 1989) and used a CuAr arc lamp to establish an accurate wavelength scale. The arc spectrum was fitted with a fifth-order polynomial to 15 lines which gave an rms of 0.05 pixel. The individual spectra were normalized to the continuum level using a polynomial fit to the continuum regions and the spectra were placed onto a heliocentric wavelength scale.

The left-hand panel of Figure 1 shows the LT+RISE photometry of CSS 41177 folded on its 2.78 hr orbital period and binned by a factor of two. Since both eclipses are detected, the light curve information alone constrains the system inclination and the radii scaled by the binary separation.

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To measure the system parameters we fitted the light curve using a code written to produce models for the general case of binaries containing a white dwarf (see Copperwheat et al. 2010 for details). The program subdivides each star into small elements with a geometry fixed by its radius as measured along the direction of centers toward the other star. From an initial set of parameters defined by the user, the code produces model light curves which are initially fitted to the RISE data using Markov chain Monte Carlo (MCMC) minimization to produce a set of covariances. We then ran another MCMC minimization using these covariances to define the parameter jumps. To produce the final parameters and their errors, we followed the procedures described in Collier Cameron et al. (2007). The parameters needed to define the model were: the mass ratio, q = M₂/M₁, the inclination, i, the radii scaled by the binary separation, R₁/a and R₂/a, the surface brightness ratio, S₁/S₂, quadratic limb-darkening coefficients for both the white dwarfs, the time of mid eclipse, T₀, and the period, P.

We corrected all the times to Barycentric Dynamical Time (TDB). We initially fitted each light curve individually in order to measure mid-eclipse times for the three primary eclipses recorded. The second eclipse recorded was set to cycle number zero. Our measured eclipse times are shown in Table 1. The zero point of the ephemeris and the period were allowed to vary in our final fit. We kept the mass ratio fixed as 1.0 (the light curves of this well-detached system are independent of the mass ratio). We determined the quadratic limb-darkening coefficients from a pair of white dwarf atmosphere models with temperatures of 21,100 K and 10,500 K and log g = 7.3 (see Section 3.2) folded through the RISE filter profile (Gänsicke et al. 1995). We determined coefficients of a = 0.105 and b = 0.228 for the primary and a = 0.176 and b = 0.288 for the secondary for I(μ)/I(1) = 1 − a(1 − μ) − b(1 − μ)², where μ is the cosine of the angle between the line of sight and the surface normal; we kept these values fixed. The inclination, scaled radii, and surface brightness ratio were allowed to vary.

The right-hand panel of Figure 1 shows the best fit to the RISE photometry around the eclipses. The two eclipses constrain the system inclination to 892 ± 03. The constraints on the two scaled radii are shown in the left-hand panel of Figure 3. We also update the ephemeris to

$$\mathrm{MJD(BTDB)} = 55618.926447(9) +\, 0.116\,015\,49(5) E,$$

which is consistent with the ephemeris of Drake et al. (2010).

We used data from the Sloan Digital Sky Survey (SDSS) to determine the temperatures of the two white dwarfs in CSS 41177. Initially, we fitted its SDSS spectrum as a single white dwarf using the models of Koester et al. (2005), which results in a T_{eff, 1} = 21, 535 ± 214 K and log g = 7.36 ± 0.44. To determine the temperature of both stars better, we fitted the SDSS ugriz magnitudes with a composite of two DA white dwarf models, with the additional constraint of the surface brightness ratio S₁/S₂ = 2.90 ± 0.23 in the LT/RISE band, as measured from the light curve. This fit results in T_{eff, 1} = 21, 100 ± 800 K and T_{eff, 2} = 10, 500 ± 500 K, where the quoted errors are estimated, as they correlate with the uncertainties in the surface gravities. The best-fit composite model reproduces well the observed Galaxy Evolution Explorer (GALEX) ultraviolet fluxes.

The left-hand panel of Figure 2 shows the trailed spectrum of CSS 41177 around the Hα line. The non-LTE absorption core from the primary (hotter) star is clearly visible. There is also a weaker absorption component moving in anti-phase with the absorption from the primary star originating from the secondary (cooler) star. No other features are visible in the GMOS spectra.

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Initially we fitted just the primary star's absorption. For each spectrum, we fitted the line using a combination of a straight line plus a Gaussian. However, the radial velocity measured does not represent the true radial velocity of the primary star as the absorption from the secondary star causes a slight underestimation. Nevertheless, we used these velocities to correct out the motion of the primary star. We then averaged the shifted spectra to obtain the rest-frame spectrum of the primary star which we then subtracted from all the observed spectra. The result is shown in the right-hand panel of Figure 2, clearly showing the absorption from the secondary star.

We fitted the absorption from the secondary star in the same way as the primary star. This too underestimated the true velocity since the primary star's contribution had not been completely removed. However, both fits provide us with an initial starting point for K₁ and K₂, the radial velocity amplitudes of both stars. In order to measure accurate radial velocities, both absorption components need to be fitted simultaneously. We fit both lines together by simultaneously fitting all of the spectra. We used a combination of a straight line and Gaussians for each spectrum (including a broad Gaussian component to account for the wings of the primary star's absorption) and allowed the position of the Gaussians to change velocity according to

$$\begin{eqnarray*} &&V = \gamma + K\sin (2 \pi \phi), \nonumber \end{eqnarray*}$$

for each star, where γ is the offset in the line from its rest wavelength and ϕ is the orbital phase of the spectrum. Due to the faintness of the secondary star's absorption we keep the offset γ₂ = γ₁ to reduce the number of degrees of freedom. All other parameters were initially set to those measured from fitting the lines individually. We fitted the data using Levenberg–Marquardt minimization (Press et al. 1986). The resultant fit had a reduced χ² of 1.02. We find K₁ = 177 ± 3 km s⁻¹, K₂ = 181 ± 20 km s⁻¹, and γ = 109 ± 2 km s⁻¹. These values imply that CSS 41177 is an equal mass binary (or very close to, given the uncertainty on K₂).

The RISE photometry constrains i, R₁/a, and R₂/a. K₁ and K₂ are constrained from the spectroscopy, therefore we can determine the masses and radii of both white dwarfs using Kepler's third law,

$$\begin{eqnarray} &&\frac{G(M_1 + M_2)}{a^3} = \frac{4 \pi ^2}{P^2}, \end{eqnarray} \tag{ 1 }$$

where P is the orbital period, given the orbital separation from

$$\begin{eqnarray} &&\frac{2 \pi }{P}\, a\sin {i} = K_1 + K_2. \end{eqnarray} \tag{ 2 }$$

We determined the masses and radii for each model produced in the MCMC analysis of the light curve using the measured radial velocities and Kepler's laws. The right-hand panel of Figure 3 shows the regions of allowed masses and radii for both white dwarfs based only upon the photometric and spectroscopic constraints. Also shown are models of helium core white dwarfs with hydrogen layer fractional masses of M_H/M_* = 10⁻⁴ (solid lines) and M_H/M_* = 10⁻⁸ (dashed lines) from Benvenuto & Althaus (1998) for temperatures of 10,500 K (gray) and 21,100 K (black) which, given the large uncertainties in our measurements, are consistent with the measured masses and radii. The final system parameters are listed in Table 2.

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