A Third Hot White Dwarf Companion Detected by Kepler
aa r X i v : . [ a s t r o - ph . S R ] D ec A CCEPTED FOR PUBLICATION IN T HE A STROPHYSICAL J OURNAL
Preprint typeset using L A TEX style emulateapj v. 11/10/09
A THIRD HOT WHITE DWARF COMPANION DETECTED BY
KEPLER J OSHUA
A. C
ARTER , S
AUL R APPAPORT , & D ANIEL F ABRYCKY
Accepted for publication in The Astrophysical Journal
ABSTRACTWe have found a system listed in the
Kepler
Binary Catalog ( P orb = 3 .
273 days; Prsa et al. 2010) that wehave determined is comprised of a low-mass, thermally-bloated, hot white dwarf orbiting an A star of about2.3 M ⊙ . In this work we designate the object, KIC 10657664, simply as “KHWD3” (“Kepler Hot White Dwarf3”). We use the transit depth of ∼ ∼ R / a and inclination angleof the binary, both the ELV and DB effects are mostly sensitive to the mass ratio, q = M / M , of the binary.The two effects yield values of q which are somewhat inconsistent – presumably due to unidentified systematiceffects – but which nonetheless provide a quite useful set of possibilities for the mass of the white dwarf (either0 . ± . M ⊙ or 0 . ± . M ⊙ ). All of the other system parameters are determined fairly robustly. Inparticular, we show that the white dwarf has a radius of 0 . ± . R ⊙ which is extremely bloated over theradius it would have as a fully degenerate object, and an effective temperature T eff ≃ ,
500 K. Binary evolutionscenarios and models for this system are discussed. We suggest that the progenitor binary was comprised of aprimary of mass ∼ M ⊙ (the progenitor of the current hot white dwarf) and a secondary of mass ∼ M ⊙ (the progenitor of the current A star in the system). We compare this new system with three other white dwarfsin binaries that likely were formed via stable Roche-lobe overflow (KOI-74, KOI-81, and the inner Regulusbinary). Subject headings: techniques: photometric — stars: binaries: eclipsing — stars: binaries: general — stars:evolution — stars: variables: other — stars: subdwarfs INTRODUCTION
The exquisite photometric precision of the
Kepler missionhas led, among many other things, to the discovery of twosystems in which a hot white dwarf both transits its parentstar, and is eclipsed by it (Rowe et al. 2010; van Kerkwijket al. 2010). In the case of KOI-74, the relative transit andeclipse depths are only 5 × - and 12 × - , respectively,while the amplitudes of the ellipsoidal light variations andDoppler boosting effect have even more remarkably small am-plitudes of 1 . × - and 1 . × - , respectively. Such smalleffects are unexplored with ground-based astronomy.The discovery of transiting/eclipsing white dwarfs whichexhibit the effects mentioned above are amenable to many in-teresting system diagnostics, and can thereby be potentiallyvery revealing about the binary stellar evolution that has ledto their current configuration (e.g., van Kerkwijk et al. 2010;Di Stefano 2010). In particular, if the progenitor of the whitedwarf was initially of mass . . M ⊙ the orbital period maybe tightly correlated with the white dwarf mass (e.g., Rap-paport et al. 1995). In all cases where the current-epoch bi-nary orbital period is short (e.g., less than a month) the sys-tem almost certainly involved a phase of mass transfer fromthe progenitor of the white dwarf to what is presently the nor-mal stellar companion. In the process, a substantial amount ofthe original primary mass was transferred to its companion orlost from the system. The temperature of the white dwarf pro- [email protected] M.I.T. Kavli Institute for Astrophysics and Space Research, 70 VassarSt., Cambridge, MA, 02139 Harvard-Smithsonian CfA, 60 Garden St., Cambridge, MA, 02138 Hubble Fellow UCO/Lick, University of California, Santa Cruz, CA 95064 vides some important information on the cooling history andthe age since the mass transfer event (see, e.g., van Kerkwijket al. 2010; Di Stefano 2010).In this work, we report the discovery of a third hot whitedwarf (with T eff ≃ ,
500 K) orbiting an A star, which weherein designate as “KHWD3”. The star, KIC 10657664, wasidentified and catalogued by Prsa et al. (2010) in the
Kepler binary star catalog, although it was not identified as harboringa hot white dwarf companion. Its transit and eclipse depthsare about an order of magnitude larger than those of KOI-74and KOI-81, due mostly to the fact that the radius of the whitedwarf in KHWD3 is a factor of & Kepler light curve for this system, and the analyses whichyield the flux ratio of the two stars, the ratio of their radii, theorbital inclination angle, and the first three harmonics of thelight curve which reveal ellipsoidal light variations, an illumi-nation effect, and Doppler boosting. In §3 we determine themodel-independent parameters of the binary system includ-ing the mass, effective temperature, radius, and luminosityof the white dwarf, and the mass ratio. Inferences about theproperties of the primary star allow us to estimate its massat 2 . ± . M ⊙ . From this estimate and the measured massratio we arrive in §4 at two possible solutions for the massof the white dwarf: either 0 . ± . M ⊙ or 0 . ± . M ⊙ ,depending on whether we emphasize in our analysis the ellip-soidal light variations or the Doppler boosting effect. In §5we discuss various evolutionary scenarios that can lead to thepresent system. LIGHT CURVE ANALYSIS
The uncorrected light curve for KHWD3, downloaded from Carter, Rappaport, & Fabryckythe
Kepler public archive , is shown in Figure 1. The dataare comprised of ∼
10 days of quarter zero (Q0) data and ∼
33 days of quarter one (Q1) data . Outside of the easily-identified eclipses, the variation of the light curve is domi-nated by a low frequency trend, assumed to be a systematicassociated with the Kepler spacecraft. The remaining periodicvariation on times at or shorter than the orbital period and inphase with the conjunctions is assumed to be astrophysicalin nature (as a result of, for example, ellipsoidal light varia-tions and/or Doppler boosting). We accounted for these threeeffects in a light curve model, as described below, and deter-mined parameters of the eclipsing system and the harmoniccontent of the out-of-eclipse variation.
Light Curve Model
We modeled the eclipse events assuming two spherical starshaving radius ratio R / R , and observed flux ratio F / F , as-signing the indices of 1 and 2 to primary and secondary, re-spectively. The binary was constrained to a Keplerian orbitparameterized by a period P , a normalized semi-major axisdistance a / R , an inclination to the sky plane i , an eccen-tricity e , an argument of periastron v and a time of inferiorconjunction (i.e., at the time of the eclipse of the primary), t ic .The normalized eclipse light curve, f ( t ), was calculated tobe f ( t ) = - λ h z ( t ) / R , R R , u , v i / (cid:16) + F F (cid:17) Primary1 - λ h z ( t ) / R , R R , u , v i / (cid:16) + F F (cid:17) Secondary (1)where z ( t ) is the sky-projected separation of the centers of thetwo stellar components and λ is the fraction of the eclipseddisk blocked by the occulter, given analytically by Mandel& Agol (2002). The limb darkening coefficients u and v pa-rameterize the radial brightness profile, I ( r ), of either binarycomponent as I ( r ) I (0) = 1 - u (cid:16) - √ - r (cid:17) - v (cid:16) - √ - r (cid:17) (2)where r is the projected radial distance from the stellar center,normalized to R ⋆ The out-of-eclipse light curve was modeled as the productof a harmonic series, h ( t ), and a low order, quarter specificpolynomial, p Q ( t ) (with Q = 0 , h ( t ) = 1 + X k =1 A k sin [ k φ ( t )] + X k =1 B k cos [ k φ ( t )] (3) p Q ( t ) = N Q X n =0 C ( Q ) n ( t - t ) n (4)for a set of amplitudes ( A k , B k ) and polynomial coefficients[ C ( Q ) n ] where φ ( t ) = (2 π/ P )( t - t ic ), and t is some fixed timenear the start of Q0, N = 3 and N = 6.The full light curve model, ℓ ( t ), is given as the product ofall three variations, i.e., ℓ ( t ) = f ( t ) h ( t ) p Q ( t ).We fitted the full model to the data with R / R , a / R , F / F , u , u , P , i , e , v , t ic , A k , B k and C ( Q ) n as free param- \protect http://archive.stsci.edu/kepler/publiclightcurves.html Kepler “quarters” are continuous observing blocks separated by shorttimes in which data are downloaded from the spacecraft. Only Q0 and Q1are currently available to the public. We have utilized only the unprocessed(“raw”) data products in our analysis. eters. We fixed the quadratic limb darkening coefficient ofthe primary to v = 0 . Kepler bandpass andhaving the stellar parameters T eff = 10000 K, log g = 4 . M / H ] = - .
10 as determined for KHWD3 and tabulated inthe
Kepler
Input Catalog. We adopted a linear limb darken-ing model for the secondary by fixing the quadratic term v tozero; the inclusion of this degree of freedom had little effecton the results but was added for completeness.The continuously defined model, ℓ ( t ), was numerically in-tegrated before being compared with the long cadence Ke-pler light curve. In detail, for each measured time, t j , wetake n j uniform samples t j , k = t j + k ∆ t j - τ int /
2, separated by ∆ t j = τ int / n j , over the long cadence integration interval of τ int = 30 minutes. Then, these times were evaluated with ℓ ( t )and averaged to determine the integrated model flux, F j , attime t j : F j = (cid:18) n j (cid:19) n j X k =0 ℓ ( t j , k ) . (5)The required number of samples, n j , depends on the curvatureof the model relative to the photometric precision at a giventime; n j > n j byincreasing its value until the goodness-of-fit statistic (definedbelow) plateaued. We settled upon n j = 60 for any t j within ∼ n j = 1 elsewhere.We determined the best fit model to the data by minimizingthe χ goodness-of-fit statistic, defined as χ = X s ( F s - F s ) σ (6)where F s is the measured flux at time t s and σ is the expectedstatistical error in the flux measurements. We selected σ =130 ppm such that the reduced χ was unity for the best fitsolution.The solid curves in Figures 1, 2, and 3 trace the best fitsolutions to the data. Markov chain Monte Carlo Analysis
We determined the posterior probability distributions forthe fitted parameters by employing a Markov chain MonteCarlo (MCMC) algorithm. With the MCMC algorithm, first, arandomly chosen parameter is perturbed by an amount drawnfrom a normal distribution of some fixed width. Next, thisnew parameter and the remaining unperturbed parameters areused to compute χ [as defined in Eqn. (6)] to determinea likelihood L ′ ∝ exp( - χ / L ′ is greater than the unperturbed likelihood L then the perturbation is accepted, otherwise it is accepted witha probability of L ′ / L . After many perturbations, the result-ing “chain” of accepted parameters samples the desired jointposterior probability distribution. For a more detailed intro-duction to the MCMC algorithm, refer to the appendix of thework by Tegmark et al. (2004).We generated a chain of 2 × links, having selected per-turbation widths such that ∼
40% of jumps were accepted.The chain was checked for adequate mixing and convergenceby visual inspection, and by observing that the number oflinks was much larger than the autocorrelation length (equal to epler
Hot White Dwarf Companion 3 F IG . 1.— Kepler light curve for KIC 10657664. The solid line gives the best fit solution to the data for the model given in §2. The dashed curve is the estimatedsystematic component of the full model. The lower panel shows residuals from subtracting the best fit model from the data (in units of parts per million).F IG . 2.— Period-folded, systematic corrected and normalized Kepler light curve for KIC 10657664 near the phases of eclipse and transit. The solid lines givethe best fit solution for the model described in §2. The lower panel shows residuals from subtracting the best fit model from the data. the number of links at which the chain autocorrelation dropsbelow one half) for any selected parameter. We report, in Ta-ble 1, the 50%, 15 .
8% and 84 .
2% values (corresponding tothe median and ± R / R , F / F , a / R , i and u to demonstrate their correlations. The remaining pa-rameters, including the harmonic function amplitudes, wereweakly correlated with other parameters. Out-of-eclipse harmonic content
We measured amplitudes (being significantly different fromzero) for, in order of decreasing amplitude, the cos(2 φ ),cos( φ ), sin( φ ), sin(2 φ ) and cos(3 φ ) modes of the out-of-eclipse variation. Repeating the above analysis including onlyan integral number of orbital cycles did not modify these re-sults. The four largest modes are plotted individually for com-parison in Figure 3. The fitted values of the 6 harmonic am-plitudes are summarized in Table 1. In §3, we associate these modes with three possible phys-ical effects and describe an analysis used to determine addi-tional system parameters (such as the mass ratio q ), assumingthe validity of these associations. Correlated Noise – We have assumed a “white” noise spec-trum in both our best-fit and MCMC analyses. However, wenote that correlated noise could have a significant affect onour estimates of harmonic content. To investigate this possi-bility further, we completed a secondary analysis, as follows,whereby we determined what we consider to be the most con-servative errors on these amplitudes.We first corrected for the low-frequency trend by divid-ing the data by the best-fit estimate for p Q ( t ). Then, havingfixed the best-fit estimates for the eclipse parameters, we fit aunique harmonic model to each individual cycle (i.e., by usingonly the data from one primary transit to the next). We thendetermined the mean amplitude, for each mode, and the stan-dard deviation in that mean averaging over the 12 availablecycles.We found that the scatter across all cycles, in a given har- Carter, Rappaport, & Fabrycky F IG . 3.— Period-folded, systematic-corrected and normalized Kepler light curve for KIC 10657664 showing the harmonic content. The solid lines give the bestfit solution to the data for the model given in §2. The four highest amplitude harmonics are plotted individually. monic amplitude, was consistent with random noise (i.e., nominority of cycles was significantly biasing the mean). Themeans across all modes agree with the MCMC-determinedamplitude medians within the MCMC-determined errors. Thestandard deviations in the means were larger, as may be ex-pected in the presence of correlated noise, than the amplitudeuncertainties determined via MCMC. However, only the sin φ and cos φ modes have substantially higher uncertainties, witheach being roughly twice as large as the MCMC estimates.These inflated, and likely overestimated, errors translate intoa ∼
10% correction in the uncertainty in the secondary massestimate (as determined in §3). All other system parameterswere less significantly affected. We therefore decided to usethe MCMC error estimates in the subsequent analysis.
Eccentricity
We also report a weak detection of eccentricity for thissystem, having measured a non-zero value for e cos( v ) =0 . ± . ∼ . ∼
30 seconds. This correctioncorresponds to an error in e cos v that is smaller than the esti-mated statistical error. In §3.6.1, we comment on the conse-quences of a non-zero eccentricity in the interpretation of theharmonic modes of the out-of-eclipse light curve. LIGHT CURVE MODEL-INDEPENDENT SYSTEM PARAMETERS
From the measured values of R / R and F / F we may de-termine the mean surface brightness ratio B / B between thesecondary and primary according to F F (cid:18) R R (cid:19) = B / B ≃ exp( hc /λ kT ) - hc /λ kT ) - λ ≈ TABLE 1KHWD3 L
IGHT C URVE M ODEL P ARAMETERS
Parameter Median 84.2% (+1 σ ) 15.8% (-1 σ ) P [day] 3.273713 +0.000008 -0.000008 t ic [BJD] 2454951.85857 +0.00012 -0.00012 e cos( v ) 0.0029 +0.0005 -0.0005 | e sin( v ) | . . a Eclipse Parameters R / R [ × ] 8.100 +0.024 -0.020 F / F [ × ] 1.900 +0.002 -0.002 a / R i [deg] 84.46 +0.14 -0.18 u u unconstrained b Out-of-eclipse Harmonic Amplitudes [ × ] A [sin( φ )] 1.03 +0.04 -0.04 B [cos( φ )] 2.64 +0.04 -0.04 A [sin(2 φ )] -0.44 +0.04 -0.04 B [cos(2 φ )] -5.68 +0.04 -0.04 A [sin(3 φ )] 0.02 +0.04 -0.04 B [cos(3 φ )] -0.13 +0.05 -0.05 a e sin v is nearly unconstrained by the data (and uncorrelated with the re-maining parameters). Here we report the full range of its weakly convergedMarkov chain. b The limb darkening parameter of the secondary was limited to positive val-ues less than 0 . Kepler ). To determine this brightness ratio in prac-tice, for a given set of temperatures, we integrated each black-body spectrum over the wide
Kepler response function.The harmonic content of the out-of-eclipse light curve vari-ation for KHWD3 is presumed to be astrophysical, i.e., relatedto the binary system, rather than systematics. In the previoussection, we reported on the measured values of the harmonicamplitudes, A k and B k for k = 1, 2, or 3, where we remind thereader that the out of eclipse variation was modeled as h ( t ) = 1 + X k =1 A k sin [ k φ ( t )] + X k =1 B k cos [ k φ ( t )] . (8)Briefly, we expect that the sin( φ ) term is due to Dopplerboosting (hereafter “DB”), the cos( φ ) term is largely due tomutual illumination (hereafter “ILL”) between the two com- epler Hot White Dwarf Companion 5ponents, and the cos(2 φ ) and cos(3 φ ) terms are largely due tothe ellipsoidal shape deformation of the primary by the sec-ondary’s tidal field (so-called ellipsoidal light variation, here-after “ELV”). In this nominal model, we do not expect non-zero amplitudes for the sin(2 φ ) or sin(3 φ ) modes. Expected harmonic amplitudes
We assume a circular binary orbit throughout the followingdiscussion. A departure from this assumption is discussed in§3.6.1.The predicted amplitudes A k , B k may be decomposed into asum of effect-specific amplitudes such that A k = A DB k + A ELV k + A ILL k (9) B k = B DB k + B ELV k + B ILL k . (10)In turn, the effect-specific amplitudes are a sum of the indi-vidual contributions from each member of the binary; for ex-ample, A DB k = A DB , k + A DB , k . Doppler boosting – The apparent spectral intensity of eithercomponent changes throughout the orbit as a result of Dopplershifting, photon emission rate modulation and beaming. Thecombination of these effects is called Doppler boosting; it isdiscussed in the context of
Kepler by Loeb & Gaudi (2003)and by Zucker et al. (2007) and was detected for at least oneof the two other candidate hot white dwarf companions in the
Kepler field (KOI-74 and KOI-81; van Kerkwijk et al. 2010)and for one planetary system (CoRoT-3; Mazeh & Faigler,2010). The time variability of the Doppler boosting signalis carried by the projected radial velocity v r , (1 , where, for acircular orbit, v r , (1 , ∝ sin φ and the total contribution to theharmonic out-of-eclipse content by Doppler boosting is A DB1 = α (cid:16) v r , c (cid:17) + α (cid:16) v r , c (cid:17) = (cid:18) π aPc (cid:19) " α (cid:18) + q (cid:19) - - α (1 + q ) - F F . (11)The DB prefactor ( α ) depends on the shape of the emissionspectrum in the emitter’s rest frame such that α ≃ - h d ln F ν d ln ν i (12)where the average is weighted over the Kepler response. Ithas been empirically validated in the case of KOI-74 (Ehren-reich et al. 2010) that prefactors estimated assuming a black-body emission model are overestimated by greater than 20%for stars with T ≈ Kepler response function. For the A star primary, we selectedmodel spectra of fixed gravity (log g = 4 . α ( T ) ≈ . - . (cid:18) T - (cid:19) (13) The DB prefactors for the WD component [ α ( T )] werecomputed analogously, but assuming a simple black-bodyemission model. Ellipsoidal light variation – The mutual gravitational inter-action between members of a binary will induce nominallyprolate distortions of their surfaces having major axes alongthe line connecting their centers. As a result, the sky-projectedcross-sectional area, and consequently the observed flux, aretime variable (Morris 1985; see also Pfahl, Arras, & Paxton2008). For a circular orbit, the expected maxima of this vari-ation occur twice per orbit at the quadratures [generating alarge cos(2 φ ) amplitude]. Kopal (1959) calculated the ELVcontribution to the out-of-eclipse harmonic content as (wherewe follow the notation by Morris 1985): B ELV1 ≃ - Z (3) q (cid:18) R a (cid:19) sin i (14) B ELV2 ≃ - Z (2) q (cid:18) R a (cid:19) sin i (15) B ELV3 ≃ - Z (3) q (cid:18) R a (cid:19) sin i (16)where the prefactors Z (2) and Z (3) (Eqn. 3 in the work byMorris 1985) depend on limb-darkening and gravity darken-ing parameters with Z (2) ≈ . Z (3) / Z (2) ≈ .
05. Inour calculations, we assume a fixed linear limb-darkening pa-rameter [ u = 0 .
44; according to Sing (2009) for the
Kepler
Input Catalog stellar parameters listed in §2.1] and a gravitydarkening parameter that depends on temperature accordingto the von Zeipel law (Eqn. 10 in the work by Morris 1985).The contribution to the ELV due to the distortion of thesecondary by the primary is smaller by a factor of at least( R / R ) ( F / F ) / q ∼ .
05% and is therefore safely ignored.
Mutual illumination – It is expected that radiation arisingfrom one binary component and incident on the other willsubsequently either be scattered or absorbed and reemitted.As such, the surfaces of the components that are directly fac-ing one another will be brighter than other regions. As thebinary orbits, we perceive these phases as a modulation ofthe out-of-eclipse total flux level. The maximum illumina-tion of the primary is at inferior conjunction while that of thesecondary is at superior conjunction, contributing with oppo-site signs to a predominately cos φ mode of the out-of-eclipsevariation. If the binary is in radiative equilibrium, then all in-cident radiation (bolometric) must be absorbed and reemittedat approximately the effective temperature of the illuminatedmember (Eddington 1926). In this scenario, Kopal (1959)again provides the expected out-of-eclipse harmonic contentdue to mutual illumination as B ILL ≃ (cid:18) R a (cid:19) (cid:20) + (cid:18) R a (cid:19)(cid:21) (cid:18) T T (cid:19) (cid:18) R R (cid:19) - (cid:18) R a (cid:19) (cid:20) + (cid:18) R a (cid:19)(cid:21) BC BC (17) B ILL ≃ (cid:18) R a (cid:19) (cid:20) π + (cid:18) R a (cid:19)(cid:21) (cid:18) T T (cid:19) (cid:18) R R (cid:19) + (cid:18) R a (cid:19) (cid:20) π + (cid:18) R a (cid:19)(cid:21) BC BC (18)where BC is an approximate bolometric correction to the vi- Carter, Rappaport, & Fabryckysual (see Allen 1964) with - . BC = - . +
10 log T eff + , T eff . (19) First cut at determining the constituent masses: aninconsistency
In principle, we may solve for the system parameters M , T , R , M , T , R and a using the formalism introduced in theprevious sections (Eqns. 7, 11-19), Kepler’s third law, and theresults from the light curve analysis presented in §2. However,the illumination effect (given by Eqns. 17 and 18) is nearly in-dependent of the system masses, which are largely determinedby the Doppler boosting (Eq. 11) and ELV effects (Eqns. 14-16). If we take the simplest forms of both the DB and ELVeffects we find: A DB1 ≃ . (cid:18) π aPc (cid:19) (cid:18) q + q (cid:19) (20) B ELV2 ≃ - . q (cid:18) R a (cid:19) (21)where the coefficients α and Z (2) have been set equal totheir nominal values, i has been set to 90 ◦ without loss of ac-curacy, and q ≡ M / M . For purpose of this particular ex-ercise, we have neglected the relatively small effect of theDoppler boosting of the hot white dwarf. If we eliminate a in favor of P through Kepler’s third law, and plug in our mea-sured values for P and R / a , we find: A DB1 ≃ . × - m / q (1 + q ) / ≃ . × - (22) B ELV2 ≃ - . × - q ≃ - . × - (23)where the numerical values on the right sides of the equa-tions are taken from Table 1, and the mass of the primary, m has been expressed in solar units. Since these equationsdepend on q and only very weakly on the actual mass of theprimary, they both, in effect, independently determine q . Forany primary mass between 2 and 3 M ⊙ , the two equationsyield: q ≃ . .
090 and q ≃ .
13, respectively, depend-ing on whether we utilize the DB or ELV amplitudes to de-termine q . These are clearly inconsistent, and this remainsthe case, even when we later utilize all the terms in Eqns. (7),(11)–(19), and take full account of the statistical uncertaintiesassociated with the measure quantities. Since there is a de-pendence on M in one of the two equations, there exists aformal solution to the two equations for M and M (0.7 M ⊙ and 0.1 M ⊙ , respectively). However, this solution is not phys-ically plausible, and therefore we are left with this intriguinginconsistency in the value of q that we determine. Roughly determined parameters: T and ρ Once we decide on the mass ratio for the system, we willneed an estimate of the mass of the primary in order to deter-mine the mass of the white dwarf. There are two accessibleparameters that we can utilize to estimate the primary mass:the effective temperature, T , and the mean stellar density, ρ . T is listed for the primary star in the Kepler
Input Catalog(KIC) as 10,500 K, based on 5-color photometry. We have noreadily available estimate of the uncertainty in the KIC valuefor T , but we estimate that it could be ∼ ±
500 K. As we shallsee in §3.5, the illumination effect can be used to directly in-fer the effective temperature of the primary, and yields a result of 9500 ±
150 K, which would then be some 2 σ away fromthe KIC tabulated value. We therefore consider a range for T that includes both the KIC value and the one we derive fromthe illumination effect.The second readily, and more accurately, determined pa-rameter associated with the primary is its mean stellar density, ρ ≡ M / (4 / π R ). We point out that ρ is only a functionof the light curve-determined parameters P and a / R , with aweak dependence on q . This dependence of ρ on these pa-rameters may be found by dividing Kepler’s third law by R : (cid:18) aR (cid:19) = 13 π GP ρ (1 + q ) (24)This relation was pointed out by Seager & Mallén-Ornelas(2003) in the context of transiting exoplanets. With exoplan-ets, the value of q is much smaller than the error in a / R suchthat ρ can be determined independently of any mass infor-mation. While the same is not quite true with our binary, thevalue of q is still small enough (with q . . ρ may be determined independently of q . In particular, we findthat ρ ≈ . ± .
02 g cm - by using Eqn. (24) with q = 0and the results from our photometric analysis. We considerthis estimate to be maximal and accurate to within 20% of itstrue value. A prior on M In the previous section, we argued that T ∼ ρ . - . Given these two values, we can moti-vate a prior on M .If we assume the primary is on the zero age main sequence(ZAMS), then we may utilize the approximate formula byEggleton (2006) to estimate its mass as a function of eithereffective temperature or density. Picking the former to bein the range 9000 K . T . . M / M ⊙ . - . ρ . - , we find a mass 2.5 . M / M ⊙ . . M intersect, suggesting M ∼ . M ⊙ ; alter-natively, either the primary has aged somewhat off the ZAMSor the metallicity of the primary is different from Solar. Toinvestigate these possibilities further, we utilized the Yonsei-Yale stellar evolution isochrones (Yi et al. 2001) to deter-mine a likely mass range subject to the priors on tempera-ture and density as given above and additionally a normallydistributed prior on the metallicity, [Fe/H] = 0.0 ± . M = 2 . + . - . M ⊙ with an estimatedage for the star in the range 200–600 Myr; although, given alikely mass transfer history (see §5), the interpretation of thisage estimate is unclear.Considering the above ranges in M , we opted for a conser-vative mass prior for the primary being normally distributedabout M = 2 . M ⊙ with an rms width 0 . M ⊙ . Derived system parameters subject to the prior on M In the following analysis of the system parameters weassume that the illumination model is reasonably correct(Eqns. 17 and 18), and we utilize either the ELV effect(Eqns. 14-16) or the Doppler boosting effect (Eqns. 11 and12). Given the common prior on M (see the previous sec-tion), either effect (ELV or DB) results in an independent esti-mate of q . As discussed earlier, however, we find that these es-timates disagree with one another: ignoring DB [by nullifying epler Hot White Dwarf Companion 7the statistical weight of the measurement of the sin( φ ) mode]gives q ≃ . ± .
01, whereas ignoring ELV [by nullifyingthe statistical weight of the measurement of the cos(2 φ ) mode]gives q ≃ . ± .
01. Given this discrepancy, we opted to de-termine and report system parameters for these two scenarios separately .To determine posterior distributions for all system parame-ters, we executed another Markov chain Monte Carlo algo-rithm (as described in §2.2) subject to the likelihood
L ∝ exp( - χ T /
2) where χ T = (cid:16) ∆ R R (cid:17) σ R / R + (cid:16) ∆ F F (cid:17) σ F / F + (cid:16) ∆ aR (cid:17) σ a / R + ( ∆ i ) σ i + ( ∆ P ) σ P + ρ R / R , a / R (cid:16) ∆ R R (cid:17) (cid:16) ∆ aR (cid:17) σ R / R σ a / R + ρ R / R , i (cid:16) ∆ R R (cid:17) ( ∆ i ) σ R / R σ i + ρ a / R , i (cid:16) ∆ aR (cid:17) ( ∆ i ) σ a / R σ i + X k ( ∆ A k ) σ A k + X k ( ∆ B k ) σ B k + (cid:0) M - . M ⊙ (cid:1) (cid:0) . M ⊙ (cid:1) (25)where for any parameter “ x ”, ∆ x is the difference between themedian estimate of the parameter x and its current value (at agiven link in the Markov chain), σ x is the variance in the vari-able x and ρ x , y is the correlation coefficient between variables x and y . The current values of the harmonic amplitudes A k and B k are determined as a function of the current values of the pa-rameters M , M , T , T , R , R using Eqns. (7), (11)–(19) andKepler’s third law. The medians, variances and correlation co-efficients are fixed to their values as calculated using the jointposterior distribution resulting from the earlier analysis of thelight curve data (as described in §2).To ignore, in effect, the statistical influence of DB or ELV,we remove the terms in χ T associated with A [sin( φ )] or B [cos(2 φ )], respectively. In each scenario, a Markov chain oflength 5 × was generated and the resulting posterior dis-tributions were inspected to have adequately converged.We report the results of our analysis of the system parame-ters in Table 2. The two columns in the table are the systemparameters determined from the ELV and ILL effects, on theone hand, and the DB and ILL effects on the other. We listthe median values of the parameters, as well as the 15.4% and84.2% (corresponding to the ± T eff , masses, gravity, and so forth),the agreement between the two columns (ELV and DB dom-inated analysis) is remarkably good except for those param-eters that involve the white dwarf mass. The latter dependsheavily on the mass ratio which, in turn, is strongly depen-dent on the assumption of whether the ELV or DB amplitudescorrectly reflect their respective physical quantities.The estimated masses, temperatures and radii of the sec-ondary, in either of the two scenarios, are consistent with ahot white dwarf companion. In §5, we discuss possible evolu- TABLE 2KHWD3 M
ODEL -I NDEPENDENT S YSTEM P ARAMETERS
Parameter ELV+ILL Model DB+ILL Model a [AU] 0.061 ± ± M / M ± ± T / T ± ± M [ M ⊙ ] 2 . ± . a . ± . a R [ R ⊙ ] 1.87 ± ± T [K] 9600 ±
150 9500 ± L [ L ⊙ ] 26.2 ± ± ρ [g/cc] 0.54 ± ± g ) [cgs] 4.29 ± ± M [ M ⊙ ] 0.37 ± ± R [ R ⊙ ] 0.151 ± ± T [K] 14900 ±
300 14600 ± L [ L ⊙ ] 1.00 ± ± ρ [g/cc] 153 ±
11 109 ± g ) [cgs] 5.65 ± ± × ] A [sin( φ )] -0.48 ± b ± B [cos( φ )] 0.01 ± ± A [sin(2 φ )] -0.44 ± b -0.44 ± b B [cos(2 φ )] -0.00 ± ± b A [sin(3 φ )] 0.02 ± ± B [cos(3 φ )] 0.08 ± ± OTE . — a = Mass of the primary is assumed. b = Values did not affectthe likelihood. See § 3 for details. tionary histories leading to these two outcomes and estimatethe properties the system progenitors.
Possible missing physics
In the previous sections, we inferred system parameters forKHWD3 only after discounting certain aspects of our modelfor the out-of-eclipse harmonic content in order to resolvesome discrepancies in the measured amplitudes that exceedthe estimated statistical uncertainties. At this time we admit-tedly have no satisfactory explanation for these discrepancies,though it is comforting that they do not affect any of our sys-tem parameter determinations other than those involving thewhite-dwarf mass – and even then the white dwarf mass isuncertain by at most a factor of 2. We note that, to the bestof our knowledge, this is the first time that all three effects,DB, ILL, and ELV, have been studied at such small ampli-tudes (e.g., parts in 10 ). However, in this regard, we wouldhave surmised that the simple, classical, analytic models forthese effects should work even better at these smaller ampli-tudes than at larger amplitudes. It is also possible that thereare some additional, heretofore unexplained, physical effectswhich complement the harmonic content that we have not at-tempted to model, e.g. the O’Connell effect (see, for example,Davidge & Milone 1984). Non-zero eccentricity
We measured a tentative non-zero value for the eccentric-ity e , at weak significance, with e cos v = 0 . ± . a / R in the model expressions Carter, Rappaport, & Fabrycky F IG . 4.— Joint posterior distributions (showing 68% and 95% confidence contours) and histograms for KHWD3 light curve model and system parameters. Theupper left portion shows some results from the analysis, as described in §2, of the Kepler light curve data for KHWD3. See Table 1 for complete results. Thelower right portion shows some results from a subsequent analysis, as described in §3, to determine absolute system parameters for KHWD3 given the resultsfrom the photometric analysis. The shaded contours and histograms correspond to the scenario in which only ellipsoidal light variation and illumination wereassessed. The non-shaded contours correspond to the scenario in which only the Doppler boosting and illumination were assessed. See §3 for more details. (Eqns. 11-19) is replaced with the instantaneous separation r / R , which is a function of φ , e cos v and e sin v . We findthat DB is the only effect that can have a significantly alteredharmonic spectrum [to linear order in eccentricity and givenour restriction on e cos v ]. This alteration is entirely mani-fested in the inclusion of a sin(2 φ ) mode whose amplituderelative to the dominant sin φ mode is A A ≈ e sin v . (26)We did detect a non-zero amplitude A / A = - . ± . Departure from synchrony epler
Hot White Dwarf Companion 9We have assumed synchronous rotation between the pri-mary, secondary, and orbit when calculating the contributionto the harmonic content from ELV or ILL (as described in§3.1). Should the primary be rotating at a different rate fromthe orbital frequency, frictional drag within the primary mayforce the tidal distortion due to the secondary to lag or leadthe orbit (as is the case with the tides imposed on Earth by theMoon). The illumination effect may also lag or lead as a re-sult of both the modified phase variation of the sky-projectedcross-sectional area and additionally, depending on the depthin the stellar atmosphere at which radiant energy from the sec-ondary is deposited, from a “hot spot” that is shifted from thesubstellar point.To investigate this possibility further, we carried out ananalysis where we assumed that both the ELV and ILL ef-fects are shifted by a common variable phase from their nom-inal circulation. DB is independent of the assumption ofsynchrony and was not permitted a phase shift. Given thisadditional degree of freedom, we find that we may accountfor the measured values of all harmonic amplitudes includ-ing the relatively large, and previously unexplained, sin(2 φ )mode. However, the results of this analysis remain unphysi-cal ( M ≈ . M ⊙ and R ≈ R ⊙ ). Moreover, we deter-mined that these effects lag the orbit by 5 ◦ while reasonableexpectations for the efficiency of tidal dissipation (quantifiedvia the parameter Q ) would predict maximal phase offsets of1 / Q ∼ - . Other effects
Here, we mention two other effects not included in ourmodel that could significantly affect our interpretation of theharmonic content and/or the photometric results:• Stellar spotting – A persistent stellar spot on a syn-chronously rotating primary could induce variationswith power in multiple harmonics, most likely in thefundamental mode (competing with DB & ILL). Theamplitude of this effect depends on the size and fluxcontrast, relative to the unspotted photosphere, of thespot. While spots in tidally locked binaries may persistfor months, it is unlikely that an A star lacking a sub-stantial convective outer envelope would exhibit signif-icant spotting behavior (Strassmeier 2009). A possibleexception would be a magnetically active A star (e.g., a“peculiar” or Ap star; see Kochukhov (2010) and refer-ences therein).• Rapid rotation of the primary – In the previous section,we discussed possible phase lags in the ELV and ILLeffects as a result of non-synchronous rotation. In ad-dition, the primary may be significantly oblate as a re-sult of very fast rotation. In this case, a / R as inferredin the light curve analysis may be different from thetrue normalized semi-major axis. In particular, if thespin axes of the rapidly rotating primary and the orbitare aligned, then a / R would likely be overestimated(Barnes 2009). Additionally, very fast rotation may in-duce an unexpected stellar brightness profile as a re-sult of strong gravity darkening; both the inferred ra-dius ratio, R / R , and a / R may be inaccurate as result(Barnes 2009). However, given the short orbital periodof 3.3 days, it is most likely that the orbit and the pri-mary star have already synchronized (see, e.g., Torres,Andersen, & Gimenez 2010). Erroneous a / R ? It should be noted that a smaller value for a / R , closer to6.2 than its current estimate of 7 ± .
1, would resolve the dis-crepancy between ELV and DB amplitudes for M ≃ . M ⊙ .If this lower value where accurate, then q ≃ .
11, as wouldbe estimated using the DB amplitude alone. However, thetemperatures of the primary and secondary would be reduced,according to the ILL model, with lower values of a / R . Inparticular we find, T ≃ a / R = 6 .
15. Additionally,the inverse correlation between R / R and a / R (see Fig. 4)suggests a relatively larger white dwarf radius for a smaller a / R .We have taken care to correctly model the transit andeclipse light curves (see §2) but note that, given the extremecorrelation between inclination and a / R (see Fig. 4), smallerrors, model inaccuracies, or non-uniform data samplingmay lead to systematically inaccurate values for a / R thatvary greatly in an absolute sense.We remark that the ∼
40 data points occurring during theingress or egress phases of the secondary eclipse carry theoverwhelming majority of the statistical weight in determin-ing a / R , i and e cos( v ). While the MCMC algorithm (§2.2)will likely yield a robust measure of parameter precision andcorrelation, the accuracy of the most-likely value is affectedby correlated noise and the (sparse) distribution of data pointsduring eclipse ingress or egress. Compiling future Kepler datafor KHWD3 will most likely resolve any current statistical bi-ases. INFERRED MASS OF THE WHITE DWARF
Recall from §3 that, depending on whether our analy-sis emphasized the ELV and illumination amplitudes or theDoppler boosting and illumination amplitudes, we find that q = 0 . ± .
006 or q = 0 . ± . q , we would infer awhite dwarf mass of M = (0 . ± . (cid:18) M . M ⊙ (cid:19) M ⊙ ELV & ILL (27) M = (0 . ± . (cid:18) M . M ⊙ (cid:19) M ⊙ DB & ILL (28)(see Table 2). We discuss the consequences of these values forthe binary stellar evolution of the system in the next section. SYSTEM PROGENITORS
The presence of a white dwarf in KHWD3, coupled withthe short orbital period (3.273 days), indicate that there wasnecessarily a phase of mass transfer/loss during the prior evo-lution of this binary. One obvious constraint on the primordialbinary is that it must have had a total mass, M p + M s , equalto at least the current mass of the binary M + M ≃ . M ⊙ ,where M p and M s are the initial masses of the primordial pri-mary (the white dwarf progenitor) and the secondary, respec-tively. As is readily evident, and we explore below, this im-plies that the mass of the primordial primary was greater than1 . M ⊙ , and therefore had a radiative envelope at the timewhen mass transfer to the primordial secondary commenced.In turn, this indicates that the mass transfer from the primaryto the secondary took place, at least initially, on a thermaltimescale.Given the two different possible masses for the white dwarf(see §4), there are two slightly different evolutionary scenar-0 Carter, Rappaport, & Fabryckyios for the formation of the white dwarf. We discuss each ofthese in turn. Primordial Primaries with M . . M ⊙ Even if the donor star is more massive than the accretor, themass transfer can still proceed quite stably if the donor hasa radiative envelope. The mass ratio of the primordial pro-genitor binary, q prog ≡ M p / M s , up to which the mass transferis stable, depends on the masses and the orbital period whenmass transfer commences, but can exceed q prog ∼
2. For starswith initial mass . . M ⊙ , a degenerate He core developsand there is a nearly unique relation between the radius of theevolving star and the core mass. This leads to a tight relationbetween the mass of the remnant white dwarf, once the en-velope of the primary has been completely transferred to thesecondary and/or lost from the system, and the orbital period: P orb ≃ . × M (1 + M . + M ) / days (29)where M wd is in units of M ⊙ (see, e.g., Rappaport et al. 1995;Ergma 1996; Tauris & Savonije 1999; Lin et al. 2010). Ifwe solve this non-linear equation for the value of M wd thatmatches the current orbital period of 3.273 days, we find: M wd ≡ M ≃ . ± . M ⊙ (30)where the estimate in M wd is uncertain by ∼
10% (Rappaportet al. 1995). This mass range is marginally consistent withthe lower of the two possible masses measured for the whitedwarf in KHWD3 (0 . ± . M ⊙ ).Another major issue that needs to be addressed regardingthe white dwarf is how to explain its large radius ( R = 0 . ± . R ⊙ ) and high effective temperature ( T = 14 , ± - M ⊙ with a H-rich atmosphere of0.01 M ⊙ can remain this large and hot for ∼
150 Myr, while a0.19 M ⊙ He dwarf with a similar H-rich atmosphere can re-main thermally bloated and hot for up to 300 Myr. Even inthe case of 0.21 M ⊙ white dwarf, 150 Myr is a non-negligiblefraction of the nuclear evolution time of the ∼ . M ⊙ par-ent star. Degenerate He white dwarfs with masses & . M ⊙ would not remain hot and thermally bloated for nearly longenough to have a plausible probability of catching them insuch a state. Therefore, the lower limit of the measured rangeof 0 . ± . M ⊙ would be strongly preferred in this sce-nario. Primordial Primaries with M & . M ⊙ Stars with mass & . M ⊙ do not evolve with degeneratecores and their radius does not follow the radius–core massrelation discussed above. In a close binary system, such starsare less likely to produce a remnant He core with a massas low as ∼ M ⊙ . However, they could produce awhite dwarf that matches the more massive of our allowedsolutions, namely ∼ . ± . M ⊙ . Such non-degenerate He stars, once the envelope of the primordial primary has been re-moved, would be on the He-burning main sequence (as longas M core & . M ⊙ ), and they could be quite luminous.An analytic fitting function for the luminosity of naked Heburning stars is: L He = 1 . × M . M + . M . + . M + . L ⊙ (31)(Hurley, Pols, & Tout 2000; hereafter “HPT”). The observedluminosity of the white dwarf in KHWD3 is L ≃ . ± . L ⊙ . The He star mass that corresponds to this luminosity,according to the above fitting formula, is down near the end ofthe He-burning main sequence with M He . . M ⊙ . The ra-dius and effective temperature of such a He burning star wouldbe ∼ . R ⊙ and ∼ & . M ⊙ ; see Han et al. 2002; Han et al. 2003). Thenuclear lifetime of such a He-burning star, i.e., near the endof the He-burning main sequence, is ∼ yr. Such a starwould be at the low-mass end of what are known as subdwarfB (sdB) stars (see, e.g., Han et al. 2003, their figures 15 and17). Possibility of a Prior Common-Envelope Phase
We note that unstable mass transfer leading to the currentsystem probably could not have resulted in a common enve-lope (“CE”) phase leading to the successful ejection of theCE. The ejection of a common envelope around such a low-mass core would have resulted in a post-CE orbital period thatis considerably shorter than 3 days. The only way in whicha 3-day post-CE binary would ensue is if the pre-CE orbitalperiod were extremely long – in which case the white dwarfmass would be much higher than is observed. For an im-pressive study of such post common-envelope systems see thework by Parsons et al. (2010).
Constraints on M p , M s and P orb , init We explore here what the range of possible and likely pa-rameters the primordial progenitor binary could have been.We know three independent binary parameters of the currentsystem: M , M , and P orb . The appropriate application ofconservation of mass and angular momentum yield two con-straints – insufficient to uniquely identify, by themselves, theinitial system parameters. However, there are two other con-straints, discussed here, that are sufficient to define a relativelynarrow range in parameter space for the primordial binary.As mass is transferred from the primary to the secondary,a fraction of it, β , will be retained by the secondary, and theremainder will be lost from the system. The ejected mass willcarry away an average specific angular momentum which wedenote as α which is in units of the specific angular momen-tum of the binary. Unfortunately, we do not know a priori either α or β , but we can make an educated guess about theformer. In terms of the quantities we have already definedearlier, we can write the mass retention fraction as: β = M - M s M p - M (32) epler Hot White Dwarf Companion 11Conservation of angular momentum then yields a relation be-tween the orbital periods before ( P i ) and after ( P f ) the masstransfer phase: P i = P f (cid:18) M p + M s M + M (cid:19) (cid:18) M p M (cid:19) C (cid:18) M s M (cid:19) C (33)where the powers, C and C are defined as C = 3 α (1 - β ) - C = - α (1 - β ) /β - ∼ M ⊙ , we can constrain P i to beshorter than P f = 3 . M p & . M ⊙ ,the initial orbital period could, in principle have been longerthan 3.273 days, and the orbit could have subsequently shrunkduring mass transfer. However, for substantially longer initial P orb , the core mass would likely exceed ∼ M ⊙ .Another constraint that we can impose is that the initial or-bital separation should be large enough so that not only canboth the primordial primary and secondary fit within their re-spective Roche lobes, but that there is sufficient room for theprimary to evolve a substantial He core before the mass trans-fer commences. This constraint can be written as: R p ( M p ) = ξ f ( q ) a = ξ f ( q ) (cid:2) G ( M p + M s ) (cid:3) / (cid:18) P i π (cid:19) / (36)where ξ is the fraction of its Roche lobe that is filled by theprimordial primary of radius R p ; f ( q ) is the ratio of the Roche-lobe radius to orbital separation, a , which depends only onthe mass ratio, q ; and P i is the orbital period before the masstransfer commences. We take as a somewhat arbitrary butquite reasonable constraint: ξ . / τ τ ≃ (cid:18) M p M s (cid:19) - . . (37)We require that this ratio be smaller than ∼ . . τ /τ . . α = 1 .
0; however, we also consider valuesof α = 0 .
75 and 1.5. This is a reasonable set of bounds since,for example, the specific angular momentum parameter, α , atthe L . . M s / M p . F IG . 5.— Regions in the plane of M p and M s where the progenitor binarythat produced KHWD3 could have originated. The three shaded regions cor-respond to the angular momentum loss parameters α = 0.75, 1.0, and 1.50,from top right to lower left. The upper left and lower right boundaries of eachregion are determined by τ / τ = 0 . P orb = 3 .
273 days, while the lower left boundaries areset by allowing the primordial primary to fill no more than half of its Rochelobe. The contours are of constant mass retention fraction, β .F IG . 6.— Comparison of the four known systems with white dwarfs thatare relics of stable Roche-lobe overflow. The plotted show the orbital pe-riod vs. the white dwarf mass for KHWD3, KOI-74, KOI-81, and Regu-lus. The heavy curve is the theoretically derived P orb ( M wd ) relation, goingas ∼ M for the lowest mass white dwarfs, while the pair of dashed curvesmarks the estimated theoretical uncertainties (see text for details and Rappa-port et al. 1995). Comparison of KHWD3 with KOI-74, KOI-81, andRegulus
We end our discussion of the prior evolutionary history ofKHWD3 by comparing this system with three other whitedwarfs in close binaries that also likely formed during aphase of stable Roche-lobe mass transfer. These are KOI-74, KOI-81, and Regulus with white dwarfs in orbital peri-ods of 5.3 days, 23 days, and 40 days, respectively (Rowe etal. 1010; van Kerkwijk et al. 2010; Gies et al. 2008; Rappaportet al. 2009). The measured white dwarf masses in these sys-tems are 0 . ± . ∼ . ± . M ⊙ . Includingthe lower-mass solution for KHWD3 (i.e., 0 . ± . M ⊙ ),we find a modestly significant correlation between masses andincreasing P orb . This correlation is plotted in Fig. 6 where thesolid curve is the theoretically expected relation for progenitorstars of mass . . M ⊙ (see eq. 29 above). The white dwarfmasses are not very well determined, and there is only a smallspread in all their masses by a factor of ∼ P orb onthe mass of the white dwarf, the known systems are roughlyconsistent with this relation. More such systems, with evenbetter determined white dwarf masses, will be needed in or-der to supplement the kind of information currently being pro-vided by radio pulsar systems (Rappaport et al. 1995; Tauris1998; Thorsett & Chakrabarty 1999). SUMMARY AND CONCLUSIONS
In this work we have utilized public
Kepler data to identifyand study in detail a third hot white dwarf in a close binaryorbit. The system is formally known as KIC 10657664, andwe have given it the shorthand name “KHWD3”. The systemis comprised of a white dwarf with T eff ≃ ,
500 K orbitingan A star of mass ∼ M ⊙ . The white dwarf is extremelythermally bloated with a radius of 0 . R ⊙ , some 7 -
10 timeslarger than its degenerate radius.We have combined the transit and eclipse data as well asthe low-amplitude, out-of-transit/eclipse light curve to inferall the system parameters. The large size of the white dwarf,coupled with its high effective temperature, produces rela-tively large eclipse and transit depths ( ∼
2% and 0.7%, respec-tively). The out-of-transit/eclipse periodic light curves are ofmuch smaller amplitude (in the range of ∼ - )and are interpreted as being due to Doppler boosting, mutualillumination, and ellipsoidal light variations (due to tidal dis-tortions). The deduced system parameters are summarized inTable 2.The only ambiguity in the determination of the system pa-rameters lies in the mass of the white dwarf. Our analysisleads to two somewhat distinct, and currently unresolvable,possible solutions: 0 . ± . M ⊙ or 0 . ± . M ⊙ . Thedifficulty in distinguishing between these two possibilities forthe white dwarf mass lies in the fact that both the Doppler ef-fect and the ELV both basically determine the mass ratio, andthese two amplitudes in the light curve produce somewhat in-consistent results in this ratio (at the ∼ σ level). However,it is important to note that the remainder of the important sys-tem parameters are well determined, in spite of this particular ambiguity.We have also briefly explored what the possible progenitorsof this system might have been. The system almost certainlyformed after a phase of mass transfer from the primordial pri-mary star (now the white dwarf) to the secondary star (nowthe A star primary of the system). If the primordial primaryhad a mass less than ∼ M ⊙ , its core mass would directlydetermine the final orbital period (3.273 days), which predictsthe current white dwarf mass to be 0 . ± . M ⊙ , in modestagreement with the lower of our two solutions. Such a low-mass white dwarf (i.e., 0 . - . M ⊙ ) could remain hot andbloated for a substantial fraction (i.e., & & . M ⊙ , then it is likely that our solution for a moremassive white dwarf is the correct one. In this case, however,the white dwarf would almost certainly be undergoing nuclearburning at the current epoch. This, in turn, would imply thatthe mass would have to be low ( ≃ . - . M ⊙ ), i.e., nearthe end of the He-burning main sequence, in order to explaina luminosity of . L ⊙ .Likely primordial stars of mass M p ≃ . M ⊙ and M s ≃ . M ⊙ (see Fig. 5) would imply a highly non-conservativephase of mass transfer with perhaps half the transferred massbeing lost from the system. For this particular set of illus-trative primordial binary masses, the mass ratio is sufficientlylarge to possibly explain the high mass loss rate. We havecompared KHWD3 with the two other hot white dwarfs dis-covered with Kepler as well as the Regulus system. The initialsystem masses for Regulus (see Rappaport et al. 2009) were M p ≃ . M ⊙ and M s ≃ . M ⊙ , which are seemingly not toodifferent than for KHWD3. However, the final orbital periodof the white dwarf in the Regulus system is 40 days, which isconsiderably longer than the 3.3-day period of KHWD3. Thedifference likely lies in the fact that in the Regulus systemthe mass transfer was considerably more conservative (lead-ing to a much wider orbit; see Eqn. 33). This is quite plausiblegiven that the initial mass ratio of ∼ ∼ REFERENCESAllen, C.W. 1964, Astrophysical Quantities, The Athlone Press, Universityof LondonBarnes, J. W. 2009, ApJ, 705, 683Carter, J. A., Winn, J. N., Gilliland, R., & Holman, M. J. 2009, ApJ, 696, 241Davidge, T.J., & Milone, E.F. 1984, ApJS, 55, 571 Di Stefano, R. 2010, arXiv:1002.3009Eddington, A. S. 1926, MNRAS, 86, 320Eggleton, P. 2006, Evolutionary Processes in Binary and Multiple Stars, byPeter Eggleton, pp. . ISBN 0521855578. Cambridge, UK: CambridgeUniversity Press, 2006. epler