OBSERVATIONS OF DOPPLER BOOSTING IN KEPLER LIGHT CURVES

The transit and eclipse light curves were fitted with detailed models by R10, and we use their ephemerides and inclinations (see Table 1). For our purposes below, however, it is useful also to have the ratio R₁/a, where R₁ is the radius of the primary, and a the semi-major axis. This ratio is well constrained by the eclipse, depending on the eclipse duration t_e (at half maximum depth) and inclination i via sin²(πt_e/P)sin²i = (R₁/a)² − cos²i (Russell 1912). We measured t_e/P graphically and list the values, as well as the inferred ratios R₁/a in Table 1. Note that the derivation assumes that the stars are spherical; if they are rotating rapidly, then the constraint is mostly on the equatorial radius (see Section 2.6).

The measurement of R₁/a, combined with Kepler's law, constrains the mean density of the primary, $\overline{\rho }_1=(R_1/a)^{-3}(3\pi /GP^2[1+q])$. For both KOI 74 and KOI 81, the mass ratios q = M₂/M₁ are small, and, as discussed by R10, the densities inferred for q ≪ 1, of ∼0.45 and 0.17 g cm^-3, respectively, are consistent with those expected for main-sequence stars with masses of 2.2 and 2.7 M_☉, respectively, as inferred from their spectral types (see Table 1).

The ratio of the transit and eclipse depths _t and _e equals the ratio of the surface brightnesses. Thus, given the temperatures for the primaries inferred from their spectra, one can estimate the temperatures of the secondaries. Assuming blackbody emission, one has

$$\begin{equation} \frac{\epsilon _{e}}{\epsilon _{\rm t}} \simeq \frac{{e}^{hc/\lambda kT_1}-1}{{e}^{hc/\lambda kT_2}-1}. \end{equation} \tag{ 1 }$$

Using the numbers from Table 1 and assuming an effective wavelength 〈λ〉 = 6000 Å, we infer T₂ = 13, 000 and 17, 000 K for KOI 74 and KOI 81, respectively, with uncertainties somewhere between 5% and 10%. These temperatures are higher than inferred by R10, who assumed that the eclipses and transits measured bolometric surface brightness. (We note that, of course, one could derive more precise values by folding blackbodies or model atmospheres through the Kepler passband, but this would not necessarily be more accurate as long as one does not account properly for, e.g., gravity darkening on what may well be a rapidly rotating primary. In any case, for our purposes the above estimates suffice.)

For KOI 74, in addition to the long-term trend, eclipses, and transits, the raw fluxes show sinusoidal modulations at the orbital and half the orbital period (Figure 1). We fitted the data outside the eclipses and transits with a 6th degree polynomial and sine waves at the orbital frequency and its harmonic (and iterated once, rejecting >3σ outliers). In the bottom panel of the figure, where fluxes normalized by the polynomial component of the fit are shown as a function of phase, the modulations are very obvious. They have much larger amplitude than is seen in Figure 1 of R10v1, likely because these authors originally had removed most of the signal in their pre-display median filtering; their later version shows a larger signal, more similar to what we find. (Related, we note that the two archive light curves include a "corrected" flux. In the second, longer light curve, the signal is clearly present, but in the first, shorter one, it is not, apparently having been removed by the "correction" applied in the Kepler pipeline.)

From our fit, we infer fractional amplitudes $\mathcal {A}_1=(1.082\pm 0.013)\times 10^{-4}$ and $\mathcal {A}_2=(1.426\pm 0.018)\times 10^{-4}$ for the fundamental and the harmonic, respectively. As can be seen in Figure 1, these two fit the data quite well; although the fit is not formally acceptable (χ²_red = 1.5 for 1843 degrees of freedom), the residuals appear quite white. A Fourier transform of the residuals confirms this impression; there is no power above fractional amplitude 10⁻⁵, apart from one faint signal with $\mathcal {A}\simeq 1.6\times 10^{-5}$ and f ≃ 1.66 d^-1. In Figure 1, one sees that the maximum of the fundamental coincides with one of the maxima of the harmonic; our fit gives a phase difference of −0.009 ± 0.003 cycles (of the orbital period; the maximum of the fundamental occurring slightly earlier). The fundamental has essentially the same offset (−0.008 ± 0.002 cycles) from the descending node of the massive star, while the harmonic has maxima coincident with the nodes to within the measurement errors (0.0005 ± 0.0008 cycles).

The above precisions are remarkable, and hence we checked whether the error estimates were reliable. We found that the amplitudes do depend on how we model the long-term trend; if we reduce the polynomial to 4th degree, we find $\mathcal {A}_{1,2}=(1.034,1.430)\times 10^{-4}$; if we fit the pipeline-"corrected" fluxes in the longer lightcurve with a constant and the two sine waves, we obtain $\mathcal {A}_{1,2}=(1.076,1.515)\times 10^{-4}$. This suggests that the true uncertainty in the amplitudes is about 5% although we note that these latter fits are not as good (we find χ²_red = 1.7, and the phases do not agree as well with expectations). Obviously, the uncertainties will reduce substantially as further data are added.

Both signals have natural if, for the fundamental, unusual interpretations. We believe that the fundamental is due to orbital Doppler boosting, an effect which, as far as we are aware, was first detected and discussed by Maxted et al. (2000); it is discussed in the context of Kepler by Loeb & Gaudi (2003) and Zucker et al. (2007). Here, the precision of the Kepler data allows us to go beyond detection, and use the Doppler boosting to measure a star's radial-velocity amplitude.

Briefly, Doppler boosting occurs because as the primary orbits, its spectrum is Doppler shifted, its photon emission rate is modulated, and its emission is slightly beamed forward. Following Loeb & Gaudi (2003), we use the relativistic invariant quantity I_ν/ν³ (or I_λλ⁵), where I_ν is the specific intensity, and ν the frequency. Then, for a radial velocity v_r, the photon rate n_γ observed by a telescope with given effective area A_λ is given by

$$\begin{equation} n_\gamma = \int _\lambda A_\lambda \frac{F_\lambda }{hc/\lambda }{\;d}\lambda = \int _\lambda A_\lambda \frac{F_{\lambda ^\prime }(1+v_r/c)^{-5}}{hc/\lambda }{\;d}\lambda, \end{equation} \tag{ 2 }$$

where F_λ ≡ ∫_ΩI_λ dΩ is the observed flux (Ω is the solid angle subtended by the star; note that the integral is correct only to first order in v_r/c). For a narrow band, the fractional variation would then be

$$\begin{eqnarray} \frac{\Delta n_\gamma }{n_\gamma } &\equiv & f_{\rm DB}\frac{v_r}{c} = \left(-5-\frac{{d}\,\ln F_\lambda }{{d}\,\ln \lambda }\right)\frac{v_r}{c}\nonumber\\ &\simeq & -\frac{(hc/\lambda kT){e}^{hc/\lambda kT}}{{e}^{hc/\lambda kT}-1}\frac{v_r}{c}, \end{eqnarray} \tag{ 3 }$$

where we introduced a Doppler boost pre-factor f_DB and where the approximate equality is for blackbody emission. For KOI 74 (T = 9400 K), observed with Kepler (〈λ〉 ≃ 6000 Å), one thus expects a signal with fractional amplitude of ∼2.8(K/c), where K is the radial-velocity amplitude. For a more precise estimate, we folded a 9500 K model atmosphere (Munari et al. 2005) through the Kepler response. This yields f_DB = 2.25; the difference with the blackbody results from the fact that the model continuum is closer to Rayleigh Jeans inside the Kepler bandpass, mimicking the emission from a hotter blackbody. In this temperature range, the boost pre-factor scales approximately linearly with temperature; for T = 9000 and 10,000 K, we find f_DB = 2.37 and 2.15, respectively. Including reddening of E(B − V) = 0.15 (from the Kepler Input Catalog) has only a small effect, yielding f_DB = 2.21. Given that adding the extinction is equivalent to a relatively large change in effective area, the small change in f_DB found also means that uncertainties in the response are not important.

Given the above, using f_DB = 2.21, our observed light amplitude corresponds to a radial-velocity amplitude of 14.7 ± 1.0 km s^-1, where we assumed 5% uncertainties in both $\mathcal {A}_1$ and in f_DB. The corresponding mass function is $\mathcal {M}=0.0017\pm 0.0004\,M_\odot$. If we adopt a mass for the primary star of 2.22 M_☉ (R10; see Section 2.1), this yields M₂ ≃ 0.22 M_☉.

Turning now to the harmonic, as in R10, we interpret it as changes in the observable surface area due to tidal distortion. The expected amplitude is given by

$$\begin{equation} \mathcal {A}_2 = f_{\rm EV}\frac{M_2}{M_1}\left(\frac{R_1}{a}\right)^3\sin ^3 i, \end{equation} \tag{ 4 }$$

where f_EV is a pre-factor of order unity that depends on the limb darkening and gravity darkening. For our system, the eclipse and transit light curves yield i and R₁/a (Section 2.1). Combining these with the amplitude yields a mass ratio M₂/M₁ = 0.092/f_EV. If we again adopt a primary mass of 2.22 M_☉, and take f_EV ≃ 1.5 (see Section 2.5), we derive M₂ ≃ 0.14 M_☉, somewhat below the value obtained above using the radial velocity amplitude, but roughly consistent, given the probably larger uncertainties associated with determining the mass ratio via ellipsoidal light variations (see Section 2.6).

Apart from the signals discussed above, one might also expect to see contributions of other factors, in particular the Doppler boosting and ellipsoidal variations of the secondary, irradiation effects on both primary and secondary, and possibly influences from an eccentric orbit. Following the procedure above, we find that the fractional amplitudes for the fundamental and harmonic of the secondary, relative to the secondary's flux, are ∼10⁻³ and ∼2 × 10⁻⁷, respectively (where we used K₂ ≃ 150 km s^-1, f_DB = 1.9, R₂/a ≃ 0.0026, and f_EV = 1). Since the secondary's flux is only 0.2% of the total flux, however, the contribution to the observed signal is negligible.

For irradiation of the primary by the secondary, one expects a change in luminosity of ΔL₁ ≃ πR²₁(L₂/4πa²), corresponding to a fractional amplitude $\frac{1}{2}\Delta L_1/L=\frac{1}{2}(L_2/L)(R_1/2a)^2\simeq 3.0\times 10^{-6}$ (where L ≃ L₁ is the total luminosity). Within the Kepler passband, this should lead to a modulation of ∼1.7 × 10⁻⁶. Similarly, for the secondary, one expects a fractional amplitude of $\frac{1}{2}(R_2/2a)^2\simeq 0.8\times 10^{-6}$ in bolometric flux, and a signal of ∼0.4 × 10⁻⁶ in the Kepler passband. The two will have opposite phase; thus, the net expected signal has an amplitude of ∼1.3 × 10⁻⁶, and is phased such that maximum occurs at the time of transit. This signal may be responsible for the fact that the fundamental is slightly out of phase with what is expected from Doppler boosting; the observed out-of-phase amplitude is $\mathcal {A}_1\sin \Delta \phi _1=(9\pm 3)\times 10^{-7}$, consistent with what we derived above.

Finally, we consider possible signals from a non-circular orbit. For small eccentricity, both the Doppler boost curve and the ellipsoidal variations would no longer be pure sinusoids, but have some contribution from higher harmonics. For the easier case of the Doppler boost curve, we can constrain the eccentricity from the absence of a signal at the first harmonic that is out of phase with that expected from ellipsoidal variations (to a limit of Δϕ₂ = 0.0005 ± 0.0008 cycles; Section 2.2). We find that, at 95% confidence, $e\sin \omega =(\mathcal {A}_2/\mathcal {A}_1)\sin 4\pi \Delta \phi _2<0.03$ (where ω is the longitude of periastron, and this expression is valid in the limit of small e). One sees that one cannot constrain the case where periastron occurs at one of the nodes, since in that case the harmonic would be hidden by the ellipsoidal variations. A complementary constraint, however, comes from the fact that the mid-transit and mid-eclipse times are consistent with a circular orbit. The measured offset from half a cycle is Δϕ = 0.0003 ± 0.0008 cycles, which, at 95% confidence, implies ecos ω = πΔϕ < 0.006 (with the expression again being valid in the limit of small e). Thus, from both constraints combined, we conclude that the orbit is circular to within e < 0.03.

For KOI 81, a detailed analysis is not possible, since variations on its much longer orbital period are strongly covariant with the trends in the raw data. Furthermore, the star is clearly pulsating; a Fourier transform shows that there is signal at numerous frequencies. In order to look for orbital modulation, we fit the raw fluxes with orbital and half-orbital modulations, a 3rd degree polynomial, as well as sine waves at the five largest pulsational signals, at 0.362, 0.723, 0.962, 1.32, and 2.08 d^-1 (some of these are harmonically related, as is often the case for multi-periodic pulsators). From the fit, orbital modulation does appear to be present, as is also clear in the folded, normalized light curve, where we divided the fluxes by the fitted trend and pulsational signals (see Figure 1). Our fit yields amplitudes $\mathcal {A}_1\simeq 5\times 10^{-5}$ and $\mathcal {A}_2\simeq 4\times 10^{-5}$, which corresponds to K₁ ≃ 7  km s^-1 and q ≃ 0.2/f_EV. Because of the long orbital period, the results are very sensitive to, e.g., the degree of the polynomial used. Comparing with a fit using a 6th degree polynomial, as well as one using the "corrected" flux from the Kepler archive light curve (the latter for the second, longer light curve only), the amplitude of the fundamental changes by up to a factor of 2, while that of the harmonic changes by ∼20%. Furthermore, we find relatively large phase offsets, of ∼0.03 cycles. Given this, we will not try to infer quantities, but simply note that for a 2.7 M_☉ primary with a 0.3 M_☉ companion, the predicted amplitudes are ∼8 × 10⁻⁵ and 3 × 10⁻⁵, respectively, consistent with the observations.

We tried to verify our semi-analytical estimates above using a light curve synthesis code, which is similar to that of Orosz & Hauschildt (2000), and has been used previously to model irradiated pulsar companions (Stappers et al. 1999). The code accounts for tidal distortion and stellar rotation in the Roche approximation, includes irradiation and gravity darkening, and calculates the flux using NextGen model atmospheres (Hauschildt et al. 1999), integrated over the Kepler passband. It cannot yet deal with eclipses, and hence we only attempt to reproduce the orbital modulations.

As input parameters, we chose to use the set P_orb, t₀, i, T₁, R₁/a, K₁, P_1,rot, q, and T_irr, where the ones not mentioned before are t₀, the time of conjunction, and T_irr, the effective temperature corresponding to the secondary flux absorbed by the primary (i.e., σT⁴_irr = (1 − A)L₂/4πa², with the albedo A expected to be close to zero for a radiative star). Our main goal is to constrain q, K, and T_irr; we assume the other parameters are as inferred from the eclipses. For our fits, we also assume that the star is corotating with the orbit (P_1,rot = P_orb); we will return to this in Section 2.6. We add to the model the tiny flux contribution from the secondary (taken to be constant at the level indicated by the eclipses).

Fitting the raw data of both sources to our model, we confirmed that for KOI 81, the results depend sensitively on how one fits the long-term trend. We thus decided to focus on KOI 74. We searched over a grid in q, K₁, and T_irr, and fitted for the best 4th degree polynomial at each position. The best-fit synthetic light curve is nearly indistinguishable from the two-sine fit, and has the parameters q = 0.0689 ± 0.0009, K = 14.2 ± 0.2  km s^-1, and T_irr = 760 ± 40 K. We did not exclude outliers, and therefore our fit is poorer than the analytical one (χ²_red = 2.6); we scaled the uncertainties such that χ²_red = 1 (but did not attempt to include uncertainties in effective temperature, etc.).

These results reproduce our analytical estimates. The offset in K₁ would be even smaller if we had included reddening in our numerical model (as this affects f_DB; see Section 2.2). The mass ratio matches our estimates for f_EV = 1.34, quite close to the value of f_EV = 1.63 inferred from the tables of Beech (1989). The secondary luminosity inferred from the irradiation is 0.08 ±  0.02 L_☉ (for orbital separation a = 16.4 R_☉ and A = 0, i.e., assuming complete absorption and reradiation), consistent with our estimate from the surface brightness ratio. Below, we give our best estimates of the masses based on these results, but first we discuss possible uncertainties due to rapid rotation.

In many ways, the light curves of KOI 74 and KOI 81 are easy to model, since all effects are small and can thus adequately be treated as perturbations. The one parameter that we have not yet constrained, however, is the rotation of the primary. Since rapid rotation is expected in the context of the evolutionary scenario described below, we discuss the observational consequences in some detail.

Rotation of the primary has a number of effects. First, the constraints on inclination and radius of the primary from the eclipse and transit change: the true inclination i' will be closer to 90° than the inclination i inferred assuming a spherical star, the radius will be smaller, and the radius that is constrained is the equatorial radius R_eq (assuming aligned rotation). Furthermore, the estimate of the mean density increases to $\overline{\rho }_1^\prime \simeq (R_{\rm eq}/a)^{-3}(3\pi /GP^2[1+q])(R_{\rm eq}/R_{\rm pole})$, where R_pole is the polar radius; thus, one would infer a somewhat smaller primary mass. For KOI 74, which has an inclination very close to 90°, we conclude that it would have R_eq/a ≃ R₁/a. For KOI 81, R_eq/a would be somewhat smaller than R₁/a inferred from the eclipses.

The second effect of a rapidly rotating primary is that its pole will be hotter than the equator; hence, looking at the equator, one will underestimate its temperature, and thus its luminosity and mass. This also affects the estimates for the secondary, since the ratios of the radii (R₂/R₁), based on the transit depth, and that of the surface brightnesses (Equation (1)) will be different from their true values: the cooler equator will lead to a larger inferred R₂/R₁ and smaller inferred T₂. For the companion radius R₂, however, the change in R₂/R₁ is partly compensated by the fact that a rapidly rotating primary also would have a smaller area (R'₁ ≃ (R_eqR_pole)^1/2; see above). We have used our model code to test an illustrative case of extreme rotation, with the primary rotation rate 20 times faster than the orbital one (corresponding to vsin i ≃ 370  km s^-1), and a ratio of equatorial to polar radius R_eq/R_pole = 1.32 (similar to what is measured for Regulus; see McAlister et al. 2005). Keeping the flux-weighted temperature at 9400 K, we find equatorial and polar temperatures of ∼7800 and 11, 600 K, respectively. For this particular example, we estimate that the radius of the hot companion would be R'₂ ≃ 0.045 R_☉, i.e., increased by only 5% from that inferred for a slowly rotating primary, and its temperature T'₂ ≃ 10, 500 K.

Rapid rotation also indirectly affects the ellipsoidal variations. First, because of the different temperature distribution on a rapidly rotating primary, different regions are weighted differently, and as a result the amplitude of the ellipsoidal variations changes. From our model code, we find that for the above-mentioned case, the predicted amplitude of the ellipsoidal variations increases by a factor ∼2. The effect is strongly non-linear, however; reducing the rotation rate from 20 to 15 times faster than orbital, the amplitude increases only by a factor of 1.3.

Another indirect effect of a rapidly rotating primary may be that the inferred q does not necessarily correspond to the true mass ratio. An implicit assumption underlying the equilibrium tide is that the primary corotates with the orbit. For stars that do not rotate synchronously, the work of, e.g., Pfahl et al. (2008), suggests that the equilibrium tide may be a rather poor approximation to reality. Thus, it may be that one cannot reliably infer mass ratios from ellipsoidal variations for systems that are not tidally locked.

We note that the light curves may contain a clue to the rotation rates. In particular, for KOI 74, we found evidence for a weak modulation at ∼0.6  d, which is about 9 times faster than the orbital period, and corresponds to 30% of break-up for a 2.2 M_☉ and 1.9 R_☉ star. For KOI 81, it may be possible to infer the rotation rate from the pulsations by looking for rotational splitting (though for high rotation rates this is non-trivial).

Of course, it would be simpler and more reliable to measure the rotational broadening spectroscopically, and the spectra mentioned by R10 may already hold the answer, but we do not have access to those. Nevertheless, we note in closing a much more subtle consequence of rapid rotation, which is the photometric analog of the Rossiter–McLaughlin effect (Rossiter 1924; McLaughlin 1924). Assuming the rotation is aligned with the orbit, at the start of the transit more blueshifted light is being blocked and at the end more redshifted light. For a rotational velocity of 300  km s^-1, Doppler boosting induces a signal in the blocked light with a fractional amplitude of ∼2 × 10⁻³. This is diluted by the unblocked light, i.e., by a factor equal to the transit depth. Thus, for KOI 74 and KOI 81, one expects net signals of ∼1 × 10⁻⁵ and 3 × 10⁻⁶, respectively, which may be detectable by averaging about 100 transits.

In principle, given our inferred values of K₁ and q, combined with the inclination, one can derive both masses without further assumptions. In practice, however, these masses are very uncertain, since the uncertainties in K₁ and q enter to high powers (for small q, M₁ ∝ K³₁q⁻³ and M₂ ∝ K³₁q⁻²). Given that these uncertainties are of order 5% for K₁ and likely larger for q (see above), the 1σ fractional errors are ≳20%, and the distributions are highly non-Gaussian.

Instead, we proceeded by assuming the primary has a mass of 2.22^+0.10_−0.14 M_☉, as inferred from its spectral type and mean density (R10; see also Section 2.1). Then, using K₁ and i, we calculate a mass of 0.22 ± 0.03 M_☉ for the secondary. If instead we use the mass ratio q, inferred from the ellipsoidal variations, we find M₂ = 0.15 M_☉, with an uncertainty that is difficult to estimate because we cannot be sure the star is in corotation (see above). We will proceed by using the value inferred from K₁ (see Table 1). For KOI 81, we do not have a good constraint on K₁ (or q), but the signals that are present are consistent with a similar secondary mass, of ∼0.3 M_☉.

If the mass loss from the progenitor of the WDs were simply through stable Roche-lobe overflow, then we could assume that the current orbital periods are very close to the orbital periods at the end of mass transfer, i.e., the point at which the mass of the envelope about each current WD became too small to maintain a giant structure and collapsed.⁷ Hence, we can apply the relationship between core mass and radius to predict the mass of each WD based on the current orbital periods. Using this relationship, Rappaport et al. (1995) approximated the orbital period P_orb as

$$\begin{equation} P_{\rm orb} \approx 1.3 \times 10^{5} \frac{M_{\rm wd}^{6.25}}{\big(1+4M_{\rm wd}^{4}\big)^{1.5}} {\rm days}, \end{equation} \tag{ 5 }$$

where the WD mass M_wd is in units of solar masses. At a given P_orb, the spread in M_wd is expected to be at most ±15%.

Applying Equation (5) we find M_wd = 0.20 ± 0.03 M_☉ for KOI 74, which matches the value independently derived in Section 2. For KOI 81, this becomes M_wd = 0.25 ± 0.03 M_☉, again consistent with a low-mass WD.⁸

Stable mass transfer seems to provide a simple way to produce both these systems. The match between the WD masses from Equation (5) and from the light curve data is encouraging. Furthermore, the full binary evolution calculations of Podsiadlowski et al. (2002) produce systems with very similar orbital periods and WD masses (see their Figure 13). In Figure 3, we show examples of how the two systems could have reached their current state, using simple analytic formulae for orbital evolution (Equation (3) of Rappaport et al. 2009; see also Podsiadlowski et al. 1992). The curves in Figure 3 take initial conditions {$M_{1, \rm init}$, $M_{2, \rm init}$, P_orb,init} of {1.6 M_☉, 1.2 M_☉, 19.3 h} for KOI 74 and {1.8 M_☉, 1.3 M_☉, 31 h} for KOI 81. In both cases we assume that any mass lost from the system carries away the specific orbital angular momentum of the binary. For KOI 74, this example assumes that the mass transfer is 73% conservative, and for KOI 81 it is 91% conservative. There are certainly other possible paths to the current systems, in particular since the above ignores the possible role of magnetic braking (in evolutionary stages where either star has a convective envelope).

Zoom In Zoom Out Reset image size

The currently more massive stars are almost certain to have accreted significant amounts of matter and are therefore likely to be very rapidly rotating (see, e.g., Rappaport et al. 2009 and references therein). Their spin angular momentum could, in principle, have been lost by magnetic braking. However, typical A-type stars do not experience strong magnetic braking (see, e.g., Kawaler 1988). Alternatively, for KOI 74, tides might have been strong enough to slow the rotation of the star (and slightly widen the orbit).

Common-envelope evolution seems to provide a much less satisfactory explanation for these systems than stable mass transfer. If the envelopes of the initially more massive stars were ejected during common-envelope evolution, then using the current P_orb in Equation (5) would only yield a minimum WD mass. So, if the observational data cannot allow more massive WDs than ∼0.25 M_☉, it seems unlikely that either of these systems have experienced a significant common-envelope phase.

In addition, any scenario in which the currently more massive stars have not accreted a significant amount of mass seems difficult. The >2 M_☉ current primary stars in these systems would produce WDs more massive than ≳0.28 M_☉ if their envelopes were removed at the end of the main sequence. If those initially less massive stars had not gained matter then, at best, significant fine-tuning would be required for the initially more massive star to have both evolved off the main sequence and produced the low-mass WDs observed.

Another class of hot compact companion stars that should result in light curves similar to those in Figure 1, are hot subdwarf (sdB, sdO) stars, sometimes referred to as extreme horizontal branch stars. The calculations of Han et al. (2002, 2003) predict that sdB stars should exist in binaries of these orbital periods about A- and B-type stars (see Han et al. 2003, Figure 15). However, at the temperatures inferred for KOI 74 and 81, sdB stars would be expected to be considerably more luminous. Typical hot subdwarfs properties are anywhere between ∼20, 000 and 40,000 K and ∼3 and 100 L_☉. In addition, the inferred companion masses in KOI 74 are somewhat too low to have ignited helium, even non-degenerately; masses greater than ∼0.3 M_☉ are required. Such hot subdwarfs should be found in current and future planetary transit searches.