TRANSITS AND LENSING BY COMPACT OBJECTS IN THE KEPLER FIELD: DISRUPTED STARS ORBITING BLUE STRAGGLERS

When a white dwarf orbits a main-sequence star, the progenitor of the white dwarf must have initially been the most massive star in the binary. Let M₁(0) denote the initial mass of the primary. If the mass of the white dwarf, M_c, is smaller than that of the remnant normally produced by a star of this mass, we can infer that the evolution of its progenitor was truncated through interaction with the secondary. The interaction may have been stable, producing a gradual erosion of the primary through mass transfer. Or it may have been unstable, producing an almost explosive disruption of the primary. In either case, the mass M of the main-sequence star we observe today may be larger than its initial mass, M₂(0). Such a star is called a blue straggler, in analogy to stars in clusters that appear to be more massive than the cluster turn-off.

In this section we consider the case in which M_c is smaller than roughly 0.25 M_☉, because this applies to KOI-81 and KOI-74. (We include more massive cores in the calculations described in Sections 3 and 4.) The primary is not a fully evolved giant at the time it fills its Roche lobe. Its mass will be close in value to M₁(0), and its radius can be expressed as follows (P. P. Eggleton 1986, private communication):

$$\begin{equation} R_d=0.85\, M_\ast ^{0.85}+\frac{3700\, M_c^4}{1+M_c^3+1.75\, M_c^4}. \end{equation} \tag{ 1 }$$

For an isolated star, the value of M_* is simply the star's initial mass. In the next paragraph, we discuss its likely value in systems that undergo two-phase mass transfer. At the time when the primary first fills its Roche lobe, q = M₁/M₂ > 1. If q is larger than a critical value η, a common envelope will form and the core of the primary will spiral closer to the main-sequence companion. The final separation, a_f, can be expressed in terms of its initial separation, a_i, the value of M_c, the stellar masses M₁ and M₂ at the time of Roche-lobe filling, and an efficiency parameter α¹:

$$\begin{equation} a_f=a_i\, \Bigg (\frac{M_c}{M_1}\Bigg)\, \Bigg [1 + \Bigg (\frac{2}{\alpha \, f(q)}\Bigg)\, \Bigg (\frac{M_1-M_c}{M_2}\Bigg)\Bigg ]^{-1}, \end{equation} \tag{ 2 }$$

where f(q) represents the ratio between the radius of the donor and the orbital separation (Eggleton 1983). While the companion's mass may not change significantly during the common envelope phase, it may have increased during the interval prior to it. This is because mass transfer may have begun in the form of a gravitationally focused wind as the primary expanded to fill its Roche lobe.

There is no single value of the critical mass ratio η. This is because mass transfer can be stabilized by a combination of factors that each assume values appropriate to a specific binary. These factors include the magnitude of winds ejected from the system, the specific angular momentum carried by winds, the role of radiation, and the donor's adiabatic index. When a common envelope is avoided, stars with modest cores will donate mass during two phases. During phase 1, mass transfer occurs at a rate determined by the thermal timescale of the donor as it attempts to adjust to mass loss. Once the masses have equalized, the donor has a chance to re-establish thermal equilibrium. The donor star may continue to lose mass, but as it comes into equilibrium with a smaller mass than its initial mass, and with a core that is still modest (less than roughly 0.25 M_☉), it begins to shrink into its Roche lobe. This leads to a quiescent interval. M_* is the mass of the donor during this time of quiescence. The mass of the accretor is approximately equal to M_* as well, although the accretor may be slightly more massive than the donor at this point. Thus, during phase 1, the donor has lost an amount of mass M₁(0) − M_*, and the accretor has gained M_* − M₂(2). Generally, M_* − M₂(2) < M₁(0) − M_*, since the thermal timescales of the two stars are different.

The continued growth of the core of the first star and the accompanying stellar expansion, perhaps combined with magnetic braking, eventually cause the donor to fill its Roche lobe again. Mass transfer during the second phase is stable and ends when the donor's envelope is depleted. Depending on the time elapsed since the start of phase 1, the donor's core will generally have grown by an amount δ.

The first question to answer for KOI-81 and KOI-74 is whether each could be an end state of stable mass transfer. If the answer is "yes," the donor star would have been filling its Roche lobe until the time mass transfer ceased, and the orbital period is

$$\begin{equation} P_{{\rm orb}}= 0.372\, {\rm days} \Bigg [\frac{R_d(M_\ast, M_c)}{R_\odot }\Bigg ]^{\frac{3}{2}} \Bigg [\frac{M_\odot }{M_c}\Bigg ]^{\frac{3}{2}}. \end{equation} \tag{ 3 }$$

Given a measured value of the orbital period, Equations (1) and (3) combine to produce a set of possible values for M_* and M_c. The value of M_c should correspond to the present-day mass of the core observed today, and it can be measured through radial velocity or other methods (R2010; van Kerkwijk et al. 2010 (vK2010)). The value of M_* represents the mass of the primary, generally as it was during the quiescent interval between phase 1 and phase 2.

The black points in the top two panels of Figure 1 show the results for KOI-81 and KOI-74. The mass of KOI-81b is constrained to lie in a very narrow range, 0.215–0.219 M_☉. (Note, however, that the systematic uncertainty associated with using the functional forms in Equation (1) is larger than this formal range indicates.) The mass M_* ranges from roughly 1.6 M_☉ to 2.4 M_☉. R2010 interpreted light curve features in terms of tidal effects and found the mass of KOI-81b to be 0.212 ± 0.031 M_☉. This is consistent with our mass-transfer model. The R2010 value is shown as a cyan triangle centered on a cyan line which delineates the uncertainty limits. vK2010 used Doppler boosting and found a mass of 0.3 M_☉ to be consistent with the observations. vK2010 also used the model of steady mass transfer to predict the mass of KOI-81b and found a value of 0.25 M_☉. Their model estimate was marginally higher than ours because they used an expression for the orbital period that did not have any dependence on the donor mass (Rappaport et al. 1995). Such a form is expected to work best for core masses higher than the estimated mass of KOI-81b. Note, however, that both vK2010's expression and Equation (3) are approximate. If Kepler identifies many similar systems and mass measurements are possible for a significant subset, we will learn how to best model the radii of stars with low-mass cores.

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There is a benefit to utilizing the significant contribution of the donor mass to the radius by invoking Equation (1). Specifically, the measured value of the orbital period constrains a combination of both M_* and M_c. This provides input for binary evolution calculations which can identify all possible initial states for the binary we observe today. We start by computing the amount of mass lost by the donor during phase 2: Δ = M_* − M_c; and the fraction of this mass accreted by the blue straggler: β = (M − M_*)/Δ, where M is the present-day mass of the monitored star. We can then derive the initial state by evolving backward in time. We find the range of possible evolutionary paths by sampling a range of values for (1) η and (2) β during phase 1.² Typical results are shown in the bottom panels of Figure 1. The initial mass of the primary can be as high as ∼3 M_☉ or as low as ∼1.7 M_☉. These differences have detectable consequences, in that more mass is ejected from the binary when M₁(0) is high. Observations can therefore test the predictions of each evolutionary channel, for example the white dwarf's age and the amount of mass ejected from the binary as a function of time.

The results of KOI-74 are shown in the panels on the right-hand side of Figure 1. Using our model, we find that the mass of KOI-74b lies in the range: 0.142–0.153 M_☉. This is marginally consistent with the results of R2010 based on tidal effects (0.111^+0.034_−0.038). On the other hand, our estimate is smaller than vK2010's Doppler-boost measurements (0.22 ± 0.03 M_☉) and also lower than predicted by their period/core-mass relationship (0.20 ± 0.03 M_☉). (Note that when the core mass is smaller, the main-sequence contribution to the donor's radius is more significant.) The same general features of the range of initial stars that we discussed for KOI-81 also apply to KOI-74, but the lower mass of today's monitored star is consistent with lower initial masses for the components of the binary.

If the masses of KOI-74b and KOI-81b are in the range 0.1–0.3 M_☉, then the work we sketched above shows that there are binary evolution models in which their progenitors filled their Roche lobes until the time their envelopes were totally eroded. This strongly supports the hypothesis that KOI-71 and KOI-81 are end states of stable mass transfer. We note here, however, that Kepler can also discover binaries in which the mass of the core is too large for the donor to have been filling its Roche lobe while in its current orbit. This would be a signature that the binary was a common-envelope survivor. We could then use the current values of the orbital separation and masses to constrain the parameters of the common envelope evolution (Equation (1)).

For KOI-74b and KOI-81b, however, there may be reason to suggest that the actual masses are smaller than the values associated with stable mass transfer. R2010 place lower limits on the masses that extend into the brown dwarf regime. In addition, for KOI-74, there is only marginal overlap between R2010 and the stable mass-transfer model presented in the last section. It is therefore worth seriously considering that the mass of one or both the cores may be in the brown dwarf regime. Here we use KOI-74 as an illustrative example. We will show that a value of the core mass smaller than expected from stable mass transfer may be a signal that there was a common envelope, but that the orbit was eccentric at the time the primary filled its Roche lobe. This is expected in some triple systems. It therefore seems inevitable that, whatever the history of KOI-74 and KOI-81, binaries in which the core mass is too low will eventually be found, and the scenario we present below will apply.

First we note that, even if the core mass of KOI-74 is small, formal solutions to the stable-mass-transfer equations exist, predicting 0.005 M_☉ < M_c < 0.018 M_☉ and 1.2 M_☉ < M₁ < 2 M_☉ at the time of Roche-lobe filling. These solutions are not viable, however, because the equilibrium radius of a low-core-mass primary would decline as mass was donated in a steady, stable manner. The second term in Equation (1) would contribute little, and, although the first term might not apply exactly throughout the evolution, the trend of smaller radius with decreasing mass would be followed during phase 2. The orbital period would have to be much smaller than 5.2 days.

If, therefore, KOI-74b is a low-mass stellar core, the binary likely passed through a common envelope phase. Using Equation (2) we find that a_i > M₁/M_c. Furthermore, dynamical instability requires M₁ to be larger than roughly (1.3–1.5) × (2.2 M_☉). This predicts an initial separation so large that the primary could not have filled its Roche lobe.

This may indicate that the initial orbit was highly eccentric and that a dynamical instability was triggered during periastron. During the common envelope, the eccentricity was eliminated while the semimajor axis became smaller. Such a high eccentricity is not generally expected in a binary in which one companion is poised to fill its Roche lobe, because tidal effects will tend to circularize the orbit. If, however, KOI-74 is a triple, the Lidov–Kozai (Lidov 1962; Kozai 1962) mechanism could have produced the extreme eccentricity that led to the disruption of the progenitor of today's low-mass core. Perets & Fabrycky (2009) have considered the generation of blue stragglers via this mechanism. They focused on mergers, but the same general idea could apply to systems like KOI-74. We therefore propose that mass transfer occurred prior to the approach that triggered the common envelope. During that earlier epoch, the progenitor of the hot core made close approaches to the star we monitor today. In a process analogous to what happens in high-mass X-ray binaries, mass was transferred during these close approaches. Eventually, under the influence of a third star, the periastron distance became small enough to trigger a dynamical instability.

We used Equation (1) to compute the initial value, a_i, of the binary's semimajor axis at the time the primary filled its Roche lobe. The secondary mass was considered to be the mass observed today, and we set M_c to 0.03 M_☉,a_f = 16 R_☉, and assumed efficient envelope ejection (α = 100), so as to compute the minimum value of a_i. We found that the eccentricity was required to be large, generally >0.997. Thus, this scenario predicts the presence of a third body in the system. Its separation from the 2.2 M_☉ star Kepler is monitoring today would likely be greater than a few hundred AU (See Figure 2), so that this third star is potentially detectable today. Note that variations in transit times could also be used to detect the presence of a third body (Holman & Murray 2005). As shown in Fabrycky & Tremaine (2007) and references therein, triple stars are common, and a large fraction of close binaries are known to have another companion in a wide orbit. (See also Eggleton & Kisseleva-Eggleton 2001.) This triple-star scenario therefore seems plausible. The one feature that seems unlikely is the extreme value of the eccentricity required.

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Finally, we note another, less likely possibility. Eggleton (2002) discusses systems in which the envelope was so weakly bound to the primary that dynamical instability did not produce substantial spiral-in. This situation does not seem likely to apply to KOI-74.

The fact that different values of the mass of today's small-radius core are associated with different evolutionary scenarios points to the importance of reliable mass measurements for both KOI-74 and KOI-81. While it is intriguing that the well-sampled Kepler light curves have allowed both Doppler boosting and tidal methods to be employed, a radial velocity measurement is required. Preliminary indications are that the rotational velocities of both KOI-74a and KOI-81a are high, perhaps in the range of ∼300 km s⁻¹ (D. W. Latham 2010, private communication). While this may complicate radial velocity measurements, it could also be a clue that each has accreted matter, thereby supporting the mass-transfer model. High rotational velocities are, in fact, often associated with blue stragglers, presumably generated through mass transfer (see, e.g., De Marco et al. 2005 and Mathieu & Geller 2009). While, therefore, the high rates of rotation may make mass measurements more difficult, it is also possible that they are providing a signal that mass transfer did occur.

We can derive a rough estimate of the fraction of Kepler targets that might be orbited by transiting white dwarfs by generating a population of stars, including both singles and binaries, and then considering the evolution of each system. We identify those that will become main-sequence stars that are both (1) in a range of masses likely to be monitored by Kepler and (2) orbited by white dwarfs. We then compute the probability that the white dwarf will transit during the time of Kepler monitoring. For short orbital periods, the probability is R_*/a, where R_* is the radius of the monitored star and a is the final orbital separation. For orbital periods longer than τ_monitor, the duration of monitoring, we multiply by a factor of τ_monitor/P_orb.

We assume that 40% of all stellar systems are singles, and that the remainder are of higher multiplicity. We use a Miller–Scalo initial mass function (Miller & Scalo 1979) to select the masses of all single stars and to select the primary mass for all multiples. This has the effect of favoring low-mass stars, even though we generate the full mass spectrum. The only multiple systems we explicitly consider are binaries. We select the secondary mass as follows: M₂ = M₁ × u^0.8, where u is a random number chosen from a uniform distribution in the range 0–1. We then generate orbital separations, selecting uniformly in log space from separations as small as three times the radius of the primary, and as large as 10⁴ AU.

Primary stars will fill their Roche lobes at some point in their evolution if the orbital separation lies within roughly an AU. This range of separations encompasses a large fraction of all binaries. Roche-lobe filling will lead either to a common envelope phase or to stable mass transfer. While a large fraction of common envelopes may end in mergers, those binaries that survive will have small orbital separations and therefore high probability of yielding transits. When mass transfer is stable, the final separation may be somewhat larger than the initial separation, but most orbits are smaller than a few AU (see Figure 3).

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The paragraph above describes the key reason why transiting white dwarfs must be common: they are produced by a large fraction of all primordial binaries. We used a simple model to design a code that would allow us to quantify this and to test a wide range of input assumptions. We conducted 13 simulations. Each was characterized by (1) the value of α, (2) the value of τ_monitor, (3) the range of masses of stars monitored by Kepler, (4) the rule used to compute the final core mass for systems experiencing stable mass transfer, and (5) the ages of the stars monitored by Kepler. In this way, we were able to take into consideration the uncertainties in the input physics and to identify those results that are robust. The key question is: what fraction of all monitored systems are binaries in which a white dwarf is expected to transit a main-sequence star?

To compute the number of monitored systems, we counted members of the following sets of stars, considering only those in the appropriate range of masses: (1) all isolated stars, (2) stars that are the results of mergers, (3) stars with dwarf stellar companions having luminosity less than 0.001 their own luminosity,⁴ (4) stars in binaries so wide that the individual stars could be resolvable,⁵ and (5) stars orbited by white dwarfs. This last category includes all white-dwarf/main-sequence pairs, regardless of the formation history of the white dwarf.

Figure 3 shows the result of one such simulation. In this particular simulation, we generated 4 million stellar systems and used α = 10 and τ_monitor = 1 year. We assumed that Kepler is monitoring main-sequence stars with masses in the range 0.9–2.9 M_☉. We assumed that, for systems undergoing stable mass transfer, the final value of the core mass would lie within 10% of its initial value (δ < 0.1 M_c). There were approximately 370, 000 stars with masses within the selected range that would appear to be single stars. We then identified all binaries in which the primary star fills its Roche lobe. Those with q > 1.5 were assumed to undergo common envelope evolution. For smaller values of q, we assumed stable mass transfer and selected the final value of the primary's core mass. The number of binaries with a present-day primary having a mass in the monitored range and (1) surviving the common envelope were ∼16,000; and (2) having ended stable mass transfer were ∼35,000. Most of these will not produce transits. Taking the transit probability into account, we find that 3.6 × 10⁻³ of the monitored stars should exhibit at least one transit per year, in which the transiting object is a white dwarf that survived a common envelope. Many of these will exhibit multiple transits per year. The comparable fraction of systems with transiting white dwarfs that emerged from stable mass transfer is 5.0 × 10⁻³.

Table 1 shows a representative sample of the results. In this section we focus on the totals listed in Column 6. These totals are normalized to the case in which transits can be detected in 135, 000 monitored stars. The first two rows correspond to a single simulation, labeled "1." Simulation 1 is the one used to produce Figure 3. (α = 10, δ < 0.1, the only criterion imposed on the Kepler target stars is that their masses are between 0.9 M_☉ and 2.9 M_☉.) If this case applies, more than 1100 transiting systems would be discovered by Kepler. In the next four rows, also labeled "1," we alter only the value of α, the common envelope ejection efficiency. We find that, as the ejection efficiency declines, significantly more mergers occur. For example when α declines from 10 to unity, the numbers of mergers increase by 50% and the common-envelope survivors are found in closer orbits than those shown in Figure 3. The transit probability declines by only 15%, however, because transits are more likely for closer orbits. The value of α must decline to ∼0.1 in order for common envelope survivors to produce fewer than 100 transits per year. Thus, the result that a large number of white dwarfs should produce transits is robust with respect to changes in the common-envelope efficiency factor. In any realistic population, the efficiency of envelope ejection will be different for different systems. To determine the numbers of transits expected from common envelope survivors, we should therefore average over a range of efficiencies.

Table 1. Transiting White Dwarfs: Numbers and Lensing Effects

Method/Type	$\frac{R_E}{R_{\rm{WD}}}>2$	$0.8 < \frac{R_E}{R_{\rm{WD}}}<2$	$0.4 < \frac{R_E}{R_{\rm{WD}}}<0.8$	$\frac{R_E}{R_{\rm{WD}}}<0.4$	Total
1:CE (α = 10)	6.3	40.0	102.7	343.6	492.6
1:MT (δmax = 0.1)	3.4	26.6	58.5	589.3	677.8
1:CE (α = 3)	4.3	44.0	116.5	311.6	476.4
1:CE (α = 1)	1.3	42.8	139.4	243.8	427.3
1:CE (α = 0.3)	0.2	22.9	144.4	159.4	326.9
1:CE (α = 0.1)	0.0	0.4	30.3	61.7	92.4
1:MT (δmax = 0.2)	3.1	25.5	55.2	506.1	590.0
1:MT (δmax = δmax(Mc))	3.5	26.6	40.1	117.1	187.3
2:CE (α = 10)	4.1	36.4	94.9	445.4	580.8
2:MT (δmax = 0.1)	1.1	9.7	21.4	239.1	271.3
2:MT (δmax = 0.2)	1.0	9.3	20.4	204.5	235.2
2:MT (δmax = δmax(Mc))	1.2	10.8	16.5	50.2	78.7

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We also tested the effect of changing the rule used to compute the final core mass for the donor stars that contribute mass during stable mass transfer. We used three prescriptions: (1) the maximum value of δ is 10%, (2) the maximum value of δ is 20%, and (3) the value of δ can be as large as half the difference between the white dwarf mass expected for a star with initial mass M₁(0) and the core mass M_c at the start of mass transfer. Changing these rules does not change the numbers of stable mass-transfer binaries, but it does affect the transit probability; the larger the core mass at the time of depletion, the wider the orbit, and the smaller the transit probability. With prescription (2), the numbers of transits by white dwarfs that are the end states of stable mass transfer decreased by ∼13%. With prescription (3), the numbers of transits by the end states of stable mass transfer decreased by roughly 70%.

No single rule for selecting a final core mass can apply to all binaries. Once mass transfer begins, the donor's envelope is depleted through mass loss, and the envelope is also contributing mass to the growing stellar core. The final core mass depends on the timescales for these two competing processes. These timescales depend on the state of the system and on the evolution of the orbital angular momentum; for example, on the amount of angular momentum carried off in winds. For donors that fill their Roche lobes when their core masses are small (∼0.1–0.2 M_☉), mass transfer may deplete the envelope at a faster rate than the nuclear evolution of the core. Values of δ may be modest. For donors that fill their Roche lobes as more evolved stars, the evolution of the core is proceeding more rapidly. The rate of mass loss will also typically be high, however, and can be increased if winds carry significant angular momentum. To incorporate these effects, we could carry out a detailed evolution for each system. The uncertainty would still be large, because the physical effects that determine the binary evolution are not yet well-enough understood. To determine the numbers of transits expected by white dwarfs that are the remnants of Roche-lobe-filling donor stars, we have therefore chosen to average over the three prescriptions described above for the final mass of the core. This averaging process corresponds to considering a range of mass-transfer rates as well as evolutionary states for the donor.

The age of the monitored systems can also influence the results. In particular, the primary needs to have had enough time to evolve and produce at least a modest core, yet the star that was originally less massive should still be on the main sequence, even after gaining some mass from its companion. To take this into account, in simulations labeled "2," we considered a constant rate of star formation over an interval of 12 billion years. We randomly generated the time at which each stellar system was formed, and considered further only single stars which would not have had time to evolve during the interval from their formation until the present day. We considered only binaries in which the primary star had time to evolve, but the secondary is still on the main sequence. In this case, the fraction of transiting systems derived from common envelope evolution (stable mass-transfer evolution) was 4.3 × 10⁻³ (2.0 × 10⁻³).

We conducted additional simulations, which are not shown in the table because the results did not change significantly. For example, changing τ_monitor from 1 year to 3.5 years changes the results by only a few percent, since most of the transits occurred in binaries with orbital periods smaller than a year. (Multiple transits of the same monitored star will, however, increase the reliability of the detection.) As another example, monitoring stars in a more limited mass range (0.9–1.9 M_☉ or 1.9–2.9 M_☉) changes the results by only ∼10%.

Although we have been considering the fraction of main-sequence stars that may have white-dwarf companions, it is also useful to consider the fraction of white dwarfs that are in binaries. By looking at a volume-limited sample, Holberg et al. (2008) find that within 20 pc, 25% of the white dwarfs are in binaries. This is roughly consistent with our assumptions. Furthermore, white-dwarf binaries with properties similar to those we predict have been found outside the Kepler field. These include V471 Tauri and the Regulus system. Interestingly enough, both the systems show evidence of higher multiplicity and interesting evolutionary histories (Guinan & Ribas 2001; Gies et al. 2008; Rappaport et al. 2009).

It is useful to consider the contribution of common-envelope survivors and stable-mass-transfer binaries separately:

$$\begin{equation} N_{{\rm transit}}=N_{{\rm transit}}^{\rm{ce}} + N_{{\rm transit}}^{\rm{mt}}. \end{equation} \tag{ 4 }$$

To estimate the value of each term, we average over the results for the simulations described above. First, we note that simulation 1 is more appropriate for an old population in which the white dwarfs would have long-ago formed from the primary stars in binaries with secondaries as massive as the Kepler targets. Simulation 2 applies to intermediate-age populations. We average the results of these two simulations. In each case, we average over the common envelope efficiency factors and also over the prescription for the value of δ:

$$\begin{equation} N_{{\rm transit}}^{\rm{ce}}=\frac{{\cal N}_{{\rm monitored}}^{{\rm detect}}}{135{,}000}\, [395 \pm 32 ] \end{equation} \tag{ 5 }$$

and

$$\begin{equation} N_{{\rm transit}}^{\rm{mt}}=\frac{{\cal N}_{{\rm monitored}}^{{\rm detect}}}{135{,}000}\, [338 \pm 143]. \end{equation} \tag{ 6 }$$

Including stellar systems with higher multiplicity could increase the numbers of systems with white dwarfs in close orbits. On the other hand, lensing effects could make it difficult to identify as many as one third to one half of the events.

The calculations described above suggest that Kepler may identify transits in hundreds of mass-transfer end states. Each case provides a unique way to measure the white dwarf radius, luminosity, and temperature, and even to explore its atmosphere. This is all in addition to the standard "bag of tricks" that can be applied to any white dwarf in a binary. The ability to conduct a large number of such studies in a uniform way will determine the properties of the white dwarfs in terms of the mass lost history of their progenitors.

With regard to mass loss histories, Kepler studies will establish the relative frequency of common envelope evolution and provide useful data points for determining the best way to compute common envelope evolution in terms of the properties of the binary at the time the dynamical instability was triggered. The range of systems producing stable mass transfer will also be better understood, including the all-important fraction, β, of material that can be retained by a main-sequence accretor. Most of the Kepler stars are field stars. Studying the Kepler blue stragglers will provide empirical insight into the contribution of ordinary binary evolution to the creation of blue stragglers. This will allow the contribution of stellar interactions in clusters to be better understood, since both primordial binaries and binaries formed or altered through interactions can produce blue stragglers.

Finally, the white-dwarf/main-sequence binaries studied by Kepler will experience a second phase of interaction when the main-sequence star of today evolves and begins to transfer matter to the white dwarf. These systems will be novae, some may become double-degenerates that eventually merge, and some may even experience Type Ia supernovae, or accretion-induced collapse, through either the single-degenerate or double-degenerate channel. Kepler will identify a set of these systems that can be studied in a unique way.

We started by considering evolutionary models for KOI-74 and KOI-81. Our results agree with those of vK2010, which suggest that each system may have evolved to its present state through a process of stable mass transfer. Our approach differs from theirs by using a different prescription for the radius of the donor star prior to envelope exhaustion. Whereas vK2010 used a formula for the radius which depends on only the core mass of the donor, our formula also incorporates the dependence on its equilibrium stellar mass M_*, prior to the last phase of mass transfer. This has two effects. First, our approach tends to predict somewhat smaller values of the white dwarf mass. This will be tested through radial velocity measurements, and Equation (1) can be adjusted if needed, based on the observations. Second, it allows Kepler's period measurement to constrain both the white dwarf mass and the mass of the donor, thereby providing useful input for detailed evolutionary models.

We also take seriously the possibility that the masses of KOI-74b and KOI-81b are in the brown dwarf range, so that the formalism we and vK2010 have applied is not valid. We find an interesting new evolutionary pathway. In this case, there is a third body that pumps up the eccentricity of an inner binary. Mass transfer from the primary to the secondary occurs at periastron, in analogy to the mass-transfer process in high-mass X-ray binaries. As the orbit evolves, eventually the primary experiences a dynamical timescale instability at periastron, leading to a common envelope phase. During the common envelope, the orbit circularizes and the core of the primary spirals closer to its main-sequence companion, but avoiding a merger. The signature that such an evolution has occurred is a white dwarf in a wider orbit than expected had there been stable mass transfer of a common envelope initiated from a circular orbit. We do not know if this scenario is needed to explain either KOI-74 or KOI-81, but if it occurs in nature, Kepler may find binaries that have evolved in this way.

Whatever the nature of KOI-74 and KOI-81, their discovery inspires us to compute the frequency of transits by white dwarfs in binaries that have experienced mass transfer. In many of these cases, the companion star has gained mass from the progenitor of the white dwarf and may be considered to be a field blue straggler. We have performed a set of preliminary calculations to estimate the number of such white dwarfs that transit stars in the Kepler field. Although our model may be viewed as a "toy model," it allows us to parameterize the uncertainties involved in the complex physics needed to predict the details of the binary evolutions. We find that, under a wide range of reasonable assumptions, roughly 0.0025–0.0075 of systems of the type monitored by Kepler should exhibit white dwarf transits. It would be surprising if fewer than 100 transiting white dwarfs were discovered by Kepler, while it is possible that more than 1000 will be found. Furthermore, evidence of gravitational lensing is expected in 10%–20% of the white dwarf transits; anti-transits may be observed in 1/3–1/2 of the cases in which there are lensing signatures.⁶ The ubiquity of lensing by the white dwarfs in the Kepler sample can be understood in terms of mesolensing, the high probability of lensing associated with certain astrophysical systems, even though the total mass in lenses, as measured by the lensing optical depth, is low (see Di Stefano 2008a, 2008b.)

Our results can be compared with a calculation of the number of white-dwarf transits expected, conducted before the Kepler targets were selected (Farmer & Agol 2003). This previous work had to anticipate the characteristics of the stars that would be monitored and the detectability of transits and lensing. Farmer & Agol (2003) concluded that at least 50 transiting white dwarf binaries could be unambiguously identified. Furthermore, if detectability issues would be less problematic than they assumed for their set of target stars, the number could be increased to 500. Because the Kepler targets have been specifically selected to make planetary (hence white dwarf) transits detectable for ∼90% of the monitored stars, the higher number they derived seems more likely. Furthermore, the mass spectrum they assumed for monitored stars was broader, meaning that the efficiency for producing white-dwarf/main-sequence binaries is lower. Taking this into account, their results can be scaled to produce rough agreement with the calculations here. We emphasize that our calculations, as well as those of other population synthesis simulations, have uncertainties that can best be understood by exploring the parameter space, as described in Section 3 and summarized in Table 1.

It is possible that KOI-74 and KOI-81 are the first examples of transiting white dwarfs. The fact that they are both in close orbits is expected, since the majority of white dwarfs whose transits will be detected are in close orbits (Table 1). They both appear to produce pure transits, without obvious signs of lensing (see the discussion in Section 4.) Their discovery and the calculations above suggest several productive lines of research.

• 1.  
  There should be several, perhaps dozens, of anti-transits in the data already taken. If these can be identified and modeled, we will have the first evidence of binary self-lensing and the first lensing measurements of the gravitational masses of white dwarfs.
• 2.  
  Mass measurements of all transiting objects are crucial. Radial velocity measurements should be obtained whenever possible. Light curve fitting will also be helpful. If enough systems can be subjected to both types of analyses, we may develop more confidence in results based on light curve methods alone. When fitting the light curves, it is important to employ models that include the possible effects of lensing. Although there are potential degeneracies in fits that include a variety of effects (such as lensing, tides, Doppler boosting), mathematical methods similar to those that test for degeneracy in lensing events can be applied to transits and anti-transits (Di Stefano & Perna 1997). The result should be to identify those events that may be associated with compact objects and to derive the radius and mass of the compact object.
• 3.  
  Transiting white dwarfs can be subject to additional observations using the opportunities presented by transits to study their atmospheres. The radiation from hot white dwarfs entering or leaving eclipse can also be used to probe the atmospheres of their companions.
• 4.  
  Lensing by white dwarfs will provide unique opportunities to probe the surface of the stars they orbit.
• 5.  
  Although we cannot yet predict the orbital or mass distributions of planets, binary evolution allows us to predict general characteristics of the orbital and mass distribution of white dwarfs in close (a< a few AU) binaries. A set of more detailed calculations is needed to generate individual binaries for each set of simulations and to compute the mass-transfer history, the time at which mass transfer ended, the radius of the white dwarf, the light curves, and the characteristics of the monitored star (rotation, metallicity) that may have been influenced by accretion.
• 6.  
  By discovering orbiting white dwarfs with lower mass than expected for their measured orbital periods, Kepler can establish the contribution of three-body interactions to the formation of blue stragglers.
• 7.  
  Once a large ensemble of Kepler events exists, we will be able to compare their properties with those predicted by the theoretical work, and learn about a wide variety of outstanding issues, such as the fraction of matter retained during stable mass transfer to a main-sequence star and the efficiency of common envelope ejection in a variety of situations.
• 8.  
  Many of the mass-transfer products in which a white dwarf orbits a main-sequence star will experience future epochs of mass transfer onto the white dwarf. Some of these white dwarfs are destined to become Type Ia supernovae. Significant uncertainties in our understanding of the progenitors (Di Stefano 2010; Di Stefano & Nelson 1996) will be addressed by the Kepler results. The same is true for other outcomes, such as accretion-induced collapse. The Kepler data on the endpoints of the evolution of the primaries will provide, for the first time, a large number of reliable starting points for us to compute the next phase of evolution in close white-dwarf binaries.

Implications. Kepler was developed as a mission to discover planets. We find that, in addition, it will be a unique resource with which to study white dwarfs and mass transfer. The results will touch on stellar evolution, binary evolution, and the formation of intriguing systems such as the progenitors of Type Ia supernovae and accretion-induced collapse.