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 time scale 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 either 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 done 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 

