The stability of mass transfer in binaries with convective giant donors remains an open question in modern astrophysics. There is a significant discrepancy between what the existing methods predict for a response to mass loss of the giant itself, as
well as for the mass transfer rate during the Roche lobe overflow. Here we show that the recombination energy in the superadiabatic layer plays an important and hitherto unaccounted-for role in he donors response to mass loss, in particular on its luminosity and effective temperature. Our improved optically thick nozzle method to calculate the mass transfer rate via $L_1$ allows us to evolve binary systems for a substantial Roche lobe overflow. We propose a new, strengthened criterion for the mass transfer instability, basing it on whether the donor experiences overflow through its outer Lagrangian point. We find that with the new criterion, if the donor has a well-developed outer convective envelope, the critical initial mass ratio for which a binary would evolve stably through the conservative mass transfer varies from $1.5$ to $2.2$, which is about twice as large as previously believed. In underdeveloped giants with shallow convective envelopes this critical ratio may be even larger. When the convective envelope is still growing, and in particular for most cases of massive donors, the critical mass ratio gradually decreases to this value, from that of radiative donors.
Using our recently improved Monte Carlo evolution code, we study the evolution of the binary fraction in globular clusters. In agreement with previous N-body simulations, we find generally that the hard binary fraction in the core tends to increase w
ith time over a range of initial cluster central densities for initial binary fractions <~ 90%. The dominant processes driving the evolution of the core binary fraction are mass segregation of binaries into the cluster core and preferential destruction of binaries there. On a global scale, these effects and the preferential tidal stripping of single stars tend to roughly balance, leading to overall cluster binary fractions that are roughly constant with time. Our findings suggest that the current hard binary fraction near the half-mass radius is a good indicator of the hard primordial binary fraction. However, the relationship between the true binary fraction and the fraction of main-sequence stars in binaries (which is typically what observers measure) is non-linear and rather complicated. We also consider the importance of soft binaries, which not only modify the evolution of the binary fraction, but can drastically change the evolution of the cluster as a whole. Finally, we describe in some detail the recent addition of single and binary stellar evolution to our cluster evolution code.
