tokens: the case of mass transfer from the primary to the secondary star before the secondary expands. Figure 1. Radii (solid lines) and core masses (dashed lines) of the two stars in the binary system, (blue) and (red) as function of time. We assume that merger takes place when the secondary experiences its large and rapid expansion. Figure 2. Like Figure 1 , but for the (blue) and (red) binary system. In Figures 3 and 4 , we present the stellar models of the stars in the two binary systems we study here when the lower-mass companion experiences its large and rapid expansion. We assume that merger occurs during this phase. Figure 3. Structures of the binary systems (upper) and (lower) when the lower-mass companion experiences its large and rapid expansions at . In Table 1 we present the following stellar parameters for the two stars at the time when the more massive star (primary star) finishes its rapid expansion: luminosity, L , effective temperature, T eff , stellar masses, M * , core mass, M core , stellar radius, R * , core radius, R core , and binding energy of the envelope, E bind . In Table 2 we present the same quantities, but for the case when the secondary star finishes its rapid expansion. We assume that a merger takes place then (later we will discuss alternative merger times). Table 1 Properties of the Giant Primary Star and the Pre-giant Secondary Star When the More Massive (primary) Star Finishes Its Rapid Expansion Note. At this phase, the core of the secondary star is not relevant for our study, and we do not present its properties. The gravitational energy that the two cores release when they merge; namely, when their separation is , is given by Equation ( 10 ). The ratio of this energy to the binding energy of the two envelopes, isThis is the typical ratio for the and binary system (using the data from Table 2 ). For the M ZAMS,1 = 31.5 M ⊙ and binary system this ratio is 0.8 (using the data from Table 2 ). As the efficiency of channeling the gravitational energy of the merging cores to envelope removal is α CE 1, when the cores merge there is still bound envelope, but a large fraction of the envelope is leaving the system, and a fraction might form an extended envelope. Table 2 Properties of the Giant Stars When the Lower-mass Star in the Binary System Finishes Its Rapid Expansion Note. We assume that in many cases merger takes place during this time (see Figures 1 and 2 for the models). 4.3. The ILOT Properties Using these results and Equation ( 10 ), we can crudely estimate the total radiated energy in the ILOT. Very crudely, in many ILOTs the typical ratio of radiated to kinetic energy is E rad ≈ 0.1 E kin (e.g., Kashi Soker 2010 ). We take about half of the envelope to be ejected at the escape speed, , and the radiation to carry a fraction of of that energy. For an ejected mass of , we find . This is the general median value of the radiated energy of ILOTs (e.g., Kashi 2018 ). The photon-diffusion time from an ejected mass of
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