tokens: so the shell velocity equals the ratio of the swept-up momentum and the swept-up mass M sh ( t ) ≈ f Ω M ej + M w,sh ( t ), where the swept-up mass of the pre-dynamical wind is given by and t end = t run ( v w / v sh ) is the time required for shock to reach the outer edge of the pre-dynamical outflow. Although the time evolution of v sh is complicated by the implicit dependence of M w,sh on v sh , we find that for fiducial parameters its value varies by less than a factor of 2 from its asymptotic value of where in the final line we have taken M w ≲ M ej , f Ω = 0.3 and . In our analytic estimates that follow we assume a temporarily constant value of , where ξ ≈ 0.5–1. Emission escapes on the photon diffusion time-scale through the fast ejecta, which at the shock radius R sh ≈ v sh t is given by where we have used equation ( 3 ) for δ o = 5 and δ i = 2. If the opacity remains roughly constant in time, then equating this to the expansion time t gives the time-scale for the shock-powered emission to rise to its peak (Arnett 1982 ) where κ 0.3 ≡ κ/(0.3 cm 2 g −1 ) is normalized to a typical value (Fig. 4 ). In the second line, we have normalized the ejecta mass to a fixed fraction of the total stellar mass; the fast ejecta velocity to the binary escape speed; and the semimajor axis of the binary to 10 times the main-sequence stellar radius R ms ≈ R ⊙ ( M / M ⊙ ) 0.8 . All else being equal, equation ( 21 ) shows that mergers between more massive stars, or evolved stars with radii much larger than their main-sequence values, will result in later peaking light curves, consistent with our full numerical solutions (Fig. 5 ). However, overall the dependence on binary parameter is relatively weak, consistent with the factor of ≲5 range of observed values t pk ≈ 30–150 d for a ∼ 2–100 R ms and M ∼ 1.5–30 M ⊙ (Fig. 1 , Table 1 ). Also note that, by assuming a fixed, relatively high value of the opacity in deriving equation ( 21 ), we cannot account for cases when the shock heating is insufficient to keep the ejecta ionized (e.g. the 2 M ⊙ model in Fig. 5 ). Table 1. Properties of LRN and their stellar progenitors. References: (1) Mason et al. ( 2010 ), (2) Munari et al. ( 2002 ), (3) Tylenda et al. ( 2005 ), Afşar Bond ( 2007 ) but see Munari et al. ( 2005 ), (4) Blagorodnova et al. ( 2017 ), (5) Smith et al. ( 2016 ), (6) Mauerhan et al. ( 2015 ), (7) Mauerhan et al. ( 2017 ), (8) MacLeod et al. ( 2017a ). a Ejecta velocity inferred from optical spectra (note that this may not always represent the velocity of the bulk of the ejecta, but instead the fastest components). b Approximate peak luminosity of second peak. c Assuming the second peak was bolometrically about 1 mag fainter than the first. If the forward shock dominates the shock power (equation 7 ), then the peak luminosity of the shock-powered second light-curve peak is given by (in the limit v sh ≫ v w ) where we have used equations ( 1 ) and ( 2 ). Normalizing again to the
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