In the seconds after collapse of a massive star, the newborn proto-neutronstar (PNS) radiates neutrinos of all flavors. The absorption of electron-typeneutrinos below the radius of the stalled shockwave may drive explosions (the"neutrino mechanism"). Because the heating rate is proportional to the squareof neutrino energy, flavor conversion of mu and tau neutrinos to electron-typeneutrinos via collective neutrino oscillations (CnuO) may in principle increasethe heating rate and drive explosions. In order to assess the potentialimportance of CnuO for the shock revival, we solve the steady-state boundaryvalue problem of spherically-symmetric accretion between the PNS surface (r_nu)and the shock (r_S), including a scheme for flavor conversion via CnuO. For agiven r_nu, PNS mass (M), accretion rate (Mdot), and assumed values of theneutrino energies from the PNS, we calculate the critical neutrino luminosityabove which accretion is impossible and explosion results. We show that CnuOcan decrease the critical luminosity by a factor of at most ~1.5, but only ifthe flavor conversion is fully completed inside r_S and if there is no mattersuppression. The magnitude of the effect depends on the model parameters (M,Mdot, and r_nu) through the shock radius and the physical scale for flavorconversion. We quantify these dependencies and find that CnuO could lower thecritical luminosity only for small M and Mdot, and large r_nu. However, forthese parameter values CnuO are suppressed due to matter effects. Byquantifying the importance of CnuO and matter suppression at the criticalneutrino luminosity for explosion, we show in agreement with previous studiesthat CnuO are unlikely to affect the neutrino mechanism of core-collapsesupernovae significantly.
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