In the seconds after collapse of a massive star, the newborn proto-neutron
star (PNS) radiates neutrinos of all flavors. The absorption of electron-type
neutrinos below the radius of the stalled shockwave may drive explosions (the
"neutrino mechanism"). Because the heating rate is proportional to the square
of neutrino energy, flavor conversion of mu and tau neutrinos to electron-type
neutrinos via collective neutrino oscillations (CnuO) may in principle increase
the heating rate and drive explosions. In order to assess the potential
importance of CnuO for the shock revival, we solve the steady-state boundary
value 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 a
given r_nu, PNS mass (M), accretion rate (Mdot), and assumed values of the
neutrino energies from the PNS, we calculate the critical neutrino luminosity
above which accretion is impossible and explosion results. We show that CnuO
can decrease the critical luminosity by a factor of at most ~1.5, but only if
the flavor conversion is fully completed inside r_S and if there is no matter
suppression. The magnitude of the effect depends on the model parameters (M,
Mdot, and r_nu) through the shock radius and the physical scale for flavor
conversion. We quantify these dependencies and find that CnuO could lower the
critical luminosity only for small M and Mdot, and large r_nu. However, for
these parameter values CnuO are suppressed due to matter effects. By
quantifying the importance of CnuO and matter suppression at the critical
neutrino luminosity for explosion, we show in agreement with previous studies
that CnuO are unlikely to affect the neutrino mechanism of core-collapse
supernovae significantly.Comment: 8 pages, 3 figures, accepted to MNRA