The Livermore Supernova Explosion Code was used to calculate the effect of a massive Dirac neutrino on neutrino emission from SN 1987A in a self-consistent manner. Spin-flip interactions involving neutrino-nucleon, nucleon-nucleon and pion-nucleon collisions lead to the copious emission of sterile, right-handed neutrinos and cool the core faster than the observed neutrino emission time for Dirac masses exceeding about 3 keV. These limits are compared with the previous limits of Burrows et al.