ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: agree well with Equation ( 19 ), though there are substantial differences at lower masses. Better agreement with our models is obtained if , and that is our suggested value. The breakdown below 12 is not particularly surprising. These low-mass SNe have low energy (10 50 erg), but also low 56 Ni mass (). The combination is still appreciable even though the mass of radioactivity itself is very small. If the mass fraction of 56 Ni in a given zone, after mixing, is less than about 0.001, the decay energy, erg g 1 , of 56 Cowill, after time , be only a small fraction of the internal energy at recombination, erg g 1 , and recombination will not be greatly affected. For the very low mass explosions and mixing prescription assumed, the cobalt abundance is below this limiting value in all zones except at the very base of the hydrogen envelope. In any case the correction factor is small, less than 10, in these low-energy explosions. Combining our best fit to the no-radioactivity models with this best-fit correction factor results in good agreement with the full set of models with radioactivity included (Figure 34 ). As expected, most of the large deviations are for the lightest and heaviest stars. For small stars, the scaling relation overestimates the plateau duration because the correction term, , is too big. At high mass, there are cases with a large amount of fallback (Figure 13 )and therefore no 56 Ni production. Indeed, such short plateaus in massive stars with normal luminosities (Figure 31 ) may be an observable signature of BH formation. Figure 34. The scaling laws that showed good agreement in Figure 32 are now compared against the standard explosion models with their 56 Ni included. The models used are from the Z9.6 and N20 series. A red vertical bar represents a range of possible plateau durations calculated from the models assuming that (1) the photospheric radius had receded to 50 of its maximum value (lower bound)or (2) the whole star had recombined (upper bound). Black points result from multiplying Equation ( 14 ) with C t = 88 by the correction term, Equation ( 19 ), with C f = 50. 7.1.3. Systematics Given the generally good agreement of our SN IIp model characteristics with analytic expressions, it is worth exploring whether these systematics can be exploited to obtain insights into the masses and energies of observed events. Another interesting question is whether SNe IIp, or some subset, can be used as standard candles. Neglecting the radioactive correction, Equation ( 19 ), the luminosity and plateau duration are proportional to powers of the explosion energy, pre-SN radius, and envelope mass. These are not independent variables. The radius of the pre-SN star increases monotonically with its main-sequence mass, ,andto good approximation in the mass range where most SNe IIp occur (920 ), (Figure 35 ). Above 20 , the hydrogen envelope mass of the pre-SN star declines with owingto mass loss, but below 20 , it too has a monotonic