We selected the model parameters in order to fit the magnitude at maximum and the long-term decline rate of the individual SNe. Although less emphasis was given to the detailed fit of the light curves, we comment on the major discrepancies and the possible reasons.
In a first set of models we kept constant, and tried to obtain the best fit by varying and . In general, increasing causes a shift of the light curve to brighter absolute magnitudes, while increasing delays the time of maximum and makes the decline of the light curve less steep.
The model fitting to the V light curves of the five SNe Ia is shown in Fig. 1. The models used for the fits are as follows.
For SN 1991bg, , . This is in agreement with findings from modelling of both the photospheric and the nebular-epoch spectra (Mazzali et al. 1997), making SN 1991bg the best-studied case of a probable sub-Chandrasekhar mass SN Ia.
The V light curves of SNe 1992A and 1993L are practically indistinguishable once they have been corrected for extinction and relative distance. With the assumed distances, they reach a maximum of about -18.6 mag. The curves are fitted by a model with , , which suggests they may also be sub-Chandrasekhar events. The model has the same ratio as W7.
SN 1994D appears to be brighter than SNe 1992A and 1993L. The peak magnitude (-19.4) is well fitted by a model with , , implying that the Ni production may be higher than in `classical' SNe Ia. That SN1992A is fainter than average has also been demonstrated recently by Della Valle et al. (in preparation) who determined the distance to its parent galaxy using the method of the globular cluster luminosity function.
As can be seen from the insert of Fig. 1, a common feature of the models is that the rise to maximum is steeper and the early decline slower than observed. This could be improved by adopting different Ni distributions. In general, placing Ni further out than in the centre of the ejecta makes the light curve faster. We have computed a model using the W7 Ni distribution, which has a `hole' in the central, highest density regions. This leads to faster escape of the optical photons after maximum, and produces a faster-declining light curve, which compares well with the observed ones. Still, W7 seems to be too faint to reproduce the light curve of SN 1994D and too bright for that of SN 1992A.
It should be noted here that the V light curves of SNe 1994D, 1992A and 1993L are very similar in shape. Thus, if the relative distances and extinctions were appropriately adjusted, all three SNe could be explained with a single explosion model. On the other hand, significant observational evidence also exists in favour of there being a difference between SNe 1994D and 1992A: SN 1994D is about 0.1 mag bluer at maximum, has a lower SiII line velocity near maximum (Patat et al. 1996, Fig.10a, where SNe 1994D, 1989B and 1990N appear to form one group and SNe 1992A and 1981B another), and broader nebular lines, with the SNe just mentioned falling again into two different groupings (Mazzali et al, in prep.). Thus, intrinsic differences between SNe generally labelled as `classical' Ia are only beginning to receive the attention they deserve.
The last object we tried to fit is SN 1991T. For this SN, analyses of the early- (Mazzali et al. 1995) and late-time (Spyromilio et al. 1992) spectra suggest that the Ni mass is about 1 , and that a significant fraction of it is located in the high velocity outer part of the ejecta. If we assume , a good fit can be obtained for , of which 0.6 is in the centre and 0.5 is in the outer layers, which confirms previous findings.
Thus, it appears that a range of almost one order of magnitude in is required to fit all the objects, and of at least a factor 2 in . The range in reflects the range in absolute magnitudes at maximum rather closely.
The models with fit the observations up to 400-500 days well for all the objects except 1991bg. In the case of SN 1991bg, starting 100-150 days after explosion the model is brighter than the observations. To reconcile the model with the observations, one could further decrease , but this would lead to a very early maximum and would also cause problems for the interpretation of the spectra near maximum (Mazzali et al., 1997). Actually, a good fit to the late light curve of SN 1991bg can be obtained assuming that the opacity for positrons is much smaller than . This is shown in the bottom panel of Fig. 2, where the model is calculated for the extreme case of complete transparency of the ejecta to positrons ().
The opposite trend may be indicated by the SNe Ia 1993L and 1992A. At phases later than 400 days the observed light curves appear to decline at a rate slower than the prediction of the model with . For SN 1994D, no observations are available at these very late epochs.
Finally, the light curve of SN 1991T is compatible with . This may be real, but it may also indicate that the fraction of Ni on the outside is less than what we have assumed. Note, however, that if we distribute all the of Ni in the centre and compute a rescaled model with , this has a larger KE than W7, and therefore a lower density, so it actually declines faster than the model shown in Fig.2. Another possibility is that . We computed a model with , , which produces a light curve with a broad maximum and a slow decline. This model fits the observations reasonably well at all epochs, including the late phases. In the phase 60-150 days the model light curve is brighter than the observed one, but this is a feature common to all models shown in this paper. Even in this case, however, cannot be ruled out at 300-400 days. Clearly, SN 1991T deserves a much more detailed investigation than has been attempted here.
In principle we cannot exclude the possibility that, in addition to the radioactive decay, something else may contribute to the luminosity. In particular, given that spectra are not available at these very late epochs, we cannot entirely rule out contributions from echoes by circumstellar dust, as has been observed for SN 1991T. However, the luminosity decline of SN 1992A from 400 to 926 days ( mag/100d) is very close to the 56 Co decay rate (0.98 mag/100d,) and other energy sources do not seem to be required. As shown in Fig. 2, this is the decline rate expected in the case of essentially complete trapping of the positrons () and complete transparency to the -rays.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998