Astron. Astrophys. 328, 203-210 (1997)

4. A simple light curve model

The observed differences in the light curves of SNe Ia may be attributed to many different factors. Accurate fitting of the light curve of a particular SN Ia may indeed require an ad-hoc explosion model and fine tuning of several parameters. However, this approach makes it difficult to disentangle the relative importance of the different parameters. Since we wanted to isolate the effects of the two arguably most interesting physical parameters, and , we developed a simple Monte Carlo code for the calculation of the bolometric light curve and did not use any particular explosion model. In particular, we made no effort to fit the shape of the light curve near maximum in detail.

Starting from a given density stratification and a distribution of Ni within the ejecta, the code computes the deposition of the -rays as a function of time with an MC scheme. The -ray energy sources are given by Sutherland & Wheeler (1984), while the opacity to -rays is assumed to be gray and to have a constant value cm² g^-1, in agreement with recent calculations (Swartz et al., 1995). The energy released by the decay of ⁵⁶ Co in the form of positrons (19% of the total) can only be transformed into -rays if the positrons annihilate.

The calculations show that with the expansion the ejecta become rapidly transparent to -rays. In particular, 200 days after the explosion the fraction of the -ray energy which is deposited in a typical model is only . At this time the kinetic energy of the positrons may become the main contributor to the light curve, if a major fraction of it is deposited.

The early assumption was that positrons deposit all of their kinetic energy and annihilate almost on the spot (Axelrod, 1980). Subsequently, this idea has been questioned. For instance, Chan & Lingenfelter (1993) suggested that a significant fraction of the positrons manage to escape, possibly because of the presence of radially combed magnetic fields. On the other hand, Swartz et al (1995) argued that positrons cannot escape at all, otherwise a significant fraction of the radioactive decay energy would be lost.

Recently, it has been argued that to fit the late light curve of SNe Ia requires that the ejecta become progressively transparent also to positrons (Colgate et al., 1997), although with a longer time scale than for -rays (a typically adopted value is cm² g^-1). Describing the positron deposition by means of a positron opacity appears to be a natural approach, and we have adopted it in our light curve code. This gives us the freedom to manipulate the positron deposition as well as the transport of optical photons.

Our -ray source function for ⁵⁶ Co therefore takes the form

where erg g^-1  s^-1 is the rate of energy production from the ⁵⁶ Co decay (excluding the KE of the positrons), and are the mean lifetimes of ⁵⁶ Ni and ⁵⁶ Co, respectively, and is the positron deposition function. The positron KE accounts for an additional -ray source, which is given by

When the -rays and the positrons deposit their energy, it is assumed that after thermalization this energy emerges as optical photons, so the contribution to PdV work is negligible. These optical photons can only contribute to the bolometric luminosity when they manage to escape from the ejecta. Given the high ejecta density, especially early-on, escape is not instantaneous. This is the reason why SNe Ia reach their peak only 2-3 weeks after the explosion. The random walk of the optical photons in the expanding ejecta is also followed with an MC scheme. A gray optical opacity is assumed where only a pure scattering opacity obtains. The ejecta are assumed to be purely scattering. Time delay is accounted for by computing the time elapsed between successive scatterings, and updating the density before a new free flight is started. So finally for each optical energy packet we have a `production time', corresponding to the time of emission, i.e. the time of deposition of the -ray packet from which the optical packet originates, and an `emission time', the time when the packet eventually escapes from the ejecta. The difference between these two times becomes smaller and smaller as time goes on. The code allows us to reproduce not only the declining part of the light curve but also, at least roughly, the rising branch and the near-maximum phase. For `average' SNe Ia we find that cm² g^-1 gives a reasonable fit to the rising branch and to the maximum of the light curve, whereas the late light curve is quite insensitive to the value of this parameter.

With these assumptions, the most important parameters for the light curve modelling are the mass of synthesized ⁵⁶ Ni (), which determines the absolute luminosity, and the total mass of the ejecta (), which determines the optical depth for the radiation from the radioactive decay. affects mostly the time of maximum and its brightness, while influences the late-time behaviour. For larger the maximum occurs later, and it is fainter and broader. For smaller the decay at late times is faster.

In the past, the consensus was that and were exactly the same in all SNe Ia events, making them ideal standard candles. Estimates for were in the range 0.5 to 0.7 , while . However, there is increasing evidence that both and can be very different in different objects, although most SNe Ia appear to cluster around the historical values.

Since the SNe we have listed in Table 1 span a range of absolute magnitudes at maximum and of decline rates, it is quite likely that they cover a range of , and possibly also of . To investigate this question further, we used our MC light curve code. As we discussed above, and can be easily changed as input to the code. We took the approach of using a simple code and of not using any particular explosion model for any particular object, because literally hundreds of models, involving different and explosion mechanisms, and producing different amounts of ⁵⁶ Ni, have now been produced by various groups, making it difficult to distinguish between them on the basis of their ability to fit observed light curves.

We adopted a W7 density structure, and rescaled it according to the epoch and to the values of and . We assumed that the Ni resides at the centre of the ejecta, in rough agreement with models of SN Ia explosions. The only exception is the model for SN 1991T. In this case, analysis of the early-time spectra suggests that there is an outer Ni shell (Mazzali et al. 1995).

Homologous expansion is assumed for rescaling to the appropriate epoch. Since different values of and should lead to different kinetic energies of the ejecta, the dependence is also rescaled. If we define the ratios and , we can rescale the kinetic energy per unit mass according to .

In doing this we assume that all the KE is produced by burning to ⁵⁶ Ni, neglecting the contribution of incomplete burning to intermediate mass elements (IME) such as Si. This is a reasonable approximation as long as is similar for all values of , which is of course not necessarily true. In particular, the energy will depend not just on , but also on a third parameter, the total mass of the newly synthesized elements. Indeed, in some cases, like SN 1991bg, the mass of these elements (e.g. Si-Ca, and ) is comparable to, and probably greater than . Thus we expect our models to be an increasingly poor representation of the real properties of the SN the further departs from the reference W7 value of 0.6 .

On the other hand, most SN Ia explosion models, including models for sub-Chandrasekhar explosions (Woosley & Weaver 1994), produce functions whose shape is similar to that of W7, so in this respect our approximation should be reasonable. The velocity of a given shell in the ejecta is then rescaled according to , while the density of that shell is given by .

Note that models with a ratio have the same KE per unit mass as W7 (for which ), and their density is simply the W7 density structure times , while models for which have the same density as W7, but with each shell shifted to a velocity . Since in the photospheric epoch the apparent photosphere forms at an almost constant value of the density, this may be a useful independent tool to investigate the values of and . This approach was adopted for all sub-Chandrasekhar models, but not for the SN 1991T ones, since the observed outer distribution of ⁵⁶ Ni clearly must have a different effect on the kinetic energy. Thus, the KE for the SN 1991T model is probably underestimated.

Apart from the somewhat arbitrary rescaling of the velocity field, several other uncertainties must be kept in mind when comparing the model bolometric light curve with the observations:

1. Though we argued the V and bolometric light curves are not very different (cf. Sect. 2), they are not the same. A proper comparison would require a full NLTE light curve calculation, with the appropriate SN model. Such calculations do not exist as yet. Actually, even the bolometric light curve would require a more complex calculation than in our simple code. In particular, the opacities may change as a function of depth in the ejecta, and with time.
2. The SN may be clumpy, thus changing the deposition rate. This would also influence the nebular spectrum (Mazzali et al., in preparation).
3. The positon deposition is not well known. We discuss this point in the next section.
4. The steady-state assumption that the luminosity equals the instantaneous energy input breaks down after about 600 days, when the deposition time for the primary electrons produced by the -rays becomes long with respect to the dynamical and ionization time scales of the nebula (Axelrod, 1980). The ionization state may then be far from a steady state solution.
5. The distance and extinction to the various SNe are uncertain. By using a single source for the distances, we should at least minimize the relative errors, but the absolute numbers could change. For instance a Cepheid-based set of distances would require much bigger Ni masses.

© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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