5 Discussion

The final OIR light curve is admittedly and necessarily based on a number of assumptions, but we are confident that the basic features of the light curve, the steep decline followed by a late flattening, are real. In this section we will discuss a few possible interpretations for such an evolution.

We first note the steep exponential decline seen in the ground based observations. This fast decline appears to have started already about 65 days past the explosion (McKenzie & Schaefer 1999; S00), and the light curve continues to fall significantly steeper than the decay rate of ⁵⁶Co up to more than 500 days past explosion. There is thus no sign for a "positron phase'' in which the fully deposited kinetic energy from the positrons would dominate the light curve. Two different scenarios could account for this.

One is that the optical depth for gamma-rays, albeit decreasing, may always be large enough to dominate over the positrons. An example is shown below (Fig. 5). Alternatively, the positrons may indeed dominate at late phases, but some of the positrons are able to escape the ejecta before being thermalized.

The second characteristic to note in the very late-time light curve of SN 1998bw is the flattening of the light curve at the last phases, as observed by HST. The late-time light curves of core-collapse supernovae can be powered by several mechanisms (see e.g., Sollerman et al. 2001), as will be further discussed below.

5.1 Radioactive powering?

The contributions from different radioactive elements to late supernova light curves were discussed by Woosley et al. (1989). Only in SN 1987A has a radioactive decay other than ( ⁵⁶Ni $\rightarrow$) ⁵⁶Co $\rightarrow$ ⁵⁶Fe been unambiguously observed to power the late light curve. Also for this supernova, a flattening of the light curve was observed after about 800 days. The reason for this was actually two-fold. First, at these late phases the radioactive decay of the more long-lived nucleus ⁵⁷Co became important. Secondly, the long recombination scale allowed energy stored at earlier epochs to contribute at later phases (Clayton et al. 1992; Fransson & Kozma 1993). This is known as the "freeze-out'' effect.

For SN 1998bw, with its very high explosion energy, the discussion must also take into account that the optical depth to gamma-rays decreases very rapidly. The high explosion energy also opens the possibility for different nucleosynthesis (Nakamura et al. 2001b).

The decreasing optical depth to gamma-rays is obvious from the first part of the light curve, as mentioned above. A very simplistic model to account for this in terms of radioactive decay is presented in Fig. 5. In this model the flux from the decay of ⁵⁶Co evolves as ${\rm e}^{-t/111.3} \times \left(1 - 0.965{\rm e}^{-\tau}\right)$, where the optical depth, $\tau = {(t_1/t)}^2$, decreases due to the homologous expansion. Here, 111.3 days is the decay time of ⁵⁶Co and t₁ sets the time when the optical depth to gamma-rays is unity. Furthermore, 3.5% of the energy in these decays is in the form of the kinetic energy of the positrons, which are assumed to be fully trapped. This model can reasonably well mimic the observed light curve from day 64 to day 538 for a value of t₁ = 105 days (Fig. 5). This means that the trapping of gamma-rays is almost complete at day 64, but that the positron contribution is not dominating until after $\sim$600 days.

Figure 5: Using a very simple model for the radioactive powering of SN 1998bw, a reasonable fit to the data can be achieved. The model is described in the text. The powering of 56Co, 57Co, and 44Ti contributes at progressively later phases. The early observations marked by crosses are from P01 and shifted to a distance of 35 Mpc.

At the HST phases the light curve flattens out and can no longer be explained in terms of ⁵⁶Co. An interesting possibility is then a contribution from ⁵⁷Co, in particular since the decay rate in the HST detections seems to agree well with the decay time of ⁵⁷Co $\rightarrow$ ⁵⁷Fe.

However, the decay of ⁵⁷Co has no positron channel. A fairly large amount of ⁵⁷Ni (which quickly decays to ⁵⁷Co) must therefore be synthesized in the explosion, in order to make a significant contribution in this model.

In Fig. 5 we have included a ratio of ⁵⁷Ni/ ⁵⁶Ni which is three times larger than the ratio observed in SN 1987A. The optical depth to gamma rays is also enhanced by a factor of 2.4 as compared to ⁵⁶Ni, to allow for the smaller gamma-ray energies involved (Woosley et al. 1989). In addition, ⁴⁴Ti has also been added with a three times larger ratio. This nucleus contributes at much later phases in SN 1987A (Kozma 2000; Lundqvist et al. 2001), but can become important at relatively earlier phases in SN 1998bw as it has a significant positron channel.

Although these abundance enhancements are quite arbitrarily chosen to match the observations, we emphasize that the nucleosynthesis yields may indeed be different for very energetic and possibly asymmetric explosions. Nakamura et al. (2001b) and Maeda et al. (2001) clearly show that enhanced amounts of ⁴⁴Ti may be expelled in such circumstances. We therefore note the possibility that the late light curve of SN 1998bw can be explained in terms of radioactivity, if the abundances of the radioactive isotopes are enhanced compared to the case of SN 1987A.

Although the above model is clearly too simplistic to give accurate results, it can be used to estimate the amount of ⁵⁶Ni needed to power the supernova at a distance of 35 Mpc. With the radioactive luminosities properly included (Kozma & Fransson 1998) we have used a nickel mass of $0.3~{M}_\odot$ to power the light curve in Fig. 5.

This is close to the lower limit obtained by S00. This is indeed a lower limit, as the simple model forces the gamma-rays to be almost fully trapped at day 65, at the start of the rapid decline. The models presented in S00, with a complete calculation of the gamma-deposition for real explosion models, trapped 6-10% of the gamma-ray energy at 400 days, while the toy model presented here traps $\sim$7% at this epoch. Although detailed calculations are needed to accurately determine the nickel mass, it is clear that the estimates will be lower than in S00 simply because some of the light attributed to the supernova in that analysis was due to the complex background.

The above estimate is, however, not very far from the 0.4 ${M}_\odot$ required to power the peak of the light curve in the refined models of Nakamura et al. (2001a), especially when the different distance estimates are taken into account.

We note that a scenario where the steep light curve decline is instead due to positron escape will clearly require much higher nickel masses to account for the observed luminosity. In such a scenario the late time contribution of ⁵⁷Co is less likely, because with no positron channel an unrealistically high abundance of this isotope would be required.

5.2 Other powering mechanisms?

It must be clarified that apart from radioactive heating, several other mechanisms could power the late light curve. None of the usual suspects can really be ruled out based on the sparse observational constraints we have at the latest phases.

Interaction with circumstellar material (CSM) is not uncommon in core-collapse supernovae (e.g., Leibundgut 1994). In the context of SN 1998bw, Tan et al. (2001) proposed an association of GRB 980425 based on a model with a high pre-supernova mass loss of $\sim$few  $\times~10^{-4}$ ${M}_\odot$ yr^-1. Weiler et al. (2001) also argued for a CSM from radio monitoring of the supernova. However, the very fast wind expected from a WR progenitor can give a rather modest density even with a high mass loss rate, and the interaction need not give rise to optical emission. We note that as long as the supernova was spectroscopically monitored, no spectral signatures from CSM interaction were ever detected in SN 1998bw (S00). Although absence of evidence is not evidence of absence, this may constrain the most massive progenitor models.

If SN 1998bw formed a black hole which accretes matter, this could in principle show up in the late light curve (Fynbo et al. 2000). We note that the luminosity at late phases is close to the Eddington luminosity, which would require very high efficiency. Moreover, an accretion scenario should provide a power-law decay rate (Balberg et al. 2000), which is not observed. Black hole powering is mainly expected to be seen for supernovae with very little radioactive material, because the radioactive heating will otherwise outshine the accretion luminosity. This is quite the opposite of SN 1998bw.

Another scenario for late-time emission is a light echo, as observed in SN 1998bu (Cappellaro et al. 2001). Again, we have little information available to exclude such a possibility. We can only note that the supernova appears fairly red in the day 778 data (LP versus CL), while a light echo should reflect an early blue phase.

Finally, the late light curve might also be boosted by a freeze-out effect. As mentioned above, this was observed in SN 1987A and contributed to the light curve tail powered by ⁵⁷Co, as illustrated in Fig. 1 of Fransson & Kozma (1993). Most of the freeze-out in SN 1987A took place in the extended hydrogen envelope, which is absent in SN 1998bw. On the other hand, the faster expansion of SN 1998bw favors a freeze-out scenario. If freeze-out was indeed important at these phases, smaller yields of the long-lived nuclei would be required to power the light curve. Detailed time-dependent modeling is needed to quantify this effect.