4.1. Uncertainties in the modeling
There are several uncertainties involved in our modeling of the line fluxes. We have already checked the effect of the lifetime of 44Ti (Sect. 3.1). We have also studied the effect of photoionization, and found that it introduces rather mild uncertainties. A similar level of uncertainty is due to the distance to the supernova. This is still inaccurate to the level of (Lundqvist & Sonneborn 1999; Walker 1999), which means an uncertainty in the line flux of .
Atomic data of iron are notoriously difficult to calculate accurately. This is therefore another source of error in our modeling, especially for individual lines. Most important for the line is the collision strength of that transition, . One normally assigns an uncertainty in the collision strength for the strongest iron lines to (e.g., Verner et al. 1999). In our models we have used (Zhang & Pradhan 1995). To study the effect in detail we have tested a model with and . We find that there is not a linear scaling between and , as one might naively believe. Instead, we find from linear interpolation that a decrease in collision strength gives a lower , and a correspondingly higher estimate of . The reason for this is that the gas is slightly hotter in the model with reduced collision strength, boosting the exponential term in the collisional rate so that it somewhat counteracts the reduced collision strength. We have recently found out (A. Pradhan, private communication) that the preferred value for at the low temperatures in SN 1987A is most likely closer to , i.e., higher than we have used. Despite this, we have generously assigned an uncertainty of (upwards) in due to atomic data.
Another source of uncertainty could be the explosion model used. In these calculations we have used the abundances from the 10H explosion model. A comparison between the two models 10H (Woosley & Weaver 1986; Woosley 1988) and 11E1 (Shigeyama et al. 1988) was done in Kozma & Fransson (1998b). There it was found that the iron lines are not sensitive to the explosion model used, because the iron core mass is the same in both models. The iron core mass is set by the amount of 56Ni which is accurately determined from the bolometric light curve. The choice of explosion model thus does not seem to be a major source of uncertainty when modeling these iron lines.
In our calculations we assume a local deposition of the positrons originating from the radioactive decays of . We believe this is a good approximation since optical and near-IR light curves of Fe I and Fe II lines show that trapping must occur (Chugai et al. 1997; Kozma & Fransson 1998b). Actually, there is no obvious sign of a leakage of positrons, neither from broad-band lightcurves (KF99), nor from the optical Fe I lines at Å until the last data point at days in Kozma (1999). Although the trapping may well be fully complete, we have assigned an error to this assumption by 15%.
Another approximation in our models is the assumption of a homogeneous density in each Fe-rich shell of the model core. To test the sensitivity to this assumption, we have run a model similar to M2, with photoionization included (see Table 2), but where we have divided the mass in the Fe-rich ejecta into two components of equal mass but with different densities. The denser component is set to be nearly five times more dense than the other. Despite the significantly different density distribution in this model compared to that in M2, the differences in and between the models are small. Instead of the values listed in Table 2, the fluxes in the two-component model are 0.98 and 0.20 Jy, respecively, i.e., differs by only compared to that in M2. This indicates that the model assumption of a homogeneous density in each Fe-rich shell does not introduce a major uncertainty.
Finally, screening and cooling by dust are potential sources of error in our models. The effects of dust are examined in Kozma & Fransson (1998ab). The screening we use is discussed in Kozma & Fransson (1998b), and is based on estimates by Lucy et al. (1991) and Wooden et al. (1993). Dust from pure iron is unlikely to form, as that would cool and block out all iron line emission once the dust has formed. On the contrary, there is a wealth of iron lines from the core at late times. KF99 and Kozma (1999) find good agreement between modeled and observed broad-band lightcurves for days and spectra at days. As was pointed out also for the positron leakage, the models are also able to reproduce optical Fe I lines at even later epochs. Based on this, we believe dust effects are small enough to neglect in our estimate of .
None of the model approximations we have used appears to be uncertain enough to allow to be of the same magnitude as . The best estimate of (within the framework of our modeling) should therefore come solely from . To estimate the combined error of due to all model approximations (except for the dust distribution in the ejecta), we adopt the uncertainties , , and for photoionization, distance, atomica data and clumping, respectively. For the choice of input model and positron leakage we adopt each. This gives a combined uncertainty which is . On top of this we must add the maximum systematic error of discussed in Sect. 3.1 for the lifetime of 44Ti. With the line profile discussed in Sect. 3.3, we therefore arrive at an upper limit on which is . We note that this limit excludes the upper ends of the allowed ranges of found by Chugai et al. (1997) and KF99 (see Sect. 1). Combining our limit with the preliminary results of KF99 for the broad-band photometry, a likely range for is . We emphasis, however, that the lower limit of this range is probably more uncertain than the upper (for the reasons mentioned in Sect. 1), which is indeed indicated by the preliminary analysis of Borkowski et al. (1997).
4.2. Implications of the derived mass of 44Ti
Models for the yield of 44Ti give quite different results. This is most likely due to how the explosion is generated in the models, and how fallback onto the neutron star is treated. Timmes et al. (1996; see also Woosley & Weaver 1995) use a piston to generate the explosion, and they account for fallback in a rather self-consistent way. In their model with zero-age mass (i.e., corresponding to SN 1987A) the mass of the initially ejected 44Ti is , but of this only escapes after fallback. This is less than we argue for in Sect. 4.1, and could suggest that fallback was not important for SN 1987A, though we caution again that the lower limit found by KF99 (see also Kozma 1999) may not be very strict. If fallback is unimportant the ejected amounts of 56Ni and 57Ni would be too high in this model, typically by a factors of , judging from the effects of fallback in the model in Woosley & Weaver (1995). It should be emphasized that the variation of with in Timmes et al. (1996) is complex, and that for models with and , the calculated comes within the range we propose, albeit close to our lower limit.
The models of Thielemann et al. (1996) simulate the explosion by depositing thermal energy in the core, and they insert the mass cut artificially so that the right amount of ejected 56Ni is produced. (This effectively means that fallback is included also in these models.) Simulating the explosion in this way ensures larger entropy and thus more alpha-rich freeze-out than in Woosley & Weaver (1995). Accordingly, the ratio (where is the mass of ejected 56Ni that does not fall back) is higher in the models of Thielemann et al. than in piston-driven simulations. For example, in the model of Thielemann et al. (1996) , and , with defined in the same way as and . The values of and are close to what have been inferred for SN 1987A (Suntzeff & Bouchet 1990; Fransson & Kozma 1993). The titanium mass is slightly larger than the upper limit of the range we estimate in Sect. 4.1. So, while our estimate of cannot rule out with certainty any of the two methods used for the explosion (piston-driven or heat generated), our results could indicate that an intermediate method should be used (see also the discussion on this in Timmes et al. 1996). From models of the chemical evolution of the Galaxy, and especially the solar abundance of 44Ca, a value for closer to that of Thielemann et al. (1996) may be more correct, at least for supernovae in general.
In this context we note that a higher value of is produced in asymmetric explosions (Nagataki et al. 1998). The method of calculation employed by Nagataki et al. (see Nagataki et al. 1997) is similar to that in Thielemann et al. (1996), though the models of Nagataki et al. allow for 2-D instead of just 1-D. With no asymmetry, the models of Nagataki et al. produce for an explosion similar to SN 1987A, when the mass cut has been trimmed to . This is significantly less than Thielemann et al. (1996) despite the similar method of modeling. Applying an asymmetry by a factor of 2 between the equator and the poles, the explosion energy becomes concentrated toward the equator resulting in relatively stronger alpha-rich freeze-out, which increases to (for ). This is outside our observationally determined range and could indicate that asymmetry was not extreme in SN 1987A, especially if is close to the limit derived by Borkowski et al. (1997). We note that a piston-driven calculation could perhaps allow for asymmetry, as such models give very small values of in 1-D.
A direct way to estimate in supernovae is to observe the gamma-ray emission from the radioactive decay of 44Ti. The 1.156 MeV line associated with the decay of 44Ti has only been detected in two supernova remnants (and no supernovae): Cas A (Iyudin et al. 1994; The et al. 1996) and the newly discovered gamma-ray/X-ray source GRO J0852-4262/RX J0852.0-4622 (Iyudin et al. 1998; Aschenbach 1998). While in the latter is not yet well-known, in Cas A was (Görres et al. 1998). In the models of Nagataki et al. (1998), the value of at the upper end of this range would indicate that Cas A exploded asymetrically. Our estimate of in Sect. 4.1 does not seem to indicate a high degree of asymmetry for SN 1987A, but an independent measurement of its 1.156 MeV line may be needed to be more conclusive on this point. To detect this emission from SN 1987A, instruments like INTEGRAL (Leising 1994) are required.
4.3. Interpretation of LWS observations
The two [O III] lines observed (see Sect. 2.2 and Fig. 2) can be used to estimate the average density of the emitting gas. We have used a multi-level atom to do this, using the atomic data of Mendoza (1983), Osterbrock (1989) and Aggarwal (1993). Assuming a temperature of K, we obtain a mean electron density of in the [O III] emitting gas.
The continuum can be fitted with a spectrum of the form , where is the dust optical depth. With a functional form of (where is in microns) we obtain a best fit with K, and . The fit (see Fig. 2) works well up to , where an extra source appears to add in. This could indicate the presence of cold dust. Assuming the same functional form for this emission as for the 37 K component, the temperature of the cold component is close to 10 K.
© European Southern Observatory (ESO) 1999
Online publication: June 30, 1999