3. Interpreting the SWS observations
To understand what the upper limits on the iron lines in Sect. 2.1 mean, we have made model calculations. The models are described in Sect. 3.1, and we discuss the results from the calculations in Sect. 3.2
3.1. The model for line emission from the supernova
The code we have used for our theoretical calculations is described in detail in Kozma & Fransson (1998a), and here we will only give a brief summary of the model.
The thermal and ionization balances are solved time-dependently, as are also the level populations of the most important ions.
The ions included are H I-II, He I-III, C I-III, N I-II, O I-III, Ne I-II, Na I-II, Mg I-III, Si I-III, S I-III, Ar I-II, Ca I-III, Fe I-V, Co II, Ni I-II. The following ions are treated as multilevel atoms: H I, He I, O I, Mg I, Si I, Ca II, Fe I, Fe II, Fe III, Fe IV, Co II, Ni I, Ni II. The remaining lines are solved as 2-level atoms or as level systems. For Fe I we include 121 levels, Fe II 191 levels, Fe III 112 levels and for Fe IV 45 levels. A total of approximately 6 400 lines are included in the calculations.
The radioactive isoptopes included are 56Ni, 57Ni, and 44Ti. These radioactive decays provide the energy source for the ejecta, and we calculate the energy deposition of gamma-rays and positrons solving the Spencer-Fano equation (see Kozma & Fransson  for a more detailed description of our thermalization of the gamma-rays). We have assumed that the positrons deposit their energy locally, within the regions containing the newly synthesized iron.
We have included of 56Ni (Sect. 1) and of 57Ni, which corresponds to a 57Fe/56Fe ratio equal to two times the solar ratio. This mass of 57Ni gives a good fit to the bolometric light curve when the effects of freeze-out are taken into account (Fransson & Kozma 1993). It is also consistent with observations of [Fe II] and [Co II] IR lines (Varani et al. 1990; Danziger et al. 1991), observations of the 57Co 122 keV line (Kurfess et al. 1992), and theoretical nucleosynthesis calculations (Woosley & Hoffman 1991). In our `standard' model we have used of 44Ti (e.g., Kumagai et al. 1991; Timmes et al. 1996). We have also made calculations with and .
Until recently, the lifetime of 44Ti has been very poorly known. In our calculations we have used an e-folding time of 78 years thought to be a representative value to those found from experiments. However, while completing our calculations, we were informed about the recent measurements mentioned in Sect. 1 (S. Nagataki, private communication). The new, and accurate value, years, is close to what we have used, but decreases the instantaneous radioactive energy input from the 44Ti decay by . This systematic error, shifting our estimated value of upward by a similar amount, has been included in the error analysis in Sect. 4.1.
As input for the calculations we have adopted the abundances from the 10H explosion model (Woosley & Weaver 1986; Woosley 1988). Spherically symmetric geometry has been assumed, with shells containing the major composition regions as given by the explosion model. We have used the same distribution of shells as in Kozma & Fransson (1998a) where hydrogen is mixed into the core. The iron-rich core is assumed to extend out to , outside of which we attach a hydrogen envelope, out to .
For the line transfer we have used the Sobolev approximation. This is justified for well-separated lines in an expanding medium. However, especially in the UV, there are many overlapping lines, and one can expect UV-scattering to be important. The importance of line scattering decreases with time as the optical depths decrease. The effect of line scattering is to alter the emergent UV-spectrum, but it also affects the UV-field within the ejecta. The ionization of elements with low ionization potential is sensitive the UV-field. During scattering the UV photons are shifted toward longer wavelengths, both due to a pure Doppler shift, but also because of the increased probability of splitting the UV-photons into several photons of longer wavelengths. A more accurate treatment of the line scattering is therefore expected to decrease the importance of photoionization. For the modeling of [Fe II] 25.99 , the photoionization of Fe I to Fe II is likely to introduce the largest uncertainty. To check the importance of this effect we have in some models simply switched off the photoionization. A more comprehensive line transfer modeling is discussed in KF99.
3.2. Model calculations
The results of our calculations can be seen in Table 2. We tabulate line fluxes for four strong lines for three values of : (M1), (M2) and (M3). For each value of we show results for models with and without photoionization. In all six models a simple form of dust absorption was assumed (Kozma & Fransson 1998b). The line fluxes in Table 2 are for a box line profile between , and assuming a distance to the supernova of 50 kpc.
Table 2. Modeled line flux in Jy at 4 000 days.
From Table 2 it is evident that [Fe II] 25.99 is by far the strongest line in our simulations, and that its flux scales roughly linearly with , as expected from Sect. 1. From power-law fits to the results of the calculations described in Table 2, we obtain when photoionization is included, and nearly the same without photoionization (). The second strongest line is [Fe I] 24.05 . Its dependence on is weaker: and , respectively. Other lines from SN 1987A are far too weak for our ISO observations.
The stronger dependence of on than for is due to ionization: while a higher boosts the relative fraction of Fe II, , the relative fraction of Fe I, , decreases. For example, in models with photoionization, the average value of in the Fe-rich gas increases from to when is increased from to , while decreases from to . The fact that the 26 line is so much stronger than the 24 line despite the roughly similar relative fractions of Fe I and Fe II is just a result of the much larger collision strength of the Fe II line than for [Fe I] 24.05 .
Table 2 shows that switching off photoionization does not have a dramatic effect on the line fluxes. In the case of M2, decreases by and increases by when we turn off photoionization, simply due to a minor shift in ionization between Fe I and Fe II. As we will discuss in Sect. 4.1, there are other uncertainties which are of the same magnitiude.
3.3. Mass of 44Ti from [Fe II] 25.99
With our results it is straightforward to estimate the upper limit on . Using Jy from Sect. 2.1, and our power-law fit to the results in Table 2 (Sect. 3.2), we find an upper limit which is with (without) photoionization included. These masses are for a box-shaped line profile, and therefore provide conservative limits. However, there is good reason to believe that the line profile should be similar to those for the lines observed by Haas et al. (1990) days after the explosion. The strongest line observed by Haas et al. was [Fe II] 17.94 and it had . Although it peaked just short off , and thus was not symmetric around the rest velocity of the supernova, it had the general appearance of the expected line profile formed by a filled sphere with constant emission throughout the sphere. For a sphere extending out to , the peak is 1.5 times higher than for the box profile used in Table 2 (which is valid for a hollow sphere), and . Using such a line profile in our upper limits on , we instead obtain with (without) photoionization included. Within the framework of our modeling, a conservative limit (i.e., the case when photoionization is unimportant) for a plausible line profile is therefore . In Fig. 1, we have included the expected line emission for such a model with this limiting mass of 44Ti. We will evaluate this limit on in Sect. 4.1.
© European Southern Observatory (ESO) 1999
Online publication: June 30, 1999