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3.1. The Spectral Shape Corrections

Merging the space-borne and ground-based data sets calls for careful consideration of the passbands used. The differences in the SN flux measured by two different realizations of the same passband, evaluated in magnitudes, have been called “spectral shape corrections,” or S-corrections (Stritzinger et al. 2002; Krisciunas et al. 2004; Pignata et al. 2004). We generalize here the use of this name to the difference between the flux measured by a ground-based passband and its space-borne counterpart.

Computing the S-corrections requires precise knowledge of the passbands’ throughput and, ideally, spectra of SN 1994I at the specific dates of the photometry. The spectroscopic evolution of SN 1994I was closely followed (Filippenko et al. 1995; Clocchiatti et al. 1996a), although the last spectrum available was taken only ∼147 days after maximum light, more than 130 days before the HST photometry. Regarding the passbands, those of the HST WFPC2 are well characterized,16 but three slightly different realizations of the ground-based BV(RI)_C system were used by Richmond et al. (1996). Hence, as a model for the ground-based system, we computed the S-corrections as if the standard system were that of Bessell (1990), the one to which Richmond et al. (1996) attempted to transform. We estimated an uncertainty for these S-corrections by using different realizations of the Bessell passbands at assorted optical–near-IR imagers at the Cerro Tololo Interamerican Observatory and the Kitt Peak National Observatory. Note that we transformed the curves of Bessell (1990) from energy-based to photon-based units, and, for the other ground-based passbands, we factored in a typical CCD quantum efficiency, the spectral response of two aluminum-coated surfaces, and the typical response of the sky telluric absorption bands if needed.

With this database of passbands and spectra, we explored whether it was better to transform the photometry from the space-borne to the ground-based system, or vice-versa. We found that because spectra are available at the times of the ground-based observations, it was preferable to convert the ground-based photometry to the space-based system. We computed S-corrections for the ground-based photometry using the spectra of Filippenko et al. (1995) and Clocchiatti et al. (1996a), and passbands as described above. We fit smooth polynomials to their time variation and used them to interpolate the S-correction and its uncertainty for the Richmond et al. (1996) observations. The results are included in Table 2.

3.2. Interstellar Extinction

All of the photometric and spectroscopic observations of SN 1994I indicate that it was heavily reddened by interstellar matter in M51. Different approaches to estimating A_V have included comparison with detailed theoretical models of light curves (Iwamoto et al. 1994) and spectra (Baron et al. 1996; 1999; Millard et al. 1999), and high-resolution spectroscopy of the narrow absorption lines produced by the gaseous phases of the interstellar matter (Ho & Filippenko 1995). Values of A_V from 0.9 up to 3.1 have been inferred, with a preferred range roughly between 1 and 2. We follow Richmond et al. (1996) in adopting A_V = 1.4 ± 0.5 mag.

This value of A_V must next be translated into extinction for the different broadband filters. Typically an interstellar reddening law is adopted and scaled by the value of A_V, and the monochromatic fluxes obtained from the photometry are then corrected, applying the extinction value that the adopted law prescribes for the effective wavelength of the passbands. This approach, however, may lead to systematic errors when applied to objects with strong and variable emission lines. The effect is typically small, but it is systematic with time, tending to change the overall shape and/or slope of a light curve. Since the extinction affecting SN 1994I is large, the issue merits consideration.

Again, the series of spectra of Filippenko et al. (1995) and Clocchiatti et al. (1996a) allow us to address this issue. Using these spectra, a typical interstellar extinction law (Cardelli et al. 1989), with R_V = 3.1, the transmittance of the passbands described earlier, and typical instrumental sensitivities, it is possible to compute the extinction affecting the spectra and its variation with time. The estimated extinction values and their uncertainties are included in Table 2.

The lack of a spectrum of SN 1994I some 280 days after maximum affects the estimates of interstellar extinction at the time, but the effect is small. Most of the variations that strongly affect the S-corrections at late times (changes of the emission lines in the near-IR) have a minor impact on the extinction.

We computed the extinction for the passbands of the HST WFPC2, fit smooth polynomials to the late-time trends, and extrapolated those from the latest observed spectrum out to the latest photometric point. To account for the uncertainty of the extrapolation, we increased the typical uncertainty of the fits by a factor of 3 for the last point. The extinction computed for the HST observation is included in Table 1.

The uncertainties due to the unknown details of the ground-based standard system realization and the lack of a very late-time spectrum are much smaller than the uncertainty of the overall extinction adopted (ΔA_V ≈ 0.5 mag according to Richmond et al. 1996). The changes in the light curves associated with the variation of the broadband extinction with time fall well within ΔA_V as well. However, computing the extinction from the evolving spectra and the expected passbands is the correct procedure for nonstellar spectra, and provides light curves that, for a given value of A_V, are free from the systematic effects related to the time variation of the extinction in each passband.

Two additional sources of uncertainty should be mentioned. First, the interstellar extinction law may not be the standard one. What can be said in this case is that, if the interstellar extinction toward SN 1994I follows the functional form of Cardelli et al. (1989) with a different value of R_V the difference will be smaller than the uncertainty in A_V, even for the more extreme values of R_V that they consider. Second, it is possible that at least a fraction of the extinction is produced in the local environment of the supernova. Since this local environment will be eventually affected by the evolution of the ejecta, a change of the foreground extinction with time would be conceivable. With the data available, it is not possible to set quantitative limits to this possibility.

5.1. Simple Comparisons

The light curves in Figure 5 illustrate the wide range that SNe Ic span in absolute brightness, and show that the diversity cuts across observational borders proposed to classify them, like the supernova or hypernova categories. Figure 5 also shows that SN 1994I was fairly luminous reaching more than 50% of the brightness of normal Type Ia SN. A foreground extinction a few tenths of a magnitude larger, which is consistent with model light curves, spectra, or high-resolution spectra of the narrow Na D absorption lines, would make it as bright as SN 1992A. The BV(RI)_C light curves also confirm that, similar to the behavior close to maximum light, the timescale for brightness variation of SN 1994I at late times was fast. The early exponential tail of SN 1994I is faster than that of the other Type Ic SNe, and comparable to that of the Type Ia SN.

Qualitative comparison with the late-time light curves of SNe 1990B, 1998bw, and 2002ap indicates a mismatch between the trend suggested by the last ground-based photometric points and the HST observation. SN 1994I shows a sharp downward change of slope in the exponential tail after ∼140 days, not seen in the other SNe Ic. The rate of decay measured from the data between ∼80 and ∼140 days after explosion is 43% slower than that measured using the points between ∼40 and ∼80 days, and the change happens very fast, between 80 and 115 days after explosion. By contrast, the decline slope of SN 1990B is essentially constant at these times, that of SN 1998bw changes by less than ∼28% between earlier than 80 days after explosion and later than 250 days after explosion, and that of SN 2002ap changes by ∼7% in this same time period.

If the trend suggested by the last two ground-based photometric points were correct, then something fairly dramatic happens to the BV(RI)_C light curve between ∼140 and ∼300 days after maximum light. One possibility is the so-called “infrared catastrophe” that shifts energy from the optical–near-IR passbands into the far-IR (Fransson & Chevalier 1989; Kozma & Fransson 1998), though it is not expected at such relatively early epochs in SNe Ic. Another possibility is that the brightness at 300 days after maximum is affected by a different amount of foreground local extinction, as would happen if the ejecta of SN 1994I formed dust. This is, however, not consistent with the small, or nill, change in broadband colors.

Another possibility is that the latest ground based photometric points are affected by contamination from background light from M51. Richmond et al. (1996) did image matching and subtraction before performing photometry, but the background of SN 1994I was particularly bright and had a steep gradient over the spatial scales sampled by the PSF of the ground-based images. Thus, residual contamination in the ground-based images could be present.

5.2. Comparison with a Simple Model

When γ-ray radiative transfer can be approximated by a pure absorption process (Sutherland & Wheeler 1984), the radioactive nickel assumed to be located at the center of the ejecta, accounting for a small fraction of the total mass (i.e., a point-source model for the γ-ray emission), and the SN ejecta modeled by a spherically symmetric distribution of matter in free expansion, the optical depth that extinguishes the γ-rays decreases with the inverse of time squared (e.g., Clocchiatti & Wheeler 1997). In this case, the energy deposited by γ-rays can be approximated by where τ_Ni is the e-folding time for the decay of ⁵⁶Co, and T₀ is the characteristic time for a given ejecta structure, which depends on its mass M, energy E, density profile, and the absorption cross section per unit mass (κ) that matter presents to γ-rays. For a given density profile, T₀ ∝ (κM²/E)^1/2. Soon after maximum light, when radiative equilibrium is established, the above assumptions are expected to hold. Then, in addition, between 100% and ∼96% of the radioactive energy is released in γ-rays, and most of them are trapped in the ejecta. In these conditions, E_γ gives a good representation of the energy available to heat the ejecta and equation 1 can be fitted to bolometric light curves to estimate the parameter T₀. At later times, as the fraction of γ-rays that escape from the ejecta increases, the ∼4% of energy released in the form of positrons needs to be considered. In what follows, we will consider the two extreme possibilities for thermalization of the positron annihilation energy: either they are completely trapped in the ejecta, or they escape.

Assuming that the BV(RI)_C light curve of SN 1994I, between ∼40 and ∼80 days after explosion, is a good representation of the bolometric light curve, we fit the light curve to determine T₀ ≈ 65 days, if the energy from the ⁵⁶Co positrons is completely deposited, or T₀ ≈ 67 days, if the positrons escape from the ejecta without interacting. The BV(RI)_C luminosity with these fitted simple models is shown in Figure 6.

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Fig. 6.— BV(RI)_C light curve of SN 1994I, the simple models given by eq. 1, and model CO21. See the electronic edition of the PASP for a color version of this figure.

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The simple models are very good at fitting the data between ∼40 and ∼80 days after explosion, suggesting some connection with reality, and they do a reasonably good job at ∼300 days. Since the (bolometric) model implies ∼50% more energy than the BV(RI)_C observation, the data taken at face value would be consistent either with full trapping of positrons from ⁵⁶Co and a small bolometric correction at ∼300 days, or with a smaller (even null) bolometric correction and a more complex process of positron transfer that traps ∼50% of their total energy. It should be noted, though, that the simple model without positron trapping falls below the observed light curve. The simplest interpretation of this observation is that SN 1994I did trap a significant fraction of the positrons.

The last two points from ground-based photometry (between ∼100 and ∼120 days after explosion) are missed by both fits. We note, in addition, that it is not possible to obtain a good fit for the simple models of equation 1 when all the ground-based observations of the exponential tail (between ∼40 and ∼120 days after explosion) are included, even if the HST point at 300 days is excluded. The problem is the fast change in the slope of the BV(RI)_C light curve between ∼80 and ∼115 days after explosion. Fitting this break requires including an additional free parameter, like two different values of T₀, which could represent a model of two different mass components each of them with different expansion properties. As a comparison, it is possible to successfully fit equation 1 to the exponential decay of the light curves of all the other SNe Ic shown in Figure 518.

5.3. Simple Comparison with Hindsight From the Simple Model

In § 5.1 we compared the derived light curves of several SNe. Although it is illustrative to see how different SN light curves behave in a plot like that of Figure 5, it is not completely correct to compare the evolution of different ejecta on a calendar-time basis. What we would like to do is to plot the light curves using units that are more physically meaningful.

From a few tens to a few hundreds of days after explosive nucleosynthesis, the evolution of the bolometric light curves is governed by two main time scales: τ_Co, the e-folding time for the radioactive decay of⁵⁶Co into ⁵⁶Fe, and T₀, the characteristic time for the decrease of the γ-ray optical depth (eq. 1). They are, respectively, the timescale for energy production and the timescale for energy deposition. This suggests that the natural energy scale to compare late-time light curves of different SNe is L_Bol(t_n)/ exp(-t_n/τ_Co), where t_n is measured since nucleosynthesis, and that the natural scale to compare the time evolution of the ejecta is t_e/T₀, where t_e should be measured from the start of free expansion. Because, at the times of interest, the difference between t_n and t_e is negligible, either of them may be used. We fitted the models of equation 1 to the light curves of all SNe Ic in Figure 5, and then normalized the energy and time axes according to the previous prescription. The result is shown in Figure 7.

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Fig. 7.— BV(RI)_C late-time light curves of SNe Ic 1994I, 1998bw, 2002ap, and 1990B, normalized according to the prescriptions of § 5.3. Points mark the observations of SN 1994I. Solid symbols indicate the region used to fit the timescale parameter T₀. A constant was added to the light curves of SNe 1994I, 1998bw, 2002ap, and 1990B to make them match the solid points of SN 1994I.

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It is reassuring to confirm that such a simple scaling brings some “order” to Figure 5. When the energy is normalized by the input ⁵⁶Co source, and time by the natural scale for γ-ray deposition (responsible for most of the energy available to the ejecta from soon after maximum until the supply of ⁵⁶Co is exhausted), all of the late-time light curves follow approximately the same track. It is possible to bring them into close agreement just by normalizing the total amount of ⁵⁶Ni ejected by the explosion (i.e., the additive constant in the logarithm of the energy used in Fig. 5).

This comparison shows that, in units of the natural timescale of evolution of the γ-ray optical depth, the late-time point of SN 1994I presented in this paper (∼280 days after maximum) is even more extreme than those observed for SN 1998bw and SN 2002ap, which had light curves extending beyond ∼500 days after maximum. Also, it shows that, in comparison with equation 1, the three very late light curves are consistent with a flattening after ∼3T₀, a time at which the γ-ray optical depth becomes negligible, and most of the γ-rays escape into space. For SN 1994I, the exact time at which the flattening occurs is unknown due to the lack of observations. However, both at 3T₀ ≈ 200 days, and at 300 days corresponding to the last observation, a modest fraction of ⁵⁶Co remains in the ejecta, and the ∼4% of the radiated energy that its decay releases in the positron channel becomes a significant source.

Regarding the late-time ground-based photometry, the scaling of Figure 7 confirms the arguments given before. The last two points depart from the common locus, and probably overestimate the luminosity of SN 1994I due to background contamination.

5.4. Insights From a Detailed Model

Iwamoto et al. (1994) presented a model of SN 1994I which, without fine tuning of the structure, provides a good fit to the early part of the UBV(RI)_C light curve, from ∼6 days before up to ∼40 days after maximum light. Their preferred configuration is a C + O core that would result from the evolution of a massive star in an interacting binary system that underwent two mass-transfer episodes. Depending on the initial parameters of the binary, at the moment of explosion the pair consists of the presupernova and a neutron star, white dwarf, or low-mass main-sequence companion (Nomoto et al. 1994).

Iwamoto et al. (1994) fitted the light curves of SN 1994I provided by Schmidt (1994, private communication) with the explosion of a star that entered the main sequence with ∼15 M_⊙, became a ∼4.0 M_⊙ He star after the first mass transfer, and a 2.1 M_⊙ C + O core after the second (model CO21). A simultaneous fit of the monochromatic and UBV(RI)_C curves required an interstellar reddening of E(B - V) = 0.45 mag, and a distance of 6.92 Mpc (m - M = 29.2 mag), in good agreement with the values used to determine the BV(RI)_C light curve presented here.

Would model CO21 fit the very late-time HST photometry? The question is relevant because previous attempts at fitting SN light curves over long time spans have uncovered a problem. When the models are good at fitting the maximum light they tend to be too dim on the tail; conversely, if they are good at fitting the late-time decay, they tend to give a maximum broader and/or brighter than observed. Solving this problem has typically meant introducing free parameters in the density structure of the model (Ensman & Woosley 1988; Tomita et al. 2006).

Based on the explosion model CO21 (Iwamoto et al. 1994), and using the same radiative transfer code, we calculated the monochromatic light curves, but here we extend the computations to 300 days after explosion. The assumptions and approximations used in the code are described in more detail in (Iwamoto et al. 1994) and (Höflich et al. 1993). A hypothesis relevant to our discussion below is that all positrons are thermalized locally and cannot escape. Because of the limitations of the code in the treatment of forbidden lines and their time dependence, we assumed the energy distribution to be unchanged after day 120. This assumption is consistent with the observations because the color indices scarcely change after about 100 days (Table 2 and Fig. 4). The assumption prevents us from testing overall migration of flux toward the IR, as may result from the formation of dust. The evolution of the colors, however, suggest that neither dust formation, nor other change in the foreground extinction by matter in the circumstellar region, do occur in this SN.

From the monochromatic fluxes, we calculated the magnitudes in each relevant HST passband. We checked that the synthetic F439W and F555W values agree with the B and V of Iwamoto et al. (1994) within a few hundredths of a magnitude. The small differences can be attributed to adjustments in the frequency grid and differences in the HST filter functions and the Bessell system, since S-corrections were not applied in this test.

From the monochromatic fluxes at the effective wavelength of the passbands, we built a BV(RI)_C light curve using the same procedure as in the observations (see § 4). We compare the theoretical light curves of model CO21 with the observations in Figure 6. Both the bolometric and BV(RI)_C luminosities are plotted. They are very similar, indicating that the model concentrates most of the energy within the passbands observed. Overall, the model luminosities agree well with the observations, especially in the post-maximum decay, although some details merit further comment.

At maximum, the model results are dimmer than the observations, while the same model better fits the UBV(RI)_C in Iwamoto et al. (1994). One reason for that is the distance. The observed BV(RI)_C light curve presented here is based on an observed distance of 8.39 Mpc (Feldmeier et al. 1997), while, for the same foreground extinction, CO21 is consistent with a distance 1.47 Mpc shorter in Iwamoto et al. (1994). The shorter distance will help our model BV(RI)_C light curve to fit the maximum better, but will give a brighter than required postmaximum decay. This highlights the importance of using extensive data sets when comparing models to observations.

The BV(RI)_C light curve of CO21 shown in Figure 6 does fit most of the observed BV(RI)_C light curve of SN 1994I. First, the distance independent width of the observed maximum, and then the distance dependent brightness of the postmaximum decline and exponential decay. Fitting the peak-to-tail contrast will require some additional fine tuning.

A possible hint is that the model UBV(RI)_C light curve does a better job at fitting the peak-to-tail contrast of the BV(RI)_C observations, indicating that the model spectrum overestimates the fraction of total flux emerging in the ultraviolet at early times. In any case, the problem of CO21 is opposite to the traditional peak-to-tail contrast problem: if the maximum of the BV(RI)_C model is arbitrarily shifted to match the observed one, then the model light curve results brighter than the observed, not dimmer. Studying the peak-to-tail contrast problem in SN light curves is beyond the scope of this paper, so we will let the issue rest there.

The late-time tail of the detailed model, which includes a realistic approximation to the γ-ray transfer and deposition, follows closely the simple deposition function given by equation 1, with full trapping of the positrons from the decay of ⁵⁶Co. Since CO21 also assumes full trapping of positrons, this is an indication that the γ-ray transfer is well represented by the simple approximations that went into equation 1. Hence, the conclusion stated above that about half of the positrons are able to deposit their energy within the ejecta is supported by the more elaborate model as well.