Supernova 1993J: One year later

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PHYSICS

ELSETIER

REPORTS

Physics Reports 256 (1995) 23-35

Supernova 1993J: one year later E. Baron a, PH. Hauschildt b, T.R. Young

a

a Dept. of Physics and Astronomy, University of Oklahoma, 440 U? Brooks, Rm. 131, Norman, OK 73019-0225, USA b Dept. of Physics and Astronomy, Arizona State University, Tempe, AZ 85287-1504, USA

Abstract Supernova 19935, a Type IIpec supernova, was discovered on March 28 in the nearby galaxy M8 1. The spectra displayed strong Balmer lines establishing its classification as a Type II supernova. About 26 days after explosion the Ha profile became anomalous, signaling the presence of significant amounts of helium. Additionally, the light curve displayed an anomalous double peak. We review the observations and interpretation of Supernova 1993J through the use of light curve fitting and synthetic spectra.

1. Introduction While in 1993 only 27 supernovae were discovered, small compared to the discovery of more than twice as many in each of the previous two years, the lack of quantity was more than made up for by the discovery of SN 19935 in the nearby galaxy M81. Not only was SN 19935 one of the brightest supernovae of this century, but it was also well-positioned in the Northern Hemisphere, which makes it one of the most well observed supernovae to date. The supernova has been detected and followed in the UV by the IUH [ l] and by HST [2,3], in the optical and infrared, in the radio [4,5], and in the X-ray band [ 6-81, Due to it being 70 times more distant than SN 1987A, the detection of neutrinos and gamma rays was not possible. SN 19935, like SN 1987A, was a Type II supernova showing strong Balmer lines in its spectra. Also like SN 1987A, the light curve of SN 19935 was anomalous (e.g. Ref. [ 9]), showing a rapid decline from maximum followed by a secondary rise, powered by the radioactive decay of 56Ni. In addition, about 26 days after explosion, the spectra of SN 19935 began to deviate significantly from those of typical Type II supernovae, in that the Ha emission profile showed a “double peaked” structure that was due to the effects of the He I A6678 line [ 10-121. SN 19935 well deserves the classification Type IIpec. The supernova was discovered on March 28.86 1993 [ 131, the first detection was on March 28.30 [ 141 and there are upper limits of V = 16.0 mag. [ 151 on March 27.91, and R = 17 mag. [ 161 on March 25.6. The time for the shock wave to traverse the star is - 0.1-0.2 days. Baron et al. [ 171, 0370-1573/95/$!9.50

@ 1995 Elsevier Science B.V. All rights reserved

SSDZ0370-1573(94)00099-9

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using the expanding photosphere method, estimate the explosion date as March 28Sf 1 days. Wheeler et al. [ 181 have estimated the explosion date to be March 28.0 f 0.1 days and de Vaucouleurs [ 191 has revised this to March 28.2 f 0.1 days. Taking all the evidence into account the explosion date is probably best taken as March 28.1 f 0.3 d. Efforts to identify the progenitor star are hindered by the fact that one arcsecond corresponds to 17 pc at the distance to M81 [ 201. The best estimates are that the progenitor was a G8-KO star located in an OB association [21,22]. Such a progenitor would imply a main sequence mass of N 17 M. [ 211, which corresponds to a helium core mass of about 4-5 M,. The total mass ejected should be in the range 2-3.5 M,, assuming that a typical mass neutron star was produced in the explosion. The only way to have such large amounts of mass loss is via a binary interaction [ 231, unless the progenitor identification is in error and the progenitor was significantly more massive such that it could lose most of its envelope in a Wolf-Rayet phase [ 241.

2. Observations Fig. 1 displays the UBVR light curve of SN 19935 along with the decay curve of 56Ni for comparison. The data displayed in the figure are the average values of all the data obtained and disseminated electronically by T. Kato [25] and M. Richmond. Bolometric light curves have been produced by Schmidt et al. [ 261, Lewis et al. [ 271, Ray et al. [ 281, and Richmond et al. [ 291. 8.0

I

I

I

* Ni-Co decay curve

16.0

I (

20.0

40.0 60.0 Days Since March 28.1 UT

Fig. 1. The UBVR light curve of SN 19935. We have taken the explosion 56Ni is shown for comparison.

80.0

1

date to be March 28.1 UT. The decay curve of

E. Baron et al./Physics Reports 256 (1995) 23-35

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In addition to optical and infrared observations, SN 19935 has been observed in the X-ray band by ASCA, ROSAT, and the OSSE instrument on CGRO. ROSAT detected X-ray emission from SN 19935 on April 3.4 with a luminosity of 2.9 x 1O39erg s-’ and 1.6 x 1O39erg s-l in mid-May in the 0.1-2.4 keV band [7]. ASCA detected SN 19935 with a luminosity of 5 x 1O39erg s-’ in the l-10 keV band [ 61. The OSSE instrument detected SN 19935 in the range 50-150 keV during observations on days 8-14 and days 22-35 [ 81. This corresponds to luminosity of 5.5 x 1040 erg SK’ on day 11, and 3 x 104’ erg s-i on day 27. SN 19935 has also been observed at radio wavelengths; in fact, it is probably the best observed radio supernova to date [4,5] and although the observations do not exactly fit a standard Chevalier [ 291 model of circumstellar interaction, they can be reconciled with a model where the wind density falls off less steeply than re2, i.e. in a model where iiZ/v, varies with time, where il? is the mass loss rate and v, is the wind velocity. The best fit is for PcsM c( T-’ where s - 1.5, and PcsM is the density of the unshocked circumstellar medium [ 301. Both the radio and X-ray emission are well understood as the result of the interaction of the supernova shock wave with the pre-existing circumstellar medium which is to be expected if the progenitor star has lost - 10 MD of hydrogen [ 30,311.

3. Light curve models The preponderance of efforts to model the light curve has focused on a binary progenitor model to account for the low mass of the ejecta that is required [ 22,28,32-351. A single star model in the form of a - 30 M, Wolf-Rayet star has been suggested by Hoflich et al. [ 241, but it requires a large reddening, E(B - V) N 0.8 which seems unlikely. Most estimates of the reddening favor values in the range E( B - V) M 0.1-0.2 [ 201. We have attempted to constrain the model parameters for the progenitor by fitting the observed bolometric light curve. We have used a distance to M81 of 3.6 Mpc [ 361 and assumed a reddening of E( B - V) = 0.2. The details of the calculation are described elsewhere [37,38]. Briefly the models were calculated using a spherically symmetric, LTE, flux limited diffusion hydro code, with Cox-Stewart opacities and a variable floor opacity to mock up the effects of lines and nonthermal ionization. Gamma-ray deposition is calculated using a simple, fast, and reasonably accurate approximation. We find that the light curve of SN 19935 can be fit with a total ejected mass Mej = 1.9-3.5 Ma (we have removed from the original helium core a 1.6 M, neutron star). In order to fit the light curve we find we can compensate for the variation of total ejected mass by varying the mass of the hydrogen envelope, M “, the explosion energy E, the nickel mass MNi, and the amount of nickel mixing. Certain changes were found to be correlated: as the ejected mass is decreased the hydrogen envelope mass must be increased; additionally, the explosion energy must be decreased and the Ni mass must be increased. The range of acceptable parameters is displayed in Table 1. The low mass model (Mea = 1.9 M,) displayed in Fig. 2 has an envelope mass MH = 0.35. This value of MH, large compared to our other models, compensates for the low total ejected mass since hydrogen, with more electrons per unit mass, is more efficient at trapping y-rays than other elements. The high nickel mass is also required to make up for the inefficient y-ray trapping of the small ejected total mass. In this model there is no mixing of the Ni into the ejecta which brightens the tail

E. Baron et al./Physics Reports 256 (1995) 23-35

26 Table 1 The ranges independent

for the model (see text).

parameters

of SN 19935 from our light curve fitting. Variations

in the parameters

are not

3.15-5.0 Ma 1.9-3.5 M@ 0.10-0.35 M@ 0.10-0.14 Ma 0.75-1.35 x 105’ erg 2.0-4.0 x IOr cm

MHL? Mcj

MH MNi

E R

-19.0 A Schmidt et al. Richmond et al. 0 Lewis et al. l

-18.0

z Iz

-16.0

-15.0

Days

Fig. 2. A model light [9], and Lewis et al. mass is 0.14 MO, the a reddening of E( B -

curve is compared to bolometric light curves constructed by Schmidt et al. [27], Richmond et al. [27]. The total ejected mass is 1.9 MD, the mass of the hydrogen envelope is 0.35 MO, the 56Ni explosion energy is E = 1.0 x 10” ergs, and the initial radius is R = 3 x 1Or3 cm. We have assumed V) = 0.2, and a distance to M81 of 3.6 Mpc [ 361.

but decreases the peak brightness and delays the peak. The initial observations constrain the initial progenitor radius to be 3 f 1 x lOi cm [37,38] as displayed in Fig. 3. The initial radius determines the width of the initial peak, with larger radii leading to brighter and broader light curves. The width of the light curve can be constrained to be narrower than would be inferred from the the time of the first upper limit point, but wider than would be inferred from the time of the first detections. This in turn constrains the initial radius, if we also demand that the models fit the first few bolometric points. Further confirmation for these radii comes from the possible observed progenitors [ 211. In Fig. 4 we plot an HR diagram for these two limiting radii assuming a distance to M81 of 3.6 Mpc. The box shows the position of the observed progenitor [ 211, which agrees well with our radius constraint. Others workers derive radii in the range l-4 x 1013 cm [ 33,351.

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-18.0

n

-16.0

-14.0

0

v

-12.0

V 1

I 0.0

I I I I I ! 1 1 i I I f I I I

0 Lewis et al. (1994) V F. Garcia IAUC 5731 n F. Garcia IAUC 5731 V J. Merlin IAU 5737 0 A. Neely IAU 5737

I 2.0

4.0

Days Since March 28.1 UT

Fig. 3. The earliest observations taken together with the bolometric light curve provide constraints on the initial radius of the progenitor star. The model calculations have initial progenitor radii, R = 2.0 x 1013 cm (dashed line) and R = 4.0 x 1013 cm (solid line). The triangles are upper limits and no bolometric corrections have been applied to the early observations of Garcia, Neely, and Merlin.

Fig. 4. An HR diagram showing the position of the possible observed progenitor radii derived from the light curve [37,38].

(box)

[21], and the limits of the initial

E. Baron et aLlPhysics Reports 256 (1995) 23-35

28

-18

M BOl

-16

-12

O%" 0

200

100

300

Days Fig. 5. The instantaneous luminosity from the decay of 0.07, 0.10, and 0.14 Ma of 56Ni is compared with bolometric light curve of SN 19935 [9,27]. The slope of the observed light curve is steeper than the 56Ni decay at late times due to escape of gamma rays from the supernova.

The light curve may also be used to derive a lower limit to the ejected nickel mass. The second peak of the light curve is produced by the diffusion of heat deposited by the decay of radioactive 56Ni and 56Co. After this secondary peak the light curve is powered almost solely by the instantaneous release of radioactive energy and its slope depends on how much of the gamma ray energy is deposited in the material. To first order, we may equate the total energy emitted by a given mass of nickel with the observed bolometric luminosity of the supernova just when the slope of the light curve changes after the second peak, at around 40 days. In Fig. 5 we plot the bolometric light curve of SN 19935 and the bolometric luminosity of 0.07, 0.10, and 0.14 M, of 56Ni. With our assumed distance and reddening, the lower limit is MNi = 0.10 M,, in agreement with other estimates [ 22,391.

4. Spectra The spectral coverage of SN 19935 has been superb. The La Palma Group [ 271 provided an invaluable service to the community by making their spectra freely available in electronic form. Nearly nightly spectra of SN 19935 were also obtained by the Beijing Astronomical Observatory 2.16-m telescope [ 401. UV spectra of SN 19935 have been published in Refs. [ 1,2], optical spectra have been published in Refs. [ 17,18,26,27,40-451 and synthetic spectra have been calculated in Refs. [ 2,3,17,41,42]. Fig. 6 displays the time evolution of the spectra. High resolution spectra provide additional evidence for a circumstellar medium. IUE observed strong N V AA1238-1242A lines [ 1 ] with a width of - 1000 km s-i. Due to the high degree of ionization it is likely that these lines have formed close to the supernova and been accelerated by

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E. Baron et al./Physics Reports 256 (1995) 23-35

2000.0

4000.0

6000.0 h (Angstroms)

8000.0

I 10000.0

Fig. 6. The time evolution

of the optical spectra of SN 19935. The March 30 and optical portion of the April 15 spectra were obtained at Lick Observatory [ 12,171, the UV was obtained by HST [ 2,3,17], the April 7 and June 13 spectra were obtained at Lowell Observatory [ 17,411 and the April 29, May 9, and August 15 spectra were obtained at McDonald Observatory [ 421.

the radiative precursor to the shock wave [30]. In the optical, narrow lines of Ha, He II A4686A, [Fe X] A6374, [Fe XI] A7892, and [Fe IV] A5301, were observed [43,45-471. While the width of Ha was comparable to that of the N V lines, the iron lines were considerably slower, with widths of - 50 kms-‘. The earliest spectrum displayed in Fig. 6, is nearly featureless with only very weak Balmer features. Baron et al. [ 171, who calculated NLTE synthetic spectra, were able to fit the continuum quite well, but in order to wash out the Balmer features required a very steep density profile p c( Y-“, with it = 50. Even with such a steep atmosphere the Balmer features were still too strong and the earliest spectra are still not well understood. Additional evidence for a steep density profile comes from models of the radio and X-ray observations. The epochs of the radio turn-ons at different wavelengths are sensitive to the value of tz. Fransson, Lundqvist, and Chevalier [30] find that n M 30 provides the least spread in the turn-on times and that 12 5 15 is ruled out by the observations. In order to fit the X-ray observations, Fransson et al. [30] find that the reverse shock is likely to be affected by radiative losses, rather than being adiabatic, which also requires y1 2 20. ROSAT observed SN 19935 on November l-2, 1993 (about 220 days after explosion) and found an X-ray temperature of N 0.5 keV. If the emission of this soft X-ray flux originates from the reverse shock, this implies y1w 25-33 [ 30,481. By April 7, there are strong well-developed Balmer features as well as Fe II, Ca II, and the He I A5876A line. Baron et al. [ 17,411 find that the He/H ratio is l-10 by number and that mixing of M 0.1 M, of 56Ni provides a good fit to the observed spectrum. Fig. 7 shows our best fit to the observed spectrum [ 17,411. The model parameters (described in detail in Ref. [ 171) are: the helium

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E. Baron et al./Physics

Reports 256 (1995) 23-35

’

I

I

8

2000.0

4000.0

6000.0

8000.0

0.0

101O(10.0

Wavelength (angstroms) Fig. 7. April 7, synthetic spectrum is compared to the observed spectrum taken at Lowell Observatory by Mark Wagner and Scott Austin [41]. The model parameters are (see text for a description): He/H = 1 (Y = 0.8), Ter = 6500 K, RO = 10” cm, uo = 12000 km s-‘, and ve = 1000 kms-‘.

to hydrogen number ratio, He/H = 1 (Y = O.S), the effective temperature which parameterizes the total luminosity in the co-moving frame, T,E = 6500 K, the reference radius, Ro = 1015 cm, the velocity of the material at the reference radius, u. = 12000 km s-l, and the e-folding velocity which parameterizes the density profile p cc exp (-U/U,), u, = 1000 km s-l. This is the first optical spectrum we have computed that shows clear evidence of helium with a strong He I A5876 line and a clear notch in the Ha emission peak which is due to He I A6678. The density profile is less steep than required by the earlier spectra, the equivalent power-law index at the photosphere is N = 12, although good fits are obtained by models having a very steep density profile with an effective power-law index at the reference radius N = 23 [ 411. Even though the hydrogen to helium ratio is equal by number (Y = 0.8), the emission and absorption of the He I A5876 line is too weak. In addition, the observed notch in the top of the Ha, profile, if real, can only be produced by the He I A6678 line. We have also calculated models where helium is enhanced by a factor of 10 over hydrogen by number. These models give reasonable fits to the He I A5876 line and even show some evidence of an asymmetry in the Hcu emission. Further enhancing helium, to 100 times hydrogen, does a much better job of fitting the He I A5876 line, particularly in emission, but the Ha emission is now far too weak; however, the “notch” due to He I A6678 is in the right place, with roughly the right strength. It is likely that the observed spectrum is due to a non-uniform composition, with the photosphere having receded into the helium layer of the star and the Hcu emission being formed by a shell containing hydrogen further out. In the model calculation displayed in Fig. 7 the Ha absorption is too strong, even though the

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Reports 256 (1995) 23-35

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emission profile is well fit. The synthetic spectrum does reproduce the features just blueward of Ha which are likely due to blends of Fe II lines. The first evidence of a major anomaly in the spectra of SN 19935 occurred on about April 22, 1993 [ 10,121. The April 29 spectrum shows that the emission peak of Ha is being eaten away by a strong He I A6678W line. Swartz et al. [42] obtain good fits to He I lines on April 29 and May 9, including the He I Al .08~ and A2.06~ lines, with a He/H number ratio of about 2, and about 0.1 M. of 56Ni. In addition, they identify 0 I lines h7774, [ 0 I] A6300, and [ 0 I] A5577W. The forbidden oxygen lines have also been discussed by Wang and Hu [40] who have invoked clumping in the ejecta - 4 months after the explosion to explain the observed blue shifts. On June 13, the spectrum has become more complicated. The supernova is making the transition to a nebular spectrum although there is still a well developed continuum. All of the Balmer lines are still strong and the “double-peaked’ structure of Her persists. The He I A5876, A6678, and A70658, lines are prominent, but the He I A587613 1’ me is likely blended with the NaD doublet. The photosphere has receded into the helium shell and there is clear evidence for non-uniform compositions, i.e. a hydrogen rich shell, at this time [ 171. The spectrum on August 15 is interesting, there is still evidence for a continuum 139 days after explosion and while Ha is now lost in the [0 I] A6300W emission, HP and Hy are still evident. By September 17, 1993, the spectrum is now dominated by nebular emission lines [ Foltz 1994, private communication], but there is a broad shoulder redward of the [0 I] A6300A emission which is likely to be evidence of hydrogen. On November 14, 1993 there is clear evidence of broad Ha [ Filippenko and Matheson 1994, private communication] . By May 4, 1994 the flux in Hcu was greater than in the [ 0 I] or Ca II] doublets. On June 4, 1994 the total flux in Ha has increased since May 4, 1994 and the peak intensity in Ha, was larger than for Ca II] at 7291 and 7324A [49]. These observations suggest [49,50] the hydrogen shell is being excited by soft X-rays coming from the reverse shock. Fig. 8 compares the photospheric velocity of our light curve model, with that from our synthetic spectra. The light curve calculation is LTE, flux limited diffusion, so the poor agreement at late times is expected. The spectral models for the earliest spectra may be too slow, which is again not surprising since the velocities are not well determined in the early spectra. It is heartening that the velocities are in agreement in April where our synthetic spectra and distance estimates are also quite good. The 56Ni required in both the light curve and spectral modeling is consistent. Our synthetic spectra predict the absolute flux. Combining the absolute flux with broad band photometry we use a version of the expanding photosphere method [ 3,17,41,51-531 to derive a distance to the supernova. We use only the results of our good quality fits for the month of April, in which case our distance estimate is ,u = 28.2f0.4 mag. (D = 4.4f0.8 Mpc), within the uncertainties of the Cepheid measurement, ,X = 27.8 f 0.2 mag. (D = 3.6 f 0.4 Mpc). It is reasonable to restrict ourselves to these high quality fits, since the April models are the only ones where we can truly claim to have fit the entire spectrum and not just a portion; the far UV is not well fit on April 15, however. In the application of the expanding photosphere method one wishes to give lower weight to early spectra, since a longer time baseline minimizes the error due to uncertainties in the explosion date. The best fits to the observed spectra, with the longest time baseline, are for the epoch April 13-15, and that is where the distance is in closest agreement with the Cepheid value. In fact, confining ourselves to that epoch alone, our result is p = 28.0 f 0.1 mag. (D = 4.0 f 0.2 Mpc), in excellent agreement with the Cepheid value. The distance to SN 19935 has also been estimated using the

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Reports 256 (1995) 23-35

j 40

60

80

100

Days Since March 28.1 UT

Fig. 8. The photospheric velocity from our hydrodynamical calculation of the light curve (solid line) is compared to velocities obtained from synthetic spectra (open circles).

angular expansion rate determined from VLBI observations, expansion speed to give D = 4.0 f 0.6 Mpc [ 541.

combined

with the optically

derived

5. Binary progenitor All workers modeling the light curve conclude that the progenitor star had a low mass, extended hydrogen envelope. Envelope masses range from 0.20 Mo [22,35,37,38] to 0.75-0.90 [ 33,34,39]. Hoflich et al. [24] find an envelope mass of - 3.25 M, while Woosley et al. [35] argue that envelopes as massive as 0.9 M, will produce light curves more similar to SNe II-L, rather than SN 19935. Broad band colors are calculated by Hoflich et al. [ 241 for the first 40 days and Woosley et al. [35] for the first - 90 days of their models. Both do reasonably well in comparison to the observations for the first 40 days, but after that the models of Woosley et al. cannot reproduce the near constant value of B - V, which they attribute to NLTE effects. Almost all modelers have invoked a binary companion to produce the necessary mass loss, since the progenitor would not be expected to lose nearly enough mass by radiatively driven stellar winds. Additional evidence for a binary companion comes from the likely identification of the progenitor with an OB association. Polarization studies of SN 19935 show an average continuum polarization of the supernova of 1.6 It 0.1% at a position angle of 49 f 3 deg, which corresponds to an axis ratio of 3 : 2 in the core if the polarization occurs in the core and not the envelope [ 551. It is difficult to obtain such strong core deformations either by differential rotation [56] or by aspherical explosions or aspherical winds [ 20,571. It is also possible that the polarization arises from an aspherical envelope with an axis ratio of 1 : 1.1-3, if it is oblate and viewed nearly edge on. Higher axis ratios are required

E. Baron et al./Physics Reports 256 (1995) 23-35

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if it is prolate [55]. Woosley et al. [35] note that the star may very well have been exceeding its Roche lobe at the time of explosion and they estimate axis ratios of 3:2, but Trammel et al. [ 553 state that the aspherical envelope is more difficult to reconcile with the data than an aspherical core. The VLBI data show the ejecta to be expanding without deceleration and to be quite spherical [ 54,581, although there may be some evidence for asphericity at later times [ 591. Using the optical spectra the expansion velocity is estimated to be - 19 000 km s-’ [ 541, consistent with the maximum velocities inferred at early times from both our light curve and synthetic spectra modeling.

6. Discussion Several authors [ 12,32,35] have designated SN 19935 as a Type IIb supernova [60] in analogy to Type Ib supernovae, which like Type Ia supernovae lack Balmer lines in their spectra, but also lack prominent Si II lines. Type Ib supernovae are further distinguished from Type Ic supernovae by displaying strong He I lines near maximum light (which SNe Ic lack), however the Ib/c spectra are indistinguishable at late times (- 6 months after maximum). The prototypical SN IIb is SN 1987K [ 611, which owing to its position near the sun at maximum light has poor temporal coverage of both spectra and photometry. The maximum light spectra of SN 1987K exhibit a feature which may be due to either He I A5876 or to NaD (or a combination), but the late time spectra are clearly very similar to the late time spectra of SNe Ib/c, dominated by calcium and oxygen lines. In all the SN 19935 spectra we have seen, Balmer lines are never absent. While the HLYis lost in [ 0 I] emission in August (see Figure 6), H/3 and Hy are clearly present. The N 6 month spectra of SN 1987K do not extend blue enough to determine whether HP or Hy are present, however, the N 8 month spectra of SN 1987K still exhibit no evidence of Hcu, whereas it is clearly present at roughly the same epoch in SN 19935. While SN 19935 may well be quite similar to the original theoretical model proposed by Woosley et al. [60] for SNe IIb, the supernova classification system is still an observational one and for that reason we believe that SN IIpec is the best designation for SN 19935.

Acknowledgments E.B. thanks Sterling Colgate for encouragement and many enjoyable and enlightening discussions over the years. We thank David Branch, Robert Cumming, Lisa Ensman, Alex Filippenko, David Jeffery, Brian Schmidt, Giora Shaviv, Sumner Starrtield, and Rainer Wehrse for helpful discussions, and Hong Bae Ann, Scott Austin, Alejandro Clocchiatti, Alexei Filippenko, Peter Garnavich, David Jeffery, Jim Liebert, Tom Matheson, and Mark Wagner for providing spectra electronically in advance of publication. This work was supported in part by NASA grant NAGW-2999, a NASA LTSA grant to ASU, and by NSF grant AST-9115061. Some of the calculations in this paper were performed at the NERSC, supported by the U.S. DOE, and the San Diego Supercomputer Center supported by the NSF; we thank them for a generous allocation of computer time.

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