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MODELING THE LIGHTCURVES OF TYPE Ia SUPERNOVAE 1
W. D. VACCA
Institute for Astronomy
2680 Woodlawn Dr., Honolulu, HI 96822, USA
AND
B. LEIBUNDGUT
ESO
Karl­Schwarzschild­Strasse 2, D­85748 Garching, Germany
Abstract.
In order to investigate non­uniformity in the luminosity evolution of Type Ia super­
novae, we fit the lightcurves with a multi­parameter empirical model. The model provides
a quantitative method of analyzing the lightcurves of Type I supernovae and a convenient
and continuous representation of the photometric data. Using photometry of the well­
observed event SN 1994D we demonstrate the application of the model and construct a
bolometric lightcurve. The rise time of the bolometric light curve can be estimated and
compared to current theoretical models. By applying this model to a large number of SN
Ia lightcurves in various filters, we can explore the variations among individual events and
derive systematic correlations. We find that the initial decline rate after maximum light
correlates with the decline rate at late phases. Such correlations can be used to constrain
theoretical models.
1. Introduction
The lightcurves of Type Ia supernovae (SNe Ia) reveal the temporal evolution of the radia­
tive flux in photometric passbands. The apparent uniformity of the lightcurves of SNe Ia
was recognized in the very earliest studies (Minkowski 1961, Zwicky 1965). Barbon, Ciatti,
& Rosino (1973) noted the similarity of the B band lightcurves of 38 SN I and generated
a composite lightcurve. Doggett & Branch (1985) used these and other data to synthe­
size a mean Type I lightcurve; they also constructed average lightcurves for Type II SNe.
Leibundgut (1988) computed average magnitudes, as a function of time past maximum
light, from the photometric data for the best observed SN I events and constructed general
``template'' lightcurves for Type Ia and Ib supernovae in several filter bands. Covering the
phase from about 5 days before until 110 days after maximum light, these templates have
been widely used and provide a means of estimating the time and magnitudes of maximum
light for events with poorly sampled data. That such general descriptions of the luminosity
evolution of SNe Ia provide fairly good representations of the observed data (Leibundgut
et al. 1991a) indicates a substantial degree of homogeneity among the members of this SN
1 To appear in the Proceedings of the NATO Advanced Study Institute on Thermonuclear Supernovae,
eds. R. Canal, P. Ruiz­Lapuente, & J. Isern, (Dordrecht: Kluwer Academic Publishers)

2 W. D. VACCA AND B. LEIBUNDGUT
subclass. This realization, together with the observed similarity of the spectra, has led to
the adoption of SNe Ia as standard candles for cosmological distance measurements and
the derivation of the cosmological expansion parameter H 0
(e.g. Kowal 1968, Tammann
& Leibundgut 1990, Hamuy et al. 1995, Vaughan et al. 1995).
However, as more and better data on Type Ia SNe have become available with improved
detectors, dedicated searches, and extensive follow­up observations, it has become clear
that variations among SNe Ia do indeed exist (Phillips et al. 1987, Frogel et al. 1987,
Filippenko et al. 1992, Leibundgut et al. 1993, Phillips 1993, Maza et al. 1994, Hamuy et
al. 1995). The lightcurves of SN Ia are not identical and some supernovae have displayed
pronounced deviations from the means represented by the templates. The most prominent
examples are SN 1986G (Phillips et al. 1987, Leibundgut 1988), SN 1991T (Phillips et
al. 1992), and SN 1991bg (Filippenko et al. 1992, Leibundgut et al. 1993). Phillips (1993)
demonstrated that the difference (in magnitudes) between the brightness at the peak and
that at 15 days after the maximum varies considerably among SNe Ia events. In addition,
he found a correlation between this difference (\Deltam 15
) and the absolute magnitude reached
at maximum. This has far­reaching implications for the use of SNe Ia as standard candles.
Variations in luminosities and decline rates have also been found by Hamuy et al. (1993,
1995) using the extensive and homogeneous sample of SNe Ia lightcurves obtained as
part of the Calan/CTIO supernova search. A correlation between \Deltam 15
and M max is also
observed in this sample, although the slope of the relation is slightly different from that
determined by Phillips (1993). An independent analysis, which uses the SN Ia template as
a basis function to which time­dependent corrections are applied to account for variations
among the observed lightcurves, yields a similar result for the same data set (Riess et al.
1995; their Figure 1).
In order to investigate the differences in SNe Ia lightcurves, a quantitative description
of their shapes is needed. We have adopted a simple analytic expression, incorporating a
few free parameters, and used it to fit the photometric data of all known SNe Ia (Vacca &
Leibundgut 1996). While not based on any particular physical model, this purely empirical
form accounts for the various phases of SNe I lightcurves and provides a continuous repre­
sentation of the data. The fitting procedure allows us to obtain objective and quantitative
measurements of a set of parameters characterizing the lightcurves of individual super­
novae, including the maximum magnitude, the time of maximum, and the decline rates
at various phases. In addition, an estimate of the error in the observed photometry or the
errors in the fitted parameters can be obtained. While several recent analyses of SNe Ia
lightcurves are based on a set of basic shapes and deviations therefrom (Hamuy et al. 1995,
Riess et al. 1995), our method assumes only that we can approximate lightcurves with a
smooth function of the adopted form. Thus, we can independently test the assumption
of a continuous sequence of lightcurve shapes in the analysis of photometric data. Here
we briefly describe our model and outline some of its applications. A detailed publication
of the fits to the observed SN lightcurve data will be forthcoming (Vacca & Leibundgut
1996). We are currently improving the model and extending its application to additional
types of SNe.
2. The Empirical Model
Leibundgut et al. (1991a) presented an atlas of all SNe I lightcurves known as of 1984.
The shapes of both the observed lightcurves and the template curves suggest a simple
analytical form for Type I lightcurves. We model the evolution of the observed brightness
(in magnitudes in a given filter band) as a function of time with a Gaussian atop a linear
decay. The late­time decline is fitted by the line while the peak phase is represented by
the Gaussian. A second Gaussian is introduced to fit the secondary bump or maximum

SN Ia LIGHTCURVES 3
which is observed in the V , R, and I lightcurves (see below). To account for the rising
branch of the lightcurves we multiply this function by an expression which rises sharply
and approaches unity near the peak of the first Gaussian. This simple approximation,
with only a few free parameters, yields a surprisingly accurate representation of observed
lightcurves in several filter passbands. In order to test the model, we fitted the U , B, and
V template lightcurves tabulated by Leibundgut (1988, see also Cadonau et al. 1985) using
a least­squares technique. We found that the model can easily and accurately reproduce
the templates to within about 0.05 mag in the U and V filters and within about 0.01 mag
in the B filter over the entire temporal range.
One of the most useful aspects of a continuous model of the lightcurve shapes is that
additional interesting quantities and parameters can be objectively derived from it. In
particular, we derive the time of maximum brightness, T max , the peak apparent magnitude,
m max , the magnitude difference between the peak and 15 days after the peak, \Deltam 15 ,
the slope of the intial decline after maximum, s 1 , the slope of the late­time decline, s 2 ,
and the time and magnitude difference between the peak and the ``inflection point'' in
the lightcurve, \Deltat infl: and \Deltam infl: , respectively. This inflection point is defined to be
the time at which the contribution of the Gaussian to the total apparent magnitude is
equal to 0.10 mag. The values of these parameters derived from our fits to the templates
in each photometric band are given in Table 1. The overall systematic accuracy of the
fit procedure can be judged from the entries in Table 1; the templates were specifically
constructed to have T max;B = 0:0 and B max = 0:0, T max;V = 2:5 and V max = \Gamma0:02, and
T max;U = \Gamma2:8 and U max = \Gamma0:08. Clearly, this fitting technique can provide a fairly
accurate representation of the observed lightcurves.
TABLE 1. Derived Parameters for SN Ia Lightcurves
Band Tmax mmax s1 \Deltam 15 s2 \Deltat inf l: \Deltam inf l:
(day) (mag) (mag/day) (mag) (mag/day) (days) mag
Templates
U \Gamma2:76 \Gamma0:05 0.128 1.23 0.025 36.8 3.20
B 0.06 0.00 0.115 1.11 0.016 37.3 2.87
V 2.53 0.03 0.070 0.61 0.026 34.4 1.74
SN 1981B
U 4670.7 a 11.83 0.140 1.36 0.035 33.1 3.19
B 4672.9 12.05 0.113 1.14 0.010 41.6 3.08
V 4673.4 11.96 0.067 0.59 0.023 42.4 2.12
SN 1989B
U 7564.4 a 12.37 0.154 1.65 0.026 29.1 3.03
B 7565.2 12.32 0.127 1.28 0.015 34.9 2.93
V 7566.0 11.93 0.065 0.60 0.024 42.3 2.09
SN 1994D
U 9432.2 a 11.24 0.178 1.94 0.027 29.4 3.48
B 9432.9 11.85 0.139 1.47 0.017 33.8 3.12
V 9433.4 11.90 0.054 0.83 0.031 27.4 1.71
a JD \Gamma2440000
By performing a least­squares fit of a parametric model to the lightcurve data (as
opposed to simply connecting the scattered data points with a spline, for example), spu­
rious fluctuations due to random observational errors are minimized, errors on all derived

4 W. D. VACCA AND B. LEIBUNDGUT
quantities can be determined, and an estimate of the goodness­of­fit of the model to the
data points can be obtained. Furthermore, the fits yield a far more objective means of
quantifying the parameters describing the lightcurves than the templates can provide. As
an example we list the lightcurve parameters found for three well observed SNe Ia in Table
1. For these SNe, the intervals between the times of maximum light in the different filters
are smaller than those adopted in the templates. In particular, the fits yield times of max­
imum in V which are within a day of those in B for all three supernovae. Furthermore,
the values of the s 1 , and the decline parameters \Deltam 15 and \Deltam inf l: are found to be clearly
correlated with filter wavelength, decreasing as the wavelength increases.
The residuals from the fits also provide a means of identifying subtle, yet system­
atic deviations from the mean evolutionary behavior represented by the templates. For
example, when fitting the V lightcurves of some well­observed, recent SNe Ia with our
single­Gaussian model, we found a small but significant ``bump'' in the residuals, corre­
sponding to a second peak in the lightcurve data, at about 15 \Gamma 20 days after the primary
maximum (see also Ford et al. 1993, Suntzeff 1995). This peak is similar to the secondary
maxima observed in the R and I lightcurves of SNe Ia in general (Elias et al. 1981). To
reproduce this feature of the lightcurves, we incorporated a second Gaussian in the model;
the fits to the observed V , R and I lightcurves of SNe Ia improve dramatically when this
additional component is included.
3. Application to SN 1994D
3.1. MODEL FITS
In Figures 1 and 2, we show a representative example of the model fits to the U , B, V ,
R, and I lightcurves of SN 1994D. Three data sets, from Richmond et al. (1995), Patat
et al. (1995), and the RGO supernova data archive (e.g., Lewis et al. 1995), have been
combined in these figures. The residuals from the fits are shown below each panel. In the
U , B, and V filters, the models fit the observed lightcurve data much better than the
templates can over the entire range of observations. Several parameters derived from our
fits to the U , B, and V lightcurve data of SN 1994D are presented in Table 1. (Results
for SN 1981B and SN 1989B are also presented in Table 1 for comparison.) Note that
immediately after maximum light SN 1994D declined faster than the template in all three
filters. The secondary peaks in the V , R, and I lightcurves are well matched by our model
fitting function. This secondary peak is stronger and occurs progressively later at longer
wavelengths; the secondary maximum occurs at approximately 20 days (V ), 22 days (R)
and 24 days (I) after the B maximum. This trend is continued in the J (25 days) and H
(29 days) lightcurves of other type Ia supernovae (Elias et al. 1981, Leibundgut 1988). In
K the second peak occurs about 20 days after maximum.
3.2. A BOLOMETRIC LIGHTCURVE
As shown by Suntzeff (1995), the U through I photometric bands comprise nearly 80% of
the total radiation energy emitted by a SN Ia. However, few events have been observed
in all of these filter bands. Accurate and extensive observations in all filters do exist for
SN 1994D, however, and the construction of a ``nearly bolometric'' lightcurve for this
event is possible. We used our fitting procedure to model the observed lightcurves of SN
1994D in all five photometric bands, and generated a ``nearly bolometric'' lightcurve. The
construction of the bolometric lightcurve is facilitated by our continuous model of the
individual filter lightcurves. The resulting bolometric lightcurve is shown at the bottom
of Figure 2. The secondary peak observed in the V , R, and I data carries through to the
bolometric lightcurve. A similar feature has been found in the bolometric lightcurve of SN

SN Ia LIGHTCURVES 5
Figure 1. The U , B, and V lightcurves of SN 1994D. The circles are the observed data points. The solid
lines are our best fits; the dashed lines are the templates of Leibundgut (1988) normalized to the time and
apparent magnitude at maximum in B, as given by Richmond et al. (1995). The residuals of the fit are
shown below each panel.
1992A (Suntzeff 1995). This is an important characteristic which will have to be matched
by theory. We then fitted the bolometric lightcurve with our model and determined its
parameters; the model can reproduce the bolometric lightcurve to within 0:01 mag over
the entire range of observations. The rise time, defined as the difference (in days) between
the time of maximum brightness and when the supernova was 30 magnitudes fainter than
peak brightness, was found to be ¸18 days for the bolometric lightcurve. The average of
the rise time determinations for the individual filter curves is the same, 18.2 \Sigma 1.4 days.
This is very similar to the value determined for SN 1990N (¸ 19 days; Leibundgut et
al. 1991b). The comparison between this estimate of the rise time and the predictions
from various theoretical models (Khokhlov et al. 1994) yields a very interesting result.
All direct detonations and deflagrations yield rise times which are much shorter than that
found for SN 1994D. Delayed detonations may have rise times as slow as 15 days, but only
(unrealistic) detonations with a damping envelope have rise times as long as that observed
for SN 1994D. The relatively slow rise times of SN 1990N and SN 1994D provide strong
constraints on the theories of supernova explosions.
4. Statistics
The primary reason for adopting an analytic model to parametrize the lightcurves of SNe
Ia was to investigate the uniformity of the lightcurves. The results provided by our fits

6 W. D. VACCA AND B. LEIBUNDGUT
Figure 2. The R, I, and bolometric lightcurves of SN 1994D. The circles are the observed data points.
The solid lines are our best fits. The residuals of the fit are shown below each panel.
allow us to determine quantitative parameters for many supernovae (most of which have
sparsely sampled data), which can then be analysed with statistical techniques in order
to examine the variation among SNe Ia lightcurves and gain some insight into the general
nature of the events. Using the results of our fits, we can also search for correlations among
a large number of lightcurve shape parameters.
We have fitted the lightcurves of all SNe I and SNe Ia with adequate photometric data
(Vacca & Leibundgut 1996). A sample of our findings is shown in Figure 3, where we plot
the values of \Deltam 15
and the initial decline rate s 1
against those of the late­time decline rate
s 2
for the B and V filter bands. We confirm the range of \Deltam 15
found by Phillips (1993). We
also find a substantial range in the values of s 1
and s 2
. Since the late phase of the lightcurve
is powered by the thermalization of the fl--rays from the radioactive source, the time scale
of which depends only on the column density (Leibundgut & Pinto 1992), this variation
in s 2
indicates differences in the combination of ejecta mass and explosion energy among
the individual SNe. Although there is considerable scatter, trends appear to be present
in all four plots. In addition, we find that \Deltam 15
and s 1
are extremely tightly correlated.
Such correlations make it possible to determine \Deltam 15
and the luminosity correction even
for those SNe Ia discovered several days after maximum.
5. The Color­Color Plot
With our model fits to the U , B, and V lightcurves we can also follow the continuous
evolution of SNe Ia in the color­color diagram. The evolution in the U \Gamma B vs. B \Gamma V
plane for three well­observed SNe is shown in Figure 4. Note the characteristic shape of

SN Ia LIGHTCURVES 7
Figure 3. The correlation between \Deltam 15 and s2 , and between s1 and s2 , as derived from our model fits
to B and V band photometric data for SNe Ia. The open circles denote values derived from the templates
of Leibundgut (1988).
the curves. The loops occur approximately at the time of the inflection in the lightcurves.
Because of this characterstic shape, it is possible that the color­color diagram for SNe
Ia can provide rough estimates of the reddening toward various objects. For example, if
we assume the two events were nearly identical, then Figure 4 indicates that SN 1989B
experienced approximately one magnitude of reddening in V relative to SN 1981B, a
result which is in agreement with the analysis of Wells et al. (1994). It is also clear that
SN 1994D was unusually bright in U (see also Patat et al. 1995 and Richmond et al.
1995). Furthermore, it can be seen that between about 25 and 40 days past maximum
light, SNe Ia have colors similar to those of fairly cool blackbodies (T eff = 4000 \Gamma 5000 K).
Moreover, the spectra of SNe Ia at this time appear to be very similar (e.g., Leibundgut
et al. 1993). These results suggest that the inflection point may represent a phase in the
lightcurve more appropriate than the maximum for simple reddening determinations. After
40 days the spectra are dominated by emission lines, rather than continuum emission, and
a comparison with blackbodies is not physically meaningful.
6. Conclusions
The assumption that SNe Ia form a homogenous set of events is the basis for the use of these
objects as cosmological standard candles. However, much evidence has become available
over the last few years which invalidates this assumption. We have developed an objective
way to quantify variations in the lightcurves of SNe Ia by a simple fitting technique.
Through an accurate description of the irregularly sampled data an objective comparison
of individual events can be performed. By investigating the observed variations in the
lightcurves we will be able to constrain the physical models of SNe Ia and also establish
the degree of suitability of these events for cosmological applications. For supernovae with
adequate observations we can also simplify the construction of bolometric lightcurves, thus
forming a link between theory and observations.
We have presented some preliminary results of our fitting procedure. We detect a

8 W. D. VACCA AND B. LEIBUNDGUT
Figure 4. The evolution of three SNe Ia in the color­color diagram. The solid diagonal line denotes the
location of blackbodies of various effective temperatures. The reddening vector is also shown.
secondary bump in the bolometric lightcurve of SN 1994D, similar to that found in the
bolometric lightcurve of SN 1992A. The rise time of the bolometric curve excludes models
which exhibit sharp increases to maximum light. The color­color plot may become an
independent way to find reddening towards supernovae, if it can be established that SNe Ia
cool to a similar temperature after the peak phase. From a sample of available lightcurves
of SNe Ia we find a correlation between the early decline rate (or \Deltam 15
) and the late
decline rate. Such a correlation is an implicit assumption in the fitting techniques employed
elsewhere. The range of late­time decline rates is surprisingly large and directly reflects
differences in the ejecta mass, the explosion energies, or a combination of the two among
individual events. Clearly the lightcurves of SNe Ia are providing a wealth of information
that we are only beginning to extract.
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