The light curves are analyzed using a descriptive model (Vacca & Leibundgut 1996). For each supernova the observed light curve in each filter is fit with an empirical model consisting of a Gaussian (for the peak phase) atop a linear decay (late-time decline), a second Gaussian (to model the secondary maximum in the V , R , and I band light curves), and an exponentially rising function (for the pre-maximum segment). The functional form of the fit is:
The first Gaussian and the decline are normalized to the phase , while the second Gaussian occurs at a later phase . The exponential cutoff function for the rise has a characteristic time and a separate phase zero-point . The amplitudes of the two Gaussians, and , as well as the intercept of the line, , the slope, , the phases and , and the widths, and , the characteristic rise time and its phase are free parameters in the fit. Each filter light curve is fitted individually and independently using a -minimization procedure to determine the best-fit values of the parameters. Other parameters which characterize the shape of the light curve, such as the time of maximum brightness or , can then be derived from the best-fit model. Although the model is a completely empirical description of the general shape of SNe Ia light curves, we note that theoretical models (Pinto & Eastman 2000a) predict a Gaussian shape for the peak in models with constant opacity and buried well within the ejecta.
Fig. 1 shows the example of the fit to the R light curve of SN 1992bc. The various components of the fit are displayed. The exponential function (solid line) rises steeply to unity, modeling the rapid brightness increase of the supernova. The decline rate (dotted line) is set by the long tail beginning about 50 days past maximum. The first Gaussian (dashed line) fully describes the maximum phase, as further demonstrated in the inset. The second Gaussian (dashed-dotted line) reproduces the "bump" in the light curve. The bottom panel of Fig. 1 displays the residuals of the fit with the observational error bars. It is clear that the function is an accurate, continuous description of the data. The small systematic undulations of the residuals indicate that the fit is not perfect and can be used to make detailed comparisons between individual supernovae. We have fitted this model to the filter light curves of more than 50 supernovae. These fits will be presented in forthcoming papers.
The fits produce objective measures of the magnitude and date of maximum, the extent of the peak phase, the amplitude and extent of the secondary peak, the late decline rate, and an estimate of the rise time. The accuracy of these parameters depends strongly on the number and quality of the observations in each phase. While in most cases the late decline can be derived fairly easily, the rise time is undetermined when no pre-maximum observations are available. It is possible to derive other light curve shape descriptions, such as , for each filter objectively and independently without comparison to templates.
Other functional forms could be imagined for the fit. However, the late decline (linear in magnitudes) and the steep rise can be assumed to have simple functional forms. Fitting polynomials or spline functions, for example, could not produce the transition to the linear decline as observed, but would simply transfer the undulations of the early phases to the late decline and fail to match the observed linear decline. Furthermore, such functions would not provide a small set of adjustable parameters which can be compared between different objects.
Nevertheless, the adopted functional form is not a perfect representation of all optical filter light curves. When fitting the I light curves of some SNe Ia we found that the model had difficulties to match the observations with the same accuracy as in the other filters. This can be seen in Fig. 2. The slope of the I band light curve decline (dotted line) is so steep that the model cannot reproduce the observed light curve. This steep slope causes the function to overshoot the observed values near the second maximum and the fitting program inverts the second Gaussian to produce the observed minimum in the light curve. If the fitting routine is constrained to fit only positive Gaussians, the exponential rising function (solid line) is shifted to late times and depresses the linear decay (dotted line) before the second maximum; in addition the first Gaussian (dashed line) is enhanced and shifted to earlier times. This implies that, for the I band, the fitted parameters describing the first Gaussian (i.e. its magnitude, its center and its standard deviation) cannot be used to characterize the first maximum, even if the light curve appears fitted well. In spite of this problem, the I band light curves of the SNe in our sample can all be reasonably well fit by the model and the data can be represented by the continuous fits. This also means that the derived light curve parameters, e.g. decline rates, are meaningful.
This steep decline in the I band is a rather unexpected result. The physical explanation behind this behavior is that the I filter light curve is dominated by a rapidly decreasing flux component and significant flux redistribution takes place in the evolution. Similar results have been found by Suntzeff (1996) in the observations of SN 1992A and Pinto & Eastman (2000b, see also Eastman 1997) in theoretical models.
The fitting procedure provides an estimate of the goodness-of-fit, as well as the associated uncertainties on the fit parameters. Uncertainties on the derived quantities depend strongly on the quality and number of data points. If there are no data before 5 days prior to maximum, e.g., the rise time cannot be determined reliably. We have estimated the uncertainties using a Monte Carlo method. We constructed synthetic data sets with the same temporal sampling as the observed light curve. The magnitude at each point was computed from the best fit model, with the assumption of a Gaussian probability distribution whose width was given by the observational uncertainty. The standard deviations of the observed data points range from 0.03 to 0.14 magnitudes. In this manner, 2000 synthetic data sets were simulated. Each synthetic data set was fit and the frequency distribution for each model parameter was constructed. Two examples of the resulting distributions (the time of B maximum and the B magnitude at maximum for SN 1992bc) are shown as histograms in Fig. 3. The differences between the mean and the best fit is less than 0.03 days for the time of maximum and less than 0.01 magnitudes for the peak brightness. In this example the skewness and kurtosis are negligible. Non-zero values of skewness and kurtosis provide an indication of the unreliability of the derived fit parameters.
Occasionally the standard deviations obtained for a fit parameter from the fitting procedure differed significantly from the values obtained from the Monte Carlo simulations. This arises because the standard deviations derived from the fit are calculated from the diagonal elements of the curvature matrix, while the standard deviations derived from the Monte Carlo simulations result from actual frequency distributions. (Both standard deviations should be the same for a perfect light curve, i.e. well sampled data, where the brightness differs from the analytical shape of the light curve by only a small error. We confirmed our code with this test.) In all cases we adopted the uncertainties given by the Monte Carlo simulations.
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000