## 3. Fitting filter light curves## 3.1. MethodThe 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 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
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
This steep decline in the ## 3.2. Uncertainties in the model parametersThe 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
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 |