The light curve in supernova modeled by a continuous radioactive decay of $^{56}$Ni

The UVOIR bolometric light curves are usually modeled by the radioactive decay. In order to model more precisely the absolute/apparent magnitude versus time relationship the continuous production of radioactive isotopes is introduced. A differential equation of the first order with separable variables is solved.


Introduction
The production of 56 Ni , see [1], in the last phase of the stellar evolution has been predicted by [2,3,4]. After this theoretical prediction the radioactive decay was used as an explanation for the observations of the light curve of supernova (SN), see among others [5,6,7,8,9,10]. At the same time the decay of 56 Ni produces a straight line in the absolute/apparent magnitude versus time relationship of the light curve which does not corresponds to the observations. We briefly recall that such a relationship presents a concavity. In order to explain this discrepancy between theory of decay and astronomical light curve we have developed a simple model for the continuous 56 Ni production. In this paper Section 2 derives and solves the differential equation which models the continuous production of 56 Ni and Section 3 shows the application of this new model to the light curve of two SNs.

The continuous production of radioactive isotope
The decay of a radioactive isotope is modeled by the following equation where τ is a constant and the negative sign indicates that dN is a reduction in the number of nuclei , see [11]. The integration of this differential equation of the first order in which the variables can be separated gives : where N 0 is the number of nuclei at t = 0. The half life is T 1/2 = ln(2) τ . The absolute magnitude version of the previous formula is where M is the absolute luminosity, C and k are two constants. This means that we are waiting for a straight line for the absolute magnitude versus time relationship. The continuous production of radioactive nuclei is modeled by the following equation where P , the production, and α, the exponent, are two adjustable parameters.
In this differential equation of the first order the variables can be separated and the solution is where the initial condition N(0) = N 0 has been used. The absolute magnitude version of the previous formula is (2) + k ln(5) ln(2) + ln (5) , where M is the absolute magnitude and C and α two constants.

Astrophysical applications
We plot the decay of the light curve of SN 2001el , which is of type Ia, adopting a distance modulus of 31.65 mag, see [12], the nuclear decay which according to equation (3) is a straight line, and the theoretical curve of the continuous production of radioactivity as represented by equation 6, see Figure 1. Another example is represented by SN 2001ay , the so called "the most slowly declining type Ia supernova", which has distance modulus of 35.55 mag and is of type Ia, see [8]. Figure 2 reports the light curve, the nuclear decay of the isotope 56 Ni and the continuous production of the isotope 56 Ni.

Conclusions
In conclusion the continuous production of 56 Ni during the evolution of a SN is here modeled introducing two parameters α and P , see eqn. (4). The solution of this differential equation of the first order with variables which can be separated has been derived, see eqn.