The radioactive contribution to the light curve follows the analytic estimation of Dado & Dar (2015) exactly, taking as inputs from our models the Ni mass (Table 1), the ejecta asymptotic velocity, Vej, and the ejecta mass, $\mathcal {M}_{\rm ej}$. All these quantities are approximate and not the result of a sufficiently long and detailed calculation. We estimate an upper bound to the ejecta mass as the difference between the stellar mass at the brink of collapse, M0, and the PNS mass at tpb = tf (i.e. $\mathcal {M}_{\rm ej}:=M_0-M_{{\small PNS}}(t_{\rm f})$). We note that $\mathcal {M}_{\rm ej}$ is significantly greater than the mass unbound at tpb = tf (Mej; see Fig. 3) or the mass enclosed by the shock at that time (Msh, e; Table 1). At tpb = tf the explosion is ongoing and both Mej and Msh, e are still increasing until they reach their final values. The asymptotic ejecta velocity is estimated as $V_{\rm ej}=\sqrt{2E_{\rm k}/\mathcal {M}_{\rm ej}} = \sqrt{2 (\mathcal {T}_{{\small PNS}}f_{\rm er} + E_{\rm exp})/ \mathcal {M}_{\rm ej}}\approx \sqrt{2 \mathcal {T}_{{\small PNS}}f_{\rm er} / \mathcal {M}_{\rm ej}}$, where we allow for the possibility that only a fraction fer of the PNS rotational energy can be converted into kinetic energy of the ejecta, Ek. The last approximation results from the fact that the $\mathcal {T}_{{\small PNS}}$ ($\gt 10^{52}\,$erg) is one order of magnitude (or more) greater than Eexp ($\lt 2\times 10^{51}\,$ erg; see Table 1). Typical ejecta velocities in all our models are $V_{\rm ej} \approx (0.4-1.6)\times 10^{9}\, \rm{cm\,s}^{-1}$. The corresponding luminosity reads (Dado & Dar 2015)
(8)$$\begin{eqnarray}
L_{\mathrm{nuc}}(t)=\frac{{\rm e}^{-t^{2} / 2 t_{\mathrm{r}}^{2}}}{t_{\mathrm{r}}^{2}} \int _{0}^{t} x {\rm e}^{x^{2} / 2 t_{\mathrm{r}}^{2}} \skew{4}\dot{E} \, {\rm d} x ,
\end{eqnarray}$$where
(9)$$\begin{eqnarray}
t_{\rm r}=\sqrt{ \frac{3\mathcal {M}_{\rm ej}f_{\rm e} \sigma _{\rm T}}{8\pi m_{\rm p} c V_{\rm ej}} },
\end{eqnarray}$$σT is the Thompson cross-section, mp is the proton mass, c is the speed of light, fe ≈ 0.275 is the fraction of free electrons, and $\skew{4}\dot{E}=\skew{4}\dot{E}_\gamma + \skew{4}\dot{E}_{e^+}$, where
(10)$$\begin{eqnarray}
\skew{4}\dot{E}_{\gamma } &amp;=&amp; \frac{M_\mathrm{Ni}}{{\rm M}_{\odot }}\left[7.78 A_{\gamma }^\mathrm{Ni} {\rm e}^{-t / 8.76 \mathrm{~d}}\right. \\
&amp;&amp;\left.+\,1.50 A_{\gamma }^\mathrm{Co}\left[{\rm e}^{-t / 111.27 \mathrm{~d}}-{\rm e}^{-t / 8.76 \mathrm{~d}}\right]\right] 10^{43} \mathrm{erg}\, \mathrm{s}^{-1}
\end{eqnarray}$$is the power supplied by gamma rays, and
(11)$$\begin{eqnarray}
\skew{4}\dot{E}_{\mathrm{e}^{+}} = \frac{M_\mathrm{Ni}}{{\rm M}_{\odot }} A_{\mathrm{e}} \left[{\rm e}^{-t / 111.27 \mathrm{~d}}-{\rm e}^{-t / 8.76 \mathrm{~d}}\right] 10^{43} \mathrm{erg}\, \mathrm{s}^{-1}
\end{eqnarray}$$is the power supply in the form of the kinetic energy of positrons. Ae ≈ 0.05 is the ratio of the energy released as positron kinetic energy and as gamma-ray energy in the decay of 56Co. $A_\gamma ^\mathrm{Ni}$ and $A_\gamma ^\mathrm{Co}$ are the absorbed fractions of the gamma-ray energy in the SN ejecta from the decay of 56Ni (decay time of 8.76 days) and 56Co (111.27 days), respectively. They are computed as $A_\gamma \approx 1-{\rm e}^{-\tau _\gamma }$, with $\tau _\gamma =3\mathcal {M}_{\rm ej}\sigma _t/(8\pi m_p V^2_{\rm ej}t^2)$, and $\sigma _t=9.5\times 10^{-26}\,$ cm2 ($\sigma _t=8.7\times 10^{-26}\,$ cm2) for 56Ni (56Co).
Citation: Dado & Dar (2015)
Citation function: Uses; Uses
Citation: Dado & Dar 2015
Citation function: Uses; Uses