Authorea

LBV Build-up Timescale and Quiescent Luminosity.

\label{sec:LBVbuildup}

To explore some of the physical implications of an LBV classification for the two HFF14Spo events, we first make a rough estimate of the total radiated energy, which can be computed using the decline timescale \(t_{2}\) and the peak luminosity \(L_{\rm pk}\):

\begin{equation} \label{eqn:Erad} \label{eqn:Erad}E_{\rm rad}=\zeta t_{2}L_{\rm pk},\\ \end{equation}

where \(\zeta\) is a dimensionless factor of order unity that depends on the precise shape of the light curve\cite{Smith:2011b}. Note that earlier work\cite{Smith:2011b} has used \(t_{1.5}\) instead of \(t_{2}\), which amounts to a different light-curve shape term, \(\zeta\). Adopting \(L_{\rm pk}\) \(\approx 10^{41}\) erg s\({}^{-1}\) and \(t_{2}\) \(\approx\)1 day (as shown in Fig. \ref{fig:PeakLuminosityDeclineTime}), we find that the total radiated energy is \(E_{\rm rad}\approx 10^{46}\) erg. A realistic range for this estimate would span \(10^{44}<E_{\rm rad}<10^{47}\) erg, due to uncertainties in the magnification, bolometric luminosity correction, decline time, and light-curve shape. These uncertainties notwithstanding, our estimate falls well within the range of plausible values for the total radiated energy of a major LBV outburst.

The “build-up” timescale\citep{Smith:2011b} matches the radiative energy released in an LBV eruption event with the radiative energy produced during the intervening quiescent phase,

\begin{equation} \label{eqn:trad} \label{eqn:trad}t_{\rm rad}=\frac{E_{\rm rad}}{L_{\rm qui}}=t_{2}\frac{\xi L_{\rm pk}}{L_{\rm qui}},\\ \end{equation}

where \(L_{\rm qui}\) is the luminosity of the LBV progenitor star during quiescence.

The HFF14Spo events are not resolved as individual stars in their quiescent phase, so we have no useful constraint on the quiescent luminosity. Thus, instead of using a measured quiescent luminosity to estimate the build-up timescale, we assume that \(t_{\rm rad}\) for HFF14Spo corresponds to the observed rest-frame lag between the two events, roughly 120 days (this accounts for both cosmic time dilation and a gravitational lensing time delay of \(\sim\)40 days). Adopting \(L_{\rm pk}=10^{41}\) erg s\({}^{-1}\) and \(t_{2}=2\) days (see Figure \ref{fig:PeakLuminosityDeclineTime}), we infer that the quiescent luminosity of the HFF14Spo progenitor would be \(L_{\rm qui}\approx 10^{39.5}\) erg s\({}^{-1}\) (\(M_{V}\approx-10\) mag).