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luminosity and decline time for each \spock event, we combine the linear fits to the light curves (shown in Figure~\ref{fig:LinearLightCurveFits}) with the predicted range of lensing magnifications (Figure~\ref{fig:LensModelContours}. (Figure~\ref{fig:LensModelContours}). For any assumed value for the time of peak brightness, the light curve fits give us an estimate of the ``observed'' peak magnitude and a corresponding rise-time and decline-time measurement. We then convert this extrapolated peak magnitude to a luminosity (e.g., $\nu L_\nu$ in erg s$^{-1}$) by first correcting for the luminosity distance assuming a standard \LCDM cosmology, cosmology \citep{Planck:2016}, and then accounting for an assumed lensing magnification, $\mu$. At the end of all this, we have a grid of possible peak luminosities for each event as a function of magnification and time of peak (or, equivalently, the decline time). Figures~\ref{fig:PeakLuminosityDeclineTimeWide} and \ref{fig:PeakLuminosityDeclineTime} show the resulting

events are representative of a single system (or a homogeneous class) then the most likely peak luminosity and decline time (the region with the most overlap) would be $L_{\rm pk}\sim10^{41}$ ergs/s and $t_3\sim1.8$ $t_2\sim1.8$ days. %In Figure~\ref{fig:PeakLuminosityDeclineTimeWide} we also demarcate %regions of the luminosity--decline time phase space occupied by known

magnitudes of $\sim$-7 to -16 mag in optical bands, and lasting for tens to hundreds of days. In some cases the LBV progenitor does indeed culminate with a final true core collapse SN event \citep[e.g.]{Mauerhan:2013, \citep[e.g.][]{Mauerhan:2013, Tartaglia:2016} % Foley:2007,Pastorello:2007,Smith:2010b,GalYam:2009}. Although most giant LBV eruptions have been observed to last much

images 11.1, 11.2 and 11.3. In 11.1 and 11.2 these measurements were extracted from the central pixel at the location of each of the two \spock events. Assuming the lensing magnification here ranges from $\mu=10$ to 100 (see Section~\ref{sec:LensingModels}, Section~\ref{sec:LensingModels}), this corresponds to a size on the source plane between 6 and 600 pc$^2$. For host image 11.3 we report the stellar population properties derived from the pixel at the center of the galaxy, because the lens models do not

from the SED fitting, and do not attempt to quantify potential systematic biases. Such biases could arise, for example, from color differences in the background light, which is dominated by the cluster galaxies and varies significantly across the MACS0416 \MACS0416 field. Such a bias might shift the absolute values of the parameter scales for any given host image (e.g., making the galaxy as a whole appear bluer, more massive and younger). However, the gradients across any single

\AA. Figure~\ref{fig:MUSEOIISequence} shows the observed \ionline{O}{[ii]} lines at 10 positions along the length of the arc, which comprises images 11.1 and 11.2. At each position the lines were extracted using apertures with a radius of 0.6\arcsec, 0\farcs6, so adjacent extractions are not independent, but extractions at the two \spock locations have no overlap. Each extraction has been normalized to show a peak line flux at unity, so that the line profiles and the doublet

\spocktwo---appeared in Hubble Space Telescope (\HST) imaging collected in January and August of 2014, respectively (Figure \ref{fig:SpockDetectionImages}). These observations were centered on the galaxy cluster \MACS0416\ \fullmacs0416 (hereafter, MACS0416) \MACS0416) and were collected as part of the Hubble Frontier Fields (HFF) survey (HST-PID:13496, PI:Lotz), a multi-cycle program for deep imaging of 6 massive galaxy clusters and associated ``blank sky'' fields observed in parallel (Methods \ref{sec:Discovery}). Combining the \HST imaging and lens models of the MACS0416 \MACS0416 gravitational lens leads to three key observables for the \spock events: (1) they are both more luminous than a classical nova, but less luminous than almost all supernovae (SNe), reaching a peak luminosity of roughly

\item{\it WSLAP+:} Created with the {\tt WSLAP+} software \citep{Sendra:2014}: Weak and Strong Lensing Analysis Package plus member galaxies (Note: no weak-lensing constraints used for this MACS0416 \MACS0416 model). Interactive online model exploration available at \url{http://www.ifca.unican.es/users/jdiego/LensExplorer}. \item{{\it Z-LTM:} A model with strong- and weak-lensing constraints, built using the ``light-traces-mass'' (LTM) methodology \citep{Zitrin:2009a,Zitrin:2015}, first presented for MACS0416 \MACS0416 in \citet{Zitrin:2013a}.} \end{itemize} \bigskip

peak brightness corresponds to a longer rise time and a faster decline timescale, independent of the specific model used. Changes to the arbitrary constraints we placed on these linear fits do not substantially affect the these results.For example, the relationship between peak brightness and decline time is not strongly affected by adjusting the assumed maximum post-peak magnitude or changing the number of pre-peak data points used for the rising light curve fits. At any assumed value for the time of peak brightness this linear interpolation gives an estimate of the peak magnitude. We then convert

bright O or B star with mass of order 10 \Msun. Depending on its age, the size of such a star would range from a few to a few dozen times the size of the sun. The net relative transverse velocity would be on the order of a few 100 km/s, km s$^{-1}$, which is comparable to the orbital velocity of stars within a galaxy or galaxies within a cluster. In the case of a smooth cluster potential, the timescale $\tau$ for

which aims to identify and study explosive transients found in the HFF and related programs. The FrontierSN team discovered \spock\ in two separate HFF observing campaigns on the galaxy cluster \MACS0416\ (hereafter, MACS0416). \MACS0416. The first was an imaging campaign in January, 2014 during which the MACS0416 cluster field was observed in optical bands using the Advanced Camera for Surveys Wide Field Camera (ACS-WFC). The second concluded in August, 2014, and imaged

PSF model was defined using \HST observations of the G2V standard star P330E, observed in a separate calibration program. A separate PSF model was defined for each filter, but owing to the long-term stability of the \HST PSF we used the same model in all epochs. All of the aperture and PSF fitting photometry was carried out using the {\tt PythonPhot} software package \citep{Jones:2015}. \citep{Jones:2015}.\footnote{\url{https://github.com/djones1040/PythonPhot}} \subsection{Host Galaxy Spectroscopy}\label{sec:Spectroscopy}

To interpret the observed light curves and the timing of these two events, we use six independently constructed cluster mass models to determine the impact of gravitational lensing from the MACS0416 \MACS0416 cluster (Methods \ref{sec:LensingModels}). The lensing scenario that has been consistently adopted for this cluster assumes that the arc in which the \spock events appeared consists of two mirror images of the

consistent with each other, and if both events are representative of a single system (or a homogeneous class) then the most likely peak luminosity and decline time (the region with the most overlap) would be $L_{\rm pk}\sim10^{41}$ ergs/s and $t_2\sim1.8$ days. $t_2\sim1$ day. As shown in Figure~\ref{fig:PeakLuminosityDeclineTime}, the relatively low peak luminosities and the very rapid rise and fall of both \spock

white dwarf stars as Type Ia SNe, and the heterogeneous class of core collapse SNe. Less well-understood classes are also ruled out, such as Superluminous SNe \citep{Gal-Yam:2012,Arcavi:2016}, Type Iax SNe \citep{Foley:2013a}, \citep{Li:2003,Jha:2006a,Foley:2013a}, fast optical transients \citep{Drout:2014}, Ca-rich SNe \citep{Filippenko:2003,Perets:2011,Kasliwal:2012}, and Luminous Red Novae \citep[also called intermediate luminosity red

problem for all of the catastrophic stellar explosions discussed so far is that they cannot explain the appearance of {\it repeated} transient events. As shown in Table~\ref{tab:LensModelPredictions}, none of the MACS0416 \MACS0416 lens models predict an 8 month time delay between appearances in host galaxy images 11.1 and 11.2, suggesting that \spocktwo is not a time-delayed image of \spockone \citep[as was the case for the 5th image of SN Refsdal;][]{Kelly:2015a,Kelly:2016}. Neither the kilonova nor the .Ia explosion models can accommodate repeated events from the same source. The kilonova progenitor systems are completely disrupted at explosion. For .Ia events, even if an AM CVn system could produce repeated He shell flashes of similar luminosity, the period of recurrence would be of order $10^5$ yr, making these effectively non-recurrent sources. Thus, to reconcile any cataclysmic explosion model with the two observed \spock events we would need to invoke a highly serendipitous occurrence of two unrelated peculiar explosions in the same host galaxy in the same year. It This is not unheard of for common SN classes \citep[e.g.][]{Poznanski:2009}, but it is difficult to quantitatively assess the likelihood of such an occurrence, occurrence for fainter rapid transients, as there are no measured rates of .Ia or kilonovae. In a study of very fast optical transients with the Pan-STARRS1 survey, \citet{Berger:2013b} derived a limit of $\lesssim0.05$ Mpc$^{-3}$ yr$^{-1}$ for transients reaching $M\approx -14$ mag on a timescale of $\sim$1 day. This limit, though several orders of magnitude higher

rapid recurrence period for a RN system. All RNe in our own galaxy have recurrence timescales ranging from 15--80 years \citep{Schaefer:2010}. The fastest measured recurrence timescale belongs tothe Andromeda galaxy nova M31N 2008a-12, which has exhibited a new outburst every year from 2009-2015 \citep{Tang:2014,Darnley:2014,Darnley:2015,Henze:2015,Henze:2015a}. Although this M31 record-holder demonstrates that very rapid recurrence is

would be on the order of hours or days (Methods \ref{sec:Microlensing}), which is comparable to the timescales observed for the \spock events. Gravitational lensing is achromatic as long as the size of the source is consistent across the SED. spectral energy distribution (SED). This means that the color of a caustic crossing transient will be roughly constant. Using simplistic linear interpolations of the observed light curves (Methods \ref{sec:LightCurves}) we find that the inferred color curves for both \spock events are marginally consistent with this expectation of an unchanging color (Methods \ref{sec:ColorCurves}). In the baseline lensing scenario adopted above---where a single critical curve subtends the \spock host galaxy arc---these events

\subsection{X-ray Non-detections}\label{sec:Xray} The MACS0416 \MACS0416 field was observed by the SWIFT X-Ray Telescope (XRT) and UltraViolet/Optical Telescope (UVOT) in April 2013. No source was detected near the locations of the \spock events (N. Gehrels, private communication). The field was also observed by the Chandra X-ray

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