Theoretical Models of Optical Transients. I. A Broad Exploration of the Duration–Luminosity Phase Space

Without a heating source, the light curves are entirely defined by the homogeneous solution to Equation (2) for an expanding photosphere:

$$\begin{eqnarray}&&L(t)={L}_{0}{e}^{-\left(\tfrac{{t}^{2}}{{t}_{{\rm{d}}}^{2}}+\displaystyle \frac{2{R}_{0}t}{{v}_{\mathrm{ej}}{t}_{{\rm{d}}}}\right)},\end{eqnarray} \tag{ 4 }$$

where R₀ is the progenitor radius and ${L}_{0}\approx {E}_{\mathrm{KE}}/2{t}_{{\rm{d}}}$ is the initial luminosity from the explosion. Although the bolometric luminosity is monotonically decreasing, the R-band light curve rises and then declines as the photosphere expands, and the peak of the blackbody SED evolves from shorter to longer wavelengths through the optical regime.

Rather than varying the progenitor radius across several orders of magnitude, we focus on three specific regimes that sample the full range of reasonable scenarios: white dwarfs (WDs; ${R}_{0}\sim 0.01$ ${R}_{\odot }$, ${M}_{\mathrm{ej}}\sim 0.1\mbox{--}1$ ${M}_{\odot }$), Wolf-Rayet/blue-supergiant-like stars (BSG; ${R}_{0}\sim 10$ ${R}_{\odot }$, ${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$), and red-supergiant-like stars (RSG; ${R}_{0}\sim 500$ ${R}_{\odot }$, ${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$). Luminous blue variables (LBVs), known for their eruptive mass-loss events, have radii intermediate between the BSG and RSG progenitors, ${R}_{0}\sim 10\mbox{--}100$ ${R}_{\odot }$.

In Figure 1, we plot a sample of simulated light curves and the DLPS of each progenitor type, randomly sampling from uniform distributions of ejecta mass and logarithmically in kinetic energy (${E}_{\mathrm{KE}}\sim {10}^{49}\mbox{--}{10}^{51}$ erg). As expected, we find the general trend that larger progenitors produce longer-duration and more luminous transients. The upper and lower bounds to the quadrilateral-like areas each of these models occupy are defined by our chosen energy limits, while the vertical (duration) boundaries are set by our chosen ejecta mass limits. Only compact (WD) progenitors produce transients with durations ${t}_{\mathrm{dur}}\lesssim 1$ day, all of which have a low luminosity (${M}_{{\rm{R}}}\gtrsim -12$ mag). In contrast, the larger progenitors (BSG/RSG) only produce transients with longer durations (${t}_{\mathrm{dur}}\gtrsim 10$ days) that are brighter (${M}_{{\rm{R}}}\gtrsim -10$ mag). Thus, there is an overall clear positive correlation between luminosity and duration for this type of explosion.

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The effects of the main parameters (${E}_{\mathrm{KE}}$ and ${M}_{\mathrm{ej}}$) are explicitly shown in Figure 1. For a constant E_KE, the transients become longer and somewhat brighter with increasing M_ej. On the other hand, transients become shorter and brighter for increasing values of E_KE given a constant value of M_ej. Therefore, the brightest (dimmest) transients with no central heating have large (small) kinetic energies and small (large) ejecta masses for a fixed value of R₀. We find that these trends are generally true in all other heating sources as well.

One of the most important and well-studied heating sources, responsible for the bulk of thermonuclear and stripped-envelope core-collapse SNe, is the radioactive decay of ⁵⁶Ni (and ⁵⁶Co) synthesized in the explosion (see Arnett 1979, 1980, 1982; Chatzopoulos et al. 2012, etc.). The input luminosity is given by

$$\begin{eqnarray}&&{L}_{\mathrm{in}}(t)={M}_{\mathrm{Ni}}[{\epsilon }_{\mathrm{Co}}{e}^{-t/{\tau }_{\mathrm{Co}}}+({\epsilon }_{\mathrm{Ni}}-{\epsilon }_{\mathrm{Co}}){e}^{-t/{\tau }_{\mathrm{Ni}}}],\end{eqnarray} \tag{ 5 }$$

where M_Ni (the initial nickel mass) is the only free parameter of this heating source. The energy generation rates of ⁵⁶Ni and ⁵⁶Co (${\epsilon }_{\mathrm{Ni}}=3.9\times {10}^{10}$ erg s⁻¹ g⁻¹ and ${\epsilon }_{\mathrm{Co}}=6.8\times {10}^{9}$ erg s⁻¹ g⁻¹) and the decay rates (${\tau }_{\mathrm{Ni}}=8.8$ days and ${\tau }_{\mathrm{Co}}=111$ days) are known constants.

The radioactive decay of ⁵⁶Ni powers objects spanning a broad range of properties. We explore four regimes in this work: SNe Ia, SNe Ib/c, pair-instability SNe, and Iax-like SNe.

3.2.1. Type Ia SNe

WDs can explode as SNe Ia after thermonuclear ignition (Hoyle & Fowler 1960), although there is ongoing debate about whether this ignition arises from pure deflagration, delayed detonation, or other mechanisms (see, e.g., Khokhlov 1991; Arnett & Livne 1994; Phillips et al. 2007). Although it is unclear whether the progenitors of SNe Ia are single- or double-degenerate systems, SNe Ia occupy a narrow range of the DLPS owing to their homogeneity.

SNe Ia have low ejecta masses (${M}_{\mathrm{ej}}\approx 1.4$ ${M}_{\odot }$), with a relatively large fraction of this ejecta being radioactive ⁵⁶Ni (${f}_{\mathrm{Ni}}\sim 0.3\mbox{--}0.5$). Rather than modeling these light curves using simple blackbody SEDs, we use an empirical relation described by Tripp & Branch (1999) and Betoule et al. (2014):

$$\begin{eqnarray}&&{t}_{\mathrm{dur}}=35s\ \,\ \mathrm{days},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{M}_{\mathrm{peak}}=-19.4+1.4(s-1),\end{eqnarray} \tag{ 7 }$$

where s is the stretch of the light curve, for which we provide a range of 0.6 to 1.2 (roughly matching the range explored in Guy et al. 2005). We use the canonical (s = 1) template from Nugent et al. (2002) to extract a template R-band light curve to stretch. As a consistency check, we find that this relation roughly agrees with that found using the R-band templates from Prieto et al. (2006).

Using this relation, we find the well-known result that brighter (dimmer) SNe Ia have longer (shorter) durations. SNe Ia are constrained to a small subset of the DLPS, with durations of ${t}_{\mathrm{dur}}\sim 25\mbox{--}50$ days and ${M}_{{\rm{R}}}\sim -18$ to −20 mag. In Figure 2, we plot the duration–luminosity relation and several SNe Ia from the Open Supernova Catalog (OSC; Guillochon et al. 2017b; see caption for details). To summarize, SNe Ia occupy a narrow portion of the DLPS, with no sources having durations of ${t}_{\mathrm{dur}}\lesssim 20$ days.

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3.2.2. SNe Ib/c

SNe Ib/c occur when stripped-envelope massive stars undergo core collapse at the end of their lives and are identified by their hydrogen-free (and helium-free in the case of SNe Ic) spectra. SNe Ib/c contain a relatively low fraction of ⁵⁶Ni (${f}_{\mathrm{Ni}}\sim 0.01\mbox{--}0.15$) in their ejecta (${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot };$ Drout et al. 2011). In this class, we include the parameter ranges for both normal and broad-lined SNe Ib/c and sample uniformly across their typical kinetic energies (${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{52}$ erg).

Our simulated light curves (Figure 3) span a wide range in both absolute magnitude (${M}_{R}\sim -16\ \mathrm{to}\ -19$ mag) and duration (${t}_{\mathrm{dur}}\sim 10\mbox{--}120$ days). The brightest (dimmest) transients have the largest (smallest) ⁵⁶Ni masses (${f}_{\mathrm{Ni}}{M}_{\mathrm{ej}}$), while the longest (shortest) durations are largely set by the ejecta velocity ($v\approx {[{E}_{\mathrm{KE}}/{M}_{\mathrm{ej}}]}^{1/2}$). Due to this positive correlation, the shortest-duration transients (${t}_{\mathrm{dur}}\sim 10$) also have the lowest luminosities (${M}_{{\rm{R}}}\sim -16$ mag). We find essentially no transients with ${t}_{\mathrm{dur}}\lesssim 10$ days. Such transients would require faster ejecta velocities than typically observed in these sources.

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We compare our generated light-curve properties to samples from the literature (Drout et al. 2011; Taddia et al. 2015). The observed sample generally overlaps our simulated DLPS, although our models extend to shorter durations of ${t}_{\mathrm{dur}}\sim 10\mbox{--}20$ days, which have not been observed.

3.2.3. Pair-instability SNe

It is predicted that stars with $M\sim 140\mbox{--}260$ ${M}_{\odot }$ will reach sufficiently high core temperatures to produce electron–positron pairs, leading to a loss of pressure and a resulting thermonuclear runaway and explosion that leaves no remnant (Barkat et al. 1967). Due to the large ejecta masses and kinetic energies, the optical light curves are expected to be both bright and of long duration (Kasen et al. 2011; Dessart et al. 2013). We expect pair-instability SNe (PISNe) to have similar (extending to slightly larger) ⁵⁶Ni fractions to SNe Ib/c (${f}_{\mathrm{Ni}}\sim 1 \% \mbox{--}30$%) but to have much larger ejecta masses (${M}_{\mathrm{ej}}\sim 50\mbox{--}250$ ${M}_{\odot }$, with the lower masses representing stripped progenitors) and kinetic energies (${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{53}$ erg); see Kasen et al. (2011).

In Figure 4, we show a sample of simulated PISN light curves and the associated DLPS. Compared to SNe Ib/c, the PISNe typically have longer durations (${t}_{\mathrm{dur}}\sim 100\mbox{--}400$ days) and higher luminosities (${M}_{{\rm{R}}}\sim -18$ to −22 mag). Like the other ⁵⁶Ni-decay powered models, the durations and luminosities are positively correlated, with the shortest-duration transients (${t}_{\mathrm{dur}}\sim 100$) being the least luminous (${M}_{{\rm{R}}}\sim -18$ mag).

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We compare our results to more detailed calculations by Kasen et al. (2011) and Dessart et al. (2013). The first-order properties of the light curves are in rough agreement (Figure 4), although the Kasen et al. (2011) models allow for less energetic/luminous explosions. By allowing energy and ejecta mass to vary independently, our model explores a larger parameter space than comprehensive PISN models in which the progenitor masses and kinetic energies are linked.

3.2.4. Ultra-stripped SNe/Iax-like SNe

Although ultra-stripped SNe and Iax SNe have some similar properties, they emerge from distinct physical scenarios, and more detailed simulations predict unique spectral features. Ultra-stripped SNe are theorized to arise from helium star–neutron star binary systems that undergo significant stripping of the helium envelope (Tauris et al. 2015; Moriya et al. 2016). Iax SNe define a loose observational class that is spectroscopically similar to SNe Ia, although they are dimmer in optical bands (Foley et al. 2013).

Ultra-stripped SNe and Iax-like SNe have a high nickel content (${f}_{\mathrm{Ni}}\sim 0.1\mbox{--}0.5$) similar to SNe Ia but have lower ejecta masses (${M}_{\mathrm{ej}}\sim 0.01\mbox{--}1$ ${M}_{\odot }$) and kinetic energy (${E}_{\mathrm{KE}}\sim {10}^{49}\mbox{--}{10}^{51}$ erg) compared to SNe Ib/c. In Figure 5 we present a sample of Iax-like SN light curves and the associated DLPS. We find a tighter positive correlation between duration (${t}_{\mathrm{dur}}\sim 10\mbox{--}50$ days) and peak magnitudes (${M}_{{\rm{R}}}\sim -16$ to −19 mag) compared to the SNe Ib/c owing to the narrower ranges of f_Ni and M_ej. Unlike the other ⁵⁶Ni-decay powered models, the Iax-like model can produce a small fraction of transients with durations ${t}_{\mathrm{dur}}\lesssim 10$ days (with a shortest duration of ${t}_{\mathrm{dur}}\sim 1$ week), but these transients are also the dimmest (${M}_{{\rm{R}}}\sim -16.5$ mag).

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We compare our models to the Iax SN sample from Foley et al. (2013) and a sample of short-duration transients from Drout et al. (2014) in Figure 5. Our models do not account for the lowest-luminosity observed objects, which likely have lower ⁵⁶Ni and ejecta masses than our model ranges. Furthermore, the realizations extend to longer durations (${t}_{\mathrm{dur}}\sim 30\mbox{--}60$ days) than seen in current observations.

3.2.5. General Trends

The effects of the kinetic energy, ejecta mass, and nickel fraction on all ⁵⁶Ni-powered models are explored in Figure 6. Unsurprisingly, the unique free parameter of this engine, M_Ni, exclusively impacts the brightness of the transient with no impact on its duration. M_ej and E_KE have degenerate and opposing effects on the light-curve parameters (the same effect as in the adiabatic case, Figure 1). For a given kinetic energy, larger ejecta masses lead to longer and dimmer transients as the diffusion process becomes less efficient. For a given ejecta mass, larger kinetic energies lead to shorter and brighter transients owing to the resulting larger velocities. Thus, the shortest-duration transients have small ejecta masses and large kinetic energies, and vice versa for the longest-duration transients. The brightest (dimmest) transients have large (small) nickel masses, corresponding to either large (small) nickel fractions or ejecta masses. We specifically find that ⁵⁶Ni heating cannot power transients with durations ${t}_{\mathrm{dur}}\lesssim 1$ week, unless their peak luminosities are also small, ${M}_{{\rm{R}}}\gtrsim -16.5$ mag.

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Neutron-rich ejecta from binary neutron star or black hole–neutron star mergers are expected to undergo r-process nucleosynthesis owing to the neutron-rich ejecta from either the initial merger event or a remnant disk outflow (Li & Paczyński 1998; Metzger et al. 2010). The radioactive decay of r-process products provides a heating source, while the synthesis of Lanthanides provides a high opacity (Barnes & Kasen 2013). The ejecta masses are expected to be small, ${M}_{\mathrm{ej}}\sim {10}^{-3}\mbox{--}0.1$ ${M}_{\odot }$ (Li & Paczyński 1998; Metzger 2016). The input luminosity can be parameterized by (Korobkin et al. 2012; Metzger 2016)

$$\begin{eqnarray}{L}_{\mathrm{in}}(t) & = & 4\times {10}^{18}{\epsilon }_{\mathrm{th}}(t){M}_{\mathrm{rp}}\\ & & \times {\left[0.5-{\pi }^{-1}\arctan \left(\displaystyle \frac{t-{t}_{0}}{\sigma }\right)\right]}^{1.3}\mathrm{erg}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 8 }$$

where M_rp is the mass of the r-process material, ${t}_{0}=1.3\,{\rm{s}}$ and $\sigma =0.11\,{\rm{s}}$ are constants, and ${\epsilon }_{\mathrm{th}}(t)$ is the thermalization efficiency (Barnes et al. 2016; Metzger 2016) parameterized as

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{th}}(t)=0.36\left[{e}^{-0.56t}+\displaystyle \frac{\mathrm{ln}(1+0.34{t}^{0.74})}{0.34{t}^{0.74}}\right].\end{eqnarray} \tag{ 9 }$$

Because kilonovae have not yet been conclusively observed (with the potential exception of GRB 130603B, Berger et al. 2013a; Tanvir et al. 2013; and GRB 050709, Jin et al. 2016), there are a number of uncertainties in the light-curve properties. Notably, the optical opacity of the Lanthanide-rich ejecta is unknown owing to the complex structure of their valence f-shells. Early work assumed that Lanthanide-rich material had opacities similar to that of iron-peak elements, leading to bluer transients (Li & Paczyński 1998). Recent work suggests that Lanthanide-rich material will have an optical opacity $\kappa \sim {10}^{2}\mbox{--}{10}^{3}$ times larger than that of iron-peak elements (Barnes & Kasen 2013; Barnes et al. 2016). However, it is possible that both cases exist, if binary neutron star mergers leave a neutron star remnant with a survival timescale of $\gtrsim 0.1\,{\rm{s}}$ (Kasen et al. 2015). We consider these two possibilities in our models by generating two sets of light curves, following parameter ranges from Metzger (2016): one with a fixed $\kappa =0.2$ cm² g⁻¹ (a "blue" kilonova, similar to that originally explored by Li & Paczyński 1998), and one with a variable κ sampled logarithmically in the range $\kappa \sim 10\mbox{--}200$ cm² g⁻¹ (a "red" kilonova). For each group, we logarithmically sample from ejecta masses of ${M}_{\mathrm{ej}}\sim {10}^{-3}\mbox{--}{10}^{-1}\,{M}_{\odot }$, uniformly sample from ejecta velocities of ${v}_{\mathrm{ej}}\sim 0.1c\mbox{--}0.3c$, and fix the r-process mass fraction f_r = 1 (Metzger 2016). We additionally choose the geometric factor $\beta =3$ to calculate the diffusion timescale, following Metzger (2016).

A sample of these models and their associated DLPS are shown in Figure 7. Both classes are dim (${M}_{{\rm{R}}}\gtrsim -15$ mag) and of short duration (${t}_{\mathrm{dur}}\lesssim 5$ days). The red kilonovae are dimmer (${M}_{{\rm{R}}}\sim -7$ to −13 mag) than the blue kilonovae (${M}_{{\rm{R}}}\sim -13$ to −15 mag). Both subclasses have similar average durations of ${t}_{\mathrm{dur}}\sim 2$ days, although a large fraction of the red kilonovae have even shorter durations of ${t}_{\mathrm{dur}}\lesssim 1$ day. We note that although red kilonovae are expected to last ∼1 week in the near-infrared, the transients are short-lived and dim in the R band even if the ejecta is Lanthanide-poor. As with the ⁵⁶Ni-powered models, the duration and peak luminosities are positively correlated.

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In Figure 7 we explicitly show the effects of κ and M_ej on the light-curve properties. Increasing opacity makes the transient shorter and dimmer in the optical, while larger ejecta masses increase the duration and luminosity. The shortest (longest) duration transients have large (small) values of κ and small (large) values of M_ej. The brightest kilonovae have small opacity values, and vice versa for the dimmest kilonovae. We find that kinetic energy has a similar effect on the light curves to that seen in transients lacking a central heating source (see Section 3.1 and Figure 1); larger (smaller) kinetic energy leads to more (less) luminous transients with shorter (longer) durations.

Finally, we compare our simple model with more detailed calculations from Metzger (2016) and Barnes & Kasen (2013). Our models are in rough agreement with the detailed calculation, in both duration and luminosity. However, our models include even dimmer kilonovae (${M}_{{\rm{R}}}\gtrsim -11$ mag), which have low ejecta masses. We conclude that r-process heating in the context of compact object mergers can lead to short-duration transients ($\lesssim \mathrm{few}$ days) but that these transients are invariably dim ($\gtrsim -15$ mag).

Young magnetars, or highly magnetized neutron stars, can power optical transients as they spin down and deposit energy into the expanding ejecta (Kasen & Bildsten 2010; Woosley 2010; Metzger et al. 2015). For a dipole field configuration, the input luminosity is given by

$$\begin{eqnarray}&&{L}_{\mathrm{in}}(t)=\displaystyle \frac{{E}_{{\rm{p}}}}{{t}_{{\rm{p}}}}\displaystyle \frac{1}{{\left(1+\tfrac{t}{{t}_{{\rm{p}}}}\right)}^{2}},\end{eqnarray} \tag{ 10 }$$

where ${E}_{{\rm{p}}}={I}_{\mathrm{NS}}{{\rm{\Omega }}}^{2}/2$ is the initial magnetar rotational energy, described by the moment of inertia (I_NS) and angular velocity of the neutron star (Ω), and t_p is the spin-down characteristic timescale:

$$\begin{eqnarray}&&{t}_{{\rm{p}}}=(1.3\times {10}^{5})\displaystyle \frac{{P}_{\mathrm{spin}}^{-2}}{{B}_{14}^{2}}\ {\rm{s}},\end{eqnarray} \tag{ 11 }$$

where ${P}_{\mathrm{spin}}=5{\left(\tfrac{{E}_{{\rm{p}}}}{{10}^{51}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-0.5}$ ms is the spin period and B₁₄ is the magnetic field in units of 10¹⁴ G. Recently, the magnetar model has been used to explain Type I superluminous SNe (SLSNe; Quimby et al. 2011; Dessart et al. 2012; Gal-Yam 2012; Nicholl et al. 2013, 2017b).

In this work we explore magnetar-powered transients with spin periods ${P}_{\mathrm{spin}}\sim 1\mbox{--}10$ ms, magnetic fields $B\sim {10}^{13}\mbox{--}{10}^{15}$ G, ejecta masses ${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$, and kinetic energies ${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{52}$ erg. These parameter ranges are designed to span realistic values where magnetar spin-down can be the dominant power source. Large spin periods of ≳10 ms and low magnetic fields of $B\lesssim {10}^{13}$ will result in low input power, and the transients will likely be dominated by ⁵⁶Ni decay (see Section 3.10). We additionally eliminate unphysical models with ${E}_{\mathrm{KE}}\mbox{--}{E}_{\mathrm{SN},\min }\gt {E}_{{\rm{p}}}$, where ${E}_{\mathrm{SN},\min }={10}^{51}$ erg is the minimum energy required to leave a neutron star remnant. This condition removes models in which most of the rotational energy feeds into ejecta expansion rather than radiation. The magnetar-powered models and the associated DLPS are shown in Figure 8. While increasing spin periods lead to dimmer transients, the transient luminosity is actually optimized at intermediate values of B₁₄ that depend on P_spin when the spin-down timescale roughly matches the diffusion timescale.

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There are several notable features caused by these dependencies in the magnetar DLPS. First, the paucity of long-duration and low-luminosity transients reflects the lower bound of our magnetic field range. In contrast, the luminosity upper limit is set by the lower bound on the spin period, which we set at the maximal neutron star spin (1 ms). There is also an absence of shorter-duration transients with ${M}_{{\rm{R}}}\sim -18$ to −20 mag. The upper boundary of this void is set by our magnetic field lower limit, while the lower boundary is set by the lower ejecta mass limit; all of the transients below this void have low magnetic field strengths. The effects of P_spin and B on the magnetar light curves are shown explicitly in Figure 8. We conclude that magnetar-powered transients are typically luminous (${M}_{{\rm{R}}}\lesssim -19$ mag) with long durations (${t}_{\mathrm{dur}}\gtrsim 30$ days).

Finally, we compare our DLPS with a sample of Type I SLSNe from the literature and with detailed models by Kasen & Bildsten (2010). We find that the majority of our realizations agree with the observed population (${M}_{{\rm{R}}}\sim -19$ to −23 mag and ${t}_{\mathrm{dur}}\sim 20\mbox{--}200$ days). Additionally, we also reproduce the lower-luminosity models (${M}_{{\rm{R}}}\sim -17$ to −19 mag) explored by Kasen & Bildsten (2010).

Several types of optical transients, including SNe IIn and LBV outbursts, display clear signs of interaction between their ejecta and dense surrounding CSM. Properties such as narrow hydrogen and metal emission lines, bright Hα luminosities, and considerable X-ray/radio luminosities can be explained by a shock propagating through a CSM (Chevalier & Fransson 1994; Matzner & McKee 1999). Similarly, bright, blue, and short-duration transients have been linked to shock breakout from dense CSM "cocoons" (Chevalier & Irwin 2011; Drout et al. 2014; Arcavi et al. 2016). Because CSM interaction can describe an expansive range of transients, we consider two primary regimes: SN-like, with ejecta masses (${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$) and kinetic energies (${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{52}$ erg) typical of SNe IIn, and outburst-like, with low ejecta masses (${M}_{\mathrm{ej}}\sim {10}^{-3}-1$ ${M}_{\odot }$) and wind-like velocities ($v\sim {10}^{2}\mbox{--}{10}^{3}$ km s⁻¹), typical of intermediate-luminosity optical transients (ILOTs; including LBV outbursts and Type IIn precursors).

Many semianalytical models have been created to describe optical light curves powered by shock heating (Chatzopoulos et al. 2012; Moriya et al. 2013b; Smith 2013; Ofek et al. 2014b). Most of these models follow the same formalism presented by Chevalier (1982) and Chevalier & Fransson (1994) and track a shock through the CSM as it thermalizes the large kinetic energy reservoir (Chevalier 1982; Chevalier & Irwin 2011; Dessart et al. 2015). Due to the current uncertainty in the analytical models available, we explore two interaction models described by Chatzopoulos et al. (2012) and Ofek et al. (2014b) and discuss their key differences. We specifically use the Ofek et al. (2014b) model for CSM shock breakout transients and an altered Chatzopoulos et al. (2012) model for both SN-like and outburst-like transients. The details of these models are presented in the Appendix. In the subsections below, we discuss the input luminosities and model parameters.

3.5.1. Shock Breakout from a Dense CSM

Shock breakouts (SBOs) from dense CSM winds surrounding massive stars have been used to describe SNe IIn and other bright, blue transients (see, e.g., Margutti et al. 2013; Ofek et al. 2014b). This model assumes that the forward shock from the ejecta–CSM interaction radiates efficiently (${t}_{{\rm{d}}}=0$) such that

$$\begin{eqnarray}&&{L}_{\mathrm{in}}={L}_{\mathrm{obs}}=2\pi \epsilon {\rho }_{\mathrm{CSM}}({r}_{\mathrm{sh}}){r}_{\mathrm{sh}}^{2}{v}_{\mathrm{sh}}^{3},\end{eqnarray} \tag{ 12 }$$

where $\epsilon =0.5$ is an efficiency factor, ${\rho }_{\mathrm{CSM}}({r}_{\mathrm{sh}})$ is the density of the CSM as a function of the shock radius r_sh, and ${v}_{\mathrm{sh}}={{dr}}_{\mathrm{sh}}/{dt}$ is the shock velocity. The shock radius and velocity depend on the geometry of the explosion ejecta and CSM. Here we assume that the CSM is distributed as a wind-like profile, ${\rho }_{\mathrm{CSM}}(r)\propto {r}^{-2}$. The ejecta density profile is described as a broken power law, with an outer profile of ${\rho }_{\mathrm{ej}}(r)\propto {r}^{-n}$, where n is a free parameter. We find that the inner profile has little effect on the light curves, and thus we assume a flat inner profile. (See the Appendix for details.)

Thus, the free parameters are the ejecta density index (n), the kinetic energy of the explosion (E_KE), the ejecta mass (M_ej), the inner radius of the CSM (R₀), and the CSM density at R₀ (${\rho }_{\mathrm{CSM}}$). We sample over the following ranges: $n\sim 7\mbox{--}12$, ${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{52}$ erg, ${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$, ${R}_{0}\sim 1\mbox{--}{10}^{2}\,\mathrm{au}$, and ${\rho }_{\mathrm{CSM}}\sim {10}^{-17}\mbox{--}{10}^{-14}$ g cm⁻³. We then eliminate realizations with mass-loss rates $\dot{M}\equiv 4\pi {R}_{0}^{2}{\rho }_{\mathrm{CSM}}{v}_{{\rm{w}}}\lt {10}^{-6}$ ${M}_{\odot }$ yr⁻¹, assuming a wind velocity of ${v}_{{\rm{w}}}\sim {10}^{2}$ km s⁻¹. This cut corresponds to the lower end of expected RSG mass-loss rates (Smith 2014). Our parameters therefore correspond to mass-loss rates of $\dot{M}\sim {10}^{-6}\mbox{--}{10}^{-2}$ ${M}_{\odot }$ yr⁻¹, roughly matching the range of mass-loss rates of RSGs, yellow super giants, and LBVs (Smith 2014).

We note that our range of R₀ values extends beyond the radii of most progenitor stars. However, it is possible that R₀ is the location of a so-called cool dense shell formed by an earlier eruption and not always representative of the progenitor radius (Smith 2016).

Finally, we note that our range of n is representative of ejecta density profiles inferred for degenerate progenitors ($n\sim 7;$ Colgate & McKee 1969) and RSG progenitors ($n\sim 12;$ Matzner & McKee 1999). Although previous work has typically set $n\sim 12$, Chevalier & Irwin (2011) suggest that the shallower portions of density profiles may play larger roles in the ejecta–CSM interaction, so we leave n as a free parameter. Finally, we remove events with $v\gt {\rm{15,000}}$ km s⁻¹.

We show sample light curves and the associated DLPS in Figure 9. Our models span a wide range in both luminosity (${M}_{{\rm{R}}}\sim -13$ to −19 mag) and duration (${t}_{\mathrm{dur}}\sim 10\mbox{--}{10}^{3}$ days). Like most of our models, luminosity and duration are positively correlated. The shortest-duration transients have ${t}_{\mathrm{dur}}\approx 10$ days and peak brightness of ${M}_{{\rm{R}}}\gtrsim -17$ mag. The duration is largely determined by the mass-loss rate, with higher (lower) mass-loss rates leading to the longest (shortest) duration transients. The luminosities of the brightest transients are set by our minimum value of n and maximum velocities, while the luminosities of the dimmest transients are set by the minimum velocities (∼4 × 10³ km s⁻¹) of our parameter ranges. We explicitly show the effects of each free parameter in Figure 9. As shown in the figure, larger values of ${\rho }_{\mathrm{CSM}}$ actually lead to less luminous transients in the optical. This is due to the fact that large values of ${\rho }_{\mathrm{CSM}}$ lead to hotter effective temperatures that actually decrease the visible luminosity assuming a blackbody SED.

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In Figure 9 we also compare our simulated distribution to the distribution of all well-sampled Type IIn SNe listed on the OSC at the time of writing, a sample of short-duration transients from PanSTARRs (Drout et al. 2014), and a sample of rapidly rising, bright transients from Arcavi et al. (2016). We note that neither of the latter two samples is claimed to be from CSM SBO models; however, the CSM model is able to roughly reproduce the peak luminosities and durations of the rapid transients from Drout et al. (2014). The simulated shock breakout models generally produce dimmer transients than observed and allow for longer-duration transients (${t}_{\mathrm{dur}}\gtrsim 400$ days).

3.5.2. Ejecta–CSM Interaction with Diffusion

We now explore the generalized problem of ejecta–CSM interaction with diffusion assuming a stationary photosphere (Chatzopoulos et al. 2012). This model is similar to that in Section 3.5.1, but in this case we consider the reverse-shock contribution and a stationary photosphere (see the Appendix). The input luminosity is given by

$$\begin{eqnarray}{L}_{\mathrm{in}} & = & \epsilon {({\rho }_{\mathrm{CSM}}{R}_{0}^{s})}^{\displaystyle \frac{n-5}{n-s}}{t}^{\displaystyle \frac{2n+6s-{ns}-15}{n-s}}\\ & & \times [{C}_{1}\theta ({t}_{\mathrm{FS}}-t)+{C}_{2}\theta ({t}_{\mathrm{RS}}-t)],\end{eqnarray} \tag{ 13 }$$

where s, C₁, and C₂ are geometric parameters of the CSM, and $\theta (t)$ is the Heaviside function that controls the input times for the forward (t_FS) and reverse (t_RS) shocks. There are seven free parameters of the model: s, n, R₀, E_KE, M_ej, ${\rho }_{\mathrm{CSM}}$, and the total CSM mass (M_CSM). We set s = 0 for "shell-like" CSM models and s = 2 for "wind-like" CSM models.

We place a number of additional physical constraints on these models:

• 1.  
  We require the photospheric radius to be within the CSM shell: ${R}_{0}\leqslant {R}_{\mathrm{ph}}\leqslant {R}_{\mathrm{CSM}}$.
• 2.  
  We require the CSM mass to be less than the ejecta mass: ${M}_{\mathrm{CSM}}\leqslant {M}_{\mathrm{ej}}$.
• 3.  
  We require the velocity of the ejecta ${v}_{\min }\leqslant {v}_{\mathrm{ph}}\,\equiv \sqrt{10{E}_{\mathrm{KE}}/3{M}_{\mathrm{ej}}}\leqslant {v}_{\max }$, where ${v}_{\min }=5000$ km s⁻¹ and ${v}_{\max }={\rm{15,000}}$ km s⁻¹ for SN-like sources and ${v}_{\min }=100$ km s⁻¹ and ${v}_{\max }=1000$ km s⁻¹ for outburst-like sources.
• 4.  
  We require the diffusion time (t_d) through the CSM to be less than the shock-crossing time through the CSM (t_FS). If this were not the case, the light curve would exponentially decline as in the case of adiabatic expansion (Section 3.1; see the shell-shocked model described by Smith & McCray 2007). Moriya et al. (2013a) and Dessart et al. (2015) argue that the optical depths in typical CSMs are significantly lower than the regime of a shell-shocked model, implying that ${t}_{{\rm{d}}}\lt {t}_{\mathrm{FS}}$.

Finally, we choose reasonable parameter ranges for SN- and outburst-like sources. For both subclasses, we sample logarithmically from ${R}_{0}\sim 1\mbox{--}100\,\mathrm{au}$ and ${\rho }_{\mathrm{CSM}}\sim {10}^{-17}\mbox{--}{10}^{-14}$ g cm⁻³, typical ranges in SN IIn studies (Moriya et al. 2013a; Dessart et al. 2015). For the SN-like models, we explore both shell-like (s = 0) and wind-like (s = 2) CSM profiles. For the outburst-like models, we only explore wind-like CSM profiles.

For SN-like transients, we sample logarithmically from ${E}_{\mathrm{KE}}\sim {10}^{51}\mbox{--}{10}^{52}$ erg and ${M}_{\mathrm{CSM}}\sim 0.1\mbox{--}10$ ${M}_{\odot }$ and uniformly from ${M}_{\mathrm{ej}}\sim 1\mbox{--}10$ ${M}_{\odot }$. Simulated light curves and the DLPS of our models are shown in Figure 10 for both shell-like and wind-like CSM profiles. In the shell-like case, the light curves decline rapidly following peak brightness owing to our use of the Heaviside function to abruptly discontinue the input luminosity once the forward and reverse shocks have traversed the CSM. In the wind-like case, these light curves are smoother owing to the continuous $\rho (r)\propto {r}^{-2}$ CSM profile.

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One notable difference between the shell-like and wind-like models is the range of peak magnitudes, with shell-like models (${M}_{\mathrm{peak}}\sim -21$ to −24 mag) spreading a narrower range than the wind models (${M}_{\mathrm{peak}}\gtrsim -23$ mag) for the same range of physical parameters. This is likely due to the fact that ${L}_{\mathrm{in}}\propto \tfrac{2n+6s-{ns}-15}{n-s}$, or ${L}_{\mathrm{in}}\mathop{\propto }\limits_{\sim }{t}^{0.7}({t}^{-0.3})$ for s = 0 (s = 2) assuming n = 12, a typical value for RSGs (Chevalier 1982). In other words, the input luminosity is always decreasing in the wind-like model, while it actually increases in the shell-like model for $t\lt {t}_{\mathrm{FS}}$. This leads to brighter transients in the shell-like case.

There is little correlation between duration and luminosity for both shell-like and wind-like models owing to the complicated effects of the multiple parameters. In Figure 11 we show how the free and derived parameters affect the wind-like CSM models. We highlight several global trends. The brightest transients typically have the largest mass-loss rates, optically thick CSM masses (${M}_{\mathrm{CSM},\mathrm{th}}$), and photospheric radii (R_ph), and vice versa for the dimmest transients. The shortest-duration (${t}_{\mathrm{dur}}\sim 10$ days) transients with relatively high luminosities (${M}_{{\rm{R}}}\lesssim -16$ mag) have small CSM masses, although most of this CSM is optically thick.

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There are two low-luminosity (${M}_{{\rm{R}}}\gtrsim -15$ mag) "branches" in the wind-like DLPS: one extending to shorter durations (${t}_{\mathrm{dur}}\lt 20$ days), and the other at ${t}_{\mathrm{dur}}\sim 100$ days. The dearth of models between these branches is due to the fact that models within this area of phase space have optically thin CSM masses that are eliminated by our physical constraints. Realizations in the shorter-duration branch have larger CSM masses, mass-loss rates, and inner CSM radii compared to the branch at ∼100 days. Realizations in the shorter-duration branch have more peaked light curves owing to thinner shells at larger radii, while those in the longer-duration branch have flatter light curves. We note that no transients with SN-like properties have been observed to date in either branch.

As with previous classes, we find that the transients with shortest durations and SN-like luminosities have ${t}_{\mathrm{dur}}\approx 15$ days. Transients with shorter durations (down to ${t}_{\mathrm{dur}}\approx 15$ days) all have low luminosities of $\gtrsim -14$ mag.

Finally, we compare our DLPS to the sample of SNe IIn and other objects as in Section 3.5.1. The wind-like DLPS largely overlaps with the sample, while the shell-like DLPS is only able to reproduce the brightest SNe IIn.

For outburst-like transients, we sample logarithmically from $v\sim {10}^{2}\mbox{--}{10}^{3}$ km s⁻¹ and ${M}_{\mathrm{ej}}\sim 0.001\mbox{--}1$ ${M}_{\odot }$ and assume s = 2. The corresponding kinetic energy limits are ${E}_{\mathrm{KE}}\sim {10}^{44}\mbox{--}{10}^{49}$ erg. These limits were chosen to roughly match the velocities of LBV eruptions and explore a full range of the lowest-luminosity transients (Humphreys & Davidson 1994).

Sample light curves and the DLPS for outburst-like transients are shown in Figure 12. These models span a large range in both duration (${t}_{\mathrm{dur}}\sim \mathrm{few}-100$ days) and luminosity (${M}_{{\rm{R}}}\sim 0$ to −16 mag). The light-curve properties generally follow the same trends as the SN IIn models. Short-duration transients (${t}_{\mathrm{dur}}\lesssim 10$ days) are less luminous (${M}_{{\rm{R}}}\gtrsim -14$ mag). The parameter trends shown in Figure 11 also hold for ILOT-like models.

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Finally, we compare the simulated DLPS to a number of events from the literature, including LBV outbursts and Type IIn precursor events (see caption for details). In general, our models cover plausible timescales and magnitudes for ILOTs with signs of CSM interaction and overlap with many known objects.

SNe IIP are explosions of red supergiants with masses of $\approx 8\mbox{--}17\,{M}_{\odot }$ that have retained their hydrogen envelopes (Smartt et al. 2009). Following the explosion, a shock wave ionizes the hydrogen envelope. The characteristic flat, "plateau" phase of their optical light curves is powered by hydrogen recombination as the expanding ejecta cools to ∼5000 K at an approximately constant radius (i.e., the photosphere recedes in Lagrangian coordinates). The duration of the plateau phase is determined by the extent of the hydrogen envelope and kinematic properties of the blast wave. Following the plateau is a rapid decline in luminosity to a predominately ⁵⁶Co-powered tail (Arnett 1980; Weiler 2003).

The bolometric and optical light curves of SNe IIP have been studied extensively (e.g., Patat et al. 1994; Hamuy 2003; Kasen & Woosley 2009; Sanders et al. 2015; Rubin et al. 2016). Multizone semianalytical and numerical models generally reproduce the observed light curves (Popov 1993; Kasen & Woosley 2009) and provide scaling relations for both the plateau durations and luminosities. Here, we use these theoretical scaling relations in conjunction with empirical trends found by Sanders et al. (2015) to construct R-band light curves, neglecting contributions from both the shock breakout and ⁵⁶Ni radioactive decay. While shock breakout should primarily affect the early light curve, significant amounts of ⁵⁶Ni can extend the plateau duration. However, recent work has shown that ${M}_{\mathrm{Ni}}/{M}_{\mathrm{ej}}\lesssim 0.01$ (Müller et al. 2017), so we choose to ignore this contribution.

To construct light curves, we first assume instantaneous rise times. In reality, SNe IIP have rise times that range from a few days to a week (Rubin et al. 2016). This is a minor effect given the long plateau durations. We then use the bolometric scaling relations derived in Popov (1993) to estimate both the peak R-band luminosity (L_p) and duration (t_p) of the plateau phase:

$$\begin{eqnarray}&&{L}_{{\rm{p}}}=1.64\times {10}^{42}\displaystyle \frac{{R}_{0,500}^{2/3}{E}_{51}^{5/6}}{{M}_{10}^{1/2}}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{t}_{{\rm{p}}}=99\displaystyle \frac{{M}_{10}^{1/2}{R}_{0,500}^{1/6}}{{E}_{51}^{1/6}}\ \mathrm{days},\end{eqnarray} \tag{ 15 }$$

where ${R}_{\mathrm{0,500}}$ is the progenitor radius in 500 ${R}_{\odot }$, E₅₁ is the kinetic energy in 10⁵¹ erg, and M₁₀ is the ejecta mass in 10 ${M}_{\odot }$. Here we have assumed that the R-band bolometric correction is negligible (approximately true during the plateau; Bersten & Hamuy 2009). Additionally, the blackbody SED has a temperature of 5054 K (the ionization temperature of neutral hydrogen), and the opacity is $\kappa =0.34\,{\mathrm{cm}}^{2}$ g⁻¹. Following Sanders et al. (2015), we assume that the light curve reaches a maximum and then monotonically declines during the plateau phase. The decline rate is strongly correlated to the peak luminosity and is parameterized by (Sanders et al. 2015)

$$\begin{eqnarray}&&L(t)=\displaystyle \frac{{L}_{{\rm{p}}}}{{e}^{(-13.1-0.47{M}_{{\rm{R}}})}t},\end{eqnarray} \tag{ 16 }$$

where M_R is the peak magnitude. Finally, we assume that for $t\gt {t}_{{\rm{p}}}$ the light curve drops off instantaneously. This assumption is justified by our definition of duration (within 1 mag of peak), which is minimally impacted by the late-time behavior of the light curve. We generate light curves by sampling uniformly from ejecta mass (${M}_{\mathrm{ej}}\sim 5\mbox{--}15$ ${M}_{\odot }$) and progenitor radii (${R}_{0}\sim 100\mbox{--}1000$ ${R}_{\odot }$) and logarithmically in kinetic energy (${10}^{50}\mbox{--}5\times {10}^{51}$ erg).

The simulated DLPS and sample light curves are shown in Figure 13. The transient durations (${t}_{\mathrm{dur}}\sim 40\mbox{--}150$ days) and luminosities (${M}_{{\rm{R}}}\sim -16$ to −19 mag) are negatively correlated. The upper luminosity boundary reflects our M_ej upper limit. In Figure 13 we also explore the effects of the progenitor radius and the ejecta mass on the model light curves. Larger progenitor radii lead to brighter and longer-duration transients as a result of the fixed photosphere. Increasing the ejecta mass produces less luminous and longer-duration transients. Thus, the brightest (dimmest) Type IIP models have large (small) radii and small (large) ejecta masses. The longest (shortest) transients have large (small) radii and ejecta masses. The shortest-duration events have ${t}_{\mathrm{dur}}\approx 40$ days and high luminosities of ${M}_{{\rm{R}}}\approx -19$ mag.

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In Figure 13 we compare our generated light-curve properties to samples from PanSTARRs (Sanders et al. 2015) and the Palomar Transient Factor (PTF; Rubin et al. 2016). It is worth noting that both samples contain so-called SNe IIL, which are spectroscopically similar to SNe IIP but decline linearly in magnitude more rapidly than most SNe IIP. Both Sanders et al. (2015) and Rubin et al. (2016) find no evidence that SNe IIP and IIL arise from separate progenitor populations, so we also choose to keep SNe IIL in the observed sample. These samples largely overlap with our generated light curves.

Following a gamma-ray burst (GRB), the interaction of the relativistic jet with the CSM leads to a long-lived afterglow powered by synchrotron radiation (Sari et al. 1998). The afterglow emission can also be detected for off-axis sight lines (an "orphan" afterglow; Rhoads 1997; Rossi et al. 2002; van Eerten et al. 2010). There are two types of GRBs, long-duration (resulting from core collapse of stripped massive stars; Woosley 1993) and short-duration (likely produced by neutron star binary mergers; Berger 2014). The energy scale of short GRBs (SGRBs) is about 20 times lower than for long GRBs (LGRBs), and their circumburst densities are at least an order of magnitude lower (Berger 2014; Fong et al. 2015).

Here we explore both LGRB and SGRB afterglow models. Rather than generating analytical models, we use the publicly available⁴ broadband GRB afterglow model presented by van Eerten et al. (2010). This model calculates broadband SEDs of both on- and off-axis GRB afterglows using a high-resolution two-dimensional relativistic hydrodynamics simulation. Typical LGRB values of isotropic energy ${E}_{\mathrm{iso}}={10}^{53}$ erg, jet half opening angle ${\theta }_{\mathrm{jet}}=11\buildrel{\circ}\over{.} 5$, circumburst medium density ${n}_{0}=1$ cm⁻³, accelerated particle slope p = 2.5, accelerated particle energy density fraction of thermal energy density ${\epsilon }_{{\rm{e}}}=0.1$, and magnetic field energy density as a fraction of thermal energy density ${\epsilon }_{{\rm{B}}}=0.1$ are assumed. From this model, we can then generate the parameter space of long and short GRBs using the scaling relation presented in Van Eerten & MacFadyen (2012):

$$\begin{eqnarray}{L}_{\nu ,\mathrm{obs}} & \mathop{\propto }\limits_{\sim } & {\epsilon }_{{\rm{B}}}^{1/2}{n}_{0}^{-1/2}{E}_{\mathrm{iso}}\\ {t}_{\mathrm{dur}} & \mathop{\propto }\limits_{\sim } & {\displaystyle \frac{{E}_{\mathrm{iso}}}{{n}_{0}}}^{1/3}.\end{eqnarray} \tag{ 17 }$$

Our simulated model, using the van Eerten et al. (2010) parameters and assuming that the afterglow is first observed $\approx 0.5$ days after the GRB, is shown in Figure 14. As the orientation becomes increasingly off-axis, the R-band transient becomes dimmer and of longer duration. We also plot scaled versions of this model, assuming ${n}_{0}\sim 1\mbox{--}10$ cm⁻¹ and ${E}_{\mathrm{iso}}\sim (0.3\mbox{--}3)\times {10}^{53}$ erg for LGRBs and ${n}_{0}\sim 0.01\mbox{--}0.1$ cm⁻¹ and ${E}_{\mathrm{iso}}\sim (0.3\mbox{--}3)\times {10}^{51}$ erg for SGRBs. Both the SGRB and LGRB models span a wide range of durations (${t}_{\mathrm{dur}}\sim 1\mbox{--}1000$ days) and luminosity (${M}_{{\rm{R}}}\sim -2$ to −21 mag for SGRBs and ${M}_{{\rm{R}}}\sim -12$ to −16 mag for LGRBs). The duration and luminosity are tightly negatively correlated, with the shortest-duration events being the brightest. The only events with ${t}_{\mathrm{dur}}\lesssim 10$ days are on-axis, which are known to be rare. For off-axis sight lines, the luminosity drops rapidly as the duration increases such that at $\approx 2{\theta }_{{\rm{j}}}$ the afterglow is comparable to SNe in terms of timescale and luminosity. For larger angles, the events are much dimmer than SNe with longer durations.

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We additionally plot a region corresponding to the $1\sigma$ observed properties of a sample of rest-frame R-band afterglows of on-axis LGRBs from the BAT6 sample (after removing LGRBs with early flares; Melandri et al. 2014). The sample is in general agreement with the afterglow model, with short durations (${t}_{\mathrm{dur}}\sim 0.4\mbox{--}2$ days) and bright luminosities (${M}_{{\rm{R}}}\sim -22$ to −25 mag), as expected for these on-axis events.

Tidal disruption events (TDEs) occur when a star passes near a supermassive black hole (SMBH) and becomes tidally disrupted (Frank & Rees 1976; Hills 1988). About half of the star's mass forms an accretion disk around the SMBH, leading to an optically bright transient that lasts for weeks to months depending on the system's characteristics (Guillochon et al. 2009). A number of complications arise when modeling these transients, including the complex 3D geometry of the system, the hydrodynamics forming the accretion disk, the possibility of existing CSM surrounding the event, and reprocessing of the disk emission by outflowing gas (Guillochon et al. 2014). We therefore present basic scaling relations for the durations and luminosities of TDEs.

Assuming that the accretion rate onto the SMBH is less than the Eddington limit, the peak bolometric luminosity of the transient scales as (Stone 2013)

$$\begin{eqnarray}&&{L}_{\mathrm{peak}}\propto \dot{M}\propto {M}_{\mathrm{BH}}^{-1/2}{M}_{* }^{2}{R}_{* }^{-3/2},\end{eqnarray} \tag{ 18 }$$

where $\dot{M}$ is the peak accretion rate of the disrupted star calculated at the tidal radius, M_BH is the black hole mass, and M_* and R_* are the star's mass and radius, respectively. The peak accretion rate is typically near the SMBH Eddington accretion rate, which leads to a plateau at the corresponding Eddington luminosity. Super-Eddington accretion will likely lead to an outflow of material (see, e.g., Alexander et al. 2016).

There are three timescales that potentially affect the transient duration: the diffusion time (t_d), the viscous time (t_ν), and the timescale of peak fallback accretion (t_peak). In most cases, the diffusion timescale is small relative to at least one of the other two (Guillochon et al. 2009; Guillochon & Ramirez-Ruiz 2013). Assuming a low disk viscosity, the duration of the transient will be proportional to (Lodato 2012)

$$\begin{eqnarray}&&{t}_{\mathrm{dur}}\propto {t}_{\mathrm{peak}}\propto {M}_{\mathrm{BH}}^{1/2}{M}_{* }^{-1}{R}_{* }^{3/2}.\end{eqnarray} \tag{ 19 }$$

For canonical parameters (a Sun-like star and ${M}_{\mathrm{BH}}={10}^{6}$ ${M}_{\odot }$), this duration is about 40 days.

If the accretion rate is near-Eddington, the light curve will plateau for a duration roughly corresponding to (Stone 2013)

$$\begin{eqnarray}&&{t}_{\mathrm{dur}}\propto {t}_{\mathrm{edd}}\propto {M}_{\mathrm{BH}}^{-2/5}{M}_{* }^{1/5}{R}_{* }^{3/5}.\end{eqnarray} \tag{ 20 }$$

For canonical parameters, this corresponds to a duration of about 750 days.

Lost in these scaling relations is the fact that more massive black holes cannot disrupt less massive stars, because the tidal radius will be inside the horizon. The limiting SMBH mass (i.e., the Hills mass, M_H) ${M}_{{\rm{H}}}=(1.1\times {10}^{8}{M}_{\odot }){R}_{* }^{3/2}{M}_{* }^{-1/2}$ (Hills 1988) is proportional to ${M}_{* }^{0.7}$, assuming ${R}_{* }\propto {M}_{* }^{0.8}$ for main-sequence stars (Demircan & Kahraman 1991).

The scaling relations in Equations (18) and (19), along with a sample of TDEs from the literature, are shown in Figure 15. The majority of these transients follow the scaling relation with black hole mass, with the notable exception of extremely luminous ASASSN-15lh (Dong et al. 2016). A rapid spin rate and large black hole masses were necessary to explain the unique optical light curve of this claimed TDE (Leloudas et al. 2016; Margutti et al. 2016; van Velzen 2017). From the sample of observed objects and the above scaling relations, it is clear that TDEs are not expected to produce short-duration ($\lesssim 20$ days) transients.

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In this section we enumerate additional types of transients that are either observed in small numbers or only hypothesized to exist, but whose physical models we do not explore in detail.

3.9.1. Accretion-induced Collapse

As an accreting WD approaches the Chandrasekhar limit, it can collapse into a neutron star (Bailyn & Grindlay 1990; Nomoto & Kondo 1991; Fryer et al. 1999) with a rotationally supported disk with mass ${M}_{\mathrm{disk}}\lesssim 0.1$ ${M}_{\odot }$ (Dessart et al. 2006). The disk will then accrete onto the neutron star and eventually unbind as free nucleons recombine to form He. The radioactive heating of this ejecta is predicted to produce a fast (${t}_{\mathrm{dur}}\sim 1$ day) and dim (${M}_{{\rm{R}}}\sim -13.5$ mag) optical transient (Metzger et al. 2009; Darbha et al. 2010). These are somewhat dimmer and of shorter duration compared to those of the ⁵⁶Ni models explored in Section 3.2. We plot several models from Darbha et al. (2010) in Figure 16.

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3.9.2. Sub-Chandrasekhar Models/.Ia SNe

3.9.3. Ca-rich Transients

3.9.4. Electron-capture SNe

3.9.5. Luminous Red Novae

3.9.6. Classical Novae

Novae have a rich observational history owing to their high observed rate and utility as standardizable candles (Della Valle & Livio 1995). They occur when H-rich matter accretes onto a WD from a binary companion and the surface undergoes thermonuclear ignition (Gallagher & Starrfield 1978). We use the empirical maximum magnitude relation with decline time (MMRD) to place classical novae in the DLPS diagram. The MMRD relates the V-band peak magnitude with the decline time, t₂ (t₃), or the time to dim by two (three) magnitudes from peak. We approximate the duration as twice the time is takes to fall by one magnitude (2t₁). However, t₁ is not often reported in studies of the MMRD and can be much faster than the naive assumption of ${t}_{2}/2$ or ${t}_{3}/3$. We approximate t₁ by assuming that ${t}_{3}\mbox{--}{t}_{2}={t}_{2}\mbox{--}{t}_{1}$, or that the light curve decays linearly between t₃ and t₂. We use the relation from Capaccioli et al. (1990) to transform between t₂ and t₃ and solve our above equation for t₁:

$$\begin{eqnarray}&&{t}_{1}=0.31{t}_{2}-1.9\,\mathrm{days}.\end{eqnarray} \tag{ 21 }$$

We then use the MMRD relation measured by Della Valle & Livio (1995) to estimate the peak magnitude:

$$\begin{eqnarray}&&{M}_{{\rm{R}}}=-7.92-0.81\arctan \displaystyle \frac{1.32-\mathrm{log}{t}_{2}}{0.23}.\end{eqnarray} \tag{ 22 }$$

Nova R-band light curves tend to be brighter and of longer duration than V-band ones (Cao et al. 2012), but we do not make an explicit correction for this. The modified MMRD relation described above is shown in Figure 16.

3.9.7. Other Theoretical Merger Models

Until now we have assumed that each transient class is powered by a single energy source. In reality, we expect SN-like explosions to have multiple heating sources. We specifically expect newly synthesized ⁵⁶Ni within SN ejecta.

In this section, we consider light curves generated from a combination of two power sources: ⁵⁶Ni decay and magnetar spin-down. We generate these light curves by adding the input luminosities from both contributions and diffuse the input luminosity through the expanding ejecta using MosFIT.

Using the same parameter distributions as in Sections 3.2 and 3.4, we generate R-band light curves for the combined power sources and show these distributions in Figure 17. The joint DLPS generally overlaps with the distribution of solely magnetar-powered transients, although the low-luminosity (${M}_{{\rm{R}}}\gtrsim -16$ mag) transients are missing, since in these cases the heat input from ⁵⁶Ni decay dominates over the magnetar heating.

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We separate the models based on the dominant (contributing $\geqslant 50$%) heat source at peak (R-band) luminosity and find that almost all transients brighter than ${M}_{{\rm{R}}}\sim -19.5$ mag are dominated by magnetar spin-down; conversely, all transients fainter than this value are dominated by ⁵⁶Ni. The transition between dominating power sources is mainly controlled by the magnetic field of the magnetar. All models dominated by ⁵⁶Ni have B ≲ 5 × 10¹³ G. At lower luminosities, the presence of a newly formed magnetar will not be apparent photometrically.

We next explore combined ⁵⁶Ni decay and ejecta–CSM interaction. The diffusion processes for these two models are different: the input luminosity from the ⁵⁶Ni decay diffuses through the ejecta and optically thick CSM (${M}_{\mathrm{ej}}+{M}_{\mathrm{CSM},\mathrm{th}}$), while the ejecta–CSM input luminosity diffuses through ${M}_{\mathrm{CSM},\mathrm{th}}$. We assume that these two components evolve independently and add together their final luminosities.

Typical light curves for the case of s = 2 (wind) and their distribution in the DLPS are shown in Figure 18. As in the case of combined ⁵⁶Ni decay and magnetar spin-down, no transients brighter than $\sim -19.5$ mag are dominated by the ⁵⁶Ni input luminosity at peak. Unlike the transients solely powered by ejecta–CSM interactions, we find no transients with durations $\lesssim 10$ days and no transients with ${M}_{{\rm{R}}}\gtrsim -14$ mag, because in such cases the timescale and luminosity are determined by radioactive heating.

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The dominance of CSM interaction over ⁵⁶Ni is mainly controlled by the mass-loss rate, $\dot{M}$. Assuming ${v}_{\mathrm{wind}}\sim 100$ km s⁻¹, CSM interaction dominates when $\dot{M}\gtrsim {10}^{-3}$ ${M}_{\odot }$ yr⁻¹. This is consistent with LBV mass-loss rates (Smith 2014).

In this section, we enumerate a number of insights that can be gained from the preceding analysis of the DLPS. We focus on the overlap of our predicted models with the observed populations and the regions of phase space that these models occupy.

We begin with the adiabatic expansion models that lack any internal heating source. The largest progenitors (RSG-like) can produce luminous (reaching ${M}_{{\rm{R}}}\sim -18$ mag) transients on timescales similar to those of SNe (${t}_{\mathrm{dur}}\sim 20\mbox{--}100$ days). In fact, the RSG models span a similar range of peak magnitudes to the Type IIP/L models. The BSG-like subclass lies within the luminosity gap (${M}_{{\rm{R}}}\sim -10$ to −15 mag) with durations similar to those of SNe (${t}_{\mathrm{dur}}\sim 20$ days), and the WD-like subclass has nova-like luminosities (${M}_{{\rm{R}}}\sim -7$ to −11 mag) and much shorter durations (${t}_{\mathrm{dur}}\sim 1$ day). In reality, these models will likely be paired with some radioactive heating or an additional heating source and therefore represent lower limits in both duration and luminosity. We can see from these models that for massive progenitors we expect transients that last $\gtrsim 10$ days. In contrast, compact object (or stripped) progenitors can reach the extreme limits of this phase space and generate faster transients, but only at low luminosities.

Next, we discuss our simulated DLPS of ⁵⁶Ni-powered transients. In the simulated and observed populations of SNe Ib/c, there is a dearth of short-duration (${t}_{\mathrm{dur}}\lesssim 20$ days) and long-duration (${t}_{\mathrm{dur}}\gtrsim 80$ days) transients. These timescales correspond to transients with small ejecta masses/high velocities and large ejecta masses/low velocities, respectively. From the literature, we find only two well-observed SNe with durations $\gtrsim 70$ days (also shown in Figure 3): iPTF15dtg (Taddia et al. 2016) and SN 2011bm (Valenti et al. 2012). Both objects require large ⁵⁶Ni and ejecta masses to explain their extended light curves, suggesting intrinsically rare massive progenitors (Valenti et al. 2012; Taddia et al. 2016). Longer-duration transients are seen in the PISN models with larger ejecta masses and kinetic energies. However, the low-metallicity progenitors of PISNe are expected to be found at high redshift, meaning that observed PISNe are likely to be even slower (time dilated by $1+z$) and redder. Shorter-duration transients are seen in the Iax-like models owing to their lower ejecta masses, although very few of these transients (observed or simulated) have ${t}_{\mathrm{dur}}\lesssim 10$ days.

When considering the radioactive decay of ⁵⁶Ni as a heating source for short-duration transients, it is important to note that it is largely the ratio of the ejecta mass to the nickel mass that limits the light-curve parameters. In reasonable physical models, it is unlikely that ${f}_{\mathrm{Ni}}\gtrsim 0.5$ (although a few SNe Ia with higher nickel fractions have been observed; e.g., Childress et al. 2015). This means that, regardless of the amount of ⁵⁶Ni within the ejecta, the timescale of the transient will typically be set by M_ej (and other factors). This fact—the low ⁵⁶Ni fraction in physical models—essentially eliminates luminous, short-duration transients powered by ⁵⁶Ni. For example, this is why ⁵⁶Ni fails as the main power source for SLSNe, which have relatively short durations given their high luminosities (Kasen & Bildsten 2010; Nicholl et al. 2013).

The kilonova models, powered by r-process decay, lie in a unique area of the DLPS, with short durations (${t}_{\mathrm{dur}}\,\sim$ few days) and low luminosities (${M}_{{\rm{R}}}\sim -8$ to −16 mag). Their short durations coupled with low luminosities follow the general trend seen in the stripped SN models. Although there is currently large uncertainty in the opacity of Lanthanide-rich ejecta (Barnes et al. 2016), all of the models are below typical SN luminosities and durations. Our red models in particular span to even dimmer events than those explored in Barnes & Kasen (2013) and Metzger (2016), consistent with the recent result by Wollaeger et al. (2017). We additionally note that a brighter and longer-lived, magnetar-powered kilonova has been recently proposed (Yu et al. 2013; Metzger & Piro 2014; Siegel & Ciolfi 2016), which was not explored in this work. Such a kilonova could peak at $\sim {10}^{44}\mbox{--}{10}^{45}$ erg s⁻¹, with a duration of several days, although it would represent a small fraction of the kilonova population.

We next examine the magnetar models explored in Section 3.4. We find that the models span a broad range in both duration (${t}_{\mathrm{dur}}\sim 20\mbox{--}250$ days) and luminosity (${M}_{{\rm{R}}}\sim -16$ to −23 mag). Our models reproduce both the detailed theoretical predictions and observed light curves of SLSNe I. However, the SLSN I light curves span a narrower range of the DLPS, primarily at the bright end. This indicates that at least those magnetar-powered events have a narrower range of parameters than explored in this work, as suggested recently by Nicholl et al. (2017b). Transients that have weak contributions from the magnetar's spin-down are likely dominated by ⁵⁶Ni decay (as discussed in Section 3.10) and are classified as normal SNe Ib/c. We also note that several Type I SLSNe have been accompanied with early-time bumps with several-day durations and SN-like luminosities (Leloudas et al. 2012; Nicholl & Smartt 2016), which were not explored in this paper. The origin of these bumps is currently unknown, although several theoretical explanations have been posed (e.g., Kasen et al. 2016; Margalit et al. 2017).

Our ejecta–CSM interaction models span the widest range of the DLPS of the models presented here, due to both a large number of free parameters (which may not be independent, as assumed) and the simplifying assumptions used (Chatzopoulos et al. 2012). One of the most striking features is the difference between the wind-like and shell-like CSM geometries, with shell-like models producing brighter transients with somewhat shorter durations (${t}_{\mathrm{dur}}\sim 100$ days for wind-like vs. ${t}_{\mathrm{dur}}\sim 50$ days for shell-like). Although wind-like models can reproduce both low- and high-luminosity SNe IIn, the shell-like models with SN-like ejecta masses and kinetic energies do not extend to the luminosities of normal SNe IIn. Focusing on the wind-like models, we find that luminosity and duration are positively correlated at shorter (${t}_{\mathrm{dur}}\lesssim 20$ days) durations, with no models brighter than ${M}_{{\rm{R}}}\sim -14$ mag in this regime.

We note that Chatzopoulos et al. (2013) find that, when fitting SLSNe with the semianalytical model used in this work, both s = 0 and s = 2 can generally be used to find acceptable fits, but the models lead to substantially different explosion parameters. For the normal SNe IIn, Moriya et al. (2014) estimated the CSM profile (s) from the post-peak light curves of 11 SNe IIn and found that most showed $s\sim 2$. This implies that the shell model (s = 0) is less physical for at least the Type IIn events.

The heterogeneous group of transients discussed in Section 3.9.5 span a broad range of the DLPS, but their phase space is not particularly unique. Many of the models with likely compact object progenitors (e.g., Ca-rich transients and sub-Chandrasekar models) are confined to a small area similar to the Iax-like models we explored in Section 3.2.4. The electron-capture SN models overlap with the SNe Ib/c and IIP, and the LRNe are broadly consistent with the CSM interaction outburst-like models. Only the accretion-induced collapse models and classical novae extend to novel regions of the DLPS at ${t}_{\mathrm{dur}}\lesssim 10$ day durations, but invariably with low luminosities (${M}_{{\rm{R}}}\gtrsim -14$ mag).

Finally, we focus on our combined models with radioactive decay coupled to either magnetar spin-down or ejecta–CSM interaction. In the ejecta–CSM interaction case, the addition of ⁵⁶Ni decay eliminates both short-duration (${t}_{\mathrm{dur}}\lesssim 10$ days) and dim (${M}_{{\rm{R}}}\gtrsim -14$ mag) transients that are otherwise produced by this model. The former is due to the fact that the decay of ⁵⁶Ni dominates the CSM interaction light curves, eliminating the artificial cutoffs to the input luminosities. Additionally, there is a clear separation of transients that are dominated by ⁵⁶Ni decay or CSM interaction/magnetar spin-down in the DLPS around ${M}_{{\rm{R}}}\sim -19.5$ mag. In the ejecta–CSM interaction case, this separation roughly coincides with where the estimated mass-loss rate of the progenitor star roughly matches typical LBV mass-loss rates ($\dot{M}\sim {10}^{-3}$ ${M}_{\odot }$ yr⁻¹; Smith 2014), and where many identified SNe IIn lie. In the case of magnetar spin-down, the separation occurs at B ∼ 5 × 10¹³, about the cutoff for expected magnetar magnetic field strengths (Zhang & Harding 2000).

In this section, we summarize the overarching results from the DLPS analysis and highlight several regions of interest. To begin, we present all of the classes simulated in this paper (excluding TDEs and the "other transients" in Section 3.9) in Figure 19. An interactive version of this plot, which can be used to compare user-uploaded transients to the complete DLPS, is available online.⁵

There is substantial overlap among models, especially in the range ${t}_{\mathrm{dur}}\sim 10\mbox{--}100$ days and ${M}_{{\rm{R}}}\sim -18$ to −20 mag. Interestingly, this is also where most observed transients lie in the DLPS. A wide range of explosion physics and internal heating sources lead to similar optical light-curve properties, in part due to similar kinetic energies and ejecta masses. This highly populated regime highlights the fact that the abundance of observed transients with ∼month-long durations and SN-like luminosities is likely due not to observational biases but to a reflection of the underlying physics. But what about the more extreme areas of the DLPS?

We begin by focusing on fast (${t}_{\mathrm{dur}}\lesssim 10$ days) and bright (${M}_{{\rm{R}}}\lesssim -18$ mag) transients within our explored models. We find essentially no models that can produce transients in this regime, with the exception of on-axis GRB afterglows. Luminous and fast optical transients cannot be powered by radioactivity, magnetar spin-down, or CSM interaction; however, they can be powered by relativistic outflows. Relativistic sources (with ${\rm{\Gamma }}\gtrsim$ a few) like GRBs are rare compared to other optical transients. For example, the GRB volumetric rate at $z\lesssim 0.5$ is only 0.1% of the core-collapse SN rate (Dahlen et al. 2004; Wanderman & Piran 2010). Given this low rate and current lack of other physically motivated models, we argue that this portion of the DLPS is, and will continue to be, sparsely populated as a result of intrinsically rare physics.

A number of heating sources can produce transients that are fast (${t}_{\mathrm{dur}}\lesssim 10$ days) but invariably dim (${M}_{{\rm{R}}}\gtrsim -14$ mag), including novae, adiabatic explosions of WDs, r-process kilonovae, and CSM interaction models. However, most of these models require unique combinations of parameters, mainly very low ejecta masses, and represent a small fraction of the DLPS explored. Therefore, short-duration transients seem intrinsically rare, even at lower luminosities.

At the other extreme, we find several models that can produce exceptionally luminous transients (${M}_{{\rm{R}}}\lesssim -22$ mag), including ⁵⁶Ni decay (in the context of PISNe), magnetar spin-down, GRB afterglows, and ejecta–CSM interactions. TDEs may also reach these high luminosities (as seen in the case of ASASSN-15lh; Margutti et al. 2016). All of these models require extreme parameters to reach such bright luminosities, implying that such events are intrinsically rare. However, these luminous transients are invariably of long duration (${t}_{\mathrm{dur}}\gtrsim 50$ days).

The dimmest transients (${M}_{{\rm{R}}}\gtrsim -14$ mag) are generated from adiabatic explosions of WDs, off-axis GRB afterglows, outburst-like ILOTs, classical novae, and r-process kilonovae, with a broad range of durations $({t}_{\mathrm{dur}}\sim 1\mbox{--}300$ days). Of these, few lie in the intermediate-luminosity gap between the brightest classical novae and dimmest SNe (${M}_{{\rm{R}}}\sim -10$ to −14 mag). Due to the low rates of GRB afterglows and kilonovae, the most commonly discovered class in this gap will likely be powered by CSM interaction in the context of massive star eruptions (rather than explosions) as inferred for the small sample of known ILOTs (e.g., Kochanek et al. 2012).

To summarize, we find three sparse regimes of the DLPS: (i) bright and fast transients (${t}_{\mathrm{dur}}\lesssim 10$ days and ${M}_{{\rm{R}}}\lesssim -16$ mag), (ii) intermediate-luminosity transients (${M}_{{\rm{R}}}\approx -10$ to −14 mag) across all durations, and (iii) luminous transients (${M}_{{\rm{R}}}\lesssim -21$ mag). Of these, the most sparsely occupied by theoretical models is the first. On the other hand, the typical parameter ranges for SNe (i.e., ${t}_{\mathrm{dur}}\sim 10\mbox{--}100$ days and ${M}_{{\rm{R}}}\sim -18$ to −20 mag) contain a number of overlapping models, consistent with the fact that most observed optical transients lie within this regime.

Until now we have investigated theoretical models of transients that occupy the DLPS. The observed DLPS of transients will be modified by each class's volumetric rate and luminosity function (which we will explore in a follow-up paper). In this section, we will consider the effects of a given survey's parameters (cadence and area) on the observed DLPS. We perform a simple calculation to explore the effect of a transient's luminosity and duration on its survey discovery potential, or its relative discovery rate assuming a constant volumetric rate (${ \mathcal R }$) for every transient, in a flat cosmology (${{\rm{\Omega }}}_{{\rm{M}}}=0.3;$ ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.7;$ ${H}_{0}=70$ km s⁻¹ Mpc⁻¹).

The number (${ \mathcal N }$) of transients of a certain luminosity and duration discovered in a given magnitude-limited survey is proportional to

$$\begin{eqnarray}&&{ \mathcal N }\propto {\int }_{z=0}^{z={z}_{\mathrm{lim}}}\epsilon { \mathcal R }{dV},\end{eqnarray} \tag{ 23 }$$

where z_lim is the redshift where the apparent magnitude of the transient is equal to the limiting magnitude of the survey. The parameter represents a detection efficiency of the survey, defined by our heuristic equation:

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{1}{1+{e}^{-({t}_{\mathrm{dur}}(1+z)-{N}_{{\rm{D}}}{t}_{\mathrm{cad}})}},\end{eqnarray} \tag{ 24 }$$

where t_dur is the transient duration in its rest frame, t_cad is the survey cadence, and N_D is a penalty term to simulate the need for multiple data points to "detect" a transient; here we choose ${N}_{{\rm{D}}}=3$ for illustrative purposes. The chosen efficiency function goes to one when ${t}_{\mathrm{dur}}\gg {t}_{\mathrm{cad}}$ and to zero when ${t}_{\mathrm{dur}}\ll {t}_{\mathrm{cad}}$. When ${t}_{\mathrm{dur}}={N}_{{\rm{D}}}{t}_{\mathrm{cad}}$, the efficiency is 0.5.

In Figure 20, we assume a limiting R-band limiting magnitude of 24.5 (matched to LSST) and calculate ${ \mathcal N }$ for a given transient's absolute magnitude and duration assuming a constant volumetric rate and ignoring any k-corrections. We find that the expected detection rates of transients drop off exponentially with decreasing luminosity, as well as with shorter duration as it approaches the survey cadence. Specifically, this simple example demonstrates the fact that, even with a relatively high cadence, a wide-field survey will detect 100–1000 times more SN-like transients (${M}_{{\rm{R}}}\lesssim -18$ mag) compared to ILOTs (${M}_{{\rm{R}}}\sim -10$ to −14 mag). Similarly, a survey with a cadence of several ($\approx 3$) days will detect 10–100 times more transients with SN-like durations ($\sim 20\mbox{--}30$ days) compared to transients with short durations ($\lesssim 10$ days). To counter these facts, one could design a survey with a faster survey cadence, but there is a trade-off between a survey's cadence, depth, and coverage area. A high survey cadence requires a much smaller coverage area, even with a large field of view. A more efficient approach to search for dim transients may be a targeted survey of nearby galaxies.

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Bringing together the above conclusions with those of the previous section, we conclude that quickly evolving transients are invariably dim. Both characteristics lead to diminishing survey potential and therefore lower observed rates. In contrast, bright (${M}_{{\rm{R}}}\lesssim -22$ mag) transients tend to have longer durations and are therefore easier to observe. However, their volumetric rates are low and have not typically been found in large numbers in untargeted surveys. Between these two regimes, SN-like transients with relatively bright luminosities and intermediate durations typically have higher volumetric rates, allowing them to be some of the most commonly observed extragalactic phenomena in wide-field surveys.

If we assume that the shock deceleration is small, ${{dv}}_{\mathrm{sh}}/{dt}=0$, and that the geometric factors ${\beta }_{F}={\beta }_{R}=1$, we can recover the bolometric luminosity solution of Ofek et al. (2014b) from Equation (25):

$$\begin{eqnarray}&&L=\epsilon \displaystyle \frac{{{dE}}_{\mathrm{KE}}}{{dt}}=\epsilon \displaystyle \frac{\epsilon }{2}\displaystyle \frac{{{dM}}_{\mathrm{sw}}}{{dt}}{v}_{\mathrm{sh}}=2\pi \epsilon {\rho }_{\mathrm{CSM}}({r}_{\mathrm{sh}}){r}_{\mathrm{sh}}^{2}{v}_{\mathrm{sh}}^{3}.\end{eqnarray} \tag{ 29 }$$

Furthermore, we assume that this shock efficiently diffuses through the CSM and has an effective diffusion time ${t}_{{\rm{d}}}=0$ (i.e., the input luminosity is equal to the observed luminosity; Ofek et al. 2014b).

The temperature can then be estimated as (Chevalier & Irwin 2011; Ofek et al. 2014b)

$$\begin{eqnarray}&&T(t)={\left(\displaystyle \frac{18}{7a}{\rho }_{\mathrm{CSM}}{v}_{\mathrm{fs}}^{2}\right)}^{1/4},\end{eqnarray} \tag{ 30 }$$

where a is the radiation constant.

A.1.1. Full CSM Interaction Solution

If we loosen the assumptions made to reproduce the light-curve solution from Ofek et al. (2014b), we will reproduce the generalized solution presented by Chatzopoulos et al. (2012). To do this, we calculate the contributions to the total luminosity from both the forward and reverse shocks and diffuse this input luminosity through the CSM.

We explore both shell-like and wind-like CSM profiles (s = 0 and s = 2, respectively) and leave the inner radius of the CSM as a free parameter (R₀). By also allowing the total mass of the CSM to be a free parameter (${M}_{\mathrm{CSM}}$), we can define the total radius of the CSM as

$$\begin{eqnarray}&&{R}_{\mathrm{CSM}}={\left(\displaystyle \frac{3{M}_{\mathrm{CSM}}}{4\pi q}+{R}_{0}^{3}\right)}^{\tfrac{1}{3}}.\end{eqnarray} \tag{ 31 }$$

We can further define the photospheric radius as

$$\begin{eqnarray}&&{R}_{\mathrm{ph}}=\displaystyle \frac{2\kappa q}{3}+{R}_{\mathrm{CSM}},\end{eqnarray} \tag{ 32 }$$

where $\kappa =0.34$ g cm⁻³. R_CSM and R_ph will be important for setting physical constraints on our generated models.

The input luminosity arises from the conversion of the forward and reverse shocks' kinetic energy into radiation, which can be described as

$$\begin{eqnarray}{L}_{\mathrm{in}}(t) & = & \displaystyle \frac{2\pi \epsilon }{{(n-s)}^{3}}{g}^{n\tfrac{5-s}{n-s}}{q}^{\tfrac{n-5}{n-s}}{(n-3)}^{2}\\ & & \times (n-5){\beta }_{F}^{5-s}{A}^{\tfrac{5-s}{n-s}}{t}^{\tfrac{2n+6s-{ns}-15}{n-s}}\theta ({t}_{\mathrm{FS}}-t)\\ & & +2\pi \epsilon {\left(\displaystyle \frac{{{Ag}}^{n}}{q}\right)}^{\tfrac{5-n}{n-s}}{\beta }_{R}^{5-n}{g}^{n}\\ & & \times {\left(\displaystyle \frac{3-s}{n-s}\right)}^{3}{t}^{\tfrac{2n+6s-{ns}-15}{n-s}}\theta ({t}_{\mathrm{RS}}-t).\end{eqnarray} \tag{ 33 }$$

θ is the Heaviside step function, which designates which components (the forward or reverse shocks) are contributing to the total luminosity based on the shock termination times:

$$\begin{eqnarray}&&{t}_{\mathrm{FS}}={\left|\displaystyle \frac{(3-s){q}^{\tfrac{3-n}{n-s}}{({{Ag}}^{n})}^{\tfrac{s-3}{n-s}}}{4\pi {\beta }_{F}^{3-s}}\right|}^{\displaystyle \frac{n-s}{(n-3)(3-s)}}{M}_{\mathrm{CSM}}^{\displaystyle \frac{n-s}{(n-3)(3-s)}},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{RS}}={\left(\left(\displaystyle \frac{{v}_{\mathrm{ph}}}{{\beta }_{R}}\right)\left({\displaystyle \frac{q}{{{Ag}}^{n}}}^{\displaystyle \frac{1}{n-s}}\right){\left(1-\displaystyle \frac{(3-n){M}_{\mathrm{ej}}}{4\pi {v}_{\mathrm{ph}}^{3-n}{g}^{n}}\right)}^{\tfrac{1}{3-n}}\right)}^{\displaystyle \frac{n-s}{s-3}}.\end{eqnarray} \tag{ 35 }$$