THE MOST LUMINOUS SUPERNOVAE

Motivated by the recent discovery of many ultra-luminous supernovae (ULSN), including, controversially, the extreme case of ASASSN-15lh (Dong et al. 2016; see also Brown 2015), the limits of several scenarios often invoked for their interpretation are considered. These include colliding shells, pair-instability supernovae (PISN), and newly born magnetars (e.g., Quimby et al. 2011; Gal-Yam 2012). Each of these energy sources will give different results when occurring in a stripped core of helium or carbon and oxygen (Type I) or a supergiant (Type II), and both cases are considered. All calculations of explosions and light curves use the one-dimensional implicit hydrodynamics code KEPLER (Weaver et al. 1978; Woosley et al. 2002) and employ presupernova models that have been previously published.

The more extreme case of "relativistic supernovae"-either supernovae with relativistic jets or the explosion of super-massive stars that collapse because of general relativistic instability (Fuller et al. 1986; Chen et al. 2014) is not considered here. These are rare events with their own distinguishing characteristics.

Any explosive energy that deposits before the ejecta significantly expands will suffer severe adiabatic degradation that will prevent the supernova from being particularly bright. An upper bound for prompt energy deposition in a purely neutrino-powered explosion is $\sim 2\mbox{--}3\times {10}^{51}\;\mathrm{erg}$ (Fryer & Kalogera 2001; Ugliano et al. 2012; Pejcha & Thompson 2015; Ertl et al. 2016), which is capable of explaining common supernovae (Sukhbold et al. 2016), but not the more luminous ones. In a red, or worse, blue supergiant, the expansion from an initial stellar radius of, at most, 10¹⁴ cm, to a few times 10¹⁵ cm, where recombination occurs, degrades the total electromagnetic energy available to $\lesssim {10}^{50}\;\mathrm{erg}$. Even in the most extreme hypothetical case, where a substantial fraction of a neutron star binding energy, ∼10⁵³ erg, deposits instantly, the light curve is limited to a peak brightness of approximately ${10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ (neglecting the very brief phase of shock break out).

This can be demonstrated analytically and numerically. Adopting the expression for plateau luminosity and duration from Popov (1993) and Kasen & Woosley (2009), as calibrated to numerical models by Sukhbold et al. (2016), SNe II have a luminosity on their plateaus of

$$\begin{eqnarray}&&{L}_{{\rm{p}}}=8.5\times {10}^{43}\;{R}_{0,500}^{2/3}{M}_{\mathrm{env},10}^{-1/2}{E}_{53}^{5/6}\;\mathrm{erg}\;{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$$

where ${R}_{\mathrm{0,500}}$ is the progenitor radius in 500 ${R}_{\odot }$, ${M}_{\mathrm{env},10}$ is the envelope mass in 10 ${M}_{\odot }$, and ${E}_{53}\lesssim 1$ is the prompt explosion energy in units of 10⁵³ erg. The approximate duration of the plateau, ignoring the effects of radioactivity, is given by

$$\begin{eqnarray}&&{\tau }_{{\rm{p}}}=41\;{E}_{53}^{-1/6}{M}_{10}^{1/2}{R}_{0,500}^{1/6}\;\mathrm{days}.\end{eqnarray} \tag{ 2 }$$

This plateau duration is significantly shorter than common supernovae due to the much higher energies considered.

These relations compare favorably with a model for a 15 ${M}_{\odot }$ explosion calculated with an assumed explosion energy of $0.5\times {10}^{53}\;\mathrm{erg}$ (Figure 1). Here the red supergiant presupernova stellar model from Woosley et al. (2007) had a radius of 830 ${R}_{\odot }$ and an envelope mass of 8.5 ${M}_{\odot }$. The estimated luminosity on the plateau from Equation (1) is $6.7\times {10}^{43}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ and the duration from Equation (2) is 46 days. The corresponding KEPLER model in Figure 1 had a duration of ∼45 days and a luminosity of $6.6\times {10}^{43}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ at day 25. The total energy emitted is approximately ${L}_{{\rm{p}}}{\tau }_{{\rm{p}}}$ or $3\times {10}^{50}{E}_{53}^{2/3}{R}_{0,500}^{5/6}$ erg.

Zoom In Zoom Out Reset image size

Similar limits apply to PISN in red supergiant progenitors. Again, maximum explosion energies are $\lesssim {10}^{53}$ erg (Heger & Woosley 2002). For the most extreme, rarest case, ${M}_{10}\approx 20$, ${R}_{\mathrm{0,500}}\approx 5$, and ${E}_{53}\approx 1$, Equations (1) and (2) imply plateau luminosities near $5\times {10}^{43}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ for about 200 days. These values are consistent with the KEPLER models given in Scannapieco et al. (2005). The total radiated energy is $1\times {10}^{51}$ erg. Most of the radioactivity decays during the plateau. Since the decay energy is substantially less than the explosion energy, the modification of the light curve during its bright plateau is not appreciable.

We conclude that ULSN must be energized by a power source that deposits its energy well after the original explosion. The known delayed energy sources are radioactivity, colliding winds, and pulsars.

The most prolific sources of ⁵⁶Ni are PISN. The rarest, most massive PISN produces, at most, 50 ${M}_{\odot }$ of the ⁵⁶Ni in an explosion with a final kinetic energy of $9\times {10}^{52}\;\mathrm{erg}$ (Heger & Woosley 2002). This large production only occurs for the most massive helium cores (∼130 ${M}_{\odot }$), which very nearly collapse to black holes. The total energy available from the decay of a large amount of ⁵⁶Ni is substantial,

$$\begin{eqnarray}&&{E}_{\mathrm{dec}}\;\approx 2.4\times {10}^{51}\;\left(\displaystyle \frac{{M}_{\mathrm{Ni}}}{50{M}_{\odot }}\right)(3{e}^{-{\rm{t}}/{\tau }_{\mathrm{Co}}}+{e}^{-{\rm{t}}/{\tau }_{\mathrm{Ni}}})\;\mathrm{erg},\end{eqnarray} \tag{ 3 }$$

where ${M}_{\mathrm{Ni}}$ is the ⁵⁶Ni mass in ${M}_{\odot }$, and ${\tau }_{\mathrm{Co}}=111\;{\rm{d}}$ and ${\tau }_{\mathrm{Ni}}=8.7\;{\rm{d}}$ are mean lives of ⁵⁶Co and ⁵⁶Ni. For a nickel mass of ∼50 ${M}_{\odot }$ the total energy is nearly 10⁵² erg. However, most of this energy is lost during the adiabatic expansion to peak.

For a star that has lost its hydrogen envelope, an approximate estimate when the PISN light curve peaks is given by equating the effective diffusion timescale, ${t}_{{\rm{d}}}$, to age. This gives a time of peak luminosity, ${t}_{{\rm{p}}}$, of

$$\begin{eqnarray}{t}_{{\rm{p}}} & = & {\left(\displaystyle \frac{3\kappa }{4\pi c}\right)}^{1/2}\;{\left(\displaystyle \frac{{M}_{\mathrm{ej}}^{3}}{2{E}_{\mathrm{exp}}}\right)}^{1/4}\sim 177\;{\left(\displaystyle \frac{{M}_{\mathrm{ej}}}{130{M}_{\odot }}\right)}^{3/4}\\ & & \times {\left(\displaystyle \frac{{E}_{\mathrm{exp}}}{{10}^{53}\mathrm{erg}}\right)}^{-1/4}\;\mathrm{days},\end{eqnarray} \tag{ 4 }$$

where ${M}_{\mathrm{ej}}$ is the ejecta mass in ${M}_{\odot }$ and ${E}_{\mathrm{exp}}$ is the explosion energy in erg. Considering the similarity of high velocity and iron-rich composition to SNe Ia, an opacity $\kappa \approx 0.1\;{\mathrm{cm}}^{2}\;{{\rm{g}}}^{-1}$ is assumed. Arnett's Rule (Arnett 1979) then implies a maximum luminosity of

$$\begin{eqnarray}&&{L}_{{\rm{p}}}\approx 8\times {10}^{44}\ \left(\displaystyle \frac{{M}_{\mathrm{Ni}}}{50{M}_{\odot }}\right){e}^{-{t}_{{\rm{p}}}/{\tau }_{\mathrm{Co}}}\;\mathrm{erg}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 5 }$$

Only the luminosity due to the decay of ⁵⁶Co is included here, since for t $\sim {t}_{{\rm{p}}}$, most of the ⁵⁶Ni will have already decayed. For the fiducial values of ${t}_{{\rm{p}}}$ and ${M}_{\mathrm{Ni}}$, the peak luminosity is then roughly $1.5\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, which compares favorably with models in which the hydrodynamics and radiation transport are treated carefully (Kasen et al. 2011; Kozyreva & Blinnikov 2015) and with the analytic models of Chatzopoulos et al. (2013).

Assuming that the total emitted energy by an SN I is

$$\begin{eqnarray}&&{E}_{\mathrm{rad}}\approx \displaystyle \frac{1}{2}{L}_{{\rm{p}}}{t}_{{\rm{p}}}+{E}_{\mathrm{dec}}({t}_{{\rm{p}}}),\end{eqnarray} \tag{ 6 }$$

and using the fiducial values, the approximate upper bound on the total luminous energy in a PISN-Type I is $2.6\times {10}^{51}$ erg, about one-quarter of the total decay energy.

4.1. Generic Models

Observations show that a substantial fraction of ULSN, especially those of Type IIn, are brightened by circumstellar interaction (e.g., Kiewe et al. 2012). In some cases, this interaction can be extremely luminous (Smith et al. 2010, 2011a). The necessary mass loss is often attributed to prior outbursts of the star as a luminous blue variable (LBV; e.g., Smith et al. 2011b) or a pulsational-pair-instability supernova (PPISN, Woosley et al. 2007), though other possibilities, e.g., common envelope (Chevalier 2012a), are sometimes invoked.

The luminosity of colliding shells is limited by their differential speeds, their masses, and the radii at which they collide. If the collision happens at too small a radius where the ejecta is still very optically thick, colliding shells become another variant of "prompt explosions" (Section 2). On the other hand, if the collision happens at too large a radius, the resulting transient has a longer timescale, lower luminosity, and may not emit chiefly in the optical (Chevalier & Irwin 2012b). In practice, these constraints limit the radius where the shells collide and produce a bright optical transient to roughly 10¹⁵–10¹⁶ cm. A similar range of radii is obtained by multiplying typical collision speeds, ∼5000 km s⁻¹, by the duration of a ULSN, ∼100 days.

Chevalier & Irwin (2012b) give a maximal "cooling luminosity" for colliding shells in which most of the dissipated energy goes into light (see also Equation (1) of Smith et al. 2010),

$$\begin{eqnarray}&&L\lesssim \;2\pi {r}^{2}\rho {v}_{\mathrm{shock}}^{3}=0.5\displaystyle \frac{\dot{M}}{{v}_{\mathrm{wind}}}{v}_{\mathrm{shock}}^{3}\end{eqnarray} \tag{ 7 }$$

where $\dot{M}$ is the pre-explosive mass-loss rate with speed ${v}_{\mathrm{wind}}$, and ${v}_{\mathrm{shock}}$ is the shock speed of the explosive ejecta impacting that "wind." Narrow lines in the spectra of SNe IIn, including some very luminous ones (Kiewe et al. 2012), imply pre-explosion wind speeds of a few hundred to 1000 km s⁻¹. At those speeds, and given that the light curve is generated at $r\sim {10}^{15}\mbox{--}{10}^{16}\;\mathrm{cm}$, the relevant time for the mass loss is a few years before the final explosion. The velocity of the shock is ${v}_{\mathrm{shock}}\approx \sqrt{2E/M}$, where E is the explosion energy of mass M. Here we normalize it to 10⁹ cm s⁻¹ as in Chevalier & Irwin (2012b), though it implies a very energetic explosion. The luminosity from the collision is then

$$\begin{eqnarray}&&L\approx 3.1\times {10}^{44}\displaystyle \frac{{\dot{M}}_{-1}}{{v}_{\mathrm{wind},7}}{v}_{\mathrm{shock},9}^{3}\;\mathrm{erg}\;{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 8 }$$

where ${\dot{M}}_{-1}$ is the mass-loss rate a few years before the explosion normalized to 0.1 ${M}_{\odot }$ per year, ${v}_{\mathrm{wind},7}$ is in ${10}^{2}\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${v}_{\mathrm{shock},9}$ is in units of ${10}^{4}\;\mathrm{km}\;{{\rm{s}}}^{-1}$. Typical values for the outbursts that produce very bright SNe IIn are ${\dot{M}}_{-1}=1$, ${v}_{\mathrm{wind},7}=1$ to 10, and ${v}_{\mathrm{shock},9}=0.5$ (Kiewe et al. 2012) implying peak luminosities near $5\times {10}^{43}$ erg s⁻¹. The large ejection in η-Carina in the 1840s ejected 12 ${M}_{\odot }$ moving at ${v}_{{\rm{wind,7}}}$ up to 6.5 (Smith 2008).

It is the mass of the shell into which a supernova of given energy plows that matters most (van Marle et al. 2010). We are unaware of any models other than PPISN (Section 4.2) or 10 ${M}_{\odot }$ stars (Woosley & Heger 2015a) that eject solar masses of material just years before dying. If a generous upper limit of 10 ${M}_{\odot }$ between 10¹⁵ and 10¹⁶ cm (M₁₀ = 1) is adopted, ${\dot{M}}_{-1}/{v}_{\mathrm{wind},7}$ is $\lesssim \;3\;{M}_{10}\;{R}_{16}^{-1}$, where R₁₆ is the outer edge of the interaction region in 10¹⁶ cm units. For a shock speed of ${v}_{\mathrm{shock},9}=0.5$ and an event duration of 100 days, ${R}_{16}\sim 0.5$. The maximum luminosity is then ${10}^{45}{v}_{\mathrm{shock},9}^{3}{M}_{10}{R}_{16}^{-1}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\approx 3\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$. More generally, the maximum luminosity is $L\ ={10}^{45}{\tau }_{{\rm{SN,100}}}^{-1}{M}_{10}{v}_{{\rm{shock,9}}}^{2}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, where ${\tau }_{{\rm{SN,100}}}$ is the duration of the brightest part of the light curve. This limit is sufficient to accommodate all ULSN that may be powered by collisions and is consistent with the theoretical results of van Marle et al. (2010).

It might be possible to raise this limit by invoking slightly greater shock speeds or shell masses, though the former requires extremely energetic supernovae. Accelerating a shell of 10 ${M}_{\odot }$ to 10⁹ cm s⁻¹ requires an explosion energy of at least 10⁵² erg and 100% conversion efficiency. This is considerably more than neutrinos can provide and already indicates a source that is, at heart, rotationally powered. However, it may be that having high mass-loss rates removes sufficient angular momentum to inhibit the formation of rapidly rotating iron cores. Even energetic PISN do not develop speeds of 10,000 km s⁻¹ in a significant part of their mass. Moreover, PISN are burning carbon radiatively in their centers during the last few years of their life and, except for PPISN (Section 4.2), experience no obvious instability that would lead to the impulsive ejection of 10 ${M}_{\odot }$.

With considerable uncertainty, we thus adopt an upper limit for colliding shells of $3\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ and a total radiated energy of ${\tau }_{\mathrm{SN}}L\sim 3\times {10}^{51}$ erg. For bare helium cores, which are not PPISN and clearly not LBVs, the values are likely to be much smaller because of the smaller shell masses, but existing models, do not allow a specific estimate.

4.2. Pulsational-pair Instability Supernovae

With some tuning, the energy deposited by a young magnetar in the ejecta of a supernova can significantly brighten its light curve (Maeda et al. 2007; Kasen & Bildsten 2010; Woosley 2010). The model has been successfully applied to numerous observations of Type Ic ULSN (e.g., Howell et al. 2013; Inserra et al. 2013; Nicholl et al. 2013) and magnetars seem to be a natural consequence of the collapse of rapidly rotating cores (e.g., Mösta et al. 2015).

The rotational kinetic energy of a magnetar with a period of ${P}_{\mathrm{ms}}=P/\mathrm{ms}$ is approximately ${E}_{{\rm{m}}}\approx 2\times {10}^{52}{P}_{\mathrm{ms}}^{-2}$ erg, where ${E}_{{\rm{m}},\mathrm{max}}\approx 4\times {10}^{52}$ erg is the rotational energy for an initial period of ∼0.7 ms. Usually this period is restricted to $\gt 1$ ms, because of rotational instabilities that lead to copious gravitational radiation. However, Metzger et al. (2015) have recently discussed the possibility that the limiting rotational kinetic energy could exceed 10⁵³ erg, depending on the neutron star mass and the equation of state and here we adopt that value as an upper bound. This energy reservoir can be tapped through vacuum dipole emission, which is approximately ${E}_{{\rm{m}}}/{t}_{{\rm{m}}}\approx {10}^{49}{B}_{15}^{2}{P}_{\mathrm{ms}}^{-4}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, where ${B}_{15}=B/{10}^{15}$ G is the dipole field strength at the equator, and ${t}_{{\rm{m}}}=2\times {10}^{3}{P}_{\mathrm{ms}}^{2}{B}_{15}^{-2}$ s is the magnetar spin-down timescale. A magnetic dipole moment B(10 km)³ is adopted, and an angle of $\pi /6$ between the magnetic and rotational axes has been assumed. Combining these relations, one obtains the temporal evolution of the rotational energy and magnetar luminosity as ${E}_{{\rm{m}}}(t)={E}_{{\rm{m}},0}{t}_{{\rm{m}}}/({t}_{{\rm{m}}}+t)$ and ${L}_{{\rm{m}}}(t)={E}_{{\rm{m}},0}{t}_{{\rm{m}}}/{({t}_{{\rm{m}}}+t)}^{2}$.

The peak luminosity can be estimated using the diffusion equation and ignoring the radiative losses in the first law of thermodynamics (Kasen & Bildsten 2010):

$$\begin{eqnarray}&&{L}_{{\rm{p}}}=\displaystyle \frac{{E}_{{\rm{m}},0}}{{t}_{{\rm{d}}}}\left[\xi \;\mathrm{ln}\left(1+\displaystyle \frac{1}{\xi }\right)-\displaystyle \frac{\xi }{1+\xi }\right],\end{eqnarray} \tag{ 9 }$$

where $\xi ={t}_{{\rm{m}}}/{t}_{{\rm{d}}}$ is the ratio of spin-down to effective diffusion timescales. The term inside square brackets has a maximum at $\xi \approx 1/2$, obtained by solving $d({L}_{{\rm{p}}}{t}_{{\rm{d}}}/{E}_{{\rm{m}},0})/d\xi =0$. This implies an optimal field strength for maximizing the peak luminosity that is

$$\begin{eqnarray}&&{B}_{15}{| }_{{L}_{{\rm{p}},\mathrm{max}}}\simeq 66{P}_{\mathrm{ms},0}{t}_{{\rm{d}}}^{-1/2}.\end{eqnarray} \tag{ 10 }$$

That is, for a given combination of ${P}_{\mathrm{ms},0}$ and ejecta parameters-${M}_{\mathrm{ej}},{E}_{\mathrm{sn}},\kappa$, the brightest possible peak luminosity is obtained for this field strength.

The maximum peak luminosity is then ${L}_{{\rm{p}},\mathrm{max}}\simeq {E}_{{\rm{m}}}/10{t}_{{\rm{m}}}$. For the limiting initial spin of ${P}_{\mathrm{ms},0}=0.7\;\mathrm{ms}$, the corresponding field strength is $B\approx 4\times {10}^{13}$ G (for $\kappa =0.1\;{\mathrm{cm}}^{2}\;{{\rm{g}}}^{-1}$, ${M}_{\mathrm{ej}}=3.5$ ${M}_{\odot }$, and ${E}_{\mathrm{SN}}=1.2\times {10}^{51}\;\mathrm{erg}$), and the limiting peak luminosity is ${L}_{{\rm{p}},\mathrm{max}}\approx 2\times {10}^{46}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$. Here ${E}_{{\rm{m}}}\gg {E}_{\mathrm{SN}}$; therefore, unless one invokes even lower $\kappa$, ${M}_{\mathrm{ej}}$ and much larger ${E}_{{\rm{m}}}$, any transient with brighter observed luminosity will be hard to explain by the magnetar model (Figure 2).

Zoom In Zoom Out Reset image size

Using a version of Arnett's Rule (e.g., Inserra et al. 2013), ${L}_{{\rm{m}}}({t}_{{\rm{p}}})={L}_{{\rm{p}}}$, the time for ${L}_{{\rm{p}}}$ is ${t}_{{\rm{p}}}={({E}_{{\rm{m}}}{t}_{{\rm{m}}}{L}_{{\rm{p}}}^{-1})}^{1/2}-{t}_{{\rm{m}}}$. For the maximal luminosity, the corresponding peak time is then ${t}_{{\rm{p}},\mathrm{max}}\simeq 2.2{t}_{{\rm{m}}}$. This can be used to estimate the limiting radiated energy in the same way as in Equation (6) to find that

$$\begin{eqnarray}&&{E}_{\mathrm{rad},\mathrm{max}}\simeq 0.4{E}_{{\rm{m}},0}.\end{eqnarray} \tag{ 11 }$$

Any observation with a total radiated energy of ${E}_{\mathrm{rad}}\gt 4\times {10}^{52}$ erg will be nearly impossible to explain by the magnetar model. A more conventional value and one that fits ASASSN-151 h (Figure 2), is 2 × 10⁵² erg. This is within a factor of two of the limiting magnetar kinetic energy inferred for gamma-ray bursts by Mazzali et al. (2014).

To illustrate these limits, a series of magnetar-powered models based upon exploding CO cores (from Sukhbold & Woosley 2014) was calculated to find a best fit to the light curve of ASASSN-15lh. In each case, soon after bounce, the magnetar deposited its energy in the inner ejecta at a rate given by the vacuum dipole spin-down rate. The top panel of Figure 2 shows the best fitting model, which employs a magnetar with an initial period of 0.7 ms and magnetic field strength of $2\times {10}^{13}$ G, illuminating the ejecta in the explosion of a 14 ${M}_{\odot }$ CO core (${M}_{\mathrm{ej}}\approx 11.2{M}_{\odot }$).

These magnetar parameters agree well with the previously published fits, but the ejecta masses are different. An ejecta mass of 15 ${M}_{\odot }$ was obtained by Dai et al. (2016) because they used simple semi-analytical models, which ignored all dynamical effects and deviated from hydrodynamic calculations most when ${E}_{{\rm{m}}}\gg {E}_{\mathrm{sn}}.$ An ejecta mass of only 3 ${M}_{\odot }$ was used in Metzger et al. (2015), as they applied the same simple semi-analytical model for the early release of the data spanning only ∼60 days. That fit would not work for the later data shown in Figure 2, and the ejecta would become optically thin at an early time. Bersten et al. (2016) limited their models to small He-cores (8 ${M}_{\odot }$), and their model does not fit the broad peak of ASASSN-15lh well.

Magnetars can also illuminate bright, long-lasting SNe II. The bottom panel of Figure 2, shows the same magnetar that was applied for the fit to ASASSN-15lh, now embedded inside the remnant of a 15 ${M}_{\odot }$ red supergiant progenitor. Because the ejecta mass is much larger, the light curve is fainter and much broader. The ejecta stays optically thick for nearly four months. Much like the radioactive decay extends the plateau duration by causing ionization, magnetar-deposited energy also significantly extends the optically thick period.

Table 1 summarizes the maximum luminosity and total luminous energy for the models considered. Given the various approximations made, the numbers are probably accurate to a factor of two in most cases except for shell "collisions" where definitive models are lacking (Section 4.1). In all but the magnetar-powered models, the peak luminosities are a few times 10⁴⁴ erg s⁻¹ and peak integrated powers are near 10⁵¹ erg. This is gratifying since most "superluminous supernovae" are within those bounds (e.g., Nicholl et al. 2015).

Table 1.  Limiting Peak Luminosities and Radiated Energies

	Model	${L}_{{\rm{p}}}\;(\mathrm{erg}\;{{\rm{s}}}^{-1})$	${E}_{\mathrm{rad}}\;(\mathrm{erg})$
Type II	prompt	1 × 1044	3 × 1050
	PISN	5 × 1043	1 × 1051
	collisions	3 × 1044	3 × 1051
	PPISN	1 × 1044	1 × 1051
	magnetar	1 × 1045	9 × 1051
Type I	PISN	2 × 1044	3 × 1051
	PPISN	3 × 1043	1 × 1051
	magnetar	2 × 1046	4 × 1052

Download table as:  ASCIITypeset image

For point-like explosions, which include PISN of Type II, the prompt energy injection is typically degraded by a factor of ∼100 by adiabatic expansion, so even obtaining 10⁵¹ erg of light requires an explosion that strains the limits of both neutron star binding energy (core-collapse supernovae) and thermonuclear energy (PISN). The upper bound for PISN-Type I is also well determined by both analytic scaling rules and numerical models.

For supernovae whose light comes from colliding shells, the constraints are less accurate due to lack of knowledge about the masses of the shells involved and the supernova explosion energies in cases where large impulsive mass loss occurs just before the star dies. The limit in the table assumes shock speeds less than 5000 km s⁻¹ and shell masses less than 10 ${M}_{\odot }$. Estimates for PPISN are more precise because the mass of the helium core needed to make luminous optical supernovae is highly constrained. In order for the duration of the pulses to be years and not months or centuries, the helium-core mass must be in the range of 50–55 ${M}_{\odot }$ and that restricts the energy of the secondary pulses and supernova.

Magnetars are a special case. The limits come from using a simple dipole formula in a situation where it has not been observationally tested and assuming what some would regard as a high limiting rotational energy for neutron stars. Rotation can tap an energy reservoir almost as great as the binding energy of the neutron star and deposit it over an arbitrarily long timescale—depending on the choice of magnetic field strength. Thus the optical efficiency for converting rotational energy to light can be (forced to be) very high.

It is interesting though that the upper bounds for magnetar-powered light curves are so high. This implies a possible observable diagnostic. Supernovae that substantially exceed $3\times {10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ for an extended period and that have total luminous powers far above $3\times {10}^{51}$ erg should be considered strong candidates for containing an embedded magnetar. Similarly, "supernovae" that exceed the generous limits for magnetar power given in Table 1 may not be supernovae at all.

ASASSN-15lh (Dong et al. 2016) is an interesting case in this regard. Figure 2 shows that it can, barely, be accommodated by a magnetar model and Table 1 says it must be a magnetar, if it is a supernova (Brown 2015).

We thank Alex Heger for his contributions in developing the KEPLER code and Chris Kochanek, Iair Arcavi, Matt Nicholl, and Todd Thompson for useful comments. This work was supported by NASA (NNX14AH34G). We also acknowledge a NASA contract supporting the "WFIRST Extragalactic Potential Observations (EXPO) Science Investigation Team" (15-WFIRST15-0004), administered by the GSFC.