Models for the Unusual Supernova iPTF14hls

Supernova iPTF14hls was discovered by the iPTF survey in September, 2014 (Arcavi et al. 2017). Although it was initially sparsely sampled, the failure of the supernova to decline marked it for more intensive investigation. In January, 2015, iPTF14hls was determined to be a Type II supernova. For the next year, the supernova displayed an irregular light curve with multiple episodes of brightening during which the luminosity varied by about 50%. The total energy emitted in light during the first 600 days was about 2.2 × 10⁵⁰ erg, making iPTF14hls a luminous but not particularly superluminous supernova. There is also some evidence of a supernova-like transient that happened at the same site in 1954. The long duration of iPTF14hls precludes the supernova from being purely a recombination event, like most Type IIp supernovae, because the required mass and energy are too great. Neither is a radioactive energy source the explanation because no radioactivity with the appropriate half life is produced with sufficient abundance in any model.

Therefore, two possibilities remain: (1) iPTF14hls was collisionally powered, its light coming from shells of matter that collided over a long period of time, possibly augmented by a central source; or (2) the event was an extreme example, in terms of duration and spectrum, of a magnetar-illuminated supernovae. Both possibilities were suggested by Arcavi et al. (2017), with the underlying cause for the shell ejections in case 1 attributed to a pulsational pair-instability supernova (e.g., Woosley et al. 2007; Woosley 2017b). Another variation on case (1) was suggested by Soker & Gilkis (2018), who claim that iPTF14hls was the result of a common envelope interaction between a neutron star and a giant star. The initial merger ejects part of the giant's envelope, which is later impacted by jets from the accreting neutron star when it merges with the core.

Here, cases (1) and (2) are explored in some detail. First, the outcome of an ordinary supernova interacting with a dense wind is considered (Section 2). Given a free choice of wind parameters, one can easily find an approximate fit to the global light curve and the high velocity (hydrogenic) component of the spectrum. However, the origin of the lower velocity iron lines is less clear in a one-dimensional model (although see Andrews & Smith 2018, and Section 3.3). Even more problematic, the reason why a star would lose roughly a solar mass of high velocity material during the last few decades of its life is not obvious unless the star dies as a pulsational pair-instability supernovae (PPISN).

Section 3 thus considers PPISN explanations for iPTF14hls. The models are of three sorts: (1) those that might be capable of producing both the 1954 and 2014 transients but which leave a stellar core that has not yet collapsed (Section 3.1); (2) slightly lighter stars where the pulsing activity is restricted to the last decade of the star's life and which experience iron core collapse while the light curve is still in progress (Section 3.2); and (3) hybrid models that invoke an asymmetric explosion with one component due to the collapse of the iron core to a compact object (Section 3.3). Each model has strengths and weaknesses. All of the models are able to explain the approximate duration and brightness of the light curve and, since each can occur in nature, may ultimately appear as iPTF14hls-like supernovae, if not as iPTF14hls itself. However, each PPISN model has difficulty explaining the high velocity (8000 km s⁻¹) H_α component of iPTF14hls.

Magnetar models are considered in Section 4. It is not difficult to find a two parameter fit that approximately tracks the overall bolometric light curve. The very bright initial display of the magnetar is masked by the overlying star and adiabatically degraded, which alleviates a concern voiced by Arcavi et al. (2017). For the 20 ${M}_{\odot }$ supernova model that we have considered, most of the pulsar energy is invisible for the first 100 days. Models with greater explosion energies and more mass loss give higher speeds, which may be necessary to explain the spectrum. If the same neutron star is responsible for accelerating the helium core to 4000 km s⁻¹ and making the light curve, then magnetic field decay must be invoked. While magnetar models are potentially successful at explaining many characteristics of iPTF14hls (Dessart 2018), they would not, without augmentation, give the spectroscopic features that are characteristic of the circumstellar medium (CSM) interaction that has recently been reported by Andrews & Smith (2018) and would not, in any obvious way, produce a transient in 1954.

In Section 5, the strengths and weaknesses of the various models are summarized and some suggestions are made for future observations.

Where stellar and supernova models are employed, they have all been calculated using the KEPLER code (Weaver et al. 1978; Woosley et al. 2002) using physics described in Sukhbold et al. (2016), Woosley (2017b), and Sukhbold et al. (2018).

CSM interaction is interesting, both as a generic model for iPTF14hls (Andrews & Smith 2018) and as a way to set some useful fiducial characteristics for later use. iPTF14hls emitted at least 2.2 × 10⁵⁰ erg of light over a period of approximately 600 days (Arcavi et al. 2017) (additional radiation may not have been optical), implying an average, although variable, luminosity ∼5 × 10⁴² erg s⁻¹. In the CSM interaction model, this light is emitted as the outer layers of a supernova plow into a lower density medium at a radius where conversion of kinetic energy to optical radiation is efficient, i.e., ∼10¹⁵ −10¹⁶ cm. An independent estimate of the radius of the CSM comes from the ∼600 day duration of the event times the highest velocity maintained throughout the event in the spectrum, ∼8000 km s⁻¹ for H_α, or $\gtrsim 5\times {10}^{16}$ cm. To ensure that the velocity in the spectrum is not significantly slowed during the observations, there must be at least several times more mass in the interacting part of the ejecta than in the CSM. The kinetic energy in the ejecta must also be at least several times what was seen in radiation or no kinetic energy would be left over after 600 days. A kinetic energy $E\,\gtrsim 5\times {10}^{50}$ erg is implied. The observed speeds throughout iPTF14hls were 4000 km s⁻¹ (for iron) to 8000 km s⁻¹ (for hydrogen), so the kinetic energy implies a mass for the impacting ejecta of at least one to several solar masses. The presupernova may have ejected a much larger mass that moved more slowly. It should be noted that these are the characteristics of just the working surface of the shock, and are minima for the total ejected mass and energy. The mass of the swept up CSM would be comparable but less to avoid excessive deceleration.

A solar mass that exists at 5 × 10¹⁶ cm implies either an explosive mass loss or a steady loss rate ≳0.01v₇ ${M}_{\odot }$ y⁻¹ where v₇ is the wind speed in units of 100 km s⁻¹. This mass loss rate must persist for 100/v₇ years. Again the actual mass loss could be larger because only the inner, slowest moving matter will interact and produce the high luminosity. Ejecting one solar mass at 1000 km s⁻¹ requires 10⁴⁹ erg.

2.1. Approximations and Models

Some approximate scaling relations will be useful. Consider the simplest case of an ejected shell with mass, M_eject, and energy, E, that encounters a previously ejected shell with mass, M_CSM, outer radius, R, and negligible inner radius. Chevalier (1982a) has studied the general case in which both the CSM and homologously coasting supernova ejecta have initial densities that depend on a power law of the radius, ${\rho }_{\mathrm{CSM}}\propto {r}^{-s}$ and ${\rho }_{\mathrm{ejecta}}\propto {r}^{-n}$, respectively. He gives useful scaling relations for the radii and masses of the shocked CSM and ejecta as a function of time. Chevalier (1982a, 1982b) considered the special case s = 2, which corresponds to a stellar wind with constant speed and mass loss rate. The simple case will be adopted here, although the real situation may be different, especially when the mass loss is explosive as in a PPISN.

Then, the unknowns are M_eject, E, n, M_CSM, and R. Actually, it is the ratio M_CSM/R that matters and not the individual terms so long as the radius of the shock remains bounded by R. This happens because, for the special case s = 2, ρ r² = constant =q = M_CSM/(4πR). As Chevalier notes, and as will later be confirmed here, the interaction region is thin and so one can, to good approximation, assign a single radius, r(t), to the forward and reverse shocks (Chevalier 1982b),

$$\begin{eqnarray}&&r(t)={\left[\displaystyle \frac{2{U}^{n}}{(n-4)(n-3)q)}\right]}^{1/(n-2)}{t}^{(n-3)/(n-2)}.\end{eqnarray} \tag{ 1 }$$

Here, U is a constant that is used to normalize the density distribution in the supernova ejecta, so that at time, t, and radius, r, ρ_ejecta(r, t) = t⁻³ (r/(t U))⁻ⁿ. For the special case n = 7, Chevalier (1982c) gives

$$\begin{eqnarray}&&{r}^{(7)}=0.823{\left(\displaystyle \frac{{E}^{2}}{{M}_{\mathrm{eject}}q}\right)}^{2/5}\,{t}^{4/5},\end{eqnarray} \tag{ 2 }$$

where the superscript 7 refers to the assumption n = 7. Normalizing to conditions we will find appropriate for iPTF14hls, E =4 × 10⁵⁰ erg, M_eject = 1.0 ${M}_{\odot }$, M_CSM = 0.4 ${M}_{\odot }$, and R_CSM =4.5 × 10¹⁶,

$$\begin{eqnarray}&&{r}^{(7)}=1.4\times {10}^{16}\ {t}_{7}^{4/5}\ \mathrm{cm},\end{eqnarray} \tag{ 3 }$$

where t₇ is time in units of 10⁷.

As Chevalier notes, n = 7 is appropriate to Type Ia supernovae and n = 12 may be a better choice for Type II supernovae occurring in red supergiants. One must then make a choice for U in Equation (1). Based upon observations of Type II supernovae, Chevalier (1982b) suggests that U is a few times 10⁹. This is consistent with the models of Woosley & Heger (2007). Here, U = 2.5 × 10⁹ is adopted, giving

$$\begin{eqnarray}&&{r}^{(12)}=8.5\times {10}^{15}\,{t}_{7}^{9/10}\ \mathrm{cm}.\end{eqnarray} \tag{ 4 }$$

The use of n = 12 becomes increasingly inappropriate, even for Type II supernovae, as one goes deeper into the ejecta. The CSM is also unlikely to have the ideal s = 2 distribution of a wind at constant speed and mass loss rate, so either formula would suffice. Here, we will use r⁽¹²⁾.

Continuing to assume a thin shell with only a single radius and speed, the shock velocity is the derivative of the radius,

$$\begin{eqnarray}&&{v}_{s}^{(12)}=7700\,{t}_{7}^{-1/10}\ \mathrm{km}\ {{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 5 }$$

The bolometric luminosity is given by

$$\begin{eqnarray}&&L=2\pi {r}^{2}{\rho }_{\mathrm{CSM}}\,{v}_{s}^{3}=2\pi {{qv}}_{s}^{3}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\ 0.5{M}_{\mathrm{CSM}}/{R}_{\mathrm{CSM}}\,{v}_{s}^{3}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&=\ 4.0\times {10}^{42}\,{t}_{7}^{-3/10}\ \mathrm{erg}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 8 }$$

where the superscript n = 12 has been omitted. The very slow evolution of the velocity and luminosity in these equations resemble that seen in iPTF14hls, and is a consequence of the steep power-law dependence of the density in the supernova ejecta. For the n = 7 case, velocity and luminosity would have declined a bit more steeply as t^−1/5 and t^−3/5, respectively. Because matter deeper in the ejecta interacts at later times, one expects n to decrease and the rate of decline to increase.

To test these approximations, a 15 ${M}_{\odot }$ supernova with a total explosion kinetic energy of $2.4\times {10}^{51}$ erg (Woosley & Heger 2007) was surrounded by a low density shell of 0.4 ${M}_{\odot }$ consisting of hydrogen and helium with an outer radius of 4.5 × 10¹⁶ cm. Within the shell, ρr² was a constant, implying q = 1.41 × 10¹⁵ g cm⁻¹. This value for q will prove a useful constraint for successful CSM models throughout this paper. The presupernova star, without the artificial CSM, had a total mass of 12.79 ${M}_{\odot }$, 8.52 ${M}_{\odot }$ of which was its low density hydrogenic envelope with radius 5.65 × 10¹³ cm. The unconfined 15 ${M}_{\odot }$ supernova developed 6.6 × 10⁵⁰ erg in its outer 1.0 ${M}_{\odot }$ of ejecta but, including the CSM, a third of that energy was radiated during the first 600 days, so this is close to the estimate that was used in developing Equation (3).

The resulting light curve is shown in Figure 1. The event is particularly bright during the first 100 days when iPTF14hls was not well sampled because the underlying supernova contributes appreciably to the CSM interaction in producing the total luminosity. This contribution would be greatly reduced if the progenitor had been a blue supergiant (BSG). It would also have been slightly reduced or shortened if the hydrogen envelope were less massive or the explosion energy was smaller. After 100 days, the light curve agrees well with Equation (8) and with iPTF14hls for the fiducial parameters.

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Figure 2 shows the density and velocity evolution for this model 280 days after core collapse. It is interesting that this long bright supernova can be powered by the interaction of only about 1 ${M}_{\odot }$ of ejecta with a modest kinetic energy. Also interesting is the pile up of the ejected matter and swept up CSM into a thin, dense shell. Half way into the supernova, 0.67 ${M}_{\odot }$ has piled up in a thin shell with a density roughly five orders of magnitude greater than the medium into which it is moving. This shell moves with nearly uniform velocity and justifies the assumption of a single radius for the forward and reverse shocks made in the analytic approximation. In a multi-dimensional calculation this shell would be unstable and spread over a region $\delta r/r\,\gtrsim$ 10% but a large density contrast would still persist (Chen et al. 2014). The KEPLER code cannot accurately calculate the properties of the photosphere for a thin shock wave in matter that has recombined and is interacting in a region optically thin to electron scattering. It is assumed throughout this paper that most of the radiation comes out in the optical waveband. iPTF14hls was not bright in radio or X-rays (Arcavi et al. 2017). The flux is well determined in the model from momentum conservation, but not its temperature. One can speculate, however, that the photosphere lies within this dense fast moving shell. If fully ionized, the shell would be optically thick and the surroundings quite thin. Perhaps that accounts for the broad hydrogen lines in the spectrum (Chevalier & Fransson 1994). Clearly much work remains to be done on the radiation transport.

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While the qualitative agreement with the light curve and speed of the fastest moving hydrogen with what was observed in iPTF14hls are good, there are several deficiencies in this simple model. The light curve lacks the flares that are seen in the observations, although an inhomogeneous CSM could be invoked. The 4000 km s⁻¹ spectral features go unexplained in the simple isotropic model, although a two component, anisotropic explosion with slower speeds at some angles might remedy that (Andrews & Smith 2018). A large increase in the mass of the circumstellar shell (q in Equation (1)) would be required to slow the speed by a factor of two in some directions. The shell mass assumed, 0.4 ${M}_{\odot }$, was already substantial since the minimum radius is already set by the duration of the event. The desired variation might be more easily achieved by invoking an anisotropic explosion; i.e., changing U.

The possible pre-explosive outburst in 1954 would also require an additional explanation but perhaps most challenging is the lack of a clear explanation of just how such a massive circumstellar shell came to be ejected just before the supernova.

2.2. Constraints on Gravity Wave-driven Mass Loss

PPISN occur in non-rotating stars from 80 to 140 ${M}_{\odot }$ that do not lose a large fraction of their helium cores (35 to 65 ${M}_{\odot }$) before dying (Woosley et al. 2007; Woosley 2017b). The number of pulses, their energy, and the total duration increase with mass and range from days to millennia. The total kinetic energy in all pulses can approach $4\times {10}^{51}$ erg but only for the most massive models with very long durations. For durations in the range of one to several years, as measured from the first pulse until iron core collapse, the initial mass is in the range 105 to 115 ${M}_{\odot }$ and the total kinetic energy, about 1 × 10⁵¹ erg. The energy in individual pulses is less. For durations of a century, the mass range extends to 120 ${M}_{\odot }$ and up to 2 × 10⁵¹ erg may be available. Except for the initial pulse that ejects most of the star's envelope, the light curves of PPISN are completely powered by colliding shells of matter. There is no contribution from radioactivity or recombination. PPISN light curves are, thus, an example of CSM interaction.

Several possible PPISN scenarios for iPTF14hls are considered here. In the first, the 1954 outburst (Arcavi et al. 2017) must be explained, as well as the long event starting in 2014. In the second, the historic outburst is left to other causes and we focus on events whose total duration is just a few years. In the third case, a PPISN occurs in conjunction with an anisotropic terminal explosion generated when the star's iron core collapses to a compact object. In all cases, the hydrogen lines are produced by a shock impacting the inner, slower moving edge of the ejected envelope. Therefore, those models where the envelope was ejected many decades before tend to be fainter because of the lower density.

Table 1 summarizes the models of all of the classes that are examined in this paper. For the PPISN models, E_m is the kinetic energy of the matter ejected in pulse m and τ_m is the time in years between that pulse and the final collapse of the iron core. For the models considered here, m =  2 or 3.

Table 1.  Models for iPTF14hls

Model	MZAMS	MHe	Menv	E1	E2	E3	τ1	τ2	τ3	Comment
	(${M}_{\odot }$)	(${M}_{\odot }$)	(${M}_{\odot }$)	(1050 erg)	(1050 erg)	(1050 erg)	(Years)	(Years)	(Years)
S15	15	4.27	8.52	24	⋯	⋯	⋯	⋯	⋯	Ordinary SNII + CSM
B120	120	54.70	11.11	7.50	6.79	⋯	63.0	44.1	⋯	BSG PPISN, 2 pulse
T115	115	52.93	11.35	7.50	5.27	3.56	1198	1152	1152	RSG PPISN, long delay
T115A	115	50.47	29.00	4.55	4.04	2.30	4.17	2.44	0.21	RSG PPISN, short delay
T110A	110	49.68	18.73	4.72	1.66	1.22	12.3	2.31	2.29	RSG PPISN, short delay
T110B	110	49.50	34.12	5.15	2.20	0.76	2.92	0.20	0.15	RSG PPISN, short delay
20A	20	6.17	9.76	11.9	0.8	⋯	⋯	⋯	⋯	magnetar, B const
20B	20	6.02	6.34	11.2	1.0	⋯	⋯	⋯	⋯	B const, $\dot{M}$*2
20C	20	5.93	4.56	12.2	1.1	⋯	⋯	⋯	⋯	B const, $\dot{M}$*2.5
20D	20	5.83	1.58	10.3	0.4	⋯	⋯	⋯	⋯	B const, $\dot{M}$*3.3
20E	20	5.83	1.58	10.3	9.2	⋯	⋯	⋯	⋯	B decay, low-E magnetar
20F	20	6.17	9.76	11.9	139	⋯	⋯	⋯	⋯	B decay, hi-E magnetar
18BSG	18	7.39	9.54	12.5	1.0	⋯	⋯	⋯	⋯	Blue supergiant

Note. For the magnetar models, E₁ is the energy input by the piston and E₂ the kinetic energy input by the magnetar.

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3.1. Models for iPTF14hls That Could Give a Transient in 1954

For this class of model, pulsing activity must span many decades and the actual death of the star occurs long after pulses have ceased. The main sequence mass range is on the higher end, 115-120 ${M}_{\odot }$ and the helium core mass is 53-55 ${M}_{\odot }$. A star remains at the site of iPTF14hls, shining at approximately the Eddington luminosity, (10⁴⁰ erg s⁻¹), and it will continue to do so for decades to centuries before the iron core finally collapses, possibly uneventfully to a black hole of about 50 ${M}_{\odot }$. This could be an observational signature of the model.

3.1.1. A Two-pulse Model

The simplest PPISN model is one with only two pulses. The first pulse ejects most of the envelope. The second, much later, pulse ejects the rest of the envelope and part of the helium core. The mass from this second ejection collides with the slowly moving inner edge of the first, illuminating a bright supernova by CSM interaction. Still later, the remainder of the core completes silicon burning and collapses to a black hole.

Woosley's (2017b, Table 1) Model B120 is a relevant example. This star was, by construction, a BSG derived from a 120 ${M}_{\odot }$ main sequence model. The star died with a residual hydrogen envelope with mass 11.1 ${M}_{\odot }$ and radius, 6.1 ×10¹² cm surrounding a helium core with mass 54.7 ${M}_{\odot }$. The first pulse ejected 9.8 ${M}_{\odot }$ of that envelope with an energy 7.5 × 10⁵⁰ erg producing a supernova. 1.3 ${M}_{\odot }$ of hydrogen-rich material remained bound. Nineteen years later, a second strong pulse ejected 5.1 ${M}_{\odot }$ with an energy of 6.8 × 10⁵⁰ erg. The peak speed at the leading edge of this second ejection was 7300 km s⁻¹, which declined to 5000 km s⁻¹ 1.0 ${M}_{\odot }$ into the ejecta. This 1 ${M}_{\odot }$ of matter had a kinetic energy of 4.0 ×10⁵⁰ erg. The inner 0.4 ${M}_{\odot }$ of the ejected envelope with which this interacted was contained within a radius of 4 × 10¹⁶ cm. These numbers are similar to the fiducial values required in Section 2.1 to describe iPTF14hls, although a slightly more energetic pulse 2 would be preferable. However, 19 years is also too short to explain the 1954 transient. Meanwhile, 44 years after the second pulse, the remaining 50.9 ${M}_{\odot }$ core collapsed to a black hole.

Figures 3 and 4 show the light curves resulting from the two pulses. The first explosion is relatively faint because of the small radius of the BSG progenitor. Arcavi et al. (2017) reported an absolute magnitude of ≈−15.6 for the 1954 outburst at the site of iPTF14hls but noted that this was a lower bound to the peak luminosity. This magnitude corresponds to a luminosity of roughly 5 × 10⁴¹ erg s⁻¹, in reasonable agreement with the plateau of the model. This luminosity only lasted for about a month in the model though and it would have been fortuitous to have detected it. Figure 3 also shows a bright tail on the light curve after day 50. Not only is this emission fainter than the observations in 1954 require but it is probably also an overestimate for the model. This late emission results from the fallback and accretion of the innermost ejecta onto the remaining star. If the matter were fully ionized, the Eddington limit would be near 10⁴⁰ erg s⁻¹. The matter falling back in the code had recombined and its opacity was low, thus the effective Eddington luminosity was high.

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Another possible source of luminosity is the interaction of the ejected envelope in the first supernova with mass lost before the explosion. The outer 0.1 ${M}_{\odot }$ of the material ejected by the first pulse moves with speeds 7000 to 12 000 km s⁻¹ and contains 7 × 10⁴⁹ erg. Taking a shock speed of 10,000 km s⁻¹, as is typical for such a small mass, a presupernova mass loss rate 10⁻⁴ ${M}_{\odot }$ y⁻¹, and a wind speed of 100 km s⁻¹, the luminosity from CSM interaction would have been $\approx 0.5\dot{M}{v}_{\mathrm{shock}}^{3}{v}_{\mathrm{wind}}^{-1}=3\times {10}^{41}$ erg s⁻¹, near the observed value, for years. However, it is not obvious that this radiation would be mostly in the optical.

The solid line in the first panel of Figure 4 shows that the light curve of the second supernova (resulting from the second pulse) continues for years after reaching a peak value of 5.3 × 10⁴² erg s⁻¹. The material close to the exploding core was not very finely zoned in the calculation—a few hundredths of ${M}_{\odot }$—and the rise time would have been shorter in a more finely zoned model. The density distribution in the slowly moving innermost layers of the ejected envelope scales roughly as r⁻¹ to r⁻². Its velocity increases radially outwards from a few hundred to 1000 km s⁻¹. The approximation s = 2 used in developing Equation (1) is thus applicable. The density in the outer interacting regions of pulse 2 obeys an approximate power law $\rho \propto {r}^{-n}$ with n ≈ 4, much shallower than usually assumed for core-collapse supernova. The shock velocity varied from 7300 km s⁻¹ at the onset to 5600 km s⁻¹ on day 275 to 4600 km s⁻¹ on day 600. At those same times the shock interaction radius moved from 7 × 10¹⁵ cm to 2.3 × 10¹⁶ cm to 3.5 × 10¹⁶ cm. The bottom panel of Figure 4 shows the velocity and radius near peak light.

In two sensitivity studies, the interval between the first and second explosions was increased to 57 years, compatible with the 60 year interval observed for the 1954 transient, and the velocity of the second pulse was increased. The dashed line in the top panel of Figure 4 shows that the lower circumstellar density resulting from the longer delay, by itself, makes the light curve too faint. A much brighter light curve, which is more compatible with the observations, results in this same model if the velocity in just the outer solar mass of ejecta in pulse 2 is increased by 50%. As expected from Equation (8), allowing the ejected envelope to expand by an additional factor of three decreases $q={M}_{\mathrm{CSM}}/{R}_{\mathrm{CSM}}$ by three and decreases the early luminosity by that factor. The timescale for decline is slower due to the decreased density in the factor U in Equation (1), so at late times the difference is less. Increasing the velocity raises the luminosity by roughly a factor of v_s³, or 3.4. A similar multiplication of the pulse speed in the standard model with pulse interval 19 years would also have raised the solid line by the same factor, although this is not plotted. In the high velocity, long interval case (triple-dot dashed line in the figure) the speed of the highest material, just inside the reverse shock was still 8000 km s⁻¹ at day 600 and its radius was 5.4 × 10¹⁶ cm. The unmodified pulse 2 had a kinetic energy of 6.8 × 10⁵⁰ erg and the one with artificial velocity increased, 1.2 × 10⁵¹ erg, which strains the limits expected for PPISN in this mass range but may be feasible (Section 5).

In summary, simple two-pulse models like B120 can explain the 1954 transient as well as the long duration and luminosity of iPTF14hls but struggle to produce the bright luminosity and high H_α velocity. An artificial adjustment to the velocity can remedy the situation but requires the energy to be doubled in the outer solar mass of pulse 2. Although brighter light curves would also result if the interval between pulses 1 and 2 was shortened, that would mean giving up a PPISN solution for the 1954 transient and having a slower shock speed, especially at late times. The models calculated here give smooth light curves and lack the irregularity of iPTF14hls. However, clumpy ejecta could be invoked and might be reasonable. The two pulses may not have been perfectly isotropic. The reverse shock in the first pulse might have caused some mixing in the envelope. Each pulse actually includes multiple subpulses as the core rings after the explosion. It is only the innermost, slowest part of the ejected envelope that participates in making the light curve and the density there is sensitive to events at the mass cut.

Making the entire event with a single pulse gives no natural explanation for the two velocity components seen in the spectrum, even though matter moving at 4000 km s⁻¹ is certainly present. Might models with more pulses fare better, or is there really just one shell seen to different depths? Interestingly the 4000 km s⁻¹ point in this model is located in hydrogen-deficient material, just inside the outer edge of the former helium core.

3.1.2. A Three Pulse Model

Model T115 (Woosley 2017b), based on a RSG progenitor, also has a light curve that resembles iPTF14hls but had three pulses, the latter two in rapid succession 46 years after the first. While the total time between the first pulse and the final collapse of the iron core was 1198 years, most of that time was spent in the final contraction to stable silicon core burning during which no additional pulses occurred.

The presupernova radius of Model T115 was 1.2 × 10¹⁴ cm; its luminosity, 9.8 × 10³⁹ erg s⁻¹; total mass, 64.28 ${M}_{\odot }$; and helium core mass, 52.93, ${M}_{\odot }$. These masses differ slightly from those in Table 2 of Woosley (2017b) because the model was rerun for this paper with a slightly different surface boundary pressure and zoning. The original model had a helium core of 53.09 ${M}_{\odot }$ and pulsed for only 17 years instead of 46 years, showing the strong sensitivity of pulse intervals to small changes in the model. The first pulse in revised Model T115 ejected 10.4 ${M}_{\odot }$ of envelope with a kinetic energy of 7.5 × 10⁵⁰ erg and a typical speed of about 2000 km s⁻¹ but with a range from 0 to 7000 km s⁻¹. Similar to Model B120 of the previous section, 0.9 ${M}_{\odot }$ of envelope with hydrogen mass fraction 0.20 was not ejected in the initial outburst. The light curve from this initial explosion is given in Figure 5.

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In this case, the first supernova was too bright to have been be the 1954 transient, unless a very substantial bolometric correction is applied or the event was accidentally sampled during its steep decline. After 50 days there is again a poorly determined tail on the light curve due to fallback. The blue line in Figure 5 shows the effect of reducing the radius of the progenitor to BSG-like proportions with a radius 6.8 × 10¹² cm (Model B115 of Woosley 2017b), and the red dashed line, the effect of increasing the opacity (κ_min = 0.01 cm² g⁻¹) in the matter that falls back. CSM interaction could again provide an enduring luminosity, especially if a low wind speed, ∼10 km s⁻¹ is invoked for the RSG progenitor. For a shock speed of 7000 km s⁻¹ and mass loss rate 10⁻⁴ ${M}_{\odot }$ y⁻¹, the luminosity would be ∼10⁴² erg s⁻¹.

Forty-six years later, after ejecting most of its envelope, Model T115 experienced a second stage of thermonuclear instability during which two additional shells of 5.6 ${M}_{\odot }$ and 3.5 ${M}_{\odot }$ were ejected in an interval of 130 days. This added a kinetic energy of 8.9 × 10⁵⁰ erg–5.3 × 10⁵⁰ erg in the second pulse and 3.6 × 10⁵⁰ erg in the third. The leading edge of pulse 2, located in hydrogen-rich matter (X_H = 0.2, X_He 0.8), which initially moved at about 6500 km s⁻¹. Its interaction with the slower moving material from the prior envelope ejection with speeds ∼300–500 km s⁻¹ at a radius of ∼10¹⁶ cm produced an enduring luminosity ∼10⁴² erg s⁻¹ (Figure 5 and Figure 6).

Figure 5 also shows the observed light curve of iPTF14hls from Arcavi et al. (2017). The black points are their published data. The green data points are their R-band magnitudes at earlier times, scaled to the early bolometric points (I. Arcavi 2018, private communication). The data have been arbitrarily adjusted so that the first R-band observation occurs about 100 days after the second pulse. In a multi-dimensional calculation, the peak at 0 days in the figure, which is produced by pulse 3, would be broader and fainter due to the instability of the shell snowplowed by pulse 2. If the shell fragmented, then additional structure more like that seen in the observations might be produced.

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The model parameters are in the range of the analytic model (Section 2.1) but the velocity and CSM density are too low. The CSM comes again from the first pulse and, although 10.6 ${M}_{\odot }$ was ejected, at the time of the second pulse, only 0.1 ${M}_{\odot }$ was within 5 × 10¹⁶ cm, moving with a speed less than ∼300 km s⁻¹. Meanwhile, 0.3 ${M}_{\odot }$ was within 1.2 × 10¹⁷ cm with a speed less than 700 km s⁻¹. This is too little by a factor of about three. The density near the supernova would have been higher if the initial pulse had less energy, if the mass of the envelope were greater, or if the interval between pulses 1 and 2 was shortened. The outer 0.1 ${M}_{\odot }$ of the matter ejected by pulse 2 initially moved at 6700 km s⁻¹ and the outer 1.0 ${M}_{\odot }$ had a kinetic energy of 2.7 ×10⁵⁰ erg and an average speed of 5200 km s⁻¹. This is again too slow by about 50%.

During the interval between pulses 2 and 3, the leading edge of pulse 2 moved to 6 × 10¹⁵ cm. Meanwhile, the third pulse, with a leading edge speed near 4000 km s⁻¹, overtook the slower moving ejecta from the second pulse (Figure 6), producing the bright delayed peak in the light curve (Figure 5). The integral of the light radiated over the period shown was 2.4 × 10⁵⁰ erg, showing the efficient conversion of the kinetic energy of the second and third pulses (i.e., 8.9 × 10⁵⁰ erg) into radiation. This is also the total light observed in iPTF14hls and this is a success of the model. Although a better treatment of the radiation transport is needed before anything definitive can be said about the spectrum, one should note the presence, after the third pulse, of two shells bounded by two shocks with different characteristic speeds not so different from what was observed, 6000 km s⁻¹ and 4000 km s⁻¹.

Reasonable adjustments to the model can bring the light curve and hydrogen velocity more in line with observations. Increasing the velocity by 50% in the outer ejecta of pulse 2, just the part moving over 4000 km s⁻¹, adds $6.2\times {10}^{50}$ erg to the kinetic energy of that pulse and gives the modified light curve in Figure 5. The shock speed at the outer edge of pulse 2 in hydrogen-rich material now declines from 9000 km s⁻¹ on day 100 to 7200 km s⁻¹ on day 600, which is similar to what was observed. The shock bounding pulse 3 decreased in speed from 4000 to 3000 km s⁻¹ during the same period, which is also consistent with observations.

Like B120, Model T115 is a reasonable approximation to iPTF14hls if one is allowed a reasonable modification of the energy of the second pulse. However, several potential deficiencies remain. The separate 4000 km s⁻¹ component does not appear until after the third pulse and has a different history than the 8000 km s⁻¹ component. The supernova was not spectroscopically sampled during the first 100 days; so, perhaps this is not a problem, but the overall curve is still too smooth. The integral under the light curve is correct but its shape is wrong, which may reflect deficiencies in the 1D model. In addition to the symmetry breaking conditions mentioned for Model B120, the matter through which the shock generated by pulse 3 passes in Model T115 has experienced mixing due to the Rayleigh–Taylor instability (Chen et al. 2014). In particular, it may be clumpy and have angular and radial variations, which would act both to broaden the peak and make the light curve more irregular, like the data (Figure 5).

3.2. Models With Shorter Delays and Prompt Black Hole Formation

PPISN models that produce long lasting light curves like iPTF14hls were all previously supernovae that ejected most of their envelope in a bright display not very long before iPTF14hls itself. Furthermore, iPTF14hls is made by subsequent pulses running into that ejected envelope, and into each other. Envelopes that were ejected more recently (well after 1954) have expanded less and have a greater q = M_CSM/(4π R_CSM) in Equation (8). CSM interaction will more easily give a brighter light curve. The lower mass also leads to more pulses within the duration of the light curve, giving a more irregular history resembling iPTF14hls.

Consider Woosley's (2017b, Table 1 here) Model T115A. Despite the similarity in name, this was a different 115 ${M}_{\odot }$ star than Model T115 in Section 3.1.2. Its helium core was 50.47 ${M}_{\odot }$ instead of 52.93 ${M}_{\odot }$ and its hydrogen envelope, more massive, 29 ${M}_{\odot }$. The initial explosion was thus more tamped and the envelope expanded more slowly. The model had three pulses. The first one (actually a pair of pulses in quick succession) ejected most of the hydrogen envelope with a kinetic energy of 4.55 × 10⁵⁰ erg, leaving a remnant of 53.1 ${M}_{\odot }$, including about 2.6 ${M}_{\odot }$ of hydrogen envelope. The initial light curve (not shown) resembled the first explosion in Figure 5. Then, 630 days later, a second strong pulse ejected an additional 5.4 ${M}_{\odot }$ with kinetic energy 4.05 × 10⁵⁰ erg. This included the residual hydrogen envelope plus the outer edge of the helium core. Then, 815 days after that, a third and final pulse ejected an additional 2.0 ${M}_{\odot }$ of helium core with kinetic energy 2.3 × 10⁵⁰ erg. The total kinetic energy in all three pulses was thus 1.09 × 10⁵¹ erg, about 4 × 10⁵⁰ erg of which was radiated as light (Figure 7). Then, 77 days after this final pulse, the star's iron core collapsed, probably to a black hole, while the light curve was still in progress (at 890 days in Figure 7).

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The initial rise in Figure 7 is due the second pulse encountering the inner edge of ejected envelope at a radius of ∼10¹⁵ cm. The sharp peak about 200 days later is not a new pulse but is the same pulse encountering a thin shell of material piled up by the reverse shock from the first pulse. In reality, this shell would have mixed and not be so thin. The peak at 200 days would be broader but contain the same radiated energy. The third and final pulse happened at day 815 on the plot and an additional spike is generated at day 1490 when the piled up shell from pulse 2 running into pulse 1 is encountered by pulse 3. At this point, the core has already collapsed and, baring additional activity generated by compact object formation, no more mass ejection occurs.

The rapid variability in this light curve resembles that of iPTF14hls, although it is perhaps too variable and lasts too long. Mixing in the ejecta would greatly reduce the variability (Chen et al. 2014). Smoothing by $\delta t/t\,\gtrsim$ 10% is reasonable and would improve the agreement with observations. Shortening the interval between pulses 2 and 3 by a factor of two also leads to a light curve more like iPTF14hls (the dashed line in Figure 7). This is also a reasonable adjustment given that the core temperature following pulses, to which neutrino cooling is very sensitive, may not be precisely determined. Typical speeds in the colliding shells are 2000–4000 km s⁻¹. This might be adequate to explain the slow component seen in the iron lines of iPTF14hls but no hydrogen moved faster than 5000 km s⁻¹. Despite its many attractive features, baring some additional input of energy (e.g., Section 3.3), this otherwise promising model is ruled out by the lack of high velocity hydrogen.

Model T110A was similar but the pulses occurred in more rapid secession because of its lower helium core mass, resulting in a light curve that was more continuous. The helium core mass was 49.7 ${M}_{\odot }$, surrounded by an envelope of 20 ${M}_{\odot }$. A first pulse (4.72 × 10⁵⁰ erg) ejected the hydrogen envelope 10 years prior to the final two pulses that, in rapid succession, ejected an additional 5.2 ${M}_{\odot }$ of mostly helium with an additional 2.88 × 10⁵⁰ erg. Collision of the ejected matter with the inner edge of the previously ejected envelope, which initially had a speed of only a few hundred km s⁻¹, gave the fainter light curve in the first panel of Figure 8. The model glowed continuously with supernova-like luminosity for over 1000 days. Typical interaction radii were 0.6 to 2 × 10¹⁶ cm. Over the course of the light curve, the shock speed declined from 4000 to 1600 km s⁻¹. Most of the time it was near 2000 km s⁻¹. The velocity of the hydrogen just outside the shock was ∼1000 km s⁻¹.

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This shock speed is far too slow and the light curve is too faint to be iPTF14hls. A brighter, shorter (but still 700 days long) light curve results if the velocity of the matter ejected by this second set of pulses is increased by a factor of 1.5, corresponding to an energy increase of $3.5\times {10}^{50}$ erg, which is well within reach of the PPISN model. Now the shock speeds are 6000 to 3200 km s⁻¹ but the higher speed only lasts a short time and may not have been observed.

Model T110A is not unique. The second panel in Figure 8 shows that Model T110B, with essentially the same helium core mass, 49.50 ${M}_{\odot }$, but larger envelope mass, 34 ${M}_{\odot }$, has a similar light curve. Apparently, when stars with the right helium core mass, roughly 50 to 55 ${M}_{\odot }$ for the physics in the KEPLER code, die in stars with substantial envelopes, they frequently make supernovae whose light curves resemble iPTF14hls.

In summary, lighter models that do not attempt to explain the 1954 transient also give light curves that, with minor adjustments, agree with the brilliance, duration, and variability of iPTF14hls. They can also explain the time history seen in the iron lines. Unfortunately, all of the models that have been examined so far fail to give the high velocity that is seen throughout the event for the Balmer lines of hydrogen. In this regard, they do even less well than the longer duration, more energetic events that might also explain the 1954 transient (Section 3.1).

3.3. Anisotropic Models with Terminal Explosions

All PPISN models thus far have been one-dimensional and they have assumed the formation of an inert black hole once pulsational activity ceases. With the additional freedom of angular dependence and the added energy of a terminal explosion, one can construct a broader range of models. Invoking disparate conditions at different angles for the low and high velocity components makes it easier to have high velocity hydrogen.

The chief uncertainty here is how an energetic, anisotropic flow would develop in a thermonuclear model that is inherently isotropic. Rotation is the obvious explanation. There is adequate angular momentum in some PPISN models for the iron core to become a millisecond magnetar, although such models must avoid ever becoming supergiants (Woosley 2017b). However, it would be very hard to reverse the inflow of collapse and eject everything outside the neutron star. One would also expect 40 ${M}_{\odot }$ of oxygen and heavy elements to eventually make their presence known. For the time being, if a terminal, anisotropic explosion is to occur in a PPISN, then it seems more natural to invoke black hole formation. The black hole would have a mass of about 45 ${M}_{\odot }$ and could be rotating rapidly with a Kerr parameter ∼0.1. Polar outflows could develop from the accretion of even a small amount of matter (MacFadyen et al. 2001; Quataert & Kasen 2012; Woosley & Heger 2012; Dexter & Kasen 2013), provided that the necessary magnetic fields could be generated near the event horizon of the rapidly rotating hole. Accreting 0.1 ${M}_{\odot }$ with 1% efficiency for conversion of rest mass into outflow could power a 10⁵¹ erg outflow. It is of note that this would make iPTF14hls a close relative of gamma-ray bursts with jets of similar energy but greater baryon loading and perhaps less collimation. The main difference here, aside from the large black hole mass, is the presence of solar-mass shells of matter with which the accretion energized outflow can interact. Yuan et al. (2018) have reported transient high energy gamma-radiation coincident with the time and location of iPTF14hls, which might be taken as evidence for energetic jets or at least strong circumstellar interaction. Unfortunately, the energy above 200 MeV that is implied by their observations, about 10⁵¹ erg, is not easily understood in any current supernova model and there is also an active galaxy in the same error box.

In addition to the uncertain physics of such a terminal explosion, its timing is a problematic issue. For two components to appear in the spectrum of the same event, they must commence close together. Iron core collapse needs to follow swiftly on the heels of the final pulses. Although there are PPISN where this is the case, there is usually at least a few week's delay as the star goes through a final stage of stable silicon shell burning (Table 1). There is also a question of how to input the energy from a terminal explosion into the code. If it comes from polar outflows driven by accretion on a massive black hole, then the energy might be in the form of a small mass moving at semi-relativistic speed. Retaining the kinetic energy of this small mass and not promptly radiating it away as something resembling a gamma-ray burst afterglow requires that the shells with which it interacts still be optically thick.

Consider Model T110A (Figure 8; Section 3.2) that ejected most of its hydrogen envelope 12 years before two final pulses and the collapse of its iron core (Table 1). Unfortunately, in this model the iron core collapse did not occur until 830 days after the last pulse. Consequently, adding a high velocity component in just the final few hundred days would not explain the observations. In Model T110B, the delay between the final pulse and core collapse was just 55 days. The difference was a slightly higher central temperature, 1.36 × 10⁹ K versus 1.13 × 10⁹ K immediately after the pulses. However, a high velocity component in Model T110 gave too bright and too brief a light curve because the hydrogen envelope had not expanded enough. The first supernova was too close to core collapse. Model T115B (Woosley 2017b) had a strong final pulse just 12 days before core collapse. Its inner solar mass had already turned to iron after that pulse, so silicon shell burning was of limited duration.

To show what might happen in a model that is capable of producing comparable luminosities in the large angle and polar outflows, attention was focused on Model T110A. Energetic explosions were introduced just 10 days after the final pulse Figure 8. However, these were not explosions of the core. The remaining 44.6 ${M}_{\odot }$ core was excised, presumably to make the black hole, and 0.1 ${M}_{\odot }$ of the matter in the inner edge of the last shell to be ejected given a high speed corresponding to energies of 24, 12, and 6 ×10⁵¹ erg. Since this was a one-dimensional simulation, these are the equivalent isotropic energies and are assumed to only pertain to some small solid angle, assumed here to be 10% of the sky. So the 12 ×10⁵¹ erg case, for example, only requires an energy input of 1.2 × 10⁵¹ erg.

The resulting light curves are shown in Figure 9. They are very bright, just 10% of the luminosity of the 12 ×10⁵¹ erg model could power the observed light curve of iPTF14hls. The velocity history (Figure 9) is also in reasonable agreement with what was observed for the H_α line, especially if the event was not observed in the first 100 days. The small mass of the hyper-velocity ejecta impacts the two shells ejected by the final two pulses and sweeps them up. Because the collision occurs in a region that is still marginally optically thick, kinetic energy is conserved and is not promptly radiated away. Models in which the energy was injected at day 60 fared less well and only the most energetic case retained enough kinetic energy after the break out transient to power a light curve like iPTF14hls—again assuming a solid angle of 10% of the sky.

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In summary, a two component PPISN model with a terminal explosion is contrived but could potentially explain the major features of iPTF14hls. The model requires both the production of a mildly relativistic jet, for which a physical basis is lacking, and a core collapse that happens very shortly after the last pulse of the PPISN. A similar model was proposed by Woosley (2017b) to explain superluminous supernovae but there the timing of the core collapse was not so constrained by the need to produce a long light curve with constant velocity components. In addition, in Woosley's (2017b) models, the entire helium and heavy element core within the given solid angle was ejected and not just a small mass outside a black hole. A similar model for iPTF14hls in which an accreting neutron star powers a jet that interacts with matter recently ejected by a common envelope interaction has been proposed by Soker & Gilkis (2018).

Magnetars are neutron stars with unusually strong magnetic fields compared with radio pulsars. Typically, a magnetar has a (dipole) field strength ≳10¹⁴ G. The strong magnetic field of the neutron star, in theory, is a consequence of its very rapid rotation at the time of its birth (Duncan & Thompson 1992). The existence of magnetars and their key role in explaining soft gamma-ray repeaters and anomalous X-ray pulsars is beyond doubt (Kaspi & Beloborodov 2017). The time during which the magnetar shines brightly is short compared with pulsars and, given their numbers and spatial distribution, more than 10% of all neutron stars are probably born with these strong fields and, presumably, rapid rotation. Rapidly rotating magnetars are also the central engine in a leading model for gamma-ray bursts (e.g., Usov 1992), where they are required to have magnetic fields greater than 10¹⁵ G and powers $\gtrsim {10}^{50}$ erg s⁻¹. Ordinary pulsars and ultra-powerful magnetars −B ∼ 10¹⁴ G and P ∼ few ms should exist somewhere between these extremes of rotation rate and field strength. For these characteristics, a light curve similar to iPTF14hls is a natural consequence.

Due to the lack of a deeper understanding, the magnetar energy and power of a magnetar just after its birth are generally approximated using the same two-parameter equations as employed for much older pulsars:

$$\begin{eqnarray}E & = & \displaystyle \frac{1}{2}I{\omega }^{2}\\ & \approx & 2\times {10}^{52}{P}_{{\rm{m}}{\rm{s}}}^{-2}\,{\rm{e}}{\rm{r}}{\rm{g}},\end{eqnarray} \tag{ 9 }$$

where E is the rotational energy, I, the moment of inertia (≈10⁴⁵ g cm²), and P_ms, the period in ms. The approximate energy loss for dipole radiation is given by the Larmor formula (e.g., Lang 1980),

$$\begin{eqnarray}\displaystyle \frac{{dE}}{{dt}} & = & \displaystyle \frac{2}{3{c}^{3}}{({{BR}}^{3}{\rm{S}}{\rm{i}}{\rm{n}}\alpha )}^{2}{\left(\displaystyle \frac{2\pi }{P}\right)}^{4}\\ & \approx & {10}^{49}{B}_{15}^{2}{P}_{{\rm{m}}{\rm{s}}}^{-4}\,{\rm{e}}{\rm{r}}{\rm{g}}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 10 }$$

Here, B₁₅ is the surface dipole field in 10¹⁵ Gauss, R ≈ 10⁶ cm is the neutron star radius, and α is the inclination angle between the magnetic and rotational axes, taken arbitrarily to be 30 degrees. This equation may be integrated to give the energy and power at time t (Woosley 2010),

$$\begin{eqnarray}\displaystyle \frac{dE}{dt} & = & {\left(\displaystyle \frac{dE}{dt}\right)}_{0}{(1+t/{t}_{p})}^{-2}\\ {t}_{p} & = & 2000\,{P}_{ms0}^{2}{B}_{15}^{-2}\,{\rm{s}}.\end{eqnarray} \tag{ 11 }$$

Similar equations have been given by Kasen & Bildsten (2010) with a different choice of inclination angle. Their equations are recovered if B in the above equations is divided by $\sqrt{2}$.

These equations have the appeal of a simple physical model that can be adjusted using the two parameters to fit almost any smooth light curve, so long as the emitted radiation and wind is thermalized inside the expanding star and emitted chiefly in the optical. However, they are too simple, especially at very early times when the neutron star and its crust are rapidly evolving. The same magnetic field and rotation needed to eject the accreting matter and power a supernova, a power of at least 10⁵⁰ erg s⁻¹ (more if appreciable ⁵⁶Ni is to be synthesized), would rapidly sap the rotational energy and leave the magnetar powerless during the months needed to power the light curve. Thus, we expect that B in the above equation is not a constant at early times.

Arcavi et al. (2017) found a good overall fit to the light curve of iPTF14hls using the formulae of Kasen & Bildsten (2010) with B = 7 × 10¹³ G and an initial rotational period of 5 ms (E = 8 × 10⁵⁰ erg). Using Equation (11) gives a similar good fit for an initial rotational energy of 6 × 10⁵⁰ erg and B = 4 ×10¹³ G. Figure10 and Table 1 show the result when a magnetar with these properties is embedded in a supernova derived from a star with solar metallicity and a mass of 20 ${M}_{\odot }$ on the main sequence. With a normal mass loss rate, the presupernova star (Table 1) had a total mass of 15.93 ${M}_{\odot }$. The helium core mass was 6.17 ${M}_{\odot }$ and the rest of the star was a low density, hydrogen-rich envelope. Model 20A was a red supergiant with luminosity 5.7 × 10³⁸ erg s⁻¹ and radius 7.42 × 10¹³ cm. Other RSG progenitors had similar properties. This star, which exploded with a piston at 1.82 ${M}_{\odot }$ (the base of the oxygen shell where the entropy per baryon reaches 4.0), has been previously published in Woosley & Heger (2007). The explosion produced 0.14 ${M}_{\odot }$ of ⁵⁶Ni and had a final kinetic energy of 1.2 × 10⁵¹ erg. The light curve of this rather standard Type II-P supernova model is shown as the faint dotted–dashed line in the top panel of Figure 10.

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In the lower panel of Figure 10, the red line with the broad peak at 150 days is the light curve that results when power from the standard magnetar that was defined above, shown as the dashed line, is embedded in Model 20A. Somewhat like the energy from ⁵⁶Co decay in SN 1987A, the bottled up magnetar energy diffuses out, producing a delayed peak. Since the magnetar energy inflates a radiation-dominated bubble whose boundary, in 1D, has a high density contrast with its surroundings, this diffusion time may not be well represented in the models. The diffusion peak would be delayed and broader if the matter remained spread out over a larger volume (Chen et al. 2016). Once that wave of radiation diffuses out, the bolometric light curve approaches steady state with the magnetar power and no further peaks occur. Even though the bolometric light curve is not badly represented by Model 20A, especially if the supernova was discovered shortly after its explosion (Figure 11), the model's velocity, shown as the red line in the second panel of Figure 10, is, for the most part, slower than the speeds observed in iPTF14hls. Only the outer 0.1 ${M}_{\odot }$ moves faster than 6000 km s⁻¹ and only 0.001 ${M}_{\odot }$ is faster than 8000 km s⁻¹.

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For comparison, Figure 10 and Figure 11 also show the light curve and final velocity for an 18 ${M}_{\odot }$ blue supergiant model (Model W18 of Sukhbold et al. 2016) with the same embedded magnetar. A similar model has been calculated by Dessart (2018). The late time light curve is identical to Model 20A but owing to its small initial radius, the early light curve of Model 18BSG is much fainter and not as good a match to the early behavior of iPTF14hls. The highest velocities are achieved in a larger amount of mass—0.37 ${M}_{\odot }$ moves faster than 6000 km s⁻¹ and 0.07 ${M}_{\odot }$ faster than 8000 km s⁻¹.

It may be easier to satisfy spectral constraints if the explosion is more energetic or the presupernova star has lost more of its hydrogen envelope before exploding. The expansion of the helium core is then not so tamped by and the small mass of the envelope expands faster. Models 20B, 20C, and 20D in Figure 10 and Table 1 had mass loss rates 2.0, 2.5 and 3.3 times the standard value and ended with total masses of 12.36, 10.41 and 7.41 ${M}_{\odot }$, helium core masses of 6.02, 5.93, and 5.83 ${M}_{\odot }$, and hydrogen envelope masses of 6.34, 4.56 and 1.58 ${M}_{\odot }$. These three stars were also exploded with pistons at the location where the dimensionless entropy was 4.0 and had final kinetic energies near 1.1 × 10⁵¹ erg (Table 1). Without an embedded magnetar, the light curves of these models (not shown) are briefer and just a bit brighter than the standard explosion, essentially the green curve in Figure 10 before the rapid rise at ∼25 days. With the standard magnetar (0.6 × 10⁵¹ erg, 4 × 10¹³ G), the light curves are similar to Model 20A but due to the smaller mass envelopes, the peak from magnetar break out is fainter and happens earlier. However, the hydrogen in these models moves much faster. In Models 20B, 20C, and 20D, 0.002, 0.02, and 0.26 ${M}_{\odot }$ of hydrogen-rich material now moves faster than 8000 km s⁻¹.

While other radii, masses of stars, explosion energies, and magnetar properties could be explored, the properties of these three 20 ${M}_{\odot }$ models are probably generic. Without greatly increasing the explosion energy, the helium core will not move much faster. The magnetar properties are essentially fixed by observations after break out. The radius only affects the initial light curve and, to some extent, the peak velocity of the ejecta. Perhaps something else is needed to explain the near constancy of of H_α and iron speeds in the spectrum?

All of the magnetar-based models have so far assumed a constant magnetic field of 4 × 10¹³ G. If the field is allowed to evolve, then a different velocity structure results for the supernova. If the magnetic field and rotational energy are bigger early on, then the resulting explosion is more energetic and has a faster moving helium core. Consider a case where the magnetar field strength and rotational energy are so great early on that its energy deposition is not a perturbation on some other undefined energy source but is the cause of the explosion. During its first 10–1000 s, the magnetar will evolve rapidly. Damping of the initial differential rotation and neutrinos may have already launched a successful explosion (Akiyama et al. 2003) but the magnetar continues to deposit energy after that, adding to the kinetic energy of the explosion. The shock crossing time for the helium core is a relevant timescale for the supernova, or about 100 s. A relevant time for the neutron star might be the interval necessary for the crust to form, which is typically estimated at minutes to hours (S. Reddy 2018, private communication). Perhaps the neutron star forms with a powerful field generated by convection which then decays until the crust forms and a residual field is locked in?

To explore this speculation, the explosion of the same 20 ${M}_{\odot }$ star was modeled again while assuming a large magnetic field, $2\times {10}^{15}$ G during the first 10⁴ s but only 4 × 10¹³ G thereafter. The initial rotational energy of the neutron star was either 2 × 10⁵¹ erg (Model 20E) or 15 × 10⁵¹ erg (Model 20F) but in both cases that rotational energy has decayed to about 6 × 10⁵⁰ erg after 10⁴ s. In the high energy case, half of the initial energy was deposited in 650 s. In the low energy case, half the energy was deposited in 5000 s. The actual timescales are not so relevant so long as: a) most of the energy is deposited in a few helium core expansion timescales, and b) the neutron star retains 6 × 10⁵⁰ erg of rotational energy after a few weeks.

For Model 20E, the light curve is very similar to the case with the same mass loss and constant magnetic field, Model 20D, the blue and green light curves in Figure 10. Doubling the final kinetic energy does not greatly affect the light curve but the velocity profile is changed in an interesting way. The hydrogenic envelope expands only slightly faster but the helium and heavy element core now move much faster, at a nearly constant speed close to 4000 km s⁻¹. This reflects the fact that the energy deposited in the core after the initial shock has exited is greater that from the shock itself. As a result the entire core is compressed into a thin shell. The inner 2 ${M}_{\odot }$ of ejecta moves with speeds between 4200 and 4400 km s⁻¹. By 10⁷ s, when the explosion is well into the coasting phase, the entire helium and heavy element core (4.1 ${M}_{\odot }$ of ejecta) is compressed into a shell with radius 4.2–5.5 × 10¹⁵ cm moving at 4200 to 5500 km s⁻¹ (Figure 12). If this shell were ionized—which it was not in the present study, perhaps due to an overly simple treatment of radiation transport—, then it would be optically thick.

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Having opened the door to the possibility of a time-varying field, it is interesting to explore the limits. Arcavi et al. (2017) suggest that a kinetic energy of order 10⁵² erg and a mass ∼10 ${M}_{\odot }$ is needed to explain the evolution of the Balmer lines in iPTF14hls. As Model 20C in Figure 10 shows, even 2 ×10⁵¹ erg is inadequate to give the high observed speeds if the progenitor explodes with a large envelope mass. Although Model 20D retains the large envelope mass and the fiducial low energy magnetar after 10⁴, during the first 10⁴ s it incorporates a magnetar with an initial rotational energy of 15 × 10⁵¹ erg (1.2 ms period) and magnetic field strength 2 × 10¹⁵ G. A similar large energy might also be produced in the form of jets if the explosion is rotationally powered (Soker & Gilkis 2017). Now the hydrogen envelope moves so fast that its average speed is near 8000 km s⁻¹, although with a wide spread. That part of the helium and heavy element core that did not end up in the neutron star moves at a nearly constant 4000 km s⁻¹ in a thin shell with high density contrast (Figure 12). In the KEPLER calculation, the large radius of this matter implies a temperature below that required to ionize helium and the electron scattering opacity is low; however, if this material had an effective opacity near 0.1 cm² g⁻¹, then this shell would be marginally optically thick at 10⁷ s. Matter outside log r =15.73 in Model 20C and log r = 15.65 for Model 20D and in Figure 12 is hydrogen rich.

In summary, given the liberty to adjust the explosion time, the overall fit to the bolometric light curve is pretty good for all the magnetar models considered, although the 1D simulations have difficulty explaining multiple peaks in the observations. These peaks might be a consequence of instabilities at the boundary of the bubble created inside the expanding supernova by the magnetar radiation and wind. This boundary is known to be Rayleigh–Taylor unstable, resulting in a clumpy filamentary structure (Chen et al. 2016; Kasen et al. 2016; Blondin & Chevalier 2017). While this shell remains optically thick, the escape of radiation could be irregular. Alternatively, the magnetar itself might experience superflares (Kaspi & Beloborodov 2017). However, to explain a peak with integrated luminosity ∼10⁴⁹ erg, these flares would need to be about 1000 times more energetic than ever seen before; e.g., in the event of 1979 March 5. Both these possibilities are speculative and would need to be reinforced by future studies; e.g., of the light curve for a multi-dimensional model.

A possible problem with the magnetar models considered here is that the slowest moving hydrogen in all but the high mass, low B-field case (Model 20A) is above 3500 km s⁻¹. Although mixing might reduce this value somewhat for Model 20B, the need to have both 8000 km s⁻¹ hydrogen the first 600 days, which rules out Model 20A, and 1000 km s⁻¹ hydrogen at late times (Andrews & Smith 2018) is highly constraining.

As this paper was submitted, a preprint by Dessart (2018) appeared that cast a much more favorable light on magnetar models for iPTF14hls. Using a blue supergiant model similar to 18BSG here,¹ Dessart calculated a multi-band light curve and spectral history that very closely resembles what was observed. This dramatic good agreement greatly strengthens the case for a magnetar explanation. However, it is still to be understood if such a model is compatible with the narrow spectral lines that are seen at late times by Andrews & Smith (2018), how the early light curve is explained, and the origin of the additional peaks in the observed light curve (Figure 11). The initial light curve might be better understood in the context of a red supergiant model. The additional peaks might reflect multiple break outs at different angles, as would only be seen in a multi-dimensional model (e.g., Chen et al. 2016).

Continuous emission at 10⁴² −10⁴³ erg s⁻¹ lasting 600 days or more, perhaps the defining characteristic of iPTF14hls, is easily achieved in a variety of models, including an ordinary supernova happening in a dense CSM medium (Section 2; Andrews & Smith 2018), PPISN (Section 3; Woosley et al. 2007; Andrews & Smith 2018; Arcavi et al. 2017; Woosley 2017a, 2017b), and magnetar-based models (Section 4; Arcavi et al. 2017; Dessart 2018); see Figures 1, 4, 5, 7, 8, 10, and 11. However, satisfying the spectroscopic constraints is more difficult.

Generic models for CSM interaction (Section 2.1) require a circumstellar mass of about 0.4 ${M}_{\odot }$ with an outer radius of 5 × 10¹⁶ cm impacted by an ejected shell of about 1.0 ${M}_{\odot }$ with kinetic energy near 5 × 10⁵⁰ erg. The velocity of the shock at the interface after 600 days is about 6500 km s⁻¹ (Equation (5)). At late times, the luminosity declines as t^0.3 (Equation (8)) and the velocity, as t^−0.1, although both of these scaling relations depend upon uncertain density distributions in the ejecta and CSM that could vary with time. Early on, one would expect narrow lines in the spectrum characteristic of the CSM but the pile up of matter in a dense shell moving with the shock speed into an optically thin medium might result in the higher velocity dominating the spectrum. The light curve in Figure 1 assumes a CSM density that varies smoothly as r⁻². Irregularities in the mass loss rate or angle dependence would be required to give structure to the light curve.

Providing a CSM of order 0.4 ${M}_{\odot }$ at a few ×10¹⁶ cm could prove difficult for binary interaction models or wave-driven mass loss (Section 2.2). The necessary timescale for the mass loss is unnaturally short for the former and long for the latter. An exception can be found in a star with mass near 10 ${M}_{\odot }$ that ejects its hydrogen envelope a couple of years before core collapse (Woosley & Heger 2015), although the explosion energies of such low mass stars may be inadequate to produce the observed light curve (Sukhbold et al. 2016).

The origin of distinct spectroscopic components at 4000 and 8000 km s⁻¹ is unclear in the simplest spherically symmetric CSM interaction model. Calculations of the spectra of e.g., the model in Figure 2 are needed to clarify this issue. Modifications to the circumstellar mass density (Andrews & Smith 2018), mass loss history, or supernova central engine could give asymmetric explosions with the different velocities coming from ejecta at different angles.

PPISN, which are just an extreme case of CSM interaction, have several attractive features (Andrews & Smith 2018; Arcavi et al. 2017; Woosley 2017a). Numerous models with initial masses in the range 105–120 ${M}_{\odot }$ are capable of producing continuous light curves with supernova-like brightnesses spanning 500, or even 1500 days (Table 1; Figures 4, 5, 7, and 8). The total energy radiated in light, 2–3 × 10⁵⁰ erg, is the same as in iPTF14hls, Some models, such as Figure 8, show variability due to multiple pulsations and collisions with piled up material from several previous pulses. Others are capable of producing transients decades before iPTF14hls, possibly even in 1954 (Figure 5; Figure 3; Section 3.1). Each PPISN model could, with minor modification, produce the characteristic 4000 km s⁻¹ seen in the iron lines of iPTF14hls. Some, with multiple pulses, could also produce multiple spectroscopic components (Figures 5, 7, and 8). All would occur preferentially in star-forming regions with low metallicity. They are a phenomenon that must happen in nature, albeit rarely, provided only that stars die with the necessary helium cores masses, 50–54 ${M}_{\odot }$. Unfortunately, no one model, here at least, has provided a complete explanation of iPTF14hls without some artificial adjustment of the pulse energies and timing. Particularly problematic was the high velocity hydrogen ($\gtrsim 8000$ km s⁻¹) seen in H_α.

Part of the difficulty is that PPISN models are not easily matched to individual events. Their repeated outbursts amplify small differences in the initial model. The pair neutrino loss rate at 10⁹ K, a relevant post-pulse temperature, depends on temperature to the 14th power. Slight variations in the temperature following a pulse, due, for example, to a minor change in the amount of fuel burned in the previous pulse, have a large effect on the interval between pulses and the light curve. Convective mixing in the core between pulses causes some uncertainty, such as when to mix and at what rate. The passage of a shock wave through the outer layers of the helium core leaves the matter that is not ejected in a state of thermal disequilibrium. Since the temperature there is too cool for neutrino emission, the matter remains stuck in a distended state (Figure 13) that might not be accurately calculated in one dimension. Both the interpulse period and shock hydrodynamics are sensitive to the density distribution in this matter. For these reasons, the interval between pulses and pulse energies were often adjusted in this paper to explore the consequences.

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Given the large masses involved, PPISN energies are relatively anemic and their characteristic speeds are thus slow. The typical total kinetic energies for the relevant mass range are 0.7–1.5 × 10⁵¹ erg (Table 1), shared among two or three pulses. Even with roughly 50% conversion of kinetic energy to light, it is difficult for PPISN to explain light curves totaling more than a few times 10⁵⁰ erg. The maximum kinetic energy in pulse 2 was 6.8 × 10⁵⁰ erg for Model B120 shared by 5.1 ${M}_{\odot }$ of ejecta. This is adequate for the light curve of iPTF14hls, especially if the interval between pulses 1 and 2 was adjusted, but it is too little to boost the necessary solar mass or so of hydrogen-rich material to speeds over 8000 km s⁻¹ (Arcavi et al. 2017). In several cases, increasing the energy of the second pulse by less than 10⁵¹ erg made the difference between an acceptable model and one that lacked sufficient high velocity hydrogen. Given the uncertainties described above, is that a reasonable variation?

The fundamental energy limit on PPISN models is that the pulses exhaust carbon, oxygen and neon within the inner 5 ${M}_{\odot }$ of the presupernova star. In other words, the silicon plus iron core of a PPISN at iron core collapse is, empirically, always near 5 ${M}_{\odot }$ for stars in the relevant mass range (Woosley 2017b). Most of this material is silicon. The iron that is there is mostly made after the pulses are over. Burning a mixture of 80% oxygen and 20% neon to one of 70% silicon and 30% sulfur generates $4.6\times {10}^{17}$ erg g⁻¹, or about 5 × 10⁵¹ erg for 5 ${M}_{\odot }$. Although most of that energy is lost to neutrinos during the interpulse intervals, an overall energy budget of 2 × 10⁵¹ erg would accommodate all the artificial modifications made in this paper.

Several varieties of PPISN models were explored, characterized by their potential ability to explain both the 1954 transient and iPTF14hls (Section 3.1), success at making only iPTF14hls (Section 3.2) with a more recent unobserved supernova, and hybrid models that invoked both a PPISN and a terminal explosion (Section 3.3). In the first case, the interval between the beginning of pulsations and iron core collapse was much longer than the period of pulsational instability. The remnant of the latest explosion would still be a star shining with a luminosity ∼10⁴⁰ erg s⁻¹, perhaps for centuries to come. This is probably too faint to detect in iPTF14hls and it might be confused with circumstellar interaction but it is worth keeping mind for future discoveries. Other models that did not make the 1954 transient produced a collapsed remnant, presumably a black hole of 40–45 ${M}_{\odot }$, either while the light curve was still active or shortly thereafter. The models that made a transient in 1954 were more successful at explaining both the light curve of iPTF14hls (Figures 4 and 5) and its high velocity hydrogen, especially if the velocity in the second pulse was increased by 50%. Models that did not attempt to make the 1954 transient had more structure and, in some cases, lasted longer but were less energetic and their hydrogen-rich ejecta was even slower.

The most speculative class of PPISN models that was considered was a hybrid (Section 3.3), in which a PPISN is accompanied by some sort of asymmetric terminal explosion when the iron core collapses. This terminal explosion could possibly produce a magnetar, which would open up a very broad parameter space but here the model briefly explored invoked black hole accretion (see also Dexter & Kasen 2013; Woosley 2017b). The black hole would be unusual in that it would be about 45 ${M}_{\odot }$, and not the usual several solar masses invoked in common supernovae or the collapsar model for gamma-ray bursts. The matter that accretes would thus come from farther out in the star, where there would be more angular momentum; however, whether that would be adequate to produce a jet remains to be demonstrated. This would need to be a subset of an already rare model where the collapse to iron core occurred within a month or so of the final pulses of the pair instability. Given the necessary condition of previously ejected shells within 10¹⁵ cm being impacted by a polar outflow with equivalent isotropic energy near 10⁵² erg, a very luminous display is generated with high characteristic speeds (Figure 9). A solid angle of 10% with actual explosion energy near 10⁵¹ erg might suffice to explain the high velocity component of iPTF14hls.

If powered by a magnetar (Section 4; Arcavi et al. 2017; Dessart 2018), then the early light curve favors a red supergiant progenitor (Figure 11; Dessart & Audit 2018) and the kinematics, a large energy to envelope mass ratio; i.e., Models 20C—20F but not Models 20A, 20B, or 18BSG (Figure 10; Table 1). In other words, the star needs to have lost most of its hydrogen envelope or experienced a very energetic explosion. In moderately energetic explosions, even for stars with low mass envelopes, the mass of hydrogen moving over at 8000 km s⁻¹ is small (e.g., 0.02 ${M}_{\odot }$ in Model 20C) and could have difficulty sustaining a strong presence in the spectrum for 600 days (Arcavi et al. 2017). However, Dessart (2018) has recently shown that a simple magnetar model with a continuous spread of velocities can produce both a spectral history and light curve in very good agreement with iPTF14hls, although some additional CSM interaction may need to be invoked to explain the irregular light curve and narrow lines in the late time spectrum.

A magnetar model or any model with a central energy source, might also help to explain the existence of a well-defined photosphere at late times (Arcavi et al. 2017; Dessart 2018). If the magnetar emission is rich in hard radiation, then the supernova would resemble an expanding optically thick H II region in which the magnetar continued to ionize a cloud of hydrogen around it. If the photosphere were bounded by the ionization temperature of hydrogen at the densities appropriate to the model, about 6000 K, then the thermal photospheric radius would stay around a few times 10¹⁵ cm, as observed. For CSM models, including PPISN, the light comes from the shock interaction or from a moderately optically thick region just outside of it. With a radius of a few times 10¹⁶ cm and a luminosity a few times 10⁴² erg, the effective emission temperature would only be about 2000 K. The color temperature would need to be quite different from the effective temperature to agree with observations.

The presence of two enduring characteristic speeds, 4000 km s⁻¹ for the iron lines and 6000–8000 km s⁻¹ for hydrogen, is suggestive, although certainly not proof, of two different velocity scales in the problem. A novel magnetar model was thus explored in which the neutron star was born rotating very fast but with a strong magnetic field that decayed appreciably during the first few hours of its life. Something like this must occur if the magnetar is to energize both the prompt explosion and the light curve, although the timescales are uncertain. The delayed energy injection resulted in the entire ejected core of helium and heavy elements piling up in a relatively thin shell moving with constant speed ∼4000 km s⁻¹. Velocities in the hydrogen envelope exceeded 8000 km s⁻¹ in a substantial fraction of the mass. However, in any magnetar model, the concurrent need for high velocity hydrogen during the first 600 days and slow, $\lesssim 1000$ km s⁻¹ hydrogen at late times (Andrews & Smith 2018) may be problematic.

Further observations and calculations will help to clarify the actual nature of iPTF14hls. The three classes of models predict very different remnants. CSM interaction in an otherwise ordinary supernova would presumably leave an ordinary neutron star in a shellular like remnant. PPISN predict either a Wolf–Rayet like stellar remnant with a luminosity of 10⁴⁰ erg s⁻¹ or a black hole that might be accreting and emitting hard radiation. The most specific predictions come from the magnetar model which predicts, three years after the explosion, a pulsar with a bolometric luminosity of 1.3 × 10⁴² erg s⁻¹, period, 10.5 ms, and field strength, near 4 × 10¹³ G. Five years after the explosion, the luminosity and period would be 6.3 × 10⁴¹ erg s⁻¹ and 12.3 ms. Eventually, an appreciable fraction of the power should be radiated in non-optical wavelengths. While the spectrum of a three year old rapidly rotating magnetar is unknown, continued X-ray monitoring at this level is recommended because we could be witnessing the birth of an anomalous X-ray pulsar or even a ultra-luminous X-ray source.

In the CSM models, which include PPISN, the shock waves generating the light should slow with time and this should be reflected in the spectrum. PPISN models predict a characteristic speed of 1000–3000 km s⁻¹ for the envelope ejected in the first pulse (depending on both the pulse energy and the mass of the remaining envelope). This is intriguing given the evidence for such slowly moving material in spectra at very late times (Andrews & Smith 2018). The supernova speed would eventually saturate near that value. In a more common supernova, slower speeds that are characteristic of the pre-explosive wind might eventually appear; however, the supernova itself would retain its high speed.

In the PPISN models, the mass that was ejected consists chiefly of hydrogen, helium, carbon, oxygen, nitrogen, neon and magnesium. Heavy elements such as silicon, calcium, and iron are confined to what existed in the envelope of the star when it was born. No new iron or intermediate mass elements are ejected. Even in the hybrid models with terminal explosions, the heavy elements are presumed to collapse into the black hole. In an ordinary supernova or magnetar model, heavy elements would have been ejected, including a solar mass or so of slowly moving oxygen. Freshly synthesized silicon, calcium, and iron might contribute to the spectrum.

In terms of theory, all of the models presented here would be more definitive with a better treatment of radiation transport. Representative examples are available to those wanting to further explore their spectrum and light curve.

This research has been supported by the NASA Theory Program (NNX09AK36G). The author acknowledges illuminating exchanges with Iair Arcavi, Jim Fuller, and Nathan Smith.