What powered the unusual supernova iPTF15eov?

Stockholm University Department of Astronomy

Bachelor thesis

Spring 2019

What powered the unusual supernova iPTF15eov?

Samuel Gullin Supervisor:

Dr. Ragnhild Lunnan

Abstract

I study in this bachelor thesis the unusual supernova iPTF15eov, from a sample of broadlined type Ic supernovae, analysed by Taddia et al. (2019). This supernova stands out in the sample as very bright and slow, while being spectroscopically classified as a Ic-bl. This high peak brightness, surpassing the common threshold of −21 mag for superluminous supernovae, motivates comparing 15eov to SLSNe, as well as SNe Ic-bl, to determine which of the two classes it better belongs to and if it could be a transition object between the two classes. Due to time constrints, I have not analysed the spectra, but rather looked at the data from Taddia et al. (2019). It is found to be similar to SLSNe in more or less all eplored aspects, especially so in absolute magnitude, temperature, and g-band fall time, while being only notably similar to SNe Ic-bl in g-band rise time, and velocity (although the velocity evolution is much more slowly declining, like a SLSN).

Fits are made of two models for powering mechanism; the generally accepted power source for SNe Ic-bl, the radioactive decay of⁵⁶Ni; and the proposed power source of some SLSNe, the spin-down of a heavily magnetised, quickly spinning neutron star (a magnetar). The only Ni model best fit out of several tries, that was consistent with the low tail, yielded an ejecta mass Mej ≈ 2.6 M and a Ni mass MN i≈ 2.3 M. Such a high ratio of Ni to ejecta mass is almost certainly impossible. The magnetar spin-down model fit yielded two good fits; one with an impossible Mej ≈ 60 M ejecta mass, and one with Mej ≈ 8.4 M, B = 0.9 · 10¹⁴ G, and P0 = 5.8 ms. This second fit (found by manually adjusting an automatic fit) is similar to other magnetar spin-down fits of SLSNe by De Cia et al. (2018), but with a higher magnetic field strength. I find that 15eov more closely resembles a SLSN than a SN Ic-bl, in the properties I have examined.

Contents

1 Introduction 2

1.1 What are supernovae? . . . 2

1.2 Supernova classification . . . 4

1.3 iPTF15eov . . . 6

2 The physics of powering a supernova 6 2.1 Radioactive heating . . . 11

2.2 Magnetar spin-down . . . 12

3 Analysis 13 3.1 How is our supernova different? . . . 13

3.1.1 Rise and fall . . . 13

3.1.2 Expansion velocities and temperatures . . . 16

3.2 Powering the light curve . . . 16

3.2.1 Could it be Ni decay? . . . 18

3.2.2 Could it be a magnetar? . . . 19

4 Discussion and conclusions 21

1 Introduction

This bachelor thesis investigates one particular supernova, iPTF15eov, from a sample of the iPTF survey analyses by Taddia et al. (2019), that was classified as a SN Ic-bl. The goals of the project are to determine if this supernova is well described by its classification or if it has more in common with superluminous supernovae, as well as examine possible powering mechanisms for its light curve. The comparison to SLSNe, and the motivation to such a project as this, is because the peak absolute bolometric magnitude is over the threshold of what is traditionally considered superluminous, which is much brighter than expected of a SN Ic-bl.

This paper begins by explaining what supernovae are, what system of categorising them has grown over the decades, and what the problems with iPTF15eov are. A lot of the introductory theory is owed to Branch & Wheeler (2017), and Stevenson (2014).

The next section handles the physics theory of powering supernovae, explaining two models from Kasen (2017) in more detail, and the underlying maths to be done. Then, the data of iPTF15eov is compared to samples of SNe and the two described models are fitted and the resulting parameters are also compared to samples of SNe. Lastly, there is a discussion section, handling the results and conclusions drawn from them.

1.1 What are supernovae?

For almost two thousand years, people have observed bright events in the sky, appearing like newborn stars and then disappearing again after just weeks or months. These supernova events have always stood out among the rapidly moving comets and seemingly never-fading stars in the night sky, some even visible during daytime. One of the earliest recorded supernovae was observed in East Asia 1054 A.D. It was visible (during the night) for 21 months, and is now known as the Crab Nebula (Branch & Wheeler, 2017).

All (main-sequence) stars start out by burning H into He, and then, as the temperature increases, will eventually reach a phase of burning He into C and O. More massive stars (M & 8M) will continue having more phases C, Ne, O and Si burning. Other elements are involved in each of the burning processes, but they are “named” after the main contributors. As each phase burns, its product will provide fuel for the next process. The burning starts in the core and moves outward, but even in the outer, less dense parts of the star, there is still burning happening. This leads to a layered structure (fig. 1), with Si burning depositing Fe and Ni into the now inert core. Fe is the final product of regular stellar processes, as it would take more energy to burn Fe than it would yield. Note that even if a star is heavy (and hot) enough to initiate C burning, that doesn’t necessarily mean it will end up with Si burning, leaving fewer layers (Rauscher & Patk´os, 2011; Stevenson, 2014).

Figure 1: Schematic picture of the layered structure of a massive star, from Rauscher & Patk´os (2011)

There are (mainly) two different processes that then lead stars to explode into supernovae: thermonu- clear explosion and core-collapse. No matter which way a supernova explodes, it will still look the same on the surface level – a star blew up and is now surrounded by a cloud of material – but it affects very much what kind of supernova it becomes. Contrary to what one might think, these two processes are not what make the difference between the first distinction of SNe into type I and type II.

As a star explodes, it throws massive amounts of its material out around itself, commonly called ejecta, but it doesn’t leave nothing behind, like regular explosives. In fact, only in a few cases will the star explode completely and not leave a remnant.

The first of the two mechanisms usually requires a white dwarf star in a binary system, where it can increase its mass by material transfer from its companion, e.g. a red giant. Eventually, the white dwarf may become very hot and dense. Under these conditions, thermonuclear burning is unstable, and a shock wave or combustion will explode it into a supernova. Since white dwarfs are the late stages of low mass, main-sequence stars (like our sun, which will become a white dwarf after a period spent as a red giant in a few billion years), with very little H left in the outer layers, it will not show much, if any, H as a supernova. This has made them prime candidates for SNe type Ia progenitors for a long time.

There is also the possibility of two white dwarfs in a binary system merging, and detonating, or that the white dwarf might collapse, rather than just thermonuclearly explode, after accretion – under the right circumstances – if it was a very massive one to begin with. (Branch & Wheeler, 2017) (Stevenson, 2014) The second mechanism involves only one star, although this does not mean it can’t be in binary, that is so massive that it undergoes all of the burning phases outlined above, and end up with a Fe core. It is worth mentioning that it is not 100 % pure Fe, there’s also some Ni produced in the reactions, or other contaminants – we are talking about an incredibly hot ball of plasma continuously fusing elements into others, so there are no hard edges. As Fe can only burn by consuming energy, it will weaken the pressure

that supports the core and lead to it imploding. As the core implodes, the shells above it are “free” to fall inwards, but then bounce back and are ejected, forming the supernova. Depending on the mass of the progenitor star, the core will either turn into a neutron star – where the electrons and protons have been forced to fuse together, making neutrons – or a black hole, a fascinating kind of object in need of (even) more research (Stevenson, 2014).

The exact method behind this bounce is complicated and not completely known, but it has become clear that neither a simple shock wave or neutrinos produced in a large quantity pushing the shells would suffice for its complete ejection. More complex models have been suggested. Adam Burrows (University of Arizona) published a model in 1993, that “demonstrated that violent heating could drive convection- like plumes that ascended from above the edge of the neutron star”, but it may be unable to explain more massive stars. Another suggested model (from the same university) involves the unevenly and violently imploding material making the star core wiggle and blast off material as well as kicking the core through the material, as has been observed. (Stevenson, 2014) Some of these more massive core-collapse supernova (types IIb and Ib/c) progenitors are able to shed most their outer layers (envelopes) of H, and sometimes He, due to winds or the presence of a binary partner, and are called hydrogen-poor supernovae or stripped-envelope supernovae. Their light curve shapes are similar to SNe Ia, but less bright and often shows signs of an asymmetric explosion (Pian & Mazzali, 2016).

1.2 Supernova classification

Just like there is not just one type of star, but rather many differing factors affecting their lifetimes, processes and properties, supernovae come in several different flavours. The current system of grouping supernovae is based just on what we can see, i.e. the light emitted by the explosion and its remnant, and the light absorbed by the ejected material. Looking for which elements are present, and their general quantities, is the main way of gathering information about a supernova, but the shape of the light curve and its feauters are also used to find out about a lot of properties of the supernova and its progenitor, i.e. the object that is the cause of the explosion.

The main, and first, method used to differentiate between SNe is spectroscopy, i.e. measuring the spectra of light emitted and absorbed by a supernova. By this SNe were first grouped into two different categories, types I and II as mentioned above; those that showed hydrogen in their spectra were named type II, and those that didn’t are were named type I. (Stevenson, 2014)

It soon became clear, however, that there were finer differences, and subdivisions of the types have been subsequently added. Type I supernovae were divided into Ia, Ib and Ic, depending on the elements present in the spectra; type II were divided into II-P and II-L, depending on the shape of the LC.

More types have also been added, like type .Ia (point one a) cleverly named from the luminosity being approximately 0.1 times that of normal type I SNe; type IIn that show narrow H emission lines; type Ibn that show narrow He lines; and type IIa that initially look like type Ia but later show prominent H lines (Stevenson, 2014). As mentioned in Pian & Mazzali (2016), some parties suggest splitting type IIb into two types depending on the mass of H present in the envelope.

Figure 2: Hierarchy of supernova categories from Branch & Wheeler (2017)

More types are added as new supernovae are found that don’t fit in as well with the existing types, or some other properties are found to not correlate perfectly with them. One such type is Ic-bl, where

’bl’ stands for broad-line; a type like Ic with no H or He, where very high expansion velocities lead to broad spectral lines. They are commonly (as the only SN type) associated with long, soft gamma-ray bursts. Another, newer type of SN that was only defined at the beginning of the century is what is called superluminous SNe. At first they were defined just as SNe with a peak absolute magnitude brighter than

−21 (Gal-Yam, 2012), corresponding to 10⁴⁴erg s⁻¹, but it is turning out to be more complicated still.

They are often 10 to 100 times as bright as normal Ia SNe, and about one ten thousandth as common as normal core-collapse SNe (Stevenson, 2014), and usually not found in the normally SN-dense parts of galaxies where people start looking – hence their recent discovery.

Since this primary definition of SLSNe doesn’t relate to the physical properties of the supernova other than the resulting luminosity, some people advocate redefining them in a different manner. Howell (2017) defines SLSNe as “luminous SNe which cannot be explained by the power sources fuelling traditional (Types I and II) supernovae[...]”, but admits that this leads to the uncertainty of whether to include type IIn as SLSNe, since some are indeed very luminous but the type has been defined and in use since the 80’s.

This is, of course, not enough, and Howell (2017) continues to describe different types of SLSNe. A distinction similar to that of normal SNe can be made between H-poor (SLSNe-I) and H-rich (SLSNe- II). SLSNe-I have similar spectra to SNe Ic, but are more luminous and evolve slower, and are distinct from SNe Ic-BL; SLSNe-II are much more rare, and poorly understood. There might be yet another type coming up, so far known as “SN 2007bi-Likes” which are similar to SLSNe-I but with about half as slowly declining light curves and are theorised to have temperatures high enough to allow for e⁻-e⁺production.

It is not known whether SLSNe and “normal” SNe approach each other continuously or they form distinct groups, especially with the arbitrary defenition of being brighter than −21 mag. SLSNe are usually found in low metallicity and low luminosity galaxies (Branch & Wheeler, 2017), which might be

a clue to their formation. If there is a continuous spectrum of SNe between normal-luminous and super- luminous (in terms of all relevant parameters, not just luminosity), there should obviously be transtition objects, which motivates examining events like 15eov.

1.3 iPTF15eov

So, as one might imagine, there are SNe found which are initially misclassified, or plainly difficult to classify, as they become outliers in any category. One such event is iPTF15eov (henceforth abbreviated to 15eov), from the intermediate Palomar Transient Factory (iPTF), continuing the Palomar Transient Factory (PTF; Rau et al. 2009).

15eov was included in a sample paper by Taddia et al. (2019), studying 34 supernova events from these two surveys. Using the primary method of SNe classification, spectroscopy, these were all classified as type Ic-bl, being the largest such sample from an untargeted survey. Taddia et al. (2019) analyse the light curves (visible in fig. 3 and fig. 4), among other things, and calculate many properties of these events. However, while 15eov is classified as Ic-bl, it is excluded from several calculations and fits and is mentioned many times as being ’peculiar’. In fig. 5, the lumniosity and bolometric magnitude is plotted without the sample from Taddia et al. (2019).

Just looking at the light curve, in comparison to the others in the sample, you can tell that it is an outlier. It is far more luminous (as is clear in fig. 4), and broader than the rest (as is clear in fig. 3), and Taddia et al. (2019) mentions that “if we use the classification code presented by Quimby et al. (2018), the spectra of iPTF15eov are better matched by SLSN-I than SN Ic spectra[...]”. Since it is right on the threshold of -21 mag for the traditional SLSN definition, it would be interesting to see how it compares in other aspects.

The goal of this project has been to analyse the data of 15eov to assess its classification as a SNe Ic-bl and similarities to SLSNe, by comparing its characteristics and looking into the parameters of possible powering mechanisms.

2 The physics of powering a supernova

There are several ways in which a supernova light curve can be powered by its progenitor. As with any explosion, there is a shock wave with a lot of energy at the forefront of the supernova, but the ejected material is now free to expand and cool, and does so adiabatically, so most of the shock wave’s energy is lost there (Branch & Wheeler, 2017). A schematic image of a supernova explosion is fig. 6, showing the shock wave and what happens behind it.

The majority of the luminosity has to come from processes that more slowly releases energy and heats the ejecta. If a SN has a lot of H in its envelopes (types IIP/L), the H will be ionised by the shock wave. Later, when the material cools down, there will be a phase of recombining where energy is radiated almost at a constant temperature of 5500 K, appearing as a plateau of constant luminosity in the light curve (hence the ’P’ in IIP). For types Ia/b/c, where the progenitor radius is comparatively small (about 10⁸to 10¹¹ cm), the adiabatic expansion cools down the material too much for them to be able to be as luminous and long as they are without any other power sources. (Branch & Wheeler, 2017).

The required energy source for stripped-envelope (including type Ic-bl) SNe is known to be the ra- dioactive decay of ⁵⁶Ni. At later times (years to decades later) the decay of⁴⁴Ti may contribute with hard X-rays and γ-rays for types Ib/c SNe, while the decay of⁵⁵Fe could provide energy for type Ia SNe, but they are not significant early on. Core-collapse SNe, on the other hand, form compact objects out of the progenitor core. One such possible object is a neutron star, and it has been proposed that the spin-down of such a star, would be able to provide energy to the SN, powering its light curve. To release such amounts of energy, it would be required to have an initial period in the millisecond range and a very high magnetic field of about 10¹⁴ G; a star known as a magnetar. (Branch & Wheeler, 2017)

Another idea explained by Kasen (2017) is that a star could lose a large amounts of material decades before the SN explosion and then the SN ejecta would collide with this after the explosion and produce thermal energy. Massive core-collapse stars have sometimes been observed to lose a lot of mass before

Figure 3: Simultaneous plot of light curves of SNe Ic-bl, scaled to the same peak brightness, from Taddia et al. (2019)

The whole sample of Ic-bl is plotted simultaneously to show differences (and similarities) in rise and fall times;

note how 15eov is much broader than the rest of the sample.

Figure 4: Simultaneous plot of light curves of SNe Icbl, after corrections (K and extinction) from Taddia et al. (2019)

The whole sample of Ic-bl is plotted simultaneously to show differences (and similarities) in brightness; note how 15eov is much brighter than the rest, about 1 mag brighter than the second brightest

Figure 5: Luminosity and bolometric magnitude plot of 15eov

The luminosity has been fitted with Gassian Processing, described in sec. 3.1. The data is shown in both plots as points with errorbars.

exploding, but it has not been conclusively explained how this happens. They mention that a shell of several solar masses, close to that of the ejecta, at about 10000R could produce a luminous light curve. This is not a very plausible candidate for 15eov however, since such an interaction would yield characteristically narrow emission lines – the opposite of what differentiates SNe Ic-bl from the rest.

For this project, we have explored the first two of these powering mechanisms, radioactive heating and magnetar spin-down. Kasen (2017) provides a method for calculating the expected light curve for a semi-analytic model of a supernova that dates back to Arnett (1982). In this model we assume that the ejecta is a spherically symmetric, homologously expanding, radiation dominated gas bubble, with constant opacity, and that the progenitor’s initial radius R0 is negligible to the expanding material.

Kasen (2017) defines the timescale tsn, the time at which the optical depth falls below c/v (where c is the speed of light), as

t_sn= 3 4π

κM_ej vc

1/2

≈ 14.5κ^1/2_0.1M₁^1/2v₉^−1/2days (1) with κ0.1 = κ/0.1 cm²g⁻¹ being the opacity, M1= Mej/M being the ejecta mass, and v9 = v/10⁹ cm s⁻¹being the ejecta’s characteristic expansion velocity, all scaled to typical values for SNe I. The light curve is then analytically derived to be

Lsn(t) = e^−(t/2tsn)2 Z t

0

Ldepe^(t0/2tsn)2(t/tsn)dt⁰ (2) where Ldepis the energy deposition rate of the powering mechanism.

However, I had trouble reproducing the results in Kasen (2017), so I looked into the equations from Appendix A¹of Valenti et al. (2008), where they are using the same method to fit SNe I light curves, and derived the integral expression myself, resulting in only a slightly different pair of equations:

1Though there is an error here as well; the fraction given in the equation for τmshould be ^3M

3 ej

10E_k instead of ^10M

3 ej 3E_k

Figure 6: Schematic image of a supernova shockwave from Stevenson (2014)

Lsn(t) = 2e^−(t/tsn)2 Z t

0

Ldepe^(w/tsn)2 w

t²_sndw (3)

t_sn=

 κ β · c

^1/2 3M_ej³ 10Ek

!^1/2

= κM_ej βcv

^1/2

(4)

where β =≈ 13.8 is a constant of integration, and Mej, κ, v are again the ejecta mass, opacity and expansion velocity, respectively. The main difference is that extra 1/tsn, that is missing from this integral in Kasen (2017), which is clear both from their eq. (8) on the page preceding the definition and the fact that the time-units cancel out properly if you add it, and the different timescale (originally called τm

in Valenti et al. (2008)) which is the more common definition. We will then use different expressions for Ldep for the different powering mechanisms, but only in the case of an instantaneous deposition of energy, Ldep(t) = Edepδ(t−tdep) is this analytically solvable, so we will have to use a numerical integration (Kasen, 2017).

I have limited the parameters, mainly due to time restrictions, treating the opacity κ = 0.07 cm²g⁻¹ as a constant of the same value as in Taddia et al. (2019), and the expansion velocity v = 17695· km s⁻¹ as a constant set to the velocity at r-band peak, calculated by Taddia et al. (2019) using the method of Fe II line measuring from Modjaz et al. (2016).

2.1 Radioactive heating

Normal thermonuclear and stripped-envelope, core-collapse supernovae – i.e. types Ia/b/c – have light curves predominantly powered by radioactive decay. In the explosion, the heavier elements on the period table are synthesised, many of which are radioactive. Branch & Wheeler (2017) explains that isotopes with a one-to-one ratio of neutrons and protons burn quickly during the explosion, so that this ratio is unchanged, and the main fuel of both core-collapse stars and white dwarfs are isotopes (of Si, and O and C, respectively) with an equal number of protons and neutrons. The most tightly bound element with this ratio is⁵⁶Ni, and it is then produced if the burning is at or near a “nuclear-statistical-equilibrium”.

The large quantities of ⁵⁶Ni decay with a half-life of 6.1 days to⁵⁶Co, then, with a half-life of 77.7 days, to the stable⁵⁶Fe, while radiating energetic γ-rays and positrons that heat up the ejecta, providing a thermal source of energy for the luminosity. This decay is in a sweet spot, not depositing the energy too early during the initial expansion, and not too late when the ejecta has become dilute. (Bersten &

Mazzali, 2017)

As stated in Kasen (2017), a normal amount of Ni to produce for core-collapse SNe would be around 0.01M to 0.1M, while SNe Ia and SNe Ic-bl would produce more around 0.6M. It has been shown that exploding 100M main-sequence stars could potentially produce around 6Mof Ni, and it has been predicted that the thermonuclear explosions of stars with masses in the range 140M to 240M could completely explode the star and produce up to 40 M of Ni, but no examples of such SNe are given, should they exist. These extreme masses of Ni would lead to incredibly long light curve decay times of hundreds of days.

The energy output of this radioactive decay of⁵⁶Ni and ⁵⁶Co is then given by

L56N i(t) = 2 · 10⁴³MN i

M



3.9e^{−t/τN i}+ 0.678

e^−t/τCo− e^{−t/τN i}

ergs s⁻¹ (5)

where τ_{N i} = 8.8 days and τ_Co = 113.6 days are the decays lifetimes of ⁵⁶Ni and ⁵⁶Co respectively (Kasen, 2017).

Inserting Ldep = L56N i into (3) and calculating the light curve for some different parameters is done in fig. 7, to show how doubling and halving their values affect the light curve’s shape. Increasing the timescale lowers and smooths the peak, as well as shifts it forward, increasing the rise time. Increasing the Ni mass, on the other hand, heightens the peak and leads to a slower decay of the tail. This is not surprising as, from (3) and (5), Lsn∝ Mni.

Figure 7: Light curves for different parameters of the Ni decay model tsn is the timescale in days, and mniis the ejected Ni mass in units of solar masses

2.2 Magnetar spin-down

The proposed alternative to radioactive heating, for energising the material of a superluminous supernova, is as mentioned the spin-down of a magnetar. This long-lived engine would continuously emit energy to the ejecta over months or years, showing a slower luminosity decline than that of radioactively powered SNe. Kasen (2017)

Core-collapse stars with lower masses, about 25–40 M, are not heavy enough to collapse into a black hole, but creates another interesting object. As the electron degeneracy pressure (arising from the inability for fermions to occupy the same quantum states as another, described in the Pauli exclusion principle) is the last barrier against collapse to fail, electrons and protons are forced to merge, creating neutrons. To conserve momentum, it has to speed its rotation to initially a few hundred revolutions per second, up as it shrinks from approximately Earth-sized to having a radius on the scale of 10 km. The outermost crust of the neutron star will be made up of iron nuclei and electrons, while the parts further inside have less and less iron and more neutrons. The core 3 km or so are not certainly known, it could be made up of neutrons with free quarks and gluons, or maybe some type of exotic matter. Neutron stars also have very strong magnetic fields, about 10¹²times the magnetic field on Earth, that accelerates particles in the surrounding ejecta and produces shining beams of light that can be spotted as pulsars, giving us a reliable way to measure the spin period of such stars with telescopes (the remnant of the Crab nebula, for instance, is a pulsating neutron star with a period of approximately 33 ms). It is not known why, but in about 10 % of these events, the resulting neutron star has a much stronger magnetic field, about 100 times as strong as those of normal neutron stars. It is assumed, however, that internal convection of the hundreds of billions of degrees hot star would play a part in driving its formation. These magnetars are observed to have spin periods in the tenths of second range, since the immense power of their magnetic field induces currents (and thus a counteracting field) in the surrounding plasma from the supernova, that provides a sort of friction, and further heats the material. Some models suggest that the magnetar would blow enormous bubbles of hot plasma from this interaction, and once they collide with the ejecta there would be a great addition of energy to power the most luminous events, potentially giving their luminosity curves a second peak. More powerful magnetars would blow these bubbles more quickly, and so the peaks would be closer together, and perhaps difficult to tell apart (Stevenson, 2014;

Branch & Wheeler, 2017).

So, to catch all this in a formula, Kasen (2017) models the “rate of rotational energy loss for a fully aligned force-free wind” (Contopoulos et al., 1999; Spitkovsky, 2006), rather than the “classical expression for a magnetic dipole spinning in a vacuum” that relies on an inclination angle between the magnetic

field axis and the rotation axis. The difference is a factor of about 3.5 for an inclination angle of 45^◦, but according to Kasen (2017), the first option probably better describes a magnetar. The luminosity is assumed to equal the loss in rotational energy, and so the deposition function is derived to be

Lmag(t) =Emag

tm

l − 1

(1 + t/tm)^l (6)

where E_mag is the initial rotational energy, l is the magnetic multipole number (here, as often, taken to be a dipole, l = 2). t_m≈ 0.5 B₁₄⁻²P₀² is the characteristic spin-down timescale for a neutron star with initial spin period P0 ms, and magnetic field strength B14· 10¹⁴ G, under the assumptions that the star’s radius Rmag= 10 km and mass Mmag= 1.4M are of typical values. (Branch & Wheeler, 2017; Kasen, 2017)

The total kinetic energy Emag stored in the star’s rotation is given by (Kasen, 2017)

E_mag= IΩ²

2 ≈ 2.5 · 10⁵²(1 ms)²

P₀² (7)

where I is the moment of inertia, Ω is the angular frequency, P0 is the initial period, and M is the mass of the magnetar. Putting 6 and 7 together, we quite simply have

L_mag(t) ≈ B₁₄² · 2.5 · 10⁵²(ms)²

0.5 P₀⁴(1 + 2B²₁₄t/P₀²)² (8) Inserting Ldep= Lmag into (3) and calculating the light curve for some different parameters is done in fig. 8, to show how doubling or halving their values affect the light curve’s shape. As can be seen from (3) and (8), Lsn is not as simply proportional to any of its parameters as in the Ni decay model, but goes roughly as 1/(P₀²+ 2t)² or B14/(1 + 2B₁₄² t)². Increasing tsn (and thus Mej) stretches the light curve downwards and shifts the peak forwards, flattening the whole shape; decreasing it instead pulls the peak upwards and backwards. Showing the same relative behaviour as for the Ni decay model (they are dependent on tsn in the same way, after all). Changing the magnetic field behaves in much the same way, but reversed, and looks to be much more sensitive for smaller values, as the two larger values have roughly the same peak luminosity. Increasing the initial spin period does little more than scale the curve down, and shift the peak very slightly forwards; decreasing it does the opposite.

3 Analysis

As briefly explained in sec. 1.3, 15eov does not ideally fit its classification as a Ic-bl. Since it has a very high luminosity (enough to be considered superluminous), it is interesting to compare it to SLSNe in different regards, and see if it woudl be better described as a SLSN itself, or perhaps as a transition object, indicating a continuous link between the two classes. Or, maybe, it is not very similar at all to SLSNe and is just a very luminous Ic-bl and that classification needs to be expanded.

Using the luminosity data from Taddia et al. (2019), I have calculated some different properties of 15eov to compare it to samples of SLSNe from De Cia et al. (2018); Lunnan et al. (2018); Liu et al.

(2017); and Nicholl et al. (2017), and samples of SNe Ib/c/c-bl from De Cia et al. (2018); Modjaz et al.

(2016); and Taddia et al. (2019). The interest lies in comparing it to SLSNe, and other Ic-bl, specifically, since these are the two classifications of SNe 15eov could possibly belong to.

3.1 How is our supernova different?

3.1.1 Rise and fall

Using Python’s Gaussian Processing package george, I fitted the light curve²and found the supposed peak time. Using this method yields an array representation (of a limited resolution greater than the data) of

Figure 8: Light curves for different parameters of the magnetar model

t sn is the timescale in days, b is the magnetic field in 10¹⁴G, and p is the initial rotational period in ms

what is essentially a function µ(t) that is the numerical fit of the original function, and a corresponding array of σ(t) that is the standard deviation. Thus, the functions f (t) = µ(t) + σ(t) and g(t) = µ(t) − σ(t) would ideally encapsulate the original function with a 1σ (approximately 68.2 %) confidence.

De Cia et al. (2018) included in their paper a plot of the times for SNe to rise by 1 mag to, and fall by 1 mag from their respective peaks against absolute g-band magnitude, comparing a sample of SLSNe to a sample of SNe Ib/c/c-bl. I have included this as fig. 10, with the addition of 15eov.

By making a similar george fit for the g-band magnitude data, visible in 9, I could calculate the g-band peak magnitude mg, and then the absolute g-band peak magnitude Mg, according to

Mg= mg− DM − Ag (9)

D_M = 5 log₁₀(d/10) (10)

where DM is the distance modulus at distance d = 233.3 · 10⁶pc (given in Taddia et al. (2019)), and Ag is the g-band extinction. Ag ≈ 2.036 was calculated by NASA/IPAC (2019), “using the Schlafly &

Finkbeiner (2011) recalibration of the Schlegel et al. (1998) extinction map”. I calculated the error on Mg through regular error propagation, assuming the error on the extinction was ∆Ag= 0.0005 and that the error on the distance modulus ∆DM = 5∆d/(d · log₂10) , i.e. ∆M_g²= ∆m²_g+ ∆D²_M + ∆A²_g, giving me an Mg= −21.42 ± 0.172 mag.

With this, I calculated the rise time, trise = 16.9^+5.9_−1.4 days, and fall time, tf all = 40.6^+0.39_−0.52days, for 15eov in the g-band. The values were calculated by solving µ(t) − Mg,peak− 1 = 0, and the uper and lower bounds by solving µ(t) ± σ(t) − Mg,peak− 1 = 0 Note that the data in and of itself does not suffice, as it does not stretch back far enough in time for this, so it had to be extrapolated (hence the large uncertainty on the rise time).

The light curves of the two supernovae closest in fig. 10 (PTF10vqv and PTF11dij)have been plotted

2Choosing to use the “Matern-3/2 kernel [that] is stationary kernel where the value at a given radius r²is given by k(r²) = (1+√

3r²) exp(−√

3r²)” – george documentation, 2019-05-28 (https://george.readthedocs.io/en/latest/user/kernels/). This has been used successfully to fit other supernova light curves (Inserra et al., 2018).

Figure 9: Plot of the absolute g-band magnitude and the george fit

The dots show the g-band magnitude data, and the green area is the fit function µ ± σ from the Gaussian Processing. The x marks the (global) peak of the fitted function.

Figure 10: Magnitude vs. rise and fall times, comparison to super-luminous supernovae

Note that 15eov might be difficult to find in the right panel, since the errorbars are very small, but it is located at around Mg= −21.4 mag, tf all= 35.4 days

Figure 11: Absolute bolometric magnitude of 15eov, 11vqv, and 10dij

together with that of 15eov in fig. 11, showing that its light curve is indeed very similar to that of iPTF10vqv but presumably a shorter rise time.

3.1.2 Expansion velocities and temperatures

Analysing the spectra for 15eov and comparing it to other SNe is out of scope for this porject, but some of the key quantities influencing the spectra are expansion velocities and temperatures, which are examined below.

As mentioned briefly in sec. 2, Taddia et al. (2019) calculated the expansion velocities of 15eov, by using the method by Modjaz et al. (2016). I haven’t touched these numbers, but plotted them in fig. 12, in comparison to some SLSNe and SNe Ic-bl. It is somewhat clear that while the values of the velocity are more typical for the quicker SNe Ic-bl, the general shape more closely matches the SLSNe, remaining approximately constant. The the difference between the largest (mean) value 20600 km/s and the lowest 17000 km/s, is only about 3600 km/s, while the errors are 1500 km/s.

Taddia et al. (2019) also included measurements of the temperatures of the sample of SNe Ic-bl. Like with the velocities, I have just plotted them together with temperatures of a sample of SLSNe from Lunnan et al. (2018) in fig. 13, highlighting 15eov. It follows the trend of SLSNe, with a high initial temperature quickly falling, ending up between the SLSNe and SNe Ic-bl late-time temperatures. Two data points have been removed from the plot, as they were most certainly too high and poor measurements.

3.2 Powering the light curve

In order to further pin down the characteristics of 15eov, I have looked into two different possible pow- ering mechanisms, explained in section 2. I used the standard scipy Python package and its numerical integration tools to calculate the light curves according to the equations 3, 5, and 8 layed out in sections 2.1 and 2.2.

I made basic, iterative grid searches in looking to minimise the reduced chi-squared statistic χ²_ν as defined below:

Figure 12: Expansion velocities of 15eov, together with samples of SLSNe and SNe Ic-bl

Figure 13: Temperatures of 15eov, together with samples of SLSNe and SNe Ic-bl The errorbars on all but 15eov have been removed to increase visibility.

χ²=X

i

(Oi− Mi)²

σ²_i (11)

χ²_ν =χ²

ν (12)

where O is the observation data, M is the model data, σ² is the variances, ν = n − m is the degrees of freedom; the number of observations minus the number of parameters fitted.

The limits for the grid search are “soft”, i.e. it is possible for the grid search to overstep the initial boundaries if any one iteration finds the (current) best fit to be close enough to either. Once each iteration with boundaries ai ≤ x ≤ biis finished, the new boundaries are set as ai+1= x−0.5∆x, bi+1= x+0.5∆x, where x is the current best value and ∆x is the step size, which is divided by a set number each iteration.

It is worth noting that I offset the data by 2 days, to counteract a small “notch” feature at the beginning of the curves these models produce (it is small, but noticable if examined closely), probably due to some technical limit. The data looks more like it is rising very quickly already at the first data point, rather than easing into it, so I thought it justified to do so, but the offset is the same for all models and is not counted as a variable parameter.

3.2.1 Could it be Ni decay?

For the Ni decay model, equations 3 and 5, there are two parameters to iterate over: tsn and MN i. Though I use tsn as a parameter, Mej is calculated for every tsn (through 4) and shown in each plot, as it is a more tangible metric, and also compares nicely to the second parameter MN i. Important to note here is that I have not enforced any restraints to have the model make physical sense. In the real world, a supernova simply cannot produce ejecta with a total mass lower than the mass of Ni contained within it (unless it were able to somehow produce negative mass as well), but I did not have the automatic fitting algorithm check for this.

Figure 14: Four fits of Ni-decay models

tsn is the timescale in days, and m is the Ni mass in solar masses. The light curve of 15eov is shown as black points, while Lsn(t) for various parameters is shown as a solid, blue line. In the top-left panel (A), a best fit focusing on the peak (t < 60 days) is shown. In the top-right panel (B), the range is extended to 80 days, and in the lower-left panel (C), the fit is of the whole range. In the lower-right panel (D), the best fit from Taddia et al.

(2019) is shown. For each parameter set, except the one in D, the reduced chi-squared statistic, from 11, of that fit is shown.

There are four attempts to fit a Ni decay model to 15eov shown in fig. 14. The first parameter combination (fig. 14A) is the best fit of the peak area (where t < 60 days), done with the soft limits (1 day ≤ tsn≤ 100 days, 0.1 M≤ MN i≤ 10 M) but the plot for this extends to cover the whole data set.

As is clear from the figure, the tail is way too high and declines way too slowly, resulting in a very large χ²_ν = 30.5. The resulting parameters were tsn ≈ 13.3 days ⇐⇒ Mej ≈ 7.0 M, and MN i ≈ 5.3 M, which is physically possible.

Widening only the fitting area to 80 days gave much the same result, but the light curve is lowering itself to better fit the points at the end, with tsn≈ 11.8 days ⇐⇒ Mej≈ 5.4 M, MN i≈ 4.4 M, and χ²_ν ≈ 8.2, which is also physically possible.

Fitting then over the whole range further shrinks the light curve, yielding a much better fit if the reduced chi-squared statistic is to be believed. The parameters were t_sn≈ 8.1 days ⇐⇒ Mej≈ 2.6 M, M_{N i} ≈ 2.3 M, and χ²_ν ≈ 2.0, which is still physically possible and a pretty good fit, by the numbers.

The problem here is obviously that it is predominantly fitting the curve to the last few points, where the errors are much smaller. This would give some sort of upper limit on the amount of synthesised Ni, however, as any M_{N i} & 2.3 would shine too brightly in the tail. Taddia et al. (2019) found that the average value for the Ni mass is 0.31 ± 0.16 M, which is much, much lower than this.

Taddia et al. (2019) provided a fit with Mej= 6.1 M, MN i= 5.1 Mof the same model, from Arnett (1982), up to about 60 days after peak brightness, shown also in the figure. Their results are not far from my own, but they might have used a different method to fit to the curve, and probably a different offset.

3.2.2 Could it be a magnetar?

For the Ni decay model, equations 3 and 8, there are three parameters to iterate over: t_sn, B₁₄, and P₀. Again, M_ej is written in the plots, rather than t_sn, for much of the same reason.

Fitting a magnetar model to 15eov proves to be, in a way, much easier. The grid search was done over (8 days ≤ t_sn≤ 16 days, 10¹³G ≤ B ≤ 1.5 · 10¹⁵ G, 2 ms ≤ P₀≤ 20 ms), and the best fit yielded t_sn ≈ 13.1 days if f M_ej ≈ 7.8 M, B ≈ 10¹⁴ G, and P₀ ≈ 6.2 ms. The reduced chi-squared statistic for this, however, was below 1 at χ²_ν ≈ 0.50, indicating overfitting. I felt (personally, with my human feelings), that the curve should be nudged a bit, so I experimented ith some manual fits, and felt more happy with tsn≈ 14.6 days ⇐⇒ Mej ≈ 9.8 M, B = 0.9 · 10¹⁴G, and P0= 5.8 ms, giving me χ²_ν ≈ 0.63.

To try to avoid this possible overfitting, I changed the grid search algorithm to look to minimising

|χ²_ν− 1| instead, and got served a solution with χ²_ν ≈ 1.0, where tsn≈ 12.3 days ⇐⇒ Mej≈ 6.99 M, B ≈ 10¹⁴ G, and P0≈ 6.5 ms, which is better by the numbers, but I still think my manual fit looks the best, especially when considering the points between the peak and day 60. It is possible that it more poorly fits the first few points, which alos have low errors, as well as being too high for the last, third last and fourth last points.

Figure 15: Three magnetar model fitted to 15eov

The dashed line is the first attempt at fitting the magnetar spin-down model, minimising χ²ν(from eq. 11). The solid line is the second attempt, minimising |χ²_ν− 1|, instead. The dotted line shows what my human opinion

“feels” is right.

These are all shown in fig. 15, with the two overfitting tries being dashed and dotted. Unlike in the case for the Ni decay model, there’s no real need for compromising; the magnetar spin-down model looks to fit 15eov very well.

The parameters for my manual fit were then plotted in a diagram comparing 15eov to other magnetar fits of SLSNe done by De Cia et al. (2018), in fig. 16. The two SNe most similar to 15eov in rise and fall time (fig. 10 and fig. 11) are highlighted, but are not as similar to 15eov in these properties as previously.

Their errorbars are simply the minimum and maximum values, assuming that the expansion velocity v is in the range (10000 km/s, 20000 km/s), as I could not find a specific number for this in De Cia et al.

(2018). 15eov lands close to several other SLSNe, especially in the B vs. M_ej plot. The light curve for those same parameters is plotted together with light curves made from the parameters of PTF10vqv and PTF11dij in fig. 17. Here, it is clear that “the magnetar model does not describe well the light curve of PTF11dij, both for the late-time decay and the early rise”, as De Cia et al. (2018) puts it. However, the light curve for PTF10vqv is similar to 15eov, despite the different parameters. This hints at the possibility of this not being a unique solution for 15eov. As we are grid-searching with three parameters,

Figure 16: Magnetar fits from De Cia et al. (2018), with the addition of 15eov 15eov is plotted twice with two “good” fits. PTF10vqv and PTF11dij are highlighted.

and quite bluntly so, where each iteration only “zooms in” on the one currently best combination, it is entirely possible that it could miss a different good solution.

So, I did another grid search around the values for PTF10vqv, (22.9 days ≤ t_sn≤ 36.9 days, 10¹³ G ≤ B ≤ 4 · 10¹⁴ G, 2.22 ms ≤ P₀≤ 4.22 ms), and got a second possible solution of χ²_ν≈ 1.0: t_sn≈ 36 days

⇐⇒ M_ej ≈ 58 M, B = 1.4 · 10¹⁴ G, and P₀= 3.0 ms, which is also plotted in fig. 17 and fig. 16. The problem with this solution is the enormous ejecta mass of almost 60 M. As stated above, Stevenson (2014) mentions that stars below 40 M are not heavy enough to collapse into black holes, which implies that this is not a possible magnetar model (the same goes for the poor fit of PTF11dij, with a 70 M ejecta); no SN in the sample from

citetdecia2018 even goes beyond 40 M.

4 Discussion and conclusions

To summarise: it is clear that 15eov resembles a SLSN in more sense than one. The first, and most clear, similarity is the absolut peak magnitude being over the traditional limit of what is considered superluminous, −21 mag. The second most noticable thing is that the light curve shape is broader than the rest of the sample. 15eov has a longer rise time than most SNe Ic-bl and many SNe Ib/c, while being on the quicker than many of the SLSNe, but the fall time is closer to the average fall time of the SLSNe, slower than most SNe Ib/c and all SNe Ic-bl. The temperature data very closely matches that of other SLSNe, and the expansion velocity is very high – typical of SNe Ic-bl – but retains an almost constant value with a slow decline, something more typical of SLSNe.

SNe Ic-bl are generally accepted to be powered by the decay of synthesised Ni, but that model can’t very well fit both the slow and broad luminosity peak and the quick post-peak falloff. It is possible that the ejecta could lose energy by becoming transparent to γ-rays, so that not all energy from the Ni goes to power the luminosity, or by some other means power the peak with a slow-rising and suddenly falling method, on top of the more conservative Ni estimates of around 2.3 M.

Though, it is looks more likely that it is powered by a magnetar formed during the collapse – a power source proposed for SLSNe – as several “good enough” fits are easy to find by hand, and a grid search

Figure 17: Calculated, fitted light curves for 15eov, PTF10vqv, and PTF11dij

The light curves predicted by the magnetar model fits for PTF10vqv and PTF11dij from De Cia et al. (2018), and two versions for 15eov. The luminosity data fro 15eov is plotted in grey points.

revealed a very good fit (and one very good, but unlikely possible fit). The parameters of these fits puts 15eov yet again in the neighbourhood of SLSNe, though still not perfectly blending in. While both the magnetic field strength and ejected mass fit snugly with the rest, it has an initial spin period that is higher than all other SNe with simliar ejecta masses.

The spectra identifies 15eov as a type I, meaning that it shows little or no H. Maybe it could have lost it to an even larger companion, or possibly through some other method, but there is little else than this and its spectra’s broad features it clearly has in common with other SNe of this classification.

Ideally, one would of course want to have larger samples to compare to, not to mention more 15eov- likes to see how abrupt this transition is. SLSN as a type is relatively new, and very rare events to begin with, but then they also tend to show up in low luminosity and low metallicity galaxies, that are not as well represented in untargeted surveys. To better analyse the samples we have, more properties could be compared – mainly the spectra, that were not directly analysed at all in this project due to time constraints, rather than proxies like temperature and velocities. Then, the spectroscopical similarities between SLSNe and SNe Ic-bl could be explored, and used to see where an event like 15eov would land in this cmoparison. Perhaps looking through existing data for extra luminous SNe Ic-bl instances, or fast, Ic-bl-like SLSNe could provide more insight.

One could also do a more rigorous fitting of the models, with more variable parameters, and with different assumptions for more advanced models. The grid search method I used has the very big flaw of only closing in on a single value, missing any potential non-uniqueness of the solutions found (as seen with the magnetar model). It was also quite rough, in the sense that each iteration shrunk the search interval by a large factor to save time, especially when exploring with more parameters. Another smaller problem is that the Python integration actually hit an overflow error at about e⁷¹⁰. It did not seem to matter for these fits, but some potential light curves exploded to infinity. With more time, computing power, or formula rewriting, this could be avoided.

The bulk of the problem with our current SN system, is having several different properties to classify SNe, many of which turn out to not be mutually exclusive. 15eov could be described as a superluminous Ic broad-line, but perhaps it is better to introduce new classifications that better encapsulate the core

properties of the members of each class. There will always be outliers, strange stars exploding under unusual conditions or on the border of what is typical for otherwise similar stars, but once we find enough of these it is probably time to revise our old systems.

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