# Bare Collapse, Formation of Neutron Star Binaries and Fast Optical
Transients

Ron Mor1 Eli Livne1 Tsvi Piran,1
1Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem,
91904, Israel E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>mail<EMAIL_ADDRESS>
(March 2022)

###### Abstract

“Bare collapse", the collapse of a bare stellar core to a neutron star with a
very small mass ejection links two seemingly unrelated phenomena: the
formation of binary neutron star (BNS) systems and the observations of fast
and luminous optical transients. We carried out calculations of the collapse
due to electron-capture of both evolutionary and synthetic isentropic bare
stellar cores. We find that the collapse results in the formation of a light
$\sim 1.3\textup{M}_{\odot}$ neutron star and an ejection of $\sim
0.1\textup{M}_{\odot}$ at $\sim 0.1c$. The outer shell of the ejecta is
composed of 56Ni that can power an ultra-stripped supernova. The models we
explored can explain most of the observed fast optical flares but not the
brightest ones. Collapse of cores surrounded by somewhat more massive
envelopes can produce larger amounts of 56Ni and explain brighter flares.
Alternatively, those events can arise due to interaction of the very energetic
ejecta with winds that were ejected from the progenitor a few days before the
collapse.

###### keywords:

Neutron stars, Supernovae, Binary Pulsars

##  1 Introduction

Numerous evidence accumulated over the years for the existence of two channels
for neutron star (NS) formation (Podsiadlowski et al., 2004; Piran & Shaviv,
2005; van den Heuvel, 2007, 2011; Beniamini & Piran, 2016; Tauris et al.,
2017). The main channel involves the collapse of a massive star whose envelope
is ejected in a powerful supernova (SN). If more than half of the mass of a
binary system is ejected during the SN of the secondary, as would be the case
in the SN of a massive star with a NS companion, the binary will be disrupted
unless the collapse involves a kick in the right direction. The resulting
binary systems will have a large eccentricity and a significant proper motion.
However, two thirds of BNS systems have low eccentricity and small proper
motion (Beniamini & Piran, 2016). This requires a second formation channel
that operates in the majority of BNS systems. In this channel, the second
neutron star forms without a significant mass ejection and with no kick, e.g.
via an electron-capture SNe (Podsiadlowski et al., 2004). We refer to this
channel as “bare collapse".

The discovery of the binary pulsar system PSR J0737-3039 (Burgay et al., 2003;
Lyne et al., 2004) confirmed this picture (Piran & Shaviv, 2005). The orbital
parameters of the system (small separation, low eccentricity and location in
the Galactic plane) implied that the pulsar J0737-3039B had a very small
$m<1.4-1.5\textup{M}_{\odot}$ progenitor and it was born with very little,
$<0.1\textup{M}_{\odot}$, mass ejection and with no kick111Detailed
calculations suggested that the progenitor of pulsar B, just before the
collapse, had a mass of $\approx 1.37\textup{M}_{\odot}$ surrounded by a
tenuous envelope of lighter elements of $0.1$-$0.16\textup{M}_{\odot}$
(Dall’Osso et al., 2014). Observations of the predicted very small proper
motion, $10$ km/sec (Kramer et al., 2006; Deller et al., 2009), confirmed this
scenario. The rate of these events, as inferred from the fraction of binary
pulsars from the total number of pulsars suggests that bare collapses require
unique progenitors (occurring for example in a very narrow mass range or
specific metallicity) or unique process taking place during the binary
evolution prior to the collapse (Tauris et al., 2017).

The very small mass ejection inferred in particular in PSR 0737-3039, but also
indirectly in the majority of Galactic binary pulsars, suggests that these
events can lead to fast optical transients powered by 56Ni radioactive decay.
For large brightness and short duration, the requirements are that a
sufficiently large fraction of the ejected mass is 56Ni and that the ejection
velocity is large enough. Such fast optical transients (see e.g. Drout et al.,
2014; Arcavi et al., 2016; Pursiainen et al., 2018) are characterised by a
fast rise time of a few days and a peak luminosity comparable but typically
somewhat lower than regular SNe. The fast rise time limits the mass ejection
involved and in many cases the inferred mass is smaller than the amount of
56Ni needed to explain the luminosity (Arcavi et al., 2016). However, these
estimates assume a regular SN behavior. Bare collapses ejecta might be
different.

In this work we simulate the collapse of a bare stellar core for both
evolutionary (Jones et al., 2013; Tauris et al., 2015) and isentropic initial
conditions. We use a modified version of the hydrodynamics code Vulcan (Livne,
1993), that includes both nuclear reaction chain and neutrino transport. We
show that these bare collapse produces neutron stars while causing low mass
ejection, hence consistent with the formation channel of most BNS systems. We
focus on calculations of the ejected mass and its composition and velocity.
Using these results we estimate the optical transient that arises from the
56Ni decay within this ejecta. The paper is structured as follows. We begin
with a review of previous work in §2. This work is divided to two groups. The
first deals mostly with the collapse and the subsequent mass ejection while
the second deals with the nucleosynthesis and the resulting ultra-stripped SN
light curve. We continue in §3 with a description of our methods. In §4 we
present our results concerning the collapse and the nucleosynthesis within the
ejecta. In §5 we describe the resulting optical transient and compare it to
observations. We discuss the results in §6. We conclude and summarize in §7.
In Appendix A we discuss the effect of neutrinos.

##  2 A brief sumamry of earlier work

Electron capture supernova (ECSN) takes place when electron capture, i.e.
$e+p\to n+\nu_{e}$, reduce the electron degeneracy pressure in a O-Ne-Mg
degenerate core of a massive star leading to collapse. In accretion induced
collapse (AIC), a white dwarf (WD) accretes mass from a companion star, until
it reaches the Chandrasekhar mass and collapses. Depending on details of the
progenitor, AIC results in either a neutron star and a possible ejection of a
small fraction of the star’s mass, or in a type Ia supernova.

From a computational point of view, but not from an astrophysical one, the
ECSN of a bare core in which almost all the stellar envelope was lost via
winds or due to interaction with a companion prior to the collapse, is very
similar to AIC (when it does not result in a type Ia SN). In both cases, a
progenitor of approximately Chandrasekhar mass collapses, and once the
collapse begins the triggering mechanism is forgotten and the collapse is
driven by electron capture at the center. The questions concerned with bare
collapse deal both with the nature of the progenitor and the collapse process
itself. We divide the discussion accordingly.

###  2.1 Progenitor models

Determination the progenitors for bare collapse is a complicated stellar
evolution issue. The critical phase occurs at the very last stages of the
stars’ lifetimes that are evolving at extreme speed. We don’t address this
question in this work and we use evolutionary initial configurations by Jones
et al. (2013); Tauris et al. (2015) as well as isentropic progenitor models.

Nomoto (1984, 1987), evolved helium cores of massive stars in the range of
$\sim 8$-$10\textup{M}_{\odot}$. In all cases, an O-Ne-Mg core was formed. In
the $M_{\mathrm{tot}}=10.4\textup{M}_{\odot}$ case, the core exceeded the
critical mass for neon ignition ($1.37\textup{M}_{\odot}$). However, in the
lighter cases, the cores did not reach neon ignition. Instead, electron
capture took place and the systems ended up in ECSNe.

Jones et al. (2013) revisited the evolution of $8$-$10\textup{M}_{\odot}$
stars. They evolved stars with initial mass
$8.2,8.7,8.75,8.8,9.5,12\textup{M}_{\odot}$. Their calculations showed that
the $8.2\textup{M}_{\odot}$ star evolved into a stable O-Ne-Mg WD, while the
$9.5\textup{M}_{\odot}$ and $12\textup{M}_{\odot}$ stars ended their lives in
type II Fe core collapse supernovae (CCSNe). It was unclear if the
$8.7\textup{M}_{\odot}$ star evolved into a stable WD or induced an ECSN.
However, the $8.75\textup{M}_{\odot}$ and $8.8\textup{M}_{\odot}$ stars ended
as stripped bare O-Ne-Mg cores and collapsed in ECSN. These results suggest
that the promising range of single star progenitors for stripped ECSNe is
narrow, with initial mass $8.7\textup{M}_{\odot}\leq M\leq
9.5\textup{M}_{\odot}$

Tauris et al. (2015) presented a systematic investigation of the progenitor
evolution leading to ultra-stripped supernovae, i.e. SNe whose progenitors are
stellar cores with extremely low helium envelope mass
$<0.2\textup{M}_{\odot}$. Progenitors of this kind can exist in close binaries
in which tidal stripping by the companion star alters the evolution
significantly (see e.g. Podsiadlowski et al., 2004). The initial masses of the
stars considered by Tauris et al. (2015) are, therefore, significantly smaller
than the initial masses discussed for single stars. Tauris et al. (2015)
evolved systems of a $2.5$-$3.5\textup{M}_{\odot}$ He-star with a
$1.35\textup{M}_{\odot}$ NS companion, with different orbit periods. They
found that ECSN only occurred in a limited range of progenitor mass,
$M_{\mathrm{He}}=2.60$-$2.95\textup{M}_{\odot}$, depending on the orbit’s
period.

###  2.2 Collapse simulations

The collapse of Chandrasekhar mass degenerate cores was calculated earlier
mostly in the context of AIC. However, from a computational point of view, AIC
is similar to our scenario. Woosley & Baron (1992) considered a progenitor of
C-O white dwarf of initial mass $1.1\textup{M}_{\odot}$ which was obtained
from Nomoto Nomoto (1986). The WD slowly accreted mass up to approximately the
Chandrasekhar limit and then collapsed. The calculations didn’t reach a very
late stage in the simulation, hence, they found only a tiny amount of ejected
mass ($\approx 10^{-4}\textup{M}_{\odot}$) during the prompt phase of the
collapse, but they estimated that the neutrino-driven wind will eject about
$0.01\textup{M}_{\odot}$.

Using the same progenitor, Fryer et al. (1999) found later mass ejection of
$0.1$-$0.3\textup{M}_{\odot}$ under various assumptions on input physics:
equation of state (EOS), neutrino physics and relativistic effects. They found
that apart from the EOS, non of the parameters significantly changed the
amount of ejected mass, that varied by factors of at most $3$. As for the
effects of the EOS, they showed that changes in the EOS explain the difference
between their results (as well as other similar results such as those of
Hillebrandt et al. (1984) and Mayle & Wilson (1988)) of $\sim
0.1\textup{M}_{\odot}$ mass ejection, and the results of Woosley & Baron
(1992) which used the same progenitor but failed to eject a significant amount
of mass by the shock mechanism222This specifically refers to the shock
mechanism, which caused only $10^{-4}\textup{M}_{\odot}$ ejected.

Most recently, Sharon & Kushnir (2020) used a modified version of the Vulcan
1D code Livne (1993) to calculate AIC, of a synthetic Chandrasekhar-mass star,
with an isentropic core in a hydrostatic equilibrium, focusing on an accurate
treatment of the EOS. They found an ejected mass of a few $\times
10^{-2}\textup{M}_{\odot}$ with an outflow velocity of $0.15$-$0.3c$, and the
ejecta was composed mainly of 56Ni . With no neutrino transport the initial
$Y_{e}=0.5$ was kept throughout their calculations, and as we will see later
this results in all the ejecta being 56Ni .

As we focus on mass ejection it is worth noting that these earlier studies
revealed three main mechanisms for mass ejection in ECSN and AIC (Fryer et
al., 1999). First, in the prompt mechanism, the converging collapse shock
bounces off the core, and the resulting diverging shock ejects the outer
shells. Second, in the delayed-neutrino mechanism, the bounced shock initially
stalls. However, after $20$\- $200$ms, the shock revives due to neutrino
heating and drives mass ejection. Lastly, in the neutrino driven wind
mechanism, neutrino emission by the the newly formed hot proto-neutron star is
absorbed in the outer (less dense) layers of the star, causing mass ejection.

###  2.3 Nucleosynthesis and light curve calculations

A different route to understanding ultra-stripped SNe is to investigate the
long-term expansion of the ejecta and compute the resulting light curve. Such
works do not follow the dynamics of the progenitor through its collapse.
Instead, in such works a large amount of energy is injected to the outer shell
of the progenitor, which then drives the mass ejection. The detailed
nucleosynthesis and the light curve are computed in post-processing methods
after the hydrodynamic calculation. This method is often referred to as
explosive nucleosynthesis simulations.

Moriya et al. (2017) have used this method to compute the nucleosynthesis and
light curve by the collapse of an ultra-stripped progenitor computed by Tauris
et al. (2013). The progenitor was initially a $2.9\textup{M}_{\odot}$ He star,
evolved as a binary of a $1.35\textup{M}_{\odot}$ NS. Moriya et al. (2017)
found that 56Ni of mass $\approx 0.03\textup{M}_{\odot}$ was formed in the
ejecta, and that the rise time of the bolometric light curve was $5$-$10$
days. The progenitor they used is similar to one of the progenitors we use by
Tauris et al. (2015).

Most recently, Sawada et al. (2021) have performed explosive nucleosynthesis
simulations of C-O progenitors of core mass $1.45$-$2\textup{M}_{\odot}$
computed by Suwa et al. (2015). For the lighter progenitors, they found that
$0.01$-$0.02\textup{M}_{\odot}$ of 56Ni was synthesized in the ejecta, and the
light curves had rise times of a few days.

##  3 Methods

###  3.1 The overall scheme

We use the Vulcan code (Livne, 1993), with some modifications, to carry out
one-dimensional simulations, including hydrodynamics, neutrino transport and
nuclear burning. The hydrodynamics is non-relativistic, and its scheme is
explicit and Lagrangian. We used an adaptive mesh refinement (AMR) mechanism
to allow a dynamical refinement of the mesh in the important regions. AMR was
mainly used to decrease the resolution of the newly formed NS after bounce,
and to increase the resolution in the ejected mass at late times. The initial
grid in our standard simulations consists of 2842 cells, about half of them
describe the inner $\sim 1.2\textup{M}_{\odot}$ which eventually remains
bound, and the remaining cells describe the outer envelop. In this initial
grid, the cell mass was equal to $10^{-3}\textup{M}_{\odot}$ for most of the
star, where the cells at the outer region of the progenitor were increasingly
smaller in mass, as low as $5\times 10^{-5}\textup{M}_{\odot}$. At later
times, once the PNS has formed and no longer affects the ejecta, we gradually
reduce the resolution of it to $\sim 100$ cells. We checked convergence by
reducing the resolution with respect to two main parameters, first increasing
the size of the numerical cells in the initial grid, and secondly increasing
the maximal allowed size of a numerical cell in the AMR mechanism. Reducing
the resolution by a factor of few with respect to these parameters did not
change our qualitative results. For the isentropic progenitor (see §4.1), in
either resolution a PNS was formed, with ejection of $\sim
0.1\textup{M}_{\odot}$, with similar composition. Quantitatively, the results
varied by up to $10$-$20$%.

###  3.2 The Equation of state

The densities in the collapse vary from333At late times of the expansion of
the ejecta, even lower densities are obtained. $\sim
10^{5}{\mathrm{g}}~{}{\mathrm{cm}^{-3}}$ to $\gtrsim
10^{14}{\mathrm{g}}~{}{\mathrm{cm}^{-3}}$. We use two different EOSs for the
different thermodynamical regimes. For high densities, namely the part of the
progenitor which eventually results as part of the NS, we use a tabulated EOS
for nuclear material. The EOS is based on tables originally provided by Shen
et al. (1998a, b) 444This EOS was also used in Sharon & Kushnir (2020).. The
EOS table (compiled by O’Connor & Ott (2010)) utilizes the relativistic mean
field (RMF) theory, and calculates the EOS for homogeneous nuclear matter, as
well as for inhomogeneous matter using the Thomas-Fermi approximation. The
matter is assumed to be in NSE, and to be comprised of a mixture of neutrons,
protons, alpha-particles, a single species of heavy nuclei, and leptons. For
the lower densities, namely the part of the progenitor which is later ejected,
we use the EOS of degenerate electrons gas with free ions.

###  3.3 Neutrino transport

The neutrino transport scheme (based on an unpublished work of Eli Livne)
solves the transport equation implicitly in the co-moving frame, adequately
treating transparent regions and opaque ones. The scheme is inherently built
to approach the diffusion method for opaque regions, and the discrete-
ordinates method $S_{n}$ for transparent regions. For the $S_{n}$ method, we
usually used $5$ azimuthal partitions, i.e. $n=5$, but we checked $n>5$ as
well and found no significant differences in the results. The neutrinos cross-
sections were taken from Burrows et al. (2006). We used $18$ energy groups for
the neutrinos, with the bin cetners between $0.5$-$286.3$ MeV, with a
logarithmic spacing of the size of the energy bins. The size of the $k$’th
energy bin is $\approx 1.29^{k-1}$ MeV.

In our simulations, we considered only $\nu_{e}$ and $\bar{\nu}_{e}$. We
estimate that the effect of the $\nu_{\nu}$ and $\nu_{\tau}$ flavors to be
insignificant for the results we are interested in. These neutrinos are not
involved in nuclear interactions that change the electron fraction.
Furthermore, their absorption cross sections are weaker, reducing their impact
on the ejecta. We expect that the main effect of these neutrinos would be
enhancing the cooling rate in the late phase of the process, once the neutron
star forms.

###  3.4 Nuclear chain

Nuclear burning was calculated using reactions rates based on Rauscher &
Thielemann (2000, 2001) with a network of $54$ isotopes. We validated the
nuclear burning results using SkyNet (Lippuner & Roberts, 2017) as discussed
in §4.1.1.

###  3.5 The progenitors

Numerous possible progenitors may evolve in different stellar evolution
environments, during the last period of star’s lifetime as discussed in §2.1.
We do not solve this question here, but rather consider two different types of
progenitors, namely synthetic isentropic progenitors, and evolutionary
progenitors obtained from detailed stellar evolution studies. We show that the
results for both types are similar. This allows us to use the isentropic
progenitors as a generic model.

Our first type of progenitor is isentropic Chandrasekhar-mass white dwarfs in
hydrostatic equilibrium. For our standard simulations we used a progenitor
with a central density of $2.2\times 10^{10}{\mathrm{g}}~{}{\mathrm{cm}^{-3}}$
and a central temperature of $1.8\times 10^{9}\mathrm{K}$. Fig. 1 shows the
thermodynamic properties of the progenitor.

A second progenitor was evolved by Jones et al. (2013) from He core of a
$8.75\textup{M}_{\odot}$ star (see §2.1). It is a degenerate core of mass
$1.37\textup{M}_{\odot}$, most of it composed mainly of 16O and 20Ne, where
the outer $0.02\textup{M}_{\odot}$ is composed mainly of 12C with some 16O,
20Ne and 24Mg. Fig. 1 shows the initial profile. It has a central density of
$2.27\times 10^{10}{\mathrm{g}}~{}{\mathrm{cm}^{-3}}$ and a central
temperature of $1.75\times 10^{9}\mathrm{K}$, similar to our standard
isentropic star. At its initial state, the progenitor is at the onset of
collapse after electron capture has very mildly started at its center. The
initial electron fraction of the progenitor is slightly smaller than $0.5$ for
most of the star. The material has small inwards velocities, but the kinetic
energy is negligible compared to the gravitational energy of the resulting NS.
As a test, we performed minor adjustments to the initial profile so it would
be in hydrostatic equilibrium. The results of the collapse of this adjusted
profile were very similar to the results of the original profile, and we do
not discuss it further in this work.

We also consider a third progenitor calculated by Tauris et al. (2015). This
progenitor arises as a result of a binary evolution, as discussed in §2.1. It
is considerably less dense compared to the other progenitors we examined, with
the central density lower by a factor of $\sim 100$ than the central density
of the progenitor found by Jones et al. (2013) for the single evolution.
Figure 1 shows the density and temperature profiles of the main three
progenitors we used in this work.

Figure 1: Initial density (full lines) and temperature (dashed lines) for the
isentropic progenitor (blue), and for the evolutionary progenitors of Jones et
al. (2013) (orange) and Tauris et al. (2015) (green). The density profiles for
the isentropic progenitor and for the progenitor of Jones et al. (2013) are
almost identical.

##  4 Results

###  4.1 Isentropic progenitor

We discuss the main results of our simulation for the isentropic progenitor.
The star is initially in hydrostatic equilibrium. As electron captures occurs
spontaneously at the center, a region of low $Y_{e}$ expands from the center
of the star outwards, and the value of $Y_{e}$ decreases in time as electron
capture keeps occurring. This causes an instantaneous reduction of the
pressure due to the removal of degenerate electrons. This process continues
and induces the collapse of material towards the center of the star.
Eventually, material bounces back from the center as nuclear densities are
achieved, launching a diverging shock wave that ejects the outermost layers.

Fig. 2 depicts mass elements trajectories as a function of time. We see that
the star starts its collapse immediately. At $t\approx 0.12$ sec the bounce
occurs and the diverging shock wave forms. Shock breakout occurs at $t\approx
0.2$ sec and the outermost layer of the star is ejected. The remnant is a
compact star of mass $\approx 1.24\textup{M}_{\odot}$ which is initially at
radius $\approx 40$ km. It is a proto-neutron star (PNS) which is still very
hot, and so has a rather large radius. It takes for the PNS tens of seconds to
achieve a standard NS radius, and we will discuss the late time evolution of
the PNS in §4.1.2.

Figure 2: Trajectories of mass elements for the isentropic progenitor. Bounce
and shock breakout occur at $t\approx 0.12$ sec and $t\approx 0.2$ sec,
respectively. Approximately $0.08-0.1\textup{M}_{\odot}$ is ejected off the
star immediately, and additional $0.02-0.04\textup{M}_{\odot}$ is ejected over
$\approx 1\mathrm{\ sec}$.

####  4.1.1 The ejected mass

About $0.08-0.1\textup{M}_{\odot}$ is ejected immediately due to the shock, at
$t\approx 0.2$ sec. Later on, a smaller amount of
$0.02-0.04\textup{M}_{\odot}$ is ejected over a period of $\approx 1$ sec in a
decreasing rate555Additional $3\times 10^{-3}\textup{M}_{\odot}$ are ejected
at later times, mostly around $\sim 2$ sec, (not shown in Figure 2) due to
additional neutrino driven wind.. We attribute this ejection to neutrino
absorption and the neutrino driven wind mechanism. Fig. 3 shows the velocity
profiles of the ejected mass at different times. We see that the ejected mass
accelerates up to time $\approx 0.7$ sec, and then expands homologously. Most
of the ejected mass travels at a small range of velocities, starting from
$1.5\times 10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$ at the inner regions of
the ejecta, up to $3\times 10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$ for the
outermost regions. A tiny amount of mass (about $10^{-3}\textup{M}_{\odot}$)
expands at up to about $7\times 10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$.

Figure 3: Velocity vs mass coordinate at different times. The ejecta
accelerates during the first $\approx 0.7$ sec and then expands homologously,
with velocities between $0.05c$ and $0.1c$ for most of the ejecta.

The ejecta contains at the time of shock breakout, $t\approx 0.2$ sec, only
neutrons, protons and alpha particles. The temperatures behind the shock, at
this time, are between $\approx 3$ MeV for the inner parts of the ejecta and
$\approx 1$ MeV for the outer parts. The temperatures are so high that nuclei
heavier than 4He disintegrate. The densities of the inner regions of the
ejecta are at this time, a few times
$10^{10}{\mathrm{gr}}~{}{\mathrm{cm}^{-3}}$, and these regions are comprised
mostly of neutrons. Upon shock breakout, the material expands and cools.
Starting from the outermost layer, and proceeding inwards as time progresses,
heavier elements, in particular ${}^{56}\mathrm{Ni}$, form as the ejecta
expands and so its density and temperature decrease. This stage occurs at
densities of $\sim 10^{7}\mathrm{g}/\mathrm{cm}^{3}$ and temperatures of $\sim
0.5$ MeV, slightly larger for the inner ejecta compared to the outer ejecta.
For each mass element, the burning phase is very short. The composition
freezes out once the temperature and density drop slightly below this range.

At $t\approx 0.6$ sec the composition of all of the ejecta freezes. The outer
$\approx 0.02\textup{M}_{\odot}$ of the ejecta has $Y_{e}\approx 0.5$ (see
Fig. 4). This region has much lower density than the rest of the ejecta.
Therefore, electron capture is substantially lower and $Y_{e}$ does not change
from its initial $0.5$ value. 56Ni is the most abundant isotope in NSE
composition of relatively dense matter with $Y_{e}=0.5$ (partially because its
nucleus has the same number of protons and neutrons). Hence 56Ni forms in this
outermost region. 56Ni was created only at this very outer shell of the
ejecta, which has important observational implications (see §5). The total
56Ni mass was $\approx 0.02\textup{M}_{\odot}$. The composition of the ejected
mass is shown in Fig. 5, where we focus on the outermost part of the ejecta,
which is the region that 56Ni was formed at and that was validated using
SkyNet (see later). In a small inner shell we see a peak of 58Ni. The rest of
the ejecta, which is most of it, has $Y_{e}\approx 0.4$ (see Figure 4). In our
Vulcan simulations 56Fe is created in this low $Y_{e}$ region, but detailed
nucleosynthesis calculations using SkyNet (see later) show that the
composition in this region is more complicated and comprises of many iron
group isotopes.

Vulcan employs a rather large network of 54 isotopes. However, it is clearly
somewhat limited. We validated the Vulcan results using SkyNet (Lippuner &
Roberts, 2017), which includes 7836 isotopes and about 93000 reactions. We
used the thermodynamic trajectories of different mass elements from the Vulcan
simulation as an input to SkyNet, which solved the nuclear reaction equations.
To account for the neutrino history of the mass elements, we assumed that the
neutrinos were emitted at a time dependent rate from a source at the origin,
as determined by the neutrino luminosity computed in our Vulcan simulations.
The source is specified by a simplistic assumption of a Fermi-Dirac
distribution with zero chemical potential, with an average energy of
$k_{B}T_{\mathrm{source}}$, where the temperature determining the distribution
was taken to be time dependent as well and is defined as follows. At the early
stages after the collapse of up to $\lesssim 1$ sec, when the nucleosynthesis
in the ejecta occurs, the PNS is qualitatively divided to two regions - an
inner dense core of densities over $10^{14}{\mathrm{g}}/{\mathrm{cm}^{3}}$,
and an outer envelope of much smaller densities that still accretes onto the
PNS (see Figure 6). We assume that the inner dense core does not contribute to
the neutrino flux at these early times as it is opaque, while the outer region
is transparent. Therefore, we average the temperature only over the outer
region with densities $<10^{14}{\mathrm{g}}/{\mathrm{cm}^{3}}$, and define the
time dependent source temperature by
$T_{\mathrm{source}}^{4}(t)=\int_{\mathrm{outer\
NS}}T^{4}(t,m)dm/\int_{\mathrm{outer\ NS}}dm$ with respect to the accumulating
mass coordinate. This assumption mimics the presence of the remnant PNS, which
is the source of neutrinos during the time that nucleosynthesis takes place.

We obtained an overall excellent match of $Y_{e}$ between our Vulcan
simulation and the SkyNet simulation, as shown in Figure 4. Fig. 5 shows a
comparison of the mass fraction of 56Ni,58Ni,4He and 56Fe (the main isotopes
that were obtained in the Vulcan simulations) in the outer
$0.0225\textup{M}_{\odot}$ of the ejecta, as calculated by the Vulcan and the
SkyNet simulations. SkyNet produces a similar composition in this outermost
region, with 56Ni being the dominant isotope, confirming Vulcan’s results. In
both Vulcan and SkyNet simulations, the region with $Y_{e}\approx 0.5$ is
precisely where 56Ni is formed666Since $Z/A$ for 56Ni is $0.5$ it is the most
abundant isotope in fluid elements where $Y_{e}\approx 0.5$. However, in
places where $Y_{e}$ is smaller, 56Ni is barely present if at all., and the
two programs agree on the size of this region and its the composition, so
correspondingly the total amount of 56Ni in the ejecta is the same.

Figure 4: Electron fraction profile of the outer $0.1\textup{M}_{\odot}$ of
the ejecta, by the Vulcan simulation (blue) and the SkyNet simulations
(orange). The match is overall excellent, and the region of $Y_{e}\approx 0.5$
corresponds to the region where 56Ni is formed.  Figure 5:  The abundance of
56Ni,58Ni,4He and 56Fe for the outer $0.225\textup{M}_{\odot}$ of the ejecta,
by our full Vulcan simulation (full lines) and by SkyNet (dashed lines). The
results are almost identical for the 56Ni,58Ni and 4He abundance. 56Fe that
appears in the Vulcan results does not form in the SkyNet simulations in which
the lower-$Y_{e}$ region comprises of many different iron peak isotopes that
are not part of our standard Vulcan nuclear network.

####  4.1.2 The Neutron Star

The relevant time scale for the PNS to achieve equilibrium is much longer than
the typical time scale for the ejection of the outer stellar layers or for the
nucleosynthesis in the ejecta. To explore the later evolution as the PNS turns
into a standard NS, we ran a longer simulation with two main changes. First,
the ejecta was removed at $t=0.4$ sec. This is late enough for the dynamics of
shock breakout and the mass ejection to have occurred, so the PNS and the
ejecta are largely decoupled777Late neutrino flux influences the ejecta but
not the PNS. at this stage and we may focus only on the PNS. Second, the
resolution was reduced to allow simulating long physical times with a
reasonable run time. In this section we show results from our standard
simulation up to $t=5$ sec, and from this modified simulation for $5<t\lesssim
30$ sec.

Figure 6 depicts the density and temperature profiles of the PNS for different
times. The inner $\approx 0.7$-$0.8\textup{M}_{\odot}$ approaches nuclear
density almost immediately. This critical point of $m\approx
0.8\textup{M}_{\odot}$ is where bounce occurred during the collapse at
approximately $120$ msec. The outer part of the PNS is initially at densities
lower by $1$-$2$ orders of magnitude. Even after $\sim 1$ sec there are still
some dynamics at this region and matter keeps falling towards the PNS and its
density increases. The temperatures range from $3$ MeV to $22$ MeV peaking at
$m\approx 0.7\textup{M}_{\odot}$-$0.8\textup{M}_{\odot}$, where the bounce
took place and where we saw a qualitative change in behaviour of the density
profile. The temperatures keep increasing for a while as gravitational energy
is released due to the contraction of the PNS that is turning into an ordinary
NS.

The dynamics of the PNS keep going for a long period. The radius of the PNS,
that was approximately $40$ km at $t\approx 1$ sec, decreases to $\lesssim 15$
km at the latest time of our simulation $t\approx 30$ sec, see Figure 7. The
general picture at these late times, which we explain shortly, is similar to
the known theory of PNS (see e.g. Burrows & Lattimer, 1986). Neutrinos
escaping from the outer shells of the PNS reduce the neutrino radiation
pressure and allow the outermost layers of the PNS to accrete on the inner PNS
core. This causes the increase in temperature at early times. Next, the hot
and opaque PNS heats inwards, while electron capture continuously reduces the
electron fraction, until a hot NS is formed. Finally, after tens of seconds
have passed, the hot NS cools down by neutrino emission and the NS becomes
more compact. Note that our simulations included only $\nu_{e}$ and
$\bar{\nu}_{e}$. Inclusion of the $\nu_{\mu}$ and $\nu_{\tau}$ flavors could
accelerate the cooling process (but won’t change the qualitative behaviour).

Figure 6:  Density (top) and temperature (bottom) vs mass for the PNS, at
different times. It takes a few seconds for the outer region ($\approx
0.5\textup{M}_{\odot}$) of the PNS to accrete and reach nuclear densities. At
the first few seconds, the temperature increases as the PNS evolves and
contracts. Then, the PNS heats from the middle towards the center, while
cooling off its outer boundary. The profiles up to time $5$s were taken from
our standard simulation. Later profiles were taken from a similar simulation
tailored to allow simulating long physical times with a reasonable accuracy
(see text).  Figure 7: Accumulating mass vs radius, at different times. It
takes many seconds for the PNS to reach an ordinary NS radius. The profiles at
all times are taken from our lower-resolution simulation (see text), but the
results are the same as in our standard simulation.

###  4.2  An evolutionary single-star progenitor

Next we simulated the collapse of an evolutionary progenitor calculated by
Jones et al. (2013). The collapse of this progenitor resulted in the formation
of a NS and a small amount of mass ejection, qualitatively and quantitatively
very similar to the results discussed in §4.1. The velocity, mass, composition
and electron fraction of the ejecta are very similar for both types of
progenitors. This similarity is reassuring and shows the robustness of our key
result, which is an ejected shell of $\approx 0.13\textup{M}_{\odot}$
traveling at $0.1c$, where the outer $0.02\textup{M}_{\odot}$ shell is
composed of $77\%$ 56Ni.

Trajectories of mass elements as a function of time are shown in Fig. 8. A
remnant of $1.23\textup{M}_{\odot}$ is left. Approximately
$0.14\textup{M}_{\odot}$ is ejected, most of it immediately after bounce. The
ejecta eventually expands homologously with velocities of about $2$-$3\times
10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$ (see Fig. 9). As in the case of the
isentropic progenitor, a tiny amount of mass travels at much larger
velocities.

The composition of the outermost region of the ejected mass at the final time
of the simulation, long after it froze out, is shown in Fig. 10. As in the
isentropic star case, we see an outer shell of $\approx
0.02\textup{M}_{\odot}$ which is composed mostly of 56Ni, followed by a small
shell of 58Ni. Further in, there is a large bulk of 56Fe, although, as
discussed in §4.1.1, this 56Fe bulk is in fact composed of various other iron
group isotopes. As before, the outer shell in which the 56Ni lies has an
electron fraction $Y_{e}\approx 0.5$, while the main bulk has lower electron
fractions $Y_{e}\approx 0.4$.

Figure 8: Trajectories of mass elements for the evolutionary single-star
progenitor of Jones et al. (2013). Approximately $0.14\textup{M}_{\odot}$ is
ejected due to the gravitational collapse, leaving a PNS of mass
$1.23\textup{M}_{\odot}$. The results are very similar to the case of the
isentropic progenitor. Figure 9: Velocity profiles of the outermost region of
the ejecta at different times, for the evolutionary progenitor of Jones et al.
(2013). The results are similar to those of the isentropic star.  Figure 10:
The final composition of the ejecta for the evolutionary progenitor of Jones
et al. (2013). The results are similar to those of the isentropic star.

###  4.3 An evolutionary binary-star progenitor

We also simulated the collapse of a progenitor calculated by Tauris et al.
(2015) which was evolved as a binary companion of a NS. This progenitor has a
slightly larger mass of $\approx 1.45\textup{M}_{\odot}$ (including its
tenuous envelope). In their work, Tauris et al. (2015) evolved the progenitor
using the BEC code, which is usually suitable to follow the evolution only up
to a few tens of years prior to the onset of gravitational collapse (Moriya et
al., 2017). Continuing the evolution, using different codes, shows that the
density of the progenitor increases significantly until it collapses (c.f.
Müller et al. (2018)). Consequently, the initial configuration described in
Tauris et al. (2015) is much larger and has a significantly lower density
compared to the other cases we studied. Therefore, the rate of electron
capture in the center of the star was not sufficient to induce its collapse in
our simulations , when we used this initial configuration. Still, using this
progenitor is interesting as it will demonstrate that our results are not very
sensitive to the initial conditions assumed. To calculate the collapse,
instead of continuing its evolution we simply induced the collapse by
providing the progenitor with an inwards velocity, resulting in kinetic energy
of $\sim 10^{50}$erg. We also managed to induce the collapse in a different
simulation of this progenitor, by artificially reducing the value of $Y_{e}$
to $0.2$ for the inner $5\times 10^{-2}\textup{M}_{\odot}$ of the progenitor,
at time $0$. The collapse took a little longer to occur in this case (bounce
occurred at $t\approx 2.5$s), but the results were very similar. Trajectories
of mass elements as a function of time are shown in Figure 11. Due to the
large radius of this progenitor, the collapse occurs on a longer time scale
and bounce is only at $t\approx 0.7$s. The outer $\approx
0.05\textup{M}_{\odot}$ of the star, which is a tenuous
${}^{4}\mathrm{He}$-${}^{12}\mathrm{C}$-${}^{16}\mathrm{O}$ envelope at a
radius of $\sim 10^{5}$km, did not move significantly. Shock breakout did not
occur during the entire simulation and is expected at time $\approx 10$sec.
Once it occurs this outer region is expected to be ejected as well.

Figure 11: Trajectories of mass elements for the evolutionary binary-star
progenitor of Tauris et al. (2015). The collapse was induced by providing the
progenitor with an inwards velocity. A progenitor of mass $\approx
1.45\textup{M}_{\odot}$ collapses, and a PNS of mass $1.24\textup{M}_{\odot}$
is formed. The outer layers of the progenitor, as well as the tenuous envelope
which did not move significantly during the collapse, are ejected. Shock
breakout did not occur during the entire simulation and is expected at time
$\approx 10$ sec. The results are similar to other progenitors we studied.

Still, despite all of these differences, the collapse resembles the cases
studies in §4.1-§4.2. A NS of $\approx 1.24\textup{M}_{\odot}$ formed. The
electron fraction is $Y_{e}\approx 0.5$ for the outer parts of the ejecta, and
$Y_{e}\approx 0.4$ for the inner parts. At the final time of our simulation,
the divergent shock is located at mass coordinate $m\approx
1.36\textup{M}_{\odot}$. Therefore, as shown in Figure 12, the composition at
the outermost part is the original, 16O-20Ne-24Mg up to mass $\approx
1.39\textup{M}_{\odot}$, and 4He-12C-16O for the envelope. Below it there is a
region which did go through nuclear burning but did not reach nickel. In the
inner parts, the composition is similar to previous cases, with 56Ni at the
outer zones ($\approx 3\times 10^{-2}\textup{M}_{\odot}$ in this case), a
narrow peak of 56Ni, and then a some other iron peak elements where
$Y_{e}\approx 0.4$ (see §4.1.1). The velocities range between $1$-$2\times
10^{9}{\mathrm{cm}}/{\mathrm{sec}}$ for most of the ejecta. These are somewhat
smaller, yet comparable to those found in the previous cases.

Figure 12:  The composition of the ejecta vs the accumulating mass coordinate
for the Tauris et al. (2015) progenitor. The divergent shock is located at
mass coordinate $m\approx 1.36\textup{M}_{\odot}$, and so the outer region is
at its initial composition. Below it there is a region which burnt partially
and did not reach nickel. In the inner parts the composition is similar to
previous cases, with 56Ni at the outer zones ($\approx 3\times
10^{-2}\textup{M}_{\odot}$ in this case) and a narrow peak of 58Ni (see
§4.1.1). The 56Ni abundance is shown in a thick green curve.

We conclude that stars which evolve in binaries may also go through bare
collapse and form a NS with similar mass, while ejecting mass with comparable
properties, hence inducing a similar observed signal.

###  4.4  Different isentropic progenitors

We used the isentropic progenitor as a model to study the collapse of
different stars, focusing on the effect of varying the progenitors’ mass.

We found that lighter progenitors with mass smaller than the Chandrasekhar
mass by up to $\approx 0.1\textup{M}_{\odot}$ (with central density and
temperature as in §3.5) may still collapse, with similar ejecta properties. In
particular, this allows for the formation of even lighter NSs. On the other
hand, light enough stars are stable and do not collapse nor produce NSs. The
particular point of transition from stable to unstable WD (given accurate EOS,
reaction rates and neutrino cross sections) is unclear and should be further
investigated.

Turning to heavier progenitors, we considered progenitors of approximately
Chandrasekhar mass, with an outer envelope added on top. These progenitors are
the same as the original Chandrasekhar-mass isentropic progenitor up to some
point which is approximately at $m_{0}\approx 1.35\textup{M}_{\odot}$, namely
slightly below the original boundary. From this point, we add an outer uniform
shell whose physical properties are the same as those of the mass element at
$m_{0}$. To allow envelopes of different densities, we simply change $m_{0}$
by a small amount. Some examples of initial configurations of heavier
progenitors are shown in Figure 13.

Figure 13:  Initial density for two modified $1.5\textup{M}_{\odot}$
progenitors. One with a higher density envelope (blue) and another with an
intermediate density envelope (orange), compared to the Chandrasekhar mass
isentropic progenitor (green). The density is the same as the isentropic
progenitor until some point, $m_{0}$, slightly below $1.35\textup{M}_{\odot}$.
From this point, we add an outer uniform shell whose physical properties are
the same as those of the mass element at $m_{0}$. To allow envelopes of
different densities, we simply change $m_{0}$ by a small amount.

We note that adding the mass in this way can have significant and non trivial
impact on the fate of the progenitor. For example instead of going through
ECSN it may explode as a type Ia SN. We don’t consider these possibilities
here. Instead, we only try to shed some light on the possible consequences of
collapse of heavier progenitors assuming ECSN occurs in these initial
conditions, without discussing the extent to which these conditions are valid
to describe realistic more massive ECSN progenitors. As such these results
should be taken with a grain of salt and clearly the question what is the
structure of heavier progenitors should be explored more extensively. With all
that being said, one possible realistic scenario for ECSNe of WDs with
envelopes could rise from the merger of two WDs, as suggested by Lyutikov &
Toonen (2019); Lyutikov (2022) (others may include the partial shedding of a
heavier envelope in the pre-collapse evolution due two stellar wind or
interaction with a binary). So although these simplified models are artificial
from a stellar evolution point of view, they can give us indications on what
happens in realistic configurations of this nature.

For progenitors with envelopes whose density is above a certain value, our
simulations resulted in a collapse leading to similar ejecta properties, mass
and composition, as in §4.1, independent of the mass. Specifically, for the
particular case of $\rho_{env}\approx
10^{7}\frac{\mathrm{g}}{\mathrm{cm}^{3}}$, and envelope mass up to
$0.25\textup{M}_{\odot}$, we found mass ejection of
$0.11$-$0.14\textup{M}_{\odot}$, of which $2$-$3\times
10^{-2}\textup{M}_{\odot}$ was composed of 56Ni , at ejecta velocities of
$1$-$3\times 10^{9}{\mathrm{cm}}/{\mathrm{sec}}$. Trajectories of mass
elements as a function of time, and the composition of the ejecta, for a
$\approx 1.5\textup{M}_{\odot}$ progenitor, are shown in Fig. 14 and Fig 15
respectively. The resulting NS is, of course, correspondingly heavier. Fig. 16
shows that the NS mass increases linearly with the progenitor mass with slope
approximately $1$. As the mean mass for NSs in BNS systems is $\approx
1.33\textup{M}_{\odot}$ (Özel & Freire, 2016), these calculations show that
bare collapse is consistent with the formation of BNS systems in this regard
as well.

Figure 14:  Trajectories of mass elements for a progenitor with a high density
envelope, and total mass $\approx 1.5\textup{M}_{\odot}$. The progenitor
undergoes collapse similar to previous cases, with $\approx
0.14\textup{M}_{\odot}$ ejected. This results in the formation of a
$1.37\textup{M}_{\odot}$ NS, which is heavier compared to all cases studied in
previous sections. There is a slight early expansion of the very outer mass
element near the boundary of the progenitor, caused by the initial progenitor
being out of hydrostatic equilibrium.  Figure 15:  The final composition of
the ejecta for a progenitor with a high density envelope, and total mass
$\approx 1.5\textup{M}_{\odot}$. The 56Ni mass, $\approx 3\times
10^{-2}\textup{M}_{\odot}$, and the composition are similar to those found in
previous studied cases. Figure 16: The remnant NS mass vs the progenitor mass.
It is approximately linear with slope $1$, meaning that excess mass in the
progenitor stays bound as part of the NS. This occurs for progenitors with
dense enough envelopes.

In the other limit, where the envelope density is very small (in particular
lying at large radii), the envelope does not affect the collapse or the
resulting NS significantly. Instead, the envelope is simply ejected, without
undergoing nuclear burning.

In the intermediate density regime, we were able to find cases where the
envelope was on the one hand dense enough to collapse (and in particular reach
the suitable thermodynamic state for nuclear burning), but on the other hand
is light enough for most of the mass that was added to eventually be ejected
upon the bounce. Figure 19 depicts the 56Ni mass as a function of the
progenitor mass, for the particular case of $\rho_{env}\approx
10^{6}{\mathrm{g}}/{\mathrm{cm}^{3}}$. It can be seen that about $1/2$ of the
added mass is converted to 56Ni . The largest progenitor we considered, with a
mass of $1.66\textup{M}_{\odot}$, produces $16\times
10^{-2}\textup{M}_{\odot}$ of 56Ni , $5$-$8$ times larger than the amount
formed in the Chandrasekhar-mass progenitors. We also validated these
nucleosynthesis results using SkyNet, by the method discussed in §4.1.1, and
found an excellent agreement. For these progenitors, the mass of the formed
PNS was $1.25$-$1.3\textup{M}_{\odot}$. Trajectories of mass elements as a
function of time, and the composition of the ejecta, for a $\approx
1.5\textup{M}_{\odot}$ progenitor, are shown in Fig. 17 and Fig 18
respectively. The amount of 56Ni formed in this particular intermediate
density regime progenitors , $9\times 10^{-2}\textup{M}_{\odot}$, is
considerably larger from all previous simulations of ECSNe. For instance,
Sawada et al. (2021) argued that is difficult to achieve 56Ni mass larger than
$5\times 10^{-2}\textup{M}_{\odot}$. The particularly high amount of 56Ni in
our simulations may arise from the specific envelope density we have chosen
(together with the simplified model of uniform envelope density). While such
models may not describe correctly realistic ECSN progenitors, they still
suggest that some unique progenitors could synthesize more 56Ni than
previously estimated.

Figure 17:  Trajectories of mass elements for a progenitor with an
intermediate density envelope, and total mass $\approx 1.5\textup{M}_{\odot}$.
The progenitor undergoes collapse similar to previous cases and result in the
formation of a $1.29\textup{M}_{\odot}$ NS. Most of the additional envelope
mass is ejected, yet some of it remains bound and the NS is slightly heavier
compared to the cases studied in previous sections. In this intermediate
density envelope density case, the 56Ni mass is significantly larger. Figure
18:  The final composition of the ejecta for a progenitor with an intermediate
density envelope, and a total mass $\approx 1.5\textup{M}_{\odot}$. The 56Ni
mass is $\approx 9\times 10^{-2}\textup{M}_{\odot}$, significantly larger than
previously studied cases. The nucleosynthesis results were also validated
using SkyNet. Figure 19: The 56Ni mass in the ejecta vs the progenitor mass.
About $1/2$ of the added mass is ejected as 56Ni. The nucleosynthesis results
were also validated using SkyNet. This occurs for progenitors with envelopes
of a limited range of densities.

##  5  Observational Signature

###  5.1 Fast Transients

The gravitational collapse of bare degenerate cores results in many cases in
the ejection of $\sim 0.1\textup{M}_{\odot}$, at about $0.1c$. The outermost
layer of this ejecta is composed of $\sim 0.02\textup{M}_{\odot}$ of 56Ni. The
decay of this 56Ni will produce an ultra-stripped SN with very different
characteristic luminosity and rise time fro a typical SN. We turn now to
consider the observational characteristics of such an outer thin shell of
radioactive nickel, as it expands homologously. We use a simple model that
gives an order of magnitude estimate of the observed bolometric luminosity,
peak time and temperatures.

Consider a shell of mass $M$, radius $R$ and width $\Delta R$, that is moving
homologously so that $R=vt$ and $\Delta R\propto t$. Let
$M_{{}^{56}\mathrm{Ni}}$ be the mass of 56Ni within the shell and $\kappa$ the
opacity of the shell. The optical depth across the shell is given by

$\tau=\int_{R-\Delta R}^{R}\kappa\rho dr\approx\frac{\kappa M}{4\pi(vt)^{2}}\
,$ (1)

where we have assumed that the opacity is constant, the density does not
depend too strongly on $r$, and $\Delta R\ll R$. Since the 56Ni is arranged in
a thin shell this reduces $\tau$ by a factor of $3$, compared to a sphere.

Radiation escapes at time $t$ only from regions whose optical depth
$\tau^{\prime}$ satisfies ${c}/{\tau^{\prime}}>v$. At early times, the
effective observable mass is $M_{\mathrm{obs}}(t)={4\pi cvt^{2}}/{\kappa}$ and
the bolometric light curve rises like $t^{2}$. The luminosity peaks when
radiation can escape from the whole shell, namely at:

$t_{\mathrm{peak}}=\sqrt{\frac{\kappa M}{4\pi cv}}\approx 0.5~{}{\rm
days}~{}\frac{\kappa_{-1}^{1/2}M_{-2}^{1/2}}{v_{-1}^{1/2}}\ ,$ (2)

where we use the notation $Q_{x}=Q/10^{x}$ in cgs units but masses are
measured in $\textup{M}_{\odot}$, velocities in $c$ and time in days.

For $t_{\mathrm{peak}}\ll\tau_{{}^{56}\mathrm{Ni}}$ the peak luminosity is:

$L_{\mathrm{peak}}\approx\frac{M_{{}^{56}\mathrm{Ni}}\epsilon_{{}^{56}\mathrm{Ni}}}{\tau_{{}^{56}\mathrm{Ni}}}\approx
7.7\times 10^{41}\times M_{{}^{56}\mathrm{Ni},-2}{\mathrm{erg}}/{\mathrm{s}}\
,$ (3)

where $\epsilon_{{}^{56}\mathrm{Ni}}\approx 2.96\times 10^{16}$ erg/g is the
energy released per gram of decaying 56Ni , $\tau_{{}^{56}\mathrm{Ni}}\approx
8.8$ day is the mean lifetime of 56Ni , and
$M_{{}^{56}\mathrm{Ni},-2}=M_{{}^{56}\mathrm{Ni}}/(10^{-2}\textup{M}_{\odot})$.
The effective temperature,

$T_{\mathrm{peak}}=\Big{(}\frac{L_{\mathrm{peak}}}{4\pi(vt_{\mathrm{peak}})^{2}\sigma}\Big{)}^{1/4}\approx
1.6\times
10^{4}~{}\kappa_{-1}^{-1/4}v_{-1}^{-1/4}\Big{(}\frac{M_{{}^{56}\mathrm{Ni}}}{M}\Big{)}^{1/4}\
K\ ,\ $ (4)

corresponds to a UV/blue signal. The spectrum will show lines according to the
specific composition typical for bare collapse, which is mostly iron group and
lighter elements. We expect that different bare collapses should have
comparable spectra.

After the peak, the luminosity decreases due to a combination of the decrease
in the amount of the decaying 56Ni , and the leakage of the decay products out
from the expanding system. The latter is more important, leading to a
significantly faster decline than the one arising from the 56Ni lifetime. We
expect the whole system to be optically thin long before we can see the
typical radioactive decay time of 56Ni .

A typical value of $M_{{}^{56}\mathrm{Ni}}=0.02$-$0.03\textup{M}_{\odot}$ of
56Ni is sufficient to explain a fast transient with a peak luminosity of
$\approx 2\times 10^{42}{\mathrm{erg}}/{\mathrm{s}}$ or equivalently a peak
absolute magnitude of $\approx-17$. The largest values of 56Ni that we have
found of $\approx 0.16\textup{M}_{\odot}$ yields a peak absolute magnitude of
$\mathcal{M}_{\mathrm{peak}}\approx-19$. The typical rise time of these events
is 1-2 days.

Combining Eqs. (2) and (3) we obtain the ejecta velocity needed for a given
rise time, and peak absolute magnitude:

$v_{-1}=0.25\kappa_{-1}\frac{M_{-2}}{t_{\mathrm{peak},1}^{2}}\approx
3.2k_{-1}\frac{M}{M_{{}^{56}\mathrm{Ni}}}\frac{L_{\mathrm{peak},43}}{t_{\mathrm{peak},1}^{2}}\
.$ (5)

The minimal required velocity to obtain a transient of given peak time
$t_{\mathrm{peak}}$ and peak absolute magnitude $\mathcal{M}_{\mathrm{peak}}$
for a given $M_{{}^{56}\mathrm{Ni}}$ is shown in Fig. 20 , assuming
$\kappa_{-1}=1$. This minimal velocity is obtained when
$M=M_{{}^{56}\mathrm{Ni}}$. Also shown is the dependence of
$\mathcal{M}_{\mathrm{peak}}$ on $t_{\mathrm{peak}}$ for different 56Ni
masses. Together, this outlines the region in which fast flares can be
explained by bare collapses. These values can explain many of the fast
transients and in particular most of the “golden sample" of Drout et al.
(2014).

Figure 20: Contour map of the minimal required velocity for given peak time
and peak absolute magnitude , obtained for a given outer shell of
$M_{{}^{56}\mathrm{Ni}}$, namely $M=M_{{}^{56}\mathrm{Ni}}$ and
$\kappa_{-1}=1$. The black curves (full lines) correspond to the
values$v/c=0.001,0.003,0.01,0.03,0.1,0.3,1$. Each such curve defines the most
luminous events theoretically possible by our model, for a given velocity $v$.
The black dashed lines show the value of $\mathcal{M}_{\mathrm{peak}}$ for
$M_{{}^{56}\mathrm{Ni}}=0.02,0.03,0.05,0.1,0.16,0.25\textup{M}_{\odot}$. The
bold full curve corresponds to $v/c=0.1$ which is the typical velocity for
most of the ejecta, and the bold dashed line corresponds to
$M_{{}^{56}\mathrm{Ni}}=0.16\textup{M}_{\odot}$ which is the highest 56Ni mass
obtained in our simulations. Also shown in red are some of the PS1 fast
transients (Drout et al., 2014, as circles) and AT 2018cow (as a star).

###  5.2 Other possible observational signatures

####  5.2.1 Very luminous fast transients

The luminosity arising from 56Ni decay in bare collapses can explain many of
the fast transients. For example it is sufficient to explain the “golden
sample" discussed by Drout et al. (2014). However, it falls short of the
brightest ones like AT 2018cow whose peak luminosity is $\sim 4\times
10^{44}$erg/s (Perley et al., 2019) or the brightest fast transients reported
by (Drout et al., 2014) that would require more than $0.3\textup{M}_{\odot}$
of 56Ni .

The ejecta of $\sim 0.1\textup{M}_{\odot}$ travelling at $\sim 0.1c$, has a
significant kinetic energy of the order of $\sim 10^{51}$ erg. This is
comparable to the kinetic energy of a typical (much more massive) SN. Tapping
even a small fraction of this energy can lead to signal that explains the
brightest transients. This could happen if on a time scale of a day or so, the
ejecta collide with material in the proximity of up to $\approx 0.1$ light day
from to the progenitor. Such material could be mass ejected from the
progenitor just prior to the collapse. While we cannot demonstrate that such
mass ejection takes place on this time scale, given the essential extensive
mass loss of progenitors for bare collapse and the fast evolution during the
last stages before the collapse, this is clearly possible. Assuming the
material with which the ejecta collides had a velocity $v_{\mathrm{w}}$, it
would have needed to escape the progenitor up to $\approx
10~{}[v_{\mathrm{w}}/({3000\mathrm{km/sec)}}]^{-1}$ days prior to the
collapse.

If the mass of the ejecta, $\sim 0.1\textup{M}_{\odot}$, is smaller than the
mass with which it collides, this collision could transfer a large fraction of
the kinetic energy to thermal energy. This is about $100$-$1000$ times larger
than the total energy released by the radioactive decay of the 56Ni. This
could easily power even the very rare brightest events. It is important to
note that much smaller mass is needed for this scenario to happen, as compared
to a similar situation with standard SNe which eject much more mass. Moreover,
this scenario can fit nicely with the evolution of bare collapse progenitors
which lose most of their envelope.

####  5.2.2 The Remnants

In the long run, bare collapse events produce a SN remnant (SNR) like
signature. As the ejecta expands it sweeps up mass. Initially it is coasting
at a constant velocity and as it accumulates more and more mass the luminosity
increases like $t^{2}$. The Sedov-Taylor (ST) phase begins at

$t_{\mathrm{ST}}=\big{(}\frac{3M_{\mathrm{ej}}}{4\pi\rho_{\mathrm{ISM}}v_{\mathrm{ej}}^{3}}\big{)}^{1/3}\approx
40~{}{\rm
years}~{}~{}{M_{\mathrm{ej,-1}}^{1/3}{n_{\mathrm{ISM}}^{1/3}{v_{-1}}}}\ ,$ (6)

where $n_{\mathrm{ISM}}$ is the interstellar medium (ISM) density. At this
stage the ejecta have passed through a mass of ISM equal to $M_{\mathrm{ej}}$.
The energy dissipation rate is maximal at $t_{\mathrm{ST}}$. As the mass of
the ejecta is much lower than the mass of a SN ejecta and its velocity is
faster, $t_{\mathrm{ST}}$ is shorter by about a factor of $\approx 10$, than
in a regular SN. This implies that the peak bolometric luminosity of this
bare-collapse SNR, $\approx 10^{42}$ erg/sec, is larger than the peak
luminosity of a regular SNR by a similar factor. This luminosity is comparable
to the peak luminosity of a rare SNR arising from a superluminous SN. Over
even longer time scale the Sedov-Taylor phase continues and it will be
impossible to distinguish this remnant from a regular SNR. In this phase all
that matter is the total energy and the external density and the initial
ejecta mass and velocity are not relevant.

##  6 Discussion

We have shown that the collapse of a bare stellar core of around Chandrasekhar
mass, results in a light, $\approx 1.2$-$1.35\textup{M}_{\odot}$, neutron star
and ejection of $\approx 0.1$-$0.2\textup{M}_{\odot}$ at $\approx 0.1$c. We
considered two evolutionary configurations that were derived from stellar
evolution simulations by Jones et al. (2013) and by Tauris et al. (2015).
Similar results were obtained for an isentropic initial configuration of hot
white dwarf, and even when a light envelope of $\leq 0.35\textup{M}_{\odot}$
was added to it.

Naturally this neutron star is rather light. This is consistent with the
observation that the mean mass of the neutron stars in BNS systems is $\approx
1.33\textup{M}_{\odot}$ (Özel & Freire, 2016). With such a small mass ejected,
a binary composed of this progenitor and a companion neutron star would remain
bound with small eccentricity and small proper motion. The double pulsar PSR
J0737-3039 is a prototype of a system that formed in this way (Piran & Shaviv,
2005). This is also consistent with the observations that BNS systems with
lower eccentricity and lower CM motion, that are expected to form in this
channel, involve lighter neutron stars (van den Heuvel, 2007, 2011).

The collapse ejects $\sim 0.1\textup{M}_{\odot}$ of iron group elements with
an outer shell composed mostly of 56Ni . This outermost region where 56Ni was
formed had $Y_{e}=0.5$, consistently with the equal number of protons and
neutrons in this isotope. This radioactive ejecta results in fast transients
for two reasons. First, the ejecta velocities are $\approx 2$-$3\times
10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$. As the rise time of the light
curve is proportional to ${v_{\mathrm{ej}}}^{-1/2}$, this leads to faster
transients by a factor of $\approx\sqrt{3}$, as compared to traditional SNe
with $v_{\mathrm{ej}}=10^{9}{\mathrm{cm}}~{}{\mathrm{sec}^{-1}}$ (see e.g.
Arcavi et al., 2016). Second, the 56Ni lays in a shell at the outermost layer
of the ejecta. The rise time determined by the expansion of a shell, rather
than a sphere, is faster by a factor of up to $\sqrt{3}$. Moreover, the rise
time was found to depend on the 56Ni mass, rather than the entire ejecta mass.
Together these two factors combine to shorten the transients by a factor of
$\approx 3$, for a given 56Ni mass. For the particular ejecta we obtained in
our simulations, we found rise times of $0.5$-$2$ days.

Chandrasekhar mass progenitors, both isentropic and evolutionary (i.e. evolved
in detailed stellar evolution schemes), induce an ejection of
$0.02-0.03\textup{M}_{\odot}$ of 56Ni, resulting in a peak luminosity of
$\approx 2\times 10^{42}{\mathrm{erg}}/{\mathrm{sec}}$ or equivalently a peak
absolute magnitude of $\approx-17$. This can explain the lower end of the fast
flares. Even higher amounts of 56Ni ejecta were found in (synthetic)
progenitors that included an additional envelope. The largest amount of 56Ni
that we have found was $0.16\textup{M}_{\odot}$, corresponding to a peak
absolute magnitude of $-19$, which already explains a large fraction of the
observed fast transients (Drout et al., 2014).

If the ejecta collides with a massive shell that was ejected within a few days
prior to the collapse then the large kinetic energy of the ejecta, which is of
order $10^{51}$erg, can be tapped (see e.g Moriya et al., 2013; Blinnikov,
2017; De et al., 2018; Fox & Smith, 2019; Leung et al., 2021). This can
explain the brighter end of the transients. In fact the combination of the
radioactive signal and interaction of the fast ejecta with earlier outflow can
explain double peaked fast transients, like iPTF 14gqr-SN 2014ft (De et al.,
2018).

Eventually the ejecta will interact with the surrounding ISM and produce a
SNR. The SNR will peaks on a time scale of $\sim 40$ years with a peak
luminosity larger than a typical peak luminosity of a regular SNR by a factor
of almost ten. However, these are rare events and it will be difficult to
catch on in such a phase. The longer term (Sedov phase and later) of this SNR
will be similar to a regular SNR.

##  7 Conclusions

Analysis of the properties (eccentricity and proper motion) of Galactic binary
pulsars has shown that most neutron stars in these systems formed with minimal
mass ejection and without a kick velocity (Beniamini & Piran, 2016). Motivated
by these observations we have explored here bare collapses that arise when a
stellar core that has lost all its envelope undergoes electron capture and
collapse. Our calculation begin with stellar progenitors (evolutionary and
synthetic) at the end of their life, we calculate the collapse that arises due
electron capture and following the bounce and mass ejection. Nucleosynthesis
and neutrino transport calculations are carried out during the hydrodynamic
calculation. The nucleosynthesis is confirmed later with more detailed
calculations using a much larger nuclear network.

Our main findings are:

  * •

Bare collapse forms a light neutron star within the observed range of mass for
NSs in BNS systems. When this collapse takes place in a the binary, due to the
small mass ejected the binary remains bound, in almost circular orbit, and
will have a small kick velocity.

  * •

The collapse ejects $\sim 0.1\textup{M}_{\odot}$ of iron group elements with
an outer shell composed mostly of 56Ni . Typical velocities of the ejected
shell are $2$-$3\times 10^{9}$cm/sec.

  * •

The typical 56Ni mass in the ejecta is $\sim 0.02\textup{M}_{\odot}$. This
result is consistent with previous explosive nucleosynthesis simulations (e.g.
Moriya et al. (2017); Sawada et al. (2021)).

  * •

For Chandrasekhar mass progenitors, This radioactive ejecta results in a fast
transient whose peak absolute magnitude is up to $\approx-17.2$ and whose rise
time is as short as $\approx 1$ day.

  * •

The addition of a small envelope around the core can increase proportionally
the 56Ni mass in the ejecta. This could create even brighter transients. The
progenitor models used for this finding are somewhat artificial. However, they
demonstrate the possibility of producing more 56Ni than estimated in earlier
studies.

  * •

The kinetic energy of the ejecta, $\sim 10^{51}$ erg can power even brighter
events if the progenitor ejected a significant wind a few days prior to the
collapse.

  * •

The resulting SNR would have an earlier and brighter peak than a regular SNR.
However, at a later stage it will resemble a regular SNR.

Our results demonstrate that bare collapse is a valid mechanism for neutron
star formation and in particular for the formation of neutron star binaries.
They also demonstrate that these events can be observed as fast bright
transients. With transient searches getting better and better sky and temporal
coverage it will be interesting to compare the BNS and fast transients
statistics to see if the two are consistent.

Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

We thank Iair Arcavi, Assaf Horesh, Samuel Jones and Roni Waldman for helpful
remarks. The research was supported by an advanced ERC grant TReX.

##  Appendix A The effect of neutrinos

As discussed in §2.2, there is a common understanding that there are three
main mechanisms for mass ejection in ECSNe; the prompt, the delayed-neutrino,
and the neutrino-driven wind mechanisms. There has been extensive research
comparing between the first two; Various different 1D simulations have shown
that when neutrino physics is included, the bounce shock is stalled and a
prompt explosion fails (Fryer et al., 1999; Baron et al., 1987a, b; Mayle &
Wilson, 1988; Hillebrandt et al., 1984; Kitaura et al., 2006). In most cases,
neutrino heating was able to revive the shock on a time scale of $\sim 100$
ms, comparable to the neutrino diffusion time scales. Consequently, there is a
delay by a similar amount between the time of bounce and the time of shock
breakout. It is suggested that this time delay should be between $20$-$200$ ms
(Fryer et al., 1999). Other simulations (Fryer et al., 1999; Sharon & Kushnir,
2020) showed that when neutrino physics is not included, a prompt explosion
succeeds and shock breakout occurs only a few ms after bounce.

In this short appendix we revisit this topic. We ran two identical simulations
of our isentropic Chandrasekhar mass progenitor, once when neutrino transport
was enabled and once when it was turned off. In the latter case, the electron
fraction remains constant during the entire simulation. Without neutrinos, the
progenitor was stable in our simulations. To induce the collapse, we
artificially reduced the value of the electron fraction $Y_{e}$ to $0.2$ for
the inner $5\times 10^{-2}\textup{M}_{\odot}$ of the progenitor, at time $0$.
To allow a fair comparison, we did the same for the simulation with neutrinos
enabled, even though it would have still collapsed without this adjustment.

Trajectories of mass elements as a function of time are shown in Figures 21\-
22 for the cases without and with neutrinos, respectively. In the case without
neutrinos shock breakout occurs only a few ms after bounce, indicative of the
prompt mechanism. Contrary, in the case with neutrinos there is a clear delay
of nearly $150$ ms between bounce and the ejection of the outermost layer,
indicative of the delayed neutrino mechanism. The amount of delay agrees with
previous works. The inclusion of neutrinos increases the amount of
ejecta888Note that in the simulations shown in this appendix the ejecta mass
is smaller than the ejecta mass obtained in §4. This occurs due to the fact
that here the initial progenitor is modified to have lower $Y_{e}$ in the
center at the beginning of the simulation., which could be a part of the
reason why some previous works report less ejecta than us, c.f. Sharon &
Kushnir (2020).

Figure 21: Trajectories of mass elements for the isentropic progenitor, with
neutrino physics turned off. The collapse is induced by reducing the value of
$Y_{e}$ to $0.2$ at the inner $5\times 10^{-2}\textup{M}_{\odot}$ of the
progenitor, as it is otherwise stable in our simulations. Bounce and shock
breakout occur only a few ms apart, indicative of the prompt mechanism. Only
$\approx 0.05\textup{M}_{\odot}$ is ejected, a smaller amount compared to the
case with neutrinos enabled, and comparable to the amount of ejecta in the
work of Sharon & Kushnir (2020). Figure 22: Trajectories of mass elements for
the isentropic progenitor, with neutrino physics enabled. The initial value of
$Y_{e}$ was reduced to $0.2$ at the inner $5\times 10^{-2}\textup{M}_{\odot}$
of the progenitor, as in Figure 21, for a fair comparison. Bounce occurs
slightly earlier compared to the case without neutrinos, because electron
capture expedites the collapse. There is a clear delay of nearly $150$ ms
between the time of bounce and shock breakout, indicative of the delayed-
neutrino mechanism. The amount of delay agrees with previous works. The amount
of mass ejected is larger compared to the case without neutrinos.

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# A survey on making skylines more flexible

Cem Cebeci  Politecnico di Milano
Milan, Italy
<EMAIL_ADDRESS>
###### Abstract

Top-$k$ queries and skylines are the two most common approaches to finding the
most interesting entries in a homogeneous multi-dimensional dataset. However,
both of these strategies have some shortcomings. Top-$k$ queries are very
challenging to specify precisely and skylines are not customizable to specific
scenarios, on top of having unpredictable output cardinalities. We describe
some alternative methods aimed at adressing the shortcomings of top-$k$
queries and sklyines and compare all approaches to illustrate which of the
desired properties each of them possesses.

Keywords: top-$k$ query, skyline, multi-dimensional optimization

##  1 Introduction

Selecting the most interesting entries from a multi-dimensional dataset is an
important operation in today’s recommender systems [8][15]. In order to
perform this task, the two most commonly used approaches are top-$k$ queries
[7] and skylines [1].

Given a multi-dimensional dataset $R$ and a scoring function
$f:R\rightarrow\mathbb{R}$, the top-$k$ query on $R$ with $f$ returns the $k$
tuples $r_{1},...,r_{k}\in R$ for which $f(r_{i})$ is smaller than any tuple
not included in the query result. Top-$k$ queries are expressive and precise
and we have fast algorithms [5][6] to compute them. However, defining ”most
interesting” as a function of dataset attributes proves to be a significant
challenge, even though techniques such as crowdsourcing [4] provide some
tangible results. Additionally, query results may differ significantly with
slight alterations on $f$ depending on the dataset, which poses a considerable
problem given the natural inaccuracy we have in determining $f$.

On the other hand, a skyline query returns the tuples in a dataset that are
not Pareto dominated. In other words, it picks the tuples $r$ for which there
are no other tuples $t$ such that $t$ is no worse than $r$ for all attributes
and better than $r$ for at least one attribute. It’s worth noting that tuples
in the skyline are the top-1 result for some scoring function. The notion of a
skyline can be extended to a $k$-skyband, the tuples that have less than $k$
tuples Pareto dominating them, or equivalently, top-$k$ results for some
scoring function.

Skylines do not suffer from the same challenge as top-$k$ queries do since
they require no scoring function to be specified. However, due to the same
reason, they are not in any way customizable. There is no way to alter a
skyline query to fit a user’s preferences, which is crucial in recommender
systems. Another shortcoming of skyline queries is that their output
cardinality is unbounded. In fact, for datasets with high dimensionality, the
skyline may comprise the whole dataset [10], which renders the skyline
operator useless. In general, a high number of results is undesirable in
skyline queries.

To address the shortcomings of skylines and top-$k$ queries, multiple
alternative operators have been introduced. In this paper, we describe some of
those proposed solutions and investigate how they relate to each other in
Section 2. We compare the different approaches in Section 3 and conclude in
Section 4.

##  2 Proposed Solutions

###  2.1 Flexible Skylines

As mentioned before, one of the major shortcomings of skylines is
customization. The skyline of a dataset does not depend on any kind of
parameter, only on the dataset itself. As a result, the skyline query can not
be customized in any way. $F$-skylines are a generalization of the skyline
that introduces a parameter to the query while compromising very little from
the simplicity of skyline queries [3].

Before we define what an $F$-skyline is, let us define $F$-domination. A tuple
$r_{1}$ $F$-dominates another tuple $r_{2}$ if and only if $r_{1}$ is at least
as good as $r_{2}$ for all scoring functions in $F$ and better than $r_{2}$
for at least one scoring function. In formal terms:

$r_{1}\prec r_{2}\iff\forall f\in F.\ f(r_{1})\leq f(r_{2})\land\exists f\in
F.f(r_{1})<f(r_{2})$

If $F$ is the set of all monotonic functions, $F$-domination reduces to Pareto
domination.

Given a dataset $R$ and a set of monotone scoring functions $F$, the two
flexible skyline operators are defined as follows:

$ND(R,F):=\\{r\in R\ |\ \nexists t\in R.\ t\prec_{F}r\\}$ $PO(R,F):=\\{r\in R\
|\ \exists f\in F,\forall r^{\prime}\in r.\ f(r)\leq f(r^{\prime})\\}$

The first operator, $ND$, denotes all non $F$-dominated tuples of $R$ and the
second operator, $PO$, denotes the tuples in $R$ that are optimal for at least
one of the functions in $F$. When $F$ is the set of all monotonic functions,
both $ND(R,F)$ and $PO(R,F)$ converge to the skyline of $R$. However, in the
general case, they are not necessarily equal [3].

$ND$ and $PO$ are both customizable since they have an additional parameter,
$F$, that enables customized queries. In addition, they partially address the
other shortcoming of skylines, they lower the output cardinality for non-
universal choices of $F$. One can show $PO(R,F)\subseteq ND(R,F)\subseteq
SKY(R)$ for all choices of $R$ and $F$. Thus guaranteeing neither $ND$, nor
$PO$ will have a greater cardinality than the skyline. Experimental results
[3] depict that the cardinality of $ND$ is dramatically lower than the skyline
even with only a few constraints on $F$ and the cardinality of $PO$ is even
lower.

Customizability and control over result cardinality, the two properties
flexible skylines provide, are already available in top-$k$ queries, although
flexible skylines do not require precisely defined scoring functions as
top-$k$ queries do. As an example, consider a simplified universe of functions
comprising only weighted sums over the attributes with normalized weights,
functions of the form $f(r)=\sum{w_{i}r_{i}}$ where $r_{i}$ denotes an
attribute in the schema of $R$. A top-$k$ query would need to specify a
precise scoring function (or equivalently, a precise vector $w$) but a
flexible skyline allows less precise queries such as
$F=\\{f(r)=w_{1}r_{1}+w_{2}r_{2}\ |\ w_{1}>2w_{2}\\}$. Such constraints are
significantly easier to obtain [12]. For instance, if a user prefers the tuple
$(2,3)$ to $(1,6)$, we can argue:

$2w_{1}+3w_{2}<w_{1}+6w_{2},\ i.e.,$ $w_{1}<3w_{2}$

Flexible skylines merge the idea of scoring functions of top-$k$ queries with
traditional skylines. This is further illustrated by the fact that choosing
$F$ to be the set of all monotonic functions reduces both operators to the
traditional skyline but choosing it to contain a single monotonic function
reduces them to the top-1 query result. By combining the two approaches,
flexible skylines are both easy to specify and customizable at the same time.

The two flexible operators are discussed only for skylines here for brevity,
but [11] and [2] extend this notion to $k$-skybands to provide a more
generalized framework.

###  2.2 Output Size Specified (OSS) operators

Another solution approach [10] to the same problem introduces the notion of
$\rho$-dominance and two operators that utilize this relation to provide types
of queries that are not as precise as top-$k$ queries but still have query
customization and smaller output cardinality than the skyline. In fact, these
operators allow specifying the cardinality of their output as a parameter.

These operators only use weighted sums over attributes with normalized
preferences as scoring functions. They define a new kind of dominance, much
like $F$-dominance. Given an estimate preference vector $w$ and tuples
$r_{1},r_{2}$, $r_{1}$ $\rho$-dominates $r_{2}$ if the following conditions
are satisfied:

  1. 1.

$\forall v\in\Delta^{d-1}.\
|w-v|\leq\rho\implies\sum_{i}{v_{i}{r_{1}}_{i}\leq\sum_{i}{v_{i}{r_{2}}_{i}}}$

  2. 2.

$\exists v\in\Delta^{d-1}.\
|w-v|\leq\rho\land\sum_{i}{v_{i}{r_{1}}_{i}<\sum_{i}{v_{i}{r_{2}}_{i}}}$

In other words, they relax the normalized preference vector $w$ up to a
distance of $\rho$ and check dominance for every preference vector inside the
resulting hypersphere. If a $r_{1}$ performs at least as good as $r_{2}$ for
all vectors and better than $r_{2}$ for at least one, $r_{1}$ $\rho$-dominates
$r_{2}$, much like the concept of domination in flexible skylines.

The first output size specified operator is called $ORD$, it is Output size
specified, has relaxed input and is dominance oriented. Given an estimate
preference vector $w$, an output size $m$ and a dataset $R$, $ORD(R,w,m)$
denotes $m$ tuples in $R$ that are not $\rho$-dominated for the minimum value
of $\rho$ that allows $m$ tuples.

The second output size specified operator is $ORU$, again because it is Output
size specified, has relaxed input and it is utility-oriented. $ORU(R,w,m)$
denotes $m$ tuples in $R$ that are optimal for at least one preference vector
with a maximum distance of $\rho$ from $w$.

Both of these operators can be further generalized by relaxing $ORD$ to
contain tuples that are dominated by fewer than $k$ others and relaxing $ORU$
to contain tuples that are in the top-$k$ result for at least one vector.

This approach is very similar to flexible skylines. In fact, for any
preference vector $v$ and distance $\rho$, weighted sums with weight vectors
inside the hypersphere centered around $v$ with radius $\rho$ specifies a set
of functions $F$. In fact, for any dataset $R$, non-$\rho$-dominated tuples in
$R$ are precisely $ND(R,F)$ and tuples that are optimal for at least one
vector are precisely $PO(R,F)$.

###  2.3 UTK queries

Another parallel method that addresses the problem of relaxing the input
preference vector is uncertain top-$K$ (UTK) queries [11]. Similar to OSS
operators, these queries only consider weighted sums over the attributes as
scoring functions. Likewise, each UTK query operates on a set of possible
preference vectors rather than a single vector and these possible preference
vectors form a convex polytope in the $d-1$ dimensional simplex. Instead of
specifying the region of possible vectors via a central vector and a radius,
UTK queries allow the region to be specified as a parameter on its own.

[11] defines two different UTK operators, $UTK_{1}$ and $UTK_{2}$. Given a $d$
dimensional dataset $R$, a desired output cardinality $k$ and a preference
region $P\subseteq\Delta^{d-1}$, $UTK_{1}(k,R,P)$ contains the tuples in $R$
that are in the top-$k$ result for a preference vector in $P$ and
$UTK_{2}(k,R,P)$ partitions $P$ so that vectors in each partition produce the
same top-$k$ result and labels the partitions with the top-$k$ results they
correspond to.

UTK queries can express a larger family of scoring functions than OSS
operators since every $d-1$ dimensional hypersphere is a $d-1$ dimensional
polytope but the converse is not true. However, flexible skylines can express
an even larger family of scoring functions since UTK is still limited to
functions that are linear in the tuple attributes.

The sets of tuples computed by $UTK_{1}$ queries correspond to the tuples
computed by $PO$ and $ORU$ operators for matching inputs. Since $UTK$ queries
only check optimality and not domination, $ND$ and $ORD$ can not be computed
by $UTK$ queries.

###  2.4 $\epsilon$-skylines

$\epsilon$-skylines [14] is one of the earlier approaches to addressing the
shortcomings of top-$k$ and skyline queries. Similar to OSS operators,
$\epsilon$-skylines only consider weighted sums as scoring functions. They use
the notion of $\epsilon$-dominance, which, given a dataset $R$, a normalized
preference vector $w$ and a constant $\epsilon\in[-1,1]$ is defined as:

$r_{1}\prec_{\epsilon}r_{2}\iff(\forall i.w_{i}{r_{1}}_{i}\leq
w_{i}{r_{2}}_{i}+\epsilon)\ \land\ (\exists i.{r_{1}}_{i}<{r_{2}}_{i})$

it is a relaxed version of Pareto dominance where the dominant tuple is
allowed to be worse in some attributes up to an additive constant $\epsilon$.
In fact, when we pick $\epsilon=0$, $\epsilon$-dominance reduces to Pareto
dominance.

The $\epsilon$ skyline of a dataset for a preference vector $w$ and constant
$\epsilon$ is simply the set of non-$\epsilon$-dominated tuples. Unlike
traditional skylines, $\epsilon$-skylines are customizable in both preferences
and output cardinality. Since $\epsilon$ is scaled with the weights $w_{i}$
while checking dominance, changing $w$ enables the operator to respond to the
preferences of a user. The impact of $\epsilon$ on dominance decreases as the
weight $w_{i}$ of an attribute increases.

Regarding the output cardinality, $\epsilon$-skylines do not provide a precise
number but allow controlling the cardinality via changing $\epsilon$. For
positive $\epsilon$, every tuple’s $\epsilon$-dominance region is larger than
their Pareto dominance region, resulting in a smaller number of non-dominated
tuples. Conversely, negative $\epsilon$ shrink the $\epsilon$-dominance
regions and result in a larger number of non-dominated tuples. In fact, for
datasets with traditional skylines that contain at least two tuples,
$\epsilon=-1$ produces the whole dataset and $\epsilon=1$ produces the empty
set[14].

###  2.5 Representative Skylines

The approaches described so far have focused on customizing the query somehow
to include user preferences. Representative skylines [9][13] take an
orthogonal approach, they reduce the cardinality of the skyline directly, by
picking $k$ representative tuples from the skyline. The original
implementation [9] picks the tuples that maximize the number of non-skyline
tuples dominated by the picked tuples. A variant [13] of this implementation
picks the tuples that minimize the maximum distance between a non-picked
skyline tuple and any picked tuple.

Figure 1: An example skyline illustrating drawbacks

Both versions have some drawbacks [13]. Consider the example in Figure 1, the
skyline of the given dataset is $\\{r_{1},r_{2},r_{4},r_{6},r_{8},r_{9}\\}$.
If we optimize the number of dominated non-skyline tuples and pick $k=3$, we
get $\\{r_{4},r_{6},r_{8}\\}$. These two tuples are very similar in the trade-
off they offer between attributes $a_{1}$ and $a_{2}$. This subset fails to
represent the overall skyline since it does not include any other types of
trade-off in the skyline and focuses on a small cluster of tuples. On the
other hand, minimizing the maximum distance gives $\\{r_{1},r_{4},r_{9}\\}$.
This selection presents other trade-offs much better, but it fails to capture
that most of the tuples in the skyline have a trade-off similar To that of
$r_{4}$.

Representative skylines have the benefit of requiring no user input. They
provide a way to limit output cardinality, with different diversification
techniques, without compromising the simplicity of skyline queries. On the
other hand, since they admit no input other than the dataset itself, they are
not customizable save for the output cardinality.

##  3 Comparison

In this section, we discuss how the various proposed solutions compare to each
other, as well as traditional top-$k$ queries and skylines. We analyze which
problems each solution can address by inspecting four properties: input
flexibility, customizability, control over cardinality and ranked output.
These properties will be discussed one by one in the following subsections.
This discussion is summarized in Table 1.

| Input Flexibility | Customizability | Cardinality Control | Ranked Output
---|---|---|---|---
Top-$k$ queries | No | Yes | Yes | Yes
Traditional Skylines | Yes | No | No | No
Flexible Skylines | Yes | Yes | Partial | No
OSS Operators | Yes | Partial | Yes | No
UTK Queries | Yes | Partial | Yes | No
$\epsilon$-skylines | Yes | Partial | Partial | No
Representative Skylines | Yes | No | Yes | Yes
Table 1: Properties of different approaches

###  3.1 Input flexibility

A query’s input is said to be flexible if it does not need to be precise and
if small changes in the input do not cause significant changes in the output.
Traditional skylines and representative skylines are the obvious superior
choices for input flexibility since they have no input but the dataset itself.
Thus, no additional parameter needs to be defined to make these queries.
$\epsilon$-skylines are a bit less flexible, they require a value for
$\epsilon$ and a weight vector but the weight vector does not affect the
output as directly as it does in some other approaches. OSS operators,
flexible skylines and UTK queries are somewhat less flexible because they
require some specifics about either the estimate preference weights or the
family of scoring functions. Finally, top-$k$ queries are easily the least
flexible of the queries we consider, they require a precisely defined scoring
function which affects the output dramatically.

###  3.2 Customizability

A query is said to be customizable if it can adapt to the specific
requirements of a user or a scenario. Traditional skylines are not
customizable in any way, they simply return the same set of tuples for every
scenario. Representative skylines, however, are mildly customizable since the
output cardinality can be controlled. $\epsilon$-skylines, OSS skylines and
UTK queries have noticeably higher customizability since they allow weight
vectors to represent preferences over the attributes, though they only allow
preferences to be expressed in the form of weighted sums. Top-$k$ queries and
flexible skylines are fully customizable, they can be configured to return any
tuple in the skyline.

###  3.3 Control over cardinality

The approaches are clearly divided into three groups on this property:
Traditional skyline queries offer no control over the output cardinality.
$\epsilon$ skylines and flexible skylines provide methods to increase or
decrease the cardinality but offer no precise control. Lastly, top-$k$
queries, representative skylines, UTK queries and OSS operators have a
parameter to specify the output cardinality.

###  3.4 Ranked output

Top-$k$ queries and representative skylines provide a ranking between the
tuples they return. None of the other approaches have this property.

##  4 Conclusion

In this paper, we mentioned the commonly used strategies to select the most
interesting tuples in homogeneous multi-dimensional datasets together with
their well-known shortcomings. Then, we described some strategies that aim to
address the shortcomings of the conventional methods: Flexible skylines,
output size specified operators, UTK queries, $\epsilon$-skylines and
representative skylines. Finally, we compared these proposed solutions with
each other and the traditional strategies, pointing out which issues each one
can address.

Future work on flexible skylines could include estimating the output
cardinality or limiting it by construction similar to what output size
specified operators do. Discovering other families of scoring functions that
are representative of users’ preferences and can be efficiently computed by
flexible skylines could also be worthwhile research. Ranking the items
contained in flexible skylines and computing the skyline tuple-by-tuple could
be other directions to improve on this notion.

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