Singularity-Free Superstar as an Alternative to Black Hole and Gravastar

doi:10.4236/jmp.2013.47A1001

Paper Menu >>

Journal Menu >>

Journal of Modern Physics, 2013, 4, 1-6

http://dx.doi.org/10.4236/jmp.2013.47A1001 Published Online July 2013 (http://www.scirp.org/journal/jmp)

Singularity-Free Superstar as an Alternative to

Black Hole and Gravastar

Ding-Yu Chung1, Volodymyr Krasnoholovets2

1P.O. Box 180661, Utica, USA

2Institute of Physics, National Academy of Sciences, Kyiv, Ukraine

Email: [email protected]om, [email protected]

Received April 21, 2013; revised May 23, 2013; accepted June 25, 2013

mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work

is properly cited.

ABSTRACT

Singularity-free superstar is proposed as a model for the collapse of large stars and for GRBs, and as an alternative to

black hole and gravastar. Similar to a superconductor, a superstar contains extreme force fields that have non-zero mo-

mentum and non-zero wavelength to prevent the inactivation of force field at absolute zero and singularity (infinite in-

teracting energy) at infinite density, respectively, based on the uncertainty principle. Emerging only at an extremely low

temperature above absolute zero or an extremely high density below infinite density, extreme force fields are short-

range, and located in between a particle and its ordinary force fields (electromagnetic, weak, strong, and gravitational

forces) to prevent the inactivation of force fields at absolute zero and singularity (infinite interacting energy) at infinite

density in ordinary force fields. Extreme force fields are manifested as the bonds among electrons in a superconductor

and among atoms in a Bose-Einstein condensate. When the stellar core of a large star reaches the critical extreme den-

sity during the stellar collapse, the stellar core is transformed into the super matter core with extreme force fields and

ordinary force fields without singularity. A pre-superstar contains the super matter core, the ordinary matter region, and

the thin phase boundary between the super matter core and the ordinary matter region. The stellar collapse increases the

super matter core by converting the in falling ordinary energy and matter from the ordinary matter region into the super

matter, and decreases the ordinary matter region. Eventually, the stellar breakup occurs to detach the ordinary matter

region and the phase boundary from the super matter core, resulting in GRB to account for the observed high amount of

gamma rays and the observed complex light curves in GRBs. Unlike black holes and gravastars that lose information,

singularity-free superstars that keep all information exist.

Keywords: Black Hole; Superstar; Gravastar; Extreme Force Field; Uncertainty Principle; Singularity; Space Structure;

Collapsar; Gamma Ray Burst; Neutron Star; Pair Instability Supernova; Stellar Breakup

1. Introduction

Black hole has been a standard model for the collapse of

a large star. Singularity in black hole remains contentious.

Gravastar (gravitational vacuum star) [1] by P. O. Ma-

zurand and E. Mottolais is a model for the collapse of a

large star without singularity. In gravastar, quantum ef-

fects would change space-time around a collapsing star,

initiating a radical phase transition like when liquid water

becomes ice, for the in falling matter. For gravastar, the

phase transition involves the transformation into a “gravi-

tational vacuum” with an interior de Sitter condensate

surrounded by a Bose-Einstein condensate (BEC) bubble,

similar to the transformation of a cloud of atoms into one

huge “super-atom”, a BEC at an extremely low tempera-

ture above absolute zero degree. The BEC is prevented

from complete collapse by the interior de Sitter conden-

sate exerting a balance pressure outwards on the conden-

sate. A thin phase boundary (shell) for the phase transi-

tion is in between the interior region and the exterior

region.

In this paper, singularity-free superstar is proposed as

a model for the collapse of large stars and GRBs (gamma

ray bursts). Superstar is an alternative to black hole and

gravastar, and is similar to superconductor that involves

extreme force fields that are derived from the space struc-

ture [2-4]. An extreme force field in a superstar is an

alternative to a gravitational vacuum in a gravastar. Ex-

treme force fields have non-zero momentum and non-

zero wavelength to prevent the inactivation of force field

D.-Y. CHUNG, V. KRASNOHOLOVETS

at absolute zero and singularity (infinite interacting en-

ergy) at infinite density, respectively. Emerging only at

an extremely low temperature above absolute zero or an

extremely high density below infinite density, extreme

force fields are short-range, and located in between a

particle and its ordinary force fields (electromagnetic,

weak, strong, and gravitational forces) to prevent the

inactivation of force fields at absolute zero and singular-

ity (infinite interacting energy) at infinite density in or-

dinary force fields. Extreme force fields are manifested

as the bonds among electrons in a superconductor and

among atoms in a Bose-Einstein condensate.

A superstar is formed by the transformation of the

stellar core of a large star into the super matter core with

extreme and ordinary force fields without singularity at

an extremely high density during the stellar collapse.

Superstar relates to space structure, the uncertainty prin-

ciple, extreme force field, black hole, gravastar, neutron

star, supernova, collapsar, GRB, and pair instability su-

pernova.

2. The Space Structure and Extreme Force

Field

The conventional explanation of the hidden extra space

dimensions is the compactization of the extra space di-

mensions. For example, six space dimensions become

hidden by the compactization, so space-time appears to

be four dimensional. Bounias and Krasnoholovets [5]

propose another explanation of the reduction of >4D

space-time into 4D space-time by slicing >4D space-time

into infinitely many 4D slices surrounding the 4D core

particle. Such slicing of >4D space-time is like slicing

3D object into 2D object in the way stated by Michel

Bounias as follows: “You cannot put a pot into a sheet

without changing the shape of the 2-D sheet into a 3-D

dimensional packet. Only a 2-D slice of the pot could be

a part of sheet”.

One way to describe the slicing of space dimension is

to have the space structure consisting of attachment

space and detachment space. Attachment space is the

space attaching to an object, such as 3D attachment space

attaching to a pot described above. Detachment space is

the space cutting attachment space into numerous attach-

ment space slices, such as detachment space cutting 3D

attachment space into numerous 2D attachment space

slices. Such 2D attachment space slices with attached

objects, therefore, are separated by 2D detachment space

gaps without attached objects.

In this paper, the digital space structure consists of at-

tachment space (denoted as 1) and detachment space

(denoted as 0). Attachment space attaches to object per-

manently with zero speed. Detachment space detaches

from the object at the speed of light. Attachment space

relates to rest mass, while detachment space relates to

kinetic energy. Different stages of our universe have dif-

ferent space structures.

The combination of attachment space (1) and detach-

ment space (0) brings about three different space struc-

tures: miscible space, binary partition space, and binary

lattice space as below:













combination

attachment detachment

space space

10binary lattice space,

10miscible space, or

10binary partitionspace







Binary lattice space, (1 0)n, consists of repetitive units

of alternative attachment space and detachment space.

Thus, binary lattice space consists of multiple quantized

units of attachment space separated from one another by

detachment space to account for ordinary force fields

(electromagnetic, strong, weak, and gravitational force

fields). In miscible space, attachment space is miscible to

detachment space, and there is no separation of attach-

ment space and detachment space to account for special

relativity. Binary partition space, (1)n (0)n, consists of se-

parated continuous phases of attachment space and de-

tachment space to account for quantum mechanics and

extreme force field.

As described in Reference [2], at the beginning of the

current universe, the 10d particle universe was sliced into

six particles: 9d, 8d, 7d, 6d, 5d, and 4d equally by mass.

Detachment space (0) involves in the slicing of dimen-

sions. Attachment space is denoted as 1. For example,

the slicing of 10d particles into 4d particles is as follows:





 



4,6

slicing

10d attachment space

4d core6typesof4d units

attachment force fieldsin

space

inary lattice space







The two products of the slicing are the 4d-core at-

tachment space for core particle and 6 types of 4d quan-

tized units for ordinary force fields. The 4d core attach-

ment space surrounded by 6 types of many (j) 4d-quan-

tized units corresponds to the core particle surrounded by

6 types of many small 4d particles. The force fields are

the force fields with binary lattice space.

The uncertainty principle for quantum mechanics is

expressed as follows:

D.-Y. CHUNG, V. KRASNOHOLOVETS 3





The position, x, and momentum, p, of a particle cannot

be simultaneously measured with arbitrarily high preci-

sion. The uncertainty principle requires every physical

system to have a zero-point energy (minimum momen-

tum) greater than zero, and to have a maximum energy

equal or less than the energy at the minimum wavelength

as the Planck length. The uncertainty principle has non-

zero momentum and non-zero wavelength. In terms of

the space structure, detachment space relating to kinetic

energy as momentum is σp, and attachment space relating

to space (wavelength) for a particle is σx. Neither detach-

ment space nor attachment space is zero in the uncer-

tainty principle.

Quantum mechanics for a particle follows the uncer-

tainty principle. It is proposed that at the extreme condi-

tions of absolute zero and infinite density, the binary

lattice space for ordinary force fields (electromagnetic,

strong, weak, and gravitational force fields) follows the

certainty principle





10

 



  

extremely lowtemperatureorhighdensity

44 44

,, ,

core ordinary forces

particlesinbinary lattice space

01 01

extreme forces

core

in binary partition

particles

space

mnk

nk nknk







xp instead of the un-

certainty principle. At absolute zero with infinitesimal

movement, all detachment space (momentum) in binary

lattice space virtually ceases to exist, so the binary space

as the force field collapses into infinite attachment space

(wavelength) with infinitesimal momentum, resulting in

the inactivation of force field. At infinite density to pro-

duce infinite interacting energy (infinite momentum) from

the interaction among particles, all attachment space (rest

mass) in the binary lattice space virtually ceases to exist,

so the binary lattice space as the force field collapses into

infinite detachment space (momentum) with infinitesimal

wavelength, resulting in singularity as infinite interacting

energy.



To prevent the inactivation of force fields at absolute

zero and singularity (infinite interacting energy) at infi-

nite density in ordinary force fields requires the presence

of the special force fields that follow the uncertainty

principle. The special force fields are “extreme force

fields” that are in binary partition space. Binary partition

space has one continuous detachment space and one con-

tinuous attachment space. The binary partition space for

extreme force fields follows the uncertainty principle.

Neither detachment space nor attachment space is zero in

the binary partition space. To follow the uncertainty prin-

ciple, extreme force fields have non-zero momentum and

non-zero wavelength to prevent the inactivation of force

field at absolute zero and singularity (infinite interacting

energy) at infinite density, respectively.

At the critical temperature above absolute zero and the

critical extreme density below infinite density, extreme

force fields emerge in between particles and their ordi-

nary force fields (electromagnetic, strong, weak, and gra-

vitational force fields) to prevent the inactivation of force

fields at absolute zero and singularity (infinite interacting

energy) at infinite density in ordinary force fields as fol-

lows:



 



 







ordinary forces

inbinary lattice

space

The whole system of core particles, extreme force

fields, and ordinary force fields never reaches absolute

zero and infinite density. Extreme force fields do not

change the normal properties of ordinary force fields in

the same system.

All extreme force fields are identical and short-range,

and are the dominant force fields over ordinary force

fields in the interior of core particles, such as supercon-

ductor and Bose-Einstein condensate. The Meissner ef-

fect is explained by the outward pressure of extreme

force fields to eject applied magnetic fields from the in-

terior of the superconductor as it transitions into the ex-

treme force fields at nearly absolute zero temperature. In

the BCS theory of superconductivity [6], the supercon-

ducting current is explained as a super fluid of Cooper

pairs, pairs of electrons interacting through the exchange

of phonons. In the explanation by extreme force, the

Cooper pairs correspond to pairs of electrons interacting

through the exchanges of extreme force bosons. The su-

perfluid property of a Bose-Einstein condensate can also

be explained by the atoms interacting through the ex-

changes of extreme force bosons. More details of super-

conductivity by extreme force field are described in Ref-

erence [3].

3. Neutron Star and Supernova

The formation of neutron star involves the core collapse

of a large star. When a star with the initial mass of about

8 to 25 solar masses depletes its nuclear fuel, it has no

outward radiation pressure to support its bulk. The core

of the star collapses into a neutron star by fusing elec-

trons and protons into neutrons, sending out huge num-

bers of neutrons. The neutrino shock wave from these

neutrinos causes a violent expulsion of the surrounding

material, resulting in supernova [7]. The collapse in

terms of the compression from a large size progenitor to

D.-Y. CHUNG, V. KRASNOHOLOVETS

a very small neutron star leads to a fast-rotating neutron

star with a high angular momentum and a strong mag-

netic field.

4. Collapsar, GRB, and Pair Instability

Supernovae

Gamma-ray bursts (GRBs) are the flashes of focused

gamma rays associated with extremely energetic explo-

sions that have been observed in distant galaxies. The

energy of a GRB is approximately equal to turning a star

like the Sun into pure energy. GRB can be explained

typically by collapsar (collapsed star) [8] that refers to a

specific model for the gravitational collapse of a fast-

rotating star, resulting in a stellar mass black hole.

In the collapsar model, when a star with initial mass

about 25 to 90 solar masses collapses into a fast-rotating

black hole, the black hole immediately begins to pull in

more stellar material, and very quickly a rotating disk of

material as black hole accretion disk (BHAD) forms. The

inner portion of the disk spins around the superstar at

near light speed. With rotating conducting fluids, the

BHAD creates a strong magnetic field. Because the inner

portion of the BHAD is rotating more quickly than the

outer portion, the magnetic field lines twist violently.

This causes a jet of material to blast outward at almost

the speed of light perpendicularly to the BHAD. The jet

contains matter and antimatter in the form of electrons,

positrons, and protons. The gamma rays are produced by

the “internal shocks” as the collisions of the shells of

matter and energy pushed by the jet.

One of the problems in the collapsar model of GRB is

to explain how some gamma-ray bursts may convert as

much as half or more of the explosion energy into

gamma-rays [9]. Another problem in the collapsar model

of GRB is to explain the complexity of the light curves of

GRBs [10]. The duration of observable emission can

vary from milliseconds to tens of minutes. The numbers,

the shapes, and the intensities of the peaks in the light

curves vary. No two light curves are identical.

A hypernova [11] is a type of supernova with energy

much higher than standard supernovae. One of the mod-

els for hypernova is pair instability supernova. Pair-in-

stability supernova occurs in stars with an initial mass

range from around 130 to 250 solar masses. The stellar

core is occupied by gamma rays whose outward pressure

keeps the star from collapse by the inward gravity. Elec-

tron-positron pairs can be created from gamma rays, re-

sulting in the reduction of outward pressure by the de-

crease of gamma rays. This outward pressure drop leads

to a partial collapse, resulting in an accelerated thermo-

nuclear burning in a runaway thermonuclear explosion

which blows the star completely apart without leaving a

star remnant behind. The result is a hypernova.

For the star with an initial mass of 100 to 130 solar

masses as in Eta Carinae [12], the partial collapse is not

large enough to cause a runaway thermonuclear explo-

sion. The thermonuclear explosion to only leads to the

ejection of a part of outer layer. The repetition of the par-

tial collapse finally depletes enough mass, resulting in a

normal supernova. For a star with an initial mass higher

than about 250 solar masses, the energies from the ther-

monuclear reactions are absorbed in photodisintegration.

The stellar collapse continues without explosion.

5. Superstar

When a star with initial mass of about 25 to 90 solar

masses collapses, the huge amount of collapsing materi-

als allows the neutrino shock wave to have a weak or no

supernova. The stellar collapse by inward gravity con-

tinues. The fast-rotating star resulted from the stellar col-

lapse creates a strong magnetic field.

The stellar core is a small size gamma ray core. The

outward pressure from this small size gamma ray core is

too weak to stop the stellar collapse. The stellar collapse

continues. When the stellar core reaches the critical ex-

treme density by the stellar collapse, the stellar core is

transformed into the super matter core with extreme force

fields that prevent singularity in ordinary force fields by

infinite density.

For a pre-superstar, the ordinary matter region is out-

side of the super matter core. The phase boundary is in

between the super matter core and the ordinary matter

region. The phase boundary is for the phase transition

from ordinary matter to super matter. As in the Meissner

effect to repel applied magnetic field in short-range, the

super matter core exerts an outward pressure to repel the

phase boundary in short-range with the strength propor-

tional to the total mass of the super matter.

During the stellar collapse, the in falling energy that

reaches the phase boundary is stored first as gamma rays

in the phase boundary that is repelled by the super matter

core. The in falling matter particles that reach the phase

boundary are also stored first in the phase boundary. The

density in the phase boundary is less than the critical

D.-Y. CHUNG, V. KRASNOHOLOVETS 5

extreme density. The further stellar collapse that in-

creases the density in the phase boundary to the critical

extreme density converts the gamma rays and matter

particles in the phase boundary into super matter parti-

cles that move to the super matter core. The whole con-

version process then starts over again. As a result, the

stellar collapse increases the super matter core, and de-

creases the ordinary matter region.

Eventually, the ordinary matter region becomes small,

and the inward gravity of the ordinary matter region is

too weak to allow the ordinary matter region and the

phase boundary to attach to the super matter core that

repels the phase boundary. The result is the stellar breakup

to detach the phase boundary and the ordinary matter

region from the super matter core. The stellar breakup

starts from the phase boundary that is repelled by the

super matter core. During the stellar breakup, the de-

tached phase boundary and ordinary matter region that

are broken into pieces by the fast-rotation become the

superstar accretion disk (SAD) as the black hole accre-

tion disk (BHAD) in the collapsar model of GRB. With

the additional energy from the gamma rays in the phase

boundary, the SAD contains much higher energy than the

BHAD. As the BHAD, the SAD produces a jet of mate-

rial in the forms of electrons, positrons, and protons to

blast outward at almost the speed of light perpendicularly

to the SAD. The higher energy SAD produces the higher

energy jet than the jet from the BHAD. The higher en-

ergy jet from the SAD produces more gamma rays from

the internal shocks than gamma rays produced from the

jet from the BHAD.

The different parts of the ordinary matter region in a

compact superstar break up nearly simultaneously, so the

GRB duration is short [13], and there is only one light

peak in the light curve. The different parts of the ordinary

region in a large superstar do not break up at the same

time, so the GRB duration is long, more than one light

peak are in the light curve, and different light peaks are

different in intensities and shapes. The short duration

GRBs have the average about 0.3 seconds and the long-

duration GRBs have the average about 30 seconds [14].

The observed high conversion of the explosion energy

into gamma rays in a superstar breakup comes from the

SAD that contains higher energy than the BHAD. For the

light curves of GRBs, the additional complication that is

not in the collapsar model is from the complex stellar

breakup in a pre-superstar. Therefore, the superstar model

of GRB solves the two problems of the collapsar model

of GRB for the high conversion of the explosion energy

into gamma rays and the complex light curves in GRBs.

After the stellar breakup, the remnant is a pure superstar

with only the super matter core. A pure superstar with a

high gravity hinders the emission of light.

For the stellar breakup of a non-rotating superstar, en-

ergies are not focused by the magnetic field of a fast-

rotating superstar. The stellar breakup is similar to a su-

pernova.

For a star with an initial mass of 100 to 130 solar

masses, the stellar core is a medium size gamma ray core

that has a strong outward pressure to stop the stellar col-

lapse and to prevent the formation of the super matter

core. The core collapse by pair instability leads to a ther-

monuclear explosion, but not a runway thermonuclear

explosion. The thermonuclear explosion leads to the ejec-

tion of a part of the ordinary matter region. For a star

with an initial mass of 130 to 250 solar masses, the stel-

lar core is a large size gamma ray core that has a strong

outward pressure to stop the stellar collapse and to pre-

vent the formation of the super matter core. The core col-

lapse by pair instability leads to a runway thermonuclear

explosion, resulting in a hypernova (pair instability su-

pernova) without any star remnant. For a star with an ini-

tial mass of higher than about 250 solar masses, photodis

integration prevents thermonuclear explosion, resulting in

continuing stellar collapse to convert the stellar core into

the super matter core for a supermassive pre-superstar.

From outside, black holes, gravastars, and superstars

look the same. From inside, they are different in terms of

information. An extreme force field that prevents singu-

larity in gravity is an alternative to a gravitational vac-

uum (with the equation of state p = −ρ) in a gravastar. In

a gravastar, a gravitational vacuum is located in one spe-

cific region. In a superstar, extreme force fields are not in

one special region. The phase boundary in superstar is an

alternative to the phase boundary in gravastar for the

phase transition with equation of state p = + ρ between

the interior region and the exterior region (with the equa-

tion of state p = ρ = 0). In a gravastar, in falling matter

that hits the phase boundary is converted into energy by

proton decay, adding to the energy of the space-time va-

cuum within the phase boundary. Some information such

as baryon number conservation is lost during the transi-

tion from the exterior region to the interior region. In a

black hole, all information other than the total mass,

charge, and angular momentum is lost. In a superstar, all

ordinary force fields in the super matter core are recov-

erable under ordinary condition, so no ordinary informa-

tion is lost in a superstar.

Black holes and gravastars lose the information about

ordinary force fields, while superstars keep all informa-

tion about ordinary force fields. Quantum mechanics is

built on the principle that information cannot be lost. Vio-

lating this basic principle of quantum mechanics, black

holes and gravastars do not exist. In compliance with this

basic principle, superstars exist.

6. Summary

It is proposed that the digital space structure consists of

D.-Y. CHUNG, V. KRASNOHOLOVETS

attachment space (denoted as 1) for rest mass and de-

tachment space (denoted as 0) for kinetic energy. At-

tachment space attaches to object permanently with zero

speed, and detachment space detaches from the object at

the speed of light. The combination of attachment space

and detachment space brings about the three structures:

binary lattice space, miscible space, and binary partition

space. Binary lattice space, (1 0)n consists of repetitive

units of alternative attachment space and detachment

space to account for ordinary force fields. In miscible

space, attachment space is miscible to detachment space

without separation to account for special relativity. Bi-

nary partition space, (1)n (0)n, consists of separated con-

tinuous phases of attachment space and detachment space

to account for quantum mechanics and extreme force

fields. Through the detachment space, a higher dimen-

sional particle in attachment space is sliced into infinitely

surrounding a lower dimensional core attachment space,

resulting in a particle surrounding by force field in the

form of binary lattice space. Under the extreme conditions

of extremely low temperature or extremely high density,

extreme force fields in binary partition space emerge.

Singularity-free superstar is proposed as a model for

the collapse of large stars and for GRBs, and as an alter-

native to black hole and gravastar. Similar to a super-

conductor, a superstar contains extreme force fields that

have non-zero momentum and non-zero wavelength to

prevent the inactivation of force field at absolute zero

and singularity (infinite interacting energy) at infinite

density, respectively, based on the uncertainty principle.

Emerging only at an extremely low temperature above

absolute zero or an extremely high density below infinite

density, extreme force fields are short-range, and located

in between a particle and its ordinary force fields (elec-

tromagnetic, weak, strong, and gravitational forces) to

prevent the inactivation of force fields at absolute zero

and singularity (infinite interacting energy) at infinite

density in ordinary force fields. Extreme force fields are

manifested as the bonds among electrons in a supercon-

ductor and among atoms in a Bose-Einstein condensate.

When the stellar core of a large star reaches the critical

extreme density during the stellar collapse, the stellar

core is transformed into the super matter core with ex-

treme force fields and ordinary force fields without sin-

gularity. A pre-superstar contains the super matter core,

the ordinary matter region, and the thin phase boundary

between the super matter core and the ordinary matter

egion. The stellar collapse increases the super matter

core by converting the in falling ordinary energy and

matter from the ordinary matter region into the super

matter, and decreases the ordinary matter region. Even-

tually, the stellar breakup occurs to detach the ordinary

matter region and the phase boundary from the super

matter core, resulting in GRB to account for the observed

high amount of gamma rays and the observed complex

light curves in GRBs. Unlike black holes and gravastars

that lose information, singularity-free superstars that keep

all information exist.

REFERENCES

[1] P. O. Mazur and E. Mottola, Proceedings of the National

Academy of Sciences of the United States of America, Vol.

111, 2004, pp. 9545-9550. doi:10.1073/pnas.0402717101

[2] D. Chung and V. Krasnoholovets, Scientific Inquiry, Vol.

8, 2007, pp. 165-182.

www.iigss.net/scientific_inquiry/2007-12/3-krasnoholove

ts.pdf

[3] D. Chung and V. Krasnoholovets, Progress in Physics,

Vol. 4, 2006, pp. 74-77.

http://www.ptep-online.com/index_files/2006/PP-06-14.P

[4] D. Chung and V. Krasnoholovets, Journal of Modern

Physics, Vol. 4, 2013, pp. 27-31.

[5] M. Bounias and V. Krasnoholovets, The International

Journal of Systems and Cybernetics, Vol. 32, 2003, pp.

945-975.

[6] J. Bardeen, L. N. Cooper and J. R. Schrieffer, Physical

Review, Vol. 108, 1957, pp. 1175-1205.

doi:10.1103/PhysRev.108.1175

[7] S. Woosley and H.-T. Janka, Nature Physics, Vol. 1,

2005, pp. 147-154. doi:10.1038/nphys172

[8] A. I. MacFadyen and S. Woosley, Astrophysical Journal,

Vol. 524, 1999, pp. 262-289. doi:10.1086/307790

[9] Y. Fan and T. Piran, Monthly Notices of the Royal Astro-

nomical Society, Vol. 369, 2006, pp. 197-206.

doi:10.1111/j.1365-2966.2006.10280.x

[10] J. I. Katz, “The Biggest Bangs,” Oxford University Press,

Oxford, 2002, p. 37.

[11] G. S. Fraley, Astrophysics and Space Science, Vol. 2,

1968, pp. 96-114. doi:10.1007/BF00651498

[12] A. Kashi and N. Soker, New Astronomy, Vol. 14, 2008,

pp. 11-24. doi:10.1016/j.newast.2008.04.003

[13] E. Nakar, Physics Reports, Vol. 442, 2007, pp. 166-236.

doi:10.1016/j.physrep.2007.02.005

[14] C. Kouveliotou, et al., Astrophysical Journal Letters, Vol.

413, 1993, pp. 101-104. doi:10.1086/186969