Quark Stars as inner engines for Gamma Ray Bursts?
ABSTRACT A model for Gamma ray bursts inner engine based on quark stars (speculated to exist in nature) is presented. We describe how and why these objects might constitute new candidates for GRB inner engines. At the heart of the model is the onset of exotic phases of quark matter at the surface of such stars, in particular the 2-flavor color superconductivity. A novel feature of such a phase is the generation of particles which are unstable to photon decay providing a natural mechanism for a fireball generation; an approach which is fundamentally different from models where the fireball is generated during collapse or conversion of neutron star to quark star processes. The model is capable of reproducing crucial features of Gamma ray bursts, such as the episodic activity of the engine (multiple and random shell emission) and the two distinct categories of the bursts (two regimes are isolated in the model with \sim 2 s and \sim 81 s burst total duration). Comment: 8 pages, 3 figures, new and more appropriate title. Major changes in the text (aspects of the models discussed in more details), better quality Figure 1 and Figure 2 and added Figure 3, version to appear in Astronomy&Astrophysics
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ABSTRACT: We discuss whether the winding-up of the magnetic field by differential rotation in a new-born quark star can produce a sufficiently-high, energy, emission rate of sufficiently long duration to explain long gamma-ray bursts. In the context of magnetohydrodynamics, we study the torsional oscillations and energy extraction from a new-born, hot, differentially rotating quark star. The new-born compact star is a rapid rotator that produces a relativistic, leptonic wind. The star's torsional oscillation modulates this wind emission considerably when it is odd and of sufficient amplitude, which is relatively easy to reach. Odd oscillations may occur just after the formation of a quark star. Other asymmetries can cause similar effects. The buoyancy of wound-up magnetic fields is inhibited, or its effects are limited, by a variety of different mechanisms. Direct electromagnetic emission by the torsional oscillation in either an outside vacuum or the leptonic wind surrounding the compact object is found to be insignificant. In contrast, the twist given to the outer magnetic field by an odd torsional oscillation is generally sufficient to open the star's magnetosphere. The Poynting emission of the star in its leptonic environment is then radiated from all of its surface and is enhanced considerably during these open episodes, tapping at the bulk rotational energy of the star. This results in intense energy shedding in the first tens of minutes after the collapse of magnetized quark stars with an initial poloidal field of order of 10**14 Gauss, sufficient to explain long gamma-ray bursts. Comment: 16 pages, accepted by Astronomy and AstrophysicsAstronomy and Astrophysics 11/2008; · 4.48 Impact Factor
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ABSTRACT: In recent years, considerable interest has been stimulated by many applications of fractional calculus in astrophysics. Motivated by recent advances of the statistical mechanical description of degenerate matter gas and fractional statistical physics, we discussed the fractional formulation of the white dwarf stellar dynamical problem. Our approach is based on the familiar definition of the Riemann–Liouville fractional integral operator of order 0Applied Mathematics and Computation 11/2011; 218:2837-2849. · 1.60 Impact Factor
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ABSTRACT: The cooling history of a quark star in the colour superconductive phase is investigated. Here we specifically focus on the two-flavour colour (2SC) phase where the novel process of photon generation via glueball (GLB) decay has already been investigated. The picture we present here can, in principle, be generalized to quark stars entering a superconductive phase where similar photon generation mechanisms are at play. As much as 1045–1047 erg of energy is provided by the GLB decay in the 2SC phase. The generated photons slowly diffuse out of the quark star, keeping it hot and radiating as a blackbody (with possibly a Wien spectrum in gamma-rays) for millions of years. We discuss hot radio-quiet isolated neutron stars in our picture (such as RX J185635–3754 and RX J0720.4–3125) and argue that their nearly blackbody spectra (with a few broad features) and their remarkably tiny hydrogen atmosphere are indications that these might be quark stars in the colour superconductive phase where some sort of photon generation mechanism (reminiscent of the GLB decay) has taken place. Fits to observed data of cooling compact stars favour models with superconductive gaps of Δ2SC∼ 15–35 MeV and densities ρ2SC= (2.5–3.0) ×ρN (ρN being the nuclear matter saturation density) for quark matter in the 2SC phase. If correct, our model combined with more observations of isolated compact stars could provide vital information to studies of quark matter and its exotic phases.Monthly Notices of the Royal Astronomical Society 02/2006; 367(1):231 - 237. · 5.52 Impact Factor
arXiv:astro-ph/0103022v3 20 Mar 2002
Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
Quark Stars as inner engines for Gamma Ray Bursts?
Rachid Ouyed and Francesco Sannino
Nordic Institute for Theoretical Physics, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
Abstract. A model for Gamma ray bursts inner engine based on quark stars (speculated to exist in nature) is
presented. We describe how and why these objects might constitute new candidates for GRB inner engines. At
the heart of the model is the onset of exotic phases of quark matter at the surface of such stars, in particular the
2-flavor color superconductivity. A novel feature of such a phase is the generation of particles which are unstable
to photon decay providing a natural mechanism for a fireball generation; an approach which is fundamentally
different from models where the fireball is generated during collapse or conversion of neutron star to quark star
processes. The model is capable of reproducing crucial features of Gamma ray bursts, such as the episodic activity
of the engine (multiple and random shell emission) and the two distinct categories of the bursts (two regimes are
isolated in the model with ∼ 2 s and ∼ 81 s burst total duration).
Key words. dense matter – Gamma rays: bursts – stars: interior
A central problem contributing to the Gamma-ray bursts
(GRBs) mystery is the unknown nature of the engine pow-
ering them (Kouveliotou et al. 1995; Kulkarni et al. 1999;
Piran 1999a; Piran 1999b). Many have been suggested but
it is fair to say that we are still far from a definite an-
swer. Regardless of the nature of the engine, however, it is
widely accepted that the most conventional interpretation
of the observed GRBs result from the conversion of the ki-
netic energy of ultra-relativistic particles to radiation in
an optically thin region. The particles being accelerated
by a fireball mechanism (or explosion of radiation) taking
place near the central engine (Goodman 1986; Shemi &
Piran 1990; Paczy´ nski 1990).
The first challenge is to conceive of circumstances that
would create a sufficiently energetic fireball. Conversion of
neutron stars to quark stars (Olinto 1987; Cheng & Dai
1996; Bombaci & Datta 2000) has been suggested as one
possibility. Other models also involve the compact object
element; such as black holes (Blandford & Znajek 1977)
and coalescing neutron stars (Eichler et al. 1989; Ruffert
& Janka 1999; Janka et al. 1999). We show in this work
that the plausible existence of quark stars combined with
the onset of a newly revived state of quark matter - called
color superconductivity - in these objects offers a new way
of tackling the GRB puzzle (Ouyed 2002). Here we will
argue that quark stars might constitute new candidates
for GRB inner engines.
Send offprint requests to: firstname.lastname@example.org
Quark matter at very high density is expected to be-
have as a color superconductor (see Rajagopal & Wilczek
2000 for a review). Associated with superconductivity is
the so-called gap energy ∆ inducing the quark-quark pair-
ing and the critical temperature (Tc) above which thermal
fluctuations will wash out the superconductive state. A
novel feature of such a phase is the generation of glueball
like particles (hadrons made of gluons) which as demon-
strated in Ouyed & Sannino (2001) immediately decay
into photons. If color superconductivity sets in at the sur-
face of a quark star the glueball decay becomes a natural
mechanism for a fireball generation.
The paper is presented as follows: In Sect. 2 we briefly
describe the concept of color superconductivity in quark
matter. Glueball formation and their subsequent two-
photon decay is described. Sect. 3 deals with quark stars
and the onset of color superconductivity at their surface.
In Sect. 4, we explain how GRBs are powered in this pic-
ture and show that variability (multiple shell emission) is
inherent to the inner engine. We isolate two GRB regimes
in Sect. 5 associated with small and massive quark stars.
The model’s features and its predictions are summarized
in Sect. 6 while a discussion and conclusion follows in Sect.
7 where the model’s assumptions and limitations are high-
2. Color Superconductivity
While in this paper we deal mostly with the astrophysics
aspect of the model, we nevertheless give a brief overview
of color superconductivity and the glueball-to-photon de-
2Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?
Fig.1. A schematic representation of a possible QCD
phase diagram (Rajagopal & Wilczek 2000). At high tem-
perature and density, matter is believed to be in a quark-
gluon plasma phase (QGP). The hadronic phase lies in
the region of low temperature and density. At very high
density but low temperature, when nuclei melt into each
other, it has been suggested that a color superconductive
phase might set in. 2SC denotes a 2-flavor color super-
conductive regime. The arrow depicts a plausible cooling
path of a HQS surface leading to the onset of color super-
cay process which leads to the fireball. The interested
reader is referred to Ouyed & Sannino (2001) for the un-
derlying physics. For a recent review see Sannino (2002)
2.1. 2-flavour color superconductivity
A reasonable Quantum Chromo-Dynamics (QCD) phase
diagram (in the µ−T plane, where µ is the chemical poten-
tial simply related to matter density) is shown in Fig. 1.
At high temperature and density, matter is believed to
be in a quark-gluon plasma phase (QGP). The hadronic
phase lies in the region of low temperature and density. At
high densities but low temperatures, when nuclei melt into
each other, it is now believed that a color superconductive
phase sets in. This phase is characterized by the formation
of quark-quark condensate. In the 2-flavor color supercon-
ductivity (2SC) the up and down quark come into play
during pairing. Furthermore, 2SC is characterized by five
out of the eight gluons acquiring mass. We refer the inter-
ested reader to Rajagopal & Wilczek (2000) for a review
of the dynamical properties of 2SC.
2.2. Light GlueBalls
The 3 massless gluons in the 2SC phase which bind into
light glueballs (LGBs) together with the quarks up and
down constitute the 2SC phase mixture. In Ouyed &
Sannino (2001) we studied certain properties of these
LGBs. Among the properties relevant to our present study
we found, i) The LGBS decay into photons with an asso-
ciated lifetime of the order of 10−14s; ii) The mass of the
LGBs is of the order of 1 MeV.
3. Quark stars
We now turn to study the astrophysical consequences
when such a state sets in at the surface of quark stars.
As such, we first assume that quark stars exists in nature
(further discussed in Sect. 7.1) and constitutes the first
major assumption in our model.
3.1. Hot Quark stars
We are concerned with quark stars born with surface tem-
peratures above Tc. We shall refer to these stars as “hot”
quark stars (HQSs) in order to avoid any confusion with
strange stars which are conjectured to exist even at zero
pressure if strange matter is the absolute ground state of
strong interacting matter rather than iron (Bodmer 1971;
Witten 1984; Haensel et al. 1986; Alcock et al. 1986; Dey
et al. 1998).
We borrow the language of the MIT-bag model for-
malism at low temperature and high density to describe
HQSs (Farhi & Jaffe 1984). This gives a simple equation
P = b(ρ − ρHQS)c2,(1)
where b is a constant of model-dependent value (close to,
but generally not equal, to 1/3 of the MIT-bag model),
and ρHQSis the density at zero-pressure (the star’s surface
density). We should keep in mind that Tc/µ ≪ 1 as is
Features of HQSs are - to a leading order in Tc/µ - iden-
tical to that of strange stars. The latter have been studied
in details (Alcock et al. 1986; Glendenning & Weber 1992;
Glendenning 1997). Of importance to our model:
i) The “surface” of a HQS is very different from the sur-
face of a neutron star, or any other type of stars. Because
it is bound by the strong force, the density at the surface
changes abruptly from zero to ρHQS. The abrupt change
(the thickness of the quark surface) occurs within about
1 fm, which is a typical strong interaction length scale.
ii) The electrons being bound to the quark matter by
the electro-magnetic interaction and not by the strong
force, are able to move freely across the quark surface
extending up to ∼ 103fm above the surface of the star.
Associated with this electron layer is a strong electric field
(5×1017V/cm)- higher than the critical value (1.3×1016
V/cm) to make the vacuum region unstable to sponta-
neously create (e+,e−) pairs.
iii) The presence of normal matter (a crust made of
ions) at the surface of the quark star is subject to the enor-
mous electric dipole. The strong positive Coulomb barrier
prevents atomic nuclei bound in the nuclear crust from
coming into direct contact with the quark core. The crust
is suspended above the vacuum region.
iv) One can show that the maximum mass of the crust
cannot exceed Mcrust≃ 5 × 10−5M⊙set by the require-
ment that if the density in the inner crust is above the neu-
tron drip density (ρdrip≃ 4.3 × 1011g/cc), free neutrons
will gravitate to the surface of the HQS and be converted
Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?3
to quark matter. This is due to the fact that neutrons
can easily penetrate the Coulomb barrier and are readily
3.2. Cooling and 2SC layer formation
The HQS surface layer might enter the 2SC phase as illus-
trated in Fig. 1. In the QCD phase diagram (Fig. 2), (ρB0,
TB0) is the critical point beyond which one re-enters the
QGP phase (the extent of the 2SC layer into the star). The
star consists of a QGP phase surrounded by a 2SC layer
where the photons (from the LGB/photon decay) leaking
from the surface of the star provides the dominant cooling
source. This picture, as illustrated in Fig. 2, is only valid if
neutrino cooling in the 2SC phase is heavily suppressed as
to become slower than the photon cooling. Unfortunately,
the details of neutrino cooling in the 2SC phase is still a
topic of debate and studies (Carter & Reddy 2000; Schaab
et al. 2000 to cite only few). One can only assume such
a scenario which constitutes the second major assump-
tion in our model. In Sect. 7.2, we discuss the remaining
alternative when photon cooling is dwarfed by neutrino
3.3. LGBs decay and photon thermalization
The photons from LGB decay are generated at energy
Eγ < Tc and find themselves immersed in a degener-
ate quark gas. They quickly gain energy via the inverse
Compton process and become thermalized to Tc. We es-
timate the photon mean free path to be smaller than few
hundred Fermi (Rybicki & Lightman 1979; Longair 1992)
while the 2SC layer is measured in meters (see Sect. 5.2).
A local thermodynamic equilibrium is thus reached with
the photon luminosity given by that of a black body radi-
Lγ= 3.23 × 1052ergs s−1(RHQS
10 MeV)4. (2)
The energy for a single 2SC event is thus
∆ELGB= δLGBM2SCc2, (3)
where M2SC = δ2SCMHQS is the portion of the star in
2SC. Here, δ2SC depends on the star’s mass while δLGB
represent the portion of the 2SC that is in LGBs (intrinsic
property of 2SC; see Ouyed & Sannino 2001). The emis-
sion/cooling time is then
with ǫ = δ2SCδLGB.
4. Powering Gamma-Ray Bursts
4.1. Fireball and baryon loading
The fireball stems from the LGB decay and photon ther-
malization. The photons are emitted from the star’s sur-
face into the vacuum region beneath the inner crust (∼ 103
Fig.2. The episodic emission as illustrated in the QCD
phase diagram. The 2SC front spreads deep inside the
star and stops at B0 before re-entering the QGP phase.
Following photon cooling, heat flows from the core and
re-heats the surface. The star then starts cooling until
A1 is reached at which point the stage is set for the
2SC/LGB/photon process to start all over again (A1→
B1) resulting in another emission.
fm in size). Photon-photon interaction occurs in a much
longer time than the vacuum region crossing time. Also,
the cross-section for the creation of pairs through interac-
tions with the electrons in the vacuum region is negligible
(Rybicki & Lightman 1979; Longair 1992). The fireball
energy is thus directly deposited in the crust. If its en-
ergy density, aT4
c(with a being the radiation density con-
stant), exceeds that of the gravitational energy density in
the crust, energy outflow in the form of ions occurs. One
can show that the condition
where ρcrustis the crust density and G the gravitational
constant, is equivalent to
30 Mev)4> (MHQS
ρdrip) , (6)
which is always true if Tc> 30 MeV. The fireball is thus
loaded with nuclei present in the crust. More specifically,
it is the energy transfer from photons to electrons which
drag the positively charged nuclei in the process. Note that
the 2SC layer is not carried out during the two-photon de-
cay process because of the star’s high gravitational energy
density: ρHQS/ρdrip>> 1.
4.2. Episodic behavior
The star’s surface pressure is reduced following photon
emission1. A heat and mass flux is thus triggered from
the QGP phase to the 2SC layer re-heating (above Tc) and
destroying the superconductive phase. The entire star is
1The pressure gradient in the 2SC layer is ∆p ∝ (8 − 5)T4
(Farhi & Jaffe 1984) where the massless gluons (3 out of 8)
have been consumed by the LGB/photon process.
4Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?
now in a QGP phase (5 gluons→ 8 gluons at the surface)
and hence the cooling process can start again. This corre-
sponds to the transition [ρ(B0),T(B0)] → [ρHQS,T(B0)]
in the QCD phase diagram (thermal adjustment). The
stage is now set for the 2SC/LGB/photon process to start
all over again resulting in another emission. For the subse-
quent emission, however, we expect the system to evolve
to point B1 generally located at different densities and
temperatures than B0 (see Fig. 2). The cycle ends after
N emissions when ρ(BN) ≃ ρHQS.
The time it takes to consume most of the star (the glue
component) by this process is
≃ 1 s (MHQS
which is representative of the engine’s activity. The above
assumes quick adjustment of the star following each event,
but is not necessarily the case for the most massive stars.
4.3. Multiple shell emission
The episodic behavior of the star together with the result-
ing loaded fireball (we call shell) offers a natural mecha-
nism for multiple shell emission if Tc< 30 MeV. Indeed
from Eq. (6) a higher Tc value would imply extraction
of the entire crust in a single emission and no loading of
the subsequent fireballs. Clearly, Tc < 30 MeV must be
considered if multiple ejections are to occur2.
The fraction (f) of the crust extracted in a single event
∆Mcrust= fMcrust. (8)
The shell is accelerated with the rest of the fireball con-
verting most of the radiation energy into bulk kinetic en-
ergy. The corresponding Lorentz factor we estimate to be,
where we used Eq. (3) and Eq. (8). ǫ and f depend on the
star’s mass and characterize the two emission regimes in
4.4. Shell-shell collision
The Lorentz factor for the nthshell is
2Even if Tc turns out to be greater than 30 MeV, in which
case the entire crust will be blown away (Eq. (6)), one can
imagine mechanisms where crust material is replenished. By
accretion, for instance, if the HQS is part of a binary. There
are also geometrical considerations where asymmetric emis-
sion/ejection can occur due to the rapid rotation of Quark
Stars; here only a portion of the crust is extracted at a time.
This aspect of the model requires better knowledge of the con-
ditions and environments where HQSs are formed.
)(1 − ǫn−1)
(1 − fn−1)Γshell,n−1. (10)
The ratio Γshell,n/Γshell,n−1 is a function of ǫ (which de-
pends on the details of the cooling process and the spread
of the 2SC front) and f (mostly related to the density
in the crust). With the two parameters varying from one
emission to another, the ratio can be randomly greater or
less than 1. As such, the shells will have random Lorentz
factors and random energies. Faster shells will catch up
with slower ones and will collide, converting some of their
kinetic energy to internal energy.
5. The two regimes
When the inner crust density is the neutron drip value,
one finds a minimum mass star of ∼ 0.015M⊙. For masses
above this critical value, the corresponding crusts are thin
and light. They do not exceed few kilometers in thickness.
Matter at the density of such crusts is a Coulomb lattice
of iron and nickel all the way from the inner edge to the
surface of the star (Baym et al. 1971). For masses below
0.015M⊙, the crust can extend up to thousands of kilo-
meters with densities much below the neutron drip. This
allows us to identify two distinct emission regimes for a
given Tc(< 30 MeV).
5.1. Light stars (MHQS< 0.015M⊙)
These are objects whose average density is ∼ ρHQS
HQSρHQS). The 2SC front extends deeper
inside the star (δ2SC ∼ 1). The star can be represented
by a system close to A0in Fig. 2. Each of the few emis-
sions (defined by ǫ) is thus capable of consuming a big
portion of the star. Furthermore, the entire crust mate-
rial can be extracted in a few 2SC/LGB/photon cycles
Using Eq. (7), the few emissions lead to
ttot ≃ fraction × tengine
≃ fraction × 0.25 s (
0.01 M⊙)(1 km
where ttot is representative of the observable time which
takes into account the presence of the crust.
5.2. Massive stars (MHQS≥ 0.015M⊙)
The surface density of a massive star being that of a light
star (ρHQSgiven by P = 0 in Eq. (1)), defines a standard
unit in our model. In other words, the mass of the 2SC
layer in a massive star case is
∆M2SC,m≃ M2SC,l, (12)
where “m” and “l” stand for massive and light, respec-
tively. It implies
3(1 km 5 km)3≃ 0.003 .(13)
Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?5
For a typical star of 5 km in radius, we then estimate a
2SC layer of about 15 meters thick (much larger than the
photon mean free path thus justifying the local thermal
equilibrium hypothesis). Equivalently,
¯ ǫ =
≃ (1 km
5 km)3≃ 0.01 ,(14)
where ¯ ǫ is the average value. This naturally account for
many events (or N fireballs). The average number of fire-
balls with which an entire star is consumed is thus
¯ ǫ≃ 100 .
Since most of the crust is at densities close to the neu-
tron drip value, Eq. (6) implies that only a tiny part of
the crust surface material (where ρcrust<< ρdrip) can be
extracted by each of the fireballs. This allows for a con-
tinuous loading of the fireballs.
The total observable time in our simplified approach
ttot≃ tengine= 1 s(MHQS
We isolated two regimes:
(i) Light stars ⇒ short emissions.
(ii) Massive stars ⇒ long emissions.
It appears, according to BATSE (Burst and Transient
Source Experiment detector on the COMPTON-GRO
satellite), that the bursts can be classified into two distinct
categories (Kouveliotou et al. 1993): short (< 2 s) bursts
and long (> 2 s, typically ∼ 50 s) bursts. The black body
c) inherent to our model puts stringent con-
straints on the value of Tcwhich best comply with these
observations. Using Tc≃ 10 MeV, from Eq. (11) and Eq.
(16) we obtain in the star’s rest frame
ttot≃ 81 s (MHQS
for massive stars (suggestive of long GRBs), and
ttot≃ 2 s (
0.01 M⊙)(1 km
for light stars (suggestive of short GRBs). There is a clear
correlation (almost one to one) between the observed burst
time and the time at which the source ejected the specific
shell (see Figure 3 in Kobayashi et al. 1997, for example).
Note that Tc≃ 10 MeV implies that only a portion of the
crust is extracted. This is also consistent with our previous
assumption (Tc< 30 MeV) and subsequent calculations.
When Tc≃ 10 MeV, Eq. (6) gives ρcrust/ρdrip≃ 1/16.
For an appropriate crust density profile (using the equa-
tion of state given in Baym et al. 1971), from Eq. (8) we
find¯f ≃ 0.01. This implies (making use of Eq. (14))
Γshell= 2 × 105(
For massive stars then3
0 < Γshell< 2 × 105.(20)
For light stars, where both ǫ and f are close to
unity, Γshell ≃ 105. The shells are also heavier than in
the massive stars case. We thus expect stronger shocks
from the shell-shell collision resulting in harder bursts.
Combined with our previous results, this is suggestive of
(i) Light stars ⇒ short and hard bursts.
(ii) Massive stars ⇒ long and soft bursts.
Eq. (17) and Eq. (18) is simply Eq. (7) rescaled to the
appropriate object size. We separated two regimes due to
intrinsic differences in the engine and the crust. From the
engine point of view, massive stars generate many more
emissions when compared to light ones, and no substantial
reduction of the engine time is expected because of the
omni-presence of the crust. Another important difference
is related to the physics of the multiple re-adjustments
following each event which is more pronounced for very
massive stars. The latter among other factors is related to
ǫ which can vary from one event to another.
6. Features and predictions
6.1. GRB energies
The maximum available energy is when the heaviest HQS
(MHQS,max≃ 2M⊙) is entirely consumed. That is,
ELGB,max≃ 4 × 1054ergs . (21)
The corresponding GRB energy is
EGRB,max≃ 1.6 × 1054ergs ,(22)
where we used a fiducial conversion efficiency of 40% (Sect.
Since MHQS,min< 0.015M⊙we conclude that,
ELGB,min< 3 × 1052ergs ,(23)
EGRB,min< 1.2 × 1052ergs .(24)
6.2. GRB total duration
From Eq. (17) and Eq. (18) we have
ttot≃ 81 s ,(25)
for typical massive stars, and
ttot≃ 2 s ,(26)
3Note that a relativistic loaded fireball is not necessarily
achieved since Γshell< 100 at times. Gaps in the GRBs spectra
are thus expected according to our model.
6Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?
for typical light stars.
Our estimate of the duration time for the massive
star case should be taken as a lower limit. As we have
said, a complete model should take into account star re-
adjustments. Nevertheless, we can still account for a wide
range in GRB duration by an appropriate choice of differ-
ent values of the mass and radius.
6.3. Peak duration vs energy
The peak duration (tp) is related to the time measured by
a clock on the shell via (Fenimore & Ramirez-Ruiz 1999),
where csis the speed of the shock front crossing the shell
leading to the burst. We find
In the above we used Eq. (9) and Mshell = ∆Mcrust =
crust∆Xcrustρcrust. The expression on the right is ∝
∆tcool as can be seen from Eq. (4) and in our model is
constant for a given star. The only parameter which is
directly linked to the shell dynamics and energetics is cs.
Shock physics gives (Ouyed & Pudritz 1993)
s∝ Eint., (29)
where Eint.is the shell’s internal energy gained during the
collision observed as the peak’s energy (Ep). That is,
in reasonable agreement with the power law dependence
extracted from temporal vs energy structure in GRBs (in-
dex that is between −0.37 and −0.46; Fenimore et al.
6.4. Shell dynamics
Take a shell of thickness ∆Xcrustto be extracted from the
crust. The upper surface of the shell is extracted first while
its lower surface lags behind by (c − vshell)t ≃ ct/(2Γ2
(t is the time to eject the entire shell in the star’s rest
frame). Taking into account mass conservation and the
fact that ∆Xshell = 2Γ2
shell∆Xcrust, it is straightforward
Interestingly enough, these are the required conditions (in-
cluding the result from Eq. (20)) in the internal shock
model which lead to the highest (up to 40%) conver-
sion efficiency and the most desirable temporal structure
(Kobayashi et al. 1997; Mochkovitch et al. 1995).
7. Discussion and Conclusion
7.1. Existence and formation of quark stars
In the last few years, thanks to the large amount of fresh
observational data collected by the new generation of X-
ray and γ-ray satellites, new observations suggest that the
compact objects associated with the X-ray pulsars, the X-
ray bursters, particularly the SAX J1808.4-3658, are good
quark stars candidates (see Li et al. 1999). While these
observations/measurements are hints that such objects
might exist in nature it remains to explain their forma-
tion. More importantly to our model, the bimodal mass
distribution remains to be explained.
7.1.1. Massive stars
For the massive stars the conversion of neutron stars to
quark stars is one plausible scenario (Cheng & Dai 1996;
Ma 1996; Bombaci & Datta 2000). They could also form
via the direct mechanism following a supernova collapse
where the core collapsed to a stable quark matter instead
of neutron matter (Gentile et al. 1993; Dai et al. 1995).
Both mechanisms would lead to the formation of quark
stars (strange stars to be more specific) with masses in
the solar mass range.
7.1.2. Light stars
The formation of small quark stars has already been
discussed (early discussions can be found in Alpar
1987; Glendenning et al. 1995; see also Chapter 10.5 in
Glendenning 1997) although these remain less understood
than the massive ones. In the case of 4U 1728-34 (where
a mass of much less than 1.0M⊙ was derived; Bombaci
1999), it seems that accretion-induced collapse of white
dwarfs is a favored formation mechanism. If the quark
star formed via the direct conversion mechanism then it
required too much mass (at least ∼ 0.8M⊙to be ejected
during the conversion).
How and why stars in the 0.01M⊙range would form
remains to be explained. Our arguments were solely based
on theoretical considerations related to the critical density
in the inner crust (neutron drip) as to differentiate be-
tween small stars with thick and heavy crust versus stars
with thinner and lighter crusts.
7.2. Neutrino cooling and HQSs
If neutrino cooling is shown to remain efficient in the 2SC
phase (for comparison of cooling paths between quark
stars and neutron stars and the plausible effects of 2SC
on cooling we refer the interested reader to Schaab et al.
1997; Blaschke et al. 2000; Blaschke et al. 2001), we would
be left with the scenario where the entire HQS enters the
2SC phase, in which case the 2SC/LGB/photon process
(the fireball) occurs only once and inside the entire star.
Here, one must involve more complicated physics (such as
Rachid Ouyed and Francesco Sannino: Quark Stars as inner engines for Gamma Ray Bursts?7
that of the crust) to account for the episodic emissions
so crucial to any model of GRBs. It is not clear at the
moment how to achieve this and is left as an avenue for
7.3. 2SC-II stars
The 2SC/LGB/photon process might proceed until one is
left with an object made entirely of 2SC. We name such
objects 2SC-II stars4which are still bound by strong in-
teractions (their density is constant ∼ ρHQS). 2SC-II stars
carry an Iron/Nickel crust left over from the GRB phase.
The crust mass range is 0 < M2SC,crust < 5 × 10−5M⊙
depending on the efficiency of crust extraction/ejection
during the GRB phase.
BATSE observes on average one burst per day. This
corresponds, with the simplest model - assuming no cos-
mic evolution of the rate - to about once per million years
in a galaxy (Piran 1999a). In the Milky way we thus ex-
pect up to 105of 2SC-II stars. Nevertheless, they are tiny
enough (M ≤ 10−2M⊙, R ≤ 1 km) to be difficult to de-
Fig.3. The Mass−Radius plane derived in our model us-
ing few existing GRBs with known energies and total du-
ration. The solid curve shows the MHQS=4π
equation for ρHQS≃ 9ρN.
4The “II” in 2SC is a simple reminder of the final state of
the star, namely the 2SC with only 5 gluons.
7.4. The Mass−Radius Plane
Take observed GRBs with known energies and total du-
ration. From the burst total energy EGRB ≃ 0.4Mc2we
derive the mass while the burst total duration (ttot) gives
us the radius (using Eq. (16) with Tc≃ 10 MeV). In Fig. 3
we plot the resulting Mass−Radius. Note that while neu-
tron stars, can only exists above a certain mass (∼ 0.1M⊙;
Baym et al. 1971), there is no lower limit to the mass of
quark stars. These would be bound by the strong interac-
tion even in the absence of gravity.
The solid curve shows the MHQS=4π
tion which is a reasonable approximation for quark stars.
While the GRB data set used is limited nevertheless it
seems to support the idea that extremely compact objects
(M ∝ R3) are behind GRBs activity within our model.
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