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Spinar Paradigm

and
Central Engine of the All Types Gamma Ray Bursts


V.Lipunov1,2,3 & E.Gorbovskoy1,2,3

1Sternberg Astronomical Institute, Moscow, Universitetsky pr. 13, Moskow
119992, Russia.
2Moscow State University, Moscow, Universitetsky pr. 13, Moskow 119992,
Russia.
3Moscow Union "Optic", Moscow, Valilov str 5/3, Moskow 119334, Russia.

A spinar is a quasi-equilibrium collapsing object whose equilibrium is
maintained by the balance of centrifugal and gravitational forces and whose
evolution is determined by its magnetic field. The spinar quasi equilibrium
model recently discussed as the course for extralong X-ray plateu in GRB
(Lipunov & Gorbovskoy, 2007).
We propose a simple non stationary three-parameter collapse model with
the determining role of rotation and magnetic field in this paper. The
input parameters of the theory are the mass, angular momentum, and magnetic
field of the collapsar. The model includes approximate description of the
following effects: centrifugal force, relativistic effects of the Kerr
metrics, pressure of nuclear matter, dissipation of angular momentum due to
magnetic field, decrease of the dipole magnetic moment due to compression
and general-relativity effects (the black hole has no hare), neutrino
cooling, time dilatation, and gravitational redshift.
The model describes the temporal behavior of the central engine and
demonstrates the qualitative variety of the types of such behavior in
nature.
We apply our approach to explain the observed features of gamma-ray
bursts of all types. In particular, the model allows the phenomena of
precursors, x-ray and optical bursts, and the appearance of a plateau on
time scales of several thousand seconds to be unified.

1.Introduction.

The interest toward magneto-rotational collapse has increased
appreciably in recent years in connection with the gamma-ray burst problem.
It is now believed to be highly likely that long gamma-ray bursts may be
associated with the collapse of a rapidly rotating core of a massive star
and short gamma-ray burst are most likely to be results of the coalescence
of neutron stars, which can be viewed as the collapse of a rapidly rotating
object. We already pointed out in our earlier papers (Lipunova, 1997,
Lipunova & Lipunov, 1998) the likely multivariate nature of, e.g., the
coalescence of two neutron stars or neutron stars and black holes
("mergingology"), which may give rise to various forms of the temporal
behavior of gamma-ray bursts. This is possibly corroborated by the recent
complex classification of gamma-ray bursts (Gehrels et al., 2006).
Moreover, observations of the so-called precursors and x-ray flare
certainly point to the complex nature of the operation if their central
engines (Lazzati, 2005; Chincarini et al., 2007). ROTSE (Quimby et al.,
1996a,) and MASTER (Lipunov et al., 2007) facilities observed optical
flares in a number of cases.
All this triggers (mostly numerical) theoretical studies of collapse
with the dominating role of rotation. Numerous attempts have been
undertaken in order to incorporate effects due to rotation and magnetic
fields in numerical computations, which are very difficult to understand
intuitively and at the same time are extremely approximate because of the
complex nature of the problem (Gehrels et al., 2006, Moiseenko et al.,
2006; Duez et al., 2005, 2006).
Recently, (Lipunov & Gorbovskoy 2007) showed that spinar paradigm
naturally explains not only the phenomenon of early precursors and bursts,
but even extraordinarily long x-ray plateaux.
In this paper we propose a pseudo-Newtonian theory of collapse based on
a simple analytical model, which allows the maximum number of physical
effects to be incorporated.
We use our model to interpret the data of observations of precursors
(Lazzati, 2005), X-ray flares (Chincarini ey al., 2007), and some
interesting gamma-ray bursts.

2. The Spinar Model.

The importance of incorporating magneto-rotational effects in collapse
models was first pointed out in connection with the problem of quasar
energy release and evolution (Hoyle and Fowler, 1963; Ozernoy, 1966;
Morison, 1969; Ozernoy and Usov, 1973), and that of the ejection of
supernova shells (Bisnovaty-Kogan; 1971, LeBlance & Wilson 1970).

In particular, it was pointed out that the collapse of a star having
substantial angular momentum may be accompanied by the formation of a quasi-
static object - a spinar - whose equilibrium is maintained by centrifugal
forces. Ostriker (1970) and Lipunov (1983) assumed the existence of low-
mass spinars with close-to-solar masses. Lipunov (1987) made a detailed
analysis the spin-up and spin-down of spinars in the process of accretion.
Lipunova (1997) developed a spinar model incorporating relativistic
effects (which include the disappearance of magnetic field during the
formation of a black hole), gave an extensive review of the research on the
spinar theory, and tried to apply the spinar model to the gamma-ray event.
A spinar can be viewed as an intermediate state of a collapsing object
whose lifetime is determined by the time scale of dissipation of the
angular momentum. As Lipunova & Lipunov (1998) pointed out, the centrifugal
barrier could explain the long (from several seconds to several hours)
duration of the process of energy release in the central engines of gamma-
ray bursts. It is remarkable that as it loses angular momentum a spinar
(unlike, e.g., a radio pulsar) does not spind down, but, on the contrary,
spins up and this effect results in the increase of luminosity, which is
followed by the luminosity decrease because of the disappearance of
magnetic field, relativistic effect of time dilatation, and gravitational
redshift near the event horizon.
Lipunova (1997) analyzes a model of a spinar in vacuum, which is
justified for two neutron stars. However, in the case of a collapse of a
core of a massive star the spinar is surrounded by the star's envelope and
matter outflowing from its equator. We analyzed the interaction of a spinar
with the ambient plasma in our earlier paper Lipunov (1987), from where we
adopt the law to describe the dissipation of the spinar angular momentum .
Finally, Lipunov & Gorbovskoy (2007) developed a stationary spinar
model, which allows for relativistic effects and maximum possible
dissipation of the angular momentum of the spinar.
Below we abandon the quasi-stationary analysis and construct a non-
stationary model of rotational collapse.

3. Spinar scenario of magneto-rotational collapse.
Collapse of a rapidly rotating core.

Let us now qualitatively analyze the magneto-rotational collapse of a
stellar core of mass Mcore and effective Kerr parameter (Thorne et al.,
1986)
[pic] (1),


(here I = k Mcore R02 is the moment of inertia of the core; ( is the
angular velocity of rotation, and c and G are the speed of light and
gravitational constant, respectively), and magnetic energy Um.
In the case of conservation of the core angular momentum (which, of
course, will be violated in our scenario), a remains constant.
Let [pic]be the ratio of the magnetic energy of the core to its
gravitational energy:
[pic] (2).


The total magnetic energy can be written in terms of the average
magnetic field B penetrating the spinar:
[pic]=(1/6)B2/R3 (3)


Note that in the approximation of magnetic flux conservation (ВR2 = const),
the magnetic-to-gravitational energy ratio remains constant during the
collapse:

[pic]= const , [pic] without considering general-relativity effects.
Let the initial Kerr parameter a0 > 1. In this case, direct formation
of a black hole is impossible and the process of collapse breaks into
several important stages (see Fig.1.):

A. Loss of stability by the core and free fall
The time scale of this stage is on the order of the free-fall time
[pic], (4)


where RA is the initial radius of the stellar core. Energy is
virtually not radiated during the collapse, and gravitational energy
transforms into kinetic, rotational, and magnetic energy of the core.
B. Halt of the collapse by centrifugal forces.
Centrifugal forces stop free-fall collapse at the distance where
[pic] (5)


It follows from this that the initial spinar radius is approximately
equal to:
RB=a2GMcore/c2=a2Rg/2 (6)


In this process, half of the gravitational energy is released:
[pic] (7)


if the energy is sufficient to "penetrate" the stellar envelope, i.e.,
if the momentum imparted to a part of the shell exceeds the momentum
corresponding to escape velocity. Let a part of the energy be converted
into the energy of the jet [pic]
[pic] (8)


In this case a burst of hard radiation occurs.
We now substitute the burst energy (formula (7)) and spinar radius (6)
into condition (8) to derive the "penetration" condition for the first
jet:
[pic] (9)


where Vp is the escape velocity at the surface of the stellar
envelope. In real situations Vp = 2000-3000 km/s , [pic] , and almost
everything is determined by the jet opening angle. This simple estimate
shows that the first penetration is highly likely even in the case of a
large jet opening angle.
Because of the axial symmetry, the burst must be directed along the
rotation axis and have an opening angle of [pic]. The duration of this
stage is determined by the time it takes the jet to emerge onto the
surface (Rshell/c~10-30s) and the character of cooling governed by the
structure of the primary jet and envelope.
The character of the spectrum is determined by the gamma factor of the
jet.
The newly formed spinar then evolves until its collapse without losing
its axial symmetry

C. Dissipative evolution of the spinar

The spinar contracts as its angular momentum is carried away. Note that
this process is accompanied by the increase of the velocity of rotation and
luminosity of the spinar. At the same time, the magnetic dipole moment
decreases and the luminosity stops increasing and begins decreasing. The
energy release curvr acquires the features of a burst.
The duration of this stage is determined by the moment of forces that
carry away the angular momentum of the collapsar. In real situations
turbulent viscosity and magnetic fields may play important part in the
process.

The corresponding dissipation time scale (the spinar life time) is:
[pic] (10)


where Ksd is the characteristic torque of dissipative forces. It is clear
that under the most general assumptions about the character of magnetic
field the spin-down torque must be proportional to the magnetic energy of
the spinar:
[pic] (11)


where [pic] is the dimensionless factor that determines how twisted
magnetic field lines are via which the angular momentum is dissipated.
Correspondingly, the total time scale of the dissipation of angular
momentum (spinar lifetime (9)) is equal to:
[pic] (12)


D. Second burst.
Energy is released during dissipation, and the rate of this process
increases progressively until general relativity effects --- redshift and
disappearance of magnetic field come into play.
As the luminosity increases, at a certain time instant the conditions
of shell penetration (similar to condition (8)) become satisfied:

[pic] (13)


A second jet appears whose intensity reaches its maximum near the
gravitational radius. Note that the effective Kerr parameter tends to its
limiting value for the extremely rotating Kerr black hole: a(1.
The maximum luminosity can be written in terms of the dissipation of
rotational energy near the gravitational radius:
[pic] (14)


It is better to write the condition of the penetration for the second jet
in terms of pressure inequality:
[pic] (15)


Note that [pic] is the so-called natural luminosity.
Of course, formula (13) does not include gravitational redshift, decay
of magnetic field, etc.
The time scale near the maximum is:
[pic] (16)


Further fate of the star depends on its mass. If the mass exceeds the
Oppenheimer--Volkoff limit the star collapses into a black hole. Otherwise
(Lipunova & Lipunov, 1998) a neutron star forms, which cools after 10
seconds, continues to spin down in accordance with the following formula
[pic] (17)


where[pic] is the magnetic dipole moment and [pic] is the radius of the
light cylinder,
and radiates as a common pulsar. In the case of constant magnetic field
the luminosity of the pulsar should decrease in accordance with the
following law:
[pic] (18)


In the case of a coalescence of two neutron stars or a neutron star and
a black hole the first stage (stage A) is very short, because the «fall»
begins at a distance of several gravitational radii. Because of gravity-
wave losses the components of the binary first approach each other to the
radius of the last stable orbit and then merge to form a spinar. A small
burst may occur at the time of stellar merging immediately before the
spinar forms. This burst has the energy of:

[pic] (19)


The qualitative picture of magneto-rotational collapse considered here
can be illustrated by the following scheme (see Fig. 2.) in the coordinates
Um and a - the effective Kerr parameter.
The proposed scenario allows easy interpretation of the precursors and
burst. In the case of large angular momentum (a>>1) the initial radius is
large and, correspondingly, the energy release rate is low, allowing stage
B to be interpreted as a precursor.
In the case of low angular momentum (a>~1) the initial spinar radius is
close to several gravitational radii and stage B must be interpreted as a
gamma-ray burst, whereas the subsequent spinar burst D must be interpreted
as a flare event.
It is remarkable that the time interval between the two bursts is
always determined by the duration of dissipation of angular momentum (12),
and, consequently, a rest-time measurement immediately yields a relation
between the Kerr parameter and the fraction of magnetic energy:
[pic] (20)

where [pic] .
Correspondingly, the characteristic magnetic field at the collapse time
(near Rg) is equal to:


[pic] (21)


where [pic].
The proposed scenario allows the observed variety of gamma-ray bursts,
precursors, and flares to be reduced to just two parameters: magnetic field
and initial angular momentum. In the case of weak magnetic field and large
angular momentum (the bottom-right corner) the first burst is weak (because
of the high centrifugal barrier) and the resulting jet does not penetrate
the stellar envelope - there are no precursors to be observed. This is
followed by slow collapse (magnetic field is weak), which results in a weak
x-ray rich burst. As the initial angular momentum decreases (we move
leftward in the diagram) the energy released at the centrifugal barrier
increases and the jet becomes capable of «penetrating» the stellar
envelope. The first burst should act as a precursor. The precursor should
be separated from the gamma-ray burst, because the time scale of the
dissipation of angular momentum is long in the case of a weak field. As
angular momentum decreases (we move further leftward along the horizontal
direction) the precursor energy increases and at a>~1 the precursor energy
exceeds 1051-52 erg and it shows up as a gamma-ray burst with the
subsequent collapse of the spinar leading to an X-ray plateau event (the
bottom-left corner Lipunov & Gorbovskoy, 2007). If we further move toward
increasing magnetic field (up), the subsequent collapse of the spinar
should lead to an X-ray flare event. In the case of even stronger magnetic
field, the flare approaches a gamma-ray burst, its energy grows and the
flare itself becomes a part of the gamma-ray burst (the top-left corner).
If we move rightward, angular momentum grows and the first flare loses
energy and becomes a precursor close to the second flare, which, in turn,
actually becomes a gamma-ray burst.
In the case of very large angular momentum (the top-right corner) the
energy of the precursor is insufficient for penetrating the envelope and we
have a burst without satellites. The duration of energy release increases
with decreasing strength of magnetic field and the burst becomes softer (we
return to the bottom-right corner).

4. One point pseudo-Newtonian Spinar Model
Nonstationary model of magneto-rotational collapse.

The aim of our model is to provide a correct qualitative and
approximate description of magneto-rotational collapse, which would allow
us to follow the evolution of the rate of energy release of the collapsing
object and demonstrate the diverse nature of the central engine. Note that
the spinar is born and dies in a natural way as a result of the solution of
nonstationary problem.
Let us assume that at the initial time instant we have a rotating
object (it may be a core of a massive star that has become unstable, or a
merged neutron star, or the massive disk around a black hole). The object
has the mass of М, radius Rcore, angular momentum [pic], dipole momentum
[pic], and Kerr parameter [pic].

a. Dynamic Equation

We write the equation of motion in the post-Newtonian approximation:

[pic] (24)


where Fgr is the gravitational acceleration, Fc, the centrifugal
acceleration, and Fnuclear, the pressure of matter.
Several attempts have been made to propose a pseudo-Newtonian potential
to simulate the Kerr metrics (see Artemova et al., 1996 ). In our model we
use effective acceleration in the form proposed by Mukhopadhyay (2002) for
particles moving in the equatorial rotation plane:


[pic] (25)

where[pic].This formula corresponds to the potential of Paczynski & Wiita
(1980) for a nonrotating black hole.
Next terms:

[pic] (26)

[pic] (27)


Pressure of gas, which includes thermal pressure, can be written as
kinetic energy of particles computed using relativistic invariant
(Zel'dovich, Blinnikov, Shakura 1980):

[pic] (28)


The second and third terms under the radical sign allow for the
pressure of degenerate gas and thermal energy, respectively.

Let us now rename constant b:
[pic] (29)


We actually use the formula for the pressure of partially degenerate
Fermi gas with the contribution of thermal pressure. It is clear that the
equation of real nuclear matter cannot be described by such a simple
formula. However, we managed, by fitting appropriate values of constant b,
to obtain neutron stars with quite plausible parameters (see Appendix 1).
By varying constant b we can, in particular, vary the Oppenheimer-Volkoff
limit for cool nonrotating neutron stars. We put MOV = 2 Solar Mass in this
paper for cool nonrotating neutron stars.
Of course, one must bear in mind that the real Oppeheimer-Volkoff limit
depends both on the velocity of rotation of the neutron star and on its
thermal energy (Friedmann et al., 1985 ). In our model this dependence is
qualitatively consistent with the numerical results obtained earlier.
We finally introduce dissipative force Fdiss:
[pic] (30)


It is clear from physical viewpoint that after reaching the centrifugal
barrier the core undergoes extremely strong oscillations with a time scale
of 1/[pic]. This process is accompanied by the redistribution of angular
momentum and complex nonaxisymmetric motions, which must ultimately result
in the release of half of the gravitational energy and formation of a quasi-
static cylindrically symmetric object - a spinar. A detailed analysis of
this transition is beyond the scope of our simple model. We just introduce
a damping force assuming that its work transforms entirely into heat so
that our model correctly describes the total energy release during the
formation of the spinar, but is absolutely unable to describe the temporal
behavior at that time. We actually assume that:

[pic] (31)


Throughout this paper, [pic]=0.04 unless otherwise indicated.

b) Angular momentum loss equation

The decrease of the angular momentum of the spinar (collapsar) is due
to the effect of magnetic and viscous forces. In this paper we assume that
dissipation of angular momentum is due to the effective magnetic field. In
this case, the breaking torque in a disk-like object is equal to (see
Lipunov, 1992)

[pic], (32)

where [pic] and [pic] - z and [pic] are the components of magnetic field.
We now introduce the magnetic moment [pic] of the spinar. Hereafter,
for the sake of simplicity, we write our equations as if the spinar had a
dipole magnetic field. However, our equations remain unchanged if we simply
use some average magnetic field of the spinar and characterize this field
by the spinar magnetic energy Um mentioned above. This is true for the
breaking torque that we use below.
Let [pic], where [pic] is dipolar strength of the magnetic fields. The
breaking torque is then equal to (see Lipunov, 1987, 1992 see below)
[pic], (33)


where [pic] and Rt is the characteristic radius of interaction between
the magnetic field and ambient plasma:

Rt = RAlfven is the Alfven radius (Propeller)

[pic] is the corotation radius (Accretor)
(34)

[pic] is the radius of the light cylinder (Ejector)

In the case of a spinar the Alfven radius is smaller than or on the
order of the stellar radius and is of little importance in the situation
considered.
In the case of a collapsing core the effective interaction radius must
be close to the corotation radius, which, in turn, is close to the spinar
radius in accordance with tits equilibrium condition. Therefore the
retarding torque can be written as:
[pic] (35)


And the corresponding dissipation time scale is:
[pic] (36)


Hence the equation of variation of the spinar angular momentum becomes
(Lipunov, 1987):

[pic] (37)



The retarding torque written in this form gives the absolute upper
limit for the possible spin-down of the spinar.

If the mass of the spinar is below the Oppenheimer-Volkoff limit, a
neutron star forms ultimately, which spins down in accordance with the
following magnetodipole formula:

[pic] (38)


c. Magnetic Field Evolution

As is well known (Ginsburg and Ozernoy, 1963) magnetic field must
disappear in the process of collapse.
In the Newtonian approximation in the case of magnetic-flux
conservation, the dipole moment behaves as:
[pic] (39)


With relativistic effects taken into account, magnetic field vanishes
not at zero, but when the star reaches the event horizon. Manko and
Sibgatullin (1992) computed the evolution of the dipole magnetic field of a
rotating body (in the Kerr metrics).
We can use the following simple formulas as the first approximation:
[pic] (40)

Here Rmin is the equatorial radius of the event horizon. Given that
R0>>Rmin , this formula correctly describes the behavior of the dipole
moment and yields zero magnetic field at the event horizon.

However, this law implies too fast decrease of magnetic field and we
use the following modified law of magnetic-field decay adopted from
Ginsburg and Ozernoy (1963):
[pic] (41)


where [pic] and xmin is the radius of horizon for current Kerr
parameter.

In this paper we neglect the effects of generation of magnetic fields.

d). Energy losses.

The release of energy in the process of collapse is initially due to
the dissipation of kinetic energy of the impact onto the centrifugal
barrier and to spinar spin-down due to magnetic forces:
[pic] before the formation of the spinar
(42)
[pic] after the formation of the spinar
(43)

Where invariably Rmin=Rc if the core mass exceeds the Oppenheimer-
Volkoff limit.
A distant observer would record lower luminosity because of
gravitational redshift and time dilatation.
We adopt the following observed luminosity:

[pic] (44)


where [pic] is the time dilatation function - the ratio of the clock
rate of reference observers to the world time rate at the equator of the
Kerr metrics (Thorne et al., 1986):
[pic] (45)


If the core mass is below the Oppenheimer-Volkoff limit, the spinar
ultimately evolves into a neutron star and its luminosity is given by the
following magnetodipole formula:

[pic] (46)


We finally consider the case where rotation is so slow that the spinar
does not form at all.
In this case direct collapse occurs. We pointed out above that Lipunova
(1997) was the first to address the problem of electromagnetic burst with
the allowance for general relativity effects. In the case of direct
collapse rotational motion is of no importance, because the star makes less
than a single rotation before it is under the event horizon. However, this
case is characterized by large radial variation of the dipole moment:

[pic] (47)



To convert this value into the observed luminosity we must take into
account gravitational redshift and the Dopple effect due to the emitter
falling in a virtually Schwartzshieldian metrics (Lipunova, 1997).

5. Collapse of a massive core (M > MOV).

Let us first consider the case where the core mass exceeds
substantially the Oppenheimer-Volkoff limit. We adopt the initial core
mass of 1000Rg as the initial conditions for our set of differential
equations. Figure 3 shows the computed variation of the radius, Kerr
parameter, and average magnetic field for several arbitrary initial core
parameters as functions of proper time (without the allowance for the time
dilatation factor). Diagram 4 shows the computed evolution of the central
engine for a wide range of models. Let us emphasize several important
points. First, the collapse of such cores ends by the formation of an
extremely rotating Kerr black hole. Of course, this event shifts to
infinitely distant time in the rest frame.
The diagram fully corroborates our qualitative scenario (Fig. 2) and
demonstrates a large variety of the time scales and energies of precursors,
gamma-ray bursts, and flares. The results of computations of the energy
release in direct collapse (a0<1) confirms the short duration and low power
of the flare. Note that the total energy does not exceed 10-4Mc2 for almost
all values of magnetic field. Evidently, in this case the appearance of
jets and of the gamma-ray burst phenomenon is difficult to imagine. Such a
collapse would rather result in a common supernova event.
However, the events acquire an increasingly dramatic turn with
increasing moment. At a0>1 centrifugal forces sooner or later exceed the
gravitational forces, halt the collapse to give time and opportunity for
enormous energy of about ~0.1Mc2 to be radiated during the halt of the
collapse. In this case a spinar is born and the relativistic jet penetrates
the envelope of the star and triggers a gamma-ray burst. The magnitude of
the first burst depends on the initial spinar radius exclusively, which to
a first approximation is determined only by the moment, as is evident from
the diagram. All systems in the same column have the same burst energy.
Magnetic field then takes the reigns of government and determines the rate
of dissipation of angular momentum and becomes the main factor to determine
further evolution of the core. As magnetic moment decays, the core radius
decreases and magnetic luminosity increases until it reaches its maximum
(at R~Rg) whose magnitude is determined by the magnetic field exclusively.
This is also evident from the diagram. After that the luminosity decreases
abruptly because of relativistic effects near the event horizon (decay of
the field, gravitational redshift, and time dilatation). It is the ratio
of the energies of the first and second flare that determines the entire
zoo (all the variety) of flare, precursor, and burst events. In the extreme
case of a strong magnetic field ([pic]?Eq21 COPY?) and comparatively small
momentum (1 short time interval of ~1-10s. It is thus impossible in this case to
separate the burst from the precursor or flare and we must view the event
as a double gamma-ray burst.
If we move rightward on the diagram in the direction of increasing
momentum the initial spinar radius increases (for systems with ~100Rg large
precursors), the gravitational energy released decreases, and the first
flare becomes weaker. We thus fall into the domain of precursors: ([pic]~10-
2-10-4, 10 magnetic field, the greater is the separation between the precursor and the
gamma-ray burst. Luminosity remains virtually constant between the
precursor and the gamma-ray burst.
If, on the other hand, we move downward from the domain of double gamma-
ray bursts, thereby decreasing the magnetic-field strength, increasing the
time interval between the primary and secondary bursts, and decreasing the
intensity of the second burst, we come into the extended domain of gamma-
ray bursts ([pic]~10-4-10-7, 2 comparatively small, the spinar has an initial radius of ~10Rg and the
first burst must be very powerful. Magnetic field, however, is weak and the
power of the second burst would suffice only to produce X-ray flares. In
the case of too weak fields ([pic]10-7) the second burst is virtually
absent, allowing some bursts (e.g., GRB070110 and GRB050904, see below for
details) to exhibit a long (~104) x-ray plateau.
Finally, the bottom-right corner is occupied by the systems where the
energy of neither the first nor the second flare is too low for a gamma-ray
burst. These cores ([pic]10-6, a0>14) may produce either an x-ray burst
with a precursor or unusual supernovas.

6. Collapse of a rapidly rotating intermediate-mass core (M>~MOV)
(supranova case).

Как отмечалось (Lipunova, 1997; Lipunova & Lipunov, 1998; Vietri & Stella,
(1998) - a "supranova" scenario), поскольку предел Оппенгеймера-Волкова для
вращающейся нейтронной звезды выше, то возможно временное образование
массивной нейтронной звезды, которая потеряв свой момент вращения
коллапсирует в черную дыру.

As it was marked (Lipunova, 1997; Lipunova & Lipunov, 1998; Vietri &
Stella, (1998) - a "supranova" scenario), as Oppenheimer-Volkoff limit for
fast rotating neutron star is higher, then massive NS temporal formation
is possible. Having lost its rotational moment

, the star collapse in to the black hole.

Этот случай мы и рассмотрим в данной части статьи. Для примера, рассмотрим
коллапс быстровращающегося ядра с массой 2.2 массы Солнца. Напомним, что
для определенности мы используем уравнение состояния с пределом
Оппенгеймера- Волкова 2.0 массы солнца (для невращающейся нейтронной
звезды).

For example let us consider a fast rotating core collapse with mass
2.2[pic]. We should remind that, for distinctness, we use the state
equation with Oppenheimer-Volkoff limit equal to 2.0 Solar masses (for non-
rotational neutron star).

Фактически, это означает, что на последних стадиях эволюции спинара его
равновесие поддерживается не только центробежными силами, но и ядерными
силами и, возможно, тепловым давлением.

Practically, that means that, spinar equilibrium at last stages of
evolution is maintained both centrifugal and nuclear forces and may be by
thermal (heat) pressure.


На рис. 5. показан результат расчета коллапса такого ядра. Тяжелая
нейтронная звезда образуется и живет примерно 100 секунд. При этом, ее
магнитовращательная светимость после начального плато, начинает падать
поскольку потери вращательного момента не приводят к ускорению вращения
(ядерное давление не позволяет сильно сжиматься нейтронной звезде -
спинару) .

Fig.5 demonstrates the result of that core collapse calculation. The heavy
neutron star exists for about 100 seconds. As magnetic momentum looses
lead to rotation acceleration, its magneto-rotating luminosity after
initial plateau begins to decrease (the nuclear pressure doesn't allow
neutron stars-spinar to compress strongly ).

Однако около 100 секунды эффективный керровский параметр становится меньше
единицы и релятивиские эффекты приводят к прямому коллапсу нейтронной
звезды в черную дыру.

But after near 100 seconds effective Kerr parameter becomes less than unity
and relativistic effects result in rapid direct collapse of neutron star
into the black hole.

7. Collapse of a rapidly rotating low-mass core (M
The collapse of a low-mass star is ultimately halted by the pressure of
degenerate matter. However, even in this case fast rotation plays important
part. In a number of cases, a neutron star does not form directly, but
first a spinar, which then transforms into a neutron star losing angular
momentum. Such a collapse does not end by abrupt decrease of luminosity (as
in the cases considered above), but has a long tail: L~t-2.
In this example we consider the collapse of a 1.5[pic] core into a
neutron star (Fig.6). The process results in the formation of a neutron
star of radius ~8.5Rg (38km). Thus the problem acquires yet another
characteristic radius - RNS (the nonrotating neutron star radius).
If centrifugal forces less then nuclear pressure (RNS>RSpinar), and
this is quite possible with strong fields and small angular momenta, the
neutron star forms directly and the light curve has only one maximum
followed by a t-2 decrease due to uniform dissipation of the angular
velocity of the NS. Such systems are located in the bottom-left corner of
diagram 6. ([pic]10-3,a0<6).
If RNS However, it does not resemble the collapse of a massive core. This is due
to the fact that if the radius of the NS is ~10Rg no second burst is to be
expected near Rg. Hence we have no systems with precursors and gamma-ray
bursts occur only in systems with small initial momenta (a<12)
Some systems with intermediate rotation and strong field ([pic]10-
4,6 systems with small angular momentum, because in these cases the height of
the plateau exceeds that of the flare.
In other cases the energy of any flare is hardly sufficient for it to
penetrate the envelope, and supernovas are observed.

8. Statistical properties of precursors, flares, and gamma-ray bursts.

An analysis of BATSE data (Lazzati, D., 2005) shows that up to 20
percent of long gamma-ray bursts have precursors preceding the trigger time
by up to 200 s. Chincarini et al., (2007) found about 30 flux increase
events (optical flares) from Swift observatory data.
There are no more doubts that at least a substantial part of these
phenomena are associated with the peculiarities of the operation of the
"central engine".

8a. Precursors.

Numerous observations of gamma-ray bursts show a complex structure in
their temporal behavior, which is impossible to explain in terms of a
single burst, formation of a jet, and development of a system of shocks in
this jet. For example, the model associated with the emergence of the tip
of the bow shock onto the star's surface (Ramirez-Ruiz et al., 2002; Waxman
& Meszaros (2003)) can explain precursors that are close to the time of the
gamma-ray burst (GRB-time), but not the early precursors preceding the main
GRB by 100-200 s (Xiang-Yu Wang & Meszaros, 2007). The latter authors
proposed a model where early precursors appear as a result of the fallback
of a part of the star's shell.
Our proposed scenario naturally explains the phenomenon of precursors
and flares. In the case of large angular momentum (a>>1) the initial radius
is large and, correspondingly, the energy release is low, allowing stage B
to be interpreted as a precursor phenomenon.
In this case, the following condition must evidently be satisfied:

[pic] (48)




where Tpre, TGRB, F90, and ( are the observed fluence, and the jet
opening angle of the gamma-ray burst or precursor, respectively. Note that
the SLOPE of the latter relation does not depend on the redshift of the
gamma-ray burst.

In the model considered a precursor is defined as the initial energy
release when centrifugal forces halt the collapse of the core (stage B ) in
the case where
GM2/R<

8b. X-Ray Flares.

If the initial spinar radius RB is small, the energy release at the
time of its formation is sufficient to produce a gamma-ray burst, and hence
the first flare should be interpreted as a gamma-ray burst, whereas the
secondary release of energy by the spinar can be interpreted as a flare. In
this case the energy of the gamma-ray burst is approximately equal to:

EGRB=[pic] (50)



The burst luminosity is
[pic] (51)



where (g and (g are the magnetic moment and angular velocity at the
distance of Rg , respectively:

[pic]
[pic]

Hence the time gap between the gamma-ray burst and the flare is equal
to the spin-down time of the spinar at the maximum radius:
[pic]


Simple substitutions yield the following relation between the observed
flare parameters:
[pic] (52)


We now substitute the observed quantities into the latter formula to
obtain:

[pic] (53)


where Fflare is the maximum flux during the flare and FluenceGRB is the
total fluence of the gamma-ray burst.
In the latter relation EGRB is the only quantity that depends on the
distance to the gamma-ray burst. We can therefore plot the observed
relation FluenceGRB/Fflare =function(tflare) . Figure 9 shows the observed
relation and the relation simulated in our model. We adopt experimental
data from Chincarini et al. (2007). The straight line shows approximate
analytical relation (53) :
Figure 9b shows our computed models for the collapse of 7-solar mass
star with parameters:
2 10-7<[pic]<10-2
One can see that the observed and theoretical spectra show similar
trends: they both grow toward (temporally) distant flares with a comparable
scatter. The scatter is mostly due to the large factor, and the difference
between the mean values is due to the following ratio
[pic]
When converting XRT observations we assumed that 1 count/s=10-10
(Sakamoto et al. 2006, GCN Report 19.1 02Dec06)
One must bear in mind, when comparing the observed and simulated
points, that BAT and XRT soft-ray detectors operate in different energy
intervals. XRT observations are made in the energy interval 0.3-10 keV,
where absorption may be important. In addition, the observed fluxes during
flares must be multiplied by a factor of five to seven, because the
spectrum has a power-law form and is much wider than the XRT energy
channel. Moreover, part of the flares (especially those with delays < 100
s) can also be explained by the emission of a system of shocks (reverse
shock Chincarini et al., (2007)).
All this leads us to conclude that the slope and scatter of the average
theoretical and observational curves agree well with each other and the
absolute vertical shift may be due to the differences of the directivity
diagrams of the gamma-ray burst and optical flare, soft x-ray extinction,
and extrapolation of the power-law spectrum to a wider energy interval.

9. An extraordinary long X-ray plateau GRB070110 and GRB050904.

Two of several hundred gamma-ray bursts - GRB070110 and GRB050904 - do
not fit the common scenario of the X-Ray afterglow formation. Both bursts
exhibit a long plateau with a rest-frame duration of 6000-7000 s. Troja et
al. (2007) associated such a long activity with the specifics of the
central engine and, in particular, with the formation of a neutron star
after the collapse of a low-mass core (with the mass below the Oppenheimer-
Volkoff limit).
We fully agree that such an unusual behavior of the X-ray afterglow is
due to the central engine, but we believe that the plateau appears not as a
result of the radiation of the neutron star, but as a result of the
activity of a spinar with anomalously weak magnetic field.
An hypothesis Troja, E., et al. (2007) associates the plateau with the
collapse
producing a neutron star - a radio pulsar - whose activity becomes
appreciable during
the fading of the afterglow. We believe this interpretation of the plateau
to be too far
fetched. The intensity of the magnetodipole radiation, which is typical for
radio pulsars,
decreases with time as t-2. The abrupt termination of the plateau stage
remains completely
unexplained in the young pulsar model. The authors of this hypothesis point
out that the
decrease of luminosity could be a result of generation of the magnetic
field. However, the
last assumption makes the model too complicated.
A plateau with a slight increase and abrupt decrease of luminosity
appears naturally in the spinar paradigm.
In our scenario a plateau is a flare with weak magnetic field. In other
words, in this case the gamma-ray burst corresponds to the halt of the
collapse by centrifugal forces at radius RB , and the plateau is an
extended flare due to magneto-rotational losses.

We try to explain the following observed quantities (Troja et al.,
2007) GRB070110:


Table1. Isotropic energy parameters and rest-frame durations of two
GRB.
|GRB |Fluence |Plateau |X-ray |Total plateau |
| | |duration |luminosity |energy |
|070110 |1.3 1053 erg |~8000 s |1048erg/s |1052 erg |
|050904 |1.3 1054 erg |~8000 s |2 1049erg/s |2 1053 erg |



Let us first make some approximate estimates. The initial Kerr
parameter is equal to:

[pic]
The initial spinar radius is:

Rs =a2GM/c2=Ѕa2Rg

The energy of the gamma-ray burst is

EGRB=ЅGM2/Rs=Mc2/2a2
We derive from this relation the Kerr parameter:

a = EGRB/Mc2

The characteristic plateau duration is determined by the time scale of
the loss of the spinar angular momentum:

[pic]
The luminosity of the plateau at its maximum computed without the
allowance for relativistic effects is:
[pic]

We now use the observed plateau time to derive the parameters of the
collapse:
[pic]

Figure 10 shows the theoretical curve of the evolution of the
luminosity of the central engine. This spinar light curve shows a
characteristic plateau whose luminosity and duration are totally consistent
with spinar parameters Umag/Ug = 10-7 and Kerr parameter a=2.0
The plateau appears naturally in the spinar model and it is a typical
feature for the collapse of a core with small angular momentum and weak
magnetic field.

To illustrate these points, we computed a theoretical light curve
artificially supplying additional self-similar radiation in accordance with
the following law (Fig.11):

F=Ftheory+ C1 t-2+ C2 t-1

10) GRB 060926

X-ray flares can sometimes also be observed at optical wavelengths. Let
us try to explain the phenomenon of such a flare in the gamma-ray burst GRB
060926, where an optical flare was discovered along with the x-ray flare.
We choose this burst not only because we want to illustrate how spinar
paradigm works for flares observed both at x-ray and optical wavelengths,
but also because the optical radiation of this burst was discovered by
MASTER group whose members include the authors of this paper.
Optical observations of the gamma-ray burst GRB 060926 recorded by
Swift gamma-ray observatory (Holland,S et al 2006) were performed with
MASTER telescope operating in an automatic mode under good weather
conditions (Lipunov et al 2006). The first exposure started at 16:49:57 UT
2006-09-26, 76 s after the gamma-ray burst was recorded. We found in the
first and subsequent coadded frames an optical transient with the following
coordinates:

[pic]= 17h35m43s.66
[pic]= 13d02m18s.3
err = +-0s.7''

which agree with the coordinates of the optical transient discovered by
Holland et al. (2006) within the errors of our observations. The results of
the corresponding photometry yielded the first data points on the light
curve.
We found an optical flare event - after a short decrease the brightness
began to rise beginning with the 300th second and reached its maximum near
500--700 s. Synchronous X-ray flux measurements with Swift XRT show a
similar event (see Fig.12). Note that the absorption determined from x-ray
data corresponds to a column density of nH=2.2 1021 cm-2 of which nH = 7
1020 cm-2 is Galactic absorption (Holland et al. 2006). Given the redshift
z=3.208, the total absorption in our band is equal to three magnitudes. We
naturally assume that the dust-to-hydrogen ratio is the same as in our
Galaxy. A comparison of our optical measurements with the x-ray fluxes
measured by Swift XRT (Holland et al. 2006) allowed us to determine the
slope of the spectrum, which we found to be constant within the errors and
equal to ?=1.0+-0.2:

F~E-? [erg/cm2 s eV]

The spectrum obtained agrees with the x-ray spectrum within the errors
(Holland,S et al 2006).

Such a phenomenon was already observed at least in several cases:
GRB060218A z=0.03 (Quimby et al, 2006a, GCN4782) at the 1000th second,
GRB060729 z=0.54 at the 450th second ( Quimby et al., 2006b,c GCN
5366,5377), GRB060526 z=3.21 at the188th second (Dai X. et al 2007), and
also during the bursts GRB990123, GRB041219a, GRB060111b (Wei D.M., 2007) ,
etc.
Note that the gamma-ray burst that we discuss here has a redshift of
3.208 (V.D'Elia et al GCN5637). Figure 12b-e shows the results of optical
and X-ray observations of the flare and of the theoretical computations of
a spinar with parameters a0=7.6 and [pic].
Note also that redshift dilates all time intervals by a factor of
(1+z) and therefore we show all experimental light curves reduced to the
rest-frame. We hence have to explain a flare at the ~100th second, which is
about 50 times weaker than the gamma-ray burst as we illustrate in Fig.12.

11) Discussion

Our proposed non-stationary model developed in terms of the Spinar
Pardigm of magneto-rotational collapse is physically transparent. It takes
into account all the main relativistic effects and allows their impact on
the operation of the central engine and on the accompanying events to be
estimated. It goes without saying that this model cannot replace precise
magnetohydrodynamic computations, but it evidently helps to choose the
inevitable simplifications for such computations.
The central assumption in our model is that dissipation of the angular
momentum of the collapsing core is due to magnetic field. It is clear that
turbulent viscosity and generation of Alfven waves may play important part
in the real situation. However, on the one hand, no simple physical model
has so far been developed for these events and, on the other hand, the
magnetic field that we introduce can be viewed as some effective parameter
describing viscous loss of momentum. We point out that although we use
dipole moment in our set of equations of motion, they actually do not
assume the dipole nature of the magnetic field. This remarkable
circumstance is due to the fact that the spin-down torque [pic] that we use
here coincides with the energy of magnetic field, [pic], for a spinar whose
radius is equal to the corotation radius R = Rc. By the way, this fact
proves that we adopted maximally effective spin-down magnetic moment.
To reduce the number of initial hypotheses, we never allowed for the
possible generation of magnetic field (Kluznuzk & Ruderman, 1998) as a
result of differential rotation of the collapsing core. On the other hand,
generation of magnetic field can be easily incorporated into the
approximation employed. It can be done should theory clearly disagree with
observations.
There are other phenomena capable of complicating the picture described
above. For example, the spinar may at a certain time break into two objects
during the collapse of a rotating core (Berezinski et al.,1988; Imshennik,
1992). We do not yet consider the second possibility, which, in principle,
may result in the appearance of several flares or precursors around the
gamma-ray burst.
The problem of precursors and flares requires a separate explanation.
On the one hand, we stress that close precursors and flares may result from
the presence of a complex system of shocks in the relativistic jet. On the
other hand, the phenomenon of multiple precursors can be easily explained
by the oscillations of the newborn spinar at the centrifugal barrier. We
artificially suppressed these oscillations by introducing a special
dissipative force with the dissipation time scale parameter. As we showed
above, we can obtain up to 10 precursors for a single gamma-ray burst if we
choose the dissipation time scale to be one order of magnitude longer than
the period of spinar rotation (Fig.13).

However, the description of fine effects lies beyond the scope of this
paper.
We assume that interpreting shock events accompanying the gamma-ray
bursts in terms of a simple two-parameter scheme is an important step
toward understanding the operation of the central engines of the gamma-ray
bursts.

We are grateful to the Russian Foundation for Basic Research for
having discontinued the financial support of our experimental studies of
gamma-ray bursts with the first Russian MASTER robotic telescope and
thereby giving us time to write this paper. We are grateful to Pavel
Gritsyk for discussions and assistance in computations.


Appendix 1. Parameters of neutron stars with equation of state (28).

To choose the most appropriate constants in approximate equation of
state (28), we analyze the global properties of neutron stars in accordance
with dynamic equation (24), which in the static case transforms into the
following simple equation:

[pic] (A1)


Figure 14 shows the dependence of the radius of a nonrotating neutron
star on its mass for various values of parameter b, which appears in our
equation of state. First, we see natural decrease of the star's radius with
increasing mass, which is typical of self-gravitating configurations with
equilibrium maintained by the pressure of ideal degenerate gas. However,
this is of minor importance for us compared to the fact that the radii of
neutron stars of reasonable (1.5---3[pic]) masses lie within reasonable
limits: from 20 to 100 km. Figure 15 shows the dependence of the
Oppenheimer-Volkoff limit on our parameter b. The available orthodox model
equations of state for neutron stars predict that the maximum mass of a
neutron star is 1.5---3[pic]. This corresponds to the following interval of
parameter b:

[pic]

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Figures and captions:
[pic]

Figure.1. Schematic view of the collapse of the rapidly rotating magnetized
core of a massive star. Gray and black shaded areas show the envelope and
core of the star, respectively. Before the collapse the size of the star is
on the order of several solar radii and its iron core is one hundred times
smaller (stage A). During the collapse centrifugal forces increase most
rapidly, resulting in the formation of a spinar (stage B). Its formation is
accompanied by anisotropic release of energy. Because of the dissipation of
angular momentum the spinar decelerates and contracts (stage C). Its
luminosity increases and a new jet forms whose energy release reaches its
maximum near the gravitational radius. Depending on the core mass, the
process results in the formation of a neutron star or an extremely rotating
black hole.


[pic]

Fig.2. Qualitative variation of the characteristics of a gamma-ray
burst and the accompanying phenomena shown on the magnetic field - Kerr
parameter diagram.


[pic]


Fig. 3. Computation of the collapse of a 7 solar mass core with Kerr
parameter a0=6 and
magnetic-to-gravitational energy ratio ?m=10-4. From top to down:
energy release as viewed by an infinitely distant observer, radius, Kerr
parameter, and the average magnetic field strength.











Fig 4. Operation of the central engine. Results of the computation of
energy release (luminosity-time in logarithmic coordinates) during the
collapse of 7[pic] stars

[pic]

Fig. 5. Computation of the collapse of a 2.2 [pic] star. The initial
Kerr parameter and initial magnetic-to-gravitational energy ratio are equal
to [pic] and [pic], respectively.


[pic]

Fig. 6. Computation of the collapse of a 1.5 [pic] star. The initial
Kerr parameter and initial magnetic-to-gravitational energy ratio are equal
to [pic] and [pic], respectively.


[pic]

Fig. 7. Operation of the central engine. Results of the computation of
energy release (luminosity-time in logarithmic coordinates) in the process
of the collapse of a core into a 1.5 [pic] neutron star with different
values of the Kerr parameter (a) and initial magnetic-to-gravitational
energy ratio ([pic]). We set the initial core radius equal to 1000Rg. The
first and second flares correspond to the formation of the spinar and
neutron star, respectively. At the end of the process, energy release
always begins to obey the magnetodipole law corresponding to the spin-down
of the neutron star, i.e., the pulsar. Computation of the collapse of a 1.5
[pic] star. The initial Kerr parameter and initial magnetic-to-
gravitational energy ratio are equal to [pic] and [pic], respectively.




[pic]

Fig.8 The gamma-ray burst time multiplied by the gamma-ray-burst to
precursor energy ratio as a function of precursor time. The filled circles
show BATSE data (Lazzati, 2002) and the data for two outsending bursts: a
short (GRB041116) and a long (GRB 060124) one. We use the fluence data and
assume that the opening angles of the precursors are equal to those of the
corresponding gamma-ray bursts. The crosses show the simulated gamma-ray
bursts with precursors computed for a 7 [pic] core. The effective Kerr
parameter varied from 7 to 20 , and magnetic field, from10-2 to 10-6.


[pic]

Fig. 9. The observed GRB fluence-to-peak-luminosity ratio as a function
of flare time (9a) based on the data of Lazzati (2005) supplemented with
two interesting bursts GRB 060124 (Romano, P et al 2006) and GRB 041116
(Golenetskii, et al GCN2835). Theoretical ratio for simulated gamma-ray
bursts (9b) . In our computations we assume that the core mass is equal to
7 [pic] the Kerr parameter varies from 2 to 7, and magnetic energy lies
between 0.01 to 10-7.
The solid line is based on bursts, it corresponds rather accurately to
equation [pic].

[pic]

Fig. 10 Computed energy release during the process of collapse with low
angular momentum and weak magnetic field.

[pic]

Fig .11. Systematic light curve of the long curve on the x-ray plateau
for the model of cores with parameters: ???


Я бы заменил на это
[pic].

[pic]


Fig 12. Light curve and theoretically computed luminosity, Kerr
parameter radius, and magnetic field for GRB060926. The initial parameter
for theoretical computation is a0=7.6 and [pic].


[pic]

Fig. 13. Computed collapse of a 7 [pic] core. Qualitative illustration
of the fine structure of the temporal behavior of the gamma-ray burst or
multiple precursors.

[pic]

Figure 14 shows how the radius of a nonrotating neutron star depends on
its mass for various values of parameter b that appears in our equation of
state.


[pic]

Fig. 15. The Oppenheimer-Volkoff limit as a function of parameter b
that appears in our equation of state NN for a nonrotating neutron star.


[pic]

Fig. 16. The Oppenheimer-Volkoff limit as a function of the velocity of
rotation of the neutron star (in the units of the Kerr parameter).

-----------------------
NS

BH

D

C

B

A








e

d

b

a

c