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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 thu