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Spinar Paradigm and Gamma Ray Bursts Central Engine
V.M. Lipunov 1
2 3

1,2,3

and E.S. Gorbovskoy

1,2,3

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

Accepted 5 Septemb er 15. Received 2007 August 31; in original form 2007 June 21

ABSTRACT

A spinar is a quasi-equilibrium collapsing ob ject 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 was recently discussed in the context of 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 flares, and the appearance of a plateau on time scales of several thousand seconds to be unified. Key words: black hole physics -- Physical Data and Processes, gravitation -- Physical Data and Processes, magnetic fields -- Physical Data and Processes, relativity -- Physical Data and Processes, gamma-rays: bursts -- Sources as a function of wavelength, gammarays: theory -- Sources as a function of wavelength

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 ob ject. 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 flares, but even extraordinarily long x-ray plateaux.


2

V.Lipunov and E.Gorbovskoy
star having substantial angular momentum may be accompanied by the formation of a quasi-static ob ject - 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 ob ject 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 spin-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 . Recently, 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.

[ht] Figure 1. Schematic view of the collapse of the rapidly rotating magnetized core of a massive star. Gray and black shaded areas show the envelop e and core of the star, resp ectively. 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. Dep ending on the core mass, the pro cess results in the formation of a neutron star or an extremely rotating black hole.

3

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.

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) a0 I 0 c 2 GMcor (1)
e

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

2 (here I = kMcore R0 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 m be the ratio of the magnetic energy of the core to its gravitational energy:


Spinar Paradigm and Gamma Ray Bursts Central Engine

3

exceeds the momentum corresponding to escape velocity. Let a part of the energy be converted into the energy of the jet ( Ej = EB ) EB 2 > B M vj
shell

2GM Rshell

(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: 1 < a0 <
[ht] Figure 2. Qualitative variation of the characteristics of a gammaray burst and the accompanying phenomena shown on the magnetic field -- effective Kerr parameter diagram.

1 Mcore C 2 B MS hell Vp

(9)



m



GM

Um 2 cor e

/R

(2)
A

The total magnetic energy can be written in terms of the average magnetic field B penetrating the spinar: U
m

=

B2 4 R3 = 8 3

1 B2R 6

3

(3)

Note that in the approximation of magnetic flux conservation (R2 = const), the magnetic-to-gravitational energy ratio remains constant during the collapse: m = const, Um R-1 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 TA =
3 RA 100s GMcore

where Vp is the escape velocity at the surface of the stellar envelope. In real situations M 1 Vp = 2000 - 3000 km/s, M core 10 - 1 , and 3 S hell 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 2 B . The duration of this stage is determined by the time it takes the jet to emerge onto the surface (Rshell 10 - 30s) and the character of cooling governed by the structure of the primary jet and envelope. The gamma factor of a jet emerging at the surface of the star can be approximately estimated using the energy conservation law (see a review by Granot, 2007): E
j et 2 2 B M shell

c2 and E

1/2 -1 50 B

Mshell M

-1/2

(10)

(4)

Here E50 = EB /1050 erg/s - jet energy. The character of the spectrum is determined by the gamma factor of the jet. If the initial Kerr parameter is large (a 1) then energy EB Mcore c2 and the emerging jet is nonrelativistic allowing the event in question to be viewed as a precursor like it was done by Ramirez-Ruiz et al. (2002) and Wang & Meszaros (2007). Its spectrum can be estimated by the blackbody formula (eq. 16 in Wang & Meszaros, 2007): T 15L
1/8 50

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 2 RB = GMcor 2 RB
e

R

-1/4 11

K ev

(5)

It follows from this that the initial spinar radius is approximately equal to: RB = a2 GM
cor e

/c2 = a2 Rg /2

(6)

In this process, half of the gravitational energy is released: EB = GM 2 GM 2 1 GM 2 - = 2M 2RB RB 2RB 2a
cor e

c

2

(7)

if the energy is sufficient to "penetrate" the stellar envelope, i.e., if the momentum imparted to a part of the shell

If the initial Kerr parameter is close to unity then the energy of the burst is high and the jet acquires a high gamma factor after penetration so that the flare should be interpreted as a gamma-ray burst. Although the jet that penetrates the star may be subrelativistic, however, a higher gamma-ray factor jet is to flood the already formed channel (the central engine continues to operate!). It is this evolved jet that should produce the gamma-ray burst provided that the spinar size is close to the gravitational radius. We do not discuss the parameters of the jet, because this issue been addressed repeatedly by different authors ( see reviews by Granot (2007) and Piran (2005)). Only future numerical computations will make it possible to accurately determine the degree of anisotropy, i.e., 2 e.g., the jet B . However, here we try to estimate the degree of anisotropy by determining the fraction of the spinar surface occupied by open field lines. Let us assume for a moment


4

V.Lipunov and E.Gorbovskoy
As the luminosity increases, at a certain time instant the conditions of shell penetration (similar to condition (8)) become satisfied: ED 2 > D M c
shell

that the spinar has a dipole moment equal to Е. Let us determine the Alfven radius RAlf ven of the jet from the condition of the balance of the jet ram pressure and magnetic-field pressure: LB Е2 2 8 R (B R2 c)
6

2

(11)

GM Rshell

(15)

We use this formula to derive the Alfven radius: R
Alf v en



1/2 B

(cЕ2 /2Lb )

1/4

We further assume that Е = B R3 /2, where B is the intensity of magnetic field at the pole of the spinar, to obtain R
Alf v en

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: L
D

3 з 108 cm(B /0.01)
15

1/2

B

1/2 15

R

1/2 7

R7 L

-1/4 50

R

7

=

Here B15 = B /10 Gs, R7 = RB /10 cm. It is evident that all field lines passing inside this radius are closed. We use the approximation of the dipole field line equation to determine the size of polar regions enclosing open field lines:
polar

7

m c M Rg 2 M c2 G

5

(16)

It is better to write the condition of the penetration for the second jet in terms of pressure inequality: GM LD > cR2 R4
2 D 2

(17)

(RB /R

Alf v en

)

1/2

0.03(B /0.01)

1/4

B

1/4 15

R

1/4 7

L

-1/8 50

Thus only 0.1% of the spinar surface participate in the formation of the jet, implying a very high degree of anisotropy of the process considered. 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 curve 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: tC = IB /K
sd

1 Note that c5 = 1059 erg /s is the so-called natural luminosity. G Of course, formula (15) does not include gravitational redshift, decay of magnetic field, etc. The time scale near the maximum is: T
D



M

cor e

Rg 2
m

U

=

GM a c3 m

3

(18)

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 K = Е2 /R
3 l

(19)

where Е is the magnetic dipole moment and Rl = c/ 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: L= Е2 t 3 Rl
-2

(12)

(20)

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: K
sd

= t U

m

(13)

where t 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: tC I GMcore a 3 Um c m t
3 0

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: E =
2 2 G (M1 + M2 )2 GM1 GM2 - - 0.1(M1 +M2 )c2 (21) RB R1 R2

(14)

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.

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 flares. In the case of large angular momentum (a 1) the initial radius is large and, correspondingly, the


Spinar Paradigm and Gamma Ray Bursts Central Engine
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 (14), and, consequently, a rest-time measurement immediately yields a relation between the Kerr parameter and the fraction of magnetic energy: tc5 m 10 = = a5 GMcore t 0
-6

5

M t 10M

-1 2 t

(22)

where t2 = t/100s. Correspondingly, the characteristic magnetic field at the collapse time (near Rg ) is equal to: B= Rg Rcor
-2 e

m GM 2 2З1015 GsЗ 4 6Rcore

1/2 -6

Mcor M

e

-3/2

(23)

where -6 = m /10-6 . 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. Let us considered firstly two upper line of the diagram (Fig.2). In the case of weak magnetic field and large angular momentum (the right side of the middle line) 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 xray rich burst. As the initial angular momentum decreases (we move leftward in the diagram along middle line) 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 middle line) the precursor energy increases and at a > 1 the precursor energy exceeds 1051-52 erg and it shows up as a gammaray burst with the subsequent collapse of the spinar leading to X-Ray flare or an X-ray plateau event (the bottom-line Lipunov & Gorbovskoy, 2007) with more weak field. 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 topright 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 come to the bottom-right corner) and turn into isolated long X-ray plateau.

[ht] Figure 3. Computation of the collapse of a 7 solar mass core with effective 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, effective Kerr parameter, and the average magnetic field strength.

4

ONE POINT PSEUDO-NEWTONIAN NONSTATIONARY SPINAR MODEL OF THE 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 ob ject 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.


6

V.Lipunov and E.Gorbovskoy
F
diss

Let us assume that at the initial time instant we have a rotating ob ject (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 ob ject has the mass of , radius Rcore , angular momentum I , dipole momentum Е0 , and Kerr parameter a0 . a). Dynamic Equation We write the equation of motion in the post-Newtonian approximation: d2 R =F dt2
gr

=-

1

dR dt

(30)

+ Fc + F

nuclear

+F

diss

(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 pseudoNewtonian 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: F
gr

It is clear from physical viewpoint that after reaching the centrifugal barrier the core undergoes extremely strong oscillations with a time scale of 1/ . 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 ob ject -- 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: = 2 / (31)

Throughout this paper, = 0.04 unless otherwise indicated.

=-

GM (x2 - 2ax + a2 )2 x3 ( x(x - 2) + a)2

(25)

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 ob ject is equal to (see Lipunov, 1992)


where x = 2R/Rg . This formula corresponds to the potential of Paczynski & Wiita (1980) for a nonrotating black hole. Next terms: Fc = 2 R F
nuclear

(26) P 1 dP dr R (27)

=

K=
Rmin

Pressure of gas, which includes thermal pressure, can be written as kinetic energy of particles computed using relativistic invariant (Zel'dovich, Blinnikov, Shakura 1980): P ( c4 + b
2/3

Bz B dS 1 = 4 2



Bz B R dR,
Rmin

(32)

+ (Q/M )2 - c2 )

(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: b= 4 3
2/3

G2 M

4/3 C lass

(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 = 2M 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 :

where Bz and B -- z and are the components of magnetic field. We now introduce the magnetic moment Е 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. Е Let Bz B = t Bd , where Bd = R3 is dipolar strength of the magnetic fields. The breaking torque is then equal to (see Lipunov, 1987, 1992 see below) K=
t

Е2 , 3 Rt

(33)

where t 1 and Rt is the characteristic radius of interaction between the magnetic field and ambient plasma: Rt = R Rt = R Rt = R
Alf v en is the Alfven radius (Prop eller) 1/3 c = GM is the corotation radius (Accretor) 2 c l = is the radius of the light cylinder (Ejector)

(34)

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


Spinar Paradigm and Gamma Ray Bursts Central Engine
equilibrium condition. Therefore the retarding torque can be written as: K= t Е2 2 t Е2 =3 GM RB (35)

7

We can use the following simple formulas as the first approximation: Е=Е
0

And the corresponding dissipation time scale is: IB 2 TC = 2 3 Е /RB (36)

R - Rmin /2 R0 - Rmin /2

(40)

Hence the equation of variation of the spinar angular momentum becomes (Lipunov, 1987): Е2 t Е2 dI =- 3 =- dt Rc GM
2

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): ЕЕ
0

(37)

Some authors (Woosly, 1993; Narayan et al., 2001) consider a situation where accretion continues onto the newborn black hole at a rate of up to 10-1 M/yr. In just the same way accretion may continue onto the spinar. The equation of the variation of the angular momentum of the an accreting spinar was first derived by (Lipunov, 1987 equation 123):
З dI t Е2 2 Е2 =- 3 =- + M GM R dt GM Rc

R0 R

2

(x0 ) (x)
x2

(41)

where (x) = xmin + 2min + ln 1 - xmin and xmin is x x x2 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: L0 = L0 = 1 M dR dt
2

where M is the disk-accretion rate. It was shown in the same paper that accretion dose not change dramatically spinar evolution and hereafter we neglect accretion. The effect of accretion should always be important if the accretion time is much shorter than the time scale of dissipation of angular momentum, tacctretion Mcore < TC However, in this case З
M

З

the very process of accretion is the process of the formation of the spinar. Note that our scenario differs substantially from that of Woosley (1993), who considers accretion to be of importance, because it is the process that determines the energetics of the gamma-ray burst. A spinar is a collapsing (but not a collapsed!) stellar core. In other words, a spinar is by itself an "accretion disk". Of course we may complicate the model in the future, but we prefer to stop our coarse (but physically transparent) approximation here and ignore accretion. 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: dI t Е =- 3 dt Rl
2

before the formation of the spinar (42) (43)

Е2 after the formation of the spinar 3 Rmin

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: L


= 2 L

0

(44)

where 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): = x2 + a2 -