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The Astrophysical Journal, 665:L97-L100, 2007 August 20
2007. The American Astronomical Society. All rights reser ved. Printed in U.S.A.

AN EXTRA LONG X-RAY PLATEAU IN A GAMMA-RAY BURST AND THE SPINAR PARADIGM V. Lipunov1,
2, 3

and E. Gorbovskoy

1, 2, 3

Received 2007 May 3; accepted 2007 June 25; published 2007 August 6

ABSTRACT The recently discovered gamma-ray burst GRB 070110 displayed an extraordinar y X-ray afterglow with Xray radiation--i.e., an X-ray plateau--obser ved for 20,000 s. We show that the obser ved properties of the plateau can be naturally interpreted in terms of the model with a spinar--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. If this model is true, then for 1 hr, the Swift X-ray telescopes recorded radiation from an object with a size smaller than the Schwarzschild radius! Subject headings: black hole physics -- gamma rays: bursts -- stars: magnetic fields -- stars: rotation
1. INTRODUCTION

After 3 years of operation of the Swift space obser vator y (Gehrels et al. 2004), it has become evident that the temporal behavior of many gamma-ray bursts (GRBs) exhibits such features as delayed flares (Chincarini et al. 2007) and early precursors (Lazzati 2005), which can in no way be reconciled with the instantaneous point explosion model and which are indicative of a long (compared to the duration of the gamma-ray burst) time of operation of the central engine (Gehrels et al. 2006; Wang & Meszaros 2007). ДД The recently discovered gamma-ray burst GRB 070110 displayed an extraordinar y X-ray afterglow with X-ray radiation-- i.e., an X-ray plateau--obser ved for 20,000 s (Troja et al. 2007). The gamma-ray burst GRB 050904 exhibits a similar behavior (Cusumano et al. 2006). Such a behavior demonstrates the central engine's long activity and gives an insight into the mechanisms of its operation. We show that the obser ved properties of the plateau can be naturally interpreted in terms of a spinar paradigm. The spinar paradigm has a transparent physical meaning, which opens up a way toward successfully understanding GRBs and their accompanying events and allows for their variety to be reduced to two physical parameters: initial angular momentum and initial magnetic field.
2. SPINAR PARADIGM

the late stages of the evolution of binar y systems in which a binar y helium star forms with an orbital period of less than 1 day (Tutukov & Cherepashchuk 2003). The rate of such events is about 10 4 yr 1 per 1011 M, (Bogomazov et al. 2007). This rate agrees well with the obser ved gamma-ray burst rate if we take into account the narrow-beamed nature of the radiation of a gamma-ray burst. Population synthesis of a merging of neutron stars yields a similar rate of 10 4 yr 1 per 1011 M, for these events (Lipunov et al. 1987). Consider now the magnetorotational collapse of a stellar core of rest mass Mcore, radius RA, and effective Kerr parameter (Thorne et al. 1986) a0 { Iq 0 c 1 1, 2 GMcore (1)

The prolonged activity of the central engine of gamma-ray bursts was earlier predicted (Lipunova 1997; Lipunova & Lipunov 1998; Vietri & Stella 1998) to be a result of the spinar formation. The lifetime of the spinar is determined by the rate of dissipation of its angular momentum as a result of the interaction of the magnetic field of the spinar with the ambient plasma. This is the essence of the spinar paradigm. The origin of the spinar paradigm dates back to the 1960s when the importance of allowing for magnetorotational effects in the process of collapse was first clearly understood. Spinars were first invoked to analyze the energy release by an evolution of quasars (Hoyle & Fowler 1963; Ozernoy 1966; Ozernoy & Usov 1973) and by an ejection of supernovae shells (LeBlanc & Wilson 1970; Bisnovatyi-Kogan 1971). A spinar may form in two ways: via a collision of two neutron stars (Lipunova & Lipunov 1998) or via the core collapse of a massive star. Finally, a situation may arise during
1 2 3

Moscow State University, Vorobjovy Gor y, 119899 Moscow, Russia. Sternberg Astronomical Institute, Moscow, Russia. Moscow Union "Optic," Moscow, Russia.

2 where I p kMcore RA is the moment of inertia of the core; c and G are the speed of light and gravitational constant, respectively; and k is a dimensionless constant, which we set equal to unity for the sake of simplicity. Parameter a remains constant if the angular momentum of the core is conser ved (the condition that is evidently violated in our scenario). However, in any case, direct collapse is impossible in such a situation, because the Kerr parameter of a black hole cannot exceed unity. Let a m be the ratio of the magnetic energy Um of the core to the gravi2 tational energy of the core: a m { Um /(GMcore /R). The total magnetic energy can be written in terms of the average magnetic field B that penetrates the spinar, Um p 1 B 2R 3. In the 6 approximation of the conser vation of magnetic flux (BR 2 p const), the ratio of the magnetic energy to gravitational energy remains constant throughout the collapse: a m p const, Um R 1. The collapse process breaks down into several important stages (see Fig. 1). After the loss of stability, virtually free-fall 3 contraction begins with a timescale of tA p ( RA/GMcore )1/ 2 11 100 s, where RA 10 cm is the initial radius of the stellar core (Wang & Meszaros 2007, eq. [15]). During the collapse, ДД the gravitational energy is hardly radiated but is transformed into the kinetic, rotational, and magnetic energies of the core. The rotational energy can be easily seen to grow faster than the gravitational energy, Uspin Iq 2/2 R 2, and the collapse stops (Fig. 1b) near the radius determined by the balance of 2 the centrifugal and gravitational forces: q 2RB p GMcore /RB . It follows from this that the initial radius of the spinar is approximately equal to RB p a 2 Rg /2 (here Rg p 2GMcore /c 2 is 0 the gravitational radius of the core). In this case, half of the gravitational energy is released: EB GM 2/2 RB p (1/ 2a 2 )Mcore c 2. 0

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The luminosity remains virtually constant and equal to L plateau p (a m /a 5 )(c 5/G ), while t K tc! Thus, even in the New0 tonian approximation, the spinar model predicts a plateau whose parameters can be estimated from the latter two formulae. If the magnetic field is sufficiently strong and if tC is small, the spinar produces an X-ray flare. General relativity effects begin to show up as the spinar radius approaches the gravitational radius. In particular, the magnetic field of the collapsar begins to vanish, in full agreement with the black-hole-no-hair theorem (Thorne et al. 1986). The general relativity evolution for the magnetic field of the collapsar has been computed repeatedly by several researchers (Ginzburg & Ozernoy 1964; Kramer 1984; Manko & Sibgatullin 1992). The results of the Ginzburg & Ozernoy (1964) computations can be approximately modified. The following formula correctly describes the behavior of magnetic energy at large distances from the gravitational radius and yields zero magnetic field at the event horizon: Um p U0 x 0 y( x 0 ) , x y( x ) (4)

Fig. 1.--Schematic view of the collapse of the rapidly rotating magnetized core of a massive star. The gray and black shaded areas show the envelope and core of the star, respectively.

where y( x ) p 1/x (1/ 2 x 2 ) ln (1 1/x ); here x p R/2 Rg. The second important group of effects consists of the reference-frame drag in the metric of the rotating body and the relativistic effects due to the close location of the event horizon. We use the post-Newtonian approximation for the centrifugal force in the Kerr metric to allow for the latter two effects (Mukhopadhyay 2002): gp GM( x 2 2ax 1/ x 3[x 1/ 2 ( x 2)
2

Because of the axial symmetr y, the burst must be collimated along the axis of rotation and must have an opening angle of QB. If a 2 100, the energy of the first explosion exceeds sig0 nificantly the bounding energy of the stellar shell, and a relativistic jet easily comes out. The duration of this stage is determined by the time it takes the jet to reach the surface and by the nature of the cooling, which in turn is determined by the structure of the initial jet and the shell, and ranges from several seconds to several hundred seconds. The nature of the spectrum is determined by the Lorentz factor of the jet (Wang & Meszaros 2007). The spinar that forms the core interiors ДД begins to lose its angular momentum due to magnetic viscosity and starts to radiate its rotational energy. The angular velocity of the spinar increases like that of a satellite whose velocity increases as it decelerates in the upper layers of the atmosphere. The spinar contracts as its angular momentum is carried away under the influence of the maximum possible magnetic torque (Lipunov 1992): dIq/dt
R

a2 )2 , a 2 ]2

x p R/2 Rg .

(5)

The cur ve of energy release acquires the features of a burst, which can be approximately described by the following set of elementar y equations: q 2R p g, dIq/dt p L p a 2L C p a 2Um q, Um , (6)

dt p dt/a,

Here L and t are the luminosity and time in the reference frame of an infinitely distant obser ver, and a is the duration function--the ratio of the rate of the clock of reference obser vers to the universal time rate at the equator of the Kerr metric (Thorne et al. 1986): x
2

ap B r 2prdr 4p
2

a x
2

2

a

2

2x ,

(7)

Um .

(2)

The timescale of the dissipation of angular momentum is tC Iq/Um p GMcore /c 3a m. During this process, the velocity of rotation of the spinar increases, and the spinar luminosity not only does not decrease but actually increases (L p qdIq/dt p Um q R 5/ 2). If computed without the allowance for relativistic effects, the spinar light curve has the form Lp am c5 (1 a5 G 0 t/tC )
3/ 5

.

(3)

which vanishes (a r 0) at the horizon of the extremely rotating black hole, R r Rg /2. As the luminosity increases, the condition of shell penetration becomes satisfied at a certain instant of time. A second jet appears whose intensity reaches its maximum near the gravitational radius. In this case, the effective Kerr parameter tends to its limiting value for a Kerr black hole a r 1. The subsequent fate of the star depends on its mass. If the mass exceeds the Oppenheimer-Volkoff limit, the star collapses into a black hole (BH), other wise a neutron star (NS) forms and continues to radiate in accordance with the magnetodipole formula L t 2 (see, for example, Lipunov 1992).


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Fig. 2.--Classification of GRBs and surrounding events in terms of the spinar paradigm. In the case of a weak magnetic field and a large angular momentum (the bottom right curve), the first burst is weak (because of the high centrifugal barrier) and soft. It is followed by a slow collapse (the magnetic field is weak), which results in a weak X-ray burst. In the case of a small angular momentum (the bottom left curve), the energy released at the centrifugal barrier is large, and the burst appears as a gamma-ray burst, whereas the second burst, which corresponds to the collapse of the spinar, is again weak and soft and shows up as a distant X-ray burst. In the case of an even weaker magnetic field, the second flare behaves as an extra long X-ray plateau. In the case of a stronger magnetic field, the flare becomes more like a gammaray burst; its energy increases, and the flare itself becomes part of a gammaray burst (the top left curve). As we move rightward, the angular momentum increases, and the first flare loses its energy to become a close precursor of the second flare, which in the case of a strong magnetic field becomes a powerful gamma-ray burst. The collapse of a core with an effective Kerr radius smaller than unity results in a supernova explosion.

The spinar paradigm allows for the obser ved variety of gamma-ray bursts, precursors, and flares to be reduced to just two parameters: the magnetic field and the initial angular momentum (Fig. 2).
3. X-RAY PLATEAU EXPLANATION

What is a plateau in the spinar paradigm? The simple answer is that an extra long X-ray plateau is an X-ray flare protracted for several thousand seconds because of the weak magnetic field of the spinar. Figure 3a shows the rest-frame light cur ve of the gamma-ray bursts GRB 070110 and GRB 050904 adopted from Troja et al. (2007). Both bursts had large redshifts (z p 2.5 and z p 6.6, respectively), and therefore the obser ver sees the duration of the corresponding plateaux to be 3-7 times longer than their rest-frame durations. Both bursts exhibited extra long plateaux that ended abruptly at 8000 s. Note that the energy was computed in isotropic approximation. We therefore did not strive to achieve the exact coincidence of luminosities, especially since we do not know the factor of conversion of the energy released into the X-ray flux. Moreover, we have no information about the detailed structure of the beam pattern, and therefore both bursts can be explained in terms of the same model provided that we see them at different angles. However, the most important factor is the duration of the plateau. Figures 3b-3e show the exact solution for the set of burst equations (eq. [6]); this solution agrees well with the obser ved plateau events for GRB 070110 and GRB 050904. Both pla-

Fig. 3.--(a) Combined Swift (BAT and XRT) light cur ves of GRB 050904 (z p 6.29) and GRB 070110 (z p 2.35) in the source rest frame (Troja et al. 2007). (b-d) Results of the computation of the spinar luminosity, radius, and effective Kerr parameter, respectively. (e) Mean magnetic field evolution calculated under the approximated general relativistic model (see eq. [4])


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teaux can best be described in terms of the model of the collapse of a 7 M, star with an initial rotation energy fraction of a m p 1.0 # 10 7 and an initial Kerr parameter a p 2.0. With these parameter values, the above scenario should appear as follows. The loss of stability of the rapidly rotating star results in the formation of a spinar and the release of energy EB (1/ 2a 2 ) Mcore c 2 4.5 # 10 53 ergs. The complex process of the emergence of a high Lorentz factor relativistic jet onto the surface produces an approximately 100 s-long gamma-ray burst. After the jet comes out to the surface, an afterglow (t 2) appears that is due to the cur vature effect (Kumar & Panaitescu 2000). Then, after 300 s, most of the energy is radiated by the bow shock, which is decelerated in the stellar wind of the progenitor star (t 1; Troja et al. 2007). During this process, the spinar continues to radiate at a virtually constant luminosity, which shows up at 1000 s, when the afterglow has faded significantly. The luminosity remains virtually constant after ward. After a small increase in luminosity due to the compression of the spinar, the plateau terminates, with the luminosity dropping abruptly, the spinar radius becomes smaller than the gravitational radius, and the spinar is now located inside the ergosphere of the future black hole! The abrupt increase in the gravitational redshift and the effects of the disappearance of magnetic field result in the abrupt decrease in the spinar luminosity, which continues to fall for about 900 s until the intensity becomes lower than the luminosity of the bow shock.

During all this time, the spinar is inside the ergosphere, Rg /2 ! R ! Rg. It seems that mankind has never before come so close to the event horizon! Since real gamma-ray bursts are located at large redshifts, the time inter vals measured with our Earth clocks are a factor of (1 z) longer than the corresponding rest-frame time inter vals, providing us with an opportunity to study the collapse inside the ergosphere for up to 2500- 9000 s, i.e., virtually during an entire hour! Figure 3b shows the synthetic light cur ve computed with the allowance for the afterglow, L p C1 t 2 C 2 t 1 L , which agrees well with the obser ved light cur ve. The small flare at 5000-10,000 s after the period of steep decay in GRB 070110 is the result of the energy input from spinar to bow shock. After end of the plateau, the level of the afterglow emission must change, but with some delay. The model of a magnetized spinar demonstrates how the parameters of the obser ved plateau depend on the physical parameters of the progenitor. Our model naturally explains other, fainter events in all types of gamma-ray bursts, which we will discuss in a separate paper. We are grateful to the MASTER (Lipunov et al. 2006) team for discussions and assistance in computations. We would like to thank B. Zhang and an anonymous referee for useful comments.

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