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A Model for Non High Energy Gamma Ray Bursts and Sources of Ultra High Energy Cosmic Rays

As a progenitor of NHE (Non High Energy) GRBs, we propose a (C+O) star and a neutron star (black hole) close binary system. Since the (C+O) star is tidally locked, a new born pulsar should be a milli second pulsar (MSP). Dynamo will generate a superstrong magnetic field(SSM). The beam of the energy from the super strongly magnetized milli second pulsar can punch a hole in the supernova ejecta . Then the beam of gamma rays flows out of the ejecta. If we observe this beam from the lateral direction, it will be NHE-GRB. In this model the pulsar with the spin period about 1sec should exist in SN1998bw so that the searches for this pulsar in all the wave length are urgent. SSM-MSPs can accelerate protons up to Ultra High Energy. If a few percent of the beam energy of SSM-MSP in NHE-GRB event is in the form of UHECRs, the flux of the observed UHECRs can be explained. Then along the direction of each UHECR, a supernova remnant should be found. It is suggested that NHE-GRB is the progenitor of the soft gamma ray repeater which has also the superstrong magnetic field and is in the supernova remnant.


Introduction
Galama et al. [1] reported the discovery of very luminous Type Ic supernova SN1998bw in the BeppoSAX Wide Field Camera error box of GRB 980425 [2], which occurred within about a day of the gamma ray burst. The duration of GRB 980425 is ∼30sec and the burst fluence is (4.4 ± 0.4) × 10 −6 erg/cm 2 . Since no burst of emission is detected above 300keV, GRB 980425 belongs to a so called NHE(Non High Energy) burst [3] which is characterized by the lack of fluence above 300keV, lower peak flux and the apparent isotropic and homogeneous distribution. A quarter ( ∼250 events/y) of GRBs is NHE bursts. [3] The host galaxy of SN1998bw is a spiral galaxy at a distance of ∼ 40Mpc. The chance probability of the coincidence is ∼ 10 −4 . [1] Motivated by the coincidence of Type Ic SN with GRB, Wang and Wheeler [4] examined six recent SN Ib/c for which an outburst epoch can be estimated with some reliability and found that all are correlated in time and space with BATSE gamma ray bursts. Since the joint probability of all six correlations is 1.5 × 10 −5 and no such correlation exists for SN Ia and SN II, Wang and Wheeler [4] proposed that all gamma ray bursts are associated with Type Ib/c supernovae. In this paper I follow the proposal by Wang and Wheeler. I assume that all NHE-GRBs are associated with Type Ib/c supernovae, which should be confirmed in future observationally. I propose a model that a (C+O) star and a neutron star(black hole) close binary is a progenitor of NHE gamma ray burst and argue the details of the model.
In §2 a formation scenario of a (C+O) star and a neutron star (black hole) close binary system is discussed. In §3 I discuss a supernova explosion of the (C+O) star which yields a super strongly magnetized milli-second pulsar(SSM-MSP) . In §4 I explain how SSM-MSP emits NHE-GRB. §5 will be devoted to discussions and astrophysical implications of our SSM-MSP GRB model.

Formation of a (C+O) star and a neutron star(black hole) close binary system
Similar to the scenario of formation of double neutron star systems [5,6] we assume that our (C+O) star-neutron star(black hole) binary is formed as follows [7] : 1) Consider a binary system composed of two main-sequence stars 1 and 2 of mass ∼ 25 M ⊙ which has a helium core mass ∼ 8M ⊙ , 2) Roche lobe overflow of star 1, which becomes a helium star 1, 3) the first supernova explosion of helium star 1 to form a neutron star 1, 4) Roche lobe overflow of star 2 which leads to a spiral-in of neutron star 1 into star 2 . The system shrinks and finally it consists of a neutron star 1 and a helium star 2. Here we note that if the super Eddington accretion is possible [8] a neutron star 1 becomes a black hole of mass ∼ 2.4M ⊙ in this spiral-in phase [9]. 5) During the presupernova evolution of helium star 2, the helium envelope expands as its core contracts. The radii of star 2 with mass 3 ∼ 8M ⊙ are 4 ∼ 1R ⊙ [10,6]. So if the Roche radius of star 2 is smaller than 4 ∼ 1R ⊙ , a spiral-in of neutron star(black hole) 1 occurs again. Now the system consists of (C+O) star 2 and neutron star (black hole) 1. The mass of the (C+O) cores are 6.0 and 3.8 M ⊙ for mass of helium stars 8 and 6M ⊙ , respectively [10], while the radius of the (C+O) star is ∼ 0.1R ⊙ . Since the radii of helium stars of mass 8 and 6M ⊙ are 1.3 and 1.9 R ⊙ , respectively, at the onset of Roche lobe overflow, the separation after the spiral-in of neutron star(black hole) 1 will be significantly smaller than 1R ⊙ , say, 0.2R ⊙ . Such a (C+O) star in a close binary may be a progenitor of Type Ic supernova such as SN1994I [11].

Super Strongly Magnetized Milli-Second Pulsar
Now I discuss an evolution of our close binary system with separation a ∼ 0.2 R ⊙ which consists of a (C+O) star of mass M 2 ∼ 6 M ⊙ and a neutron star(black hole) of mass M 1 = 1.4(2.4)M ⊙ [7]. There are three important time scales in this system: 1) t ev : the evolution time of the (C+O) star. They are 100 yr and 450yr for M 2 = 6 and 3.8 M ⊙ , respectively [10].
2) t gw :coalescing time of the binary due to the emission of gravitational waves which is given by 3) t synch :the tidal synchronization time which is given by [12] t synch ∼ IΩ orb N , where I, Ω orb and N are the moment of the inertia of the (C+O) star 2, the orbital angular velocity and the tidal torque, respectively. N is estimated as where R and α are the radius of star 2 and the angle between the orbital separation vector and the tidal bulge, respectively. α is expressed by the viscous damping time t vis as Therefore t synch becomes Since the convective core exists in the (C+O) star [10], t vis is estimated as where v conv and l m are velocity of convection and the mixing length, respectively. Putting v conv ∼ 10 5 cm s −1 and l m ∼ 0.2R , we have Since we have the inequality as the (C+O) star 2 is tidally locked in the carbon burning phase and the supernova explosion of star 2 occurs with essentially the same separation a. In tidally locked binary the spin angular velocity is the same as the orbital one so that the collapsing central 1.4M ⊙ core of star 2 will have the spin angular momentum J estimated as where I core is the moment of inertia of the central 1.4M ⊙ core. Inserting I core of 10 50 ∼ 10 51 gcm 2 in the carbon burning phase [10], we have This angular momentum is comparable to that of 1 ms pulsar (∼ 6 ×10 48 g cm 2 s −1 ), which means that the new born pulsar after the supernova explosion of star 2 is necessarily a milli-second pulsar. Duncan and Thompson [13,14] argued α−Ω dynamo theory for such a rapidly rotating new born pulsar and concluded that such a milli second pulsar may have a superstrong magnetic field of ∼ 10 16 G within 10s after the formation. Although Duncan and Thompson considered the accretion induced collapse, the same argument can apply to our (C+O) star. Recently such a superstrong magnetic field (∼ 10 15 G) has been identified in a soft gamma ray repeater SGR1806-20 [15] with the age ∼ 10 4 y. This observation supports a model of soft gamma ray repeaters based on super strongly magnetized neutron stars [16,17]. Therefore even relying only on the observational results, it may be possible to consider a superstrong magnetic field as strong as ∼ 10 16 G in a new born milli-second pulsar. Considering both theoretical suggestions and observations of such a superstrong magnetic field, we assume here that after the explosion of our (C+O) star a milli-second pulsar with the superstrong magnetic field (B ∼ 10 16 G) is formed in ∼ 10 s. Here note that in vacuum there exists the maximum electric field (E c = 2πm 2 e c 3 /he = 4.4×10 13 e.s.u) while the maximum magnetic field does not exist since the vacuum with such a strong field is shown to be unstable only for (E 2 − B 2 ) > 0. [18] (See page 677 of reference 18.)

Supernova explosion of a (C+O) star and NHE gamma ray bursts
Now star 2 explodes as a Type Ic supernova with the ejecta of mass M e = 2 ∼ 5M ⊙ . The column density (Σ) of the ejecta at time t from the explosion is given by where E 51 is the explosion energy in the unit of 10 51 erg. Recently Woosley, Eastman and Schmidt [19] made light curve models for SN1998bw and found that a 6M ⊙ (C+O) star model with the unusually large explosion energy of ∼ 2 × 10 52 erg can fit the light curve quite well. They also found that both shock break-out and relativistic shock deceleration in circumstellar material fail to produce a GRB of even the low luminosity(∼ 10 48 erg) inferred for GRB980425. This means that the source of gamma rays should be inside the ejecta. Since the time difference between the supernova explosion and GRB 980425 is one day or so, from Eq. (12) the column density at t ∼ 1day is so high that any gamma rays emitted from the central new pulsar, neutron star or black hole can not reach the surface of the ejecta unless there is a hole in the ejecta. This reminds us the mysterious spot in SN1987A [20,21] which appeared 5-7 weeks after the explosion of Type II SN1987A at ∼ 0.06 arc-sec away from SN1987A. To explain mysterious spot in SN1987A, at least four mechanisms were examined to punch a hole in the ejecta [22,23]. These mechanisms can apply to SN1998bw/GRB980425. They are; 1. Due to the existence of the secondary (a neutron star or black hole in this case) the density in the direction shadowed by the companion will be low [22].
2. Rayleigh-Taylor instability in the ejecta may make a hole [22].
3. The relativistic jets formed at the core collapse punch holes in the ejecta [23].
4. Beams of energy from the super strongly magnetized milli second pulsar may punch holes in the ejecta [23].
In the first mechanism the relativistic plasma from the pulsar gradually inflates a bubble within the remnant. Since the remnant is punctured, the relativistic plasma from the over-pressured bubble would squirt into the surrounding. However the time needed for this mechanism to operate would be much longer than ∼ 30s unless the direction of the hole coincides with the direction of the beam from the pulsar or from the black hole with the accretion disk. In the second mechanism, the chance probability of such a hole would be very low for t∼30 s. A chance coincidence of the hole and the direction of the beam is also needed.
In the third mechanism the direction of the hole should agree with the direction of the spin of the pulsar or the rotating axis of the black hole. General relativistic numerical simulations [24] have demonstrated that a rapidly rotating core collapses first along its rotation axis. Collapse continues along the equatorial directions, then bounces along the rotation axis to form jets. In one example, a 1.4M ⊙ stellar core modeled as a γ = 3/4 polytrope produced jets with mass 0.007M ⊙ , kinetic energy 5 × 10 51 erg and average velocity 0.6c. Since the opening angle of this jet is a few degree, it is possible to punch a hole at the first 10 seconds or so. The total angular momentum in this model 2.4 × 10 49 gcm 2 s −1 is four times larger than 1ms pulsar. Although this kind of jets may also be formed in MHD rotating core collapses [25] , in general, with the increase of the angular momentum the kinetic energy of the explosion decreases since the bounces occur at the lower density and weaker gravitational potential, while SN1998bw has the unusually large kinetic energy. We therefore consider the fourth mechanism mainly in this paper.
For the typical angular momentum in the core of our (C+O) star corresponding to a milli-second pulsar, ∼ 6 × 10 48 gcm 2 /s, the collapse is essentially spherically symmetric because the ratio of the centrifugal force to the gravity is ∼0.2 even for the neutron star radius. A milli-second pulsar has the rotational energy(E rot ) given by , where I 45 and P ms are the moment of inertia in the unit of 10 45 gcm 2 and the spin period in milli-second, respectively. The spin period and the luminosity (L) of the pulsar are given by L = 3.8 × 10 51 erg/sB 2 16 R 6 6 (( where P i , B 16 , R 6 and t are the initial spin period, the magnetic field in unit of 10 16 G , the radius of the neutron star in unit of 10 6 cm and the time after the bounce of the core, respectively. The expansion velocity (V exp ) of the ejecta is given by For SN1998bw the observed expansion velocity is ∼ 2 × 10 9 cm/s [26]. This means that E 51 ∼ 20 for M e ∼ 5M ⊙ . This unusually large explosion energy is consistent with a model of light curves [19]. The shock will reach the surface of the (C+O) star(∼ 10 10 cm) in ∼ 5s. Since the magnetic field will be amplified to ∼ 10 16 G within 10s [13], the superstrong magnetic field pulsar begins to operate after or just when the shock breaks out. The solid angle extended by the beam from the pulsar is given by [27] Then the energy of ∼ 10 52 erg would be injected into 5% of the ejecta in ∼ 10s. If this energy is converted to the kinetic energy of the ejecta within the cone of the beam, the velocity (V cone ) would be This is much larger than V exp so that a hole is punched out in the ejecta. If, however, E 51 ∼ 1 and M e ∼ 5M ⊙ as usual , V exp ∼ 5 × 10 8 cm/s and the shock will reach the surface in ∼ 20s so that the pulsar activity would start before the shock breaks out. Then a part of the energy of the beam would be used to expand the ejecta as a whole so that an effective explosion energy would increase and we may have V exp ∼ 10 9 cm/s finally. In this case V cone will decrease but as far as V cone > V exp ∼ 2 × 10 9 cm/s , we may regard the hole is punched out. For this the minimum energy needed is ∼ 2 × 10 51 erg so that almost all the rotational energy of the milli-second pulsar might be used to the expansion energy of the ejecta. Note here that the unusually large explosion energy of SN1998bw is the same as the rotational energy of the milli-second pulsar. This may not be a chance coincidence. In any case a hole would be punched out by the beam of the pulsar in the first 10 s or so.
We assume here that at t ∼ 10s the hole is punched out. Then the period of the pulsar is ∼ 2ms and the rotational energy is ∼ 5 × 10 51 erg. The energy beam of the pulsar will flow out from the hole. This situation is similar to the millisecond pulsar models of GRBs based on the accretion induced collapse of magnetized white dwarfs [28,29]. The beam of the energy would be finally converted to the beam of gamma rays [28,29]. As a whole ∼ 5 × 10 51 erg of gamma rays would be emitted in a cone of the solid angle ∼ 0.025 × 4π. The chance probability that the line of the sight is in the cone is ∼ 0.025. We will observe the beam of the gamma rays, in general, from the lateral direction so that only gamma rays scattered by the thin matter blown off from the ejecta in the first 10s will come along the line of sight .
In Compton scattering the energy of the scattered photon (ω) is given by where θ is the scattered angle and ω 0 is the incident energy. Even for ω 0 ≫ m e c 2 , unless θ is small such that ω 0 (1 − cos θ) ≪ m e c 2 , ω < m e c 2 . That is , the energy of the scattered gamma ray should be smaller than 500keV. This explains the lack of fluence above 300keV for NHE-GRBs. The scattered photon is more or less distributed isotropically, that is, it is observed from almost all the direction and the total energy would be much smaller than ∼ 10 51 erg because only a small fraction of the beamed gamma rays would be scattered by the very low density plasma. This is also consistent with the total energy (∼ 10 48 erg) of GRB980425. About a fourth of gamma ray bursts(∼250 events/y) is NHE-GRBs [3] . The event like GRB980425 will not be observed if it occurred at 20 times farther ∼ 800 Mpc. The number of galaxies within 800Mpc is ∼ 2 × 10 7 and the event rate of TypeIb/Ic supernova is ∼ 10 −3 /y/galaxy [31] so that the event rate of TypeIb/Ic supernova within 800Mpc is ∼ 2 × 10 4 /y. This suggests that only a fraction (∼ a few%) of TypeIb/Ic supernova becomes NHE gamma ray burst which is different from the suggestion by Wang and Wheeler. [4] This is , however, consistent with our model. Depending on the initial separation and initial mass of the binary, only a fraction of the binary yields the super strongly magnetized milli-second pulsar.
After an NHE-GRB event the luminosity of the pulsar is given by For SN1998bw, the above luminosity is much smaller than the observed luminosity so that the existence of the pulsar will not affect the light curve of the supernova.

Discussion
It is not clear at present that our model can explain HE(High Energy) GRBs and its afterglow also. However if we observe NHE-GRB in our model along the beam direction and if we regard that the energy is emitted isotropically, then the total energy amounts to ∼ 2 × 10 53 erg, which is comparable to the maximum observed energy of GRBs [30]. We here argue the event rate of NHE-GRBs up to z ∼ 3. Adopting that Ω M ∼ 0.2 and Ω Λ ∼ 0.8 and H 0 =65km/s/Mpc, the total number of NHE-GRBs up to z ∼ 3 is ∼ 10 5 /y. If NHE-GRBs observed along the cone of the pulsar beam look like HE GRBs, the event rate of HE GRBs would be ∼ 10 3 /y, which is comparable to the event rate of HE GRBs.
The maximum energy of a proton accelerated by the SSM-MSP is given by [27] E max = 7.5 × 10 21 eV R 3 6 B 16 ( This is larger or comparable to the maximum energy of the cosmic rays so far observed. Such a high energy cosmic ray can propagate only up to the distance of ∼ 100Mpc due to the production of pion through the collision with the microwave background radiation. Then the number density of UHECRs(Ultra High Energy Cosmic Rays E > 10 20 eV) is estimated as n cr ∼ 10 −30 cm −3 . [32] If UHECRs are supplied by NHE-GRBs , the total energy of UHECR per one NHE-GRB-event is estimated as ∼ 10 50 erg. This energy is only 0.05% of the total energy of the milli-second pulsar and 2% of the energy emitted by the pulsar after the hole is punched out in our model. Very recently Agasa [33] team reported possible clustering in position of UHECRs, which may be in favor of GRB model of UHECRs [34]. Since the time broadening of the pulse of UHECRs due to the intergalactic magnetic field is more than 100 y [32], the arrival time of UHECR does not coincide with the GRB event. However in our model there should exist the supernova remnant in the direction of each UHECR if the time broadening is shorter than the maximum age of the observable supernova remnant. The relation between soft gamma ray repeaters (SGRs) and NHE-GRBs in our model is quite interesting. NHE-GRBs in our model is a strongly magnetized pulsar in the supernova remnant with the event rate of 10 −4 ∼ 10 −5 events/y/galaxy. This is exactly the same as the characteristics of soft gamma ray repeaters. This suggests that progenitors of SGRs are NHE-GRBs in our model.