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On the Induced Gravitational Collapse of a Neutron Star to a Black Hole by a Type Ib/c Supernova

It is understood that the Supernovae (SNe) associated to Gamma Ray Bursts (GRBs) are of type Ib/c. The temporal coincidence of the GRB and the SN represents still a major enigma of Relativistic Astrophysics. We elaborate here, from the earlier paradigm, that the concept of induced gravitational collapse is essential to explain the GRB-SN connection. The specific case of a close (orbital period<1 h) binary system composed of an evolved star with a Neutron Star (NS) companion is considered. We evaluate the accretion rate onto the NS of the material expelled from the explosion of the core progenitor as a type Ib/c SN, and give the explicit expression of the accreted mass as a function of the nature of the components and binary parameters. We show that the NS can reach, in a few seconds, the critical mass and consequently gravitationally collapses to a Black Hole. This gravitational collapse process leads to the emission of the GRB.

The systematic and spectroscopic analysis of the Gamma Ray Burst (GRB)-Supernova (SN) events, following the pioneering discovery of the temporal coincidence of GRB 980425 (Pian et al. 2000) and SN 1998bw (Galama et al. 1998), has evidenced the association of other nearby GRBs with Type Ib/c SNe (see e.g. Della Valle 2011;Hjorth & Bloom 2011, for a recent review on GRB-SN systems). It has been also clearly understood that the most likely explanation of SN Ib/c, which lack Hydrogen (H)/Helium (He) in their spectra, is that the SN core progenitor star, likely a He, CO, or a Wolf-Rayet star, is in a binary system with a compact companion, a Neutron Star (NS) (see e.g. Nomoto & Hashimoto (1988); Iwamoto et al. (1994) and Fryer et al. (2007); Yoon et al. (2010) for more recent calculations).
In the current literature there has been the attempt to explain both the SN and the GRB as two aspects of the same astrophysical phenomenon. Hence, GRBs have been assumed to originate from a specially violent SN process, a hypernova or a collapsar (see e.g. Woosley & Bloom 2006, and references therein). Both of these possibilities imply a very dense and strong wind-like CircumBurst Medium (CBM) structure. Such a dense medium appears to be in contrast with the CBM density found in most GRBs (see e.g. Fig. 10 in Izzo et al. (2012b)). In fact, the average CBM density, inferred from the analysis of the afterglow, has been shown to be in most of the cases of the order of 1 particle cm −3 (see e.g. Ruffini 2011). The only significant contribution to the baryonic matter component in the GRB process is the one represented by the baryon load (Ruffini et al. 2000). In a GRB, the electron-positron plasma, loaded with a certain amount of baryonic matter, is expected to expand at ultra-relativistic velocities with Lorentz factors Γ 100 (Shemi & Piran 1990;Piran et al. 1993;Meszaros et al. 1993). Such an ultrarelativistic expansion can actually occur if the amount of baryonic matter, quantifiable through the baryon load parameter, B = M B c 2 /E e + e− , where M B is the engulfed baryon mass from jorge.rueda@icra.it, ruffini@icra.it the progenitor remnant and E e + e− is the total energy of the e + e− plasma, does not exceed the critical value B ∼ 10 −2 (see Ruffini et al. 2000, for details).
In our approach we have assumed that the GRB consistently has to originate from the gravitational collapse to a Black Hole (BH). The SN follows instead the complex pattern of the final evolution of a massive star, possibly leading to a NS or to a complete explosion but never to a BH. There is a further general argument in favor of this explanation, namely the extremely different energetics of SNe and GRBs. While the SN energy range is 10 49 -10 51 erg, the GRBs are in a larger and wider range of energies 10 49 -10 54 erg. It is clear that in no way a GRB, being energetically dominant, can originate from the SN.
There are however scenarios for GRB-SN systems that invoke a single progenitor, e.g. the collapse of massive stars (see e.g. Zhang 2011, for a recent review). In these models the core of the star must rotate at very high rates in order to produce, during the gravitational collapse, a collimated (e.g. jet) emission with a beaming angle θ j . In this way, an event with observed isotropic energy E iso , corresponds to an actual energy released at the source reduced by the beaming factor Woosley & Bloom 2006;Zhang 2011, and references therein). Outstandingly small beaming factors of order f p ∼ 1/500, corresponding to jet angles θ j ∼ 1 • , are needed to bring the most energetics GRBS with E iso ∼ 10 54 erg to standard energies ∼ 10 51 erg (Frail et al. 2001). However, observational evidence of the existence of so narrow beaming angles in GRBs, as suggested by these models, is inconclusive (see e.g. Covino et al. 2006;Sato et al. 2007;Burrows et al. 2009).
We explain the temporal coincidence of the two phenomena, the SN explosion and the GRB, within the concept of induced gravitational collapse (Ruffini et al. 2001(Ruffini et al. , 2008. In the recent years we have outlined two different possible scenarios for the GRB-SN connection. In the first version (Ruffini et al. 2001), we have considered the possibility that the GRBs may have caused the trigger of the SN event. For the occurrence of this scenario, the companion star had to be in a very special phase of its thermonuclear evolution (see Ruffini et al. 2001, for details).
More recently, we have proposed (Ruffini et al. 2008) a different possibility occurring at the final stages of the evolution of a close binary system: the explosion in such a system of a Ib/c SN leads to an accretion process onto the NS companion. The NS will reach the critical mass value, undergoing gravitational collapse to a BH. The process of gravitational collapse to a BH leads to the emission of the GRB (see Fig. 1). In this Letter we evaluate the accretion rate onto the NS and give the explicit expression of the accreted mass as a function of the nature of the components and the binary parameters.
We turn now to the details of the accretion process of the SN material onto the NS. In a spherically symmetric accretion process, the magnetospheric radius is given by (see e.g. Toropina et al. 2012) R m = B 2 R 6 /(Ṁ √ 2GM NS ) 2/7 , where B, M NS , R are the NS magnetic field, mass, radius, andṀ ≡ dM/dt is the mass-accretion rate onto the NS. We now estimate the relative importance of the NS magnetic field on the accretion process. At the beginning of a SN explosion, the ejecta moves at high velocities v ∼ 10 9 cm s −1 , and the NS will capture matter at a radius approximately given by R sph cap ∼ 2GM/v 2 . For R m << R sph cap , we can neglect the effects of the magnetic field. In Fig. 2 we have plotted the ratio between the magnetospheric radius and the gravitational capture radius as a function of the mass accretion rate onto a NS of B = 10 12 Gauss, M NS = 1.4M ⊙ , R = 10 6 cm, and for a flow with velocity v = 10 9 cm s −1 . It can be seen how for high accretion rates the influence of the magnetosphere is negligible.
We therefore assume hereafter, for simplicity, that the NS is non-rotating and neglect the effects of the magnetosphere. The NS captures the material ejected from the core collapse of the companion star in a region delimited by the radius R cap from the NS center where v rel,ej is the ejecta velocity relative to the orbital motion of the NS with v ej the velocity of the ejecta and v orb the orbital velocity of the NS, where a is the binary separation. Here we have assumed that the velocity of the SN ejecta v ej is much larger than the sound speed c s of the material, namely that the Mach number of the SN ejecta satisfies M = v ej /c s >> 1, which is a reasonable approximation in the present case. The orbital period of the binary system is where M SN−prog is the mass of the SN core progenitor. The NS accretes the material that enters into its capture region defined by Eq. (1). The mass-accretion rate is given by (see Bondi & Hoyle 1944, for details) where the parameter ξ is comprised in the range 1/2 ≤ ξ ≤ 1, ρ ej is the density of the accreted material, and in the last equality we have used Eqs. (1) and (2). The upper value ξ = 1 corresponds to the Hoyle-Lyttleton accretion rate (Hoyle & Lyttleton 1939). The actual value of ξ depends on the properties of the medium on which the accretion process occurs, e.g. vacuum, wind. In Fig. 1 we have sketched the accreting process of the SN ejected material onto the NS. The density of the ejected material can be assumed to decrease in time following the simple power-law (see e.g. Chevalier 1989) where, without loss of generality, we have assumed that the radius of the SN ejecta expands as r ej = σt n , with σ and n constants. The velocity of the ejecta is thus v ej = nr ej /t. If the accreted mass onto the NS is much smaller than the initial mass of the ejecta, i.e. M acc (t)/M ej (0) << 1, one can assume M ej (t) ≈ M ej (0) and thus the integration of Eq. (4) gives where F = t −3(n+1) −4n(2n − 1)t 4n √ kt 2−2n + 1 2 F 1 1 2 , 1 n−1 ; n n−1 ; −kt 2−2n − k 2 n 2 − 1 t 4 + 2k(n − 1)(2n − 1)t 2n+2 + 4n(2n − 1)t 4n k 3 (n − 1)(n + 1)(3n − 1) with k = v 2 orb /(n σ) 2 , 2 F 1 (a, b; c; z) is the Hypergeometric function, and t acc 0 is the time at which the accretion process starts, namely the time at which the SN ejecta reaches the NS capture region (see Fig. 1), i.e. so ∆M(t) = 0 for t ≤ t acc 0 . The above expression increases its accuracy for massive NSs close to the critical value, since the amount of mass needed to reach the critical mass by accretion is much smaller than M ej . In general, the total accreted mass must be computed from the full numerical integration of Eq. (4).
We turn now to the maximum stable mass of a NS. Nonrotating NS equilibrium configurations have been recently constructed by Belvedere et al. (2012) taking into account the strong, weak, electromagnetic, and gravitational interactions within general relativity. The equilibrium equations are given by the general relativistic Thomas-Fermi equations, coupled with the Einstein-Maxwell system of equations, the Einstein-Maxwell-Thomas-Fermi system of equations, which must be solved under the condition of global charge neutrality. The strong interactions between nucleons have been modeled through the exchange of virtual mesons (σ, ω, ρ) within the Relativistic Mean Field (RMF) model, in the version of Boguta & Bodmer (1977). These self-consistent equations supersede the traditional Tolman-Oppenheimer-Volkoff ones that impose the condition of local charge neutrality throughout the configuration.
The uncertainties in the behavior of the nuclear equation of state (EOS) at densities about and larger than the nuclear saturation density n nuc ≈ 0.16 fm −3 , lead to a variety of EOS with different nuclear parameters. A crucial parameter in this respect is the so-called nuclear symmetry energy, E sym = d 2 (E/n b )/dδ 2 δ=0,n nuc , where E is the nuclear matter energy-density, and δ = (n n − n p )/n b is the asymmetry parameter with n n , n p , n b = n p + n p the neutron, proton, baryon densities; we refer to (Tsang et al. 2012) for a recent review. The symmetry energy is relevant for the determination of the value and density dependence of the particle abundances (e.g. n n /n p ratio) in the NS interior (see e.g. Müther et al. 1987;Kubis 2007;Sharma & Pal 2009;Hebeler et al. 2010;Loan et al. 2011). The differences in the behavior of E sym (n) for different nuclear EOS models and parameterizations lead to a variety of NS mass-radius relations and consequently to different values the NS critical mass M crit and the corresponding radii (see e.g. Gandolfi et al. 2012). We have plotted in Fig. 3 the behavior nuclear symmetry energy for the RMF parameterizations used in (Belvedere et al. 2012) in a wide range of baryon densities expanding from n b ∼ 0.7n nuc all the way up to high densities n b ∼ 10n nuc found in the cores of NSs; we have included in the legend the values of the mass and radius of the critical neutron star configuration, M crit and R. Concerning our induced gravitational collapse scenario, the precise value of the time needed for the NS to reach M crit by accretion of the SN material depends, for fixed binary parameters (M NS , a, M prog , v ej ), on the adopted EOS which lead to a specific value of M crit .
The high and rapid accretion rate of the SN material can lead the NS mass to reach the critical value M crit . This system will undergo gravitational collapse to a BH, producing a GRB. The initial NS mass is likely to be rather high due to the highly non-conservative mass transfer during the previous history of the evolution of the binary system (see e.g. Nomoto & Hashimoto 1988;Iwamoto et al. 1994, for details). Thus, the NS could reach the critical mass in just a few seconds. Indeed, Eq. (4) shows that for an ejecta density 10 6 g cm −3 and ejecta velocity 10 9 cm s −1 , the accretion rate might be as large asṀ ∼ 0.1M ⊙ s −1 .
The occurrence of a GRB-SN event in the scenario presented in this Letter is subjected to some specific conditions of the binary progenitor system, such as a short binary separation and orbital period P < 1 h. This is indeed the case of GRB 090618 (Izzo et al. 2012b) and GRB 970828 (Izzo et al. 2012a), which we are going to analyze within the framework presented here in forthcoming publications. In addition of offering an explanation to the GRB-SN temporal coincidence, the considerations presented in this Letter leads to an astrophysical implementation of the concept of proto-BH, generically introduced in our previous works on of GRBs 090618, 970828, and 101023 (see Izzo et al. 2012b,a;Penacchioni et al. 2012, respectively). The proto-BH represents the first stages, 20 t 200 s, of the SN evolution.
It is also worth noticing that the condition B 10 −2 on the baryon load parameter of a GRB (Ruffini et al. 2000) might constrain on the binary separation a for the occurrence a GRB-SN event. When the NS reaches the critical mass, the distance between the location of the front of the undisturbed SN ejecta and the NS center should be << a, otherwise the emitted e + e − plasma in the GRB might engulf a large amount of baryonic matter from the SN ejecta, reaching or even overcoming the critical value B ∼ 10 −2 .
It is appropriate now to discuss the possible progenitors of such binary systems. A viable progenitor is represented by X-Ray Binaries such as Cen X-3 and Her X-1 (Schreier et al. 1972;Wilson 1972;Tananbaum et al. 1972;Leach & Ruffini 1973;Gursky & Ruffini 1975;Rawls et al. 2011). The binary system is expected to follow an evolutionary track (see Nomoto & Hashimoto 1988;Iwamoto et al. 1994, for details): the initial binary system is composed of main-sequence stars 1 and 2 with a mass ratio M 2 /M 1 0.4. The initial mass of the star 1 is likely M 1 11M ⊙ , leaving a NS through a corecollapse event. The star 2, now with M 2 11M ⊙ after some almost conservative mass transfer, evolves filling its Roche lobe. It then starts a spiral in of the NS into the envelope of the star 2. If the binary system does not merge, it will be composed of a Helium star and a NS in close orbit. The Helium star expands filling its Roche Lobe and a non-conservative mass transfer to the NS, takes place. This scenario naturally leads to a binary system composed of a CO star and a massive NS, as the one considered in this Letter.
We point out that the systems presenting a temporal coincidence of GRB-SN form a special class of GRBs: (1) There exist Ib/c SNe not associated to a GRB, e.g. the observations of SN 1994I (Immler et al. 2002) and SN 2002ap (Soria et al. 2004. Also this class of apparently isolated SNe may be in a binary system with a NS companion at a large binary separation a and long orbital period P (3), and therefore the accretion rate (4) is not highly enough to trigger the process of gravitational collapse of the NS. A new NS binary system may be then formed and lead the emission of a short GRB in a NS merger after the shrinking of the binary orbit by gravitational waves emission.
(2) There are GRBs that do not show the presence of an associated SN. This is certainly the case of GRBs at large cosmological distances z 0.6 when the SN is not detectable even by the current high power optical telescopes. This is likely the case of GRB 101023 .
(3) There is the most interesting case of GRBs that do not show a SN, although it would be detectable. This is the case of GRB 060614 (Caito et al. 2009) in which a possible pro-genitor has been indicated in a binary system formed of a white dwarf and a NS, which clearly departs from the binary class considered in this Letter. Finally, there are systems originating genuinely short GRBs which have been proved to have their progenitors in binary NSs, and clearly do not have an associated SN, e.g. GRB 090227B (Muccino et al. 2012;. Before closing, we like to look to the problem of the remnants of the class of GRBs considered in this Letter. It is clear that after the occurrence of the SN and the GRB emission, the outcome is represented, respectively, by a NS and a BH. A possible strong evidence of the NS formation is represented by the observation of a characteristic late (t = 10 8 -10 9 s) Xray emission (called URCA sources, see Ruffini et al. (2005)) that has been interpreted as originated by the young (t ∼ 1 minute-(10-100) years), hot (T ∼ 10 7 -10 8 K) NS, which we have called neo-NS (see Negreiros et al. 2012, for details). This has been indeed observed in GRB 090618 (Izzo et al. 2012b) and also in GRB 101023 ). If the NS and the BH are gravitationally bound they give origin to a new kind of binary system, which can lead itself to the merging of the NS and the BH and consequently to a new process of gravitational collapse of the NS into the BH. In this case the system could originate a yet additional process of GRB emission and possibly a predominant emission in gravitational waves.
We thank the anonymous referee for many suggestions which have improved the presentation of our results.