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Progenitor model of Cosmic Ray knee

Primary energy spectrum of cosmic rays exhibits a knee at about $3$ PeV where a change in the spectral index occurs. Despite many efforts the origin of such a feature of the spectrum is not satisfactorily solved yet. Here it is proposed that the steepening of the spectrum beyond the knee may be a consequence of mass distribution of progenitor of cosmic ray source. The proposed speculative model can account all the major observed features about cosmic rays without invoking any fine tuning to match flux or spectra at any energy point. The prediction of the proposed model regarding primary composition scenario beyond the knee is quite different from most of the prevailing models of the knee and thereby can be discriminated from precise experimental measurement of the primary composition.


I. INTRODUCTION
Ever since their discovery more than a hundred years back now, the origin of cosmic rays has been one of the central question of physics. But despite many efforts so far there is no consistent and complete model of the origin of cosmic rays.
The energy spectrum of cosmic rays provides important clues about their origin. The most intriguing feature of the energy spectrum is that though it extends a wide range of energies, from sub GeV to at least 3 × 10 20 eV (the highest energy observed so far), it can be well represented by a steeply falling power law for energies above the solar modulated one. However, the spectrum has a knee around 3 PeV where it steepens sharply and it also has an ankle at an energy about 3 EeV where it flattens again to its pre-knee slope. It is relatively easier to interpret the flattening of the spectrum above the ankle as the eventual superseding of a harder cosmic ray component which is sub-dominant at lower energies. In contrast the feature of knee is more difficult to explain. The existence of the knee in the spectrum is definitely an important imprint of the true model of origin of cosmic rays and hence a proper explanation of the knee is expected to throw light on the problem of cosmic ray origin.
Several mechanisms have been proposed so far to explain the knee which include leakage from the galaxy due to reduced efficiency of galactic magnetic field to confine the cosmic ray particles within galaxy [1], rigidity dependent acceleration scenario [2], nuclear photodisintegration at the sources [3], dominant contribution of the cosmic ray flux at the knee by a nearby source [4], as a result of cosmic ray acceleration by a variety of supernovae [5], a change in the characteristics of high energy interactions [6]. In the framework of Cannonball model [7], the knee corresponds to maximum energy gained by nuclei by elastic magnetic scattering of am-bient ISM particles in the Cannonball while reacceleration of cosmic rays by Cannonballs of other supernova explosions causes the extra steepness above the knee. None of these models is, however, free from problems. For instance, if the knee corresponds to break in proton spectrum, either because it is the maximum energy to which proton can accelerate in a galactic cosmic ray source or due to begin of proton leakage from the galaxy at this energy, then maximum energy of iron flux from the same sources/due to leakage from galaxy should be around 10 17 eV. Hence a special variety of supernovae or some other type of galactic/extra-galactic source has to be invoked as generator of cosmic rays between ∼ 10 17 eV and the ankle or galactic extra-galactic transition should occur at around 10 17 eV. The problem with such a proposal is that it requires fine-tuning to match both the flux and the energy at the point of taking over. The same fine-tuning argument is also applicable against the model involving variety of supernova for the cosmic ray origin till the ankle or the nuclear photo-disintegration at the source explanation of the knee. A problem with the single source proposal is that it requires a background cosmic ray spectrum with smoothly changing spectral index origin of which is not known. The change in interaction scenario at the knee energy has not received any support from the LHC experiment against the expectations. The dilemma of the knee thus still continues.
It is generally believed that the cosmic rays below around 10 18 eV are of galactic origin whereas those having energies above this energy are extragalactic though there are also suggestions for lower transitional energies [8,9]. Among the galactic sources the remnants of supernova (SN) explosions are considered as the most potential candidate [8], at least below the knee energy. Pulsars and pulsar nebula are other potential galactic sources whereas probable extra-galactic sources include Gamma Ray Bursts (GRBs), Active Galactic Nuclei (AGN). Whatever may be the sources, there is little doubt that they are products of stellar evolution process. And an interesting fact is that the zero age mass spectrum (ZAMS) of stars also exhibits power law behavior [10,11]. This immediately suggests that the cosmic ray energy spectrum might have some connection with the mass distribution of progenitor of their sources. In the present work we explore the idea and propose a hypernova model for the cosmic ray origin in which the knee of the primary cosmic ray energy spectrum at ∼ 3 PeV is a consequence of mass distribution of progenitor of cosmic ray sources.
The organization of the article is as follows. The model proposed in this work is presented in the next section. The outcome of the present model is discussed in section 3. Finally the results are concluded in section 4.

II. THE PROPOSED MODEL
During the late nineties a new class of SN has been discovered which have distinctly large explosion energies compared to previously known SNe and are often called as hypernova (HN). The observations suggest that the kinetic energy released from HN explosion could be an order or two higher than that of an ordinary SN. The association of long-duration Gamma Ray Burst (GRB) with hypernova explosion has been observed over the last two decades [16]. Here we propose a model of origin of cosmic rays in which HNe are the sole class of major cosmic ray sources in the galaxy.
The basic conjectures of the present model are the followings: 1) Cosmic rays at least up to the ankle energy are produced either in gravitational explosions (core collapse) of massive stars those lead to formation of black holes rather than neutron stars/white dwarf or in accretion on to black holes. Here we consider hypernova/gamma ray bursts (GRBs) as most probable candidate sources of cosmic rays. Active Galactic Nuclei (AGNs) are another viable possibility. No other type of galactic or extragalactic source dominates at least up to the ankle energy.
2) Particles are accelerated by expanding shock waves up to a maximum energy E max . The maximum attainable energy E max is, however, not the same for all the sources (of same kind) but depending on energy released in explosion/accretion, it has a range. The minimum E max that is possible for the cosmic ray sources is equal to the knee energy. We shall argue in the following section that the correspondence of minimum E max with the knee energy is quite plausible and suggestive.

A. Hypernovae and the Progenitor connection
The energy involved in generation of shocks, relativistic or non-relativistic, in stellar atmosphere is most likely gravitational in origin. There are two different viable mechanism, explosion and accretion, by which an astrophysical source may release a large amount of energy which is used to generation of high energy cosmic ray particles.
Perhaps one distinction between a HN and an ordinary SN is whether a black hole (BH) or a neutron star (NS) is formed in the stellar evolution processes. Through stellar core collapse progenitor stars with M < 20M ⊙ are supposed to give rise to a neutron star or white dwarf whereas stars more massive than 20 to 25M ⊙ form a black hole [12] though such end point fate also depends on metallicity [13]. The formation of white dwarf or neutron star is usually associated with supernova explosion. The masses of white dwarfs and neutron stars have to be within the Chandrasekhar limit and Openheimer-Volkof limit respectively. Consequently the energy released in all ordinary supernova explosions are nearly the same. Since black hole has no such upper mass limit the energy released in core collapse of massive stars leading to black holes should depend on the mass of the progenitor star.
The gravitational collapse of massive stars to black holes involves some complex, still poorly understood aspects of stellar physics. In the collapsar mechanism [14], a black hole is formed when the collapse of a massive star fails to produce a strong supernova explosion, leading to ultimate collapse into a black hole. If the stellar material falling back and accreting onto the black hole has sufficient angular momentum, it can hang up, forming a disk. This disk, by neutrino annihilation or magnetic fields, is thought to produce the jets which finally results in Active Galactic Nuclei (AGN) or GRB/hypernova. An uniform sphere of mass M and radius r possesses a gravitational potential energy ∼ GM 2 /r. Hence in a core collapse equivalent energy should be released. When a BH of mass M o (M o ≡ M/2 for maximum energy of extraction [15]) is formed, its radius may be taken as the Schwarzschild radius of the star i.e. r = 2M o and thereby total gravitational energy released ǫ should be proportional to M . Such a relation can also be established directly from the black hole mass-energy relation [15] which is an important result in understanding the physics of black hole. According to this formula, in the gravitational collapse of a spherical mass distribution of rest mass M leading to formation of black hole, the energy of extraction out of the collapse will be M c 2 /2 [15].
The theoretical mass-energy relation receives some support from the Hypernovae (HNe) explosion energy data. Hypernovae are not fully understood yet [16]. Those observed to date have generally been Type Ib/c SNe. The estimation of mass of the HNe progenitors, obtained from fitting of the optical light curve and spectra, indicates that they are very massive. Moreover, explosion energy is found larger for larger progenitor's mass. The variation of the explosion energy with the progenitor's mass has been studied from the estimated data. So far six HNe have been discovered, SN 1997ef [17], SN 1998bw [18], SN 1999as, SN 1997dq [19], SN 2002ap [20] and SN 2003dh [21], out of which the spectral coverage of SN 1999as is not extensive and the properties of this HN is not accurately known. Hence it is not included in the analysis. The parameters of HNe as estimated in different works are, however, not unique. But the general trend is that the explosion energy increases rapidly with the progenitor's mass as displayed in figure 1. The mass of the ejected material also found to increase in similar manner [22]. Instead of collapse and resulting explosion, large amount of energy also can be released through accretion process. The Eddington luminosity limit, the maximum steady-state luminosity that can be produced is given by L ed = 4πGM m p c/σ τ where M is the mass of the black hole, m p is the proton mass and σ τ is the Thomson cross section. The luminosity is thus also proportional to the mass of the black hole.
During the final stages of stellar evolution, a massive star losses a significant amount of mass. But if a black hole is formed stellar material likely to fall back and accreting onto the black hole [14]. The mass of the final produced black hole is thus expected to increase linearly with the mass of progenitor, and hence the distribution of released energy is expected to follow the mass distribution of progenitors.

III. 4. OUTCOMES OF THE PROPOSED MODEL
We are now in a position to test the proposed model against the observational features of cosmic rays.

A. The cosmic Ray Luminosity:
Earlier Sveshnikova demonstrated that hypernovae can satisfy the power requirement for accelerating all galactic cosmic rays [5] assuming the rate of hypernovae is about 10 −4 y −1 . The conventional estimate of cosmic ray lumi-nosity of our galaxy is ∼ 5 × 10 40 erg s −1 . The average energy released in a HN explosion is around 10 53 erg. The rate of HN explosion has been estimated from observations as ∼ 10 −5 yr −1 [23] which is an order smaller than the required rate. However, the true rate may be an order higher than the estimation because i) the observed rate of Type 1b and 1c SNe is around 10 −3 yr −1 . Radio observations suggest that about 5% SN 1b/1c can be produced in GRBs [24] and thus in hypernovae which implies an event rate nearly 10 −4 yr −1 . ii) The rate of stars having M > 20M ⊙ is 2 × 10 −3 yr −1 . Hence an event rate of at least 10 −4 yr −1 for the HNe seems quite likely. iii) HNe are detected only recently, during the last few years and within a short span of time a few HNe have been detected. So it is likely that HNe explosions are more frequent than the present estimates. The power from the HNe in the galaxy is, thus, ∼ 3 × 10 41 erg s −1 satisfying the power requirement for accelerating all galactic cosmic rays. Extragalactic origin of cosmic rays is usually considered as unlikely on the energetic grounds. However, such a problem can be circumvented by employing flux trapping hypothesis as proposed in [25]. Hence the possibility of AGN as the sole kind of dominate source of cosmic ray source cannot be totally ruled out from energetic consideration.

B. The Maximum attainable Energy:
The maximum energy that a particle with charge Ze can be attained in a bulk magnetized flow on a scale R s , with velocity cβ s and magnetic field B is [26] E max = ZeBΓ s β s R s where Γ s is the Lorentz factor of the relativistic shock wave. This value of E max is a factor Γ s larger than that obtained from the Hillas condition. In the standard scenario the acceleration of cosmic rays occurs at (non-relativistic) shocks of isolated supernova remnants (SNRs). The maximum energy that can be attained by a cosmic ray particle in an ordinary SNR when the remnant is passing through a medium of density GeV which is falling short of the knee by about one order of magnitude. Energy released in HN explosion is at least an order higher than that in SN explosion. Moreover, as stated before, for relativistic shock acceleration E max will be a factor Γ s higher where Γ s is the Lorentz factor of the relativistic shock wave. In a HN scenario, a fraction of all kinetic energy carries a debris ejected with the largest Lorentz factor thereby generating gamma ray emission in the form of burst (GRB), but the bulk of ejecta is less relativistic or even sub-relativistic. Note that if ∼ 10M ⊙ is given ∼ 10 54 erg then a typical velocity is 10 10 cm i.e. c/3. The GRB observations (considering hypernova GRB correlation) suggest minimum Γ s is just few tens [27]. Hence the minimum E max for HNe should be of few PeV that we identify as the knee energy.
An important question for HN origin of cosmic rays is whether or not E max could reach the ankle energy. Unlike almost constant energy released in SN explosions, energy output in hypernovae/AGN varies and it may go at least up to 2 order high from its minimum value. The GRB observations also support such a view. Γ s can be as high as few thousands or even more [27]. And such high energy events are expected to occur in more rarefied medium. Hence it is very likely that the maximum E max will exceed the ankle energy.
For AGN minimum E max is about 3 PeV [28] which is the knee energy and the maximum E max can be many order higher than that owing to the wide range of luminosities of AGNs.
So both hypernova and AGN satisfy the maximum attainable energy requirement of the model in a generic way.

C. Energy Spectrum:
In the proposed hypernova model cosmic rays are accelerated in diffusive relativistic shock acceleration. The energy spectrum of accelerated particles in each source is, therefore, given by a power law with γ is around 2.2, and A is the normalization con- where E min and E max are respectively the minimum and maximum attainable energies of cosmic ray particles in the source).
All the sources do not have the same E max . As per the arguments presented in the preceding sub-section, the minimum possible E max is E knee . Due to diffusive propagation of cosmic rays through the interstellar medium the slope of the resulting spectrum should be steepen to ∼ 2.7 till the knee of the spectrum and the knee should be a sharp one as observed. Above the knee the spectrum will be modified due to the distribution of E max . To get the spectrum beyond the knee we need to obtain maximum energy distributions of the cosmic ray sources from the mass distribution of their progenitors. The calculation involves a sequence of steps. Using the expression for explosion energy as function of progenitor mass as obtained in the previous section, we convolve the resulting explosion energy-progenitor mass relation with the initial mass function of the progenitors to obtain explosion energy distribution. Subsequently using the relation of maximum energy that a cosmic ray particle may attain in relativistic shock acceleration process with explosion energy, we derive the maximum energy distribution for main cosmic ray sources. Using such distribution we obtain the energy spectrum of cosmic rays beyond the knee.
The stellar initial mass function, or distribution of masses with which stars are formed can be represented by a declining power law with the universal (Salpeter) value of the exponent α = −2.35 over the whole mass range above 3M ⊙ [10,11]. Since explosion energy (ǫ) scales linearly with M , the expected explosion energy distribution of massive progenitor stars, is also represented by dn dǫ ∝ ǫ −α . The Lorentz factor of relativistic shock is nearly equal to the initial Lorentz factor of the jet i.e. Γ s ∼ γ o . The relativistic shock waves must carry a significant fraction of the explosion energy which subsequently convert to energies of cosmic rays. Hence Γ s should be proportional to explosion energy. On the other hand E max is also proportional to Γ s . So for the hypernova/AGN model, . Therefore, the number of sources having E max ≥ E is j(E max ≥ E) ∝ E −α+1 max . As the minimum E max of a source is equal to E knee all such sources will contribute to cosmic ray flux when cosmic ray energy is below or equal the knee energy. However, for energies above the knee (E > E knee ) only sources having E max ≥ E will contribute. The resultant cosmic ray spectrum above the knee will be Therefore beyond the knee the spectrum should be steepen by 0.35 in spectral index as observed. Note that the difference in power of energy by one between the above equation and the Eq.(3) of Kachelriess and Semikoz [29], where power law distribution of maximum attainable energy of sources was assumed, is due to the fact that our normalization constant A is proportional to the explosion energy (and hence to maximum attainable energy) unlike the explosion energy independent normalization constant as adopted in [29].

IV. DISCUSSION
According to the proposed model, cosmic rays below and just above the knee region are produced in HNRs of comparable progenitor's mass. Hence there should not be any abrupt change in mass composition through the knee unlike the case of SNR model of cosmic ray origin. In this model higher energy particles originate from the sources of heavier progenitor. Since HNE is the last stage of evolution of massive stellar objects, the composition is unlikely to change much for HNE of heavier progenitors. Therefore, the resulting composition of HNE accelerated cosmic rays is expected to remain almost unaltered with energy or may become slightly heavier at higher energies.
Several Extensive Air Shower (EAS) measurements (till now the study of cosmic rays above 1 PeV is of indirect nature via the EAS observations) have been made to determine the mass composition of cosmic rays in the concerned energy region but the measurements have not yielded mutually consistent results yet due to the weak mass resolution of the measured shower observables [30]. Very recent findings by the KASCADE-GRANDE collaboration about the existence of a Fe-knee around 80 PeV along with the heavier dominated composition scenario [31] together with earlier results of KASCADE experiment for a proton knee at 3 PeV [32] do not support the composition picture predicted by the proposed model. In fact most of findings [33] based on electron content relative to muon content (or vice versa) in EAS suggest that composition becomes heavier with energy beyond the knee though Haverah Park and few other observations (particularly underground muon telescopes) [34] found opposite trend of mass composition. Mass composition estimated from measurement of the depth of shower maximum through observation of Cerenkov [35] or fluorescence radiation [36][37][38] differ significantly from those obtained with muon to electron content ratio [30,39]. Importantly in the overlapping energy region around 1 EeV, the composition scenario inferred from the KASCADE-GRANDE or ICETOP findings of mixed composition with nearly same contribution from proton and Iron [32] is not in agreement with a proton dominated chemical composition as emerged from the observations of Pierre Auger Observatory [36], HiRes [37] and Telescope Array [38]. This only shows the difficulty in estimating primary masses from air shower experiments that rests on comparisons of data to EAS simulations with the latter requires hadronic interaction models as input which are still uncertain to a large extent at present. The mass composition picture of primary cosmic rays is thus still inconclusive in the PeV and higher energy region. It is expected that the mass composition scenario predicted by the present model will motivate newer experiments, exploiting both e/m and optical techniques, to establish unambiguous cosmic ray mass composition in the knee region and in particular to confirm the KASCADE-Grande results including the Fe-knee.
In summary, the proposed speculative hypernova model of origin of cosmic rays can account all the major observed features about cosmic rays without any serious contradiction to observational results. The knee of the energy spectrum has been ascribed as the consequence of the mass distribution of progenitor of cosmic ray source. Such a philosophy seems applicable to the Cannonball model of cosmic ray origin replacing the original proposal of second order Fermi acceleration of cosmic rays by Cannonballs of other SN explosions as the cause of spectral steepening above the knee [7]. Precise measurement of primary mass composition can be used to discriminate the proposed model from most of the standard prevailing models of cosmic ray knee.