Bhadra, A

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29th International Cosmic Ray Conference Pune (2005) 6, 133136Origin of the Knee in the Cosmic Ray Energy SpectrumA.BhadraUniversity of North Bengal, Siliguri, WB 734430 INDIAPresenter: A. Bhadra (aru bhadra@yahoo.com), ind-bhadra-A-abs3-he12-oralPrimary energy spectrum of cosmic rays exhibits a knee at about PeV where a change in the spectral indexoccurs. Despite many efforts the origin of such feature of the spectrum remains unresolved. Here it is proposedthat the steepening of the spectrum beyond the knee may be a consequence of mass distribution of progenitorof cosmic ray source.1. IntroductionThe energy spectrum of cosmic rays is the main source of information about their origin. The most importantfeature of the energy spectrum is that though it extends a wide range of energies, from sub GeV to at least 	eV (the highest energy observed so far), it can be well represented by a steeply falling power law forenergies above the solar modulated one. However, the spectrum has a knee around PeV where it steepenssharply and it also has an ankle at an energy about EeV where it flattens again. It is easier to interpret theflattening of the spectrum above the ankle as the eventual superseding of a harder cosmic ray component whichis sub-dominant at lower energies. In contrast the feature of knee is more difficult to digest.It is generally believed that the cosmic rays below the ankle are of galactic origin whereas those having energiesabove the ankle are extragalactic though there are also claims for lower transitional energies. Among thegalactic sources the remnants of supernova explosions are considered as the most potential candidate [1],at least below the knee energy. Gamma Ray Bursts (GRBs) /Cannonballs [2] are also recently evolved aspossible sources of cosmic rays of all energies. Whatever may be the sources, there is little doubt that theyare products of stellar evolution process. And an interesting fact is that the ZAMS mass spectrum of stars alsoexhibits power law behavior. This immediately suggests that the cosmic ray energy spectrum could have someconnection with the mass distribution of progenitor of their sources. In the present work we explore the ideaand propose a model of origin of cosmic rays in which the steepening of the spectrum beyond the knee is aconsequence of mass distribution of progenitor of cosmic ray sources.2. The proposed modelThough a number of arguments favor the model of supernova (SN) origin of cosmic rays but it also suffers fromsome persistent problems [2, 3]. In particular, it seems that the maximum energy that a cosmic ray particle mayachieve in diffusive shock acceleration process can hardly reach the ankle energy for an ordinary supernovaremnant (SNR). Hence a special variety of SNe or some other type of source has to be invoked as generator ofcosmic rays between the knee and the ankle. The problem with such a proposal is that it requires fine tuning tomatch both the flux and the energy at the point of taking over.During the last few years a new class of SNe has been discovered which have distinctly large explosion energiescompared to previously known SNe and are often called as hypernovae (HNe). The observations suggest thatthe kinetic energy released from HN explosion could be as much as order of two higher than that of an ordinarySN. In a recent model of origin of cosmic rays by Sveshnikova [4], HNe are considered as the dominant cosmicray sources below the knee energy whereas type II SNe are assumed as major sources beyond the knee. Herewe propose a model of origin of cosmic rays in which HNe are the sole class of major cosmic ray sources inthe galaxy. The basic features of the present model are:134 A. BhadraSN 2002apSN 1997efSN 1997dqSN 2003dhSN 1998bwMass of Progenitor (in solar mass unit)Explosion energy (in foe)204020 25 30 35 40Figure 1. Explosion energy versus progenitor’s mass for hypernovae1) Hypernovae are the main sources of galactic cosmic rays. No other variety of SNe or any other type ofgalactic source dominates at any energy range.2) Particles are accelerated by expanding shock waves from hypernovae up to a maximum energy  . Theminimum  that is possible for HNe corresponds to the knee energy.3. The Progenitor connectionPerhaps 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. It is generally believed that stars with   give rise toa NS whereas stars more massive than to ﬀ form a BH. However, it is unlikely that all BH formationevents lead to a HN because this would overproduce HNe by a large factor. Probably only stars with rotatingBH give rise to HNe.The estimation of mass of the HNe progenitor, obtained from fitting of the optical light curve and spectra, indi-cates that they are very massive. Moreover explosion energy is found larger for larger progenitor’s mass. Thevariation of the explosion energy with the progenitor’s mass has been studied from the estimated data. So farsix HNe have been discovered, SN 1997ef [5], SN 1998bw [6], SN 1999as, SN 1997dq [7], SN 2002ap [8] andSN 2003dh [9], out of which the spectral coverage of SN 1999as is not extensive. As a result the properties ofthis HN is not accurately known. Hence it is not included in the analysis. The parameters of HNe as estimatedin different works are, however, not unique. But the general trend is that the explosion energy increases rapidlywith the progenitor’s mass. The variation can be described by a power law ﬁﬂﬃ! #" with $ is around &%(Figure 1). The mass of the ejected material also found to increase in similar manner [10].Whether there is any upper mass cutoff of the stars is not conclusively known yet. A recent study on massdistribution in a massive and apparently young cluster Arches near the galactic centre indicate that the maxi-mum mass that a star can possess in the Galaxy is aroundsolar mass [12]. This corresponds to a explosionenergy of around ('*)erg which is larger by an order than the observed maximum kinetic energy in a HNexplosion and thus may appear questionable. However, it needs to keep in mind that though HNe explosionsgenerate enormous amount of kinetic energies, they are discovered only recently.The stellar initial mass function, or distribution of masses with which stars are formed can be approximatedby a declining power law [13, 14]. The exact slope of the mass function, however, has been subject of intenseOrigin of the Knee in the Cosmic Ray Energy Spectrum 135discussion. For higher mass range (mass greater than 10 solar mass) the distribution is found steeper than theSalpeter slope of +-,/.%  and is better represented by +-,0.%21 [15]. The expected explosion energydistribution of HNe, thus, can be represented by35476ﬁ8ﬂﬃ:9; <ﬁ>=@?ﬂAﬃ(1)where B is expected to be around  % C( .4. The Cosmic Ray SpectrumIn the present model cosmic rays are accelerated in sub-relativistic or in mildly relativistic shocks produced inHN explosions. Hence the energy spectrum of accelerated particles is given by D8ED8F G=IHwhere J is around to &%. Due to diffusive propagation of cosmic rays through the interstellar medium slope of the resultingspectrum would be steepen to  GK%21 .However, all HNe do not have the same   . Contribution of a single hypernova is given byLMLONPﬁ ﬂAﬃQSRUT 4 (VXWZY 9(2)As explosion energy increases the generation of accelerated particles by a hypernova also increases. How-ever, the frequency of the hypernovae of larger energies is small and is governed by the mass distribution ofprogenotor. The maximum energy roughly goes as  \[ ﬁ^]_ﬂﬃV-]_a`bdc . So the distribution of   followsfrom that of distribution of progenitor’s mass as3546e^9 f=hgwhere i is around K% j . Consequentlythe slope of the energy spectrum would be steeper by  % j after the knee.5. DiscussionAs shown by Sveshnikova [4], hypernovae can satisfy the power requirement for accelerating all galacticcosmic rays. In doing so the rate of hypernovae is assumed as kU=Ilnm(o(=]which seems larger than theobserved rate of  ='mpo=][16]. However, HNe are detected only recently, during last few years and itappears from the detection rate that HNe explosions are more frequent than that obtained from the present data.Moreover, the observed rate of Type 1b and 1c SNe is aroundU=)mpo(=]. The rate of stars having 6is qU=)mpo(=]. Hence an event rate of at leastU=rlempo=]for the hypernovae seems very likely.The main problem is, however, the maximum energy attainable in the acceleration process. In the standardscenario of cosmic ray origin, the acceleration occurs at the shocks of isolated SNRs. The maximum energythat can be attained by a cosmic ray nuclei in the process when the remnant is passing through a medium ofdensity3ﬂtsW=)is [17]ue^Nj'vxwuyﬃU']zoT|{]*_wb}c5{=]*_a`w3ﬂ =)sW=){=]*_)w~ {z(3)The above expression is valid for HNe also as they belong to a class (SN type 1c) of SNe. An explosion energyof around']erg can give  e^ equal to knee energy for realistic values of the other relevant parameters.The main question is whether or not u could reach the ankle energy. The  increases very slowly withmass and it is difficult to reach the ankle energy. However, as the explosion energy increases the shock velocitybecomes relativistic and hence the above expression for maximum energy does not valid in such situation. Somore analysis is required before any final conclusion.136 A. BhadraAccording to the present model, cosmic rays below and just above the knee region are produced in HNRs ofcomparable progenitor’s mass. Hence there should not be any abrupt change in mass composition through theknee. The observations appear to support such a scenario [18]. The higher energy particles originated fromthe source of heavier progenitor, thus resulting composition is expected to become heavier as observed. Theother features of cosmic ray composition scenario also favors for massive progenitor of cosmic ray sources asadopted in the present model.6. ConclusionRecent observations clearly indicate that HNe are associated with GRBs. Hence there is a possibility that en-ergy released in GRBs are also related with the progenitor’s mass and the present explanation of steepening ofthe spectrum beyond the knee may also be applicable for the model of GRB origin of cosmic rays.7. AcknowledgementWe thank Professors C. L. Fryer and S. E. Woosley for helpful discussions.References[1] V. L. Ginzburg and S. I. Syrovatskii, The Origin of Cosmic Rays, Macmillan, NewYork (1964)[2] A. Dar, astro-ph/0408310 (2004); A. De Rujula, astro-ph/0412094 (2004)[3] E. Parizot et al., 27th ICRC 6, 2070 (2001); E. Parizot, preprint astro-ph/0501274[4] L. G. Sveshnikova, Astron. Letts. 30, 41 (2004)[5] P. A. Mazzali et al., ApJ, 545, 407 (2000)[6] K. Iwamoto et al, Nature 395, 672 (1998)[7] R. Knop et al., IAU Circ. 7128 (1999)[8] P. A. Mazzali et al., ApJ, 614, 858 (2004)[9] P. A. Mazzali et al. ApJ 572, L61 (2002)[10] P. A. Mazzali et al., ApJ 599, L95 (2003)[11] K. Nomoto et al., AIP Conf. ser. 332, 374 (2005)[12] D. F. Figer, Nature 34, 192 (2005)[13] E. E. Salpeter, ApJ 121, 161 (1955)[14] P. Kroupa, Science 295, 82 (2002)[15] J. M. Scalo, Fund. Cosmic Phys. 11, 1 (1986); P. Kroupa et al., MNRAS, 262, 545 (1993); de la Marcoset al., New Astron. 9, 475 (2004)[16] Ph. Podsiadlowski et al., ApJ 607, L17 (2004)[17] E. G. Berezhko et al., JETP, 82, 1 (1996)[18] A. Bhadra and S. Sanyal, 29th ICRC, (2005)

Origin of the Knee in the Cosmic Ray Energy Spectrum

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