Origin of Spectral Hardening of Secondary Cosmic-Ray Nuclei
aa r X i v : . [ a s t r o - ph . H E ] F e b MNRAS , 1β8 (2021) Preprint 26 February 2021 Compiled using MNRAS L A TEX style ο¬le v3.0
Origin of Spectral Hardening of Secondary Cosmic-Ray Nuclei
Norita Kawanaka , β and Shiu-Hang Lee Department of Astronomy, Graduate School of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan Hakubi Center, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
26 February 2021
ABSTRACT
We discuss the acceleration and escape of secondary cosmic-ray (CR) nuclei, such as lithium, beryllium and boron, producedby spallation of primary CR nuclei like carbon, nitrogen, and oxygen accelerated at the shock in supernova remnants (SNRs)surrounded by the interstellar medium (ISM) or a circumstellar medium (CSM). We take into account the energy-dependentescape of CR particles from the SNR shocks, which is supported by gamma-ray observations of SNRs, to calculate the spectraof primary and secondary CR nuclei running away into the ambient medium. We ο¬nd that if the SNR is surrounded by a CSMwith a wind-like density distribution (i.e., π CSM β π β ), the spectra of the escaping secondary nuclei are harder than those of theescaping primary nuclei, while if the SNR is surrounded by a uniform ISM, the spectra of the escaping secondaries are alwayssofter than those of the escaping primaries. Using this result, we show that if there was a past supernova surrounded by a densewind-like CSM ( βΌ . Γ β π β yr β ) which happened βΌ . Γ yr ago at a distance of βΌ . βΌ
200 GV that have recently been reported by AMS-02.
Key words: acceleration of particles β cosmic rays β shock waves β supernova remnants
Cosmic-ray (CR) secondary nuclei such as lithium, beryllium, andboron are considered to be produced via spallation of heavier nucleisuch as carbon, nitrogen, and oxygen, which are mainly producedat Galactic supernova remnants (SNRs), during their propagationin the interstellar medium (ISM). The amount of secondary CRnuclei produced per primary nucleus is proportional to the gram-mage traversed by the primaries in the ISM. Therefore, the ο¬uxesand rigidity dependence of secondary CRs are considered to be aprobe of the propagation of CRs in our Galaxy. The boron-to-carbon(B/C) ratio has been measured up to βΌ TeV per nucleon by AMS-02(Aguilar et al. 2016b), and it has been shown that above the rigid-ity π of 65 GV the B/C ratio can be well ο¬tted with a power law, β π Ξ with the index Ξ β β . Ξ isasymptotically equal to β / π· as a function ofrigidity is proportional to π / . However, AMS-02 later reported thatthe spectra of CR lithium, beryllium, and boron deviate from a singlepower law above 200 GV in an identical way and that they hardeneven more than the primary CRs (Aguilar et al. 2018) which have al-ready been known to harden at higher rigidities from measurementsby PAMELA (Adriani et al. 2011), CREAM (Ahn et al. 2010), andAMS-02 (Aguilar et al. 2015a,b; Aguilar et al. 2017).Several scenarios have been discussed to account for thedeviation of the index of the secondary-to-primary ratio athigh rigidities from what is expected from the diο¬usion co-eο¬cient in the bulk of the ISM, including the eο¬ect of β E-mail:[email protected] propagation in the Galaxy (Cowsik & Burch 2010; Tomassetti2012; Cowsik et al. 2014; Cowsik & Madziwa-Nussinov 2016;Tomassetti 2015), re-acceleration of secondary CRs (Bresci et al.2019; Yuan et al. 2020), contribution from diο¬erent kinds ofsources (Kawanaka & Yanagita 2018; Boschini et al. 2020; Niu2020) and the eο¬ect of acceleration of secondary CRs in su-pernova remnants (SNRs) (Mertsch & Sarkar 2009; Ahlers et al.2009; KachelrieΓ & Ostapchenko 2013; Mertsch & Sarkar 2014;Cholis & Hooper 2014). These models are proposed mainly to ex-plain the positron fraction measured by PAMELA and AMS-02(Adriani et al. 2009; Aguilar et al. 2013), the antiproton-to-protonratio measured by AMS-02 (Aguilar et al. 2016a), or the spectralhardening of proton and helium (Adriani et al. 2011; Aguilar et al.2015a,b). Especially, in the last scenario, secondary CR nuclei areproduced inside the SNRs via the spallation of primary CR nucleibeing accelerated at the SNR shock, and are accelerated at the sameshock. Since the spectra of secondary CR nuclei produced in thisway are harder than that of primary CR nuclei, if their contributionsare comparable to the background CR ο¬uxes above a certain rigid-ity, the rigidity dependence of the secondary-to-primary ratio woulddeviate from what is expected from simple ISM diο¬usion. Very re-cently, Mertsch et al. (2020) performed a comprehensive study onthis scenario by searching for the best ο¬t parameters to account forthe recent AMS-02 measurements of CR protons, helium, positrons,anti-protons, and primary/secondary nuclei. However, they discussthe spectra of secondary CRs that are shock-accelerated and advectedto the downstream (i.e., inside the SNRs), and they do not considerthe spectra of CRs escaping from the SNRs, which are what weobserve.In this paper we discuss the energy-dependent escape of pri-mary and secondary CRs that are accelerated in the SNR shock, Β© N. Kawanaka & S.-H. Lee and predict their energy spectrum assuming that they come froma local SNR. The escape of CR particles from an SNR hasbeen intensively investigated to interpret the observed CR spec-tra and the gamma-ray emission feature of SNRs interacting withmolecular clouds (Ptuskin & Zirakashvili 2005; Reville et al. 2009;Gabici et al. 2009; Caprioli et al. 2010; Ohira et al. 2010, 2011,2012, 2016; Ohira & Ioka 2011; Drury 2011; Kawanaka et al. 2011;Bell et al. 2013). If CRs escape the SNR in an energy-dependentway, the spectrum of escaping CRs would be steeper than that ofCRs trapped inside the SNR (Ohira et al. 2010; Caprioli et al. 2010),which may account for the steepness of the observed CR spectral in-dex. Therefore, it is worth investigating theoretically the production,acceleration, and escape of secondary CRs in an SNR, and comparingthem with observational data. In this work, we consider CR accelara-tion in SNRs not only within a uniform ISM but also those surroundedby a dense circumstellar medium (CSM). Recent intense optical andnear-infrared transient searches have revealed that many SNe showsigns of strong interaction between their ejecta and the CSM sur-rounding them. There are various classes of CSM-interacting SNe:Type IIn, Ibn, Ia-CSM, type I superluminous SN, etc (for a com-prehensive review, see Smith 2017). The origin of such dense CSMis still uncertain, but many observations indicate that their progeni-tors had expelled the stellar material shortly before their explosions,and the mass-loss rates estimated from their observational proper-ties are up to βΌ β π β yr β (assuming their wind velocities are βΌ
100 km s β ), continuing for decades (Moriya et al. 2014). The non-thermal emissions of photons or neutrinos due to the strong interac-tions between an SN ejecta and a dense CSM have been investigated insome studies(Murase et al. 2019; Wang et al. 2019; Matsuoka et al.2019; Cristofari et al. 2020; Matsuoka & Maeda 2020), but the CRproduction and escape in such a system have not been discussed yet.In Section 2, we describe our model of CR production inside aSNR surrounded by a CSM and energy-dependent escape of primaryand secondary CRs. In Section 3 we solve the transport equationsfor primary and secondary CR nuclei to evaluate the spectra of theescaping CRs, compare them with observations, and discuss ourresults in light of stellar evolution scenarios. We conclude our workin Section 4. Here we describe how CR particles accelerated at the SNR shockescape the source into the ISM depending on their energy, followingOhira et al. (2010). In the context of the diο¬usive shock acceleration(DSA) theory (Blandford & Eichler 1987), particles can go backand forth across the shock front and gain kinetic energy becausethey are scattered by the turbulent magnetic ο¬eld around the shock.Especially, the turbulence in the upstream (i.e. the outside of the SNRshock) may be ampliο¬ed by the streaming instability caused by theaccelerating particles themselves (Bell 1978; Lucek & Bell 2000).Since the turbulence is generated only in the vicinity of the shockfront, if accelerated particles have suο¬ciently high energy, they canescape the SNR into the far upstream region without being trappedby the turbulent magnetic ο¬eld. Using the diο¬usion coeο¬cient ofaccelerated particles as a function of their momentum π· ( π ) , theshock velocity π’ sh , and the distance from the shock beyond whichthe turbulence is negligible π , the escape condition for a particleaccelerated at the shock can be described as π· ( π ) π’ sh & π, (1) where the left hand side represents the diο¬usion length of a particlewith momentum π . Zirakashvili & Ptuskin (2008) investigated thegeneration of magnetohydrodynamic (MHD) turbulence driven bythe non-resonant streaming instability using numerical MHD calcu-lations, and show that π should be the same order as the radius of theSNR, π sh .As a SNR evolves with time, the magnetic ο¬eld around the shockdecays and the speed of the SNR shell slows down. As a result, theescape condition Eq.(1) also evolves with time. Especially, if the dif-fusion length of a particle with a speciο¬c momentum increases fasterthan the SNR radius ( βΌ π ), the required momentum of particles forescaping the SNR decreases with time. In other words, CR particlesaccelerated at the SNR shock can escape the SNR to far-upstreamin an energy-dependent way. In the following subsections, we willevaluate the energy distribution of CR particles produced in a SNRtaking into account the spallation of primary CRs, the production ofsecondary CRs, and their energy-dependent escape. Here we present our formalism to derive the energy distributions ofprimary and secondary CRs produced in the SNR and their escapingο¬uxes as functions of time. In the following discussion, we focuson the CR nuclei of lithium, beryllium, boron, carbon, nitrogen, andoxygen. Among them, only oxygen is regarded as a pure primaryelement, and other lighter elements are partly or entirely producedvia spallation of heavier nuclei during their propagation.We assume that the CR particles can be regarded as test particlesduring DSA in a SNR. Letting the shock front be at π₯ =
0, thestationary transport equation for the distribution functions of CRnuclei π π ( π₯, π ) ( π represents the type of nuclei) in the shock restframe is π’ ( π₯ ) π π π ππ₯ = πππ₯ (cid:20) π· π ( π ) π π π ππ₯ (cid:21) + π ππ’ππ₯ π π π π π β Ξ π π π + π π + π’ β π π πΏ ( π₯ ) πΏ ( π β π ) , (2)where π’ ( π₯ ) is the ο¬uid velocity, π· π ( π ) is the diο¬usion coeο¬cientfor a nuclei of π with momentum π , Ξ π is the total spallation rate ofa nuclei π (i.e. Ξ π = Γ π> π Ξ π β π ), π π is the source term due to thespallation of parent particles, and π π is the injection rate of a nuclei π at the shock front (the injection momentum is π ). Considering thatthe kinetic energy per nucleon of a nucleus is conserved before andafter the spallation, π π is given by4 ππ ππππ π π π ( π ) = Γ π< π Ξ π β π π π ( π π ) , (3)where π π ( π π ) = ππ π π ( π )( ππ π / ππ π ) ( π π is the momentum of anucleus π with kinetic energy of π π ) is the kinetic energy distributionfunction of nuclei π , and Ξ π β π is the rate at which a nucleus π isproduced via spallation of a heavier nucleus π . The ο¬uid velocity isgiven by π’ ( π₯ ) = ( π’ β ( π₯ < ) π’ + ( π₯ > ) , (4)where π’ β = π’ sh and π’ + = π’ sh / π are constants, and π is the shockcompression ratio, which is assumed to be equal to 4 (the strongshock limit, ignoring non-linear eο¬ects from CR feedback) hereafter.We then solve the transport equation (2) by imposing the following MNRAS000
0, thestationary transport equation for the distribution functions of CRnuclei π π ( π₯, π ) ( π represents the type of nuclei) in the shock restframe is π’ ( π₯ ) π π π ππ₯ = πππ₯ (cid:20) π· π ( π ) π π π ππ₯ (cid:21) + π ππ’ππ₯ π π π π π β Ξ π π π + π π + π’ β π π πΏ ( π₯ ) πΏ ( π β π ) , (2)where π’ ( π₯ ) is the ο¬uid velocity, π· π ( π ) is the diο¬usion coeο¬cientfor a nuclei of π with momentum π , Ξ π is the total spallation rate ofa nuclei π (i.e. Ξ π = Γ π> π Ξ π β π ), π π is the source term due to thespallation of parent particles, and π π is the injection rate of a nuclei π at the shock front (the injection momentum is π ). Considering thatthe kinetic energy per nucleon of a nucleus is conserved before andafter the spallation, π π is given by4 ππ ππππ π π π ( π ) = Γ π< π Ξ π β π π π ( π π ) , (3)where π π ( π π ) = ππ π π ( π )( ππ π / ππ π ) ( π π is the momentum of anucleus π with kinetic energy of π π ) is the kinetic energy distributionfunction of nuclei π , and Ξ π β π is the rate at which a nucleus π isproduced via spallation of a heavier nucleus π . The ο¬uid velocity isgiven by π’ ( π₯ ) = ( π’ β ( π₯ < ) π’ + ( π₯ > ) , (4)where π’ β = π’ sh and π’ + = π’ sh / π are constants, and π is the shockcompression ratio, which is assumed to be equal to 4 (the strongshock limit, ignoring non-linear eο¬ects from CR feedback) hereafter.We then solve the transport equation (2) by imposing the following MNRAS000 , 1β8 (2021) ardening of Secondary CR Nuclei boundary conditions: ( i ) lim π₯ ββ π π = lim π₯ β+ π π , (5) ( ii ) lim π₯ ββ π π π = , (6) ( iii ) (cid:12)(cid:12)(cid:12) lim π₯ β+β π π (cid:12)(cid:12)(cid:12) < β , (7) ( iv ) (cid:20) π· π ( π ) π π π ππ₯ (cid:21) π₯ = β π₯ = + = ( π’ + β π’ β ) π π π π, π π + π’ β π π πΏ ( π β π ) , (8)where condition (i) means that the distribution functions should becontinuous across the shock, and condition (ii) means the free escapeof CR particles from the outer boundary. Condition (iv) comes fromthe integration of Eq.(2) across the shock front ( π₯ = π₯ = π , π π, ( π ) . Hereafter we solve Eq.(2) for the relativisticregime, i.e., the kinetic energy per nucleon π π is greater than a fewGeV / n, so that we can approximate π β π΄π π / π .Following Mertsch & Sarkar (2009) we can solve for the energydistribution function of nuclei π , π π ( π π ) = ππ π π π ( π )( ππ π / ππ π ) separately in the downstream ( π₯ >
0) and the upstream ( π₯ < π + π = Γ π > π πΈ ππ π π π π₯ / , (9) π β π = Γ π > π πΉ ππ π π π π₯ / + πΊ π , (10)where π π = π’ + π· π (cid:18) β q + π· π Ξ π / π’ + (cid:19) , (11) π π = π’ β π· π (cid:18) + q + π· π Ξ π / π’ β (cid:19) , (12)and πΈ ππ and πΉ ππ are determined recursively as πΈ ππ = β Γ π π Ξ π β π πΈ ππ π· π π π β π’ + π π β Ξ π ( for π > π ) , (13) πΈ ππ = π π, β Γ π>π πΈ ππ , (14) πΉ ππ = β Γ π π Ξ π β π πΉ ππ π· π π π β π’ β π π β Ξ π ( for π > π ) , (15) πΉ ππ = π π, β Γ π>π πΉ ππ ( β π β π π π / ) β π β π π π / , (16) πΊ π = π π, (cid:18) β β π β π π π / (cid:19) β Γ π>π πΉ ππ π β π π π / β π β π π π / β π β π π π / , (17)where π π, = ππ π π π, ( π π )( ππ π / ππ π ) . The diο¬erential equation forthe distribution function at the shock front π π, can be derived fromthe boundary condition (iv) as π π π π, π π = β π· π ( π’ β β π’ + ) (cid:20)(cid:18) π π β π β π π π / β π π (cid:19) π π, + Γ π>π ( Λ πΉ ππ π π β β π β π π π / β π β π π π / π π ! β Λ πΈ ππ (cid:0) π π β π π (cid:1))ο£Ήο£Ίο£Ίο£Ίο£Ίο£» + π’ β π’ β β π’ + π π πΏ ( π β π ) , (18) where Λ πΈ ππ = πΈ ππ ππ π ( ππ π / ππ π ) , (19)Λ πΉ ππ = πΉ ππ ππ π ( ππ π / ππ π ) . (20)The diο¬erence between Mertsch & Sarkar (2009) and this study isthe position of the (eο¬ective) escape boundary: the former assumesthat π π should damp at π₯ = ββ , while we impose the outer boundarycondition π π = π₯ = β π . When the acceler-ation timescale is much shorter than the spallation timescale (i.e., Ξ π π· π / π’ β βͺ
1) and the escape boundary is very far from the shockfront (i.e., π β π π π / βͺ π₯ = β π as π π ( π ) = π’ β π π | π₯ = β π β π· π ( π ) π π π ππ₯ (cid:12)(cid:12)(cid:12)(cid:12) π₯ = β π = β π· π ( π ) Γ π > π π π πΉ ππ π β π π π / , (21)which is the function of time through the size of the escape boundary, π . The energy spectrum of CR particles escaping the SNR per unittime is given by ππ esc ,π ( π π ) ππ‘ β ππ Β· π (cid:18) π΄π π π (cid:19) | π π ( π )| ππ π ππ π , (22)where π sh is the radius of the SNR shock.To understand the nature of the escape ο¬ux π π ( π ) , let us assumethat the π -nuclei are purely primary (i.e., they are not produced viaspallation of heavier nuclei), and that the loss due to spallation isnegligible. In this case, the escape ο¬ux can be described as β π π ( π ) = π’ β π π, ( π ) exp ( π’ β π / π· π ) β , (23)Here we assume Bohm-type diο¬usion, in which the mean free pathof a charged particle is proportional to its Larmor radius, inside theSNR: π· π ( π ) = π π π π πππ΅ , (24)where π΅ is the magnetic ο¬eld strength and π π is the gyro factor. Wecan see that the momentum at which the absolute value of the escapeο¬ux of the π -nuclei attains its maximum value is given by π = π π,π β‘ π’ β ππππ΅πΎπ π π , (25)and the maximum value of the escape ο¬ux is β π π ( π π,π ) = π’ β π π, ( π π,π ) π πΎ β , (26)(Caprioli et al. 2009; Ohira et al. 2010). Noting that the distribu-tion function at the shock front behaves as π π β π β πΎ when π βͺ π π,π where πΎ = π’ β /( π’ β β π’ + ) = π /( π β ) , and that π π, ( π ) β exp (β π / π π,π ) when π β« π π,π , the particles acceleratedat the SNR shock with momentum π π,π can escape the SNR mosteο¬ciently, i.e., π π,π can be regarded as the maximum momentum ofthe escaping particles.To evaluate the resulting CR spectra, we need a model for thespatial diο¬usion coeο¬cient, π· π ( π ) , which in general depends ontime because of the evolution of the magnetic ο¬eld around the shock,and the primary CR injection rate at the shock, π π . In the next section, MNRAS , 1β8 (2021)
N. Kawanaka & S.-H. Lee we describe a phenomenological model to determine these quantitiesbased on analytic formulae of SNR evolution and the observed spectraof Galactic CRs.
Neglecting the back reaction of particle acceleration, the dynamicsof the SNR shock can be well described by analytic solutions. Whenan SNR is surrounded by the ISM or a CSM, the SNR expandsfreely until it sweeps up a mass comparable to its own mass fromthe surrounding medium, and afterward the SNR shell is deceleratedand expands in a self-similar way. This is so called the Sedov-Taylorphase, and the evolution of the SNR shell radius is determined bythe explosion energy of a supernova πΈ SN , the ejecta mass π ej , andthe number density of the ambient medium π . In the case with theuniform ISM, the SNR shell radius is given by π sh = π S (cid:18) π‘π‘ S (cid:19) / (27)where π S and π‘ S are the Sedov radius and Sedov time, respectively,and they are given by π S = .
59 pc (cid:18) π ej π β (cid:19) / (cid:18) π . β (cid:19) β / , (28) π‘ S =
450 yr (cid:18) πΈ SN erg (cid:19) β / (cid:18) π ej π β (cid:19) / (cid:18) π . β (cid:19) β / , (29). While in the case that the ambient density proο¬le is wind-like (i.e. π β π β ), the SNR shell radius is given by π sh = π S (cid:18) π‘π‘ S (cid:19) / (30)where π S = .
26 pc (cid:18) π ej π β (cid:19) (cid:18) Β€ π β π β yr β (cid:19) β (cid:18) π£ π€
100 km s β (cid:19) , (31) π‘ S =
178 yr (cid:18) πΈ SN erg (cid:19) β / (cid:18) π ej π β (cid:19) / (cid:18) Β€ π β π β yr β (cid:19) β Γ (cid:18) π£ π€
100 km s β (cid:19) , (32)where Β€ π and π£ π€ are the mass loss rate and wind velocity, re-spectively. Here we use the density proο¬le for a steady wind, i.e., π = Β€ π /( ππ π π£ π€ π ) .We here adopt a phenomenological model proposed byGabici et al. (2009) (see also Ohira et al. 2010), which is based on theassumption that Galactic SNRs are responsible for CRs with energybelow the knee ( βΌ . eV). In this model it is assumed that themaximum momentum of a CR particle π π,π accelerated at the SNRis limited by its escape during the Sedov-Taylor phase, and that themaximum energy of an escaping particle, βΌ ππ π,π , at the beginningof the Sedov phase is equal to the knee energy ( βΌ . eV). Usinga variable π to describe the evolution of an SNR (e.g., the SNRβsage, the shock radius of an SNR, etc.), we assume that the maxi-mum momentum of an escaping CR particle decreases with time as π π,π β π β πΌ , and that the spectrum of CRs accelerated at the SNRis proportional to π π½ π β π (i.e., CRs inside the SNR have a power-law spectrum with index of π and their total number increases withtime). One can then see that the spectrum of CRs escaping the SNRis proportional to π β π + π½ / πΌ . This means that if πΌ and π½ are positive(i.e., the maximum momentum decreases and the total number ofaccelerated CRs increases with time), the spectrum of escaping CRs become steeper than that inside the SNR. Here πΌ and π½ are the phe-nomenological parameters that are determined so that the resultingCR spectra are consistent with observations. Following Ohira et al.(2010), we hereafter adopt π sh as π , and ο¬x the parameters as πΌ = . π½ = .
1. Actually, with these parameters, the maximum energyof escaping CRs evolves from the knee energy at the beginning ofthe self-similar phase to 1 GeV at the end of the Sedov phase, whenthe radius of the SNR shell becomes 10 times larger than π π , andthe spectral index of escaping primary CRs would be similar to whatis inferred from observations, assuming that the diο¬usion coeο¬cientin the ISM, π· ISM ( π ) , is proportional to a power of rigidity, π .One can determine the diο¬usion coeο¬cient and the normaliza-tion factor of the escaping CR ο¬ux in the following way. First,the time dependence of π π,π ( β π β πΌ ) is related to the evolutionof the diο¬usion coeο¬cient, which depends on the ampliο¬cationand decay of magnetic ο¬eld and the turbulence around the shock(Ptuskin & Zirakashvili 2003, 2005; Yan et al. 2012). Assuming theBohm diο¬usion inside the SNR, one can determine the evolution ofthe diο¬usion coeο¬cient in the SNR using the relation in Eq.(25).Assuming π β π sh , we can describe the evolution of the diο¬usioncoeο¬cient at a speciο¬c momentum as β π πΌ β / in the case withuniform ISM, while β π πΌ + / in the case with the wind-like ambientdensity proο¬le. Second, the normalization factor of the spectrum ofCRs accelerated at the SNR ( β π π½ ) is related to the CR injection rateat the shock, π π . If we assume that π π evolves with time as β π π½ β² where π½ β² is a constant, since the escaping CR spectrum per unit timeis expressed as Eq.(22), where π ( π ) is proportional to π’ sh π π, , onecan see that the time-integrated spectrum of escaping CRs would be β« ππ‘ ππ esc ,π ππ‘ β π π½ β² + π πΎ β π,π β π β( πΎ β )β π½ β²+ πΌ , (33)where we use π π,π β π β πΌ . Considering that the power-law indexof the on-site CR spectrum is βΌ πΎ β
2, we can see how π π shouldincrease with time as π½ β² = π½ β . (34) In this study, we try to explain the observed CR spectral hardeningabove βΌ
200 GV by introducing a single local SNR surrounded bythe ISM or a dense CSM. The propagation of CR particles in theISM can be described by the diο¬usion equation, πππ‘ π π, ISM ( π, π, π‘ ) = π· ISM ( π )β π π, ISM ( π, π, π‘ ) + π π ( π, π, π‘ ) , (35)where π π, ISM ( π, π, π‘ ) is the distribution function of CR particles in theISM at a distance π from the source and the time π‘ , with momentum π , and π π ( π, π, π‘ ) is the CR injection rate from the source, which islocated at π = π π β πΏ ( π ) ). In our case, this diο¬usion equationcan be solved as π π, ISM ( π, π, π‘ ) = β« π‘ ππ‘ β² ππ sh ( π‘ β² ) π | π π ( π )|( ππ· ISM π‘ ) / exp (cid:18) β π π· ISM π‘ (cid:19) . (36). The observed spectra can then be obtained by summing this distri-bution function and the background ο¬ux due to the myriad of sourcesin the Galaxy. MNRAS000
200 GV by introducing a single local SNR surrounded bythe ISM or a dense CSM. The propagation of CR particles in theISM can be described by the diο¬usion equation, πππ‘ π π, ISM ( π, π, π‘ ) = π· ISM ( π )β π π, ISM ( π, π, π‘ ) + π π ( π, π, π‘ ) , (35)where π π, ISM ( π, π, π‘ ) is the distribution function of CR particles in theISM at a distance π from the source and the time π‘ , with momentum π , and π π ( π, π, π‘ ) is the CR injection rate from the source, which islocated at π = π π β πΏ ( π ) ). In our case, this diο¬usion equationcan be solved as π π, ISM ( π, π, π‘ ) = β« π‘ ππ‘ β² ππ sh ( π‘ β² ) π | π π ( π )|( ππ· ISM π‘ ) / exp (cid:18) β π π· ISM π‘ (cid:19) . (36). The observed spectra can then be obtained by summing this distri-bution function and the background ο¬ux due to the myriad of sourcesin the Galaxy. MNRAS000 , 1β8 (2021) ardening of Secondary CR Nuclei -1 Ο * E . [ r e l a t i v e ] KE/nuc [GeV]
Li x 300Be x 300B x 300C x 1.5N x 13O
Figure 1.
Time-integrated energy spectra of CR nuclei escaping the SNRsurrounded by a uniform interstellar medium. The ambient density is assumedto be 0 . β . -1 Ο * E . [ r e l a t i v e ] KE/nuc [GeV]
Li x 50Be x 50B x 50C x 1.5N x 13O
Figure 2.
Time-integrated energy spectra of CR nuclei escaping the SNRsurrounded by a wind-like circumstellar medium. The mass loss rate and windvelocity are assumed to be 3 Γ β π β yr β and 100 km s β , respectively. Fig. 1 depicts the time-integrated spectra of CRs that have escapedthe SNR surrounded by a uniform ISM whose density is 0 . β ,and Fig. 2 depicts those escaped the SNR surrounded by a wind-likemedium whose mass loss rate and velocity are 3 Γ β π β yr β and 100 km s β , respectively. In both cases, the explosion energyand ejecta mass of the supernova are assumed to be 10 erg and 3 π β respectively, and the elemental abundance in CRs are assumedto be identical to that of the background CRs. In the former case,the spectra of secondary nuclei (lithium, beryllium, and boron) aresofter than those of the primary nuclei (carbon and oxygen), whilein the latter case the spectra of secondary nuclei are a bit harder thanthose of primary nuclei. These spectral features and the diο¬erencebetween two cases can be interpreted in the following way. Since thesecondary CR nuclei escaping the SNR are produced by the spalla-tion of primary CR nuclei being accelerated at the shock, the numberof CR particles escaping the SNR per unit time is proportional to π’ β π pr π amb π‘ int , where π pr is the distribution function of their parentnuclei accelerated at the shock ( π π½ ), π amb is the density of the am- bient gas (constant, or β π β ), and π‘ int is the time during which theirparent nuclei interact with the ambient gas before escape. Especially,in the escape-limited regime, π‘ int is equal to their acceleration timein the SNR, βΌ π· π / π’ β . Since we are assuming Bohm diο¬usion in theSNR, π· π β π , the intrinsic spectral index of secondary nuclei insidethe SNR would be π β
1. On the other hand, the normalization factorof the secondary CR ο¬ux evolves with time as β π· π / π’ β Β· π π½ in thecase of a uniform ISM and β π· π / π’ β Β· π β Β· π π½ in the case of a wind-like ambient medium. Taking into account the time-dependence ofthe shell expansion velocity ( π’ β β π β / in the case of uniform ISMand π’ β β π β / in the case with the wind-like ambient medium) andthe diο¬usion coeο¬cient (see the previous section), we can evaluatethe spectral index of the distribution function of secondary CR nucleiescaping the SNR per unit time as ( π β ) + β / + π½ + ( πΌ + / ) πΌ = π + π½ + πΌ , (37)in the case of uniform ISM, and ( π β ) + β / + π½ β + ( πΌ + / ) πΌ = π + π½ β πΌ , (38)in the case of wind-like ambient medium. Now we can see that inthe former case the spectrum of secondary nuclei is softer, while inthe latter case it is harder compared to that of the primary nuclei.This diο¬erence is caused mainly by the diο¬erence in the density pro-ο¬le of the ambient medium. Generally, the CR particles with higherenergy would escape the SNR earlier, so the time for primary CRswith higher energy to produce secondary CRs due to the interac-tion with ambient matter would be shorter. This makes the numberof escaping secondary CRs with higher energy smaller when theambient matter distribution is uniform. By contrast, when the SNRis surrounded by a wind-like CSM, the primary CRs escaping intothe ISM earlier can interact with larger amount of matter than thoseescaping later, and then the number of secondary CRs produced bythem is enhanced. As a result, the energy spectrum of secondaryCRs escaping the SNR would be harder than that of primary CRs.Note that this spectral hardening of secondary nuclei is essentiallydiο¬erent from what have been studied in Mertsch & Sarkar (2009)and Mertsch & Sarkar (2014), in that they discussed the accelerationof secondary CRs produced inside the SNR assuming that the max-imum energy of accelerated CRs is limited by the age of the SNR.In this scenario, the primary CRs with higher energy can interactwith ambient medium longer and produce more secondary CRs thanthose with lower energy. However, they did not take into accountthe energy-dependent escape of primary and secondary CRs, or thenon-uniform distribution of ambient medium, both of which havebeen implied by recent observations of SNRs and SNe.According to the recent measurements of secondary CR nuclei byAMS-02 (Aguilar et al. 2018), the spectra of lithium, beryllium, andboron are hardened at βΌ
200 GV as has been observed for primaryCR nuclei, but they harden more than the primaries. To account forsuch a feature, we will introduce the CR contribution from a pastlocal SN. In this context, the case with a wind-like CSM, which canmake the spectra of secondary CRs harder, is more relevant. In thefollowing discussions, therefore, we will show results for the wind-like CSM case only. In addition to the elements shown in Figure 2,we also show the spectra of protons and helium nuclei expected fromthe SNR in the next section.
In our scenario, we assume that the ο¬ux of CR nuclei observedby AMS-02 is a superposition of two components: one is from a
MNRAS , 1β8 (2021)
N. Kawanaka & S.-H. Lee Ο * R . [ G V . m - s r - s - ] R [GV] pHeCNO Γ Figure 3.
Comparisons of our model spectra of CR oxygen, nitrogen, andcarbon with AMS-02 data. The contributions from our hypothetical pastCSM-interacting SN are also shown by thin lines.age 1 . Γ yeardistance 1 . ergejecta mass 3 π β mass loss rate 2 . Γ β π β yr β wind velocity 100 km s β Table 1.
Properties of the past local SN we introduced to explain the AMS-02data. local SNR with a wind-like CSM that is supposed to reproduce thehardenings observed in the CR nuclei spectra at βΌ
200 GV, andthe other is the background ο¬ux due to average SNRs without denseCSM (i.e., these SNRs do not produce hard secondary CRs eο¬cientlyinside themselves). As for the ο¬rst component, we can calculate itsο¬ux using Eq.(36), choosing the parameters such as the distance andage of the SNR, total CR energy, CSM properties, abundance ratioin CRs, etc. On the other hand, the background ο¬ux of π -th nuclei N π is given recursively by N π ( π π ) = Γ π< π Ξ π β π N π ( π π ) + R π esc ,π ( π π ) / π esc ,π ( π π ) + Ξ π , (39)where R β .
03 yr β is the Galactic SN rate and π esc ,π ( π π ) is thetimescale for an π -th nucleus with kinetic energy per nucleon of π π to escape from the Galaxy, which is modeled using an expression of π· ISM , π esc ,π β π» π· ISM ( π΄ π π π / π π ) , (40)where π» β π· ISM ( π ) = π· ( π / ) . where π· = Γ cm s β .Figure 3 depicts the observed spectra of proton, helium, carbon,nitrogen, and oxygen nuclei, and Figure 4 depicts the observed spec-tra of lithium, beryllium, and boron nuclei as functions of rigidityreported by AMS-02, ο¬tted by our model. Figure 5 depicts the pre-dicted ratios of lithium to carbon, beryllium to carbon, and boron tocarbon in the observed CRs as functions of rigidity compared with -1 Ο * R . [ G V . m - s r - s - ] R [GV]
LiBeB
Figure 4.
Comparisons of our model spectra of CR boron, beryllium, andlithium with AMS-02 data. The contributions from our hypothetical pastCSM-interacting SN are also shown by thin lines. the ratios reported by AMS-02. In making these plots, we assumea local SNR surrounded by dense wind-like CSM with parameterssummarized in Table 1. As for the elemental abundance in CRs, weassume that the fractions of carbon, nitrogen, and oxygen are threetimes as much as those in the background CRs. We can see that thehard component appearing above βΌ
200 GV in each spectrum is wellexplained by introducing this local SNR. The important point hereis that we can ο¬t not only the spectra of primary CRs (i.e., oxygenand carbon) but also that of secondary CRs (i.e., lithium, berylliumand boron) by taking into account the production, acceleration, andescape of secondary CRs inside a local SNR surrounded by a denseCSM. Another remarkable thing is that our model can be tested bythe secondary to primary ratios at rigidity of & βΌ
200 GV are dominated by the contribution from a hypothet-ical SNR in our model, the predicted secondary-to-primary ratios inthis rigidity range would reο¬ect the hardening of secondary CRs es-caping the SNR. The secondary-to-primary ratios above βΌ TeV willbe explored by future experiments such as AMS-100 (Schael et al.2019), which will provide a critical test for the existence of such alocal SNR with dense wind-like CSM.
While it is beyond the scope of this study to discuss in depth about theprogenitor nature and detailed mass loss mechanism of the proposedlocal SNR, it is instructive to brieο¬y account on its possible origin,and the feasibility of invoking such a local source without violatingcurrent observations. There exists a few possible scenarios for an en-hanced mass loss rate during a certain period prior to core-collapse,such as giant eruptions at the envelope, envelope stripping in a bi-nary system by Roche-lobe overο¬ow (RLOF) to a companion starand a possible common envelope phase, and so on (see, e.g., Smith2014). The hypothetical close-by SNR invoked here in particular canpossibly be the result of a past SN of type Ib/c, originating from theexplosion of a Wolf-Rayet (WR) star in a close binary system (e.g.,Yoon et al. 2010; Dessart et al. 2012, and reference therein). For ex-
MNRAS000
MNRAS000 , 1β8 (2021) ardening of Secondary CR Nuclei -2 -1 R [GV] model Li/CAMS-02 10 -3 -2 -1 R [GV] model Be/CAMS-02 10 -2 -1 R [GV] model B/CAMS-02
Figure 5.
Li/C ( left ), Be/C ( center ), and B/C ( right ) ratios in the observed CRs reported by AMS-02 along with our model predictions. ample, a RLOF phase that lasts for βΌ yrs with a mass loss rateof βΌ β Msun/yr and a velocity βΌ
100 km/s can result in a denseCSM extending up to a radius π π€ βΌ π β typical for SN Ib/c, and for simplicity a π β density proο¬lefor the CSM structure and the mass loss parameters in Table 1, theSedov radius π π of the SNR is about 0 . This is much smallerthan π π€ , and the transition to Sedov phase happened at an age ofroughly 10 yrs old. Therefore, the escape of the highest energy CRsmust have occurred when the SNR blastwave was still in the midstof interacting with the dense CSM material, which is consistent withour picture above.On the other hand, the required chemical enhancement of heavymetal abundances in the CRs produced by this local accelerator is alsoconsistent with a stripped envelope SN origin, in that the CSM in thevicinity of the ejecta of a type Ib/c progenitor is expected to be rich inmetals in comparison to that around a SN IIP from the explosion of ared super-giant star (e.g., Yoon 2015). The blast wave colliding withthis metal-rich CSM shell in the early phase after SN should thenbe able to accelerate these heavy ions to high energies . This inter-action between the blastwave and a chemically-enriched CSM shellhas been suggested by recent observations of type Ibn SNe (e.g., seereview in Smith 2017), or even the hypothetical type Icn that may bedetected in the future. Meanwhile, at a current age of 1 . Γ yrs, theshock should have long died out already due to radiative loss, and theSNR should have merged with and become indistinguishable fromthe surrounding ISM, rendering it invisible in various wavebands at adistance > In this paper, we investigated the eο¬ect of the production of secondaryCR nuclei at supernova remnants by the diο¬usive shock accelerationmechanism on the CR population measured near the Earth using an We are ignoring any anisotropy and episodic history of the mass loss fromthe progenitor here. We note that ejecta of SN Ib/c can reach trans-relativistic velocities rightafter explosions, for which particle acceleration at the shock can be mod-elled self-consistently using methods such as Monte-Carlo simulations (e.g.,Ellison et al. 2013; Warren et al. 2015). A more accurate approach with treat-ment of DSA at trans-relativistic shocks will be done in a future work. analytic approach. By including nuclear spallation eο¬ects and ac-celeration locally at the acceleration sites, we predicted the spectraof the accelerated ions escaping from the SNR shocks. Our resultsshow that the secondary CR nuclei escaping the SNR have a softerspectral index compared to the primary CRs when the SNR is sur-rounded by uniform ISM, while they have a harder spectral indexwhen they are surrounded by the CSM with a wind-like densityproο¬le. We show that, by introducing a past and relatively close-bySN event surrounded by a wind-like CSM, the current CR measure-ments including the spectral hardening recently discovered above afew 100 GV in ion species up to oxygen can be successfully repro-duced. We suggest that this past local accelerator can be a SN of typeIb/c with its progenitor enclosed by a CSM from pre-SN mass losshighly enriched in heavy metals. Our model also predicts a charac-teristic spectral ο¬attening of CR secondary-to-primary ratios, suchas Li/C, Be/C and so on, above a rigidity βΌ ACKNOWLEDGEMENT
N.K. acknowledges support by the Hakubi project at Kyoto Univer-sity. S.H.L. acknowledges support by JSPS Grant No. JP19K03913and the World Premier International Research Center Initiative(WPI), MEXT, Japan.
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