On the Positron Fraction and Cosmic-Ray Propagation Models
aa r X i v : . [ a s t r o - ph . H E ] S e p XVI International Symposium on Very High Energy Cosmic Ray InteractionsISVHECRI 2010, Batavia, IL, USA (28 June 2 July 2010) On the Positron Fraction and Cosmic-Ray Propagation Models
B. Burch and R. Cowsik
Washington University in St. Louis, St. Louis, MO 63130, USA
The positron fraction observed by PAMELA and other experiments up to ∼
100 GeV is analyzedin terms of models of cosmic-ray propagation. It is shown that generically we expect the positronfraction to reach ∼ . ∼
10 GeVis generated through spallation of cosmic-ray nuclei in a cocoonlike region surrounding the sources,and the positrons of energy higher than a few GeV are almost exclusively generated through cosmic-ray interactions in the general interstellar medium. Such a model is consistent with the bounds oncosmic-ray anisotropies and other observations.
I. INTRODUCTION
The recent observation of the positron fraction incosmic rays by PAMELA [1] has created much ex-citement because of its possible connection with theannihilation or decay of dark matter in the Galaxyor with a variety of astrophysical processes (see [2]for references to these discussions). These suggestionswere prompted by the recognition that the energy de-pendence of the positron fraction cannot be fitted bythe comprehensive propagation model (solid line inFig. 1) developed by Moskalenko and Strong (M-S)[3, 4]. In Fig. 1, we show the PAMELA observationsof the ratio, R , of the positron flux to that of the totalelectronic component in cosmic rays along with ear-lier observations [5–7]. The PAMELA measurementshave been called anomalous as they do not conform tothe predictions of the M-S model. Accordingly, newmodels of cosmic-ray propagation have been discussed(see references in [2]).General arguments based on cosmic-ray propaga-tion models indicate that the positron fraction shouldincrease at high energies and asymptotically reach avalue of ∼ . γ -ray astronomy has shown that cosmic rays generatedin the sources suffer nuclear interactions in the prox-imity of the sources [8–15]. This has a strong bearingon the models of cosmic-ray propagation in that ifa fraction of the B/C ratio observed in cosmic rays,especially at energies below ∼
10 GeV, is generatedin a dense cocoonlike region surrounding the sources,then the contribution from spallation in the generalinterstellar medium would have a flat or a weak de-pendence on energy. Such a model [16–18] is shownto fit the PAMELA observations and to be consistentwith the high degree of isotropy observed in cosmicrays at high energies [4, 19–21].
II. POSITRON FRACTION AT HIGHENERGIES
The asymptotic value of the positron fraction is es-timated by noting that cosmic rays observed near theEarth are accelerated in a set of discrete sources dis-tributed over the Galaxy [22, 23], which acceleratemostly electrons rather than positrons, as the Galaxyis made up of matter rather than antimatter. Duringthe diffusive transport, the electronic component suf-fers loss of energy due to synchrotron radiation andinverse-Compton scattering on the microwave back-ground and other photons. As this loss increasesquadratically with energy as bE , the spectrum of theelectronic component is sharply cut off at high en-ergies. Solutions to the diffusion equation [18, 23],which include the energy losses by electrons, yield aspectrum that cuts off as F e ( E, r n ) ∼ exp − br n EE x κ ( E x − E ) ! . (1)Here, r n is the distance to the nearest source, E x isthe maximum energy up to which the sources acceler-ate electrons, the diffusion constant κ ≈ cm s − ,and b ≈ . × − GeV − Myr − . Thus even for avery large value of E x , the directly accelerated elec-tron spectrum is cut off at E b ≈ κ/ ( br n ) ≈ n /kpc) . The cutoff in the spectrum at ∼ ∼
200 pc ofthe solar system. If this is taken to be the typicalspacing between the sources in our Galaxy, then weexpect about 10 sources in this disk within a radius of ∼
15 kpc [23]; accordingly, each of these sources needonly to generate a very small fraction of the cosmicray luminosity of the Galaxy, on the average.We do not expect the secondary electrons andpositrons to exhibit such a cutoff because, unlike thediscrete sources of primary electrons, the source func-tion for the secondary component extends from thenearest proximity to the solar neighborhood to far-offdistances. The secondary positrons and electrons are
C61
XVI International Symposium on Very High Energy Cosmic Ray InteractionsISVHECRI 2010, Batavia, IL, USA (28 June 2 July 2010)
FIG. 1: The positron fraction measured by PAMELA andearlier measurements are shown. Gradient drifts in solarmodulation may account for some of the difference in thedata at
E <
10 GeV [1]. The prediction of the positronfraction expected in the M-S model is shown as a solid lineand in the NLB model as a dashed line. generated through the π ± → µ ± → e ± decay chain,the pions being produced in high-energy interactionsof cosmic rays with the matter in interstellar space,both of which are distributed rather smoothly, with-out large overall gradients. Accordingly, the effects ofthe energy loss are less severe and the index of the sec-ondary electron spectrum at low energies is the sameas the source spectrum, which is the same as that ofthe nucleon spectrum [25–27]. At high energies, thesecondary spectra of positrons and electrons steepensby one additional power. F s ( E ) ∼ E − β f or E ≪ E c F s ( E ) ∼ E − ( β +1) f or E ≫ E c (2)where β = 2 .
65 and E c ∼ −
200 GeV.This spectrum of the secondary electronic com-ponent will progressively dominate over that gener-ated by the discrete sources. This implies that atvery high energies, the positron fraction simply corre-sponds to that in the production process in the high-energy collisions of cosmic rays. The fact that the p/n fraction in primary cosmic rays is greater thanunity favors the production of e + over e − , reflectingthe slightly greater production of π + compared with π − . Whereas the theoretical calculations [26, 27] yield F s + /F s − ≈ . −
2, the direct observations of µ + /µ − produced by cosmic rays in the Earth’s atmosphereyields F s + /F s − ≈ . E > ∼ R s ( E ) → F s + ( E ) F s + ( E ) + F s − ( E ) ≈ . . (3)We may expect that such a large value of R will bereached at E >
FIG. 2: The observed
B/C ratio [29–35] with the predic-tions from the M-S and NLB models are shown.
III. ENERGY DEPENDENCE ATMODERATE ENERGIESA. Residence Time of Cosmic Rays
There are two classes of models for cosmic-ray prop-agation with which to explain the measurements of theprimary and secondary nuclei in cosmic rays as showin Fig. 2. In the M-S model, the secondary produc-tion is distributed throughout the Galaxy, and the ob-served decrease with energy of the ratio of secondaryto primary nuclei is explained by an effective residencetime of cosmic rays in the Galaxy decreasing with en-ergy [3, 4, 36]. This decrease may be parameterizedbeyond a few GeV/n by τ L ( E ) ∼ τ L ( E + E ) − ∆ ≈ τ L ( E + E ) − . (4)where E ≈ ≈ . . ≤ ∆ ≤ . τ L ≈ . T , and E and E are expressed in GeV. Models of this class, which maybe approximated by a leaky-box (LB) model [25, 37],produce a nuclear secondary to primary ratio such asthat given by the dotted lines in Fig. 2. Note herethat the LB model approximates the predictions ofthe M-S model also shown in Fig. 2. The secondclass of models takes explicit account of significantsecondary production in dense regions in the vicinityof the primary cosmic-ray sources. Such a model maybe realized as a nested leaky-box (NLB) [16, 17]. B. Including Spallation in the Source Regions
In the NLB model, it is assumed that subsequentto the acceleration, the cosmic rays spend some timein a cocoon-like region surrounding the sources, inter-acting with matter and generating some of the secon-daries, mainly at low energies. Such interactions will
C61
VI International Symposium on Very High Energy Cosmic Ray InteractionsISVHECRI 2010, Batavia, IL, USA (28 June 2 July 2010) π ◦ → γ de-cay and could be observed by space-borne gamma-raytelescopes like FERMI [38, 39]. Since, according tothe arguments summarized in Section 2, the averageluminosity of a cosmic-ray source is rather low, theirgamma-ray emission will be detected only in somefavorable cases. The effective residence time in thecocoon, τ c ( E ), is energy dependent, with the higherenergy particles leaking away more rapidly from thecocoon. After they leak out of the cocoon into theinterstellar medium, the cosmic rays at all energiesup to several hundred TeV reside for an effective time τ G before they escape from the Galaxy. In the NLBmodel, the observed energy dependence of the nu-clear secondary to primary ratio is fit with an energy-dependent leakage time τ c ( E ) out of the cocoon andwith a leakage time τ G out of the Galaxy that is in-dependent, or nearly independent, of energy up to ∼ τ c ( E ) and τ G mayconveniently be parameterized as τ c ( E ) ∼ τ C E ǫ − δlogE , τ G ∼ constantτ N = τ c ( E ) + τ G . (5)Here, the lifetimes τ C , τ G , and τ N are in T unitsand take on values τ c ≈ .
24 and τ G ≈ .
08 when E is expressed in GeV, with the parameters ǫ = − . δ = 0 .
13. The cocoon should have a high den-sity so that adequate spallation might take place inthe short amount of time that the cosmic rays spendaround their sources. Circumstellar envelopes, darkclouds, molecular clouds, and giant molecular com-plexes are some of the candidates that may serve ascocoons. These have widely ranging densities, from ∼ cm − down to ∼ cm − [40], and the cosmicrays need to spend anywhere from 10 yr to 10 yr inthese regions to generate the requisite B/C ratio at ∼ − cm s − .Both LB and NLB models can provide adequatefits to the nuclear secondary to primary ratios ob-served to date, even though the difference betweenthem becomes progressively larger at higher energies.Whereas the LB models require an effective galacticresidence time, τ L ( E ), progressively decreasing withenergy, the NLB models fit the data on cosmic-raynuclei with a constant residence time τ G at high en-ergies. Accordingly, LB models predict cosmic-rayanisotropies that increase with increasing energy, inconflict with the observations [4, 19, 20]. In con-trast, NLB models predict constant anisotropies upto several hundred TeV, consistent with the obser- vations as shown in Fig. 3. To be specific, the ex-pected anisotropies, δ ( E ), are inversely proportionalto the effective residence time of cosmic rays in theGalaxy. Accordingly, the anisotropy in the NLBmodel δ NLB ( E ) is given by δ NLB ( E ) = τ LB ( E ) τ G δ M − S ( E ) ≈ GeVE ! ∆ δ M − S (∆ , E ) . (6)Here, δ M − S (∆ , E ) refers to the anisotropy in the M-S model calculated for the two values ∆ ≈ . ≈ . τ L ( E ) and τ G intersect in Fig. 2. When theirestimates are rescaled for the NLB model, accordingto Eq. 6, the expected levels of anisotropy becomeconsistent with the observational limits [4, 19–21].We can also directly estimate the anisotropy param-eter δ NLB using the standard formula in cosmic-rayliterature [4] δ NLB = 3 κ ∇ ρcρ ≈ κh c ≈ × − (7)where h ≈ ≈ × cm is the scale height ofthe distribution of the cosmic rays above the Galac-tic plane and κ ≈ cm s − is the diffusion con-stant of cosmic rays in the interstellar medium. Thisanisotropy is shown in Fig. 3 with an uncertainty of ∼ ∼ ∼ s n ( E ) represent the spec-trum of nuclei accelerated by the source by writtenas s n ( E ) = s n E − α . (8)Since in the LB model these nuclei have a lifetime τ L ( E ) and an interaction lifetime ( cn H σ int ) − , thespectrum of cosmic-ray nuclei in the interstellar spacebecomes F n ( E ) = s n ( E ) τ L ( E )1 + cσ int n H τ L ( E ) (9) ≈ s n ( E ) τ L ( E ) = s n τ L ( E + E ) − ∆ = s n τ L E − α − ∆ for E ≫ E . C61
XVI International Symposium on Very High Energy Cosmic Ray InteractionsISVHECRI 2010, Batavia, IL, USA (28 June 2 July 2010)
FIG. 3: Measurements of the cosmic-ray anisotropy fromvarious compilations [4, 19–21]. Also plotted are the pre-dictions from models in Moskalenko and Strong (MS) [4]and the results from Eq. 6 (CB). The gray region showsthe predicted anisotropy from Eq. 7.
Here ( cσ int n H ) − is the effective mean free path forthe loss of cosmic rays at a particular energy throughnuclear interactions. In order to match the observedspectrum of cosmic rays with F n ( E ) = F E − β in theLB model, we need to set α + ∆ = β ≈ .
67. Thus s n ( E ) = s n E − β +∆ ∼ E − . . (10)The calculation of the source spectrum in the NLBmodel, s nc ( E ) = s c E − ξ , is a two-step process. Thespectral density inside the cocoons is the product of s nc ( E ) and the leakage lifetime inside the cocoon: F c ( E ) = s nc ( E ) τ c ( E ) = s nc ( E ) τ c E ǫ − δ log E . (11)These leak out into the interstellar space at a rateinversely proportional to the leakage lifetime from thecocoon so that F n ( E ) = F c ( E ) 1 τ c ( E ) = s nc ( E ) = s c E − ξ . (12)Thus to match with the observed spectrum of cosmicrays with F n ( E ) ∼ E − β we need ξ = β . This meansthat the observed cosmic rays have spectra identicalto that accelerated by the sources, especially at highenergies where losses due to ionization and nuclearinteractions are small in the source regions. C. Derivation of the Positron Fraction
In assessing the positron fraction in the NLB mod-els, we note that the secondary nuclei, such as B ,which are generated by the spallation of primary nu-clei like C , have the same energy per nucleon as theirprogenitors. By contrast, positrons carry away, on theaverage, only about 5% of the energy per nucleon oftheir nuclear progenitors [3, 26, 27]. This implies that even though a significant amount of B is generatedthrough spallation within the cocoon, very little pro-duction of positrons at energies beyond 5 GeV occursthere. This is because the progenitors of the positronswith E ≥ sourcefunction for the positrons to be the same – it is simplyproportional to the product of the observed spectrumof the cosmic-ray nuclei and the density of the inter-stellar medium and has the spectral form s E − β . Be-low ∼
100 GeV, where the radiative energy losses arenot significant, the observed positron fluxes would bethe product of this source function and the residencetime of cosmic rays in the Galaxy, τ L ( E ) or τ G , asrelevant to the model under consideration.The calculation of the positron ratio in the twoclasses of models is straightforward when we note thatits source spectrum S + G ( E ) in the interstellar mediumis generated through nuclear interactions [26] and hasa nearly identical spectrum to that of the parent nu-clei, F E − β , except that it is shifted down in energyby a factor η ≈ .
05 and multiplied by the rate ofnuclear interactions S + G ( E ) ≈ σ in n H F η β − E − β . (13)Here, σ in is the inclusive cross section for the produc-tion of π + , which carries off a fraction η of the energyper nucleon of the primary cosmic-ray nucleus. Thefactor η β − in Eq. 13 accounts for the shift in theenergy and the change in the energy bandwidth whentransforming from the spectrum of the primary nucleito that of the positrons. The source function is thesame for both the M-S and NLB models. In the NLBmodel, there is an additional small contribution S + c due to positron generation from nuclear interactionsin the cocoon. Taking the expression for the spectraldensity for the nuclei in the cocoon from Eq. 11 shows S + c ( E ) ≈ cσ in n Hc s nc η − ǫ + β − δlog ( E/η ) × E ǫ − β − δlog ( E/η ) . (14)However, this contribution is entirely negligible be-yond a few GeV. Therefore the steady state spectra F + LB ( E ) and F + NLB ( E ) are essentially given by theproduct of the source function and the effective life-time of the positrons in the Galaxy. At energies below ∼
100 GeV the radiative losses are small and the effec-tive lifetimes in the two models are essentially givenby the leakage lifetime τ L ( E ) or τ NLB ≈ τ G , respec-tively. Thus the positron spectra in the two modelsare given by F + LB = S + G ( E ) τ L ( E ) ∼ E − β ( E + E ) − ∆ ∼ E − ( β +∆) ; (15) F + NLB ( E ) = S + G ( E ) τ G ∼ E − β . (16) C61
VI International Symposium on Very High Energy Cosmic Ray InteractionsISVHECRI 2010, Batavia, IL, USA (28 June 2 July 2010) F + LB and F + NLB by the spectral intensities of the total elec-tronic component in cosmic rays. A recent compila-tion of the observations of the total electronic com-ponent can be found along with a smooth fit to thedata that includes a slight enhancement in the inten-sities below ∼ R NLB ∼ S + ( E ) τ G /F ± ( E ) ∼ E − . /E − . ∼ E . (17) R M − S ∼ S + ( E ) τ L /F ± ( E ) ∼ E − . E − ∆ /E − . ∼ E − . (18)We see in Fig. 1 that the NLB model shows thepositron fraction increasing with energy at high en-ergies and the M-S model shows a declining positron fraction at high energies. IV. CONCLUSIONS
We see that the nested leaky-box model pro-vides a satisfactory fit to the PAMELA observations.This analysis obviates the need for exotic sourcesof positrons, suggested by comparison between thePAMELA data and the M-S propagation model, andshows that the data may be accounted for by NLBpropagation models. Since NLB models also relievethe anisotropy problem encountered in the LB/M-Sclass of models and qualitatively accommodate the ob-servations of π gamma rays from regions near cosmic-ray sources, we conclude that the rising positron frac-tion observed by PAMELA is the natural result ofcosmic-ray interactions in the interstellar medium. Acknowledgments
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