Structures in the cosmic ray energy spectra
aa r X i v : . [ a s t r o - ph . H E ] O c t Structures in the cosmic ray energy spectra A. D. Erlykin a,b and A. W. Wolfendale b ( a ) P N Lebedev Physical Institute, Moscow, Russia( b ) Department of Physics, Durham University, Durham, UKSeptember 21, 2018 Abstract
All the components of Cosmic Rays (CR) have ’structure’ in their energy spec-tra at some level, ie deviations from a simple power law, and their examination isrelevant to the origin of the particles. Emphasis, here, is placed on the large-scalestructures in the spectra of nuclei (the ’knee’ at about 3 PeV), that of electronsplus positrons (a shallow ’upturn’ at about 100 GeV) and the positron to electronplus positron ratio (an upturn starting at about 5 GeV).Fine structure is defined as deviations from the smooth spectra which alreadyallow for the large-scale structure. Search for the fine structure has been per-formed in the precise data on positron to electron plus positron ratio measuredby the AMS-02 experiment. Although no fine structure is indicated, it could infact be present at the few percent level.
Starting with the ’all-particle spectrum’ it has been known for many decades that asimple power law does not represent the spectrum, rather, the power law exponentincreases at several PeV ( the ’knee’) and falls again at several EeV (the ’ankle’). Bothhave been examined in great detail, but without a concensus as to their detailed origin.Here, as for the all-particle spectrum, we restrict attention to the knee and its’ apparent’sharpness’.Turning to electrons, their energy spectrum has a distinctive shape, with increasingagreement that, when plotted as logIE vs logE , there is a flattening at about 100GeV and a downturn commencing at about 1000 GeV. Interest centres on the role ofa single source, a pleasing result in view of the lukewarm reaction to our early ’SingleSource Model’ for the knee [1]. Surely, the single source responsible for the flatteningin the electron plus positron spectrum has to be different from that responsible for theknee, but the principle is the same: a single source is providing structure. Corresponding author: tel +74991358737E-mail address: [email protected]
In view of its’ history (it was discovered in 1958 [8]) this feature is the most studied. Inour detailed analysis using the developed SNR model [9], a model in which the standardFermi acceleration mechanism was used for acceleration by the SNR shock, we derivedthe age-distance diagram for the SNR which could be responsible for the formation ofthe knee, ie our ’single source’ [10]. The 95% confidence level encompassed distance:250 to 400 pc and age: 85 to 115 ky. Surprisingly, perhaps, the area occupied by theage-distance contours for an SNR giving a knee sharper, or more pronounced, than thatobserved is not much bigger than this, assuming that the energy injected by the SNRinto CR is the standard 10 erg. The mean number of SNR expected in the requiredage-distance region is about 0.01-0.02, ie the probability to find SNR within this region,which gives the sharp knee-like structure, is about 2%.If we relax the requirement of the sharpness and look for somewhat smoother devi-ations from the simple power law then the examination of simulated spectra presentedin [11] ( anomalous diffusion with α = 1 ) gives the probability of such features ofabout 28%. In view of the difficulty in both measuring the knee and in interpreting it( diffusion characteristics, etc ) we see no reason to doubt the conclusion that a singlesource is responsible. In [12] we presented a model in which, as for protons, electrons are accelerated by theshock in the SNR, the SNR then being distributed in the Galaxy randomly in spaceand time. The well-known steeper energy spectrum for electrons than for protons wasexplained by way of an energy-dependent Mach number for the shock. It must beremarked, however, that this feature is not yet fully understood.Inspection of our model’s prediction for the shape of the electron energy spectrum[12] shows a wide range of spectra with a median intensity ( logIE ) which falls slowly2ith energy to about 100 GeV, beyond which it falls rapidly. This is in contrast withthe observed electron plus positron spectrum which rises slightly or flattens at about100GeV and falls with modest rapidity above 1000 GeV. It should be noted that theATIC spectrum [3] has remarkable structure in the range 100 to 1000 GeV, but this isnot shared by other measurements, eg Fermi LAT [5]. Examination of [12] indicates thatabout 24% of the predicted spectra have the necessary or stronger large-scale spectralstructures. A number of experiments have shown an upturn in the positron fraction, as mentionedalready. The most precise are due to : PAMELA [2], Fermi-LAT [5] and AMS-02 [6].Many authors have suggested that a pulsar is responsible , but, in a very recent work[13] , we have proposed the SNR presumed to be the predecessor of the pulsar Geminga.In this case, the positrons come from radioactive ejecta from the SN; the positrons arethen accelerated by the SNR shock in the usual way. An advantage of this mechanismis that the acceleration efficiency of the near-1 MeV positrons is very high.In a manner similar to that for the origin of the knee ( § D and age T : 250 < D <
320 pc and 170 < T <
If anti-protons are generated and accelerated in SNR, by the interactions of protonsand heavier nuclei with the interstellar medium (ISM) within the remnant, then theirspectrum should show large-scale structure, in the form of an upturn in the ¯p/p ratio.Interestingly, neutrons produced in ¯n,n - pairs in the interactions will augment the ¯pflux. The neutrons (and ¯n) have the advantage of escaping the magnetic trapping,which may be strong in the early remnant, before decaying into p and ¯p.The only measurements extending as far as hundreds of GeV are those from PAMELA[14], which finish at 180 GeV, although the errors are large at the high energies. Themeasured ratio at 100 GeV is (1 . ± . · − , but this includes a rather uncertainbackground contribution [14] so that a single source contribution, or upturn in the ratio,cannot yet be determined. All that can be said at present is that the ¯p/p ratio hintsat a flattening above 10 GeV, which, if confirmed and after subtraction of the rapidlyfalling background and accounting for a steep proton spectrum in the denominator of¯p/p ratio, would suggest the onset of a finite single source contribution.Its absence would suggest that ¯p are not produced by the local SNR, unlike thepositrons which we hypothesise to come from the radioactive SN ejecta. At present theexpected ’cross-over energy’, where SNR-generated ¯p equal background, is not clear.3 Fine Structure in the Positron Fraction
By ’fine’ rather than ’large-scale’ structure we mean anything in the spectrum, orparticle (positron) fraction, that is over and above the first order fit, suggested by theAMS-02 collaboration [6]. This fit was the sum of two simple expressions: a power lawfor those positrons coming largely from CR interactions in the ISM and an exponentiallymodified power law for positrons from a local source. The energy range is from 1 to 300GeV, ie logE, GeV : 0 to 2.5. Below 10 GeV , solar modulation is important and thuswe divide the data into two parts: logE, GeV =0 to 1.25 and logE, GeV = 1.25 to 2.5.The lower part might show fine structure due largely to solar modulation, whereas thehigher energy part might indicate Galactic effects, associated with the finite number ofsources contributing to the CR flux.In an attempt to determine at least an upper limit to the fine structure in thepositron fraction we have examined the ’precise’ AMS-02 data in some detail, as follows.Fits to the ratio of the measured positron fraction to that obtained by its fit with theAMS-02 suggested function were made for the two halves of the data for various degreesof polynomial function ’n’ from 2 to 9 with particular emphasis on 2 and 9 themselves.Thus, we are searching for fine structure within the already allowed - for large-scalestructure. Clearly, it will be small, otherwise it would have been commented on already.Figure 1:
The ratio of the positron fraction measured in the AMS-02 experiment to its fit by theAMS-02 suggested function. Errors of the ratio are statistical. Full lines in both halves of the energyrange show the weighted fit of the ratio by the 2-degree polynomial function: a + a X + a X , where X = logE . The quality of the fit is shown by the values of the reduced χ . Dashed lines indicate thestructure, which is of the same shape of the 2-degree polynomial, but has P ( χ ) = 0 . Figure 1 shows the results for n = 2 (full lines) and the derived chi-square valuewith its’ significance for both halves of the studied energy range. Indicated errors of the4atio are statistical. Systematic errors are weakly energy dependent and cannot haveirregular behaviour with the energy. In this analysis we did not take them into accountsince they cannot help to reveal the possible fine structure of the energy spectra.The weighted polynomial best-fit has a reasonable significance at lower energiesand the very large value of the probability at high energies indicates that even thestatistical errors shown may have been overestimated. The limit for a 5% probability offit is shown by the dashed lines. The 5% level is the usual value for acceptance of a fitas being just non-allowable. It will be noted that the ’fine structure’ of this shape couldreach about 8% maximum at least in the high energy part of the range and still not bedis-allowed by the data. We presented these dashed lines just as examples illustratingthe non-negligible probability of deviations from the AMS-02 fit at the level of the fewpercent.Figure 2 shows the situation for n = 9. The best-fit has again a very high probabilityin both halves of the range. This fit has some interesting features although none is asyet significant. The 5% probability limits are shown by the dashed lines. They showthat the amplitude of ’undulations’ in some restricted regions are still allowed to be ashigh as 17% by the data. Again the dashed lines here are given as possible examplesof the fit which still have the allowable probability.Figure 2:
The same as in Figure 1, but the ratio is fitted by the 9-degree polynomial function. Theposition of the minimum in the ATIC electron plus positron energy spectrum [3] is indicated by thearrow. Its approximate coincidence with the upward excursion of the positron fraction from the regularmodel calculation is an interesting feature. However, due to the low statistics, it should be regardedrather as a hint for the possible fine structure.
As remarked already, the lower energy region is the province of solar modulation, whichis known to be dependent on the charge sign of the CR and on the time of observation,5y way of the solar magnetic field polarity [eg [15]]. The difference between the positronfraction measured by AMS-01 [16] and PAMELA [2] in the different periods of solaractivity can be caused by this phenomenon [7].Although precise measurements of AMS-02 in the lower energy region are quiteconsistent with their fit by two-term function ( background plus single source, χ /ndf =28 . / , P ( χ ) = 0 .
586 ) it is worth-while to estimate the upper limits of the possiblefine structure consistent with these measurements. As can be seen in Figure 1 themaximum contribution of the 2-degree polynomial-like structure which still has the 5%confidence level of consistency with data points is about 0.011. The same estimate forthe narrower structures which appear in the 9-degree polynomial fit gives a maximumcontribution of only 0.035.It is seen that the present high precision measurements does not allow the presenceof fine structure greater than about 3-4% in the GeV energy region due to solar windmodulation. Later measurements should be of adequate accuracy to enable the effectof time-dependent modulation to be studied, using factors such as those in [17].
We now discuss the higher energy region. It is here that the ATIC results [3] haverelevance, in that the measured electron plus positron spectrum has considerable struc-ture. The energy of about 220 GeV at which an apparent minimum in the electron+ positron intensities appeared is indicated in Figure 2. It would have been expectedthat, if positrons were uncorrelated, there would have been a maximum at this energyin the positron fraction. However, statistical errors of the measured positron fractionare too high to confirm the anti-correlation between the ATIC minimum of the elec-tron plus positron intensity and the AMS-02 maximum of the positron fraction. OtherATIC minima are at higher energies which do not overlap with the range of AMS-02measurements.
Both SNR and pulsars are candidates for CR origin and a distinction between them isnot a trivial problem. However, spectral structure ( or fine structure ) can be useful,particularly for the electron component which, because of energy losses, comes predom-inantly from ’local’ regions ( within a kpc or so ) and thus from a smaller number ofCR sources.A comparison can be made between our electron spectra from the random SNRmodel [12] and the prediction for pulsars [7, 18]. It is immediately apparent that theprediction for SNR are ’smoother’ than those for pulsars. Over the range 100-2000 GeV,the SNR model has rarely excursions in ’intensity’ ( logIE ) bigger than 30% whereasfor pulsars there are four excursions with a mean of 40%.The reason for the difference is self-evident. For SNR, in the model, at least, uniqueenergy spectra are emitted from the SNR when the SNR ’bubble’ bursts, and thespectral structure arises from propagation effects alone, ie contributions from SNR ofdifferent ages at different distances. For pulsars, the flatter ’emission spectra’ have6aximum energies depending on their ages, with, conventionally, sharp cut-offs, andthese fluctuations are added to those due to propagation.Comparing the structure for electrons and protons, it is useful to examine the rangeof intensities as a function of energy ( for the same propagation model ) from our work[11, 12], which relates to 50 independent samples. A large range suggests more structurethan is the case for a small range. It is found that the ranges for electrons and protonsare similar to about 1000 GeV, above which the range for electrons increases morerapidly. A similar result is apparent for the degree of ’oscillation’ of the spectra - thatfor electrons is singularly large in the next decade of energy. However, the ’degree’ ofoscillation is hard to quantify and this is why the range of intensities is considered. Thisbehaviour follows from the fact that electron losses increase as the energy squared andthe transit time from source to observer; the actual spatial and temporal distributionof nearby sources is therefore critical. We have analysed the available results on the energy spectra of CR particles fromthe standpoint of the ’structures’ in their energy spectra. Large-scale structures areregarded as differences from simple forms, which point to the existence of a single SNR.The model adopted is that introduced by us [11, 12] involving CR origin from randomlysituated SNR from which CR diffuse. The probability of seeing such structure is ∼ ∼
20% for electrons and positrons. For anti-protons, measurements ceaseat the energy at which structure might be expected to show itself.; more extendedmeasurements might show an upturn in the ¯p/p ratio.Fine structure is defined as deviations from the smooth spectra which already allowfor the large-scale (single source) structure. The precise positron fraction data [6] aretaken as an example. The datum is taken as the two components: background plussingle source. The polynomial fits are taken as examples: n = 2 and n = 9, and the dataare divided into equal energy ranges: logE, GeV = 0 − .
25 and 1 . − .
50. Althoughno fine structure is indicated , it could be present at the few percent level. For the lowerenergy band solar modulation effects, which are charge dependent, should be detectablewhen temporal and somewhat better statistical data are available.For the higher energy range, models are not yet available for the fine structureexpected as a result of detailed source mechanisms (eg SNR or pulsars) and irregularitiesof CR diffusion, but these will come. Again, although the positron fraction data arestatistically precise, fine structure could be present at a few percent level ( up to 8%for n = 2 and up to 17% for n = 9 as an example ).
Acknowledgements
The authors are greatful to the Kohn Foundation for financial support.