7/27/2019 Tang & Ng.pdf

1/17

Calculation of the equilibrium antiproton spectrum

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

1983 J. Phys. G: Nucl. Phys. 9 227

(http://iopscience.iop.org/0305-4616/9/2/015)

Download details:

IP Address: 128.252.102.225

The article was downloaded on 26/06/2013 at 17:54

Please note that terms and conditions apply.

View the table of contents for this issue, or go to thejournal homepage for more

ome Search Collections Journals About Contact us My IOPscience

http://iopscience.iop.org/page/termshttp://iopscience.iop.org/0305-4616/9/2http://iopscience.iop.org/0305-4616http://iopscience.iop.org/http://iopscience.iop.org/searchhttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/journalshttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/contacthttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/myiopsciencehttp://iopscience.iop.org/contacthttp://iopscience.iop.org/page/aboutioppublishinghttp://iopscience.iop.org/journalshttp://iopscience.iop.org/collectionshttp://iopscience.iop.org/searchhttp://iopscience.iop.org/http://iopscience.iop.org/0305-4616http://iopscience.iop.org/0305-4616/9/2http://iopscience.iop.org/page/terms

7/27/2019 Tang & Ng.pdf

2/17

J. Phys. G : Nucl. Phys. 9 (1983) 227-242. Printed in Great Britain

Calculation of the equilibrium antiproton spectrumL C Tan and L K N gPhysics Department, University of Hong K ong, Hong Kong

Received 20 September 1982

Abstract. A thorough discussion of the equilibrium antiproton spectrum prediction for theusual leaky-box model, the closed-galaxy model of Rasmussen and Peters and the revisedclosed-galaxy model of Peters and Westergaard is presented. It is found that the effect of th eantiproton non-annihilation inelastic interaction is significant even for the leaky-box model, inthat the predicted equilibrium antiproton spectrum does not show a strong low-energy cut-offsimilar to that existing in the antiproton pro duction s pectrum due to kinematic limitations.

1 . IntroductionRecently, a serious challenge to the usually adopted picture of the origin and propagationof primary cosmic rays has arisen from the contradictory observations of the interstellarantipr oton (0) flux. The statistically significant da ta of G olden et al(1 979) and Bogomolovet a1 (1979 ) have shown th at, a t p energies of several GeV , the measured p flux is evidentlyhigher than that predicted by many author s (see the review by Stephens 1981a) using theleaky-box model with escape parameters derived from experimental data on cosmic-raynuclei. It is also noted that an apparently large p flux was predicted by Badhwar et a1(1975 ) which a ppea red to be consistent with the d at a measured above. Hen ce it is essentialas a first step to find out the reasons for the differences in the various theoreticalpredictions. In so doing, we have fo und tha t the prediction of Badhwar et al is in erro r dueto the inaccurate nuclear physics data used and a computational mistake (Tan and Ng198 la) . This is also confirmed by the recent work of Gaisser a nd M auger (1982).Whilst a number of authors (e.g. Cowsik a nd G aisser 1981, Gaisser et a1 1981,Protheroe 1981, Stephens 1981b) have been attempting to reconcile with the datameasured above by using other models such as the modified nested leaky-box model, therevised closed-galaxy model, etc, a new measurement of interstellar ps performed byBuffington et a1 (198 1) at p energies of a few hundred MeV has brought a more seriousdifficulty to its interpretation. The antiproton to proton ratio, p/p, reported by Buffingtonis only slightly less than tha t pre viously found a t higher energies, althoug h it is comm onlyadmitted that the predicted p spectrum should have a characteristic shape with a low-energy cut-off due to the kinematic properties of p production (Gaisser and Levy 1974).This shar p divergence between observ ation and prediction has compelled some authors togive up the traditional concept that interstellar ps ar e generated in collisions of the primarycosmic rays with the interstellar medium (ISM) in our galaxy, and to replace the modelsbased o n the traditional conc ept by more exotic ones such as the antimatter galaxies a ndbaryon symmetric cosmology model (Stecker et al 1981) and the exploding primaevalblack-hole model (K iraly et al 198 1).0305-4616/83/020227 + 16$02.25 0 983 The Institute of Physics 227

7/27/2019 Tang & Ng.pdf

3/17

228 L C Tan a n d L K NgIn or der to judg e whe ther the existing traditional con cep t can still be used to explain thedata observed, a more exact calculation of the equilibrium p spectrum prediction for themodels is essential. In the existing model calculations, there are a number of points whichindeed need to be carefully examined (see the recent comments of Gaisser and Mauger1982), especially at very low energies of several hundred MeV. Moreover, even if onewants to explore any other possible p origin, an exact knowledge of the backgroundsecondary p flux is still nec essa ry.Hence, as a second step in our interstellar p investigation, we have p erformed a m oreexact calculation on the equilibrium p spectrum, starting from an improved para-metrisation of th e p invariant cross section (Tan and Ng 1982). The calculations for theeffect of p ionisation loss, annihilation and no n-annih ilation inelastic interactions h ave beentreated as exactly as possible. We have found that the effect of the p non-annihilationinelastic interaction is very significant. Even for the usual leaky-box model, our predictedlow-energy p flux is evidently higher than th at repor ted previously by variou s au tho rs. As aconsequence, the so called kinematic cut-off does not exist in the equilibrium p spectrum.

In this pape r we will repo rt o ur impro ved calculation of the equilibrium p spectrum.

2. Production rate of interstellar antiprotons

2.1. New param etrisation of p inva riant cross section in p-p collisionsUsing a new scaling variable, we have reparametrised the p invariant cross section in p pcollisions (see Tan and Ng 1982). For the sake of completeness, we present the mainformu lae here.

The radial variable XR s expressed asXR = $ / E m k , p (2.1)

where E denotes the total energy of a particle, the asterisk expresses a qua ntity measuredin the centre-of-mass system (CMS), the subscripts p and p refer to quantities belongingto a ntiprotons a nd their pare nt pro tons, respectively, andE ; ~ ~ , ~(s-lii:c4 + m;c4)/2$, (2.2)

$ eing the energy available in the C MS , mp the proton m ass and fix 3m, t he minimummass of the undetected particle system consistent with q uantum -num ber conservation.Th e rad ial-scaling limit of the p invariant cross sections, ( E d30/d3p)RS,p,s reached by4- 0 GeV a nd can be parametrised aswhere

and pt, is the transverse mom entum of p.

7/27/2019 Tang & Ng.pdf

4/17

Calculation of the equilibrium antiproton spec trum 229W e have introduced a new scaling variable,

AX, =XR -XRdnwhere

2 4 112 E *X R m i n =(P?,pC2 + m pc / ma x , p .The ratio of the low-energy (LE) invariant cross section(E d3 /d3p)LE,

to its radial-scaling limit ca n be expressed a s( E d3a/d3p)LE,p/(E d3a/d3p)RS,p- 1

= 6.25 x x (exp(-0.592Q) + 49 3 exp(-5.40Q))X (exp(6.08 + 2 . 5 7 h x ~ 7 . 9 5 h x i )- 1) eXp[3 .00hX~(3 .09- Q ) ] (2.6)

where Q = &- 4m ,c2, 4m,c2 being the & value at p production threshold.According to our result, the p invariant cross section extrapolated to near the pproduction threshold is evidently higher than that previously obtained (e.g. by Stephens198 IC).2.2. Calculation o f p production ra teFrom the p invariant cross section (Ed3a/d3p)p t is easy to calculate the p differentialproduction cross section in p p ollisions, i .e.

ep maxdap (Ep ,EP)/dEp = 2nlpminI,,@ d3dd3P ) , de, (2.7)where 8, is the an gle of p emission in the labo ratory system. Thus the p production rate forthe collision of cosmic- ray proton s with interstellar hydro gen a tom s is

where j , is the differential flux of interstellar protons.In order to calculate the integral in equation (2.7), a suitable coordinate system inmomentum space must be selected. Here we use the polar coordinate system (pp e,)following Stephens (1981~). hus for a given E , all variables in equation (2.7) can beexpressed as a function o f pp a nd Op only , i.e.

(2.9)where PCc and yc are respectively the velocity and Lorentz factor of the CMS. Further, weonly need t o determine the limits of integration for 0, as follows.Fr om kinematic considerations, we have

where ppc s the p velocity andtan 0, =s in O,*/[yc(cos ; +pc/p$)]. (2.11)

7/27/2019 Tang & Ng.pdf

5/17

230 L C T a n a n d L K N gFrom equations (2.9), (2.10) an d (2.1 l), we find tha t un der kinematic limitations thepermitted range of 0, is a function of E,. Fo r a typical case of Ep= 24.0 GeV, which is theminimum median energy of Ep for p production, the allowable region of 0, is given by themeshed region in figure 1. The uppe r a nd lower bound aries of this region a re determined bythe curves of XR= 1 and of XR =0.322 at Ep= 24.0 GeV) respectively. Hence ,substituting X , = 1 into equation (2.10), we obtain e,,, from equation (2.1 1). W eintegrate equation (2.7) from 0, = 0 to 0, but with the restriction that (E d3u/d3p), = 0for X, 60 GeV (2.12)

E D ( G e V I

Figure 1. The kinematic limitation on 8, related to XR a t E , = 2 4 . 0 GeV (i.e. XR,,,,"=0.322). T he full curves are iso-XR lines, the chain curves are iso-O$ lines and the brokencurve is the upper limit of integration of equation (2 .7) given by Stephens (19 81 ~) .

7/27/2019 Tang & Ng.pdf

6/17

Calculation of the equilibrium antiproton spectrum 23 1(in units of m-' s r - ' s - ' GeV -I). Fr om equation (2.8), we have obtained qpH as shown bythe curve T I in figure 2. It com pares favourably with the estimations of Stephens (198 IC,the curve SX) and Protheroe (1 98 1, the curve P). All these curves a re very consistent withone another at both ends, i.e. at 6 kinetic energies To greater than 10 GeV and less than0.5 GeV. The difference lies only in the middle range at energies of a few GeV. It issurprising to see that the peak value of SX is greater than that of TI, since thecorresponding.values of ( E d3a/d3p)p do not show this tendency a nd the sam e jps used inboth calculations. W e have checked th e calculation procedure of Stephens and found thatthe difference originates from his wrong choice of the upper integration limit expressed bythe bro ken cu rve in figure 1, which is consistent with ou rs only at 0; of about 90. We haverecalculated qPH using his p invariant cross section formula and this is given by the cu rveSC in figure 2. SC is lower than TI and closer to P. The reason that SC and P are lowerthan T I is due to inadequate extrapolation of the corresponding p invariant cross section inthe very low-energy range.Further, we also try another, simpler interstellar p spectrum

jpI1(Tp(GeV))=2.0 x 104T;2.75 (m-* sr-l s- ' GeV-l) (2.13)fo r T p3 10 GeV. On a log plot this spectrum is a straight-line extrapolation of the high-energy segment of jpI.he corresponding qPH is shown in figure 2 a s the c urve TIL

Figure 2. The p production rate, qPH, against its kinetic energy To. S X , f rom Stephens( 1 9 8 1 ~ ) ;SC, recalculated by us using Stephens' formula for th e p invariant cross section; P,from Protheroe (1981); TI, this work using Stephens' proton spectrum (equation (2.12)); TII ,this work using a proton sp ectrum with yp= 2.75 (equation (2.16)).

7/27/2019 Tang & Ng.pdf

7/17

23 2 L C T a n a n d L K N gAfter considering the equal abun dance of the antineutrons produced which deca y to psand the co rrect ion factor &sM due t o the presence of nuclei other than proton s in thecosmic rays and atom s other than hydrogen in the ISM, we obtain the production rate ofinterstellar ps:

qp = 2 h M qpH (2.14)qp can be parametrised asqp(T,(GeV))=&sMcl TFc Z/(l+c, TFCB )m P2 s-l G eV-l g- cm2) (2.15)

fo r Tp> O . l GeV wherec, = c3 + c4 In T, + c5 ln 2 Ti,c, = c6 + c7 In Tp+ c8 In2 Ti,.

For jp Ir jpII,he values of c1 c8 are listed in table 1.Meanwhile, we have also calculated EP, he median energy of parent protonsresponsible for p production with kinetic energy T,. The variation of EP s a function of

T, is shown in figure 3. It is found tha t 8, ossesses a minimum value of about 24.0 GeVat T, -0.5 GeV. The existence of a minimum Ep is a clear reflection of kinematiclimitations, i.e. ps with smaller Tp must be produced in p p ollisions with higher parentproton energies, and the smaller value of Tp is due to the backward direction of p flight inth e C M S . E , can also be parametrised asEp(Tp (GeV ))= 15.8Ti0.26 6 9 .23TF956 GeV) for T, >0.1 GeV. (2.16)SinceE , is always greater than 20 GeV, from o ur discussion of CISM (Tan and N g 198 IC) tis found tha t &M = 1.14.

3. Calculationof the equilibrium p spectrum3.1. Basic ormulaAccording to the traditional concept that ps are produced in collisions of cosmic-raynuclei with the ISM, the equilibrium p intensity N, is calculated by solving the continuityequation under the assumption of a quasi-steady state,

where A, is the escape mean length of the cosmic rays, AY1 is the total p inelasticinteraction mean length including the p annihilation length A;, I is the ionisation loss rat eTable 1. Th e coefficients c 1 - c ~ in the parametrised p production spectra.j , C I C l c3 C4 c5 c6 C l C 8j, , 7.93 2.57 28 1 4.46 78.5 3.74 -0.529 0.0198jpl1 7.40 2.54 209 3.44 63.3 3.68 -0.545 0.0224

7/27/2019 Tang & Ng.pdf

8/17

Calculation of the equilibrium antiproton spectrum 233

rF G e V )Figure 3. Variation of the median energy E , of parent protons responsible for producing psat a kinetic energy T , with T p .

and Q p is the total p production rate including the decelerated ps after their non-annihilation inelastic intera ction with th e ISM.Thus

where qp is taken from equation (2.15), A; is the mean length of p non-annihilationinelastic interaction and Wp(Ep, 6) /dEp is the energy distribution of ps after their non-annihilation inelastic interaction.Thus from equation (3.1) we have (see Daniel and Stephens 1975)

and the interstellar p fluxjp &,Np/471. (3.4)

The calculated j p s dependent on the assumed shap e of A e Before discussing I\,,we haveto estimate the physics pa ram eter s involved in our c alculations.3.2 . Nuclear physics parameters3.2 .1 . Ionisation loss rate, I. According to Ginzburg and Syrovatskii (1964), theionisation loss rate of a particle with cha rge Z e and velocity /3c is

1 . 5 2 4 ~ 0-4z2n ( ( p ) )I= 1 1 . 2 + l n- p 2 (MeVs- )P 1 -p

(3.5)where n is the number of hydrogen atoms ~ m - ~ .e have reduced I to unit length of the

7/27/2019 Tang & Ng.pdf

9/17

234 L C Tan a n d L K NgISM by adopting NHe/NH= 1/11 (Ta n and Ng 1981c), where N, is the number density oftype-i atoms in the I S M .3.2.2. Total p inelastic interaction cross section, IS?'.rom our adopted IS M components,the relation between the interaction me an length A and the corresponding interaction crosssection cs ca n be expressed as

A(a(mb))= 1.79 x 103/a(g cm-*). (3.6)are shown in figure 4.

(3.7)

The experimental data compiled by Flaminio et a1 (1979) forOu r parametrised result for (the curve TEL in figure 4) can be expressed asa?'(T,(GeV))= 24.7(1 + 0.584T;0.115 + 0.856T;0.5") (mb)

for T, 2 0 . 0 5 G e V.3.2.3. Proton inelastic interaction cross section, 0;. Stephens (198 1c) assumed thatab =ab, which is a good approximation at high energies. Hence, as a start, we parametriseab from the data collected by Flaminio et a l( l9 79 ) and by H ayakaw a (1969). Our result isoL(E,(GeV)) =O;(HEL)

= 32.2[ 1 + O.O273U+ 0 .01 U2B (U ) ] (mb) for Tp>3 GeVd,(T,(GeV))=ab(HEL)/(l + 2.62 x 1 0 - 3 T p 9 ( m b ) f o r 0.3

7/27/2019 Tang & Ng.pdf

10/17

Calculation of the equilibrium antiproton spectrum 235where U(E ,(GeV ))=ln(& E,) and C,(T,(GeV))= 17.9 + 13.8 In T, +4. 41 ln2 T,. Thedefinition of f lu) s given in equation (2.3) and the expression for U~(HEL) s taken fromHillas (1979). O ur param etrised result is shown in figure 4 as the curve TPI .3.2 .4 . i , annihilation cross section, o r . The assumption that 4 oi is consistent withexpecta tions from qu ark models on ly at high energies (Proth ero e 198 1). A t low energies4should be substantially larger than ob for the sam e reason that the K - p inelastic interactioncross section is larger than that for K + p nteractions (i.e., presence of antiquarks in thebeam). Hence at T, 6 0 GeV, the measured data on oy have to be included in theestimation for 0;. hus we have

(3.9)el -,- , 0;.The data on 0 can be found from the compilations of Flaminio et a1 (1979) andRushbrooke and Webber (1978) up to about 10 GeV of T,. The parametrised sha pe of 0;is

(3.10)for 0.1

7/27/2019 Tang & Ng.pdf

11/17

236 L C T a n a n d L K N g(3.3) with an assumed value of Ae. The assumption of Ae is related to the propagationmodel of the cosmic rays. The predicted p flux for some existing propagation modelsworked out will be presented in the next section. Here we shall briefly describe theprocedure of our computation.The procedure adopted to solve the simultaneous equations (3.2) and (3.3) is aniterative process. Starting fr om a first estim ation of Qp =q p (i.e. neglecting the second termon the right-hand side of equation (3.2)), we can obtain the value of N p from equation(3.3) before iteration. The iteration is then carried out by substituting N p into equation(3.2), and the process is repeated ten times. In order to eliminate the effect due to somediscontinuities existing in the physical assumptions, the N p and Qp spectra obtained aftereach iteration are reparametrised before they are used in the next iteration cycle. In sodoing, it is found that convergence of the results is very rapid, and the final values arereached after only five cycles of iteration.

4. Comparison of the predicted p flux with experim ental data4.1, Experimental data on the interstellar p j l u xIn order to convert the measured value of p/p to the interstellar p flux, j o , e need t o find areasonable spectrum of local interstellar protons. Different authors chose different localinterstellar proton spec tra fairly arbitrarily. S tephens (198 IC) used the dem odulated protonspectrum of Burger and Swanenburg (1971), which is shown in figure 5 as the curve BS,and which is evidently inconsistent with the later measurements of Smith et a1 (1973) andWebber and Lezniak (1974) for T p> 5 GeV where the solar modulation effect should benegligible. H owever, the spectrum suggested by Jokipii and Kopriva (1979) (the curve JKin figure 5 ) agrees well with these data. Thus we have decided to adopt JK as our localinterstellar proton spectrum. This choice was also guided by our previous work indemodulating the mea sured electron spectrum (T an and N g 198 lb), in which we foundthat the effect of solar modulation w as often overestimated.Our reduced experimental data for jp re shown in figure 6 as full crosses. For thedatum G of Golden et a1 (1979), it should be noted that their value of p/p was reported in agiven momentum (not kinetic energy) interval, and that their correction for secondary 0sof atmospheric origin was based o n the incorrect estimation of Badhwar et a1 (1975). Forthe datu m Bo we have used the recently reported value of p/p (Bogomolov et a1 1981).Further, in demodulating the datum Bu of Buffington et a1 (1981), some otherproblems have to be taken into account.(1) There is a divergence in estimating the adiabatic deceleration parameter p.Buffington et a1 used p = 60 0 MV, b ut the neutron monitor da ta in the period October1979-July 198 0 favou r a smaller q= 350 MV (Koch et a1 1981).(2) The role of the charge-dependent drift effect is not yet clear. Jokipii and Davila(198 1) pointed out th at their previous estim ation could be overestimated. Mc Do nal d et a1(1981) found that the long time delay of cosmic-ray intensity variation between la u and23 au wa s also not c omp atible with the model of Jokipii an d Da vila in its present form.Moreover, an inconsistency also appeared in the analysis of interstellar positron da ta (T anand Ng 1981c), i.e. according to this effect, after 1971, it would be easier for positiveparticles to penetrate the inner solar system than for negative particles, but the measuredpositron flux of H ar tm an and Pellerin (1976) disappeared below 200 MeV. In view of theabove, we adopted only a very simple (but not less reliable) method to demodulate the

7/27/2019 Tang & Ng.pdf

12/17

Calculation of the equilibrium antiproton spectrum10

lo- 100 101 102T , I G e V I

237

Figure 5. Interstellar proton spectra. JK, Jokipii and Kapriva (1979); BS, Burger andSwanenburg (1971); S1 and S2 , the low-energy and high-energy segments of Stephens(1981c), respectively; TN, deduced by us on the basis of the cosmic-ray intensity gradient inthe galaxy (unpublished work). Experimental data: 0,Webber and Lezniak (1974); 0 ,Smithet al (1973) ; A,An an de t al(1968);., Ryan et al (1972) .

datum Bu, i.e. the j p datum was obtained by adding an energy increment v, to themeasured TpN 200 MeV and by applying a flux-increasing factor equal to the ratio of ourlocal interstellar proton flux at T, +q to the measured proton flux at T, = 0.2 GeV on theboard of IMP-8 just before the observation of Buffington et al. Thus for v,= 400 MV and600 MV , the j p dat a are show n in figure 6 as Bu4 and Bu6 respectively.From figure 6, our deduced j p data (G, Bo and Bu4) are consistent with theindependent estimation of Kiraly et a1 (1981) (given by the broken crosses) but not thedatum Bu6. The inconsistency of Bu6 could be due to the different interstellar protonspectra assumed.4 .2 . Predicted jl i spectrum f o r the leaky-box modelIn order to compare with the recent work of Protheroe (1981), we have calculated the

7/27/2019 Tang & Ng.pdf

13/17

238 L C T an and L K Ng

10- 100 101 102T- , i G e V )Figure 6 . Various predictions of p flux: TLBS, TLB F. TC GS, TC GF , TC GX . TR C G , o urpresent work (see text); PLB and PRCG, Protheroe (1981); SCG, Stephens (1981~).Experimental data (present work in full crosses, Kiraly et a1 (1981) in broken crosses): Bo,Bogomolov et al (1979); G, Golden et a1 (1979); Bu, Buffington et a1 (1981) withoutdemodulation, Bu4, demodulated with q = 400 MV ; Bu6, demodulated with q=600 MV .

predicted j i , or the leaky-box model using the same & values as P roth eroe , i.e.A,(p(G eV/c))= 7.0 (g cm-') for p 4 GeV/c. (4.1)=7.0(4 .0 /~ ) ' .~g cm-')In figure 6, curves PLB, T LBS and TL B F show, respectively, the result of Protheroe, andour results before (i.e., assuming Qi, = q p in equation (3.2)) and after iteration. At highenergies all three spectra are consistent with one another. At low energies, the differencebetween PLB and TLBS may be due to the lower p production spectrum used byProtheroe (see figure 2). It is apparent that at low energies the drop of TLBF with Ti, isslower than those of TLBS and PLB. This indicates that the effect of p non-annihilationinelastic interaction is significant, an d this will be d iscussed in the next subsection .4.3. Predicted j , spectrumfor the closed-galaxy model of Rasmussen and PetersLetting Ae+ CO in equation (3.3), we obtain the predicted j , for the closed-galaxy modelof Rasm ussen and Peters (1975). Th e predicted j , curves so obtained are shown in figure6, in which S C G is f rom Stephens (1981a) and TC GS and T C G F are our predic tionsbefore an d af ter iteration, respectively.It is seen th at the consistency of T C G S with S C G is good at high energies and at verylow energies. The difference in the peak values of the two curves may be explained from acomparison of the p production spectra used (see figure 2). However, the most apparentdifference between Stephen's prediction (curve SCG) and ours after iteration (curveTCGF) is that ours has a dramatic increase of the interstellar p flux, particularly at lowenergies. C omp aring with th e curve T LB F it is concluded that, as Ae increases, the effect

7/27/2019 Tang & Ng.pdf

14/17

Calculation of the equilibrium antiproton spectrum 239of th e p non-annihilation inelastic interaction becomes even more prominent. A detaileddiscussion relating t o this pheno men on will be given in the next se ction.

4.4. Predicted j p pectrum f o r the revised closed-galaxy model of Peters andWestergaardRecently, Protheroe (1981) presented a thorough discussion of the 0 flux in the revisedclosed-galaxy model of Peters and Westergaard (1977). In order to compare with hiscalculated result, we divided the primary proton spectrum (equation (2.12)) into old andyoung components in the manner recommended by Protheroe. Then from the old protoncomponent we recalculated the production spectrum of the old p component. Then anadditional contribution from the leakage of the young com ponent of ps was also included.The resultant p spec trum is the sum o f the equilibrium flux of the old p component and thatof the young p component, the latter being identical to the predicted p flux for theleaky-box model.The m ost important parameter for the revised closed-galaxy model is IC, the ratio of thetotal mass of interstellar gas in the Gala xy to tha t in the arms. F or the case IC= 00, whichis consistent with the experimental data fro m cosm ic-ray nuclei and positrons ( Protheroe1981), our calculated j p after iteration is shown in figure 6 as the curve TRC G, which iscompared with the curve PR C G taken from Protheroe (198 1). As in the case of the leaky-box model, our result is very consistent with that of Protheroe at high energies, but at lowenergies our predicted j p s appa rently higher. H owever, th e divergence is less serious tha nthat between the curves S C G and T C G F for the closed-galaxy model of Rasmussen andPeters (1 975). Th e reaso n will be given in the following discussion.

5. DiscussionSince our deduced low-energy p flux for each of the existing models is dramatically higherthan that predicted by various a uthors for each of the corresponding models, queries havebeen raised ab out the correctness of ou r calculation. The following points serve to clarifythe queries.

(1) The effects of the annihilation and the ionisation loss of Ijs which suppresses j p a tlow energies have been included in our calculation with the required precision (see 0 3).That the resultant equilibrium p spectrum shows a dramatic increase in spite of thesesuppressive effects is due to the more predominant effect of deceleration of ps after theirnon-annihilation inelastic interaction w ith the ISM.(2) The total number of ps under the curves T C G F and TL B F is significantly largerthan that under the curves TCGS and TLBS respectively. This is because TCGS andTLB S were obtained under the assumption that Qp q p in equation (3.2). This means thatall ps which have suffered from the total inelastic interaction (through the term inequation (3.3)) ar e considered to be destroyed. But in a n actual situation those which havesuffered from the non-annihilation inelastic interaction (through the 4 erm in equation(3.2)) have survived after this interaction. Therefore, the total number of ps u nd er T C G Fand T LB F is significantly larger.(3) The deceleration effect also produc es a higher p flux a t high energies, beca use it is awell known fact tha t the energy spectra of cosmic rays can extend to very high energies. Aquantitative illustration can be performed by taking a power-law approximation to the p

7/27/2019 Tang & Ng.pdf

15/17

240 L C T a n a n d L K N gspec trum at high energies, i.e.

an d equa tions (3.1) an d (3.2) respectively reduce toN p Ep)/AP'E Qp(E,)and

Hence

Substituting A V and z 2.5 into equation (4.5) at high energies, we havej,(TCGF)/jp(TCGS)= 1.6. This rough estimation is consistent with the result of ournum erical calculation (see figure 6).(4) Of the various calculations for the j p spectrum, only Stephens and Protheroe gavedetails of their calculations, The dramatic increase of th e p flux was not obtained byS te ph en s ( 1 9 8 1 ~ ) ecause he replaced the exact expression for equation (3.2) by

which is in error as the dimensions within equation (4.6) do not agree. A numericalcomparison of the contribution of his deceleration term (the second term on the right-handside of equation (4.6)) with that of ours at high energies shows that the contribution in hiscalculation is underestimated by a factor of AF'/(y,iYP)%1 o ! N o wonde1 his decelera tionterm ma kes a negligible effect.As referred to the calculation of Prothero e (198 1), it is found from figure 6 that his pspectral shape also show s a smo oth tendency toward s lower energies, though his p flux isstill lower than ours. Comparing the amount of divergence between Stephen's predictionand ou rs and th at between Protheroe's prediction and our s, we czn see that this is causedbecause Protheroe also took the p non-annihilation inelastic interaction into account.However, he used a 1/E' distribution for the decelerated p's instead of our l/T'distribution. This imp ortant difference and the difference in other param eters used acco untfor the divergence between Protheroe's result and ours.Finally, we have also calculated the equilibrium p spectrum for the closed-galaxymodel of Rasm ussen and P eters (1975) by using Stephens' assump tion that dp= dp insteadof our more precise estimation of U;, The resultant j i is shown in figure 6 as the curveTCGX, which is slightly higher than TCGF although the difference is not significant.Hence we conclude th at the d ramatic increase of the equilibrium p flux at low energies is aphysical reality. The kinematic cut-off of low-energy $s only exists in the pro duc tionspec trum , an d doe s not a pp ear i:i the equilibrium interstellar spectru m.So far we have considered the important comments of Gaisser and Mauger (1982)related to the calculation of the equilibrium p spectrum, except for the effect of the Fermimotion of the target nucleons. This effect should be negligible when compared with otherimp ortan t effects presented in this section (see also Gaisser an d Levy 1974).We shall not enter into a discussion of other existing propagation models as they are

7/27/2019 Tang & Ng.pdf

16/17

Calculation of the equilibrium antiproton spectrum 24 1unable to explain the p measurement at low energies. However, from the achievementshown in this paper, the inconsistency between the experimental data and the predictionsbased on the traditional concept is less se rious than imagined previously. Th e effect of thep non-an nihilation inelastic interaction ha s to be taken into account in any further work.

AcknowledgmentsThe authors are grateful to Professors and Dr s S A Stephens, T K Gaisser, A Buffington,R J Protheroe, A W Wolfendale and P K MacKeown for their stimulating and helpfulcomments, and to Professor D J New man for his keen interest in and s upport of this work.

ReferencesAnand K C, Daniel R R, Stephens S A, Rhowmik B, Krishna C S , Aditya P K and Puri R K 1968 Can. J.Badhwar G D, Golden R L, Brown M L and Lacy J L 1975 Astrophys. S p a c ~ ci . 37 28 3Bogomolov E A, Lubyanaya N D, Romanov V A. Stepanov S V and Shulakova M S 1979 Proc. 16th In t.- 98 1 Proc. 17th Int. Conf on Cosmic Rays, Paris 9 14 6Buffington A , Schindler S M and Pennypacker C R 1981Astrophys. J . 248 1179Burger J J and Swanenburg B N 1971 Proc. 12th In t. Con$ on Cosmic Ray s, Hobart 5 1858Cowsik R and Gaisser T K 1981 Proc. 17th Int . Con$ on Cosm ic Ra ys, Paris 2 218Daniel R R and Stephens S A 1975 Space Sci. Rev. 17 45Fishbane P M , Gaisser T K, Maurer R H and Trefil J S 1974 Phys. Rev. D 9 3083Flaminio V , Graf I F, Hansen J D, Moorhead W G and Morrison D R 0 1979 C E R N - H E R A 7 9 - 0 3Gaisser T K and Levy E H 1974 Phys. Rev. D 10 1731Gaisser T K and M auger B G 1982 Astrophys. J . Lett . 252 L57Gaisser T K, Owens A J and Steigman G 1981 IAU Symp. 94, Origin o fco sm ic Rays ed. G Setti, C SpadaGinzburg V L and Syrovatskii S I 1964 Origin o fco sm ic Rays (Oxford: Pergamon)Golden R L, Horan S , Mauger B F, Badhwar D, Lacy J L, Stephens S A, Daniel R R and Zipse J E 1979Hartm an R C and Pellerin C J 1976 Astrophys. J . 204 92 7Hayakawa S 1969 Cosmic Ray Physics (New York: Wiley)Hillas A M 1979 Proc. 16th Int. Con$ on Cosmic Rays, Kyoto 6 13Jokipii J R an d Davila J M 198 1 Astrophys. J . 248 115 6Jokipii J R and Kapriva D A 1979 Astrophys. J . 234 38 4Kiraly P, Szabelski J, Wdowczyk J and Wolfendale A W 1981 Nature 293 12 0Koch L, Perron C, Goret P, Cesarsky C J, Juliusson E, Soutoul A and Rasmussen L 1981 Proc. 17th In t.McDonald F B, La1 N, Tr ainor J H and Van Hollebeke M A I 198 1 Astrophys. J . Lett. 249 L7 1Peters B and Westergaard N K 1976 Astrophys. Space Sci. 48 21Protheroe R J 198 1 Astrophys. J . 25 1 38 7Rasmussen I L and Peters B 1975 Nature 258 4 1 2Rushbrooke J G and Webber R R 1978 Phys. Rep. 44 1Ryan M J, Ormes J F and Balasubrahmanyan V K 1972 Phys. Reo. Lett. 28 98 5Smith L H, Buffington A, Smoot G F and Alvarez L W 1973 Astrophvs. J . 180 98 7Stecker F W, Protheroe R J and Kazanas D 1981 Proc. 17th Int. Conf on Cosmic Rays, Paris 9 2 1 1Stephens S A 1981a Proc. 17th Int. Conf: on Cosm icRays, Paris 13 89- 981b Proc. 17th Int. C onf on Cosmic Rays, Paris 9 19 9- 981c Astrophys. Space Sr i. 76 87

Phys. 46 S652

Conf on Cosmic Rays, Kyoto 1 330

(unpublished)

and A W Wolfendale (Dordrec ht: Reidel) p 257

Phys. Rev. Lett. 43 1196

Conf on Cosmic Rays, Paris 2 18

7/27/2019 Tang & Ng.pdf

17/17

242 L C T a n a n d L K N gT an L C and Ng L K 1 9 8 1 a J .Phys. G: Nucl. Phys. 7 123- 98 1b J . Phys. G:Nucl. PhyS. I 1123- 9 8 1 ~. Phys. G: Nucl. Phys. I 1135- 982 Phys. Rev. D 26 1179Webber W R and Lezniak J A 1974 Astrophys. Space Sci. 30 36 1Yash Pal and Peters B 1964 K . Danske Vidensk. Selsk., Mat.-Fys. Meddr. 33 no 15