The Theoretical Power Law Exponent for Electron and Positron Cosmic Rays: A Comment on the Recent Letter of the AMS Collaboration

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by  A. Widom

The Theoretical Power Law Exponent for Electron and Positron Cosmic Rays:
A Comment on the Recent Letter of the AMS Collaboration
A. Widom and J. Swain
Physics Department, Northeastern University, Boston MA USA
Y.N. Srivastava
Physics Department, University of Perugia, Perugia IT
Inarecentletter,theAMScollaboration reportedthedetailedandextensivedataconcerningthe
distribution in energy of electron and positron cosmic rays. A central result of the experimental
work resides in the energy regime 30 GeV < E < 1 TeV wherein the power law exponent of
the energy distribution is measured to be α(experiment) = 3.17. In virtue of the Fermi statistics
5
obeyed by electrons and positrons, a theoretical value was predicted as α(theory) = 3.151374 in
1
very good agreement with experimental data. The consequences of this agreement between theory
0
2 and experiment concerning the sources of cosmic ray electrons and positrons are briefly explored.

b PACSnumbers: 23.40.-s,31.15.V-,94.05.Fg,96.50.sb,95.35.+d,95.85.Ry,98.70.Sa
e
F
8 INTRODUCTION thatproperlyaccountsfortherepulsionbetweenfermions
inherent in their quantum statistics.

]
h In a recent letter[1], the AMS collaboration reported In the concluding section, we briefly discuss the im-
p the energy distribution of the lepton sector, i.e. the en- plications of this agreement between theory and experi-
- ergyprobabilitydensityρ(E)oftheelectronandpositron ment.
p
contribution to cosmic rays. In particular, the AMS col-
e
h laboration measured the slope of the log-log plot of the
[ energydistributionρ(E)versusthesinglecosmicraypar-
STATISTICAL THERMODYNAMICS
2 ticle energy E,
v
dlnρ(E) E dρ(E)
0 α(E)= = . (1) The energy distribution of particles evaporated from
1 − dlnE (cid:20)ρ(E)(cid:21)(cid:20) dE (cid:21) compactstellar objectsis determinedby the entropyper
78 Wehavepreviouslyargued[2]thatforcosmicraysemit- evaporatedparticles(E)viaρ(E)∝exp[−s(E)/kB]. To-
gether with Eq.(1), we thereby have
0 tedasaparticlewindevaporatingfromacompactstellar
1. source, such as a neutron star, the function αFermi(E)
E ds(E) E
0 may be computed from quantum statistical thermody- α(E)= = , (6)
5 namics. Theargumentisreviewedinwhatfollowsbelow. (cid:20)kB(cid:21) dE kBT(E)
1 For fermions -in the asymptotic high energy region- we
v: predicted that the power law exponent would be given whereinthe thermodynamic relationshipT =dE/ds has
i by beenemployed. Itiscomputationallymoresimpletofind
X
the energy as a function of temperature E(T) and later
r αFermi(theory)=3.151374, (2) find the temperature as a function of energy T(E) as an
a
inversefunction. InvirtueofarelativisticidealFermigas
while in the experimental region, 30 GeV <E <1 TeV,
with a density of states per unit energy per unit volume
we have
g(ǫ), one obtains
α(experiment)=3.170 0.008(stat+syst)
±
0.008(energy scale), (3) ǫ√ǫ2 m2c4 2
± g(ǫ)= π2−¯h3c3 for ǫ≥mc ,
wherein the agreement between Eqs.(2) and (3) is more
1
than satisfactory. Incidentally, were leptons a classi- f(ǫ)= ,
exp(ǫ/kBT)+1
cal relativistic Maxwell-Boltzmann gas, the index would
ǫg(ǫ)f(ǫ)dǫ
have been
E =ǫ= . (7)
R g(ǫ)f(ǫ)dǫ
αMB =3, (4) R
For the ultra relativistic regime wherein electron mass
instead of the corresponding Fermi-Dirac index given in
Eq.(2). The difference is small and positive effects are small, i.e. mc2 ≪ kBT , mc2 ≪ E and
(E/kBT)=constant,ourtheoreticalpredictioninEq.(2)
αFD αMB +0.151374, (5) is recoveredfrom Eqs.(6) and (7).
− ≈
2
RADIATION DAMPING is the upper time limit t<t∗ 1014 sec. The time t∗ is
∼
by a large margin more than the time taken for a speed
If the energydistribution of evaporating electrons and of light signal to transverse our galaxy.
positrons from compact stellar sources is as described
above,thenthe questionarisesastowhether this energy
CONCLUSION
distribution changesappreciably due to radiationdamp-
ing as these cosmic leptons propagatefromthe source to
the laboratory detectors built within our solar system. Inthebaryonsector,theheavynuclearcosmicraypar-
Weherenotethattheaccelerationsofthesechargedpar- ticlescanbebosonsorfermions,theleastmassivecosmic
ticles due to randomcosmic electromagneticfields ofthe ray particles being protons that are fermions. Bosons
order of micro-Gauss do not radiate appreciable energy (such as bosonic nuclei) and fermions each have their
for propagation distances of galactic proportions. ownpowerlawexponentαinthe highenergyregimede-
To see what is involved, consider an electron or pending purely upon statistics. A likely source of these
positron with energy E = mc2γ moving along a circu- evaporatingcosmic raysarecompactstellarobjects such
lar arc in a magnetic field B. Due to radiationdamping, asneutronstars. Suchobjects wouldalsoradiatea copi-
it is well known that the radiation energy loss obeys[3] ousamountofelectrons(directlyorfromneutrondecay)
and electron-positron pairs (if for no other reason than
dγ 1 2 that fast chargedparticles, be they electrons or baryons,
= (γ 1), (8)
dt −τ − whenscatteringthroughanybackgroundmatter orradi-
ationwillproducesuchpairs). Itisofcentralimportance
with a characteristic time scale τ determined by
that the AMS collaboration has measured the appropri-
1 2 e2 eB 2 atefermionpowerlawexponentherecharacteristicofthe
= , lepton sector of cosmic rays.
τ 3(cid:18)mc3(cid:19)(cid:18)mc(cid:19)
10−6 Gauss 2
20
τ 5.15868 10 sec . (9)
ACKNOWLEDGEMENTS
≈ × (cid:18) B (cid:19)
With an initial energy Ei = mc2γi = mc2cothχ, the J. S. would like to thank the United States National
exact solution of the radiation energy loss is thereby Science Foundation for support under PHY-1205845. Y.
S.wouldliketothankProfessorBrunaBertucciforhelp-
t tanh(t/τ)+γi
γ(t)=coth +χ = . (10) ful discussions.
(cid:18)τ (cid:19) (cid:20)1+γitanh(t/τ)(cid:21)
It is thereby evident that
1 γitanh(t/τ) and γi 1 implies γ(t) γi. (11)
≫ ≫ ≈ [1] M. Aguilar, et. al., AMS Collaboration, Phys. Rev. Lett.
113, 221102 (2014).
For the energy range of importance in the AMS experi-
[2] A. Widom, J. Swain and Y. N. Srivastava, “Concerning
mentandforcosmicmagneticfieldsoftheorderofmicro-
the Nature of the Cosmic Ray Power Law Exponents”,
Gauss,thetimeofflightforanelectronorpositronemit- arXiv:1410.6498v1[hep- ph] 15 Oct 2014.
ted froma compactsourceto a detector within oursolar [3] L. D.Landau and E. M. Lifschitz, “The Classical Theory
system without appreciable radiationdamping of energy of Fields”, 4th edition, Pergamon Press, Oxford, 1975.