The Empirical Atmospheric Muon Bundlemodel: Mathematical Representation
Shigeru Yoshida, Aya Ishihara, and Mio OnoDecember 10, 2006
The Elbert formula [1] gives the number of muons with energies greaterthan Eµ in a bundle as follows:
ET A2 −αξ (1 − ξ)βE0 cos θAEµξ =E0Nµ =(1)where A is the mass number of primary cosmic rays with energy of E0 , θ iszenith angle of a muon bundle. The other parameters are ET = 14.5 GeV,
α=1.757, β=5.25.
One can differentiate it to obtain the differential spectrum of muons in abundle dNµ /dEµ .
At this stage of the IceCube analysis, we are not resolving individualmuons, but measuring its bundle energy resulted from summing all the muonenergies contained in the bundle. It is, therefore, useful to define the bundle
energy as follows:
EµB ≡EZ0 /A

EthdNµEµ dEµdEµ(2)The approximation using the fact that the integration above is mainly
determined in the regime of ξ  1 leads to the relatively simple formula torepresent the bundle energy asEµB = ETAαAEthcos θ α − 1E01−α+1.

(3)

We approximate a bundle event as a single muon with energy of EµB . Thisapproximation should work well in the analysis based on Npe which in turncalorimetrically measures the energy deposit that is proportional to EµB tothe first approximation. Then we can estimate the rate of muon “bundle”
by rewriting the primary cosmic ray flux as a function of EµB , i.e.,dJµB
dE0 dJCR=(E0 (EµB , cos θ, A, Eth )).
B
B
dEµdEµ dE0(4)Here dJCR /dE0 is the relatively well-measured cosmic ray spectrum thatcan be represented by the power law form κE0−γ . The formula described by
Eq 3, then gives the bundle rate. We getdJµB
dEµB

=E0 =

1 E0 dJCR(E0 )α − 1 EµB dE0cos θ α − 1 EµB
A
α ET! 1(5)α−1AEth(6)Eq. 5 provides the atmospheric muon intensity at the surface. The parameters in Eq. 6 are to be determined by the IceCube high energy datasamples. The most uncertain parameter which is a deciding factor in thenormalization to give the absolute intensity is Eth , the threshold energy ofmuons in the integral of Eq. 2. Because the IceCube detector is only sensitive
to sufficiently energetic muons, it is likely that Eth is around 1 TeV. The real
data will tell us this value.
Since the IceCube is an underground detector, we see muons after propagating inside the earth. The propagation reduces the muon energy from Eµto EµI3 . The bundle energy at the IceCube depth is given byEµB,I3 ≡EZmax

Eth,I3dNµ I3 I3
E dEµdEµI3 µ(7)The numerical calculation as the JULIeT does is able to estimate distribution of EµI3 , but the simple analytic representation is more useful in the real
data fitting to obtain the parameters such as Eth . The CEL approximation
gives
EµI3 = (Eµ + ) exp−βµ X −
(8)2

where βµ ∼ 4 × 10−6 cm2 g−1 is the inelasticity parameter due to the muoninteractions, and  ∼ 500GeV is the critical energy below which the ionization
loss dominates over the radiative interactions. X is the slant depth of themuon trajectory. Eqs. 7 and 8 give
EµB,I3 ' e−βµ X EµB −β(1 − e−βµ X )A EµB .
αE0(9)

Here EµB must be calculated for the threshold energy of muons at the IceCubedepth, Eth,I3 , which is related to the one at the earth surface by Eq. 8. One
can see that the second term represents the contribution from the ionization
loss. Because the EHE regime is E0 ≥ 1 PeV  , this term is negligible.
Note that we used the approximation here that βµ is energy independent,
which is not exactly true because of the photo-nuclear reaction. The systematics by introducing this approximation should be properly accounted in
fitting the data.

References[1] T. K. Gaisser, Cosmic Rays and Particle Physics, (Cambridge University
Press, Cambridge, England, 1990) Chapter 12.

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