In order to check a contribution from tracks, I checked several values such as the total energy loss by separating contributions both of cascades and tracks.
I used three sets of data. One is standard juliet MC, second one is corsika files generated by myself, and the final set is the corsika files generated by Henrike. The difference of the corsika files between mine and Henrike's is mainly the threshold energy for muon tracks at the generation. The threshold energy for the muon generation is 273 GeV for Henrike's and 1 TeV for mine, so in the Henrike's sample, there are more muon tracks.
Because I realized that in Henrike's sample the zenith angle is only up to 10 deg., so I also put the similar condition for other samples, namely ZA < 30 deg. The distance distribution is also a bit different, so the dist < 600m is also applied to all samples.
At first energy loss for a track is examined.
In the left plot below, I plotted dE/dX for the radiative process (pair creation + photo-nuclear interaction + brems.) (red) and ionization loss (blue) by applying dE/dX table (used in mmc) for simulated tracks.
The result is simply as same as the table, and the realtion is expressed as below.
log10(dE/dX [GeV/(g/cm^2)]) = -5.60 + 1.01*log10(track energy [GeV])
On the right plot below, sum of the cascade energy for a track is plotted.
All three MC datasets agrees with each other. There are steep fall off at around 10^3 GeV. This is because a track looses its energy by ionization loss, leaing to the less cascade output. Because of this steep fall off, the total cascade energy doesn't nearly change when I applied to 1 TeV muon generation threshold. See below.
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Theoritical energy loss Vs track energy 
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Sum of cascade energy from a track Vs the track energy 
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The cascade energy distribution for a track (10^5 GeV) is shown below. All three distributions for diffrent MC data set are reasonably agrees with each other at 10^5 GeV.
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Cascade energy distributions from a track (10^5 GeV) 
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Cascade energy distributions from a track (10^5 GeV) multiplied by the energy 
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Then, total energy loss of a bundle is checked.
The theoritical dE/dX derived as above is summed up for a bundle both for radiative process and the ionization loss. This is shown in the left plot below. As seen, the energy loss by radiative process is same for three MC datasets, even though the track distribution in a bundle is different. On the other hand, the energy loss by ionization loss is more than that by radiative process in case of Henrike's dataset because it contains many tracks.
The total cascade energy for a bundle is shown in the right plot below. The total cascade sum for all three samples roughly agrees, but corsika samples gives slightly less total cascade energy in a bundle compared to the juliet one, which I don't understand completely. (There is also a slight difference between corsika samples, but this is due to a way we define the bundle energy. See below.)
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Theoritical dE/dX sum for a bundle Vs bundle energy 
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Sum of cascade energy from a bundle Vs the bundle energy 
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The energy distributions for a bundle with a fixed energy is shown below.
Because the corsika I generated has higher threshold (1 TeV), the distribution is slightly shifted to right side in order to compensate low enrgy tracks that is ignored. (So, effectively the cosmic ray primary energy is slightly higher for my samples. See below.)
The distributions of cascades are different between juliet and corsika MC, though both gives roughly same total cascade energy sum for a bundle.
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Energy distribution in a bundle of 10^5 GeV 
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Energy distribution in a bundle of 10^6 GeV 
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I multiplied the energy.
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Energy distribution in a bundle of 10^5 GeV 
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Energy distribution in a bundle of 10^6 GeV 
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Here, I show a CR primary energy distributions for a fixed bundle energy (10^5 GeV).
As mentioned before, since the muon generation threshold is higher for my sample, the CR energy also becomes higher. The difference of peak position is about 10^0.2 ~= 60%.
This means that we should be careful to define the "bundle energy". It depends on the generation threshold. (Although the low energy threshold is not preferable for the EHE MC genration. It takes too much time.)
I also shows the track/cascade energy distributions for a bundle with the fixed CR primary energy of 10^8 GeV. As seen, the track distributions are same except the low energy side.
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Energy distribution of CR primary energy for a bundle of 10^5 GeV 
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Energy distribution in a bundle with the fixed primary energy of 10^8 GeV 
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Finally, I also put a plot showing the relation of sum of cascade energy from a bundle Vs the CR primary energy. As seen, here the cascade energy sum for a CR energy is same for corsika samples. This means that the muon generation threshold (1 TeV) doesn't affect on the total cascade energy.
However, the 1 TeV threshold for the muon generation is not enough for the tracks. As you see, we underestimate the energy a bit (~60%). Moreover, we will underestimate the multiplicity of tracks with the high threshold, and leads to a substantial underestimate for Npe (~factor 3.7).
The Npe Vs bundle energy is also shown here. Although the statistics is not enough, it's a bit difficult to check the relation between samples, but you clearly see that there is a certain amount of Npe's from bare tracks for corsika samples.
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Sum of cascade energy from a bundle Vs CR primary energy 
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total Npe Vs the bundle energy 
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multiplicity blue: juliet, red: corsika(at Chiba), green: corsika(Henrike's)
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I show here the ratio of contribution between tracks and cascades.
In the left plot below, the ratio of the theoritical dE/dX (ionization loss / loss by radiative process)is shown. As seen, in case of Henrike's corsika dataset the track contribution is 2.92 times larger than that of cascades. In case of corsika dataset I generated, it underestimate the contribution from tracks due to lack of (many) low energy tracks.
I also plotted Npe ratio between tracks and cascades (Npe(track)/Npe(cascade)) in the right plot below. In case of Henrike's dataset the contribution from track is 2.66 times higher than that from cascade in Npe.
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Ratio of total theoritical dE/dX between radiative process and ionization loss 
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Ratio of Npe between from cascade and tracks. 
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At last, the Npe distributions are shown with the observational data.
The Npe distribution for corsika data with the low energy threshold seems to agree with the observational data (though, the statistics is low).
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Npe distributions (with corsika data generated by Henrike W.) (zenith angle (FG) < 10 deg.) 
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By taking into account the tracks, the agreement between MC and data becomes better for the NDOM Vs Npe .
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NDOM Vs Npe 
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Keiichi Mase Last modified: Thu Aug 28 19:53:39 JST 2008
