# Why is the zenith angle distribution of the muons cos2(x) ?

by phys_student1
Tags: angle, cos2x, distribution, muons, zenith
 P: 83 Hello, Empirically, the flux distribution of cosmic ray muons follow cos^2(θ) where θ is angle of incidence. Looking up the papers, I did not find any clue as to why is this the case. All sources simply consider this an experimental fact. Is their any real explanation for this?
 Mentor P: 10,523 That is a common question and I never saw a good answer. It is probably just some function which is not so far away from the real distribution, without a deeper physical argument behind it.
 P: 7 First of all consider that the earth's atmosphere is a perfect sphere. The path length for θ=0 is the minimum possible. Also note that there no chance to detect a muon for θ=π/2(for this specific angle the muon would have to travel along the earth's crust, it would collide and interact with the crust's atoms with a very high probability). Those arguments state that we need a cos-like distribution. The next step is to think that the muon obeys the Bethe-Bloch formula. In simple words the energy loss per unit length, is something like 1/β^2... This implies that it is very much likely to observe a muon for small θ(where the path length is minimum) rather that large θ, in a non-linear way. If you also put in mind that the earth's atmosphere isn't a perfect sphere, but it's a "3D ellipsoidal", you are again moving away from linearity. I believe that Andersson has done some "fitting" on experimental data of cosmic ray flux, which proves the cos^2θ distribution. So it is an experimental model, as far as I know.
Mentor
P: 15,401

## Why is the zenith angle distribution of the muons cos2(x) ?

This is an approximation, and as such, you won't be able to derive it.

If cosmic rays were uniform and isotropic, you'd expect a distribution that goes more like 1/cos(theta), because there's not a lot of area with a small zenith angle compared to a large one. However, as atha points out, we have an atmosphere, so cosmic rays from the horizon are unlikely to make it to us.
 P: 83 Thanks all. Yes I also looked too much in the literature without finding any clue, all state is as empirical formula or experimental fact. Thanks for your responses.

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