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

1. Nov 27, 2013

### phys_student1

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?

2. Nov 29, 2013

### Staff: Mentor

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.

3. Nov 30, 2013

### atha

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.

4. Nov 30, 2013

Staff Emeritus
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.

5. Nov 30, 2013

### phys_student1

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.