Need some help interpreting the muon energy spectrum

[stakhanov]

I have been looking at the flux of muons (as secondary cosmic ray particles) at the Earth's surface as a function of both energy and angle from the zenith. From what I have read, the flux follows a squared cos theta relationship with angle from the zenith and several attempts have been made to accurately parametrize the energy dependence. I am using a revision of the Gaisser parametrization, dN/dE=f(theta,E) [per square cm, per second, per steradian, per GeV]. My questions are as follows:

1. Do I just integrate f(theta,E) over an energy range to find the number of muons in that range? I have done this and it doesn't give what I expect (something that looks roughly like a Maxwell-Boltzmann spectrum in form) so I want to check I am doing the right thing.

2. How do I understand the 'per GeV' part of the untis of f(theta,E)? If for example, f(theta=10,E=20) = 2x10^-5, then it means that on average 2x10^-5 muons with energy 20GeV will flow through a square cm from a solid angle of 1 steradian centred around a point at 10 degrees from the zenith. Where does the 'per GeV' come into it?

Sorry if this doesn't make sense but I'd appreciate any help.

Thanks

 

[jtbell]

I2. How do I understand the 'per GeV' part of the untis of f(theta,E)? If for example, f(theta=10,E=20) = 2x10^-5, then it means that on average 2x10^-5 muons with energy 20GeV will flow through a square cm from a solid angle of 1 steradian centred around a point at 10 degrees from the zenith.

No. It means that if you select only muons with energies from 19.5 to 20.5 GeV, that is, a range of 1.0 GeV, you will get approximately $(2 \times 10^{-5})(1.0) = 2 \times 10^{-5}$ muons per steradian at 10 degrees. If you select only muons with energies from 19.95 to 20.05 GeV, that is, a range of 0.1 GeV, you will get approximately $(2 \times 10^{-5})(0.1) = 2 \times 10^{-6}$ muons. And similarly for other energy ranges.

In general, this sort of calculation is only approximate. To get the exact number of muons in an energy range, you have to integrate over the desired energy range:

$$\int^{E_{max}}_{E_{min}} {f(\theta, E) dE}$$

Of course, you also have to integrate over a suitable angular range as appropriate.