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.

 

[sir-pinski]

1. Yes, you are correct in integrating f(theta,E) over an energy range to find the number of muons in that range. However, the result may not always look like a Maxwell-Boltzmann spectrum. This is because the energy spectrum of muons is affected by various factors such as the energy spectrum of the primary cosmic rays, the geomagnetic field, and the Earth's atmosphere. These factors can cause fluctuations and distortions in the muon energy spectrum.

2. The "per GeV" part of the units of f(theta,E) refers to the energy interval over which the flux is measured. In your example, if f(theta=10,E=20) = 2x10^-5, it means that for every 1 GeV of energy, there will be 2x10^-5 muons flowing through a square cm from a solid angle of 1 steradian centered around a point at 10 degrees from the zenith. So, the "per GeV" part is important in understanding the energy dependence of the flux.

I hope this helps clarify your questions. If you need further assistance, I would recommend consulting with a physicist or researcher who specializes in cosmic rays and muon energy spectra for more detailed explanations.