Integration of a High Energy Cosmic Ray Spectrum

Aaron Koga
Physics 305
6 Feb 2007

1. Introduction

uhecr_3.png
FIG1: High Energy Cosmic Ray Spectrum.
uhecr_pow1.png
FIG2: High energy cosmic rays corresponding to power-law 1.
uhecr_pow2.png
FIG3: High energy cosmic rays corresponding to power-law 2.
uhecr_pow3.png
FIG4: High energy cosmic rays corresponding to power-law 3.
    Cosmic rays are particles, often hydrogen and helium nuclei, that originate somewhere outside of earth.  Ultra High Energy Cosmic Rays (UHECR) are cosmic rays with extremely high energies of almost 50J.  Measurements of the differential flux of particles from a detector in Utah, plotted in FIG1, show that theses UHECR obey three power-laws. 
Power-law 1:pl1.png(1)
Power-law 2:pl2.png(2)
Power-law 3:pl3.png(3)

Each of these power-laws, which are determined by fitting the data, gives the differential flux (in units of m-2 s-1 sr-1 eV-1) as a function of energy, E.  Each power-law describing a different range of E is plotted in FIG2-FIG4. 


2.  C Program

    A C Program was written to numerically integrate the three power-laws for UHECR so that the total number of particles arriving at a detector could be calculated.  The program is shown in the appendix.  The simple program contains a function f(), which has the information to calculate the differential flux F(E) for each power-law.  Trapez(), Simpson(), and Analytic() use f() to calculate the integral using the trapezoidal approximation, Simpson's approximation, and the analytic result respectively.  The program outputs a file showing the results from the integration using these three methods.  The integrals are performed using various lower limits (E=1017  eV, 1018  eV, 1019  eV, and 1020  eV).  The final number for the integral is given in units of particle*m-2 s-1 sr-1 .


3.  Results

  

Trapezoidal Approximation (particles/year)
Simpson's Approximation (particles/year)
Analytic Result (particles/year)
lower limit E = 1017  eV
1.81(106)
1.55(106)
6.54(105)
lower limit E = 1018  eV
9.13(103)
6.09(103)
2.98(103)
lower limit E = 1019  eV
1.35(102)
1.58(102)
3.11(101)
lower limit E = 1020  eV
1.64
1.10
1.46(10-1)
TABLE1: Results from the C Program's output  
    The C Program was used to calculate the total number of particles arriving at a square detector 10 km on a side in one year for a solid angle acceptance of 1 sr.  The output from the C Program, which was given in particle*m-2 s-1 sr-1 , was multiplied by 3.076e15 to find the particles/year arriving at the detector.  These numbers are shown in TABLE1.  dE was set in the C Program for the different power-laws by trial and error.  A simple C Program was used where the number of intervals (and thus dE=(Emin-Emax)/intervals) used in the Trapezoidal and Simpson approximations were varied from 1-5000 for each power-law.  The value for dE was set by choosing the result for the integral of F(E) using the Trapezoidal/Simpson approximation that was closest to the known analytical result.

4. Conclusions

    As seen in TABLE1, the Trapezoidal and Simpson Methods do not do a good job of approximating the function.  The approximations overestimate the function by factors ranging from ~2 to ~10.  However, this over estimation is not surprising.  Because the function being integrated (F(E)) goes as the negative power of E, the function "falls" faster than either its linear approximation if the trapezoidal method or its parabolic approximation in simpson's method.  Because the actual function falls faster than its approximations, the integral is overestimated.  It seems that because of the overestimation of the Trapezoidal and Simpson Methods the approximations are useful only to get a rough idea of the integral of F(E).


5.  References

[1] P. W. Gorham, http://www.phys.hawaii.edu/~gorham/P305/UHECR.html.

    "Cosmic Rays."  http://en.wikipedia.org/wiki/Cosmic_ray.

    Castelvecchi, Davide.  "Let it Rain."  Symmetry.  http://www.symmetrymagazine.org/cms/?pid=1000063. 


Appendix

--REMOVED--