S.Klimushin SOME COMMENTS ON AMANDA-B10 ANGULAR INTENSITIES OF ATMOSPHERIC MUONS (27-Aug-2001, revised 30-Sep-2001) INTRODUCTION This note presents results of an examination of both experimental and expected (MC) angular intensities I(>=Ef,cos(theta),h) for AMANDA-B10 (1997 data sample) by means of their direct best fit with 2 parameters (A0,gamma) of the sea level spectrum using the analytic expression described in Ref.[1]. The other 3 parameters of the sea level spectrum were fixed: B0=0.037, Ecr_PI(0)=103 GeV, Ecr_Ka(0)=810 GeV (see Ref.[1]). It is clear that the Amanda data used here (as presented in Ref.[2]) are still subjected to systematic uncertainties, which are currently under investigation by the Amanda collaboration. In this context we regard this note as an "exercise" only to point out aspects of the data set and of more basic nature usually not discussed (e.g., influence of the detection threshold on the angular spectrum in absolute and relative terms). This analysis was partly inspired by a letter of P.Desiati (from 1-Aug-2001), suggesting to detail what was done in the framework of the Baikal NT-36 analysis. INPUT DATA These data were kindly supplied by P.Desiati and they are summarized in tables of Appendix A. 1. The analysis is being done for the vertical depth of h=1842.5 m of the ice (the center of the bottom part of AMANDA-B10), that corresponds (with the ice density of 0.92 g/cm**3) to 1695.1 m of the water. 2. The experimental unfolded (with CORSICA + MMC) data table with angular in- tensity relative to winter sample (table 6 of Appendix A). 17 points were taken (with cos(theta) bin of 0.05)) from cos=0.975 to cos=0.175 (that corresponds to the maximum slant depth of 9.69 km of water). All the errors are statistic and are as small as (0.7 - 3.5)% ( <1% for cos>=0.53 ). 3. The MC-simulation data table (CORSICA + MMC) with angular intensity using the same binning (table 7 of Appendix A). 15 points of both experimental and MC data samples (corresponding to cos>=0.275) are presented in Ref.[2]. NUMERICAL TOOLS 1. The best fit was done by means of HBOOK procedure HFITV ( MINUIT + least squares method). The input conditions for MINUIT were following: parameter AO : start value=0.25, start step =0.1, no limits parameter gamma: start value=2.8, start step=0.1, no limits. 2. The Confidence Level (CL) was calculated by using CERNLIB procedure PROB for corresponding number of degrees of freedom (ndf). OUTPUT DATA are given by the fitted normalization factor A0 and spectral index gamma of a sea level spectrum. The reduced chi-square (rch) normalized to the ndf as well as the confidence level (CL in %) are also presented. SOME IMPORTANT COMMENTS 1. Since the MMC-code [3] of the muon propagation (used for input data) gives almost the same results (survival probabilities, energy losses) as the MUM- code [4] (discrepancy is less than 1-2 %) the implementation of the analytic expression based on the correction factor and effective energy losses calculated using MUM seems quite correct. The accuracy of the analytical expression itself is less than 3-4 % up to slant depths of 10 km and cut off energies < 100 GeV for a wide range of spectral indices (gamma=2.6-2.9). 2. To perform the correct best fit of a sea level spectrum parameters for a given cutoff value Ef one should describe the dependence of the effective cutoff of the telescope upon the zenith angle Ef(cos,h) and recalculate the experimental angular intensity to THIS GIVEN cutoff (e.g., Ef=1 GeV) by means of: I(>=1 GeV,cos,h)_experim = I(>=Ef,cos,h)_experim * K(cos,h), (1) were coefficients K(cos,h) are given by ratio of the expected intensities K(cos,h) = I(>=1 GeV,cos,h)_expect / I(>=Ef,cos,h)_expect (2) and are presented in a table below (for h=1695.1 m of water). Table 1. Coefficients K(cos,1695.1 m) given by Eq.(2) for various values of Ef. ------------------------------------------------------------------------- zenith Cutoff Ef (GeV) angle cos 1 10 20 30 40 50 60 70 80 90 100 1.0 1.00 1.07 1.15 1.24 1.32 1.41 1.50 1.59 1.69 1.79 1.89 0.9 1.00 1.06 1.14 1.22 1.29 1.37 1.46 1.54 1.63 1.72 1.81 0.8 1.00 1.06 1.13 1.20 1.27 1.34 1.41 1.49 1.57 1.65 1.73 0.7 1.00 1.05 1.12 1.18 1.24 1.31 1.37 1.44 1.51 1.58 1.66 0.6 1.00 1.05 1.10 1.16 1.22 1.28 1.33 1.39 1.46 1.52 1.59 0.5 1.00 1.04 1.09 1.14 1.19 1.24 1.30 1.35 1.40 1.46 1.52 0.4 1.00 1.04 1.08 1.12 1.17 1.21 1.26 1.31 1.35 1.40 1.45 0.3 1.00 1.04 1.07 1.11 1.15 1.19 1.23 1.27 1.31 1.35 1.39 0.2 1.00 1.03 1.06 1.09 1.13 1.16 1.20 1.23 1.27 1.30 1.34 0.1 1.00 1.03 1.07 1.10 1.14 1.17 1.20 1.23 1.27 1.30 1.33 ------------------------------------------------------------------------- Since the description of the effective threshold function Ef(cos,h) is a really nontrivial task, it is usually omitted in the muon data analysis. So, the results presented below were obtained with the different values of effective cutoff Ef (independent on zenith angle) to look for an appropriate consistence with the fitted sea level spectrum. BEST FIT FOR EXPERIMENTAL DATA 1. The whole range of cos(theta)=(0.175-0.975), ndf=15 [ 17 (experimental bins) - 2 (parameters) ]. Table 2. ---------------------------------------------------------------------------- | 1: Only stat.err | 2: stat.err + 0.1*I | 3: stat.err + 0.2*I | | | (10% syst.err) | (20% syst.err) | ---------------------------------------------------------------------------- Ef | A0 gamma rch CL | A0 gamma rch CL | A0 gamma rch CL | | | | | 1 | 0.41 2.91 30.0 0 | 0.19 2.80 0.81 67 | 0.19 2.80 0.25 99.9 | 10 | 0.46 2.92 33.2 0 | 0.22 2.81 0.87 60 | 0.20 2.80 0.26 99.8 | 20 | 0.69 2.96 37.5 0 | 0.28 2.84 0.96 50 | 0.25 2.83 0.29 99.6 | 30 | 0.87 2.98 41.2 0 | 0.32 2.85 1.03 42 | 0.29 2.84 0.31 99.5 | 40 | 0.97 2.99 44.2 0 | | | ---------------------------------------------------------------------------- Systematic errors were introduced artificially (10 and 20% of the absolute value of the experimental intensity I) to show their influence to the results of the best fit. 2. The "horizontal" directions only: cos(theta)=(0.175-0.625), ndf=8. Table 3. -------------------------------------------------- | 1: Only stat.err | 2: stat.err + 0.1*I | | | (10% syst.err) | -------------------------------------------------- Ef | A0 gamma rch CL | A0 gamma rch CL | | | | 1 | 0.17 2.80 20.6 0 | 0.08 2.70 0.59 79 | 10 | 0.18 2.80 20.8 0 | 0.10 2.72 0.64 75 | 20 | 0.19 2.80 21.6 0 | 0.12 2.74 0.69 70 | 30 | 0.21 2.80 22.6 0 | 0.13 2.75 0.72 67 | -------------------------------------------------- 3. The "vertical" directions only: cos(theta)=(0.625-0.975), ndf=6. Table 4. ------------------------------ Ef | A0 gamma rch CL | | | 10 | 0.60 2.95 10.6 0 |<-- only stat. errors 10 | 0.82 3.00 0.03 100 |<-- stat.err + 10% syst.err ------------------------------ The fitted sea level spectrum gets so steep (with the increase of Ef) that the results are shown for Ef=10 GeV only. BEST FIT FOR MC-DATA The absolute values of MC-intensities have been substituted instead of errors at the input of HFITV because the statistic errors of the MC data are negligibly small. Table 5. ------------------------------------------------------------------------- | cos(theta) | | 0.175 - 0.975 | vert.: 0.625 - 0.975 | horiz.: 0.175 - 0.625 | ------------------------------------------------------------------------- Ef | A0 gamma rch* | A0 gamma rch* | A0 gamma rch* | | 0.001 | 0.0001 | 0.001 | 1 | 0.078 2.667 1.8 | 0.220 2.813 7.0 | 0.068 2.650 2.5 | 10 | 0.083 2.670 1.9 | 0.230 2.810 4.0 | 0.075 2.658 2.7 | 20 | 0.094 2.679 2.3 | 0.239 2.806 1.5 | 0.086 2.670 3.0 | 30 | 0.122 2.706 2.8 | 0.246 2.802 0.5 | 0.098 2.680 3.4 | 40 | 0.138 2.715 3.2 | 0.254 2.798 0.6 | 0.101 2.680 3.4 | ------------------------------------------------------------------------- DISCUSSION OF RESULTS AND CONCLUSIONS 1. It is important to point out that the substitution of Gaisser's set of 3 fixed parameters instead of those from Ref.[1] (see INTRODUCTION): B0=0.054, Ecr_PI(0)=103 GeV, Ecr_Ka(0)=810 GeV leads to the almost same results of the best fit (deviation is less than 1%). 2. The fitting of A0 and gamma using the depth-intensity curve (after correspon- ding recalculation of the angular intensity) gives almost the same results of the best fit. 3. The errors of reconstructed A0 and gamma given by MINUIT error matrix have no great sense without (correlation + CL)-analysis and range typically within (0.005-0.02) for both parameters. 4. The results improve (rch decreases) with the decrease of cutoff value Ef (even for MC-data, except the vertical directions for Ef=30 GeV). 5. The taking into account the "artificial" systematic errors results ih the decrease of values of fit parameters. The increase of systematic errors above 10% does not lead to the change of results. 6. The fit parameters A0 and gamma reconstructed using experimental intensity differ from those reconstructed for MC-intensity for all investigated cases except 2 cases only: EXPERIMENT MC 1. Ef=10 GeV, whole range of cos, 10% syst.err - Ef=10 GeV, vertical dir. 2. Ef=10 GeV, horiz.dir, 10 % syst.err - Ef=40 GeV, whole range of cos 7. The best fit using MC data for Ef=(30-40) GeV results in spectral indeces ranging within gamma=(2.706 - 2.715), i.e. close to Gaisser and that from Ref.[1] but corresponding values of fit normalization factors A0 are 15% less than Gaisser's A0=0.14 and 27% less than A0=0.175 of Ref.[1]. 8. The reconstructed sea level spectra both for experimental and MC data are steeper for vertical directions and flatter for horizontal directions. 9. For an example (only) of the influence of taking into account the angular dependence of effective cutoff the following "imaginary" function was introduced for experimental data: Ef(cos) = 60 GeV * ( 1-cos(theta) ) (3) It leads to the linear increase of Ef with cosine from Ef(0.975)=1.5 GeV to Ef(0.175)=49.5 GeV (the linear decreasing function Ef(cos) explicitly results in "un-probably" steep sea level spectra with gamma > 3). Then, the experimental intensity was recalculated to the only cutoff value Ef=1 GeV by using the expression (1). It led to the following results of the best fit (only statistic errors are taken into account):: A0=0.26, gamma=2.81, rch=18.8 ,CL=0 - for the whole range of cos A0=0.19, gamma=2.80, rch=25.6, CL=0 - for horizontal directions A0=0.23, gamma=2.81, rch=1.9, CL=8% - for vertical directions. The 10% systematic errors led to: A0=0.15, gamma=2.76, rch=0.67, CL=82% - for the whole range of cos. One should point out: - the coincidence of the reconstructed gammas for all ranges of cosines and more "realistic" values of fit parameters [closer to MACRO/LVD (A0=0.26, gamma=2.78) and SNO (A0=0.14, gamma=2.80)]. - the considerable improvement of consistence with the model for the case of whole cosine range (rch=18.8 instead of 30 and smaller deviations (see fourth column in table 9 of Appendix B)) The use of the same Ef angular function (given by Eq.(3)) for MC-data results in extremely flat sea level spectrum (A0=0.06, gamma=2.62 ) for a whole range of cos. 10. The permanent excess of the experimental intensity over the fitted one for horizontal directions with cos(theta)<=0.225 (see table 9 of Appendix B) may indicate the possible: - "unproper" unfolding - overestimation of number of muon bundles For MC-data this effect is smaller and restricted by cos(theta)<=0.175 (see table 8 of Appendix B). ACKNOWLEDGMENTS I acknowledge useful discussions with P.Desiati on the experimental procedure and with R.Wischnewski concerning a possible check of the influence of the angular dependence of the effective energy cutoff. S.Klimushin (INR, Moscow, Baikal Collaboration) is available by e-mail klim@pcbai11.inr.ruhep.ru ----------------------------------------------------------------------------- REFERENCES [1] S.I.Klimushin et al., Phys.Rev. D64, 014016 (2001) [2] P.Desiati, W.Rhode for the AMANDA Collaboration, Poster report at 27-th ICRC (2001); P.Desiati, talk at postICRC Neutrino-Telescope-Workshop, Hamburg, (2001). See also: http://www-zeuthen.desy.de/nuastro/publications/conferences/ws2001.shtml [3] D.Chirkin, W.Rhode, Proceedings of 27-th ICRC, Hamburg (2001) [4] I.A.Sokalski et al., Phys.Rev. D64, 074015 (2001) APPENDIX A. INPUT DATA (including statistic errors) Table 6. EXPERIMENTAL DATA -------------------------------------- cos I Stat.err (theta) (cm^2 s sr)^-1 (cm^2 s sr)^-1 0.175 9.68098E-11 3.47736E-12 0.225 4.01102E-10 1.07166E-11 -------------------------------------- 0.275 1.09280E-09 2.50311E-11 ^ 0.325 2.38601E-09 4.32925E-11 | 0.375 4.45520E-09 6.98960E-11 | 0.425 7.41589E-09 9.41742E-11 | 0.475 1.13327E-08 1.31362E-10 | 0.525 1.62186E-08 1.61623E-10 | 0.575 2.23642E-08 2.01198E-10 0.625 2.98714E-08 2.45529E-10 see Ref.[2] 0.675 3.91438E-08 2.96790E-10 0.725 4.93311E-08 3.42365E-10 | 0.775 6.05855E-08 4.02727E-10 | 0.825 7.40296E-08 4.60008E-10 | 0.875 8.75503E-08 5.48281E-10 | 0.925 1.04345E-07 6.26146E-10 | 0.975 1.22346E-07 8.51739E-10 | ^ Table 7. MC DATA -------------------------------------- cos I Stat.err (theta) (cm^2 s sr)^-1 (cm^2 s sr)^-1 0.175 1.26739E-10 3.49236E-12 0.225 5.36597E-10 7.18600E-12 -------------------------------------- 0.275 1.58352E-09 1.23445E-11 ^ 0.325 3.32332E-09 1.78834E-11 | 0.375 6.15211E-09 2.43318E-11 | 0.425 1.00491E-08 3.10975E-11 | 0.475 1.44082E-08 3.72364E-11 | 0.525 2.05062E-08 4.44227E-11 | 0.575 2.70533E-08 5.10239E-11 0.625 3.48370E-08 5.79006E-11 see Ref.[2] 0.675 4.33386E-08 6.45804E-11 0.725 5.31389E-08 7.15104E-11 | 0.775 6.47209E-08 7.89197E-11 | 0.825 7.58480E-08 8.54348E-11 | 0.875 8.86302E-08 9.23536E-11 | 0.925 1.01164E-07 9.86677E-11 | 0.975 1.16967E-07 1.06095E-10 | ^ APPENDIX B. THE ACCURACY OF FIT The tables below contains the deviation (in percent with sign) of observed intensity from the fitted one (calculated using reconstructed A0 and gamma for corresponding Ef and cos). For instance, "-1.563" means that the observed intensity is lower than the fitted one on 1.563%. The corresponding set of (AO,gamma) can be found in Table.1 for experimental data (for the meaning of fourth column see point 9 of sec.DISCUSSIONS) and in table.5 for MC-data. Table 8. MC-DATA ---------------------------------------------------- cos Ef=1 Gev | Ef=30 GeV whole horiz | whole vert 0.975 1.563 | 3.875 0.334 0.925 0.040 | 1.901 -1.058 0.875 0.773 | 2.150 -0.103 0.825 0.299 | 1.225 -0.293 0.775 1.105 | 1.592 0.891 0.725 -0.527 | -0.447 -0.266 0.675 -1.015 | -1.321 -0.152 0.625 -0.718 1.070 | -1.371 0.883 0.575 -1.115 0.475 | -2.068 0.525 -0.427 0.920 | -1.605 0.475 -2.845 -1.739 | -4.201 0.425 0.095 0.857 | -1.253 0.375 -1.770 -1.375 | -3.008 0.325 -4.274 -4.355 | -5.156 0.275 -0.964 -1.652 | -1.073 0.225 -1.809 -3.350 | -0.544 0.175 15.073 12.714 | 18.234 ---------------------------------------------------- Table 9. EXPERIMENTAL DATA ---------------------------------------------------- Ef=1 GeV | Ef=30 GeV cos stat. 10 % Ef | stat. err. sys.err Eq.(3)| err. | 0.975 3.569 10.097 0.589 | 4.118 0.925 2.350 8.301 0.761 | 2.606 0.875 0.591 5.923 0.186 | 0.580 0.825 0.851 5.382 1.464 | 0.608 0.775 -0.412 3.308 1.111 | -0.842 0.725 -0.597 2.173 1.479 | -1.157 0.675 -1.323 0.397 1.011 | -1.948 0.625 -3.501 -2.968 -1.173 | -4.116 0.575 -4.484 -5.343 -2.504 | -4.972 0.525 -4.720 -7.190 -3.446 | -4.940 0.475 -3.571 -7.872 -3.387 | -3.348 0.425 -2.288 -8.722 -3.606 | -1.406 0.375 -0.340 -9.281 -3.635 | 1.479 0.325 3.308 -8.526 -2.478 | 6.395 0.275 11.003 -3.844 2.373 | 15.645 0.225 26.783 9.900 15.769 | 32.838 0.175 50.350 33.825 38.698 | 56.874 ----------------------------------------------------