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 intensity 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
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
----------------------------------------------------