Space angle resolution

    

Figure 19: (left) Distribution of $\Delta \theta = \theta_{fit}- \theta_{true}$ (top) $\Delta \phi = \phi_{fit}- \phi_{true}$ (bottom) after all selection criteria applied.
Figure 20: (right) Average angle between muon and neutrino versus neutrino energy. The total angle is the muon and neutrino angle at the interaction point and the multiple scattering angle added in quadrature.

Space angle resolution is needed to address the question of appropriate bin size. From simulation of signal, the median of the space angle uncertainty is 3.3 degrees (figure 18) using the cuts in table 2. Since the sky is binned in zenith and azimuth, the corresponding error in each angular variable is determined for zenith and azimuth by the Monte Carlo (figure 19). They can be fitted by a function of the sum of two Gaussians (equation 7).

$$f(\Delta \vartheta)= A_{1} \cdot e^{-\frac{(\Delta \vartheta ... ...{-\frac{(\Delta \vartheta - \delta_{2})^{2}}{2\sigma_{2}^{2}}}$$ (7)

where A₁<> and A₂<> are the amplitudes of each Gaussian, $\delta_{1}$ and $\delta_{2}$ are the angular offset for each Gaussian, and $\sigma_{1}$ and $\sigma_{2}$ are the standard deviation of each Gaussian. The variable, $\Delta \vartheta$, corresponds to either uncertainty in zenith ( $\Delta \theta$) or in azimuth ( $\Delta \phi$). Both of which are defined as the difference between the reconstructed track and the true track in zenith or in azimuth.

The excellent fit by the two-Gaussian model suggest two classes of events. One population reconstructs well, with $\sigma_{1}$ in zenith of 1.8 degrees and 3 degrees in azimuth. The other population of events are poorly reconstructed with $\sigma_{1}$ of 5.1 degrees and 10.3 degrees respectively. The poorly reconstructed population is approximately 10% of the total sample. Also note there is a slight bias in the reconstructed zenith toward vertical by about 0.5 to 2 degrees. This systematic should not impact the analysis because it is small compared to angular scale of the sky bins, which are $2 \cdot (1.6) \cdot \sigma_{1} \simeq 6$ degrees in zenith and 10 degrees in azimuth. The analysis in this report used a binning of 24x48 (24 zenith bins across the entire sky and a maximum of 48 azimuth bins at the horizon). This corresponds to 319 approximately equal area bins in the observable northern hemisphere.

The results presented by Tim Miller at the ICRC [8] show that the absolute pointing of AMANDA is good to a few degrees in zenith and the zenith angle resolution for Monte Carlo and data agree to ``level three cuts'', which are similar cut level as our optimized cut.

It is important to note that space angle resolution ($\sim$ 3 degrees) is significantly larger than the intrinsic angular correlation between the neutrino and muon for $E_{\nu}>$ 1 TeV. Figure 20 shows that the angular deviation between neutrino and muon decreases from 0.9 degrees at 1 TeV to 0.06 degrees at 100 TeV. The measured angular resolution is detector limited so AMANDA-II should be better, and further improvements in B10 analysis may improve the angular resolution.