| | 85 | In the following \(N\) refers to a number of simulated events and \(M\) to a number of measured (excess) events. |
| | 86 | |
| | 87 | |
| | 88 | \(n(\delta E, \delta\theta)\) is the number of produced events in the energy interval \(E\in\delta E=[e_\textrm{min};e_\textrm{max}]\) and the zenith angle interval \(\theta\in\delta\theta=[\theta_\textrm{min};\theta_\textrm{max}]\). The weighted number of events \(n'(\delta E, \delta\theta)\) in that interval is then |
| | 89 | |
| | 90 | \[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \omega(E_i, \theta_j)\] |
| | 91 | |
| | 92 | Since \(\omega(E, \theta) = \rho(E)\cdot \tau(\theta) \) with \(\rho(E)\) the spectral weight to adapt the spectral shape of the simulated spectrum to the real (measured) spectrum of the source and \(\tau(\theta)\) the weight to adapt to oberservation time versus zenith angle. |
| | 93 | |
| | 94 | \[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...n}^{E\in\delta E}\rho(E_i)\cdot\sum_{j=0...n}^{\theta\in\delta\theta} \tau(\theta_j)\] |
| | 95 | |
| | 96 | The weighted number of produced events \(N'(\Delta E, \Delta\Theta)\) in the total energy interval \(E\in\Delta E=[E_\textrm{min};E_\textrm{max}]\) and the total zenith angle interval \(\theta\in\Delta\Theta=[\Theta_\textrm{min};\Theta_\textrm{max}]\) is then |
| | 97 | |
| | 98 | \[N'(\Delta E, \Delta\Theta) = \sum_{i=0...N}^{E\in\Delta E}\sum_{j=0...N}^{\theta\in\Delta\Theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...N}^{E\in\Delta E}\rho(E_i)\cdot\sum_{j=0...N}^{\theta\in\Delta\Theta} \tau(\theta_j)\] |
| | 99 | |
| | 100 | with \(N(\Delta E, \Delta\Theta)\) being the total number of produced events. |
| | 101 | |
| | 102 | The weights are defined as follow: |
| | 103 | |
| | 104 | \[\rho(E) = \rho_0\frac{\phi_\textrm{src}(E)}{\phi_0(E)}\] |
| | 105 | |
| | 106 | where \(\phi_0(E)\) is the simulated spectrum and \(\phi_\textrm{src}\) the (unknown) real source spectrum. \(\rho_0\) is a normalization constant. The zenith angle weights \(\tau(\delta\theta)\) in the interval \(\delta\theta\) are defined as |
| | 107 | |
| | 108 | \[\tau(\delta\theta) = \tau_0\frac{\Delta T(\delta\theta)}{N(\delta\theta)}\] |
| | 109 | |
| | 110 | where \(N(\delta\theta)\) is the number of produced events in the interval \(\delta\theta\) and \(\Delta T(\delta\theta)\) is the total observation time in the same zenith angle interval. \(\tau_0\) is the normalization constant. |
| | 111 | |
| | 112 | As the efficiency \(\epsilon(\delta E)\) for an energy interval \(\delta E\) is calculated as |
| | 113 | |
| | 114 | \[\epsilon(\delta E) = \epsilon(\delta E, \Delta\Theta) = \frac{N'_\textrm{exc}(\delta E, \Delta\Theta)}{N'_\textrm{src}(\delta E,\Delta\Theta)}\] |
| | 115 | |
| | 116 | and \(N'_\textrm{exc}(\delta E,\Delta\Theta)\) and \(N'_\textrm{src}(\delta E,\Delta\Theta)\) are both expressed as the sum given above, the constants \(\rho_0\) and \(\tau_0\) cancel. |
| | 117 | |
| | 118 | The differential flux in an energy interval \(\delta E\) is then given as |
| | 119 | |
| | 120 | \[\phi(\delta E) = \phi(\delta E,\Delta\Theta) = \frac{1}{A_0\cdot \Delta T}\frac{M_\textrm{exc}(\delta E)}{\epsilon(\delta E)\cdot \delta E} = \frac{M_\textrm{exc}(E)}{N'_\textrm{exc}(E)}\cdot \frac{N_\textrm{src}(E)}{A_0\cdot\Delta T}\] |
| | 121 | |
| | 122 | |
| | 123 | Where \(A_0\) is total area of production and \(\Delta T\) the total observation time. The number of measured excess events is in that energy interval is \(M_\textrm{exc}(\delta E)=M_\textrm{exc}(\delta E, \Delta\Theta)\). |
| | 124 | |
| | 125 | |
| | 126 | |
| | 127 | |
| | 128 | |
| | 129 | |
| | 130 | |
| | 131 | |
| | 132 | === Monte Carlo === |
| | 133 | |
| | 134 | |
| 87 | | \[\omega_i(\Delta \theta) = \frac{\sum_{\Delta\theta}\Delta T_i}{\sum_\theta\Delta T_i} \cdot \frac{N_0}{N_0(\Delta \theta)} \] |
| 88 | | |
| 89 | | |
| 90 | | |
| 91 | | \(n(\delta E, \delta\theta)\) is the number of produced events in the energy interval \(E\in\delta E=[e_\textrm{min};e_\textrm{max}]\) and the zenith angle interval \(\theta\in\delta\theta=[\theta_\textrm{min};\theta_\textrm{max}]\). The weighted number of events \(n'(\delta E, \delta\theta)\) in that interval is then |
| 92 | | |
| 93 | | \[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \omega(E_i, \theta_j)\] |
| 94 | | |
| 95 | | Since \(\omega(E, \theta) = \rho(E)\cdot \tau(\theta) \) with \(\rho(E)\) the spectral weight to adapt the spectral shape of the simulated spectrum to the real (measured) spectrum of the source and \(\tau(\theta)\) the weight to adapt to oberservation time versus zenith angle. |
| 96 | | |
| 97 | | \[n'(\delta E, \delta\theta) = \sum_{i=0...n}^{E\in\delta E}\sum_{j=0...n}^{\theta\in\delta\theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...n}^{E\in\delta E}\rho(E_i)\cdot\sum_{j=0...n}^{\theta\in\delta\theta} \tau(\theta_j)\] |
| 98 | | |
| 99 | | The weighted number of produced events \(N'(\Delta E, \Delta\Theta)\) in the total energy interval \(E\in\Delta E=[E_\textrm{min};E_\textrm{max}]\) and the total zenith angle interval \(\theta\in\Delta\Theta=[\Theta_\textrm{min};\Theta_\textrm{max}]\) is then |
| 100 | | |
| 101 | | \[N'(\Delta E, \Delta\Theta) = \sum_{i=0...N}^{E\in\Delta E}\sum_{j=0...N}^{\theta\in\Delta\Theta} \rho(E_i)\cdot\tau(\theta_j) = \sum_{i=0...N}^{E\in\Delta E}\rho(E_i)\cdot\sum_{j=0...N}^{\theta\in\Delta\Theta} \tau(\theta_j)\] |
| 102 | | |
| 103 | | with \(N(\Delta E, \Delta\Theta)\) being the total number of produced events. |
| 104 | | |
| 105 | | The weights are defined as follow: |
| 106 | | |
| 107 | | \[\rho(E) = \rho_0\frac{\phi_\textrm{src}(E)}{\phi_0(E)}\] |
| 108 | | |
| 109 | | where \(\phi_0(E)\) is the simulated spectrum and \(\phi_\textrm{src}\) the (unknown) real source spectrum. \(\rho_0\) is a normalization constant. The zenith angle weights \(\tau(\delta\theta)\) in the interval \(\delta\theta\) are defined as |
| 110 | | |
| 111 | | \[\tau(\delta\theta) = \tau_0\frac{\Delta T(\delta\theta)}{N(\delta\theta)}\] |
| 112 | | |
| 113 | | where \(N(\delta\theta)\) is the number of produced events in the interval \(\delta\theta\) and \(\Delta T(\delta\theta)\) is the total observation time in the same zenith angle interval. \(\tau_0\) is the normalization constant. |
| 114 | | |
| 115 | | As the efficiency \(\epsilon(\delta E)\) for an energy interval \(\delta E\) is calculated as |
| 116 | | |
| 117 | | \[\epsilon(\delta E) = \epsilon(\delta E, \Delta\Theta) = \frac{N'_\textrm{exc}(\delta E, \Delta\Theta)}{N'_\textrm{src}(\delta E,\Delta\Theta)}\] |
| 118 | | |
| 119 | | and \(N'_\textrm{exc}(\delta E,\Delta\Theta)\) and \(N'_\textrm{src}(\delta E,\Delta\Theta)\) are both expressed as the sum given above, the constants \(\rho_0\) and \(\tau_0\) cancel. |
| 120 | | |
| 121 | | The differential flux in an energy interval \(\delta E\) is then given as |
| 122 | | |
| 123 | | \[\phi(\delta E) = \phi(\delta E,\Delta\Theta) = \frac{1}{A_0\cdot \Delta T}\frac{M_\textrm{exc}(\delta E)}{\epsilon(\delta E)\cdot \delta E}\] |
| 124 | | |
| 125 | | Where \(A_0\) is total area of production and \(\Delta T\) the total observation time. The number of measured excess events is in that energy interval is \(M_\textrm{exc}(\delta E)=M_\textrm{exc}(\delta E, \Delta\Theta)\). |
| 126 | | |
| 127 | | |
| 128 | | |
| 129 | | |
| 130 | | |
| 131 | | |
| 132 | | |
| 133 | | |
| 134 | | === Monte Carlo === |
| | 137 | \[\omega_i(\Delta \theta) = \frac{\sum_{\Delta\theta}\Delta T_i}{\sum_\theta\Delta T_i} \cdot \frac{M_0}{M_0(\Delta \theta)} \] |
