Changes between Version 77 and Version 78 of DatabaseBasedAnalysis/Spectrum

Timestamp:

12/05/19 09:31:57 (5 years ago)

Author:

tbretz

Comment:

--

DatabaseBasedAnalysis/Spectrum

@@ -19 +19 @@
 Note that the exact calculation of the efficiency \(\epsilon(\Delta E)\) depends on prior knowledge of the correct source spectrum \(N_0\). Therefore, it is strictly speaking only correct if the simulated spectrum and the real spectrum are identical. As the real spectrum is unknown, special care has to be taken of the systematic introduced by the assumption of \(N_0\).
 
+=== Theta and Spectral Weights ===
+
+While the source spectrum \(N_0\) is of course independent of the zenith angle \(\theta\) of the observation, the efficiency is not, so that
+
+\[\epsilon(E) = \epsilon(E, \theta)\]
+
+In addition, the observed differential flux depends on the zenith angle, so that
+
+\[\epsilon(E, \theta) = \epsilon(E, \theta, \Delta T(\theta))\]
+
+That means that for a correct calculation of the efficiency, the number of simulated events per zenith angle interval has to match the observation time distribution. This can be achieved by applying zenith angle dependent weights \(\omega(\theta)\) to each simulated event. Similarly, the discussed requirement of the match between result spectrum and simulated spectrum can be achieved applying energy dependent weights \(\omega(E)\).
+
+As the simulated energy spectrum is independent of zenith angle, it can be expressed as
+
+\[N_0(E,\theta) = N_0\cdot \frac{\eta(E)}{\int_E\eta(E)dE}\cdot \frac{\eta(\theta)}{\int_\theta\eta(\theta)d\theta}\]
+
+with the differential energy spectrum \(\eta(E)\) and the zenith angle distribution \(\eta(\theta)\). 
+
+
 === Excess ===
+
 The number of excess events, for data and simulations, is defined as
 
@@ -39 +59 @@
 
 with \(\sigma^2(N_\textrm{sig,bg}) = N_\textrm{sig,bg}\).
+
+For the simulations where \[\sigma^2(N) = \sum\omega_i^2(E)\omega_i^2(\theta)\] this yields
+
+\[\sigma(N)^2 = \]
 
 
@@ -93 +117 @@
 As the simulated energy spectrum is independent of zenith angle, it can be expressed as
 
+\[N_0(E,\theta) = N_0\cdot \fract{\eta(E)}{\int_E\eta(E)dE}\cdot \fract{\eta(\theta)}{\int_\theta\eta(\theta)d\theta}\]
+
+with the differential energy spectrum \(\eta(E)\) and the zenith angle distribution \(\eta(\theta)\). 
+
+
+
 \[dN_0(E,\theta) = N_0\cdot d\eta(E)\cdot d\eta(\theta)\]
-
-with the differential energy spectrum \(\eta(E)\) and the zenith angle distribution \(\eta(\theta)\), for the total production range \(E_\textrm{min}\)  to \(E_\textrm{max}\) and \(\theta_\textrm{min}\) to \(\theta_\textrm{max}\)