## 5 Effective Area  
The effective area of a gamma-ray detector accounts for the detection efficiency of the instrument and the observable area. The full-energy peak effective area, \(A_{eff}\) (cm\({}^{2}\)), for a detector is determined as [38]  
\[A_{Eff}=\frac{C}{a_{c}P_{T}}4\pi s^{2}\,, \tag{7}\]  
where \(C\) is the number of counts in the photopeak, i.e., the area under the Gaussian, \(P_{T}\) is the transition probability, \(s\) is the distance (cm) between the source and the detector and \(a_{c}\) is the activity of the radioactive source (1/s). The activity is given by  
\[a_{c}=a_{o}e^{-\lambda t}=a_{o}e^{-\frac{\frac{1m(2)t}{t}}{\frac{1}{2}}}\,, \tag{8}\]  
where \(a_{o}\) is the activity of the source (s\({}^{-1}\)) at a specified time, t is the time (s) between the determination of the activity and the performance of the test campaign and \(T_{\frac{1}{2}}\) is the half-life of the source (s). As the source is emitting gamma rays in all directions but only a fraction is observable to the detector, the term \(4\pi s^{2}\) accounts for the fraction of photons emitted by the source that will actually come in contact with the detector. Figure 12 presents the effective area of the flight model of GMOD as a function of energy, where all the sources were positioned above the +Z face of the spacecraft at the CoC. The source activity must be known for the determination of an absolute effective area, which limited the sources in Table 1 that could be used to determine GMOD's effective area. Sources 1-3, 5 and 6 were used to determine the effective area of GMOD as a function of energy. Table 2 provides the position of these sources relative to the crystal centre, the \(s\) term in Equation (7). In Figure 12, the previously simulated effective area (blue) from [29] is in good agreement with the physical measurements (see Section 7.2 for the updates made to the model to represent the FM including the addition of the GMOD top reinforcement bracket (GTRB), the structural bracket the detector assembly is mounted on). The instrument's sensitivity appears to peak in the range of tens to a few hundred keV, as is desired for the detection of the prompt emission of GRBs. The effective area is lower for smaller energies as a result of gamma-ray absorption by the detector housing and spacecraft. The effective area of FM GMOD as a function of source incident angle was also investigated. As described in Section 2.2, due to time constraints, two sources, \({}^{137}\)Cs and \({}^{241}\)Am, were used together for the angular measurements. The effective area of GMOD was determined at 34 different incident angles, as listed in Table 3. Figure 13 presents the angular response of GMOD at 662 keV and Figure 14 at 59.4 keV. The angular response of GMOD at both these energies follows similar profiles. As shown in Figures 13 and 14, the calculated values of the effective area have an uncertainty of 6-7%, which is due to an uncertainty of the source activity (5% for the \({}^{241}\)Am source and 3.5% for the \({}^{137}\)Cs source) and errors in the source-to-crystal distance (about 5 mm). The measurements at polar angle 90\({}^{\circ}\) show the most diversity and the impact of the crystal geometry on the detector's performance. The area of the Y and Z faces of the GMOD crystal is 10 cm\({}^{2}\), and the X faces are 6.25 cm\({}^{2}\). The effective area of GMOD peaks at the \(-\)Y and \(+\)Y faces, which correspond to the longer faces of the crystal. The only obstacle between the source and crystal at these faces are the spacecraft panels with mounted solar arrays. At the \(-\)X face, the effective area drops by \(\sim\) 0.2 cm\({}^{2}\), as this face of the crystal is smaller. The effective area is at a minimum at the \(+\)X face as the gamma-ray photons are attenuated by the PCBs (SIPHRA and adapter boards) before reaching the crystal. For the other polar angles, the effective area tends to be at a minimum at azimuthal angle 0\({}^{\circ}\) (\(+\)X face).