## 7 Results
### Sensor Comparison  
#### 7.3.1 Limiting Magnitude  
Cameras with different capabilities and price ranges were employed for RSO and star observations during this mission. Before the mission commenced, the limiting magnitude of the stars for each sensor was estimated. This magnitude denotes the faintest star imaged by the sensor. The sensor's limiting magnitude can be computed by finding the magnitude of an object where the Signal-to-Noise Ratio (SNR) is the minimum for detectability. The magnitude of an RSO can be calculated using the irradiance equation from [36]. Equations (1) and (2) show the steps for calculating the photon-based irradiance. \[E_{RSO-Power}=1.78\times 10^{-8}\cdot 10^{-0.4m_{obj}}[\mathrm{W/m^{2}}] \tag{1}\]  
\[E_{RSO}=\frac{E_{RSO-Power}}{hc}=\frac{1.78\times 10^{-8}\cdot 10^{-0.4m_{obj} }\lambda}{hc}[\mathrm{ph/s/m^{2}}] \tag{2}\]  
In Equations (1) and (2), \(E_{RSO-Power}\) refers to the irradiance in units of power per unit area, \(E_{RSO}\) refers to the photon-based irradiance, \(m_{obj}\) is the object magnitude, \(\lambda\) is the wavelength, \(h\) is [PERSON]'s constant, and \(c\) is the speed of light. The photon-based irradiance can be used in the signal photoelectron equation from [36] as shown in Equations (3) and (4). \[e_{s}=Q_{e}\tau A\tau_{atm}E_{RSO}t \tag{3}\]  
\[e_{s}=Q_{e}\tau A\tau_{atm}t\frac{1.78\times 10^{-8}\cdot 10^{-0.4m_{obj}} \lambda}{hc} \tag{4}\]  
In Equations (3) and (4), \(e_{s}\) is the number of signal photoelectrons, \(Q_{e}\) is the sensor's quantum efficiency, \(\tau\) is the optical transmittance, \(\tau_{atm}\) is the atmospheric transmittance, A is the aperture area, and \(t\) is integration time. We reorganize Equation (4) and assume the optical and atmospheric transmittance values equal to one result in the equation for the object magnitude as shown in Equation (5). Figure 10: The graph illustrates temperature fluctuations of various components during the mission flight on 22 August 2023, from 4:52 a.m. to 9:34 a.m. (UTC). Notably, both the payload and the environment experienced significant temperature changes in the initial two hours. Subsequently, the temperature of all components stabilized as the flight coasted at the targeted altitudes, with only minor temperature fluctuations observed. \[m_{obj}=-2.5\cdot\lg\biggl{(}\frac{e_{s}hc}{1.78\times 10^{-8}Q_{e}A\lambda t} \biggr{)} \tag{5}\]  
As shown in [37], the SNR can be estimated as in Equation (6) and an updated expression for the signal photoelectrons is obtained in (7). \[SNR=\sqrt{e_{s}} \tag{6}\]  
\[e_{s}=SNR^{2} \tag{7}\]  
By substituting the expression for \(e_{s}\) as per Equation (7), the limiting magnitude expression can be updated as shown in Equation (8). \[m_{lim}=-2.5\cdot\lg\biggl{(}\frac{SNR^{2}hc}{1.78\times 10^{-8}Q_{e}A\lambda t }\biggr{)} \tag{8}\]  
Assuming a minimum SNR of 5 for ensuring reliable detection of faint objects in CCD imaging [38], Equation (9) shows the limiting magnitude equation used in this study. \[m_{lim}=-2.5\cdot\lg\biggl{(}\frac{25 hc}{1.78\times 10^{-8}Q_{e}A\lambda t} \biggr{)} \tag{9}\]  
Equation (9) is simplistic and does not account for complexities such as readout noise. Thus, it can be easily applied when there is limited sensor information as is with many commercially available cameras. Factors such as the motion of the RSO are not considered in Equation (9). This is acceptable because the equation is used to provide an estimate of the sensor performance prior to capturing images were the motion of RSOs would need to be taken into account. Figure 11 displays a plot of limiting magnitudes for a range of integration times determined using Equation (9). The vertical line indicates an integration time of 100 ms, which was the value used during the mission. The images obtained at the end of the mission from subpayloads 1 and 2 were processed with Astrometry.net [39] to compute the World Coordinate System (WCS) and the field distortions present in them.