# Modeling and Calculation of Limit Magnitude Detection of Orbital Optoelectric Tracking System
## 4 Calculation and Experiment Analysis  
### Computation Analysis  
According to the detection and tracking principle of the orbital optoelectronic tracking system and the calculation model of the limit magnitude, under different lighting conditions, the relationship between the limit magnitude and the exposure time of the CCD camera, the relationship between the limit magnitude and the signal-to-noise ratio threshold of the system, and the relationship between the limit magnitude and the dark current of the system are calculated. According to the geometric relationships of the system in Figure 2, assuming that the average wavelength of the CCD detector sensing is 580 nm, the average imaging pixel number of the target is \(M=6\) during its entire movement on the trajectory; the transmittance rate of the optics is \(\tau_{0}=0.88\); the aperture diameter of the optical lens is \(D=200\) mm; the electron number in the dark current is \(N_{d}=150e^{-}\); and the spectral quantum efficiency of the CCD detector in the average detection wavelength is \(\eta=0.65\). If the minimum detectable signal-to-noise ratio threshold of the system is \(K_{\rm min}=3.5\), [PERSON]'s constant is \(h=6.6260693\times 10^{-34}\) J s, the light velocity is \(c=3.0\times 10^{8}\) m/s, the exposure time of the CCD camera is \(t_{0}=0.3\) s, and the background illuminance is \(2\times 10^{4}\) cd/m\({}^{2}\), then according to Equation (25), the detectable limit magnitude can be calculated to be 10.4 magnitudes. If the CCD camera exposure time, the system's signal-to-noise ratio threshold, and the dark current are changed, their corresponding change curves can be obtained according to Equations (25) and (29). Figure 3a shows the relationship between the limit magnitude detection and the SNR threshold \(K_{\rm min}\), Figure 3b shows the relationship between the limit magnitude detection and the exposure time of the CCD camera, Figure 3c shows the relationship between the limit magnitude detection and the dark current of the CCD camera, and Figure 3d shows the relationship between the limit magnitude detection and the background lumination. From Figure 3a, it can be seen that lowering the signal-to-noise ratio threshold can improve the system's limit magnitude detection, which is conducive to the tracking of the target. Especially in the background environment of the relatively strong light conditions, the higher the signal-to-noise ratio, the better the limit magnitude detection performance and the more stable the system tracking target. The lower signal-to-noise ratio threshold can be selected under the conditions of the higher signal-to-noise ratio. From Figure 3b, it can be seen that the longer the exposure time of the CCD sensor, the higher the limit magnitude detection of the system, which indicates that in the tracking system, choosing a suitable exposure time can also improve the limit magnitude detection capability. From Figure 3c, it can be seen that reducing the dark current in the system can increase the limit magnitude detection, and the detection capability is improved. But from the curve change rate, the influence of the dark current change on the limit magnitude detection is small. The change in the limit magnitude detection of the system is only 0.2 magnitude when the dark current is changed from \(100e^{-}\) to \(160e^{-}\). Therefore, in order to effectively improve the limit magnitude detection, the exposure time should be increased as much as possible, and the signal-to-noise ratio threshold should be lowered. At the same time, the effective aperture of the optical lens should be increased. Figure 3d gives the relationship between the limit magnitude detection of the system and the exposure time of the CCD camera, as well as the relationship with the signal-to-noise ratio threshold under illuminated conditions. As can be seen from Figure 3d, the limit magnitude detection decreases significantly when the background light is enhanced.