# Modeling and Calculation of Limit Magnitude Detection of Orbital Optoelectric Tracking System
## 3 The Limit Magnitude Detection Capability of the Orbital Optoelectric Tracking System Modeled
### Optical Signal Characterization of Noise  
Suppose the target radiated photon noise is \(n_{1}\), the background radiated photon noise is \(n_{2}\), and CCD sensing surface dark current noise is \(n_{3}\). The target radiation photon noise is the incident photon flow caused by the random undulation of the detection circuit when the target comes into the optical detection system, and \(n_{1}=(N_{s})^{1/2}\); the equivalent photon number of the background radiation noise is \(n_{2}=(N_{b})^{1/2}\), wherein \(N_{b}\) is the number of photogenerated electrons by the background; and the dark current noise is a random process signal caused by the carrier thermal effect that is a kind of white noise, and its electron number \(n_{3}\) is equal to the square root of the number of the dark current electrons \(N_{d}\), expressed as \(n_{3}=(N_{d})^{1/2}\). The equivalent noise voltage output from the orbital optoelectronic tracking system is  
\[V_{n}=\frac{1}{C_{r}}\delta_{t}A_{V}\sqrt{N_{s}+N_{d}+N_{b}} \tag{19}\]  
Calculation Model and Analysis for Limit Magnitude of Orbital Optoelectric Tracking System 3.3.1. Limit Magnitude Model under Unillumination Conditions  
The detection capability of the orbital optoelectronic tracking system can be measured by the signal-to-noise ratio:  
\[SNR=\frac{V_{o}}{V_{n}}=\frac{\pi D^{2}\cdot t_{0}\cdot\tau_{0}\cdot\int_{ \lambda_{1}}^{\lambda_{2}}\eta(\lambda)\cdot\frac{\lambda}{hc}E_{b}(\lambda) d\lambda}{4 MA_{V}^{2}\sqrt{N_{s}+N_{d}+N_{b}}} \tag{20}\]  
From the signal-to-noise ratio Equation (20), it can be seen that if the \(SNR\) is larger, the detection performance and the detection capability of the optical detection system are stronger. But for the orbital optoelectric tracking system, in order to stably obtain the target image information at each moment on the full ballistic trajectory, it mainly relies on the detection capability of the system, in addition to the stability of the rotary mechanism. Assuming that the minimum detectable signal-to-noise ratio threshold of the system is \(K_{\text{min}}\), the inequality (21) can be established from the Equation (20).  
\[N(\lambda)\geq(K_{\text{min}^{2}}+\sqrt{K_{\text{min}^{2}}+4(N_{d}+N_{b})K_{ \text{min}^{2}}})/2 \tag{21}\]  
From the relationship between the magnitude and the luminosity, the radiant illuminance, the luminosity \(E_{\text{m}}\) of the target on the incident surface of the optical system, is obtained as  
\[E_{m}=E_{0}\cdot 2.512^{-m}lx \tag{22}\]  
where \(E_{0}\) is the luminosity of the 0 limit magnitude and \(E_{0}=2.65\times 10^{-6}lx\). The luminosity \(E_{\text{m}}\) should satisfy the condition of Formula (23)  
\[E_{m}=E_{0}\cdot 2.512^{-m}lx\geq\frac{2 Mhc}{\lambda^{*}}\cdot\frac{K_{ \text{min}^{2}}+\sqrt{K_{\text{min}^{4}}+4(N_{d}+N_{b})K_{\text{min}^{2}}}}{t_ {0}\cdot\tau_{0}\cdot\eta\cdot\tau D^{2}}\text{W}/\text{m}^{2} \tag{23}\]  
where \(\lambda^{*}\) is the average value of the detection wavelength of the system (usually \(\lambda^{*}=580\) nm), and \(\eta\) is the spectral quantum efficiency of the CCD detector at the average value of the detection wavelength.  
With the above analysis, the limit magnitude \(m\) of the system is  
\[m\leq 1.0857\cdot[-10.392-\lg(\frac{2 Mhc}{\lambda}\cdot\frac{K_{\text{min}^{2}}+ \sqrt{K_{\text{min}^{4}}+4(N_{d}+N_{b})K_{\text{min}^{2}}}}{t_{0}\cdot\tau_{0} \cdot\eta\cdot\tau D^{2}})] \tag{24}\]  
From Equation (24), it can be seen that the limit magnitude of the system is related to the signal-to-noise ratio threshold, the dark current, the background noise, and the optical lens parameters, wherein \(h\) is Planck's constant and \(c\) is the speed of light.  
If the orbital optoelectronic tracking system detects under unillumination conditions, the background noise can be neglected, which is 0, and Equation (24) can be transformed into Equation (25).  
\[m\leq 1.0857\cdot[-10.392-\lg(\frac{2 Mhc}{\lambda}\cdot\frac{K_{\text{min}^{2} }+\sqrt{K_{\text{min}^{4}}+4N_{d}K_{\text{min}^{2}}}}{t_{0}\cdot\tau_{0}\cdot \eta\cdot\tau D^{2}})] \tag{25}\]