# 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
### Optoelectric Signal Characterization of Dynamic Targets  
The dynamic target in the orbital optoelectric tracking system does not emit light itself, and the radiant brightness of the target comes from the radiation of the sun, which mainly generates brightness information on the detection system by reflecting the solar radiation. Therefore, during the target signal producing process, it is affected by the attributes of the sunlight, target, optical lens, and CCD sensing surface, so the voltage of the target signal is modeled from these aspects. Let the spectral irradiance of the sun be \(E(\lambda)\) and \(\lambda\) be the wavelength, and then the solar radiation received by the dynamic target above the element \(ds\) is  
\[dF_{1}=\int_{\lambda}E(\lambda)ds\cos\phi d\lambda \tag{10}\]  
where \(\phi\) is the angle between the sunlight direction and the normal direction of the dynamic target surface element \(ds\). If the diffuse reflection coefficient of the dynamic target surface element \(ds\) is \(c(\lambda)\), then the reflecting light luminous flux of the dynamic target surface element \(ds\) is  
\[dF_{2}=\int_{\lambda}c(\lambda)E(\lambda)ds\cos\phi d\lambda \tag{11}\]  
Considering the dynamic target surface element \(ds\) as a fully extended surface, the luminous flux of the CCD receiving surface element \(ds^{\prime}\) at an angle \(\theta\) in the normal direction and at a distance \(R\) is  
\[dF=\frac{\cos\theta}{\pi R^{2}}\int_{\lambda}c(\lambda)E(\lambda)ds\cos\phi d \lambda ds^{\prime} \tag{12}\]  
Integrating over the entire irradiated receiving surface element \(ds^{\prime}\) for a dynamic target, the detection surface illuminance corresponding to the target's spectral properties is obtained as  
\[E_{m}=\sum_{i}\frac{c(\lambda)E_{0}}{\pi R^{2}}\int_{s}\cos\theta\cos\phi ds \tag{13}\]  
where \(E_{0}\) is the illuminance of the sun reaching the detection surface and \(i\) is the serial number of a plane, sphere, column, cone, etc. Based on the optical detection principle and the illumination of the target, the spectral radiance obtained by the orbital optoelectronic tracking system is  
\[E_{1}=\int_{\lambda_{1}}^{\lambda_{2}}E_{m}(\lambda)d\lambda \tag{14}\]  
In Equation (14), \(\lambda_{1}\) and \(\lambda_{2}\) are the spectral wavelength ranges to which the imaging CCD detector responds. According to the effective aperture of the optical lens, if \(\Phi\) is the total luminous flux received in the orbital optoelectronic tracking system, it can be expressed by Formula (15). \[\Phi=\tau_{0}\cdot\int_{\lambda_{1}}^{\lambda_{2}}\frac{1}{4}\cdot\pi D^{2} \cdot E_{b}(\lambda)d\lambda \tag{15}\]  
where \(D\) is the effective aperture diameter of the optical lens if the optical transmittance of the optical lens is \(\tau_{0}\). Assuming that the CCD exposure time is \(t_{0}\), the total radiant energy acquired on the light-sensitive surface of the CCD detection element is  
\[Q=t_{0}\cdot\tau_{0}\cdot\int_{\lambda_{1}}^{\lambda_{2}}\frac{1}{4}\cdot\pi D ^{2}\cdot E_{b}(\lambda)d\lambda \tag{16}\]  
According to the principle of CCD-photocoupled detection, assuming that the spectral quantum efficiency of the CCD detector is \(\eta(\lambda)\), the number of spectral photogenerated electrons on the CCD sensing surface is  
\[N(\lambda)=\frac{1}{4M}\cdot\pi D^{2}\cdot t_{0}\cdot\tau_{0}\cdot\int_{ \lambda_{1}}^{\lambda_{2}}\eta(\lambda)\cdot\frac{\lambda}{hc}E_{b}(\lambda)d\lambda \tag{17}\]  
where \(M\) is the amount of pixels of the target imaging coverage energy of the CCD, and the voltage output from the CCD photoreceptor is  
\[V_{o}=\frac{1}{4M}\cdot\pi D^{2}\cdot\frac{e\cdot\delta_{t}\cdot t_{0}\cdot \tau_{0}}{A_{V}\cdot C_{r}}\cdot\int_{\lambda_{1}}^{\lambda_{2}}\eta(\lambda) \cdot\frac{\lambda}{hc}E_{b}(\lambda)d\lambda \tag{18}\]In Equation (18), \(\delta_{t}\) is the total charge transfer efficiency, \(A_{V}\) is the amplifier gain, \(C_{t}\) is the readout equivalent capacitance, and \(e\) is the electron charge.