# Research on Space-Based Visible Detection for Conical Space Targets
## 3 Radiation Model of Conical Space Target  
As shown in Figure 2, the radiative transfer model of the conical space target is established. The \(Z\)-axis is the direction of the cone axis pointing to the vertex, the projection of the reflected radiation vector on the coordinate plane is the X-axis, and the Y-axis forms the right-hand rule with the X-axis and the Z-axis [17]. \(r\) is the radius of the bottom of the cone, \(\sigma\) is the conical target inclination which ranges from 0 to \(\pi/2\), \(\overrightarrow{R_{s}}\) is the solar incident radiation vector received by the space target, \(\overrightarrow{R_{d}}\) is the radiation vector reflected by the space target received by the detector, \(\overrightarrow{n}\) is the normal vector of the space target surface bin, \(dA\) is the effective incident cross-sectional area of the space target, \(\Theta_{s}\) is the incident angle between \(\overrightarrow{R_{s}}\) and \(Z\)-axis, and its value range is \((0,\pi/2+\sigma)\), \(\Theta_{d}\) is the observation angle between \(\overrightarrow{R_{d}}\) and \(Z\)-axis, and its value range is \((0,\pi/2+\sigma)\), \(\Theta_{s}^{\prime}\) is the angle between \(\overrightarrow{R_{s}}\) and \(\overrightarrow{n}\), \(\Theta_{d}^{\prime}\) is the angle between \(\overrightarrow{R_{d}}\) and \(\overrightarrow{n}\), \(\varphi\) is the phase angle between the incident direction of sunlight and the direction of the optical axis of the detector, hereinafter referred to as the phase angle, while \(\alpha\) is the spatial azimuth of conical target surfel ranging from \(\alpha_{1}\) to \(\alpha_{2}\), hereinafter referred to as spatial azimuth. Figure 1: Radiation emission of various radiation sources of space targets. interval of \(M\) and \(N\) cannot make the subsequent condition of \(\varphi+M<N\) hold. Therefore, Case 2 does not exist. 3. Case 3: \(-N<\varphi-M<\varphi+M<N\) The critical condition for Case 3 is \(\varphi+M=N\), which is discussed in the following two situations: if \(M=\frac{\pi}{2}\), then \(N=\frac{\pi}{2}+\varphi\); if \(N=\pi\), then \(M=\pi-\varphi\). Therefore, in order to satisfy the condition of \(\varphi+M<N\), the value interval of N is \(\left(\frac{3\pi}{4}+\frac{\varphi}{2},\pi\right)\), and the value interval of M is \(\left(\frac{\pi}{2},\frac{3\pi}{4}-\frac{\varphi}{2}\right)\). At the same time, the value interval of M and N also satisfies the precondition of \(-N<\varphi-M\). 4. Case 4: \(-N<\varphi-M<N<\varphi+M\) One critical condition for Case 4 is \(N=\varphi+M\), as follows: if \(N=\pi\), then \(M=\pi-\varphi\), and the other critical condition is \(-N=\varphi-M\): if \(M=\pi\), then \(N=\pi-\varphi\). Therefore, in order to satisfy the condition of \(-N<\varphi-M<N<\varphi+M\), the value intervals of \(M\) and \(N\) are both \((\pi-\varphi,\pi)\). The value range of the phase angle is defined as \([0,\pi]\). According to the literature research, when the phase angle is about \(70^{\circ}\sim 90^{\circ}\), the detector has the best observation conditions for space targets [20]. Therefore, we set the phase angle as \(75^{\circ}\), \(80^{\circ}\), and \(85^{\circ}\), respectively, to explore the effect of the phase angle on the radiation intensity of the space target received by the detector. The inclination of the conical target also has a certain influence on the incident angle of sunlight. Considering the actual processing situation, the conical target inclination is generally set to \(60^{\circ}\) or \(72^{\circ}\). When the conical target inclination and the phase angle are determined, the value of M is determined only by \(\Theta_{s}\), and the value of N is determined only by \(\Theta_{d}\). The extreme value of the solar radiation intensity reflected by the space target is related to the range of spatial azimuth. The value interval of the spatial azimuth obtained in each case is different, so the extreme value of the radiation intensity, reflected by the space target received by the detector, also changes in different cases.