## 2 System Modeling  
### Solar Irradiance: Theoretical Approach  
Configuring the optimal path for O-ISLs involves extensive calculations, and considering all the satellites in a constellation can significantly increase the model's complexity. To reduce the computational load, this study proposes an approach to classify satellites affected by solar irradiance from those unaffected. Conventionally, the region on Earth illuminated by sunlight is referred to as half the surface of the Earth. However, sunlight can extend beyond the half, particularly as satellites operate hundreds of kilometers above the ground. To address this, Equations (1) and (2) are introduced as follows:  
\[\sqrt{p(p-d_{sun})(p-r_{s}-r_{E})(p-d_{max}\ )}=\frac{1}{2}\times d_{max}\times r _{E} \tag{1}\]  
\[p=\frac{1}{2}(r_{s}+r_{E}+d_{sun}+d_{max}). \tag{2}\]  
These equations numerically derive \(d_{max}\), the maximum distance sunlight can reach, accounting for a satellite's altitude. The area of a triangle can be determined using [PERSON]'s law, which defines it in terms of the semi-perimeter (\(p\)), and \(d_{max}\) is calculated by equating [PERSON]'s law with the typical triangle area formula. For this, the triangle is formed by the satellite's altitude (\(r_{s}\) ) plus Earth's radius (\(r_{s}\)), the distance from the satellite to the sun (\(d_{max}\)), and the Earth-sun distance (\(d_{sun}\)). Figure 1 illustrates this approach, and the results are shown in Figure 2. As shown in Figure 2a, representing spring and fall, and Figure 2b, representing summer, the illuminated region varies by season. The altitudes of the five shells in the inter-shell of the Starlink's satellite constellation, detailed in Section 2.3, are 540 km, 550 km, 560 km, and 570 km. The gaps between \(d_{max}\) and 1 AU are confirmed as 2678 km, 2704 km, 2729 km, and 2755 km, respectively. This analysis validates that the illuminated area encompassing the satellites (yellow) is larger than the region that covers half the Earth (orange). To optimize O-ISL paths under solar irradiance, interference effects must be accurately identified. There are corresponding scenarios to occur, including (i) the receiver operating successfully despite the solar illumination and (ii) the receiver being affected and avoiding the affected nodes. Figure 1: Definition of the maximum illumination distance using the triangle from Heron’s law. To simulate the interference effects, an SNR analysis is applied, considering the typical wavelength range used in O-ISLs. The simulation incorporates practical components, such as avalanche photodiodes (APDs) [9] and lasers [10], which focus on performance at 1550 nm. The background current (\(I_{b}\)), caused by sunlight, is defined as follows [11]:  
\[I_{b}=A_{d}M_{FoV}\int_{1000}^{1700}P_{D}^{S}(\lambda)R(\lambda)T_{sig}(\lambda )d\lambda, \tag{3}\]  
where \(P_{D}^{S}\) is the solar irradiance (W/m\({}^{2}\)/nm), \(R\) is the responsivity (A/W), \(A_{d}\) is the APD's aperture (mm\({}^{2}\)), and \(T_{sig}\) is the bandpass filter coefficient. The transmitted signal current (\(I_{x}\)) is defined as follows:  
\[I_{x}=\frac{(m+1)A_{d}M_{FoV}}{2\pi d^{2}}\int_{1000}^{1700}P_{T}^{L}(\lambda) R(\lambda)T_{sig}(\lambda)d\lambda, \tag{4}\]  
where \(m\) is the Lambertian order, \(d\) is the typical O-ISL laser propagation distance (km), and \(P_{T}^{L}(\lambda)\) is the transmission power as follows:  
\[P_{T}^{L}(\lambda)=P_{Max}^{L}(\lambda)\frac{P^{L}(\lambda)}{\int_{1000}^{1700 }P^{L}(\lambda)d\lambda}, \tag{5}\]  
where \(P_{Max}^{L}(\lambda)\) is the maximum transmission power, and \(P^{L}(\lambda)\) is the wavelength-specific power distribution.