## 2 Methodology
### Simulation Model
#### 2.3.2 Optics Module  
In remote sensing, ground sample distance (GSD) is used as a metric of image resolution and refers to the distance between the centers of digital photo pixels projected on the ground. Other terms, such as ground-projected sample interval (GSI) and ground-projected instantaneous field of view (GIFOV) may be used interchangeably [47]. This separation requirement for resolving two ground points can be translated into the satellite aperture diameter if the optics system is limited only by  
Figure 4: ReCon simulations: (**a**) 3D view; (**b**) 2D view.  
Figure 3: Walker pattern of Globalstar satellite constellation.  
diffraction, not by the lens imperfections or the pixel size [48]. Diffraction limit is illustrated as \(d^{\prime}\) in Figure 5 where the standard Rayleigh diffraction criterion limits angular resolution to:  
\[\theta=1.22\frac{\lambda}{D}\ \, \tag{4}\]  
The angular resolution (\(\theta\)) is proportional to the wavelength (\(\lambda\)) and inversely proportional to the aperture diameter (\(D\)). The same angular resolution may also be expressed in term of the radius of the first Airy ring, i.e., the distance between the first lobes from the center of point spread function:  
\[\theta=\frac{d^{\prime}}{2f}=\frac{x}{2R_{\mathrm{s}}}\ \, \tag{5}\]  
where \(f\) is the focal length, \(h\) is the satellite altitude, and \(x\) is the ground sample distance (GSD) when the satellite is pointing perpendicular to the ground (nadir direction).  
When the observation payload is oriented at the edge of its field of regard, tilted with an off-nadir angle \(\eta\), the satellite-to-ground distance increases to \(R_{\mathrm{s}}=h/\cos(\eta)\). Retaining the same resolution therefore requires a larger aperture than the nadir case, which can be calculated by replacing \(h\) with \(R_{\mathrm{s}}\) in Equation (6) [49]:  
\[D=2.44\frac{\lambda R_{s}}{x}=2.44\frac{\lambda h}{xcos\eta}\ \, \tag{6}\]  
The ground projection of a pixel (\(x\)) is also elongated into GSD per relationship \(x=\mathrm{GSDcos}(\eta)\):  
\[D=2.44\frac{\lambda h}{GSDcos^{2}\eta}\ \, \tag{7}\]  
where GSD = 1 m and \(\lambda\) = 500 nm. This aperture diameter is used to calculate the mass of the sensor payload, including the primary mirror, optical telescope assembly (OTA), imagers, and supporting mechanical and electronic components. The mass calculation is done by considering an empirical relationship between the payload mass and the aperture size, as shown in Equation (8) where the mass is in kilograms and the aperture diameter is in meters. Similarly, Equation (9) describes the relationship between the aperture size and the non-sensor platform mass of a satellite. Unlike \(m_{dry,optics}\) which increases super-linearly (exponent greater than 1) with aperture in Equation (7), \(m_{dry,non-optics}\)  
Figure 5: Optical relationship of ground sample distance (GSD) with other parameters (not to scale).  
increases almost linearly (exponent close to 1) with aperture. The data used to derive the power law relationship is provided in Appendix B[50].  
\[m_{sensor\_payload}=m_{dry,optic}=418D^{1.37}[\mathrm{kg}]\enspace, \tag{8}\]  
\[m_{platform}=m_{dry,non-optic}=754D^{1.03}[\mathrm{kg}]\enspace, \tag{9}\]