## 4 Bespoke Space System Design
### Detailed Constellation Design and Simulation  
Based on the coverage estimates made in Section 4.2, a 600 km altitude, 60 deg inclined orbit is selected as a baseline for the constellation. In order to ensure a 14 day revisit over Scotland, a repeating ground-track (RGT) orbit is designed [38]. It is calculated that for a spacecraft in a 60 deg inclined orbit, an altitude of 581.077 km will give a repeat period of 207 orbits in 14 days. At this altitude, the width covered by a single spacecraft pass at 54.63 deg latitude, the lowest latitude of interest, is 15.3 km as calculated using Equations (2) and (3), meaning eight spacecraft would be required for full coverage in 14 days. To ensure an even distribution of coverage, the spacecraft should be organised in a 'train formation' such that one edge of the swath of the first spacecraft passing 54.63 deg latitude is aligned exactly with the opposite edge of the swath of the subsequent spacecraft [39]. The time for Earth to \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Altitude, km** & **GSD, m** & **Swath Width, km** & **Width Covered at 54.63 deg Latitude, km** & **Time to Full Coverage for One** & **No. of s/c for 2-Week Revisit** & **Time to Deorbit** \\ \hline
200 & 0.29 & 2.6 & 5.2 & 270.9 & 20 & 0.003 \\ \hline
300 & 0.43 & 4.0 & 7.9 & 184.8 & 14 & 0.072 \\ \hline
400 & 0.58 & 5.3 & 10.5 & 141.7 & 11 & 0.762 \\ \hline
500 & 0.72 & 6.6 & 13.1 & 115.9 & 9 & 4.748 \\ \hline
600 & 0.86 & 7.9 & 15.7 & 98.7 & 7 & 21.18 \\ \hline
700 & 1.01 & 9.2 & 18.3 & 86.4 & 7 & 74.97 \\ \hline
800 & 1.15 & 10.6 & 21.0 & 77.2 & 6 & 224.1 \\ \hline
900 & 1.30 & 11.9 & 23.6 & 70.1 & 5 & 588.8 \\ \hline
1000 & 1.44 & 13.2 & 26.2 & 64.4 & 5 & 1397.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimated GSD, swath width, revisit times, and deorbit time as a function of altitude for a spacecraft baselined against the SkySat spacecraft. rotate such that the surface at 54.63 deg latitude will move through a distance equal to the width seen by the spacecraft at the latitude of interest, \(W\), can be estimated as  
\[t_{sep}=\frac{2\pi W}{\omega c_{\delta}} \tag{7}\]  
where \(\omega\) is the rotational rate of Earth, and \(c_{\delta}\) is the circumference of the circle of the latitude of interest. The spacecraft should then be separated through true anomaly (i.e., within the orbit plane) such that a following spacecraft will reach the true anomaly of the previous spacecraft in the time \(t_{sep}\). This separation can be calculated, in radians, as  
\[\theta_{sep}=\frac{2\pi t_{sep}}{T} \tag{8}\]  
where \(T\) is the orbit period, as calculated using Equation (4). For spacecraft in a RGT orbit at 581.077 km, this gives a required true anomaly separation between spacecraft of approximately 3.556 deg. An orbit propagator is used to estimate the coverage that could be expected from a constellation of eight spacecraft in a 581.077 km altitude, 60 deg inclined orbit, with a separation between spacecraft of 3.556 deg through true anomaly. Using Equation (5), the spacecraft are calculated to have a swath width of 7.7 km at 581.077 km altitude and the simulation is carried out for a 30 day period. The propagator uses the Gauss/Lagrange planetary equations [40] to calculate the change in the Keplerian orbit elements of the spacecraft over time. J\({}_{2}\), which is the perturbation due to an oblate central body, is the only perturbation included [41]. Drag and other forces acting on the satellite are disregarded as they are considered small when compared to the J\({}_{2}\) effect. The satellite ground track and associated coverage is calculated assuming a spherical Earth. The constants and orbit parameters used in this analysis are given in Tables 4 and 5, respectively. The epoch date of 1 June 2020 is arbitrarily chosen for this example study. Sample sections of the north, east and south of Scotland are considered to reduce computational complexity, selected on the basis of the areas of interest (Figure 2 and Table 6).