## 4 Bespoke Space System Design
### Orbit Design
This angle \(\beta\) can be calculated [33] as  
\[\beta=\tan^{-1}\left(\frac{\sqrt{\sin^{2}i-\;\sin^{2}\delta}}{\cos i-\omega\; \cos^{2}\delta}\right) \tag{2}\]where \(\omega\) is the rotational rate of the Earth, taken as 7.2921 \(\times 10^{-5}\) rad/s. The width of the latitude band covered in a single pass is then  
\[W=\frac{s}{\sin\beta}. \tag{3}\]  
It should be noted that as the latitude of interest, \(\delta\), approaches the orbit inclination, \(i\), the width covered will tend to infinite; as such the above approximation is not valid at latitudes close to \(i\). For a swath width of 6.6 km, and a spacecraft inclined at 97 deg, this gives a width covered, \(W\), at 54.63 deg latitude of 6.8 km, meaning 3417 passes would be required for full coverage at 54.63 deg latitude, assuming no overlap between passes. The orbit period, \(T\), of a spacecraft in a circular orbit is  
\[T=2\pi\sqrt{\frac{\left(R_{E}+h\right)^{3}}{\mu}} \tag{4}\]  
where \(h\) is the orbit altitude, \(R_{E}\) is the mean Earth radius, taken as 6371 km, and \(\mu\) is the standard gravitational parameter of the Earth, taken as 3.986 \(\times 10^{5}\)m\({}^{3}\)/s\({}^{2}\). A spacecraft at an altitude of 500 km would have an orbit period of 94.47 min and therefore would require approximately 224 days to achieve 3417 passes; this gives a revisit time of approximately 224 days at 54.63 deg latitude. Assuming spacecraft could be distributed to provide no overlap in coverage, 16 spacecraft would hence be required to meet the desired revisit criteria of two weeks. In order to minimise the revisit time of a spacecraft over a region, an inclination can be selected that is greater than, but close to, the most northerly latitude of interest. For Scotland, this is approximately 60 deg. For an orbit inclined at 60 deg, the width covered by a spacecraft with a swath of 6.6 km in a single pass at 54.63 deg increases to 13.1 km, as calculated using Equations (2) and (3), meaning 1761 passes are required to cover the full Earth's circumference at 54.63 deg latitude. This would take 116 days for a single spacecraft, using the orbit period calculated with Equation (4), and so nine spacecraft could meet the two-week requirement--a significant improvement in comparison to the sun-synchronous case. Increasing the spacecraft altitude will increase the swath width but decrease the GSD; this provides an opportunity to fine-tune the performance of the space system to, potentially, further reduce the number of spacecraft required. The relationship between GSD, swath width, \(s\), and spacecraft altitude, \(h\), can be expressed as  
\[\frac{p}{GSD}=\frac{L}{h}=\frac{w}{s} \tag{5}\]  
where \(w\) is the sensor width, \(p\) is the photosite pitch and \(L\) is the instrument focal length [34]. Assuming the sensor parameters \(p\), \(L\), and \(w\) remain constant, both the GSD and swath width vary linearly with altitude according to Equation (5). Estimating the GSD and swath width at a given altitude using Equation (5) gives the results shown in Table 3. The number of spacecraft that would be required to provide a two-week revisit of Scotland at each altitude for a 60 deg inclined orbit as calculated using Equations (2)-(4) are also given. Note that the number of spacecraft required is rounded up to the nearest whole and the values should be considered a minimum as eclipse periods, cloud cover, and tasking restrictions may impact the spacecraft performance. The general trend is that a higher altitude gives a reduction in the number of spacecraft required, with diminished returns at greater altitudes. If the upper limit on GSD is taken as 1 m, then the minimum number of spacecraft needed will be seven, at an altitude of \(<\)700 km. Orbit altitude will also directly impact the spacecraft on-orbit lifetime. A spacecraft in low-Earth orbit, that is with an altitude \(<\)1000 km, will experience drag due to the Earth's atmosphere. Over time this will cause the spacecraft altitude to reduce, ultimately resulting in the spacecraft naturally deorbiting.