## 4 Bespoke Space System Design
### Orbit Design  
The orbit selected for the spacecraft will greatly influence the coverage and revisit capabilities of the system. Using the SkySat parameters defined in Section 4.1 as a baseline, an orbit is selected with the intent of maximising coverage of Scotland, and an estimation is made of the number of spacecraft required within this orbit to provide the desired revisit times. Circular orbits are assumed to be used due to ease of orbit insertion and operational maintenance. Altitude is one of the most significant parameters for the orbit design as it will influence the spatial resolution and swath width of the payload instrument; a lower altitude will result in a higher resolution image, but a narrower swath width. Inclination, which describes the angle between the orbit plane and the Earth's equator, is another critical parameter in orbit design. The inclination is critical for two primary reasons. Firstly, the most northerly and southerly latitude a spacecraft will pass over is defined by the orbit inclination such that the geodetic latitude, \(\delta\), range of the spacecraft coverage can be defined as, where \(0\leq\delta\leq tan^{-1}\left(\frac{tani}{1-f(2-f)}\right)\), where \(i\) is the orbit inclination in degrees and \(f\) is the Earth flattening factor to allow for conversion from geocentric to geodetic latitude [30]. It should be noted that depending on the swath width of the imager and the pointing capabilities of the spacecraft, it may be possible to view latitudes beyond this range. Secondly, spacecraft will have the shortest revisit period over regions at the most northerly and southerly viewable latitudes. The revisit time will generally increase as the latitude of the location of interest approaches the equator. The SkySat spacecraft operate in sun-synchronous orbits. A sun-synchronous orbit is a near-polar orbit where the average rate of nodal precession (i.e., the rate at which the spacecraft orbit rotates around the Earth's axis of rotation) is the same as the Earth's average rate of orbit around the sun. This synchronicity means that the relative time of day along the satellite's ground track remains approximately the same for each orbit pass. This is especially useful for Earth observing satellites as it ensures consistent lighting conditions each time a region is observed. Furthermore, as a near-polar orbit it allows for the entirety of the Earth to be observed. However, sun-synchronous orbits generally have an inclination of 96 deg \(<i<\) 99 deg, depending on the orbit altitude, and hence provide maximum coverage in the region of 81 deg \(<\delta<\) 84 deg. In order to estimate the revisit time for a satellite in a sun-synchronous orbit over the entirety of Scotland, the time to provide full coverage of the lowest latitude of interest is calculated, assuming no overlaps between passes. The southernmost point of Scotland is the Mull of Galloway, with a latitude of approximately 54.63 deg. The circumference of the circle defined by a given line of latitude, \(c_{\delta}\), can be estimated as [31]  
\[c_{\delta}=2\;\pi\frac{R_{\bigoplus}}{\sqrt{1-\sin\delta\left[\left(1-f\right) ^{2}-1\right]}}\cos\delta \tag{1}\]  
where \(R_{\bigoplus}\) is the Earth equatorial radius, taken as 6378 km, \(f\) is the Earth flattening factor and \(\delta\) is the geodetic latitude of interest. The circumference of the Earth at 54.63 deg latitude is hence estimated as 23135 km, assuming a flattening factor \(f\) of \(3.3528\times 10^{-3}\) as given by the World Geodetic System 1984 [32]. The width of a latitude band covered by an inclined spacecraft in a single pass, \(W\), with a swath width, \(s\), will depend on the direction of motion of the spacecraft relative to the line of latitude at nadir. At the equator, the relative angle between the spacecraft direction of motion and that of the line of latitude, \(\beta\), will be \(\beta=i\), where \(i\) is the orbit inclination, while at a latitude \(\delta=i\), there will be \(\beta=0\).