# Geometric Spherical Surfaces Emulating the Geostationary Orbit at Any Latitude with Zenith Links
## 2 Geocentric Spherical Surfaces Emulating the Geostationary Orbit  
Let us consider a Walker star constellation in which all orbits are circular and polar (inclination \(90^{\circ}\)). Let \(\varphi\) (\({}^{\circ}\)) be the geocentric angle with which the \(360^{\circ}\) of a polar orbit is uniformly divided (Figure 1). Consider the aperture angle \(\theta\) (\(-3\) dB) of the satellite antenna that corresponds to \(\varphi\) and the approximate radius of the service area corresponding to this angle (Figure 1), viewed from a satellite at altitude \(h\) from the surface of a perfectly spherical Earth, angularly spaced \(\varphi^{\circ}\) from adjacent satellites to North and South. With these assumptions, the number of satellites per orbital plane is independent of the satellite altitude and is given by:  
\[n=\frac{360}{\varphi} \tag{1}\]Now let us assume that the orbital planes of the constellation are separated in longitude by the same angle \(\varphi\), therefore the adiacent orbit planes are angularly space \(\varphi^{\circ}\) to West and East. This assumption means that the distance between close satellites belonging to adjacent orbit planes is largest at the equator and \"zero\" at the poles. Viewed from an inertial observer (i.e., an observer outside the Earth), the GeoSurf constellation would appear made of satellites moving synchronously (e.g., from North to South) along the meridians. Viewed from the Earth, the satellites would be moving South-West coming from the North\(-\)East. Therefore, if we assume that at a site the connection is, at a moment, with a satellite at the local zenith, this satellite would belong to the adjacent orbit plane to the East. After the time taken by the Earth to rotate \(\varphi^{\circ}\) to the East, the connection would be with a satellite belonging to the next adjacent orbit plane to the East, and so on. In other words, as anticipated, any site at any latitude will view a distributed geostationary satellite orbiting at that latitude. In conclusion, the number of orbital planes of the GeoSurf constellation is:  
\[n_{p}=\frac{360}{2\varphi} \tag{2}\]  
Combining Equations (1) and (2), we get the total number \(N=n\times n_{p}\) of the satellites of the constellation:  
\[N=\frac{1}{2}\bigg{(}\frac{360}{\varphi}\bigg{)}^{2} \tag{3}\]  
Figure 2 shows \(N\) as a function of the geocentric angle \(\varphi\). \(N\) ranges from 2592 for \(\varphi=5^{\circ}\) to 288 for \(\varphi=15^{\circ}\). Figure 1: Geometry for finding the relationship between the geocentric angle \(\varphi\) and aperture angle \(\theta\) (the main antenna lobe, e.g., at \(-3\) dB gain) of the downlink and uplink antennas of the GeoSurf. This situation, with no overlapping of the service areas, is found only at the equator. O is the center of the Earth; AB is the approximate diameter of the service area. \(R\) is the Earth radius, the segment CD gives the (approximate) height \(h\) of the orbit from the surface. The connected satellite is at C. The triangles at left and right give the position and service area of the adjacent satellites in the same orbit plane in the North\(-\)South direction, or in adjacent orbit planes in West\(-\)East direction but only at the equator (no overlapping). Now, let us assume the fundamental hypothesis that both all satellites and the transmitters/receivers on the Earth have fixed antenna pointing (no steering), both directed to the local vertical direction, i.e., to the local zenith. Therefore, these assumptions make us emulate a receiver/transmitter located at the equator that connects to a geostationary satellite stationing at the same longitude of the site (zenith path). Any transmitter/receiver will connect to the satellite entering its service area. For example, the satellite will be seen coming from the North East if the satellite is orbiting from the North Pole to the South Pole and the site is in the Northern Hemisphere.