# Geometric Spherical Surfaces Emulating the Geostationary Orbit at Any Latitude with Zenith Links
## 1 Satellite Constellations: Many Architectures and Many Problems  
According to the altitude above the Earth surface, satellite orbits can be divided into geostationary Earth orbits (GEO), medium Earth orbits (MEO), and low Earth orbits (LEO) [1; 2; 3; 4]. A GEO satellite is deployed in the equatorial plane (\(0^{\circ}\) latitude) at about 36,000 km above the Earth's equator and it appears fixed to a fixed (or slowly moving) observer on the Earth. Because of the advantages of the fixed position, i.e., does not require steering antennas or steering arrays, wide coverage, and constant propagation delay (constant latency), GEO satellites are mostly used to implement communication and broadcasting systems. Today, however, there is great interest in providing high data\(-\)rate access for the Internet services by means of GEO satellites, and several GEO high-throughput satellite communication systems have been deployed, such as Inmarsat Global Xpress, IPStar, Viasat, Chinasat-16 [5; 6; 7; 8]. However, GEO communication systems suffer several drawbacks: (a) For real\(-\)time use, as most Internet services require, the large latency might not be tolerated; (b) the free\(-\)space attenuation is large, therefore, for mobile communications it means that the power required for transmission results in increased weight and size of user terminals, which increases the complexity and reduces the usability of mobile terminals; (c) The service is not available at high latitudes because the path elevation angle is very low--the satellite can even be below the horizon--causing large tropospheric attenuation due to water vapor, oxygen and raindrops, mostly in the form of rain and scintillation; (d) The GEO orbit is unique, therefore of limited orbital and frequency resources \(-\) because of minimum angle separation between adjacent satellites in the GEO orbit working in the samefrequency band\(-\) that limit the system coverage and capacity; (e) The cost of satellite launching and moving it to the geostationary orbit is high, although it can be minimized by launching from equatorial sites. However, the total cost of launching and positioning satellites in the geostationary orbit for global coverage may be lower than that of launching and positioning hundreds or even thousands of LEO satellites. Moreover, further costs are associated with the ground control of satellites, lower in GEO constellations, higher in LEO constellations. In general, however, it is difficult to estimate, at this stage, the total costs (in any case, the issue is far beyond the purpose of this paper) because many parameters are involved in both GEO and LEO systems, such as ground and satellite equipment, including antennas and electronic devices (receivers, transmitter), total capacity, global (LEO) and quasi-global (GEO) coverage. MEO satellites fly at altitudes ranging from 2000 km to 20,000 km [1]. The satellites are not fixed but rise and set; therefore, the transmitters/receivers must have steering antennas designed for following the satellite and changing the elevation angle. Compared to GEO fixed elevation angle, a MEO variable elevation angle affects the link budget with larger tropospheric attenuation and scintillation, and with a service area of variable size. Many satellites are required to cover the same service area continuously (24 h). The MEO advantages are shorter propagation delay and smaller free-space attenuation. Some MEO satellite communication systems are Odyssey, Inmarsat-P and O3b. LEO satellites fly at altitudes in the range from 500 to 2000 km. Because of the relatively lower latency, path loss, and production and launching costs, LEO systems are very attractive. The first global LEO communication system was the Iridium system, consisting of 66 satellites distributed in 6 orbital planes at about 800 km above the ground, mainly providing voice and low-rate data services. Other veteran LEO systems are Globalstar, with 48 satellites at the altitude of 1400 km, and Orbcom, with 36 satellites at about 750 km.