# Geometric Spherical Surfaces Emulating the Geostationary Orbit at Any Latitude with Zenith Links
## 3 Geometrical and Radio Electrical Parameters  
Let us derive the fundamental relationships among the geometrical parameters, according to the geometry of Figure 1. For the purpose of illustration, we consider four typical altitudes: \(h=35,876\) km for the geosyncronous orbits, \(h=20,000\) km for typical MEO altitude, \(h=10,000\) km for low MEO altitude and \(h=1500\) km for a LEO altitude. Given the geocentric angle \(\varphi\), the radius of the Earth \(R=6378\) km, the altitude \(h\) (km) of the satellite from the surface, the antenna \(-3\) dB angle \(\theta\) is given by:  
\[\theta=2\times arctan\Big{\{}\frac{R}{h}\times sin\Big{(}\frac{\varphi}{2} \Big{)}\Big{\}} \tag{4}\]  
For instance, when \(\varphi=15^{\circ}\), \(h=35,876\) km (GEO), we get \(\theta=2.7^{\circ}\), see Figure 3. According to [PERSON]'s third law, the period \(T\) (in seconds) of the orbit is given by:  
\[T^{2}=\sqrt{\frac{4\pi^{2}}{GM}(R+h)^{3}} \tag{5}\]  
Figure 2: Total number of satellites in the design of the GeoSurf constellation. This number is independent of the altitude of the GeoSurf constellation. \(N\) ranges from \(2592\) for \(\varphi=5^{\circ}\) to \(288\) for for \(\varphi=15^{\circ}\). where \(GM=398\),600.5 km\({}^{3}\)s\({}^{-2}\). Therefore, for \(\varphi=15^{\circ}\), for example, at the GEO altitude the time taken to cross the service area, of size about 1670 km, is 60 min, as Figure 4 shows. Figure 5 shows the gain of the antennas (satellites and Earth transmitters/receivers) versus the approximate service area diameter. The gain is independent of frequency because it is a function of the \(-3\) dB aperture angle of the main lobe, which determines the service area diameter reported in abscissa. In fact, for a parabolic antenna of diameter \(D\) (m) and efficiency \(\eta\), the gain is given by:  
\[G=\eta\pi^{2}\qty(\frac{D}{\lambda})^{2} \tag{6}\]  
Figure 4: Satellite service-area time as a function of the service area diameter for the GeoSurf Constellation. **A** refers to \(h=35,876\) km; **B** to \(h=20,000\) km; **C** to \(h=10,000\) km, **D** to \(h=1500\) km. Figure 3: Up\(-\)link, down\(-\)link and service area angle (\(-3\) dB) of the GeoSurf constellation design, as a function of the geocentric angle \(\varphi\). **A** refers to \(h=35,876\) km; **B** to \(h=20,000\) km; **C** to \(h=10,000\) km, **D** to \(h=1500\) km. Now, the \(-3\) dB antenna main lobe angle is give by:  
\[\theta^{{}^{\circ}}=70^{{}^{\circ}}\frac{\lambda}{D} \tag{7}\]  
where \(\lambda\) (m) is the wavelength. Therefore, after solving Equation (7) for \(D\) and substituting it in Equation (6), the antenna gain results indepedent of \(\lambda\), and hence of frequency, as is given by:  
\[G=\eta\pi^{2}\bigg{(}\frac{70^{{}^{\circ}}}{\theta^{{}^{\circ}}}\bigg{)}^{2} \tag{8}\]  
However, the physical dimension of the antennas (which affects satellite weight, etc., and ground station costs) depends on frequency, as Figures 6 and 7 show for the gain reported in Figure 5, as example, at 20 GHz (downlink) and 30 GHz (up-link), frequencies adopted at Ka Band, while Figures 8 and 9 show the dimensions of the antennas at 40 and 50 GHz frequencies. As these figures show, the size of the antennas is always very small, so that they can be very easily installed onboard the satellites, even onboard small ones, and easily installed at customer's premises, even if looking upwards to the zenith. \[\theta=2\times arctan\Big{\{}\frac{R}{h}\times sin\Big{(}\frac{\varphi}{2} \Big{)}\Big{\}} \tag{4}\]0 Antenna gain as a function of the service area diameter for the GeoSurf constellation. Antenna efficiency is 0.6. **A** refers to \(h=35,876\) km; **B** to \(h=20,000\) km; **C** to \(h=10,000\) km, **D** to \(h=1500\) km. \[\theta=2\times arctan\Big{\{}\frac{R}{h}\times sin\Big{(}\frac{\varphi}{2} \Big{)}\Big{\}} \tag{4}\]1 Antenna diameter as a function of the service area diameter, at 20 GHz for the GeoSurf constellation. **A** refers to \(h=35,876\) km; **B** to \(h=20,000\) km; **C** to \(h=10,000\) km, **D** to \(h=1500\) km.