Geocentric Spherical Surfaces Emulating the Geostationary Orbit at Any Latitude with Zenith Links

: According to altitude, the orbits of satellites constellations can be divided into geostationary Earth orbit (GEO), medium Earth orbit (MEO), and low Earth orbit (LEO) constellations. We propose to use a Walker star constellation with polar orbits, at any altitude, to emulate the geostationary orbit with zenith paths at any latitude. Any transmitter / receiver will be linked to a satellite as if the site were at the equator and the satellite at the local zenith. This constellation design can have most of the advantages of the current GEO, MEO, and LEO constellations, without having most of their drawbacks. Doppler phenomena are largely minimized because the connected satellite is always seen almost at the local zenith. The extra free-space loss, due to the ﬁxed pointing of all antennas, is at most 6 dBs when the satellite enters or leaves the service area. The connections among satellites are easy because the positions in the orbital plane and in adjacent planes are constant, although with variable distances. No steering antennas are required. The tropospheric propagation fading and scintillations are minimized. Our aim is to put forth the theoretical ideas about this design, to which we refer to as the geostationary surface (GeoSurf) constellation.

very difficult to adopt in LEO systems because of the very fast change in channel transfer function and the small interval of time of its passage.
The design of a satellite constellation is one of the critical factors that determine the performance of MEO and, especially, LEO satellite communication systems. The fundamental parameters of a satellite constellation are the type of orbit, the altitude of the orbital plane, the number of orbital planes, the number of satellites in each orbital plane, and the satellite phase factor between different orbit planes.
Currently, two LEO satellite constellations are mostly studied and used, the Walker Delta constellation with circular orbits and the Walker star constellation with circular near−polar orbits [1,12]. The Walker Delta constellation can adjust the inclination of the orbit according to the desired overlapping coverage of service areas. For example, the overlapping coverage at poles can be reduced, while the overlapping at the Equator can be increased. However, one of its disadvantages is that it cannot establish a stable inter-satellite link with any other satellite on adjacent orbit planes because the relative distance and position of the satellites change continuously [13]. In contrast, in the Walker star constellation, all satellites move regularly from the South Pole to the North Pole, or all in the opposite direction. Therefore, each satellite can easily connect with its neighbors both in the same plane and in the adjacent planes. This orbital characteristic simplifies the networking design. However, to avoid satellites crashing at poles when all orbit planes meet, near polar orbits with inclination ranging from 80 • to 100 • are used to replace the perfect polar orbit at 90 • [1].
Whatever solution is adopted for designing a satellite constellation, the complexity of a system composed of many satellites is large and refers to: launch of many satellites, distribution along the orbit with accurate positions, control and maintenance for all the lifespan, provision of the disposal at the end of life, and so on. The most challenging task is the control and maintenance of the constellation over a long time, which requires control of each satellite and of each satellite with respect to the others.
After [14], in this paper, we propose to use the Walker star constellation to emulate, for any latitude, the geostationary orbit with zenith paths. In other words, any transmitter/receiver, located at any latitude, can be linked to a satellite as if the site were at the equator and the satellite at its zenith. In other words, the distributed geostationary satellite concept, discussed in [14], will not be only in the equatorial plane, but in any plane parallel to the equatorial plane, from 0 • latitude to 90 • latitude, North or South, therefore covering also the poles, without an ad−hoc communication system [15].
The proposed constellation design can have most of the advantages of the current GEO, MEO and LEO constellations without having most of their drawbacks. Our aim is to put forth the main theoretical ideas about this design. The technological development necessary to make it practical should be addressed in many future studies. In the following, we refer to this design as to the geostationary surface (GeoSurf) constellation design.
After this introduction, Section 2 sets the theory of the Walker star constellation used for the GeoSurf design. Section 3 shows some straight theoretical results concerning the number of satellites, service area dimension, available flight time for single satellites, and size of antennas. Section 4 discusses the overall theoretical results by comparing the GeoSurf constellation with the current constellations and draws some conclusions.

Geocentric Spherical Surfaces Emulating the Geostationary Orbit
Let us consider a Walker star constellation in which all orbits are circular and polar (inclination 90 • ). Let ϕ ( • ) be the geocentric angle with which the 360 • of a polar orbit is uniformly divided (Figure 1). Consider the aperture angle θ (−3 dB) of the satellite antenna that corresponds to ϕ 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 ϕ • 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: (1) Figure 1. Geometry for finding the relationship between the geocentric angle and aperture angle (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.
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 that the orbital planes of the constellation are separated in longitude by the same angle , therefore the adiacent orbit planes are angularly space ° 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 ° 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: Combining Equations (1) and (2), we get the total number = × of the satellites of the constellation: (3) Figure 2 shows as a function of the geocentric angle . ranges from 2592 for = 5° to 288 for = 15°. Geometry for finding the relationship between the geocentric angle ϕ and aperture angle θ (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 that the orbital planes of the constellation are separated in longitude by the same angle ϕ, therefore the adiacent orbit planes are angularly space ϕ • 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 ϕ • 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: Combining Equations (1) and (2), we get the total number N = n × n p of the satellites of the constellation: 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. Because the satellite cannot be at the zenith all the time when crossing diagonally the service area, although it is very close to it, the link budget will lose 3 dBs on the side of the receiver, and 3 dBs on the side of transmitter when the satellite is entering/leaving the service area.
When the sites are really at the equator, with the above assumptions there is no overlapping of the −3 dBs footprint of each satellite. For all different latitudes, there will be overlapping, the larger the closer to the poles. The overcrowding of the satellites at the high latitudes is a drawback of the Walker star constellation, but this effect can be partially taken care of if the orbit planes are inclined just a little bit more (or less) than 90° [1]. In any case, as for all constellations, a careful frequency planning and frequency reuse must be considered.
The Walker star constellation, as noted in Section 1, makes easier the use of inter-satellite links both within the same orbit plane (intra links) and with adjacent orbit planes (inter links). Let us now explore, in the next section, the geometrical and radio electrical parameters of the GeoSurf design.

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 , the radius of the Earth = 6378 km, the altitude ℎ (km) of the satellite from the surface, the antenna −3 dB angle is given by: For instance, when = 15°, ℎ = 35,876 km (GEO), we get = 2.7°, see Figure 3. According to Kepler's third law, the period T (in seconds) of the orbit is given by: 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. Because the satellite cannot be at the zenith all the time when crossing diagonally the service area, although it is very close to it, the link budget will lose 3 dBs on the side of the receiver, and 3 dBs on the side of transmitter when the satellite is entering/leaving the service area.
When the sites are really at the equator, with the above assumptions there is no overlapping of the −3 dBs footprint of each satellite. For all different latitudes, there will be overlapping, the larger the closer to the poles. The overcrowding of the satellites at the high latitudes is a drawback of the Walker star constellation, but this effect can be partially taken care of if the orbit planes are inclined just a little bit more (or less) than 90 • [1]. In any case, as for all constellations, a careful frequency planning and frequency reuse must be considered.
The Walker star constellation, as noted in Section 1, makes easier the use of inter-satellite links both within the same orbit plane (intra links) and with adjacent orbit planes (inter links). Let us now explore, in the next section, the geometrical and radio electrical parameters of the GeoSurf design.

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 ϕ, the radius of the Earth R = 6378 km, the altitude h (km) of the satellite from the surface, the antenna −3 dB angle θ is given by: For instance, when ϕ = 15 • , h = 35, 876 km (GEO), we get θ = 2.7 • , see Figure 3. According to Kepler's third law, the period T (in seconds) of the orbit is given by:  Therefore, for = 15°, 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 (m) and efficiency , the gain is given by: Therefore, for ϕ = 15 • , 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.
where = 398,600.5 . Therefore, for = 15°, 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 (m) and efficiency , the gain is given by:  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 η, the gain is given by: where (m) is the wavelength. Therefore, after solving Equation (7) for and substituting it in Equation (6), the antenna gain results indepedent of , and hence of frequency, as is given by: 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.
where λ (m) is the wavelength. Therefore, after solving Equation (7) for D and substituting it in Equation (6), the antenna gain results indepedent of λ, and hence of frequency, as is given by: 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.   Figure 6. 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.      To guarantee the minimum signal-to-noise ratio required in the design, the radio frequency power onboard the satellites will largely depend on the free space distance. In other words, keeping constant the minimum signal-to-noise ratio in any possible propagation conditions, the frequency band (therefore, working always with the same tropospheric fading), the ultimate variable that To guarantee the minimum signal-to-noise ratio required in the design, the radio frequency power onboard the satellites will largely depend on the free space distance. In other words, keeping constant the minimum signal-to-noise ratio in any possible propagation conditions, the frequency band (therefore, working always with the same tropospheric fading), the ultimate variable that determines the necessary power is the (practically zenith) distance between the satellite and the ground station, and therefore it sets the satellite constellation features.

Discussion and Conclusions
At this stage of developing GeoSurf constellations, we can draw only some general remarks, summarized in Table 1. One of the main advantages of the GeoSurf design is having zenith paths worldwide. With the GeoSurf constellation, we do not need steering antennas, the tropospheric propagation fading is minimized both for the stationary fading due to water vapor and oxygen, and for rain attenuation, because at zenith the path length in the troposphere is shortest [16]. The footprint of a single satellite is always the same, therefore making the design easier. However, several footprints can overlap with possible frequency interference, to be avoided by a careful frequency plan and coding. The Doppler phenomena are largely minimized because the connected satellite would be always viewed almost at the local zenith. The extra free−space loss, due to the fixed pointing of all antennas, is 6 dBs when the satellite enters or leaves the local service area. The connections among the satellites are easier to establish because of the constant positions along an orbital plane or adjacent planes, although with variable distances. The number of satellites is much less than that of a current large constellation proposal (Starlink). The main drawback is the large number of satellites crowding at high latitudes, a problem, however, that could be eased by slightly changing the polar orbits inclination. Because the GeoSurf constellation design can be applied to any orbit altitude, we could design a large number of GeoSurf constellations, by varying the altitude. We have presented some particular and popular orbit altitudes, just to show what could be done and to orient readers. Lower altitudes could also be used, such as, for example, 600 km, suitable for the very small cube satellites [17,18]. Of course, any satellite of the GeoSurf constellation could be linked to a real geostationary satellite and many architectures are possible. The same overall design could be considered for future large−scale human settlings on Mars.
In conclusion, GeoSurf constellations could effectively replace most of the actual constellations for Internet applications. However, as any new complex satellite constellation design, its possible implementation requires many feasibility studies, especially from the point of view of space companies, a task well beyond this theoretical paper.