1、 Rep. ITU-R SA.2066 1 REPORT ITU-R SA.2066 Means of calculating low-orbit satellite visibility statistics (2006) CONTENTS Page 1 Introduction 2 2 Percentage of time and maximum duration for a low-orbiting spacecraft occupying a defined region. 2 2.1 Bounding equation for percentage of time spacecraf
2、t is in defined region 3 2.2 The maximum time a satellite spends in the beam of a ground station 3 3 Probability density function (pdf) of the position of a low-orbiting satellite on the orbit shell 5 3.1 Probability density function of interference to low-orbiting satellites caused by emissions fro
3、m FS systems 7 3.2 Probability density function of interference to FS systems caused by emissions from low-orbiting satellites 11 4 Simplified methods for calculating visibility statistics. 12 4.1 Simplified method for circular antenna beams. 13 4.2 Manual method to calculate visibility statistics 1
4、6 4.3 Comparison of the numerical results obtained using the simplified and manual methods for circular antenna beams 19 5 Means of calculating the coordinates of the intersection of two orbital planes . 20 5.1 Analysis 20 2 Rep. ITU-R SA.2066 1 Introduction The increasing use of space stations in c
5、ircular low-orbit in the space research service (and other services) necessitates the development of dynamic sharing models in which the potential interference from the space station can be treated as a time-varying function. This Report defines analytical tools for calculating visibility statistics
6、 for low-orbiting spacecraft in circular orbits (see Note 1) as seen from a specific point on the Earths surface. NOTE 1 This Report only deals with circular satellite orbits in which the orbital period is not an even multiple of the Earths rotational period. Section 2 of this Report describes the f
7、actors affecting the visibility statistics, presents a bounding equation for determining the percentage of time that a low-orbiting satellite will occupy specified regions of the orbit shell visible to an earth station, and contains summary charts giving the maximum duration a low-orbiting satellite
8、 spends in certain regions of the orbit shell as a function of several parameters. Section 3 develops the probability density function (pdf) of a satellite occupying specific locations on the orbit shell, illustrates how the pdf may be used to calculate the statistical characteristics of interferenc
9、e to low-orbiting satellites resulting from emissions from stations in the FS, and demonstrates the computation of the pdf of the interference to FS systems assuming the power flux-density (pfd) of the emissions of the low-orbiting satellites conform to a specific profile. Section 4 proposes a simpl
10、ified method to calculate the visibility statistics of earth stations or terrestrial stations using an antenna with a beam of circular cross section and also presents a manual visibility computation method based on the use of a spreadsheet to calculate the visibility statistics of earth stations or
11、terrestrial stations employing an antenna with a beam of a more complex cross section. Finally, section 5 provides a means to calculate the coordinates in inertial space of the intersection of two orbital planes. This section is particularly useful for predicting the conjunction of satellites in sun
12、-synchronous orbits whose orbital planes are offset. 2 Percentage of time and maximum duration for a low-orbiting spacecraft occupying a defined region Even for the simplest of dynamic sharing models, at least six specific system parameters must be evaluated to define precisely the primary time depe
13、ndent statistics of a low-orbit space station as seen from a location on the Earths surface. The time dependent statistics are: the longest time of passage of a space station through the main beam of a ground antenna (discussed in 3); the long-term percentage of time that the space station spends in
14、 various areas of the orbit sphere as seen from the ground station. The first statistic is important in that it defines the longest continuous duration of noise power into the ground receiving system from the space station. The second set of statistics, after convolution with transmit and receive an
15、tenna patterns, and range loss, can be used to develop interference-to-noise (I/N) relations as a function of time for the dynamic sharing model. In one sense then, I/N versus time relations can be treated in a method similar to the signal strength versus time relations derived from atmospheric prop
16、agation statistics. However, instead of a receiver experiencing change in the S/N ratio as a statistical function of time, it experiences a change in signal-to-noise-plus-interference ratio, as a statistical function of time, based upon the low-orbit space station model parameters. The specific para
17、meters which define the long-term visibility statistics of a space station in a low circular inclined orbit as seen from a receiving system on the Earths surface are: altitude of the space station, H (km); Rep. ITU-R SA.2066 3 inclination of the space station orbit, i (degrees); latitude of the grou
18、nd station, La (degrees); pointing azimuth of the ground station antenna measured from North, Az (degrees); pointing elevation of the ground station antenna measured from the local horizontal plane, El (degrees); angular area of the region of interest, A. The last parameter may take on several diffe
19、rent physical interpretations depending upon the purpose of the analysis. For instance, it may be the angular area of the main beam of the ground station antenna or it may be taken as an angular area expressed by an azimuth “width” of Az (degrees) and an elevation height expressed as El (degrees). 2
20、.1 Bounding equation for percentage of time spacecraft is in defined region The bounding equation is given below and may be used to determine the percentage of time that a low-orbit spacecraft will reside in certain regions visible to a ground station over long periods of time: 100sinsinsinsin)sin(s
21、in2(%)112 +=iLiLLT (1) where: L, L : latitude limits of the region on the orbital shell (see Fig. 1) : longitudinal extent of the region on the orbital shell, between the longitude limits of 1and 2(as seen in Fig. 1) i : inclination of the satellite orbit (all angles in rad). FIGURE 1 2.2 The maximu
22、m time a satellite spends in the beam of a ground station This section provides worst case numerical data on one aspect of frequency sharing with low-orbit, inclined orbit satellites. Such sharing is influenced by the amount of time that an “unwanted” and potentially interfering satellite appears wi
23、thin the 3 dB beamwidth of a ground station. This 4 Rep. ITU-R SA.2066 parameter is evaluated for several orbit altitudes and for two “bounding” elevations of the receiving antenna. The numerical results developed in this paper represent an upper bound on the length of time a spacecraft at a given a
24、ltitude will appear within the beam of a ground station. The time a satellite spends in a ground stations beam is a function of the beams width, the elevation of the beam and the altitude of the satellite. The worst case, i.e. when the satellite spends the maximum possible time in the beam, occurs w
25、hen the ground station is located at the equator with a beam of elevation = 0 and the satellite is traveling east along an orbit with 0 inclination. The time the satellite spends in the beam depends upon the satellites velocity relative to the velocity of the beam as it rotates with the Earth, and u
26、pon the length of the intersection of the orbit with the beam. The maximum time that a spacecraft can spend in the main beam of an antenna is shown in Figs. 2 and 3 for antenna elevations of 0 and 90 respectively, and refers to a variety of orbital altitude and beamwidths. Rep. ITU-R SA.2066 5 3 Pro
27、bability density function pdf of the position of a low-orbiting satellite on the orbit shell The position, i.e. the latitude and longitude of an orbiting satellite on the orbit shell relative to a fixed point on Earth, is a function of two independent parameters: the position of the satellite in its
28、 orbit plane; and the longitude of the observation point on Earth relative to the orbit plane. The geometry used for this analysis is shown in Fig. 4. It is assumed that the satellite is in a circular orbit at an altitude, h, the inclination of the orbit plane is i, and the period of rotation of the
29、 satellite and of the Earth are not directly related. 6 Rep. ITU-R SA.2066 The coordinate system shown in Fig. 4 is a right-handed, geocentric system with the x-y plane corresponding to the equatorial plane, and the x-axis pointing in an arbitrary direction in space (usually the First Point of Aries
30、). For simplicity, assume that the intersection of the orbit plane and the equatorial plane is the x-axis. The latitude sof the position of the satellite in space is given by: sin s= sin ssin i (2) where sis the central angle between the x-axis and the position vector of the satellite. For satellite
31、s in circular orbits, sis a linear function of time t, i.e. s= 2t / , where is the period of the orbit. Equation (2) relates the latitude of the satellite as a function of the central angle sand the orbit inclination angle i. If the central angle of the position vector of a satellite in a circular o
32、rbit is sampled at random times, the angle swill be found to be uniformly distributed between 0 and 2 radians. Stated as a probability density function p(s): =21)(sp (3) The pdf of the latitude of the position vector of the satellite may be found using a straightforward transformation technique from
33、 probability theory. It may be shown for a random variable x with pdf p(x) that undergoes the transformation y = g(x), that the pdf p(y) of the random variable y is given by: )()(.)()()(11nnxgxpxgxpyp+= (4) where: xxgxgd)(d)( = and x1, . xnare the real roots of y = g(x). Rep. ITU-R SA.2066 7 Applyin
34、g the procedure described above to equations (2) and (3) yields the pdf of the latitude of the position vector of the satellite in its orbital plane: sssip=22sinsincos1)( (5) Equation (5) represents the function that would be obtained if the latitude of the satellite were randomly sampled a large nu
35、mber of times. Inspection of equation (5) shows that the expression is defined only for real values of | s| i as expected. It may also be shown that: =iissp 1d)(6) also as expected. For the satellite to appear at a specific longitude son the orbit shell relative to the reference point on the surface
36、 of the Earth, the orbit plane must intersect the orbit shell at that longitude. The probability of this occurring is uniformly distributed over 2 radians, i.e.: =21)(sp (7) Finally, since it has been assumed that the period of the satellite and the rotation of the Earth are not directly related, th
37、e pdf of the satellite position is the joint probability of two independent events which is given by the product of the individual pdfs: ssssip=222sinsincos21),( (8) The probability P(,) of the satellite occupying the region on the orbit shell bounded by latitude s, s + sand longitude sis given by:
38、ssssoiPssss=+222sinsinddcos21),( (9) Carrying out the integration yields: +=iiPsssssinsinsinsin)sin(sin2),(112(10) 3.1 Probability density function of interference to low-orbiting satellites caused by emissions from FS systems The pdf of interference to low-orbiting satellites caused by emissions fr
39、om FS systems is a function of the geometry and the pdf of the satellite position. If the interference can be expressed as a function of the coordinates (latitude and relative longitude) of the visible orbit shell, i.e. I (s, s), then the pdf of the interference to the low-orbiting satellite p(I ) i
40、s given by: =ssssspdIIP dd),()( (11) where S indicates that the integration is to be performed over the segment of the surface of the orbit sphere that contributes a level of interference between the values of I and I + dI. 8 Rep. ITU-R SA.2066 The function I (s, s) is a complex function of a number
41、 of parameters that include the location of the FS station, the transmitter power spectral density, the directional characteristics of the transmitting antenna gain, the azimuth and elevation angle of the transmitting antenna, the altitude and orbit inclination angle of the satellite, the range to t
42、he satellite, the gain of the satellite receiving antenna in the direction of the interference and the operating frequency. An integral involving a function of this complexity is most readily handled through the use of numerical techniques. The steps used in the numerical procedure are: Step 1 : def
43、ine sand sas independent variables over the surface of the visible orbit sphere. Step 2 : define an array I (n) corresponding to the range of interest (maximum to minimum value of interference (I (s, s) where n corresponds to the number of desired increments (e.g. 0.25 dB increments) (this array wil
44、l be used to store the differential pdf). Step 3 : evaluate I (s, s) at specific values of and (this value will be used to point to a specific element n0in the array I (n). Step 4 : calculate p(s, s) dsdsand add to the value stored in I (n0). Step 5 : increment and over the surface of the visible or
45、bit sphere. Step 6 : repeat steps 3 to 5. It is noted that evaluating equation (11) numerically results in the transformation of the integral to a summation. The geometrical parameters required to evaluate I (s, s) are obtained by using a geocentric coordinate system similar to the one shown in Fig.
46、 4. The main difference is that the coordinate system rotates at the same rate and direction as the Earth. The x-y plane is the equatorial plane and the z-axis is the rotational axis of the Earth. The position of the FS station is assumed, for simplicity, to lie in the x-z plane. The scale of the co
47、ordinate system is normalized to the radius of the Earth. Therefore, any distances computed in this coordinate system must be multiplied by the radius of the Earth (6 378 km) to obtain the correct value. The normalized components of the station position vector P are given by: pp=sin0cosP (12) where
48、pis the latitude of the FS station. The direction the FS transmitting antenna is pointed, is given by a unit vector that lies in the plane of the local horizontal and is offset from the direction of North by a specified azimuth angle az. The components of the antenna pointing vector UAare given by:
49、rrrrrAU=sinsincoscoscos(13) where: ()azpr=coscossin1(14a) Rep. ITU-R SA.2066 9 =azpazpr221coscos1cossincos (14b) The values of sand sdefining the limits of the visible surface of the orbit sphere may be readily determined. The limits for sare given by: max= p+ lim, max i otherwise max= i (15a) min= p lim, max i (15b) where: lim= cos1(1/)
