REPORT ITU-R SA Means of calculating low-orbit satellite visibility statistics

Transcription

1 Rep. ITU-R SA REPORT ITU-R SA.066 Mean of calculating low-obit atellite viibility tatitic (006) CONTENTS 1 Intoduction... Page Pecentage of time and maximum duation fo a low-obiting pacecaft occupying a defined egion....1 Bounding equation fo pecentage of time pacecaft i in defined egion The maximum time a atellite pend in the beam of a gound tation Pobability denity function (pdf) of the poition of a low-obiting atellite on the obit hell Pobability denity function of intefeence to low-obiting atellite caued by emiion fom FS ytem Pobability denity function of intefeence to FS ytem caued by emiion fom low-obiting atellite Simplified method fo calculating viibility tatitic Simplified method fo cicula antenna beam Manual method to calculate viibility tatitic Compaion of the numeical eult obtained uing the implified and manual method fo cicula antenna beam Mean of calculating the coodinate of the inteection of two obital plane Analyi... 0

2 Rep. ITU-R SA Intoduction The inceaing ue of pace tation in cicula low-obit in the pace eeach evice (and othe evice) neceitate the development of dynamic haing model in which the potential intefeence fom the pace tation can be teated a a time-vaying function. Thi Repot define analytical tool fo calculating viibility tatitic fo low-obiting pacecaft in cicula obit (ee Note 1) a een fom a pecific point on the Eath uface. NOTE 1 Thi Repot only deal with cicula atellite obit in which the obital peiod i not an even multiple of the Eath otational peiod. Section of thi Repot decibe the facto affecting the viibility tatitic, peent a bounding equation fo detemining the pecentage of time that a low-obiting atellite will occupy pecified egion of the obit hell viible to an eath tation, and contain ummay chat giving the maximum duation a low-obiting atellite pend in cetain egion of the obit hell a a function of eveal paamete. Section 3 develop the pobability denity function (pdf) of a atellite occupying pecific location on the obit hell, illutate how the pdf may be ued to calculate the tatitical chaacteitic of intefeence to low-obiting atellite eulting fom emiion fom tation in the FS, and demontate the computation of the pdf of the intefeence to FS ytem auming the powe flux-denity (pfd) of the emiion of the low-obiting atellite confom to a pecific pofile. Section 4 popoe a implified method to calculate the viibility tatitic of eath tation o teetial tation uing an antenna with a beam of cicula co ection and alo peent a manual viibility computation method baed on the ue of a peadheet to calculate the viibility tatitic of eath tation o teetial tation employing an antenna with a beam of a moe complex co ection. Finally, ection 5 povide a mean to calculate the coodinate in inetial pace of the inteection of two obital plane. Thi ection i paticulaly ueful fo pedicting the conjunction of atellite in un-ynchonou obit whoe obital plane ae offet. Pecentage of time and maximum duation fo a low-obiting pacecaft occupying a defined egion Even fo the implet of dynamic haing model, at leat ix pecific ytem paamete mut be evaluated to define peciely the pimay time dependent tatitic of a low-obit pace tation a een fom a location on the Eath uface. The time dependent tatitic ae: the longet time of paage of a pace tation though the main beam of a gound antenna (dicued in 3); the long-tem pecentage of time that the pace tation pend in vaiou aea of the obit phee a een fom the gound tation. The fit tatitic i impotant in that it define the longet continuou duation of noie powe into the gound eceiving ytem fom the pace tation. The econd et of tatitic, afte convolution with tanmit and eceive antenna patten, and ange lo, can be ued to develop intefeence-tonoie (I/N) elation a a function of time fo the dynamic haing model. In one ene then, I/N veu time elation can be teated in a method imila to the ignal tength veu time elation deived fom atmopheic popagation tatitic. Howeve, intead of a eceive expeiencing change in the S/N atio a a tatitical function of time, it expeience a change in ignal-to-noieplu-intefeence atio, a a tatitical function of time, baed upon the low-obit pace tation model paamete. The pecific paamete which define the long-tem viibility tatitic of a pace tation in a low cicula inclined obit a een fom a eceiving ytem on the Eath uface ae: altitude of the pace tation, H (km);

3 Rep. ITU-R SA inclination of the pace tation obit, i (degee); latitude of the gound tation, La (degee); pointing azimuth of the gound tation antenna meaued fom Noth, Az (degee); pointing elevation of the gound tation antenna meaued fom the local hoizontal plane, El (degee); angula aea of the egion of inteet, δa. The lat paamete may take on eveal diffeent phyical intepetation depending upon the pupoe of the analyi. Fo intance, it may be the angula aea of the main beam of the gound tation antenna o it may be taken a an angula aea expeed by an azimuth width of δaz (degee) and an elevation height expeed a δel (degee)..1 Bounding equation fo pecentage of time pacecaft i in defined egion The bounding equation i given below and may be ued to detemine the pecentage of time that a low-obit pacecaft will eide in cetain egion viible to a gound tation ove long peiod of time: δλ 1 in( L + L) 1 in L T (%) = in in 100 π in in (1) i i whee: L, L : latitude limit of the egion on the obital hell (ee Fig. 1) δλ : i : longitudinal extent of the egion on the obital hell, between the longitude limit of λ 1 and λ (a een in Fig. 1) inclination of the atellite obit (all angle in ad). FIGURE 1. The maximum time a atellite pend in the beam of a gound tation Thi ection povide wot cae numeical data on one apect of fequency haing with low-obit, inclined obit atellite. Such haing i influenced by the amount of time that an unwanted and potentially intefeing atellite appea within the 3 db beamwidth of a gound tation. Thi

4 4 Rep. ITU-R SA.066 paamete i evaluated fo eveal obit altitude and fo two bounding elevation of the eceiving antenna. The numeical eult developed in thi pape epeent an uppe bound on the length of time a pacecaft at a given altitude will appea within the beam of a gound tation. The time a atellite pend in a gound tation beam i a function of the beam width, the elevation of the beam and the altitude of the atellite. The wot cae, i.e. when the atellite pend the maximum poible time in the beam, occu when the gound tation i located at the equato with a beam of elevation = 0 and the atellite i taveling eat along an obit with 0 inclination. The time the atellite pend in the beam depend upon the atellite velocity elative to the velocity of the beam a it otate with the Eath, and upon the length of the inteection of the obit with the beam. The maximum time that a pacecaft can pend in the main beam of an antenna i hown in Fig. and 3 fo antenna elevation of 0 and 90 epectively, and efe to a vaiety of obital altitude and beamwidth.

5 Rep. ITU-R SA Pobability denity function pdf of the poition of a low-obiting atellite on the obit hell The poition, i.e. the latitude and longitude of an obiting atellite on the obit hell elative to a fixed point on Eath, i a function of two independent paamete: the poition of the atellite in it obit plane; and the longitude of the obevation point on Eath elative to the obit plane. The geomety ued fo thi analyi i hown in Fig. 4. It i aumed that the atellite i in a cicula obit at an altitude, h, the inclination of the obit plane i i, and the peiod of otation of the atellite and of the Eath ae not diectly elated.

6 6 Rep. ITU-R SA.066 The coodinate ytem hown in Fig. 4 i a ight-handed, geocentic ytem with the x-y plane coeponding to the equatoial plane, and the x-axi pointing in an abitay diection in pace (uually the Fit Point of Aie). Fo implicity, aume that the inteection of the obit plane and the equatoial plane i the x-axi. The latitude ϕ of the poition of the atellite in pace i given by: in ϕ = in θ in i () whee θ i the cental angle between the x-axi and the poition vecto of the atellite. Fo atellite in cicula obit, θ i a linea function of time t, i.e. θ = πt / τ, whee τ i the peiod of the obit. Equation () elate the latitude of the atellite a a function of the cental angle θ and the obit inclination angle i. If the cental angle of the poition vecto of a atellite in a cicula obit i ampled at andom time, the angle θ will be found to be unifomly ditibuted between 0 and π adian. Stated a a pobability denity function p(θ ): 1 p ( θ ) = (3) π The pdf of the latitude of the poition vecto of the atellite may be found uing a taightfowad tanfomation technique fom pobability theoy. It may be hown fo a andom vaiable x with pdf p(x) that undegoe the tanfomation y = g(x), that the pdf p(y) of the andom vaiable y i given by: p( x1) p( xn ) p( y) = (4) g ( x ) g ( x ) whee: dg( x) g ( x) = dx and x 1,... x n ae the eal oot of y = g(x). 1 n

7 Rep. ITU-R SA Applying the pocedue decibed above to equation () and (3) yield the pdf of the latitude of the poition vecto of the atellite in it obital plane: 1 coϕ p( ϕ ) = (5) π in i in ϕ Equation (5) epeent the function that would be obtained if the latitude of the atellite wee andomly ampled a lage numbe of time. Inpection of equation (5) how that the expeion i defined only fo eal value of ϕ i a expected. It may alo be hown that: i p( ϕ )dϕ = 1 (6) i alo a expected. Fo the atellite to appea at a pecific longitude λ on the obit hell elative to the efeence point on the uface of the Eath, the obit plane mut inteect the obit hell at that longitude. The pobability of thi occuing i unifomly ditibuted ove π adian, i.e.: 1 p ( λ ) = (7) π Finally, ince it ha been aumed that the peiod of the atellite and the otation of the Eath ae not diectly elated, the pdf of the atellite poition i the joint pobability of two independent event which i given by the poduct of the individual pdf : 1 coϕ p( ϕ, λ ) = (8) π in i in ϕ The pobability P( ϕ, λ) of the atellite occupying the egion on the obit hell bounded by latitude ϕ, ϕ + ϕ and longitude λ i given by: Caying out the integation yield: λ 1 coϕdλ dϕ P( ϕ, λ) = (9) π in i in λ P( ϕ, λ) = π in o ϕ + ϕ ϕ 1 in( ϕ + ϕ ) in in i ϕ 1 in ϕ in i 3.1 Pobability denity function of intefeence to low-obiting atellite caued by emiion fom FS ytem The pdf of intefeence to low-obiting atellite caued by emiion fom FS ytem i a function of the geomety and the pdf of the atellite poition. If the intefeence can be expeed a a function of the coodinate (latitude and elative longitude) of the viible obit hell, i.e. I (ϕ, λ ), then the pdf of the intefeence to the low-obiting atellite p(i ) i given by: (10) P( I) di = p( ϕ, λ )dϕ dλ (11) whee S indicate that the integation i to be pefomed ove the egment of the uface of the obit phee that contibute a level of intefeence between the value of I and I + di.

8 8 Rep. ITU-R SA.066 The function I (ϕ, λ ) i a complex function of a numbe of paamete that include the location of the FS tation, the tanmitte powe pectal denity, the diectional chaacteitic of the tanmitting antenna gain, the azimuth and elevation angle of the tanmitting antenna, the altitude and obit inclination angle of the atellite, the ange to the atellite, the gain of the atellite eceiving antenna in the diection of the intefeence and the opeating fequency. An integal involving a function of thi complexity i mot eadily handled though the ue of numeical technique. The tep ued in the numeical pocedue ae: Step 1 : define ϕ and λ a independent vaiable ove the uface of the viible obit phee. Step : define an aay I (n) coeponding to the ange of inteet (maximum to minimum value of intefeence (I (ϕ, λ )) whee n coepond to the numbe of deied incement (e.g. 0.5 db incement) (thi aay will be ued to toe the diffeential pdf). Step 3 : evaluate I (ϕ, λ ) at pecific value of ϕ and λ (thi value will be ued to point to a pecific element n 0 in the aay I (n)). Step 4 : calculate p(ϕ, λ ) dϕ dλ and add to the value toed in I (n 0 ). Step 5 : incement ϕ and λ ove the uface of the viible obit phee. Step 6 : epeat tep 3 to 5. It i noted that evaluating equation (11) numeically eult in the tanfomation of the integal to a ummation. The geometical paamete equied to evaluate I (ϕ, λ ) ae obtained by uing a geocentic coodinate ytem imila to the one hown in Fig. 4. The main diffeence i that the coodinate ytem otate at the ame ate and diection a the Eath. The x-y plane i the equatoial plane and the z-axi i the otational axi of the Eath. The poition of the FS tation i aumed, fo implicity, to lie in the x-z plane. The cale of the coodinate ytem i nomalized to the adiu of the Eath. Theefoe, any ditance computed in thi coodinate ytem mut be multiplied by the adiu of the Eath (6 378 km) to obtain the coect value. The nomalized component of the tation poition vecto P ae given by: coϕ in ϕ p P = 0 (1) p whee ϕ p i the latitude of the FS tation. The diection the FS tanmitting antenna i pointed, i given by a unit vecto that lie in the plane of the local hoizontal and i offet fom the diection of Noth by a pecified azimuth angle θ az. The component of the antenna pointing vecto U A ae given by: U A = coϕ coϕ in ϕ coλ in λ (13) whee: ϕ = in 1 ( coϕ coθ ) p az (14a)

9 Rep. ITU-R SA λ 1 in ϕ p coθaz = co (14b) 1 co ϕ θ p co az The value of ϕ and λ defining the limit of the viible uface of the obit phee may be eadily detemined. The limit fo ϕ ae given by: ϕ max = ϕ p + ϕ lim, ϕ max i othewie ϕ max = i (15a) ϕ min = ϕ p ϕ lim, ϕ max i (15b) whee: ϕ lim = β = co 1 (1/β) 1 + h / e h : altitude of the atellite e : adiu of the Eath. If ϕ min < i, then the low-obiting atellite i not viible at the FS tation. Fo an abitay value of ϕ between the limit ϕ min and ϕ max, the limiting value of the elative longitude λ min and λ max on the viible egment of the obit phee ae given by: λ max = λ min = co 1 coϕlim in ϕ p in ϕ coϕ p coϕ (16) Given value fo ϕ and λ between the limit deived above, the ange to the atellite and the angle between the diection the FS tation antenna i pointed and the diection to the atellite i mot eaily obtained uing vecto analyi. Specifically, the vecto to the atellite R i given by: R = S P (17) whee P i the poition vecto of the FS tation a given by equation (1), and S i the poition vecto of the ample point of the location of the atellite on the obit phee given by: coϕ in ϕ coλ S = β coϕ in λ (18) The nomalized ange to the atellite R i given by the quae oot of the um of the quae of the component of the ange vecto given in equation (17). The off-axi angle to the atellite i obtained uing the cala poduct of the antenna pointing vecto U A (whoe nomalized component ae given by equation (13)) and the ange vecto R. The off-axi angle ϕ off-axi i given by: ϕ = 1 R U A off axi co (19) R

10 10 Rep. ITU-R SA.066 Recommendation ITU-RF.699 et foth the efeence adiation patten to be ued fo FS tation tanmitting antenna with D / λ f < 100 and fo antenna with D / λ f > 100, whee D i the diamete of the antenna and λ f i the wavelength at the opeating fequency. The efeence adiation patten to be ued fo the eceiving ytem on the low-obiting atellite will be aumed to be iotopic. Uing the above aumption: I( ϕ, λ ) = P G T T ( ϕ off axi πr ) (4 ) G Rλ f (0) whee: P T : tanmitte powe (o powe pectal denity) G T (ϕ off-axi ) : tanmitting antenna gain in the diection of the ample point (ϕ, λ ) G R : λ f : eceiving antenna gain at the ample point in the diection of the FS tation wavelength of the opeating fequency R : ange (in the ame dimenion a λ f ) fom the FS tation to the ample point (i.e. R e ). Uing the pocedue decibed ealie in thi ection, the pdf of the intefeence to a low-obiting atellite caued by emiion of an FS tation i obtained uing equation (11) and (0). An example cae ha been evaluated to illutate the eult that ae obtainable uing the analytical pocedue decibed above. Fo thi cae it ha been aumed that: the FS tation i located at 38 N latitude; the antenna gain i 50 dbi; the azimuth angle of the antenna i 90 ; the opeating fequency i 050 MHz; the tanmitte powe pectal denity at the input to the antenna i 0 db(w/1 khz); the atellite i in a cicula obit at an altitude of 800 km; the inclination of the obit plane i 90 ; and the atellite ue an iotopic eceiving antenna with a gain of 0 dbi. The eult of the analyi ae hown in Fig. 5. The olid cuve i the pobability denity function of intefeence eceived by a low-obiting atellite. The dahed cuve give the cumulative pobability that the intefeence exceed a pecific value. Fo example, the olid cuve how that the pdf of the intefeence being on the ode of 150 db(w/1 khz) i about Similaly, the dahed cuve indicate that the pobability of intefeence exceeding 170 db(w/1 khz) i about 1 10 o 1%.

11 Rep. ITU-R SA Pobability denity function of intefeence to FS ytem caued by emiion fom low-obiting atellite The appoach ued to compute the pdf of intefeence to FS tation caued by emiion fom lowobiting atellite i a mino extenion of the appoach decibed in the peviou ection. In thi cae, the incident intefeence at the FS tation i aumed to confom to value of pfd that ae pecified a a function of the elevation angle at the FS tation. The tep of the pocedue decibed in 3.1 ae ued. The computation of I (ϕ, λ ) become: f λ I ( ϕ, λ ) = ρ( δ) GT ( ϕoff axi) (1) 4π whee ρ(δ) i the pectal pfd, δ i the elevation angle and the othe paamete ae a defined peviouly. Equation (19) i ued to compute the off-axi angle and the pectal pfd i given by: 154 ρ( δ) = ( δ 5) 144 db(w /(m db(w /(m db(w /(m 4 khz)) 4 khz)) 4 khz)) fo fo fo 0 δ < 5 5 δ < 5 5 δ < 90 () The elevation angle i obtained uing the cala poduct of the ange vecto R and the FS tation poition vecto P. Noting that co (90 δ) = in δ, then: 1 [ R P] δ = in (3) R

12 1 Rep. ITU-R SA.066 An example ha been evaluated fo thi intefeence cae. Hee it ha been aumed that: the FS tation i located at 38 N latitude; the eceiving antenna gain i 35 dbi; the azimuth angle of the antenna i 90 ; the opeating fequency i 50 MHz; the incident pectal pfd at the FS tation i given by equation (); the atellite i in a cicula obit at an altitude of 800 km; and the inclination of the obit plane i 90. The eult of the analyi ae hown in Fig. 6. The olid cuve i the cumulative pobability that the intefeence exceed a paticula value. Figue 6 how that the pobability of the intefeence exceeding 167 db(w/4 khz) i on the ode of Simplified method fo calculating viibility tatitic The exact method to calculate viibility tatitic fo a atellite in a cicula obit, and whoe obital peiod i incommenuate with the otational peiod of the Eath, may be detemined with the aid of equation (8). Thi equation, which give the pdf of a atellite occupying a location at a pecified latitude ϕ and longitude λ on the obital hell i epeated a equation (4): whee: p(ϕ, λ ): ϕ : pdf 1 coϕ P( ϕ, λ ) = (4) π in i in ϕ geocentic latitude on the obital hell of inteet

13 Rep. ITU-R SA λ : elative geocentic longitude on the obital hell of inteet i: inclination of the obital plane with epect to the equatoial plane. The pobability that a atellite i within a bounded aea of the obital hell, fo example, and i viible within the 3 db beamwidth of a eceiving antenna, i given by a uface integal: 1 coϕ P( ϕ, λ ) = dϕdλ π in i in ϕ The geneal olution of equation (5) fo an abitaily defined aea on the obital hell i difficult. Howeve, fo the pactical cae of a cicula antenna beam, cetain aumption lead to a implified olution. Thi cae i conideed in 4.1. Fo a econd pactical cae, whee the antenna beam i eithe cicula, o pehap of a omewhat moe complicated hape, a numeical method i decibed in Simplified method fo cicula antenna beam Two implifying aumption may be made to equation (5) to obtain an accuate etimate of the pobability that a atellite will be viible. The pactical cae i an eath tation o a teetial tation that ue a elatively high gain antenna with a cicula beam pointed at a fixed azimuth and elevation angle. The fit implifying aumption involve the denominato of the integand in equation (5). If the vaiation of the value of the denominato i mall ove the ange of the latitude of inteet on the obital hell, then the following implification may be made 1 1 P( ϕ, λ ) = co ϕdϕdλ π in i in Φ whee: 1 / in i in Φ epeent a weighting facto evaluated fo Φ S to be applied to the uface integal. (A will be hown late, Φ S i taken a the latitude of the cente of the egion of inteet.) The integand i geatly implified by thee aumption ince it become imply the encloed uface aea A S on a unit phee, and the pobability educe to: (5) (6) 1 AS P( ϕ, λ ) = (7) π in i in Φ The baic geomety poblem to be olved i to detemine, A S, the aea of the inteection of a cone (the cicula antenna beam) and a phee (obital hell). Thi i facilitated with the econd et of aumption. When the angula dimenion of the cone i ufficiently mall, the poblem become the inteection of a cone and a plane that i nomal to the phee at the cente of the inteection. It i well known that the inteection eult in an ellipe, which fo a unit phee, encompae an aea: whee: θ a : θ b : A S = πθ a θ b (8) emimajo axi of the ellipe emimino axi of the ellipe, both angle being meaued in ad. The aea A S may be calculated with the aid of Fig. 7. Figue 7 how an eath/teetial tation at point P on the x-axi of a 3-dimenional, coodinate ytem. The boeight of the tation antenna i pointed towad point P in the x-y plane at an elevation angle of δ 0. R i the ange fom the tation

14 14 Rep. ITU-R SA.066 to P. The geocentic angle between the tation poition vecto P and P i θ 0. The majo axi of the ellipe, lie in the x-y plane and the mino axi of the ellipe lie in a plane nomal to the x-y plane. The majo axi of the ellipe may be detemined uing a imple elationhip between the elevation angle and the cental angle: coδ θ = co 1 δ (9) β whee: θ: cental angle δ: elevation angle β = 1+h/ e ; h: altitude of the atellite of inteet e : adiu of the Eath. Fom equation (9), it may eadily be hown that θ a, the emimajo axi of the ellipe on the unit phee i: 1 1 co( δ0 ϕ3 / ) 1 co( δ0 + ϕ3 / ) θa = co co + ϕ3 (30) β β whee ϕ 3 = the beamwidth of inteet of the antenna (uually the 3 db beamwidth); and, the othe paamete ae a peviouly defined. FIGURE 7 Geomety to detemine the emimajo and emimino axe of the ellipe eulting fom the inteection of the cicula beam and the obital hell

15 Rep. ITU-R SA Refeing to Fig. 7, the emimino axi of the ellipe i detemined by fit computing the ac S b, which lie in the plane nomal to the x-y plane. The econd tep i to detemine the cental angle coeponding to the ac S b. The cental angle i the emimino axi of the ellipe on a unit phee. Thu, but, R S e = β S e b S co R = b e S e δ 0 ϕ 3 b in δ 0 (31a) (31b) = βθ (31c) ϕ3 1 θ = β δ δ b co 0 in 0 (31d) β The value fo A S i detemined fom equation (8), (30) and (31d). Note that θ a, θ b and ϕ 3 mut be expeed in ad: A π = 4 ϕ3 co β β 1 co δ co 0 ( δ ϕ / ) co( δ + ϕ / ) 0 β 3 in δ 0 co 1 0 β 3 + ϕ3 (3) The latitude of the intecept of the boeight of the antenna of the eath/teetial tation Φ S i detemined in the following way. The geomety i hown in Fig. 8. The tation of inteet i now located in the x-z plane of a geocentic coodinate ytem at a latitude of ϕ p. The antenna pointing angle ae given in tem of the azimuth angle θ az in the clockwie diection fom Noth and the elevation angle δ 0 elative to the local hoizontal plane. Shown in Fig. 8 i an oblique pheical tiangle with ide a, b and c which ae oppoite the angle α, θ az and γ. The paamete of the oblique pheical tiangle ae elated to the phyical paamete by: b = π/ Φ c = π/ ϕ P a = co 1 (β 1 co δ 0 ) δ 0 (33a) (33b) (33c) The latitude at which the boeight of the antenna inteect the unit phee and the elative longitude of the point of inteection ae given by the Law of Coine fo Side of an oblique pheical tiangle: in Φ S = in ϕ P co a + co ϕ P in a co θ az (34a) co a in ΦS in ϕp co α = (34b) coφ coϕ Note that λ S i the angle between the two plane nomal to the x-y plane and which contain the ac b and c. With thi obevation, λ S i obtained fom the law of coine when Φ S = ϕ P = 0: S P λ S = α (35)

16 16 Rep. ITU-R SA.066 FIGURE 8 Geomety to detemine the latitude and longitude of the intecept of the tation antenna, given the azimuth and elevation angle, and, the elative altitude of the atellite obit 4. Manual method to calculate viibility tatitic Equation (9) how that the pobability of a atellite occupying a mall egion on the obital hell bounded by latitude ϕ ϕ /, ϕ + ϕ / and longitude λ i given by: 1 P( ϕ, λ) = π λ ϕ + ϕ / 0 ϕ ϕ / in coϕ i in ϕ dϕ dλ Futhe, a hown by equation (10), caying out the integation ove the aea yield: P ( ϕ, λ) λ = π in 1 in ( ϕ + ϕ / ) in( ϕ ϕ / ) in i in Thi ugget that the pobability ove a lage, and pehap moe complicated aea could be evaluated by the eie: P ( ϕ λ) 1 in i ( ϕ + ϕ / ) in( ϕ ϕ / ) (36) (37) λ j 1 in k 0 1 k 0, = in in (38) π in i in i j, k whee ϕ k ae tip in latitude of height ϕ 0 and of longitudinal extent λ j uch that they ae encloed within the bounday on the obital hell by the inteection of the antenna beam of inteet and the obital hell. The implementation of thi technique i bet explained uing an example. Figue 9 how a typical inteection of an eath tation cicula antenna beam with the obital hell. The paamete fo thi example ae given in Table 1. The olution of equation (38) i implemented uing a peadheet. A quae aay coniting of cell i et up that epeent the longitude in the x diection and latitude in the y diection on the obital hell. The latitude and longitude of the

17 Rep. ITU-R SA cente of the aay coepond to the latitude and longitude of the inteection of the antenna boeight with the obital hell. It alo epeent the latitude and longitude of the poition vecto of the atellite when the atellite i aligned with the boeight of the antenna. Thu, each of the othe cell in the aay epeent the poible latitude and longitude of the poition vecto of the atellite. It emain to detemine which of thoe cell ae within the aea encompaed by the eath tation antenna beam of inteet ϕ 3. Thi i accomplihed with the aid of vecto a hown in Fig. 10. Figue 10 how the poition vecto of the eath tation o teetial tation P, the poition vecto of the atellite S at an abitay location, and the ange vecto R S. Since the poition vecto of the tation and the atellite ae eithe known o aumed, the ange vecto i detemined fom: R S The tation and atellite poition vecto ae given by: = S P (39) P = coϕ 0 in γ P P (40a) coϕ S = β coϕ in ϕ coλ in λ S S (40b) The key to detemining if a paticula atellite location lie within an aea cicumcibed by the tation antenna beam i the cala poduct of the boeight ange vecto and the ange vecto aociated with the aumed atellite poition vecto. The angula offet between the two vecto i: 1 1 ϕ j, k = co RS 0 R j, k (41) R S0 R j, k whee: ϕ j,k : off axi angle fo the j-th value of λ and the k-th value of ϕ R S0 : boeight ange vecto : ange vecto fo the j-th value of λ and the k-th value of ϕ. R j, k Fo a cicula antenna beam, if ϕ j,k ϕ 3 /, then the atellite will appea within the beamwidth of inteet. If thi condition i met fo the paticula cell, then the value in the cell i et to 1, if the condition i not met, the value of the cell i et to 0. Thu, fo each ow of the cell aay, it i only neceay to um the numbe of 1 in a ow and to multiply that numbe by the facto given in equation (38). Summing the value thu obtained fo each ow ove the 41 ow of latitude yield the etimate fo the pobability that a atellite will appea within the pecified beamwidth of the tation antenna. The tep ize in both latitude and longitude ae paamete that ae manually enteed into the peadheet. Value ae elected to enue that the eulting aea, a hown in Fig. 9, i fully contained within the aay. In othe wod, the cell at the exteme in latitude and longitude all contain 0, and eulting aea i ufficiently lage to enue the accuacy of the numeical olution.

18 18 Rep. ITU-R SA.066 TABLE 1 Example paamete and eult Eath tation latitude = 40 Latitude of inteection (Φ S ) = Eath tation longitude = 0 Longitude of inteection (λ S ) = 8.88 Antenna azimuth angle = 105 Latitude tep ize = 0.03 Antenna elevation angle = Longitude tep ize = Satellite altitude = 400 km Pobability of viibility = % Satellite inclination = 51.6 Beamwidth of inteet = 7 FIGURE 9 Latitude and longitude of the inteection of a cicula beam with the obital hell (ee Table 1)

19 Rep. ITU-R SA FIGURE 10 Detemining the latitude and longitude on the obital hell which ae encompaed by the eath tation antenna beam of inteet 4.3 Compaion of the numeical eult obtained uing the implified and manual method fo cicula antenna beam Table how epeentative eult fo ix example cae. In each cae it wa aumed that the atellite of inteet wa in an 800 km obit, inclined by 8 o with epect to the equatoial plane. It wa futhe aumed that the obital peiod and the peceion of the node wee incommenuate with the otation ate of the Eath. A a conequence, in a eie of tial, the location of the atellite on the obital hell i a andom event. A Table how, the eo between the eult obtained uing the two method wa le than 0.4% fo the ix cae. Cae TABLE Compaion of the eult obtained fo the pobability of viibility fo ix example cae uing the implified method and the manual method of calculation Station location Antenna data Pobability of viibility Latitude (degee) Longitude (degee) Azimuth angle (degee) Elevation angle (degee) Beamwidth of inteet (degee) Simplified method (%) Manual method (%) Relative eo (%)

20 0 Rep. ITU-R SA Mean of calculating the coodinate of the inteection of two obital plane Intefeence event involving an eath tation and two o moe atellite fequently occu at the point of cloet appoach of the two atellite. One cae of paticula impotance concen Eath obeving atellite in un-ynchonou obit. Thei obit ae uually at the ame altitude, but thei obital plane ae offet. If no conideation ha been given to phaing the atellite in thei epective obital plane, it i poible fo the atellite to actually co, one in font of the othe. If thi occu while one atellite i being tacked by an eath tation, it can eult in the eath tation being captued by the othe atellite and tating to tack the othe atellite. Thi intefeence event lead to a lo of data fo the peiod between the lo of lock on the deied atellite and ubequent eacquiition of the deied atellite. The latitude and elative longitude in inetial pace whee thi will occu i elatively eay to calculate. 5.1 Analyi The geomety of the inetial coodinate ytem i hown in Fig. 11. Thee ae two obital plane. The fit, which i inclined with epect to the x-y plane by I 1, i offet fom the econd by λ 1. The x-axi lie in the econd obital plane which i inclined by I with epect to the x-y plane. The latitude of the inteection of the two plane i deignated by ϕ 0. The path of the atellite on the unit obital phee ae hown by the ac a and b. It i well known fom pheical tigonomety that thee i a imple elationhip between the latitude of a location on an inclined cicula obit and the cental angle. Thu, fo plane No.1: and fo plane No. At the point of inteection, ϕ 1 = ϕ. Theefoe: in ϕ 1 = in bin I1 (4a) in ϕ = in a in I (4b) in b in I1 = in a in I (43) Futhe, fom the Law of Sine fo oblique pheical tiangle: in λ in γ 1 in a = in I 1 in b = in(180 I ) (44) Alo, fom the Law of Coine fo angle: Equation (45a) may be olved fo γ: co γ = co I λ (45a) 1 co(180 I) + in I1 in(180 I) co 1 γ = co (co I 1 co I + in I1 in I co λ1) (45b) 1

21 Rep. ITU-R SA FIGURE 11 Geomety to detemine the latitude and longitude of the inteection of two plane Conequently, the latitude of inteection i given by: ϕ 1 in λ 1 = in in I 1 in (46) in γ 0 I The longitude of inteection i obtained in the following way. Fom equation (44), the cental angle a i given by: Futhe: 1 in λ 1 a = in in I1 (47a) in γ 1 ( tan a I ) λ0 = tan co (47b) Table 3 give eveal example of the latitude and longitude of inteection of the obital plane in an inetial coodinate ytem. Fo convenience, the ight acenion of the acending node (RAAN) fo atellite No. i aumed to be the x-axi. Note that the latitude and longitude of the inteecting plane i the latitude and longitude of the point whee the atellite will co if the altitude of the atellite obit ae the ame.

22 Rep. ITU-R SA.066 Cae RAAN (degee) TABLE 3 Example of the inteection of offet obital plane Satellite No. 1 Satellite No. Inteection Inclination (degee) RAAN (degee) Inclination (degee) Latitude (degee) Longitude (degee)