Analytical Effective Method for Verification of a Satellite Pass over a Region of the Earth Surface Atanas Marinov Atanassov

Transcription

1 Analyical Effecive Mehod for Verificaion of a Saellie Pass over a Region of he Earh Surface Aanas Marinov Aanassov Solar Terresrial Influences Insiue, Bulgarian Academy of Sciences, Sara Zagora Deparmen, P.O. Box 73, 6000 Sara Zagora, Bulgaria Absrac. An analyical mehod is proposed in his work for verificaion wheher an arificial earh saellie during is orbial moion passes over a region of he earh surface. The mehod is based on undisurbed Keppler s approximaion of he orbi and approximaion of he region by a circular segmen S. In order o define he siuaional condiion, a conic surface is used wih apex in he earh cenre, cuing ou he circular segmen. The angens of he conical surface wih Keppler s plane deermine he ime inervals in which he saellie race on he earh surface occurs inside he segmen S. The ransformaion of hese angens in he plane of Keppler s orbi and he deerminaion of heir crossing poins wih Keppler s ellipse lies in he basis of he examined mehod. 1. Inroducion. A number of cases exis when, during space experimens, i is necessary o know he ime of a saellie pass over a definie region of he earh surface. Thus, for example, in synchronous saellie and ground-based measuremens, i is imporan when he saellie passes over a definie erriory where he ground-based saion is locaed. When problems of meeorological characer are solved on he basis of saellie informaion, i is significan when he saellie is going o pass over a definie erriory or a meeorological srucure (cyclone cenre, fron). The soluion of many oher problems, conneced wih he sudy of he earh surface from space is conneced wih he deerminaion of he emporal inerval pass over a specific region. This is necessary in some of he cases for experimens planning. In oher cases, he analysis is needed o schedule he seances for receiving saellie informaion. In boh cases his is imporan for he qualiy of he conduced experimens, and from economical poin of view. The problem for deermining a saellie pass over a definie geographic region has a sandard soluion. I is obained on he basis of he imiaion modelling by selecing a proper geomerical model for region V which deermines he siuaional condiion. The discreizaion of he soluion of he arificial earh saellie moion equaion and he respecive analysis, as concerns he model of he region, allow o deermine wheher he saellie passes over he region as well as he momens of crossing is borders. For he equaion of he arificial earh saellie moion in geo-equaorial coordinae sysem (GeCS) we have: d r (1) m = f k, d

2 wih iniial condiions r 0 d r d r (0) = r (0), =, where r is he saellie radius- d d vecor; m - is mass and - he ime. The specific form of (1) reflecs he acceped moion model. The soluion of (1) can be obained on he basis of analyical or numerical mehods [1,]. In any case, a discreizaion of he soluion of (1) is obained: () r, r, r,..., r, n Usually () is obained in GeCS or in orbial co-ordinae sysem (OCS). I is necessary o ransform he soluion of (1) ino Greenwich co-ordinae sysem (GrCS): r ( GKS ) (3) ( GrKS ) = αgrg. r In (3) αgrg is he ransformaion marix [3]. Problems exis in which region V is resriced by a complex ouline conour (for example, a sae border). There are known mehods o presen V and o solve he problem for crossing is borders by he sub-saellie race [4]. Wihin he erms of differen problems, he approximaion of region V by a circular spherical segmen of he earh surface is compleely sufficien and subsaniaed boh physically and of geomerical poin of view. The applicaion of such a simplifying siuaional condiion in he discreizaion of he soluion of he arificial earh saellie moion equaion requires also considerable compuaion ime.. Formulaion of he Problem. We shall examine he considered region of he earh surface as a spherical segmen S (Fig. 1). I is cu ou of he earh surface by a sraigh circular cone wih angle ψ beween he axis and he generan and is apex is in he earh cenre. The crossing poin of he cone axis wih he earh surface has Greenwich co-ordinaes ( λ, Θ ). Therefore, he segmen can be described by he following parameers angle ψ, earh radius R and he Greenwich co-ordinaes λ and Θ, i.e. S ( Ψ, R, λ, Θ). Moving along wih he earh surface, he cone angens wih he plane of Keppler s orbi a is wo sides a momens 1 and. (Fig. ). Beween he wo momens 1 and, he Keppler s plane and he conic surface inercross. This means ha par of he Keppler s ellipsis is also resriced wihin he limis of he conic surface and ha i is locaed over segmen S.

3 Figure 1. A region of he earh surface, presened by a circular segmen. We shall discuss an approach, allowing o obain momens ~ 1 and ~ when he saellie crosses he cone generans τ 1 and τ which angen wih he Keppler s orbi. Figure. The spherical segmen crosses Keppler s plane beween momens 1 and ( 3 and 4, respecively); vecors τ 1 и τ ( τ3 and τ4, respecively) deermine he generans, by which he conic surface angens wih Keppler s plane.

4 The relaion beween he inervals ( 1, ) and ( ~ 1, ~ ) on he ime axis shows wheher he arificial earh saellie passes over segmen S (Fig. 3). If he wo inervals inercross, hen he condiion for passing over he examined segmen is fulfilled. ~ Figure 3. Differen cross-secions beween inervals ( 1, ) and ( 1, ), in which he siuaional condiion is execued. ~ 3. Consrucion of an algorihm. Le's assume ha segmen S forms a angen wih K. For disance δ from he cenre of S o K we can wrie down [5]: n (4) (Rc - x ) = δ = sin ψ. R n or (4') n 0.Rc = sin ψ. R where n 0 is he null vecor of K, Rc -is he radius-vecor of he segmen middle and R = Rc - he Earh radius. The radius-vecor of he spherical segmen cenre Rc can be presened in he following way: Xc = R sin Θ.cos[ ωz ( 0)] (5) Yc = R Zc = R sin Θ.sin[ ωz ( cos Θ 0)] In (5) ω is he Earh angular roaion velociy and 0 is appropriaely seleced epoch (for example, he momen when he arificial earh saellie passes hrough he orbi perigee). If we subsiue (5) in (4') we'll obain:

5 (6) A cos ϕ + Bsinϕ + C = 0, where n. sin Θ, B = n.sin Θ, C = sin ψ n. cos Θ, ϕ = ω ( ). A = x y z 0 By solving (6) we deermine Rc a he angening momens 1 and as well as he very momens. Thus, for he angen vecor we can wrie down: (7) τ = ( R c n ) n Vecor τ is deermined in (7) in GeCS. We make a ransformaion of τ (8) τ (OKS) = α OGe. τ (GеКS) In (8) he ransformaion marix αoge has he following form [3]: α 11 = cosω. cosω - sin ω. cos i. cosω α 1 = cosω. sin Ω + sinω. cos i. cosω α 13 = sin ω. sin i α 1 = -sin ω. cosω - sinω.cos i. sin Ω α = -sin ω. sin Ω + sinω.cos i. cosω α 3 = cosω. sin i α 31 = sin Ω. sin i α 3 = -cos Ω. sin i α 33 = cos i in OCS [3] : Afer deerminaion of he angen vecor τ in K, we can deermine is crossing poins wih Keppler's ellipse in OCS: ( ξ+ c) η (9) + = 1, η = k. ξ a a (1 e ) In he second equaion of sysem (9) k signifies he angen s coefficien in OCS. The following relaion exiss beween he orbial co-ordinaes (ξ,η) and he eccenric anomaly E [1]: ξ = a ( cos E - e ) (10), η = a 1 e.sin E where a is he large orbial semi-axis, e - is he eccenriciy. On he oher side, on he basis of Keppler's equaion we can wrie down: (11) = 0 + ( E - e. sin E ) / λ Afer we find ou he eccenric anomaly E in (10) and subsiue i in (11), we deermine he momens when he saellie crosses he specified angens. 4. Esimaion of he Mehod. The explained mehod is analyical and i is presened by final formulae. I is reduced o a single applicaion of he respecive calculaion procedure wihin he limis of one saellie circle. Afer correcion of he orbial elemens, he procedure can be repeaed for he nex inerval of ime.

6 The examined mehod is based on a siuaional condiion whose geomerical model is reduced o he deerminaion of angens τ 1 and τ in GeCS. The ransformaion of he angens in OCS is equivalen o he ransformaion of he siuaional condiion in he orbial plane [6]. A srucural approach is applied for he mehod algorihmizaion. Based on a programme complex for siuaional analysis, developed for soluion of he problems in [6], i was necessary o add wo new sub-programmes for ensuring he reaed siuaional problem. This means ha he developmen of algorihms for siuaional analysis, based on he ransformaion of he siuaional condiions o Keppler's plane is faciliaed by he presence of common sub-problems. In our case and for hese in [6] his is he crossing of a sraigh line wih Keppler's ellipse. The following cases are possible for one Earh roaion around is axis: - wih sufficien orbial inclinaion equaion (6) has four roos which leads o deerminaion of four angens conneced wih wo crossings of segmen S wih Keppler's plane; - wih smaller orbial inclinaion equaion (6) has wo soluions which deermine wo angens, corresponding o one crossing of segmen S wih K ; - wih small orbial inclinaion segmen S doesn' cross K. The correcion of he orbial elemens of each saellie circle on he basis of he seleced model of disurbances allows o apply he presened approach for siuaional analysis wihin coninuous inerval of ime. The mehod is applicable in he cases when Keppler's approximaion in he erms of he saellie's circle period is admissible wih a view o he solved problem. For solving pracical problems in many cases his is execued. R e f e r e n c e s 1. А б а л а к и н, В., Е. А к с е н о в., Е. Г р е б е н н и к о в., Ю. Р я б о в., Справочное руководство по небесной механике и астродинамике, М., "Наука", 1971, 584с.. Э л ь я с б е р г, П., Введение в теорию полета искуственных спутников Земли., М, "Наука", Э с к о б а л, П., Методы определение орбит., М., "Мир", Навигационное обеспечение полета орбитального комплекса "Салют-6" - "Союз"-"Прогресс", Под ред. Б. Петрова, М., "Наука", К о р н, Г., Т. К о р н, Справочник по математике. М. "Наука", A a n a s s o v, A., Mehods for Siuaional Analysis Based on he Transformaion of he Siuaional Condiion Towards he Plane of Keppler's Orbi. Comp. Rend. Acad. Bulg. Sci., 45, 199, 6, p