Journal of Geodesy (1997) 71: 749±767
Theory of satellite ground-track crossovers M. C. Kim Center for Space Research, The University of Texas at Austin, Austin, Texas, 78712-1085, USA Tel: +1 512 471 8121; fax: +1 512 471 3570; e-mail: [email protected]
Received: 19 November 1996 / Accepted: 12 May 1997
Abstract. The fundamental geometry of satellite ground The points where the ground track of a satellite in- tracks and their crossover problem are investigated. For tersects itself on the surface of the earth are called (single- idealized nominal ground tracks, the geometry is gov- satellite) crossover points. When the intersections are erned by a few constant parameters whose variations from ground tracks of two distinct satellites, these are lead to qualitative changes in the crossover solutions. usually called dual-satellite crossover points. As a means On the basis that the theory to locate crossovers has not of obtaining relevant measurements at the same geo- been studied in suf®cient detail, such changes are graphic location separated in time, crossover points hold described in regard to the number of crossover solutions a signi®cant usage in satellite geodesy, especially in the in conjunction with their bifurcations. Employing the realm of satellite oceanography associated with the ap- spinor algebra as a tool for establishing the ground- plications of satellite altimetry. The satellite-altimeter track crossing condition, numerical methodologies to crossover dierence, which is de®ned as the dierence in locate crossovers appearing in general dual-satellite two satellite altimeter ranges interpolated at a crossover ground-track con®gurations are also presented. The point along their respective ground tracks, provides pe- methodologies are applied to precisely determined culiar time-dierenced oceanographics. While the altim- orbital ephemerides of the GEOSAT, ERS-1, and eter crossover dierences have been also used in the TOPEX/POSEIDON altimeter satellites. evaluation of an altimeter time-tag bias (Schutz et al. 1982), in the re®nements of satellite orbits (Cloutier 1983; Sandwell et al. 1986; Born et al. 1986; Moore and Roth- Key words. Satellite á Ground track á Crossover well 1990; Hsu and Geli 1995; Kozel 1995), in the cali- Bifurcation á Spinor bration of a gravity ®eld model (Klokocnik and Wagner 1994), in the estimation of the altimeter inferred sea state bias (Gaspar et al. 1994), in the improvement of a plan- etary topography model (Stottlemyer 1996), etc., the computational eciency of locating crossovers cannot be overemphasized as one of routine tasks for such a wide 1 Introduction range of applications. Techniques for the determination of crossover loca- In the geodetic applications of an arti®cial satellite as a tions have been introduced in several studies. For single- platform for active remote sensors, associated measure- satellite crossovers, Hagar (1977) presented a computa- ments are frequently sampled along the ground track of tional ¯owchart to obtain circular-orbit ground-track the satellite's nadir point, which is also referred to as the crossovers taking into account the secular perturbation subsatellite point. For instance, a satellite-borne altim- eects of the earth's oblateness. Hereinafter such an eter measures the distance between the satellite and its idealized circular orbit, its geocentric projection onto the nadir point so that the measurement, called the satellite surface of a uniformly rotating spherical earth model, and altimeter range, can be used in the mapping of the the associated crossover points will be referred to as the physical shape of the mean sea surface (Kim 1993) and nominal orbit, nominal ground track, and nominal in the monitoring of time-varying oceanographics, such crossover points, respectively. As the nominal orbit and as the oceanic tides (Ma et al. 1994), the oceanic eddy its ground track can be eectively described by a few variabilities (Shum et al. 1990b), the global mean sea constant parameters, a set of fundamental mathematical level rise phenomena (Nerem 1995), etc., when the relations between the latitude, longitude, and ground- satellite's orbital trajectory is determined by indepen- track crossing times can be easily established. Therefore dent tracking to the satellite (Tapley et al. 1994). Rowlands (1981) and Shum (1982) presented iterative 750 positioning of actual single-satellite crossovers starting mapping of the satellite's orbital trajectory onto a from nominal crossovers to utilize altimeter crossover prescribed earth-surface model such as a reference dierences in their orbit adjustment and determination ellipsoid of revolution (Torge 1991) or a spherical studies. As also summarized in Schrama (1989) and harmonics geoid model (Shum 1982). While de®nitions Spoecker (1991), the general procedure of single-satellite of the nadir mapping and the earth surface can be crossover locating has been based on the search for in- dependent on a speci®c problem and earth satellites are tersections between piecewise ground tracks segmented under the in¯uence of a variety of perturbations, the by ascending and descending passes (i.e., by semirevolu- nominal ground track is appropriate analytically to tions), and mainly consists of two steps of the prediction investigate the fundamental geometry of ground tracks. of nominal crossovers and the determination of actual In the rotating earth-®xed geocentric equatorial crossovers by interpolating discrete subsatellite points Greenwich-meridional reference frame (hereinafter, the projected from satellite trajectories. As well as the actual earth-®xed frame), the orbital angular motion of a sat- determination step, the analytical prediction step itself ellite can be represented by the geocentric unit vector needs numerical iterations due to the inherent transcen- dental relation between the latitude and the dierence of x cos / cos k r 1 crossing timings as discussed by Colombo (1984). Such a r^ 8 y 9 8 cos / sin k 9 two-step procedure has also been adopted in ®nding dual- r r <> z => <> sin / => satellite crossovers. For a particular combination of two nominal orbits, Santee (1985) employed spherical trian- cos s cos # cos i sin s sin # :> ;> :> ;> gular models to make an initial guess of simulated dual- cos s sin # cos i sin s cos # 1 8 9 satellite nominal crossovers followed by iterative re®ne- > > ments by updating inclinations. More or less general < sin i sin s = dual-satellite crossover-®nding techniques, all based on where:> x; y; z denotes the Cartesian;> components of the the semirevolutions search, have been also discussed by satellitef positiong vector r associated with its geocentric Shum et al. (1990a), and by Moore and Ehlers (1993). radius r, latitude /, and longitude k, s is the argument of Given orbital ephemerides of actual satellites and a latitude and represents the satellite's orbital rotation, # proper de®nition of ground tracks, the crossover locating is the longitude of the orbital ascending node measured problem is to determine the ground-track crossing time- from the Greenwich meridian and represents the tag pairs. Strictly speaking, the solutions are intersections rotation of the satellite's orbital plane with respect to of two unrelated curves. Depending on the relevant orbit the rotating earth, and i is the inclination of the orbital geometries, the number of dual-satellite crossovers can be plane. For a nominal ground track, i becomes constant a priori unknown. There can exist more than one inter- and the two rotations s and # become uniform variables section (or no intersection) in a semirevolution ground- resulting in their linear dependence, track pair and therefore one may lose some crossovers at the prediction step. Even for the counting of crossovers in # ke qs 2 the nominal single-satellite con®guration, we found some aspects overlooked in the previous studies. On the basis where ke represents the longitudinal origin of the that the theory to locate crossovers has not been surveyed nominal ground track, and q denotes the angular in sucient detail and that we need a more systematic velocity ratio between the two uniform rotations. With approach to the problem, this paper presents both ana- the linear dependence we can consider s as a time-like lytical and numerical aspects of locating crossovers in the continuously increasing independent variable (the cases of the general dual-satellite scenario which include along-track location in the nominal ground track) and the single-satellite cases as a subset. We ®rst investigate # as a dependent variable which is continuously the fundamental geometric structure of ground tracks. decreasing, as the rotation of any orbital plane with With some relevant mathematical tools described in the respect to the earth is always westward (i.e., q > 0). Appendix, we establish the ground-track crossing con- Nominal ground tracks are thus on the three-di- dition, followed by numerical strategies for ®nding them. mensional parametric space of 0 i 180; We will provide an answer to the predictability on the 0 < q < ; 0 k < 360 . Apart from the longitudi- 1 e number of crossover solutions. With some treatments on nal origin ke, the manner in which a nominal ground the issue of the nadir mapping of satellite ephemerides track is laid down on the spherical earth is primarily onto the reference ellipsoid of revolution, actual appli- dependent on i and q. While i determines the latitudinal cations of the developed techniques to the precisely de- bounds of a nominal orbit and its ground track, it is also termined orbital ephemerides of the historic GEOSAT the parameter to classify conveniently the longitudinal and the currently orbiting ERS-1 and TOPEX/POSEI- direction of orbital motions. We say that an orbit is DON altimeter satellites will be also presented. posigrade when 0 i < 90 , is retrograde when 90 < i 180, and is circumpolar when i 90. How- ever, such a simple classi®cation is valid only when or- 2 Nominal ground tracks bits are viewed from the inertial frame. When orbits are viewed from the earth-®xed frame, in which ground The ground track of an earth-orbiting arti®cial satellite tracks are actually depicted, we need careful accounting is the time-trace of its subsatellite point: a nadir for the relative rotational motions between the orbital 751 planes and the earth. For a nominal orbit the relative rotation can be eectively quanti®ed by the parameter q. To investigate the behavior of a nominal ground track, using Eqs. (1) and (2), we ®rst formulate the spin- axis component velocity of the unit vector r^,
z 0 2 2 /0cos / sin i cos s sgn cos s cos / cos i r p3 and the spin-axis component angular momentum,
x y 0 y x 0 2 k0cos / r r r r cos i q q sin2 i 1 cos2 s 4 cos i q cos2 / Á where the prime indicates dierentiation with respect to s, and the sgn function takes 1, 0, or 1 depending on the sign of its argument. Equations (3) and (4) can be combined to yield Fig. 1. Longitudinal direction of nominal ground tracks classi®ed in q; cos i space 1 d/ cos2 / cos2 i Æ 2 5 cos / dk cospi q cos / when cos i < 0, f when cos i > 0), q which represents the instantaneous slope of the nominal 1f cot2 i . In the region1q < cos i 1, the associat-1 ground track on the unit sphere. Note that for given ed ground tracks are always eastward f > 1 . In the values of and , the slope is a double-valued function 1 i q two regions q < cos i 1and 1 cos i < 0, the as- of the latitude /. sociated ground tracks are always westward f < 0 . The latitudinal motion of the nominal ground track Hereinafter these two regions are respectively referred to is bounded in cos / cos i and is j j as the posigrade westward region and the retrograde westward region, where the ``posigrade'' and ``retro- northward when cos s > 0 grade'' indicate the longitudinal direction with respect to stationary when cos s 0 6 the inertial frame (the ``eastward'' and ``westward'' are 8 > used for the earth-®xed frame). In the remaining region < southward when cos s < 0 1 of 0 < cos i < min q; q (where 0 < f < 1), the asso- as:> the sign of /0 in Eq. (3) will indicate. Meanwhile the ciated ground tracks change the longitudinal direction; Á 2 1 longitudinal motion can be somewhat complex. As the eastward at higher latitudes of cos / < q cos i and 2 1 sign of k0 in Eq. (4) will indicate, the nominal ground westward at lower latitudes of cos / > q cos i.We track is will refer to this region as the eastward/westward mixed region or simply the mixed region. It can be said that 2 2 1 eastward when cos s < f cos / < q cos i ground tracks of all circumpolar orbits cos i 0 are 2 2 1 westward k0 q<0. The point q cos i 1 8stationary when cos s f Ácos / q cos i > ( f 1=2 as a limit) corresponds to the geostationary > 2 2 1
Fig. 2. Nominal ground tracks 753
(A to U) with the same value of k 180 but with generate no crossovers, its neighboring track e varying q and i as indicated by the corresponding 21 I q 11=20; i 55 has 160 crossovers showing the points (A to U) in the q; cos i space shown in Fig. 1. eect of a small change in the value of q. The two arrows in each of the nominal ground tracks J q 1=2; i 95 is a highly inclined westward track indicate the subsatellite point at the epoch s 0 and which shows four crossover points. The following three after a quarter orbital revolution s p= 2 . As the tracks (K, L, M) are for the same value of q 2=3 but tracks are periodic, we can limit our interest in a repeat with varying i. While both K i 70:5 and L i 85 period 0 s < 2Mp . are located in the mixed region, they are quite dierent The ®rst seven tracks (A±G) correspond to as far as the number of crossovers is concerned; none in q N=M 1=3 but with varying i.Ai 0or K and twelve in L. The track M i 120 is another cosi 1 is an eastward equatorial track. In the period typical westward track having the same number of of the three M orbital revolutions, the track makes two crossovers as L but with a quite dierent topological M N eastward equatorial revolutions with the net structure. eects of one westward N revolution of the orbital The next three tracks (N, O, P) correspond to node with respect to the earth. As a degenerate case, the q 1=1. N i 60 is a typical geosynchronous satellite track is con®ned to the equator, and one can say that its ground track which forms a ®gure-eight curve. The actual repeat period is not three M revolutions but one track librates both latitudinally and longitudinally and half, i.e., M= M N , revolutions. B i 60 or spending 12 h successively in each of its northern and cos i 1=2 is a typically inclined eastward ground southern loops. With q 1=1, this geosynchronous be- track. One can say that the two longitudinal revolutions havior is preserved between the geostationary case of the track A have been separated with an equatorial i 0 and the circumpolar geosynchronous case symmetry. Now the repeat period becomes three M i 90 which is shown as the track O. As the track orbital revolutions. There are three crossover points (at becomes purely retrograde westward i > 90 , the lon- the equator) where the two longitudinal revolutions in- gitude no more librates but undergoes smooth rotations tersect each other. C i 70:5 or cos i 1=3 is at the as shown in the track P i 120 . Each of the three border q cos i of the purely eastward region and the tracks has only one crossover point (as its epoch posi- eastward/westward mixed region shown in Fig. 1. The tion). track crosses the equator vertically so that the angle The last ®ve tracks (Q±U) are for q 2=1 but with between ascending and descending traces becomes 180 varying i. With the track Q i 30 we are now in the at the crossover points. With D i 85 or posigrade westward region. The track generates no cos i 0:087 we are now in the mixed region. The track crossover point. R i 60 , a cuspidal track, is a lim- is eastward (resp. westward) above (resp. below) iting case located at the border q cos i 1 of the mixed / 59:2 , at which it becomes instantaneously meri- region and the posigrade westward region. We see two j j dional k0 0 . There exist nine crossovers with the cusps where the ground-track velocity instantaneously appearance of six nonequatorial crossovers whose lon- becomes zero. With the track S i 85 , we are again in gitudes are aligned with those of equatorial crossovers. the mixed region showing a looping ground track with E i 90 or cos i 0 is a circumpolar ground track an equatorial asymmetry. The cusps of R have been which consists of inclined straight lines as both the lat- evolved into loops with the appearance of two cross- itude and longitude become linear functions of s, i.e., overs. Following another circumpolar track T with two /0 sgn cos s and k0 q. Each of the north and crossover points, the nominal ground track becomes south poles becomes a collocation point of three cross- westward again as shown in the track U i 120 . overs, and we therefore still have nine crossovers. With While all nominal ground tracks have longitudinal F i 120 or cos i 1=2 we are now in the retro- cyclic symmetry (i.e., the same pattern repeats M times), grade westward region. While there still exist nine they also have either a symmetry or an asymmetry with crossover points, the longitudes of nonequatorial respect to the equator. The equatorial symmetry occurs crossover points are no more aligned with those of the when M and N have the same parity (both odd) and the equatorial crossover points. As the last example of equatorial asymmetry occurs when M and N have the q 1=3 cases, G i 180 or cos i 1 is a westward opposite parity. While one can count M equatorial equatorial track. Like the track A, the ground-track crossovers in the same parity cases, no equatorial con®guration is again con®ned to the equator. In the crossover point exists in the opposite parity cases due to period of the three M orbital revolutions the track the asymmetry. In its repeat period, a nominal ground makes four M N westward equatorial revolutions track undergoes M latitudinal librations such that it and one can say that its actual repeat period is 3/4, i.e., crosses a zonal line, in sin / < sin i,2Mtimes (along M= M N , revolutions. The equatorial tracks each of its M ascendingj andj M descending traces). cos i 1 actually form degenerate cases, for which Meanwhile the longitudinal behavior can be diverse. In the ground-trackÆ motion becomes one dimensional. case of a retrograde westward track, there are M N The next six tracks (H±M) are examples of equa- longitudinal revolutions which consist of M revolutions torially asymmetric ground tracks characterized by the due to the satellite's orbital motion and N revolutions opposite parity of M and N. While H q 1=2; i 55 due to the net eects of the orbital plane's motion with closely corresponds to a ground track of the Global respect to the earth. Such a track crosses a meridional Positioning System (GPS) satellites (Leick 1995) which line M N times. Since the M N longitudinal revo- 754 lutions interweave so that there exist M N 1 cross- satellite nominal ground-track con®gurations, one may over latitudes and each of the crossover latitudes con- need to survey a ®ve-dimensional parametric space of tains M crossover points, the number of crossovers is i1; i2; q1 N1=M1; q2 N2=M2; D ke1 ke2 . Here D M M N 1 . In case of a purely eastward track, there represents the mutual longitudinal origin of two ground exist M N longitudinal revolutions which produce tracks, and is introduced to reduce the number of pa- M N 1 crossover latitudes and consequently rameters (from six to ®ve) by using the fact that the M M N 1 crossovers. In case of a posigrade west- relative geometry of two nominal ground tracks is in- ward track, there exist N M longitudinal revolutions variant upon a simultaneous rotation around a ®xed which produce N M 1 crossover latitudes and con- axis. In Sect. 4 we will provide a more systematic sequently M N M 1 crossovers. For the eastward/ treatment for the count of crossovers in both the single- westward mixed tracks, as will be veri®ed later, the and dual-satellite nominal ground-track con®gurations. number of crossover latitudes varies between M N 1 and M N 1 depending on the values of 3 Ground-track intersection cosjj i andj q, andj consequently the number of crossovers varies from M M N 1 to M M N 1 with the Let r1 t1 and r2 t2 be two arbitrary orbital trajectories increment M. jj j j referenced to the earth-®xed frame, where t1 and t2 are Some previous studies have also presented formulas time-tags (to be frequently omitted) used as independent to count the number of single-satellite nominal cross- variables of the two trajectories. To the extent that the overs appearing in a repeat period. Tai and Fu (1986) geocentric nadir mapping is concerned, the ground- claimed that the number equals M M N 1 if track crossing condition for the two trajectories can be cos i > 0 and equals M M N 1 if cos i < 0 (note simply written as that M and N change their notational roles in Tai and 2 Fu). Spoecker (1991) claimed M . Considering the case r^1 t1 r^2t2 10 q < 1, Cui and Lelgemann (1992) claimed M M rN 1 where r sgn cos i . Along with a sign where r^1 and r^2 are geocentric unit vectors pointing r1 error in Tai and Fu, and the complete ignorance of the and r2, respectively. earth rotation eects in Spoecker, these studies passed Let F denote the direction cosine between r1 and r2, over the fact that the crossing geometry of a single- i.e., satellite nominal ground track is more signi®cantly Ft1;t2 cos w t1; t2 r^1t1r^2t2 11 variant in the two-dimensional parametric space of Á q; cos i . The reciprocal of q has been called the where w represents the spherical angular distance ground-track parameter. Although some of its impor- between the two ground tracks. Locating ground-track tant roles have been recognized in the counting of crossovers can be considered as a two-dimensional root- crossover latitudes (King 1976; Farless 1985; Parke et al. ®nding problem of a single scalar equation Ft1;t2 1, 1987), we claim that all of these previous studies missed which replaces the vector equation in Eq. (10). Since F the existence of the eastward/westward mixed tracks and yields its maximal value 1 at crossovers, the problem can the posigrade westward tracks. be solved by using the extremal condition In Fig. 3 we overlay a pair of nominal ground tracks _ _ with i1 60; q1 1=3; k 1 180 and i2 95; Ft1 ^r1 r^2 0; Ft2 r^1 ^r2 0 12 e Á Á q2 1=2; ke2 155 . In addition to the three cross- overs of the eastward track and the four crossovers of where the subscripts t1 and t2 indicate partial dieren- tiations. Given a t1; t2 point close to a crossover the westward track, one can count ten points where the fg two tracks intersect. These intersections, ordered from 1 solution, the Newton-Raphson iteration scheme to 10 along the eastward track, are nominal examples of 1 t1 t1 ^r1r^2 ^r_1^r_2 ^r_1r^2 the dual-satellite crossover point. To fully classify dual- Á Á Á13 t2 t2 ^r_1^r_2 r^1^r2 r^1^r_2 Á Á Á may converge to the solution. The iteration fails when the determinant of the 2 2 Hessian matrix, i.e., the Gaussian curvature of F , Â
2 2 G Ft t Ft t F ^r1 r^2 r^1^r2 ^r_1^r_2 14 1 1 2 2 t1t2 Á Á Á vanishes. Á Á Á Let K be the angle between ^r_1 and ^r_2, i.e.,
^r_1 ^r_2 ^r_1 ^r_2 cos K 15 Á k At a crossover point, , using the relation r^1 r^2 ^r r^ ^r_ 2, Eq. (14) yields Á 2 2 2 2 2 2 G ^r_1 ^r_2 ^r_1 ^r_2 ^r_1 ^r_2 sin K 16 Fig. 3. Overlay of two nominal ground tracks v Á
755 where the subscript v, to be frequently omitted on the Now we present an alternative approach by repre- right-hand sides of equations, indicates evaluation at the senting the orbital states of satellites in terms of spinors, crossover point. We note that the iteration failure occurs which enable us to rewrite the ground-track crossing when the ground tracks become collinear condition in a more convenient form. At this point, the K 0 or 180 . author encourages the reader to refer to the Appendix, A converged solution of Eq. (13) does not always which presents a summary on spinors, their algebra, and yield a crossover solution (where F 1, a maximum of the two-way transformations between the vector state F , or equivalently w 0, a minimum of w) since other and the spinor state. Some equations in the Appendix, types of extrema also satisfy Eq. (12). In other words, e.g., the spinor multiplication laws in Eq. (A1), will be Eq. (12) forms only a necessary condition for a cross- frequently used in the following development. over. Figure 4 shows the variation of w in the two-di- Let e^ ; e^ ; e^ be a right-handed orthogonal triad of f x y zg mensional domain 0 s1=2p < M1 3; 0 s2=2p < unit vectors and suppose it forms the earth-®xed frame. M2 2 associated with the two nominal ground tracks We consider the spinor decomposition of the two orbital shown in Fig. 3. Among the 14 minimal points of w in position vectors the white area, 10 points correspond to crossovers which r1 Ry e^ R1; r2 Ry e^ R2 18 are numbered from 1 to 10, consistent with Fig. 3. The 1 x 2 x remaining four minima labeled by L are local minima. with One may also count 14 maxima in the gray area. There actually exist 56 extremal points including 14 saddle R1 a1 jb1e^x c1e^y d1e^z 19 points in each of the white and gray areas. The extrema R2a2jb Á2e^x c2e^y d2e^z of F can be classi®ed as Á
Crossovers F 1 Maxima H < 0 Centers G > 0 hLocal Maxima F < 1 17 Extrema F 0 hMinima H > 0 r hSaddles G < 0
where H denotes the mean curvature of F given by where a1; b1; c1; d1 and a2; b2; c2; d2 denote the real f g f g H Ft1t1 Ft2t2 =2. With the iteration scheme, Eq. (13), components of the spinors R1 and R2 respectively, j is we need to check whether a converged solution is a the unit imaginary number, and the superscript crossover or not. With the symmetry in the numbers of denotes the spinor conjugate. y centers and saddles, the symmetry in the numbers of With Eq. (18) the ground-track crossing condition maxima and minima, and the existence of local maxima, Eq. (10) can be rewritten as less than a quarter of converged solutions will be crossovers. Ry e^ R Ry e^ R 1 x 1 2 x 2 20 r1 r2 where, for the magnitudes of the two vectors, we have
r1 Ry R1 R1Ry ; r2 Ry R2 R2Ry 21 1 1 2 2 Premultiplying R1 and postmultiplying R2y , Eq. (20) yields
e^ R1Ry R1Ry e^ 22 x 2 2 x on using Eq. (21). We introduce a new spinor
Q R1Ry A jBe^ Ce^ De^ 23 2 x y z Á Substituting Eq. (19) into the product in R1R2y in Eq. (23), which is also given in Eq. (A5), one obtains
Fig. 4. Spherical angular distance (in degree) between two nominal ground tracks 756
A a1a2 b1b2 c1c2 d1d2 r_1 g1 r1 r_2g2r2 F 1 j e^ m1 e^ ;Fyje^m2e^ r r2 z g x 2r r2zgx B b1a2 a1b2 d1c2 c1d2 1 1 1 222 32 C c1a2 d1b2 a1c2 b1d2 D d1a2 c1b2 b1c2 a1d2 24 where g's are the magnitudes of the total angular momenta, With Eq. (23), Eq. (22) yields 2 g g1g^ r1 r_1 r r^1 ^r_1 1 1 Â 1 Â 33 e^xR1R2y R1R2y e^x e^xQ Qe^x 2 De^y Ce^z 0 2 g g2g^ r2 r_2 r r^2 ^r_2 2 2 Â 2 Â Á25 and m's are the eective normal perturbations of the on using Eq. (A1). Note that the ground-track crossing satellites in the earth-®xed frame. condition is now reduced to Substituting Eq. (31) into Eq. (30), we have C t ; t 0; Dt;t 0 26 1 1 1 2 1 2 Q F 1R1Ry F 1Q t1 2 2 2 34 whereas and can be arbitrary. 1 1 A B Q R1 Ry F y Q F y We will refer to the two scalar equations in Eq. (26) t2 2 2 2 2 2 as the spinor representation of the ground-track cross- Now, substituting Eq. (32) into Eq. (34), in comparison ing condition. In the t1; t2 domain the crossovers are with Eq. (30), we obtain located at the intersections of the two nonlinear curves; 1 r_1 g1 r1 i.e., we are again working with a two-dimensional root- A A D m1 B t1 2 r r2 g ®nding problem for a set of the unknown crossover time 1 1 1 pairs. The two equations in Eq. (26) can be solved with 1r_1g1r1 the Newton- Raphson iteration scheme Bt1B2Cm1A 2r1rg1 1 35 1 1r_1g1r1 t1 t1 Ct1 Ct2 C 27 Ct1C2Bm1D t2 t2 Dt Dt D 2r1 r1g1 1 2 1r_1g1r1 with DDAm1C t12rr2 g 111 C c_1a2 d_1b2 a_1c2 b_1d2 and t1 _ _ 1 r_2 g2 r2 Ct2 c1a_2 d1b2 a1c_2 b1d2 A A D m B 28 t2 2 2 2 r2 r2 g2 Dt d_1a2 c_1b2 b_1c2 a_1d2 1 1r_gr _ _ B2B2Cm2A Dt2 d1a_2 c1b2 b1c_2 a1d2 t2 22 2r2r g2 2 36 obtained on dierentiating the last two equations in 1r_2g2r2 Eq. (24). It is quite noteworthy that the condition of Eq. Ct2C2Bm2D 2r2 rg2 (26) is necessary and sucient, and therefore a converged 2 solution of the spinor-version iteration scheme Eq. (27) 1r_2g2r2 Dt2D2Am2C always yields a crossover, in contrast to the vector- 2r2 r g2 2 version iteration scheme of Eq. (13). Equation (27) fails when the determinant of the 2 2 Jacobian matrix, as similar to Eq. (A34). Â With Eqs. (35) and (36), at crossovers, where J C D D C 29 C D 0, the Jacobian in Eq. (29) becomes t1 t2 t1 t2 1 g1 g2 i.e., the Jacobian of the crossing condition Eq. (26), Jv 2 2 AB 37 vanishes. 2 r1 r2 To further analyze the Jacobian, we take partial which, as will be veri®ed, vanishes when the ground derivatives of Q in Eq. (23), i.e., tracks become collinear in accordance with Gv in Eq. (16). _ y Q R1R At1 jBt1e^x Ct1e^y Dt1e^z We consider the two rotating satellite-®xed orbital t1 2 30 _ frames (relative to the rotating earth-®xed frame); QR1RyAjB Áe^ Ce^ De^ t22t2t2 x t2 y t2 z _ _ r^1 e^x r^2 e^x For the derivatives ÁR1 and R2y , from Eqs. (A44) and 1 1 (A45), we have g^1 r^1 R1y e^y R1; g^2 r^2 R2y e^y R2 8 Â 9 r1 8 9 8 Â 9 r2 8 9 < g^1 = < e^z = < g^2 = < e^z = _ 1 _ 1 R1 F 1R1; Ry Ry F y 31 38 2 2 2 2 2 : ; : ; : ; : ; with The spinor algebra enables us to write 757
1 between the Jacobian (appearing in the spinor-version g^1 g^2 g^1g^2 R1ye^zR1R2ye^zR2 Á hir1r2 iteration scheme) and the Gaussian curvature (in the 1DE vector-version) at crossovers. ezR1R2ye^zR2R1y 39 r1r2 1 DE1 4 Number of crossovers e^ Qe^ Qy A2B2C2D2 r r z z rr 1 2 12 The ground-track crossing condition derived in the DE Á on using Eq. (23). In Eq. (39), the bracket denotes the previous section is exact and general as far as the scalar operator in which the product of two spinors is geocentric nadir mapping is concerned. Given orbital commutative as applied to R1y and e^zR1R2y e^zR2. In similar ephemerides of two satellites in their ®nite mission ways, one can obtain the remaining eight direction periods, the crossover locating problem is now reduced cosines between the two orbital frames to write to solve the crossing condition. How many crossovers
2 2 2 2 r^1 A B C D 2 BC AD 2 BD AC r^2 1 2 2 2 2 g^ r^1 2 BC AD A B C D 2 CD AB g^ r^2 40 8 1 Â 9 r r 2 38 2 Â 9 g^ 1 2 2 BD AC 2 CD AB A2 B2 C2 D2 g^ < 1 = < 2 = 4 5 : ; : ; which generalizes Eq. (A18). With the relation exist? And when do they (pairs of time-tags) occur? Strictly speaking, the number of solutions can be a y 2 2 2 2 Q Q R1R2y R2R1y r1r2 A B C D 41 priori unknown as highly perturbed, irregularly maneuvered satellites can generate signi®cantly com- Eq. (40) yields r^1 r^2, and plicated ground-track patterns, and there is nothing 22 special about any crossover point from either satel- g^1 r^1 1AB2AB g^2r^2 Â 22 2 2 Âlite's point of view. Let us consider instead the g^1 AB2AB A B g^2 v vnumber of crossovers in nominal ground tracks. In 42 practice, as most of geodetic satellites have near- circular orbits, nominal ground tracks often have at crossovers C D 0 . topologically equivalent structure to true ground With the relations tracks. Nevertheless, no satisfactory answer has been g1 g2 given to the question regarding the number of cross- ^r_1 g^ r^1; ^r_2 g^ r^2 43 r2 1 Â r2 2 Â over solutions between two nominal ground tracks 1 2 (even for single-satellite nominal ground tracks as
Eq. (42) states that, at crossovers, the angle between g^1 discussed before). _ _ and g^2 is same as that of ^r1 and ^r2. (The ground-track We ®rst consider the single-satellite nominal ground- crossing angle is essentially formed by two osculating track crossing by representing the spinor Q in terms of orbital planes instantaneously passing through the cross- Euler angles which are less useful than spinor compo- over point. When observed from the earth-®xed frame, nents for numerical purposes but superior in providing the orientations of the orbital planes can be eectively some qualitative insights. Substituting Eq. (A48) into represented by the unit vectors g^1 and g^2.) We can write Eq. (24), with some trigonometry one can derive A2 B2 2AB cos K ; sin K 44 A pr1r2 sin s cos i sin # cos s cos i cos # v A2 B2 v A2 B2 Bpr1r2 sin s sin i sin # cos s sin i cos # so that the correct quadrant of the crossing angle Kv is Cpr1r2 cos s sin i sin # sin s sin i cos # de®ned in 0; 360 . With Eq. (44), Eq. (37) yields Dpr1r2 cos s cos i sin # sin s cos i cos # 48 1 g1 g2 Jv sin Kv 45 4 r1 r2 with
Meanwhile, with Eq. (43), Eq. (16) can be written as 1 1 s s1 s2 ;ii1i2; Æ 2 Æ2 2 1 Æ Æ 49 g1 g2 ##1#2 2 Gv 2 2 sin Kv 46 r1 r2 From the last two equalities in Eq. (48), with i1 i2 i, We therefore obtain a simple relation q1 q2 q, ke1 ke2 ke,and# qs , the ground- track crossing condition C 0 and D 0 yield 4J 2 G v 47 v r r cos s sin qs sin i 0 50 1 2 758 and cos s sin qs cos i sin s cos qs 51 Meanwhile we restrict our interests to a repeat period which corresponds to the two-dimensional domain 0 s2 < s1 < 2Mp 0 < s