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Journal of Geodesy (1997) 71: 749±767

Theory of 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

z 0 2 2 /0cos / sin i cos s sgn cos s cos / cos i r      p3  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 dierentiation 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  cospi 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 region1q < 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 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  f Ácos / > q cos i orbit whose ground track is motionless over a point on the equator 0 . > 7 / ; k ke :> Á A nominal ground track  will eventually be parallel to where its preceding trace with an oset less than some preset limit. To this extent all nominal ground tracks are re- q cos i 1 q cos i cos i peating, although the repeat interval may be as long as f 1  8 possible. Hence we restrict our interest to the cases  q sin2 i  q 1 cos i 1 cos i     where q is a rational number, i.e., is a constant which can be considered as a parametric q N=M 9 function of i and q.   Based on Eqs. (7) and (8), in Fig. 1 we classify the where M and N are positive coprime integers. With Eq. longitudinal direction of nominal ground tracks in the (9), which is a resonance condition between s and #, the two-dimensional parametric space of 0 < q < ; nominal ground track becomes periodic and repeats its 1 cos i 1 . There are four regions separated 1 by trace after M orbital revolutions of the satellite, or q cos i f 1 , q cos i 1 f 0 , cos i 0 f 0 , equivalently, N nodal days. In order to visualize the and surrounded   by cosi 1 f , cos i 1 topological evolution of nominal ground tracks, in Fig. 2 f when q < 1; f when q > 11 , q 0(f we geographically display 21 nominal ground tracks 1 1    752

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. eect  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 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 dierent  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 0or K and twelve in L. The track M i 120 is another cosi 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 dierent topological M N eastward  equatorial revolutions with the net structure. eects 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-trackÆ  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 eects 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 iandjq, 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 eects 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 dieren- 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 ÁÁ  vanishes. ÁÁÁ 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 R2a2jbÁ2e^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^ m1 e^ ;Fyje^m2e^      r  r2 z  g x 2rr2zgx B b1a2 a1b2 d1c2 c1d2 1 1 1 222   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 eective 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 m1 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 Bt1B2Cm1A 2r1rg1 1 35 1 1r_1g1r1  t1 t1 Ct1 Ct2 C 27 Ct1C2Bm1D t2 t2 Dt Dt D  2r1r1g1   1 2   1r_1g1r1 with DDAm1C t12rr2g 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 _ _ B2B2Cm2A Dt2 d1a_2 c1b2 b1c_2 a1d2 t2 22   2r2rg2 2 36 obtained on dierentiating the last two equations in 1r_2g2r2  Eq. (24). It is quite noteworthy that the condition of Eq. Ct2C2Bm2D 2r2rg2 (26) is necessary and sucient, and therefore a converged 2 solution of the spinor-version iteration scheme Eq. (27) 1r_2g2r2 Dt2D2Am2C always yields a crossover, in contrast to the vector- 2r2rg2 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); QR1RyAjBÁe^ Ce^ De^ t22t2t2 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  hir1r2 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 DEÁ 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 222 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 eectively 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

2 2 2 1 cos sÃ cos qsÃ KÃ q KÃq 058  cos s sin qs   Ã Ã Á which indicate the range of the extremun KÃ. In other words, the function K is monotone when K < 0 and 1 1 when min q; q

Fig. 5. The s1; s2 domain for single-satellite nominal crossover 2M solutions of s and the L solutions of s ) in the two- solutions  dimensional domain of 0 < s < Mp; 0 s < 2Mp .    759

Since we are working in its semidomain 0 < s when both ground tracks are circumpolar, it is essen- < Mp; s s < 2Mp s , we say that the number tially a quadratic equation and may yield 0, 1, or 2 real of single-satellite   nominal crossovers is ML. solutions depending on the parametric values. The As we have seen in the two-dimensional parametric existence of a solution (or solutions) implies that the space of single-satellite cases (which forms a subspace of corresponding parametric point is in bifurcating regions the ®ve-dimensional parametric space of dual-satellite where the number of crossovers can be changed as the cases), changes may occur in the number of crossover parameter values vary. 1 solutions in the mixed region 0 < cos i < min q; q . In Fig. 7 we display eleven i1; i2 sections of the  These changes correspond to ``bifurcations'' i1; i2; q1; q2 space by varying the values of q1; q2 . The Á   (Guckenheimer and Holmes 1986, p. 117): the term eleven points (A±K) in the q1; q2 section at the right  originally used to describe the splitting of equilibrium lower corner depict the approximate values of q1; q2  solutions in a family of dierential equations at para- corresponding to the eleven i1; i2 sections. In each of  metric points or regions where the Jacobian derivative of the i1; i2 sections, bifurcation regions are striped the dierential equations has a zero eigenvalue. In our (horizontally,  vertically, or in both ways). The hori- case, the splitting of crossover solutions occurs where zontally striped regions correspond to the existence of the Jacobian of the ground-track crossing condition the 0 crossing angle solution, and the vertically striped Eq. (26) is singular. We know that the Jacobian vanishes regions correspond to the existence of the 180 crossing when the ground-track crossing angle becomes 0 or angle solution. While both roots exist in the regions 180. In other words, as it can be quite naturally un- where the horizontal stripes and vertical stripes cross, no derstood, births and deaths of crossover solutions can real root exists in the white areas, which will be called occur at parametric regions where the two ground tracks regular regions. The lines labeled a±g are can be collinear. Such regions will be referred to as bi- a. sin i2 i1 q1sin i2 q2 sin i1 furcating regions.  We now return to the ground-track slope equation b. sin i2 i1 q1sin i2 q2 sin i1 (5). One can write the ground-track collinear condition   c. cos i1 cos i2 by  d. cos i1 cos i2 64 1 d/ 1d/   60 e. q2 cos i2 1 cos / dk 1cos / dk 2    f. q1 cos i1 1  which yields g. Y 2 4XZ 0  2 2 2 cos / cos i2 cos i1 q1 cos / 61 at which the characteristics of the parametric space can 2 2 2 be signi®cantly changed. pcos / cosÁi1 cos i2 q2 cos /  Æ In each of the eleven sections, there essentially exist where,p in the dual sign,Á ``plus'' is for the 0 crossing four regular regions (left, right, lower, and upper white angle cases (i.e., both tracks are north-going or south- areas) which are separated by the diagonal and antidi- going) and ``minus'' is for the 180 crossing angle cases agonal bifurcating regions. We focus our discussion on (i.e., one track is north-going and the other south- the section A q1 N1=M1 1=3; q2 N2=M2 1=2     going). It is noteworthy that D is absent in Eq. (61), and which is typical for two satellites below the geosyn- we can therefore work with a four-dimensional para- chronous altitude (i.e., q < 1). In each of the four reg- metric space of i1; i2; q1; q2 to search the bifurcating ular regions, the number of crossovers is invariant. Since  regions. As the roles of two tracks can be exchanged, we one can consider the four limiting points i1 0;  can set q2 q1. Taking squares of both sides, Eq. (61) i2 90 , i1 180; i2 90 ,i190; i2 0 ,i1        can be written as 90; i2 180 as representative points of the four reg- ular regions, the corresponding numbers of crossovers 6 4 2 X cos / Y cos / Z cos / 0 are readily summarized as 2M2 M1 N1 ;2M2M1N1,    jj  2 2 2 2M1M2N2;2M1M2N2, respectively. For in- with max cos i1; cos i2 cos / 1 62 jj    stance, with i1 0; i2 90 each of the M2 revolu-   and ÈÉ tions of the circumpolar ground track crosses with each of the M1 N1 revolutions of the eastward equatorial X = q2 q2 track twicejj (one along its ascending trace and one along 1 2 its descending trace). In the bifurcating regions, the 8Y =2q2cos i2 q1 cos i1 22 2 2 number of crossovers cannot be easily guessed and may >qcos i1 q cos i2 63 >2 1  depend on the value of the remaining parameter D.In >2 2 2cosi1 cos i2 q1 cos i2 q2 cos i1 >  ground tracks, such a small change can be easily intro- > Equation:> (62) is a cubic equation in cos2 /. Excluding duced as the parameters experience perturbed varia- the peculiar triple solution cos2 / 0, which can occur tions.  760

Fig. 7. Examples of i1; i2 parametric sections 

1=2 1=2 1=2 Appendix R RRy RyR a2 b2 c2 d2 j j         A7 Satellite state in spinor space  The spinor R can be decomposed into R R^ where R^ is a Let e^x; e^y; e^z and e^u; e^v; e^w be right-handed orthogo- ^ j j f g f g unimodular spinor (i.e., R 1. nal triads of geocentric unit vectors which form the The spinor algebra isj equivalentj  to the quaternion rotating earth-®xed frame and the non- rotating inertial algebra, which has long been known as the ideal tool to frame, respectively. In the spinor algebra, which forms a describe rotational motions of rigid bodies. In practice, subalgebra of the so-called Cliord algebra [geometric the four components of a unimodular spinor correspond algebra in Hestenes (1983)], such a triad corresponds to to the Euler parameters (Goldstein 1980, p. 155). For the Pauli spin matrices (Goldstein 1980, p.156) and instance, the earth's rigid body rotation can be eec- yields the spinor multiplication laws (Cartan 1981, p.44; tively represented by Landau and Lifshitz 1977, p.202), e.g., e^x e^u e^u e^ e^ e^ e^ e^ e^ 1 x x y y z z e^ E^y e^ E^ x e^ A8    A1 8 y 9 8 v 9 e 8 v 9 e^ e^ je^ e^e^ (in cyclic order)  e^  e^  Â e^  x y  z  y x < z = < w = < w = where j is the unit imaginary number. where: ;E^ is a: unimodular; spinor: which; accounts for the Let v1 b1e^ c1e^ d1e^ and v2 b2e^ c2e^ precession, nutation, spin axis rotation and polar  x  y  z  x  y d2e^z be two arbitrary vectors. With the multiplication motion of the earth (Lambeck 1988), and xe is the laws of Eq. (A1), one can readily verify the relation corresponding angular velocity vector. There is a complete analogy between the orbital an- v1v2 v1 v2 j v1 v2 A2  Á  Â  gular motion of a satellite and the rotational motion of a rigid body. Let s ue^u ve^v we^w represent the orbital between the spinor product, the scalar product, and the position vector of a satellite  in the geocentric inertial vector product of the two vectors. Equation (A2) can be frame. Let S a j be^u ce^v de^w be a spinor based also represented by on the triad e^ ;e^ ; e^ . As it can be directly veri®ed f u v wg v v 1 v v v v using the multiplication laws of Eq. (A1), which is also 1 2 2 1 2 2 1 applicable to e^ ; e^ ; e^ , the spinor algebra enables us Á  1   u v w v1 v2 jv1v2 v2v1 A3 f g Â 2   to write The spinor space is a combination of the scalar and s Sye^uS A9 vector spaces such that a spinor R a j v and its     conjugate Ry a j v, where v be^x ce^y de^z, are with de®ned by the four real-valued scalar components 2 2 2 2 a; b; c; d . It is equivalent to the two-dimensional u a b c d ; v 2 bc ad ; w 2 bd ac f g        complex vector space appearing in quantum mechanics A10 in conjunction with the wave functions (Greiner 1994, p.  306). We will frequently denote the scalar part of R with Let s be the magnitude of s. We also have R , i.e., 1=22 h i s s u2 v2 w2 SSSySyS 1 jj   A11 R a Ry R Ry A4 2222  h i h i2    aÁbcd  where the bracket will be called the scalar operator. hi On dierentiating Eq. (A9), we have Let R1 a1 j v1 and R2 a2 j v2 be two arbitrary spinors. With Eq. (A2), one obtains _ _ s_ Sye^ S Sye^ S A12  u  u  R1R2y a1 jv1 a2 j v2 which can be expanded to relate the scalar components     _ _ _ a1a2 v1 v2 j a2v1 a1v2 v1 v2 A5 of s u_e^u v_e^v w_ e^w and those of S a_ j be^u c_e^v   Á   Â         d_e^w . One can actually write which can be further expanded in terms of the real   components a ; b ; c ; d and a ; b ; c ; d to yield u u_ ab c d a2a_ 1 1 1 1 2 2 2 2 the equationsf given in Eq.g (24) inf the main text.g v v_ dcb a b2b_ 2 3 2 32 3 A13 The products of the two spinors are in general not w w_  cd ab c2c_  commutative, e.g., R1R2y R2y R1. But they are commuta- 6 s s_ 7 6abc d76d2d_7 6 6 7 6 76 7 tive inside the scalar operator, i.e., with Eq. (A5), we have 4 5 4 54 5 on using the equations in (A10), (A11), and their _ R1R2y R2yR1 a1a2 v1 v2 dierentiations. As the six components of s and s h ih i  Á constitute the orbital state in the vector space, the eight a1a2 b1b2 c1c2 d1d2 A6      components of S and S_ constitute the orbital state in the For the norm of the spinor R, we use a positive scalar spinor space. While Eq. (A13) stands for the spinor-to- 764 vector state transformation, we need more information where the four-parametric representation of the 3 3 to write its inverse transformation (note that s and s_ are transformation matrix can be also found in classicalÂ auxiliaries and the 4 4 matrix is inherently singular). dynamics textbooks, e.g., Goldstein (1980, p. 153). We As the vector s canÂ be decomposed into its dilation s denote the nine elements of the transformation matrix and rotation ^s, i.e., s s^s, where ^s is the unit vector by T11; T12; ...;T33. These can be easily obtained from  pointing s, the spinor S can be also decomposed into its the vector state s; s_ , as they correspond to the f g ^ ^ dilation S and rotation S^, i.e., S S S^. As shown in Cartesian components of s; h s; sh. Note that the  transverse unit vector h^ ^s isÂ pointing towards the Hestenes (1983), the unimodular spinor S^ can be de- Â velocity of the radial unit vector composed into the Eulerian rotations, i.e., _ h ^ ^s 2 h ^s A19 ^ 1 1 1  s Â  S exp je^w2s exp je^u2i exp je^w2X A14      which itself is not a unit vector. We have where the three Euler angles X; i; s (Goldstein 1980, p. _ h 146) correspond, respectively, to the longitude of the ^s 2 A20 ascending node measured from the equinox, the incli- j js  nation, and the argument of latitude of the orbit. With Based on Shepperd (1978), an ecient means of the relation S s1=2, we can write computing the four components of S from the trans-  formation matrix is to write

4a2 2T T s 2   4ad T12 T21; 4bc T12 T21 4b 2T11 T s    8 2   and 4ab T23 T32; 4cd T23 T32 A21 > 4c 2T22 T s 8     > 2   4ac T31 T13; 4bd T31 T13 < 4d 2T33 T s <      > : :> 1=2 1 1 1 1 with T T T T : the trace of the transformation S s cos 2s je^w sin 2s cos 2i je^u sin 2i 11 22 33      matrix. Several ways exist to obtain the four unknowns, cos 1X je^ sin 1X A15 Â 2  w 2   e.g., which can be expanded to yield the spinor components T s T23 T32 a s1=2 cos 1 X s cos 1i; b s1=2 cos 1 X s sin 1i a  ; b  2   2  2  2 Æ 4  4a 1=2 1 1 1=2 1 1 r c s sin X s sin i; d s sin X s cos i T31 T13 T12 T21  2  2  2   2 c ; d A22 A16  4a  4a   showing their correspondence to the Euler parameters where the sign of a cannot be determined but can be (Goldstein 1980, p. 155) and to the Cayley-Klein selected freely assuming that we are working with a one- parameters (Whittaker 1937, p. 12). to-one correspondence, or isomorphism, between s and Let h be the orbital angular momentum vector of the the pair S. One may say that S is a double-valued satellite, i.e., functionÆ of the transformation matrix. While the solution given in Eq. (A22) is singular when a 0, we h hh^ s s_ s^s s_^s s^s_ s2^s ^s_ A17    Â  Â   Â  can write three more solutions by changing the compo- nent to be used as the divisor. At least one nonzero where the scalar h and the unit vector h^ represent the component exists as long as s 0. For numerical magnitude and direction of h, respectively. Now we purposes, the optimal choice among6 the four possibil- introduce a new triad ^s; h^ ^s; h^ , which consists of the ities corresponds to the largest among absolute values of radial, transversal, andf normalÂ g unit vectors of the , whose magnitude order is same as that of rotating satellite-®xed orbital frame. The spinor algebra a; b; c; d . enables us to write the transformation T ; T11; T22; T33

2 2 2 2 ^s 1 e^u 1 a b c d 2 bc ad 2 bd ac e^u ^ ^ ^  2 2  2  2  h ^s Sy e^v S 2 bc ad a b c d 2 cd ab e^v A18 8 Â^ 9  s 8 9  s 2   2 2  2  2 38 9  < h = < e^w = 2 bd ac 2 cd ab a b c d < e^w = 4     5 : ; : ; : ; 765

Now, to obtain the components of the time derivative on using Eqs. (A27) and (A30). Now, with Eq. (A33), S_ , we consider the Eulerian angular velocity vector of one can expand Eq. (A32) to yield the four components the orbit expressed as (Hestenes 1983) of S_ , ds di dX 1 s_ h s _1s_hs ^ ^ a_ a 2 d n b ;bb2cna xs h f e^w A23  2 s s h 2ssh  dt  dt  dt  1s_hs_1s_hs c_2Áscs2bnhd;d2Ásds2anhcA34 with   ToÁ obtain the right-hand sidesÁ in Eq. (A34) we need h^ sin i sin X e^u sin i cos X e^v cos i e^w A24 to know n. However, having the same instantaneous    position and velocity as the true orbit, we can set n 0, and which relates the spinor state to the osculating Keplerian orbit. From Eq. (A34) one can extract f^ cos X e^ sin X e^ A25  u  v  ^ ca_ db_ ac_ bd_ 0 A35 where f represents the unit vector toward the ascending    . For the time derivatives of the Euler and angles, we have the Gauss variation of parameter equations (Battin 1987, p.488) s2 ba_ ab_ dc_ cd_ n A36 dX s sin s di s ds h s sin s cos i    h  n ; n cos s; n dt  h sin i dt  h dt  s2 h sin i The relation in Eq. (A35) is called the spinor gauge A26 condition (Hestenes and Lounesto 1983) and also  appears in the Kustaanheimo-Stiefel transformation where n represents the normal component of the satellite (Stiefel and Scheifele 1971; Deprit et al. 1994). The acceleration. two relations of Eqs. (A35) and (A36) compensate the Substituting Eqs. (A24±A26) into Eq.(23), with some two redundancies in the spinor state. manipulation, one can obtain So far we have established the two-way transforma- tion between the vector and spinor states observed in the 1 h s x Sy e^ n e^ S A27 inertial frame. The transformation is general and s  s s2 w  h u   equivalently applicable to states for the earth-®xed frame. To clarify, let us ®rst denote the orbital position or and velocity vectors of a satellite observed in the earth- h s ®xed frame by r xe^x ye^y ze^z Rye^xR and r_ x_e^x A28 _   _    xs 2 n y_e^ z_e^ Rye^ R Rye^ R, where R a jbe^ ce^  s  h   y  z  x  x   x y de^ is a spinor based on the rotating earth-®xed triad on using the ®rst equality in Eq. (A18). Meanwhile, the  z e^ ; e^ ; e^ . Note that r s but their Cartesian compo- velocity vector s_ can be decomposed into (Hestenes and x y z nents are dierent, i.e., x; y; z u;v;w , as they are Lounesto 1983) referencedÈÉ to the earth-®xedf g6f frame andg the inertial s_ frame, respectively. Also note that r_ s_ as the time s_ s_^s x s s 1 j x s s x 1 Wys sW 6   s Â  s 2 s s2   derivatives are observed in the two dierent reference frames. The dierence is the eects of the earth rotation, A29  i.e., with r_ s_ xe s A37 s_  Â  W j xs A30 The total angular velocity (to be denoted by x ) of the s r    orbit observed in the earth-®xed frame combines the Comparing Eqs. (A12) and (A29), we obtain the time orbital angular velocity and the earth-rotation eect in derivative, the sense of taking their dierence, i.e.,

S_ 1 S W A31 xr xs xe A38  2    _ which can be also written as and therefore the total angular momentum g r r is due to the orbital angular momentum h s s_ andÂ the  Â S_ 1G S A32 earth-rotation eect, i.e.,  2  g h r x r h r2x r x r A39 with  Â e Â  e  Á e  1 s_ h s on using Eq. (A37). G S W Sy j e^ n e^ A33 Substituting Eq. (A28) into Eq. (A38) and using  s  s  s2 w  h u   Eq. (A39), we can write 766

g r where x m A40 r  r2  g  # X h A49 with   is the argument of ascending orbital node measured r s m n r^ x A41 from the Greenwich meridian. g  h Á e  where m can be called the normal component acceler- References ation of the satellite eective in the rotating earth-®xed frame. Battin RH (1987) An introduction to the mathematical methods of Note that the two two-way transformations astrodynamics. American Institute of Aeronautics and Astro- u; v; w; u_; v_; w_ a;b;c;d;a_;b_;c_;d_and x; y; z; x_; y_; z_ nautics, New York \$a;b;c;d;a_;b_;c_;d_are equivalent under  Eq. (A41) Born GH, Tapley BD, Santee ML (1986) Orbit determination \$ Á using dual crossing arc altimetry. Acta Astronaut 13±4:157±163 which is nothing but r^ xr r^ xs r^ xe. We can ÁÁ  Á Á Cartan E (1981) The theory of spinors. Dover, New York write the counterparts of Eqs. (A30)±(A36) simply by Cloutier JR (1983) A technique for reducing low-frequency, time- using new variables. Let dependent errors present in network-type surveys. J Geophys Res 88:659±663 r_ Colombo OC (1984) Altimetry, orbits and tides. NASA Technical V j x A42  r  r  Memorandum 86180, Goddard Space Flight Center Cui C, Lelgemann D (1992) On cross-over dierences of the radial then, orbital perturbations as functions of force model parameters. In: Geodesy and physics of the earth, International Union of R_ 1 R V A43 Geodesy and Geophysics. Springer, Berlin Heidelberg New  2  York and Deprit A, Elipe A, Ferrer S (1994) Linearization: Laplace vs. Stiefel. Celestial Mech Dyn Astron 58:151±201 Farless DL (1985) The application of periodic orbits to TOPEX R_ 1 F R A44  2  mission design. AAS 85±301:13±36 Ferziger JH (1981) Numerical methods for engineering application. with Wiley, New York Gaspar P, Ogor F, Le Traon P-Y, Zanife O-Z (1994) Estimating 1 r_ g r the sea state bias of the TOPEX and POSEIDON altimeters F R V Ry j e^z m e^x A45  r  r  r2  g  from crossover dierences. J Geophys Res 99:24981±24994  Goldstein H (1980) Classical mechanics (2nd edn) Addison-Wesley, and so on. Reading Mass The vector state r; r_ is independent of the eective Grafarend EW, Lohse P (1991) The minimal distance mapping of normal acceleration m which changes R_ (but, not R). the topographic surface onto the (reference) ellipsoid of revo- ÈÉ lution. Manuscr Geod 16:92±110 From Eq. (A41), we note that m is caused by n and the Greiner W (1994) Quantum mechanics, an introduction (3rd edn) radial component of x (i.e., r^ x ). For n one may re- e Á e Springer, Berlin Heidelberg New York ¯ect the eects of any perturbation, e.g., the earth's Guckenheimer G, Holmes P (1986) Nonlinear oscillations, dy- oblateness with namical systems, and bifurcations of vector ®elds. Springer, 2 Berlin Heidelberg New York le re z Hagar H (1977) Computation of circular orbit ground trace in- n 3J2 cos i A46  r2 r r  tersections. Engineering Memorandum 315±29, Jet Propulsion where l is the gravitational parameter of the earth, r is Laboratory e e Hestenes D (1983) Celestial mechanics with geometric algebra. the mean radius of the equator, and J2 is the second Celestial Mech 30:151±170 zonal harmonic of the geopotential. Again, for the Hestenes D, Lounesto P (1983) Geometry of spinor regularization. purpose of state transformation, we can work with an Celestial Mech 30:171±179 osculating Keplerian orbit, i.e., we can set n 0. Like n, Hsu SK, Geli L (1995) The eect of introducing continuity con-  ditions in the constrained sinusoidal crossover adjustment r^ xe does not change r; r_ and R. However, the Á _ method to reduce satellite orbit errors. Geophys Res Lett component of xe normal to r changes R and R as well as _ ÈÉ 22:949±952 r (but, not r). We consider a simpli®ed earth-rotation Kim MC (1993) Determination of high-resolution mean sea surface model uniformly rotating along the e^w-axis, i.e., and marine gravity ®eld using satellite altimetry. Ph.D Disser- x x e^ x e^ A47 tation. Dept Aerospace Engineering and Engineering Mechan- e  o w  o z  ics, The University of Texas at Austin where xo represents the constant rate of the Greenwich King JC (1976) Quantization and symmetry in periodic coverage mean to be denoted by h. Then, as the patterns with applications to earth observation. J Astronaut Sci counterpart of Eq. (A16), the components of R can be 24:347±363 Klokocnik J, Wagner CA (1994) A test of GEM-T2 from GEO- expressed as SAT crossovers using latitude lumped coecients. Bull Geod 68:100±108 a r1=2 cos 1 # s cos 1 i; b r1=2 cos 1 # s sin 1 i  2   2  2  2 Kozel BJ (1995) Dual-satellite altimeter crossover measurements c r1=2 sin 1 # s sin 1 i; d r1=2 sin 1 # s cos 1 i for precise orbit determination. Ph.D. Dissertation. Dept  2  2  2   2 Aerospace Engineering and Engineering Mechanics, The Uni- A48 versity of Texas at Austin  767

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