GEOMETRY OF OPTIMAL COVERAGE FOR TARGETS AGAINST A SPACE BACKGROUND SUBJECT TO VISIBILITY CONSTRAINTS

Transcription

1 AAS GEOMETRY OF OPTIMAL COVERAGE FOR TARGETS AGAINST A SPACE BACKGROUND SUBJECT TO VISIBILITY CONSTRAINTS Belinda G. Marchand and Chritopher J. Kobel INTRODUCTION The optimal atellite coverage problem traditionally refer to maximizing the viibility of target againt an Earth background. In thi tudy, the focu hift to target againt a pace background under certain contraint. Specifically, the goal of thi tudy i to identify a ytematic approach for determining the optimal altitude that maximize enor viibility of an area of pace encloed within an upper and lower target altitude range. It i further aumed that enor viibility below the local horizon i diminihed due to atmopheric or environmental factor. Thi environmental contraint i repreented mathematically by optimizing the area covered above the local horizon but within the target altitude hell of interet for a fixed enor range. In the coure of thi development, geometrical argument are employed to identify an analytical expreion for the coverage area. A graphical analyi tool, developed for thi tudy, i employed in demontrating the variou geometrical arrangement and viualizing the optimal configuration. The goal of thi tudy i the development of a ytematic approach to identify the optimal atellite altitude needed to maximize the enor viibility ubject to a pecific et of contraint. Firt, the target of interet are aumed to exit above a lower and below an upper target altitude hell. Furthermore, it i aumed that the enor capabilitie are diminihed by atmopheric and environmental factor. Thi i mathematically repreented by contraining the area calculation to exclude the region below the local horizon. Traditionally, optimizing atellite coverage refer to maximizing the viibility of target againt an Earth background [-]. Thi i often referred to a the optimal Below the Horizon or Below the Tangent Height (BTH coverage problem. Specifically, the optimal BTH coverage problem eek to identify the minimum number of atellite required to achieve continuou coverage and maximize the viible area below the local horizon. The local horizon refer to an imaginary line that originate at the atellite and i tangent to the urface of the Earth, alo referred to a the tangent line (TL. In the preent invetigation, Aitant Profeor, The Univerity of Texa at Autin, 0 E. 4 th St., Autin, TX 787; Member of the Technical Staff Senior, The Aeropace Corporation. Project Leader Senior, The Aeropace Corporation, 350 E. El Segundo Blvd., M4-948, El Segundo, CA Approved for Public Releae, 07-MDA-3004 ( Dec 07, 008 The Aeropace Corporation

2 however, the area of interet pertain to target viible againt a pace background [3-4], termed Above the Horizon or Above the Tangent Height (ATH coverage. Specifically, thi invetigation eek to etablih analytical mean of maximizing the viible ATH coverage given enor range contraint and a erie of altitude band. The approach preented i geometrical in nature. That i, geometrical argument [5] are employed to etablih an analytical expreion for the coverage area and the optimal atellite altitude i identified numerically through a grid earch of the olution pace. From thi grid earch, the optimal atellite height i eaily identified for a given et of contraint. The reult of the current invetigation offer great inight into the evolution of the coverage area a a function of thee parameter and contraint. The inight gained from thee reult erve a a tepping tone to more complex analye involving additional contraint. Traditional Method for Optimal Coverage A previouly tated, the traditional problem of optimal coverage refer to maximizing the viibility of target againt an Earth background while minimizing the number of atellite in the contellation []. Identifying the atellite height that lead to maximum BTH coverage depend on the enor capabilitie. However, overall, the computation i traightforward. For intance, conider the illutration in Figure, where θ denote the coverage angle, R repreent the range hell (RS radiu, R e i the Earth radiu, and r denote the radial ditance from the center of the Earth to the atellite. It i evident, from Figure, that the ground area covered by a ingle atellite i maximized when the coverage angle i maximized for a given enor range. While the BTH problem focue on target againt an Earth background, early tudie into maximizing the viibility of target againt a pace background led to invetigation on how to maximize the ATH coverage are within a precribed altitude hell [4]. To illutrate thi, conider the image in Figure. In thi cae, r tar repreent the target altitude of interet, and r t denote the radiu of the Tangent Height Shell (THS. The tangent height i an idealized verion of the extent of the atmophere, or any environmental element that might diminih or interfere with the nominal viibility of the atellite target detection enor. Thu, thi concept i introduced a a mean of incorporating thee limitation in the preent analyi. R r R e θ Figure - BTH Coverage Geometry

3 R r r tar θ in R r t θ out r tar Figure - ATH Coverage Geometry For a fixed enor range, idealized here a a pherical hell, optimizing the ATH coverage implie maximizing the volume that exit above the tangent height cone and below the target altitude hell. For implicity, the preent analyi i trictly two-dimenional and focue on coverage area rather than coverage volume computation. However, it i undertood that due to the ymmetry of the geometrical aumption, the volume of coverage i imply the revolution of the coverage area by 360 degree. The orientation of the tangent line (TL i determined by the atellite altitude. For a given target altitude and tangent height, it i poible to geometrically identify the minimum enor range required for ATH coverage to exit, namely R R. Similarly, R + R repreent the enor range beyond which the coverage area reache a plateau. Thi i evident from the variation in the haded area of Figure 3. Thu, in the traditional ene, maximum ATH coverage i achieved when the atellite altitude i elected uch that R= R+ R. In the preent tudy, the goal i to maximize ATH coverage within an altitude band defined by a lower altitude hell, r l, and an upper altitude hell, r u. Both of thee hell exit above the tangent height hell. Thu, the problem i to maximize the area of interection between the lower target altitude hell (LTAS, the upper target altitude hell (UTAS, the tangent line (TL, and the range hell (RS. In thi cae, the determination of the optimal atellite altitude i neither intuitive nor traightforward and require additional conideration of the geometry of coverage. SHELL INTERSECTIONS The geometry of coverage depend on a number of key interection defined by the problem parameter: atellite altitude, upper and lower target altitude, tangent height, and enor range. In the algorithm preented here, the calculation of the coverage area i a function of the location of thee key interection. 3

4 r >r >r 3 R = atellite effective range = contant R tar = R e + h tar = target radiu = contant h tan = tangent height = contant R r R r R tar r 3 θ h tan R tar θ R θ 3 R e R tar DECREASING SATELLITE ALTITUDE W/ITH FIXED EFFECTIVE RANGE Figure 3 - Optimal ATH Coverage Geometry The reult of thi tudy reveal that there i no ingle cloed form equation that define the coverage area. At bet, a piecewie continuou function may be defined, but there are many pecial cae to conider that affect the calculation. Alo, the repreentation of the coverage area i not unique, a there are multiple way of decompoing the coverage geometry into fundamental component. Figure 4 illutrate a ample geometry that depict one of the poible interection cenario. Thi illutration i ueful in etablihing the baic notation employed in thi tudy. The definition for each of the key interection poible are ummarized in Table, and graphically illutrated in Figure 4. At mot, 4 interection are poible. The only interection that are guaranteed to exit for all parameter are (T, T, (L A, L A, (U A, U A. The remaining interection may or may not exit depending on the atellite altitude and the ize of the RS. For example, (U B, U B exit only if the atellite i above the UTAS. Similarly, (L B, L B exit only if the atellite i above the LTAS. Alo, (A, A and (B, B exit if the RS interect the LTAS and UTAS, repectively. Naturally, it i poible to identify value of the atellite altitude and enor range for which thee interection do not exit. Table Decription of Shell Interection Interection U A, U A L A, L A L B, L B U B, U B T, T A, A B, B Decription The outhern interection of the TL with the UTAS The outhern interection of the TL with the LTAS The northern interection of the TL with the LTAS The northern interection of the TL with the UTAS The interection of the TL with the RS The interection of the RS with the LTAS The interection of the RS with the UTAS 4

5 Figure 4 Shell Interection A previouly mentioned, the reult of thi tudy reveal that a piecewie continuou function may be identified to define the coverage area. The reulting compoite function i continuou, though not necearily mooth. A revealed by thi invetigation, the range of altitude over which the function i mooth depend on where the RS interect the TL and where that interection (T, T fall relative to the interection of the LTAS and UTAS with the TL. Thi, in turn, depend on the atellite altitude and the ize of the RS for a given UTAS, LTAS, and THS. The definition of thee interection are employed in thi tudy to define the coverage area aociated with a given atellite altitude. A rotating coordinate ytem i elected for thi determination uch that the y-axi i the line from the center of the Earth to the atellite, and the x-axi i on the plane of the orbit of the atellite and perpendicular to the y-axi, a illutrated in x, y = 0, r. Figure 4. A uch, the location of the atellite in thi rotating coordinate frame i given by ( ( 5

6 Interection of RS with UTAS and LTAS The interection of the RS with the UTAS (B, B i eaily computed by imultaneouly olving the mathematical relation that define the UTAS and the RS. At the point where the RS and UTAS interect, the following equation mut be atified, ( B B x + y r = R ; The olution to thi ytem of equation i given by y B ( ru + r R = ; r Of coure, the ymmetrical nature of the arrangement implie that x = x ; B B A imilar reult applie for interection at A and A, y A Interection of the TL with the LTAS ( rl + r R y x + y = r ( B B u x = r y ( B u B = y. (3 B B = ; xa = r l ya (4 r x = x, y = y. (5 A A A A The tangent line (TL i defined by the poition of the atellite (, x y, and the point of interection with the tangent height circle. The coverage angle aociated with thi point i defined, in Figure 4, a r t θt = co. (6 r Baed on thi definition, the lope of TL i given by, yt y m =, (7 x t x where, xt = rtinθt; yt = rtco θt. (8 Let ( xl, y A/ B L denote the carteian coordinate of L A/ B A or L B. Then, the interection of the LTAS with the TL i given by the olution to the following ytem of equation, x y r + = ; ( L A/ B L A/ B l Since x = 0 and y = r, ubtitution of (7 into (9 lead to y = m x x + y (9 L A/ B L A/ B ( m xl mrxl ( r rl = 0. (0 A/ B A/ B The above quadratic i applicable to two different point, (A and (B; thu, there are really a total of two quadratic equation. Only one of thee equation i guaranteed to have a real et of olution. The other may or may not have a olution, depending on whether or not the atellite i above the LTAS. 6

7 At a minimum, there are two olution, ( xl A, y L and ( A LA, LA However, there are four olution ( xl, y L, ( xl, y L, ( xl, y L and ( L, L x y, when the atellite i below the LTAS. x y, if the atellite i A A A A B B B B above the LTAS. For the right hand ide interection on Figure 4, it i clear that m < 0. Thu, if r < r l the outhern interection of the tangent line with the LTAS occur when x L A ( ( ( + m mr + 4m r 4 + m r r = l. ( If r > r l, a northern interection exit and occur at a maller value of the x-coordinate, namely In either cae, and x L B Interection of the TL with the UTAS ( ( ( + m mr 4m r 4 + m r r = x l L A/ B L A/ B. ( y = mx + r, (3 = x ; L A/ B L A/ B y = y. (4 L A/ B L A/ B In the event the RS interect the UTAS, the coordinate of interection are eaily identified by adapting equation ( and (3 to the interection with the UTAS, x U A x UB ( ( ( + m ( ( ( + m mr + 4m r 4 + m r r = u mr 4m r 4 + m r r = u, (5, (6 Once again, ymmetry implie that y = mx + r. (7 U A/ B U A/ B Interection of the TL with the RS x = x ; U A/ B U A/ B y = y. (8 U A/ B U A/ B The interection of the TL with the RS are identified a T and T. The carteian coordinate of T and T are determined from the following ytem of equation, ( T T x + y y = R ; The olution to thi ytem of equation i determined a y = mx + y. (9 T T for x = R T x + m ; = x, y T T y = mx + r (0 T T = y. ( T T 7

8 COMPOSITE COVERAGE AREA COMPONENTS A previouly mentioned, thi tudy reveal that the coverage area cannot be reduced to a imple generalized equation. At bet, a piecewie continuou function may be defined to ae the effective coverage area. The primary reaon for thi complication i that the definition of the coverage area equation change with the location of the interection point ( T, T relative to the interection point U A/ B, U A/ B, L A/ B, and L A/ B. Thi, in turn, depend on a variety of factor including the atellite altitude, the enor range, the LTAS, the UTAS, and the THS. To etablih a ytematic way of obtaining a piecewie continuou repreentation, the computation of the coverage area i divided into fundamental geometrical element, mainly triangle, arc egment, and combination thereof [5]. The effective coverage area i then contructed a a compoite of thee fundamental area element. Thi ection ummarize the coverage area of the fundamental hape employed in contructing the piecewie continuou coverage area function. Triangle The coverage area function depend trongly on the Carteian coordinate of each of the fundamental interection previouly defined. Thu, it i convenient to define the area of a triangle a a function of the emiperimeter becaue thi quantity depend only on the length of each ide of the triangle. The length of each ide of a triangle are eaily identified from Carteian coordinate uing the ditance between vertice. The area of a triangle, Α, in term of the emiperimeter, i given by ( abc,, = ( a( b( c Α, ( where a, b, and c denote the length of each ide and i the emiperimeter, ( a+ b+ c =. (3 Arc Segment The area of an arc egment, Α Σ, i given by the difference between the area of a ector of a circle, and the area of a triangle a illutrated in Figure 5. The equation for the area of the arc egment illutrated in Figure 5 i given by c T r 3 u φ Α Σ ( ru, ct = φr co 3 u. (4 SECTOR TRIANGLE c T 3 B B r u r u φ Figure 5 - Area of Arc Segment 8

9 In Figure 5, the angle φ i determined a c T3 φ = in, (5 ru Where c T 3 denote the chord of the arc egment, and ct 3 ru φ 4ru ct 3 co = =. (6 ru ru Subtitution of (5 and (6 into (4 lead to, c T c 3 T3 ( ru, ct = r in 3 u ( 4ru ct 3 Α Σ. (7 ru 4 Teardrop Sector Traditionally, the area of a circular ector i defined a where λ = co ( RPP, A π = λr, (8 PP PS PS PS PS denote the angle interior to the ector, PP repreent the chord, R i the radiu of curvature, and S define the origin. A geometrical object derived from the concept of a circular ector, i the teardrop pattern illutrated in Figure 6. In thi cae the haded area i denoted certain range of parameter when computing the coverage area. A π (9. Thi type of pattern i ueful over a Figure 6 Area of a Teardrop Pattern 9

10 The area of a teardrop pattern can be expreed a a um of two fundamental component; a triangle and an arc egment, or it complement. In thi cae, Compoite Triangle A π ( rrpp,, ( AΣ ( ( A Σ ( A R, PP, R + r, PP ; R< r + r = A R, PP, R + r, PP ; R r + r. (30 In the coure of deviing geometrical argument to compute the area of coverage, two type of compoite triangle are often encountered. In thi cae, a compoite triangle refer to a three ided geometrical hape whoe ide conit of any combination of line egment and pherical arc. Figure 7 illutrate one poible type of compoite triangle that i ometime encountered in the proce of computing the effective coverage area. In thi invetigation, thi type of compoite triangle i denoted by Λ. Thee will be dicued in more detail in later ection. L A T A Figure 7 Compoite Triangle In the example illutrated in Figure 7, the radiu of TL i r l, and the radiu of BT i R. The area, Α Λ, of thi compoite triangle i a function of the area of TL, BT, and the triangle defined by vertice B, T, and L. Since the Carteian coordinate of thee vertice are known, we find that, TL TL Α Σ ( rl, TL = rl in 4rl TL, r l 4 (3 BT BT Α Σ ( RBT, = R in 4R BT R 4, (3 ( BT, BL, TL = ( BT ( BL ( TL Α, (33 0

11 Thu, the equation that decribe the area of the compoite triangle in Figure 7 i given by where, and ( rl, R, BT, BL, TL = ( BT, BL, TL ΑΣ( rl, TL + ΑΣ( R, BT Α Α Λ (, (34 = BT + BL + TL, (35 ( ( BT = x x + y y, (36 B T B T ( ( BL = x x + y y, (37 B L B L ( ( TL = x x + y y. (38 T L T L Another type of compoite triangle, Λ, i illutrated in Figure 8. Thi type of geometry i typically viible when the atellite croe one of the target altitude hell, LTAS or UTAS. The area of the geometry illutrated below i determined a ( rl, L S, LL, LS = ( L S, LL, LS Σ ( rl, LL Α Α Α. (39 Λ S L B L A Figure 8 - Compoite triangle Interection of Two Circle Figure 9 illutrate one poible cenario for the area of interection between the RS and the LTAS, Α RS LTAS. However, the formulation preented here i alo applicable to the area of interection between the RS and the UTAS, Α RS UTAS. For the interection of the RS with the LTAS, OA = rl and SA = R. The haded area Figure 9 i eaily determined by computing the area of two arc egment, per the formula previouly preented. Adapting equation(7 to thi particular cae yield, ( r, R, AA = ( r, AA + ( R, AA Α Α Α. (40 RS LTAS l Σ l Σ

12 Figure 9 - Area of Interection Between Two Circle However, thi equation i not applicable to all poible type of interection between the two circular hell. The actual value of the coverage area depend on where the two circle interect, and the location of thee point relative to S and O. Figure 0 illutrate four poible cae relevant to thi invetigation that mut be conidered in computing the area of interection between two circular hell. Figure 0(a and Figure 0(b illutrate the two poible cenario for R < rl, while Figure 0(c and Figure 0(d correpond to R rl. Figure 0(a i further aociated with r > rl R, while Figure 0(b i aociated with r rl R. Similarly, Figure 0(c correpond to r > R rl, while Figure 0(d implie r R rl. Table ummarize the general formula for the determination of the area of interection between two circle, within the cope of the aumption employed in thi invetigation. CONDITION FOR NO COVERAGE In defining the range of altitude to conider, for a given et of parameter, it i only neceary to identify the minimum allowable atellite altitude and the altitude at which the atellite provide no ATH coverage. In thi tudy, the minimum allowable altitude i defined a the tangent height. The maximum altitude of interet correpond to the point at which r > r u while T = UB and T = UB. A oberved from Figure, no part of the range circle exit within the UTAS and above the TL at or above thi critical altitude, where the radial ditance from the planet origin to the atellite i defined a r 3. Geometrically, thi critical altitude i eaily identified a, ( 3 u t t r = R+ r r + r. (4

13 Figure 0 Area of Interection Between RS and LTAS Table Area of Interection of Two Circle Condition Area of Interection ( rl > R and r > rl R or ( rl R and r > R rl ( R, AA A ( r, AA Α = A + (4 RS LTAS Σ Σ l ( rl > R and r rl R RS LTAS = π R AΣ( R, AA + AΣ( rl, AA Α (43 ( rl R and r R rl RS LTAS = π rl AΣ( rl, AA + AΣ( R, AA Α (44 3

14 x R r 0.5 r u 0 r t Figure Critical Altitude for no ATH coverage, for r > r u SINGLE SATELLITE ATH COVERAGE AREA The table and image below ummarize the calculation of the coverage area for all cae of interet. There are a total of 8 cae to conider in computing the coverage area, not including pecific ubcae introduced by equation (4-(44. Thee are ummarized in lit form below. The ymbol implie the quantity i empty, thu, AA = implie that the RS doe not interect the LTAS and BB = implie the RS doe not interect the UTAS. Note that there i no unique way of repreenting the coverage area for a pecific cae. Thu, the reult preented here repreent one poible repreentation, though care wa taken in electing a formulation that would implify the calculation while allowing for a ytematic approach to contructing the piecewie continuou coverage area function. When the atellite i below the LTAS and above the THS, the function that decribe the coverage area can be divided into three egment: TS < LAS, LAS TS < UAS, and UAS TS. Even within each of thee three cae, the calculation of the coverage area need to be adjuted according to the condition defined in Table 3. Thu, the coverage area function, for r t r < r l, can take on up to five different form; (a, (b.i, (b.ii, (c.i, and (c.ii. Table 4 and Table 5 illutrate a imilar relation for r l r < r u and ru r < r 3. Baed on the condition in Table 3 through Table 5 the compoite piecewie continuou coverage area function, for each of the three poible altitude range, are ummarized in Table 6 through Table 8. TS LAS x 0 4 Table 3 Coverage Area Subcae for r t r < r l < A A L S T S < U S UAS TS (a (b.i: AA (c.i: AA (b.ii: AA = (c.ii: AA = 4

15 < B A TS LB S Table 4 - Coverage Area Subcae for r l r < r u L S T S < L S LAS TS < UAS UAS TS (a (b (c.i: AA (d.i: AA (c.ii AA = (d.ii: AA = < B B TS UBS Table 5 - Coverage Area Subcae for ru r < r 3 U S T S < L S LBS TS < LAS LAS TS < UAS UAS TS 3(a 3(b 3(c 3(d.i: AA 3(d.ii: AA = 3(e.i: AA 3(e.ii: AA = Cae Area Table 6 Coverage Area for r t r < r l (a Α=ΑUTAS RS Α LTAS RS (b.i Α=ΑUTAS RS ΑLTAS RS Α Λ ( Rr, l, TL A, TA, AL A (b.ii Α=ΑUTAS RS π rl Α π ( R, TT +Α π ( rl, R, LA LA (c.i Α=ΑUTAS RS ΑLTAS RS ΑΛ ( rl, R, TLA, TA, ALA + Α Λ ( ru, R, TU A, TB, BU A (c.ii Α= πru πrl Απ ( ru, R, UAUA + Α π ( rl, R, LALA 5

16 Cae Area (a Α=ΑUTAS RS Α π ( R, TT Table 7 Coverage Area for r l r < r u (b Α=ΑUTAS RS ΑLTAS RS Α Λ ( rl, LB S, LB LB, LBS (c.i Α=ΑUTAS RS ΑLTAS RS ΑΛ ( rl, LB S, LB LB, LBS Α Λ ( rl, R, TLA, TA, AL A (c.ii Α=ΑUTAS RS ΑLTAS RS ΑΛ ( rl, LB S, LB LB, LBS Α π ( R, TT +Α π ( rl, R, LA LA (d.i Α=ΑUTAS RS ΑLTAS RS ΑΛ ( rl, LB S, LB LB, LBS Α ( rl, R, TLA, TA, ALA Λ Α=Α+ΑΛ ( ru, R, TU A, TB, BU A (d.ii Α= πru πrl Α π ( ru, R, UAUA +Απ ( rl, R, LALA Α ( rl, LBS, LBLB LBS Λ Cae Area 3(a Α=0 Table 8 - Coverage Area for r l r < r u 3(b Α=ΑUTAS RS Α π ( RTT, +Α Λ ( ru u, BSU, BUB, UBS 3(c Α=ΑUTAS RS Α Λ ( rl, LB S, LB LB, LBS +Α Λ ( ru, UBS, UBUB, UBS 3(d.i Α=ΑUTAS RS Α Λ ( rl, LB S, LB LB, LBS +Α ( ru, UBS, UBUB, UBS Λ Α=Α ΑΛ ( rl, R, TLA, TA, AL A 3(d.ii Α=ΑUTAS RS π rl Α Λ ( rl, LB S, LB LB, LBS +ΑΛ ( ru, UBS, UBUB, UBS Α ( R, TT π Α=Α+Απ ( rl, R, LALA 3(e.i Α=ΑUTAS RS ΑLTAS RS Α Λ ( rl, LB S, LB LB, LBS +Α ( ru, UBS, UBUB, UBS Λ Α=Α Α Λ ( rl, R, TLA, TA, ALA +ΑΛ ( ru, R, TU A, TB, BU A 3(e.ii Α= πru πrl Απ ( ru, R, UAUA + Απ ( rl, R, LALA Α ( rl, LBS, LBLB, LBS Λ Α=Α+ΑΛ ( ru, UBS, UBUB, UBS 6

17 The coverage area aociated with each of thee cae i illutrated in Figure 4. To demontrate how one arrive at each of the equation in Table 6 through Table 8, conider cae (d.i. The coverage area equation for cae (d.i can be decompoed into everal area element, including ΑUTAS RS, ΑLTAS RS, ΑΛ, and Α Λ. Figure 5 illutrate how the compoite total coverage area i contructed from thee element. The content of Table 6 through Table 8 define an objective function to an optimization proce that eek to identify the optimal altitude that maximize the contrained ATH coverage area. An example of thi proce i preented in the next et of example. Figure - Cae Scenario Figure 3 - Cae Scenario 7

18 Figure 4 - Cae 3 Scenario Figure 5 - Coverage Area Calculation for Cae (d.i EXAMPLE Figure 6(a depict the coverage area a a function of atellite altitude for a fixed enor range. A expected, the coverage area i nullified a r r 3. In thi particular example, the maximum coverage area i achieved when the atellite altitude i 353 km, for R = 5000 km, h l = 000 km, h u = 5000 km, and h t = 00 km. The correponding geometrical configuration i illutrated in Figure 6(b. It i intereting to note that, unlike the BTH cae, maximizing θ doe not lead to maximum coverage area. Alo, unlike the traditional ATH cae, R+ R doe not offer the maximum coverage area for a fixed altitude band. In fact, the reult of thi tudy indicate that there i no obviou relation between the coverage angle and the maximum coverage area. Thu, the methodology preented here i an important tep toward identifying optimal contellation for maximum ATH altitude band limited coverage, and alo for the future incluion of additional contraint. 8

19 EXAMPLE A more complete analyi of the optimal olution pace i preented in Figure 7 (a-b. Thee figure depict contour of optimal atellite altitude a a function of enor range and upper target altitude. In thi particular example, the lower target altitude i held fixed at 600 km to implify the viualization of the optimal altitude trend. Figure 7(b i of particular interet a it reveal two ridge line that eparate the central region of the contour from the exterior region. Numerical examination of thee ridge line reveal that the lower ridge line correpond to the et of optimal olution aociated with a critical interection between La, T, and A. By ymmetry, at that ame altitude, there i alo an interection between La, T, and A. Similarly, the upper ridge line i repreentative of optimal olution aociated with the critical interection of Ua, T, and B. It i intereting to note that, in the exterior region outlined by thee ridge line, the relation between the maximum coverage area and the enor and upper target altitude, eem relatively linear, unlike that oberved in the interior region. While the ignificance of thee trend i of interet for future invetigation, it i not the focu of the preent tudy. Every deign problem i linked to a different et of contraint and parameter, all of which affect the trend outlined in thee Figure. Thu, it would be premature to draw any globally definitive concluion from thee reult. However, the reult preented here do offer a proof of concept for the propoed methodology. The algorithm and graphical tool preented are an important firt tep toward the future goal of thi tudy. Specifically, the development of algorithm that can ait the deigner in identifying the minimum number of atellite and the optimal arrangement required to maximize the coverage of target againt a pace background over a prepecified altitude and latitude band. x 0 7 Area of Coverage (km 4 3 X: 353 Y: 4.05e Altitude (km θ out (deg Altitude (km ( a ( b Figure 6 ATH Coverage Area v. Altitude 9

20 (a (b Figure 7 - Sample Optimal Height Surface CONCLUSIONS The preent tudy focue on the identification of the optimal atellite altitude neceary to achieve maximum ATH coverage over an altitude limited band with atmopheric or environmental viibility contraint. Unlike earlier reult regarding optimal BTH and ATH coverage, no direct correlation exit between the coverage angle and the maximum coverage area. The coverage area, in thi cae, i a nonlinear piecewie continuou function that require pecial geometrical conideration. Thi invetigation reveal that the key to computing the coverage area in thi more complex cae lie in the proper identification of the mutual interection between the tangent line with the range hell, the upper target altitude hell, and the lower target altitude hell. Many pecial cae are identified and dicued in thi tudy. The end reult i a compoite nonlinear objective function that relate the bounded ATH coverage area to the atellite altitude. Future tudie, building on thee reult, will focu on optimal contellation deign and the incorporation of other contraint, uch a latitude band of interet. ACKNOWLEDGEMENTS Thi work wa performed at The Aeropace Corporation. The author would like to thank Bill Adam and Tom Lang, of The Aeropace Corporation, for their valuable input during the coure of thi invetigation. REFERENCES. Walker, J.G., Satellite Contellation, Journal of the Britih Interplanetary Society, Vol. 37., PP , Lang, T.J., Optimal Low Earth Orbit Contellation for Continuou Global Coverage, AAS/AIAA Atrodynamic Specialit Conference, Victoria, B.C., Canada, Aug. 6-9, Rider, L., Optimal orbital contellation for global viewing of target againt a pace background, Optical Engineering, Vol. 9, No., Kobel, C., LOS, ATH, Same Theta Coverage Geometry, The Aeropace Corporation, Internal Preentation. 5/6/ Fewell, M.P. Area of Common Overlap of Three Circle, Maritime Operation Diviion, Defene Science and Technology Organization, Technical Note, DSTO-TN-07. 0