Gabbard Diagram Formation: The General Theory for Elliptical Orbits

Transcription

1 Avancs in Aospac Scinc an Applications. Volum 8, Numb (018), pp Rsach Inia Publications Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits Robt C. Rynols 1, Ajun Tan* an Maius Schamschula Dpatmnt of Physics, Alabama A & M Univsity, Nomal, AL 3576, U. S. A. 1 VP fo Avanc Pogams, STAR Dynamics Inc., Hillia, OH 4306, U.S.A. Abstact In a pvious stuy, th thoy of Gabba iagam fomation fo fagmnts of a satllit bakup in cicula obit was psnt incluing th X fomation, th quations of th apog an pig lins, th fobin zon, th sma of points abov an blow th X, an th quation of th hypbolic nvlops of th points obtain. Th magnitus of th limiting vlocity ptubations in th own-ang an aial ictions w stimat fom that iagam. Th cunt stuy xtns th thoy of fomation of th Gabba iagam to satllit fagmntations in lliptical obits. Th slops of th apog an pig lins a obtain an th conition of paalllism of th lins tmin. Th pnncs of th Gabba iagam on th ccnticity of th obit an tu anomaly of th fagmnting satllit a iscuss. Th sults uc to thos fo cicula obits fo th limiting cas of vanishing ccnticity. 1. INTRODUCTION Th Gabba iagam is a simpl yt usful tool now-a-ays mploy in vitually all satllit fagmntation stuis. It was invnt by John Gabba, thn at Noth Amican Aospac Dfns Comman (NORAD/ADCOM) in th aly yas of satllit fagmntation histoy [1]. It plots th apog an pig hights of th fagmnting satllit an its fagmnts against thi obital pios. Constucting th Gabba iagam of any satllit fagmntation vnt is invaiably th fist o of businss to lan o confim about th spcifics of th vnt,.g., th natu of th obit, th location of th fagmntation point, th ictionality an intnsity of th fagmnts spa, tc. In Th Satllit Fagmntation Catalog [], Gabba iagams of ach an all fagmntation vnt a shown. Gabba iagams of cicula, naly cicula, lliptic o highly lliptic obits hav chaactistic fatus pning upon th location of th fagmntation in obit. Illustativ xampls of th vaious typs of Gabba iagams a foun in fncs [3 5].

2 100 Robt C. Rynols, Ajun Tan an Maius Schamschula Fo fagmntation fom a cicula obit, th Gabba iagam has th shap of an inclin X fom by a hoizontal staight lin an a staight lin having a positiv slop, th two lins intscting at th cooinats of th fagmnting satllit (P0, h0). Th ight han si of th X, to th ight of (P0, h0), is gnally cat by posiga impulss to th fagmnts (own-ang vlocity chang v > 0) whil th lft han si of th X is nomally gnat by toga impulss to th fagmnts (v < 0). Ially, fo an isotopic fagmntation of a satllit in a cicula obit, th Gabba iagam will hav a symmtic spa about (P0, h0). In th absnc of aial vlocity changs (v), th apog an pig points li on th two staight lins. Th ffct of v is to shift points abov an blow th apog an pig lins, cating a halo - lik appaanc contain btwn two hypbolic nvlops abov an blow th apsial lins. Th angula spac btwn th two lins is, ially voi of fagmnts an psnts a fobin zon. Th Gabba iagam os not pn upon th coss-ang vlocity changs (vx).. Gabba Diagam Fomation fo Satllit Fagmntation in Cicula Obits Fo th poblm at han, it is foun suitabl to viw th fomation of th Gabba iagam fo fagmntation in cicula obits fist an gnaliz it to lliptical obits thupon. In th Gabba iagam, th inpnnt vaiabl is th obital pio P of th fagmnting satllit an its fagmnts. By Kpl s hamonic law, P is ictly lat to th smi-majo axis a, of th satllit: P 4 3 a (1) wh μ is th gavitational paamt of th Eath. Taking iffntials, on gts 3P P a () a Thus, a chang in th smi-majo axis of a fagmnt is ictly popotional to th chang in its obital pio. P is also lat to th spcific total ngy of th obit E by th lation: Taking iffntials, likwis, w gt P (3) 3 E 3P 3P P E ( E) (4) E E Sinc E is ngativ fo boun Kplian obits, E is a positiv quantity. Thus P is also ictly lat to E. Equations () an (4), thfo val that P, a, an E, a all ictly lat to on anoth.

3 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits 101 Dynamical vaiabls in spac must b fin in an intial systm of cooinats. A spaccaft local intial systm can b obtain fom an Eath-cnt intial systm by tanslation an otations. An appopiat systm to stuy th th othogonal componnts of th vlocity ptubation of a fagmnt is th following: (1) th aial componnt v in th iction of th local vtical; () th own-ang componnt v along th local hoizontal lin in th plan of th obit; an (3) th coss-ang componnt vx in th iction of th obital angula momntum of th fagmnting satllit. In that systm, th vlocity of th pant satllit is (v, v, 0); an that of th fagmnt is (v + v, v + v, vx). Whas v an v pouc changs in th obital lmnts of th fagmnt in th pant s obital plan, an thfo affct th Gabba iagam, vx alts th plan of th fagmnt but os not affct th Gabba iagam. Th ol of th latt is hncfoth isga in this stuy. Th spcific total ngy of th fagmnt pio to th fagmntation (whn it was still a pat of th fagmnting satllit) is: 1 E v v wh μ is th gavitational paamt of th Eath an is th aial istanc of th fagmntation point fom th cnt of th Eath. Th spcific total ngy of th fagmnt upon fagmntation is: 1 E E v v v v v x Th chang in spcific ngy of th fagmnt is, fom Eqs. (5) an (6): 1 E v v vv v v vx (7) Sinc th fist tm on th ight of Eq. (7) is ovwhlmingly gat than th st, paticulaly fo a na-cicula obit, w can wit fo obits of small ccnticitis: (5) (6) E v v (8) Thus, fo all pactical puposs, v ictats E an thfo P by vitu of Eq. (4). In gnal, v an v both pouc changs in th smi-majo axis a an ccnticity in accoanc with Lagang s plantay quations [6]: an sin a v n 1 a 1 1 sin v 1 v (10) na a v (9)

4 10 Robt C. Rynols, Ajun Tan an Maius Schamschula In th abov quations, a is th smi majo axis of th pant satllit, its ccnticity, n its man motion, an θ its tu anomaly at th fagmntation point. Th changs to th apog hight ha an pig hight hp a givn by [7]: an h a h p a a 1 (11) a a 1 (1) spctivly. Substituting fom Eqs. (9) an (10), an stting = 0, w obtain fo cicula obits: an h h p a sin v v (13) n n sin v v (14) n n If v > 0, th fagmnt attains a high ngy obit, an th chang in pio P > 0. Howv, th Hohmann tansf pincipl ictats that hp is unaffct by v.. Thus Eqs. (13) an (14) a -wittn as an h a sin v v (15) n n h sin (16) n p v Sinc v alon alts a [vi Eq. (8)] an hnc P, in th absnc of v, th pig points li on a hoizontal staight lin an th apog points li vtically abov th pig points on a staight lin with positiv slop. Th ffct of v is thn to mov th apog an pig points abov an blow th apog an pig lins, spctivly by th sam istanc givn by th last tms of Eqs. (15) an (16), spctivly. This scibs th constuction of th ight han si of th Gabba iagam. If, on th oth han, v < 0, th fagmnt loss ngy, an th chang in pio P < 0. In this cas, th Hohmann tansf pincipl assus that ha mains th sam. Thn Eqs. (13) an (14) a -wittn as an h sin (17) n a v

5 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits 103 sin hp v v (18) n n In this cas, th apog points li on a hoizontal staight lin an th pig points li vtically blow th apog lins on th inclin staight lin. Th ffct of v is again to mov th apog an pig points abov an blow th apog an pig lins, spctivly by th sam istanc givn by th last tms of Eqs. (17) an (18). This xplains th constuction of th lft han si of th Gabba iagam. It is asy to s that th pig lin of th ight han si is intical to th apog lin of th lft han si, sinc both a hoizontal lins passing though th sam point (P0, h0). It has also bn shown that th apog lin on th ight han si an th pig lin on th lft han si of th Gabba iagam hav th sam slop [7]: h a P hp a 4a (19) P P 3P An, sinc both lins pass though (P0, h0), th two lins a coincint. l 3. Gabba Diagam Fomation fo Satllit Fagmntation in Elliptical Obit Gabba iagams fo satllit fagmntations in lliptical obits hav mo vaity but a lss spctacula than thos in cicula obits. Howv, th basic pincipls of fomation main th sam. In all cass, th a two apsial lins which mak th bounais of th fobin zons : (1) a hoizontal lin on which on apsis is locat; an () a slant lin on which th oth apsis is situat. Th slant lin has a small upwa cuvatu in consqunc of Eq. (). Th two lins gnally intsct whn xtn. Th intsction point maks th pio of a cicula obit fo bakup at that altitu. Whth a fagmnt gains o loss pio pns upon whth th total vlocity of th fagmnt incass o cass lativ to that of th pant. But sinc this pns lagly on th own-ang vlocity ptubation v on account of Eq. (8), fagmnts to th ight of th intsction point nomally gain in ngy, whas thos to th lft of th intsction point nomally los ngy. Fo fagmnts to th lft of th intsction point (low ngy fagmnts), th low lin maks th maximum pig fo a fagmnt with that pio; an fo fagmnts to th ight of th intsction point (high pio fagmnts), th upp lin maks th maximum apog fo a fagmnt with that pio. Gabba iagams fo satllit fagmntations in lliptical obits pn on two pimay factos: (1) ccnticity of th fagmnting satllit s obit; an () tu anomaly of th satllit at th point of fagmntation. Th cas scnaios a iscuss in this stuy: (1) whn fagmntation occus at apog o pig; () whn fagmntation occus btwn th apsial points; an (3) whn fagmntation taks plac in a highly ccntic obit (.5).

6 104 Robt C. Rynols, Ajun Tan an Maius Schamschula 3.1. Fagmntation occus na an Apsial Point Whn a satllit in an lliptical obit fagmnts na its pig, Hohmann tansf pincipl ictats that th pig hights of th fagmnts main th sam whil th apog hights incas o cas accoing to whth th ownang vlocity ptubation is posiga o toga. Th Gabba iagam is simila to th ight han si of that of a fagmntation in a cicula obit. Fo an isotopic fagmntation, th a appoximatly qual numb of points to th lft an ight of th pant satllit s location. If th satllit fagmnts na its apog, on th oth han, th apog points of th fagmnts main th sam by th Hohmann tansf pincipl; an th pig hights incas o cas in accoanc with posiga o toga vlocity ptubation in th own-ang iction. Th Gabba iagam in this cas smbls th lft han si of that of th X pattn fo cicula obits. Fig. 1. Gabba iagams of K48 an K5 fagmnts. Figu 1 shows th Gabba iagams of both of th abov cass of fagmntations. It ptains to th fist Sovit ASAT tst against a liv tagt in obit. In Octob 1968, Cosmos 48 (K48) was launch into a na-cicula obit (ccnticity =.005) to sv as th tagt satllit. A fw ays lat, Cosmos 5 (K5) was launch into an lliptical obit (ccnticity =.104). At th nzvous location (which was plann

7 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits 105 to coinci with th apog of K48 an th pig of K5), th intcpto satllit K5 was libatly xplo into ov 100 tackabl fagmnts, on of which thn fagmnt th tagt into fou tackabl pics in consqunc [8 11]. Th combin Gabba iagams of th K48 an K5 fagmnts (Fig. 1) xhibit an X fom about th nzvous altitu of 535 km an pio of 94 min. Th K5 fagmnts li on th ight hin si of th X whil th K48 fagmnt a locat on th lft han si, with th latt showing obvious signs of atmosphic cay. Th K5 fagmnt points li almost vnly on ith si of th fagmnting pant. Th K48 obit was naly cicula, vn though th tagt was na its apog. 3.. Th Gnal Thoy of Gabba Diagam Fomation Whn th stiction of cicula obit ( = 0) is lift, on can poc with th gnal thoy of Gabba iagam fomation. W continu with th notion that th stuctu of th apsial lins a tmin by v s alon an th sma of points fom th lins a th sult of v s. Th apog an pig lins a tmin by th own-ang componnts in Eqs. (11) an (1): an h h a p a a 1 (0) a a 1 (1) wh a an a givn by Eqs. (9) an (10). Th sults can b xpss in tms of th tu anomaly θ of th fagmnting satllit by substitution by th quation of th obit: Aft consiabl algba, on aivs at: an h h a p 1 n 1 cos 1 n 1 cos a 1 () 1 cos 1 cos cos 1 cos 1 cos cos 1 cos v v (3) (4) Equation (3) an (4) fin th apog an pig lins of th fagmnts, spctivly.

8 106 Robt C. Rynols, Ajun Tan an Maius Schamschula Th ffcts of th aial componnts of th vlocity ptubations of th fagmnts a foun fom th following: an h h a p Upon similaly simplifying, w obtain: an h h a p a a 1 (5) a a 1 (6) 1 1 sinv (7) n sinv (8) n 1 Equations (7) an (8) show that th apog an pig points a shift abov an blow th apog an pig lins, spctivly fo ascning mo of th satllit (0 < θ < π). Thus, th gion btwn th apog an pig lins constituts th fobin zon fo upwa motion of th fagmnting satllit only. Fo ownwa motion of th satllit, th aas of th fobin zons will li abov th apog lin an blow th pig lin. Figu is an xampl of th latt cas. It is th Gabba iagam of Himawai Dlta scon stag ockt which fagmnt on 14 July 1977 whil scning. Fig.. Gabba iagam of Himawai Rockt fagmnts.

9 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits 107 Equations (3), (4), (7) an (8) compis th gnal thoy of Gabba iagam fomation fo fagmntation in an lliptical obit. Thy uc to Eqs. (15) (18) as a spcial cas whn Slops of th Apog an Pig Lins Fom th gnal poptis of a Kplian llips, w hav [vi Eqs. (11) an (1)]: h h a (9) a p Thus h h a p a 4a m (30) P P P 3P wh m is a constant. If αa an αp a th slop angls of th apog an pig lins, spctivly, thn tan tan m (31) a Equation (31) constituts a thom of consvation of slops of th apsial lins. Th sults of 3.1 show that: (1) Whn θ = 0, αa = m, αp = 0; an () Whn θ = π, αa = 0, αp = m. Thus, as th tu anomaly θ is vai btwn 0 an π, αa cass as αp incass. At som point, th two slop angls bcom qual, an th apog an pig lins bcom paalll. Th conition of paalllism is givn by αa = αp. Equating Eqs. (3) an (4), an simplifying, on gts a quaatic quation in cosθ: p giving: 1cos 1cos 0 (3) cos (33) 1 Rtaining th upp sign only (th low sign givs an unphysical solution), w obtain th tu anomaly θ0 fo which th apog an pig lins hav th sam slop: cos o 0 cos (34) 1 In Eq. (34), th fist solution cospons to th ascning mo of th fagmnting satllit an th scon solution cospons to th scning mo. Th sults fo vaious ccnticitis of th satllit obit a shown in Tabl I. Th pnnc on th ccnticity is highly vint.

10 108 Robt C. Rynols, Ajun Tan an Maius Schamschula Tabl I. Tu anomaly θ at which slops of apog an pig lins qual as functions of ccnticity Eccnticity θ0 (ascning) θ0 (scning) o 64.9 o o o o 53.8 o o 48.1 o o 43.0 o o o o o o 8.37 o o 1.31 o Figu 3 picts th vaious scnaios of Gabba iagam fomation. Th pnnc on th tu anomaly of th fagmnting satllit is vint. Th vaiation with ccnticity changs th valu of θ0, but not th gnal fatus. Figu 4 (fom [1, 13]) btays two of th fiv scnaios in Fig. 3. Th lft panl of Fig. 4 (fom [13]) is th Gabba iagam of K374 fagmnts which svs as an xampl fo th upp ight panl of Fig. 3. Likwis, th ight panl of Fig. 4 (fom [1]) is th Gabba iagam of K49 fagmnts which cospons to th low lft panl of Fig. 3. Both K49 an K374 w intcpto satllit which fail in thi ASAT missions against th tagts K48 an K373 spctivly, an w xplo on comman. Th ccnticity of K49 s obit was.1088 an th tu anomaly at bakup point was stimat as o [1]. Likwis, th ccnticity of K374 s obit was.1039 an th tu anomaly was calculat to b 86. o [13]. Both sts of infomation a consistnt with Tabl I an Fig.3.

11 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits 109 Fig. 3. Gabba Diagams fo vaious tu anomalis of th fagmnting satllit.

12 110 Robt C. Rynols, Ajun Tan an Maius Schamschula Fig. 4. Gabba iagam of fagmnts of K374 an K59 kill satllits.

13 Gabba Diagam Fomation: Th Gnal Thoy fo Elliptical Obits Fagmntation in highly Elliptical Obit Whn a satllit baks up in a highly lliptical (ccnticitis of.5 an abov), th Gabba iagam of th fagmnts is usually unspctacula an unintsting. Th slop of th pig lin is usually fa small than that of th apog lin, an th is a lag gap btwn th two. As a sult, th pig lin lis at th bottom of th iagam with th apog lin situat high abov it. Figu 5 (fom [14]) is an xampl of such a Gabba iagam which blong to th fagmnts of a Biz-M ockt boy (U.S. Satllit Numb 8944) which suff a massiv xplosion in 19 Fbuay 007. It was in a highly ccntic obit having ccnticity of Figu 5 inclus th fist 106 fagmnts catalog though 1 Januay 011. Fig. 5. Gabba iagam of fist 106 catalog fagmnts of Biz-M ockt (8944). 4. CONCLUDING REMARKS Sinc th awn of satllit fagmntation stuis, th Gabba iagam has bn an inispnsabl tool of analyzing satllit fagmntations in obit. Yt, no thoy of its fomation ha v bn untakn until th authos publish thi spcial thoy of Gabba iagam fomation fo satllit fagmntations in cicula obits [7]. Th cunt pap psnts th gnal thoy of Gabba iagam fomation fo satllit fagmntations in lliptical obits. This pap, in ffct, conclus th thoy of Gabba iagam fomation fo satllit fagmntations in obit.

14 11 Robt C. Rynols, Ajun Tan an Maius Schamschula REFERENCES [1] J.R. Gabba, Explosion of Satllit an oth Dlta Scon Stag Rockts, NORAD/ADCOM Tch. Mmo, 81-5 (1981). [] N.L. Johnson, E. Stansby, D.O. Whitlock, K.J. Abcomby & D. Shoots, Histoy of On-Obit Satllit Fagmntations, NASA/TM (008). [3] R.D. Culp, Analysis of th Oigins of Dbis Clous, Tlyn Bown Engg. Rpt. Contact SC 7460 (1986). [4] N.L. Johnson & D.S. McKnight, Atificial Spac Dbis, Kig Publ. Co., Malaba, Floia (1991). [5] M. Matny, [6] L. Miovitch, Mthos of Analytical Dynamics, McGaw-Hill, Nw Yok (1970). [7] A. Tan, R.C. Rynols & M. Schamschula, NOAA-16 Satllit Fagmntation in Obit: Gnsis of th Gabba Diagam an Estimation of th Intnsity of Bakup, Av. Aospac Sci. Appl., 7, (017). [8] N.L. Johnson, Atificial Satllit Bakup (Pat ): Sovit Anti-Satllit Pogam, J. Bit. Intplant. Soc., 36, (1983). [9] P.B. Stas, Militaization of Spac, U.S. Policy, , Conll Univsity Pss, Ithaca (1985). [10] P.B. Stas, Spac an National Scuity, Bookings Institution, Washington, (1987). [11] Siiqi, A.A. Th Sovit Co-obital Anti-satllit Systm: A Synopsis, J. Bit. Intplant. Soc., 50, 5-40 (1997). [1] A. Tan, V. Ewas, & M. Schamschula, Fagmnts Analyss of th Sovit Anti- Satllit Tsts Roun 1, Av. Aospac Sci. & Appl., 4, 1-33 (014). [13] A. Tan, V. Ewas, & M. Schamschula, Fagmnts Analyss of th Sovit Anti- Satllit Tsts Roun, Av. Aospac Sci. & Appl., 4, (014). [14] A. Tan, S. Shng & M. Dokhanian, Vlocity Ptubations Analysis of th Biz- M Rockt (Aabsat-4A), Av. Aospac Sci. Appl., 4, 1-9 (014).