DESIGNING TRAJECTORIES IN A PLANET-MOON ENVIRONMENT USING THE CONTROLLED KEPLERIAN MAP

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1 AAS DESIGNING TRAJECTORIES IN A PLANET-MOON ENVIRONMENT USING THE CONTROLLED KEPLERIAN MAP Py Gv d S R Dgg f ff j w v dff m b dd by bkg p pbm mp -bdy pbm. T pp, d pd -bdy pp vd db y, d pvd d b f vg mpd d pd- pp. W d pbm f dgg f-ff m-m b j Jp-Ep-Gymd ym w f m. Pvy, dvd md g f-pm (.. z f) j m ym, b j bd fb d vy g f m vvd. T m f pp db mdgy w xp dyg f dym f w -bdy pbm,.. Jp-Ep-Spf d Jp-Gymd-Spf, g Hm pvg Kp mp ppxm dvd d g m p fm f V g by w m-ffg bd pvy. INTRODUCTION Lw gy pf j m-m b f Jp ym b bd by g mp gvy by m j w b 1, 2, 3 p dy d f g. T pm v b xpd by ppyg q fm dym ym y ym f bd dd m. 4, 5, 6, 7, 8 O dg j w pdmd f d p, m f f fm H g. Ug pp, w b dbbd M-M Ob (MMO), f pf b v m f y dd d, d f fyby g y d. T pp d wk w w xg q, g py jy dg pb f ggv m p-m vm. T pp q fxb pf b md pd f v, d b md v y g. Pvy, 9 v b b f pm j Jp- Ep-Gymd ym, b j bd fb d vy g Gd Sd, Egg S d M, Vg T. A Pf, Egg S d M, Vg T.

2 f m vvd. T f pp db mdgy g yy dvd Kp Mp 10 b j w by w mf-fg, g m p fm f V. T pp gd fw. W f db Kp Mp, w y d p f fdg dd j. W vw fmwk f bkg dw m-m pbm 2 3-bdy pbm d fdg z-f j g fm m. Ug fmwk, w db mdgy f gg w gy j b mpd b m g m p. W d dff bw f mp d m-f-fg f fmy f j bd g m md. KEPLERIAN MAP FOR EVOLUTION UNDER NATURAL DYNAM- ICS OF PCR3BP E mp, w w pp Pé mp ( Kp mp), pd mp f g f pp ω g fm d Kp gy K, (ω, K ) (ω +1, K +1 ). T mp fm ( ω+1 K +1 ) ( ) ω 2π( 2K = +1 ) 3/2 K + µf(ω ; C J, K).., mp f yd A = S 1 R f. T mp md pf -Kp b b bdy f m, w pf pbd by m bdy f m µ. T f pf w pb mdd mpv kk pp pg, pd kk f f, Fg 1(), w (µ, C J, K) dd bf pm. T mp p w dym f f q f m; my, p p, w Fg 1(b), dy vd by f b d, bw w d z. T m pyy v mmj x pd f v x d f Kp gy K, w = 1/(2K). T kk f bd fm mp w bgg kk vd f vy w g f pp v. If pp gy d f pb, p g gv kk, d f pp gy bd pb, kk pv. Ug m md bv, w pp kk mp f w pf m,.. w m-mj x f pb. I f pp kk mp, v pv kk, pp d b gy d f pb ( pp d b gy π). Smy, f gv kk, pp d b gy bd pb ( pp d b gy m π). T gg pp vd f mp dg f w gy j, pfy bw m Jp m ym. Mp gvy ky py mm w d b xpd f f m. 1 F (1)

3 ω/π () (b) ω/π () Fg 1 () T gy kk f f v. ω f yp v f pm. (b) T d p p f Kp mp. T mmj x [= 1/(2K)] v. g f pp ω w f pm µ = , CJ = 2.995, = 1/(2K ) = 1.35 ppp f pf Jp-C ym. T d w k y d fwd f 104, pdg w pp w pdg b d d. () Upp p: p p jy w p mkd w g d f p w q. Lw p: fg p pj fm f jy. Jp d C w p, d C b dd. T d pf mg fm g m mmj x, kpg pp d gy p. B pf d C b Jp -kw.

4 xmp, jy fm E Jv ym, j gzg b f m y m C, mg g f fm b w g pp m. T w Fg 1() b p p d fg p. Fm b gy g C, pf b pd b d m. T f p b d yd b p p, w Fg 2() (f d f b mp,,.g., Rf 6 ). Fwd bkwd m d b vy Kp mp, pd Pé f-f-. T g p g, Fg 2(b), x fm jv b x C. I f bkwd Pé f d b f p b T dvg f dg y w-dm mp ppd f m g f d -bdy q f m w ppy d mp my ppb p p p ymp w mp. 11 F xmp, pv wk w mp b ppd, vg x f f f mg bw b f dff mmj x. T b d by pf fm E Jv ym. A pf w jy j gz b f m y m C mg g f fm b w g pp m. PATCHED THREE BODY APPROXIMATION T P3BA dd by R. 1 d m f pf fd f bd, dd w m,.g., Jp d m, M. W jy f pf m b f M, pb f pf m wy fm py Kp m b Jp dmd by M. I, w y pf m w mdd by Jp-M -pf d -bdy pbm. W -bdy pbm, w k dvg f p p b f p d p, w b d w mvm bw b. B b d b, d dym d w m, mp f dg f MMO jy. T m f pf gv fd f bd Jp, Gymd, d Ep ppxmd by w gm f py bdy m, d -bdy md. T jy gm f bdy ym, Jp-Gymd-pf, pppy pd gm Jp-Ep-pf -bdy ym. F gm f py bdy m, v mfd b f L 1 d L 2 bd b (dg pd b) d wd wy fm mpy p d m. P f b d by b mdg mvm bw b. Dd mvm bw b w w pf v g g b. W pf m, mdd -bdy ym, b wby w -bdy ym, w w p -bdy md w ym. T g fd b jy m f-bdy md. Evd gg g vy gd, 1 v f -bdy md.

5 Σ: P S Pp Jv M P Jp Jv M B Cp L2 () K Ex ω/π (b) Fg 2 () A pf P d b f gv p b w fd f gg fm b b Jp b b m. T pf y d b w bdy b v mfd f pd b b L2. T -dm b, md p f dvd j, w pjd fg p. A w f f b w Σ, P mp pp x m. (b) T my mpd f x Σ, w m mp pm bf. Spf w x w bqy p p m d pbg m. T v x Kp gy K f b b Jp. Pvy,9 dbd mdgy b f pm j f MMO, w p f Kp Mp. Dg -m f w w v m d f m, Jp w d f w p, w my Fg 3, w M2 m. I f p, f dm pbm bm f fdg ppp f Jp-M1 -pf pbm w jmp bw b w M1,.., pfmg GA d pjv.1 M1 pb y gf v m p f pf jy pjv (A Fg

6 3()). T ff f M 1 mp mp pf, qv V b f M 1. () (b) Fg 3 I-m f v gvy. () T b f w Jv m w. Up xg m (M 1 ) pf-f, pf pd d d bdy ff p b b Jp. T pf g gvy fm m w p g pjv (dd A). T v fyby xb gy m pf/m gmy b pf b - w m b pd d f m m b m p b m. O pf b m gzg b f m, M 2 ( f, gzg b f M 2 L 2 p), m bm dm pb. T pf b w dd E. (b) T pf w v gvy fm M 2 pjv (P ), w - d pp. T pf by pd by M 2. T pjv dd v M 2 b, f, b f M 2 L 2. W m GA b vd w M 2 w ppp gmy M 2 bm dm pb d bq GA w b w M 2 y. W p d, pf b by pd by m. 6 T f pf jy w pf pb w fm bg dmd by m M 1 bg dmd by M 2 d wg b. A k b mv d b y ff w. T f pb wg b wg g f P3BA, Fg 4. I g f p f f p d pd- ppxm, w gd m dg gdg w w bdy f md f pf Kp m. T k f g f j g fm -Gymd -Ep Jv-

7 Fg 4 Sm jy - p wg v g. V pp/pp mkd x b w mpfd g Kp Mp f w bdy ym. T fg 5,pdd fm pp, w vdy f pd bdy ppxm. Fg 5 w m-mj x m y, g fm x fm Gymd p by Ep. Fg 5b w m y - p, y wg p fw gy w gm. Fg 5 d Fg 5d w -bdy gy y f p f Jp-Gymd- P d Jp-Ep-P ym. T m =0 f p w w p w. Cy, gy f fm ym f 0 d gy f ym m f 0, f pf pd bdy pp b d b j. A METHOD FOR DESIGNING TRAJECTORIES USING CONTROLLED MAP T d Kp mp w pd mp f g f pp ( pp) ω g fm d Kp gy K, F : A U A (( ) ) ( ) ( ) ω ω+1 ω 2π( 2K F, K = = +1 ) 3/2 (2) K +1 K + µf(ω ) + α w U = [ mx, mx ], mx 1. T m α = α(c J, K) ppxmd. T gy mpyd g dd j w-fd: vv p w m g pd d m-mj x v f pf, d f p, w w g pf g f p p. T f w-pgd gy d fwd-bkwd pp 12 wk b y f bg bw g p d g g. A w md, d vd fm Fg 1b, p

8 () (b) () (d) Fg 5 Tjy fd g Pd T Bdy Appxm. ). Smmj x m y b). Tjy - p. )Jb C f J-G-P ym d). Jb C f J-E-P ym p f pbm ppd w bg g mxd p p, d, fwd-bkwd pp wd b ff f pp. T gm bd f g g m-mj x(.. ) f vy m g f v f pp g(.. g), Fg 1. W dp py f gg w fw, w pp g w gg b g g m-mj x. T f gm f g bdy ym fw:

9 D z, d f w dw d ff q. I f w d _ m x Y I y A A+ g? N Ap p y. _ =0. A y q d g w A? N C q m m z p v j mp Y C d g w g b d f w_ p, w m m mf. A d w k q g v y y. I f w d g. Fg 6 T gm 1). A (ω, ), w fwd mx p w = 0, w mx d v p w mx. If y f d w g f g, bd A+ g f g d, bd A, w (2 d 2b, pvy).e, w d mpy y p (., = 0). T z f b g(. A d A+ ) dy pp pm δω. 2). Idy, f f f d bv A+, w w ppy mv wy fm gbg A g. If (w < mx ) A+, w q v x. T dm (.. [, ] [, ]... m) y dzd d w b g q g fm dz. If m q f bg A, w q w f bg m gbd f ωp. H ωp v f ω w d mxmm d m-mj x v. T, g vy y w.. q, w wk v ω ωp. O d, f q f bg A, w d mv f pp g d mmz. T pmz b dd by m dz pd bv. 2b). If f f A, w g dz f dm, d q m wy bf. Svy y d g f ω ωp. 3). O d v f d, w w f-. F b dd by fwd-bkwd md, w g f f f b v mfd f pd b d L2 w P pp.

10 () Fg 7 Smp jy dgd g gm. () P f Sm-mj x V Pp g. (b) Tm y f m-mj x. T pf pdy v g f g d m-mj x. (b) I Fg 7 w w mp jy bd by bv md gm, f Jp-Ep-Spf ym. N f pppy md p, pf v g f g d f m-mj x. W w gm w fmwk f pd bdy ppxm. W d pp mp g jy fm Gymd wg g, d p w jy bd by g d pp mp w d p d Ep. A mp jy f w Fg 8. T pf mp jy g 160 m/ f f 1.7 y, w dd 116 v d Jp (pp/pp pg). T m k f m 10% f k f pvy w pm (z) f jy f m f bdy ym. Tdff Bw F d Tm-f-Fg W w d dff bw m-f-fg d f(). T m f f d f pvdg V xpd b pp pm δw p p, pm dd z f g p p w vy ppd, dd pvy. M pv xpd d m-f-fg p m. I Fg 9, w w p g m-f-fg V f dff f g bdy ym (Jp-Ep-Spf). Sv dm d w k = 1.5, d w w p f w f d, k m d m f x

11 () Fg 8 Tjy f Jp-Ep-Gymd ym g pd bdy pp. () Tm y f m-mj x. b). Sm-mj x V y p w -bdy gy bkgd. (b) Fg 9 P wg dff bw d d mb f qd f w dff d dg p d Ep f Jp-Ep-Spf PCR3BP g, f dff v f δω. T gv m f f qd f f m. T b f ym xpd b m f pd bdy ym.

12 SUMMARY AND CONCLUSION W v pd mdgy b fm f w f j w m-f-fg f bdy pbm g y Kp mp. T d y mp, d w fmwk f pd bdy pp, p dg p dby, mpd g f q f m, d b d b ppp w p d pd g mmj x. T pp p gg gd ppxm qky, w v p m pd d--d jy dg md. W g w d m p qd mp m gv m fm. A pb x wk wd b d w p md d f d p md dd pp. REFERENCES [1] S. D. R, W. S. K, M. W. L, d J. E. Md, Dg f M-M Ob, 13 AAS/AIAA Sp Fg M Mg, P, P R, Fby Pp N. AAS [2] G. J. Wff, A Ivg f Jp G M Ob Tjy, AAS/AIAA Adym Sp Cf, Bg Sky, M, Ag Pp N. AAS [3] S. D. R, T py p wk, Am S, V. 94, 2006, pp [4] C. C. Cy, Lw gy b d -bdy pbm, SIAM J. App. M., V. 16, 1968, pp [5] E. A. Bb d J. K. M, S-pbd E--M f w b p, J f Gd, C d Dym, V. 16, 1993, pp [6] W. S. K, M. W. L, J. E. Md, d S. D. R, H bw pd b d m, C, V. 10, 2000, pp [7] W. S. K, M. W. L, J. E. Md, d S. D. R, Lw gy f M, C M d Dym Amy, V. 81, 2001, pp [8] E. Bb, Cp Dym d C M C M: W App C f Lw Egy Tf. P Uvy P, [9] S. D. R d P. Gv, F-pm j p-m vm g mp gvy, 20 I Sympm Sp Fg dym, 2007, pp. NASA/CP [10] S. D. R d D. J. S, Mp gvy, p, d p d -bdy pbm, SIAM J Appd Dym Sym, V. 6, N. 3, 2007, pp [11] J. D. M, Symp mp, v pp, d p, Rv. Md. Py., V. 64, 1992, pp [12] C. G. T.Sb, E. O d J. Yk, Ug d j g, Py. Rv. L., V. 65, 1990, pp