Australian Journal of Basic and Applied Sciences. An Error Control Algorithm of Kowalsky's method for Orbit Determination of Visual Binaries

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1 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: AENSI Journls Austrlin Journl of Bsic nd Applid Scincs ISSN: Journl hom pg: An Error Control Algorithm of Kowlsky's mthod for Orbit Dtrmintion of Visul Binris M. A. Shrf,,3 A.S. Sd, 4 A.A. Alshry Dprtmnt of Astronomy, Fculty of Scinc, King Abdulziz Univrsity, Jddh, Sudi Arbi Dprtmnt of Astronomy, Ntionl Rsrch Institut of Astronomy nd Gophysics, Ciro, Egypt 3 Dprtmnt of Mthmtics, Prprtory Yr, Qssim Univrsity, Buridh, Sudi Arbi 4 Dprtmnt of Mthmtics, Collg of Scinc for Girls, King Abdulziz Univrsity, Jddh, Sudi Arbi A R T I C L E I N F O Articl history: Rcivd 5 August 04 Rcivd in rvisd form 9 Sptmbr 04 Accptd 9 Novmbr 04 Avilbl onlin Dcmbr 04 Kywords: Error nlysis; visul binry strs; Kowlsky s mthod; computtionl lgorithm. A B S T R A C T In this ppr, n rror control lgorithm of Kowlsky s mthod for orbit dtrmintion of visul binry systms is lbortd. Th lgorithm hs th dvntg tht it includs nw full nlyticl formul for vluting th probbl rrors of th orbitl lmnts in trms of th corrsponding rrors of th lliptic prmtrs of th pprnt orbit. Tht llows mor globl viw of th nsmbl of solutions nd ccurt clcultions to th lmnts. Morovr vrious sign mbiguitis rsulting from squring th including xprssions r trtd nwly. Th nlyticl rsults togthr with th rmovl of th mbiguitis provid th bsis for nw lgorithm of clculting orbitl lmnts of visul binry systms with thir rror nlysis. Applictions of th lgorithm r givn for two visul binry systms, th first for ADS 998 systm of unit wighting, whil th scond is ADS 0780 s n xmpl of unqul wighting. 04 AENSI Publishr All rights rsrvd. To Cit This Articl: M. A. Shrf, A. S. Sd, A. A. Alshry, An Error Control Algorithm of Kowlsky's mthod for Orbit Dtrmintion of Visul Binris. Aust. J. Bsic & Appl. Sci., 8(7): , 04 INTRODUCTION Binry str systms r vry importnt in strophysics bcus th clcultions of thir orbits llow th msss of thir componnt strs to b dirctly dtrmind. Sinc mor thn cntury go, stronomrs hv bn working on dvloping mthmticl mthods for orbit dtrmintion of binry strs. Th first solution to th problm of clculting th lliptic orbit of visul binry systm givn sufficint dt ws suggstd by Thil-Inns -Vn dn Bos (883, 96 nd 93). This is followd by Cid (958 nd 960) who dvlopd dirct mthod involving thr complt obsrvtionl dt nd n incomplt obsrvtion nmly 3 obsrvtions. Docobo (985) nd Docobo t l. (99) usd Cid s mthod for stblishing n nlyticl mthod tht is pplicbl vn whn only rltivly short nd linr rc of th orbit hs bn msurd. Pourbix (994), nd Pourbix nd Lmpns (997) dvlopd mthod bsd on function tht quntifis th distnc btwn obsrvd nd computd position. Th simultd nnling mthod ws succssfully usd to minimiz it; in this wy th bst orbit is tht on which minimizs th doptd function. Shrf t l. (00), nd Nouh nd Shrf (0) dvlopd n lgorithm for orbit nd mss dtrmintion of visul binris. Th lgorithm uss n optiml point (, ), which minimizs spcific function nd dscribs th vrg lngth btwn th lst squr solution nd th xct on. Brnhm (005) invstigtd th problm of clculting th pprnt orbit of visul binris. H prsntd solution bsd on dvlopd mthmticl tchniqu for optimiztion known s smi-dfinit progrmming (SDP). This solution offrs th dvntgs of bing nlyticl nd clculting uniqu llips tht stisfis th obsrvtions. Rcntly, Kowlsky's mthod ws usd s prliminry computtion of orbitl lmnts of visul binris by Priur t l. (00). Olvi c nd Cvtkovi c (004) usd n improvd vrsion of Kowlsky's mthod for clculting th orbits of 0 intrfromtric binry systms. Th bsic mthmticl tool of Kowlsky;s mthod is th lst squrs mthod. Although th lst-squrs mthod is th most powrful tchniqus tht hs bn dvisd for th problms of strosttistics in gnrl (Andron nd Hurn, 03), it is t th sm tim xcdingly criticl. This is bcus th lst-squrs stimt suffrs from th dficincy of th mthmticl optimiztion tchniqus tht giv point stimts; th stimtion Corrsponding Author: A.S Sd, Dprtmnt of Mthmtics, Prprtory Yr, Qssim Univrsity, Buridh, Sudi Arbi, E-mil: Sd65@gmil.com

2 64 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: procdur hs not bn built-in dtcting nd controlling tchniqus for th snsitivity of th solution to th optimiztion critrion of to b minimum. As rsult, thr my xist sitution in which thr r mny significntly diffrnt solutions tht rduc th vrinc σ to n ccptbl smll vlu. At this stg w should point out tht () th ccurcy of th stimtors nd th ccurcy of th fittd curv r two distinct problms; nd () n ccurt stimtor will lwys produc smll vrinc, but smll vrinc dos not gurnt n ccurt stimtor. It is wll known tht th lowr bounds for th vrg squr distnc btwn th xct nd th lst-squrs vlus is which my b lrg vn if σ is vry smll, dpnding on th mgnitud of th minimum ignvlu, σ / λ λ min min, of th cofficint mtrix of th lst-squrs norml qutions. Unlss dtcting nd controlling this sitution, it is not possibl to mk wll-dfind dcision bout th rsults obtind from th pplictions of th lst squrs mthod, nd hnc of Kowlsky's mthod. In th prsnt ppr, w introduc n rror control lgorithm nd computtionl dvlopmnts of Kowlsky's mthod through thr folds. Th first is th dtrmintion of th typ of th orbit, dirct or rtrogrd nd this will b th subct of Subsction 3.. Th scond fold is th computtionl dsign of th orbitl lmnts from th fiv lliptic prmtrs tking du ccount th typ of th orbit nd othr difficultis ncountrd by th dirct us of Kowlsky's mthod nd this will b th subct of Subsction 3.. Finlly, nw full nlyticl formul of th probbl rrors of th orbitl lmnts in trms of th corrsponding rrors of th lliptic prmtrs A, H, B, G, F nd this will b th subct of Subsction 3.3. Fitting Th Binry Dt At Unqul Wight: Obsrvtions of visul binry systm produc polr msurmnts (, ) for th position of th scondry str rltiv to th primry str. Whr, is th rdil distnc in sconds of rc nd is th position ngl in dgrs s msurd stwrd from north. In th first stp of th nlysis, th qution of th pprnt llips ws trnsformd to (x,y) coordints whr x ws north t 0 o nd y st t 90 o. Thus x cos, y sin. () In Crtsin coordints, th gnrl qution of conic sction cn b writtn s: Ax Hxy By Gx Fy 0. (-) Eqution (-) rprsnts n llips if nd only if th following conditions r stisfid: 0, 0 nd. S 0 Rl (-) whr A H G H B F G F A H H B ; S A B ;. (-3) From th obsrvd dt on cn obtin th cofficints by using lst squrs mthod (LSM). Th LSM rquir to minimiz th rsidul function givn by N i w Ax Hx y By Gx Fy i i i i i i i whr w i is th wight of th point i, it is n intgr nd is tkn qul to th numbr of obsrvtion nights of th point i,nd N is th numbr of obsrvtions. is function of th prmtrs A, H, B, G nd F. Sinc th minimum of th squr sum of th rsidul dfind by occurrd whn th prtil diffrntil of ( A, H, B, G, F) with rspct to A, H, B, G nd F r ll idnticlly qul to zro. So w gt fiv lgbric qutions which cn b solvd to obtin th prmtrs A, H, B, G nd F. In this rspct w dvlopd lgorithm for th dtrmintion of th pprnt orbit constnts A, H, B, G, F, togthr with thir rror nlysis. Th lgorithm uss th most gnrl cs of dt t unqul wighs. Applictions of th lgorithm r illustrtd in Appndix A for two visul binry systms, th first for ADS 998 systm of unit wighting ( Lobo, 994), whil th scond is dvotd for th systm of unqul wighting (Coutu, 98). Onc th vlus of ths cofficints r obtind togthr with thir probbl rrors, th orbitl lmnts nd thir probbl rrors cn b drivd s in th following sction. (3)

3 64 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: Dtrmintion Of Th Orbitl Elmnts And Thir Probbl Error: Kowlsky's mthod is ssntilly n nlyticl mthod. Smrt systmtizd this mthod in grt dtil in his book (Smrt, 958). Th rltions btwn th orbitl lmnts nd th lliptic prmtrs r found in Smrt's book nd mny othr txts. In th prsnt sction w shll stblish, nw computtionl dvlopmnts of Kowlsky's mthod through thr folds. Th first is th dtrmintion of th typ of th orbit dirct or rtrogrd, nd this will b th subct of Subsction 3.. Th scond fold is th computtionl dsign of th orbitl lmnts from th fiv lliptic prmtrs tking du ccount th typ of th orbit nd othr difficultis ncountrd by th dirct us of Kowlsky's mthod nd this will b th subct of Subsction 3.. Finlly full nlyticl formul of th probbl rrors of th orbitl lmnts in trms of th corrsponding rrors of th lliptic prmtrs A,H,B,G,F. nd this will b th of Subsction Th typ of th orbit: Comput th diffrnc, in th position ngls, so If, 0, thn th orbit is rtrogrd. If, 0, thn th orbit is dirct. 3. Th orbitl lmnts from th fiv lliptic prmtrs:. Th longitud of th scnding nod is obtin from (FG H) tn F G A B (4-) nd th dscnding nod from p d. (4-). Th orbitl inclintion i nd th smi-ltus rctum p, r found from F G (A B) /, (5-) whr F G A B cos, (5-) whil i is computd from tn i tn (p (p ) ) for rrogrd motion, for dirct motion. 3. Th rgumnt of pripsis is found from (Fcos G sin )cosi tn (Fsin G cos ) - (Fcos G sin )cosi tn (Fsin G cos ) for rrogrd motion for dirct motion (5-3) (6) (Do not cncl th signs in th numrtor nd dnomintor for rson which will b clrifid soon) 4. Th ccntricity is obtind from Gsin Fcos sin pcosi 5. Th smi mor xis is found from nd p s, p (8) (7)

4 643 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: Th doubl of th rl constnt is found from N N S, whr N is th numbr of th obsrvtionl dt (, ), Sk ( xky k ykxk )/ tk, nd nd y x x r givn from Eqution() s : cos ; y sin ;,,, N., thn th priod P (yr) is found in trms of, nd i s P cosi (9) 7. Th mn motion n is found, from n. (0) P 8. Th tru nomly v is found for ny point whr is vilbl using tn( v ) tn( )sci. () In computing v two conditions should b tkn into ccount: () th qudrnt of th ngls nd (b) th typ of th orbit. Ths conditions r illustrtd through th following computtionl squnc: A- v tn [tn( )sci] C- E- B-Dtrmin th qudrnt II of (v TT v TT ) for rrogrdmotion for dirct motion D- Dtrmin th qudrnt JJ of TT v v v v / / if if JJ II JJ II if JJ II 9. Th ccntric nomly E is found from E tn tn(v / ) 0. Th mn nomly M is obtind from Kplr's qution. () M E sine. (3). Th tim of th pristron pssg T is found from T ( nt M)/ n. (4) In ordr to void odd rsults s for xmpl: T bing b lss thn th vlu of th first tim t of obsrvtion or T bing grtr thn th lst tim t N, critrion must b implmntd in such wy tht t T tn. Th vlu of T is clcultd s th rithmticl mn of T vlus which stisfy this critrion.

5 644 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: Nots: Th utiliztion of th bov computtionl formul nds som uxiliry usr dfind functions, of ths functions r th following i- Computtion of tn y/x : Whn th invrs function of tn is tkn n mbiguity riss which hs to b clrd up. Th ngl tn y/ x lis in th first qudrnt if x 0, y 0, in th scond qudrnt if x0, y 0, in th third qudrnt if, x0, y 0 finlly in th fourth qudrnt if x0, y 0.To obtin th corrct qudrnt, dd or subtrct D y / x, z tn D If x0, y 0 put Q z If x0, y0 put Q z If x0, y0 put Q z If x0, y0 put Q z or from th ngl Q tn (y/ x) th lgorithm my b s: Now it sms clr why w do not cncl th signs in th numrtor nd dnomintor of Eqution 5 (6),bcus tn givs n ngl 4.55 r o ,whil tn givs n ngl r o tht is,lthough tn tn 5. Thrfor, it should xist in th gnrl progrm usr dfid procdur rctn[y/x] (sy) to implmnt th bov lgorithm. ii- To rduc n ngl x in rdin to th intrvl [0, ]. Du to th rithmtic oprtions, n ngl x (sy) my tks ngtiv vlu or vlu. hv to rduc th ngl x in th rng [0, π]. In this cs w 3.3 Anlyticl formul of th probbl rror of th orbitl lmnts: Considr function ( c );,,, m of c's. Lt th stimtd vlus of c r ~ c ;,,,m probbl rrors m nd thir probbl rrors r c is givn s (Smrt 958) c ~ c c c,thn th probbl rror du to th corrsponding known, (5) c ~.Now, sinc th orbitl lmnts r functions of th whr th prtil drivtivs r vlutd t lliptic prmtrs A,B,.., which could b dtrmind togthr with thir probbl rrors using lgorithm. Thn w cn find out th probbl rror of ch orbitl lmnt using th rltions of Subsction 3. with Eqution (5), doing so, w gt ( FG H ) A ( FG H ) B ( AG G( B F G ) FH ) F X ( F( A B F G ) GH ) G ( A B F G ) H Y ( A B F G ) 4( FG H) X Y sc ( A B 4( F F G G ( A B F G ) tn )) 3 P ( A B 4F F 4 G G ) ( A B F G ) i 4 P P ( P )

6 645 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: G cos if F cos ig sin i( G cos F sin ) Q ( F cos G sin ) i ( F G ) cos i S ( Gcos F sin ) cos i( F cos Gsin ) QS (csc ( P cos i cos F P cos i sin G P cos i( G cos F sin ) ( Fcos G sin ) ( P sin i i cos i( P P cot )) 4 P ( ) P ( ) 4 P 4 ( ) cos i ( cos i ( ) sin i i ( ) W dvlopd lgorithm for th dtrmintion of th orbitl lmnts nd thir probbl rrors, its output r displyd through two points: A mssg indicts th typ of th orbit Orbitl lmnts of th visul binry systm nd thir probbl rrors Th output of lgorithm is illustrtd in Appndix B for th sm binris of Appndix A. Conclusion: In concluding, th prsnt ppr introducd n Error control lgorithm of Kowlsky's mthod for orbitl lmnts of visul binry systm. Th dvntg hr is, th constructd lgorithm includs full nlyticl formul for th probbl rrors of th orbitl lmnts in trms of th corrsponding rrors of th lliptic prmtrs of th pprnt orbit. This will llow mor globl viw of th nsmbl of solutions nd ccurt clcultions to th lmnts. Morovr vrious sign mbiguitis rsulting from squring th including xprssions r trtd nwly. Th lgorithm is pplid for two visul binry systms, th first for ADS 998 systm of unit wighting (Tbl I), whil th scond is ADS 0780 s n xmpl of unqul wighting (Tbl II). Th constnts of th pprnt orbits r clcultd with thir probbl rrors using lgorithm, whil th scond lgorithm is dvotd for dtrmining th typ of th givn orbit (dirct or rtrogrd) nd clculting th orbitl lmnts with thir rror nlysis. Th rsiduls for th two systms r givn in figurs nd rspctivly. Th nlyticl rsults togthr with th rmovl of th mbiguitis provid th bsis for nw lgorithm of clculting orbitl lmnts of visul binry systms with thir rror nlysis. Appndix A: Exmpls of lgorithm : - Considr th dt st for ADS 998 of unit wighting ( Lobo, 994) writtn s follows Tbl I: Dt st for ADS 998 of qul unit wighting ( Lobo, 994) Dt ('') ( o) () Th constnts nd thir probbl rrors A H B G W

7 Rsidul 646 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: F () Ths cofficints rprsnt n lliptic orbit r (3) Th probbl rror of th fit is (4) Th cofficint of corrltion (5) Confidnc intrvl for ch cofficint For A is ( , ) For H is ( 3.746, 3.50) For B is ( , ) For G is (.98,.48) For F is ( , ) (6) Plot for th rsiduls (7) tim(yrs) Fig. : rsidul for th systm ADS 998 of qul unit wighting (Tbl I) - Considr th dt st for visul binry systm (Coutu, 98) of unqul wighting listd s follows Tbl II: Dt st for ADS 0780 of unqul wighting (Coutu, 98) Dt ('') ( o) Appliction of lgorthim to ths dt yilds () Th constnts nd thir probbl rrors A W

8 Rsidul 647 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: H B G F () Ths cofficints rprsnt n lliptic orbit (3) Th probbl rror of th fit is (4) Th cofficint of corrltion r (5) Confidnc intrvl for ch cofficint For A is ( ,.955) For H is (.45,.49) For B is (0.494, ) For G is (0.4968,0.3948) For F is (.5469,3.0079) (6) Plot for th rsiduls tim (yrs) Fig. : rsidul for th systm ADS 0780 of unqul unit wighting (Tbl II) Appndix B: Exmpls of lgorithm : - Appliction of lgorithm for xmpl of Appndix A yilds () Th motion is rtrogrd () Th orbitl lmnts nd thir probbl rrors i d P T Appliction of lgorithm for xmpl of Appndix A yilds () Th motion is dirct () Th orbitl lmnts nd thir probbl rrors i d P

9 648 M. A. Shrf t l, 04 Austrlin Journl of Bsic nd Applid Scincs, 8(7) Novmbr 04, Pgs: T REFERENCES Andron, S., M. Hurn, 03. Msurmnt rrors nd scling rltions in strophysics: rviw, Sttisticl Anlysis nd Dt Mining, 6(): Brnhm, R.L., Jr., 005. Ap. J., 6: Cid, R., 958. AJ, 63, 9 Cid, R., 960. Rv. Acd. Cincis. Zrgoz. Sr 3, Tomo XV, Fsc Coutu, P., 98. Obsrving Visul Doubl Strs, th MIT Prss, Cmbridg, Msschustts, London Docobo, J.A., 985. Clst. Mch., 36: 43. Docobo, J.A., J.F. Ling, C. Prito, 99. In H. A. McAlistr, W. I. Hrtkopf (ds), Complmntry Approchs to Doubl nd Multipl Str Rsrch, ASP Confrnc Sris, 3: 35, 0. Lobo, D.C., 994. Ppr prsntd to XX ~ ~ Brsilirs R uni o Annul d Socidd Astronomic SAB,Cmpos d Jordo, SP, gosto Nouh, M.I., M.A. Shrf, 0. J. Ap. A., 33(4): Olvi c, D., Z. Cvtkovi c, 004. A&A 45: Pourbix, D., 994. Astron. Astrophys., 90: 68. Pourbix, D., P. Lmpns, 997. In J. A. Docobo, A. Elip, H. McAlistr (ds), Visul Doubl Strs: Formtion, Dynmics nd Evolutionry Trcks, Kluwr Acdmic Publishrs, Dordrcht, 383. Priur, J.-L., M. Scrdi, L. Pnscchi, R.W. Argyl, M. Sl, 00. Mon. Not. R. Astron. 407: Shrf, M.A., M.I. Nouh, A.S. Sd, 00. Rom. Astron. J., (): 3-3. Smrt, W.M., 958. Combintion of Obsrvtions, Cmbridg Univrsity Prss Thil, T.N., 883. Astron. Nchr., 04: 45. Vn dn Bos, W.H., 96. Union Obs. Circ., : 354. Vn dn Bos, W.H., 93. Union Obs. Circ., 86: 6.