Jounal of Modn Physics, 01, 3, 1966-1971 http://dxdoiog/10436/jmp01314 Publishd Onlin Dcmb 01 (http://wwwscipog/jounal/jmp) An Eccntic Divation of th avitational-lns Effct Yoonsoo Bach Pak, l-tong Chon 1 aculty of Scinc, Koa Advancd nstitut of Scinc and Tchnology, Dajon, South Koa Koan Acadmy of Scinc and Tchnology, Songnam, South Koa Email: itchon@hotmailcom civd Sptmb 1, 01; visd Octob 8, 01; accptd ovmb, 01 ABSTACT Th gavitational-lns ffct is intptd in th famwok of th wtonian mchanics gading th photon of h ngy h as a copuscl with a tiny mass of m W calculat it s path bndd by th gavitational foc na c th sufac of th sun Effcts of dak matt hav also bn valuatd Kywods: avitational-lns Effct; Cuvd Spac; Dak Matt 1 ntoduction Th avitational-lns Effct (LE) povids on of th powful mthods to invstigat physical poptis of stlla objcts This ffct is actually th sult divd by th nal Thoy of lativity (T) and, thfo, dynamical undstanding is almost impossibl without basic knowldg of th T, which is usually taught in th gaduat cous of univsity vthlss, it might b instuctiv to div th LE on th basis of th wtonian mchanics which is foundd ath upon ou xpintial vnts ndd, th bnding of th light path is oiginatd in th spac-tim stuctu inducd by th gavitational fild, and, thfo, it may not b undstandabl in th famwok of th classical mchanics Sinc th light (photon) dos not cay any mass, th bnding of its path cannot b takn plac by th gavitational foc in th classical wtonian mchanics Howv, if th photon is intptd as a copuscl with E a mass, m, wh E is th photon ngy c and th copuscl is assumd to mov always at th spd of light, c, without any vaiation of m, th path of th photon can b bndd by th gavitation of th fnc stlla objct vn in th classical wtonian dynamics Along this lin, th bnding angl of th light passing though na th sola sufac shall b calculatd and ou mthod of calculation will b xtndd to th systm nvlopd in dak matt halo Divation of th avitational-lns Effct Lt us consid th gavitational-lns ffct causd by th sun Whn th photon is gadd as a copuscl with a mass m, th gavitational foc of th sun acting on it yilds M m f, (1) wh is th gavitational constant, M is th sola mass and is th distanc btwn th sun and th copuscl Th motion of th copuscl can b obtaind by solving th wtonian quation of motion [1] ts obit is actually hypbolic nstad of taking such a tatmnt, lt us ty to div th hypbolic obit of th copuscl on th basis of th gomtic popty of th conic sction Th ccnticity of th hypbola is gnally dfind as P, () P H wh P and a distancs of an abitay point P on th hypbola fom th focus and fom th dictix, Q, spctivly, as is shown in igu 1 On of two focuss in th hypbola is assighnd by at which th sun is locatd n oth wods, th tac of th point P which satisfis th lation () daws a hypbola with th ccnticity Lt P b wh th angl uns clockwis aound th focus, thn is givn as s cos, (3) wh Ao is th adius of th sun and s BA o Accodingly, Equation () yilds (4) s cos Copyight 01 Scis
Y B PAK, -T CHEO 1967 Y Q Q O H H O θ O B A O X igu 1 Hypbolic obit of a copuscl : focus, Q Q o oigin, : dictix, : tangnt lin Th sun is at Sinc P 0 at 0, w hav O : th s (), Thus, by solving Equation (4) with spct to th quation of th hypbola can b obtaind in th fom 1, 1 cos which is xactly idntical to th sult obtaind by solving dictly th wtonian quation of motion [1] o th paticl with its mass, m moving with th vlocity v, th angula momntum, L m v, (7) is always consvd and th total ngy of th systm, E, is wh mv E k L k m, (6) (8) k M m Th solution of Equation (8) with spct to positiv 1 is 1 mk EL 1 1 L m k 1 (9) o v c, th angula momntum and th total ngy of th systm at 0 can b obtaind by placing by in Equations (7) and (8) Thn, cipocal of th fist panthsis in Equation (9) is qual to th numato of Equation (6), and, thfo, w find th valu of th ccnticity as c M 1 (10) This sult can also b obtaind fom th scond panthsis in Equation (9) t is asily povd to b c And it cosponds to th dnominato of Equa- M tion (6) at 0, which is 1 Thn, w obtain th sam sult as that in Equation (10) On th oth hand, th quation of a hypbola in ctangula coodinats whos oigin is locatd at th cnt of two focuss can b xpssd as x y 1 (11) a b A diffnc btwn distancs of an abitay point on th hypbola fom ach focus is always qual to a This is a basic substanc of th hypbola om Equation (11), w know immdiatly that th distanc OAo in igu 1 is a And, thus, th distanc btwn two focuss is a Lt x1, y 1 b th coodinats of a cossing point btwn th hypbola and th staight lin ppndicula to th X-axis at th focus x 1,0 Thn, making us of th substanc of th hypbola as wll as th Pythagoas thom, w can asily find y1 (1) a n addition, fom Equation (6), it is obvious that π y1 1 Making this sult qual to Equation (1) givs a a, which is actually valu of x 1 and, consquntly, a (13) 1 Substituting valus of x 1 a and y1 1 in Equation (11), on can find 1 b (14) 1 Sinc th quation of th asymptot of th hypbola is x a y b 0, (1) Copyight 01 Scis
1968 Y B PAK, -T CHEO th angl of th asymptot,, is found as actan b a actan 1 (16) Thus, th bnding of th light path, i th gavitational-lns ffct, is π π a ctan 1 (17) With valus, 710 km, M 0 3 c 310 km/s and 6710 km /s kg, tain and 4710, 30 10 kg, w ob- (18) 0878, (19) which is in agmnt with th sult obtaind by J von Soldn using a compltly diffnt fomalization in 1801 [,3] Howv, this sult is unfotunatly just a half of that obtaind in th T [4] Th missing facto may b fd to as lativistic facto Q O Y Q H H A θ Q B A O O P P H O X 3 wtonian Divation of th lativistic acto n contxt of th concpt that th stuctu of th spac is associatd with th gavitational fild, it is convincd that th spac aound th sun must b cuvd by its gavitation Moov, it should b makd that th light has natu to tavl always along th dg of th spac whatv it is staight o cuvd Th cuv in igu has bn th path of th copuscl moving und opation of th gavitation Howv, if it is intptd as th dg of th cuvd spac inducd by th gavitational fild, th photon will tavl along this cuv without any influnc of th foc Lt us now calculat th path of th copuscl moving in this cuvd spac whn th gavitational foc acts dictly on it As is shown in igu, th dictix Q in th gula spac is now bndd into Q by th sam amount of cuvatu as th dg of th sufac of th cuvd spac amly, Q is th tanslatd hypbola of th cuv Thn, th abitay point P on th hypbola is shiftd to P Thus, th cuv should b th obit of th copuscl whn th gavitational foc dictly acts on it in th cuvd spac All points, P, P, H o, H A and H a on th hoizontal lin As th staight lin of dictix, Q is bndd into th cuv Q th distanc is shiftd to, i A A Whn th distanc o is xpssd by u, i o u, w hav, of cous, H AH u, and, thfo, PP ought to b u uthmo, th distanc AoB is xpssd by s, th distanc A should also b, i s n accodanc with s A igu Paths of th copuscl Q : tangnt lin, Q : dictix in th cuvd spac, Q Equation (), th ccnticity should b A o : oiginal dictix P (0) Whn is xpssd as and A P P is considd, w hav A P P P su A =, 1 cos (1) wh s can b obtaind fom th scond lin of Equation (1) bcaus P Ao fo 0, and, thn, 0 Ao and u 0 To lad th final xpssion in Equation (1), w hav also usd that fo 0 o PP o u cos () Solving Equation (1) with spct to, w obtain 1 1 cos (3) Copyight 01 Scis
Y B PAK, -T CHEO 1969 placing by, i, w obtain th standad xpssion of th hybola (4) 1 () 1 cos This quation can also dictly b divd in th gula spac wh th ccnticity, is givn by th dfinition quivalnt to thos wittn in Equations () and (0) as P, (6) in which th point H is shown in igu 3 Whn distancs AoB and BB a xpssd by s and x, i AoB s and BB x, distancs and a dscibd as and s cos, (7) BB (8) x s cos At 0, Equation (6) yilds, (9) x s bcaus of 0 and, thus, 1 1 x, (30) wh s is usd Thfo, 1 1 s cos cos Solving Equation (31) with spct to (31), w find th sult xactly idntical to Equation () Ou sult in Equation (4) xplicats that th ccnticity,, of th hypbolic obit of th copuscl in th cuvd spac yilds just a half of th ccnticity,, obtaind in th gula spac, whn th gavitational foc is dictly acting on it, i 310 (3) Y O H A O Y H H Q B B Q W θ L P A O W igu 3 Paths of th copuscl W and W tots of th hypbola and Q and Q : dictics of th hypbola and, spctivly Thus, th avitational-lns ffct in this cas can b divd fom th fomula, Equation (17), povidd is placd by, π actan 1 Th numical valu is, now, found as X : asymp- 17 (34) (33) This valu is twic of valu obtaind in Equation (19) and ags with th sult obtaind by Einstin [4] in th T Th T holds th concpt that th photon tavls along th sufac of th cuvd spac vn without intacting with th souc of gavitation yilding th cuvatu in th spac, whil th physical wold dscibd h is that th path of th copuscl moving along th sufac of th cuvd spac oiginatd fom th gavitation is focd to bnd by th gavitational foc acting dictly on it Although th spac cuvatu in ou cas is just a half of th sult obtaind by th T, th copuscl vntually kps moving along th sam path as that pdictd by th T 4 Th avitational-lns Effct by th Dak Matt Halo f th dak matt xists in suounding of th galaxy as a halo, th gavitational-lns ffct will b inducd by it n Copyight 01 Scis
1970 Y B PAK, -T CHEO od to simplify th calculation, lt us consid a 11 sphical galaxy with th mass M 10 M and 4 th adius 10 ly compltly covd by th dak matt (DM) with a thicknss DM 100 ly, i th adius of th sph including th DM is DM 101 Th dnsity of th dak matt is assumd to b as lag as 10 tims that of th galaxy (i) Without any dak matt, th LE of th copuscl passing by th galaxy sufac is found as 611 by using Equa- tions (9), (4) and (33), povidd M and a placd by M and (ii) Whn th copuscl passs bsid th sufac of th dak matt, th LE is obtaind as 789, which is calculatd by placing M and by M M DM and DM in Equa- tion (9) (iii) Th light can pntat into th DM without scattd and snak by th sufac of th galaxy Th path of th copuscl in th gion outsid th DM can b calculatd with Equations (9), (4) and (33) by assuming that th total mass, M MDM, is insid th sph of th adius Th obit is dfinitly a hypbola On th oth hand, th path of th copuscl insid th DM is abl to obtain by a numical mthod xplaind blow Lt us consid a copuscl which stats fom th galaxy sufac, namly 0 and, and dashs in th dak matt until unning away fom th DM gion Lt th copuscl b locatd at in a ctain momnt aft it statd fom th galaxy sufac, i a distanc,, fom and an angl,, aound th galaxy cnt Accoding to th aussian Thom, th gavitational potntial outsid a sph can b dtmind by th total mass insid th sph Thus, th total mass insid a sph with th adius paticipats in fomation of a hypbolic obit of th copuscl at that momnt mak that th location of th copuscl,, is situatd actually at th common point on th sph sufac and th hypbolic obit Sinc th copuscl migats in th DM, it foms vy momnt a diffnt sph with a diffnt adius and, accodingly, th total mass insid th sph will chang simultanously amly, th hypbola to which th copuscl on th sufac of th sph fomd in ach momnt blongs is not th sam on anymo as fomd in th pvious momnt Th tac of all ths points which th copuscl occupid vy momnt yilds a cuvd lin t may not b a simpl hypbola but a slightly dviatd on Such a physical pocss might b xplicitly dscibd with following quations Th total 1 mass of th systm is M M with 3 DM 1, (3) 1 DM wh fo D M is th distanc of th copuscl fom th cnt of th galaxy whn it is locatd at th angl aound th focus Th ccnticity at this position is and, thn, fo c c 1 1 1, M M, DM 1 1 cos 1 1 cos otic that is also a function of (36) (37) nally, it is possibl to solv Equation (37) analytically with spct to Howv, it is not asy and, thus, w solv Equation (37) numically with spct to th valu of on-by-on fo ach angl duing th copuscl passs though th DM gion Th obit of th copuscl found in this mann shows slightly dviatd fom a simpl hypbola, paticulaly in th gion of th dak matt Actually, th gion of th DM in which th light is moving holds only within 16 dgs aound th galaxy cnt, i 8 8 inally, ou sult found fo th LE is 797, which is slightly lag than that of th cas (ii) This sult is obviously undstandabl bcaus of that th clos th obit of th copuscl is to th gavitational cnt, th lag th LE bcoms Conclusions t has bn shown that th LE could b divd in th famwok of wtonian mchanics if th photon w gadd as a copuscl with th tiny ffctiv mass, h m, which was basd on th quantum thoy and c th Spcial Thoy of lativity Th facto aising fom th T has bn divd by intoducing th concpt of th cuvd spac du to th gavitational fild and its nomal action Of cous, all ths pocdus would b xpdintial Howv, it would hlp snio high school and undgaduat univsity studnts to comphnd th physical stuctu of th LE Sinc xistnc of th dak matt is not optically visibl bcaus it dos not intact with th light, th LE is known as a powful mthod to obsv it Th psnt wok invstigats th LE by a sphical galaxy compltly covd by th dak matt Th sults a dfinitly in dtctabl ang t is tu that on can obtain infomations of th dak matt by invstigating th LE 6 Acknowldgmnts This wok has bn caid out und th KAST Mnto Copyight 01 Scis
Y B PAK, -T CHEO 1971 Pogam 010 EEECES [1] J B Maion, Classical Dynamics of Paticl and Sys- tm, Acadmic Pss, w Yok, 1970 [] J von Soldn, Blin Astonomischs Jahbuch, 1801, pp 161-17 [3] P Lnad, Zu Wassfallthoi d witt, Annaln d Physik, Vol 370, o 1, 191, pp 93-604 doi:10100/andp191370106 [4] A Einstin, Di undlag d Allgminn lativittsthoi, Annaln d Physikm, Vol 34, o 7, 1916, pp 769-8 doi:10100/andp191634070 Copyight 01 Scis