A Redshift Phenomenon in Relativistic Binary System

Transcription

1 J. Basic. Appl. Sci. Rs. 3(8) TxtRad Pulicatin ISSN Jurnal f Basic and Applid Scintific Rsarch A Rdshift Phnmnn in Rlativistic Binary Systm A.B.Mrcs 34 Dpartmnt f Astrnmy Natinal Rsarch Institut f Astrnmy and Gphysics Hlwan Cair Egypt. Cntr f thrtical physics (CTP) British Univrsity in Egypt (BUE). 3 Egyptian Rlativity Grup. 4 Ladr f th grup f xtra-slar plants grup in Kttamia Cntr f Scintific Excllnc fr Astrphysics and Spac Scinc ABSTRACT In prvius trials similar tchniqus hav n usd t study th slar lim ffct phnmnn y using Schwarzschild slutin and Lns-Thirring Fild. A mr gnral frmula fr th lim ffct f rapidly rtating stars was dn y th authr. In this wrk a mdifid Curzn xact slutin fr Einstin s fild quatins has n usd t study th rd-shift f a inary systm. In this tratmnt it is assumd that th primary star is massiv with rspct t th scndary n and its cntr f mass is cincidnt t th cntr f rtating plar crdinat systm. Th scndary star is assumd t rtat arund th primary star and Earth s srvr rtats with th Earth. A gnral thrtical frmula fr th rdshift f inary systm is taind. This frmula may usful in studying diffrnt cass f inary systms and may thrugh th light n third dy f sm inary systms. KEYWORDS: Binary Pulsars Rdshift Rlativistic Binary Stars Curzn Slutin Axial Symmtry. INTRODUCTION Th discvry f th first dul-nutrn-star inary PSR 96+3 y Huls-Taylr and many thr inary systms as PSR B93+6 and PSR B534+ intrducd th pssiility f strng srvatinal tsts f gravity in th strng filds. As it is wll knwn that th rcivd signals frm clstial jcts ar ur surcs f knwldg aut thm. Th rcivd signals frm ths jcts cm t us via diffrnt carrirs as phtns nutrins gamma ray and x-ray. Fr this rasn w cnsidr hr n f th imprtant phnmnn i.. th rdshift phnmnn fr phtns and a similar shift fr thr carrirs. Th prlm f th lim ffct phnmnn fr th slar disk has n studid thrtically y using th gnralizd rdshift frmula givn y Mikhail t al. (00). Many authrs hav attmptd t find satisfactry intrprtatin fr this ffct thrtically in th fram wrk f GR (cf. Mikhail t al. (00) Wanas t al. (008)). Th gnralizd frmula f th rdshift nt nly usd t study th lim ffct in th slar disk ut als t study th rdshift f raidly rtating stars (Mrcs (03)) and th rdshift f static inary systm (Wanas t. al (0)). Wald and Wanas (00) cnsidrd a mdifid Curzn slutin fr Einstin fild quatins t dscri th fild f inaris. A mr gnral frmula fr rd-shift fr inary systm was writtn in th cas f a statinary srvr n th Earth surfac. Th aim f this papr is driving a mr gnral frmula fr th rdshift f (rlativistic) inary systms. W ar ging t us an xact axial symmtric slutin f Einstin's fild quatins and th Krmak McCra and Whittakr (KMW) thrms n null gdsics t driv a mr gnral frmula fr rdshift f rlativistic inary systm. This has n dlivrd y using th adjustd Curzn givn y Wanas t al. (0) ut in ur tratmnt w cnsidrd th cntr f rtating plar crdinat systm and th cntr f massiv primary star ar cincidnt. Als th rtatinal mtin f th scndary cmpnnt f th inary systm arund its cntr f mass is cnsidrd in ur calculatins as wll as th rtatin f th srvr with th Earth arund its axis is cnsidrd. Gnrally ur tratmnt is mr gnral than thr tratmnts. In th first sctin a rviw aut dvlpd Krmack Mcra and Whittakar Frmula is givn. In th scnd sctin th mdifid Curzn slutin is displayd. Th calculatd valus fr vctrs and null-vctrs alng th wrld lins f stars and Earth ar shwn in th third sctin. Th gnral frmula fr th rdshift f inary systm and cnclusin ar givn in th last tw sctins. * Crrspnding Authr: A.B. Mrcs Dpartmnt f Astrnmy Natinal Rsarch Institut f Astrnmy and Gphysics Hlwan Cair Egypt..mail:fadymrcs@htmail.cm

2 Mrcs 03 DEVELOPED KERMACK MECREA AND WHITTAKER FORMULA Mikhail t al. (00) dvlpd th frmula givn y Krmack Mcra and Wittakr (KMW) (933) t suital fr masuring th ttal rdshift f a clstial jct. This was instad f using th frmula fr studying csmlgical rdshift nly. In what fllws a shrt rviw aut th dvlpd frmula will givn. Lt us cnsidr tw pints S and S n th wrld lin (W s ) f a clstial jct in th spac tim. Th tw pints S and S ar cnsidrd as th mittrs f signals n clstial jct at wavlngths and rspctivly. An srvr O is mving in th fild f th clstial jct and its wrld lin is (W ) rciving th signals mittd y S and S. As it is wll knwn th rcivd signals dpnd n th lin f sights. If it is assumd nw that thr ar tw null-trajctris T r and T passing radially and liqu twn th srvr pint O n its wrld lin and S and S n clstial jct rspctivly. Lt th rcivd wavlngths frm S and S at O ar and rspctivly. By using KMW 0 0 thrms and can xprssd as 0 0 S 0 O S 0 O () () whr is th unit vctr alng th wrld lin f th mittr W s and is th dirctinal unit-vctr alng th wrld lin f th srvr W at a crtain instant. And is th transprt null-vctr alng th liqu null trajctry T and is th transprt null-vctr alng th radial null trajctry T r. Th suffixs O S and S dnt that th xprssins twn th rackts ar valuatd at O S and S rspctivly. By cnsidring that th tw pints S and S ar idntical s Th rdshift f th signals is givn y: (3) Z 0 0. (4) By using () and () th rdshift can xprssd as Z S S O O (5) A MODIFIED CURZON SOLUTION Curzn s xact slutin f Einstin s quatins is an xact axial symmtric statinary slutin. Wanas t al. (0) mdifid Curzn slutin t dscri th gravitatinal fild f th tw-dy systm and calculat th rdshift f spctral lins frm inary systm. Thy hav adjustd Curzn mtric in such way t dscri th gravitatinal fild f a inary systm. Thy wrt Curzn s mdifid mtric in th plar crdinat systm ( r t ) in th frm

3 J. Basic. Appl. Sci. Rs. 3(8)-7 03 ds g rr dr g d gd gtt dt gdd gt ddt g t ddt (6) whr th mtric functins ar givn y g rr cs sin ( t ) cs ( t ) g r cs sin cs ( t ) r sin sin( t ) cs cs( t ) g cs sin cs ( t ) r sin sin( t ) cs cs( t ) gt cs sin cs ( t ) cs sin ( t ) cs ( t ) g r sin cs sin cs ( t ) g t cs sin ( t ) cs ( t ) r sin cs sin cs ( t ) g tt r sin cs sin ( t ) cs ( t ) cs sin cs ( t ) / cs sin cs ( t ) and th ptntial functins ar m m r t r r whr r t m m r r r ( r a) a r ( sin sin( t )) r ( r a) a r ( sin sin( t )) m m cs sin cs ( t ) 4 4 r r r mm r a a r r ar th distancs twn a ptntial pint and th pint masss m and m rspctivly and a is th distanc twn m and m. GENERAL ASSUMPTIONS FOR CALCULATING BINARY STARS REDSHIFT W ar ging t us th mdifid vacuum slutin f Curzn t dscri th fild f cmpact inaris spcially in th cas f dul inaris whr thr is n mattr r nn-gravitatinal filds ar prsnt in th rgins twn th tw pulsars r utsid thm. In rdr t calculat th rdshift f th pulss cming ut frm a inary systm and using quatin (5) w assum that: - Th gravitatinal fild f th primary star is axially symmtric and will givn y (mdifid Curzn) whil th fild f th scndary is a sphrically symmtric (Schwarzschild fild). - Th mass f th scndary star is vry small with rspct t th primary n. Th scndary rtats arund th primary in a circular rit f radius (a) at an angular vlcity V. 3- Th srvr at th Earth s surfac is rtating with th Earth arund its axis at a rtatinal vlcity V. Th Earth s srvr srvs th primary star radially whil h srvs th scndary in an liqu lin f sight. 3

4 Mrcs Th inary systm is at a distanc () frm th Earth s srvr. 5- Th Earth s fild is assumd t Schwarzschild sphrical symmtric fild. Frm th prvius assumptins it is clar that w ar ging t calculat th valus f th tw null-transprt vctrs and alng th null-gdsic trajctris T r (radial and passing thrugh primary star) and T (liqu and passing thrugh scndary star). Th tw unit vctrs and will calculatd alng th wrld lins f th primary scndary stars and at th Earth s wrld lin rspctivly. In th nxt sctin w ar ging t calculat th valus f vctr and null-vctrs rquird t chang in rdshift. GENERAL FORM FOR THE SOLUTION OF EQUATION OF MOTION As it is assumd in th prvius sctin that th fild f th th scndary star and th Earth ar sphrically symmtric thn th transprt vctrs alng th wrld lin f th scndary star and th null vctrs & at th Earth and th unit vctr alng Earth s wrld lin can valuatd in Schwarzschild fild. As it is wll knwn th rdinary Schwarzschild s spac-tim is givn y S ds dr r ( d sin d ) dt (7) m GM whr m and M is th mass f th scndary in CGS units. Th quatin f mtin f a fr r c tst particl can put (cf. Adlr t al. 975 p. 54) int th frm d x dx dx 0 (8) dp dp dp whr P is th affin paramtr charactrizing th trajctry f th particl which is xisting in th Schwarzschild fild thn th slutin f quatin (8) y using (7) is in gnral frm a st f diffrntial quatins givn y Errr! Bkmark nt dfind.: dr dp E r d 0 / (0) dp d l dp r dt () dp whr l ar tw cnstants f intgratin and E is a paramtr taks th fllwing valus: 0 fr a phtn ; E fr a matrial particls As it is assumd that th rtatinal vlcity f th scndary star arund th primary is V thn w can writ d dt V d dt V V dp dp ( a) () (9) 4

5 J. Basic. Appl. Sci. Rs. 3(8)-7 03 m whr ( a) m is th mass f th scndary stars. a Th cmpnnts f th unit-vctr alng th scndary star's wrld lin ar dr d d dt dp dp dp dp 0 0 V ( a) ( a) Sinc. (3) Hnc using (7) and (3) w gt a V ( a) (4) 4 V a ( a) 4 V a ( a) Sinc th srvr is rtating arund th Earth s axis f symmtry at a vlcity V fllwing th sam st f quatins frm (9) t () taking int ur cnsidratin that thn th unit vctr alng its wrld lin is 4V V 4 V whr m. Nw w ar ging t valuat th null vctrs & and th transprt vctr (5) f th primary star assuming that th gravitatinal fild f this star is axially symmtric and is givn y th mdifid Curzn mtric (6) w hav s a a a a V a V a a a V a V 0 a a a a a V V a V a a a V a V a a a a a a V V a V a V a a V a (6) w V V s V V 0 V V V V V V V V V V (7) 5

6 Mrcs 03 Nw w can calculat th valus f th transprt null-vctr & alng th null-trajctris Tr and T0 at th primary star S and Earth. Using & t xprss thir valus at Earth and & at star. Th valus f ths null- vctrs ar givn y: s s ( ) C ( ) ( ( )) C 00 (8) ( a C ( ( a)) r C 00 s ( a) D D L L L a a a a L D a D a a L a L s a a a (9) (0) () THEORETICAL RELATION OF REDSHIFT Hr w fund th valu f rdshift f a inary systm n f thm is massiv and th crdinat systm is fixd at its cntr and rtat with crdinat systm at an angular vlcity ω whil th scndary star is f small mass with rspct t primary and rtats arund th primary at a rtatinal vlcity V. If w us th quatins frm (4) t () in quatin (5) w hav Z a a L V a a V a a V 4 4 a a 4 a V a a V V a V a v a a a V + a v a V a V a 3 a a 3 a V a V a v a a a V a V a a V V D V L V L () whr L and D ar cnstant f intgratins. This rlatin sms t cmplicatd ut all its trms can dtrmind asily frm srvatins. CONCLUSION Th quatin () rprsnts a thrtical gnral rlatin fr calculating th rdshift f a rlativistic inary systm in which th primary star is mr massiv with rspct t th scndary n. Als it is assumd that th cntr f th plar crdinat is cincidnt t th cntr f mass f primary and th primary rtat with th 6

7 J. Basic. Appl. Sci. Rs. 3(8)-7 03 crdinat systm. Th scndary star f th systm is rtating arund th primary and als th srvr is assumd t rtat with Earth arund its axis. This situatin is t nar t th tru situatin f a inary systm in ur Galaxy. If it is assumd that th crdinat systm is at rst this rlatin will tnd t a vry asy frm dpnds nly n th masss f th tw stars sid th rtatinal spds f scndary star and th srvr. Althugh th mdifid Curzn slutin is a suital fr studying a inary pulsars (Wanas t al. (0)) ut frm th rlatin () and th assumd cnditins n can us it in cas f inary systms. This rlatin can adaptd als t study a systm f inary stars systm apprximatly qual in mass and th f thm rtat arund th cntr f mass f that systm. Mrvr th taind rlatin may hlp t study th gravitatinal radiatin y cmpact inaris. Furthrmr this may usd t study th rstrictd thr dy prlm. Cnsidring th mtin f a third pint-lik particl in th fild f a inary systm withut applying th prturatin tchniqu that may hlp in slving many prlms in clstial mchanics. REFERENCES. Adlr R. Bazin M. and Schiffr M. 975 INTRODUCTION TO GENERAL RELATIVITY d d. Mc Graw Hill pp. 54. Krmack W.O. Mc Cra W.H. and Wittakr E.T. 933 Prc. Ry. Sc. Edin Mikhail F.I. Wanas M.I. and Mrcs A.B. 00 Astrphys. Spac Sci. 80 (3): pp Mrcs A.B. Jun 03 accptd t pulishd in NRIAG JAG. It is a pr-rviwd scintific jurnal pulishd y Elsvir. 5. Wald El Hanafy and Mamduh Wanas 00: RELATIVISTIC GRAVITY AND COMPACT BINARY OBSERVATIONS: A Cmplt Guid t Binary Pulsar Analysis a Stp twards an xact slutin f Tw Bdy Prlm in Gnral Rlativity LAP LAMBERT Acadmic Pulishing pp Wanas M.I Awadalla N.S. and El Hanafy W.S. 0 arxiv: v5 [gr-qc]. 7. Wanas M.I Mrcs A.B. and El Gamal S.I Lns-Thirring Fild and th Slar Lim Effct Prc. MERIEM-IAU rginal Mting (): pp ( 7