General Relativistic Orbital Effects in Compact. Binary Stars (Solution by the Method

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1 Wold Jounal of Mechanics, 17, 7, ISSN Online: 16-5 ISSN Pint: 16-49X Geneal Relativistic Obital Effects in Copact Binay Stas (Solution by the Method of Celestial Mechanics) Linsen Li School of Physics, Notheast Noal Univesity, Changchun, China How to cite this pape: Li, L.S. (17) Geneal Relativistic Obital Effects in Copact Binay Stas (Solution by the Method of Celestial Mechanics). Wold Jounal of Mechanics, 7, Received: Septebe 5, 17 Accepted: Decebe 6, 17 Published: Decebe 9, 17 Copyight 17 by autho and Scientific Reseach Publishing Inc. This wok is licensed unde the Ceative Coons Attibution Intenational License (CC BY 4.). Open Access Abstact Petubation ethods ae eployed to calculate tie vaiation in the obital eleents of a copact binay syste. It tuns out that the sei-ajo axis and eccenticity exhibit only peiodic vaiations. The longitude of peiaston and ean longitude of epoch exhibit both secula and peiodic vaiation. In addition, the elativistic effects on the tie of peiaston passage of binay stas ae also given. Fou copact binay systes (PSRJ77-9, PSR191+16, PSR154+1 and MX-7) ae consideed. Nueical esults fo both secula and peiodic effects ae pesented, and the possibility of obseving the is discussed. Keywods Geneal Relativistic Obital Effect, Copact Binay Stas 1. Intoduction In the wake of unceasing developent in the post-newtonian celestial echanics, at pesent, the eseach on the post-newtonian effects has been exhibited gadually due to the fact that the degee of accuacy of astonoical obsevation ipoves unceasingly. Hence seveal authos devoted thei eseach to this subject and the scopes (Bubeg, (197, 1985) [1] [], Rubinca (1977) [], Soffel (1987, 1989) [4] [5], Ioio (5) [6]). These authos eseach ainly the post- Newtonian effect of the obital eleents of planets and atificial satellites in the sola syste. The lagest post-newtonian effects have been exhibited in the obits of binay systes, especially in the copact binay systes. So, the eseach in the post-newtonian obital effect of the copact binay stas is ipotant and DOI: 1.46/wj Dec. 9, 17 6 Wold Jounal of Mechanics

2 eaningful. Hence soe authos eseach this subject fo theoy and obsevation. Such as, Will (1981, 6) [7] [8], Daou & Deuelle (1985, 1986) [9] [1], Schāfe & Wex (199) [11], Wex (1995) [1]. Calua (1997) [1], Ioio (7) [14] studied this subject fo theoy. On the othe hand, Bugay et al. () [15], Konacki () [16], Kae et al. (5) [17], and Weisbeg & Taylo (5) [18] eseached the post-newtonian effect fo the peiaston shift of copact binay stas (PSRJ77-9, PSR PSR154+1) fo obsevation. These eseaches ae vey inteested in the theoetical and obseving aspects. This pape pesents the post-newtonian effect on all obital eleents of the copact binay stas on the theoetical aspect. The eseach ethod of this pape is diffeent fo pevious studies that this pape uses the ethod of petubation theoy in the celestial echanics.. R, S and W Coponents fo the Geneal Relativistic Acceleations in the Two-Body Poble The elative acceleation of two-body with the Post-Newtonian Paaetes is given by Will (1981) [7] a PN X = ( γ β) γv ( α ζ ) ( 6 α α α ) v ( 1 α ) ( v n) ( X v) 1 1 µ µ µ v µ + ( γ ) ( α1 α) hee = 1+, µ =, n = X, X = n =, ṫ =, v = = n+ f λ, v = + f X v n, ( v n) = v = = = ( ) X v v = ( n v) v = ( ) = + f = + ( )( n λ ) ( n f λ ). hee f denotes the tue anoaly. n is a unit vecto in the adial diection and λ ae unit vectos in the obital plane. n is diected along the adial diection, and λ is pependicula to n. In the equation denotes G and the ight side should be ultiplied by c. G is gavitational constant and c is the speed of light. In this pape we eseach the geneal elativistic effect. In the geneal elativistic case the Post-Newtonian paaetes α1 = α = α =, β = 1, γ = 1, ζ =. (Will 1981) [7]. The Equation (1) can be witten (1) () DOI: 1.46/wj Wold Jounal of Mechanics

3 a X µ µ µ v v ( X v) v µ + 4. ( v n) PN = () Using the elative expessions the Equation () ay be witten a ppn µ µ µ = ( n) ( + f ) + ( n f λ ) µ (4) hee bolace denotes vecto. We esolve the acceleation a into a adial coponent Rn, a coponent Sλ, noal to Rn and a coponent W noal to the obital plane. i.e, a = Rn+ S + W( n ) λ λ, n λ = N (the unit vecto noal to the obital plane). On copaison with the expession (4), we get thee scala acceleative coponents R, S and W µ µ µ R= 4 1+ ( + f ) + µ 4 + µ S = 4 f (5) W = Substituting the following foulas of the poble of two body into the above foula (Sat, 195) [19] We obtain naesin f dt 1 e f = = na 1 e, =, n a =. (6) µ 4+ 7 µ e sin f µ p R= + 1+, p µ = 4 sin, S e f W =. hee p a( 1 e ) =. An independent vaiable dt is tansfoed to an independent vaiable in the Gaussian petubation equations (Bouwe & Cleence, 1961 [], Roy, (7) DOI: 1.46/wj Wold Jounal of Mechanics

4 1988) [1] by using the second foula of the Equations (6), we get the petubation equations with an independent vaiable tue anoaly f da a p = R esin f S, + ( 1 ) e de 1 = R sin f S ( cos E cos f ), na + + d ω 1 = R cos f + S 1+ sin f, nae p dε R e d ω = +, 4 na 1 e 1+ 1 e ( ω + f ) di cos = na 1 e ( ) 4 ( ω f ) Ω sin = 4 + d na 1 e W, W, (8) whee ω is the longitude of peiaston, E is the eccentic anoaly and ε is the ean longitude at epoch.. Integation fo the Petubation Equations and Its Petubation Solutions Substituting R, S and W fo expessions (7) into the petubation Equation (8) na = + =, we obtain by using Keple s thid law ( 1 e ) 1 da µ 1µ µ µ = 7 e e sin f 5 4 e sin f e sin f de µ 47 µ µ µ = e f e f e f p sin 5 4 sin sin, d ω µ 1 µ e µ µ = cos 5 4 cos cos, ep e f e f e f di dω = =, (9) ε µ µ µ 6 4 ecos f f a 1 e p d 9 7 e 1 1 = + d e e d ω. Integating the above Equation (9), we obtain the petubation secula and peiodic solutions DOI: 1.46/wj Wold Jounal of Mechanics

5 µ 1 µ δ a= e e f f + 8 ( 1 e ) ( 1 e ) 7 cos cos 1 µ 1 µ e ( cos f cos f) e ( cos f cos f), 8 µ 47 µ δ e= + e f f a 8 7 cos cos 1 µ e ( cos f ) 1 µ cos f e cos f cos f, 8 µ 1 µ δω = e f f e sin f sin f a e e 8 ( 1 ) 1 µ 1 µ 5 4 e( sin f sin f) + e ( sin f sin f), 8 µ µ δε = a 1 e δλ ( f f ) e ( E E ) µ e 8 e( sin f sin f ) + δω, 1+ 1 e δε, ( t) = nt + = δω=, = 1+ δi (1) whee λ denotes the ean longitude of peiaston, E denotes the eccentic anoaly. In the last integal expession, we have used the next integal aleady: f= a e E d 1 d e sin f e 1 f d = f d + ecos ff d. p p 4. The Secula and Peiodic Vaiation of the Obital Eleents 1) The secula vaiation pe cycle (evolution) By letting f =, f = π, E =, E = π, the peiodic tes ae disappeaed and one obtains the secula vaiables pe cycle (evolution): a = ( c cycle ) e = (/cycle ) ω = 6π ( ad cycle ) a( 1 e ) π µ µ 1 ε = ( 1 e ) a 1 e 6π e ( ad cycle ) a( 1 e ) 1+ 1 e λ = ( π + ε) ( ad cycle ) i = Ω = ( ad cycle ) (11) DOI: 1.46/wj Wold Jounal of Mechanics

6 whee µ = G( M + ) The autho Li (1) [4] obtained the foulas fo the post-newtonian effect on the tie vaiation of peiaston passage of binay stas. In that pape we change the sybol of the fist expession of (5) (in the case of elativity) as the sybol of the pesent pape, which is µ 1 µ µ τ = πa e 51 4 e s Rev ) The secula vaiable ate:. (1) a = e = i =Ω= ( ad y) ε = ε P ( ad y) λ = + ω = ω P [ π P ε P] ( ad y) τ = τ P ( sd) (1) whee the peiod P is denoted in unit of day ) The peiodic vaiation of aplitudes: In the expessions (1) all tes ae the peiodic vaiable tes except fo all secula tes. Hee we list the axial and inial aplitudes of the peiodic tes fo sei-ajo axis, a and eccenticity, e fo the expessions (1). Fo the sei-ajo axis: µ 1 µ A e e 8 A ax = 7 + ; ( 1 e ) in = +. ( e ) 41 µ e (14) Fo eccenticity: E E µ 47 µ = + 7 e ; a( 1 e ) 8 ax in = + ( e ) 8a 1 µ e. (15) 5. Nueical Results fo Fou Copact Binay Systes We use the foulae (11) - (1) to calculate the secula of the geneal elativistic secula effect on the obital eleents of fou copact binay systes. It is convenient to educe the foulas (11) - (1) to pactical units 1, and a ae denoted by the unit in sola ass M1 ( ), M ( ) and sola adius A( R ), and P is denoted by the unit in day = 864 s, G = , (c. g. s), c = 1 1 c/s. The foulae (11) - (1) can be witten by taking the secula effect DOI: 1.46/wj Wold Jounal of Mechanics

7 ( M1+ M) ω = 6 K ad cycle A ( 1 e ) ( M ) K M1+ µ µ 1 ε = ( 1 e ) A 1 e 6K( M1+ M) e A 1 e 1+ 1 e ( π ε ) ( ad cycle ) ( ad cycle ) λ = + i = Ω = ( ad cycle ) 1 1 µ 1 µ τ = A ( M1+ M) e M + M M1+ M µ e ( s Rev ). M1+ M 6 hee K = πgm c R = (c, g, s)., This pape chooses fou copact binay systes: PSR191+16, PSR154+1, (16) PSRJ77-9 and a black hole M X-7 as an exaple. Fo these copact binay stas, thei data fo P, a, e, M and ae etieved fo Bugay et al. () [15], Konacki et al. () [16], Kae et al. (5) [17], Willes et al. (4) [] and Oosz et al. (7) []. Thei data ae listed in Table 1. Substituting these data in Table 1 into foulas (16) and (1) and (14) - (15), we obtain the nueical esults fo the peiodic and secula vaiation of the obital eleents of fou copact binay stas in Tables -4. Table 1. The data of fou copact binay stas. Copact binay stas P(d) A(R ) e M 1 ( ) M ( ) Ref M x Oosz et al. (7) [] PSR J Willes et al. (4) [] Bugay et al. () [15] Kaa et al. (5) [17] PSR Willes et al. (4) [] PSR Konacki et al. () [16] Willes et al. (4) [] Table. Peiodic vaiation of the aplitudes of the obits of sei-ajo axis and eccenticity. Copact binay stas Sei-ajo axis Eccenticity 5 6 A ax ( k) A in ( k) E ax ( 1 ) E in ( 1 ) M X PSR J PSR PSR DOI: 1.46/wj Wold Jounal of Mechanics

8 It can be seen fo Table that the axiu aplitude of sei-ajo axis is the black hole binay syste M X-7 and the axiu eccenticity is PSR It can be seen fo Table that the axiu secula vaiable of longitude pe peiod is PSRJO77-9. The longest tie of the peiaston passage is M-7. It can be seen fo Table 4 that the axiu secula vaiable ates of longitude is PSRJ77-9. The longest tie of the peiaston passage is MX Discussion and Conclusions 1) The copaison of the theoetical esults with the obsevable esults. The theoetical esults in this pape as copaed with the obsevable esults given by seveal authos fo thee copact binay stas ae listed in the Table 5. It can be seen fo the above Table 5 the theoetical esults ae vey close to the obseved esults. Table. Secula vaiation of the obital eleents of fou copact binay stas pe cycle (Revolution). Copact binay stas a ( c Re) e/re ω ("/Re) ε ("/Re) τ ( s Re) M X-7 16".68 " PSR J PSR PSR [Note] The sybol denotes ac-second: Rev denotes Revolution (cycle). Table 4. Secula vaiable ates of the obital eleents of fou copact binay stas pe yea. Copact binay stas ȧ ("/y) ė ("/y) ω ε ("/y) (deg/y) ("/y) (deg/y) τ ( s y) M X PSR J PSR PSR Table 5. Copaison of the theoetical esults with the obsevable esults. Copact binay stas The theoetical values ω (deg/y) The obsevable values ω (deg/y) Authos fo poviding Data PSRJ Kae et al. (5) [17] Bugay et al. () [15] PSR Weisbeg & Taylo (5) [18] PSR Kohacki et al. () [16] DOI: 1.46/wj Wold Jounal of Mechanics

9 ) The possibility of obseving effects. In the sola syste the advance of peihelion of Mecuy ay be obseved by the ecent instuent. We can see fo Table 4 that axial value of the advance of PSRJ77-9 is 6116" pe yea which coespond to 145" tie value of advance of peihelion of Mecuy in sola syste. The value of the advance of peiaston of copact binay sta is lagest than that of Mecuy in sola syste. Theefoe the effects of the advance of copact binay stas can be obseved too. We have fou conclusions: a) The copact binay stas ae the best objects fo studying the post-newtonian effects on the obits b) Although thee ae no secula vaiation fo the sei-ajo axis and eccenticity, thee is axial aplitude of the peiodic tes fo sei-ajo axis, such as A ax = k. c) The longitudes of peiaston and the ean longitude at epoch exist both secula and peiodic vaiable tes, and the axial values fo ω and ε aive at 16.95/y and 9.68/y fo PSRJ77-9 espectively. d) The longest tie of peiaston passage is inute pe yea fo black hole M X-7. This coesponds to ove seconds pe a day. Refeences [1] Bubeg, V.A. (197) Relativistic Celestial Mechanics. Nauk, Moscow. (In Russian) [] Bubeg, V.A. (1985) Essential Relativistic Celestial Mechanics. Ada Hilge, Bistol. [] Rubinca, D.P. (1977) Geneal Relativity and Satellite Obit: The Motion of a Test Paticle in the Schwazchild Metic. Celestial Mechanics and Dynaical Astonoy, 15, [4] Soffel, M.H., Rude, H. and Schneide, M. (1987) The Two-Body Poble in the (Tuncated) PPN Theoy. Celestial Mechanics and Dynaical Astonoy, 4, [5] Soffel, M.H. (1989) Relativity in Celestial Mechanics, Astonoy and Geodesy. Spinge, Heidelbeg. [6] Ioio, L. (5) On the Possibility of Measuing the Sola Oblateness and Soe Relativistic Effects fo Planetay Ranging. A & A, 4, [7] Will, C.M. (1981) Theoy of Expeient in Gavitational Physics. Cabidge Univesity Pess, London, New Yok, New Rochelle, Melboune, Sydney. [8] Will. C.M. (6) The Confontation between Geneal Relativity and Expeient. Living Reviews in Relativity, 9, [9] Daou. T. and Deuelle, N. (1985) Geneal Relativistic Celestial Mechanics 1. The Post-Newtonian Motion. Annales de l Institut Heni Poincaé, 4, [1] Daou, T. and Deuelle, N. (1986) Geneal Relativistic Celestial Mechanics 11. The Post-Newtonian Tiing Foulas. Annales de l Institut Heni Poincaé, 44, 6-9. [11] Schāefe, G. and Wex, N. (199) Second Post-Newtonian Motion of Copact Bina- DOI: 1.46/wj Wold Jounal of Mechanics

10 ies. Physics Lettes A, 174, [1] Wex, N. (1995) The Second Post-Newtonian Motion of Copact Binay-Sta Systes with Spin. Classical and Quantu Gavity, 1, [1] Calua, M., et al. (1997) Post-Newtonian Lagangian Planetay Equation. Physical Review D, 56, [14] Ioio, L. (7) The Post-Newtonian Mean Anoaly Advance as Futhe Post- Kepleian Paaete in Pulsa Binay Systes. Astophys & Space Scince, 1, 1-5. [15] Bugay, M., et al. () An Inceased Estiate of the Mege Rate of Double Neuton Stas fo Obsevation of a Highly Relativistic Syste. Natue, 46, [16] Konacki, M., Wolszczan, A. and Stais, I.H. () Geodetic Pecession and Tiing of the Relativistic Binay Pulsas B154+1 and PSR The Astophysical Jounal, 589, [17] Kae, M., et al. (5) The Double Pulsas A New Test Relativistic Gavity. In: Rasio, F.A. and Stais, I.H., Eds., ASP Confeence Seies, 8, [18] Weisbeg, J.M. and Taylo, J.H. (5) The Relativistic Binay Pulsas B Thity Yeas of Obsevations and Analysis. In: Rasio, F.A. and Stais, I.H., Eds., ASP Confeence Seies, 8, 5-1. [19] Sat, W.M. (195) Celestial Mechanics. London, New Yok, Toonto, 19. [] Bouwe, D. and Cleence, G.M. (1961) Methods of Celestial Mechanics. Acadeic Pess, New Yok, London. [1] Roy, A.E. (1988) Obital Motion, Ada Hilge, Bistol and Philadelphia, 19. [] Willes, B., Kalogea, V. and Henninge, M. (4) Plusa Licks and Spin Tilts in the Close Double Newton Stas PSR J77-9. PSR B and PSR The Astophysical Jounal, 616, [] Oosz, J.A., et al. (7) A Sola Mass Black Hole in an Eclipsing Binay in the Nealy Spial Galaxy M. Natue, 449, [4] Li, L.-S. (1) Post-Newtonian Effect on the Vaiation of Tie of Peiaston Passage of Binay Stas in Thee Gavitational Theoies. Astophysics and Space Science, 7, DOI: 1.46/wj Wold Jounal of Mechanics