arXiv:1401.6530v1 [astro-ph.HE] 25 Jan 2014 stesi eido h pulsar). the of period spin the is trs otecne fpla.I at hstm elapse time this fact, In pulsar. as of measured frame center be reference the can the to in rest measured at time homogeneous the etasomdt enaoay ¯ anomaly, mean to transformed be pnpoetn oLS eoe ydirection by denoted LOS, to projecting upon e oetatteaglrmmnu ftepla fPRJ2051-0827 PSR of pulsar t the in of laboratory momentum better angular pulsars the th makes in left extract only account, shift not to into Doppler effect state net Doppler the velocity cannot Consequently, second motion each broken. binary an at is in red-shift phase signal time orbital net the propagation the to that the shift velocity granted Doppler taking for of s orbital taken effect varying periodic pa the been with that has the this it system with In And binary changes orbit. a actually systems. In frequency pulsar orbital shift. signal of different Doppler change residual totally term mechanism, at long the occurring and phenomena 4U1627-67 binary, X-ray mass httetu anomaly, true the that where 676,7 1627-67, srn and astron, usrt h bevrcagswith changes observer the to pulsar time, e e flight, of hsc faceinds fXrybinaries. X-ray of disc accretion of physics in h eoiyo h inlpoett h ieo sight of line the to project signal the (LOS), of velocity the tion, en- waves previous during time observer. source the wave the by the since countering move because fields to occurs time countless It has to 1842. applied in discovered been its its to has relative which moving observer source, an for event periodic other h ie sarsl,tepleo rgnlfrequency, original of pulse the result, a As time. the sditdwt epc oa bevrby, observer an to respect with drifted is 2 cos sh cetiiyo h orbit, the of eccentricity he is rpittpstuigL using 2018 typeset 25, Preprint August version Draft h uedrvtv fobtlpro bevdi iaypla S B PSR pulsar binary in observed period orbital of derivative huge The eas fsc ria oin h itneo the of distance the motion, orbital a such of Because nabnr ytmwt eidcsga nobtlmo- orbital in signal periodic with system binary a In or wave a of frequency in change the is Doppler-shift ) VDNEO EIULDPLRSITO HE USR,PRB12 PSR PULSARS, THREE ON SHIFT DOPPLER RESIDUAL OF EVIDENCE 1 / 2 f t i which (in ] ,wihcrepnst iedlyof, delay time a to corresponds which ), n . ∆ v p ujc headings: Subject ν hne ihteobtlpaeo h inlall signal the of phase orbital the with changes , K steui etro ieo sight, of line of vector unit the is = × f ssemi-amplitude, is v 10 stetu nml,wihi ucinof function a is which anomaly, true the is · c 50 n a z c p gcm p 1. ν = stesm-ao xso h pulsar), the of axis semi-major the is e 1 t A colo hsc,Hahn nvriyo cec n Techn and Science of University Huazhong Physics, of School INTRODUCTION = T 2 = r E f /s tl mltajv 5/2/11 v. emulateapj style X sin Kν perdi q1adE. can Eq.2 and Eq.1 in appeared , c nP epciey hc r fiprac ntesuyo tutr o structure of study the in importance of are which respectively, , tr:nurnXry:bnre—-as niiul(S B1259 (PSR individual binaries—X-rays: neutron—X-rays: stars: S 25-87 S 832,PRJ6113,HrX1 e X- Cen X-1, 1907+09) Her 4U J0631+1036, X-1, PSR Vela 1657-415, 0823+26, OAO PSR J2051-0827, PSR i e [cos( sin( (where K ω ω ω n steavneo peri- of advance the is + 2 = + 2 = n r f f (2) ) = 1 = πa + ) πt/P a p p , e sin c (1 2 cos , b stespeed the is where , 3 iPn Gong Bi-Ping − i/ rf eso uut2,2018 25, August version Draft ... z ω e [ Notice . cP and , 2 (1) ] ) / b (1 1+ (1 ABSTRACT t J2051-0827 ν P − e is , 1 ..Yan Z.Q. , unyo iaypla ytm? system pulsar binary a of quency nabnr ytm h retm esrdb nob- an by measured time true the is system, server binary a In negligible. is out of out which counting mass motion), (after center proper observer the the the to between large system corre- binary separation a delay, the the such time to constant As a sponding produces ago. nowadays discrepancy years measured time pulsar thousand Earth, binary the happens a to actually of kpc of phase distance orbital of the usually Eq.2. are of pulsars Since delay granted. is for Roemer taken Eq.1 of been of effect has effect conclusion the Doppler This the to orbit, equivalent one thus in out cancelled R ity ahrta h time the than rather h iaysystem. binary the h ria hs esrdb nosre is words, observer other an In by measured signal. pulse phase the orbital to the shift Doppler tional nta of instead h w uss eta oiin1ad2 a ec the reach can 2, and 1 position at instantaneously. sent observer pulses, two the ftepla nobtlmto.I n ae h time, the takes one If motion. position the orbital (where determine in which pulsar pulsar, the the of of center the to 0 P hti h e ope hf,eg,t h us fre- pulse the given, to a e.g., shift, For Doppler net the is What hra,tenncntn iedlyi nte story. another is delay time non-constant the Whereas, reconsidered. be to needs equivalence an such However usiuigthis Substituting .. ehv w oa times, local two have we E.g., nsc ae h w inl,wihsn ihatime a with sent which signals, two the case, such In b V lesito nfl ri iceb acle u,so out, cancelled be circle orbit full an of blue-shift d ∆ nobti o-eo nesadn hsNewtonian this Understanding non-zero. is orbit an ( ν eido S 25-87 emttlyunrelated totally seem J2051-0827, PSR of period t rjce oln fsgta ieetpae of phases different at sight of line to projected , ets fgaiainleet,btas losus allows also but effects, gravitational of test he eacmltdoe h ria eid However, period. orbital the over accumulated be ymtyo h eoiydsrbto vrthe over distribution velocity the of symmetry e n ec ∆ hence and ) ( k , t ) ≤ e,te r ipyitrrtdb h same the by interpreted simply are they per, 1 = dt 1 gassnigt nosre,tedito the of drift the observer, an to sending ignals f 296,tetru eesldslyn nlow on displaying reversal torque the 1259-63, 2 ( lg,Whn407,China 430074, Wuhan ology, ,s httebu n e-hf a be can red-shift and blue the that so 0, = t , ). × ) sosre’ ie hni mle that implies it then time, observer’s as 2), 10 t ehv ria phase, orbital have we , 43 t gcm obs ν t t obs eciigteobtlpaeof phase orbital the describing , ( noE. ie h reobserva- true the gives Eq.1 into t 2 .Teitgaint ∆ to integration The ). 96,412-7ADPSR AND 4U1627-67 59-63, n h crto ico 4U of disc accretion the and ; = t + t eto tr n the and stars neutron f 6,4U1627-67, -63, z c 1 and ,G 1+4, GX 3, t 2 enda rest at defined , f ( f t ( ,veloc- ), t + ν ( t z/c is ) (3) t k ) 2 discrepancy say, t2 −t1 =1.0s, are thought to be detected As analysed above, the time t is actually the local time as ν(t1) and ν(t1 +1.0) respectively. at rest to the center mass of the pulsar. At this very Whereas, the true situation is that the second signal time, the orbital phase of the pulsar is f(t), whereas, the is actually detected as, ν(t1 +1.0+ δz/c), where δz/c = pulse signal sent at this phase is measured as ∆ν(t+z/c) z2/c − z1/c. instead of ∆ν(t). It is this additional time that changes the distribution The discrepancy between the two shift of pulse fre- of the projected velocity with respect to the observer. quency, ∆ν(t + z/c) − ∆ν(t), is equivalent to O-C E.g., for a pulsar in circular motion around its companion (observation-calculation). Expand ∆ν(t + z/c) in Taylor star, it will have 1/4 orbital period of blue shift (close series, ∆ν(t + z/c) = ∆ν(t) +ν ˙(t) · (z/c)+ ... Then the to), and 1/4 orbital period of red shift (away) at the integration of the second term at right hand side gives, two sides of the point of closest distance to the observer, Pb z Pb z Pb z˙ without considering the propagation time. But when the ν˙(t) dt = ∆ν(t) dt − ∆ν(t) dt (6) additional time is included, the turning point of the blue Z0 c Z0 c Z0 c and red shift will change its position. And thus, the distribution of the projected velocity is no longer uniform Apparently, the first term at right hand side of Eq.6 with respect to the observer. equals zero, considering the trigonometric function of Consequently, net Doppler shift to the frequency of Eq.1 and Eq.2, and the second term is equivalent to Eq.4, − pulse is given(Gong 2005), which is actually ∆ν(t + z/c) ∆ν(t). The variation of such a discrepancy versus time is as shown in panel b of 1 Pb z xK e2 Fig. 2. ∆ν = ∆ν(t)d(t + )= νπ(1 − ) (4) The accumulated change of pulse frequency corre- Z Pb 0 c Pb 4 sponds to a timing residual as shown in the bottom panel Notice that it is this additional time delay dz/c that of Fig. 3, which is obtained by, makes the shift of pulse frequency of Eq.4 non-zero. This ∆ν′ result means that in every orbital period the signal fre- Res(tk)= Res(tk−1)+ (tk − tk−1) (7) quency of one object in the binary system is shifted by ν ′ a value described by Eq.4, which is a Newtonian second where ∆ν = ∆ν(tk−1 + zk−1/c) − ∆ν(tk−1). Doppler effect. The curve with steps in the bottom panel of Fig. 2 Such a residual Doppler shift affects three typical bi- stems from accumulation of the Doppler residual at each nary pulsars, PSR B1259-63, 4U1627-67 and PSR J2051- orbit. The magnitude of each step actually corresponds 0827 differently. In the following sections, they will be to frequency shift given by Eq. 5. analysed one by one. Cut off such a jump directly at the passage of preces- sion of periastron, we have timing residual versus time 2. PSR B1259-63 (orbital phase), as shown in panel b of Fig. 3. PSR B1259-63 was discovered in a large-scale high- As shown in Fig. 3b, the dashed horizontal line corre- frequency survey of the Galactic plane (Johnston sponds to the level of 10 ms timing residual, which crosses 1992). It is a radio pulsar in orbit about a mas- each peak with a time scale of around 40 days. This well sive, main-sequence B2e star. With an orbital period consistent with the facts that the emission is absent for of 1237 d and companion mass of > 10M⊙, the 23yr nearly 40 days during the passage of periastron. Panel data(Shannon et al. 2013) indicates a significant orbital c displays the timing residual upon cut the part absent period decay, 1.4(7) × 10−8, while the timing resid- during the passage of periastron. ual versus orbital phase has similar shape as ten years Moreover, panel b of Fig. 3b suggests that the closer ago(Wang et al. 2004). the data to periastron, the larger the magnitude of the The huge derivative of orbital period corresponds to a peak detected. Therefore, the reported glitch at MJD 50690.7(Wang et al. 2004) should be the epoch, which time residual in one orbit, δTo = P˙bPb ≈ 1(s). Can residual Doppler shift reproduce such a signifi- detects pulses with shortest time interval with respect to cant timing residual ? This can be tested easily by the passage of periastron. −1 Comparing the observational(Wang et al. 2004; putting x = 1296.3s, Pb = 1236.7d, ν = 20.9s and e = 0.87(Wang et al. 2004; Shannon et al. 2013), into Shannon et al. 2013) and simulated timing residual, the residual Doppler shift in one orbit(Gong 2005), as shown in panel a and c in Fig. 3 respectively, there are still deviation between panel a and panel 2 c, because panel a reduces the steps by assuming xK e − − ∆ν = νπ(1 − )=1.0 × 10 7(s 1) (5) glitches(Wang et al. 2004), which is not a constant in Pb 4 timing residual; while panel c is obtained by subtracting Therefore, the predicted additional time residual in every a constant timing residual. orbital period is δT ≈ ∆νPb/ν ≈ 0.6s, which explains Consequently, the strange timing behaviour of the residual, δTo ≈ 1(s), corresponding to the significant this pulsar can be interpreted with available binary derivative of orbital period(Shannon et al. 2013), with parameters(Shannon et al. 2013), and without intro- a flying color. ducing any additional parameters. Further, the timing residual versus orbital phase can Notice that the amplitude of timing residual originat- 3 also be tested. Input t into Eq.1, obtains phase, velocity, ing in Shapiro delay in a orbit is given, r = GMc/c ∼ and hence frequency shift of pulsar at moment t, from 10−5s, which is much less than that of the residual which the variation of ∆ν(t) versus time is obtained as Doppler effect of 1s in the case of PSR B1259-63. shown in panel a of Fig. 2. If such a residual Doppler shift is evident in PSR 3

B1259-63, which has a orbital period of years, what about As shown by Eq. 8, the magnitude of δν is a binary of orbital period as short as tens of minutes ? determined by Pb, sin i, and ap, in which Pb is given by observation directly, while sin i and ap ≡ 3. 4U 1627-67 aMc sin i/(Mc) (where M = Mp + Mc) are also The accreting-powered pulsar 4U 1626-67 with constrained by observations(Middleditch et al. 1981; a pulse period of 7.66 s, was discovered by Levine et al. 1988). Uhuru(Giacconi et al. 1972). Although orbital motion We can make Mc and i as free parameters, and use Eq. has never been detected in the X-ray data, pulsed op- 8 to fit both the amplitude and time scale of the observed tical emission reprocessed on the surface of the sec- change of pulse frequency(Camero-Arranz et al. 2010). ondary revealed, and thus confirmed the 42 minutes or- Obviously, the variation of δν is due the change of i, bital period(Middleditch et al. 1981). This pulsar sys- which is in turn originated in the S-L coupling as shown tem is recognized as a low mass X-ray binary (LMXB), in the bottom left of Fig. 1. As the time scale of variation with an extremely low mass companion of 0.04M⊙ for i of i is determined by Eq. 9, this set constraint not only = 18deg(Levine et al. 1988). on the companion mass, but also on the spin angular After the steady spin-up observed during 1977 to momentum around the pulsar, S. 1989, the torque reversal occurred during 1990 June, with Pb = 42min and Mp = 1.4M⊙, we find that the this pulsar began steadily spinning down(Chakrabarty best orbital inclination (average), companion mass and 1998). Interestingly, after about 18 yr of steadily spin- spin angular momentum are i = 0.3rad, Mc = 0.16M⊙, ning down, the accretion-powered pulsar 4U 1626-67 and S =6.9 × 1050gcm2/s, respectively as shown in Ta- experienced a new torque reversal at the beginning of ble 1. 2008(Camero-Arranz et al. 2010). Such a spin angular momentum is much larger than The evolution of pulse frequency with two abrupt that of NS and WD, which suggests that it stems from “torque reversal”, can be understood in the context of accretion disc around the pulsar rather than the pulsar the residual Doppler effect at long-term. itself. The existence of such a disc is supported by the The residual Doppler shift of Eq.4 predicts a change of X-ray emission lines(Camero-Arranz et al. 2012). pulse frequency in an orbital period (with e ≈ 0), With the fitting parameters of Table 1, the resultant integration of Eq.8 by time is shown at the top panel 2 2 2π apν 2 of Fig. 4, which exhibits a constant increase of pulse ∆ν = sin i (8) −10 P 2c2 frequency ofν ˙ =8.9 × 10 Hz/s, due to Eq.8 is always b positive. The observations(Camero-Arranz et al. 2010) corre- The constant increase of pulse frequency have been sponds to the change of δν in dozens of years, containing cancelled by the spin-down of pulsar spin. However, the numerous orbital period, Pb = 42min. Such a long-term cancellation is not perfect, the positive (residual Doppler variation can be obtained by the integration of sin2 i of shift) overwhelms the negative (spin-down) a little, so Eq.8 by time. that the frequency variation vs time has a overall trend The spin-orbit coupling of binary system is likely re- of increase, as shown in panel b of Fig. 4. sponsible for the variation of i, as shown in the bottom Such an increase of δν predicts that next peak of δν left panel of Fig. 1 . vs time, immediately after that of 2000 (MJD 54500), Under such a coupling effect, both the spin and or- should occur around MJD 62140, and the amplitude of bital angular momentum precess around the total an- which must be higher than that of MJD 54500. This gular momentum. The precession of the spin angular prediction can be tested soon. momentum, which is so called geodesic precession, re- On the other hand, the panel d of Fig. 4 shows the sults in the change of pulse profile which have been spin-orbit coupling induced variation of i, which reaches observed in a number of pulsar binaries, e.g., PSR its maximum and minimum around MJD 44600 and MJD 1913+16(Weisberg & Taylor 2002; Konacki et al. 2003). 51500 respectively. Accordingly, x ≡ ap sin i/c, varies as And the variation of the orientation of the orbital shown in panel c of Fig. 4, and the discrepancy of which plane, represented by the angular momentum of the or- can be up to 10 times in magnitude. This explains dif- bit, L, leads to evolution of the orbital inclination angle, ferent x measured by different authors at different times, i, denoting the misalignment angle between L and LOS, x =0.36 ± 0.10s(Middleditch et al. 1981; Levine et al. as shown in the bottom left of Fig.1. 1988) and x =8ms − 3ms(Chakrabarty et al. 1997). The time scale of such a spin-orbit coupling depends The maximum value of x should appear again around on the orbital period. E.g., for PSRJ0737-3039 of or- MJD 58450, as shown in panel c of Fig. 4, which is 5 bital period of 2.4 hours, the long-term spin-orbit cou- years after Jan 1, 2014 (of MJD 56658), this prediction pling effect is about 70 years, and for PSR 1913+16 will test the model from another aspect. with orbital period of 7.7 hours, the time scale is 300 The spin-orbit coupling process is actually the preces- years. The precession of orbital plane of 4U1627-67 is sion of both vectors, L and S around the total vector J, calculated(Barker & O’Connell 1975), with L and S are at opposite side of J instantaneously, as shown in the bottom left panel of Fig. 1. Therefore, their GS(4+3M /M ) Ω= c p Sˆ (9) misalignment angle with LOS varies differently with the 2c2a3(1 − e2)3/2 precession, one misalignment angle at maximum corre- sponds to the minimum of the other. where S and Sˆ are the magnitude and the unit vector of E.g., the misalignment angle between the spin angu- the spin angular momentum around the pulsar respec- lar momentum, S and LOS reached the minimum at tively. around MJD 44630, as shown in panel e of Fig. 4. And 4 since the spin angular momentum vector usually aligns We first analyses the magnitude of variation of the two with the outflow of a X-ray binary system, the mini- values, x and Pb. mum misalignment angle corresponds to the strongest From the observation(Lazaridis et al. 2011), the vari- effect of Doppler boosting, which automatically explains ation amplitude of ∆x in about 2000 days is of 10−4. the flares of 4U1627-67 in early 1980s (of MJD around This corresponds to a variation of sin i of 10−4 in the 44630). same time interval, since the change of i is the most prob- Later, the increase of such a misalignment angle weak- able origin of the variation of x, as shown in Fig.1. ens the Doppler boosting, and hence prevents the flares The usual recognized the companion mass of from being observed afterword. 0.05M⊙(Lazaridis et al. 2011) is actually difficult to sat- According to panel e of Fig. 4, in around MJD 58460, isfy x =0.045lt-s and i = 40◦ simultaneously. It is found the misalignment angle will return to the level like the that a companion mass of Mc =1.0M⊙ not only satisfies early 1980s again. In other words, this predicts that the the observational constraint, x =0.045lt-s, in the case of ◦ flaring stage like the early 1980s will happen in 2019. i = 33 , but also explains the variation of both x and Pb. By the prediction of the bottom panel of Fig. 4, the By the observed binary parameters(Lazaridis et al. misalignment angle starts increasing at MJD 44630, till 2011), Pb = 0.1day, and assuming Mp = 1.4M⊙ and the turning point around MJD 51570. This predicts a Mc = 1.0M⊙, we have semi-major axis of the binary, flux decrease during MJD 44630-51570. And the flux a = 5.9 × 109cm, through Kepler’s third law. And will start increasing after MJD 51570, as indicted by the putting these parameters into Eq.8 obtains δν = 4.5 × bottom panel of Fig. 4. This is well consistent with the 10−7 sin2 i(Hz), which corresponds to an additional time X-ray light-curve observed(Camero-Arranz et al. 2010), delay of, which shows that the enhancement of flux density of −5 2 4U1627-67 occurs surely before MJD 54000. δT = δνPb/ν =1.8 × 10 sin i , (s) Moreover, as shown in panel b of Fig. 4, the in each orbital period. Dividing such a δT by P , we turning point of δν is between MJD 54000-56000, b have the first derivative of the orbital period, P˙b =0.2 × this again is consistent with the observed evolution of −8 2 δν(Camero-Arranz et al. 2010), in which the turning 10 sin i. point is surely after MJD 54000. Expanding it in Taylor series, Therefore, the new model not only explains the ampli- P˙ =0.2 × 10−8[sin2 i + ∆i sin 2i + O(∆i2)] tude and time scale of δν, x, and flares of 4U1627-67, b 0 0 but also their turning points and correlation. All of The first term at the right hand side of above equation is −10 ◦ these are interpreted by a simple scenario of the binary a constant, predicting P˙b ≈ 6×10 in the case i0 = 33 . system, the effect of residual Doppler shift. It is more And the second term actually predicts a variable P˙b of difficult to understand if all these happen by chance. The ∼ 10−13 (with ∆i = 10−4). future correlation of these three parameters, are clearly ˙ predicted in Fig. 4, some of which can be further tested If the constant part of Pb (first) is completely absorbed by spin parameters like ν andν ˙, then one would observe a soon. ˙ Further more, the spin angular momentum of the ac- varying Pb with equal amplitude of variation, as given by cretion disc around the pulsar inferred by the residual the second term in the equation above. However, down Doppler shift is of importance in the understanding of trend in the top panel of Fig.5(Lazaridis et al. 2011) the timing and emission of X-ray binaries. suggests that the constant part of P˙b is not completely −12 Moreover, two radio pulsars, PSR 0823+26 and removed. The net value of the two terms is P˙b ∼ 10 . PSR J0631+1036 exhibit an abrupt change of timing Consequently, at the time interval of T=1000 days (1× 4 residual(Baykal et al. 1999; Yuan et al. 2010). And 10 orbits), the change of orbital period is δPb = P˙bT ∼ other X-ray pulsars, Her X-1, Cen X-3, GX 1+4, 1 × 10−8, which consists with the observational change OAO 1657-415, Vela X-1(Bildsten et al. 1997), and 4U of Pb(Lazaridis et al. 2011). 1907+09(Inam et al. 2009), exhibit torque reversal as In fact, the variation of sin i at the level 4U 1627-67, which suggest that they may stem from the 10−4(Lazaridis et al. 2011) implies a ratio between same mechanism as 4U 1627-67 does. spin and orbital angular momentum of S/L ∼ 10−4 4. PSR J2051-0827 (assuming a right angle between S and L). The orbital angular momentum is L = µωa¯ 2(1 − PSR J2051-0827 is the second eclipsing millisecond pul- 2 1/2 49 2 −1 sar system. Long-term timing observations have shown e ) ≈ 4.5 × 10 gcm s (whereω ¯ = 2π/Pb). This predicts a minimum spin angular momentum of 4.5 × secular variations of the projected semi-major axis and 45 2 −1 the orbital period of the system. These two varia- 10 gcm s . The means the minimum moment of iner- tia of the pulsar is about 2 × 1043gcm2. tions has been interpreted separately. The variation −4 of the former has been interpreted as S-L coupling in Moreover, as shown in Fig.1, S/L ∼ 10 requires that the misalignment angle between the L and J is of order the binary system, whereas, and the change in the lat- −4 ter is explained by tidal dissipation leading to vari- 10 , and the precession scenario is that both L and S ation in the gravitational quadrupole moment of the precesses about J rapidly. companion(Lazaridis et al. 2011). Then precession velocity of the orbital plane The residual Doppler effect provides simple and uni- can be treated as(Barker&O’Connell 1975; fied mechanism that interprets both variations without Apostolatos et al. 1994; Wex et al. 1999), introducing any additional parameter to this binary sys- 3Gω¯(M + µ/3) tem. Ω= c =3.9 × 10−8s−1 2c2a(1 − e2) 5

TABLE 1 one. It is the large amplitude of variation of δν at each Different fitting parameters for 4U 1627-67. periastron passage that makes it different (which is a short-term effect, at the time scale of the orbital period). 50 2 MS Mc(M⊙) λ(deg) λLJ S(1 × 10 gcm ) And for 4U 1627-67, the residual Doppler shift in- a 1.9 0.126 18.0 34.4 9.2 duced timing residual makes the pulse frequency change b 1.9 0.251 14.4 18.0 9.3 steadily for 18yr (for ordinary binary pulsars the time scale is at least one order magnitude larger), which can be , which well consistent with that required by observation, hidden in the parameter, e.g.,ν ˙, before the sign changes 2π/(2.7 × 103 × 86400) = 3 × 10−8s−1. that reveal its identity. In summary, at the expanse of increase companion As a result, most of the binary pulsar system, with mass to 1.0M⊙ (without introducing any new parame- binary period not so large and so less, would not be so ter), both the amplitude and time scale of the variation lucky as the three pulsar systems that display so distin- of x and P are interpreted. guished timing properties so that the new effect can be b extracted. 5. DISCUSSION Thus, understanding residual Doppler shift sheds new Such a residual Doppler shift should exist in all binary light on pulsar timing. On the other hand, the counting pulsars, and the timing residual produced by it is much out this effect is of importance not only in the test of larger than that of Shapiro delay in each orbital period. effects of general relativity like Shapiro delay, Einstein And differing from the effect of Shapiro delay, residual delay, relativistic precession, but also in pulsar timing Doppler shift can accumulate over time, which results in array using pulsar timing to detect gravitational wave. puzzling timing behaviours at the time scale much longer In addition, obtaining the spin angular momentum of than the orbital period. the pulsar and the accretion disc around the compact Where does it go in ordinary binary pulsars ? The object are also of importance in the understanding the constant part of the effect can be absorbed by either nature of both pulsar and X-ray binaries. ν orν ˙ (or both). E.g., for PSR B1259-63, its residual Doppler shift given by Eq.5 divided by the orbital pe- We thank Manchester, R.N., Ni, W.T., Wang,N., Peng, riod yields,ν ˙ =1 × 10−15Hz/s, which is much less than Q.H., Zou, Y.C., Yuan, J.P., and Zhang, M. for help- the amplitude of its spin down,ν ˙ = −1 × 10−11Hz/s ful discussion. This research is supported by the Na- observed(Shannon et al. 2013). Therefore, its con- tional Natural Science Foundation of China, under grand tribute toν ˙ is not easy to figure out from the observed NSFC11373018, and Beyond the Horizons 2012.

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0 2000 4000 6000

(a)

0 /

-4

-2x10

(b)

-7 )

2x10 /

(

0

0.8

(c)

0.4

0.0 Timingresidula (s)

0 2000 4000 6000

MJD-47900

Fig. 2.— Simulation of the normal Doppler and resid- ual Doppler on PSR B1259-63. (a) Input t into Eq.1, ob- tains Doppler-shift to the pulse signal at different time. (b) Shows the discrepancy between Doppler-shift with and with- out propagation time, ∆ν(t + z/c) − ∆ν(t). (c) Timing resid- ual originated in accumulated Doppler shift, represented by the curve with steps.

Fig. 3.— Comparison between observed and simulated tim- ing residual versus time of PSR B1259-63. The top panel is the observed timing residual of(Wang et al. 2004). The middle panel displays the result after counting out the steps in accumulated Doppler shift, as shown in panel c of Fig.2, which is obtained by cut off two neighbouring steps at exactly the periastron. The dashed horizontal line corresponds to the level of 10 ms timing residual, which crosses each peak with a time scale of around 40 days. This well consistent with the facts that the emission is absent for nearly 40 days during the passage of periastron. 7

Fig. 4.— Fitting results of 4U 1627-67. (a) The resultant integration of Eq.8 by time. (b) The change of δν after the constant increase of δν is cancelled by the spin-down of pulsar spin ofν ˙ = −8.9 × 10−10Hz/s. The dots correspond to obser- vational data. (c) Predicted evolution of the projected semi- major axis of the pulsar, x. The dashed horizontal line at the bottom corresponds to the level of x = 10ms, the MJD corre- sponds to such levels can be found. And upper dashed hori- zontal line corresponds to the level of x = 50ms. (d) Predicted evolution of orbital inclination, i, the variation of which is de- termined by the effect of S-L coupling. (e) Predicted evolution of the misalignment angle between spin angular momentum of the accretion disc and the LOS. The dots correspond to observational flux change(Camero-Arranz et al. 2010).