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On the time delay in binary systems Angelo Tartaglia1,2, Matteo Luca Ruggiero1,2 and Alessandro Nagar1,3 1 Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 23, 10129 Torino, Italy 2 INFN, Sezione di Torino, Via Pietro Giuria 1 Torino, Italy 3Departament d’Astronomia i Astrof´ısica, Universitat de València, Edifici d’Investigació, Dr. Moliner 50, 46100 Burjassot (València), Spain (Dated: February 7, 2008) The aim of this paper is to study the time delay on electromagnetic signals propagating across a binary stellar system. We focus on the antisymmetric gravitomagnetic contribution due to the angularmomentumofoneofthestarsofthepair. Consideringapulsarasthesourceofthesignals, the effect would be manifest both in the arrival times of the pulses and in the frequency shift of their Fourier spectra. We derive the appropriate formulas and we discuss the influence of different configurationsontheobservabilityofgravitomagneticeffects. Wearguethattherecentlydiscovered PSRJ0737-3039 binarysystemdoesnotpermitthedetectionoftheeffectsbecauseofthelargesize 5 of theeclipsed region. 0 0 PACSnumbers: 4.20.-q95.30.Sf, 2 n a I. INTRODUCTION gravitomagneticeffectshasbeenextremelylimitedsofar. J Although a number of proposals for experimental tests 8 Gravitomagnetic effects are perhaps the most elusive of gravitomagnetism have been put forward during the 2 phenomenapredictedbyGeneralRelativity(GR).These past decades [2, 3], the only one presently under way is effects are originatedby the rotationof the sourceof the the Gravity Probe B (GPB) mission [4], which is cur- 2 v gravitational field, which gives rise to the presence of rently collecting scientific data to verify both the Lense 9 off-diagonal g0i terms in the metric tensor. The grav- Thirring and the Schiff [5] precessions of orbiting gyro- 5 itational coupling with the angular momentum of the scopes. Other existingexperimentaltests ofgravitomag- 0 source is indeed much weaker than the coupling with netism are: 1 mass alone (gravito-electricinteraction). Considering an 0 lunar laser ranging [6]; axisymmetric stationary configuration, we may compare • 5 the relevantterms of the metric tensor by looking at the laser ranging of terrestrial artificial satellites LA- 0 • / ratio GEOS and LAGEOS II [7] . c q g0φ Ratio(4)canbelessunfavorablewheneverrisapproach- ε= , (1) r- g00 ing the Schwarzschild radius of the source: this can be g the case of a source of electromagnetic (e.m.) signals : whereapolarnon-coordinatedbasisisassumedwithunit orbiting around a compact, collapsed object, as it may v formsω0 =cdtandωφ =rdφ. Almosteverywhereinthe i happen in a compact binary system where (at least) one X universe a weak field approximationis acceptable, hence of the stars is a pulsar. r Theaimofthispaperistodiscussthetimedelayinthe a RS g00 =1 , (2) propagationof e.m. signals in GR. It is well known that − r the curvatureof spacetime produces a delay inthe prop- aR g = S sin2θ , (3) agation time of light with respect to a flat environment 0φ r2 (Shapiro delay), a phenomenon that has been measured where R = 2GM/c2 is the Schwarzschild radius of the within the solar system [8, 9, 10]. S In the presence of a rotating source, a specific grav- source, M being its mass; we have defined a = J/(Mc) itomagnetic contribution to the delay is also expected, where J is the source angular momentum. In the equa- which would show up as an asymmetry in the time of torial plane (θ =π/2), Eq. (1) reads flight of the signals. In Ref. [11] it was proposed to look aR aR for this effect in the vicinity of the Sun. However, the S S ε= r(r R ) ≃ r2 . (4) magnitude of the effect is really tiny, 10−10 s from op- − S posite sides of the solar disk. Other p∼roposalspertain to EvaluationofEq.(4)atthe surfaceofthe Sun(the most the measurement of the frequency shift induced on e.m. favorableplaceinthesolarsystem)givesε 10−12,thus signals by the gravitomagnetic field of the Sun [12]. ∼ evidencing the weakness of the gravitomagnetic versus As we pointed out above, a more favorable situation the gravitoelectric interaction. couldbeexpectedinacompactbinarysystem. Presently, The smallness of ε is the reason why, though having just a few of them are known [13], but all are interesting been suggested from the very beginning of the relativis- laboratories for testing GR. Among the known systems, tic age [1], the explicit verification of the existence of the recently discovered PSR J0737-3039 [14] presents a 2 aR y favorableconfigurationandisparticularlyappealingalso S g = , (8) because both stars are pulsars. The data collection is 0x − r3 goingonandmaybesomeinterestingresultscanbefound aRSx g = . (9) alsowithrespecttogravitomagneticeffects: forexample, 0y r3 it has been recently argued that both the precession of iii) The trajectory of light rays is assumed to be a thespinningbodiesandthespineffectsontheorbitcould straight line. Actually, there is a bending, whose effects be measured in this system [15]. are usually assumed to be negligible [16], [23]. Inthis paperwederivethegravitationaltimedelayon iv) The center of mass of the system is at rest with e.m. pulses in a binary system, focusing on the gravito- respect to the observer on Earth. magneticcontribution,andproposehowitsconsequences v) The orbit of the source of the signals (i.e., object could be revealed. The corresponding frequency shift is ) around the center of mass of the system is circular also briefly discussed. O1 [24]. The organizationof this paper is as follows. In Sec. II vi) Thesizeofthebinarysystemismuchsmallerthan wedescribethegeometricalbackgroundandthehypothe- the distance from the observer on Earth. ses assumed to calculate the time delay. In Sec. III we vii) As a result ofthe proper motionof object , the review the properties of the binary system PSR J0737- O2 originofthereferenceframeismovingwithrespecttothe 3039, pointing out its relevance for experimental tests centerofmassofthesystemandthenwithrespecttothe of GR. In Sec. IV we apply the developed formalism to observer. However, we shall assume that this motion is a PSR J0737-3039-like binary system and in Sec. V we slowenoughnottoappreciablychangetheexpression(7) present our conclusions. of the metric. In practice, from the viewpoint of the observer, the propagation of light through the system is II. TIME DELAY FROM A BINARY SYSTEM described as a series of static snapshots. Under these conditions, weidentify the positionofthe sourceofthesignalswiththespacecoordinates(x ,y ,z ) We shall refer to two objects, composing the binary s s s and that of the observer with (x ,y ,z ). According system, with notation and . Object is sup- s obs obs O1 O2 O2 to hypothesis iii), the trajectory of the e.m. beam is a posedtobe rotating(e.g. arotatingneutronstar)andis straight line then the source of the gravitomagnetic field. The other object (e.g. the radio-pulsar) plays the role of the O1 x = xs , (10) source of e.m. beams. Thus, we shall focus on time- z = z +(y y )tanχ. (11) delay observed on the e.m. signals emitted by 1 when s − s O they experience the field generatedby . Furthermore, O2 Each beam, represented by a dashed line in Fig. 1, lies we shall consider observers that are far away from the in a plane parallel to yz, thus source of the gravitational field, so that they do not feel its effects, e.g. Earth-based observers. In this case, the 0=g c2dt2+g h2dy2+2g cdydt, (12) 00 yy 0y coordinatetimecorrespondstothepropertimemeasured by the observers. and in the components of the metric tensor (6)-(9) we The derivation of the expressions relative to such a can replace r2 =h2y2+2ky+w2, with configurationreliesonsomestandardassumptions,listed from i) to vii) in the following. h = 1+tan2χ, (13) i) We choose Cartesian coordinates, whose origin is located on the mass which is the source of the gravito- k = q(zs ystanχ)tanχ, (14) − magneticfield(i.e.,object ): thez-axisisalignedwith O2 w = (z y tanχ)2+x2 . (15) the direction of the angular momentum −→J of the source s− s s q (see Fig. 1), the x-axis is orthogonal to the line of sight According to the standard approach to the time delay fromEarthandthey-axisisorthogonaltoboth;weshall problem[16], wesolveEq.(12)fordt/dy,thenthe result refertoxyz astothe “gravitomagnetic”referenceframe. is integrated along the trajectory of the ray; i.e., Consequently, the line element reads 1 yobs ds2 =g c2dt2+g dx2+g dy2 t = dy (r R )−1 00 xx yy flight S c − +g dz2+2g cdxdt+2g cdydt. (5) Zys zz 0x 0y aR x a2R2x2 ii) The gravitational field is in any case weak enough ×(− rS2 s +r r4S s +(r2−RS2)h2). to admit the approximation (r = x2+y2+z2) (16) RS p g00 = 1 , (6) When the propagation is “on the left” of the oriented − r R projection of −→J in the sky, with respect to the observer S gxx = gyy =gzz =−1− r , (7) (xs > 0, see Fig. 1), the first term in the parenthesis is 3 due to the angular momentum of the source reads x yobs aR x aR h2y+k yobs s S s S Z z yz tJ =− c r3 dy = c k2 h2w2 r . xs+yz Zys − (cid:12)(cid:12)ys(20) (cid:12) (cid:12) Y Θ The quantity xs changessignif the e.m. sourceis onthe left or on the right of the rotating body with respect to χ the observer; as a result, t can have opposite signs on from the e.m. source xs Φ to the observer opposite sides accordingly.J Focusing onthe geometrypeculiar ofa binarysystem, oneisexpectingy tobethesumofatime-independent x y obs part,y ,correspondingtothedistancefromthecenterof 0 X massofthesystemtotheobserver,andatime-dependent part,acontributionoscillatingintime due to the orbital motion of object ; condition v) implies that this orbit 2 O tooisacircumferenceofradiusR . Theamplitudeofthe 2 oscillation is of the order of magnitude of the size of the binary system, just as k and w. Since for condition vi) the size of the system is much smaller than the distance from Earth, Eqs. (19) and (20) are simplified as FIG. 1: Gravitomagnetic reference frame xyz. The origin is located on O2 and the z-axis is aligned with the direction of itsangular momentum. TheCartesian referenceframe XYZ RS 2y0h2 t ln is located in O2 and XY identifies the orbital plane of the M ≃ c h2ys+k+hrs binary system. 1 hR2 + S 2c√h2w2 k2 − negative;ontheopposite,whenthepropagationisonthe 2h2y2 2(k+h2y2) arctan 0 arctan s , right (xs <0), the sign is positive. × √h2w2 k2 − √h2w2 k2 Fortheweakfieldconditionofii), Eq.(16)canbefur- (cid:26) − − (cid:27)(21) ther expanded in powers of R and a, up to their prod- S x aR h2y +k s S s uct, i.e., up to second order. Second order is necessary t h , (22) J ≃ c k2 h2w2 − r to describe the gravitomagnetic interaction; in addition, − (cid:18) s (cid:19) it should be noticed that, for collapsed objects (as for a where we have set h2y2 +2ky +w2 = r2 = R2; R is star like the Sun) it is reasonably R a, so that the s s s S ∼ the distance between the two stars in the pair, which secondorderterminR cannotbe simplyneglected. By S is constant according to hypothesis v). By restoring an performing suchan expansion,the integralof Eq.(16) is explicit notation, we have divided into four terms, which may be further grouped into three contributions to the total time of flight as R 2y S 0 t = t +t ln M M1 M2 ≃ c (y cosχ+z sinχ+R)cosχ δt t =t +t +t , (17) s s flight 0 M J ≡ 1 R2 + S (23) where 2c R2 (y cosχ+z sinχ)2 s s − 1 yobs h q t = hdy = (y y ) (18) 0 obs s π 2(y cosχ+z sinχ) c Zys c − arctan s s , × 2 − R2 (y cosχ+z sinχ)2 s s represents the pure geometric term, − x aR q cosχ  1 S t = hR yobs 1 + RS dy tJ ≃ − c R R+yscosχ+zssinχ . (24) M c S r 2r2 Zys (cid:18) (cid:19) R h2y +k+hr = S ln obs obs (19) A. The time dependent part c h2y +k+hr s s + 1 hRS2 arctan 2 k+h2y2 yobs In Eqs. (18), (23), and (24) we are interested in the 2c√h2w2−k2 (cid:18) √h2w2−k2(cid:19)(cid:12)ys time dependent part, which is implicit in xs, ys and zs, (cid:12) since the position of the e.m. source at each successive (cid:12) is the mass delay up to second order, where w(cid:12)e have “snapshot” is different because of the orbital motion of defined r r(y ) and r r(y ); the contribution . Byassumptionv),theorbitof iscircular. Letus obs obs s s 1 1 ≡ ≡ O O 4 startbyexpressingthepositionof withrespecttoan- 1 O TABLE I: A toy model for system PSR J0737-3039. We other reference frame (called XYZ, see Fig. 1) centered chooseω∼7×10−4 s−1 andR=2R2 ≃109 m. From left to in suchthat X =Rcosωt andY =Rsinωt, where ω O2 right,thecolumnscontainwhichstarofthepairisthesource is the orbitalangular velocity of the pair and the X axis of the gravitomagnetic field, the Schwarzschild radius RS of isidentifiedbytheintersectionbetweentheorbitalplane O2 andtheanglesthatidentifythegeometrical configuration of the system and the gravitomagnetic equatorial plane of the system. of . We call Φ the angle between the X-axis and the 2 O x-axis; Θ identifies the tilt angle between the axis of the O1 O2 RS (m) χ Θ Φ orbitandthe angularmomentum−→J of 2. The “gravit- A B 1.6×103 0◦ 0◦ 0◦ O omagnetic” coordinates of O1 expressed with respect to B A 1.7×103 50◦ 50◦ 0◦ the xyz frame read ξ =cosΦcosψ cosΘsinΦsinψ , s − η =sinΦcosψ+cosΘcosΦsinψ , (25) s ζ = sinΘsinψ , s − where, for convenience, we have introduced the reduced coordinates ξ = x /R, η = y /R, ζ = z /R, and we s s s s s s have defined ψ =ϕ+α: ϕ=ωt is the orbital phase and ativistictheoriesand, inprinciple,itcouldbe usefulalso α = arctan(cotΦ/cosΘ). We can measure times from for measuring gravitomagneticeffects on time delay. Let the configuration x =0, y >0 (conjunction). s s us then briefly review its most important physical fea- From Eq. (18), the time-dependent contribution t∗ in 0 tures [14]. The two pulsars, J0737-3039A and J0737- flatspacetimereads(hereafter,starredquantitiesreferto 3039B(hereaftersimplyAandB),haveperiodsP =23 A the time-dependent contributions to t ) flight ms and P = 2.8 s; they revolve about each other in a B ct∗ 2.4-hrorbitofsignificanteccentricity(0.088);thesepara- 0 cosχ=sinΦcosψ+cosΘcosΦsinψ , (26) tionofthetwoobjectsistypically9 105 km. Theorbital r R · − plane is viewed nearly edge-on from the Earth, with an that is, a harmonic oscillation whose amplitude corre- inclination angle of i = 87◦ 3◦. It has been possible sponds to the time the beam takes to cross the system. to detect a huge rate of peria±stron advance, ω˙ = 16.88◦ The mass-dependent term is more involved, since it is yr−1, whichis aboutfour times the one ofPSR1913+16 composedbyafirstorderandasecondorderterm. Both [17]. If this effect is entirely due to GR, from the obser- ofthemcanbefactorizedintoan“amplitude”,containing vations carried out so far it has been possible to estab- the size of the system and the Schwarzschild radius of lish that MA = 1.337(5)M⊙ and MB = 1.250(5)M⊙. In , and a pure geometrical part. It is given by the sum tO∗2 =t∗ +t∗ , with addition, due to the collision of A’s wind with B’s mag- M M1 M2 netosphere it seems very likely that the spin axis of B is ct∗ aligned with the orbital angular momentum of the sys- M1 = ln(ηscosχ+ζssinχ+1) , (27) tem [18, 19]. On the other hand, observations show that R − S A is almost an aligned rotator (angle between A’s mag- ct∗ 1 M2 = (28) netic and rotation axes 5◦ [18]), but with its spin axis RS2 2R 1 (η cosχ+ζ sinχ)2 substantiallymisaligned∼withtheorbitalangularmomen- s s − tumby 50◦. Inaddition,thesystemhastheimportant q ∼ π 2(η cosχ+ζ sinχ) featurethat,for27s,AiseclipsedbyB’smagnetosphere. s s arctan . SuchdurationoftheeclipsewasusedinRef.[20]toplace × 2 − 2 1 (ηscosχ+ζssinχ) alimitof18.6 103kmonthesizeoftheeclipsedregion. − ×  q  This region is much bigger than the expected physical Eventually, from Eq. (20), the contribution to the time linear dimensions of an actual neutron star ( 10 km): delay due to the angular momentum of O2 reads the typical features of the observed signals see∼m to sug- ctJ R ξs gestthatthe eclipseis dueto the absorptionofthe radio . (29) aR cosχ ≃−(1+η cosχ+ζ sinχ) emission from A by a magnetosheath surrounding B’s S s s magnetosphere [19, 21, 22]. These time-dependent parts of t would be visible in flight the sequence of the arrival times of the pulses from the source . 1 O Because of (i) the alignment of the orbital plane with the line of sight, (ii) the fact that B eclipses A and (iii) III. THE BINARY SYSTEM PSR J0737-3039 thespinaxisofBisprobablyperpendiculartotheorbital plane,the configurationofthe systemcouldbe favorable TherecentlydiscoveredbinarysystemPSRJ0737-3039 for studying the gravitomagneticeffects on signals prop- hasprovedto be animportantlaboratoryfortesting rel- agation. 5 106 TABLEII:Thefirsttworowsshowtheorderofmagnitudeof the contributions to the time-dependent part of the time of flight of e.m. signals emitted by object O1 and propagating in the gravitational field generated by O2. The bottom row contains the order of magnitude of the contributions to the (relative) frequency shift on thesignals emitted by star A. 105 O1 O2 AO02 (s) AOM21 (s) AOM22 (s) AOJ2 (s) A B −3 −5×10−6 4×10−12 6×10−13 B A −4.7 −5.6×10−6 4.8×10−12 −3.6×10−12 104 DO2 DO2 DO2 DO2 0 M1 J1 J2 A B −21×10−4 −3.7×10−9 −4×10−15 4×10−15 103 3.14 3.1402 3.1404 3.1406 3.1408 3.141 3.1412 3.1414 3.1416 IV. APPLICATION TO A PSR J0737-3039-LIKE MODEL A. The time delay Let us specify the simple model outlined above using values of the parameters similar to those of the actual PSRJ0737-3039: starB isactingasthe sourceofgravit- omagnetism( B and A,see TableI), R 109 FIG. 2: The function τ(ϕ) (measured in picoseconds) versus 2 1 m and ω 7 O10≡−4s−1. TOhe a≡ngularmomentum o∼fstar the orbital phase of star A in proximity of the occultation ∼ × position ϕ = π. The horizontal dashed line corresponds to B is aligned with the orbital angular momentum of the thedetectability threshold of 10−8 s. system(Θ=0)andwechoosethemostfavorableconfig- uration with Φ = 0 (i.e. α = π/2). The e.m beams are thus propagating in the orbital plane (χ = 0). Suppos- ing that the progenitorstar was only a little bigger than i.e., in seconds the Sun, and that most of the angular momentum was preserved during the collapse, we can assume a 103m. With these hypotheses, Eqs. (26)-(29) read ∼ τ(ϕ) 10−12 sinϕ . (35) ≃ 1+cosϕ t∗ = Bcosϕ, (30) 0 A0 t∗ = B ln(1+cosϕ), (31) The function τ(ϕ) is shown in Fig. 2 (solid line) close to M1 AM1 1 π 2cosϕ the conjunction position. If we suppose that the thresh- t∗M2 = ABM2 sinϕ 2 −arctan sinϕ , (32) old for detecting the change in the arrival rate of the | |(cid:18) | |(cid:19) signalsisforinstanceat10−8 s,thenonlythepartofthe sinϕ t = B (33) graph above the horizontal dashed line is useful. This J AJ1+cosϕ means that the access to the interesting region would where the numerical coefficients O2 with B give be possible only if the minimum impact parameter was A O2 ≡ smaller than 1.8 102 km. Since the typical radius of theorderofmagnitudeoftheeffect(seeTableII). Theϕ- ∼ × aneutronstaris 10km,wecanexpecttheappropriate dependentpartsinEqs.(31)and(33)becomebiggerand ∼ conditions to be satisfied in a double pulsar binary sys- biggerclosetotheconjunctionposition(ϕ=π;i.e.,when tem. However, this is not the case of PSR J0737-3039: the impact parameter is zero), but this divergence has infact, in this systemthe minimum impactparameteris no physicalmeaning because the actualcompact objects notgivenbytheradiusofobjectB,butratherbythesize havefinitedimensionsandthebeamcannotpassthrough of the opaque area of the magnetosheath surrounding B the center of B. itself, that is 1.8 104 km; i.e. two order of magnitude Let us suppose that it is possible to identify conjunc- × bigger than the detectability threshold. tion (ϕ = π) and opposition (ϕ = 2π) points in the sequence of arriving pulses. Since the geometric and mass The same kind of analysis can be performed by ex- termsaresymmetricwithrespecttoconjunctionandop- changing A with B; i.e., 2 A and 1 B, although O ≡ O ≡ position, whereas t is antisymmetric, for 0 ϕ π we in the actual system the B pulses are extremely weak J have ≤ ≤ and aleatory. In this case, we must choose Θ = 50◦ and χ = 50◦ (see Table I) since only the e.m. beams prop- τ(ϕ)=δt∗(ϕ) δt∗(2π ϕ)=2t , (34) J agating in the orbital plane can be seen by the observer − − 6 on Earth. In this case we have where ξ˙ , η˙ and ζ˙ are obtained from Eqs. (25). Here s s s it becomes clear that the relative frequency shift is a t∗ = Acosϕ, (36) complicatedfunctionoftimethatreadsoutasaperiodic 0 A0 t∗ = A ln(1 ncosϕ) , (37) modification in the frequency spectrum of the signal. M1 AM1 − Applying theaboveequationsto ourPSRJ0737-3039- like system, we just presentresults for pulses emitted by t∗ = A 1 star A. In this case, we have M2 AM2 2 1 (ncosϕ) − δν q = Bsinϕ, (46) π mcosϕ ν (cid:12)0 D0 arctan , (38) (cid:12) × 2 − 1−(ncosϕ)2 δνν(cid:12)(cid:12) =DMB11+sincoϕsϕ −y˙s∂ystM2 , (47)  sinϕ q  (cid:12)M tJ = AAJ 1+ncosϕ , (39) δνν(cid:12)(cid:12)(cid:12) =DJB11+cocsoϕsϕ +DJB2(1+sicnoϕsϕ)2 , (48) (cid:12)J where n = 0.17 and m = 0.34, from the evaluation of (cid:12) (cid:12) Eqs. (26)-(29) employing the parameters of Table I. In where th(cid:12)e coefficients O2, with B, give the order 2 D O ≡ this case, we obtain of magnitude of the effect and are listed in Table II. All the contributions (including δν/ν that is then τ(ϕ) 7.2 10−12 sinϕ . (40) needless to make explicit) are odd in ϕ|Me2xcept for the ≃− × 1+ncosϕ one proportional to B, which is even. Summing shifts DJ1 symmetricwithrespecttotheoppositionpointweobtain Since n < 1, the denominator never vanishes, so that the am|o|unt of the effect is at most 10−11 s. This configuration is thence less favorable t∼han the previous δν (ϕ)+ δν (2π ϕ)= 8 10−15 cosϕ . (49) ν ν − − × 1+cosϕ one, because a lower detectability threshold is required. B. The frequency shift V. CONCLUSIONS Let us consider the Fourier spectrum of an e.m. beam In this paper, we have discussed the effects of the propagating into a gravitational field: the time-delay ef- gravitationalinteractiononthetimedelayofelectromag- fect we have discussed so far corresponds, in the fre- netic signals coming from a binary system composed by quency domain, to a frequency shift of each harmonic a radio-pulsar and another compact object. In particu- componentofthe signal. Since the periodT ofeachhar- lar, we have evidenced that the behavior of the gravito- monic is much shorter than all the other characteristic magneticcontribution,nearthe occultationofthe radio- time-scalesofthesystem,wecanwriteitsrelativechange pulsar by its companion, is antisymmetric while the ge- in period, hence in the frequency ν, as ometric and mass contribution are symmetric, thus sug- gesting a possible way for decoupling the effects. δν δT = = t˙0+t˙M +t˙J , (41) The recently discovered binary pulsar system PSR ν − T − J0737-3039, lends itself for the study of the time delay, (cid:0) (cid:1) where the overdot stands for derivative with respect to because (i) the orbital plane almost contains the line of the coordinate time. For each contribution, we have sight,(ii)thestarBeclipsesAand(iii)thespinaxisofB seemstobe alignedwiththeorbitalangularmomentum. δν h However,we have arguedthat the gravitomagneticef- = ω (R R)sinϕ, (42) 2 ν c − fect on time delay still remains extremely small in this (cid:12)0 δν(cid:12)(cid:12) RS η˙scosχ+ζ˙ssinχ system: under reasonable assumptions on the mass and (cid:12) = , (43) angular momentum of the sources of the gravitational ν c 1+η cosχ+ζ sinχ (cid:12)M1 s s field,thepossibilitytorevealtheeffectcriticallydepends (cid:12) δν(cid:12) on the configuration of the system and on the minimum ν (cid:12) = −(y˙s∂ys +z˙s∂zs)tM2 (44) impact parameter achievable for the e.m. ray. Because (cid:12)(cid:12)M2 of the existence of a large opaque region represented by δν(cid:12) aR ξ˙ cosχ (cid:12) = S s a magnetosheath surrounding PSR J0737-3039B,the ef- ν (cid:12)J cR (cid:26)1+ηscosχ+ζssinχ fective impact parameter is much bigger than the actual (cid:12) linear dimension of a neutron star, so that the magni- (cid:12) ξ η˙ cos2χ+ζ˙ sin2χ (cid:12) s s s tude of the gravitomagnetictime delay is smaller than a , (45) −(1(cid:16)+η cosχ+ζ sinχ)(cid:17)2 reasonable detectability threshold. s s (cid:27) 7 ACKNOWLEDGMENTS The activity of A.N. in València is supported by the Angelo Della Riccia Foundation. [1] J. Lense and H. Thirring, Phys. Z. 19, 156 (1918); the physics, Springer-Verlag, Berlin 1984. EnglishtranslationavailableinB.Mashhoon,F.W.Hehl [17] J.H. Taylor, et al.,Nature277, 437 (1979). and D.S. Theiss, Gen. Rel. Grav. 16, 711 (1984). [18] P. Demorest, et al.,astro-ph/0402025. [2] V.B.Braginsky,C.M.CavesandK.S.Thorne,Phys.Rev. [19] J. 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