Table Of ContentOn 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 Val`encia,
Edifici d’Investigaci´o, Dr. Moliner 50, 46100 Burjassot (Val`encia), 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 se-
quence 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 Val`encia is supported by the
Angelo Della Riccia Foundation.
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