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ANDASTROPHYSICS

23.2.1996

Gravitational waves from pulsars: emission by the magnetic

eld induced distortion

S. Bonazzola and E. Gourgoulhon?

Departement d'Astrophysique Relativiste et de Cosmologie (UPR 176 du C.N.R.S.), Observatoire de Paris,

Section de Meudon, F-92195 Meudon Cedex, France

e-mail : bona,gourgoulhon@obspm.fr

Received 6 November 1995 / Accepted 20 February 1996

Abstract. The gravitational wave emission by a distortedrotating uid star is computed. The distortion is supposedto be symmetric around some axis inclined with respectto the rotation axis. In the general case, the gravitationalradiation is emitted at two frequencies: and 2, where is the rotation frequency. The obtained formul are ap-plied to the specic case of a neutron star distorted byits own magnetic eld. Assuming that the period deriva-tive _P of pulsars is a measure of their magnetic dipolemoment, the gravitational wave amplitude can be relatedto the observable parameters P and _P and to a factor which measures the eciency of a given magnetic dipolemoment in distorting the star. depends on the nuclearmatter equation of state and on the magnetic eld distri-bution. The amplitude at the frequency 2, expressed interms of P , _P and , is independent of the angle be-tween the magnetic axis and the rotation axis, whereas atthe frequency , the amplitude increases as decreases.The value of for specic models of magnetic eld distri-butions has been computed by means of a numerical codegiving self-consistent models of magnetized neutron starswithin general relativity. It is found that the distortionat xed magnetic dipole moment is very dependent of themagnetic eld distribution; a stochastic magnetic eld or asuperconductor stellar interior greatly increases with re-spect to the uniformly magnetized perfect conductor caseand might lead to gravitational waves detectable by theVIRGO or LIGO interferometers. The amplitude modu-lation of the signal induced by the daily rotation of theEarth has been computed and specied to the case of theCrab pulsar and VIRGO detector.

Key words: gravitation magnetic elds gravitationalradiation stars: neutron pulsars: general numericalmethods: spectral

Send oprint requests to: E. Gourgoulhon? author to whom the proofs should be sent

1. Introduction

Rapidly rotating neutron stars (pulsars) might be an im-portant source of continuous gravitational waves in the fre-quency bandwidth of the forthcoming LIGO and VIRGOinterferometric detectors (cf. Bonazzola & Marck 1994 fora recent review about the astrophysical sources these de-tectors may observe). It is well known that a stationaryrotating body, perfectly symmetric with respect to its ro-tation axis does not emit any gravitational wave. Thusin order to radiate gravitationally a pulsar must deviatefrom axisymmetry. Various kinds of pulsar asymmetrieshave been suggested in the literature: rst the crust of aneutron star is solid, so that its shape may not be nec-essarily axisymmetric under the eect of rotation, as itwould be for a uid: deviations from axisymmetry aresupported by anisotropic stresses in the solid. The shapeof the crust depends not only on the geological historyof the neutron star, especially on the episode of crystal-lization of the crust, but also on star quakes. Due to itsviolent formation (supernova) or due to its environment(accretion disk), the rotation axis may not coincide witha principal axis of the neutron star moment of inertia andthe star may precess (Pines & Shaham 1974). Even if itkeeps a perfectly axisymmetric shape, a freely precessingbody radiates gravitational waves (Zimmermann & Sze-denits 1979). Neutron stars are known to have importantmagnetic elds; magnetic pressure (Lorentz forces exertedon the conducting matter) can distort the star if the mag-netic axis is not aligned with the rotation axis, which iswidely supposed to occur in order to explain the pulsarphenomenon. Another mechanism for producing asymme-tries is the development of non-axisymmetric instabilitiesin rapidly rotating neutron stars driven by the gravita-tional radiation reaction (CFS instablity, cf. e.g. Schutz

1987) or by nuclear matter viscosity (cf. e.g. Bonazzola,Frieben & Gourgoulhon 1996 and references therein).

In the seventies it was widely thought that the solidpart of neutron stars was large (see Sect. III of Goldreich1970). A rotating rigid body whose rotation axis does notcoincide with a principal axis of the moment of inertiamust have a precessional motion. Therefore, there havebeen a number of studies about the precession of neutronstars and the resulting gravitational wave emission (Zim-mermann 19781, Zimmermann & Szedenits 1979, Zimmer-mann 1980, Alpar & Pines 1985, Barone et al. 1988, deAraujo et al. 1994). However, modern dense matter cal-culations reveal that neutron star interiors are completelyliquid (see e.g. Haensel 1995). Since the crust representsonly 1 to 2% of the stellar mass (Lorenz et al. 1993), itappears that most of the neutron star is in a liquid phase.In this case, the star is not expected to precess, or at leastits precession frequency must be reduced by a factor 105

as compared with the solid case (Pines & Shaham 1974).

The gravitational radiation from a rotating uid starhas not been studied as much as that from a solid star.The only reference we are aware of is the work by Gal'tsov,Tsvetkov & Tsirulev (1984) and by Gal'tsov & Tsvetkov(1984) about the gravitational emission of a rotating New-tonian homegeneous uid drop with an oblique magneticeld. These authors have shown that the gravitationalwaves are emitted at two frequencies: the rotation fre-quency and twice the rotation frequency. They did not giveexplicit formul for the two gravitational wave amplitudeh+ and h but have instead computed the response ofa heterodyne detector (consisting in a rotating dumbbellorientated at right angles to the direction of propagationof the wave) directly from the Riemann tensor.

In the present article, we compute the gravitational ra-diation from a rotating uid star, distorted along a certaindirection by some process (for example an internal mag-netic eld), the angle between this direction and the rota-tion axis being arbitrary (Sect. 2). We do not assume thatthe star is a Newtonian object: it can have a large gravita-tional eld described by the theory of general relativity |which is relevant for neutron stars. As an illustration wegive the specic example of a Newtonian incompressible uid with a uniform internal magnetic eld (Sect. 3). Weconsider also the (more general) case where the deforma-tion is due to an internal magnetic eld which is also re-sponsible for the _P of the pulsar (electromagnetic braking)(Sect. 4). Using the numerical code presented in (Bocquet,Bonazzola, Gourgoulhon & Novak 1995) for magnetizedrotating neutron stars in general relativity, we computethe gravitational emission for a specic 1:4M neutronstar model and for various congurations of the internalmagnetic eld. Finally, we derive the response of an inter-ferometric detector to the gravitational signal, taking into

1 the results presented in this reference are erroneous and arecorrected in Zimmermann & Szedenits (1979)

account the daily change induced by the Earth rotationin the relative orientation of the detector arms and thesource (Sect. 5).

2. Generation of gravitational waves by a rotating

uid star

2.1. Thorne's quadrupole moment

Neutron stars are fully relativistic objects, so that thestandard quadrupole formula for gravitational wave gen-eration | which is derived for non relativistic sources (cf.Sect. 36.10 of Misner, Thorne & Wheeler 1973, hereafterMTW) | does not a priori apply. However Ipser (1971)has shown that the gravitational radiation from a slowlyrotating and fully relativistic star is given by a formulastructurally identical to the weak eld standard formula,provided that the involved multipole moments are denedin an appropriate way.

The leading term in the gravitational radiation eld isthen given by the formula2:

hTTij =2G

c41

r

P ki P l

j 1

2PijP

kl

Ikl

t r

c

; (1)

where r is the distance to the source and Pij := ij rirj=r

2 is the tensor of projection transverse to the line ofsight. Eq. (1) holds for highly relativistic sources providedthat Iij is the mass quadrupole moment dened as thequadrupolar part of the 1=r3 term of the 1=r expansion ofthe metric coecient g00 in an asymptotically Cartesian

and mass centered (ACMC) coordinate system (Thorne1980). This latter is a very broad coordinate class, whichincludes the harmonic coordinates. The precise denitionof Iij is the following one. The space around a neutronstar of mass M , \typical" radius R and angular velocity can be divided in three regions:

the strong-eld region: r < a few GM=c2; the weak-eld near zone: max(R; a few GM=c2) < r <c=

the wave zone : r > c=

In the wave zone, retardation eects are importantwhereas in the weak-eld near zone, the gravitationaleld can be considered as quasi-stationary. Note thatfor the fastest millisecond pulsar to date, PSR 1937+21,c= 80 km, whereas R 10 km and if its mass isM = 1:4M, GM=c2 2 km. Hence, even in this extremecase, the weak-eld near zone is well dened. A coordinatesystem (t; x1; x2; x3) is said to be asymptotically Cartesian

and mass centered (ACMC) to order 1 if the metric admitsthe following 1=r expansion (r := xix

i) in the weak-eld

2 The following conventions are used: Latin indices

(i; j; k; l; : : :) range from 1 to 3 and a summation is to be per-formed on repeated indices.

2

near zone (Sect. XI of Thorne 1980):

g00 = 1 + 2M

r+1

r2

+1

r3

3 Iijx

ixj

r2+ 1i

xi

r+ 2

+ O

1

r4

(2)

g0i = 4iklJk xl

r3+ O

1

r3

(3)

gij = ij +3ij

r+

1

r2

2ijk

xk

r+ 4ij

+ O

1

r3

; (4)

where 1, 2, 3ij, 4ij, 1i and 2ijk are some constants.The coecients M and Jk in the above expansion are thestar's total mass and angular momentum. They are inde-pendent of the (xi) coordinate choice, provided that theyreduce to usual Cartesian coordinates at the asymptoti-cally at innity. On the contrary, the coecient Iij is notinvariant under change of asymptotically Cartesian coor-dinates: it is invariant only with respect to the sub-classof ACMC coordinates. This quantity is called the mass

quadrupole moment of the star. As discussed in Sect. XIof Thorne (1980) (cf. also Thorne & Gursel 1983), Iij isa at-space-type tensor which \resides" in the weak-eldnear zone, which means that it can be manipulated as atensor in at space. This tensor is symmetric and trace-free. If the star has a gravitational eld weak enough tobe correctly described by the Newtonian theory of grav-itation, Iij is expressible as minus the trace-free part ofthe moment of inertia tensor Iij :

Iij = Iij + 1

3I kk ij ; (5)

Iij being given by

Iij :=

Zxkx

k ij xixjd3x : (6)

In particular, Iij coincides at the Newtonian limit with thequantity Iij introduced in MTW. For highly relativisticcongurations, Iij can no longer be expressed as someintegral over the star, as in Eqs. (5)-(6); it must be readfrom the 1=r expansion of the metric coecient g00 in anACMC coordinate system. In Appendix A, we give thetransformation from a quasi-isotropic coordinate system,usually used in relativistic studies of rotating neutron stars(cf the discussion in Sect 2 of Bonazzola, Gourgoulhon,Salgado & Marck 1993), to an ACMC coordinate system.

2.2. Slightly deformed rotating star

Very tight limits have been set on the deformation ofneutron stars by pulsar timings: the relative deviationfrom axisymmetry is at most 103 (cf. Table 1 of Newet al. 1995). This number is certainly overestimated byseveral orders of magnitude for it is obtained by assum-ing that all the observed period derivative _P is due to

the gravitational emission, whereas it is much more likelyto be due to the electromagnetic emission. Indeed, thepresent measurements of the braking index n of pulsars aren = 2:509 0:001 for the Crab, n = 2:01 0:02 for PSR0540-69 and n = 2:837 0:001 for PSR 1509-58 (cf. Mus-limov & Page 1996 for references). Thus they are closerto n = 3 (magnetic dipole radiation) than to n = 5 (grav-itational radiation, cf. Table 9.1 of Manchester & Taylor1977).

For such small deformations, we make the assumptionthat the quadrupole moment Iij can be linearly decom-posed into the sum of two pieces:

Iij = Irotij + Idistij ; (7)

Irotij is the quadrupole moment due to rotation: Irotij = 0 if

the conguration is static. Idistij is the quadrupole momentdue to some process that distorts the star, for examplean internal magnetic eld or anisotropic stresses for thenuclear interactions.

The fact that the star does not precess impliesthat there exists some ACMC coordinate system x =(t; x; y; z) such that the components Irotij in this coordi-

nate system do not depend upon x0 = t and take thediagonal form

Irotij =

0@I

rotzz =2 0 00 Irotzz =2 00 0 Irotzz

1A : (8)

The line x = y = 0 is the rotation axis. For a small angularvelocity , Irotzz is a quadratic function of .

Let us make the following assumptions about the dis-tortion process:

1. there exists some privileged direction, i.e. two of thethree eigenvalues of Idistij are equal, i.e. there existsin the weak-eld near zone a coordinate system (notnecessarily ACMC) x = (t; x; y; z) such that the com-ponents Idist

ijare

Idistij

=

0@I

distzz =2 0 00 Idistzz =2 00 0 Idistzz

1A : (9)

2. the deformation is rotating with the star, i.e. the weak-eld near zone vector ez associated with the coordinatez is rotating at the angular velocity at a xed angle from the weak-eld near zone vector ez associatedwith the coordinate z (ez is along the rotation axis)(cf. Fig. 1):

ez = cos ez + sin [cos'(t) ex + sin'(t) ey] ; (10)

with

'(t) = (t t0) : (11)

In particular, the above assumptions are satised by amagnetic eld symmetric with respect to some axis: ez isthen parallel to the magnetic dipole moment vectorM.

3

φ = Ω t

αi

zz

x

yn TheEarth

Ω

Fig. 1. Geometry of the distorted neutron star.

2.3. Application of the quadrupole formula

Inserting Eq. (7) into the quadrupole formula (1) and us-ing the time constancy of Irotij , we get

hTTij =2G

c41

r

P ki P l

j 1

2PijP

kl

Idistkl

t r

c

: (12)

In order to apply this formula, one must rst expressIdistij in the ACMC coordinates (t; x; y; z), from its compo-

nents (9) in the coordinates (t; x; y; z). The transformation(t; x; y; z) ! (t; x; y; z) is simply the composition of tworotations (cf. Fig. 1): one of angle around an axis ex0

in the (ex; ey) plane and rotating with the star, and oneof angle '(t) = (t t0) around ez. The transformationmatrix is then:

P =

0@cos'(t) sin'(t) 0sin'(t) cos'(t) 0

0 0 1

1A

0@1 0 00 cos sin0 sin cos

1A :(13)

The tensor transformation law writes Idist = P Idist tP , where Idist is the matrix formed by the componentsIdistij and Idist is the matrix formed by the components

Idistij

[Eq. (9)]. Performing this matrix product leads to

Idistij =1

2Idistzz

0@ 3 sin2 sin2'(t) 1

3=2 sin2 sin 2'(t)3 sin cos sin'(t)

3=2 sin2 sin 2'(t) 3 sin cos sin'(t)3 sin2 cos2'(t) 1 3 sin cos cos'(t)3 sin cos cos'(t) 3 cos2 1

1A : (14)

In order to apply the quadrupole formula, the secondderivative with respect to t of this expression must be

taken. One obtains, using Eq. (11),

Idistij =3

2Idistzz 2 sin

0@2 sin cos 2'(t)2 sin sin 2'(t)

cos sin'(t)

2 sin sin 2'(t) cos sin'(t)2 sin cos 2'(t) cos cos'(t) cos cos'(t) 0

1A : (15)

The next step consists in taking the transverse tracelessprojection of Idistij . Let us denote by i the angle betweenthe neutron star's rotation axis ez and the direction n

from the star's centre to the Earth (i is called the line of

sight inclination) (cf. Fig. 1). Without any loss of general-ity, we can choose the ACMC coordinates (t; x; y; z) suchthat n lies in the (ey; ez) plane. The components with re-spect to (t; x; y; z) of the transverse projection operatorPij = ij ninj are then

Pij =

0@1 0 00 cos2 i sin i cos i

0 sin i cos i sin2 i

1A : (16)

From Eqs. (16) and (15), the computation of the right-hand side of the quadrupole formula (12) is straightfor-ward, though somewhat tedious. The result is

hTTij = h+ e+ij + h e

ij ; (17)

with

e+ij =

0@1 0 00 cos2 i sin i cos i

0 sin i cos i sin2 i

1A ; (18)

eij =

0@ 0 cos i sin i

cos i 0 0 sin i 0 0

1A (19)

and

h+ = h0 sinh12cos sin i cos i cos (t t0)

sin1 + cos2 i

2cos 2(t t0)

i(20)

h = h0 sinh12cos sin i sin (t t0)

sin cos i sin 2(t t0)i; (21)

where

h0 := 6Gc4Idistzz

2

r: (22)

Note that in the above expressions, Eq. (11) has been sub-stituted for '(t) and that the retardation term r=c hasbeen incorporated into the constant t0.

4

2.4. Discussion

From the formul (20)-(21), it is clear that there is nogravitational emission if the distortion axis is aligned withthe rotation axis ( = 0 or ). If both axes are perpendic-ular ( = =2), the gravitational emission is monochro-matic at twice the rotation frequency. In the general case(0 < jj < =2), it contains two frequencies: and 2.For small values of the emission at is dominant.

It may be noticed that Eqs. (20)-(22) are structurallyequivalent to Eq. (1) of Zimmermann & Szedenits (1979)(hereafter ZS), although this latter work is based on adierent physical hypothesis (Newtonian precessing rigidstar). In order to compare precisely Eqs. (20)-(22) andEq. (1) of ZS, some slight re-arrangements must be per-formed. First, the ellipticity dened by ZS is linked toour Izz by I1 = 3=2 Izz [cf. Eq. (5)], this explains thefactor 2 in front of the right-hand side of Eq. (1) of ZS in-stead of the factor 6 in Eq. (22). Other apparent dier-ences are actually due to dierent conventions: the originof time in ZS is the instant when the equivalent of the de-formation axis is at its farthest position from the observer,whereas in our case it corresponds to its nearest position.One must then make t! t+ in order to compare the twoformul. Moreover the choice for the matrices e+ij and e

ij

are exactly opposite in both approaches [cf. Sect. III.B ofZimmermann (1980)], so that the transforms h+ ! h+and h ! h must also be performed. When all thisis done, Eqs. (20)-(22) and Eq. (1) of ZS appear to haveexactly the same structure. However, the physical signi-cance is dierent: the frequency ! which appears in Eq. (1)of ZS diers from the pulsar frequency by the body-frameprecessional frequency whereas in Eqs. (20)-(22), is ex-actly the pulsar frequency. The angle of ZS, which intheir Eq. (1) takes the place of our angle in Eqs. (20)-(22), is the angle between the total angular momentum J

and the star's third principal axis, whereas our is theangle between the rotation axis and the direction of thedistortion, which even in the Newtonian case, does not co-incide with any of the principal axis of the body (exceptin the non-rotating case).

As special cases of Eqs. (20)-(22), one may recover re-sults previously published in the literature. For instance,the case of a triaxial star rotating about a principal axisof its moment of inertia tensor can be obtained by set-ting = =2 in Eqs. (20)-(22). The result can be com-pared with Eqs. (48) and (54) of Thorne (1987), noticingthat f in these equations is = and that Ixx and Iyyof Thorne (1987) are related to our Izz by Ixx = Izz=2and Iyy = Izz. The two formul appear then to be identi-cal, as expected. If in addition to = =2, the inclinationangle i is set to zero, Eqs. (20)-(22) reduce to the for-mula used in the recent work by New et al. (1995) [theirEq. (5)]. Note that in the two studies mentionned above,the gravitational waves are emitted at the frequency 2only, due to = =2 or equivalently due to the fact that

the rotation axis coincides with a principal axis of themoment of inertia tensor. Let us stress again that in our(more general) case, the gravitational radiation containstwo frequencies: and 2.

2.5. Numerical estimates

In order to describe the star deformation by a dimension-less quantity, let us introduce instead of Idistzz the ellipticity

:= 32

Idistzz

I; (23)

where I is the moment of inertia with respect to the ro-tation axis, dened as

I := J= ; (24)

J being the star angular momentum. The denition (24)is valid even in highly relativistic cases, provided that thestar distortion is small | as we suppose throughout thiswork. Indeed, in this case the conguration is essentiallyaxisymmetric and the angular momentum J is well de-ned (cf. e.g. the discussion in Wald (1984), p. 297). Thefactor 3=2 in Eq. (23) is introduced in order to recoverthe classical denition of the ellipticity at the Newtonianlimit (see e.g. Shapiro & Teukolsky 1983).

Let us introduce also the rotation period P = 2=,since observational data about pulsars are usually pre-sented with P instead of .

Inserting Eq. (23) in expression (22) for the character-istic gravitational wave amplitude leads to

h0 =162G

c4I

P 2 r: (25)

Replacing the physical constants by their numerical valuesresults in

h0 = 4:21 1024hmsP

i2hkpcr

ih I

1038 kgm2

ih

106

i:(26)

Note that I = 1038 kgm2 is a representative value for themoment of inertia of a 1:4M neutron star [see Fig. 12 ofArnett & Bowers (1977)].

For the Crab pulsar, P = 33 ms and r = 2 kpc, so thatEq. (26) becomes

hCrab0 = 1:89 1027h I

1038 kgm2

ih

106

i: (27)

For the Vela pulsar, P = 89 ms and r = 0:5 kpc, hence

hVela0 = 1:06 1027h I

1038 kgm2

ih

106

i: (28)

For the millisecond pulsar3 PSR 1957+20, P = 1:61 msand r = 1:5 kpc, hence

h1957+200 = 1:08 1024h I

1038 kgm2

ih

106

i: (29)

3 We do not consider the \historical" millisecond pulsar PSR1937+21 for it is more than twice farther away.

5

At rst glance, PSR 1957+20 seems to be a much morefavorable candidate than the Crab or Vela. However, inthe above formula, is in units of 106 and the very lowvalue of the period derivative _P of PSR 1957+20 impliesthat its is at most 1:6 109 (New et al. 1995). Hencethe maximum amplitude one can expect for this pulsar ish1957+200 1:7 1027 and not 1:08 1024 as Eq. (29)might suggest.

3. The incompressible magnetized uid example

As an illustration, let us consider the specic exampletaken by Gal'tsov, Tsvetkov & Tsirulev (1984) (hereafterGTT) and Gal'tsov & Tsevtkov (1984). In their work, apulsar is idealized as a rigidly rotating Newtonian bodymade of an incompressible uid and endoved with a mag-netic eld which is uniform inside the star and dipolaroustide it, the magnetic dipole moment being inclined byan angle with respect to the rotation axis.

The rotation rate is supposed to be far from the mass-shedding limit so that the departure from spherical sym-metry is small. Moreover, the magnetic energy is assumedto be much lower than the rotational kinetic energy, whichis satised by realistic congurations. Under these hy-potheses, the star takes the shape of a (quasi-spherical)triaxial ellipsoid. The gravitational potential can be thencalculated from the analytical formul of Chandrasekhar(1969). The shape of the ellipsoid is deduced from the rstintegral of the equation of motion (including the magneticpressure) and the magnetic eld matching conditions atthe stellar surface. It is found that the ellipsoid is deter-mined by the equation (ij + aij)X

iXj = R2, where (i)R is the mean radius of the (quasi-spherical) ellipsoid, (ii)Xi are Cartesian coordinates in a co-moving frame, i.e. ro-tating at the angular velocity with respect to an inertialframe and (iii) the aij are given by4

aij =15

2

ij

!2J+450322

MiMj

R8!2J: (30)

In this expression, is the constant mass density of thestar, !J =

p4G is the Jeans frequency, i and Mi are

respectively the components of the angular velocity vec-tor and the components of the magnetic dipole momentwith respect to the Xi coordinates: i = (0; 0;) andMi = (0;M sin;M cos). By diagonalizing the matrixaij given by Eq. (30), one obtains the principal axes of theellipsoid and the values of the three semi-axes, a1, a2 anda3. From these quantities the moment of inertia tensorIij can be computed by evaluating the integral (6) in theframe of the principal axes. The quadrupole moment Iij is4 Note that the value of aij presented in GTT [theirEq. (2.15)] is erroneous: the i part has the wrong sign and

the Mi part should contain a R8 factor instead of the R4.

This latter error has been corrected in Gal'tsov & Tsvetkov(1984), but not the former one.

then immediately deduced via Eq. (5). Transforming theresult in the inertial frame leads to the form (7) of Iij, in-cidently demonstrating this formula in the particular caseunder consideration, with the form (8) for Irotij with

Irotzz = R52

3G(31)

and with the form (14) for Idistij with

Idistzz = 0M2

162GR3: (32)

The moment of inertia with respect to the rotation axis ofthe homogeneous star is I = 8R5=15, so that Eq. (23)gives the ellipticity:

=45

64

B2pole

0G2R2: (33)

Note that in this formula, the magnetic dipole momentMhas been expressed in terms of the North pole magneticeld Bpole = 0=(4) 2M=R3.

For numerical estimates, let us take typical values forneutron stars: Bpole = 109 T, M = 1:4M and R = 10km. Then ' 6:0 1011. This is a very tiny value,which leads to a gravitational wave amplitude of onlyh0 31030 for P = 10 ms and r = 1 kpc [cf. Eq. (26)].However the above model is a very simplied one. It canbe expected that relaxing the assumptions of (i) incom-pressible uid, (ii) Newtonian gravity and (iii) uniforminternal magnetic eld, may lead to a greater value of .

4. Magnetic eld induced deformation

4.1. Emission formula

The situation considered in the preceding section is a verysimplied one. However, one may consider that the ob-tained form of the deformation, Eq. (32), is qualitativelythe same for realistic magnetized neutron star models,i.e. a compressible perfect uid obeying a \sophisticated"equation of state resulting from nuclear physics calcula-tions, and involving general relativity. More precisely, weconsider that the magnetic eld induced deformation is aquadratic function of the amplitude of the magnetic dipolemoment,M, as in Eq. (32):

= M2

M20

; (34)

where M0 has the dimension of a magnetic dipole mo-ment in order to make the coecient dimensionless. Letus choose M2

0 = (4=0)GI2=R2 where R is the circum-

ferential equatorial radius of the star5. Hence

= 0

4

M2R2

GI2: (35)

5 For Newtonian stars, R is simply the equatorial radius; for

relativistic stars, R is the length of the equator, as measuredby a locally non rotating observer, divided by 2.

6

Provided the magnetic eld amplitude does not take (unrealistic) huge values (> 1014 T), this formula is certainlytrue, even if the magnetic eld structure is quite compli-cated, depending upon the assumed electromagnetic prop-erties of the uid: normal conductor, superconductor, fer-romagnetic... The coecient measures the eciency ofthis magnetic structure in distorting the star. In the fol-lowing, we shall call the magnetic distortion factor. Forthe simplied model considered in Sect. 3 (incompressible uid, uniform internal magnetic eld), = 1=5.

As argued in Sect. 2.2, the observed spin down of radiopulsars is very certainly due to the low frequency magneticdipole radiation. M is then linked to the observed pulsarperiod P and period derivative _P by [cf e.g. Eq. (6.10.26)of Straumann (1984)]

M2 =4

0

3c3

82IP _P

sin2; (36)

where is the angle between the magnetic dipole momentM and the rotation axis. For highly relativistic congura-tions, the vectorM is dened in the weak-eld near zone(cf. Sect. 2.5 of Bocquet et al. 1995), so is . InsertingEqs. (35) and (36) into Eq. (25) leads to the gravitationalwave amplitude

h0 = 6R2 _P

crP sin2; (37)

so that formul (20) and (21) become

h+ = 6R2 _P

crP

h sin i cos i2 tan

cos(t t0)

1 + cos2 i

2cos 2(t t0)

i(38)

h = 6R2 _P

crP

h sin i

2 tansin (t t0)

cos i sin 2(t t0)i: (39)

Equation (37) can be cast in a numerically convenientform:

h0 = 6:481030

sin2

h R

10 km

i2hkpcr

ihmsP

ih _P

1013

i:(40)

As a check of this equation, let us consider again the sim-plied case of Sect. 3. The uniformly magnetized homo-geneous Newtonian star with M = 1:4M, R = 10 km,Bpole = 109 T and P = 10 ms has a moment of inertiaI = 1:1 1038 kgm2, a factor = 1=5, and magneticdipole momentM = 5:0 1027 Am2. From Eq. (36), thecorresponding period derivative is _P = 2:21012 sin2 .Equation (40) gives then h0 = 3 1030 for r = 1 kpc, inagreement with the result obtained in Sect. 3.

The main feature of the emission formul (38)-(39) isthat the amplitude of the gravitational radiation at thefrequency 2 does not depend upon the (unknown) angle.

4.2. Discussion

Among the 706 pulsars of the catalog by Taylor et al.(1995, 1993), the highest value of h0 at xed , andR, as given by Eq. (40), is achieved by the Crab pulsar(P = 33 ms, _P = 4:21 1013, r = 2 kpc), followedby Vela (P = 89 ms, _P = 1:25 1013, r = 0:5 kpc)and PSR 1509-58 (P = 151 ms, _P = 1:54 1012, r =4:4 kpc):

hCrab0 = 4:08 1031h R

10 km

i2

sin2(41)

hVela0 = 1:81 1031h R

10 km

i2

sin2(42)

h1509-580 = 1:50 1031h R

10 km

i2

sin2(43)

h1957+200 = 4:51 1037h R

10 km

i2

sin2: (44)

We have added to the list the millisecond pulsar PSR1957+20 (P = 1:61 ms, _P = 1:68 1020, r = 1:5 kpc)considered in Sect. 2.5. From the above values, it appearsthat PSR 1957+20 is not a good candidate. This is notsuprising since it has a small magnetic eld (yielding alow _P ). Even for the Crab and Vela pulsars, which havea large _P , the h0 values as given by Eqs. (41), (42) are,at rst glance, not very encouraging. Let us recall thatwith the 1022 Hz1=2 expected sensitivity of the VIRGOexperiment at the 30 Hz frequency (Bondu 1996; see alsoFig. 9 of Bonazzola & Marck 1994), the minimal ampli-tude detectable within three years of integration is

hmin 1026 : (45)

Comparing this number with Eqs. (38)-(39) and (41)-(42),one realizes that in order to lead to a detectable signal, theangle must be small and/or the distortion factor mustbe large. In the former case, the emission is mainly at thefrequency . From Eqs. (38)-(39), the gravitational waveamplitude can even be arbitrary large if ! 0. However,if is too small, let say < 102, the simple magneticbraking formula (36) certainly breaks down. So one can-not rely on a tiny to yield a detectable amplitude6. Thealternative solution is to have a large . Let us recall thatfor an incompressible uid with a uniform magnetic eld, = 1=5 (Sect. 3). In the following section, we give the coecients computed for more realistic models (com-pressible uid, realistic equation of state, general relativ-ity taken into account) with various distribution of themagnetic eld.

4.3. Numerical results

We have developed a numerical code to compute the de-formation of magnetized neutron stars within general rel-

6 From the observed pulse prole and polarization of pulsars,

values of can be inferred; they spread all the range between0 and =2 (Lyne & Manchester 1988, Rankin 1990)

7

Fig. 2. Magnetic eld lines generated by the current distri-bution corresponding to the choice f = const. The thick line

denotes the star's surface. The distortion factor of this cong-

uration is = 1:01.

Fig. 3. Magnetic eld lines generated by a current distribu-

tion localized in the core of the star and corresponding to the

choice f = const. The thick line denotes the star's surface and

the dashed line the external limit of the electric current distri-

bution. The distortion factor corresponding to this situation is

= 0:70.

ativity. This code is an extension of that presented in Bocquet et al. (1995) (hereafter BBGN). The main improve-ments are (i) the use of an arbitrary number of grids to de-scribe the stellar interior, which allows a greater diversityof magnetic eld congurations, and (ii) the possibility ofa type I superconductor interior. We report to BBGN fordetails about the relativistic formulation of Maxwell equa-tions and the technique to solve them. Let us simply recallhere that the obtained solutions are fully relativistic andself-consistent, all the eects of the electromagnetic eldon the star's equilibrium (Lorentz force, spacetime curva-ture generated by the electromagnetic stress-energy) be-ing taken into account. The magnetic eld is axisymmetricand poloidal. The numerical technique is based on a spec-tral method (numerical details can be found in Bonazzolaet al. 1993).

Thanks to the splitting (7), we do not need to takeinto account the rotation to compute the Idistij inducedby the magnetic eld. Consequently we consider static

magnetized neutron star models. The reference (non-magnetized) conguration is taken to be a 1:4M staticneutron star built with the equation of state UV14+TNIof Wiringa, Fiks & Fabricini (1988). This latter is a mod-ern and medium stiness equation of state (cf. Sect. 4.1.2of Salgado et al. 1994). The circumferential radius isR = 10:92 km, the baryon mass 1:56M, the momentof inertia I = 1:23 1038 kgm2 and the central value ofg00 is 0.36, which shows that such an object is highly rel-ativistic. Various magnetic eld congurations have beenconsidered; the most representative of which are presentedhereafter.

4.3.1. Normal case

Let us rst consider the case of a perfectly conductinginterior (normal matter, non-superconducting). As dis-cussed in BBGN, the electric current distribution j can-not be arbitrary in order to lead to a stationary con-guration: it must be related to the covariant ' com-ponent of the electromagnetic potential vector, A', byj' jt = (e + p)f(A), where e and p are respectivelythe proper energy density and pressure of the uid andf is an arbitrary function. The simplest magnetic cong-uration is given by the choice f(x) = const. It results inelectric currents in the whole star with a maximum valueat half the stellar radius in the equatorial plane. The cor-responding magnetic eld distribution is shown (in coor-dinate space) in Fig. 2. The resulting distortion factor is = 1:01, which is above the 1=5 value of the uniformmagnetic eld/incompressible uid Newtonian model con-sidered in Sect. 3, but still very low. This is not surprisingsince the magnetic eld conguration has a very simplestructure: it is certainly not the conguration which max-imizes the deformation at xed magnetic dipole moment.

8

Changing the function f does not lead to a dramaticincrease in : for f(x) = 1(x exp(x2) + 2), we get = 1:07 and for f(x) = 1 cos(x=2), = 1:29.

Our multi-grid code allows to study localized distri-bution of the electric current. Figure 3 corresponds to theelectric current distribution given by f = const from r = 0up to r = 0:5 req, where req is the r coordinate of theequator, and to no electric current for r > r. The distor-tion factor is only = 0:70. This can be understood sincein the absence of electric current, there is no Lorentz forcein the outer part of the star. For electric currents concen-trated deep in the stellar core, the situation is more favor-able. Indeed, since in the regions where j = 0 the magneticeld B falls o as r3, a moderate value ofM can cor-respond to an important value of B at the stellar centre,leading to a substantial deformation of the core, that grav-itationally in uences the rest of the star. For instance, forr = 0:1 req, we get = 5:86.

The opposite situation corresponds to electric currentslocalized in the neutron star crust only. Figure 4 presentsone such conguration: the electric current is limited tothe zone r > r = 0:9 req (f(x) = const:). The resultingdistortion factor is = 8:84.

Fig. 4. Magnetic eld lines generated by a current distribu-

tion localized in the crust of the star and corresponding to thechoice f = const. The thick line denotes the star's surface and

the dashed line the internal limit of the electric current distri-

bution. The distortion factor corresponding to this situation is = 8:84.

Fig. 5.Magnetic eld lines generated by a current distribution

exterior to a type I superconducting core. The thick line de-

notes the star's surface and the dashed line the external limit ofthe superconducting region. The distortion factor correspond-

ing to this situation is = 157.

Fig. 6. Magnetic eld lines generated by some electric currentrotating in one direction for 0 r 0:3 req and in the opposite

direction for 0:9 req r req. The thick line denotes the

star's surface and the dashed lines the limits of the electriccurrent distributions. The distortion factor corresponding to

this situation is = 5:7 103.

9

4.3.2. Type I superconductor

Let us consider the case of a superconducting interior, oftype I, which means that all the magnetic eld has beenexpulsed from the superconducting region. In the cong-uration depicted in Fig. 5, the neutron star interior is su-perconducting up to r = 0:9 req. For r > r, the matteris assumed to be a perfect conductor carrying an electriccurrent which corresponds to f(x) = const. The resultingdistortion factor is much higher than in the normal case: = 157. For r = 0:95 req, is even higher: = 517.

4.3.3. Counter-rotating electric currents

The above values of , of the order 102103, though muchhigher than in the simple normal case (Sect. 4.3.1), arestill too low to lead to an amplitude detectable by the rstgeneration of interferometric detectors in the case of theCrab or Vela pulsar [cf. Eqs. (41), (42) and (45)]. It is clearthat the more disordered the magnetic eld the higher ,the extreme situation being reached by a stochastic mag-netic eld: the total magnetic dipole moment M almostvanishes, in agreement with the observed small value of _P ,whereas the mean value of B2 throughout the star is huge.Note that, according to Thompson & Duncan (1993), tur-bulent dynamo amplication driven by convection in thenew-born neutron star may generate small scale magneticelds as strong as 3 1011 T with low values of Bdipole

outside the star and hence a large .In order to mimic such a stochastic magnetic eld, let

us consider the case of counter-rotating electric currents,namely j is given by (i) f(x) = f1 for 0 r r1, wheref1 is a constant, (ii) f(x) = f2 for r

2 r req, where f2is a constant with the opposite sign than f1 and (iii) noelectric current for r1 < r < r2. Figure 6 corresponds tothe case r1 = 0:3 req, r

2 = 0:9 req and f1=f2 = 5:45. Theresulting distortion factor is = 5:7 103. The value ofmagnetic dipole moment,M = 5:11027 Am2, is similarto that of the Crab pulsar (assuming 1), but theamplitude of the magnetic eld at the star's centre, Bc =5:71012 T = 5:71016 G is 104 times higher than thepolar value deduced from a simple dipole model. Clearlyfor such congurations can be made arbitrary large byadjusting the parameters f1 and f2.

4.3.4. Type II superconductor

It is not clear if the protons of the neutron star interiorform a type I (Sect. 4.3.2) or a type II superconductor(P. Haensel, private communication). In the latter case,the magnetic eld inside the star is organized in an arrayof quantized magnetic ux tubes, each tube containing amagnetic eld Bc 1011 T (Ruderman 1991). Besides,the neutrons constitute a super uid, with quantized vor-tices. As the neutron star spins down, the neutron vorticesmigrate away from the rotation axis. As discussed by Rud-erman (1991, 1994), the magnetic ux tubes are forced to

move with them. However, they are pinned in the highlyconducting crust. This results in crustal stresses of the or-der BcB=20 [Ruderman 1991, Eq. (10)], where B is themean value of the magnetic eld in the crust (B 108 Tfor typical pulsars). This means that the crust is submit-ted to stresses 103 higher than in the uniformly dis-tributed magnetic eld considered in Sect. 4.3.1 (compareBcB=20 with B

2=20). The magnetic distortion factor should increase in the same proportion. We have not doneany numerical computation to conrm this but plan tostudy type II superconducting interiors in a future paper.

5. Signal received by an interferometric detector

Due to the weakness of the expected gravitational signalfrom pulsars, long integration times, typically of the or-der of the year, are required to extract the signal out ofthe noise. For such a long observing time, the motion ofthe Earth must be taken into account for it modies theposition of the pulsar with respect to the antenna pat-tern of the interferometric detector. It results in a mod-ulation of both the amplitude and the frequency of thesignal (Jotania, Valluri & Dhurandhar 1995). Note that ifone is searching for a known pulsar, the frequency mod-ulation (Doppler shift) can be obtained readily from theradio observations.

In this section, we examine the daily modulation of thesignal amplitude due to the Earth's rotation. The specialcase of an interferometric detector situated on the equatorwith arms symmetrically placed about the North-Southdirection and a gravitational wave coming from the North-ern celestial pole has been treated by Jotania et al. (1995).

5.1. Beam-pattern factors

Let (e~x; e~y; e~z) be the orthonormal frame associated withthe gravitational wave: e~z is perpendicular to the waveplane and parallel to the \line of sight", being orientatedfrom the neutron star centre to the Earth centre; e~x is theunit vector of the wave plane which is perpendicular to thestar's rotation axis. In the (e~x; e~y; e~z) frame the transversetraceless gravitational wave is expressible from its two po-larization modes h+ and h as given by Eqs. (20)-(21) or(38)-(39):

h = h+ (e~x e~x e~y e~y) + h (e~x e~y + e~y e~x) :(46)

The response of an interferometric detector of theVIRGO/LIGO type to the above gravitational wave de-pends upon the relative orientation of wave frame withrespect to the detector's arms. Let (eX ; eY ; eZ) be theorthonormal frame such that eZ is perpendicular to thedetector plane, pointing toward the zenith and eX andeY are unit vectors along the two detector's arms. Let(;;) be the three Euler angles which specify the po-sition of the wave frame (e~x; e~y; e~z) with respect to the

10

XY

z

z

~

~

y

y

~

~

~

~

x

x

z

y

i

Ω

Θ

Ψ

Φ D

W

W

S

Fig. 7. Relative orientation of various orthonormal frames introduced in the text: (ex;ey;ez) is the neutron star inertial

(non-rotating) frame, associated with the ACMC coordinates (x;y; z); (ex;ey) dene the plane S perpendicular to the rotation

axis. (e~x;e~y;e~z) is the frame associated with the gravitational wave received on Earth; (e~x;e~y) dene the wave plane W. Notethat e~x = ex. (eX ;eY ;eZ) is the interferometric detector frame; (eX ;eY ) dene the detector plane D.

detector frame (eX ; eY ; eZ) (cf. Fig. 7). The signal mea-sured by the detector is

h(t) = F+(;;)h+(t) + F(;;)h(t) ; (47)

with the following beam-pattern factors [cf. e.g.Eqs. (103)-(104) of Thorne (1987) or Eq. (7a) of Dhurand-har & Tinto (1988)] :

F+(;;) =1

2(1 + cos2) cos 2 cos 2

cos sin 2 sin 2 (48)

F(;;) =1

2(1 + cos2) cos 2 sin 2

+cos sin 2 cos 2 : (49)

The Euler angles (;;) are not constant because of themotion of the detector with respect to the source inducedby the Earth's diurnal rotation and revolution around theSun. To compute their variations let us introduce the \ce-lestial sphere frame" (eX0 ; eY 0 ; eZ0) such that eZ0 is alongthe Earth's rotation axis, pointing toward the North pole,eX0 and eY 0 are in the Earth's equatorial plane, eX0 point-ing toward the vernal point (i.e. eX0 is along the inter-section of the Earth's equatorial plane with the Earth'sorbital plane). For time scales of the order of the year,(eX0 ; eY 0 ; eZ0) can be considered as xed with respect tothe (approximatively) inertial frame containing the solarsystem barycentre and the neutron star centre. The waveframe (e~x; e~y; e~z) is also xed with respect to this iner-tial frame. Then let (0;0; ) be the Euler angles which

specify the position of the wave frame with respect to thecelestial sphere frame (cf. Fig. 8). 0 and 0 are simplyrelated to the equatorial coordinates of the neutron staron the celestial sphere (the right ascension7 and thedeclination ) by (cf. Fig. 8):

0 = +

2and 0 =

2: (50)

The triad (e~x; e~y; e~z) is related to the triad (eX0 ; eY 0 ; eZ0)by

e~i = A~iJ 0 eJ 0 ; (51)

with the orthogonal matrix [cf. e.g. Eq. (4-46) of Goldstein(1980) with the substitution (50)]

A~iJ 0 =

0@ sin cos cos sin sin sin sin cos sin cos

cos cos

cos cos sin sin sin cos sin cos sin sin sin cos cos cos

sin cos sin

1A : (52)

Let (eX00 ; eY 00 ; eZ00) be the orthonormal frame linkedto the geographical location of the detector site, such thateZ00 = eZ is the local vertical, eX00 is in the North-Southdirection and eY 00 is in the West-East direction (cf. Fig. 9).The position of the wave frame (e~x; e~y; e~z) with respect to

7 a bar is put on to distinguish it from the angle betweenthe magnetic and rotation axis introduced earlier in the text.

11

Z’

X’

Y’

z~

y~

x~Θ’

’Φ α_

_δ

ψ

pulsar

celestial pole

Fig. 8. Relative orientation of the gravitational wave frame

(e~x;e~y;e~z) and the celestial sphere frame (eX0 ;eY 0 ;eZ0).

Φ

λa

X

Y

N

EW

S

pulsarwaveplane

Fig. 9. Orientation of the detector arms with respect to thelocal North-South and West-East directions. The dashed line

is the projection of the gravitational wave propagation direc-

tion on the horizontal plane. The depicted conguration cor-

responds to that of the VIRGO detector ( = 161o 330).

the cardinal frame (eX00 ; eY 00 ; eZ00) is determined by thethree Euler angles (;00;) where and are the sameangles as those relative to the detector frame and enteringEqs. (48)-(49). The angle 00 is related to by

= 00 + ; (53)

where is the azimuth of the X-arm of the detector, i.e.the angle between the South direction and the X-arm, as

measured in the retrograd way. is directly related tothe azimuth a of the source by

00 = a

2: (54)

The minus sign which occurs in this relation comes fromthe fact that the azimuth is measured westwards from theSouth, hence in the inverse trigronometric way. Puttingtogether relations (53) and (54) we obtain

= a+

2: (55)

Let (00; '00; 00) be the Euler angles which specify the po-sition of the cardinal frame (eX00 ; eY 00 ; eZ00) with respectto the celestial sphere frame (eX0 ; eY 0 ; eZ0). One has im-mediatly 00 = =2 because eX00 is orientated in theNorth-South direction. 00 is simply related to the latitudel of the detector site by

00 =

2 l : (56)

'00 is linked to the local sidereal time T (i.e. the anglebetween the local meridian and the vernal point) by

'00 = T +

2: (57)

T can be expressed in terms of the sidereal time at Green-wich at 0h UT, TGreen(0), and the local UT time t by

T (t) = t+ TGreen(0) L ; (58)

where = 1:00273790935 15o=h (Meeus 1991) is a con-version factor frommean solar time to sidereal time (hence accounts for the revolution of the Earth around the Sun)and L is the longitude of the detector site (the minus signin front of L comes from the fact that the geographicallongitudes are measured positively westwards). The triad(eX00 ; eY 00 ; eZ00) is related to the triad (eX0 ; eY 0 ; eZ0) by

eI00 = BI00J 0 eJ 0 ; (59)

with the orthogonal matrix

BI00J 0 =

0@sin l cos T sin l sin T cos l sin T cosT 0

cos l cosT cos l sinT sin l

1A : (60)

Now the wave frame (e~x; e~y; e~z) is related to the triad(eX00 ; eY 00 ; eZ00) by

e~i = C~iJ 00 eJ 00 ; (61)

with the orthogonal matrix

C~iJ 00 =

0@ cos cos 00 cos sin 00 sin sin cos 00 cos sin 00 cos

sin sin 00

cos sin 00 + cos cos00 sin sin sin sin sin 00 + cos cos00 cos cos sin

sin cos00 cos

1A :(62)

12

From equations (51), (59) and (61), the matrix C is givenby the product

C = A tB : (63)

Performing the right-hand-side product and identifyingeach matrix element by those of expression (62) leads tothe following trigonometrical relations

cos = cos cos l cosH(t) sin sin l (64)

sin cos a = cos sin l cosH(t) sin cos l (65)

sin sin a = cos sinH(t) (66)

sin sin = cos cos l sinH(t)

sin sin cos l cosH(t) + sin cos sin l (67)

sin cos = cos cos sin l

+sin cos l sinH(t) cos sin cos l cosH(t); (68)

where we have used the relation (54) between 00 and a

and have introduced the local hour angle of the source:

H(t) = T (t) = t + TGreen(0) L : (69)

From the above relations the beam-pattern factorsF+(;;) and F(;;) can be computed for anyinstant t, given the position (l; L) of the detector on theEarth, its orientation , and the position (; ) of thesource on the sky, as well as the angle that the vec-tor e~x forms with the intersection line of the wave planeand the Earth's equatorial plane. First one computes thelocal hour angle H(t) by Eq. (69), and the angle byEq. (64). The angle is computed by means of Eqs. (55),(65) and (66). Finally, the local polarization angle isdeduced from Eqs. (67) and (68).

5.2. Signal from a rotating magnetized neutron star

According to Eqs. (38)-(39), the gravitational wave sig-nal from a rotating neutron star slightly deformed by itsmagnetic eld is

h+(t) = h+1 cos(t t0) + h+2 cos 2(t t0) (70)

h(t) = h1 sin (t t0) + h2 sin 2(t t0) ; (71)

with

h+1 = ~h0sin i cos i

2 tan(72)

h+2 = ~h0(1 + cos2 i)=2 (73)

h1 = ~h0sin i

2 tan(74)

h2 = ~h0 cos i (75)

~h0 := 6R2

cr

_P

P(76)

Note that in Eqs. (70) (71) t0 accounts for a dierent timeorigin between the neutron star frame (where Eqs. (38)-(39) have been derived) and the detector frame. Formallyt0 accounts also for the propagation delay r=c.

According to the above formul, when the location(; ) of the source is known, the signal h(t) measured bythe detector depends on four a priori unknown parameters:

the inclination angle i of the line of sight with respectto the rotation axis of the star;

the angle between the magnetic axis and the rotationaxis;

the polarization angle (cf. Figs. 7 and 8) : measuresthe orientation of the major axis of the ellipse formedby the projection of the neutron star's equator on theplane of the sky;

the time t0 such that t0 r=c is the instant whenthe magnetic dipole moment vectorM is in the planeformed by the rotation axis and the line of sight, andwhen the scalar productM e~z is positive.

The signal measured by the detector can be written withthe explicit dependence upon these parameters:

h(t) = F+

t

tSD;

h+1(i; ) cos(t t0)

+F

t

tSD;

h1(i; ) sin(t t0)

+F+

t

tSD;

h+2(i) cos 2(t t0)

+F

t

tSD;

h2(i) sin 2(t t0) (77)

where tSD is the duration of one sidereal day: F+ and Fare periodic functions of t=tSD, with period one.

Let us introduce the amplitudes of the signal at thefrequencies and 2 respectively:

A1 :=qF 2+ h

2+1 + F 2

h21 (78)

A2 :=qF 2+ h

2+2 + F 2

h22 (79)

The daily variation of A1 and A2, computed fromEqs. (48)-(49) and (64)-(69), is represented in Figs. 10-12for the specic case of the VIRGO detector [L = 10o 300,l = +43o 400 and = 1610 330 (P. Hello, private com-munication)] and the Crab pulsar ( = 5h 34min, =+22o 010). Note that generally the amplitude of the signalat both and 2 is maximum within a few hours of theinstant when the Crab pulsar crosses the local meridian(local sidereal time T = 5h 34min). Figs. 10-12 show thatthe measure of the time-varying amplitude at one of thetwo frequencies and 2 allows to determine the polar-ization angle and the inclination angle i. If the signalis recorded at both frequencies, the angle between themagnetic axis and the rotation axis can be determined aswell. Note that this angle is a fundamental parameter forthe theory of pulsar magnetospheres.

13

Fig. 10. Relative amplitude of the signal at the frequency (A1, top) and 2 (A2, bottom) as a function of the local UT timet (assuming a vanishing local sidereal time at 0 h UT), for the specic case of the VIRGO detector and the Crab pulsar. The

polarization angle is = 0o. Each plot corresponds to a given value of the inclination angle i. Various lines in the same plot

correspond to various values for the angle between the magnetic axis and the rotation axis: solid line: = 15:00o, dashedline: = 33:75o, dot-dash-dot-dash line: = 52:50o, dotted line: = 71:25o, dash-dot-dot-dot line: = 90:00o. ~h0 is dened

by Eq. (76).

6. Conclusion

We have considered the gravitational radiation emitted bya distorted rotating uid star. The distortion is supposedto be symmetric with respect to some axis which does notcoincide with the rotation axis. The gravitational emis-sion takes place at two frequencies: and 2, where is the rotation frequency, except in the particular casewhere the distortion axis is perpendicular to the rotationaxis (only the frequency 2 is then present). As an appli-cation, the magnetic eld induced deformation is treated.If, as usually admitted, the period derivative, _P , of pul-sars is a measure of their magnetic dipole moment, thegravitational wave amplitude can be related to the observ-able parameters P and _P of the pulsars and to a factor which measures the distortion response of the star to agiven magnetic dipole moment. depends on the nuclear

matter equation of state and on the magnetic eld distri-bution. The amplitude at the frequency 2, expressed interms of P , _P and , is independent of the angle be-tween the magnetic axis and the rotation axis, whereas atthe frequency , the amplitude increases as decreases.

Using a numerical code generating self-consistent mod-els of magnetized neutron stars within general relativity,we have computed the deformation for explicit models ofthe magnetic eld distribution and a realistic equationof state. It appeared that the distortion at xed mag-netic dipole moment depends very sensitively on the mag-netic conguration. The case of a perfect conductor in-terior with toroidal electric currents is the less favorableone, even if the currents are concentrated in the crust.Stochastic magnetic elds (that we modeled by consider-ing counter-rotating currents) enhance the deformation by

14

Fig. 11. Same as Fig. 10 but for the polarization angle = 30o.

several orders of magnitude and may lead to a detectableamplitude for a pulsar like the Crab. As concerns super-conducting interiors | the most realistic conguration forneutron stars | we have studied numerically type I su-perconductors, with a simple magnetic structure outsidethe superconducting region. The distortion factor is then 102 to 103 higher than in the normal (perfect conduc-tor) case, but still insucient to lead to a positive detec-tion by the rst generation of kilometric interferometricdetectors. We have not studied in details the type II su-perconductor but have put forward some argument whichmakes it a promising candidate for gravitational wave de-tection. Due to the complicated microphysics involved intype II superconductors we delay their study to a futurepaper. We also plan to study the deformation induced bya possible ferromagnetic solid interior of neutron stars, aswell as the eects of a strong toroidal internal magneticeld.

Regarding the reception of gravitational waves from apulsar by an interferometric detector, we have computedthe amplitude modulation of the signal induced by the di-

urnal rotation of the Earth. By inspecting the wave form,and assuming the position of the pulsar to be known (thepulsar can be recognized by its period), one can determinethe inclination angle of the line of sight with respect to thepulsar rotation axis, as well as the orientation of the pul-sar equatorial plane. Moreoever, by comparing the waveforms at the frequencies and 2, the angle betweenthe rotation axis and the magnetic axis can be determined.

Pulsars may be good candidates for the detection bythe forthcoming VIRGO and LIGO interferometric detec-tors. A frequently invoked mechanism for gravitationalemission concerns asymmetries of the neutron star solidcrust and the resulting precession. In this article, we haveexamined instead the bulk deformation of the star inducedby its own magnetic eld. For some congurations of themagnetic eld (stochastic distribution, type II supercon-ductor), the deformation may be large enough to lead toa detectable signal by VIRGO, with the total magneticdipole moment (or equivalently the surface magnetic eld)keeping its (relatively small) observed value. The positivedetection of gravitational waves from pulsars would lead

15

Fig. 12. Same as Fig. 10 but for the polarization angle = 60o. The case = 90o is identical to = 0o (Fig. 10).

to some constraints on the internal magnetic eld distri-bution, which would be of great interest for the theories ofpulsar magnetospheres. This would constitute an exampleof a signicant contribution of gravitational astronomy toclassical astrophysics.

Acknowledgements. We warmly thank Francois Bondu andPatrice Hello for useful discussions and for checking the calcu-

lations presented in Sect. 5. We are also indebted to Sreeram

Valluri for his careful reading of the manuscript. The numerical

calculations have been performed on Silicon Graphics work-

stations purchased thanks to the support of the SPM depart-

ment of the CNRS and the Institut National des Sciences del'Univers.

A. From QI to ACMC coordinates

Most studies of stationary rotating neutron stars makeuse of quasi-isotropic (QI) coordinates (t; r0; 0; ') (cf. thediscussion in Sect. 2 of Bonazzola et al. 1993). In thesecoordinates, the spatial components of the metric tensor

have the following asymptotic behavior:

gr0r0 = 1 +1

r0+11 cos 2

0 + 10

r02+O

1

r03

(A1)

g00 =

1 +

1

r0+11 cos 2

0 + 10

r02+O

1

r03

r02(A2)

g'' =

1 +

3

r0+30

r02+O

1

r03

r02sin2 0 ; (A3)

where 1, 3, 10, 11 and 30 are some constants. Bycomparison with the denition (4), it appears that suchcoordinates are not ACMC to order 1: in order to be so,the 1=r0

2term in gr0r0 and g00 should not contain any

cos 20, i.e. 11 should vanish. It can be seen easily that thefollowing coordinate transformation leads to an ACMCcoordinate system (t; r; ; '):

r0 = r +11 cos

2

r(A4)

0 = 11 cos sin

r2: (A5)

16

By computing the g00 component in the (t; r; ; ') coordinates from the components g00 in the (t; r0; 0; ') co-ordinates and by identication with Eq. (2), one obtainsthe following value of Thorne's mass quadrupole moment:

Ixx = Iyy = 12Izz (A6)

Izz =4

9( 2 M11) (A7)

Iij = 0 if i 6= j ; (A8)

where (i) 2 is the coecient of cos 20 in the 1=r0

3term of

the 1=r0 expansion of the metric component g00 in the QIcoordinates (t; r0; 0; ') and (ii) M is half the coecientof 1=r0 in the same expansion (M is nothing else than thetotal gravitational mass of the star).

To summarize, Thorne's quadrupole moment compo-nent Izz can be computed via equation (A7) by readingo the coecients 11 and 2 in the 1=r

0 expansions of themetric components in the QI coordinates.

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