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On the gravitational radiation of an oblique rotator
PHYSICS LETTERS
Volume 103A, number 4
ON THE GRAVITATIONAL
RADIATION
2 July 1984
OF AN OBLIQUE ROTATOR
D.V. GAL’TSOV Moscow State University, I 17234, Moscow, USSR and V.P. TSVETKOV Kalinin State University, 170013,
Kalinin, USSR
Received 26 April 1984
The magnetic hydrodynamical model of an oblique rotator with arbitrary inclination angle is given. It is shown that in its gravitational radiution spectra there are the first and the second harmonics of the rotation frequency, the first being completely due to the newtonian stresses. Generation of the first harmonic is not accompanied by precession of the magnetic axis. Polarisation of gravitational radiation is also calculated for both harmonics.
The recent discovery of fast pulsars with millisecond pei-iods [ 1,2] provides new grounds for the projects aimed at the detection of monochromatic gravitational waves emitted by neutron stars. Though for PSR 1937 + 214 the observed deceleration parameter seems to be small for such an experiment (f/T lo-l9 s-l), one can hope to find new objects with greater values of ?/T. It should also be noted that the gravitational radiation power may actually be underestimated if it is based on deceleration considerations only. As is known from the theory of rotating selfgravitating fluid configurations, there are figures (Jacobi ellipsoids) that even accelerate [3] under the influence of gravitational radiation, the energy being supplied by the newtonian energy of the body. The above arguments inspired us to recalculate the gravitational radiation of a neutron star using a more elaborate model of the oblique rotator than has been done by Ostriker and Gunn [4] and later workers. We take as a model a rotating self-gravitating fluid drop with the internal magnetic field inclined to the rotation axis at an arbitrary angle (Y(earlier ones were taken with CY= 0 [S] and (Y= 71/2 [4]). Our model is solved analytically in perturbation theory with the small parameters the ratio of the rotation frequency to the Jeans frequency w/wJ (of = 4nGp, the fluid
density p assumed to be constant) and the ratio of the magnetic energy of the body B2R3 to its newtonian energy Gp2R5, i.e. fl= B2/(oipR2) < 1. It is believed [4] that the gravitational radiation of a rotating neutron star must have the frequency 20 (doubled rotation frequency). When a possible mechanical precession of the star is taken into account [6], the first harmonic of the rotation frequency may also appear. However, observations do not seem to confirm the existence of precession [7] which should influence also the electromagnetic signals. In the absence of precession the standard model [4] predicts only the second harmonic in the gravitational radiation spectra from pulsars. However, these considerations are based on a purely mechanical model in which the contribution of the magnetic field to the total angular momentum of the body is not taken into account. Actually the mechanical and electromagnetic angular momenta of the pulsar are not conserved separately, and hence an effect which imitates mechanical precession may appear without any change of the inclination angle of the magnetic axis with respect to the rotation axis. Our model accounts accurately for such effects. In addition we do not apply the ordinary quadrupole radiation formula to calculate the gravitational radiation power but instead we calculate the contributions 193
Volume 103A, number 4
from the kinetic term, the newtonian and magnetic stresses and anisotropic pressure separately in order to analyse the physical origin of the first and second harmonics. The calculation along these lines has led to the following results: (i) For values of the inclination angle (Yf (0, fn) there are two frequencies w and 2w in the gravitational radiation spectrum (for (IL= 0 or $71there is only the frequency 2w [4]). (ii) The contribution to the first harmonic is given completely by the newtonian stresses, while the contribution to the second harmonic is composed in equal parts by the kinetic term and the newtonian stresses. It should be emphasized that the presence of the first harmonic in the gravitational radiation spectrum is not accompanied by the precession of the pulsar magnetic axis and consequently does not involve any modulation of electromagnetic signals. Owing to the non-conservation of purely mechanical angular momentum, the axis of rotation in our model does not coincide with any of the principal axes of inertia of the body, while the angle between the rotation and magnetic axes is conserved. Consider the rotating self-gravitating drop of homogeneous fluid of quasispherical shape possessing an internal homogeneous magnetic field Bin = 2c/R3, matched with the dipole field B out = [3(jl*r)r
-p2]r-5
)
(1)
in the near external region. In the absence of rotation the drop is assumed to be spherical with radius R, and its deformation caused by rotation will be described by the equation (r2 - R2 + aiiXiXj>= 0 ,
(2)
where Uij Q 1 is a small dimensionless deformation tensor. The total stress-energy-momentum tensor of the system will be the sum of four contributions: T/l” = P,” +b/l” +tfiv +7/l,
3
(3)
where p,, is the kinetic term, b,, the magnetic field stresses, tpv the newtonian stresses and 7PV the anisotropic pressure tensor. In the following we need only the space components of these quantities pii =
PViVi
+p6ii
,
tii = (1/4nG)[@,i@,i 194
bii = -( 1/4n)(BiBi -
2 July 1984
PHYSICS LETTERS
!i~ii(V4)21 9
- iSiiB2) , (4)
where p = const is the density of the fluid, p the pressure,u(r, t) the velocity field, @the newtonian gravitational potential and G the gravitational constant. The tensor of anisotropic pressure will be calculated using the equilibrium condition in the rotating reference frame. From the conservation equation aTpv/axv = 0 [the derivative is not covariant since the gravitational stresses have already been taken into account in (3)!] we obtain the following equilibrium conditions p + p$ - ipv2 + B2/8n = const , (a/axi)(7ij
- BiBj/4~) = 0 .
(5)
The newtonian potential can be obtained using the standard general expression for the case of a triaxial ellipsoid, in the quasispherical case we are interested in @in = wJ2(-iR2
+ ir2 - &ar2 + ~aijXiXj + iaR2), (6)
inside the body and 4 out = w:R3 [-1/3r + (a/6r)(l - R2/5r2) + (R3/10r5)aijxixj]
,
(7)
outside of it (U = tr aij). According to eq. (5) and the Maxwell equations, at the surface Z of the drop the following boundary conditions must hold
TijI~nj=(l/4n)(-Bo,,
i +Bin i)(B*n)
9
(8)
where n is the outer normal vector. In the first nonzero approximation in terms of the small quantities a/aJ and fl the deformation tensor aij will be obtained as follows Qij=-(l5/2w:)wiwi
+(45R-8/8nw:p)/_+~j.
In the same approximation tensor will have the form Tij = (3R-8/2K)[pi/+r2
the anisotropic
- 6ij(lr.r)2 J .
(9) pressure
(10)
As it is seen from eq. (8) our drop will in general have the form of a triaxial ellipsoid with its rotation axis inclined with respect to the magnetic axis. (Note that in account of the deformation (9) the magnetic field inside the body will not be strictly homogeneous but
PHYSICS LETTERS
Volwne 103A, number 4
acquire small corrections, responsible for matching with the external field (1). However, these corrections are of higher order and can be neglected in our further calculations). Now we turn to the calculation of the Riemann tensor describing the gravitational radiation of the rotating magnetised drop in the wave zone. The response of the presumed detector is determined by the components Rojui which in the transverse traceless gauge have the simple form ROiOj = -:a2hijlat2
.
JTii(k,
,
(12)
n) exp(ik*n
- inwt)
2
,
(13)
and an analogous transformation for hii, and introducing the standard polarisation tensors e$, t$ orthogonal to the vector k, we get for the two independent polarisation states the following solutions of eq. (12) in the wave zone h+,x = _ 4G _R il=_-m 5 T+*X(noR/R,
n) exp[-ino(t-R)],
(14) where R is the radius-vector from the pulsar to the observation point. The total gravitational radiation intensity can be expressed through the quantities T+*x as foilows f =;
ij n-
(rz~~)~(lT~l~ f lT+12).
For the case of a non-relativistic
n) ,
(16)
to put eq. (15) into the usual quadrupole form. Then the standard calculation gives for the total intensity the sum of two harmonics, n = 1 and n = 2, 4w6/10GM2)
sin2cu cos2a,
12, = (8~ 4w6/.5GM2) sin4a,
t)
= F
Ti&k, H) = ;(nw)2(a2/akiakj)To0(k,
r,=Cu
where Tii in the r.h-.s. is the space part of the total stress tensor (3) including newtonian stresses inside the body and in the near external region and magnetic stresses in the same regions (in our scheme we do not need to match explicitly the gravitational and the electromagnetic fields in the near and far zones, because the contributions of the near outer stresses to the retarded solution of eq. (12) are convergent), Using for Tij the Fourier-transformation T&
surface it is sufficient to calculate the Fourier transform (13) in the long-wavelength approximation talcing exp(ik*r) = 1. Note that if we are not interested in the calculation of the separate contributions to hji from the terms in the sum (3) we can use the conservation condition in the form [8]
(11)
The metric perturbations hii are to be determined from the usual wave equation
qhii = 16nGTij
2 July 1984
(1%
velocity of the pulsar
(17) (18)
where M is the total mass of the body. As it is seen from eq. (17) lw = 0 for (Y= 0 and (Y= n/2, but Z, # 0 for all other values of CY.For small cr the first harmonic dominates. Now we want to show that the radiation at the first harmonic is completely due to the contribution of the newtonian stresses in the sum (3). For this purpose we calculate Ti&k, n) from eq. (13) explicitly as the sum of separate contributions from pii, bij, tij and rij. In first-order terms of w/t_dJ and fi we find that pii gives no contribution, while the other terms mutually cancel: rii(k, n) = 6tq(k, n) = -$bq(k,
n)
2n =
d(wt) s
0
ViPj
-
10nR3
.
(19)
This cancellation is due to the absence of the dipole term in the gravitational radiation power. All contributions of second order are of quadrupole type. The calculation shows that in this order the magnetic stresses and anisotropic pressure gives zero contribution while the contributions coming from the kinetic term and the newtonian stresses are of the form
(20)
Volume
103A, number
PHYSICS
4
where /A,,i, ~Ecliare magnetic moment components parallel and perpendicular to the rotation axis. In the coordinate frame having its z-axis aligned with the axis of rotation, the components of the magnetic moment are pL = /_tsin 0 (cos ot, sin at) ,
p,, = p cos a
eqs.
(20),
R’ = -w2(h1 + 2ah2(
(22)
where the dimensionless amplitudes h 1 and h2 are connected with the intensities of gravitational radiation at the first and the second harmonics by the relations h, = (uR)-‘(~OGI,/~)~/~
,
h, = (wR)-1(5G12w/8)1/2
.
(22)
that the kinetic term pij has non-zero Fourier-transform only if n = 2, i.e. it contributes only to the second harmonic. The newtonian stresses cii have non-zero Fourier-transform for both n = 1 and n = 2, the contribution with n = 1 being exactly equal to that contained in (17). Hence the generation of the first harmonic can be considered as the effect of newtonian stresses (however one should not attach too much meaning to this statement because the separation of the total Tq into the sum of nonconserved quantities (3) leads to the loss of gauge invariance of the separate contributions into hii). It is interesting that the contributions of the mass and newtonian stresses to the second harmonic of gravitational radiation appear to be exactly equal. Now we give the final expressions for the projections of the Riemann tensor R’x = -fa2h’>xlat2 corresponding to two independent polarization states From
sin 20 cos wt/2fi
The use of the polarization properties of radiation in the geterodyne detection project leads to an effective enhancement of the sensibility of the detector [9]. References [ 1] M.J. Rees, R.D. Blanford [ 21 [ 31 [4] [ 5) [6] [7] [8]
1 + cos20) cos 2wt) , [ 91
196
(24)
it is obvious
RX = w2(h1 sin 13sin at/& t 44~0s
2 July 1984
LETTERS
Oh, sin 2wt),
(23)
and J.P. Ostriker, Nature 300 (1982) 615. V. Boriakoff, R. Buccheri and F. Eauci, Nature 304 (1983) 417. M. Rees, R. Ruffini and J. Wheeler, Black holes, gravitational waves and cosmology (London, 1974). J.P. Ostriker and J.E. Gunn, Astrophys. J. 157 (1969) 1395. S. Chandrasekhar and E. Fermi, Astrophys. J. 118 (1953) 116. M. Zimmermann and Szedenits Jr., Phys. Rev. 20D (1979) 351. G.R. Huquenin, J.H. Taylor and D.L. Helfand, Astrophys. J. 181 (1973) L139. D.V. Gal’tsov. Yu.V. Grats and V.I. Petukhov, Gravitational radiation from electrodynamical systems (Moscow Univ. Press, Moscow, 1984). D.V. Gal’tsov, V.P. Tsvetkov and A.N. Tsirulev, Zh. Eksp. Theor. Fiz. (1984). to be published.
Volume 103A, number 4
ON THE GRAVITATIONAL
RADIATION
2 July 1984
OF AN OBLIQUE ROTATOR
D.V. GAL’TSOV Moscow State University, I 17234, Moscow, USSR and V.P. TSVETKOV Kalinin State University, 170013,
Kalinin, USSR
Received 26 April 1984
The magnetic hydrodynamical model of an oblique rotator with arbitrary inclination angle is given. It is shown that in its gravitational radiution spectra there are the first and the second harmonics of the rotation frequency, the first being completely due to the newtonian stresses. Generation of the first harmonic is not accompanied by precession of the magnetic axis. Polarisation of gravitational radiation is also calculated for both harmonics.
The recent discovery of fast pulsars with millisecond pei-iods [ 1,2] provides new grounds for the projects aimed at the detection of monochromatic gravitational waves emitted by neutron stars. Though for PSR 1937 + 214 the observed deceleration parameter seems to be small for such an experiment (f/T lo-l9 s-l), one can hope to find new objects with greater values of ?/T. It should also be noted that the gravitational radiation power may actually be underestimated if it is based on deceleration considerations only. As is known from the theory of rotating selfgravitating fluid configurations, there are figures (Jacobi ellipsoids) that even accelerate [3] under the influence of gravitational radiation, the energy being supplied by the newtonian energy of the body. The above arguments inspired us to recalculate the gravitational radiation of a neutron star using a more elaborate model of the oblique rotator than has been done by Ostriker and Gunn [4] and later workers. We take as a model a rotating self-gravitating fluid drop with the internal magnetic field inclined to the rotation axis at an arbitrary angle (Y(earlier ones were taken with CY= 0 [S] and (Y= 71/2 [4]). Our model is solved analytically in perturbation theory with the small parameters the ratio of the rotation frequency to the Jeans frequency w/wJ (of = 4nGp, the fluid
density p assumed to be constant) and the ratio of the magnetic energy of the body B2R3 to its newtonian energy Gp2R5, i.e. fl= B2/(oipR2) < 1. It is believed [4] that the gravitational radiation of a rotating neutron star must have the frequency 20 (doubled rotation frequency). When a possible mechanical precession of the star is taken into account [6], the first harmonic of the rotation frequency may also appear. However, observations do not seem to confirm the existence of precession [7] which should influence also the electromagnetic signals. In the absence of precession the standard model [4] predicts only the second harmonic in the gravitational radiation spectra from pulsars. However, these considerations are based on a purely mechanical model in which the contribution of the magnetic field to the total angular momentum of the body is not taken into account. Actually the mechanical and electromagnetic angular momenta of the pulsar are not conserved separately, and hence an effect which imitates mechanical precession may appear without any change of the inclination angle of the magnetic axis with respect to the rotation axis. Our model accounts accurately for such effects. In addition we do not apply the ordinary quadrupole radiation formula to calculate the gravitational radiation power but instead we calculate the contributions 193
Volume 103A, number 4
from the kinetic term, the newtonian and magnetic stresses and anisotropic pressure separately in order to analyse the physical origin of the first and second harmonics. The calculation along these lines has led to the following results: (i) For values of the inclination angle (Yf (0, fn) there are two frequencies w and 2w in the gravitational radiation spectrum (for (IL= 0 or $71there is only the frequency 2w [4]). (ii) The contribution to the first harmonic is given completely by the newtonian stresses, while the contribution to the second harmonic is composed in equal parts by the kinetic term and the newtonian stresses. It should be emphasized that the presence of the first harmonic in the gravitational radiation spectrum is not accompanied by the precession of the pulsar magnetic axis and consequently does not involve any modulation of electromagnetic signals. Owing to the non-conservation of purely mechanical angular momentum, the axis of rotation in our model does not coincide with any of the principal axes of inertia of the body, while the angle between the rotation and magnetic axes is conserved. Consider the rotating self-gravitating drop of homogeneous fluid of quasispherical shape possessing an internal homogeneous magnetic field Bin = 2c/R3, matched with the dipole field B out = [3(jl*r)r
-p2]r-5
)
(1)
in the near external region. In the absence of rotation the drop is assumed to be spherical with radius R, and its deformation caused by rotation will be described by the equation (r2 - R2 + aiiXiXj>= 0 ,
(2)
where Uij Q 1 is a small dimensionless deformation tensor. The total stress-energy-momentum tensor of the system will be the sum of four contributions: T/l” = P,” +b/l” +tfiv +7/l,
3
(3)
where p,, is the kinetic term, b,, the magnetic field stresses, tpv the newtonian stresses and 7PV the anisotropic pressure tensor. In the following we need only the space components of these quantities pii =
PViVi
+p6ii
,
tii = (1/4nG)[@,i@,i 194
bii = -( 1/4n)(BiBi -
2 July 1984
PHYSICS LETTERS
!i~ii(V4)21 9
- iSiiB2) , (4)
where p = const is the density of the fluid, p the pressure,u(r, t) the velocity field, @the newtonian gravitational potential and G the gravitational constant. The tensor of anisotropic pressure will be calculated using the equilibrium condition in the rotating reference frame. From the conservation equation aTpv/axv = 0 [the derivative is not covariant since the gravitational stresses have already been taken into account in (3)!] we obtain the following equilibrium conditions p + p$ - ipv2 + B2/8n = const , (a/axi)(7ij
- BiBj/4~) = 0 .
(5)
The newtonian potential can be obtained using the standard general expression for the case of a triaxial ellipsoid, in the quasispherical case we are interested in @in = wJ2(-iR2
+ ir2 - &ar2 + ~aijXiXj + iaR2), (6)
inside the body and 4 out = w:R3 [-1/3r + (a/6r)(l - R2/5r2) + (R3/10r5)aijxixj]
,
(7)
outside of it (U = tr aij). According to eq. (5) and the Maxwell equations, at the surface Z of the drop the following boundary conditions must hold
TijI~nj=(l/4n)(-Bo,,
i +Bin i)(B*n)
9
(8)
where n is the outer normal vector. In the first nonzero approximation in terms of the small quantities a/aJ and fl the deformation tensor aij will be obtained as follows Qij=-(l5/2w:)wiwi
+(45R-8/8nw:p)/_+~j.
In the same approximation tensor will have the form Tij = (3R-8/2K)[pi/+r2
the anisotropic
- 6ij(lr.r)2 J .
(9) pressure
(10)
As it is seen from eq. (8) our drop will in general have the form of a triaxial ellipsoid with its rotation axis inclined with respect to the magnetic axis. (Note that in account of the deformation (9) the magnetic field inside the body will not be strictly homogeneous but
PHYSICS LETTERS
Volwne 103A, number 4
acquire small corrections, responsible for matching with the external field (1). However, these corrections are of higher order and can be neglected in our further calculations). Now we turn to the calculation of the Riemann tensor describing the gravitational radiation of the rotating magnetised drop in the wave zone. The response of the presumed detector is determined by the components Rojui which in the transverse traceless gauge have the simple form ROiOj = -:a2hijlat2
.
JTii(k,
,
(12)
n) exp(ik*n
- inwt)
2
,
(13)
and an analogous transformation for hii, and introducing the standard polarisation tensors e$, t$ orthogonal to the vector k, we get for the two independent polarisation states the following solutions of eq. (12) in the wave zone h+,x = _ 4G _R il=_-m 5 T+*X(noR/R,
n) exp[-ino(t-R)],
(14) where R is the radius-vector from the pulsar to the observation point. The total gravitational radiation intensity can be expressed through the quantities T+*x as foilows f =;
ij n-
(rz~~)~(lT~l~ f lT+12).
For the case of a non-relativistic
n) ,
(16)
to put eq. (15) into the usual quadrupole form. Then the standard calculation gives for the total intensity the sum of two harmonics, n = 1 and n = 2, 4w6/10GM2)
sin2cu cos2a,
12, = (8~ 4w6/.5GM2) sin4a,
t)
= F
Ti&k, H) = ;(nw)2(a2/akiakj)To0(k,
r,=Cu
where Tii in the r.h-.s. is the space part of the total stress tensor (3) including newtonian stresses inside the body and in the near external region and magnetic stresses in the same regions (in our scheme we do not need to match explicitly the gravitational and the electromagnetic fields in the near and far zones, because the contributions of the near outer stresses to the retarded solution of eq. (12) are convergent), Using for Tij the Fourier-transformation T&
surface it is sufficient to calculate the Fourier transform (13) in the long-wavelength approximation talcing exp(ik*r) = 1. Note that if we are not interested in the calculation of the separate contributions to hji from the terms in the sum (3) we can use the conservation condition in the form [8]
(11)
The metric perturbations hii are to be determined from the usual wave equation
qhii = 16nGTij
2 July 1984
(1%
velocity of the pulsar
(17) (18)
where M is the total mass of the body. As it is seen from eq. (17) lw = 0 for (Y= 0 and (Y= n/2, but Z, # 0 for all other values of CY.For small cr the first harmonic dominates. Now we want to show that the radiation at the first harmonic is completely due to the contribution of the newtonian stresses in the sum (3). For this purpose we calculate Ti&k, n) from eq. (13) explicitly as the sum of separate contributions from pii, bij, tij and rij. In first-order terms of w/t_dJ and fi we find that pii gives no contribution, while the other terms mutually cancel: rii(k, n) = 6tq(k, n) = -$bq(k,
n)
2n =
d(wt) s
0
ViPj
-
10nR3
.
(19)
This cancellation is due to the absence of the dipole term in the gravitational radiation power. All contributions of second order are of quadrupole type. The calculation shows that in this order the magnetic stresses and anisotropic pressure gives zero contribution while the contributions coming from the kinetic term and the newtonian stresses are of the form
(20)
Volume
103A, number
PHYSICS
4
where /A,,i, ~Ecliare magnetic moment components parallel and perpendicular to the rotation axis. In the coordinate frame having its z-axis aligned with the axis of rotation, the components of the magnetic moment are pL = /_tsin 0 (cos ot, sin at) ,
p,, = p cos a
eqs.
(20),
R’ = -w2(h1 + 2ah2(
(22)
where the dimensionless amplitudes h 1 and h2 are connected with the intensities of gravitational radiation at the first and the second harmonics by the relations h, = (uR)-‘(~OGI,/~)~/~
,
h, = (wR)-1(5G12w/8)1/2
.
(22)
that the kinetic term pij has non-zero Fourier-transform only if n = 2, i.e. it contributes only to the second harmonic. The newtonian stresses cii have non-zero Fourier-transform for both n = 1 and n = 2, the contribution with n = 1 being exactly equal to that contained in (17). Hence the generation of the first harmonic can be considered as the effect of newtonian stresses (however one should not attach too much meaning to this statement because the separation of the total Tq into the sum of nonconserved quantities (3) leads to the loss of gauge invariance of the separate contributions into hii). It is interesting that the contributions of the mass and newtonian stresses to the second harmonic of gravitational radiation appear to be exactly equal. Now we give the final expressions for the projections of the Riemann tensor R’x = -fa2h’>xlat2 corresponding to two independent polarization states From
sin 20 cos wt/2fi
The use of the polarization properties of radiation in the geterodyne detection project leads to an effective enhancement of the sensibility of the detector [9]. References [ 1] M.J. Rees, R.D. Blanford [ 21 [ 31 [4] [ 5) [6] [7] [8]
1 + cos20) cos 2wt) , [ 91
196
(24)
it is obvious
RX = w2(h1 sin 13sin at/& t 44~0s
2 July 1984
LETTERS
Oh, sin 2wt),
(23)
and J.P. Ostriker, Nature 300 (1982) 615. V. Boriakoff, R. Buccheri and F. Eauci, Nature 304 (1983) 417. M. Rees, R. Ruffini and J. Wheeler, Black holes, gravitational waves and cosmology (London, 1974). J.P. Ostriker and J.E. Gunn, Astrophys. J. 157 (1969) 1395. S. Chandrasekhar and E. Fermi, Astrophys. J. 118 (1953) 116. M. Zimmermann and Szedenits Jr., Phys. Rev. 20D (1979) 351. G.R. Huquenin, J.H. Taylor and D.L. Helfand, Astrophys. J. 181 (1973) L139. D.V. Gal’tsov. Yu.V. Grats and V.I. Petukhov, Gravitational radiation from electrodynamical systems (Moscow Univ. Press, Moscow, 1984). D.V. Gal’tsov, V.P. Tsvetkov and A.N. Tsirulev, Zh. Eksp. Theor. Fiz. (1984). to be published.