Upload
d-m-sedrakian
View
212
Download
0
Embed Size (px)
Citation preview
194
Astrophysics, Vol. 49, No. 2, 2006
0571-7256/06/4902-0194 ©2006 Springer Science+Business Media, Inc.
GRAVITATIONAL RADIATION OF SLOWLY ROTATING NEUTRON STARS
D. M. Sedrakian, M. V. Hayrapetyan, and M. K. Shahabasyan UDC: 524.354.6-423
The gravitational rotation of slowly rotating neutron stars with rough surfaces is examined. The source of thegravitational waves is assumed to be the energy transferred to the crust of the star during irregular changesin its angular rotation velocity. It is shown that individual pulsars whose angular velocity regularly undergoesglitches will radiate a periodic gravitational signal that can be distinguished from noise by the latest generationof detectors. Simultaneous recording of a gravitational signal and of a glitch in the angular velocity of apulsar will ensure reliable detection of gravitational radiation.
Keywords: stars: neutron: gravitational radiation
1. Introduction
The observed approach of the components of the binary system PSR 1913+16 owing to gravitational effects has
served as an indirect proof of the existence of gravitational waves [1]. Collisions or close approaches of astrophysical
objects can also be possible sources of gravitational waves, as can the gravitational collapse of a massive star leading
to the formation of a neutron star or a black hole [2]. These phenomena, however, are quite rare and unpredictable, so
the probability of detecting gravitational radiation from them is low. Thus, recently more attention has been devoted
to the possibility of gravitational radiation from individual compact objects, neutron stars and white dwarfs, which can
be observed continuously. Given this, the purpose of modern gravitational wave detectors based on earthbound (LIGO
and VIRGO) and space based (LISA) interferometers is the direct detection of gravitational waves.
It is generally known that an isolated neutron star will emit gravitational waves if its quadrupole moment varies
with time. Various mechanisms leading to a time dependence for the quadrupole moment of a star have been examined
[3-9]. Quadrupole fluctuations of a star that has been flattened because of its rotation have been discussed and the
intensity and characteristic damping time of the radiation calculated in Ref. 3. It was shown that the quasiradial pulsations
of neutron stars can be damped in a few seconds, so an energy source within the star is required to maintain them
Original article submitted October 26, 2005. Translated from Astrofizika, Vol. 49, No. 2, pp. 221-229 (May 2006).
Erevan State University, Armenia; e-mail: [email protected]
195
continuously. In Ref. 4 it was assumed that the source of gravitational radiation might be the energy owing to deformation
of a rotating neutron star. The deformation energy of the star is released during braking as a result of the magnetic dipole
radiation of the star. The deceleration is not continuous and is accompanied by a buildup of critical stresses and
“starquakes”, which also excite quasiradial pulsations in the star. Yet another source for the generation and maintenance
of quasiradial pulsations was pointed out in that paper: the energy released during glitches and fluctuations in the angular
velocity of pulsars. It was assumed that during irregular changes in the angular velocity, part of the energy of rotation
is transferred to the crust of a neutron star through the excitation of quasiradial pulsations. Estimates of the amplitude
h of the gravitational waves from the Vela and Crab pulsars for an earthbound observer show that it is still below the
limits of sensitivity of existing gravitational radiation detectors, and this has been confirmed by recent measurements [10].
The various modes of the pulsations and precession of neutron stars have been studied taking the superfluidity
of the inner layers of the star into account [5-9]. A study of the long-lived precession of a triaxial neutron star showed
that the amplitude of the gravitational waves for an earthbound observer is less than 10-30 [8], substantially below the
sensitivity threshold of planned detectors. The possibility of gravitational radiation in different modes of the pulsations
of a two-component superfluid system in a neutron star has been examined [6,7]. However, it has been shown [9] that
these modes [6,7] of the pulsations of the material cannot change the quadrupole moment of the star; thus, these pulsations
cannot be accompanied by gravitational wave emission.
The possibility of gravitational radiation from a neutron star in the shape of a triaxial ellipsoid of rotation has
been considered [5]. A star might acquire such a shape as a result of stresses arising during crystallization of the crust
as the star slows down and because of roughness in the form of “mountains” which develop during crystallization of the
star’s surface. Calculations show (see below), however, that the characteristic dimensions of the surface nonuniformities
of a neutron star cannot exceed 10-1 cm because of the star’s strong gravitational field. The nonuniformities are small
in size and, if we assume that the star radiates gravitational waves owing to the energy of rotation or the energy of
deformation, then the mechanisms for gravitational radiation considered in Refs. 4 and 5 can be sufficiently effective only
for millisecond pulsars. Because of the rapid rotation of these stars, the most flattened ones also have a large store of
rotational and deformational energy. Converting these forms of energy into gravitational radiation over the entire lifetime
may provide radiation of sufficient intensity to be measured by gravitational wave detectors. Thus, millisecond pulsars
are one of the likely sources of gravitational waves for the new generation of detectors.
Most pulsars, however, have rotational velocities much lower than those of millisecond pulsars. We can expect
gravitational waves of measurable amplitude from slowly rotating neutron stars only for a powerful source; that is,
conversion of the source energy into gravitational radiation must take place in a short time. This might happen during
glitches and fluctuations in the angular rotation velocity during which the angular velocity undergoes relative changes
on the order of 96 1010 −− −Ω∆Ω ~ and its derivative, on the order of 42 1010 −− −ΩΩ∆ ~&& . As the star slows down, an
instability builds up in it which, when removed, leads to the transfer of part of the rotational energy of the inner layers
to the crust. Because of this energy, the crust of the star is suddenly accelerated and quasiradial pulsations are excited
in the star. Subsequently, owing to constant fluctuations in the angular velocity, undamped quasiradial pulsations develop
in the star which are accompanied by continuous transfer of rotational energy to the star’s crust.
The purpose of this paper is to study the gravitational radiation of slowly rotating neutron stars which are
undergoing quasiradial pulsations. Here the source of the gravitational radiation is assumed to be the energy released
in the course of irregular changes in the angular velocity- glitches in the angular velocity and the “noise” between
196
glitches. In Section 2 we calculate the intensity of the gravitational radiation and the amplitude of the gravitational waves
for an earthbound observer for quasiradial pulsations of a neutron star with a rough surface, i.e., when “mountains” exist
on its surface. In Section 3 the dimensions of the roughness on the surface of a neutron star are estimated and in Section
4 the possibility of detecting gravitational waves from slowly rotating pulsars during glitches and the subsequent
relaxation of the rotational velocity is discussed.
2. Gravitational radiation of a pulsating neutron star with a rough surface
Let us consider a neutron star with a rough surface that is rotating slowly with angular velocity Ω<<Ωcr, where
Ωcr is the maximum angular velocity of rotation for a given configuration corresponding to escape of matter from the
equator. As it rotates a neutron star is flattened owing to centrifugal forces so that its polar and equatorial radii differ.
We assume that because of the presence of mountains on the surface the cross section of the neutron star’s equatorial
surface is an ellipse. An object of this sort has a nonzero quadrupole moment. If we assume, as well, that the principal
fluctuation mode of the material, i.e., quasiradial pulsations, is excited because of constant fluctuations in the angular
velocity and wobbling, then the quadrupole moment of the star also depends on time. Thus, a neutron star with a rough
surface that undergoes quasiradial pulsations is a gravitational radiation source. The intensity of the gravitational
radiation is given by [11]
, 45
25 αβ= D
c
GJ &&& (1)
where
( )∫ αββααβ δ−ρ= dVrxxD 23 (2)
is the quadrupole moment tensor for the mass. With these pulsations the coordinates αx vary as
( ), sin10 txx ωη+= αα (3)
where η is the relative amplitude of the oscillations and ω is their frequency. It is assumed that 1<<η and is independent
of the radial and angular coordinates. Substituting Eq. (3) in Eq. (2), we obtain the time dependent quadrupole moment
of the pulsating star:
( ) ( ), sin210 tDtD ωη+= αβαβ (4)
where 0αβD is the quadrupole moment of the star without the pulsations. Since the star we are considering rotates slowly,
i.e., Ω<<Ωcr, the polar and equatorial radii differ little. We can also assume that the density of matter in the star depends
only on the spherical radius, i.e., ( )rρ=ρ . The coordinate system has its origin at the center of mass and the X axis is
directed along the direction of a “mountain,” while the Z axis is directed toward the poles of the star. Then for the nonzero
components of 0αβD we have the following relationships:
. 2
000 xxzzyy
DDD −== (5)
Substituting Eq. (4) in Eq. (1) and using Eq. (5), we obtain the following for the intensity of the gravitational radiation:
197
, coscos15
8 20
220625
tJtDc
GJ zz ′ω=′ωωη= (6)
where cRtt 0−=′ and
, 15
8 206250 zzD
c
GJ ωη= (7)
where R0 is the distance from the star to the observation point. As will be shown below (Section 3), the dimensions of
the surface roughness are small compared to the star’s radius. If we introduce the ellipticity e according to the definition
( ),
2
ba
ba
+−=ε (8)
where a and b are the semiaxes of the ellipse at the equatorial cross section, then we have b = R and ( )ε+= 1Ra , where
R is the star’s radius. Then Eq. (7) for J0 can be reduced to
, 135
32 20
26250 I
c
GJ εωη= (9)
where ( )∫ρπ=R
drrrI0
40 4 is the moment of inertia of the star with respect to its center.
The two types of linearly polarized plane waves emitted by the object are defined by h+ and h
x, which obey the
following expressions:
( ) ( ), 32
1
04 zzyyzzyy DDRc
Ghhh &&&& −−=−=+ (10)
and
, 3
2
04 yzyz DRc
Ghh &&−==× (11)
where R0 is the distance of the neutron star from an earthbound observer. If the wave vector of the gravitational wave
forms an angle Φ with the direction of the X axis, then, by rotating the system of coordinates so that the direction of
the wave vector coincides with the X axis, for h+ and h
x we can write
, sinsinsin3
40
20
2
04
thtIRc
Gh ′ω=′ωΦεηω=+ (12)
and
, 0=×h (13)
where averaging over the angles Φ yields the amplitude h0 of the gravitational wave in the form
. 3
20
2
040 IRc
Gh εηω= (14)
Thus, if we specify the energy of the source of the gravitational radiation, i.e., specify the value of J0, then Eqs. (7) and
(14) show that we can obtain an expression for the amplitude h0 of the gravitational wave in the following form:
198
. 8
15123
0
00 ω
=c
GJ
Rh (15)
For a given value of J0, Eq. (9) also yields η:
. 32
1351 50
03 G
cJ
Iεω=η (16)
Our calculations apply only if η << 1.
3. Estimates of the dimensions of the surface roughness of a neutron star
As Eqs. (9) and (15) show, the intensity of the gravitational radiation depends on the ellipticity ε. In order to
estimate this quantity, we have to know the characteristic dimensions of the surface roughness of a neutron star. According
to the generally accepted picture of the internal structure of neutron stars, the crust of a star consists of inner (Aen-phase)
and outer (Ae-phase) parts [12]. The Ae-phase consists mainly of Fe5626 nuclei and a degenerate gas of normal electrons.
The iron nuclei form a volume-centered crystalline lattice, thereby creating the solid outer crust of the neutron star. The
density of matter in the Ae-phase ranges from 104 to 1011 g·cm-3. Now we can estimate the maximum height of a mountain
that can withstand its own weight in the gravitational field of a neutron star [13]. It is known that the elastic modulus
of the crust of a neutron star with respect to transverse vibrations is
, 40
22
a
eZY = (17)
where a0 is the internuclear separation. The maximum shear stress is roughly 0.01 times this value, i.e.,
. 100 4
0
22
a
eZS= (18)
The height of the mountain can be estimated as
, cm 10 31
12
ρ=ρ
=gg
SH (19)
where ρ is the density of the surface layer and g is the acceleration of gravity on the surface. After finding the height
of a mountain on the surface, Eq. (8) gives us the expression RH=ε for the ellipticity, which we shall use below to
estimate the intensity and amplitude of the gravitational waves.
4. Gravitational radiation during glitches in the angular velocity of slowly rotating neutron stars
As mentioned above, glitches in the angular velocity of pulsars on the order of 96 1010 −− −Ω∆Ω ~ and in its
derivative on the order of 42 1010 −− −ΩΩ∆ ~&& are observed. We assume that one of the possible sources for the
199
generation and sustainment of quasiradial pulsations may be glitches together with constant fluctuations in the star’s
angular velocity. It may be assumed that part of the energy involved in accelerating a neutron star during irregular changes
in its angular velocity is transferred to the crust of the star through excitation of quasiradial pulsations and is then removed
by gravitational radiation. In glitches an energy
, Ω∆Ω=∆ IW (20)
is transferred to the star’s crust, where I is the moment of inertia of the star. The power delivered to the crust is given
by
, ΩΩ∆=
ΩΩ∆ΩΩ=ΩΩ∆=∆
&
&&
&
&&&& WIIW (21)
where W& is the steady state rate of loss of rotational energy by the neutron star in its secular deceleration. In deriving
Eq. (21) we have used the fact that ( )( ) 1<<Ω∆Ω∆ΩΩ& . We assume that all the energy of the quasiradial pulsations
is radiated in the form of gravitational waves. Thus, we must assume that
. 0 WJ &∆= (22)
In order to establish whether gravitational waves from a slowly rotating neutron star can be detected, let us estimate
the intensity of the possible gravitational radiation for the case of the pulsar Vela PSR 0833-45. The angular rotation
velocity and the rate of slowing down of this pulsar are Ω = 70 s-1 and 1010−=Ω& s-2, while the distance to it is
r = 0.3 kpc. It is also known that during glitches Ω& varies at most by a relative amount 210−ΩΩ∆ ~max
&& , while after
relaxation Ω& undergoes changes on the order of 410−ΩΩ∆ ~&& . It is clear from Eqs. (21), (22), and (15) that in order
to determine J0 and h
0 we also have to know the moment of inertia of the neutron star and the frequency of the quasiradial
pulsations of the matter in the star. Equilibrium configurations of neutron stars have been studied [14,15] in connection
with determinations of their integral characteristics and the conditions for their stability. Based on the standard model
of a neutron star from Ref. 14 with a central density 1510=ρc g·cm-3, then the mass of the star is roughly ¤
M.M 31=
and its radius is R = 10 km, while the moment of inertia is on the order of I = 1045 g·cm2. According to Ref. 15, for
¤M.M 50> the frequency w of the pulsations depends only weakly on the star’s mass and varies from 2·103 to
104 s-1. For our chosen model of neutron stars, the frequency of the pulsations is roughly 3105⋅=ω s-1. Using the above
values for the mass M and radius R, Eq.(19) yields the height of a mountain on the surface. Taking the density of the
surface layer to be 104 g/cm3, we obtain H = 0.6 cm. Then the ellipticity of the star, defined as RH=ε , will be
ε = 6·10-7. We now turn to Eqs. (9) and (14) to estimate the intensity of the gravitational radiation and the amplitude
of the gravitational waves. For the maximum intensity J0, Eq. (9) gives J
0max=7·1034 erg/s and Eq. (14) gives
h0max
= 4·10-27. Usually after hundreds of days from the time of a glitch fluctuations of the angular velocity are established
with 410−ΩΩ∆ ~&& . Then the intensity J0 = 7·1032 erg/s and the amplitude h
0 = 4·10-28. Note also that, according to
Eq. (16), η = 10-4<<1, which confirms the validity of our calculations.
The gravitational radiation from the oscillations of a neutron star was discussed in an earlier paper [4], but for
a star deformed by rotation. A comparison of the present value of h0 and that obtained in Ref. 4 shows that they are of
the same order of magnitude. Thus, a neutron star can radiate away the energy of a glitch in the form of gravitational
200
waves at the oscillation frequency ω owing to the flattening caused by rotation and to surface roughening.
In conclusion, we note that isolated pulsars whose angular velocity regularly undergoes glitches are another class
of objects which are candidate gravitational radiation sources. We also point out that glitches take place in the Vela and
Crab pulsars an average of once in two-three years. This time is sufficient for isolating a periodic gravitational signal
from the noise and detecting it. Simultaneous recording of a gravitational signal and a glitch in the angular velocity
of a pulsar (such as Vela and Crab) would ensure reliable detection of the gravitational radiation.
We acknowledge with gratitude the financial support of ANSEF grant No. N05-PS-astroth-811-78 and CRDF/
NFSAT grant No. ARP2-3232/YE-04.
REFERENCES
1. J. H. Taylor and J. M. Weisberg, Astrophys. J. 253, 908 (1982).
2. S. L. Shapiro and S. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects
[Russian translation], Vol. 2, Mir, Moscow (1985).
3. Yu. L. Vartanyan and G. S. Adzhyan, Astron. zh. 54, 1047 (1977).
4. D. M. Sedrakian, M. Benacquista, K. M. Shahabasyan, A. A. Sadoyan, and M. V. Hayrapetyan, Astrofizika 46, 545
(2003).
5. M. Zimmerman, Phys. Rev. D 21, 891 (1980).
6. D. I. Jones and N. Andersson, Mon. Notic. Roy. Astron. Soc. 324, 811 (2001).
7. N. Andersson and C. T. Comer, astroph/0101193 (2001).
8. A. D. Sedrakian, I. Wasserman, and J. Cordes, Astrophys. J. 524, 341 (1999).
9. A. D. Sedrakian and I. Wasserman, Phys. Rev. D 63, 024016 (2000).
10. B. Abbott, M. Kramer, A. G. Lyne et. al., gr-qc/0410007 (2004).
11. L. D. Landau and E. M. Lifshitz, Field Theory [in Russian], Nauka, Moscow (1972).
12. G. S. Saakyan, Equilibrium Configurations of Degenerate Gas Masses [in Russian], Nauka, Moscow (1972).
13. F. J. Dyson and D. Ter-Haar, Neutron Stars and Pulsars [Russian translation], Nauka, Moscow (1973).
14. V. V. Papoyan, D. M. Sedrakian, and E. V. Chubaryan, Astron. zh. 49, 750 (1972).
15. H. H. Harutyunyan, D. M. Sedrakian, and E. V. Chubaryan, Astron. zh. b, 1216 (1972).