Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~qfthep04/2004/Proceedings2004/Sadoyan_QFTHEP04_353_356.ps Äàòà èçìåíåíèÿ: Thu Sep 17 22:04:20 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 08:28:08 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: massive stars

Should it be possible to reconstruct the equation of state from
gravitational radiation of single superdense sources?
Avetis Abel Sadoyan
Yerevan State University
Alex Manoogian 1, 375025 Yerevan, Armenia
Abstract
This work is done in the framework of CRDF/NFSAT Collaboration on ''Gravitational Waves
from Neutron Stars with superfluid Component''. Calculating Gravitational wave parameters from
superdense configurations with di#erent equations of state we found that there is one to one cor­
respondence between equation of state and GW parameters. So we conclude that there should be
possible to guess the equation of state from gravitational waves of compact stelar objects.
1 Introduction
We do expect that in the next couple of years a new window to astrophysical objects will be opened:
gravitational wave detectors of second generation will start to function with desired accuracy. Along
with the main challenge to detect a gravitational wave, if succeeded, detectors may provide valuable
information on the type and main parameters of the source, that can not be obtained in other ways. The
small interaction of gravitational waves with matter will allow us to look for waves coming directly from
cores of superdense compact objects providing direct information about Equation of state, size, type,
rotating frequencies of the last. Isolated single compact objects with superfluid interiors undergoing
selfsimilar oscillations are going to be promising sources for gravitational waves[1]. We are considering a
special regime, for rotating configuration where ''deformation energy'' is going to feed gravitational waves
through quasi­radial oscillations.
2 Quasi­radial Oscillations
Quasi­radial oscillations of rotating compact objects were investigated in the early 1970's [6, 4] where the
frequency spectrum of the fundamental oscillation mode for maximally rotating compact objects was de­
termined. These stars are oblate due to their rotation and consequently they have a non­zero quadrupole
moment. The oscillations add a time dependance to the quadrupole moment [8]. The oscillation is de­
scribed by assigning each mass element a time dependent coordinate given by x# = x 0
# (1+# sin #t) where
# # 1 and a constant. Thus, the reduced quadrupole moment is given by:
Q## = # # # x#x # - 1/3### x 2
# d 3 x # Q 0
## (1 + 2# sin #t) (1)
where Q 0
## are the components of the quadrupole moment of the rotating oblate compact object in
equilibrium and we have neglected terms of order # 2 . Taking the axis of rotation to lie along the z­axis,
the non­zero components of the quadrupole moment obey:
Q 0 = -Q 0
zz = 2Q 0
xx = 2Q 0
yy . (2)
The power emitted in gravitational radiation is given by:
J = G
5c 5
# # # #
d 3
dt 3 Q## # # # #
2
, (3)
and consequently one obtains:
J = 6G
5c 5 # 2 # 6
# # Q 0
# # 2 cos 2 #t # = J 0 cos 2 #t # (4)
where the retarded time is t # = t - r/c for a source at distance r.
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To determine the wave form and the angular distribution of the radiation, we rotate to coordinates
in which the wave vector lies along the z­axis and use the transverse­traceless gauge. Consequently,
h+ = 1/2 (h xx - h yy ) = 3GQ 0 ## 2
c 4 r
sin 2 # sin #t # hâ = h xy = 0, (5)
where # is the angle between the wave vector and the axis of rotation of the compact object. We can
express the strain amplitude in terms of the power by combining Eq. 4 with Eq. 5 to obtain:
h+ = # 15GJ 0
2c 3
1
r#
sin 2 # sin #t # . (6)
If an energy source can be found to drive the pulsations, the rate at which power is put into the vibrations
can be combined with the lifetime of the energy source to estimate the strain amplitude from an individual
white dwarf and thus from the galactic population as a whole. We discuss a possible mechanism in the
next section.
3 Quadrupole Moments and Oscillation Frequencies
Equilibrium configurations of neutron stars and quasiradial oscillations were discussed in the papers [6, 4]
in connection with integral characteristics and stability conditions.
# c(15) M/M
o# max (kHz) I (44) # (kHz) N (57) Q 0
(43) W r(52) # - # 0 W g(53)
0.546 0.7815 5.97 4.92 3 1.51 7.68 0.877 0.082 1.79
1.14 1.3737 8.37 9.85 5 1.76 9.73 3.45 0.068 1.46
2.44 1.7127 12.3 11.1 7 2.02 30.7 8.40 0.070 1.30
In the table the results of those calculations are presented for a real baryon gas equation of state.
It is necessary to use a known equation of state and central density # c to find the following integral
characteristics, mass (M ), moment of inertia (I), maximum rotational
frequency(# m ), and quadrupole
moment (Q 0
zz ). The frequency of adiabatic quasiradial oscillation (#) is also given. All values are
calculated in a general relativistic expansion about the rotation
frequency# to
order# 2 . Calculations
have shown [6, 4] that as the mass increases from 0.1 M o to 0.5 M o the oscillation frequency # quickly rises
from 10 Hz up to 2 kHz. For M > 0.5 M o the oscillation frequency shows a very slow increase from 2 kHz
up to 10 kHz. We note that the oscillation frequencies are calculated for the nonrotating configurations,
however calculations in [6, 4] show that there is very little change in # when going to the maximally
rotating configuration. The oblateness of the star and consequently the value of the quadrupole moment
Q 0
zz do depend on the rotation of the neutron star and the values shown in Table are calculated for the
maximally rotating configuration. Since the quadurople moments are calculated
to# 2 order we can find
the appropriate values for any rotation rate using Table above and scaling
by(# /# m ) 2 .
4 Continuous Energy Sources
To determine the gravitational wave amplitude h 0 for oscillating neutron stars, it is necessary to find
a permanent source of energy to sustain undamped oscillations and therefore to estimate the intensity
J 0 . One possible source of energy is the deformation energy of the neutron star. For rotating neutron
stars, the surfaces of constant density are rotating elipsoids. During the spin­down, these surfaces tend
towards sphericity. Because the crust is a crystalline solid, the process of spin­down will be accompanied
by starquakes which will relieve the stress built up in the core and drive quasiradial oscillations. We
propose that part of the deformation energy is converted to gravitational radiation in this process. The
deformation energy of the star can be calculated in the following way [7]:
W def = (M -M 0 ) c 2
-W
r(# , (7)
where M and M 0 are the rotating and non­rotating masses for the same baryon number, respectively.
The kinetic energy of rotation is W
r(# =
I# 2 /2. Using results obtained in [5] for the gravitational
coe#cient of packing # for rotating and non­rotating configuration, we find:
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# = 1 - M
mN
(8)
# 0 = 1 -
M 0
mN
(9)
where m is the baryon mass and N is the baryon number. The di#erence in masses #M = M - M 0 is
then:
#M = (# - # 0 ) mN. (10)
In Table above the coe#cient of packing as well as all other values are calculated for a maximally
rotating configuration. We obtain values for lower rotation rates as above scaling with the
factor(# /# m ) 2 .
Therefore the deformation energy for a rotating star with angular
velocity# is given by:
W def (# =
## # m
# 2
W
def(# m ). (11)
If part of the deformation energy is radiated as gravitational waves during the lifetime of the pulsar,
the gravitational wave intensity J 0 can be estimated by:
J 0 # #
W def
# , (12)
where
#
=# 2|
—# |
(13)
is the lifetime of the pulsar, and # # 1 is the branching ratio indicated the fraction of energy converted
to gravitational radiation. Thus, using the previous equations, we obtain:
J 0 #
2## |
—# |# 2
m
W
def(# m ). (14)
We can now estimate the intensity and amplitude of gravitational radiation from neutron stars if we know
the appropriate parameters that appear in Eq. 14. We take for a model neutron star # c = 1.14â10 15 gcm -3
corresponding to a mass M = 1.4 M o , and read the following values from Table
above:# m = 8.4â10 3 Hz
and W
def(# m ) = 1.8 â 10 53 erg. Substituting these values into Eq. 14, we find:
J 0 #
## |
—# | # 5.1 â 10 45 erg s 2 . (15)
5 Glitches
We now turn to burst signals coming from glitches. It is known that the angular velocity of pulsars is
continuously decreasing due to magneto­dipole radiation. On top of this continuous change there are
irregular fluctuations and jumps (known as glitches) in the angular velocity on the order
of## /# #
10 -6 - 10 -9 as well as its time derivative ( —
# on the order of |#
—# /
—# | # 10 -2 - 10 -4 [2]. Henceforth
we suggest that glitches are possible sources of gravitational radiation. It is possible that during a glitch
some of the rotational energy is transmitted to the crust and excites harmonic oscillations. We then
assume that the energy transmitted to the crust is then emitted as gravitational radiation. The energy
of spin­up during the glitch is given by:
#W =
I#3 . (16)
The power which is transmitted to the crust is then determined by:
# —
W = I
—# =
I# —# #
—# —# , (17)
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where we have ignored both —
I and
—# ## as both of these terms are very small. To obtain an upper limit
on the strength of the gravitational radiation, we assume that all the power transmitted to the crust is
then emitted in gravitational radiation, so that J 0 = # —
W . Using Eq. 6, we find the amplitude for the
gravitational wave to be:
h 0 = # 2 â 10 -9 (erg s) -1/2 cm # # # —
W
#r
. (18)
We now estimate an upper limit on the gravitational radiation using the Vela pulsar. Using obser­
vational values for the Vela pulsar, we take the following
values:# # 70 s -1 , |
—# | # 10 -10 s -2 , and
r # 0.3 kpc. After the glitch, the angular velocity gradually relaxes back to its original value. Usually,
after a hundred days from the moment of the glitch the behavior of the angular velocity begins to fluc­
tuate with relative changes in
—# on the order of |#
—# /
—# | # 10 -4 [2]. Using the values for I , #, and Q 0
from our model above and using the values of # —
# and # #
—# /|
—# | # max
one can find:
J 0max # 7 â 10 34 erg s -1 (19)
h 0max # 6.8 â 10 -27 (20)
.
It is clear that after the glitch J 0 and h 0 will decrease at the same rate as given in Eq. ?? for |#
—# /
—# |.
For fluctuations of the angular velocity, we have #
—# /|
—# | # 10 -2 # #
—# /|
—# | # max
. Therefore we can see
that the intensity J 0 is two orders of magnitude less the value in Eq. 19 and the amplitude h 0 one order
of magnitude less than the value given in Eq. 20.
We can also estimate the value of #. Substituting the values for J 0max , #, and Q 0 into Eq. 4 we find
# # 1.6 â 10 -4 .
We will estimate the amplitude of the gravitational radiation for the Crab pulsar as well. We take the
same model above and
use# # 190 s -1 , |
—# | # 2.65 â 10 -9 s -2 , # #
—# /|
—# | # max
# 5 â 10 -3 , and r # 2 kpc
[21]. We then find J 0max # 2 â 10 36 erg s -1 , h 0max # 10 -26 , and # # 10 -3 .
6 Conclusions
The numbers estimated in the paper are showing that with gravitational wave detectors of new generation
we could register gravitational waves from pulsars and white dwarfs. The comparison of gravitational
wave amplitude and frequency will give us information on equation of state of the superdense object.
Acknowledgements
This work is supported by CRDF award AP2­3207 and 12006/NFSAT PH067­02.
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