European Journal of Alternative Education Studies
ISSN: 2501-5915
ISSN-L: 2501-5915
Available on-line at: www.oapub.org/edu
10.5281/zenodo.57099
Volume 1│Issue 1│2016
AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM
COMPACT BINARY NEUTRON STAR
Bringen, B.1*i, Bakwa, D. D.2, and Girma, D. P.3
Department of Integrated Science, School of Sciences,
1
Kashim Ibrahim College of Education, Maiduguri, Nigeria
Department of Physics, Faculty of Natural Sciences, University of Jos, Nigeria
2
McAulleyMemorial Secondary School, Ballang, Shipang, Pankshin,
3
Plateau State, Nigeria
Abstract:
This research is concerned with the mysteries of neutron stars and the quest for
gravitational waves. Neutron stars are anticipated sources of gravitational waves, and
are expected to be detectable within the next decade using kilometre-scale laser
interferometry especially the aLIGO. We estimate the range of amplitudes that the
waves may have if a neutron star spirals inside a giant star in the end phase of binary
evolution. The signal of the calculated values with a peak gravitational-wave strain of h
~ 5.749 x 10−22 – 3.992 x 10-22 matches the strain amplitude sensitivity range h ~ 10-23 - 1018
of LIGO and VIRGO for the inspiral and merger of a pair binary systems. The results
obtained in this research as compared to aLIGO’s values of GW detection has proved
beyond reasonable doubt that NS are promising candidates; which imply that there are
more neutron stars (binary systems) out there than expected.
Keywords: amplitude, gravitational wave, neutron stars, compact binary
Introduction
The Gravitational Waves (GWs) are a prediction of Einstein’s General theory of
relativity; when Einstein first proposed general relativity in 1916, an initial piece of that
theory was the existence of GW (Maggiore, 2013). The GWs are essentially the ripples of
space time itself. Einstein did not think of it as a force, but as a curvature of space time,
where he thinks of membrane or cushion where in the centre of it put a bouncing ball,
Copyright © The Author(s). All Rights Reserved
Published by Open Access Publishing Group ©2015.
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AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
the cushion curves inwards; if you put a plane marble at the edge, the plane marble falls
in towards the bouncing ball, that was Einstein’s way thinking of gravity, gravity is
geometry. Gravity is not static, GW are essentially radiated by very massive objects,
when these objects accelerated and they go out ward from that object and out into the
universe (Maggiore, 2008).
If for example, considering some source in the sky, like a pair of Neutron Stars
orbiting each other, as they orbit each other, they radiate GW and those GWs are
travelling from the source towards the observer at the speed of light (Kostas, 2002).
Gravitational waves alternately stretch and squeeze space-time both vertically and
horizontally as they propagate. Test particles in the presence of a passing gravitational
wave will experience gravitational tidal forces that alternately stretch and squeeze
along orthogonal axes in the plane perpendicular to the direction of propagation. The
tidal deformations preserve the area enclosed by a ring of test particles, so a measure of
the strength is the relative fractional deformation, or dimensionless strain amplitude, h
= 2∆L/L, where L is the length and ∆L is the change in length.
The emission of gravitational radiation dissipates the kinetic or internal energy of
the star so that the final product of the evolution is an axisymmetric and/or non-rotating
object. Then, each source can be characterized by a decay time, the time during which
gravitational waves are emitted (it is determined, for instance, by the spin down rate for
rotating neutron stars). Here, we are interested in those sources which radiate in the
sensitivity band of laser interferometeric gravitational wave observatory (LIGO): they
essentially involve compact objects, namely neutron stars. The number of neutron stars
in the Galaxy has been estimated to be of the order of 10 9, with a comparable number of
stellar mass black holes (van Paradijs, 1995).
In binary neutron star system in VIRGO cluster, a 1000 times further than the
Hulse – Taylor binary so allowing a further 1000 times out; hence, the strain intended
to measure goes to 10-21; so the goal of LIGO is to try to measure GW strains at the level
of 10-21 using kilometre long detector (Asi, 2015). The relative displacement to be
measured then corresponds to about 10-18m. The efforts of LIGO is still to measure
something at a level a 1000 time smaller than a proton using a sophisticated
interferomtric detector such as the Advance LIGO. A laser interferometer is an
alternative choice for GW detection, offering a combination of very high sensitivities
over a broad frequency band.
This research considers the possibilities that neutron stars may emit radiation at
certain amplitude detectable by advance detectors.
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Source of Gravitational Waves
Gravitational waves are emitted in many processes involving isolated stars or binary
systems. The emission of gravitational radiation determines an evolution of the emitting
source (due to the gravitational radiation reaction) so that the amplitude and the
frequency of the waves change with time.
LIGO has gathered a full year of data at its design sensitivity, monitoring
displacements a thousand times smaller than the size of a proton (Kelvin, et al, 2015;
www.ligo.org). Reaching this design sensitivity was a great achievement, and was
aided by the formation of a large international collaboration of over 500 people from 35
institutions. LIGO’s frequency band is ~ 10 − 2000Hz, which corresponds to the last
few minutes of the inspiral of binary neutron stars or black holes of a few solar masses,
visible to LIGO out to ~ 15 mega parsecs (www.ligo.caltech.edu). Astrophysical sources
in this band besides compact object (neutron star) inspirals and mergers include
spinning neutron stars in our Galaxy, supernovae, stochastic waves from processes in
the early Universe (inflation, phase transitions, etc.) and the large discovery space of
unexpected sources and effects in the universe. LIGO can observe neutron star binary
inspirals out to a distance of ~ 20Mpc ~ 6 × 1020km, which includes the thousands of
galaxies in the Virgo cluster. The fact that no events have been seen yet has been used to
place upper limits on the event rates. For binary neutron stars, statistical analyses based
on the observed number of progenitor binary star systems indicated an event detection
rate of between 1/3000 per year to 1/8 per year (Acernese, for LIGO Scientific
Collaboration, 2015).
Interferometric Gw Detectors
Interferometers are investigative tools used in many fields of science and engineering.
They are called interferometers because they work by merging two or more sources of
light to create an interference pattern, which can be measured and analysed; hence
"Interfere-ometer". The interference patterns generated by interferometers contain
information about the object or phenomenon being studied. They are often used to
make very small measurements that are not achievable any other way. This is why they
are so powerful for detecting gravitational waves--LIGO's interferometers are designed
to measure a distance 1/10,000th the width of a proton (www.ligo.caltech.edu).
The current networks using lasers comprised of detectors in the US, the Laser
Interferometric Gravitational wave Observatory (LIGO), the German- UK detector (GE
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AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
0600) in Germany, the French – Italian detector (VIRGO) near Pisa, Italy and the
Japanese detector KAGRA (www.ligo.caltech.edu).
A laser interferometer is an alternative choice for GW detection, offering a
combination of very high sensitivities over a broad frequency band. Suspended mirrors
play the role of “test-particles”, placed in perpendicular directions. The light is reflected
on the mirrors and returns back to the beam splitter and then to a photo detector where
the fringe pattern is monitored (www.sciencedirect.com.article).
Figure 1: Basic design of the LIGO interferometers
Source: www.ligo.caltech.edu
LIGO consists of two perpendicular, 4-km “arms” as depicted in Figure 1. A laser beam
is fired into a beam splitter that sends half the light down one of these arms, and half
down the other. The mirrors then reflect the light back the way it came, and the beam
splitter combines the two beams back into one, sending the combined beam to a
detector. LIGO carefully tunes the lengths of the detector arms so that the light from the
arms almost completely cancels out, or undergoes destructive interference, when the
reflected beams recombine back at the beam splitter. However, if the arm lengths
change slightly due to a passing gravitational wave, then the differences in length will
introduce a small difference in phase between the beams from different arms. The
waves that would have cancelled each other at the beam splitter will now travel
different path lengths and end up producing some light at the detector. It is this
interference property of light that is exploited by LIGO to detect gravitational waves
(Gregory, 2010). When a gravitational wave passes by, the stretching and compressing
of space causes the arms of the interferometer alternately to lengthen and shorten, one
getting longer while the other gets shorter, and then vice-versa.
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AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
Neutron Star Characteristics
Neutron stars are the most compact objects that can be directly observed. They have a
mass of the order of 1 − 2Mʘ and a 10 − 20 km radius, such that the average density
equals or exceeds the nuclear matter density of 2 x 1014 g cm−3. In fact, if neutron stars
were compressed to just a third of their size, they would be black holes, which can only
be detected indirectly. Due to their extreme densities, neutron stars are unique
laboratories for particle physics.
A neutron star can be formed through the accretion induced collapse of a white
dwarf in a binary system. Also in this case the collapse gives rise to a supernova;
depending on the amount of energy released the white dwarf either completely
disrupts or transforms into a neutron star (Nomoto & Kondo 1991).
Compact objects-white dwarfs, neutron stars, and black holes are “born” when
normal stars “die,” that is, when most of their nuclear fuel has been consumed. All
three species of compact object differ from normal stars since they do not burn nuclear
fuel, they cannot support themselves against gravitational collapse by generating
thermal pressure. Instead, white dwarfs are supported by the pressure of degenerate
electrons, while neutron stars are supported largely by the pressure of degenerate
neutrons. Black holes, on the other hand, are completely collapsed stars-that is, stars
that could not find any means to hold back the inward pull of gravity and therefore
collapsed to singularities. With the exception of the spontaneously radiating “mini”
black holes with masses M less than 1015 g and radii smaller than a Fermi, all three
compact objects are essentially static over the lifetime of the Universe. They represent
the final stage of stellar evolution.
The second characteristic distinguishing compact objects from normal stars is
their exceedingly small size (Thorset, & Chakrabarty, 1999). Relative to normal stars of
comparable mass, compact objects have much smaller radii and hence, much stronger
surface gravitational fields.
Range of Gravitational Wave Amplitude
This research considered the masses of some neutron stars with same radius of 10 Km
but different masses and developed a model that could test observation of the
amplitude range which these stars (J1518 + 409, B1534 + 12, B1913 + 16, B2127+11C, and
B2303 + 46) could possibly be within the detectors sensitive band.
The fundamental geometrical framework of relativistic metric theories of gravity
is space-time, mathematically described as a four-dimensional manifold whose points
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AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
are called events. Every event is labelled by four coordinates
coordinates
( = 1,2,3)
coordinate time (
=
(µ = 0, 1, 2, 3);
give the spatial position of the event, while
the three
is related to the
, where c is the speed of light). The choice of the coordinate
system is quite arbitrary and coordinates transformations of the form
=
(
)
are
allowed. The motion of a test particle is described by a curve in space-time. The distance
ds between two neighbouring events, one with coordinates
coordinates
tensor
(
, can be expressed as a function of the coordinates via a symmetric
+
)=
and the other with
(
),
i.e.,
(1)
=
This is a generalization of the standard measure of distance between two points in
Euclidian space. For the Minkowski space-time (the space-time of special relativity),
≡
.
=
(−1,1,1,1)
(2)
The symmetric tensor is called the metric tensor or simply the metric of the space-time.
The information about the degree of curvature (i.e., the deviation from flatness) of a
space-time is encoded in the metric of the space-time. According to general relativity,
any distribution of mass bends the space-time fabric and the Riemann tensor
a function of the metric tensor
(that is
and of its first and second derivatives) is a measure of
the spacetime curvature. The Riemann tensor has 20 independent components. When it
vanishes the corresponding space-time is flat. In this thesis, we will consider mass
distributions, which we will describe by the stress-energy tensor
(
).
Einstein’s gravitational field equations connect the curvature tensor (see 3) and
the stress-energy tensor through the fundamental relation:
=
−
=
(3)
This means that the gravitational field, which is directly connected to the geometry of
space-time, is related to the distribution of matter and radiation in the universe.
is
the Ricci tensor and comes from a contraction of the Riemann tensor, R is the scalar
curvature, while
is the Einstein tensor, k = 8πG/c4 is the coupling constant of the
theory and is the gravitational constant. The vanishing of the Ricci tensor corresponds
to a space-time free of any matter distribution. However, this does not imply that the
Riemann tensor is zero. As a consequence, in the empty space far from any matter
distribution, the Ricci tensor will vanish while the Riemann tensor can be nonzero; this
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means that the effects of a propagating gravitational wave in an empty space-time will
be described via the Riemann tensor.
The equations of structure are obtained using gravitational waves in the theory
of linearized gravity. Within this structure, it is assumed that space-time, described by
the metric tensor, is approximately flat. In other words, it is decomposed into the flat
Minkowski metric
and some contribution
=
(4)
+
Since the space-time is approximately flat, this contribution must be small. As a result,
in calculating physically significant quantities only terms up to linear order in
will be
kept
(5)
|| µ || << 1
Equation 2 is not a simple expression to work with; it would be suitable to reduce the
Einstein equation. One way of doing this is to cease working with hµν and instead work
with an expression that is known as the trace-reversed metric which is defined as
−
=
(6)
and its name stems from the fact that its trace is the opposite of the original metric.
Choosing the trace-reverse of
and the Lorentz condition,
(7)
=
We find the linearized Einstein Field Equation and can be written elegantly as
□
=−
(8)
Where
□=
=
is the flat-space d’Alambertian, and
⁄
Therefore,
□
=−
⁄
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AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
In order to study the propagation of gravitational waves and their interaction with test
masses, we are interested in the governing equations outside the source, i.e.
□
=
:
(10)
=0
In the vacuum case, the Lorenz gauge condition alone is not enough to fix the gauge
freedom. Further inspection shows that the Lorenz gauge condition is not violated by
imposing = 0, then
≡
, and
=
. The Lorenz condition then becomes:
(11)
=
(12)
=
This means that
corresponds to the static (time-independent) part of the
gravitational field; the time-varying gravitational degrees of freedom, the GW itself, is
contained in the time-dependent components
.
By via exploiting the gauge degrees of freedom we have set:
=
(13)
(14)
=
=
(15)
This set of conditions defines the transverse-traceless gauge (TT gauge), the most
convenient gauge to express gravitational waves outside the source. The general
complex solutions of Equation (10) are plane-wave solutions:
=
with
=
,
(16)
the wave vector. In the TT-gauge this general expression can be rewritten
as:
=
Where
and
(17)
are the polarization tensor and vector respectively.
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The Einstein equation (9) can readily be solved with the use of method of Green’s
functions.
The retarded Green’s function is given by
)=−
( −
⃗ ⃗
−
|⃗ ⃗ |
−
(18)
This of course satisfies the defining relation for a Green’s function, namely:
□
( −
)=
−
( )(
(19)
)
The usefulness of such a function resides in the fact that the general solution to an
equation such as (8) can be written as:
=−
( −
∫
)
( )
(20)
Substituting (17) into (19) yields
=
( ,
∫
)/
|
(21)
|
Assuming that the GWs are generated by a weak source and observed at large distance
from the source, i.e.
>> , where R is the characteristic size of the source. In this case,
one can perform the standard multipole expansion of the denominator analogous to the
expansion of the electromagnetic field at large distance from the source:
|
In the limit
|
≈ +
(22)
⇒ ∞ the asymptotic solution of the linearised field equations therefore is:
=
( − ,
∫
that in linearized theory
)
(23)
fulfills the flat-space conservation law, i.e.,
=
and
sources therefore move on geodesics in flat Minkowski space. Applying the
conservation law, we find that
∫
=
∫
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Assuming a standard stress-energy tensor, the tt-component denotes the rest-mass
energy density of the source µ(x). We can now define the second moment of mass or
moment of inertia tensor:
=∫
( ,
(25)
)
Hence, the asymptotic solution describing GWs generated by weak sources is found to
be:
( − )
=
(26)
Equation (16) is rather instructive:
i)
It shows that GWs are generated by accelerated sources similar to
electromagnetism where accelerated charges generate EM radiation,
ii)
The radiation obeys a
-fall-off, which implies that GWs generated by
astrophysical sources are indeed weak when they reach ground-based detectors (far
from the source), and
iii)
Gravitational radiation is of quadrupolar nature as the conservation laws do not
permit monopole and dipole gravitational radiation.
If the motion inside the source is highly non-spherical, then a typical component
of
( − ) will (from Equation (24)) have magnitude
.
., where
.
is the non-
spherical part of the squared velocity inside the source. So one way of approximating
any component of Equation (26) is:
=
.
(27)
Suppose a neutron star of radius R spins with a frequency f and has an irregularity, a
bump of mass m on its otherwise axially symmetric shape. Then the bump will emit
gravitational radiation again at frequency 2f because it spins about its centre of mass, so
it actually has mass excesses on two sides of the star, and the non-spherical velocity will
be just
.
=
. The radiation amplitude will be,
=
(28)
But,
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=
(
)
(29)
=
Substituting (29) into (28) yields,
=
16
.
=
=
.
=
16
4
4
Which implies that,
(30)
=
Let us make some estimates based on this formula (30) using the different masses of
neutron stars for gravitational wave emission.
Therefore, h may be obtained using the form in (30) thus as:
=
Putting the parameters for neutron star(s), J1518+4904 we have;
m = 1.56Mʘ=1.56x2x1030kg = 3.120x1030kg
R = 10km=10x1000m=10000m = 104m
r = 15Mpc=15x1000000x3.086x1016m = 4.65x1023m
G = 6.673x10-11 Nm2/Kg-2
C = 3x108 m/s
=
( .
(
) ( .
×
)( .
.
=
.
×
×
×
= .
×
×
)( ×
)
)
Therefore,
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which is dimensionless. However, the strain amplitude, h, for NSs B1534 + 12, B1913 +
16, B2127+11C, and B2303 + 46 were equally calculated using the same formula in
equation (30) and presented in table (1)
Results
The calculated results of the strain amplitude estimate for neutron stars (J1518 + 409,
B1534 + 12, B1913 + 16, B2127 + 11C, and B2303 + 46) are presented in table 1.
Table 1: Strain Amplitude Estimate for Neutron Star
Star (Source)
M(x 1030 Kg)
R (m)
r (x 1023m)
h
1.56
104
4.650
5.749x10-22
B1534 +12
1.339
104
4.650
4.236x10-22
B1913 +16
1.441
104
4.650
4.905x10-22
B2127 +11C
1.349
104
4.650
3.992x10-22
1.30
104
4.650
4.318x10-22
J1518 + 4904
B2303 +46
Discussion
In this research, we have been able to estimate the amplitude for gravitational waves
detection where the GW candidate (Neutron star) can be detected within the sensitivity
of the ground based detectors set by Advanced LIGO (aLIGO). Gravitational waves can
be emitted by many systems, but, to produce detectable signals, the source must consist
of extremely massive objects moving at significant fraction of the speed of light. The
main source is a binary of two objects; these include compact binaries made up of two
closely orbiting staler-mass objects, such as white dwarfs, Newton stars or black holes.
Advanced LIGO’s frequency band is ~ 10 -2000 Hz, which corresponds to the last few
minutes of the inspiral of binary neutron stars of a few solar masses, visible to aLIGO
out to ~ 15 megaparsecs (Mpc). Astrophysical sources in this band besides compact
object (neutron star) inspirals and mergers include spinning neutron stars in our
Galaxy. aLIGO can observe neutron star binary inspirals out to a distance of ~ 20Mpc ~
6 × 1020km, which includes the thousands of galaxies in the Virgo cluster. Hence, the
estimated amplitude in table (1) reveals that the amplitude was found to be h, ~ 10-22
and such waves generated within these ranges are targets for the ground-based
detectors. The sensitivity of ground based detectors is fundamentally limited at low
frequencies because they cannot be shielded from time-varying curvature fluctuations
due to the environment.
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Conclusion
The landmark discovery that was reported by the Advanced Laser Interferometer
Gravitational-Wave Observatory (Advanced LIGO) team, confirming months of rumors
that have surrounded the group’s analysis of its first round of data has been actualized.
Astrophysicists say the detection of gravitational waves opens up a new window on the
universe, revealing faraway events that can’t be seen by optical telescopes, but whose
faint tremors can be felt, even heard, across the cosmos.
The detection ushers in a new era of gravitational-wave astronomy that is
expected to deliver a better understanding of the formation, population and galactic
role of Neutron Stars and black holes; super-dense balls of mass that curve space-time
so steeply that even light cannot escape. When these stars spiral toward each other and
merge, they emit a “chirp”: space-time ripples that grow higher in pitch and amplitude
before abruptly ending. The chirps that LIGO can detect happen to fall in the audible
range, although they are far too quiet to be heard by the unaided ear. Physicists are
already surprised by the number and strength of the signals detected so far, hence, the
results obtained in this research as compared to LIGO’s values of GW detection has
proved beyond reasonable doubt that NS are positive candidates; which imply that
there are more neutron stars and black holes out there than expected. Compact binaries,
binary star systems in which each member is a neutron star or black hole are currently
the best understood sources of GWs.
References
1.
Asi, J. (for LIGO Scientific Collaboration) (2015). Classical Quantum Gravity32.
074001
2.
http://www.sciencedirect.com/article/pii Retrieved on 15th June, 2016: 09.22pm.
3.
Gregory, M. H. (For LIGO scientific Collaboration) (2010). Advanced LIGO: The
next generation of gravitational wave detectors; Class Quantum Grav.27.
4.
Kostas, D. K ( 2002). Gravitational Waves. Encyclopaedia of Physical Science and
Technology. 3rd Edition, Volume 7.Accedemic Press, University of Thessalonica, 54124,
Greece.
5.
LIGO homepage: http://www.ligo.caltech.edu/ Retrieved on 04th June, 2016:
11.45pm.
6.
Nomoto and Kondo (1991). Thermonuclear Burning on Accreting Neutron Star.
Astophysical Journal. 367 L19.
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Bringen, B., Bakwa, D. D., and Girma, D. P. –
AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR
7.
Maggiore, M. (2008). Book Review. Gravitational Waves, volume I: Theory and
Experiments Classical and Quantum Gravity. Volume 25. Issue 20. DOI 10. 1088/0264 –
9381/25.
8.
Maggiore, M. (2013). Gravitational Waves vol. I: Theory and Experiments.
Oxford University Press. ISBN 978- 0-19-857074-5
9.
Thorset, S. E. and Chakrabarty, D. (1999). Neutron Stars with sub millisecond
Periods. Astrophysical Journal. 512, 288
10.
van Paradijs, J. (1995). The lives of the neutron stars, Kluwer Academic Publisher,
Dordrecht
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