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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. 49 Bringen, B., Bakwa, D. D., and Girma, D. P. – 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. European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 50 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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 European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 51 Bringen, B., Bakwa, D. D., and Girma, D. P. – 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. European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 52 Bringen, B., Bakwa, D. D., and Girma, D. P. – 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 European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 53 Bringen, B., Bakwa, D. D., and Girma, D. P. – 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 European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 54 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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, □ =− ⁄ European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 (9) 55 Bringen, B., Bakwa, D. D., and Girma, D. P. – 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. European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 56 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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 ∫ = ∫ European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 (24) 57 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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, European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 58 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR = ( ) (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, European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 59 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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. European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 60 Bringen, B., Bakwa, D. D., and Girma, D. P. – AMPLITUDE RANGE OF GRAVITATIONAL WAVES FROM COMPACT BINARY NEUTRON STAR 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. European Journal of Alternative Education Studies - Volume 1 │ Issue 1 │ 2016 61 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 Creative Commons licensing terms Authors will retain the copyright of their published articles agreeing that a Creative Commons Attribution 4.0 International License (CC BY 4.0) terms will be applied to their work. Under the terms of this license, no permission is required from the author(s) or publisher for members of the community to copy, distribute, transmit or adapt the article content, providing a proper, prominent and unambiguous attribution to the authors in a manner that makes clear that the materials are being reused under permission of a Creative Commons License. Views, opinions and conclusions expressed in this research article are views, opinions and conclusions of the author(s). 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