The Gravitational Radiation Emitted by a System Consisting of a Point Particle in Close Orbit around a Schwarzschild Black Hole

Affiliation(s)

Department of Mathematical Sciences, University of South Africa, Pretoria, South Africa.

Department of Mathematical Sciences, University of South Africa, Pretoria, South Africa.

ABSTRACT

We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the -function to model the matter density of the particle. To model the whole problem, we apply the second order differential equation obtained elsewhere for a dynamic thin matter shell around a Schwarzschild black hole. The only thing that changes on the equation is the quasi-normal mode parameter which now represent the orbital frequency of the particle. We compare our results with that of the standard 5.5 PN formalism and found that there is a direct proportionality factor that relates the two results, i.e. the two formalisms.

We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the -function to model the matter density of the particle. To model the whole problem, we apply the second order differential equation obtained elsewhere for a dynamic thin matter shell around a Schwarzschild black hole. The only thing that changes on the equation is the quasi-normal mode parameter which now represent the orbital frequency of the particle. We compare our results with that of the standard 5.5 PN formalism and found that there is a direct proportionality factor that relates the two results, i.e. the two formalisms.

Cite this paper

A. Kubeka, "The Gravitational Radiation Emitted by a System Consisting of a Point Particle in Close Orbit around a Schwarzschild Black Hole,"*Journal of Modern Physics*, Vol. 3 No. 10, 2012, pp. 1503-1515. doi: 10.4236/jmp.2012.310186.

A. Kubeka, "The Gravitational Radiation Emitted by a System Consisting of a Point Particle in Close Orbit around a Schwarzschild Black Hole,"

References

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[4] M. Ishii, M. Shibata and Y. Mino, “Black Hole Tidal Problem in the Fermi Normal Coordinates,” Physical Review D, Vol. 71, No. 4, 2005, Article ID: 044017. doi:10.1103/PhysRevD.71.044017

[5] D. Lai and A. G. Wiseman, “Innermost Stable Circular Orbit of Inspiraling Neutron-Star Binaries: Tidal Effects, Post-Newtonian Effects, and the Neutron-Star Equation of State,” Physical Review D, Vol. 54, No. 6, 1996, pp. 3958-3964. doi:10.1103/PhysRevD.54.3958

[6] M. C. Miller, “Prompt Mergers of Neutron Stars with Black Holes,” Astrophysical Journal, Vol. 626, No. 1, 2005, p. L41. doi:10.1086/431583

[7] B. Mashhoon, “On Tidal Phenomena in a Strong Gravitational Field,” Astrophysical Journal, Vol. 705, 1975, pp. 705-716. doi:10.1086/153560

[8] B. Carter and J. P. Luminet, “Tidal Compression of a Star by a Large Black Hole,” Astronomy & Astrophysics, Vol. 121, 1983, pp. 97-113.

[9] B. Carter and J. P. Luminet, “Mechanics of the Affine Star Model,” Monthly Notices of the Royal Astronomical Society, Vol. 212, 1985, pp. 23-55.

[10] W. H. Lee, “Newtonian Hydrodynamics of the Coalescence of Black Holes with Neutron Stars—III. Irrotational Binaries with a Stiff Equation of State,” Monthly Notices of the Royal Astronomical Society, Vol. 318, No. 2, 2000, pp. 606-624. doi:10.1046/j.1365-8711.2000.03870.x

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[12] S. Kobayashi, P. Laguna, E. S. Phinney and P. Meszaros, “Gravitational Waves and X-Ray Signals from Stellar Disruption by a Massive Black Hole,” Astronomy & Astrophysics, Vol. 615, No. 2, 2004, p. 855. doi:10.1086/424684

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[17] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, “Quasiequilibrium Black Hole-Neutron Star Binaries in General Relativity,” Physical Review D, Vol. 75, No. 8, 2007, Article ID: 084005. doi:10.1103/PhysRevD.75.084005

[18] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, “Black Hole-Neutron Star Binaries in General Relativity: Effects of Neutron Star Spin,” Physical Review D, Vol. 72, No. 4, 2005, Article ID: 044008. doi:10.1103/PhysRevD.72.044008

[19] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, Physical Review D, Vol. 74, 2006, Article ID: 041502(R).

[20] J. A. Faber, T. W. Baumgarte, S. L. Shapiro and K. Taniguchi, “General Relativistic Binary Merger Simulations and Short Gamma-Ray Bursts,” Astrophysical Journal, Vol. 641, No. 2, 2006, p. L93. doi:10.1086/504111

[21] J. A. Faber, T. W. Baumgarte, S. L. Shapiro, K. Taniguchi and F. A. Rasio, “Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption,” Physical Review D, Vol. 73, No. 2, 2006, Article ID: 024012. doi:10.1103/PhysRevD.73.024012

[22] F. Loffler, L. Rezzollas and M. Ansorg, “Numerical Evolutions of a Black Hole-Neutron Star System in Full General Relativity: Head-On Collision,” Physical Review D, Vol. 74, No. 10, 2006, Article ID: 104018. doi:10.1103/PhysRevD.74.104018

[23] M. Shibata and K. Uryū, “Merger of Black Hole-Neutron Star Binaries: Nonspinning Black Hole Case,” Physical Review D, Vol. 74, 2006, Article ID: 121503(R).

[24] M. Shibata and K. Uryū, “Merger of Black Hole-Neutron Star Binaries in Full General Relativity,” Classical and Quantum Gravity, Vol. 24, No. 12, 2007, p. S125. doi:10.1088/0264-9381/24/12/S09

[25] C. F. Sopuerta, U. Sperhake and P. Laguna, “Hydro-without-Hydro Framework for Simulations of Black Hole-Neutron Star Binaries,” Classical and Quantum Gravity, Vol. 23, No. 16, 2006, p. S579. doi:10.1088/0264-9381/23/16/S15

[26] B. C. Barish and R. Weiss, “LIGO and the Detection of Gravitational Waves,” Physics Today, Vol. 52, No. 10, 1990, p. 44. doi:10.1063/1.882861

[27] A. Coory, A. J. Farmer and N. Seto, “The Optical Identification of Close White Dwarf Binaries in the Laser Interferometer Space Antenna Era,” Astrophysical Journal Letters, Vol. 601, No. 1, 2004, p. L47. doi:10.1086/381780

[28] L. Lehner, “Gravitational Radiation from Black Hole Spacetime,” Ph.D. Thesis, University of Pittsburg, Pittsburg, 1998.

[29] H. Bondi, M. J. G. van der Burg and A. W. K. Metzner, “Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems,” Proceedings of the Royal Society A, Vol. 269, No. 1336, 1962, pp. 21-52. doi:10.1098/rspa.1962.0161

[30] R. K. Sachs, “Gravitational Waves in General Relativity. VIII. Waves in Asymptotically Flat Space-Time,” Proceedings of the Royal Society A, Vol. 270, 1962, pp. 103-126.

[31] N. T. Bishop, R. Gómez, L. Lehner, M. Maharaj and J. Winicour, “High-Powered Gravitational News,” Physical Review D, Vol. 56, No. 10, 1997, pp. 6298-6309. doi:10.1103/PhysRevD.56.6298

[32] N. T. Bishop, “Linearized Solutions of the Einstein Equations within a Bondi-Sachs Framework, and Implications for Boundary Conditions in Numerical Simulations,” Classical and Quantum Gravity, Vol. 22, No. 12, 2005, p. 2393. doi:10.1088/0264-9381/22/12/006

[33] N. T. Bishop and A. S. Kubeka, “Quasinormal Modes of a Schwarzschild White Hole,” Physical Review D, Vol. 80, No. 6, 2009, Article ID: 064011. doi:10.1103/PhysRevD.80.064011

[34] E. Poisson, “Gravitational Radiation from a Particle in Circular Orbit around a Black Hole. I. Analytical Results for the Nonrotating Case,” Physical Review D, Vol. 47, No. 4, 1993, p. 1497. doi:10.1103/PhysRevD.47.1497

[35] M. Sasaki and H. Tagoshi, “Analytic Black Hole Perturbation approach to Gravitational Radiation,” Living Reviews in Relativity, Vol. 6, 2003, p. 6.

[1] K. A. Postnov and L. R. Yungelson, “The Evolution of Compact Binaries Star Systems,” Living Reviews in Relativity, 9, 2006, p. 6. http://www.livingreviews.org/lrr-2006-6

[2] S. Chandrasekhar, “Ellipsoidal Figures of Equilibrium,” Yale University Pres, New Heaven, 1969.

[3] L. G. Fishbone, “The Relativistic Roche Problem. I. Equilibrium Theory for a Body in Equatorial, Circular Orbit around a Kerr Black Hole,” Astrophysical Journal, Vol. 185, 1973, pp. 43-68. doi:10.1086/152395

[4] M. Ishii, M. Shibata and Y. Mino, “Black Hole Tidal Problem in the Fermi Normal Coordinates,” Physical Review D, Vol. 71, No. 4, 2005, Article ID: 044017. doi:10.1103/PhysRevD.71.044017

[5] D. Lai and A. G. Wiseman, “Innermost Stable Circular Orbit of Inspiraling Neutron-Star Binaries: Tidal Effects, Post-Newtonian Effects, and the Neutron-Star Equation of State,” Physical Review D, Vol. 54, No. 6, 1996, pp. 3958-3964. doi:10.1103/PhysRevD.54.3958

[6] M. C. Miller, “Prompt Mergers of Neutron Stars with Black Holes,” Astrophysical Journal, Vol. 626, No. 1, 2005, p. L41. doi:10.1086/431583

[7] B. Mashhoon, “On Tidal Phenomena in a Strong Gravitational Field,” Astrophysical Journal, Vol. 705, 1975, pp. 705-716. doi:10.1086/153560

[8] B. Carter and J. P. Luminet, “Tidal Compression of a Star by a Large Black Hole,” Astronomy & Astrophysics, Vol. 121, 1983, pp. 97-113.

[9] B. Carter and J. P. Luminet, “Mechanics of the Affine Star Model,” Monthly Notices of the Royal Astronomical Society, Vol. 212, 1985, pp. 23-55.

[10] W. H. Lee, “Newtonian Hydrodynamics of the Coalescence of Black Holes with Neutron Stars—III. Irrotational Binaries with a Stiff Equation of State,” Monthly Notices of the Royal Astronomical Society, Vol. 318, No. 2, 2000, pp. 606-624. doi:10.1046/j.1365-8711.2000.03870.x

[11] W. H. Lee, “Newtonian Hydrodynamics of the Coalescence of Black Holes with Neutron Stars—IV. Irrotational Binaries with a Soft Equation of State,” Monthly Notices of the Royal Astronomical Society, Vol. 328, No. 2, 2001, pp. 583-600. doi:10.1046/j.1365-8711.2001.04898.x

[12] S. Kobayashi, P. Laguna, E. S. Phinney and P. Meszaros, “Gravitational Waves and X-Ray Signals from Stellar Disruption by a Massive Black Hole,” Astronomy & Astrophysics, Vol. 615, No. 2, 2004, p. 855. doi:10.1086/424684

[13] S. Rosswog, R. Speith and G. A. Wynn, “Accretion Dynamics in Neutron Star—Black Hole Binaries,” Monthly Notices of the Royal Astronomical Society, Vol. 351, No. 4, 2004, pp. 1121-1133. doi:10.1111/j.1365-2966.2004.07865.x

[14] T. W. Baumgarte, M. L. Skoge and S. L. Shopiro, “Black Hole-Neutron Star Binaries in General Relativity: Quasiequilibrium Formulation,” Physical Review D, Vol. 70, No. 6, 2004, Article ID: 064040. doi:10.1103/PhysRevD.70.064040

[15] P. Grandclément, “Accurate and Realistic Initial Data for Black Hole-Neutron Star Binaries,” Physical Review D, Vol. 74, No. 12, 2006, Article ID: 124002. doi:10.1103/PhysRevD.74.124002

[16] P. Grandclément, “Erratum: Accurate and Realistic Initial Data for Black Hole-Neutron Star Binaries,” Physical Review D, Vol. 74, 2007, Article ID: 129903(E).

[17] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, “Quasiequilibrium Black Hole-Neutron Star Binaries in General Relativity,” Physical Review D, Vol. 75, No. 8, 2007, Article ID: 084005. doi:10.1103/PhysRevD.75.084005

[18] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, “Black Hole-Neutron Star Binaries in General Relativity: Effects of Neutron Star Spin,” Physical Review D, Vol. 72, No. 4, 2005, Article ID: 044008. doi:10.1103/PhysRevD.72.044008

[19] K. Taniguchi, T. W. Baumgarte, J. A. Faber and S. L. Shapiro, Physical Review D, Vol. 74, 2006, Article ID: 041502(R).

[20] J. A. Faber, T. W. Baumgarte, S. L. Shapiro and K. Taniguchi, “General Relativistic Binary Merger Simulations and Short Gamma-Ray Bursts,” Astrophysical Journal, Vol. 641, No. 2, 2006, p. L93. doi:10.1086/504111

[21] J. A. Faber, T. W. Baumgarte, S. L. Shapiro, K. Taniguchi and F. A. Rasio, “Dynamical Evolution of Black Hole-Neutron Star Binaries in General Relativity: Simulations of Tidal Disruption,” Physical Review D, Vol. 73, No. 2, 2006, Article ID: 024012. doi:10.1103/PhysRevD.73.024012

[22] F. Loffler, L. Rezzollas and M. Ansorg, “Numerical Evolutions of a Black Hole-Neutron Star System in Full General Relativity: Head-On Collision,” Physical Review D, Vol. 74, No. 10, 2006, Article ID: 104018. doi:10.1103/PhysRevD.74.104018

[23] M. Shibata and K. Uryū, “Merger of Black Hole-Neutron Star Binaries: Nonspinning Black Hole Case,” Physical Review D, Vol. 74, 2006, Article ID: 121503(R).

[24] M. Shibata and K. Uryū, “Merger of Black Hole-Neutron Star Binaries in Full General Relativity,” Classical and Quantum Gravity, Vol. 24, No. 12, 2007, p. S125. doi:10.1088/0264-9381/24/12/S09

[25] C. F. Sopuerta, U. Sperhake and P. Laguna, “Hydro-without-Hydro Framework for Simulations of Black Hole-Neutron Star Binaries,” Classical and Quantum Gravity, Vol. 23, No. 16, 2006, p. S579. doi:10.1088/0264-9381/23/16/S15

[26] B. C. Barish and R. Weiss, “LIGO and the Detection of Gravitational Waves,” Physics Today, Vol. 52, No. 10, 1990, p. 44. doi:10.1063/1.882861

[27] A. Coory, A. J. Farmer and N. Seto, “The Optical Identification of Close White Dwarf Binaries in the Laser Interferometer Space Antenna Era,” Astrophysical Journal Letters, Vol. 601, No. 1, 2004, p. L47. doi:10.1086/381780

[28] L. Lehner, “Gravitational Radiation from Black Hole Spacetime,” Ph.D. Thesis, University of Pittsburg, Pittsburg, 1998.

[29] H. Bondi, M. J. G. van der Burg and A. W. K. Metzner, “Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems,” Proceedings of the Royal Society A, Vol. 269, No. 1336, 1962, pp. 21-52. doi:10.1098/rspa.1962.0161

[30] R. K. Sachs, “Gravitational Waves in General Relativity. VIII. Waves in Asymptotically Flat Space-Time,” Proceedings of the Royal Society A, Vol. 270, 1962, pp. 103-126.

[31] N. T. Bishop, R. Gómez, L. Lehner, M. Maharaj and J. Winicour, “High-Powered Gravitational News,” Physical Review D, Vol. 56, No. 10, 1997, pp. 6298-6309. doi:10.1103/PhysRevD.56.6298

[32] N. T. Bishop, “Linearized Solutions of the Einstein Equations within a Bondi-Sachs Framework, and Implications for Boundary Conditions in Numerical Simulations,” Classical and Quantum Gravity, Vol. 22, No. 12, 2005, p. 2393. doi:10.1088/0264-9381/22/12/006

[33] N. T. Bishop and A. S. Kubeka, “Quasinormal Modes of a Schwarzschild White Hole,” Physical Review D, Vol. 80, No. 6, 2009, Article ID: 064011. doi:10.1103/PhysRevD.80.064011

[34] E. Poisson, “Gravitational Radiation from a Particle in Circular Orbit around a Black Hole. I. Analytical Results for the Nonrotating Case,” Physical Review D, Vol. 47, No. 4, 1993, p. 1497. doi:10.1103/PhysRevD.47.1497

[35] M. Sasaki and H. Tagoshi, “Analytic Black Hole Perturbation approach to Gravitational Radiation,” Living Reviews in Relativity, Vol. 6, 2003, p. 6.