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G S Bisnovatyikogan - One of the best experts on this subject based on the ideXlab platform.
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parameters of innermost stable Circular Orbits of spinning test particles numerical and analytical calculations
Gravitation & Cosmology, 2016Co-Authors: Yu O Tsupko, G S Bisnovatyikogan, Paul I JefremovAbstract:The motion of classical spinning test particles in the equatorial plane of a Kerr black hole is considered for the case where the particle spin is perpendicular to the equatorial plane.We review some results of our recent research of the innermost stable Circular Orbits (ISCO) [1] and present some new calculations. The ISCO radius, total angular momentum, energy, and orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of the Kerr parameter a and investigate their dependence on both black hole and test particle spins. Then we describe in detail how to calculate analytically small-spin corrections to the ISCO parameters for an arbitrary values of a. The cases of Schwarzschild, slowly rotating Kerr and extreme Kerr black holes are considered. The use of the orbital angular momentum is discussed. We also consider the ISCO binding energy. It is shown that the efficiency of accretion onto an extreme Kerr black hole can be larger than the maximum known efficiency (42%) if the test body has a spin.
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parameters of innermost stable Circular Orbits of spinning test particles numerical and analytical calculations
arXiv: General Relativity and Quantum Cosmology, 2016Co-Authors: Yu O Tsupko, G S Bisnovatyikogan, Paul I JefremovAbstract:The motion of classical spinning test particles in the equatorial plane of a Kerr black hole is considered for the case where the particle spin is perpendicular to the equatorial plane. We review some results of our recent research of the innermost stable Circular Orbits (ISCO) [P.I. Jefremov, this http URL. Tsupko and G.S. Bisnovatyi-Kogan, Phys.Rev. D 91 124030 (2015)] and present some new calculations. The ISCO radius, total angular momentum, energy, and orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of the Kerr parameter $a$ and investigate their dependence on both black hole and test particle spins. Then we describe in details how to calculate analytically small-spin corrections to the ISCO parameters for an arbitrary values of $a$. The cases of Schwarzschild, slowly rotating Kerr and extreme Kerr black hole are considered. The use of the orbital angular momentum is discussed. We also consider the ISCO binding energy. It is shown that the efficiency of accretion onto an extreme Kerr black hole can be larger than the maximum known efficiency (42 %) if the test body has a spin.
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innermost stable Circular Orbits of spinning test particles in schwarzschild and kerr space times
Physical Review D, 2015Co-Authors: Paul I Jefremov, Oleg Yu Tsupko, G S BisnovatyikoganAbstract:We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable Circular Orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in detail. For a slowly rotating black hole, the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulas obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle's spin while values of energy and total angular momentum depend on it. We have also considered Circular Orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum, energy and frequency at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.
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innermost stable Circular Orbits of spinning test particles in schwarzschild and kerr space times
arXiv: General Relativity and Quantum Cosmology, 2015Co-Authors: Paul I Jefremov, Oleg Yu Tsupko, G S BisnovatyikoganAbstract:We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable Circular Orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in details. For a slowly rotating black hole the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulae obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle's spin while values of energy and total angular momentum depend on it. We have also considered Circular Orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum and energy at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.
Paul I Jefremov - One of the best experts on this subject based on the ideXlab platform.
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parameters of innermost stable Circular Orbits of spinning test particles numerical and analytical calculations
Gravitation & Cosmology, 2016Co-Authors: Yu O Tsupko, G S Bisnovatyikogan, Paul I JefremovAbstract:The motion of classical spinning test particles in the equatorial plane of a Kerr black hole is considered for the case where the particle spin is perpendicular to the equatorial plane.We review some results of our recent research of the innermost stable Circular Orbits (ISCO) [1] and present some new calculations. The ISCO radius, total angular momentum, energy, and orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of the Kerr parameter a and investigate their dependence on both black hole and test particle spins. Then we describe in detail how to calculate analytically small-spin corrections to the ISCO parameters for an arbitrary values of a. The cases of Schwarzschild, slowly rotating Kerr and extreme Kerr black holes are considered. The use of the orbital angular momentum is discussed. We also consider the ISCO binding energy. It is shown that the efficiency of accretion onto an extreme Kerr black hole can be larger than the maximum known efficiency (42%) if the test body has a spin.
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parameters of innermost stable Circular Orbits of spinning test particles numerical and analytical calculations
arXiv: General Relativity and Quantum Cosmology, 2016Co-Authors: Yu O Tsupko, G S Bisnovatyikogan, Paul I JefremovAbstract:The motion of classical spinning test particles in the equatorial plane of a Kerr black hole is considered for the case where the particle spin is perpendicular to the equatorial plane. We review some results of our recent research of the innermost stable Circular Orbits (ISCO) [P.I. Jefremov, this http URL. Tsupko and G.S. Bisnovatyi-Kogan, Phys.Rev. D 91 124030 (2015)] and present some new calculations. The ISCO radius, total angular momentum, energy, and orbital angular frequency are considered. We calculate the ISCO parameters numerically for different values of the Kerr parameter $a$ and investigate their dependence on both black hole and test particle spins. Then we describe in details how to calculate analytically small-spin corrections to the ISCO parameters for an arbitrary values of $a$. The cases of Schwarzschild, slowly rotating Kerr and extreme Kerr black hole are considered. The use of the orbital angular momentum is discussed. We also consider the ISCO binding energy. It is shown that the efficiency of accretion onto an extreme Kerr black hole can be larger than the maximum known efficiency (42 %) if the test body has a spin.
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innermost stable Circular Orbits of spinning test particles in schwarzschild and kerr space times
Physical Review D, 2015Co-Authors: Paul I Jefremov, Oleg Yu Tsupko, G S BisnovatyikoganAbstract:We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable Circular Orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in detail. For a slowly rotating black hole, the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulas obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle's spin while values of energy and total angular momentum depend on it. We have also considered Circular Orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum, energy and frequency at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.
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innermost stable Circular Orbits of spinning test particles in schwarzschild and kerr space times
arXiv: General Relativity and Quantum Cosmology, 2015Co-Authors: Paul I Jefremov, Oleg Yu Tsupko, G S BisnovatyikoganAbstract:We consider the motion of classical spinning test particles in Schwarzschild and Kerr metrics and investigate innermost stable Circular Orbits (ISCO). The main goal of this work is to find analytically the small-spin corrections for the parameters of ISCO (radius, total angular momentum, energy, orbital angular frequency) of spinning test particles in the case of vectors of black hole spin, particle spin and orbital angular momentum being collinear to each other. We analytically derive the small-spin linear corrections for arbitrary Kerr parameter $a$. The cases of Schwarzschild, slowly rotating and extreme Kerr black hole are considered in details. For a slowly rotating black hole the ISCO parameters are obtained up to quadratic in $a$ and particle's spin $s$ terms. From the formulae obtained it is seen that the spin-orbital coupling has attractive character when spin and angular momentum are parallel and repulsive when they are antiparallel. For the case of the extreme Kerr black hole with co-rotating particle we succeed to find the exact analytical solution for the limiting ISCO parameters for arbitrary spin. It has been shown that the limiting values of ISCO radius and frequency do not depend on the particle's spin while values of energy and total angular momentum depend on it. We have also considered Circular Orbits of arbitrary radius and have found small-spin linear corrections for the total angular momentum and energy at given radius. System of equations for numerical calculation of ISCO parameters for arbitrary $a$ and $s$ is also explicitly written.
Luc Blanchet - One of the best experts on this subject based on the ideXlab platform.
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energy and periastron advance of compact binaries on Circular Orbits at the fourth post newtonian order
Physical Review D, 2017Co-Authors: Laura Bernard, Luc Blanchet, A Bohe, Guillaume Faye, Sylvain MarsatAbstract:In this paper, we revisit and complete our preceding work on the Fokker Lagrangian describing the dynamics of compact binary systems at the fourth post-Newtonian (4PN) order in harmonic coordinates. We clarify the impact of the non-local character of the Fokker Lagrangian or the associated Hamiltonian on both the conserved energy and the relativistic periastron precession for Circular Orbits. We show that the non-locality of the action, due to the presence of the tail effect at the 4PN order, gives rise to an extra contribution to the conserved integral of energy with respect to the Hamiltonian computed on shell, which was not taken into account in our previous work. We also provide a direct derivation of the periastron advance by taking carefully into account this non-locality. We then argue that the infra-red (IR) divergences in the calculation of the gravitational part of the action are problematic, which motivates us to introduce a second ambiguity parameter, in addition to the one already assumed previously. After fixing these two ambiguity parameters by requiring that the conserved energy and the relativistic periastron precession for Circular Orbits are in agreement with numerical and analytical gravitational self-force calculations, valid in the limiting case of small mass ratio, we find that our resulting Lagrangian is physically equivalent to the one obtained in the ADM Hamiltonian approach.
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high order post newtonian fit of the gravitational self force for Circular Orbits in the schwarzschild geometry
Physical Review D, 2010Co-Authors: Luc Blanchet, Steven Detweiler, Alexandre Le Tiec, Bernard F WhitingAbstract:We continue a previous work on the comparison between the post-Newtonian (PN) approximation and the gravitational self-force (SF) analysis of Circular Orbits in a Schwarzschild background. We show that the numerical SF data contain physical information corresponding to extremely high PN approximations. We nd that knowing analytically determined appropriate PN parameters helps tremendously in allowing the numerical data to be used to obtain higher order PN coecients. Using standard PN theory we compute analytically the leading 4PN and the next-to-leading 5PN logarithmic terms in the conservative part of the dynamics of a compact binary system. The numerical perturbative SF results support well the analytic PN calculations through rst order in the mass ratio, and are used to accurately measure the 4PN and 5PN non-logarithmic coecients in a particular gauge invariant observable. Furthermore we are able to give estimates of higher order contributions up to the 7PN level. We also conrm with high precision the value of the 3PN
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post newtonian and numerical calculations of the gravitational self force for Circular Orbits in the schwarzschild geometry
Physical Review D, 2010Co-Authors: Luc Blanchet, Steven Detweiler, Alexandre Le Tiec, Bernard F WhitingAbstract:The problem of a compact binary system whose components move on Circular Orbits is addressed using two different approximation techniques in general relativity. The post-Newtonian (PN) approximation involves an expansion in powers of $v/c\ensuremath{\ll}1$, and is most appropriate for small orbital velocities $v$. The perturbative self-force analysis requires an extreme mass ratio ${m}_{1}/{m}_{2}\ensuremath{\ll}1$ for the components of the binary. A particular coordinate-invariant observable is determined as a function of the orbital frequency of the system using these two different approximations. The post-Newtonian calculation is pushed up to the third post-Newtonian (3PN) order. It involves the metric generated by two point particles and evaluated at the location of one of the particles. We regularize the divergent self-field of the particle by means of dimensional regularization. We show that the poles $\ensuremath{\propto}(d\ensuremath{-}3{)}^{\ensuremath{-}1}$ appearing in dimensional regularization at the 3PN order cancel out from the final gauge invariant observable. The 3PN analytical result, through first order in the mass ratio, and the numerical self-force calculation are found to agree well. The consistency of this cross cultural comparison confirms the soundness of both approximations in describing compact binary systems. In particular, it provides an independent test of the very different regularization procedures invoked in the two approximation schemes.
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the third post newtonian gravitational wave polarizations and associated spherical harmonic modes for inspiralling compact binaries in quasi Circular Orbits
Classical and Quantum Gravity, 2008Co-Authors: Luc Blanchet, Guillaume Faye, B R Iyer, Siddhartha SinhaAbstract:The gravitational waveform (GWF) generated by inspiralling compact binaries moving in quasi-Circular Orbits is computed at the third post-Newtonian (3PN) approximation to general relativity. Our motivation is two-fold: (i) to provide accurate templates for the data analysis of gravitational wave inspiral signals in laser interferometric detectors; (ii) to provide the associated spin-weighted spherical harmonic decomposition to facilitate comparison and match of the high post-Newtonian prediction for the inspiral waveform to the numerically-generated waveforms for the merger and ringdown. This extension of the GWF by half a PN order (with respect to previous work at 2.5PN order) is based on the algorithm of the multipolar post-Minkowskian formalism, and mandates the computation of the relations between the radiative, canonical and source multipole moments for general sources at 3PN order. We also obtain the 3PN extension of the source multipole moments in the case of compact binaries, and compute the contributions of hereditary terms (tails, tails-of-tails and memory integrals) up to 3PN order. The end results are given for both the complete plus and cross polarizations and the separate spin-weighted spherical harmonic modes.
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the 2 5pn gravitational wave polarizations from inspiralling compact binaries in Circular Orbits
Classical and Quantum Gravity, 2005Co-Authors: K G Arun, Luc Blanchet, B R Iyer, Mohd S S QusailahAbstract:Using the multipolar post-Minkowskian and matching formalism, we compute the gravitational wave form of inspiralling compact binaries moving in quasi-Circular Orbits at the second and a half post-Newtonian (2.5PN) approximation to general relativity. The inputs we use include notably the mass-type quadrupole at the 2.5PN order, the mass octupole and current quadrupole at the 2PN order, the mass 25-pole and current 24-pole at 1PN. The nonlinear hereditary terms come from the monopole–quadrupole multipole interactions or tails, present at the 1.5PN, 2PN and 2.5PN orders, and the quadrupole–quadrupole interaction arising at the 2.5PN level. In particular, the specific effect of nonlinear memory is computed using a simplified model of binary evolution in the past. The 'plus' and 'cross' wave polarizations at the 2.5PN order are obtained in ready-to-use form, extending the 2PN results calculated earlier by Blanchet, Iyer, Will and Wiseman.
Nan Song - One of the best experts on this subject based on the ideXlab platform.
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a pair of giant planets around the evolved intermediate mass star hd 47366 multiple Circular Orbits or a mutually retrograde configuration
The Astrophysical Journal, 2016Co-Authors: Bunei Sato, Lei Wang, Yujuan Liu, Gang Zhao, Masashi Omiya, Hiroki Harakawa, Makiko Nagasawa, Robert A Wittenmyer, Paul Butler, Nan SongAbstract:We report the detection of a double planetary system around the evolved intermediate-mass star HD 47366 from precise radial-velocity measurements at the Okayama Astrophysical Observatory, Xinglong Station, and Australian Astronomical Observatory. The star is a K1 giant with a mass of 1.81 ± 0.13 Me, a radius of 7.30 ± 0.33 Re, and solar metallicity. The planetary system is composed of two giant planets with minimum masses of 1.75 +0.20 to -0.17 MJ and 1.86 +0.16 to -0.15 MJ, orbital periods of 363.3 +2.5 to -2.4 days and 684.7 +5.0 to -4.9 days and eccentricities of 0.089 +0.079 to -0.060 and 0.278 +0.067 to -0.094, respectively, which are derived by a double Keplerian orbital fit to the radial-velocity data. The system adds to the population of multi-giant-planet systems with relatively small orbital separations, which are preferentially found around evolved intermediate-mass stars. Dynamical stability analysis for the system revealed, however, that the best-fit Orbits are unstable in the case of a prograde configuration. The system could be stable if the planets were in 2:1 mean-motion resonance, but this is less likely, considering the observed period ratio and eccentricity. A present possible scenario for the system is that both of the planets have nearly Circular Orbits, namely the eccentricity of the outer planet is less than ∼0.15, which is just within 1.4σ of the best-fit value, or the planets are in a mutually retrograde configuration with a mutual orbital inclination larger than 160°.
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a pair of giant planets around the evolved intermediate mass star hd 47366 multiple Circular Orbits or a mutually retrograde configuration
arXiv: Earth and Planetary Astrophysics, 2016Co-Authors: Bunei Sato, Lei Wang, Yujuan Liu, Gang Zhao, Masashi Omiya, Hiroki Harakawa, Makiko Nagasawa, Robert A Wittenmyer, Paul Butler, Nan SongAbstract:We report the detection of a double planetary system around the evolved intermediate-mass star HD 47366 from precise radial-velocity measurements at Okayama Astrophysical Observatory, Xinglong Station, and Australian Astronomical Observatory. The star is a K1 giant with a mass of 1.81+-0.13M_sun, a radius of 7.30+-0.33R_sun, and solar metallicity. The planetary system is composed of two giant planets with minimum mass of 1.75^{+0.20}_{-0.17}Mjup and 1.86^{+0.16}_{-0.15}Mjup, orbital period of 363.3^{+2.5}_{-2.4} d and 684.7^{+5.0}_{-4.9} d, and eccentricity of 0.089^{+0.079}_{-0.060} and 0.278^{+0.067}_{-0.094}, respectively, which are derived by a double Keplerian orbital fit to the radial-velocity data. The system adds to the population of multi-giant-planet systems with relatively small orbital separations, which are preferentially found around evolved intermediate-mass stars. Dynamical stability analysis for the system revealed, however, that the best-fit Orbits are unstable in the case of a prograde configuration. The system could be stable if the planets were in 2:1 mean-motion resonance, but this is less likely considering the observed period ratio and eccentricity. A present possible scenario for the system is that both of the planets have nearly Circular Orbits, namely the eccentricity of the outer planet is less than ~0.15, which is just within 1.4sigma of the best-fit value, or the planets are in a mutually retrograde configuration with a mutual orbital inclination larger than 160 degree.
Remo Ruffini - One of the best experts on this subject based on the ideXlab platform.
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general classification of charged test particle Circular Orbits in reissner nordstrom spacetime
European Physical Journal C, 2017Co-Authors: D Pugliese, Hernando Quevedo, Remo RuffiniAbstract:We investigate charged particles’ Circular motion in the gravitational field of a charged mass distribution described by the Reissner–Nordstrom spacetime. We introduce a set of independent parameters completely characterizing the different spatial regions in which Circular motion is allowed. We provide a most complete classification of Circular Orbits for different sets of particle and source charge-to-mass ratios. We study both black holes and naked singularities and show that the behavior of charged particles depend drastically on the type of source. Our analysis shows in an alternative manner that the behavior of Circular Orbits can in principle be used to distinguish between black holes and naked singularities. From this analysis, special limiting values for the dimensionless charge of black hole and naked singularity emerge, namely, Q/M $$=$$ 1/2, $$Q/M=\sqrt{13}/5$$ and $$Q/M=\sqrt{2/3}$$ for the black hole case and Q/M $$=$$ 1, $$Q/M=5/ (2 \sqrt{6})$$ , $$Q/M=3 \sqrt{6}/7$$ , and finally $$Q/M= \sqrt{9/8}$$ for the naked singularity case. Similarly and surprisingly, analogous limits emerge for the orbiting particles charge-to-mass ratio $$\epsilon $$ , for positive charges $$\epsilon =1$$ , $$\epsilon =2$$ and $$\epsilon =M/Q$$ . These limits play an important role in the study of the coupled electromagnetic and gravitational interactions, and the investigation of the role of the charge in the gravitational collapse of compact objects.
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general classification of charged test particle Circular Orbits in reissner nordstr om spacetime
arXiv: General Relativity and Quantum Cosmology, 2013Co-Authors: D Pugliese, Hernando Quevedo, Remo RuffiniAbstract:We investigate charged particles Circular motion in the gravitational field of a charged mass distribution described by the Reissner-Nordstrom spacetime. We introduce a set of independent parameters completely characterizing the different spatial regions in which Circular motion is allowed. We provide most complete classification of Circular Orbits for different sets of particle and source charge-to-mass ratios. We study both black holes and naked singularities and show that the behavior of charged particles depend drastically on the type of source. Our analysis shows in an alternative manner that the behavior of Circular Orbits can in principle be used to distinguish between black holes and naked singularities. From this analysis, special limiting values for the dimensionless charge of black hole and naked singularity emerge, namely, Q/M=1/2, $Q/M=\sqrt{13}/5$ and $Q/M=\sqrt{2/3}$ for the black hole case and Q/M=1, $Q/M=5/ (2 \sqrt{6})$, $Q/M=3 \sqrt{6}/7$, and finally $Q/M= \sqrt{9/8}$ for the naked singularity case. Similarly and surprisingly, analogue limits emerge for the orbiting particles charge-to-mass ratio $\epsilon$, for positive charges $\epsilon=1$, $\epsilon=2$ and $\epsilon=M/Q$. These limits play an important role in the study of the coupled electromagnetic and gravitational interactions, and the investigation of the role of the charge in the gravitational collapse of compact objects.
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equatorial Circular motion in kerr spacetime
Physical Review D, 2011Co-Authors: D Pugliese, Hernando Quevedo, Remo RuffiniAbstract:We analyze the properties of Circular Orbits of test particles on the equatorial plane of a rotating central mass whose gravitational eld is described by the Kerr spacetime. For rotating black holes and naked singularities we explore all the spatial regions where Circular Orbits can exist and analyze the behavior of the energy and the angular momentum of the corresponding test particles. In particular, we nd all the radii at which a test particle can have zero angular momentum due to the repulsive gravity eects generated by naked singularities. We classify all the stability zones of Circular Orbits. It is shown that the geometric structure of the stability zones of black holes is completely dierent from that of naked singularities.
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Circular motion of neutral test particles in reissner nordstrom spacetime
Physical Review D, 2011Co-Authors: D Pugliese, Hernando Quevedo, Remo RuffiniAbstract:We investigate the motion of neutral test particles in the gravitational field of a mass $M$ with charge $Q$ described by the Reissner-Nordstr\"om (RN) spacetime. We focus on the study of Circular stable and unstable Orbits around configurations describing either black holes or naked singularities. We show that at the classical radius, defined as ${Q}^{2}/M$, there exist Orbits with zero angular momentum due to the presence of repulsive gravity. The analysis of the stability of Circular Orbits indicates that black holes are characterized by a continuous region of stability. In the case of naked singularities, the region of stability can split into two nonconnected regions inside which test particles move along stable Circular Orbits.