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PSEUDO-NEWTONIAN TOROIDAL STRUCTURES IN SCHWARZSCHILD-DE SITTER SPACETIMES Jiří Kovář Zdeněk Stuchlík & Petr Slaný Institute of Physics Silesian University in Opava Czech Republic Hradec nad Moravicí, September, 2007 This work was supported by the Czech grant MSM

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Introduction Discription of gravity Newtonian > Newtonian gravitational potential (force) General relativistic > curvature of spacetime (geodesic equation) Pseudo-Newtonian > pseudo - Newtonian gravitational potential (force) Schwarzschild-de Sitter spacetime Geodesic motion [Stu-Kov, Inter. Jour. of Mod. Phys. D, in print] Toroidal perfect fluid structures [Stu-Kov-Sla, in preparation for CQG]

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Introduction Newtonian central GF Poisson equationGravitational potential r-equation of motionEffective potential

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Einstein’s equations Line element r-equation of motion Effective potential Introduction Relativistic central GF

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Gravitational potential Paczynski-Wiita r-equation of motion Effective potential Introduction Pseudo-Newtonian central GF

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Schwarzschild-de Sitter geometry Line element

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Schwarzschild-de Sitter geometry Equatorial plane

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Schwarzschild-de Sitter geometry Embedding diagrams Schwarzschild Schwarzschild-de Sitter

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Schwarzschild-de Sitter geometry Geodesic motion horizons marginally bound (mb) marginally stable (ms)

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Pseudo-Newtonian approach Potential definition Potential and intensity Intensity and gravitational force

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Pseudo-Newtonian approach Gravitational potential NewtonianRelativistic Pseudo-Newtonian y=0, P-W potential

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Pseudo-Newtonian approach Geodesic motion RelativisticPseudo-Newtonian

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Pseudo-Newtonian approach Geodesic motion exact determination of - horizons - static radius - marginally stable circular orbits - marginally bound circular orbits small differences when determining - effective potential (energy) barriers - positions of circular orbits

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Relativistic approach Toroidal structures Perfect fluidEuler equation Potential Integration (Boyer’s condition)

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Pseudo-Newtonian approach Toroidal structures Euler equation Potential Integration

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Shape of structure Comparison

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Mass of structure Comparison Pseudo-Newtonian mass Relativistic mass Polytrop – non-relativistic limit

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Adiabatic index y=10 -6 y= y= =5/3 9.5x x x x x x =3/2 1.8x x x x x =7/5 2.8x x x x x x10 -7 Central density of structure Comparison

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exact determination of - cusps of tori - equipressure surfaces small differences when determining - potential (energy) barriers - mass and central densities of structures Pseudo-Newtonian approach Toroidal structures

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GRPN Fundamental Easy and intuitive Precise Approximative for some problems Approximative for other problems Conclusion

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NewtonianRelativistic Footnote Pseudo-Newtonian definition

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Relativistic potential Newtonian potential Shape of structure Newtonian potential

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Thank you Acknowledgement To all the authors of the papers which our study was based on To you

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