Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
2006-02-15
Class. Quantum Grav. 24 (2007) 1775-1808
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
LaTeX file, 46 pages, typos fixed, substantial changes, version published in Classical and Quantum Gravity, Vol 24 (2007) 1775
Scientific paper
10.1088/0264-9381/24/7/007
The exact solution for the motion of a test particle in a non-spherical polar orbit around a Kerr black hole is derived. Exact novel expressions for frame dragging (Lense-Thirring effect), periapsis advance and the orbital period are produced. The resulting formulae, are expressed in terms of Appell's first hypergeometric function $F_1$, Jacobi's amplitude function, and Appell's $F_1$ and Gau$\ss$ hypergeometric function respectively. The exact expression for frame dragging is applied for the calculation of the Lense-Thirring effect for the orbits of S-stars in the central arcsecond of our Galaxy assuming that the galactic centre is a Kerr black hole, for various values of the Kerr parameter including those supported by recent observations. In addition, we apply our solutions for the calculation of frame dragging and periapsis advance for stellar non-spherical polar orbits in regions of strong gravitational field close to the event horizon of the galactic black hole, e.g. for orbits in the central milliarcsecond of our galaxy. Such orbits are the target of the GRAVITY experiment. We provide examples with orbital periods in the range of 100min - 54 days. Detection of such stellar orbits will allow the possibility of measuring the relativistic effect of periapsis advance with high precision at the strong field realm of general relativity. Further, an exact expression for the orbital period of a test particle in a non-circular equatorial motion around a Kerr black hole is produced. We also derive exact expressions for the periapsis advance and the orbital period for a test particle in a non-circular equatorial motion in the Kerr field in the presence of the cosmological constant in terms of Lauricella's fourth hypergeometric function $F_D$.
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