Nodal and periastron precession of inclined orbits in the field of a rotating black hole

Abstract

The inclination of low-eccentricity orbits is shown to significantly affect the orbital parameters, in particular, the Keplerian, nodal precession, and periastron rotation frequencies, which are interpreted in terms of observable quantities. For the nodal precession and periastron rotation frequencies of low-eccentricity orbits in a Kerr field, we derive a Taylor expansion in terms of the Kerr parameter at arbitrary orbital inclinations to the black-hole spin axis and at arbitrary radial coordinates. The particle radius, energy, and angular momentum in the marginally stable circular orbits are calculated as functions of the Kerr parameter j and parameter s in the form of Taylor expansions in terms of j to within O[j 6 ]. By analyzing our numerical results, we give compact approximation formulas for the nodal precession frequency of the marginally stable circular orbits at various s in the entire range of variation of Kerr parameter.

Observables of a Test Mass along an Inclined Orbit in a Post-Newtonian-Approximated Kerr Spacetime Including the Linear and Quadratic Spin Terms

“…This quantity has been calculated by other methods [10] whose result coincides with ours, also see [2] to leading-pN-order. Notice that one can replace the Carter-like constant C by L in the Eqs.…”

“…In this case, C = L = L z and our precession angle Φ agrees to leading pN order with Eq. (53) in [2], where it is called the nodal precession rate. Furthermore, the leading-order linear-in-a terms of ∆ Ū and ∆ Φ fully agree with the results in [7] for the case of an inclined orbital plane when referred to the test-mass limit.…”

“…Merloni in [1] calculates the Lense-Thirring precession frequency of test-masses restricted to spherical orbits (constant r) and identifies them with the frequencies of QPOs from black hole sources. Sibgatullin in [2] calculates the nodal and periastron-rotation frequencies for nearly circular, nonequatorial orbits. Comparison of the observed and theoretical frequencies, once the relation between them confirmed, also act as a tool to test the general theory of relativity in strongly gravitating regions.…”

“…Comparison of the observed and theoretical frequencies, once the relation between them confirmed, also act as a tool to test the general theory of relativity in strongly gravitating regions. Exact analytic solutions for arbitrarily oriented orbits are known in case of rotating black holes [3][4][5][6] but the Newtonian analog or the post-Newtonian (pN) expansion of the results, say periastron shift, is not straightforward, see [2]. Exact expression for the periastron advance of a test-mass in the gravitational field of a non-rotating black hole are well known, e.g.…”

Quasi-periodic X-ray oscillations in the source Scorpius X-1

“…The observed inconsistency of the RPM may stem from the fact that the orbital inclination of the test particle in the model simplification is assumed to be infinitesimal. An exact solution for an arbitrary inclination is given in Sibgatullin (2001). In such an analysis, the frequency ν HBO for fixed mass and angular momentum can change by a factor of ∼ 3 as the inclination changes from 0 to π/2 (for the marginally stable orbit at M N S = 2M ⊙ and ν φ = 1200 Hz).…”