On the different flavours of Lense-Thirring precession around accreting stellar mass black holes

Type-C QPOs in X-ray binaries have been often interpreted as a consequence of relativistic Lense-Thirring precession around a spinning black hole and they potentially offer a way to measure black hole spins and masses. The connection between relativistic precession and the resulting QPO has been made either in terms of a simplified model involving a single test particle producing the QPO, or in terms of a global model where a geometrically thick accretion flow precesses coherently as a rigid body. In this paper, we analyse similarities and differences between these two models, sometimes considered as in opposition to each other. We demonstrate that the former is the limiting case of the latter when the radial extent of the precessing flow is very small, and that solid lower limits to the black hole spin can be obtained by considering the test particle model alone. We also show that the global precession model naturally accounts for the range of frequencies observed for Type-C QPOs without the need to invoke a truncation of the inner accretion flow before it reaches the innermost stable circular orbit. Finally, we show that, in order to maintain rigid precession, the thick accretion flow should be radially narrow, and that if it extends beyond $10-10^2$ gravitational radii it aligns with the black hole spin too fast to produce a coherent QPO.