A systematic study of the phase difference between QPO harmonics in black hole X-ray binaries

We perform a systematic study of the evolution of the waveform of black hole X-ray binary low-frequency QPOs, by measuring the phase difference between their fundamental and harmonic features. This phase difference has been studied previously for small number of QPO frequencies in individual sources. Here, we present a sample study spanning 14 sources and a wide range of QPO frequencies. With an automated pipeline, we systematically fit power spectra and calculate phase differences from archival Rossi X-ray Timing Explorer (RXTE) observations. We measure well-defined phase differences over a large range of QPO frequencies for most sources, demonstrating that a QPO for a given source and frequency has a persistent underlying waveform. This confirms the validity of recently developed spectral-timing methods performing phase-resolved spectroscopy of the QPO. Furthermore, we evaluate the phase difference as a function of QPO frequency. For Type-B QPOs, we find that the phase difference stays constant with frequency for most sources. We propose a simple jet precession model to explain these constant Type-B QPO phase differences. The phase difference of the Type-C QPO is not constant but systematically evolves with QPO frequency, with the resulting relation being similar for a number of high-inclination sources, but more variable for low-inclination sources. We discuss how the evolving phase difference can naturally arise in the framework of precession models for the Type-C QPO, by considering the contributions of a direct and reflected component to the QPO waveform.