A NICER view of the corona through time-dependent Comptonization of the quasi-periodic oscillations in nine black-hole X-ray binaries

TL;DR

This study uses NICER data from nine black-hole X-ray binaries to map the evolution of the corona during outbursts by jointly fitting time-averaged spectra and QPO rms/phase-lag spectra with the time-dependent Comptonization model $vKompthdk$, constraining corona size $L$ and feedback fraction $\eta$ across spectral states. The QPO frequency increases from $\sim 0.1$ to $\sim 10$ Hz in the hard states and remains near $4$–$5$ Hz in the SIMS, while the corona contracts from $\sim 10^4$–$10^5$ km in the LHS to $\sim 10^3$ km in the HIMS and becomes compact ($\sim 4$–$8\times 10^3$ km) in the SIMS/HSS, with a flare-like expansion near state transitions. The intrinsic feedback fraction evolves in step with corona size, rising during contraction and dropping during expansion, indicating a dynamically changing disk–corona coupling across outbursts. These results provide a cohesive, cross-source view of coronal geometry evolution and its link to spectral-state transitions, complementing prior QPO- and reverberation-based studies while highlighting the complex, state-dependent energy exchange between the disk and corona.

Abstract

We present a systematic study of the evolution of the corona geometry in nine black hole X-ray binaries (BHXRBs) using archival data from NICER. We identify 171 observations exhibiting quasi-periodic oscillations (QPOs) across various spectral states and model the time-averaged energy spectra of the source, as well as the energy-dependent rms and phase-lag spectra of the QPO, with the time-dependent Comptonization model vKompthdk. This allows us to simultaneously constrain the corona size and feedback fraction during outbursts. By using the power color hue diagnostics, we identify different spectral states, and observe that the QPO frequency increases from $\sim$0.1 Hz to $\sim$10 Hz in the low-hard and hard-intermediate states (LHS and HIMS), and remains approximately constant at 4--5~Hz in the soft-intermediate state (SIMS). The corona size shows significant evolution: the corona is large ($\sim10^4$--$10^5$ km) in the LHS, contracts rapidly to $\sim10^3$ km in the HIMS, and exhibits a flare-like expansion near the HIMS-to-SIMS transition. In the SIMS and high-soft state (HSS), the corona becomes compact and stable (4000--8000~km). The feedback fraction of the corona photons increases during the periods in which the corona contracts and decreases during the periods in which the corona expands, indicating a change of the disk-corona coupling. Our results are consistent with previous QPO-based studies using vKompthdk on some individual sources. This work, however, provides the first view of the coronal evolution across outbursts for a diverse BHXRB sample, offering critical insights into coronal behavior as a function of the spectral state of the source.

Figures (15)

• Figure 1: Power-color diagram (PCD) for observations with QPOs from nine selected BHXRBs. The $x$-axis ($\log \text{PC1}$) and $y$-axis ($\log \text{PC2}$) represent the two PC components derived from the PDS. Different colors and markers correspond to different sources, as indicated in the legend. The diagram is divided into four spectral states: the LHS, HIMS, SIMS, and HSS, with boundaries marked by grey dashed lines. The hue, representing the angular position in the PC1-PC2 plane, is labeled at specific angles. The distribution of points shows how different sources evolve through various spectral states. Error bars indicate 1-$\sigma$ uncertainties.
• Figure 2: QPO frequency as a function of power color hue for 9 selected BHXRBs. Different colors and symbols represent different sources, as indicated in the legend. The vertical dashed lines separate the LHS, HIMS, SIMS, and HSS. The QPO frequency increases systematically with hue, reflecting the spectral state evolution of BHXRBs. Error bars indicate 1-$\sigma$ uncertainties.
• Figure 3: Ratio between the inner disk temperature, $kT_{\rm in}$, of diskbb and the seed photon temperature, $kT_{\rm s}$, of vKompthdk as a function of power-color hue. Grey points show individual measurements from observations of nine sources, while black points represent values averaged in bins of $10^\circ$ in hue, with error bars represent the uncertainty on the mean within each bin. Vertical dashed lines mark the boundaries between the LHS, HIMS, SIMS, and HSS.
• Figure 4: Corona size as a function of PC hue for nine BHXRBs. Different colors and symbols represent different sources, as indicated in the legend. The vertical dashed lines, located at hue values of $140^\circ$, $220^\circ$ and $300^\circ$, mark the boundaries between the LHS, HIMS, SIMS, and HSS. The error bars indicate 1-$\sigma$ uncertainties. The points with arrows indicate the 90% lower or upper limits. See Figure \ref{['fig:size_hue_bin']} for binned data.
• Figure 5: Coronal size as a function of the QPO frequency for the nine sources. The error bars indicate 1-$\sigma$ uncertainties. The points with arrows indicate the 90% lower or upper limits.