The Width-Flux Relation of the Broad Iron Kα Line during the State Transitions of the Black Hole X-Ray Binaries

The observations of varying broad iron lines during the state transition of the black hole X-ray binaries have been accumulating. In this work, the relation between the normalized intensity and the width of iron lines is investigated, in order to understand better the variation of iron lines and possibly its connection to state transition. Considering the uncertainties due to ionization and illuminating X-rays, only the effects of geometry and gravity are taken into account. Three scenarios were studied, i.e., the continuous disk model, the innermost annulus model, and the cloud model. As shown by our calculations, at given iron width, the line flux of the cloud model is smaller than that of the continuous disk model; while for the innermost annulus model, the width is almost unrelated with the flux. The range of the line strength depends on both the BH spin and the inclination of the disk. We then apply to the observation of MAXI J1631-479 by Nuclear Spectroscopic Telescope Array during its decay from the soft state to the intermediate state. We estimated the relative line strength and width according to the spectral fitting results in Xu et al., and then compared with our theoretical width-flux relation. It was found that the cloud model was more favored. We further modeled the iron line profiles, and found that the cloud model can explain both the line profile and its variation with reasonable parameters.